Enzyme Kinetics: Catalysis & Control: A Reference of Theory and Best-Practice Methods [1 ed.] 012380924X, 9780123809247

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Enzyme Kinetics: Catalysis & Control A Reference of Theory and Best-Practice Methods

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Enzyme Kinetics: Catalysis & Control A Reference of Theory and Best-Practice Methods

Daniel L. Purich Department of Biochemistry and Molecular Biology University of Florida College of Medicine Gainesville, Florida

AMSTERDAM  BOSTON  HEIDELBERG  LONDON  NEW YORK  OXFORD PARIS  SAN DIEGO  SAN FRANCISCO  SINGAPORE  SYDNEY  TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier 32 Jamestown Road, London NW1 7BY, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA First edition 2010 Copyright Ó 2010 Elsevier Inc. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+ 44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively, visit the Science and Technology Books website at www.elsevierdirect.com/rights for further information Cover Image: Two views of the X-ray crystallographic structure of the complete Thermus thermophilus 70S ribosome ˚ resolution (from Yusupov, M. M., Yusupova, G. Z., containing bound messenger RNA and transfer RNAs at 5.5-A Baucom, A., Lieberman, K., Earnest, T. N., Cate, J. H., and Noller, H. F. (2001) Science 292, 883–96 with permission). Perhaps Natures’s most complicated molecular machine, the ribosome consists of RNA and proteins that work together to accomplish the multiple structural, catalytic, and force-generating steps required for the high-fidelity synthesis and elongation of polypeptides. Although the centerpiece of the 2009 Nobel Prizes in Chemistry, the ribosome remains a major challenge for enzyme scientists and kineticists seeking to unlock its many secrets. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-380924-7 For information on all Academic Press publications visit our website at elsevierdirect.com

Typeset by TNQ Books and Journals Pvt Ltd. www.tnq.co.in Printed and bound in China 10 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1

Contents

Table of Contents Preface

xix 1.6.2.

Chapter 1. An Introduction to Enzyme Science 1.1. Catalysis 1.1.1. Roots of Catalysis in the Earliest Chemical Sciences 1.1.2. Synthetic Catalysts 1.2. Biological Catalysis 1.2.1. Roots of Enzyme Science 1.2.2. Enzyme Technology 1.3. Development of Enzyme Kinetics 1.4. The Concept of a Reaction Mechanism 1.4.1. Chymotrypsin: The Prototypical Biological Catalyst 1.4.2. Ribozymes 1.4.3. Mechanoenzymes 1.5. Explaining the Efficiency of Enzyme Catalysis 1.5.1. Stabilization of Reaction Transition States 1.5.2. Electrostatic Stabilization of Transition States 1.5.3. Intrinsic Binding Energy 1.5.4. Reacting Group Approximation, Orientation and Orbital Steering 1.5.5. Reactant State Destabilization 1.5.6. Acid/Base Catalysis 1.5.7. Covalent Catalysis 1.5.8. Transition-State Stabilization by Low-Barrier Hydrogen Bonds 1.5.9. Catalytic Facilitation by Metal Ions 1.5.10. Promotion of Catalysis via Enzyme Conformational Flexibility 1.5.11. Promotion of Catalysis via Force-Sensing and Force-Gated Mechanisms 1.6. Prospects for Enzyme Science 1.6.1. We Need Better Methods for Analyzing Enzyme Dynamics

1 5 1.6.3. 5 7 12 12 13 15 19

1.6.4.

1.6.5.

1.6.6. 20 22 23

1.6.7.

25 26

1.6.8.

27 28

1.6.9.

28 29 30 30

1.6.10.

31 32 1.6.11. 32

33 34

1.6.12.

to Understand the Detailed Mutual Changes in Both Substrate and Enzyme During Catalysis We Need New Approaches for Determining the Channels Allowing Energy Flow During Enzyme Catalysis We Need Additional Probes of Enzyme Catalysis We Need to Learn How Proteins Fold and How to Manipulate Protein Stability We Need to Develop a Deeper Understanding of Substrate Specificity We Need to Develop the Ability to Design Entirely New Biological Catalysts We Need to Define the Efficient Routes for Obtaining High Potency Enzyme Inhibitors as Drugs and Pesticides We Need to Learn More About In Singulo Enzyme Catalysis We Need to Develop Comprehensive Catalogs of Enzyme Mechanisms and to Use Such Information in Fashioning New Metabolic Pathways We Need to Understand How to Analyze the Kinetic Behavior of Discrete Enzyme-Catalyzed Reactions as Well as Metabolic Pathways in their Environment We Need to Develop Techniques that will Facilitate Investigation of Chromosomal Remodeling, Epigenetics, and the Genetic Basis of Disease and Cell Survival We Need to Develop Effective Enzyme Preparations for Use in Direct Enzyme Therapy

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37 38

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Contents

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Chapter 2. Active Sites and their Chemical Properties 53 2.1.

2.2.

2.3.

2.4.

Enzyme Active Sites 2.1.1. Most Enzymes are Proteins, which are Linear Polymers of a-Amino Carboxylic Acids 2.1.2. Active-Site Residues may be Classified with Respect to their Function(s) 2.1.3. Active Sites Typically Occupy only 2–3 per cent of the Total Volume of an Enzyme 2.1.4. Binding Energy Often Indicates the Strength of Enzyme Interactions with Substrates and Cofactors 2.1.5. The Structural Organization of Enzymes can be Considered Hierarchically 2.1.6. Enzymes Often Occur in Multiple Molecular Forms Forces Affecting Enzyme Structural Stability and Interactions 2.2.1. Electrostatic Interactions Influence Enzyme Structure and Interactions 2.2.2. Ion–Dipole and Dipole–Dipole Interactions are Specialized Electrostatic Phenomena 2.2.3. Hydrogen Bonding Mainly Plays a Compensatory Role in Stabilizing Proteins 2.2.4. Hydrophobic Interactions Play a Dominant Role in Stabilizing Most Proteins 2.2.5. Although Individually Weak, van der Waals Interactions are so Numerous that they Contribute Significantly to Overall Protein Stability 2.2.6. Some Proteins are Occasionally Stabilized by p-Cation Interactions Active-Site Diversification 2.3.1. Enzyme Diversification can be Explained Structurally 2.3.2. Catalytic Promiscuity may Explain the Emergence of Catalytically Diversified Enzymes Additional Functional Groups in Enzyme Active Sites 2.4.1. Vitamin-Based Coenzymes Increase the Chemical Versatility of Enzyme Active Sites

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2.5.

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61 63 64

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66 2.6. 69

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70 70 71

74 77

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2.7.

2.4.2. Some Enzymes Exploit Specialized Amino-Acid Residues in Catalysis Metal Ions in Enzyme Active Sites 2.5.1. A Group of Biologically Significant Metal Ions is Essential for Catalysis by Some Enzymes 2.5.2. Enzyme-Bound Metal Ion Complexes Share Structural and Chemical Features 2.5.3. The Chemistry of Metal Ion-Ligand Complexes is Dominated by the Nature of their Ligancy 2.5.4. Field-Effects Influence the Color and Magnetic Properties of Metal Ion Coordination Complexes 2.5.5. The Reaction Mechanisms of Transition Metal Complexes are Determined by their Inner- and Outer-Sphere Coordination Behavior 2.5.6. Metal Ions Form Complexes with Enzymes and/or their Substrates 2.5.7. Properties of Selected Active-Site Metal Ions 2.5.8. A Survey of Metal Ion Complexes within Selected Enzymes Reveals Key Features of Binding-Site Organization Active Sites of Enzymes Acting on Polymeric Substrates 2.6.1. Many Endonucleases Achieve their Remarkable Specificity by Means of Subsite Recognition 2.6.2. Proteases were the First Enzymes Shown to have Subsites for Interacting with their Polymeric Substrates 2.6.3. Endo-Glycosidases also Exploit Subsites to Achieve Specificity 2.6.4. Subsites Facilitate Substrate Recognition by SignalTransducing Protein Kinases Basic Organic Chemistry of Enzyme Action 2.7.1. There are Six Major Classes of Enzyme-Catalyzed Covalent Bond-Making/-Breaking Reactions 2.7.2. Carbon has Several Reactive Forms in Enzymatic Mechanisms 2.7.3. Many Enzymes Use the Same General Reaction Mechanisms

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Contents

First Discovered by Physical Organic Chemists 2.7.4. Nucleophilic Substitution is a Widely Used Reaction Mechanism in Enzyme Catalysis 2.7.5. Enzyme-Catalyzed Elimination Reaction Mechanisms have Many Precedents in Organic Chemistry 2.7.6. Enzymes are Highly Effective in Forming, Stabilizing, and Utilizing Carbanion Intermediates During Catalysis 2.7.7. Free Radicals are Formed in a Surprising Number of Enzyme-Catalyzed Reactions 2.7.8. The Versatility of Enzymes can be Illustrated by Considering a Selected Group of Reaction Mechanisms 2.8. Detecting Covalent Intermediates in Enzyme Reactions 2.8.1. Enzymes Form a Wide Range of Enzyme-Substrate Covalent Compounds, and Many are Catalytically Competent 2.8.2. Side-Reactions Often Provide Invaluable Clues About Mechanisms of Enzyme Catalysis 2.8.3. Some Enzyme-Substrate Covalent Compounds can be Chemically Trapped 2.9. Basics of Enzyme Stereochemistry 2.9.1. Definitions 2.9.2. The Cahn-Ingold-Prelog System Allows One to Assign the Absolute Stereochemical Configuration of Chiral Compounds 2.9.3. The Prochirality of Molecules may also be Specified Systematically 2.9.4. The Stereochemistry of Methyl Transfer Reactions may be Analyzed Using Enzymes of Known Stereochemistry as Reference Reactions 2.10. Electron Transfer Reactions 2.10.1. The Thermodynamic Properties of Oxidation–Reduction Reactions are Defined by Redox Potentials 2.10.2. The Redox Behavior of Complex Metalloenzymes can be Evaluated Spectroscopically by Stoichiometric Titration Techniques 2.10.3. Respiratory Chains are Comprised of Highly

vii

Coordinated Electron Transfer Reactions 2.10.4. Enzyme-Catalyzed Electron Transfer may be analyzed by Marcus Theory 2.10.5. Simple Kinetic Models can Account for the Behavior of Biological Electron Transfer Reactions 2.10.6. Several Prototypical Redox Enzymes Provide Valuable Insights into Electron Transfer Kinetics and Mechanisms 2.10.7. Enzyme Electrodes Combine the Specificity of Biological Catalysis with the Versatility of Potentiometry or Amperometry Appendix

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134 137

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Chapter 3. Fundamentals of Chemical Kinetics 3.1. 3.2. 3.3.

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143 145 145

3.4.

3.5. 146

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149 150 3.6. 154

157

3.7.

Timescale of Chemical Processes The Empirical Rate Equation Reaction Rate, Order and Molecularity 3.3.1. Reaction Rate 3.3.2. Reaction Order 3.3.3. Molecularity 3.3.4. Zero-Order Kinetics 3.3.5. First-Order Kinetics 3.3.6. Second-Order Kinetics 3.3.7. Pseudo First-Order Kinetics Basic Strategies for Evaluating Rate Processes 3.4.1. Initial-Rate Method 3.4.2. Progress Curve Analysis Composite Multi-stage (Multi-step) Mechanisms 3.5.1. Series First-Order Kinetics 3.5.2. Reversible First-Order Kinetics 3.5.3. Reversible Second-Order Kinetics 3.5.4. Rapid-Equilibrium and Steady-State Treatments 3.5.5. Rate-Controlling Steps 3.5.6. Principles of Detailed Balance 3.5.7. Thermodynamic Cycles for Evaluating Detailed Balance 3.5.8. Kinetic Equivalence and Mechanistic Ambiguity Thermal Energy: The Boltzmann Distribution Law Solution Behavior of Reacting Molecules 3.7.1. Water: A Unique Solvent for Biochemical Processes 3.7.2. Diffusion Limitations on Chemical Processes Occurring in Water

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3.7.3. Electrostatic Effects on Magnitude of Bimolecular Rate Constants 3.7.4. Reactant Desolvation 3.8. Transition-State Theory 3.9. Chemical Catalysis 3.9.1. Accelerating Rate without Altering the Equilibrium Poise 3.9.2. Nucleophilic and Electrophilic Facilitation 3.9.3. Buffer Catalysis 3.9.4. Autocatalysis 3.10. Reaction Coordinate Diagrams 3.11. Thermodynamic Principles 3.11.1. Chemical Equilibrium 3.11.2. Direction and Extent of Chemical Reaction 3.11.3. Using DDG to Define Binding Energetics 3.11.4. Alberty Treatment of Biochemical Thermodynamics 3.11.5. Some Reacting Systems are Best Analyzed by Principles of Non-Equilibrium Thermodynamics 3.12. Concluding Remarks

199 200 201 203 203 205 206 206 207 210 210 210 211 211

212 214

Chapter 4. Practical Aspects of Measuring Initial Rates and Reaction Parameters 215 4.1.

4.2.

4.3.

4.4.

Design of Initial-Velocity Enzyme Assays 4.1.1. ‘‘Activity Purity’’ is Sufficient in Most Initial-Rate Studies 4.1.2. Discontinuous and Continuous Rate Measurements 4.1.3. Each Enzyme Rate Assay has Its Own Special Set of Requirements Enzyme Purification 4.2.1. While Time-Consuming, the Task of Enzyme Purification is Often Well Founded 4.2.2. Biochemists have Developed a Powerful Battery of Techniques for Purifying Enzymes Coupled (or Auxiliary) Enzyme Assays 4.3.1. A Simple Kinetic Treatment Explains the Lag-Phase in Coupled Enzyme Assays 4.3.2. The Auxiliary Enzyme and Assay Conditions must be Suited to the Primary Enzyme Reaction Basic UV/Visible Absorption Spectroscopy 4.4.1. Absorption Spectra Depend on the Quantum States of Electron Orbitals

215 216 217 220 232

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239 240 240

4.4.2. Beer’s Law is a Quantitative Expression Linking Absorbance to Concentration 4.4.3. Some Enzyme Assays Use Alternative Substrates that are Chromogenic 4.5. Basic Fluorescence Spectroscopy 4.5.1. Fluorescence Spectra Depend on Excited-State Relaxation 4.5.2. Features of a Research-Grade Spectrophotofluorimeter 4.5.3. The Concentration of Various Metabolites may be Quantified Through Fluorescence Spectrometry 4.5.4. Biological Molecules may Contain Intrinsic or Extrinsic Fluorescent Reporter Groups 4.5.5. Fluorescence Anisotropy is a Powerful Technique For Quantifying Binding Interactions 4.5.6. Fo¨rster (Fluorescence) Resonance Energy Transfer (FRET) is an Exquisitely Distance-Sensitive Probe of Enzymes 4.5.7. Continuous Fluorescence Assays are Now Available for Pi- and PPi-Producing Reactions 4.5.8. Chemiluminescence is a Photoemissive Process Often Exploited in Enzyme Rate Assays 4.6. Measuring Reaction Rates with Isotopes 4.6.1. Stable Isotopes are Versatile Probes in Enzyme Kinetics 4.6.2. Radioisotopes Provide Extremely Sensitive Assays of Enzyme Rate Processes 4.7. Multisubstrate Kinetics and Inhibitor Kinetics 4.8. Analysis of Enzyme Rate Data 4.8.1. Enzyme Rate Data must be Appropriately Weighted 4.8.2. Quantitative Analysis of Reaction Progress-Curves can be Used to Evaluate Rate Parameters 4.8.3. Global Analysis Offers Added Advantages in Statistical Analysis 4.9. Working with ATP-Dependent Enzymes 4.10. Regenerating Nucleoside 59-Triphosphate Substrates 4.10.1. Protein and Enzyme Concentration 4.10.2. Total Protein Concentration can be Determined Quantitatively 4.10.3. Active Enzyme Concentration can be Quantified by Several Techniques

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268 270 272 275 276 276 276

Contents

4.11. Equilibrium Constant Determinations 4.11.1. Equilibrium Constants can be Evaluated in a Variety of Ways 4.11.2. The Determination of the Arginine Kinase Reaction Equilibrium Constant is an Excellent Example of a Well-Designed and Well-Executed Determination 4.12. Concluding Remarks

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281 284

Chapter 5. Initial-Rate Kinetics of OneSubstrate Enzyme-Catalyzed Reactions 287 5.1.

5.2.

Michaelis-Menten Treatment 5.1.1. Derivation of the MichaelisMenten Equation Reveals how Key Assumptions Define an Enzyme’s Initial-Rate Behavior 5.1.2. KS, Vm, Vm/KS, and [S]/KS are Rate Parameters Defining an Enzyme’s Initial-Rate Behavior 5.1.3. Several Methods for Plotting InitialRate Data are Quite Useful but have Inherent Limitations 5.1.4. The Michaelis-Menten Equation Predicts a Linear Dependence of Reaction Rate on the Concentration of Active Enzyme 5.1.5. The Quadratic Formula is Required When the Enzyme Concentration Approaches Substrate Concentration 5.1.6. Nonproductive Substrate Binding Cannot be Detected by the MichaelisMenten Treatment The Briggs-haldane Steady-State Treatment 5.2.1. Derivation of this Rate Equation Reveals Key Features of Steady-State Processes 5.2.2. Reaction Energetics Determine the Effect of Increasing Substrate Concentration on the Conversion of E þ S to E$S Complex 5.2.3. The Corresponding Reverse-Reaction Rate Equation can now be Written 5.2.4. The Haldane Relationship Constrains the Values of Key Rate Parameters for Reversible Enzyme-Catalyzed Reactions 5.2.5. The Briggs-Haldane Equation Requires that an Enzyme System Satisfies the SteadyState Assumption

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5.3.

Catalysis Involving Two or More Intermediates 5.3.1. Derivation of the TwoIntermediate Case Illustrates Why this Treatment is a More Realistic Representation of an Enzyme Mechanism 5.3.2. A Shortcut can be Taken to Derive the Steady-State Rate Equation for the Reverse Two-Intermediate Reaction Scheme 5.3.3. Multiple Internal Isomerizations are without Effect on the General Form of the Final Steady-State Rate Equation 5.3.4. The Haldane Relationship also Constrains the Relative Magnitudes of Key Rate Parameters in the TwoIntermediate Scheme 5.4. Additional Comments on Fundamental Kinetic Parameters 5.4.1. The Michaelis Constant has Several Important Implications, with Regard to Both ‘‘Substrate Affinity’’ and Substrate Specificity 5.4.2. The Turnover Number k cat Indicates Number of Substrate Molecules Converted to Product per Enzyme Active Site per Second 5.4.3. The ‘‘Specificity Constant’’ Vmax/Km or kcat/Km Indicates the Efficiency of Substrate Capture by an Enzyme 5.4.4. The Commitment to Catalysis Measures an Enzyme’s Ability to Convert the E$S Complex to E$P, as Compared to Reconversion of E$S to a Prior Enzyme Form 5.4.5. Evolution of Catalytic Proficiency 5.4.6. Internal Equilibria and Energetics of Perfected Enzymes 5.5. Reaction Progress Curve Analysis 5.6. Ribozyme Kinetics 5.7. Proteasome Kinetics 5.8. Isomerization Mechanisms 5.9. Simultaneous Action of an Enzyme on Different Substrates 5.10. Enantiomeric Enrichment and Anomeric Specificity 5.11. Simultaneous Action of Two Enzymes on the Same Substrate 5.12. Induced-Fit Mechanisms

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5.12.1. There are Many Outstanding Examples of Induced-Fit Binding Behavior 5.12.2. Induced-Fit Energetics may be Analyzed with Thermodynamic Cycles 5.12.3. Induced-Fit Behavior and Enzyme Specificity 5.12.4. Induced-Fit Behavior Represents a Considerable Challenge for ComputerBased Ligand-Docking 5.13. Kinetics of Enzymes Acting on Polymeric Substrates 5.13.1. Processive versus Distributive Mechanisms 5.13.2. Random Scission Kinetics of Endo-Depolymerases 5.13.3. Some Depolymerizing Enzymes Show Evidence of SubstrateAssisted Catalysis 5.13.4. Microarray and Phage-Display Profiles of Enzyme Specificity 5.13.5. ‘‘Hidden’’ Nonproductive Interactions in Steady-State Treatments of Enzyme Acting on Polymeric Substrates 5.14. Concluding Remarks

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6.3.

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330 331 6.4. 332 333

Chapter 6. Initial-Rate Kinetics of MultiSubstrate Enzyme-Catalyzed Reactions 335 6.1.

6.2.

Bisubstrate Kinetic Mechanisms 6.1.1. Cleland’s Notation Conveniently Represents Multi-Substrate Kinetic Mechanisms 6.1.2. There are Numerous Examples of Well-Characterized Bisubstrate Enzyme Kinetic Mechanisms Derivation Steady-State Bisubstrate Rate Equations 6.2.1. Fromm’s Systematic Method for Deriving Rate Equations is a Simple and Reliable Alternative to the More Confusing KingAltman Approach 6.2.2. The Two-Step Computer-Assisted Method is a Rapid, Automatic Way to Obtain Enzyme Rate Laws 6.2.3. Cleland’s Net Reaction Rate Method is a Simple, Elegant, and Reliable Way to Derive Rate Equations for Unbranched Kinetic Mechanisms

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6.5.

6.2.4. Theorell and Chance Defined a Special Ordered Binary Complex Mechanism for Two-Substrate Enzyme Catalyzed Reactions 6.2.5. The Steady-State Random Kinetic Mechanism is Far Too Complicated for the Unambiguous Experimental Determination of Key Rate Parameters 6.2.6. The Cha Method Assumes that Certain Reaction Mechanism Segments are at Thermodynamic Equilibrium Derivation of Rapid Equilibrium Bisubstrate Rate Equations 6.3.1. The Rapid-Equilibrium Assumption Greatly Facilitates the Derivation of Rate Laws for Bisubstrate Random Kinetic Mechanisms 6.3.2. The Rapid Equilibrium Treatment of the Ordered Sequential Kinetic Mechanism Gives Rise to What is Probably the Simplest of Multi-Substrate Rate Laws Ping Pong Bi Bi Mechanism 6.4.1. Ping-Pong Mechanisms have a Distinctive Steady-State Rate Equation that Gives Rise to Parallel-Line Patterns in 1/v-versus-1/[A] and 1/v-versus-1/[B] Plots 6.4.2. Ping Pong Enzymes Catalyze Partial-Exchange Reactions 6.4.3. Some Partial Exchange Reactions can be Mechanistically Ambiguous 6.4.4. Burst Kinetics Provide Information About RateContributing Steps in EnzymeCatalyzed Reaction Mechanisms 6.4.5. Certain Hydrolase/TransferaseType Enzymes have Distinctive Kinetic Properties 6.4.6. Substrate Inhibition Offers Insights About Ping-Pong Reactions 6.4.7. Multi-Site Ping Pong Kinetic Mechanisms Account for the Transfer of Reactant Moieties Between Substrate-Binding Pockets within Topologically Complex Active Sites Graphical and Quantitative Analysis of Bisubstrate Kinetics 6.5.1. Re-Plotting Bisubstrate Experimental Rate Data is a Useful Way to Derive Key Rate Parameters

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6.6.

6.7. 6.8.

6.9.

6.5.2. Haldane Relations can be Used to Distinguish Kinetic Mechanisms of Bisubstrate Enzyme-Catalyzed Reactions 6.5.3. The Dalziel Phi Method is a Quantitative Approach for Distinguishing Rival Bisubstrate Kinetic Mechanisms 6.5.4. Fromm’s Point-of-Convergence Method is Another Method for Distinguishing Bisubstrate Kinetic Mechanisms 6.5.5. Crossover-Point Analysis also Allows Discriminates Bisubstrate Enzyme Kinetic Mechanisms 6.5.6. Some Multi-Substrate Initial-Rate Kinetic Data can be Ambiguous Three-Substrate Enzyme Kinetics 6.6.1. There are Numerous Kinetic Schemes for Three-Substrate Enzyme Catalyzed Reactions 6.6.2. There are Several General Strategies for Reducing the Complexity of Three-Substrate Initial-Velocity Experiments Multisubstrate ‘‘ISO’’ Mechanisms Kinetic Properties of Enzymes Exhibiting Branched Transfer Pathways Concluding Comments Appendix

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363 364 366

7.2.

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368 370

372 373 374

7.3.

Chapter 7. Factors Influencing Enzyme Activity 379 7.1.

Activator Effects on Enzyme Kinetics 7.1.1. Definitions 7.1.2. Some Reversible Essential Activators Bind Before the Substrate 7.1.3. Some Reversible Essential Activators Bind After the Substrate 7.1.4. Some Enzymes Randomly Bind Essential Activators and Substrate 7.1.5. Some Essential Activators Bind to an Otherwise Unreactive Substrate 7.1.6. Some Activators Released During Catalysis Exhibit Rate-Limiting Rebinding 7.1.7. Some Non-Consumed Substrates Behave as ‘‘Pseudo-Essential Activators’’ 7.1.8. Enzyme Must Always have Basal Activity when a

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7.4.

Nonessential Activator is Absent 7.1.9. 39,59-cyclic AMP Phosphodiesterase Activation by Ca2þ-Calmodulin: A Thorough Kinetic Analysis 7.1.10. The Method of Continuous Variation Analysis may be Used to Determine Activator Binding Site Number and Affinity 7.1.11. Time-Dependent Enzyme Activation Requires Special Treatment 7.1.12. Some Agents Exhibit Biphasic Activation and Inhibition Effects Metal-Nucleotide Complexes as Substrates 7.2.1. Most ATP-Dependent EnzymeCatalyzed Reactions Require Complexation of ATP4 with a Divalent Metal Ion 7.2.2. Certain Mechanoenzymes Use Metal-Free ATP4 and GTP4, Albeit Slowly 7.2.3. Exchange-Inert Metal-Nucleotide Complexes are Powerful Mechanistic Probes pH Effects on Enzyme Kinetics 7.3.1. Many Enzymes Exhibit a Characteristic pH Optimum 7.3.2. Enzymes Often Display pHDependent Changes in Activity 7.3.3. Several Methods may be Employed to Estimate Catalytic pKa Values 7.3.4. Acetoacetate Decarboxylase Possesses a Catalytic Lysine Exhibiting an Atypical (or Perturbed) pKa Value 7.3.5. pKa Values may be Estimated on the Basis of Protein Structural Calculations 7.3.6. Enzymes Exhibit a Wide Range of pH-Dependent Behaviors 7.3.7. Some Enzymes Undergo pHDependent Changes in Mechanism 7.3.8. The pH Kinetics of Bisubstrate Enzymes can be Complex 7.3.9. Brønsted Theory Explains Important Aspects of Acid/Base Catalysis Buffer Effects on Enzyme Kinetics 7.4.1. Many Factors Influence the Choice of a Buffer 7.4.2. Biochemists Exploit Various Properties of Selected pH Buffers

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7.5.

7.6. 7.7.

7.8. 7.9.

7.4.3. Some Buffers Actively Participate in Enzyme Catalysis 7.4.4. Some Rate Studies may Require Buffers of Constant Ionic Strength Ionic Strength Effects on Enzyme Kinetics 7.5.1. Ionic Strength Defines a Solution’s Ionic Nature 7.5.2. The Debye-Hu¨ckel Treatment Explains How Ions Alter the Thermodynamic Activity of Solutes 7.5.3. Changes in Ionic Strength can Alter the Magnitude of Rate Constants 7.5.4. Ionic Strength Alters Enzyme Catalysis Profoundly 7.5.5. There are Limits on the Applicability of Ionic Strength Effect of Organic Solvents on Enzyme Activity Temperature Effects on Enzyme Kinetics 7.7.1. Temperature Often Strongly Influences Enzyme Activity and Stability 7.7.2. The Kinetics of Thermal Inactivation can be Treated Phenomenologically 7.7.3. Temperature Alters Both Equilibrium and Rate Constants 7.7.4. Many Nonlinear Arrhenius Plots can be Explained in Terms of ‘‘Rate Compensation’’ 7.7.5. The Q 10 Parameter is a SemiQuantitative Measure of an Enzyme’s Sensitivity to Changes in Temperature 7.7.6. Certain Organisms have their Own Characteristic Physiologic Temperature 7.7.7. Extremophilic Enzymes have Unusual Structural Stability 7.7.8. Some Multi-Subunit Enzymes Exhibit the Phenomenon of Reversible Cold Inactivation 7.7.9. Cryoenzymology Techniques Greatly Reduce the Rate of Enzyme Catalysis Pressure Effects on Enzyme Kinetics Effects of Immobilization on Enzyme Stability and Kinetics 7.9.1. Kinetic Behavior of MatrixImmobilized Enzymes can be Substantially Different than the Behavior of Solution-Phase Enzymes 7.9.2. Enzymes Tethered with Flow Tubes have Special Kinetic Properties 7.9.3. Enzyme Confinement may be Relevant to Cellular Conditions

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433

435 438 439

7.10. Non-Ideality Imposed by Molecular Crowding 7.11. Enzyme Action on Sequestered Substrates 7.12. Interfacial Catalysis 7.13. Proofreading Effects on Enzyme Catalysis 7.14. Kinetics of Crystalline Enzymes 7.14.1. Accurate Activity Assays of Crystalline Enzymes can be Technically Challenging 7.14.2. Time-Resolved Laue X-Ray Crystallography is Quickly Becoming a Powerful Mechanistic Tool 7.14.3. Direct Measurement of Reactant Diffusion Rates in Enzyme Crystals can be Accomplished by Video Absorption Spectroscopy 7.14.4. Cross-Linking can be an Effective Tool in Analyzing the Behavior of Crystalline Enzymes 7.15. Probing Enzyme Catalysis Through Site-Directed Mutagenesis 7.15.1. Mutations – Particularly Long-Lived Naturally Occurring Mutations – are Intrinsically Interesting 7.15.2. Early Mutagenesis Experiments Exploited Chemical Modification to Replace One Naturally Occurring Amino Acid with Another 7.15.3. Alanine Scanning Mutagenesis Often Provides Useful Clues About Essential Functional Groups in Enzymes 7.15.4. Enzyme Chemists have Adopted Efficient Strategies for Investigating Enzyme Catalysis by Site-Directed Mutagenesis 7.15.5. Triose-Phosphate Isomerase: A Case Study in Directed Mutagenesis 7.15.6. Chemical Rescue is a Method for Restoring Activity in some Mutant Enzymes 7.15.7. Site-Directed Mutagenesis Suffers Significant Limitations 7.16. Concluding Remarks

444 446 447 453 457

458

459

459

459 460

460

461

462

464

472

479 480 483

442

Chapter 8. Kinetic Behavior of Enzyme Inhibitors 485

443

8.1.

443

Scope and Significance of Enzyme Inhibition 8.1.1. Distinguishing Reversible and Irreversible Enzyme Inhibitors

485 485

Contents

8.2.

8.3.

8.4.

8.5.

8.1.2. Enzyme Inhibitors in Biomedicine 8.1.3. Broader Applications of Enzyme Inhibitors Reversible Enzyme Inhibition 8.2.1. Competitive Inhibition Requires a Substance to Bind to the Same Enzyme Form as the Substrate 8.2.2. Noncompetitive Inhibition Requires an Inhibitor to Bind to Both E and E$S Forms 8.2.3. Uncompetitive Inhibition Occurs when an Inhibitor Only Binds to E$S in One-Substrate Mechanisms 8.2.4. Inhibition can be Linear or Non-Linear 8.2.5. Cleland Developed Useful Rules for Analyzing Reversible Dead-End Inhibition 8.2.6. Some Inhibitors Act Synergistically Substrate Inhibition 8.3.1. Excess Substrate can Give Rise to Nonlinear Inhibition 8.3.2. Fromm’s Alternative Substrate Inhibition Method Distinguishes Rival Multi-Substrate Kinetic Mechanisms 8.3.3. Huang’s Constant-Ratio Alternative Substrate Inhibition Method Distinguishes Multi-Substrate Kinetic Mechanisms 8.3.4. Induced Substrate Inhibition is a Type of abortive Complex Inhibition Product Inhibition 8.4.1. The Alberty/Fromm Strategy Uses Product Inhibition Patterns to Distinguish Rival Multi-Substrate Kinetic Mechanisms 8.4.2. Product Inhibition Equations for Various Two-Substrate Kinetic Mechanism Indicate Potentially Unique Inhibition Patterns 8.4.3. Abortive Complex Formation Alters Idealized Product Inhibition Patterns for Two-Substrate Kinetic Mechanisms 8.4.4. A Foster-Neimann Plot Permits the Analysis of Progress Curves for Enzyme in the Presence of Product Inhibition 8.4.5. Product Inhibitors can Provide Valuable Clues About Multisubstrate ‘‘Iso’’ Mechanisms 8.4.6. The Metabolic Significance of Product Inhibition Merits Greater Consideration Multi-Substrate Geometric Inhibitors

xiii

486 489 489

489

501

502 504

505 505 506 506

508

510

511 512

512

513

516

520

521

521 523

8.6.

Transition-State Inhibitors 8.6.1. The Energetics of Transition-State Stabilization Explains the Considerable Inhibitory Potency of Substances Resembling the Transition State 8.6.2. There are Numerous Examples of Naturally Occurring and Synthetically TransitionState Analogs 8.6.3. High-Affinity Binding of Certain ‘‘Pro-Transition-State Analogs’’ is Triggered by Some Enzymes 8.7. Tight-Binding Reversible Inhibitors 8.7.1. Reversible Tight-Binding Inhibitors Undergo Slow Inhibitor-Induced Enzyme Conformational Changes 8.7.2. Slow-Binding Inhibitors and Slow, Tight-Binding Inhibitors are Time-Dependent 8.7.3. Dihydrofolate Reductase Inhibition by Methotrexate Illustrates Key Features of Time-Dependent Reversible Inhibitors 8.7.4. b-Site Amyloid Precursor Protein-Cleaving Enzyme Undergoes Time-Dependent Inhibition by a Statine-Based Peptide 8.8. Measures of Reversible Inhibitor Potency 8.8.1. Percent Inhibition and Degree of Inhibition 8.8.2. The IC 50 Parameter 8.9. Irreversible Enzyme Inhibition by Affinity Labeling Agents 8.9.1. Baker Advanced the Rational Design of Active-Site Directed Irreversible Inhibitors 8.9.2. A Simple Rate Equation Explains Affinity-Labeling Kinetics 8.9.3. The Presence of Substrate can Retard, but Not Block, Irreversible Inhibition 8.9.4. Site-Directed Irreversible Inhibitors may be Distinguished from Tightly Bound Reversible Inhibitors 8.9.5. pH Often Strongly Influences the Action of Irreversible Inhibitors 8.9.6. Unstable Affinity Reagents Frequently Undergo TimeDependent Deactivation 8.9.7. Quiescent Enzyme Inactivators are Special Irreversible Inhibitors 8.9.8. Syncatalytic Affinity-Labeling Agents React Synchronously with Catalysis

525

525

527

528 531

531

533

535

536 537 537 538 539

539 540

544

544 544

545 546

546

Contents

xiv

8.10. Photoaffinity Labeling of Enzyme Active Sites 8.10.1. Westheimer First Recognized the Power and Range of Photoaffinity Enzyme Reagents 8.10.2. Photochemical Reactions Exhibit Distinctive Properties 8.10.3. Even Photoaffinity Reagents Can Suffer Major Limitations 8.11. Mechanism-Based Inhibition 8.11.1. Mechanism-Based Inhibitors Proceed Along Parallel First-Order Paths 8.11.2. Mechanism-Based Inhibitors are Highly Versatile 8.11.3. Some Noncovalent Enzyme Inhibitors Resemble Mechanism-Based Inhibitors 8.12. Designing Highly Effective Enzyme-Directed Drugs 8.12.1. Drug Discovery Focuses on ‘‘Druggable’’ Enzyme Targets and Selection/Evaluation or their Inhibitors 8.12.2. Identifying and Perfecting Inhibitory Potency has Become a WellPracticed Art 8.12.3. Development of MechanismBased Inhibitors Remains a Powerful Approach 8.12.4. Schramm’s Drug Design Strategy Focuses on Discerning Subtle Differences in Enzyme Transition States and Replicating Them When Designing Inhibitors 8.12.5. Pro-Drug Development is another Viable Approach in Rational Drug Design 8.12.6. Adaptive Inhibition is a New Approach for Designing EnzymeDirected Drugs 8.12.7. Lupinski’s ‘‘Rule-of-Five Index’’ Predicts the Efficacy of Oral Drugs 8.12.8. Fragment-Based Lead Design 8.12.9. Distal-Site Drug Potentiation is Untested Approach for Improving Efficacy 8.12.10. RNA Interference is an UnderExplored Way to Deplete Target Enzymes 8.12.11. Macromolecules also Offer Promise as Enzyme Inhibitors 8.12.12. Metabolic Control Analysis may Facilitate the Evaluation of Drug Action 8.13. Concluding Remarks

547

Suggested Reading Other Authoritative Readings from Methods in Enzymology

573 574

548 548 549 550 552 556

557 558

558

561

563

563

566

567 568 569

571

571 572

572 572

Chapter 9. Isotopic Probes of Biological Catalysis 575 9.1. Utility of Isotopes in Defining Enzyme Stereochemistry 9.1.1. Vennesland and Westheimer Established the Stereochemistry of NADHDependent Hydride Transfer Reactions 9.1.2. The Stereochemistry of Nucleotide-Dependent Reactions Provides Valuable Insights into the Chemical Mechanisms of Phosphoryl and Nucleotidyl Transfer Reactions 9.2. Labeling Substrates for Isotopic Experiments 9.3. Isotope Exchange at and Away from Equilibrium 9.3.1. All Exchange Processes Obey Simple First-Order Kinetics, Regardless of the Number or Nature of Intermediate Steps in the Overall Chemical Reaction 9.3.2. The Basic Experimental Strategy Focuses on the Atoms (or Groups of Atoms) Undergoing Exchange 9.3.3. Equilibrium Exchange Rate Equations may be Derived Using Equilibrium or Steady-State Approximations 9.3.4. Boyer’s Strategy for Examining Equilibrium Isotopic Exchanges can Define the Order of Substrate Addition and Product Release 9.3.5. Early Measurements Demonstrated Isotope Exchange, Even with Reversible Reactions Away from Equilibrium or with Virtually Irreversible Enzymes 9.3.6. A Fuller Range of Isotopic Exchange-Rate Behavior can be Revealed Using Britton’s Flux Ratio Method 9.3.7. Isotopic Rate Measurements Provided the First Truly Comprehensive Analysis of Enzyme Transition-State Energetics 9.4. Isotope Trapping Method: EnzymeBound Substrate Partitioning Kinetics 9.5. Positional Isotope Exchange 9.6. Kinetic Isotope Effects

576

576

579 585 586

586

587

589

592

596

596

599 603 606 607

Contents

9.6.1. Basic Definitions and Notation 9.6.2. Primary Kinetic Isotope Effects Reflect Differences in the Respective Zero-Point Energies of Isotopomers 9.6.3. Some Kinetic Isotope Effects are Measured by Equilibrium Perturbation 9.6.4. Quantum Mechanical Hydrogen Tunneling Reveals Important Information about Reaction Barriers 9.6.5. The Magnitude of Secondary Kinetic Isotope Effects Distinguishes SN1- and SN2-type Nucleophilic Mechanisms 9.6.6. Other Reaction Steps may Alter the Observed Kinetic Isotope Effects 9.6.7. Multiple Isotopically Sensitive Steps may Influence the Magnitude of the Observed Kinetic Isotope Effect 9.6.8. Solvent Kinetic Isotope Effects (SIEs) Occur when Isotopically Labeled Solvent Molecules are Used in Rate Measurements 9.6.9. Other Cases Illustrating the Power of Kinetic Isotope Effect Measurements 9.7. Determining the Rates of Enzyme Synthesis and Degradation 9.8. Concluding Remarks Authoritative Readings from the ‘‘Enzyme Kinetics and Mechanism’’ volumes of Methods in Enzymology

xv

608

609 612

10.3.

613

616 619

622

10.4.

623 627 631 634

10.5.

635 10.6.

Chapter 10. Probing Fast Enzyme Processes 10.1. Range of Fast Reaction Techniques 10.2. Flow Techniques 10.2.1. Rapid-Mixing Continuous-Flow Methods Permit the Detection of Reaction Intermediates 10.2.2. Chance’s Stopped-Flow Technique Revolutionized the Investigation of Moderately Fast Reactions 10.2.3. NADþ Binding to Alcohol Dehydrogenase: A Case Study Illustrating the Utility of the Stopped-Flow Kinetic Measurements 10.2.4. Rapid-Scan Devices Permit Spectroscopic Analysis During Stopped-Flow Experiments 10.2.5. Rapid Mixing/Quenching Devices Permit Kinetic Analysis of a Wide Range of Chemical Reactions 10.2.6. Freeze-Quench Approaches Rely on a Sudden Drop in Temperature to Arrest Otherwise Highly Dynamic Processes

637 638 641

641

10.7. 10.8.

Chapter 11. Regulatory Behavior of Enzymes 11.1. 11.2.

642

11.3. 11.4.

645

647

649

653

10.2.7. Ribonucleotide Reductase Catalysis is Gainfully Examined by Rapid Freeze-Quench Techniques 10.2.8. ‘‘Burst-Phase’’ Kinetics also Reveal Key Features of Enzyme Action Relaxation Kinetics 10.3.1. Chemical Relaxation Encompasses a Robust Range of Techniques 10.3.2. While Conceptually Straightforward, Relaxation Rate Analysis Offers Powerful Insight into Complicated Chemical Processes 10.3.3. The Temperature-Jump Method is Probably the Most Versatile Chemical Relaxation Technique Stopped-flow and Temperature-Jump Techniques Provide Powerful Insights into Enzyme Catalysis 10.4.1. Ribonuclease 10.4.2. Aminotransferase Catalysis: A Case Study in TemperatureJump Kinetics 10.4.3. Dihydrofolate Reductase: Another Outstanding Example Other Relaxation Techniques 10.5.1. Pressure-Jump Methods Increase the Versatility of Relaxation Studies 10.5.2. Concentration Analysis (CCA) Other Rapid Reaction Methods 10.6.1. Flash Photolysis 10.6.2. Pulsed Radiolysis Data Analysis Concluding Remarks

11.5. 11.6.

Overview of Enzyme Regulation General Strategies for Measuring Ligand Binding The Hill Equation The Scatchard Equation 11.4.1. A Modified Scatchard Equation Accounts for Steric Hindrance Amongst Sites 11.4.2. The Scatchard Analysis may be Extended to Deal with Two Classes of Binding Interactions – One Strong and One Weak Wyman’s Linked Function Analysis The Monod-Wyman-Changeux Model 11.6.1. Several Key Properties of Allosteric Systems Suggested the Symmetry-Conserving MWC Model

654 655 656

658

658

666

669 669

670 671 672

672 673 674 675 677 678 680

685 685 688 691 693

693

694 694 695

695

Contents

xvi

11.7.

11.8.

11.9.

11.10. 11.11. 11.12.

11.6.2. MWC Ligand Saturation Functions are Simple Polynomials Accounting for Ligand Binding to One or Two Conformationally Distinct States of the Enzyme 11.6.3. The Saturation Function may be Generalized to Explain Ligand Binding to an Oligomer with n Symmetry-Conserved Sites 11.6.4. The MWC Model Also Accounts for the Effects of Positive and Negative Allosteric Modifiers The Koshland-Ne´methy-Filmer Model 11.7.1. The KNF Model is Rooted in Adair’s Treatment of Polyvalent Ligand Binding Interactions 11.7.2. The KNF Model Incorporates Elements of Pauling’s Site Interaction Model 11.7.3. The KNF Model Accounts for Both Positive and Negative Cooperativity 11.7.4. While not an Enzyme, Hemoglobin Provided Many Clues About Allostery 11.7.5. Negative Cooperativity Distinguishes KNF Models from MWC Models 11.7.6. Fraction-of-the-Sites Behavior: The Case of Escherichia coli Alkaline Phosphatase Other Cooperativity Models 11.8.1. Hybrid Cooperativity Models 11.8.2. The Duke, Le Nove`re and Bray Conformational Spread Model 11.8.3. V-Type Allosteric Systems Oligomerization-Dependent Changes in Enzyme Activity 11.9.1. Enzyme Self-Association can Alter Catalytic Activity 11.9.2. Some Substrates Alter Enzyme Oligomerization and Catalytic Activity Hysteresis Enzyme Amplification Cascades Substrate Channeling 11.12.1. Several Criteria Define Substrate Channeling 11.12.2. Tryptophan Synthase is an Outstanding Example of Substrate Channeling 11.12.3. The Once-Confusing Story of NADþ Transfer Between Dehydrogenases Illustrates the Need for Careful Studies on Substrate Channeling

696

698

11.12.4. Substrate Hydration may also Affect Channeling Measurements 11.13. Metabolic Control Analysis 11.14. Concluding Comments

Chapter 12. Single-Molecule Enzyme Kinetics 12.1.

699 699

12.2. 12.3.

700 12.4. 700

701 12.5. 702

703

705 707 707 708 709 709 709

711 712 713 718 720

721

722

12.6. 12.7. 12.8.

General Comments on Single-Molecule Enzyme Kinetics Demonstration of Single-Molecule Reaction Rates Kinetic Treatment of Single-Molecule Enzyme Behavior Video Microscopy 12.4.1. Kinesin Takes One 8-nm Step per ATP Molecule Hydrolyzed 12.4.2. Dark-Field Microscopy Affords Direct Observation of Microtubule Assembly/ Disassembly Dynamics Optical Tweezers 12.5.1. Optical Tweezers Facilitated Single-Molecule Studies on RNA Polymerase 12.5.2. Optical Trapping Facilitates Single-Molecule Studies of RecA Polymerization on Double-Stranded DNA 12.5.3. Actin-Based Listeria Motility Exhibits Monomer-Sized Stepping Atomic Force Microscopy Near-Field Optical Microscopy Fluorescence Microscopy 12.8.1. Epifluorescence Permits Uniform Sample Illumination 12.8.2. Fluorescence Microscopy Permits Direct Observation of Rotatory Catalysis 12.8.3. Total Internal Reflection Fluorescence Microscopy (TIRFM) Exploits Evanescent Wave Phenomena 12.8.4. Single Dihydrofolate Reductase Molecules ‘‘Blink’’ During Catalysis 12.8.5. Single-Molecule Fluorescence Facilitates Observation of Dextran Binding to Bacterial Glucosyltransferase 12.8.6. Single-Molecule Fluorescence also Provides a Way to Analyze the Conformational Dynamics of Staphylococcal Nuclease Catalysis

722 723 726

729 729 730 733 737 737

738 739

740

742

742 744 745 746 746

748

749 749

750

751

Contents

xvii

12.9.

Fluorescence Correlation Spectroscopy (FCS) 751 12.9.1. FCS Detects Emitted Light Fluctuations within Extremely Small Volumes 752 12.9.2. One- and Two-Photon FCS Provides a Highly Versatile Enzyme Probe 753 12.9.3. Basic Kinetic Theory 754 12.9.4. Proteolytic Cleavage may be Fruitfully Investigated by FCS 755 12.9.5. Endonucleolytic Cleavage has also been Examined by FCS 757 12.10. ‘‘Zero-Mode’’ Waveguides for Single-Molecule Analysis 757 12.11. Prospects 758

Chapter 13. Mechanoenzymes: Catalysis, Force Generation and Kinetics 761 13.1. 13.2.

Brief Overview of Energase-Type Reactions The Driving Force for AffinityModulated Molecular Motors 13.3. Qualitative Features of Force-Induced Noncovalent Bond Rupture 13.3.1. Bond Energetics may be Described as Potential Energy Functions 13.3.2. Kramers Developed an Insightful Bond Rupture Model 13.3.3. Noncovalent Bonding Interactions are Inherent to Mechanoenzyme Action 13.3.4. Noncovalent Bonds may be Classified as Ideal, Slip, and Catch Bonds 13.3.5. Green Fluorescent Protein Unfolding/Refolding is Force-Dependent 13.4. Keller-Bustamante Treatment of Molecular Motor Behavior 13.4.1. Motor Molecule Motions are Analyzed in Terms of State Space and the Potential of Mean Force 13.4.2. Molecular Motors Operate Stochastically

13.5.

13.6. 13.7. 13.8.

13.9. 13.10. 13.11.

761 766

13.12.

770 770 770

771

773

775

13.13.

13.4.3. Motors ‘‘Walk’’ on the Potential Energy Surface During Chemical and Positional Transitions Calcium Ion Pump: Chemical Specificity versus Vectorial Specificity Actomyosin Mechanism GTP-Regulatory Proteins AAAþ Mechanoenzymes 13.8.1. The AAAþ ATPases Possess Common Structural Elements 13.8.2. DNA Processivity Clamp Loader: An AAAþ Mechanoenzyme Gradient-Driven Mechanoenzymatic Processes ATP Synthase: Boyer’s Binding Change Mechanism Role of ATP in Protein Folding 13.11.1. GroEL/GroES is a Model ‘‘Foldase’’ System 13.11.2. GroEL/GroES Mediates an Annealing/Folding Cycle Actoclampin Molecular Motors 13.12.1. Hill-Type Mechanisms for Force Generation by Polymerizing Free-ended Filaments 13.12.2. The Actoclampin Hypothesis: Concerning the Existence and Action of Cytoskeletal Filament End-Tracking Motors 13.12.3. Processive Single-Filament End-Tracking by ActA-VASP Complex 13.12.4. Properties of Actoclampin Motors Concluding Remarks and Prospects

778

779 782 782 783 784 785 788 790 792 792 792 793

794

794

795 798 802

776

777 778

References Appendix Glossary Index

807 845 847 865

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Preface

When I first pursued bench research on enzymes, my only trusted companions were ENZYMES by Dixon & Webb and the first ten volumes of METHODS IN ENZYMOLOGY. As effective as these resources were at guiding my inexperienced hands and schooling my thoughts, they were, more often than not, insufficient – simply because researchers of that time lacked a comprehensive view of enzyme kinetics, catalysis and control. With the central metabolic pathways already well defined and with the broad outlines of molecular biology emerging, modern enzyme science was established by a generation of biochemists that included my mentors Herb Fromm and Earl Stadtman and their mentors Paul Boyer and Fritz Lipmann. Recognizing a need for a comprehensive multi-volume treatise on enzyme kinetics, and with considerable encouragement from the founding editors Nate Kaplan and Sydney Colowick as well as Academic Press president Jim Barsky, I was privileged thirty years ago to initiate what has become the six-volume Enzyme Kinetics & Mechanism sub-series in METHODS IN ENZYMOLOGY. More recently, I sought to produce a single-volume enzyme kinetics reference that might serve the needs of biochemists, molecular life scientists, as well as physical scientists and engineers with an interest in learning how enzymes work. I undertook what became a seven-year task of writing a single-authored reference that I hoped would prove to be accessible, interesting, informative, and, above all, useful. I also had in mind the needs of those who come to enzyme science from physics and engineering, and for them, I included additional basic information not typically found in enzyme kinetics books. Much of biochemistry as well as molecular and cell biology is devoted to understanding how enzymes work, simply because enzymes are the verbs in all biotic sentencesdactively playing countless roles so essential for the flow of matter, energy and information within and among cells. It follows then that, during the course of virtually any study of a cellular process, one or more enzymes will play an indispensable part, and it is that realization that motivates widespread fascination about enzyme action. Current estimates put the number of unique enzyme-catalyzed reactions in the neighborhood twenty thousand, and despite the universality of certain metabolic processes, organisms have evolved in response to their own unique set of selective pressures. In this respect, enzymes catalyzing the same reaction in any two species probably have different amino acid sequences and are likewise

unlikely to exhibit identical catalytic properties (e.g., substrate specificity, catalytic efficiency, sensitivity to metabolic end-products, etc.). With the number of species easily exceeding 300,000, there must be countless unique molecular forms of any single enzyme, say hexokinase or alcohol dehydrogenase, especially when one includes the tissue-specific isozymes found in any single organism. A mutation in any single enzyme has the potential to affect the performance of complete metabolic pathways or even impair the health of an entire organism. Thus, while one can draw general inferences about metabolism, the physiology of various life forms must to some extent reflect this robust diversity of enzymes, and the universe of enzymes available for kinetic characterization is astonishingly vast. The cardinal feature of every enzyme is catalysis, and any endeavor to characterize the properties of an enzyme requires the examination and determination of its timedependent processes. One approach for analyzing the catalytic mechanics of complex enzyme systems is to determine the chronology of discrete steps within the overall processda pursuit called ‘‘kinetics.’’ This strategy allows an investigator to assess the structural and energetic determinants of transitions from one step to the next. By identifying voids in the time-line, one considers the possibility of other likely intermediates and ultimately identifies all elementary reactions of a mechanism. Kinetics is a highly analytical and intellectual enterprise that is deeply rooted in chemistry and physics, and enzyme chemists have intuitively and inventively honed the tools of chemists and physicists to investigate biological processes. Enzyme kineticists have likewise gainfully exploited advances in physical organic chemistry, structural chemistry, and spectroscopy to dissect enzyme mechanisms into their constituent time-ordered steps. The persuasive logic of kinetics also exemplifies the rigorous application of the scientific method in the molecular life sciences, especially biochemistry and biophysics, molecular and cell biology, as well as pharmacology, immunology, and even neuroscience. In short, enzyme kinetics discloses Life’s rhythmsdfrom the virtually instantaneous photon absorption in photosynthesis to what Frost called ‘‘the slow smokeless burning of decay.’’ My goal in writing this reference book was to present and explain the kinetic principles that have advanced enzyme science so that students can understand past and current research publications and can advance the field by applying these principles and by inventing new ones. Over my thirtyxix

xx

five years of lecturing, I have attempted to give voice to the beauty of enzyme science by providing graduate students with a solid grounding in its underlying chemical principles. The latter requires comprehension of basic chemical kinetics, appreciation of the power and scope of initial-rate and fast-reaction techniques, the origins of kinetic isotope effects, as well as the grandeur of allostery. Beginning with my hand-copied notes from Herb Fromm’s semester-long course on enzyme kinetics at Iowa State University as well as the notes Earl Stadtman used in his revered biochemistry course at NIH, I developed and consolidated my own ideas about enzyme catalysis and control in Chem 242: Chemical Aspects of Biological Systems and Chem 252: Enzyme Kinetics and Mechanism, courses I presented annually over my 11-year tenure in the Department of Chemistry at the University of California Santa Barbara. Other topics were developed in BCH 6206: Advanced Metabolism; BCH 6740: Physical Biochemistry & Structural Biology; and BCH 7515: Dynamic Processes in the Molecular Life Sciences, courses offered over the past 24 years here at the University of Florida. I have also presented short courses on enzyme kinetics at several pharmaceutical firms and foreign universities. My experience is that students from remarkably diverse backgrounds can readily comprehend, appreciate, and apply the logic of enzyme kinetic theory, especially when provided with logical explanations and aided initially by step-by-step derivations. Because the principles and practices of enzyme kinetics are also of great interest to chemists, engineers and physicists, this book also presents basic background information that should allow them to fill gaps in their understanding of biochemical and organochemical principles. As they discover the molecular life sciences, an entirely new generation of biophysicists and structural biologists has emerged. To facilitate their use of this book, I have also indicated the catalyzed chemical reactions upon first mention of most enzymes. I also provided additional descriptions of biochemical phenomena and explanations of regulatory concepts that students of chemistry, engineering and physics may not otherwise encounter in their coursework. While enzyme kinetics might become a life-long and fulfilling passion for a few biochemistry students, my experiences suggest that virtually all molecular life scientists can benefit from a solid understanding of kinetic principles and their rigor in testing rival models. Learning that many students retained and used my course notes well beyond their graduate student years inspired me to write a rigorous, yet thoroughly explained, reference book. While designed as a comprehensive reference, the book may be suitable for special topics courses in enzyme kinetics and enzymology. While the scope and detail of some sections may prove to be too encyclopedic for classroom presentation, such sections should be a valuable resource in the laboratory and in preparation of research reports and publications.

Preface

Enzyme kinetics encompasses a spectrum of experimental approaches, each suited to a particular task or time domain. Irrespective of the technique, the underlying motivation is to develop quantitative models for analyzing an enzyme of interest. Model building in kinetics is manifested as a multi-reaction scheme comprised of all reacting species identified and the rate constants and equilibrium constants needed to define their interactions. Models are stressed throughout this reference. Chapter 1 introduces the history and scope of enzyme catalysis as well as theories of enzyme rate enhancement, and Chapter 2 provides a foundation in the chemistry of enzyme active sites. The next nine chapters focus on what I consider to be the core topics of enzyme kinetics: Chapter 3 on the basic principles of chemical kinetics; Chapter 4 on making enzyme rate measurements; Chapter 5 on initial-rate theory of onesubstrate enzymes; Chapter 6 on initial rate behavior of multi-substrate enzymes; Chapter 7 on a myriad of factors influencing enzyme activity; Chapter 8 on reversible and irreversible enzyme inhibition; Chapter 9 on using isotopes to uncover otherwise invisible aspects of enzyme catalysis; Chapter 10 on fast reaction techniques; Chapter 11 on enzyme cooperativity and regulatory enzyme kinetics; and Chapters 12 and 13, which are unique among enzyme kinetic books, respectively covering single-molecule enzyme kinetics and those mechanoenzymatic reactions that generate force. To provide direct access to the research literature on enzyme kinetics, over 2,600 original research reports and reviews are cited. Even so, this list of references is necessarily incomplete, and I apologize to those scientists whose outstanding contributions could not be included. I also welcome comments, corrections, needed additions, and suggestions for papers meriting further study. Wherever possible, I have attempted to give full attribution to the ideas of others, and I will be the first to say that a student of enzymology is always a student, and rarely a master. Human frailty is such that we too often do not know what we do not know, a failing evident in nearly every scholarly enterprise. For the instances where I have failed to explain an important concept adequately or have misinterpreted the findings of others, I apologize in advance and would welcome suggestions for improvement. As I pointed out in 2001 (Trends in Biochemical Sciences 26, 417), biological catalysis need not require the making/ breaking of covalent bonds: some substrate-like and product-like states differ only with respect to their noncovalent bonding interactions. Accordingly, I redefined an enzyme as a biological agent that catalyzes the making/ breaking of chemical bonds, a term that includes both covalent and noncovalent bonds. I also suggested that a new enzyme class is needed to classify nearly every so-called ATPase or GTPase reaction as specialized enzymes that transduce covalent bond energy into mechanical work. In every known case, these so-called energase reactions can be

Preface

written in terms of a mechanism having one or more energydriven, affinity-modulated binding interaction, much like the ATP dependent actoclampin motor that Professor Richard Dickinson and I recently proposed is the forcegenerating mechanism responsible for cell crawling. This reference book is the first to introduce a fully integrated treatment of energase-type mechanoenzymes and to describe how kineticists have discovered fundamental features of energase-type reactions. Over my many years writing this book, I benefitted from the advice and suggestions from many friends, especially my pre- and post-doctoral lab partners, Fred Rudolph and Charles Y. Huang, as well as Bryce Plapp, Jeremy Knowles and Dan Koshland. (Fred’s, Jeremy’s and Dan’s passing represent an immense loss for all of enzymology.) I am likewise delighted to acknowledge my University of Florida colleagues, especially Professors Linda Bloom and David Silverman as well as R. Donald Allison, my coauthor on other book projects. I likewise thank Professors Giulio Magni, Silverio Ruggieri, and Nadia Raffaelli, for making the Istituto di Biotecnologie Biochimiche at the Universita´ Polytecnica delle Marche in Ancona, Italy such a welcoming intellectual and cultural home away from home. Many ideas presented in this book were first conceived, nurtured and/or tested during what are always pleasant stays in Ancona. Finally, I am indebted to the students and postdoctoral scientists, whom I have taught and who in return have taught me, both in my laboratory as well as in classrooms at the University of California and

xxi

University of Florida. Their persistent and insightful questions have given focus and meaning to my career as a teacher, chemist, and molecular life scientist. I also note with sadness the recent passing of my postdoctoral mentor Earl R. Stadtman, a magnificent scientist who quietly imbued in all his students a fascination for enzymology and metabolic regulation. The burden of converting my manuscript into this reference book was lightened through the masterful assistance of Jacquiline Holding and Caroline Johnson at Elsevier as well as the capable copyediting of my student Matthew Neu. I am also grateful to my partner Li Lu for all of her understanding and sustaining encouragement during my struggle to write and illustrate this book. Above all, I acknowledge Professor Herbert J. Fromm, in whose laboratory I began my research career. I marvel at Herb’s work ethic, his focus and intensity, and his sustained excitement and passion for science. Herb’s seminal research on multisubstrate enzyme kinetic mechanisms and his timeless book INITIAL RATE ENZYME KINETICS inspired a generation of scientists to pursue careers in enzyme kinetics and mechanism. In recognition his high standards of personal and professional conduct as well as our fortyplus years of friendship, I humbly dedicate this book to Herb. Daniel L. Purich October, 2009 Gainesville, Florida

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Chapter 1

An Introduction to Enzyme Science Enzymes are astonishing catalysts – often achieving rate enhancement factors1 of 1,000,000,000,000,000,000! Water, electrolytes, physiologic pH, ambient pressure and temperature all conspire to suppress chemical reactivity to such a great extent that even many metabolites as thermodynamically unstable as ATP (DGhydrolysis z 40 kJ/mol) and acetyl-phosphate (DGhydrolysis z 60 kJ/mol) are inert under normal physiologic conditions. Put simply, metabolism would be impossibly slow without enzymes, and Life, as we know it, would be unsustainable.2 As a consequence, enzymes are virtual on/off- switches, with efficient conversion to products in an enzyme’s presence and extremely low or no substrate reactivity in an enzyme’s absence. At millimolar concentrations of glucose and MgATP2, for example, substantial phosphorylation of glucose would require hundreds to thousands of years in the absence of hexokinase, but only seconds at cellular concentrations of this phosphoryl transfer enzyme. Without hexokinase, there would also be no way to assure exclusive phosphorylation at the C-6 hydroxymethyl group. And even when an uncatalyzed reaction (termed the reference

1

Catalytic rate enhancement (symbolized here as 3) equals the unit-less ratio kcat/kref, where the catalytic rate constant kcat (units ¼ s1) is the catalytic frequency (i.e., the number of catalytic cycles per second per enzyme active site), and kref (units ¼ s1) is the corresponding first-order rate constant for the uncatalyzed reaction. The value of 3 will be a direct measure of catalytic proficiency (i.e., an enzyme’s ability to enhance substrate reactivity), if and only if the enzymatic and nonenzymatic reactions operate by the very same chemical mechanism, in which case the nonenzymatic reaction is called the reference reaction. Note also that the value of 3 achieved by any given enzyme need only be sufficient to assure unimpeded metabolism. In the Principle of Natural Selection, mutation is the underlying search algorithm for evolution, and any mutation that markedly improves 3 beyond that needed for an organism’s survival should be inherently unstable and subject to reduction over time. 2 The upper limit on the room temperature rate constant for nonenzymatic water attack on a phosphodiester anion, for example, is about 1015 s1, necessitating 100-million year period for uncatalyzed P–O cleavage (Schroeder et al., 2006). Depending on reaction conditions, the corresponding rate constant for hydrolysis of the bg P–O bond in MgATP2 is around 104 to 106 s1, and given that bimolecular processes obey the simple rate law v ¼ k[A][B], rates for phosphoryl group transfer reactions (e.g., MgATP2 þ Acceptor # Phosphoryl  Acceptor þ MgADP) would be suppressed even further at low micromolar-to-millimolar concentrations of acceptor substrates within most cells. Enzyme Kinetics Copyright Ó 2010, by Elsevier Inc. All rights of reproduction in any form reserved.

reaction) is reasonably fast – as is the case for the reversible hydration of carbon dioxide to form bicarbonate anion or for the spontaneous hydrolysis of many lactones – an enzyme (in this case, carbonic anhydrase) is required to assure that the reaction’s pace is compatible with efficient metabolism under the full range of conditions experienced by that enzyme. Most enzymes also exhibit rate-saturation kinetics, meaning that velocity ramps linearly when the substrate concentration is below the Michaelis constant, and reaches maximal activity when the substrate is present at a concentration that is 10–20 times the value of the Michaelis constant. In this respect, an enzyme’s action is more akin to a variable-voltage rheostat than a simple on/off switch. Biochemists recognize that substrate specificity is another fundamental biotic strategy for effectively organizing biochemical reactions into metabolic pathways. Two analogous chemical reactions can take place within the same (or adjoining) subcellular compartments simply because their respective enzymes show substrate or cofactor specificity directing metabolic intermediates to and through their respective pathways, often without any need for subcellular co-localization or enzyme-to-enzyme channeling. Substrate specificity also minimizes formation of unwanted, and potentially harmful, by-products. By controlling the relative concentrations of such enzymes, cells also avoid undesirable kinetic bottlenecks or the undue accumulation of pathway intermediates.3 Experience tells us that extremely reactive chemical species can also be sequestered within the active sites of those enzymes requiring their

3

The term intermediate has several distinctly different meanings in biochemistry. In the context of the above sentence, intermediate refers to a chemical substance that is produced by an enzyme reaction within a metabolic pathway (A / B / C / P / Q / R, where B, C, P, and Q are metabolic intermediates) and is likewise a substrate in a subsequent enzyme-catalyzed reaction in that or another pathway. In the very next sentence, intermediate refers to a enzyme-bound substrate, enzyme-bound reactive species, or enzyme-bound product formed during the catalysis (E þ S # ES1 # ES2 # EXz # EP1 # EP2 # E þ P, where ES1, ES2, EXz, EP1, and EP2 are various enzyme-bound species/intermediates) in a single enzymatic reaction. For reactions occurring in the absence of a catalyst, chemists routinely use the term intermediate to describe any reactive species Xi-1, formed during the course of chemical transformation, whether formed reversibly (i.e., Xi-1 # Xi # Xiþ1) or irreversibly (i.e., Xi-1 / Xi / Xiþ1). All such usages of intermediacy connote metastability and/or a transient nature.

1

Enzyme Kinetics

2

formation, while hindering undesirable side-reactions that would otherwise prove to be toxic. So enzyme catalysis is inherently tidy. Enzyme active sites can also harbor metal ions that attain unusually reactive oxidation states that rarely form in aqueous medium and even less often in the absence of side-reactions. The resilience of living organisms stems in large measure from the capacity of enzymes to specifically or selectively bind other ligands (e.g., coenzymes, cofactors, activators, inhibitors, protons and metal ions). Attesting to the significance of enzyme stereospecificity in the biotic world is that most metabolites and natural products contain one or more asymmetric carbon atoms. The stereospecific action of enzymes is the consequence of the fact that both protein and nucleic acid enzymes are polymers of asymmetric units, making resultant enzymes intrinsically asymmetric. It should be obvious that any L-amino acidcontaining polypeptide having even a single D-amino acid residue cannot adopt the same three-dimensional structure as a natural polypeptide. Although some enzymes utilize both enantiomers of a substrate (e.g., glutamine synthetase is almost equally active on D-glutamate and L-glutamate), proteins containing exclusively L-amino acids are produced by the ribosome’s peptide-synthesizing machinery. This outcome is the result of the stereospecificity of aminoacyltRNA synthases that supply ribosomes with activated subunits, the stereochemical requirements of peptide synthesis, as well as ubiquitinylating enzymes and proteasomes that respectively recognize and hydrolyze wrongly folded proteins. Cells also produce a range of enzymes, such as D-amino acid oxidase (Reaction: D-Amino Acid þ O2 þ H2O # 2-Oxo Acid þ NH3 þ H2O2), that remove certain enantiomers (in this case, D-amino acids) from cells. In the case of protein enzymes, certain aspartate residues are also susceptible to spontaneous racemization as well as N-to-O acyl shifts, and cells produce enzymes that recognize and mediate the repair or destruction of proteins containing monomers having improper stereochemistry. Additional metabolic pathway stability is afforded by steady-state fluxes that resist sudden changes in rate or reactant concentrations. The processes lead to the phenomenon of homeostasis, wherein reactant concentrations appear to be time invariant merely because the processes producing and destroying these reactants are so exquisitely controlled. In some respects, the behavior of the whole of metabolism appears to exceed the sum of behaviors of its individual reactions. Experience has shown that hierarchically complex, large-scale networks often give rise to emergent properties (i.e., properties of a highly integrated metabolic or physiologic system that are not easily predicted from the analysis of individual components). Beyond the coordinated operation and regulation of the many pathways comprising intermediary metabolism, other emergent properties of living systems are evident in the adaptive resilience of signal transduction, long-range actions affecting chromosomal organization, as well as cellular morphogenesis and motility.

The creation of organizationally complex neural networks, as facilitated by the capacity of single neuronal cells to engage in tens of thousands of cell–cell interactions with other neurons via synapse formation, is also thought to underlie what we sense as our own consciousness. And at all such levels, enzyme catalysis and control are inevitably needed for effective intracellular and intercellular communication. As the essential actuators of metabolism, enzymes are often altered conformationally via biospecific binding interactions with substrates and/or regulatory molecules (known as modulators or effectors) to achieve optimal metabolic control. An additional feature is the capacity of multi-subunit enzymes to exhibit cooperativity (i.e., enhanced or suppressed ligand binding as a consequence of inter-subunit cross-talk). Because enzyme structure changes can be triggered by changes in the concentrations of numerous ligands, enzymes possess an innate capacity to integrate diverse input signals, thereby generating the most appropriate changes in catalytic activity. An interaction is said to be allosteric if binding of a low-molecular weight substance results in a metabolically significant conformational change. In most cases, modulating effects are negative (i.e., they result in inhibition), but positive effects (i.e., those resulting in activation) are also known. Feedback regulation has proven to be a highly effective strategy for controlling the rates of metabolic processes. When present at sufficient concentration, a downstream pathway intermediate or product (known as a feedback inhibitor) alters the structure of its target enzyme to the extent that the inhibited enzyme exhibits little ot no activity (Scheme 1.1). Target enzymes (shown below in red) are most often positioned at the first committed step within a pathway or at a branch point (or node) connecting two or more pathways. The lead reactions are frequently highly favorable (DG 105 s1, some 63 higher the wild-type enzyme. Mutant enzymes were selected from a library of 3.4  107 mutated phosphotriesterase genes using the ingenious strategy of linking genotype and phenotype by means of in vitro compartmentalization (IVC) in water-in-oil emulsions. First, microbeads, each displaying a single gene and multiple copies of the encoded protein, were formed by compartmentalized in vitro translation. To select for catalytic properties, the microbeads were re-emulsified in a reaction buffer containing a soluble substrate, and the product and any unreacted substrate were coupled to the beads when the reaction rate assay was complete. Product-coated beads, displaying active enzymes and the genes that encode them, were detected with anti-product antibodies and selected using flow cytometry. With this completely in vitro approach, Griffiths and Tawflik (2003) were able to select for substrate recognition, product formation, rate acceleration and turnover. Kim et al. (2001) simultaneously incorporated and adjusted functional elements within an existing enzyme by inserting, deleting, and substituting several active-site loops, followed by fine-tuning of catalytic properties by means of site-directed point mutation. They successfully introduced b-lactamase activity into the ab/ba-metallohydrolase scaffold of glyoxalase II, and the re-engineered enzyme lost its original activity and gained the ability to catalyze the hydrolysis of cefotaxime with a (kcat/Km)app value of 1.8  102 M1 s1. While this specificity constant value is rather low, Escherichia coli containing the redesigned enzyme exhibited 100 greater resistance to cefotaxime. The potential for extending these efforts by combining sitedirected-mutagenesis and chemical modification to improve the specificity of enzymes, especially those used by synthetic organic chemists, should not be underestimated (Jones and Desantis, 1998) (see also Section 2.3: Active Site Diversification).

An intriguing case of substrate specificity is the carboxylase/oxygenase, the CO2-fixing enzyme that exhibits relatively slow catalysis attributed to the need to discriminate between its substrates CO2 and O2. Tcherkez, Farquhar and Andrews (2006) argued that these characteristics arise from difficulty in specific binding of the structurally featureless CO2 molecule, forcing substrate specificity for CO2 versus O2 to be determined later (i.e., in the transition state). They suggest that natural selection for greater CO2/O2 discrimination, in response to reducing atmospheric [CO2]/[O2] concentration ratios, resulted in a transition state for CO2 addition that resembles a carboxylate group. This adaptation maximizes structural differences between transition states for carboxylation and oxygenation. However, the resulting increased similarity between the structure of the carboxylation transition state and its carboxyketone product exposes the carboxyketone to the strong binding needed to stabilize the transition state, causing the carboxyketone to bind so tightly that its cleavage to products is slowed. Tcherkez, Farquhar and Andrews (2006) suggested that such apparent compromises in catalytic efficiency for the sake of specificity represent a new type of evolutionarily perfected enzyme. Substrate specificity also reinforces the idea that enzymes are ideally suited for the synthesis and/or derivitization of drugs. Consider, for example, the studies of Khmelnitsky et al. (1997) focusing on the synthesis of water-soluble forms of paclitaxel (taxol), the potent anticancer drug that binds selectively to assembled microtubules. Scheme 1.12 shows that in the absence of any selective functional group protection, these investigators identified a two-step enzymatic process for selective acylation and deacylation. There are two potentially reactive hydroxyl groups (marked in red), but thermolysin selectively transfers the adipoyl moiety to only one, thereby preventing loss of biological activity by modification of the taxane ring. Likewise, only one of the two ester-linkages (marked in blue) is cleaved by the fungal lipase. Notice that both reactions occur in polar organic solvents. There is also good reason to believe that biochemists have not as yet identified all of the physiologically significant ligands – even for those enzymes already thought to be well characterized. The search for enzyme regulatory molecules is often hit-or-miss, as evidenced by the serendipitous discovery of the pivotally important allosteric effector Fructose-2, 6-P2 as well as the recent unanticipated development of synthetic glucokinase activators. In fact, we have no way to reckon just how many central pathway activators and inhibitors remain to be discovered. Moreover, although most enzymes are first discovered and isolated through the use of a well-defined activity assay, one can never be absolutely certain that a particular substrate is the physiologic substrate or that D-ribulose-1,5-bisphosphate

Chapter j 1 An Introduction to Enzyme Science H3 C

41

O

Ph

OH

O H3C

O

NH

O CH 3 CH 3

Ph

O

OH

HO O

CH 3 O

Ph

Divinyl Adipate in tert-Amyl Alcohol

Thermolysin (Salt-Activated) H3 C

O

Ph

OH

O H3 C

O

NH

O CH 3 CH 3

Ph

O

O C O

OH

CH 2 =CH—O-C(=O)—(CH2 )3—CH 2

O

CH 3 Ph

Acetonitrile (solvent)

O

Lipase Candida antarctica H3C

O

Ph

OH

O H3 C

O

NH

O CH 3 CH 3

Ph

OH

O

O C O

HOOC—(CH 2 )3—CH 2

O

CH 3 Ph

O

Scheme 1.12

other substrates are also metabolized. Many enzymes are selective in their action toward substrates and are only rarely exhibit absolute specificity. Nowhere is this statement truer than in the identification of the primary phosphoryl-acceptor substrate for the numerous signaltransducing protein kinases. An added issue is the phenomenon of ‘‘catalytic promiscuity’’ (see Section 2.3.2), wherein a single enzyme operates by more than one catalytic mechanism, giving rise to multiple enzymatic activities. Catalytic promiscuity increases the likelihood that we have unknowingly failed to identify many physiologically important reactions. Such concerns point the need for a far more comprehensive X-ray and NMR investigation of many, many more enzymes

to define the structures of their active sites and regulatory sites at atomic resolution. Consider the fact that the Protein Data Bank (PDB) presently lists some 56,000 structures, with nearly one-fourth of human origin. Some 49,000 structures were established by X-ray techniques, with 7,000 determined by NMR and fewer than 200 by EM. Also listed in the PDB are ~2,100 nucleic acid structures, with ~1,200 from X-ray analysis, ~900 from NMR, and > KA), then the equation reduces to the Michaelis-Menten equation. The steady-state expression is the same, except that Vmax,f ¼ k5k7[ET]/(k5 þ k6 þ k7), KA ¼ k2/k1, and KS ¼ [k4(k6 þ k7) þ k5k7)/(k3(k5 þ k6 þ k7)], where [P] >> 0 and the forward rate constants are k1, k3, k5, and k7 for the E þ A / E$A, the E$A þ S / E$S$A, the E$S$A / E$P$A, and the E$P$A / E$A þ P steps, respectively, and k2, k4, k6, and k8 are the corresponding rate constants for the E þ A ) E$A, the E$A þ S ) E$S$A, the E$S$A ) E$P$A, and the E$P$A / E$A þ P steps, respectively.

[A] =

x

1 2x

3x

1/ [Substrate]

7.1.3. Some Reversible Essential Activators Bind After the Substrate In this case, the essential activator binds only to the enzymesubstrate or enzyme-product binary complexes:

S

E

A

ES

A

EAS

EAP

7.1.4. Some Enzymes Randomly Bind Essential Activators and Substrate It is instructive to consider the behavior of an essential activator whose binding does not depend on the way in which the substrate binds (i.e., the activator and substrate bind randomly with respect to each other).

P

EQ

FIGURE 7.3 Kinetic behavior of an essential activator that binds after the substrate. Notice that the slope decreases at higher constant concentration of activator.

E

Scheme 7.2

A

P

S

EA

Such a mechanism is occasionally called substrateinduced activation. If all of the binding steps are rapid relative to interconversion of E$S$A to E$P$A, the initialrate equation based on the rapid-equilibrium assumption for this scheme is: v ¼

Vm ½A½S KA KS þ KA ½S þ ½A½S

E

EAS

EAP

A

A

E

EP

P

Scheme 7.3

7.4

where KS ¼ [E][S]/[E$S], KA ¼ [E$S][A]/[E$S$A], and Vm ¼ k5[ET] in which k5 is the forward rate constant for the E$S$A / E$P$A conversion. This expression differs from that obtained for the rapid-equilibrium case, inasmuch as the denominator for the equation in the earlier scheme contains a KS[A] term, but the above equation instead contains a KA[S] term. As shown in Fig. 7.3, a double-reciprocal plot of 1/v versus 1/[S] yields a series of straight lines intersecting at a common point in the second quadrant, to the left the vertical axis. This intersection point will have the horizontal-axis coordinate: 1/[S] ¼ –1/KS; and the verticalaxis coordinate: 1/v ¼ 1/Vm. The rapid-equilibrium treatment for the case where the essential activator binds second should therefore be readily distinguishable from the case where the activator must bind first.

EA

ES

S

A

The rapid-equilibrium expression for this scheme, in which the sole rate determining step is the E$S$A # E$P$A interconversion, is: v ¼

Vm ½A½S KA KS þ KA ½S þ KS ½A þ ½A½S

7.5

where Vm ¼ k9[ET], KiS ¼ [E][S]/[E$S], KA ¼ [E$S][A]/ [E$S$A], KS ¼ [E$A][S]/[E$S$A], and k9 is the forward rate constant for the E$S$A / E$P$A conversion. As shown in Fig. 7.4, plots of 1/v versus 1/[S] or 1/[A] yield a series of straight lines. The intercept replot of the 1/v versus 1/[A] data will provide a value for KS. The slope replots will then yield KiS. In the 1/v versus 1/[S] plot, the intercept replot will provide a value for KA and Vmax,f. In plots of 1/v versus 1/[S], the lines will intersect at a common point having the horizontal-axis coordinate: 1/[S] ¼ –1/KiS;

Chapter j 7 Factors Influencing Enzyme Activity

[A]=x

385

[S]=y 2x

1

2y

1 3x

1/[Substrate]

3y

1/ [Activator]

FIGURE 7.4 Kinetic behavior of an essential activator binding randomly relative to substrate. See text for details.

and the vertical axis coordinate: 1/v ¼ (1 – (KS/KiS)/Vmax,f. Hence, the point-of-intersection will be in the second quadrant if KS >> KiS, in the third quadrant if KS 1, indicating that the enzyme is more active in the activator’s presence. If a < 1, the above reaction scheme defines the action of a partial competitive inhibitor (see Section 8.2.1b), where the analogous equation has inhibitor I in place of activator A. The copper-containing enzyme dopamine b-monooxygenase (E.C.1.14.17.1) catalyzes the conversion of dopamine to norepinephrine. The dicarboxylic acid fumarate is a nonessential activator of the enzyme under steadystate conditions. Wimalasena et al. (2002) explained this activation effect in terms of an electrostatic interaction between the amine group of the enzyme-bound substrate and a carboxylate group of fumarate. In principle, we may extend this treatment of activator effects to include multi-substrate enzyme reactions, such as the following scheme. S1+ E

7.1.8. Enzyme Must Always Have Basal Activity when a Nonessential Activator is Absent An enzyme remains active in the absence of a non-essential inhibitor, suggesting that the enzyme does not have an

S1

A S1+ A E

E S1

S2

E S1 S2

A E S1

E+P

A

A S1

k

S2

A E S1 S2

Scheme 7.7

αk

E+P

388

As the number of reacting components increases, there is reduced likelihood that the system obeys the rapidequilibrium assumption, especially when one of the reactants is at low concentration. Kinetic mechanisms of this type may be difficult to examine quantitatively, almost hopelessly so for steady-state treatments. The problem is that the steady-state rate equation for a bisubstrate enzyme in the presence of a nonessential activator A contains [S1]2 and [S2]2 terms as well as higher-order [A] terms, and the equation is too complex for graphical analysis of activation constants. It is more gainful to examine the properties of various limiting mechanisms, such as when one or both of the substrates is maintained at a high, but nonsaturating concentration. Simplification by the rapid equilibrium method of Cha (1968) eliminates the [S1]2 and [S2]2 terms, but retains higher-order A terms. Dewolf and Segel (2000) showed that, if rapid equilibrium is assumed between free E, A, and E$A and for all but one other A-binding reaction, the resulting initial-rate equation for an ordered bisubstrate enzyme is first degree in all S1, S2, and A concentration terms in the absence of products. Although the equation is an approximation, it contains five activation constants associated with M, and all can be obtained by replots and/or curve-fitting procedures. The equation also allows limiting constants to be obtained (Vmax9, K9iS19, KmS19, and KmS19) for the enzyme operating at saturating concentrations of A. A similar approach allows researchers to treat activator effects for an enzyme operating by a steady-state Ping Pong reaction. Finally, nonessential enzyme activators always partially activate enzymes, rising from a basal activity vbasal to a limiting activity vlim. Even so, the amplitudes (vlim,x – vbasal) and (vlim,y – vbasal) for two different activators x and y need not be identical, even when x and y bind at the very same activator site. One activator may be lodged within its binding site in a manner that is more conducive to catalysis. Such behavior is routinely observed in pharmacology for the action of agonists on their respective allosteric receptors, the classical case being the muscarinic acetylcholine receptor (mAChR), which itself is not an ion channel, but is instead a GTP-protein-coupled receptor that activates ionic channels via second messenger action. In such systems, each agonist binding site is located at the subunit-subunit interface within the mAChR pentamer, and various agonists interact with different sub-site binding determinants. Agonist-receptor interactions must therefore be interpreted in terms of two fundamentally different parameters: affinity (measured by 1/Kd) and efficacy (symbolized by 3). The former defines how tightly an agonist is associated with its receptor, whereas the latter measures the receptor’s physiologic response to receptor-bound ligand. The basic idea is that, although two agonists may achieve an identical degree of binding-site occupancy, they may not elicit the very same physiologic response. In this respect, affinity and

Enzyme Kinetics

efficacy are independent properties, and the extent to which agonist binding and activation are tightly coupled can be determined by considering the dissociation constant Kd, which is obtained from the mid-point in the ligand binding isotherm, and the effective concentration EC50, which is obtained from the mid-point in the dose-response curve. Ehlert (1988) suggested that efficacy can be defined as: 3 ¼ (Emax/Emax,sys){1 þ Kd/EC50}/2, where Emax,sys and Emax represent the maximal functional response (e.g., membrane depolarization in the case of the acetylcholine receptor) in the system and the maximal response of the particular ligand tested, respectively. Only when Emax ¼ Emax,sys and Kd ¼ EC50 is site-occupancy a direct measure of agonist action, such that 3 ¼ 1. This concept of ligand efficacy, which is essential when considering the differential activation by two agents occupying the same binding site, may also be helpful in analyzing how nonhydrolyzable and slowly hydrolyzing ATP analogues operate in the ATP hydrolysis-dependent mechanoenzyme mechanisms considered in Chapter 13. Many mechanoenzymes have phosphoryl-sensors that detect the adenine nucleotide phosphorylation state, often by means of hydrogen bonding to nucleotide’s b and g phosphoryl groups. Therefore, while agents like p(NH)ppA and p(CH2)ppA are ATP isosteres, their efficacy is almost certainly different from the action of ATP.

7.1.9. 39,59-cyclic AMP Phosphodiesterase Activation by Ca2D-Calmodulin: A Thorough Kinetic Analysis Calmodulin, the universal calcium ion-regulatory protein, binds four Ca2þ ions and stimulates a number of enzymes, including 39,59-cyclic-AMP phosphodiesterase and Ca2þ-calmodulin-stimulated protein kinase. The former is a key enzyme that dampens the excito-regulatory properties of the adenylate cyclase (Reaction: MgATP2 # 39,59-cyclic-AMP þ MgPPi) and cyclic-AMP-stimulated protein kinase (Reaction: MgATP2 þ Protein # Phosphoprotein þ MgADP) cascade by catalyzing cyclic-AMP hydrolysis (Reaction: 39,59-cyclic-AMP þ H2O # 59AMP). Although a two-step mechanism (i.e., Ca2þ first binds to calmodulin, and the resulting complex in turn combines with and activates its target enzyme) had been thought to account for phosphodiesterase activation by calmodulin, such a model failed to account for phosphodiesterase interactions with the various calcium ion-bound states of calmodulin. In what may be regarded as a truly classic study in metabolic regulation, Huang et al. (1981) performed an ingenious kinetic investigation of 39, 59-cyclic-AMP phosphodiesterase activation as a function of the concentration of calmodulin and Ca2þ, thus defining a general approach for treating the kinetics of target enzyme activation by calmodulin. Ca2þ concentration was

Chapter j 7 Factors Influencing Enzyme Activity

389

buffered through the use of a Ca(II)EGTA complex, based on an association constant Kformation of 1.2  106 M1. Ka

Cm(Ca2+)4 K4 3.1x10-5M

10-10 M Kb

Cm(Ca2+)3 K3 3.1x10-5M

K2 2.7x10-6M

K1 7.5x10-6M

Cm

E-Cm(Ca2+)3

E-Cm(Ca2+)2 K2’

Kd

Cm(Ca2+)1

K4’

K3’

Kc

Cm(Ca2+)2

E-Cm(Ca2+)4

E-Cm(Ca2+)1 K1’

Ke -5

10 M

E-Cm

Scheme 7.8 Scheme 7.8 illustrates a model for the interactions between phosphodiesterase and the various forms of Calmodulin$Ca2þ complexes, abbreviated Cm(Ca2þ)i. The dimeric phosphodiesterase was treated as a monomeric enzyme, because no subunit cooperativity was detectable in the binding of the two molecules of calmodulin to the enzyme. Dissociation constants K1, K2, K3, and K4 for the respective Cm$Ca2þ complexes were determined by fluorescence techniques. The activation constant Ka for Cm$Ca2þ was determined by the activation of phosphodiesterase as a function of calmodulin concentration at a saturating concentration of Ca2þ and by measurement of the on- and off-rate constants of calmodulin. Huang et al. (1981) used Scheme 7.8 solely to describe the degree of saturation of Ca2þ for Calmodulin and Enzyme$Calmodulin complexes, and sequential binding of Ca2þ was not necessarily implied. Moreover, because phosphodiesterase assays were conducted at a saturating cyclic-AMP concentration, the symbol E actually represents the Enzyme$Substrate complex. For the limiting case where the Cm(Ca2þ)4 complex is the only enzyme-activating species, the operant rate equation is: Dv Dk½E$CmðCa2þ Þ4  ¼ ½ETotal  ½E þ ½E$Cm þ ½E$CmðCa2þ Þ þ ½E$CmðCa2þ Þ2  2þ

2þ

þ½E$CmðCa Þ3  þ ½E$CmðCa

Þ4 

7.7

in which Dv ¼ (vactivated – vbasal), and Dk is the difference between the catalytic rate constants for the activated and non-activated enzyme species. To simplify the order to their analysis, the researchers imposed the condition that [CmT] >> [ET], which resulted in the following equation shown in reciprocal form:

  1 1 f1 f3 K3 ¼ þ Dv DVm f2 f2 ½CmT 

7.8

where DVm ¼ Dk[ET], f1 ¼ 1 þ [Ca2þ]/K19 þ [Ca2þ]2/ K19K29 þ [Ca2þ]3/K19K29K39 þ [Ca2þ]3/K19K29K39K49, f2 ¼ [Ca2þ]4/K19K29K39K49, and f3 ¼ 1 þ [Ca2þ]/K1 þ [Ca2þ]2/ K1K2 þ [Ca2þ]3/K1K2K3 þ [Ca2þ]3/K1K2K3K49. When the free (uncomplexed) Ca2þ concentration is held constant by use of EGTA-Ca2þ buffer, Eqn. 7.6 predicts that a double reciprocal plot of 1/Dv versus 1/[CmT] will yield a straight line, a property that was verified over a calmodulin concentration range of 20 nM to 10 mM and a Ca2þ concentration range from 0.23 to 2.1 mM. Without delving further into the experimental details, the kinetic treatment of Huang et al. (1981) convincingly demonstrated that the binding of all four Ca2þ ions is necessary for calmodulin to form an activated complex with the enzyme. The regulatory and mechanistic advantages of having four Ca2þ sites on calmodulin lie in the fact that phosphodiesterase can be fully activated from its basal level by only a 10 increase in Ca2þ concentration. This sharp increase cannot be accomplished if calmodulin had only one Ca2þ binding site, where a 100 change in saturation would require 100 change in calcium ion concentration. The existence of four Ca2þ sites on calmodulin, therefore, provides a highly effective On/Offswitch for activating or deactivating phosphodiesterase over a narrow range of Ca2þ concentration. In fact, the calcium ion concentration for 50% activation decreases with increasing calmodulin concentration, implying that, depending on the concentrations of calmodulin in a particular cell, the Ca2þ concentration required for target enzyme activation may change. The above treatment is based on the assumption that all metal ion binding steps and all interactions of calmodulin with the phosphodiesterase are in rapid equilibrium. In some cells, the sub-mM Ca2þ concentrations may make one or more steps less likely to conform to the rapid equilibrium assumption. Moreover, the complexity of interactions would be far greater in order to account for interactions of the various forms of Cm$Ca2þ complexes when the phosphodiesterase operates at sub-saturating concentrations of its substrate cyclic-AMP.

7.1.10. The Method of Continuous Variation Analysis may be Used to Determine Activator Binding Site Number and Affinity The Method of Continuous Variation derives its name from the fact that a binding interaction may be quantitatively analyzed as a function of the mole fractions XA and XB of two interacting substances, say A and B, under conditions where the sum of the molar concentrations always remains constant. (Note that XA ¼ {[A]/([A] þ [B])} and XB ¼

Enzyme Kinetics

390

{[B]/([A] þ [B])}, such that (XA þ XB) ¼ 1.) When the concentration of the resulting complex (or some quantitative measure of such complexation) is plotted in this manner, the resulting graph is called a Job Plot. While originally employed to study metal ion interactions with multivalent chelators, the Job Plot also provides useful qualitative and quantitative information about proteinligand and protein-protein interactions. If an enzyme, for example, prefers to bind the one-to-one complex AB, then the enzymatic activity will be maximal at a mole fraction XA of 0.5 (i.e., the point at which A and B are present in a one-to-one stoichiometry). Similarly, if AB2 is the active species, then the enzyme will be most active at an XA value of 0.33. In this manner, the stoichiometry of binding may be readily determined. Job Plots always have ascending and descending limbs, and the curvature of the plot near its apex provide valuable information concerning the equilibrium constant for forming the active species. Huang (1982) provided a lucid account of how the method of continuous variation can be employed to explore fundamental issues in enzymic catalysis through the use of initial-rate data or equilibrium ligand binding data. He also showed that the correct binding ratio, indicated by the stoichiometry index n, is generally obtained when the total concentration of reactants CT is much greater than the equilibrium dissociation constants for the interacting species (Fig. 7.5). The technique can also assist in distinguishing between different cooperativity models. The stoichiometry for non-cooperative binding systems varies between unity and n, as CT increases; however, those systems exhibiting positive cooperation tend to yield high

Xmax=0.5

Xmax=0.5

1.1 Complex Tight Binding 0 1.0

Reactant A Reactant B

1.1 Complex Loose Binding 1.0 0

0 1.0

Xmax=0.66

1.0

Reactant A Reactant B

1.0 0

Xmax=0.33

1.2 Complex Loose Binding 0

Reactant A Reactant B

2.1 Complex Loose Binding 1.0 0

0 1.0

Reactant A Reactant B

1.0 0

FIGURE 7.5 Job plots for various stoichiometries of enzyme-ligand interactions. Notice that the sum of [A] and [B] equals a constant value, such that the mole fraction of A equals [A]/([A] þ [B]) and the mole fraction of B equals [B]/([A] þ [B]).

n values, even at low CT. The interested reader should consult Huang et al. (2003) for examples of how the continuous variation method can provide valuable insights about enzyme catalysis and control.

7.1.11. Time-Dependent Enzyme Activation Requires Special Treatment Most enzyme activation is treated as a rapid-equilibrium process, wherein the activator is assumed to bind rapidly to free enzyme E or enzyme-substrate complex E$S to form one or more thermodynamic enzyme-bound complexes (or E$A or E$A$S), the concentration of which determines the fraction of total enzyme that is catalytic active. In many instances, the rapid-equilibrium approximation is unjustified or is tacitly assumed to hold in the absence of any evidence. More rigorous treatments of the time-evolution of enzyme activation are required to account for the coupling of slow conformational maturation to steady-state processes. In principle, slow isomerizations between two conformational states, say E and E$A (or E$S and E$A$S), differing in catalytic activity can be treated in the same way that Frieden (1968) originally developed to hysteresis or lag-phase kinetics (see Section 11.10). In this case, hysteresis in activator binding would give rise to time-dependent enzyme activation. Ray and Hatfield (1970) analyzed activation as a series of time-dependent conformational isomerizations, starting from an initially inactive state F, through intermediates Xi, to its fully active catalytic conformation CC (e.g., F # X1 # X2 # Xi # Xn–1 # CC). A series of linear differential equations can be written for systems of this type. After setting all expressions for d(Xi)/dt equal to zero, the series of equations can be solved for (CC), either by a successive substitution procedure or by the determinant method. Since d(CC)/dt does not equal to zero, the expression for CC will contain this differential, together with ET, since under the specified conditions [F] þ [CC] ¼ [ET]. After rearranging the final expression for [CC], we obtain d[CC]/dt ¼ a[CC] ¼ b[ET], where coefficients a and b are obtained by the determinant methods described in Chapter 5. They reported that the lag phase or the product burst phase of an enzymatic reaction, where such phenomena are caused by the transition of one enzyme form to another, can be treated as a steady-state process if: (a) the accumulation of product during the time interval under study does not affect the enzymatic rate; (b) concentrations of activators, inhibitors, and substrates are maintained at sufficiently high levels so that the concentrations of all intermediate enzyme forms are small with respect to the total enzyme (i.e., [Xi] > [E]Total. In this particular situation, inequality should be written as [SH1]Total >> [E] Total, indicating that the steady-state assumption is very likely to fail if the pH is too acidic or has basic extremes, such that [SH1] Total z [E] Total.

7.3.6c. Some Inhibitor Effects are pH-Dependent Many enzymes are inhibited by agents, whose potency is pH-dependent. One possible explanation for such behavior is that the state of inhibitor ionization affects its ability to inhibit a target enzyme. Scheme 7.15 describes the action of a competitive inhibitor: (a) that is only effective in its deprotonated form; and (b) that, for the sake of simplicity

407

only, there are no pH-dependent changes in the key catalytic groups on the enzyme. EI IH

Kai H

+

–

I

–

Ki E+S

I = active inhibitor IH = inactive inhibitor

Km

EX

k

E + Product

Scheme 7.15 In this scheme, the Michaelis constant Km equals [E][S]/ [E$X], the enzyme-inhibitor dissociation constant Ki equals [E][I1]/[E$I1], and the inhibitor ionization constant Kai equals [Hþ][I1]/[IH]. The rate equation then becomes:     1 1 Km 1 ½I  þ 7.27 ¼  1þ v Vm V m ½S Ki;apparent where the apparent dissociation constant Ki,apparent is given by the equation:   ½Hþ  7.28 Ki;apparent ¼ Ki 1 þ Kai Notice that Ki,apparent becomes larger as the solution becomes more acidic. It can be shown that a plot of pKi versus pH will be flat when the pH > (pKi þ 1) and will have a unit slope at low pH. Low pH results in loss of inhibition.

7.3.7. Some Enzymes Undergo pH-Dependent Changes in Mechanism An underlying tenet of pH kinetic studies is that only the degree of enzyme, or in some cases substrate, ionization changes as a function of pH. This assumption requires three more aspects that: (1) there are no changes in the chemical mechanism (e.g., collapse of a tight ion pair between a carbonium ion and a stabilizing carboxylate anion); (2) there are no changes in kinetic mechanism (e.g., rate-limiting or rate-contributing step(s) are unaffected); and (3) the enzyme remains stable and all pH effects on enzyme activity are reversible (e.g., full restoration of original catalytic activity occurs whenever an enzyme-containing solution is changed to an experimental pH and then subsequently returned to a reference pH used for standardized activity measurements). Like many other phosphomonoester hydrolases, E. coli alkaline phosphatase catalyzes both phosphomonoester hydrolysis and transphosphorylation. In the latter reaction, phosphate is transferred to a hydroxyl group of a phosphoryl acceptor substrate via the phospho-serine intermediate formed during hydrolysis. The rate-determining step for the wild-type enzyme is pH-dependent. At alkaline pH, release of the product phosphate from the non-covalent enzyme-phosphate complex determines the reaction rate,

Enzyme Kinetics

408

whereas at acidic pH hydrolysis of the covalent enzyme phosphate complex controls the reaction rate. To assess the effect of active-site electrostatics on catalysis, Martin and Kantrowitz (1999) replaced the Lys-328 with a cysteine. The rate-determining step at pH 8.0 of the mutant enzyme was altered, such that hydrolysis of the covalent Enz-P became limiting, rather than phosphate release. The transphosphorylation activity of the Lys-328-Cys enzyme was also enhanced, whereas hydrolase activity was reduced relative to wild-type enzyme. The transphosphorylase/ hydrolase ratio was 28 greater for the mutant enzyme.

Reversible Range

Irreversible Range

Return to original pH after t = x

Add base

time Enz at pHref

Enz at pHexp for x min

Enz at pHref

Martin and Kantrowitz (1999) concluded that the (þ)-charge at residue-328 is at least partially responsible for maintaining the balance between the hydrolysis and transphosphorylation activities and plays an important role in determining the rate-limiting step of E. coli alkaline phosphatase. Warning: For kinetic analysis, the effect of pH on enzyme activity must be reversible. All enzyme rate equations for analyzing pH effects on enzymes are based on the tacit assumption that pH reversibly alters enzyme rate, and any journal reviewer worth her/his salt will require such evidence before publication is warranted. Figure 7.15 illustrates a useful protocol for testing the reversibility of pH effects on an enzyme. Note that the curve for pH-dependent changes in enzyme activity should follow the same curve whether it starts at the lower pH and raises the pH or vice versa. Although the incubation period in this example is 5 min, it is important to choose a time interval that best matches the experimental conditions employed in the actual pH-kinetic measurements. Finally, the rate of each reaction step in a multi-step kinetic scheme may, in principle, involve pH-dependent changes in the enzyme or substrate. This situation gives rise to the reasonable expectation that one component reaction may be the rate-determining step at a certain pH, whereas another component reaction may limit the overall reaction rate at some other pH.

pH

Irreversible Range

7.3.8. The pH Kinetics of Bisubstrate Enzymes can be Complex

Reversible Range

Add acid

Return to original pH after t = x

Some multisubstrate enzyme-catalyzed reactions are also amenable to pH kinetic analysis. In the simplest case, there is no pH-dependent change in the kinetic mechanism (i.e., an ordered mechanism remains ordered over the pH range investigated, a Ping Pong mechanism remains Ping Pong over the pH range investigated, etc.). Consider the TheorellChance kinetic mechanism, where the enzyme is distributed among three protonic states:

time Enz at pHref

Enz at pHexp

EA

E

Enz at pHref

k1[A]

H

pH FIGURE 7.15 Demonstration of reversibility of pH effects on an enzyme. Test of reversible regain of activity after timed exposure of enzyme to various experimental pH’s, followed by return to the reference pH for immediate activity assay. A, Effect of brief exposure to high pH. B, Effect of brief exposure to low pH. In this hypothetical case, the enzyme is irreversibly denatured with loss of catalytic activity, even after only 5 min at alkaline pH. Such experiments indicate the pH range over which an enzyme should be studied without introducing the confounding effects of irreversible inactivation.

K3

K1

K2

K4 H

E H

EQ

k4[P]

H

E

k6[Q]

K6

K2 H

H

EA H

K1 k5

H

EA k2

H

K5 k3[B]

H

E

E

EQ

E

EQ H

Scheme 7.16

H

Chapter j 7 Factors Influencing Enzyme Activity

where the dashed lines are unfilled protonation sites. (Note: For generality, the charge on each species is not specified.) Because substrates and products might influence the enzyme’s affinity for a proton, there are six different thermodynamic dissociation constants: K1 ¼ [E][H]/[EH1]; K2 ¼ [EH][H]/[EH2]; K3 ¼ [EA][H]/[EAH1]; K4 ¼ [EAH1][H]/[EAH2]; K5 ¼ [EQ][H]/[EQH1]; and K6 ¼ [EQH1][H]/[EQH2]. In this case, the rate equation is:    Vm ½Hþ  K5 Ka Kia Kb ½Hþ  K1 1þ ¼1 þ þ þ þ þ þ þ v ½A ½A½B K6 K2 ½H  ½H    Kb ½Hþ  K3 þ þ þ 1þ 7.29 ½B K4 ½H  This rate equation predicts that the functional dependencies in plots of 1/v versus 1/[A] and 1/v versus 1/[B] at different pH values or v versus pH at different A and B concentrations will be affected by the values of the different acid dissociation constants. As pointed out by Fromm (1975), a plot of Ka/KiaKb versus pH will yield a line that is pH-invariant, if the k2/k3 ratio remains constant. The same would not be true for other bisubstrate rate equations. Alberty (2008) showed how pKa values for the active-site functional groups and E$S complexes can be determined from kinetic experiments on enzymes catalyzing rapidequilibrium Bi Bi mechanisms, with and without the consumption of hydrogen ions. The rapid-equilibrium condition allows for the determination of thermodynamic pKa values, the apparent equilibrium constant K9 for the reaction catalyzed, as well as the number of protons consumed in the rate-determining reaction. Experimentally determined Michaelis constants can be corrected for the pKa values of the substrates A and B and products P and Q, making it possible to obtain the pKa values of E, E$A, E$B, E$A$B, E$Q, and E$P$Q complexes. The complexity of multisubstrate pH kinetics becomes even more complicated when one or more of the following conditions apply: (a) more than one protonic state is catalytically active; (b) slow conformational changes attend changes in protonation state of any ionizable species; (c) the kinetic mechanism changes (e.g., both branches of a random mechanism operate at one pH, but only one branch operates at a different pH); (d) the reaction produces or consumes a proton that alters the protonation of active-site functional groups; (e) one or both of the substrates gains or loses a proton over the pH range of interest; (f) the interactions of a substrate with a metal ion are substantially weakened or strengthened by a pH change; and/or (g) the ionizable groups are sufficiently near to each other that there is positive or negative cooperativity (non-ideality) in the protonation reactions. Although, in principle, rate equations that account for any combination of these possibilities can be written, there is sufficient kinetic ambiguity that the likelihood of definitive conclusions is dubious.

409

7.3.9. Brønsted Theory Explains Important Aspects of Acid/Base Catalysis Most nucleophilic, electrophilic, and redox reactions are strongly influenced by pH and are also subject to acid/base catalysis. In the Brønsted-Lowry treatment, proton transfer to the substrate is an essential in acid catalysis; on the other hand, proton transfer from the substrate is an essential feature of base catalysis (Bell, 1978; Brønsted, 1928; Brønsted and Pedersen, 1924; Klumpp, 1982; Marcus, 1968). Put simply, the substrate behaves as a base in acid catalysis, but as an acid in base catalysis (Bender, 1960). Catalysis by weak acids or bases often occurs by means of their solvent-related ions that are formed upon dissociation of the acid or base, albeit incomplete dissociation. (On a mole-for-mole basis, strong acid and bases form more dissociated ions in aqueous solution.) In the simplest case, the reaction rate is insensitive to the concentration of the undissociated acid (or base). When the only effective catalytic species are ions related to the solvent, the rate enhancement is said to be the consequence of specific acid catalysis (or specific base catalysis). In many reactions, however, reaction rate depends on the concentration of all proton donors (or all proton acceptors), and the rate expressions of these processes have the following form: n X kib ½Basei ½Reactant 7.30 vGeneral-Base ¼ i¼1

vGeneral-Acid ¼

n X j¼1

kja ½Acidi ½Reactant

7.31

For example, in an acetate buffer solution, the rate equation for a general acid/base catalyzed reaction can be written as: v ¼ kcat ½S ¼ fk0 þ kH ½H3 Oþ  þ kOH ½OH  þ kHOAc ½HOAc þ kOAc ½OAc g½S

7.32

where k0 indicates the water’s action either as an acid or base, depending on the nature of the reactant. Notice that certain terms in the above rate equation will be negligible at low or high pH, and the experimenter can also carry out the reaction in the absence or presence of acetic acid (or acetate), such that: At low pH and at ½HAc ¼ 0; v ¼ fk0 þ kH ½H3 Oþ g½S

At low pH and at ½HAcs0;

v ¼ fk0 þ kH ½H3 Oþ þ kHOAc ½HOAcg½S

7.33

7.34

At high pH and at ½HAc ¼ 0; v ¼ fk0 þ kOH ½OH g½S

7.35

At high pH and at ½HAcs0;

7.36

v ¼ fk0 þ kOH ½OH  þ kOAc ½OAc g½S

Enzyme Kinetics

410

½AL  ¼

½HAtotal Ka 1þ ½H3 Oþ  ½HAtotal ½H3 Oþ  1þ Ka

7.37

7.38

where Ka ¼ [Hþ][A]/[HA]. If the reaction rate v depends on [HA], a plot of k versus pH, the curve will be a downwardlydirected sigmoidal curve (highest at low pH), and a plot of log k versus pH, the curve will be a straight line (slope ¼ 1). On the other hand, if the reaction rate v depends on [A], a plot of k versus pH, the curve will be an upwardly directed sigmoidal curve (highest at high pH), and a plot of log k versus pH, the curve will be a straight line (slope ¼ þ1). Brønsted acid and base catalysis rate laws are linear free energy relationships of the form:   kHA qKHA a 7.39 Brønsted Acid Relation: ¼ G p p   kA qKA —b Brønsted Base Relation : ¼ G 7.40 q p where: (a) the Brønsted acid and base coefficients a and b and the parameter G depend on the given reaction, solvent, and temperature; (b) the rate constant kHA depends on HA concentration; (c) KHA is the acid dissociation constant; (d) p is a statistical factor representing the number of equivalent protons in HA; and (e) q is a statistical factor representing the number of equivalent basic sites in the conjugate base A. Rate data are plotted in the form log(kHA/p) ¼ log G þ a log(KHAq/p). General acid catalysis, where reactivity is dominated by proton availability, is characterized by a straight-line plot of the log[Catalytic rate constant] versus the log[dissociation constant] for a variety of acids. Likewise, in general base catalysis, where reactivity is dominated by the unavailability of protons, a plot of the log[Catalytic rate constant] versus the log[dissociation constant] for a variety of

binding-step chemical-step bobs þ bleaving-group leaving-group ¼ b

bbinding-step ¼ -step bchemical leaving-group ¼

7.41

dðlog Ka Þ dðpKleaving-group Þ

7.42

dðlog kchemical-step Þ dðpKleaving-group Þ

7.43

logkB

½HA ¼

bases yields a straight line. The slope of such curves determines the value for the Brønsted constant, a or b, which typically ranges between zero and one. A value of zero suggests that the transition state resembles the reactants. When the Brønsted constant approaches unity, the transition state will most likely resemble the products. If catalysis depends on the behavior of a particular acid, and not just proton availability, the phenomenon is called specific acid catalysis. If catalysis depends on the behavior of a particular base, and not just proton unavailability, the phenomenon is called specific base catalysis. One limitation in applying the Brønsted theory of acid and base catalysis is that various factors, such as solvation and ionic strength, alter the strength of an acid or base, thereby causing deviations from the predicted linear dependence in plots of log kHA versus pKa or log kB versus pKa (Fig. 7.16). In fact, solvation and ionic strength may in some instances change acidity more than catalytic effectiveness. Another concern in the unambiguous application of the Brønsted theory is that an enzyme’s substrate specificity/ selectivity can obscure the intrinsic effects of acidity or basicity on the observed enzyme-catalyzed rate. In principle, the binding and chemistry steps may be parsed as follows:

logkHA

Therefore, at low pH in the absence of any acetic acid, a plot of vobs versus [H3Oþ] results in a linear plot of slope kH and an intercept k0. Once kH and k0 are determined, the experimenter can repeat the low-pH rate measurements in the presence of a defined concentration of acetic acid; the resulting plot can now be analyzed to obtain k[HAc]. Likewise, at high pH in the absence of any acetate, a plot of vobs versus [HO] results in a linear plot of slope kOH and an intercept k0. Once kOH and k0 are determined, workers can repeat the high-pH rate measurements in the presence of a defined concentration of acetate; the resulting plot can now be analyzed to obtain kOAc . If k[Ac] or k[HAc] is zero, the general-acid (or general-base) involvement may be determined by examining the pH-dependence of the reaction. The conservation equation for a prototrophic species may be expressed as: [HA]Total ¼ [HA] þ [A], such that:

pKa

pKa

FIGURE 7.16 Graphical depiction of: (A) log kHA versus pKa and (B) log kB versus pKa. Note: The plot should be plotted as log(ka/p) versus log(qKa/p), where ka is the rate constant for general acid catalysis, Ka is the acid ionization constant, p is the number of equivalent protons on the acid, and q is the number of equivalent positions where a proton can be accepted in the conjugate base. An analogous Brønsted plot is used for analyzing general base catalysis: log(kb/q) is plotted versus log(pKb/q). In the absence of a linear correlation, one may infer that the rate-determining step does not involve general acid or general base catalysis, suggesting instead that the reaction operates by alternative mechanism. Usually, the alternative mechanism is nucleophilic catalysis, which can be substantiated further experimentally by product studies since catalyst would be incorporated in final product structure.

Chapter j 7 Factors Influencing Enzyme Activity

O

H O +

–

O

411

H

H

H

OH

H

P

–

OH Reactants

O

O

–

HO

OR

P

OH H

H+

–

+

P OR

O

Products

O

H

–

+ OR

O

–O

O

Dissociative Transition State

O O +

–

HO

OR –

O

Reactants

H

O

–

P

O

P OR

H+

O

+

O

P O

–

+ OHR

OH

–

Associative Transition State

Products

Scheme 7.17 Such behavior is discussed in the outstanding report by Hollfelder and Herschlag (1995), who examined the linear free energy relationships for phosphatase catalysis to understand whether phosphoryl-transfer enzymes might adopt associative-type transition states rather than the dissociative-type transition states observed in model chemical reactions. The features of these two reaction pathways are illustrated in Scheme 7.17, where the dissociative mechanism implies the transient formation of a metaphosphate-like species. These investigators realized that important clues would emerge by comparing the linear free energy relationships for nonenzymatic and enzymatic reactions, but only if the enzyme’s binding-site specificity did not perturb the chemical rate dependence on the intrinsic reactivity of a series of substrates. They recognized that the shallow binding groove and broad specificity of E. coli alkaline phosphatase made this phosphohydrolase ideally suited for such a study. A second requirement – namely that the actual chemical step is ratelimiting, was found to be satisfied by using aryl phosphorothioates as substrates. The following observations strongly suggested that the chemical step limits kcat/Km for the phosphorothioate substrates. First, enzyme-catalyzed hydrolysis is slower by factors 100 to 10,000 for phosphorothioates than for corresponding phosphate esters. Second, different viscosity dependencies were observed for phosphorothioate and phosphate esters, again suggesting that a different step is rate-limiting for these different classes of substrates. The occurrence of a viscosity effect for the phosphate ester reactions contrasted with no viscosity effect for phosphorothioate ester reactions. Third, ortho substituents decrease kcat/Km by a factor of 100 for phosphorothioate substrates, but have no significant effect for phosphate substrates, again suggesting that a different step is rate-limiting for these two classes of substrates.

Hollfelder and Herschlag (1995) therefore determined the dependence of the rate of enzyme-catalyzed cleavage for a series of substituted aryl phosphorothioates. The large negative values of bleaving-group ¼ 0.8 for the enzymatic reaction (measured as kcat/Km) and the bleaving-group ¼ 1.1 for the nonenzymatic hydrolysis reaction suggested that there is considerable dissociative character in both the enzymatic and nonenzymatic transition states. Despite the wide specificity of alkaline phosphatase, certain substrates deviate from the LFER, underscoring that extreme care is required in applying LFERs to enzymatic reactions. The large negative value of bleaving-group suggested that alkaline phosphatase can achieve substantial catalysis via a transition state exhibiting dissociative character. As shown in Scheme 7.18, the E. coli alkaline phosphatase active site contains a constellation of two tightly bound Zn2þ ions and an arginine residue that interact directly with the transition state (E$X)z. ASP327

ASP51 HIS412 HIS370

HIS331

ASP369 2+

Zn (II)

2+

Zn (I)

O P

O R

O

–

H N

SER102

O H

+

H ARG166

O

N H

NH

Scheme 7.18

Enzyme Kinetics

412

In addition to phosphate monoester substrates, this phosphatase hydrolyzes sulfate monoesters, albeit extremely slowly. The catalytic proficiency is at least 1010 times lower, amounting to ~13 kcal/mol of additional transition-state stabilization for phosphate ester hydrolysis (O’Brien and Herschlag, 1999). Phosphate monoester substrates are dianionic, whereas sulfate monoesters are mono-anionic, raising the question for Nikolic-Hughes, Rees and Herschlag (2004) of whether the overall charge difference between these substrates would result in their being handled differently within the highly charged active site. Accordingly, they determined kcat/Km for a series of aryl sulfate ester mono-anions to derive the Brønsted coefficient b for the leaving-group, and compared the value to that obtained previously for a series of aryl phosphorothioate ester di-anion substrates. Despite differences in substrate charge, the observed Brønsted coefficients for phosphatase-catalyzed aryl sulfate and aryl phosphorothioate hydrolysis (0.76  0.14 and 0.77  0.10, respectively) are strikingly similar, with steric effects being responsible for the uncertainties in these values. NikolicHughes, Rees and Herschlag (2004) noted that aryl sulfates and aryl phosphates react via similar loose transition states in solution. These observations suggest an apparent equivalency of the transition states for phosphorothioate and sulfate hydrolysis reactions at the AP active site. NikolicHughes, Rees and Herschlag (2004) also concluded that there were negligible effects of active-site electrostatic interactions on charge distribution in the transition state. Another interesting application of the Brønsted theory presents itself in the case of phosphotriesterase (Caldwell et al., 1991), a microbial enzyme that catalyzes the hydrolysis of a broad range phosphotriester substrates, including diethyl p-nitrophenyl-phosphate (paraoxon) and diethyl p-nitrophenyl-thiophosphate (parathion). These neurotoxic cholinesterase inhibitors are very effective substrates for phosphotriesterase. Liberation of one reaction product, p-nitrophenolate anion (pKa z 7), affords a convenient continuous spectrophotometric assay at 410 nm. O Et

O

P O

NO2

H2O

O Et

O

O Et

S Et

O

P O O Et

P OH + HO

NO2

O Et

NO2

H2O

S Et

O

P OH + HO

NO2

O Et

Scheme 7.19

When the two ethyl groups in paraoxon are replaced by propyl and butyl groups, the magnitudes of the maximal velocity and Km decrease substantially.

Caldwell et al. (1991) examined the catalytic mechanism of enzymatic hydrolysis using a series of paraoxon analogues. Brønsted plots relating the pKa of the leaving group to the observed kinetic parameters, Vm and Vm/Km, are both nonlinear. Such a finding is consistent with the idea that there is a change in the rate-limiting step, perhaps from chemical to physical events, as the pKa of the leaving group decreases. The magnitude of the rate constant k3 for the bond-breaking step depends on the pKa of the leaving group phenol, as predicted by the Brønsted equation (log k3 ¼ bpKa þ C) where b ¼ 1.8 and the constant C ¼ 17.7. The magnitude of b suggested that the transition state for substrate hydrolysis resembles the product.

7.4. BUFFER EFFECTS ON ENZYME KINETICS Most biochemical reactions are attended by the consumption or liberation of protons, a fact that necessitates the inclusion of a pH buffer in most enzyme rate measurements. For maximal buffer capacity, the buffer’s pKa should be roughly the same as the desired experimental pH, thus bringing about a minimal change in sample pH. Ideally, the buffer’s pK does not change much with changes in temperature, and only controls the hydrogen ion concentration. Moreover, while investigators often fastidiously verify the chemical purity of enzyme and substrates used in a study, they often ignore the possibility that commercial buffer salts are contaminated with metal ions and other chemically reactive species. Workers should not overlook the likely possibility that the enzyme may bind a species of the buffer that alters the kinetic properties of the enzymatic site. Alberty (2006) reminds us that when the pH of a buffer is changed, the [Bufferx]/[H$Buffer(x1)] ratio (e.g., for phosphate buffer, the researcher might work in the range where H2PO3 1 and HPO3 2 predominate) must also change, and the binding of these ion species to the enzyme may introduce confounding effects if the activating or inhibitory effects of Bufferx and H$Buffer(x1) should differ. Such behavior does not typically influence the thermodynamics of enzyme-catalyzed reactions because Bufferx and H$Buffer(x1) most often do not bind to the substrates or products. Alberty suggests that we may reduce these buffer effects in both kinetics and thermodynamics by using buffers for which the ionic strength is primarily supplied by NaCl or KCl. In most cases, monovalent cations and anions are relatively weak ligands. If a buffer is found to interact strongly with the enzyme or its substrate(s), the validity of the kinetic data may be questioned. Boric acid (H3BO3) is known to bind tightly to many carbohydrates, a property that is frequently exploited in the chromatographic separation of certain carbohydrates.

Chapter j 7 Factors Influencing Enzyme Activity

Therefore, this substance should not be employed as a buffer in enzyme assays of reactions involving carbohydrate substrates. Because many enzymes are glycosylated, the presence of boric acid might be expected to affect structural and kinetic properties of glycosylated enzymes. It should also be noted that certain synthetic buffers, known as Good’s buffers (Ferguson and Good, 1980; Good and Izawa, 1972; Good et al., 1966) in honor of their innovator, have been used extensively in biochemical studies, chiefly because they are reasonably inert and generally exhibit remarkably low affinity for metal ions. Finally, the ethane sulfonate moiety of many Good’s buffers is introduced synthetically by means of nucleophilic attack on chloroethanesulfonate. Raines et al. (1986) discovered that the latter tends to polymerize to form trace amounts of oligovinyl sulfonates, which turn out to be extremely potent inhibitors of certain RNases. Oligovinyl sulfonates are a contaminant in nearly all commercial preparations of the frequently used buffer 2-(N-morpholino)ethane sulfonate (MES).

7.4.1. Many Factors Influence the Choice of a Buffer In most biochemical experiments, the buffer is by far the most abundant chemical component. Except for maintaining pH, a buffer should exert no other effect on the biochemical process. The following practical advice is offered to facilitate selection of the most suitable buffer for a particular enzyme assay or biochemical experiment. Recommendation-1. Chose a buffer with a pKa value that is within 0.3–0.5 pH units of the anticipated experimental pH. Greatest buffering capacity1 is achieved when the pH ¼ pKa, and try to avoid carrying out experiments more than one pH unit away from the pKa. Another important consideration is that a buffer’s pKa often changes with temperature. For example, the d(pKa)/dT value for Tris buffer is 0.028 pH units/ C, indicating that a pH 8.00 Tris buffer will experience a reduction of 0.14 pH units with a temperature change of 5 C. We should also bear in mind that, if the assay solution is 1

That buffering capacity is maximal when pH ¼ pKa may be demonstrated as follows. Define d[X]/d(pH) as the buffering capacity, and let a equal the ratio {[A–]/([HA] þ [A])}, where HA is the acid and A is its conjugate base. Then, d[X]/d(pH) will equal k  da/ d(pH), where k is a proportionality constant. Expressing the Henderson-Hasselbalch equation (e.g., pH ¼ pKa þ log {[A]/[HA]}) in terms of a, we get: pH ¼ pKa þ log[a/(1  a)]. Differentiating, we get: d(pH) ¼ d(pKa) þ d(log[a/(1 – a)]) ¼ (1/2.3) d(ln(a/(1  a)) ¼ da/ [2.3a(1  a)]. Therefore, da/d(pH) ¼ 2.3a/(1  a) and d[X]/d(pH) ¼ constant  2.3a/(1  a). Therefore, the quantity a/(1  a) reaches a maximum value when a ¼ 0.5, precisely where pH ¼ pKa. Because the ratio a/(1  a) changes with pH, acceptable buffering capacity is observed typically over the pH range from (pKa  1) up to (pKa þ 1)].

413

prepared and stored at one temperature but assayed at another temperature, the pH will be in error. Recommendation-2. Purchase a high-purity buffer, one that is free of metal ions and inhibitors as well as one that is transparent in the UV/visible range. Simple buffers, such as imidazole and glutamate, bind metal ions (most often Ca2þ, Mg2þ, Mn2þ as well as heavy metal ions), which may be removed by re-crystallization from a reagentgrade organic solvent. This procedure also frequently removes UV-absorbing contaminants. In some cases, passage over ChelexÔ or another cation exchanger may be necessary before a buffer is employed in studies of enzymes exhibiting exquisite sensitivity to metal ion contaminants. When the highest quality buffer is required, purchasing buffer salts that are certified for use in high-performance liquid chromatography (often indicated as ‘‘HPLC-grade’’) should be considered. Recommendation-3. Confirm that the buffer is chemically inert under the experimental conditions used in your experiments. Some highly effective buffer ions (e.g., acetate, citrate, glutamate, phosphate, and succinate) are often substrates for contaminating enzymes present in a preparation of an enzyme of interest. For example, if the enzymatic reaction of interest requires NADþ or NADPþ, then use of glutamate may result in NADH or NADPH production if glutamate dehydrogenase is a contaminant within the primary or auxiliary enzyme preparations. Other buffers, such a Tris, have reactive groups that bind metal ions and can alter the apparent formation constants for metal-nucleotide complexes. Tris is also known to be a phosphoryl acceptor for some phosphotransferases. At low pH, cacodylate buffer oxidizes thiol groups, often inactivating enzymes requiring reduced thiols for catalytic activity. Recommendation-4. Determine if the buffer interacts with the enzyme of interest or its substrates. Because buffers obviously contain ionizable groups, they can often interact electrostatically with enzymes, often altering enzyme activity or its interactions with regulatory molecules. Recommendation-5. Choose conditions that minimize buffer interactions with other essential assay components, especially metal ions. Most buffers are highly effective, even when present at 25–50 mM. At higher concentrations, workers must deal with the effects of nonspecific interactions associated with the buffer binding to enzymes. Therefore, the experimenter should estimate the amount of protons produced or taken up in the reaction of interest. Then, with the aid of the Henderson-Hasselbalch equation (e.g., pH ¼ pKa þ log {[A]/[HA]}), calculate the lowest buffer concentration needed to maintain the pH within 0.02–0.03 pH units.

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414

TABLE 7.8 pKa Values and Temperature Dependencies for Selected Buffers pKa

d(pKa)/dT pH Unit/ C

Buffer

4.64

Acetic Acid

0.0002

5.28

Succinic Acid (pK2)

0

5.80

Citric Acid (pK3)

0

5.90

2-(N-Morpholino)ethanesulfonic Acid (MES)

6.01

3,3-Dimethylglutaric Acid

6.63

Arsenic Acid (pK2)

6.67

N-(2-Acetamido)-2-aminoethanesulfonic Acid (ACES)

6.84

Phosphoric Acid (pK2)

6.86

Piperazine-N,N9-bis(2-ethanesulfonic) Acid (PIPES)

0.011 0.006

0.0001 0.02

0.0028

0.0085

6.97

Imidazole

6.98

N,N-Bis(2-hydroxyethyl)-2-aminoethanesulfonic Acid) (BES)

7.02

3-(N-Morpholino)propanesulfonic Acid (MOPS)

7.27

N-Tris(hydroxymethyl)methyl-2-aminoethanesulfonic Acid (TES)

7.39

N-2-Hydroxyethylpiperazine-N’-2-ethanesulfonic Acid (HEPES)

7.68

N-Ethylmorpholine

7.78

Triethanolamine

8.00

Tris-hydroxymethyaminomethane (Tris)

8.09

Glycylglycine

10.05

3-(Cyclohexylamino)-1-propanesulfonic Acid (CAPS)

0.020

Charge 1

2

3

1

2

2

1

2

2 0

0.016

1

0.020

1

0.015

0.014

0.022

0.02

0.031

0.028

0.032

1

1 0

0 0 1

1

Sources: Ellis and Morrison (1982); Perrin and Dempsey (1974).

Recommendation-6. Choose a buffer that is not highly temperature-dependent. Listed in Table 7.8 are some commercially available buffers, along with their reported pKa values (at 25 C, unless otherwise noted) and their corresponding d(pK a)/dT values. Avoid using those buffers that are strongly temperaturedependent.

7.4.2. Biochemists Exploit Various Properties of Selected pH Buffers Whether of protein or nucleic origin, enzymes are polyelectrolytes, and their structures, stability, and interactions are intrinsically pH-dependent. Many of their substrates are also ionic, and as described throughout this book, enzymesubstrate interactions strongly depend on both pH and ionic strength. Various activating metal ions are Lewis acids that interact with both enzymes and substrates in ways that are influenced by pH or that alter the acid-base properties of catalytic groups. Therefore, as already noted in Chapter 4, there is a compelling need to control pH, and considerable thought must be given to the choice of buffers in all enzyme experimentation, particularly because some buffers bind to enzymes, whereas others bind metals. This section describes

the roles played by buffers in enzyme catalysis, the acidbase and metal ion properties of buffers, as well as the composition and use of pH buffers of constant ionic strength.

7.4.3. Some Buffers Actively Participate in Enzyme Catalysis For certain enzymes, the buffer itself participates actively in catalysis. Such is the case for carbonic anhydrase (CA), a Zn2þ-dependent enzyme whose catalytic efficiency is so great (kcat > 106 s1) that proton transfer out of the active site actually becomes rate-limiting. To appreciate the importance of solution-phase buffer salts on deprotonation of Enz–ZnII(H2O), consider Schemes 7.20 and 7.21, which highlight the determinative role of the metal ion in the acid/base chemistry so essential for catalysis. In the first scheme, we see the general active-site constellation of three imidazole groups supplied by His-94, His-96, and His-119 that are coordinated to the essential zinc ion, which takes the oxygen of a water molecule as its fourth ligand. A proton relay is created by the action of the b-carboxylate of nearby Glu-106 and the hydroxyl group of Thr-199.

Chapter j 7 Factors Influencing Enzyme Activity Thr-199 N

H O

O H Glu-106

C O

H

O

NH

N Zn2+

O

His-119

N O

N

HN

NH His-96

His-94

Scheme 7.20 As indicated in Scheme 7.20, carbonic anhydrase’s catalytic cycle may be viewed as occurring in two-stages – first, bicarbonate formation and departure, and second, regeneration of the active-site Enz–ZnII(OH). Stage-1: CO2 + E–ZnII(OH–) E–ZnII(HCO3–) + H2O Stage-2: E–ZnII(H2O)

415

exchanges of oxygen isotope between carbon dioxide and water. Because the exchange rate law does not include proton transfer steps, the effect is strictly on kcat, whereas kcat/Km is predicted to be unaffected. Notably, 3-(N-morpholino)propane sulfonate failed to restore catalytic activity of His-64-Ala CA-II, presumably because this buffer is too large or has the wrong shape/charge to penetrate the outer reaches of the enzyme’s active site. These buffer effects are akin to chemical rescue (see Section 7.15.5), a catalysis-enhancing phenomenon observed when an added acidic/basic solute reactivates an enzyme that suffered loss of catalytic activity upon removal of an essential functional group from the enzyme’s active site as a result of random mutation or site-directed mutagenesis (Toney and Kirsch, 1989). Finally, while 10–20 mM imidazole is sufficient to deprotonate E–ZnII(H2O) in the laboratory, precisely what naturally occurring metabolite fulfills this role in vivo remains a mystery.

E–ZnII(HCO3–) E–ZnII(H2O) + HCO3–

H+E–ZnII(OH–) + (B–H)+

Scheme 7.21 The active-site Zn2þ is a remarkably good Lewis acid that activates its bound water by reducing the water pKa to ~7 in most carbonic anhydrase isozymes, and nearer to pH 5 in others. In the first stage, the zinc-bound hydroxide ion reacts with carbon dioxide to produce zinc-bound bicarbonate anion that is displaced by a water molecule. In the second stage, a water proton must leave the active site to regenerate the starting catalyst Enz–ZnII(OH). At 298 K, Kwater equals 1.821  1016 M, and at blood pH, the hydroxide concentration is well below 108 M, far too low to deprotonate Enz–ZnII(H2O) at any appreciable rate. As shown by Silverman and Tu (1975), however, deprotonation of CA can be facilitated in the presence of a suitable buffer. For carbonic anhydrase, catalysis is optimal at 10–20 mM imidazole or methyl imidazole, where the buffer’s neutral form is present at concentrations that are five orders of magnitude higher than hydroxide ion. To explore their hypothesis that the His-64 imidazole group in the CA-II isozyme can also be an essential component of the proton wire (see Section 3.7.1: Water is a Uniquely Suited Solvent for Biochemical Processes) conducting the outbound proton, Tu et al. (1989) employed site-directed mutagenesis to generate His-64-Ala CA-II which exhibited a 25 reduced kcat, with no change in kcat/Km. The authors also produced convincing evidence for the role of buffer in carbonic anhydrase catalysis through their observation of an imidazole buffer-dependent enhancement in equilibrium

7.4.4. Some Rate Studies may Require Buffers of Constant Ionic Strength Nearly all pH-dependent processes are strongly influenced by ionic charge interactions, and failure to control solution ionic strength (see Section 7.5) is the single most pervasive error in experiments aimed at determining the pKa values of acid and base groups (see Section 7.5). This situation is particularly problematic for charged substrates that interact largely via charge neutralization, and failure to maintain a constant ionic strength over a pH range may have unpredictable effects on rate constants and composite kinetic constants, especially the specificity constant Vm/Km. More effort must be taken to calculate the ionic strength of the buffer component at each pH in the range of interest. Because thermodynamic pKa values are determined at zero ionic strength, this calculation requires the use of a practical pKa value, designated as pKa *, meaning the value at the ionic strength of the experiment. Because ionic strength I depends on the charge zi and concentration Ci of each ionic species (see Section 7.5), the practical pKa may be calculated as follows: pKa * ¼ pKa þ (2z þ1){[0.5 OI]/[1 þ OI] – 0.1 I}. Ellis and Morrison (1995) provided detailed procedures and software for calculating: (a) the practical pKa values that apply to the buffer components under a chosen set of experimental conditions; and (b) the ionic strength of reaction mixtures at various pH values and hence the amount of inert electrolyte (e.g., tetramethylammonium ion) that must be added to maintain the ionic strength at a constant value. Ellis and Morrison (1995) also describe the composition of two- and three-buffer systems that can be employed over much wider pH ranges than are feasible with solutions

Enzyme Kinetics

416

7.5. IONIC STRENGTH EFFECTS ON ENZYME KINETICS

containing a single buffer component (three-component systems are described in Table 7.9). Williams and Morrison (1981) used a mixed-buffer system consisting of acetic acid, MES, and Tris in their investigation of the pH kinetics of dihydrofolate reductase. Many investigators are beginning to recognize the advantage of evaluating kinetic parameters of an enzyme at its physiologic temperature, and not just a convenient laboratory temperature. Because acetic acid has a reasonable vapor pressure, mixed-buffers containing acetic acid are likely to prove unmanageable at 37 C. Finally, although these constant ionic strength buffer systems were chosen for their applicability to pH kinetic studies to enzymes, they are also invaluable when investigating pH effects on ligand binding as well as protein oligomerization and polymerization. Given the dominant role of ionic interactions in protein interactions of nucleic acids, buffering of pH, while maintaining constant ionic strength, is probably essential in pH rate studies.

Ionic interactions strongly influence the rates of chemical reactions involving ionic species. Protein and RNA enzymes are themselves polyelectrolytes, as are many of their substrates (e.g., organic acids and bases, phosphate mono-, di- and tri-esters, nucleotides, as well as other proteins, RNA, DNA, and biomembrane components). Their structures and interactions are strongly influenced by the ionic properties of solutions containing them. The presence of small ions also affects the solubilities of proteins and other charged macromolecules as well as the solution-phase solute composition near charged surfaces of biomembranes and biominerals. It is therefore essential to examine how ions and their charge affect enzymatic activity, especially in the context of the behavior of interactions between reacting molecules. Enzyme structures are, in many respects, stabilized through highly cooperative

TABLE 7.9 Three-Component Buffer Mixtures with Virtually Constant Ionic Strength Charge on Conjugate Base

pKa at I ¼ 0.1, and T ¼ 30 C

Concentration Yielding I ¼ 0.1

Useful pH Range

Components

4.2–7.9

Acetic Acid

–1

4.64

0.05

MES

–1

6.02

0.05

N-Ethylmorpholine

0

7.68

0.10

4.2–8.0

4.2–8.2

5.8–9.1

4.2–7.9

5.7–9.1

6.25–9.7

5.0–9.1

Acetic Acid

–1

4.64

0.05

MES

–1

6.02

0.05

Triethanolamine

0

7.78

0.10

Acetic Acid

–1

4.64

0.05

MES

–1

6.02

0.05

Trishydroxymethylamino-methane

0

8.00

0.10

MES

–1

6.02

0.05

TAPSO

–1

7.49

0.05

Diethanolamine

0

8.88

0.10

Acetic Acid

–1

4.64

0.10

Bis-Tris

0

6.32

0.05

Triethanolamine

0

7.76

0.05

MES

–1

6.02

0.10

N-Ethylmorpholine

0

7.68

0.051

Diethanolamine

0

8.88

0.051

ACES

–1

6.65

0.10

Tris

0

8.00

0.052 0.052

Ethanolamine

0

9.47

Succinic Acid

–1

5.28

0.033

Imidazole

0

6.97

0.044

Diethanolamine

0

8.88

0.044

The ionic strength varies by  8% over the indicated pH range. All buffer systems (except that containing succinate) are likely to be suitable for experiments on metal ion-nucleotide complexes. From Ellis and Morrison (1995).

Chapter j 7 Factors Influencing Enzyme Activity

417

interactions, exerted locally among numerous structural elements in nearly folded proteins.

to assure the system reaches equilibrium. At equilibrium, dGforward can be written as: dGforward ¼ mA Aeq þ mB Beq ¼ mC Ceq þ mD Deq ¼ dGreverse

7.5.1. Ionic Strength Defines a Solution’s Ionic Nature

Canceling the subscripts in each term, we arrive at:

One way to express a solution’s ionic content is by means of an ionic concentration function known as the ionic strength (symbol ¼ I). This parameter, which accounts for electrostatic interactions among charged substances in solutions, gives greater weight to ions having higher ionic charge: n X ci z2i 7.44 I ¼ 0:5 i¼1

The ionic strength I of a solution (expressed in units of Molarity) is a summation taken over all ions of charge zi (i.e., 1 or þ1 for monovalent ions, 2 or þ2 for divalent ions, etc.), each present at a molar concentration ci, as indicated by the following calculated values: (a) for 10 mM NaCl: I ¼ 0.5[0.01  (–1)2 + 0.01  (+1)2] ¼ 0.01 M (b) for 10 mM Na2SO4: I ¼ 0.5[0.01  (–2)2 + 0.02  (+1)2] ¼ 0.03 M (c) for 10 mM MgSO4: I ¼ 0.5[0.01  (–2)2 + 0.01  (+2)2] ¼ 0.04 M (d) for 10 mM Na3HATP: I ¼ 0.5[0.01  (–3)2 + 0.03  (+1)2] ¼ 0.06 M The squared-term in the ionic strength equation indicates that the behavior of ions in solution depends more strongly on the net charge of the dissolved ions than on their concentrations.

7.5.2. The Debye-Hu¨ckel Treatment Explains how Ions Alter the Thermodynamic Activity of Solutes While enzyme kineticists typically investigate the dependence of reaction rate on the analytical concentration of reactant(s), thermodynamic equilibria actually depend on the chemical potential of a species. This dependence is illustrated in the relationship: dG ¼

n X i¼1

mi dni

7.46

7.45

where dG is a change in the Gibbs free energy, mi is the chemical potential of the ith species, and dni is the change in moles of that substance, using plus and minus signs to signify its addition or removal from a solution. For a reversible equilibrium, say aA þ bB # cC þ dD, workers can make a solution by initially ‘‘adding’’ substances A and B in only minute amounts (say Ae and Be) or by ‘‘adding’’ products C and D, again in minute amounts (say Ce and De)

mA A þ mB B ¼ mC C þ mD D 7.47 We may then write out the chemical potential for each species:   7.48 mA ¼ mA þ RT lnðAA =AA Þ 











mB ¼ mB þ RT lnðAB =AB Þ

7.49

mC ¼ mC þ RT lnðAC =AC Þ

7.50

mD ¼ mD þ RT lnðAD =AD Þ

7.51

where mx  is the standard chemical potential of substance x, R is the universal gas constant, expressed in calories/mole or Joules/mole, T is the absolute temperature, and Ax is the activity (or the effective concentration) of substance x under standard conditions. To relate activity Ax to the molar concentration of a component, the proportionality constant gx (called the activity coefficient) is used, such that: Ax ¼ gx[A]. Attractive and repulsive interactions affect the value of g for each substance. The Debye-Hu¨ckel Limiting Law describes how the behavior of charged solutes within a solution depends upon the presence of other similar and/or dissimilar solutes, especially those that are ionic. The Limiting Law is a thermodynamic treatment for ions in aqueous solution and is based on the following simplifying assumptions:  Electrolytes are completely dissociated, and the behavior of sparingly soluble substances and precipitates of salts are not considered.  All ions are treated as point charges, the interactions of which are dominated by Coulomb’s Law: F ¼ q1q2/Dr2. F is the force of attraction/repulsion, q1 and q2 are the charges expressed in electrostatic units, r is the distance between the ions, and D is the dielectric constant of the medium.  Long-range interactions are considered to occur in a uniform medium, and the dielectric constant and viscosity of pure water are used in calculations.  Short-range forces such as ion-dipole interactions, ion pair formation, and van der Waals attractions are considered to be unimportant.  Thermal motions of ions disrupt any oriented inter-ionic attractions that can be defined by the Boltzmann distribution law. For simple monovalent cations and anions, the Limiting Law defines how the activity coefficient gi depends on the ionic strength I, such that: pffiffi 7.52 log gi ¼ Az2i I

Enzyme Kinetics

418

where the activity A may be written as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 e2 8pe2 N A ¼   2:303 2DkB T 1;000DkB T

TABLE 7.10 Effect of Ionic Strength on Single-Ion Activity Coefficient g

7.53

Here, N is Avogadro’s number, kB is the Boltzmann constant, and T is the absolute temperature. For water, A has a value of 0.512, such that: pffiffi 7.54 log gi ¼ 0:512z2i I By introducing the mean activity coefficient (g):

gzM þzN ¼ gzMM gzNN

Ionic Strength (0.01 M)

Ionic Strength (0.05 M)

Ionic Strength (0.10 M)

Hþ

0.91

0.86

0.83

Na

0.90

0.82

0.76

Cl

0.90

0.81

0.76

Mg2þ

0.69

0.51

0.45

Ca2þ

0.69

0.49

0.41

þ

7.55

Taking the negative logarithm of each side, we obtain: ðzM þ zN Þ log g ¼ zN log gM  zM log gN

Ionic Species

TABLE 7.11 Effect of Ionic Strength on the pKa Values of Weak Acids

7.56

Substituting the Debye-Hu¨ckel limiting law into each term on the right-hand side of the equation, and then rearranging, we get: pffiffi 7.57 log g ¼ 0:512zM zN I

using the absolute values of zM and zN. To illustrate how this equation is used, the value of g for a solution of 0.l F HCl can be calculated. Note that formality F is the number of gram-formula-weights per liter, indicating explicitly how the solution was prepared, whereas molarity M specifies the solution concentration of the actual species. The value of ionic strength is: X I ¼ 1=2 ci z2i ¼ 0:5fð0:1Þð1Þ2 þ ð0:1Þð1Þ2 g ¼ 0:5 f0:2g ¼ 0:1 pffiffi log g ¼ 0:512zM zN I ¼ 0:512ð1Þð1Þð0:1Þ1=2 ¼ 0:16

Taking the antilogarithm of each side, we get g ¼ 0.69, as compared to a measured activity of 0.80. The Debye-Hu¨ckel treatment is limited to ionic strengths below 0.2. To estimate the activity coefficient more accurately, workers must consider the fact that ions are not point charges, but instead have a finite size relative to the distance over which they interact electrostatically. This approach leads to the extended Debye-Hu ¨ ckel equation:   pffiffiffiffiffiffiffipffiffi pffiffiffi m½1 þ 50:3r DT I  7.58 — log g ¼ 0:512zM zN For pure water at 298 K, the dielectric constant D is 78.5, such that: npffiffiffi pffiffi o log g ¼ 0:512zM zN m½1 þ 0:328r I  7.59

where the adjustable parameter r is the radius of the hydrated ion in Angstro¨m units. Typical effects of ionic strength on the activity coefficients of monovalent ions are illustrated in Table 7.10, and the effects of ionic strength on the pKa values of mono-, diand tri-protic acids are illustrated in Table 7.11.

Ionic Strength (0.10 M)

Ionic Strength (0.25 M)

Acid

Constant

Ionic Strength (0.00 M)

Acetic Acid

Ka

4.75

4.54

4.47

ATP

K1

7.60

6.74

6.47

K2

4.68

4.04

3.83

K1

7.18

6.53

6.33

K2

4.36

3.93

3.79

K1

6.39

5.75

5.54

K2

4.76

4.33

4.19

KSH

8.38

8.16

8.09

ADP Citric Acid Cysteine SH Group PPi

K1

9.46

8.60

8.33

K2

6.72

6.08

5.87

K3

2.26

1.83

1.69

7.5.3. Changes in Ionic Strength can Alter the Magnitude of Rate Constants The dependence of a reaction rate constant on ionic strength is often termed the primary salt effect. According to transition-state theory, a rate constant k0 in some standard solution is related to k, the rate constant in a second solution, by the equation: g g g

k ¼ k0  A B z C 7.60

g

where gA, gB, and gC are the activity coefficients of the reactants, and gz is the activity coefficient of the transitions state complex in the standard solution. Applying the DebyeHu¨ckel theory (i.e., the activity p coefficient of an ion obeys ffiffi the relationship – log gi ¼ Az2i I , where A has the value that ranges from 0.509 to 0.512 in dilute aqueous solution at 298 K, we obtain: pffiffi k 7.61 log ¼ 2AzA zB I k0

Chapter j 7 Factors Influencing Enzyme Activity

This equation predicts that the rate of a reaction between like-charged ions should proceed more rapidly as the solution’s ionic strength is increased. Likewise, the rate of a reaction between unlike-charged ions should proceed more slowly as the solution’s ionic strength is increased. The qualitative effect is called ionic screening, wherein an increase in ionic strength has the effect of surrounding each cationic species by negatively charged ions, and likewise each anionic species by a positively charged ion. It should be emphasized that this treatment also applies to reactive species containing partial charges, such as a polarized carbonyl group. Therefore, a positive ionic strength effect (i.e., increased reaction rate as I is increased) indicates that the reacting species are of like charge, and require screening to promote reactivity. A negative ionic strength effect (i.e., decreased reaction rate as I is increased) indicates that the reacting species are of unlike charge, such that screening reduces their ability to find and react with each other. Thus, when two diffusing reactants collide, they first form an encounter complex, a fleeting intermediate that more often than not fails to proceed any further along the reaction coordinate, but occasionally progresses to form a true thermodynamic complex. When the apparent equilibrium constant of a reaction can be calculated from the kinetic parameters, the equilibrium composition can be calculated. Alberty (2006) considered such behavior to be remarkable because, although all the kinetic parameters depend on the properties of the enzymatic site, the reaction’s apparent equilibrium constant and the equilibrium reactant concentrations do not. He demonstrated that, for such behavior to be observed, the effects of ionic strength and pH on both the unoccupied enzymatic site and the occupied enzymatic site must cancel when the apparent equilibrium constant from rate constants for individual steps in the mechanism is calculated. Taking ionic strength effects as an example, the substrate Michaelis constant KS ¼ [E]eq  [S]eq/[E$X]eq becomes KSChem ¼ [E]eqgE  [S]eqgS/[E$X]eqgEX, and the product Michaelis constant KP ¼ [E]eq  [P]eq/[E$X]eq becomes KPChem ¼ [E]eqgE  [P]eqgP/[E$X]eqgEX. The activity coefficients of each species are represented by gS, gP, gE, and gEP, and the activity coefficients of S and P can be estimated using the extended Debye-Hu¨ckel equation in the ionic strength range zero to about 0.35 M. The chemical equilibrium constant K for the reaction S # P is equal to the ratio of KPChem/KSChem or [P]gP/[S]gS, and the equilibrium constant is independent of ionic strength. Activity coefficients are used in chemical thermodynamics, and chemical equilibrium constants K are independent of ionic strength, whereas biochemical thermodynamics is concerned with the apparent equilibrium constant and transformed thermodynamic properties of reactants, which are functions of ionic strength in addition to temperature and pH. Alberty has emphasized that this is essential, because biochemists have used various ionic strengths in thermodynamic and kinetic

419

studies (see also Section 3.11.4: Thermodynamics of Biochemical Systems: A Primer). the simple definition of ionic strength I ¼ 0.5 P Finally, ciz2i applies to small ions, but is insufficient when dealing with the ionic strength of macromolecular polyelectrolytes (e.g., proteins, nucleic acids, sulfated carbohydrates, etc.). The main reason is that the ionic charges of polyelectrolytes are distributed throughout the macromolecule, and not concentrated, as they are in micro ions. One approach toward estimating the effective ionic strength of a macromolecular polyelectrolyte is to estimate the concentration of well-behaved low-molecular-weight electrolytes that have an equivalent effect on a system whose properties are sensitive to ionic strength. An inherent problem with this approach is nonspecific binding of enzymes to macromolecular polyelectrolytes. Bloomfield, Crothers and Tinoco (1974) discuss the polyelectrolyte behavior of nucleic acids, their osmotic properties and Donnan effects, and the influence of ionic strength both on the structure and long-range interactions of nucleic acids.

7.5.4. Ionic Strength Alters Enzyme Catalysis Profoundly It is worthwhile to consider a few published examples indicating the extent to which ionic strength affects the rates and properties of enzymatic reactions. Ionic strength strongly affects the enzyme-DNA and enzyme-RNA interactions. Such is the case for the affinity of EcoRI endonuclease for 59-GAATTC-39 recognition sequence (Jack, Terry and Modrich, 1982; Jen-Jacobson et al., 1983; Terry et al., 1983; Wright, King and Modrich, 1989). This is the palindromic site that the enzyme seeks, as it scans DNA molecules by means of positionally correlated diffusion (Terry, Jack and Modrich, 1985). The Km and kcat with pBR322 DNA substrate increased by 1,000 and 15, respectively, when the ionic strength is increased from 0.059 to 0.23 M (Wright, Jack and Modrich, 1999). By contrast, their pre-steady-state kinetic analysis showed that recognition, as well as first and second strand cleavage events occurring once the nuclease arrives at the EcoRI site, are essentially insensitive to ionic strength. The differential effects of ionic strength on Km and kcat can be rationalized in terms of nonspecific endonuclease$DNA complexes in the reaction mechanism. At low to moderate ionic strength, positionally correlated facilitated diffusion, or ‘‘sliding,’’ plays a significant role in the mechanism by which EcoRI endonuclease reaches its recognition sequence. At an ionic strength of 0.08 M, the endonuclease appears to scan about 1,300 base-pairs per DNA binding event by a sliding-type mechanism. The effective chain length scanned in this manner decreases at higher ionic strength. Baerga-Ortiz, Rezale and Komives (2000) demonstrated that at physiological ionic strength, the association and

Enzyme Kinetics

420

dissociation rate constants for the reaction of thrombin with thrombomodulin were 6.7  106 M1 s1 and 0.03 s1, respectively, yielding an overall Kd of 5 nM. Surprisingly, a relatively modest change in ionic strength from 0.1 to 0.25 M (with either NaCl or tetramethylammonium chloride) caused a 10 change in kon, without any noticeable effect on koff. Such behavior is consistent with the idea that the interaction of thrombin with thrombomodulin is completely determined by electrostatic interactions. The copper, zinc superoxide dismutases (Cu,Zn SODs) catalyze the dismutation of superoxide at diffusion-limited rates. The persistent presence of four charged residues (Lys-120, Glu/Asp-130, Glu-131, and Lys-134) in many Cu,Zn SODs suggested that these enzymes employ attractive and repulsive electrostatic steering to guide its anionic substrate to the active site (Getzoff et al., 1992). However, by replacing these residues by Leu-120, Gln-130, Gln-131, and Thr-134, Ciriolo et al. (2001) made the surprising observation that, at two different pH values and ionic strengths, the mutant enzymes had a kcat that was higher than wild-type enzyme. Such findings challenge the idea that long-range electrostatics may not be as significant as previously inferred. Ionic strength is also a factor in enzyme oligomerization. For example, the association/dissociation equilibrium of the homodimeric p66/p66 form of HIV reverse transcriptase (Reaction: 2 p66 # (p66)2; Kd ¼ [p66]2/(p66)2) is affected by many factors, including enzyme concentration, salt concentration, pH, RNA template, and nucleoside 59triphosphate substrates (Cabodevilla et al., 2001). When the concentration of p66 is below 1 mM in a buffer possessing physiological salt concentration, the enzyme is mainly monomeric and essentially inactive. Increasing the sodium chloride concentration from 0.05 to 1 M had the effect of reducing Kd from 2.0 to 0.3 mM, resulting in a two-step kinetic mechanism for reactivation:

2(p66)inactive

(p66⋅p66)inactive

(p66⋅p66)active

Scheme 7.22 The presence of poly(rA)/dT20 resulted in a 30 reduction of Kd, whereas 5 mM dTTP was without effect. Another example of a salt effect on oligomerization is described by Kim et al. (2001) for the case of bacterial maltogenic a-amylase. This glycosidase (EC 3.2.1.133) catalyzes the hydrolysis of (1 / 4)-a-D-glucosidic linkages in polysaccharides so as to remove successive a-maltose residues from the non-reducing ends of the chains (Kim et al., 1992). Addition of potassium chloride shifted the monomer/dimer equilibrium toward the monomer, lowering its activity toward b-cyclodextrin, while activating its action on soluble starch (Kim et al., 2001). Finally, it is instructive to consider the significance of ionic strength effects on photophosphorylation. Based on the chemiosmotic model (Mitchell, 1961), the free energy

generated by respiratory or photosynthetic2 electron transfer reactions is stored electrochemically as a transmembrane proton motive force (pmf) that can be parsed into contributions from electric field and DpH: pmf ¼ Dji-o þ

2:303RT DpHi-o F

7.62

where Dji-o and DpHo-i represent the electric field and pH difference, R is the universal gas constant, T is the absolute temperature, and F is the Faraday constant (9.6485  104 coulomb/mol). As pointed out by Cruz et al. (2001), the kinetic and physiological consequences of storing pmf as Dji-o and DpHo-i are clearly different. First, Dji-o and DpHo-i are not kinetically equivalent driving forces for the ATP synthesis. For the bacterial and mitochondrial ATP synthases, Dj is the more effective component for maintaining the synthase’s catalytic turnover, whereas Dj and DpH contribute equally for the chloroplast synthase (Dimroth and Matthey, 1998; Fischer and Gra¨ber, 1999; Kaim and Dimroth, 1998). Dimroth (2000) has advanced the concept that some contribution from Dj is essential for respiration. For DpH to reach a significant level, one compartment must be maintained at a substantially different pH, a condition that could lead to acid or base inactivation of key enzymes and regulatory proteins in the compartment of altered pH. Therefore, mitochondrial and bacteria limit the magnitude of DpH by keeping Dj sufficiently high enough. For plants, photic energy results in several phases of pmf storage that are observed in such experimental systems as isolated thyalkoid membranes and the giant chloroplasts of Peperomia metallia (van Kooten, Snel and Vredenberg, 1986). In chloroplasts, photic energy is captured by light-harvesting complexes (LHCs) that funnel the energy into the reaction centers of Photosystem–I (PS–I) and Photosystem–II (PS–II). The absorbed energy drives electron transfer through the series of redox carriers in the electron transport chain, leading to NADPH formation and culminating in ATP synthesis by means of a transmembrane chemiosmotic circuit. The observed levels of DGATP, as well as the activation behavior of the CF1CF0–ATP, suggest a minimum transthylakoid pmf equivalent to a DpH of ~2.5 pH units. Kramer et al. (1999) presented evidence that the pH of the thylakoid lumen does not drop below 5.8, a finding that suggests Dj can contribute to steady-state pmf. Cruz et al. (2001) extended this work significantly by demonstrating that the plant cell, quite possibly at the level of the inner chloroplast envelope, uses regulated changes in ionic strength to determine the extent to which Dj and DpH contribute to the observed pmf. Their success came by examining the effects of varying ionic strength (as added KCl) on the 2

For the plant thylakoid membrane, the ‘‘outside’’ is the stroma, and the ‘‘inside’’ is the lumen of the chloroplast.

Chapter j 7 Factors Influencing Enzyme Activity

light-induced kinetics of the electrochromic shift of lightharvesting pigments (abbreviated as ECS). When a suspension of thylakoids were illuminated with 300 mmol of photons$m2$s1 at low ionic strength (5 mM), a rapid increase in the ECS was followed by a gradual decrease to a steady-state level, termed the ECSsteady-state, which persisted for more than 40 sec. As the potassium chloride concentration is increased to 50–100 mM, the extent of ECSsteady-state decreased and vanished, indicating a strong dependence on ionic strength. The authors also provided evidence that a substantial fraction of trans-thylakoid pmf is stored as Dj under in vivo conditions. Attesting to the reasonableness of their conclusion are results of simulations of the effect of ion permeability on the parsing of transthylakoid pmf into Dj and DpH.

7.5.5. There are Limits on the Applicability of Ionic Strength Ionic strength is treated as though only the species concentration ci and its corresponding electric charge zi are determinative. Experience shows, however, that the catalytic properties of many enzymes are strongly affected by the concentration of specific ions – not just ionic strength. Such behavior, known as an explicit ion effect, is likely to be a direct consequence of an ion’s hydration energy and its coordination chemistry, as well as the microenvironment of enzyme active sites and the nature of other dissolved solutes. The Hofmeister series, for example, is an empirically derived relationship describing the systematic effects of different ions on the solubility of proteins. The Hofmeister series ranks various anions (SO42 dianion > HPO42 dianion > Acetate monoanion > Chloride ion > Nitrate) and cations (Mg2þ > Liþ > Naþ z Kþ > NH4þ) in terms of their ability to precipitate a mixture of hen egg white proteins (Hofmeister, 1888). Because protein precipitation can be treated in terms of the extent of the ions binding to water, the effective concentration of the proteins increases in the remaining ‘‘free’’ water, such that precipitation is favored energetically by release of low-entropy surface water. Hofmeister series effects are now given in terms of the ability of anions and cations to stabilize protein structure. The most stabilizing anions form strongly hydrated anions, whereas the most destabilizing are weakly hydrated anions, such that the series from most to weakly stabilizing is: Citrate3 > SO4 2 > HPO4 2 > F1 > C1 > Br > I > NO3 . The opposite is true for cations (i.e., the most stabilizing anions form weakly hydrated cations, whereas the most destabilizing are strongly hydrated cations, such as the following series: N(CH3)4 þ > NH4 þ > Csþ > Rbþ > Kþ > Naþ > Hþ > Ca2þ > Mg2þ > Al3þ. Certain ions known as chaotropes increase the solubility of nonpolar entities. These ions (e.g., Liþ, Mg2þ, Ca2þ,

421

Ba2þ, I, ClO4 , SCN, and the guanidinium cation) are thought to bring about protein denaturation by disrupting hydrogen-bonding interactions. Thus, in enzyme-catalyzed reactions, the researcher must be concerned about the chaotropic nature of various ionic species. The major intracellular monovalent cations (e.g., potassium ion, arginine guanidinium ion, histidine imidazolium ion, and lysine e ammonium ion) are thought to act as chaotropes. Collins (1997) introduced the term kosmotrope to represent any small ion of high charge density that binds water molecules more strongly than the strength of water–water interactions in bulk solvent. The major intracellular anions (e.g., phosphates and carboxylates) are kosmotropes. They differ from chaotropes, which are large ions of low charge density and only weakly bind water molecules. Collins and Washabaugh (1985) indicate that the order of ionic species eluting from a Sephadex G-10 column corresponds to the known order of effectiveness ions in the Hofmeister series on protein solubility: SO42–

≈

HPO42– > F– > Cl– > I–

≈

ClO4– > SCN–

Scheme 7.23 The order of elution indicates that ion size and charge density strongly influence the apparent molecular size of hydrated anions, hence the effect on the order of elution of these ions in gel permeation experiments. Those species eluting before chloride ion are termed polar kosmotropes to designate their ability to enhance water structure in their vicinity and to stabilize proteins. Those ions eluding after chloride ion are termed chaotropes, and they reduce local water structure and destabilize proteins. Chloride ion is regarded as having little effect on water structure or protein stability over a concentration range from 0.1 to 0.7 M (Collins and Washabaugh, 1985). When membranes, particles, and large molecules make contact with water, the polarity of the solvent promotes the formation of an electrically charged interface (Shaw, 1980). The accumulation of charge can result from at least three mechanisms: (a) ionization of acid and/or base groups on the particle’s surface; (b) the adsorption of anions, cations, ampholytes, and/or protons; and (c) dissolution of ion-pairs that are discrete subunits of the crystalline particle, such as calcium$oxalate and calcium$phosphate complexes that are building blocks of kidney stone and bone crystal, respectively. Electric charging of the surface also influences how other solutes, ions, and water molecules are attracted to that surface. These interactions and the random thermal motion of ionic and polar solvent molecules establishes a diffuse part of what is termed the electric double layer, with the surface being the other part of this double layer. The GuoyChapman model describes the properties of the diffuse region of the double layer. This intuitive model assumes that

Enzyme Kinetics

422

counter-ions are point charges that obey a Boltzmann distribution, with highest concentration nearest the oppositely charged flat surface. The polar solvent is assumed to have the same dielectric constant within the diffuse region. The effective surface potential depends on the charge density of the surface and the ionic properties of the aqueous medium and its solutes. A more realistic case allows hydrated ions only to approach the surface at a finite distance, thereby defining what is called the Stern layer to honor the model’s originator When an aqueous solution flows tangentially with respect to an electrically charged surface, there remains a thin immobilized water layer known as the stagnant layer. This layer behaves like a two-dimensional gel, in which the ions can freely move, but nevertheless behaves macroscopically as a rigid body. The plane separating the stagnant layer and the mobile fluid above it is called the slip plane, and the electrokinetic potential z (measured in volts) is the electric potential at that plane. In its simplest form, the z-potential refers to the electrostatic potential generated by the accumulation of ions at the surface of a (colloidal) particle that is organized into an electrical double-layer, consisting of the Stern layer and the diffuse layer. The tangential transport of counter-ions can be modeled by statistical distribution functions like the Poisson-Boltzmann distribution. These effects affect the transfer of solute from the bulk solution to sites on an immobile surface, such as enzyme molecules tethered to a resin, gel, or large polymer (see Section 7.9: Effects on Enzyme Kinetic Behavior). Macromolecules also have a stagnant layer around them, and if the z-potential is zero, they tend to interact and aggregate. Nucleic acids represent another challenging class of ionic macromolecules, which may be treated as high-charge density cylindrical polyanions giving rise to distinctive effects of ionic strength on their reactions with other biological molecules. Those seeking a comprehensive treatment of salt and solute effects on biopolymer equilibria should consult Record, Zhang and Anderson (1998), who present a practical guide to recognizing and interpreting polyelectrolyte effects, Hofmeister series effects, as well as osmotic effects of salts. As they rigorously demonstrate, ligand-binding equilibria and conformational transitions of DNA and double-stranded RNA exhibit characteristic functional dependencies for binding constants Kobs and melting temperatures Tm on solution electrolyte concentration that differ significantly from those observed with simple low-molecular-weight electrolytes. Such a polyelectrolyte effect must be explicitly accounted for in experiments dealing with enzyme interactions with DNA and RNA interactions (and probably ribozyme reactions as well). The diverse range of salt effects on enzymes can be analyzed by the linked function treatment of Wyman (1948). We may also apply the treatment of Timasheff (1998) accounting for the stabilizing solute effects of solutes on proteins at elevated temperature. Finally, the Poisson-

Boltzmann’s equation may be employed to describe the equilibrium distribution of the electrolyte ions around an electrically charged ‘‘macro ion’’ (e.g., protein, nucleic acid, membrane, etc.) – see Section 3.6: Thermal Energy: The Boltzmann Distribution Law.

7.6. EFFECT OF ORGANIC SOLVENTS ON ENZYME ACTIVITY Some enzymes are known to operate effectively in nearly anhydrous solvents, a finding that seems to be at odds with the idea that water is vitally important for the structural integrity of proteins. Of course, the crucial question is not whether water plays a stabilizing role (it certainly does!), but how much water is necessary for an enzyme to remain

TABLE 7.12 Effect of Organic Solvents on LipaseCatalyzed Transesterification of Tributyrin and Heptanol

Solvent

Pancreatic Lipase Yeast Lipase Mold Lipase Reaction Rate Reaction Rate Reaction Rate mmol/hr-mg mmol/hr-mg mmol/hr-mg

Hexane

5.2

4.0

0.31

Dodecane

4.0

5.5

0.34

Hexadecane

2.9

6.3

0.32

Diethyl ether

5.1

0.1

0.12

Isopropyl ether

5.1

0.55

0.20

Butyl ether

4.7

2.5

0.20

Acetonitrile

2.2

0.04

0.04

Tetrahydrofuran

2.0

0.02

0.05

Dioxane

1.4

0.01

0.04

Toluene

2.1

0.95

0.08

Pyridine

1.3

0.01

0.02

0

0

0

Dimethyl sulfoxide

0

0

0

Carbon tetrachloride

Formamide

1.5

0.6

0.12

Acetone

1.2

0.02

0.06

2-Pentanone

1.1

0.03

0.06

2-Heptanone

1.2

0.08

0.10

A freeze-dried lipase powder (10 mg) was added to 1 ml of an organic solvent containing 0.3 M tributyrin and heptanol; the mixture was shaken at 20 C, and the time-course of the reaction was followed by gas chromatography. Organic solvents contained 0.02% (wt/wt) water, except for the toluene and carbon tetrachloride, in which that water concentration is not attainable and, therefore, it was even lower (our method of measurement does not afford an exact determination in that range). The water content of the lipases was 0.5%, 6.1%, and 4.8% for porcine pancreatic (which was precipitated from pH 8.4), yeast and mold, respectively. The lipases inactivated with diethyl p-nitrophenyl phosphate exhibited no enzymatic activity in all organic solvents. From Zaks and Klibanov (1985) with permission of the authors and publisher.

Chapter j 7 Factors Influencing Enzyme Activity

stable. In principle, protein solvation effects are exerted over a short range, and the stability of an enzyme is rarely influenced by water molecules lying beyond a few layers of solvent molecules from its surface. The properties of three lipases in various organic solvents (Table 7.12) were explored by Zaks and Klibanov (1985), who observed that these lipases retained their catalytic activity in solvents that were immiscible with water, but lost activity when placed in solvents like pyridine, acetone, and formamide. The latter are known to dehydrate the protein surfaces. Zaks and Klibanov (1985) demonstrated that different lipases remain effective catalysts, even in the presence of nearly waterimmiscible organic solvents. The catalytic power exhibited by the lipases in organic solvents is comparable to that displayed in water. In addition to transesterification, lipases can catalyze several other processes in organic media including esterification, aminolysis, acyl exchange, and thio-transesterification, such that several actually only proceed to an appreciable extent in nonaqueous solvents. In hexane, porcine pancreatic lipase, for example, still catalyzed various transesterification reactions and, in doing so, obeyed Michaelis-Menten kinetics. The enzyme’s pHdependence in organic media was also bell-shaped, showing maximum coinciding with the pH optimum for enzymatic activity in water. Zaks and Klibanov (1985) suggested that such behavior could be exploited in enzyme reactors to take advantage of: (a) the high solubility of most organic compounds in nonaqueous media; (b) the potential for carrying out new reactions impossible in water because of kinetic or thermodynamic restrictions; (c) the enhanced stability of enzymes; (d) the relative facility of product recovery from organic solvents, as compared to water; and (e) the insolubility of enzymes in organic media, thus facilitating enzyme recovery. A fascinating aspect of their findings is that hydrophobic solvents actually result in higher enzymatic activity than their hydrophilic counterparts (Klibanov, 2003). This property has been attributed to the presence of a few clusters of water molecules, which are stubbornly bound to charged groups on the surface of freeze-dried enzyme molecules and are required for enzymatic function. Hydrophilic solvents tend to strip some of this essential water off enzyme molecules, thereby lowering the catalytic power. This effect can be prevented, however, with the consequent restoration of enzymatic activity, by adding small quantities of exogenous water to the solvent (Zaks and Klibanov, 1988). Klibanov (2001) reviewed the literature showing that enzymatic selectivity (including substrate selectivity/specificity, stereo-selectivity, and chemo-selectivity) can be markedly affected, and even inverted in some cases, by the solvent. These effects were observed at ambient temperature, proving that these properties are not limited to the use as cryosolvents. While pH profoundly influences enzymatic activity in aqueous solution, pH has no physicochemical meaning in

423

organic solvents (Klibanov, 2001). However, when placed in organic media, enzymes and proteins exhibit a ‘‘pH memory,’’ meaning that their catalytic activity reflects the pH of the last aqueous solution to which they were exposed. This phenomenon appears to be due to the fact that a protein’s ionogenic groups retain their last ionization state upon dehydration and subsequent placement in organic solvents. Therefore, the enzymatic activity in such media can be much enhanced, sometimes hundreds of times, if enzymes are lyophilized from aqueous solutions of the pH optimal for catalysis. Despite the fact that lipases are admittedly stable at water-lipid interfaces, later studies confirmed that waterinsoluble organic solvents permitted enzyme activity (Table 7.13). When a lyophilized enzyme (which typically retains ~0.01% water, amounting to a few hundred water molecules per enzyme molecule) is suspended in a suitable anhydrous organic solvent, enzymatic activity is frequently retained. It is important to recognize, however, that the enzyme is suspended (not dissolved) in the organic solvent phase and behaves as a heterogeneous catalyst. To facilitate diffusion of reactants into the enzyme, the freeze-dried enzyme must be mixed vigorously, usually by sonication, to obtain finely dispersed small particles. Solvent polarity is believed to be the most important factor in balancing enzyme stabilization versus enzyme inactivation by certain organic solvents (Chaplin and Bucke, 1990). Those solvents with low polarity (i.e., greater hydrophobicity) tend to be less disruptive, because they do not strip the enzyme molecules of the tightly bound water molecules that play a major structure-stabilizing role. Perhaps the best measure of polarity is log10P, the logarithm

TABLE 7.13 Some Benefits of Conducting Enzymatic Catalysis in Organic Solvents 

Increased solubility of apolar reactants and certain cofactors.



Suppression of nonenzymatic (spontaneous) reactions and side-reactions.



Enhanced stability of some enzymes.



Altered selectivity of an enzymatic toward its substrates.



Enzymatic properties affected and modulated by the molecular ‘‘memory’’ of the enzyme sample (i.e., persistent retention of pH, and other solutes present prior to addition of the apolar organic solvent).



Permanent enzyme complexes with ligands insoluble in organic solvents; hence no dissociation of water-trapped solutes, even when enzyme normally exhibits low affinity for its substrate.



Reversal of hydrolases (Reaction: R–C(]O)–O–R9 # R–COOH þ R9–OH) to favor synthesis of apolar esters.



Altered enantiomeric enrichment, depending on the nature of the solvent.

Based on Klibanov (2003).

Enzyme Kinetics

424

of the partition coefficient P ¼ Sn-Octanol/Swater, where Sn-Octanol and Swater are the respective solubilities for a substance in n-octanol and water (i.e., log P ¼ log{Sn-Octanol/ Swater}). Typical log10 P values are: butanone, 0.3; ethyl acetate, 0.7; butanol, 0.8; diethyl ether, 0.8; methylene chloride (CH2Cl2), 1.4; di-isopropyl ether, 2.0; benzene, 2.0; chloroform, 2.2; toluene, 2.7; carbon tetrachloride, 2.8; dibutylether, 2.9; cyclohexane, 3.1; petroleum ether (degree of polymerization ¼ 60–80), 3.5; petroleum ether (degree of polymerization ¼ 80–100), 3.5; heptane, 3.5; petroleum ether (degree of polymerization ¼ 100–120), 4.3; and hexadecane, 8.7 (Chaplin and Bucke, 1990). A rule-of-thumb is that enzymes are most often inactivated by solvents having log10 P value less than 2, whereas solvents with log10 P values exceeding 4 have little effect. An interesting finding is that the presence of deuterium oxide tends to enhance protein stability to a greater degree than H2O. Chymotrypsin also catalyzes peptide hydrolysis in n-octane (Zaks and Klibanov, 1986), but the substrate specificity is altered. For example, while N-acetyl-L-histidine methyl ester is cleaved 200 more slowly than N-acetyl-L-phenylalanine ethyl ester in aqueous medium, it is cleaved 20 faster in n-octane. Enzyme-catalyzed condensation of carboxylic acids with amines and alcohols in the presence of a waterimmiscible organic solvent is useful for driving the syntheses of amides and esters that are intrinsically more soluble in organic solvents (Kobayashi and Adachi, 2004). To optimize reaction conditions for maximal product yield, knowledge of the reaction equilibrium and the log10P value for amide/ester are helpful. In some cases, watermiscible organic solvents may be used, especially when enzyme stability is a consideration. For example, Castillo and Lopez-Munguia (2004) reported that the synthesis of levan, using Bacillus subtilis levansucrase, occurs in the presence of the water-miscible solvents acetone, acetonitrile and 2-methyl-2-propanol (2M2P). Enzyme activity is only slightly affected by acetone and acetonitrile, but 2methyl-2-propanol has an activating effect. The enzyme is highly stable in water at 30 C; however, incubation in the presence of 15 and 50% (vol/vol) 2M2P reduced the halflife to 23.6 and 1.8 days, respectively. This effect is reversed in 83% 2-methyl-2-propanol, where a half-life of 11.8 days is observed. The presence of methyl-2-propanol in the system increases the transfer/hydrolysis ratio of levansucrase. As the reaction proceeds with 10% (w/v) sucrose in 50/50 water/methyl-2-propanol, sucrose is converted to levan and an aqueous two-phase system (2M2P/ levan) is formed and more sucrose can be added in a fed batch mode. It is shown that high-molecular-weight levan is obtained as a hydrogel and may be easily recovered from the reaction medium. When long-chain alcohols are used as organic solvents, workers must entertain the possibility of the alcoholysis of acyl intermediates. For example, long-chain alcohols may

replace water as the acyl group acceptor in reactions catalyzed by seine proteases and phosphomonoester hydrolases. Depending on the product quantified in rate experiments, alcoholysis of acyl intermediates may give the appearance of enhanced enzymatic activity. Castro and Knubovets (2003) have discussed the correlation between activity and structure of the intact enzymes suspended in neat organic solvents, as well as modifications of natural enzymes to favor catalytic active in non-aqueous environment. Bruns and Tiller (2005) also described what they called a ‘‘nanophase-separated amphiphilic network,’’ consisting of an enzyme entrapped within its hydrophilic domains. A substrate that diffuses into the hydrophobic phase of such a network can access the biocatalyst by way of the extremely large interface. Entrapped horseradish peroxidase and chloroperoxidase exhibited dramatically increased activity and operational stability compared to the native enzymes. Lyophilized salt-enzyme preparations exhibit activities that are increased by factors as high as 35,000 relative to activities of lyophilized salt-free preparations (Lindsay, Clark and Dordick, 2004). Among the factors associated with reduced enzyme activity in organic solvents are decreased enzyme stability and partial denaturation, reduced enzyme flexibility, incompatibility of solvent and enzyme transition states, over-stabilization of the substrate in its ground-state, and dehydration of the enzyme’s water shell. Using NMR spectroscopy, Eppler et al. (2006) determined deuterium spin relaxation rates to assess how salt affects the structure of lyophilized enzymes suspended in organic solvents. Such results suggest that the presence of added salt increases in enzyme-bound water mobility, serving as a molecular lubricant to enhance enzyme flexibility. This increased flexibility may facilitate transitionstate mobility and catalysis in the salt-activated enzyme. The range of organic solvents that support enzymatic activity has been greatly expanded through the advent of nonaqueous ionic liquids composed solely of a charged organic molecule and a suitable counter-ion. Two examples are 1-butyl-3-methylimidazolium cation and 1-ethyl-3methyimidazolium cation: O

F C

F F

S O

O

F O

C

F

CH3

F

N

S O

CH3 N

N

N H2C

O

H 2C

CH2 H2C

CH3

CH3

Nonaqueous Ionic Liquids

When combined with a suitable counter-ion, such as trifluoromethanesulfonate anion above, these imidazolium

Chapter j 7 Factors Influencing Enzyme Activity

7.7. TEMPERATURE EFFECTS ON ENZYME KINETICS Of all the factors affecting enzyme catalysis, temperature is doubtlessly the most challenging in terms of achieving an unambiguous interpretation. Temperature influences virtually every aspect of a biochemical reaction, including pHdependent ionization of enzyme and substrate(s), metalligand binding interactions, enzyme conformational changes, protein oligomerization, hydrogen bonding, hydrophobic interactions, transition states, etc. Such complex behavior is to be expected on the basis of the variety of bonding interactions and their temperaturedependent enthalpy and entropy changes. (For example, from the temperature-dependent changes in the pKa (as defined by the derivative of the Gibbs-Helmholtz equation, d(pKa)/dT ¼ –DH/2.303 RT2), a temperature change of 50 C can often change the pKa of an acid or base by a full pH unit.) Additional complexity arises from the sensitivity of protein folding/unfolding to changes in temperature. In fact, enzyme activity and stability are so strongly affected by temperature that kineticists try to minimize variation in reaction temperature.

7.7.1. Temperature Often Strongly Influences Enzyme Activity and Stability The reaction time-courses depicted in Fig. 7.17 allow us to make several generalizations about the behavior of enzymes

60° C 50° C

Product Formed

cations form ionic liquids in which many proteins are ‘‘soluble.’’ Irimescu and Kato (2004) reported lipase-catalyzed enantioselective reaction of amines with carboxylic acids in ionic liquids of this type. Because ionic solutions bind water tightly, hydrolases may catalyze dehydration when carried out in ionic solutions (e.g., R–COOH þ R9– OH # R–C(O)–O–R9 þ H2OY and R–COOH þ R9–NH2 # R–C(O)–NH-R9 þ H2OY, where the symbol Y indicates depletion of water by combination with an ionic solvent). In this respect, ionic solvents may facilitate peptide bond formation. Finally, by applying two-dimensional mean-field lattice theory to model enzyme immobilization and stabilization on a hydrophobic surface containing grafted polymers, such as polyethylene glycol, Moscovitz and Srebnik (2005) concluded that: (a) short hydrophilic grafted polymers should protect surface-immobilized enzymes from unwanted adsorption and denaturation upon contact with the hydrophobic surface; (b) screening is most effective when using a combination of short- and long-chain polymer grafts; and (c) grafted hydrophilic polymers should also protect enzymes when organic solvents are needed to solubilize substrates in the bulk solution phase.

425

70° C

40° C

80° C

20° C

90° C

10° C

Time

Time

FIGURE 7.17 Time-course of product formation for an enzyme assayed at various temperatures. Note that the initial rate roughly doubles for each 10 C rise over the 10 C–40 C temperature range. At high temperature, linear product formation falls off rapidly. Adapted from Dixon and Webb (1979).

operating at ambient and elevated temperatures. First, most enzymes exhibit increased catalytic activity over a relatively narrow temperature range (typically from 5 C up to 40–50 C), activity often doubling for every 10 C rise in temperature. While obviously consistent with transitionstate models for catalysis, such behavior also accords with the idea that enzyme flexibility is of paramount significance in catalysis. Second, if briefly exposed to elevated temperatures, many enzymes resist permanent loss of catalytic activity. Such durability is enhanced by: (a) judiciously choosing the solution pH; (b) adding a stabilizing ligand (e.g., substrate, cofactor, metal ion, regulatory effector); (c) protecting the enzyme against proteolysis; and (d) the presence of other proteins, the latter affording what has been vaguely described as a protective colloid effect. Of these, pH appears to be especially critical with respect to both enzyme activity and stability. As polyelectrolytes, enzymes are least soluble at or near their isoelectric points, where thermal unfolding is highly favorable. And third, the activity of most enzymes is permanently lost above 70–80 C, despite thermophilic enzymes and a few curious exceptions (e.g., adenylate kinase withstands prolonged exposure to 100 C, even at pH 1). In their classic treatise The Enzymes, Dixon and Webb (1958) argued that an enzyme’s stability depends on the position of the transition state for irreversible denaturation relative to the transition state for enzyme catalysis. Shown in Fig. 7.18 is a profile of enzyme activity versus temperature, showing an increase in enzyme activity with rising temperature. The system is reversible within the lower temperature range, but the enzyme is irreversibly inactivated at high temperature. The position of maximal activity and the reversibility of high-temperature effects depend on the exposure period. Brief exposure shifts the curve to the left, and longer periods of exposure to elevated temperature tend to shift the activity-versus-temperature plot to the right. The diagram shown in Panel B illustrates how temperatureinduced activation of catalysis and temperature-induced

Enzyme Kinetics

426

-A-

Rev

-B-

Irrev

Denaturation Transition State

∆Edenat

∆G

Catalysis Transition State

∆Ecat Denaturation Temperature

Temperature

Ground State

advances in manipulating the thermal stability of biological catalysts can be anticipated.

7.7.2. The Kinetics of Thermal Inactivation can be Treated Phenomenologically For many enzymes, denaturation obeys first-order decay, such that: d½N ¼ —kD ½N dt

Reaction Coordinate

FIGURE 7.18 Thermal enzyme inactivation. A, Typical behavior of enzyme activity v versus temperature. B, Diagram showing how temperature-induced activation of catalysis and temperature-induced inactivation depend on the energetics of the respective transition states for the two processes. Raising the temperature in the reversible range most often results in greater enzymatic activity, and lowering the temperature returns the enzyme’s activity to its original activity. When irreversible denaturation occurs, cooling to a lower temperature will not restore catalytic activity. The defining quality of enzymes – conformational flexibility – is thus the weakness of many enzymes with respect to thermal denaturation and irreversible loss of catalytic activity.

inactivation depends on the respective transition states for the two processes. This argument ignores the likelihood that much of the higher energy of the catalytic transition state may be related to the reactant’s configuration, such that the active enzyme proves to be more stable. Kintses et al. (2006) recently constructed a temperaturejump/stopped-flow apparatus that can mix reactants on a sub-millisec timescale with temperature-jumps as great as 60 C. They showed that enzyme reactions that proceed more rapidly than the rates of denaturation can now be investigated above denaturation temperatures. The same instrument also allows us to investigate at physiological temperature the mechanisms of many human enzymes that usually exhibit significant heat instability. Finally, in view of the tremendous utility of Taq and vent DNA polymerases in the Polymerase Chain Reaction technique, it is not surprising that so much time and treasure has been devoted to modifying the thermal stability of enzymes. These efforts have yielded the highly stable haloperoxidases found in color-safe laundry bleach, and they have provided new ways to perfect high-throughput bioreactors in the chemical process industry. That said, thermal lability occasionally offers some technical advantage. A good example is the use of shrimp alkaline phosphatase in place of its calf intestinal mucosal counterpart to produce dephosphorylated DNA vectors. Because the shrimp enzyme is rapidly deactivated at 65 C (t1/2 ¼ 30 s; 100% inactivation in 5 min), its use along with subsequent heat inactivation allows the shrewd molecular biologist to obtain ligation-ready vectors without need of an intervening step to remove partially active phosphatase. Given the ability to obtain high-resolution atomic-level structures of enzymes as well as the ease of site-directed mutagenesis, significant

7.63

where N is the native enzyme (i.e., the catalytically active form), kD is the first-order denaturation rate constant, and D represents the denatured end-product(s). In this case, exposure of the enzyme to a sufficiently high temperature should bring about a predictable decrease in catalytic activity, where a plot of ln{[N]t/[N]initial versus exposure time will be linear, with a slope of –kD. Should denaturation result in partial loss of enzymatic activity, the rate will drop to a residual activity residual. In such a case, the rate law becomes: d½N ¼ kD ðvnative  vresidual Þ dt

7.64

Exposure of the enzyme to a sufficiently high temperature should bring about a predictable decrease in catalytic activity, where a plot of ln{(vt – vresidual)/(vinitial – vresidual)} versus exposure time will be linear, with a slope of –kD, and a denaturation t1/2 ¼ 0.693/kD. The more general empirical rate equation for protein denaturation can be written as follows: vD ¼ kD ½Native Enzymen þ a½Hþ  þ b½OH 

7.65

where kD is the denaturation rate constant, the reaction order is n, and coefficients a and b are parameters applying to a particular enzyme. Although most often first-order, n can be as great as 1.5 (Dixon and Webb, 1958). Table 7.14 provides several estimated values for the activation energies for denaturation of several well-known

TABLE 7.14 Activation Energies for the Denaturation of Selected Proteins Protein

pH

DEact (kcal/mol)

Myosin ATPase

7.0

70

Ovalbumin

6.9

Reference Ouellet, Laidler and Morales (1952)

130

Lewis (1926)

Pancreatic Lipase 6.0

46

McGillivray (1930)

Pepsin

6.0

150

Casey and Laidler (1951)

Trypsin

6.5

41

Pace (1930)

Chapter j 7 Factors Influencing Enzyme Activity

enzymes and proteins. In general, if the activation energy for denaturation exceeds ~40 kcal/mol, little denaturation will occur until the temperature exceeds 60–70 C. Only when the activation energy for denaturation is in the range of 20 kcal/mol will the denaturation rate become appreciable around 35–50 C. Chaplin and Bucke (1990) treated thermal denaturation as a two-step, serial deactivation process involving two irreversible steps:

E

km

Em

kinact

Einact

Scheme 7.24 Their derivation is similar to that for any series first-order process (see Section 3.5.1: Series First-Order Processes), total enzyme concentration [ET] written as the sum of [E], [E1], and [E2]. They also introduced a fractional activity term A9, equal to: (a) {[E] þ A1[E1] þ Ai[Ei]}/[ET], if E1 retains some residual enzymatic activity, or (b) simply {[E] þ A1[E1]}/[ET], if E1 is completely inactive. The coefficients A1 and A2 indicate the enzymatic activity of Em and E2 relative to E (i.e., A1 < 1, if Em is less active than E, A1 ¼ 1, if E and Em are equally active, and A1 > 1, if Em is more active than E). The resulting equation:   A1 k1  A2 k2 A9 ¼ A 2 þ 1 þ expðk1 tÞ k 2  k1   A1 k1  A2 k2 7.66 expðk2 tÞ  k2  k1 allows us to plot the time-evolution of enzyme activity at elevated temperatures. As pointed out by Chaplin and Bucke (1990), although many thermal denaturation curves can be made to fit a model involving four determined parameters (A1, A2, k1 and k2), the researcher cannot be completely confident of the uniqueness of fit or the appropriateness of the model to a particular enzyme system. Moreover, k1 and k2 are apt to exhibit their own Arrhenius-type temperature dependencies, such that k1 ¼ k10 exp(–DEact/RT) and k2 ¼ k20 exp(–DEact/RT). A more realistic case allows the enzyme E to reversibly form a metastable species Em at elevated temperature, and Em then undergoes time-dependent, irreversible decay to the inactivated enzyme Einact:

E

km

Em

kinact

Einact

k–m Scheme 7.25 where Kmeta ¼ [Einact]/[Em] ¼ km/k–m, and kinact is the first-order inactivation rate constant. Note that DGmeta ¼

427

–RT ln [Einact]/[Em] ¼ –RT ln Kmeta, such that [Einact]/[Em] ¼ exp(–DGmeta/RT); likewise, kinact exhibits Arrhenius-type temperature dependence, such that kinact ¼ ki0 exp(–DEact/ RT). Each step can, in principle, exhibit its own temperature dependence. The reversible two-step treatment is more realistic and explains how Em can convert back to E, if the enzyme solution is returned to the lower temperature. This approach also allows us to incorporate other useful features, such as the ability of a reversibly bound ligand to stabilize an enzyme against thermal denaturation. In the simplest case, such a stabilizing ligand behaves in a manner resembling the action of a reversible competitive inhibitor. The researcher can likewise treat situations in which another agent may facilitate thermal denaturation by increasing: (a) k1; (b) the concentration of Em; (c) ki; or (d) any combination of these factors. This approach also offers the opportunity to account for the ability of low or high pH to enhance the rate of thermal inactivation. Finally, most enzymes are stable for months if refrigerated, preferably stored in an ice bucket to avoid cyclical variations in temperature in the refrigerated compartment (see Section 4.1.3c: Avoiding Enzyme Inactivation). The presence of additives, such as glycerol, trehalose, sucrose, and dimethylsulfoxide, often prevents freezing of samples cooled to below 0 C for enhanced stability. Freezing of enzyme solutions should be avoided, because it causes surface tension changes and pH variation accompanies ice crystal formation. If freezing is absolutely unavoidable, the sample should be cooled quickly in liquid nitrogen or another cryogen to facilitate formation of a ‘‘glass’’ phase instead of ice crystals.

7.7.3. Temperature Alters Both Equilibrium and Rate Constants Based on the Arrhenius and van’t Hoff treatments for chemical reactivity, the temperature effect on the equilibrium constant Keq ¼ k1/k2 of a reversible reaction: A # P, where the forward rate constant is k1, and the reverse rate constant is k2, can be written as: 2:303 log Keq ¼ B 

DEact RT

7.67

For two different temperatures T1 and T2, the above equation can be rewritten as follows:   k2 DEact 1 1 7.68   2:303 log ¼  k1 R T2 T1 Arrhenius also formulated a quantitative expression for the dependence of a rate constant on temperature: k ¼ A expðDEact Þ

7.69

Enzyme Kinetics

428

Aapp High ‡

Slope = ∆HHigh

2.1 cbz-GF

In k

log10k

Aapp Low ‡

Slope = ∆HLow

1.8 cbz-GW

1.4

1/T (absolute) 33

34

35

36

(1/T ) x 104 FIGURE 7.19 Effect of temperature on the carboxypeptidase catalysis. Enzymatic hydrolysis of cbz-Glycyl-L-phenylalanine (cbz-GF) and cbz-Glycyl-L-tryptophan (cbz-GW), yielding apparent activation energies of 9.6 kcal/mol and 9.9 kcal/mol, respectively (Lumry, Smith and Glantz, 1951).

where A is the so-called pre-exponential term (which obviously must the same units as k). When rewritten in logarithmic form, we obtain: 2:303 log k ¼ A 

DEact RT

7.70

For two different temperatures, the above equation can be rewritten as follows:   DEact 1 1  7.71 2:303 log k ¼   R T2 T1 where k1 and k2 are the observed rate constants at temperatures T1 and T2, respectively. A typical linear Arrhenius-type plot is shown in Fig. 7.19 for the action of carboxypeptidase on cbz-GlycylL-Phenylalanine and cbz-Glycyl-L-Tryptophan.

7.7.4. Many Nonlinear Arrhenius Plots can be Explained in Terms of ‘‘Rate Compensation’’ Nonlinear Arrhenius plots are inherently problematic, and parameters derived therefrom are often meaningless or misleading. The slope of any given tangent in an Arrhenius plot is DHapp z/R, and the point of intersection on the y-axis is ln(Aapp). Therefore, that one of these apparent values changes, if others change, must be expected. This so-called compensation effect arises from what has been called a ‘‘sympathetic variation’’ of the activation energy with ln A, as illustrated in Fig. 7.20. The reader will note that

FIGURE 7.20 Rate compensation effects in Arrhenius treatments. A nonlinear plot of ln(k) versus 1/K inevitably results in a series of slopes, depending on the segment of the curve that is analyzed.

a large apparent activation enthalpy leads to a large apparent pre-factor and vice versa. This means that the nature of a nonlinear Arrhenius plot is inherently complex and cannot be unambiguously interpreted. There are several likely explanations for nonlinear Arrhenius-type plots. First, the reaction may exhibit a significant temperature-dependent change in the constant pressure heat capacity DCp, such that DCp/DT s 0. Second, there may be a change in the rate-determining step over the chosen temperature range. On the basis of classical transition-state theory (Section 3.6), each rate constant in a multistep mechanism has its own temperature dependence. Because steady-state rate parameters almost inevitably consist of a rate constant for more than a single elementary reaction, temperature effects on steady-state parameters (i.e., Km, Vm, and even Vm/Km) only rarely provide reliable mechanistic information. In principle, fast reaction kinetics studies can be employed to define the thermodynamic properties of each elementary reaction within a multi-step enzyme mechanism. Finally, the system may exhibit irreversible enzyme inactivation, such that [ET] is not constant over the chosen temperature range. It is therefore of paramount importance to demonstrate the reversibility of any temperature effect. The easiest way to accomplish this check is to expose the enzyme to each experimental temperature for a period equal to that required for an activity determination and then to assay activity after the enzyme samples are cooled to the original temperature. If the effects of elevated temperature are reversible, then all samples should regain the same enzymatic activity. In any case, the watchwords for interpreting nonlinear Arrhenius plots are caveat emptor! The idea that Arrhenius-type behavior can be reliably ascribed to the enthalpy of activation for a rate-determining bond-breaking step is also questionable. In many

Chapter j 7 Factors Influencing Enzyme Activity

biochemical processes, hydrophobic interactions are known to play a major role in enzyme-substrate complex formation, such that the –TDS term has a dominant effect on the Gibbs free energy change. Therefore, a temperaturedependent increase in reaction rate may reflect a bondbreaking or bond-making step. A good analogy is that, while knowledge of the CD spectrum of individual amino acid residues can, in many instances, be used to calculate the protein’s composite CD spectrum, the opposite exercise is meaningless. The following quote from a recent review by Kraut, Carroll and Herschlag (2003) makes these points succinctly: In principle, with sufficiently accurate potentials and advanced computational skills, workers could model the system and fully describe its thermodynamics. But even if this were possible, it is not clear what would be gained conceptually. The entropy change is one number, and it represents an enormously complex sum of terms. Although these brute force approaches would certainly be useful, there is a more immediate need of conceptual advances to solve this central but vexing problem. Is it possible to understand configurational entropy and/or a positioning term that describes how preorganization of the various enzymatic components provide a rate advantage? Can the mixing of thermodynamic terms in enthalpy/entropy compensation that occurs in practice be meaningfully deconvoluted?

7.7.5. The Q10 Parameter is a SemiQuantitative Measure of an Enzyme’s Sensitivity to Changes in Temperature The temperature-sensitivity of reaction rate for many chemical processes occurring in aqueous solution is such that a typical reaction experiences a doubling in rate for each 10 K rise in temperature. Such behavior, which was first noted by Harcourt (1867) in his studies of hydrogen iodide oxidation by hydrogen peroxide, also represents a useful rule-of-thumb for most enzymatic processes. The Q10 temperature quotient is the ratio of enzyme reaction rates vT þ 10/vT, where the substrates indicate two temperatures differing by 10 C. The energy of activation, DEact, for a reaction rate constant k, equals RT2{d(ln k)/dT}, where R is the universal gas constant and T is the absolute temperature. For a temperature increase of 10 C, DEact ¼ {RT(T þ 10)ln Q10}/10. Because a DT of 10 K is significantly less than 310 K (human body temperature), this relationship reduces to DEact ¼ {RT2 ln Q10}/10, confirming that the Q10 parameter should be temperature-dependent. Most enzymes have Q10 values near 2, as exemplified by fumarase (Massey, 1953), for which Q10 is 1.994 at 30 C. By contrast, rat liver mitochondrial ATPase has an extraordinarily large Q10, reaching a value of 29 at temperatures below 22 C, or 295 K (Raison, 1973). This parameter is usually evaluated at saturating concentrations

429

of substrate(s), such that temperature-dependent changes in Michaelis constant(s) are inconsequential. The Q10 value is a characteristic property of a particular gene product from a specific organism. Therefore, the Q10 value should never be used for one hexokinase from yeast to infer the temperature dependence of another hexokinase, say from rat brain. On the other hand, the Q10 value need not remain unchanged for mutant and wild-type enzymes from the same source. Finally, Wolfenden et al. (1999) reasoned that if the generalization that reaction rate doubles for a DT of 10 K were true, then the enthalpies of activation DHz would be similar for enzymatic and their nonenzymatic reference reactions, implying that the catalytic effect of enzymes arises from their ability to increase the entropy of activation. That this is not the case is indicated by the data listed in Table 7.15. They showed that very slow reactions tend to be much more sensitive to temperature than that implied by a doubling of reaction rate for a DT of 10 K, and rate enhancements for many enzyme-catalyzed reactions increase sharply with decreasing temperature. They also convincingly demonstrated that such behavior bears significantly on the expected thermodynamic properties of transition-state analogue inhibitors, and is also of interest in considering the attractive forces that are apt to drive catalysis.

7.7.6. Certain Organisms have their Own Characteristic Physiologic Temperature Living organisms thrive in a diverse range of thermal niches. The human body, for example, operates at 37.3 C, but the temperature at any given time can range from 34–42 C, depending on the heat produced through metabolic and physical activity and that lost through evaporation, radiation, convection, and conduction. The temperature of a human remains reasonably constant throughout its body, but extreme exercise (especially in bursts of anaerobic exercise) can raise the local temperature of skeletal muscle by 3–4 C. The fact that most skeletal muscle proteins are

TABLE 7.15 Activation Enthalpies for Enzyme-Catalyzed and Nonenzymatic Reactions DHz for kcat

DHz for knonenz

Yeast OMP decarboxylase

11.0

44.4

Jackbean urease

9.9

32

Reaction

Bacterial a-glucosidase

10.5

29.7

Staphylococcal nuclease

10.8

25.9

Chymotrypsin

8.6

24.4

Chorismate mutase

12.7

20.7

Source: Wolfenden et al. (1999).

Enzyme Kinetics

430

especially rich in hydrophobic amino acid side-chains probably accounts for their added stability for short-term exposure to elevated temperature. The nominal body temperature of a few other homeotherms (i.e., warm-blooded animals) are: Rhesus macaque, 38 C; whales, 35 C; horse, 38 C; cow, 38 C, guinea pig, 38 C; rabbits, sheep and cats, 38–39 C and goats, 39.5 C. The body temperature of poikilotherms (i.e., cold-blooded animals) depends on seasonal temperature variations: they bask at warm temperatures and become inactive in cold. Desert poikilotherms adjust to temperature variation on a daily basis, and insects do the same in temperate climes. Certain organisms, like the bumblebee, have specialized thermogenic mechanisms for rapidly raising the temperature of the flight muscle to its optimal temperature for initial flight. For example, simultaneous operation of phosphofructokinase (Reaction: Fructose 1-phosphate þ MgATP2 # Fructose 1,6-bisphosphate þ MgADP) and fructose bisphosphatase (Reaction: Fructose 1,6-bisphosphate þ H2O # Fructose 1-phosphate þ Phosphate) temporarily creates an unbridled ATPase (Reaction: MgATP2 þ H2O / MgADP þ Phosphate þ Heat). The temperature of the bumblebee’s flight muscle can increase by 5–8 C in a matter of only a few seconds. Bacteria operate over a much broader temperature range (Table 7.16). Psychrophilic bacteria operate over the range from 3 to 20 C (optimal at 15 C); mesophilic bacteria operate over 10 to 45 C (optimal at 35 to 40 C); moderate thermophiles operate over 40 to 70 C (optimal at 55 to 65 C); extreme thermophiles operate over 65 to 95 C (optimal at 85 to 90 C); extreme thermophilic archaeon operate over 90 to 121 C (optimal at 110 to 115 C). The essential thermostability of Taq DNA polymerase greatly facilitates the Polymerase Chain Reaction technique, and ribosomes from thermophiles permitted the first X-ray structural analysis of the cell’s protein synthetic machinery.

With mounting interest in employing enzymes in bioreactors (see Section 7.9), the eagerness of biochemists to explore thermophilic enzymes is not surprising. Given these optimal temperature ranges and in view of the aforementioned Q10 temperature coefficient, there is an obvious need to conduct enzyme kinetic studies at physiologically relevant temperatures. For many years, the temperature chosen for a particular study often depended on the laboratory temperature. Enzyme rate studies were conducted at 20 C – ambient laboratory conditions in Stockholm, Copenhagen, or Berlin, but a far cry from 37.3 C. The advent of the bimetallic thermostat and simple feedback circuitry revolutionized temperature control, allowing us to achieve the desired temperature to within 0.1 C. Even so, most investigators knowingly work at an arbitrary temperature, typically around 30 C, which includes the study of human enzymes. A major consideration is the difficulty in maintaining constant reaction volume in test tubes at or above 37 C. In the absence of a mechanical device that constantly inverts the sample, even a sealed tube will lose water vapor to the surrounding walls and ceiling of a closed vessel. In most cases, mechanical stirring is without avail, but for experiments employing test tubes and cuvettes, the use of a close fitting, polymeric foam float can greatly reduce volume loss.

7.7.7. Extremophilic Enzymes have Unusual Structural Stability Extremophiles are organisms that grow optimally in extreme environments of temperature below 10 C or above 80 C, salinity reaching that of saturated NaCl, pH below 4 or above 9, and/or pressures above 400 atmospheres. The enzymes within extremophiles must be rugged to resist denaturation or even partial unfolding (Daniel, Danson and

TABLE 7.16 Effect of Temperature on the Growth Rate of Bacteria of Special Interest Optimal Temperature for Growth ( C)

Minimal Temperature for Growth ( C)

Maximal Temperature for Growth ( C)

Escherichia coli (facultative anaerobe)

10

35–37

45

Staphylococcus aureus

10

30–37

45

Clostridium kluyveri (obligate anareobe)

19

35

37

Listeria monocytogenes (intracellular pathogen)

1

30–37

45

Bacillus flavothermus

30

60

72

Thermus aquaticus (source of Taq polymerase)

40

70–73

79

Methanococcus jannaschii

60

85

90

Sulfolobus acidocaldarius

7

75–85

90

Pyrobacterium brockii

80

102–105

115

‘‘Strain 121’’ (most thermostable organism)

unknown

unknown

121

Chapter j 7 Factors Influencing Enzyme Activity

431

TABLE 7.17 Extremophiles and Selected Applications of their Enzymes Type

Growth Conditions

Enzymes

Applications

Thermophiles Hyperthermophiles

T z 60–80 C

Proteases

Detergents, partial hydrolysis in food technology (feed, brewing and baking)

T > 80 C Glycosyl hydrolases

Psychrophiles

T < 15 C

Amylases

Starch degradation

Pullulanase

Polysaccharide debranching

Pectinases

Pectin processing

Glucoamylases

Glucose production

Glycosidases

High-maltose syrup, high-fructose syrup

Cellulases

Cellulose degradation and textiles

Chitinases

Chitin modification for food and health products

Xylanases

Paper bleaching

Lipases

Detergents

Esterases

Trans-esterification, organic biosynthesis

DNA polymerases

Molecular biology (e.g., PCR)

Dehydrogenases

Oxidation reactions

Proteases

Detergents, food and dairy applications

Amylases

Detergents and baking

Cellulases

Detergents, feed and textiles

Dehydrogenases

Biosensors

Lipases

Detergents, food and cosmetics Peptide synthesis

Halophiles

> 2.5 M NaCl

Proteases Dehydrogenases

Biocatalysis in organic media

Alkaliphiles

pH > 9

Proteases and Cellulases

Detergents, food and feed

Acidophiles

pH < 2.3

Amylases

Starch processing

Proteases and Cellulases

Feed component

Oxidases

Desulfurization of coal

Source: van den Burg (2003) with permission of the author.

Eisenthal, 2001). Because these properties commend extremophile enzymes for use in biotechnology (Table 7.17), systematic characterization of these enzymes has become a commercially important enterprise. Thermophilic enzymes exhibit high activation energies (i.e., DEact >> kBT), such that little, if any, catalytic activity is detectable at ambient temperatures. Certain steps in the catalytic reaction cycle of thermophilic enzymes require so much energy that catalysis is highly unlikely or simply impossible at room temperature. In other words, roomtemperature kinetic studies of enzymes from extremophilic organisms are akin to sub-zero degree kinetic studies (see Section 7.7.9) of enzymes that normally operate at room temperature. In one of the most thorough papers on activity-stability relationships in extremophilic enzymes, D’Amico, Gerday and Feller (2002) examined three structurally homologous a-amylases from psychrophilic, mesophilic, and thermophilic organisms in terms of their conformational stability,

heat inactivation, irreversible unfolding, activation parameters of the reaction, properties of the enzyme in complex with a transition-state analogue, and structural permeability. a-Amylase (AHA) from the Antarctic bacterium Pseudoalteromonas haloplanktis is currently the best-characterized psychrophilic enzyme. Its closest structural homolog in the mesophiles is porcine pancreatic a-amylase (PPA). Bacillus amyloliquefaciens secretes the corresponding thermostable a-amylase (BAA). Amylase activity was recorded between 3 and 25 C, using 3.5 mM 4-nitro phenyl-a-D-maltoheptaoside-4,6-Oethylidene as substrate and excess a-glucosidase as coupling enzyme in 100 mM Hepes, 50 mM NaCl, 10 mM MgCl2 (pH 7.2). Activities were recorded in a thermostatted spectrophotometer and calculated on the basis of an absorption coefficient for 4-nitrophenol of 8,980 M1$cm1 at 405 nm. To obtain the composite activation parameters, the rate data were analyzed using standard equations: (a) DGz ¼ RT {ln(kBT/h) – ln k}; (b) DHz ¼ DEact – RT; and

Enzyme Kinetics

432

TABLE 7.18 Activation Parameters of Several Amylases at 10 C a-Amylase

kcat(s1)

DEact (kcal/mol)

DGact (kcal/mol)

DHact (kcal/mol)

TDSact (kcal/mol)

AHA

294

8.9

13.8

8.3

PPA

97

11.7

14.0

11.1

5.5

BAA

14

17.4

15.0

16.8

(c) DSz ¼ (DHz – DGz)/T). The derived parameters are given in Table 7.18. The kcat values show a dramatic 20 change in catalytic activity. The apparent activation energy DEact also exhibits a strong temperature dependence.

2.9 1.8

The stability of AHA, PPA, and BAA is illustrated in Fig. 7.21 by their guanidine hydrochloride-induced unfolding transitions. Least squares analysis of DG values as a function of guanidinium chloride (GdmCl) 40

80

∆ G (kcal mol-1)

Fraction unfloded (%)

100

60 40 20 0

30 BAA 20 PPA 10

AHA

0 6

4

2

0

20

0

8

Guanudinium-HCI

60

40

100

80

Temperature (°C) 60

20 50

0 100

rel.fluo

AHA

40

Cp (kcal mol-1K-1)

rel.activity

100

0 PPA

40 20 0

BAA

40

50

20 0

0 0

20

40

60

Temperature (°C)

80

100

30

40

50

60

70

80

90

100

Temperature (°C)

FIGURE 7.21 Activity-stability relationships of a-amylases from psychrophilic, mesophilic, and thermophilic organisms. A, Equilibrium unfolding of the psychrophilic AHA (B), the mesophilic PPA (D), and the thermophilic BAA (,) as recorded by fluorescence emission. Thermodynamic parameters of guanidinium chloride-induced unfolding at 20 C are as follows: for AHA, C1/2 ¼ 0.9 M, m ¼ 4.3 kcal/mol, DGH2O ¼ 3.7 kcal/mol; for PPA, C1/2 ¼ 2.6 M, m ¼ 2.7 kcal/mol, DGH2O ¼ 6.9 kcal/mol; for BAA, C1/2 ¼ 6.0 M, DGH2O ¼ 23.8 kcal/mol. Reversible activity regain was ~85% for all three enzymes. B, Gibbs free energy of unfolding. Stability curves are shown for the (B), the mesophilic PPA (D), and the thermophilic BAA (,), as calculated from micro-calorimetric data. C, Temperature dependence of activity (upper panel) and of unfolding as recorded by fluorescence emission (lower panel). Data for the (), the mesophilic PPA (D), and the thermophilic BAA (,) are provided. Experiments were performed at similar protein concentrations (5–40 mg/ml) in 100 mM Hepes, 50 mM NaCl (pH 7.2) in the absence of added Ca2þ (~1 mM residual Ca2þ as estimated by atomic absorption). Note also the decrease in unfolding cooperativity as the enzyme stability increases (lower panel). D, Differential scanning calorimetry endotherms of a-amylases in the free state (thin lines) and in complex with the transition-state analogue acarbose (heavy lines). Tmax corresponds to the top of the transition. Figures and captions from D’Amico et al. (2002) with permission of the authors and publisher.

Chapter j 7 Factors Influencing Enzyme Activity

7.7.8. Some Multi-Subunit Enzymes Exhibit the Phenomenon of Reversible Cold Inactivation Given the dominance of hydrophobic interactions in stabilizing the tertiary and quaternary structure of many proteins,

psychrophile thermophile

E

concentrations allowed the estimation of DGH2O, the conformational stability in the absence of denaturant, using the equation: DG ¼ DGH2O – m[GdmCl]. These enzymes unfold at distinct denaturant concentrations that result in half-maximal denaturation (C1/2). They are characterized by a decrease of unfolding cooperativity (m value) and the appearance of intermediate states (BAA) as the stability increases. These observations parallel the behavior of proteins that are adapted to different temperatures, as recorded in their thermal unfolding. The magnitude of the entropic term of the enzymatic reaction decreases as the enzyme stability increases. D’Amico et al. (2002) hypothesized that the flexible and mobile active site of psychrophilic enzymes undergoes larger structural fluctuations between the free and activated states than the more compact and rigid catalytic center of stable enzymes. If this is true, and on the basis of the DSz, AHA trapped in the activated state should display larger structural differences with the free enzyme, PPA should display intermediate differences, and BAA should display minimal structural differences. To check their hypothesis, the stability of the three enzymes was recorded by differential scanning calorimetry (DSC) in the free state and in complex with acarbose, a large pseudo-saccharide inhibitor acting as a transition-state analogue. Upon acarbose binding, the psychrophilic enzyme is indeed strongly stabilized, as indicated by large increases of the melting point and of the calorimetric enthalpy DHcal. According to the same criterion, PPA is less stabilized in the transition state, whereas BAA is destabilized, as shown by the lower DHcal value without melting point alteration. Both the magnitude and sign of these variations, as well as the macroscopic interpretation (i.e., stabilized transition-state intermediate is more ordered), parallels the structural fluctuations predicted by the activation entropy. These data strongly suggest that the increased activity at low temperatures of the psychrophilic enzyme is achieved by destabilizing the active-site domain to reduce the temperature dependence of the activity, which in turn implies large structural motions upon substrate binding. The low activity of thermophilic enzymes at room temperature can be explained in the same way. The catalytic domain designed to be active at high temperature is stabilized by numerous interactions, resulting in a greater temperature sensitivity, whereas the moderate conformational changes of the rigid active site result in weak activation entropy variations (see Fig. 7.22 caption for additional comments).

433

Conformational Reaction Coordinate FIGURE 7.22 Proposed energy landscape model of folding funnels for psychrophilic and thermophilic enzymes. The upper edge of each funnel is occupied by the unfolded (random coil) conformations. Because psychrophilic enzymes typically have less proline, fewer –S–S– bonds, and more glycine clusters than mesophilic and thermophilic enzymes, the edge of the funnel for psychrophilic proteins is slightly larger (reflecting broader distribution of the unfolded state) and lies at higher energy. As folding proceeds, the free energy decreases, as does the number of conformations. By contrast, thermophilic proteins pass through intermediate states corresponding to local minima of energy. These minima are responsible for the ruggedness of the funnel slopes and for the reduced cooperativity of the folding/unfolding reaction, as demonstrated by unfolding induced by guanidinium chloride and heat. In contrast, the structural elements of psychrophilic proteins generally unfold cooperatively without intermediates, as a result of fewer stabilizing interactions and stability domains, and therefore, the funnel slopes are steep and smooth. The bottom of each funnel depicts the stability of the native state ensemble and displays significant differences between both extremozymes: (a) for a highly stable, rigid thermophilic protein, the bottom has a single global minimum (or only a few minima separated by high barriers); and (b) for an unstable and flexible psychrophilic protein, the bottom is rugged, with a large population of conformers having low energy barriers, allowing them to equilibrate. Rigidity of the native state is a direct function of barrier height. The activity-stability relationships in these extremozymes depend on the bottom properties. Indeed, it has been argued that upon substrate binding to the association-competent sub-population, the equilibrium between all conformers is shifted toward this sub-population, leading to an ensemble of active conformations. In the case of psychrophilic enzymes, such an equilibrium shift only requires modest free energy changes, with a low DH* for conformational interconversion and a large DS* for fluctuations in the wider conformer ensemble. Figure and caption from D’Amico et al. (2002) with permission of the authors and publisher.

the thermodynamics of stabilization is consistent with DH – TDS > 0 in the cold (typically 3–5 C, but, depending on the relative values of DH and TDS, can be as high as 15–18 C). For most proteins, DH – TDS < 0 at room temperature (25– 30 C) to that of warm-blooded animals (35–40 C). The idea is that a temperature change DT of ~25 C determines the sign and magnitude of DG for a transition between properly folded and disordered states of an enzyme. Such structural changes are often accompanied by a reversible change in an enzyme’s catalytic activity: if an enzyme loses activity at low temperature, that enzyme is said to exhibit cold inactivation.

434

Reversible temperature-dependent loss of enzyme activity has been observed frequently with allosteric enzymes, because these proteins must rely on slight differences in the strength of their subunit-subunit interactions to modulate catalytic activity. Because these structural rearrangements are poised delicately (i.e., DG z 0), the energetics of enzyme interactions with inhibitors and activators often determines whether an enzyme will undergo cold inactivation, and, if so, the ability of changes in substrate concentration to reverse these inhibitory and activating properties. Moreover, because the water accessibility of hydrophobic groups influences the magnitude of DS (Richards, 1977), and because certain solvents exhibit preferential solvent interactions (Timasheff, 1998), agents like glycerol and dimethylsulfoxide can greatly influence the cold inactivation behavior of a given enzyme. Escherichia coli possesses two glutamine amidohydrolases (Reaction: L-Glutamine # L-Glutamate þ NH3). Glutaminase A has an optimal pH < 5, whereas glutaminase B is optimally active above pH 7 (Prusiner, Davis and Stadtman, 1976). Like many regulatory enzymes, glutaminase B proved to be cold-labile – inactivated by cooling to 4 C, but fully reactivated upon warming. Both processes are first order with respect to time, and the DEact for activity regain upon re-warming was found to be þ5.9 kcal/mol. An obvious break in the Arrhenius plot above 20 C suggested that warming the enzyme to temperatures above 24 C results in a decreased activation. Re-warming increased the Vm and decreased the [S]0.5 for L-glutamine, without any effect on the molecular weight of the catalytically active enzyme. Borate and glutamate protected glutaminase B from inactivation by cold. Another enzyme exhibiting cold inactivation is HMG coenzyme A reductase (Reaction: 3-Hydroxy-3-methylglutaryl CoA þ 2 NADPH # Mevalonate þ CoA þ 2 NADPþ), the rate-controlling enzyme of cholesterol biosynthesis in liver. This enzyme can be released from rat liver microsomes (i.e., the vesicular subcellular fraction obtained from mechanically dispersed endoplasmic reticulum) by exposure to high salt or glycerol concentrations. Unlike the particulate enzyme, the solubilized enzyme is rapidly inactivated by chilling to 4 C. The solubilized enzyme is also protected from cold-induced inactivation by salt, but only at concentrations in the range of 4 M, similar to the salt concentration required to release the enzyme from microsome fractions. Also consistent with the theme that oligomeric regulatory enzymes often exhibit cold inactivation are the findings of Kono and Uyeda (1973), who examined the cold-induced dissociation of liver phosphofructokinase. Reactivation of cold-inactivated enzyme upon warming is attended by self-association of the enzyme into its 400kDa form. Moriyami and Nelson (1989) reported that incubation of the reconstituted proton-ATPase from chromaffin granules

Enzyme Kinetics

on ice resulted in inactivation of the proton-pumping and ATPase activities of the enzyme. Cold inactivation depended on the concentration of magnesium ion, chloride, and ATP. Approximately 1 mM ATP, 1 mM Mg2þ, and 200 mM chloride were required for maximum inactivation. Inactivation in the cold resulted in the release of 72-, 57-, 41-, 34-, and 33-kDa polypeptides from the membrane. The 72-, 57-, and 34-kDa polypeptides were identified as subunits of vacuolar proton-translocating ATPases, based on antibody cross-reactivity. Similar results were obtained with several other vacuolar proton-ATPases, including those from plant sources. Moriyami and Nelson (1989) concluded that the catalytic sector of the enzyme is released from the protonATPase complex by cold treatment, resulting in inactivation of the enzyme. Tobacco ribulose-1,5-bisphosphate carboxylaseoxygenase (3-phospho-D-glycerate carboxylyase (dimerizing), EC 4.1.1.39) is partially inactivated by cold treatment. Activity is fully restored by heating in the absence of sulfhydryl reagents and regulatory effectors. This reversible cold inactivation does not involve a gross change in subunit association state, and Chollet and Anderson (1976) concluded that the enzyme undergoes a subtle conformational change. In later work, Chollet and Anderson (1977) found that SH reactivity of the cold-inactivated protein with 5,59-dithiobis-(2 nitrobenzoate) and p-mercuribenzoate exceeded that of the reactivated enzyme. Both the coldinactivated and heat-reactivated enzymes enhanced the fluorescence intensity of 8-anilino-1-naphthalenesulfonate (ANS), with a blue-shifted spectrum. Upon re-warming, restoration of catalytic activity closely paralleled a concomitant decrease in protein-bound ANS fluorescence intensity at 25 C. Fluorescence titration experiments revealed that the decrease in fluorescence intensity upon heat-induced reactivation was due to a reduced number of hydrophobic sites available for ANS binding, and not dissociation of the ANS-protein complex. Chollet and Anderson (1977) suggested that low temperatures alter the octameric structure of the hydrophobic catalytic subunits and that complete dissociation is prevented by the enzyme’s small hydrophilic subunits. Finally, Erez et al. (1998) examined the slow, reversible cold inactivation of the tetrameric enzyme tryptophanase (Reaction: L-Tryptophan # Indole þ Pyruvate þ NH3) in the presence of potassium ion and much faster in its absence. The Trp-330-Phe tryptophanase mutant undergoes rapid cold inactivation, even in the presence of Kþ. Coldinduced inactivation of both wild-type and mutant enzyme is attended by a decrease of 420-nm circular dichroism and absorption peaks of the coenzyme pyridoxal-5 phosphate, with the absorption spectrum shifting to shorter wavelength. The spectral changes and the NaBH4 test indicate that cooling of tryptophanase leads to breaking of the internal aldimine bond, followed by release of PLP and dissociation of the apoenzyme into dimers.

Chapter j 7 Factors Influencing Enzyme Activity

435

7.7.9. Cryoenzymology Techniques Greatly Reduce the Rate of Enzyme Catalysis Even with the advent of the fast reaction techniques (see Chapter 10), the considerable speed of enzyme-catalyzed processes still poses a challenge, particularly when the experimenter wishes to observe the fastest intermediate steps. The traditional strategy has been to use alternative substrates or modified coenzymes that react more slowly. Even so, this approach rarely dispels concerns regarding a change in mechanism with a change in substrate, especially with a very poor substrate. The field of cryoenzymology (i.e., the investigation of enzyme catalysis at temperatures below 0 C) offers an entirely different approach for slowing down enzyme rate processes (Coll, Compton and Fink, 1982; Douzou, 1977; Douzou and Balny, 1977; Fink, 1977; Fink and Geeves, 1979). The chief opportunities offered by cryoenzymology experiments are that: (a) the most specific substrate(s) and cofactors can be used, thus avoiding the use of slow substrates operating by a different catalytic mechanism; (b) otherwise fleeting intermediates, often in high yield, can be trapped; (c) improvement in the signal-to-noise of spectroscopic detection is possible because slower reaction rates at sub-zero temperatures make it feasible to accumulate numerous repetitive spectral scans, thereby taking advantage of Fourier transform methods; (d) the experimenter can obtain detailed kinetic and thermodynamic data on species that inter-convert more slowly at low temperatures; and (e) slower methods may be used (e.g., X-ray crystallography, NMR, and ESR) to obtain high-resolution structural information (see also description of Laue technique in Section 10.2.4: Rapid-scan Techniques).

7.7.9a. Rationale for Kinetic Studies at Sub-Zero Temperatures An intermediate in any reaction series of the form: Reactant / Intermediate / Product, should accumulate in higher concentration if the intermediate’s decay rate can be made to be smaller than its formation rate. The Arrhenius rate law (see Section 7.7.4) reminds us that unimolecular rate constants for most elementary reactions, including those in enzyme-catalyzed reactions, have activation energies falling in the 6–20 kcal/mol range. The Arrhenius law can be rewritten to indicate k1/k2, the ratio of rate constants at temperatures T1 and T2: z

z

k1 AeDE RT1 eDE RT1 ¼ ¼ z z k2 AeDE RT2 eDE RT2

7.72

Therefore, the above expression can be used to calculate that a 100 K drop in the temperature should reduce the magnitude of rate constants by factors ranging from 104 to 1010. Moreover, because different elementary steps most

often have different activation energies, the rates within a multi-step process need not be affected to the same extent, and rarely are. The net effect is that elementary reactions within a rapid multi-step process are slowed down differentially. This property allows certain intermediates to accumulate to higher concentrations, as they are effectively trapped within the free energy wells separating the transition states of elementary reactions. At a rate constant ratio k298K/k200K of 104, an intermediate having a 1-ms half-life at room temperature may easily have a half-life of several hours at 200 K. Likewise, if k298K/k200K of 1010, an intermediate having a 1-ns half-life at room temperature may exhibit a 1-hour half-life at 200 K.

7.7.9b. Practical Considerations The temperatures required in cryoenzyme kinetics are readily achieved by use of liquid nitrogen, dry ice, or refrigerated constant temperature circulating baths. The more serious practical constraint is the choice of cryosolvent, especially with respect to an enzyme’s solubility and stability. Cryosolvents are a mixture of water and organic solvents, often consisting of 20–50% water and 50–80% methanol, dimethyl sulfoxide, dimethyl formamide, ethylene glycol, ethanol, or combinations of organic solvents (e.g., methanol/ethylene glycol). Initially, the experimenter must determine whether the solvent is compatible with the planned experimental protocol, particularly when metal ions and other ionic solutes are required for enzymatic activity. These issues are best addressed in test experiments using a reliable spectrophotometric assay of enzyme activity. Because cryosolvents often increase Michaelis constants, activity assays must be carried out at sub-saturating substrate concentrations to detect and assess the impact of solvent on Michaelis constants. Because the dielectric constant of a solvent increases as the temperature is lowered, care must be taken to assess changes in enzyme stability arising from alterations in hydrogen bonding and hydrophobic interactions. Choice of the right buffer is also an imperative, because the meaning of pH in mixed aqueous-organic solvents is somewhat uncertain. In fact, each cryosolvent has its own pH scale (often designated pH* to indicate the pH reading in a particular solvent at a given absolute temperature). One then usually assumes a linear dependence of pH* on 1/T to estimate the pH at subzero temperatures graphically by extrapolation. Many cryokinetic measurements are conducted in a stopped-flow apparatus fitted with a modified multi-jet mixer that is suitable for the mixing of high-viscosity solvents at sub-zero temperatures (Table 7.19). When using NMR or ESR techniques, it is often possible to conduct slow temperature jumps to alter the relative concentrations of trapped intermediates or to study enzyme renaturation kinetics.

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436

TABLE 7.19 Some Enzymes Studied by Cryoenzymologic Techniques Enzyme

Cryosolvent

Reference

Chymotrypsin

Methanol-Dimethylsulfoxide

Fink (1973; 1976)

Trypsin

Dimethylsulfoxide

Fink (1974)

Alkaline Phosphatase

Methanol-Gycerol

Maier, Tappel and Volman (1955)

Myosin S1 Fragment

Ethylene glycol

Trentham (1977)

Alcohol Dehydrogenase

Dimethylsulfoxide-Dimethylformamide

Fink and Geeves (1979)

Because rate reductions of 104 to 109 are attained by lowering the reaction temperature from 298 K to ~175 K, an intermediate with a lifetime in the microsecond to millisecond range may be accumulated as a result of its greatly reduced reactivity. The field should now blossom as a result of the efforts expended in developing adequate cryosolvents that do not alter the structural and catalytic properties of the enzyme and in perfecting observation techniques for accumulating, stabilizing, and examining the intermediates. In principle, the experimenter may obtain kinetic, thermodynamic, and structural information about the intermediary states. When there is no spectral change in the protein or substrate attending formation of an intermediate, environmentally sensitive reporter-group methods may be developed that have been gainfully applied at normal temperatures. By changing the temperature, it is possible to stabilize and accumulate a number of the reaction intermediates and to measure the first-order rate constants for their interconversion. The major limitation with spectral reports (certainly with reporter groups) is that the interpretation of active-site structure and chemistry is indirect, depending largely on the experimenter’s experience and intuition. In particular, the formation of acyl-enzyme, glycosyl-enzyme, and phosphoryl-enzyme intermediates must be inferred from the structural changes of the enzyme, because these intermediates are generally chromogenic. Occasionally, alternative substrates may be designed to partly surmount these difficulties. Among the significant accomplishments in cryoenzymology are X-ray diffraction studies on trapped intermediates in the crystalline state providing detailed structural information of ‘‘frozen’’ intermediates (Alber, Petsko and Tsemoglou, 1976; Fink and Ahmed, 1976; Rees et al., 1981); detection of tetrahedral intermediates in the reactions catalyzed by proteolytic enzymes (Douzou, 1977; Fink and Geeves, 1979); and correlation of sub-zero data with fast-reaction data obtained at room temperature (Fink and Geeves, 1979).

7.7.9c. Cryokinetics of Carboxypeptidase A Consider the example of carboxypeptidase A (Peptidase Reaction: N-Dansylated-peptide þ H2O # N-Dansylated

fragment-COOH þ Amino Acid; Esterase Reaction: NDansylated Ester þ H2O # N-Dansylated-fragment-COOH þ Alcohol). With these fluorescently labeled peptides and their ester analogues, carboxypeptidase A exhibits Michaelis-Menten kinetics over the temperature range from 20 to þ20 C, with kcat/Km values of 0.3  107 at 20 C to 3  107 at þ20 C. When corrected for ES formation, measurements made under conditions where [SubstrateTotal] < Km yielded data in good agreement with the usual conditions for evaluating rate constants. At sub-zero temperatures, the pre-steady-state kinetics were biphasic: ES1 formation required less than 15 msec; a second intermediate ES1 formed in a first-order reaction. All data conformed to a two intermediate scheme:

KS E+S

E.S

k1

E.P

k3 E+P

k2 Scheme 7.26 ES1 formed in a pre-equilibrium, and ES2 arising from ES1. Table 7.20 summarizes these findings with several peptide and ester substrates.

7.7.9d. Acyl-Enzyme Formation at Sub-Zero Temperatures Coll, Compton and Fink (1982) describe the reaction conditions that permit the synthesis of O-acyl-enzyme intermediates for the serine proteases elastase, chymotrypsin, and trypsin. Under normal physiologic conditions, the acyl-enzyme would be too unstable to accumulate in significant amounts. Because deacylation depends on the active-site imidazole-aspartate proton shuttle (pKa ~7), deacylation is substantially retarded at low pH. Temperature is also a powerful factor that influences the accumulation of the acyl enzyme compound. Finally, because p-nitrophenol is a particularly effective leaving group, these alternative substrates favor acyl-enzyme formation. Table 7.21 shows that substantial accumulation of acylenzyme compounds can occur at sub-zero temperatures.

Chapter j 7 Factors Influencing Enzyme Activity

437

TABLE 7.20 Cobalt and Zinc Carboxypeptidase A-Catalyzed Hydrolysis of Matched Peptide-Ester Pairs Substrate

Metal Ion

Km (mM)

kcat (s1)

k2 (s1)

K–2 (s1)

k3 (s1)

KS (mM)

Km9 (mM)

Dns-Ala-Ala-Phe

Zinc

13.5

1.2

40

3.5

1.3

100

11

Dns-Ala-Ala-Phe

Cobalt

2.8

0.6

36

0.2

0.6

154

3.2

Dns-Ala-Ala-O-Phe

Zinc

1.6

0.06

53

0.5

0.06

129

1.3

Dns-Ala-Ala-O-Phe

Cobalt

0.2

0.04

59

0.1

0.04

61

0.2

Assay conditions: –20 C for peptide and 10 C for ester in 4.5 M sodium chloride, 10 mM HEPES (pH 7.5). Calculated Km9 values were obtained using: Km9 ¼ KS (k-2 þ k3)/(k-2 þ k2 þ k3). Source: Galdes, Auld and Vallee (1983); Auld et al. (1984).

TABLE 7.21 Formation of Acyl-Enzymes at Sub-Zero Temperatures Condition

Elastase

Chymotrypsin

Trypsin

Substrate

CBZ-Alanyl-p-nitrophenyl ester

N-Acetyl-Tryptophanyl-p-nitrophenyl ester

CBZ-Lysyl-p-nitrophenyl ester

[Total Substrate], mM

3.2 and 0.75

1

1

[Total Enzyme], mM

13.4 and 5.1

6.6

28

Solvent

70% Methanol

60% DMSO

65% DMSO

pH*

7.2 and 5.7

5.7

5.7

Temperature ( C) kobserved (s–1) % Acyl-Enz formed

43 and 50

1  103 and 3  104

82 and 92

42

5  104

86

33

3  104 44

Source: Coll, Compton and Fink (1982).

Similar approaches may be applied to accumulate the tetrahedral intermediate (see Fig. 1.4). In this case, anilide substrates are most suitable because the rate-limiting step in their hydrolysis is acyl-enzyme formation, and the electronic properties of p-nitroaniline helps to stabilize the tetrahedral intermediate. With 5 mM elastase, 0.2 mM Nacetyl-alanyl-prolyl-alanyl-p-nitroanilide, in 70% methanol, pH* ¼ 9.3 (morpholine, adjusted at 25 C), 40% of the enzyme is converted to the tetrahedral intermediate at 40 C (Fink and Meehan, 1979). X-ray crystallography can be conducted at sub-zero temperatures, and cryoenzymology techniques allow enzyme chemists to prepare the corresponding acyl enzyme compounds starting with crystalline enzyme in a manner that preserves crystal integrity. The unmodified crystals are grown under typical conditions, except that one dimension of the crystal is kept thin (~0.2 mm) to facilitate diffusion of substrate and cryosolvent into the crystal interior. The crystals are dialyzed at 0 C against the cryosolvent and cooled to 50 C, followed by the anilide substrate dissolved in the cryosolvent. The whole procedure takes about a day, and the modified crystals are stable for days. This approach does not work for a-chymotrypsin, which crystallizes in an arrangement where the active site is blocked by an adjacent a-chymotrypsin molecule (Coll, Compton and Fink, 1982).

7.7.9e. Limitations A persistent concern regarding cryoenzymology is whether the same mechanism operates at sub-zero versus physiological temperatures. The experimenter can usually detect breaks in Arrhenius plots even in aqueous solutions, and with enzymes these breaks may signal changes in ratedetermining step, enzyme inactivation, changes in the enzyme’s catalytic configuration, and/or changes in the substrate. The cryosolvent effect on the pathway may add further uncertainty when comparing the results of low- and room-temperature experiments. The changes resulting from the relative interplay of enthalpic and entropic factors in each elementary reaction become another matter with some reservation. In this regard, solvation effects that could greatly alter the potentially subtle balance of even internal equilibria might be anticipated (e.g., an enzyme-bound tight ion pair of a substrate carbocation versus a covalent enzyme substrate compound, or an enzyme-bound imine versus carbanolamine form). It is also true that enzyme-enzyme interactions might take place at the high concentrations of enzyme required for study, and these interactions can be again altered by cryosolvent effects on such equilibria. In this respect, we might say that the greatest potential of the cryoenzymological methods is the description of

Enzyme Kinetics

438

mechanistic options that may influence the course of the room temperature pathway. In the light of the likely central role of enzyme dynamics in catalysis, there is good reason to believe that cryosolvents may alter an enzyme’s conformational flexibility and, in doing so, fundamentally change the likely reaction trajectory. To the degree that enzyme conformational changes depend on solvation effects, the choice of solvent may play a greater role than previously appreciated. Finally, under favorable conditions, cryoenzymology allows the experimenter to populate some of the ‘‘passes’’ (‘‘saddle-points’’ or colls) through an alpine-like reaction coordinate diagram – never the summit itself. The Boltzmann Law clearly disfavors extensive accumulation of transition-state intermediates, however low the temperature. For the present and perhaps for many years ahead, the mind’s ‘‘eye’’ still offers the best view of transition states.

7.8. PRESSURE EFFECTS ON ENZYME KINETICS The equilibrium and rate constants of most enzyme reactions are relatively insensitive to modest changes in applied pressure (Laidler, 1987; Markley, Northrop and Royer, 1996). Very high pressures tend to amplify the otherwise modest volume of reaction DV and volume of activation DVz of a reactant proceeding to the reaction’s transition state. (The pressure dependence of the reaction equilibrium constant (vKeq/vP)T is given by the well-known ClausiusClapeyron equation, and the interested reader can consult any physical chemistry textbook for further details.) In reactions that consume or produce metabolic gases (i.e., O2, CO2, N2, NO, H2, CO, and CH4), equilibrium and rate constants are apt to experience more discernible changes. Reversible protein oligomerization and polymerization reactions also exhibit appreciable DV and DVz values. Likewise, some conformational changes and protein hydration states are affected at high pressure, particularly those dominated by hydrophobic interactions. The equilibrium positions and rates of many chemical or biochemical reactions are pressure-dependent, such that: v ln KP DV o ¼  vP RT

7.73

where DV is the change in reaction volume. When DV equals zero, an applied pressure is without effect on the reaction’s equilibrium position. Likewise, according to classical transition-state theory (Evans and Polanyi, 1935; Glasstone, Laidler and Eyring, 1941), the effect of pressure on reaction rate is given by: v ln k DV z ¼  RT vP

7.74

where DVz, the change in volume in the reaction, is nonzero. Activation volume changes on the order of 20 to 50 mL/ mol occur during neutralization of electrostatic charge, because more water molecules are packed around separated ions Xþ and A than around their associated form in the transition state (Activation Reaction: (H2O)m-xXþ þ A(H2O)m-y # [(H2O)m-xXþ$$$A(H2O)m-y]z þ (x þ y)H2O). Erijmann and Klegg (1998) investigated the effect of high hydrostatic pressure on the stability of RNA polymerase molecules during transcription. RNA polymerase molecules participating in stalled or active ternary transcribing complexes do not dissociate from the template DNA and nascent RNA, even at pressures up to 180 MPa. A lower limit for the free energy of stabilization of an elongating ternary complex relative to the quaternary structure of the free RNA polymerase molecules is estimated to be 20 kcal/mol. The rate of elongation decreases at high pressure; transcription completely halts at sufficiently high pressure. The overall rate of elongation has an apparent activation volume of 55–65 mL$mol1 at 35 C. The pressure-stalled transcripts are stable and resume elongation at the prepressurized rate upon decompression. The efficiency of termination decreases at the r-independent terminator tR2 after the transcription reaction has been exposed to high pressure, suggesting that high pressure modifies the ternary complex such that termination is affected in a manner different from that of elongation. Morild (1981) lists some 135 enzyme reactions affected by pressure. A non-linear pressure dependence for kcat may result from: (a) an interdependence between the applied pressure and other parameters like temperature, solvent composition, and pH; (b) enzyme unfolding or oligomer dissociation; and/or (c) a pressure-sensitive change in the rate-limiting step (Masson and Balny, 2005). Technically, the most rigorous evaluation of DV and DVz requires knowledge of the pressure dependence of individual kinetic constants for the elementary reaction reactions comprising a multi-step reaction mechanism. In this respect, pressure effects on Km, Vmax, and Vmax/Km values are ambiguous in that these parameters are composites of elementary reaction rate constants. Hoa et al. (1990) developed a high-pressure reactor suitable for studying steady-state enzyme kinetics at 400 MPa by means of stopped-time assays. Their design permits injection, stirring, and sampling without pressure loss. The substrate or enzyme can be injected to initiate an enzymecatalyzed reaction whose progress can then be followed by measurements on samples removed from the reactor. The 15-sec dead-time for sampling allows the examination of reactions with pseudo-first-order rate constants of about 1 min1 to be monitored. Tobe´ et al. (2005) used such a device to explore the effects of hydrostatic pressure on the catalytic activity of the minimal hairpin ribozyme shown in Fig. 7.23. They applied high pressures (up to 200 MPa) on this ribozyme, well

Chapter j 7 Factors Influencing Enzyme Activity

C Helix-1 U G A Helix-2 C CC C C C AG U 3′ -U U U G U C A50 G U C AA U 5′ -A A A C A GA A C G G A 10 C G Helix-3 Loop A U A 20 G C A C A G U A U Loop A 40 A U B A A C U A G CG A U Helix-4 Hairpin CG Ribozyme U 30G U

439

NW = 78 ± 4 H2O molecules V = 17 ± 4.5 mL/mol

TransitionState V=17 ± 4.5 mL/mol

below the 1,200 MPa needed to disrupt the structure of most nucleic acids. The volume of activation DVz for the reaction was 34  5 mL/mol, similar to that commonly found with protein enzymes. Such behavior probably reflects the importance of RNA compaction during catalysis, including release of 78  4 water molecules per RNA molecule, based ˚ 3 for the molecular volume of water. on a value of 30 A Kinetic studies were also carried out under osmotic pressure and confirmed this interpretation. Northrop (1999; 2002) suggested that, by separating and quantifying pressure effects on binding from those affecting catalysis, it may be possible to retrieve additional information about the nature of reactants in the transition state. While beyond the scope of this reference work, he suggested that pressure effects and kinetic isotope effects can be combined to provide additional insights into enzyme catalysis. He offered the optimistic view that pressure promises greater versatility and wider applicability than temperature effects, isotope effects, or pH effects. For example, nearly all rate constants exhibit virtually identical temperature dependencies, making it unlikely that the experimenter can separate one reaction step from another in a multi-step mechanism. In contrast, the magnitude and sign for each step need not be identical, thus increasing the likelihood that certain steps can be uniquely identified. With yeast alcohol dehydrogenase as a model system, Cho and Northrop (1999) observed that high pressure causes biphasic effects on benzyl alcohol oxidation, as expressed in the kinetic parameter V/K. Moderate pressure increases the rate of capture of benzyl alcohol by activating the hydride transfer step, suggesting the transition state for hydride transfer has a smaller volume than the free alcohol plus the capturing form of enzyme, with a DVz of 39 mL/mol. Pressures of > 1.5 kbar decrease the rate of capture of benzyl alcohol by favoring an enzyme form that binds NADþ less tightly. Some pressure effects are linked to changes in protein oligomerization. Ruan and Weber (1988), for example, found that hexokinase P1 and P2 isozymes undergo

FIGURE 7.23 Pressure effects on catalytic RNA catalysis. A, Diagram of hairpin ribozyme showing stem-loop arrangement. The cleavage site is indicated with an arrow, and nucleotides directly involved in catalysis are marked with a dot. B, Volume changes and water movement during the hairpin ribozyme cleavage reaction. The closure of the molecule and formation of the transition state is accompanied by a positive activation volume and the release of water molecules per RNA molecule. The positive DV of the reaction suggests that after cleavage the molecule relaxes to a conformation that is less solvated than the uncleaved opened conformation. Courtesy of Tobe´ et al. (2005).

pressure-induced dissociation, as signaled by the intrinsic protein fluorescence spectral shift and by the fluorescence polarization of hexokinase extrinsically labeled by dansyl chloride. They determined that the DG of association of the monomers at atmospheric pressure was 14.2 kcal$mol1 at 20 C and 11.4 kcal$mol1 at 0 C. The observed positive DS indicated that monomer association is entropydriven, presumably overcoming the DH of hydration at the subunit interfaces.

7.9. EFFECTS OF IMMOBILIZATION ON ENZYME STABILITY AND KINETICS An immobilized enzyme is a biological catalyst that has been spatially confined by chemical cross-linking or physical entrapment. Enzymes can be chemically tethered on the surface of biocompatible beads or entrapped within microporous substances and polymeric gels (Avnir et al., 1994; Cao and Schmid, 2006; Drott et al., 1997; Gill and Ballesteros, 2002; Lei et al., 2002; Silman and Katchalski, 1966). Immobilized enzymes are employed as chemicalprocessing agents within bioreactors, which reduces contamination of pharmaceutical products and foodstuffs by enzyme immunogens, antigens, and allergens that might otherwise evoke undesired immune/allergic responses. The chief advantage of enzyme immobilization is that the enzyme can be easily separated from solution-phase substrate(s) and product(s). Another advantage is that costly enzymes are retained within bioreactors, thereby conserving the catalyst. In many cases, enzyme immobilization also confers greater stability. One explanation for such behavior is that enzymes that are free to diffuse within a solution often accumulate at the air-liquid interface, where the interfacial energy may be greater than the forces maintaining the native enzyme structure. In such cases, the enzyme may undergo denaturation, particularly when bubbles are present.

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440

Enzymes in Bioreactors

In Purified Form Soluble Batch Continuous flow

Immobilized 1. Adsorbed 2. Entrapped 3. Cross-linked

Within Cells Growing Non-growing 1. Continuous 1. In suspension 2. Batch 2. Adsorbed 3. Semi-batch Immobilized Immobilized 1. Within matrix 1. Recycling 2. On membrane systems 2. Gel support 3. Adsorbed cells

FIGURE 7.24 Bioreactor configurations employed in agrichemical and biotechnology industries. Certain bioreactors are designed to operate with the use of purified enzymes, the chief advantage being high specificity. Other bioreactor designs exploit whole cell preparations, which offer the advantages of: (a) low cost; (b) no requirement for high-purity enzymes; and (c) no need for coenzymes.

Bioreactors are batch or continuous-flow devices designed to control physical parameters (such as heat/mass transfer, pH, reactant concentration(s), etc.) that can be manipulated to alter enzyme performance to maximize product output and/or to minimize process costs (Cao, 2006; Messing, 1975). Bioreactors typically employ either purified enzymes or permeabilized cells (Fig. 7.24) in the synthesis of biopharmaceutics, foods, and sweeteners, as well as in bioremediation for the removal or passification of toxic substances. Bioreactors may be open or closed systems, depending on whether heat is able or unable to exchange with its surroundings. In an open system, the immobilized enzyme is placed in a flow cell, an arrangement that facilitates continuous throughput of reactant(s) in a manner that is conveniently controlled elecctromechanically. To overcome diffusion limitations on reaction rate, most bioreactors are stirred – either mechanically or by turbulent flow – and immobilization may aid in stabilizing enzymes from denaturation resulting from the inclusion of surfactants and upon frothing. Great attention is now being given to the development of an entirely new generation of integrated micro-fabricated bioreactor/biosensor devices (Taylor and Schultz, 1996), often referred to as a ‘‘Lab-on-a-Chip’’ or a Micro Total Analysis System. Their development involves both established and evolving nanotechnologies, including microlithography to create enzyme circuitry, micro-machining to fabricate miniature reaction vessels and mixers, Micro Electro-Mechanical Systems (MEMS) technology to control flow, and microfluidics to transfer solutions under conditions where surface tension typically prevents flow (see also Section 2.10.7: Enzyme Electrodes). The widespread use of enzyme-based bioreactors in bioprocess and biopharmaceutical chemistry has mandated the development of immobilized enzymes for improved

stability and/or reduced catalyst loss (Table 7.22). Enzymes contain numerous chemically reactive functional groups on their surface, and covalent is therefore a simple and highly reproducible process. Enzymes may also become physically adsorbed onto, cross-linked to, trapped within, or even microencapsulated into a polymeric gel or matrix. Other supporting media include porous glass, silica, synthetic biogels, celite (diatomaceous earth), bentonite, alumina, and titanium oxide. As discussed below, immobilization per se can substantially alter the kinetic behavior of immobilized enzymes. Enzymes can be attached to solid surfaces and polymers by means of two classes of cross-linking agents. Homobifunctional cross-linkers possess identical reactive functional groups (e.g., glutaraldehyde (O¼CH–(CH2)3– HC¼O) and dimethylsuberimidate (CH3–O–C(¼NH)– (CH2)6–C(¼NH)–O–CH3)) that cross-link proteins and surfaces, in this case via free e-amino groups. Such agents suffer several disadvantages. First, because enzymes often have numerous lysine residues, nonspecific intramolecular cross-linking is highly likely. Second, the presence of many reactive protein functional groups can also result in heterogeneous populations of enzymes attached to the surface at one or more sites. Third, homo-bifunctional cross-linking of functionally identical groups on beads (v–NH2) and enzyme (Enz–NH2) can yield unwanted products in yield (e.g., v–N¼CH–(CH2)3–HC¼Nv and Enz–N¼CH–(CH2)3–HC¼N–Enz rather than the desired v–N¼CH–(CH2)3–HC¼N–Enz product). Therefore, the preferred route is to use hetero-bifunctional cross-linkers X–(CH2)n¼Y, because they react with different functional groups on the enzyme and polymeric bead (or bioreactor) surface. The first reaction usually directs attack at the crosslinker’s most labile group X by functional groups on the bead (or enzyme); then, after filtration or centrifugation to remove unreacted cross-linker, the modified beads are added to a suitably buffered enzyme containing (or beadcontaining) solution, thereby allowing reaction with the cross-linker’s less-reactive group Y. Several widely used hetero-bifunctional cross-linkers contain an NH2-reactive, N-hydroxysuccinimidylester (or NHS) attached at one end and an SH-reactive, N-ethylymaleimide (NHS–(CH2)n– NEM), pyridyl–disulfide (NHS–(CH2)n–S–S–Pyr), or a-haloacetyl moiety (NHS–(CH2)n–C(¼O)CHBr or NHS– (CH2)n–C(¼O)CH2I) at the other end. Some heterobifunctional reagents contain a photo-reactive group, offering the opportunity to capture the target enzyme photochemically. Another useful reagent is the carbodiimide cross-linker EDC that directly couples carboxyl and amino groups. Haynes and Walsh (1969) originally demonstrated that the surface of colloidal particles can be used to immobilize an enzyme that had been first adsorbed electrostatically as a monolayer and subsequently cross-linked covalently with glutaraldehyde. Later work by Kurota, Kamata and Yamauchi

Chapter j 7 Factors Influencing Enzyme Activity

441

TABLE 7.22 Methods for Immobilization of Enzyme within Bioreactors Method

Comment

Carrier-binding

Enzyme is chemisorbed or physi-sorbed onto an insoluble carrier (consisting of a dextrin, agarose, or a synthetic polymer matrix such as polyacrylamide). Because no cross-linking reagent is required, adsorption tends to be less disruptive to enzyme structure or its catalytic activity. However, because the binding occurs mainly by hydrogen bonds, hydrophobic interactions, salt linkages, and van der Waals forces, the enzyme can be readily released by changes in temperature, pH, or ionic strength.

Gel Inclusion

A hydrophilic matrix is formed around the enzyme by carrying out the matrix polymerization from a watersoluble monomer (often acrylamide) in the presence of the enzyme, along with a suitable polymer cross-linker (N,N9-bis-acrylamide) to control pore size. Mechanical shearing of the resulting gel allows one to obtain the desired particle size. Gel inclusion allows free diffusion of low molecular weight substrates and reaction products; on the other hand, high-molecular-weight substrates may not be able to gain access to a matrixincluded enzyme. One may also use alginate microspheres covered with polyelectrolyte nanofilm coatings to increase enzyme retention and stability.

Ultrasonic Encapsulation

Reservoir-type microcapsules may be produced with a coaxial ultrasonic atomizer that causes midair collision of microdrops of two liquids, one consisting of a polymer solution and the other an aqueous enzyme solution (Yeo and Park, 2004). This method allowed lysozyme to be encapsulated without loss of functional integrity and to be released with near zero-order kinetics for over 50 days.

Cross-Linking

The enzyme is modified chemically by a reagent that has already been covalently linked to a supporting matrix or to another protein. For example, glass surfaces can be silanized with aminopropyl triethoxysilane in dry toluene. After rinsing with toluene and drying under argon, the surface is then reacted with maleimide-v-N hydroxysuccinimide-ester. The resulting maleimide-containing matrix is then reacted with thiol groups on an enzyme of interest.

Entrapment

The enzyme is incorporated into the interstices of a gel, channel or membrane by means of polymerization, liquid drying, or phase separation. Entrapment allows free diffusion of low molecular weight substrates and reaction products.

Glutathione-labeled Enzyme

The chimeric enzyme (a fusion protein consisting of enzyme of interest fused to glutathione S-transferase) is permitted to adsorb onto glutathione-coated beads or surfaces. This method avoids enzyme inactivation of chemical cross-linkers.

Chelation

Hexa-histidine-tagged enzyme is reacted with nitrilotriacetate-modified surface in the presence of divalent nickel or cadmium ions. This linkage is stable in the absence of EDTA or EGTA. This method avoids enzyme inactivation of chemical cross-linkers.

(1990) detailed how enzymes can be covalently immobilized after adsorption onto micro-porous ion exchanger beads, such as those employed in high-resolution HPLC columns. An example of a commercially available enzyme-immobilizing product is EupergitÔ, which is formed as a copolymer of methylacrylamide, a second acrylamide monomer possessing amine-reactive oxirane groups, and the polyacrylamide crosslinker N,N9 methylene-bis(acrylamide). This Ro¨hm GmbH product consists of 150-mm hydrophilic spheres that exhibit excellent flow characteristics. With an oxirane content of ~800 mmol/g, EupergitÔ affords the opportunity for extensive incorporation of enzymes possessing reactive 3-amino side-chain groups. Given avidin’s and streptavidin’s extreme affinity for biotin (Kd z 1013 M), enzymes may be modified with Nhydroxysuccidimidyl–NH–(CH2)n–NH–C(¼O)–Biotin or N-ethylmaleimidyl–NH–(CH2)n–NH–C(¼O)–Biotin and then combined with surface immobilized avidin or streptavidin. Because avidin or streptavidin becomes sandwiched between the bioreactor surface and the immobilized enzyme, this approach prevents direct interaction of the

enzyme and bioreactor surface. Another popular approach exploits recombinant DNA techniques to create fusion proteins by splicing the cDNA sequence of the enzyme of interest to the cDNA for a peptide or protein serving as the tether. For example, the enzyme of interest may be fused to catalytically inactive g-glutamyl-S-transferase (GST), thereby exploiting the latter’s high affinity for surface immobilized glutathione. A consensus biotinylation site into an enzyme of interest may be introduced, such that biotin may be subsequently attached to an 3-amino group of a lysyl residue through the action of a biotin ligase (Reaction: Target–NH2 þ Biotin þ MgATP2 # Target– NH–Biotin þ AMP þ PPi). There are several reasons why an immobilized enzyme is likely to differ from its solution-phase counterpart. First, immobilization may alter the enzyme’s conformation, changing one or more rate constants in a multi-step enzyme reaction. Second, the enzyme is restricted to a different physicochemical environment from that of the bulk-phase solution. And third, the behavior of substrates and cofactors may be changed as they encounter the immobilized enzyme.

Enzyme Kinetics

442

Laidler and Bunting (1980) treated two cases: (a) within a solid matrix, a condition that is typically characterized by high [Esurface], low [S], large membrane thickness l, low substrate diffusivity D9, high catalytic constant, and a small Michaelis constant; and (b) within tubes, a condition that is typically characterized by low substrate flow, low [S], large tube radius r, large tube length L, and a small Michaelis constant. Depending on the length of the chemical chain that tethers the enzyme to the surface, there is apt to be some partitioning of the enzyme within the inner and outer regions of the double-layer, such that enzyme performance is altered in these distinctly different microscopic environments. Moreover, if substrate and product diffusion are at all influenced by these interactions, workers can anticipate that changes in transport can alter bimolecular rate constants as well as catalysis.

7.9.1. Kinetic Behavior of MatrixImmobilized Enzymes can be Substantially Different than the Behavior of Solution-Phase Enzymes In an early theoretical treatment of immobilized enzyme kinetics, Sundaram, Tweedale and Laidler (1970) considered the properties of three enzyme-substrate systems. If the presence of a surface affects an enzyme’s catalytic performance, as illustrated in Fig. 7.25, then, depending on the

enzyme + substrate in aqueous solution

enzyme + substrate within a solid support

enzyme + substrate embedded in a solid support suspended in H2O

S E+S

Support

H2O =

kcat[E]tot[S] Km + [S]

E+S

E+S

=

k'cat[E]tot[S] K' m + [S]

+ Support H2O =

k'cat[E]tot[S] Km(ap) + [S]

FIGURE 7.25 Three physical environments that affect the kinetics of enzyme substrate interactions. Left, Enzyme and substrate diffuse freely in solution, and kinetics are defined by classical Michaelis-Menten kinetics. Middle, Enzyme and substrate interact on the surface, with surface-immobilized enzyme lying beneath a porous matrix support, where the substrate concentration is [S]matrix. This matrix is covered by the so-called stagnant boundary layer that resists mixing with solution-phase substrate [S]solution. In this case, substrate must first cross the stagnant layer by means of diffusion, which thus obeys mass-transfer kinetics (i.e., vtransfer ¼ ktransfer{[S]solution – [S]matrix}). The transfer through the matrix obeys Fick’s Law, such that vmatrix ¼ D{[S]inner-matrix – [S]outer-matrix}. Right, The enzyme is embedded in a solid or matrix that is immersed in a solution of substrate (Km ¼ K9m/[PF]). The dependence of reaction rate on substrate concentration is also shown. From Laidler and Bunting (1980).

fraction of total enzyme localized near the surface and in the bulk solution, the kinetic parameters Km9 and Km,app could affect the observed rate behavior. The third case in Fig. 7.25 illustrates how physical/chemical conditions restrict virtually all of the enzyme to the surface of the medium, because untethered enzyme is washed away during preparation. The relevant rate equation has approximately the same form as the Michaelis-Menten equation, such that: v ¼ 

kcat9 ½Esurface ½S Km;app þ ½S

7.75

where [Esurface] is the concentration of enzyme confined within the solid support, Km,app is the apparent Michaelis constant, and kcat9 is the kcat for the immobilized enzyme: Km;app ¼ 

Km9 FP

7.76

In this treatment, P is the partition coefficient, representing the ratio of surface concentration of substrate in the support to that in the bulk solution. Parameter F is a Thieletype function that accounts for the diffusivity D9 of the substrate within the matrix: # " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kcat9 ½Esurface  l tanh 0:5 D9 Km9 tanhðglÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ F ¼ 7.77 gl kcat9 ½Esurface  0:5l D9 Km9 where tanh is the hyperbolic tangent, l is the locally restricted diffusion length, and kcat is the apparent kcat. As pointed out by Laidler and Bunting (1980), when gl is small, diffusion is not a factor, and the value of parameter F will lie close to unity. When l is small, the substrate is readily accessible to the active sites of immobilized enzymes. (If l ¼ 0, the system behaves as though the substrate and enzyme are as freely available to each other as enzyme and substrate present in the bulk solution.) When D9 is large, the substrate is also highly mobile and can readily reach the immobilized enzyme. If [Esurface] is sufficiently low, or if the effective bimolecular rate constant is small, catalysis will be slow enough that substrate diffusion will not affect the overall catalytic performance of the enzyme. As gl becomes larger, F ¼ 1/gl, which determines the degree to which diffusion will affect reaction rate. Finally, in chemical engineering, conditions that determine whether a chemical reaction is likely to be limited by chemical reaction rate vchem or the rate of reactant diffusion vdiffusion is often found. The Damko¨hler number is a unitless parameter defined as Da ¼ vchem/(vchem þ vdiffusion). When Da < 1, the reaction is reaction rate-limited, and when Da > 1, the reaction is diffusion rate-limited. In other words, when Da > 1, the mass transfer resistance is high and rate-controlling.

Chapter j 7 Factors Influencing Enzyme Activity

7.9.2. Enzymes Tethered with Flow Tubes have Special Kinetic Properties Many bioreactors consist of hollow fiber tubes through which reactants flow and react with enzymes attached to the tube’s inner surface. Such conditions impose geometric conditions that limit enzyme-substrate interactions that can be appreciated on the basis of a treatment developed by Kobayashi and Laidler (1974). In this model, the enzyme is treated as a uniform population of enzyme molecules attached to the inner surface of the flow tube. Substrate diffusion only occurs in the lumen of the tube. In the absence of turbulence, substrate concentration is essentially uniform in the central region of the tube, but because substrate is constantly removed when within the diffusional ‘‘reach’’ of the enzyme, its concentration falls the closer the substrate approaches the tube surface. The most important phenomena affecting enzyme kinetics are: (a) mass transfer of the substrate to the surface; and (b) substrate diffusivity within the diffusion layer (Fig. 7.26). As pointed out by Laidler and Bunting, the diffusion layer’s thickness depends on the mass-transfer coefficient kL, a parameter that measures the substrate diffusion rate through the layer: rffiffiffiffiffiffiffiffiffiffi 2 3 D vf 7.78 kL ¼ 1:29  rL where D is the substrate’s diffusion coefficient, vf the flow rate, r is the tube’s radius, and L is the tube’s length. Appearance of the flow rate r in the denominator of Eqn. 7.61 indicates that the effective thickness of the diffusion layer decreases as the flow rate increases. At very high flow rate, there is little or no diffusion layer, and the entire solution can be considered to be well-mixed. At low flow

Reaction Tube Geometry Cross-Section Longitudinal-Section diffusion layer Substrate Concentration Profile

flow

tube wall immobilized enzyme molecules

FIGURE 7.26 Flow reaction tube containing surface-bound enzyme. Shown are: enzyme molecules attached to inner surface, diffusion layer, and substrate depletion profile (dotted line) near the tube’s inner surface. Adapted from Laidler and Bunting (1980).

443

rates, a substantial diffusion layer is established, and little substrate in the centermost region reaches the surface tethered enzyme. The kinetics obey an equation resembling the MichaelisMenten equation, but the apparent Michaelis constant is influenced by flow-tube geometry and other factors: rffiffiffiffiffiffiffiffiffiffi rL 3 Km;app zKm 0 þ 0:39kcat 0 ½Enzsurface 7.79 D 2 vf where km9 and kcat9 are the kinetic parameters at high flowrates, such that there are no diffusional effects. The interested reader should consult Laidler and Bunting (1980) for a detailed discussion of how flow rate affects product formation. They also provide experimental findings on the kinetic behavior of several immobilized enzymes.

7.9.3. Enzyme Confinement may be Relevant to Cellular Conditions Many enzymes frequently reside in highly hindered spaces within and between cells. Intracellular enzymes, even those tacitly treated as soluble proteins when isolated from tissues, are often bound to vesicles, cytoskeletal fibers, and the nuclear matrix. Although the term ectoenzyme refers generally to any excreted or external enzyme that acts catalytically outside its cell of origin, many ectoenzymes remain attached to the external surface of the cell’s plasma membrane or to the extracellular matrix. Enzyme confinement is therefore an important reality that is often overlooked in studies on highly purified enzymes. When surface-bound enzymes act on solution-phase substrates, their kinetics can be treated as described in the previous section. Even so, ectoenzymes residing on the outer surface of one membrane often act on substrates confined to another nearby membrane surface. Ectoenzymes localized to the outer cell surface may also experience the osmotically sensitive confines of interstitial spaces, often making intimate contact with the extracellular matrix of neighboring cells, other biological surfaces, such as bone or tooth, and even pathogens. Palmitoylated and myristoylated enzymes (including numerous GTP-regulatory proteins, or small GTPases) are immobilized on the cytoplasmic face of the peripheral membrane. Still others are attached to or located within endosomes, phagosomes, lysosomes, mitochondria, tubuloreticular elements, Golgi compartments, as well as the outer and inner surfaces of the nuclear envelope. Likewise, the nucleus presents a diffusionally restrictive environment that requires DNA, RNA, and other polymerases to percolate within and through the nuclear matrix. Even the soluble enzymes of glycolysis, gluconeogenesis, and the purine nucleotide cycle are often trapped by and/or contained within 1,000–5,000 kDa glycogen particles that continually grow or shrink,

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depending on cellular demands for storage or mobilization of glucosyl units. Barriers to diffusion may give rise to the phenomenon known as diffusional anisotropy, wherein diffusion in some directions is preferred over others. Diffusional anisotropy can be measured by many procedures. Fluorescence microscopy (Nicholson and Tao, 1993; Nicholson and Sykova, 1998) is an especially powerful technique that permits us to use video microscopy to follow the redistribution of fluorescently-tagged molecules introduced into specific cellular or extracellular spaces. With the availability of micro injection methods, such studies can be initiated by placing a bolus dose of tagged molecules with considerable spatial precision. In NMR microscopy, we take advantage of the fact that NMR signal intensities depend in part on translational self diffusion, and any field gradient affecting transverse magnetization prior to or during signal acquisition will result in loss of signal strength (Callaghan, 1991; Torrey, 1955). The cytoskeleton is a distributed cytoplasmic compartment (formed by networks of actin filaments, intermediate filaments, and microtubules) that is mechanically attached to the peripheral membrane and other organelles, including the centrosome. These cytoskeletal fibers bind countless regulatory proteins and enzymes. Janmey (1995) even estimated that the total surface area of assembled cytoskeletal fibers within a typical cell exceeds the area of its peripheral membrane by a factor of seventy! Dynamic mechanochemical anchoring of the membrane and cytoskeleton by means of filament end-tracking motors (Dickinson, Caro and Purich, 2004) provides a push-pull sensor that allows the cytoskeleton to sense and respond to cues from the extracellular environment and to respond by generating force needed for organelle motility. Because the cytoskeleton is viscoelastic, it provides a continuous mechanical coupling throughout the cell that changes as the cytoskeleton remodels (Janmey, 1998). Such mechanical effects may conduct mechanical stresses from the cell membrane to internal organelles. Beyond these mechanical features, the cytoskeleton also provides a large negatively charged surface on which several glycolytic enzymes and many signaling molecules (e.g., protein and lipid kinases, phospholipases, RNA and ribosomes, as well as GTPases and the cytoplasmic domains of specific transmembrane receptors). Janmey (1998) suggested that the resulting spatial localization and concomitant change in enzymatic activity can alter the magnitude and limit the range of intracellular signaling events. He suggests that the intimate relationship between cell shape and gene expression is thought to be mediated, at least in part, by the cytoskeleton, which is the only known cellular structure of appropriate length-scale to directly link the surface of the cell to the nucleus. Finally, the interior of assembled microtubules may be a specialized sub-cellular compartment that endows cells with the ability to immobilize and/or transport

Enzyme Kinetics

metabolites and macromolecules (Purich and Allison, 2000). Confinement imposes other constraints on the ability of two surface-bound reactants to approach and/or contact each other as well as the lateral mobility of the enzyme and its substrate on their respective membranes. Although no such treatment appears to be available for the action of a surface-bound enzyme with a substrate attached to a nearby membrane surface, a good starting place for developing the theory would be the report of Chang and Hammer (1999), who treated receptor-ligand binding as a two-step process, requiring encounter and reaction. During the encounter phase, two molecules A and B must be brought to adjacent positions, and after encountering each other, the molecules must react, the latter involving quasivibrational adjustments of receptor and ligand configurations that are assumed to be independent of transport. In their model, collision (or transport) depends on the relative motion of surfaces and the lateral diffusivities DA and DB for each molecule on the surfaces, and the encounter rate is obtained by solving the convection-diffusion equation under appropriate boundary constraints. The encounter duration is obtained from the theory of first passage times. The binding rate for components A and B is fully described by two dimensionless parameters, namely the Pe´clet number and the Damko¨hler number. The former is a dimensionless number relating the rate of advection (i.e., transport in a fluid) of a flow to its thermal diffusivity D, whereas the latter is defined as the reaction rate divided by the convective mass transport rate. In those systems involving interphase mass transport, a second Damko¨hler number defines the ratio of the chemical reaction rate to the rate of mass transfer (or contact) between the reacting phases.

7.10. NON-IDEALITY IMPOSED BY MOLECULAR CROWDING Many of the cells interior spaces, including the cytoplasm and nucleus as well as membrane-bounded organelles (e.g., mitochondria, endoplasmic reticulum, Golgi apparatus) and vesicles (e.g., endosomes, phagosomes, pinosomes, etc.), are densely packed with proteins, nucleic acids and polysaccharide (Ellis and Minton, 2003; Minton, 2003). The human erythrocyte, for example, has a hemoglobin concentration of ~380 mg/mL. Therefore, based on hemoglobin’s approximate partial specific volume (~0.74 cm3$g1), the fractional volume occupied by hemoglobin in a red cell is ~0.28 (or 28% vol/vol). Given chromosomal DNA compaction factors >3,000 (i.e., the length of an extended chain of pure DNA divided by the length of a chromosome), the density of histone and DNA packing in nucleosomes and condensed chromatin is also extremely high. Likewise, the extremely large glycogen particles formed in the cytoplasm of hepatocytes tend to trap

Chapter j 7 Factors Influencing Enzyme Activity

glycolytic and glycogen enzymes within a dense polysaccharide matrix. The formation and loss of these particles is a highly dynamic process, reflecting episodes of glycogen synthesis and phosphorolysis. Given such very high concentrations of macromolecules, Minton (1980; 1983) first described the phenomenon of macromolecular crowding (or just molecular crowding), where excluded volume effects alter the properties of macromolecular solutes. Solute, molecules are mutually impenetrable, and when macromolecules are present at a significant volume fraction, dissolution of a test macrosolute happens in a manner that depends upon the relative sizes, shapes, and concentrations of all macrosolutes in the medium (Fig. 7.27). The energetic consequences of molecular crowding often affect the thermodynamics and kinetics of many essential cellular processes (e.g., protein folding, protein-ligand interactions, activity coefficients, etc.). At the same time, such effects are minor in most biochemical studies, simply because the concentration of macromolecules is too low for crowding

A

B

FIGURE 7.27 Excluded (pink and black) and available (blue) volume in a solution of spherical background macromolecules. Depicted is a region (demarcated by the black outline of the colored square) within a solution containing spherical background macrosolutes (colored black), of radius rb, that occupy ~30% of the total volume (vtot) of the specified region. The available volume (va,T) is defined to be that part of the volume of the region which may be occupied by the center of mass of a molecule of a spherical test species T of radius rt added to the solution. If the test species is very small relative to the background species (Panel A), then the available volume, indicated in blue, is approximately equal to that part of the total volume not occupied by the background species (i.e., ~0.7 vtot). However, if the size of the test species is comparable with (or larger than) the background species (Panel B), the available volume is substantially smaller, as the center of a molecule of the test species can approach the center of any background molecule to no less than the distance, denoted by rC, at which the surfaces of the two molecules contact each other. One may visualize this restriction by drawing a circular shell with radius rC about each background molecule. Then the volume available to the test species, indicated by the blue-colored regions in Panel B, is that part of the total volume, which is not occupied by any background molecule or by any shell. It is evident upon inspection of Panels A and B that the available volume is a sensitive function of the relative sizes (and shapes) of test and background molecules and the number density of background molecules. Image and caption reproduced from Minton (2001) with permission of the author and publisher.

445

effects to be detected. In other words, the conditions used in most biochemical experiments are artificial in that they fail to disclose the nature of solute-solute and solute-solvent interactions that are most relevant to cellular metabolism and signal transduction. The high total concentration of macromolecules inside living cells is such that 5–40% of the total cell volume is physically occupied by these macromolecules. Because large globular molecules generally cannot pack as densely as low-molecular-weight substances, an even larger fraction of the total volume is effectively unavailable to other macromolecules added to an already crowded solution. For example, in a solution containing identical globular molecules at 30% vol/vol, Ellis and Minton (2003) suggest that Vm) when enzyme and substrate are situated at the lipid-water interface. (Note: Many lipases bind to more than one phospholipid molecule, as indicated by the fact that ‘‘dilution’’ by a neutral nonionic detergent (such as Triton X-100) lowers that enzyme’s affinity for the lipid surface.) For phospholipase A2 from several sources, catalytic turnover in the bulk solution is negligible (i.e., kcat*/kcat > 105).

exchange of the enzyme between vesicles is constrained to the pre-steady state. Moreover, the exchange of natural phospholipid substrates and products is very slow on the kinetic time-scale, permitting dissection of the steps (shown in Fig. 7.30B as a blue surface) for the interfacial kinetic turnover; (2) when substrate and products exchange rapidly, the interfacial chemical step can remain rate-limiting, as is the case with micelles consisting of short-chain phospholipid. This condition is not satisfied with mixed-micelles of natural phospholipid combined with neutral detergents (used as diluents), where substrate replenishment in the enzyme-containing micelles can become rate-limiting; (3) when rapidly exchanging substrate and products are partitioned in a diluent interface (where the enzyme binds), interfacial kinetic rate constants for the catalytic turnover contain information about the primary mechanism of interfacial activation. The substrate concentration sensed by the enzyme in the interface is the phospholipid number density (best approximated as the mole fraction in the interface), not the substrate concentration in the bulk solution phase. Likewise, dissociation constants and Michaelis constants for interfacial catalysis are best expressed in mole fraction units.) The substrate concentration is manipulated by altering the

Chapter j 7 Factors Influencing Enzyme Activity

451

mole fraction of lipid substrate in the membrane, which is achieved by mixing with the enzymatically inert diluent 2hexadecyl-sn-glycero-3-phosphocholine. From the decrease in observed initial velocity v0, owing to dilution of the phospholipid substrate present on the membrane surface, a Km* value of 0.3 mole-fraction was obtained according to the modified Michaelis-Menten equation: v0 ¼

kcat ½S kcat XS ¼ X S þ Km 9 1 þ Km 9

7.80

where XS < 1, and Km* is defined as the substrate molefraction in the interface at which one-half of the total enzyme is in the form E* and the other half is the sum of E*S and E*P. Phospholipase kinetics have also been analyzed in Surface-Dilution Approach (Deems et al., 1975; Dennis, 1973; Warner and Dennis, 1975), in which the substrate is a mixed micelle composed of phospholipid(s) and a nonionic surfactant (typically Triton X-100). The reaction scheme can be written as:

E

k1[A] k-1

EA

k2[S] k-2

EAS

k3 k-3

EA+P

Scheme 7.28 where E is the enzyme, A is the mixed micelle surface, and S is the phospholipid substrate present in the mixed micelle. This two-step scheme accounts for the involvement of lipidwater interface; the surface is effectively treated as an essential activator, forming EA, and the phospholipid is substrate S. The surface-dilution technique allows the experimenter to manipulate the interface activation process by changing the detergent:phospholipid ratio. With phospholipase A2, phospholipase C, and phosphatidylserine decarboxylase, the approach yielded numerical values for the association constant of each enzyme with the lipid-water interface and the binding constant of the phospholipid molecules present in the micelle. Haiker et al. (2004) employed two different experimental approaches to demonstrate reversible interfacial adsorption of pancreatic lipase (PL) to fat droplets during lipolysis. Lipid hydrolysis was measured in olive oil/gum arabic emulsions containing [14C]triolein in the presence of bile salts and lecithin at rate-limiting concentrations of porcine PL (PPL) or human PL (HPL). The use of a lipolysis rate (measured as [14C]oleate release from the labeled emulsion) was immediately reduced by around 50% when the emulsion was diluted with an equal amount of an unlabeled emulsion. Moreover, lipase-catalyzed lipolysis was rapidly and completely suppressed when a non-exchanging lipase inhibitor was included in the second emulsion. Such results are fully consistent with lipase hopping between emulsion droplets. Hopping of PL between triolein droplets stabilized with gum arabic at bile salt concentrations above the CMC

was observed only in the presence, not in the absence, of lecithin. Displacement from a membrane-water interface of active HPL by an inactive [Ser-152-Gly]-HPL mutant was studied in the presence of bile salts by measuring HPL distribution between the water phase and the oil-water interface. These and other experimental evidence led Haiker et al. (2004) to conclude that phospholipase can exchange rapidly between oil droplets. Investigators often use a Langmuir trough to explore the strength of interfacial enzyme interaction with the membrane (Verger, 1980). This specialized device, which allows one to measure the surface pressure exerted by a surface film on liquids, consists of a flat, horizontally positioned tray (often fabricated from a slab of Teflon) that is partly filled with aqueous medium. In a typical experiment, a small volume of a surface-active substance, such as one or more phospholipids dissolved in a volatile organic solvent (called the spreading solvent), is added as a small aliquot (> kelong = 0.003 s , M /kN such that k elong elong = 100,000 M −1 1 s−1 = kN PP >> k PP = 0.0001 s ,

such that

M kN PP/k PP =

10,000

M −1 3 s−1 = kN exo >> k exo = 0.2 s ,

such that

M kN exo/k exp =

15

M −1 0.4 s−1 = kN off >> k off = 0.2 s , N M such that koff/k off = 2

In such a circumstance, it is evident that elongation exhibits the highest degree of discrimination between matching and non-matching nucleotides. Structural studies of DNA polymerases confirm there is an elongation/editing equilibrium, such that bound DNA dwells about seven-eighths of the time at the polymerase (or elongation) site and one-eighth of the time at the nuclease (or editing) site (Capson et al., 1992; Eger et al., 1991; Eger and Benkovic, 1992). Although the two sites are separated by ˚ , metal ion interactions with the misa distance of 35 A incorporated nucleotide slows down the next elongation step, allowing the DNA more time to dwell at the nuclease site. Another likely target for kinetic proofreading are the aminoacyl-tRNA synthases (Reaction: AA þ tRNAAA þ MgATP2 # AA–tRNAAA þ AMP þ MgPPi), each of which must be recognized and link carboxyl-group activated amino acids to the cognate tRNA (e.g., Cys to tRNACys, Ala to tRNAAla, Try to tRNATry, etc., where the superscript indicates the tRNA corresponding to its cognate amino acid). Baldwin and Berg (1966) first entertained the editing phenomenon in their studies of isoleucyl-tRNA

synthetase. For ribosomal peptide synthesis, the correctly charged aminoacyl-tRNA then uses its trinucleotide sequence in its anticodon loop to exactly match the corresponding mRNA sequence. However, any mismatched aminoacyl-tRNA would result in the biosynthesis of a mutant protein that would be nonfunctional or, worse still, could wreak havoc on a living cell. Because the carboxyl group of an amino acid must be activated prior to its eventual linkage to tRNA, all aminoacyl-tRNA synthases operate by catalytic mechanisms that first generate highly unstable aminoacyl adenylate intermediates. Fersht, Mulvey and Koch (1975) entertained the idea that a Hopfield-like dissociation-and-nonenzymatic-hydrolysis pathway might permit aminoacyl-tRNA synthases to edit errors, should they mistakenly incorporate the wrong amino acid (termed a noncognate amino acid) into an aminoacyltRNA. Because acyl-adenylates are so unstable, they reasoned that, once released from the synthetase, these mixed anhydrides should readily undergo nonenzymatic hydrolysis. Fersht’s group carried out an exhaustive analysis of aminoacyl-tRNA synthetase editing. Mischarged aminoacylated-tRNA’s are recognized and hydrolyzed at a separate active site, consistent with the following ‘‘doublecheck’’ proofreading mechanism (Fersht, 1977):

E AA ATP tRNA

fast

PPi

E AA-AMP tRNA

slow

major editing

E AA-tRNA + AMP "mopping up"

E + AA + AMP + tRNA

Scheme 7.31 Without going into the detailed kinetic strategies for investigating the phenomenon at this point, it is useful to compare the relative rates of aminoacylation and editing observed with several aminoacyl-tRNA synthetases (Table 7.24). Those interested in aminoacyl-tRNA synthetase proofreading should consult Fersht (1998) or First (1998). The cogent analysis of Yarus (1992) showed that enzymatic editing (proofreading) requires four properties: (1) a branched pathway that diverts and recycles any incorrectly produced product; (2) constrained reliance on substrate specificity alone; (3) consumption of energy; and (4) some degree of compromise between accuracy, rate, and yield. His report also describes how Michaelis-Menten-type enzymes operating by branched kinetic pathways might produce useful proofreading properties. While a detailed consideration of kinetic proofreading is clearly beyond the scope of this section, the recent review of Francklyn (2008) identified and analyzed the common features of kinetic proofreading mechanisms exploited by DNA polymerases and aminoacyl-tRNA synthases (Schemes 7.32 and 7.33).

Enzyme Kinetics

456

Ee + DNAn+1 Step-4alt

Ee-DNAn+1 dNTP ES + DNAn

Step-1

ES-DNAn

Step-2

ES-DNAn-dNTP

Step-4 Step-3

Step-5

Ee + DNAn Step-6

ES-DNAn+1

start new polymerization round

Step-6

DNA Polymerase Proofreading

ES + DNAn+1

Scheme 7.32

Ee + tRNA–AA

Step-4alt

Ee-tRNA–AA

Step-5

Ee-tRNA + AA

AA + ATP ES + tRNA

Step-1

ES-tRNA

Step-4 Step-3 Step-2 ES-tRNA-AA–AMP ES-tRNA-AA + AMP

AA-tRNA Synthase Proofreading

Step-7

Step-8 ES-tRNA + AA + AMP

ES + tRNA–AA + AMP

Scheme 7.33

In this scheme, Es refers to the polymerase that binds its dNTP substrates bound at its synthetic site, and Ee refers to enzyme with substrates bound in the editing site. In this scheme, Es refers to the aminoacyl-tRNA synthase with its substrates bound within the synthetic site, and Ee refers to the synthase with substrates bound in its editing site. Note that Step-2 in Scheme 7.32 refers only to the binding step for the incoming dNTP, whereas the corresponding step in Scheme 7.33 includes both the amino-acid and ATP binding step as well as the adenylylation reaction. (By merging the binding and adenylylation chemistry step for the aminoacyl-tRNA synthases, Francklyn simplified the global comparison between DNA polymerases and aminoacyl-tRNA synthases.) Additional elementary steps that may follow the binding of substrates and precede chemistry have been omitted for clarity. These include domain closure and other conformational changes, which are rate-limiting in some systems (see Johnson, 2008). In Scheme 7.32, the translocation of the primer-template from the editing site to the synthetic site (Step-6) is treated for esthetic reasons as though it is in equilibrium with the species Es-DNAnþ1. As pointed out by Francklyn (2008), the true species is EsDNAn, which is the immediate product of Step-1.

Despite any obvious evolutionary relationship, the review of Francklyn (2008) shows that DNA polymerases and aminoacyl-tRNA synthases employ mechanistically similar error-correction strategies to eliminate noncognate substrates at multiple steps or checkpoints during the catalytic cycle: (a) at the stages of initial substrate recognition and binding; (b) during pre-‘‘chemistry’’ isomerization steps; (c) during the bond-making chemical step itself; and (d) by applying post-synthetic editing. Significantly, DNA polymerases and aminoacyl-tRNA synthases employ kinetic checkpoints as a form of kinetic gating that eliminates incorrect incorporation of noncognate substrates. In both systems, misacylated/misincorporated intermediates are translocated from the enzyme’s synthetic site to its editing site, where they rapidly undergo hydrolysis. Finally, there are doubtlessly other forms of kinetic proofreading that improve the accuracy of highly repetitive processes in which errors might occur and accumulate. Proofreading is also likely to play a critical role in mechanoenzymes (see Chapter 11). For example, ATP-dependent metal ion pumps probably use kinetic proofreading to suppress the indiscriminant transport of metal ions of similar or smaller ionic radii.

Chapter j 7 Factors Influencing Enzyme Activity

457

TABLE 7.24 Relative Rates of Aminoacyl-AMP Formation and Editing for Selected Aminoacyl-tRNA Synthetases Acting on Various Amino Acids Aminoacyl-tRNA Synthetase

Amino Acid Tested

Relative Rate of AA-AMP Formation

Relative Editing Rate

Absolute Editing Rate

Isoleucyl-tRNA Synthetase

Isoleucine

1

1

0.014 s1

Valyl-tRNA Synthetase

Methionyl-tRNA Synthetase

Phenylalanyl-tRNA Synthetase

Valine

0.007

43

Threonine

0.0002

23

a-Aminobutyrate

0.0003

24

Valine Threonine

1

1

a-Aminobutyrate

0.004

180

Cysteine

0.005

58

0.001

90

Methionine

1

1

Homocysteine

0.005

60

Norleucine

0.005

5

Ethionine

0.035

7

Phenylalanine

1

1

Tyrosine

0.0004

300

0.02 s1

0.04 s1

0.007 s1

Source: First (1998).

7.14. KINETICS OF CRYSTALLINE ENZYMES Perhaps the most crowded state experienced by any enzyme is that occurring upon its crystallization, whereupon the enzyme becomes partitioned into crystals and the surrounding mother liquor. To obtain crystals of adequate size and diffraction quality, crystallographers simultaneously probe a sparse matrix of solution variables (e.g., solute concentration, metal ion and anion concentrations, pH, etc.) to identify the most suitable crystallization conditions (Ducruix and Giege, 1982; McPherson, 1999). There are two ways for preparing crystalline enzymesubstrate complexes. In the first method, already formed enzyme crystals are soaked with a solution containing the substrate (or more frequently a nonreactive substrate analogue) under conditions that keep the crystals intact. If the binding interaction does not substantially alter the enzyme’s structure, then the lattice contacts that maintain the crystal will not be disrupted. In the second method, the enzyme-substrate or enzyme-analogue complex is first formed, followed by addition of solutes needed to bring about crystallization without disrupting the enzyme’s affinity for substrate or analogue. Although good experimental technique is required in either case, a positive outcome is almost always a matter of fortuity. Any changes that alter the relative stability of crystals and mother liquor are apt to alter the abundance of crystals. A typical protein crystal contains between 1013 and 1016 molecules that are precisely arranged in its defined threedimensional lattice. The crystal lattice causes the

constructive and destructive interference of scattered X-rays that results in a diffraction pattern according to Bragg’s Law, with the geometry of the diffraction pattern determined by the molecular structure of the protein as well as the symmetry and spacing of protein molecules within the crystal lattice. Structural biologists frequently seek to determine whether a substrate binds to an enzyme in the crystalline state. Such information can be used to locate the active site, to determine the constellation of active-site residues that determine enzyme specificity, to identify residues likely to participate directly in catalysis, and to observe any conformational rearrangements of the enzyme or its substrate(s) after binding has occurred. A major concern about such data is whether the substrate is lodged in a productive or nonproductive manner, and this question is not a trivial matter, inasmuch as the structural information may set the investigator on the right or wrong path toward understanding the underlying catalytic mechanism. Detection of catalytically active enzyme crystals is therefore a telling observation, and enzyme chemists have devised a variety of methods to assay enzyme activity. In an earlier study of crystalline enzyme kinetics, Vas et al. (1979) employed a Leitz UV microphotometer to analyze the reaction of crystalline glyceraldehyde 3-P dehydrogenase (Reaction: Glyceraldehyde 3-P þ NADþ þ Pi # 2,3-Bisphosphoglycerate þ NADH) with a chromogenic substrate b-(2-furyl)acryloyl phosphate in the absence and presence of NADþ. Single-crystal spectra, based on the measured intensities of unpolarized monochromatic light transmitted through a part of the crystal, were taken every

458

5–10 nm for 5-s periods. For kinetic measurements, a single crystal (typical size ¼ 0.4  0.4  0.03 mm) was placed into a 20-mL polyethylene flow-cell fitted with quartz windows, the upper of which was easily removed to position a crystal within. Reactants were added by means of capillaries that pierced the polyethylene walls, allowing ammonium sulfate solutions containing a desired reagent to flow in and out in a few seconds. A second-order bimolecular reaction between the crystalline enzyme and a reagent followed pseudo-first-order kinetics. After mounting the crystal in the microspectrophotometer flow cell, the substrate-containing medium was added, and the reaction followed at 360 nm. The following elementary steps were examined: (a) acylation of the enzyme’s catalytic thiol; (b) binding of NADþ to the acylated subunits and activation of the acyl bond; and (c) deacylation, in the presence of the nucleophilic acyl acceptor, arsenate. Although ammonium sulfate in the crystallization medium greatly reduced the rates of all investigated reactions, the latter proceeded in the crystalline state with practically the same rate constants observed in solution at high salt concentrations. Moreover, protein-protein interactions within the crystal lattice were without effect on the catalytic efficiency of active sites and do not prevent the effects of coenzyme binding to the acylenzyme. The remarkable fidelity of DNA polymerase catalysis, for example, requires the enzyme to check for correct basepair formation both at the point of nucleotide insertion and at subsequent DNA polymer elongation steps. Despite extensive biochemical, genetic and structural studies, the mechanism by which nucleotides are correctly incorporated is not known. Kiefer et al. (1998) obtained high-resolution crystal structures of a thermostable bacterial (Bacillus stearothermophilus) DNA polymerase I large fragments with DNA primer templates bound productively at the polymerase active site. The active site of the crystalline enzyme retains catalytic activity, allowing direct observation of the products of several rounds of nucleotide incorporation. The polymerase also retains its ability to discriminate between correct and incorrectly paired nucleotides in the crystal. Comparison of the structures of successively translocated complexes allowed Kiefer et al. (1998) to deduce unambiguously the structural features for the sequence-independent molecular recognition of correctly formed base pairs.

7.14.1. Accurate Activity Assays of Crystalline Enzymes can be Technically Challenging Reaction velocities are typically reduced by the limited diffusion rates for tortuous percolation of substrate into and product from those active sites arrayed deep within the crystal lattice. The solutes that drive crystallization also

Enzyme Kinetics

tend to suppress the reaction velocity by competing for substrate, by complexing weakly bound metal ions, or by reducing the substrate’s thermodynamic activity (see Section 3.7.3: Electrostatic Effects on the Bimolecular Rate Constant). Just as with initial rate assays described in Chapter 4, enzyme activity is determined by mixing the suspension of enzyme crystals with an unlabeled substrate, a chromogenic substrate, or a radioactively labeled substrate. In stoppedtime assays, the entire sample of crystals and reactants must be quenched and then analyzed by HPLC, paper chromatography, etc. In one case, Moncrief et al. (1995) found that crystalline Klebsiella aerogenes urease (Reaction: (NH2)2C]O þ H2O # CO2 þ 2NH3) had less than 0.05% of the activity observed with the soluble enzyme under standard assay conditions. The 2 M concentration of lithium sulfate present in the crystal storage buffer inhibited the soluble urease by a mixed inhibition mechanism (Ki ¼ 0.38 M for the binding of Li2SO4 to free enzyme, and Kii ¼ 0.13 M for binding to the Enz$(NH3)2C]O complex). However, the activity of crystals was less than 0.5% of the expected value, suggesting that salt inhibition does not account for the near absence of crystalline activity. Dissolution of crystals resulted in ~43% recovery of the soluble enzyme activity, demonstrating that protein denaturation during crystal growth does not cause the dramatic diminishment in the catalytic rate. Finally, the authors found that the crushed crystals exhibited only a three-fold increase in activity over that of intact crystals, indicating that the rate of substrate diffusion into the crystals does not significantly limit the enzyme activity. Moncrief et al. (1995) concluded that urease is effectively inactive in this crystal form, possibly due to conformational restrictions associated with a lid covering the active site. They also suggested that the small amounts of activity observed arise from limited enzyme activity at the crystal surfaces or trace levels of dissolved enzyme within the mother liquor. Other approaches are designed specifically for observing rates nondestructively (i.e., the crystals remain intact). It is often necessary to perform a rate measurement in conjunction with the X-ray crystallographic experiment in order to obtain a direct correlation between the reaction progress and the diffraction data. As an example, Karlsson et al. (2000) performed UV/VIS absorption studies in conjunction with X-ray excitation on a Rigaku RU100 X-ray generator to investigate the reduction of Fe–S clusters by X-ray induced photoelectrons. Klink, Goody and Schleidig (2006), for example, developed a fluorescence microspectrophotometer for kinetic crystallography. Their instrument is designed for a 0 angle excitation of light guided from a laser by means of quartz fiber optics and a suitably juxtaposed fluorescence detector. Due to the reduced spatial requirements and the need for only one objective, the system is readily adapted to many different applications. In combination with a conventional

Chapter j 7 Factors Influencing Enzyme Activity

stereomicroscope, fluorescence measurements or reaction initiation can be performed directly in a hanging drop crystallization setup. Their micro-scale spectrophotometer can be combined with most X-ray sources, normally without the need of a specialized mechanical support.

7.14.2. Time-Resolved Laue X-Ray Crystallography is Quickly Becoming a Powerful Mechanistic Tool Traditionally, X-ray crystallography has provided highresolution structures of enzyme with one or more substrates or inhibitors bound within the active site. While such data have been extremely helpful in deducing enzyme mechanisms, they provide relatively little insight into the likely barriers to catalysis. The latter requires a series of structures acquired over the course of the catalytic reaction cycle. In time-resolved X-ray crystallography, the investigator must address three critical issues (Galburt and Stoddard, 2001; Moffat, 2001; Stoddard, 2001). First, the researcher must initiate the reaction in such a way that an intermediate species accumulates uniformly throughout the crystal. Because the intensity of diffracted X-rays is proportional to the concentration of enzyme molecules in a sample, it is essential to maximize the fraction of an enzyme-bound intermediate. Second, if the desired intermediate is not inherently rate-limited or displays a lifetime that is too short to allow its visualization, then a strategy is usually adopted to stabilize or ‘‘trap’’ that intermediate. Finally, a method of X-ray data collection that is fast enough to match the lifetime of both the transient reaction intermediate and the crystal specimen must be chosen. In single-turnover experiments, all the data are collected during a single catalytic reaction cycle, and conditions are adjusted so that the reaction does not cycle repeatedly. Sufficient time must be available to minimize the chance that an intermediate might accumulate nonuniformly or not at all. The reaction is usually triggered by a photolytic event (see ‘‘Caged Substrates’’ in Section 10.6.1: Flash Photolysis), such that substrate can form abruptly on a nanosecond-to-millisecond time-scale. At the slowest experimental extreme, reactions may be initiated by diffusion of substrate into the crystal. The intermediate can often be trapped by using chemical (e.g., suboptimal reaction conditions) and physical techniques (e.g., reactions in cryosolvents maintained at temperatures well below 0 C). As discussed by Galburt and Stoddard (2001), timeresolved crystallographic studies have been particularly productive for nuclease and phosphatase enzymes, ribozymes (RNA macromolecules that catalyze chemical reactions, particularly making and breaking phosphodiester bonds), electron transport proteins, photoreactive proteins, and a variety of metabolic enzymes.

459

7.14.3. Direct Measurement of Reactant Diffusion Rates in Enzyme Crystals can be Accomplished by Video Absorption Spectroscopy To observe the structure of intermediate enzyme-reactant complexes during catalysis, the researcher must ensure the homogeneous and complete accumulation of the intermediate throughout a crystal during the collection of X-ray diffraction data. Accumulation of the intermediate is dictated by the rate of substrate diffusion throughout the crystal relative to the rate of turnover and disappearance of a rate-limited catalytic intermediate. O’Hara et al. (1995) devised a simple quantitative method for measuring substrate diffusion and binding within an enzyme crystal. The experimental set-up consists of a light source, specific band-pass filters, a crystallographic flow cell, a syringe pump, a CCD video camera and frame-grabber card, and a workstation for images acquisition and subsequent data processing. Using this technique, any diffusion and binding process in a protein crystal leading to a visible absorbance shift may be accurately monitored and quantified during data collection. This method represents an inexpensive alternative to the use of a single crystal microspectrophotometer to measure the relatively slow process of diffusion.

7.14.4. Cross-Linking can be an Effective Tool in Analyzing the Behavior of Crystalline Enzymes Recognizing that stabilized enzyme crystals offer many advantages in bioprocess chemistry, biochemical engineers have exhaustively investigated the properties of cross-linked enzyme crystals (Roy and Abraham, 2003). Cross-linking is usually accomplished with glutaraldehyde (1–5% weight/ volume), which readily permeates protein crystals. The combined action of the cross-links and intermolecular contacts in the enzyme crystal lattice stabilize the enzyme against denaturation. Compared to soluble and immobilized enzymes, cross-linked enzyme crystals exhibit much higher activity per unit volume, and owing to their insoluble nature in both organic and aqueous media, separation of the reaction product from crystals is readily accomplished by settling or filtration. Cross-linked enzyme crystals also withstand the shear forces associated with mixers, stirrers, microfilters, and pumps found in flow- and batch-type bioreactors. Cross-linked enzyme crystal suspensions can be reused over and over again, and their ability to withstand proteolysis and autolysis facilitates the use of cross-kinked enzyme crystals in hydrolytic reactions. Roy and Abraham (2006), for example, used glutaraldehyde to cross-link crystalline Trametes versicolor laccase (Reaction: Benzenediol þ O2 # Benzosemiquinone þ 2 H2O) to form insoluble crystals that are catalytically active. These

460

cross-linked enzyme crystals exhibited improved thermal stability (the t1/2 at 60 C was 123 min compared for crosslinked laccase crystals versus 24 min for the soluble enzyme). The kinetic parameters with 2,2-azino-bis(3-ethylbenz-thiazoline-6-sulfonic acid were: Km ¼ 0.86 mM and kcat/Km ¼ 3.7  103 M1$sec1, the latter indicating the low efficiency of substrate access into the crystals. Cross-linked laccase crystals displayed higher activity in hexane, toluene, iso-octane and cyclohexane. In chemical engineering, a catalyst’s productivity is defined as the ratio of mass of product per mass of catalyst (Roy and Abraham, 2003). Small amounts of cross-linked enzyme crystals are required to produce large amounts of products and/or to reduce the time required for product formation. The catalyst-to-product ratio for reactions in cross-linked enzyme crystal normally ranges from 1-to-100 to 1-to-5,000 for a single reaction cycle. If the enzyme crystals can be reused 10–20 times, the final productivity will be in the 1,000–100,000 range. For a biocatalytic process, the catalyst cost should be less than 5–10% of the product value. Depending on the enzyme, the cost of a commercially available cross-linked enzyme crystal catalyst ranges from $12,000 to $350,000/kg (Roy and Abraham, 2003). They point out that, if the productivity is ~10,000, then 100 mg cross-linked enzyme (at an approximate cost of US$5) would form 1 kg of product. The resolution of 1-phenylethanol with vinyl acetate in toluene catalyzed by cross-linked lipase crystals is the best example for cross-linked enzyme crystal productivity; in this reaction, the substrate-to-catalyst ratio is 4,600 (Roy and Abraham, 2003).

7.15. PROBING ENZYME CATALYSIS THROUGH SITE-DIRECTED MUTAGENESIS Two advances in recombinant DNA technology have forever altered the way that enzymes are investigated. First, the development of bacterial expression systems greatly increased the quantities of enzyme available for chemical and structural investigations. Because eukaryotic enzymes often require post-translational processing for proper folding and enhanced stability, implementation of yeast and Bacculovirus expression systems have expanded the range of enzymes that can be produced in high yield. Second, the advent of site-directed mutagenesis (Smith, 1997) heralded the widespread investigation of enzyme catalysis through the substitution of amino acid residues within and around active sites (Johnson and Benkovic, 1990; Plapp, 1995). Equally significant advances in X-ray crystallography and multidimensional NMR have provided invaluable information about the structure of enzyme active sites, thereby guiding the choices of residues to be probed by means of directed mutagenesis.

Enzyme Kinetics

The primary goals of enzyme mutagenesis are to: (1) identify critical residues affecting catalysis as observable changes of enzyme-substrate binding interactions; (2) establish or confirm inferences about functional groups required for catalysis; (3) evaluate proposals or models concerning the chemical events responsible for substrate conversion to product; (4) improve enzyme stability; (5) understand how enzyme control mechanisms alter catalysis; (6) understand enzyme interactions with other binding partners (i.e., regulatory subunits, enzymes catalyzing post-translational modifications, proteins determining an enzyme’s subcellular localization, etc.); (7) identify residues or motifs regulating enzyme degradation; (8) improve enzyme yields; and (9) facilitate rapid isolation, often achieved by incorporating the coding sequence for hexa-histidine tags in the recombinant cDNA. These ambitious goals have been fully achieved only in rare instances, and this section only deals with aspects of the first three.

7.15.1. Mutations – Particularly Long-Lived Naturally Occurring Mutations – are Intrinsically Interesting Any enzymes studied today represent the time averaged outcome of natural selection – a process that may be described as a biased ‘‘random walk’’ through protein structure ‘‘space’’ by means of mutations, the occurrence of which often depend on gene location and protein structural stability; but mainly on functional pressure imparted by metabolic needs and bio-specific binding interactions. Metastable mutations tend to revert, whereas gain-ofstability and gain-of-function mutations persist if there is a gene-stabilizing metabolic advantage. Protein diversification probably occurred by gene duplication, frequently giving rise to subtle changes in local structure, and only rarely punctuated by profound large-scale rearrangements. Both pathways have the potential to modify protein structure, but only the latter is likely to create novel protein folds of unusual and/or enduring stability. In this respect, sequence identity at the level of individual amino acid residues can be lost, but stable folds, motifs, and domains are retained. The above ideas fit with studies on the effects of sitedirected mutation on protein stability, from which it is now known that stability is enhanced by: (a) cumulatively increasing the hydrophobic interactions among side-chains lying deep within the interior; (b) hydrogen bonding networks that compensate for the burial of amide nitrogen atoms and carbonyls into the apolar interior; (c) reducing strain; and (d) introducing strain-free disulfide linkages, salt-bridges, etc. Any action that decreases the conformational freedom of the unfolded state should in principle reduce the entropy loss attending protein folding (Nicholson et al., 1992). Many studies have demonstrated that structurally independent mutations tend to result in additive

Chapter j 7 Factors Influencing Enzyme Activity

461

changes to protein stability, suggesting that protein stability is a global property. Biological individuality (i.e., polymorphism) need not arise from necessity, and in many cases, mutations are without any discernible functional advantage. An important corollary is that clusters of mutations showing no functional advantage are apt to define regions of proteins that are remote from the active site. Such regions may allow the introduction of mutations that can be exploited for enzyme without having to pay a penalty of reduced catalytic power. In seeking to comprehend structure-function relationships in enzyme catalysis, biochemists have long appreciated the need to identify which of these functional groups participate directly in catalysis. This challenging enterprise is best accomplished by applying more than one of the techniques summarized in Table 2.1 in search of a convergent view of likely amino-acid residues.

7.15.2. Early Mutagenesis Experiments Exploited Chemical Modification to Replace One Naturally Occurring Amino Acid with Another Polgar and Bender (1966) and Neet and Koshland (1966) developed virtually identical methods for replacing the hydroxyl group of the catalytic serine of subtilisin with a thiol group (Scheme 7.34). Their ingenious method capitalized on the enhanced nucleophilicity of the activesite serine in subtilisin, an enzyme that contains the same catalytic triad as chymotrypsin (see Fig. 1.4) but has a different overall tertiary structure. The enzyme was treated with a 20% molar excess of phenylmethane sulfonylfluoride, followed by reaction with excess thiol-acetate to generate the S-acetyl-S-enzyme. Subsequent hydrolysis yielded Ser-221-Cys-subtilisin, which is more commonly known as thiolsubtilisin.

OH

O F S CH2 O

O O O S CH2 O

HS

C CH3

O S C CH3

H2O

S H

Scheme 7.34 Neet, Nanci and Koshland (1968) observed that the modified enzyme was inactive toward normal substrates, such as typical protein substrates and simple esters, but

exhibited slight activity toward nitrophenyl esters. While substrate binding was reduced, the acylation rate by substrates with poor leaving groups was remarkably slow. One explanation for these properties is that thiol-subtilisin undergoes enhanced partitioning of the tetrahedral intermediate back to starting material. This suggestion would be consistent with the pKa values of Cys–SH and Ser–OH relative to the substrate’s leaving groups. Another possibility is that the lower nucleophilicity of the SH group might require prior protonation of the leaving group in order to form the tetrahedral intermediate. p-Nitrophenyl esters of straight-chain fatty acids are chromogenic substrates of subtilisin and thiol-subtilisin (Philipp, Tsai and Bender, 1979). Among the substrates tested, both enzymes show highest specificity with p-nitrophenyl butyrate. Subtilisin was more sensitive to changes in substrate chain length than thiol-subtilisin. The second-order acylation rate constants k2/KS for both enzymes was remarkably similar, but thiol-subtilisin deacylation rate constants and Km values are lower than the analogous subtilisin constants. Brocklehurst and Malthouse (1981) provided important evidence that the lack of high catalytic activity of thiol-subtilisin towards certain substrates may be due to a poorly positioned proton-distribution system. Although site-directed mutagenesis will be described at length in Section 7.15.5, it is useful to extend our discussion on subtilisin to the significant experiments of Carter and Wells (1988), who deleted one, two, or all three residues of subtilisin’s catalytic triad to assess the kinetic importance of their functional groups. Again with reference to the chymotrypsin mechanism (Fig. 1.4), these three groups are thought to act as a proton relay during the catalytic reaction cycle, thereby increasing the nucleophilicity of Ser-32 in the peptide bond-cleavage step and by promoting acyl-enzyme hydrolysis later in the cycle. Alanine substitutions were chosen to minimize unfavorable steric contacts and to avoid imposing new charge interactions or hydrogen bonds from the substituted side chains. The results of their experiments are compiled in Table 7.25, where the Ds indicate single, double, or triple deletions. With all three residues removed, the enzyme is a relatively feeble catalyst, such that kcat is only a few thousand times greater than the corresponding constant for uncatalyzed peptide bond cleavage. The catalyzed rate is hardly only histidine or aspartate residue is present, whereas inclusion of the serine does increase the catalyzed rate by an order of magnitude. This situation is not bettered when the intervening histidine is absent from the active site, but another order of magnitude increase in kcat/ kuncat is realized, if the serine and histidine are both present. Note also that the complete charge relay system affords a kcat/kuncat value that is five orders of magnitude greater than the corresponding parameter for any of the singledeletion mutants. To provide a fuller picture of their experimental findings, Carter and Wells (1988) also evaluated kcat/kDDD for the wild-type and mutant subtilisins.

Enzyme Kinetics

462

TABLE 7.25 Site-Directed Mutagenesis Subtilisin’s Catalytic Triad Ser-32

His-64

Asp-221

kcat/kuncat

kcat/kDDD

Mutant

Mutant

Mutant

Mutant

Wild Type

Mutant

2.7  103

Mutant

Mutant

Wild Type

Mutant

Wild Type

Wild Type

Wild Type

Mutant

Mutant

Wild Type

Mutant

Wild Type

Wild Type

Wild Type

Mutant

Wild Type

Wild Type

Wild Type

7.15.3. Alanine Scanning Mutagenesis Often Provides Useful Clues About Essential Functional Groups in Enzymes This technique, first developed by Cunningham and Wells (1989) refers to the systematic substitution of amino acid residues by alanine to identify functional side chain groups that may play a role in enzyme catalysis and/or stability. Substitution with alanine essentially removes all side chain atoms beyond the b-carbon, such that the role of side chain functional groups at specific positions can be inferred. Alanine’s side chain methyl group lacks any unusual backbone dihedral angle preferences, and is preferred over glycine (R ¼ H), which in principle might introduce unwanted conformational flexibility into the protein’s polypeptide chain. They used the technique to identify specific side chains in human growth hormone (hGH) that strongly modulate binding to the human liver hGH receptor. Sixty-two single alanine mutations were introduced at every residue within the three discontinuous segments of hGH (residues 2-19, 54-74, and 167-191) that had been implicated in receptor recognition. Alanine scanning revealed a cluster of a dozen large side chains that when mutated to alanine each showed more than a factor of four lower binding affinity to the hGH receptor. Many of these residues that promote binding to the hGH receptor are altered in homologs of hGH (such as placental lactogens and prolactins) that do not bind tightly to the hGH receptor. The overall folding of these mutant proteins was indistinguishable from that of the wild-type hGH, as determined by strong cross-reactivities with seven different conformationally sensitive monoclonal antibodies. The alanine scan also identified at least one side chain, Glu-174, that hindered binding because when it was mutated to alanine the receptor affinity increased by more than a factor of four. Gibbs and Zoller (1991) applied the same strategy to identify the functional regions and residues of a protein kinase. Clusters of the charged amino acids in the catalytic subunit of Saccharomyces cerevisiae 39,59-cyclicAMPdependent protein kinase (PKA) were systematically

DDG (kcal/mol)

1

0

2.5  103

0.9

0

2.5  103

0.9

0

3.0  103

1.1

0

4

2.3  10

8.4

~1

3.4  103

1.2

0

2.0  105

385

~3.5

1.9  1010

1.9  106

~9

mutated to alanine, producing a set of mutations that encompassed the entire molecule. Residues indispensable for enzyme activity were identified by testing the ability of the mutants to function in vivo. Active mutants were assayed in vitro, and mutants with reduced specific activity were subsequently analyzed by steady-state kinetics to determine the effects of the mutation on kcat and on Km for MgATP and for a peptide substrate. Specific residues and regions of the enzyme were identified that are likely to be important in catalysis and in binding of MgATP, functions that are common to all protein kinases. Additional regions were identified that are likely to be important in binding a peptide substrate, the recognition of which is likely to be specific to the serine/threonine protein kinases that have a requirement for basic residues around the target hydroxy-amino acid. The properties of mutants defective in substrate recognition were consistent with an ordered sequential reaction mechanism. Their work represented the first comprehensive analysis of a protein kinase by a rational mutagenesis strategy. As shown in Table 7.26 and as discussed below, replacement of active-site functional groups by the methyl group of alanine has provided powerful insights to catalysis. When such replacement results in complete loss of catalytic activity, there is a high probability that the residue participates directly in catalysis or indirectly in binding an essential metal ion or coenzyme. The bifunctional enzyme fructose-6-phosphate,2-kinase/ fructose 2,6 bisphosphatase catalyzes synthesis and degradation of the key regulatory effector fructose 2,6-bisphosphate (Fru-2,6-P2). In their investigation of potential catalytic groups in rat liver Fru-2,6-Pase, Tauler, Lin and Pilkis (1990) showed that His-258 is phosphorylated during catalysis and that mutation of the histidine to alanine resulted in complete loss of activity. Mizuguchi et al. (1999) demonstrated that mutation of the corresponding histidine (His-256) in the rat testis enzyme decreases activity by less than a factor of 10, with a kcat that was one-sixth that of the wild-type enzyme. Mutation of His-390 to Ala results in a kcat value that is one-eighth of the wild-type enzyme kcat.

Chapter j 7 Factors Influencing Enzyme Activity

463

TABLE 7.26 Effect of Alanine Replacement of Active-Site Acid, Base or Nucleophile Enzyme (Reference)

Residue

Fructose 2,6-bisphosphatase

Wild Type

(Mizuguchi et al., 1999; Sakurai et al., 2000)

His-256-Ala Glu-325-Ala

5

kcat (s1)

Km (mM)

kcat/knon

32

0.06

Nucleophile

0.005

0.88

6  107

~107

General Acid

0.002

9.1

54,000

0.34

4  106

400

0.12

2,800

1.0

0.25

1.3

310

0.13

Catalytic Role

D -3-Ketosteroid Isomerase

Wild Type

(Kuliopulos, et al., 1989; Henot and Pollack, 2000)

Asp-38-Ala

b-Lactamase

Wild Type

(Jacob, Joris and Frere, 1991)

Ser-70-Ala

Ribonuclease T1

Wild Type

General Base

Nucleophile

(Steyaert et al.,1990; Steyaert and Wyns, 1993) Glu-58-Ala

General Base

11

0.08

His-92-Ala

General Acid

0.002

0.14

Attempts to detect a phospho-histidine intermediate with the His-256-Ala mutant enzyme were unsuccessful, but the phospho-enzyme is detected in the wild-type, His 390-Ala, Arg-255-Ala, and Glu-325-Ala mutants. Tauler, Lin and Pilkis (1990) also demonstrated that the mutation of His256 induces a change in the phosphatase hydrolytic reaction mechanism. Elimination of the nucleophilic catalyst, His256-Ala, results in a change in mechanism, with His-390 likely acting as a general base that activates water for direct hydrolysis of the 2-phosphate of Fru-2,6-P2. In a related study, Sakurai et al. (2000) investigated whether Glu-325 is an acid/base catalyst. The pH-rate profile for kcat for the wild-type enzyme exhibits pKa values of 5.6 and 9.1. The pH dependence of kcat for the Glu-325-Ala mutant enzyme gives an increase in the acidic pKa from 5.6 to 6.1. Their findings are consistent with Glu-325 serving an acid/base role in the phosphatase reaction. D5-3-Ketosteroid isomerase (EC 5.3.3.1) catalyzes the rearrangement of D5-3-ketosteroids to D4-3-ketosteroids by means of a direct and stereospecific transfer of the 4b hydrogen as a proton to the 6-b position via a dienolate intermediate. O

O 5-Androstene-3,17-dione

O

O Dienolate Interrmediate

O

O 5-Androstene-3,17-dione

2  1011 ~109

9  109

8  105 ~1011

4  109 7  107

The catalytic mechanism involves an acidic residue that protonates the 3-carbonyl function of the steroid as well as a basic group needed for proton transfer. The steroid lies in a hydrophobic cavity near Asp-38, Tyr-14, and Tyr-55. In order to assess the role of these amino acid residues in catalysis, the following mutants were prepared: Asp-38 to asparagine and Tyr-14-Phe and Tyr-55-Phe. The kcat value of the Asp-38-Asn mutant enzyme is 105.6-times lower than that of the wild-type enzyme, suggesting that Asp-38 functions as the base which abstracts the 4-b proton of the steroid in the rate-limiting step. Three-fold lower Km values in all mutants indicated slightly tighter substrate binding to the more hydrophobic sites. Compared to the wild-type enzyme, the Tyr-55 Phe mutant shows only a 4-times decrease in kcat, while the Tyr-14-Phe mutant shows a 104.7 decrease in kcat, suggesting that Tyr-14 is the general acid. Mutation of the active-site base Asp-38 to the weakly basic Asn has previously been shown to result in a greater than 108-times decrease of catalytic activity. Unexpectedly, Asp38-Ala exhibited a kcat that is around 106 greater than the value for Asp-38-Asn and only about 140 less than that for wild type. Kinetic studies as a function of pH show that Asp-38 Ala-catalyzed isomerization involves two groups, with pKa values of 4.2 and 10.4, respectively, in the free enzyme, which are assigned to Asp-99 and either Tyr-14 or Tyr-55. They proposed a mechanism in which Asp-99 is the catalytic base, with stabilization of the intermediate dienolate ion and the flanking transition states provided by hydrogen bonding from both Tyr-14 and Tyr-55. By using site-directed mutagenesis, Jacob et al. (1991) replaced the active-site serine residue of b-lactamase by alanine. Surprisingly, the alanine mutant exhibited a weak but specific activity for benzylpenicillin and ampicillin. In addition, a very small production of wild-type enzyme, probably due to mistranslation, was detected, but that activity could be selectively eliminated. Both mutant enzymes were nearly as thermostable as the wild type.

Enzyme Kinetics

464

As shown in Scheme 7.35, ribonuclease T1-catalyzed transesterification (transphosphorylation) is thought to operate by a mechanism employing His-92 as a general acid that protonates the 5-oxygen leaving group and uses Glu-58 as a general base that abstracts a proton to make the 29hydroxyl an effective internal nucleophile (Heinemann and Saenger, 1982).

O N

NH

N

RNA-O

NH2

N

O

O O H

O

H

O-

P

O C

O

N

Glu-58

O

RNA

N His-92

O N

NH

N

RNA-O

N

NH2

O

O

O P

O N N

HO C Glu-58

O H

O

and His-92 in catalysis rather than in substrate binding. Plots of log(kcat/Km) versus pH for wild-type, His-40-Lys, and Glu-58-Ala RNase T1, together with the NMR-determined pKa values of the histidines of these enzymes, strongly support the view that Glu-58 and His-92 acts as the base/ acid couple. The curves also show that His-40 is required in its protonated form for optimal activity of wild-type enzyme. Steyaert et al. (1990) proposed that the charged His-40 participates in electrostatic stabilization of the transition state; the magnitude of the catalytic defect (a factor of 2,000) from the His-40-Ala replacement suggests that electrostatic catalysis contributes considerably to the overall rate acceleration. For Glu-58-Ala RNase T1, the pH dependence of the catalytic parameters suggested an altered mechanism in which His-40 and His-92 act as base and acid catalyst, respectively. The ability of His-40 to adopt the function of the general base must account for the significant activity remaining in Glu-58-mutated enzymes. Steyaert and Wyns (1993) examined the functional interplay between the His-40, Glu-58 and His-92 in ribonuclease T1 catalysis. The kinetic properties of the single His-40-Ala, Glu-58-Ala and His-92-Gln mutants were compared with those of the corresponding double and triple mutants. When His-40, Glu-58 and His-92 are mutated separately or together, they observed large effects on turnover, but only minor effects on substrate binding. The free energy barriers to kcat introduced by the single His-40-Ala, Glu-58-Ala and His-92-Gln mutations are non-additive in the corresponding His-40Ala þ Glu-58-Ala, Glu-58-Ala þ His-92-Gln and His-40Ala þ His-92-Gln double mutants; a significant dependence of the pair-wise interactions on the third residue has been observed. Based on their analysis of related triple mutant, Steyaert and Wyns (1993) concluded that the collaborative effects of His-40, Glu-58 and His-92 decrease the energetic barrier to kcat by 6.8 kcal/mol. The overall effect caused by the triple mutation is smaller than that expected from the product of the fractional kcat values resulting from the individual mutations, illustrating the limitations of using single mutants to probe the energetics of a catalytic group whose function is dependent upon interactions with others.

O RNA

His-92

Scheme 7.35 A revised mechanism was proposed in which His-40, and not Glu-58, is engaged in catalysis as a general base (Nishikawa et al., 1987). To clarify the functional roles of His-40, Glu-58, and His-92, Steyaert et al. (1990) analyzed the effects of amino acid substitutions (His-40-Ala, His-40Lys, His-40-Asp, Glu-58-Ala/Glu-58-Gln, and His-92-Gln) on the kinetics of GpC transesterification. The dominant effect of all mutations is on kcat, implicating His-40, Glu-58,

7.15.4. Enzyme Chemists have Adopted Efficient Strategies for Investigating Enzyme Catalysis by Site-Directed Mutagenesis While recombinant DNA methodology allows us to make site-specific mutations at will, random mutagenesis rarely provides much insight into enzyme catalysis. Site-directed mutagenesis has, by contrast, proven to be a powerful tool in modern enzymology (Plapp, 1995). Another excellent primer is the review Tinkering with Enzymes: What are We Learning by Knowles (1987). Another useful resource is the monograph Directed Enzyme Evolution: Screening and Selection by Arnold and Georgiou (2003). The directed

Chapter j 7 Factors Influencing Enzyme Activity

evolution of novel enzymes was also reviewed by Woycechowsky, Vamvaca and Hilvert (2007). The following recommendations are based in part on a strategy offered by Plapp (1995) in his review on maximizing the probative utility of mutagenesis experiments in studies on critical active-site residues and other aspects of enzyme catalysis. Recommendation-1: Use protein sequence databases to identify invariant amino acid residues and well-defined structural motifs. The worldwide web now provides routine access to protein sequence and analysis algorithms. These so-called proteomics servers provide nucleic acid and polypeptide sequence database for enzymes from a wide range of organisms. They also provide systematic descriptions of protein function, any already recognized domains, motifs or folds, as well as evidence of post-translational modifications, intracellular targeting sequences and their proteolytic excision sites, as well as any sequence variants arising from alternative mRNA splicing. By accommodating deletions and insertions in polypeptide sequences, various alignment programs allow the user to bring shared sequences into registration, thereby facilitating visual comparisons of these regions. Other software predicts secondary structure as well as the likelihood that a given sequence will adopt a recognized domain, motif or fold. Whenever a long stretch of amino acid residues is identical, or nearly so, for the same enzyme from many species, it may be inferred that such a region endows the enzyme with some essential property. Even so, we cannot conclude that sequence invariance is proof that this sequence is at or near the active site. Conserved sequences may instead indicate an essential structural element such as a subunit interface, a coiled-coiled motif, or a posttranslational modification site. Consider, for example, the following sequences in trioseP isomerases from several species.

YEPIWAIGTGRTPTT YEPIWAIGTGLTPTT YEPIWAIGTGKSSTA YEPIWAIGTKSAPTP YEPIWAIGTGKTATD The blue-colored letter indicates conserved residues, and the red-colored E indicates the catalytic residue Glu-165. These sequences contain Loop-6, which is common to all triose-P isomerases, and forms a lid that closes over the active site whenever substrate or a nonproductive substrate/ transition-state analogue is bound.

465

The ambiguity of sequence invariance first led enzymologists to develop better criteria for associating an invariant residue with the active site. An almost universally applied approach was the design of active-site-directed irreversible enzyme inhibitors that incorporate highly reactive moieties in substrate-like molecules (Baker, 1967; Shaw, 1970). Upon incubation with such reagents, the enzyme is covalently modified, and subsequent fragmentation, peptide mapping, and sequencing established the putative active-site residue and its position. A better approach to identify an active-site residue is possible if an enzyme of interest forms a covalent bond with a reaction intermediate, as in the case of chymotrypsin. Even so, the gold standard for identification of active-site residues is a high-resolution enzyme structure with bound substrate or substrate analogue. Recommendation-2: Obtain a high-resolution structure that provides evidence about likely catalytic residues and their exact positions in enzyme active sites. A high-resolution structure of the wild-type enzyme is essential to establish the impact of amino acid substitutions. Structures with bound substrates, ‘‘slow’’ substrates, reversible and slow, tight-binding competitive inhibitors, high-affinity transition-state inhibitors, or active-site-directed irreversible inhibitors can be particularly helpful for detecting substrate-induced conformational changes and rearrangements of active-site residues. Because many enzymes prove to be catalytically active, substrates can undergo catalytic turnover. In the case of multi-substrate enzyme-catalyzed reactions, one of the substrates may be omitted to preclude any opportunity for catalytic turn˚ scale is also extraorover. Atomic resolution on the 1-A dinarily valuable for the unambiguous visualization of side-chain functional groups and their orientation. X-ray crystallographic studies on subtilisin (Katz and Kossiokof, 1986), dihydrofolate reductase (Villafranca et al., 1987), and tyrosine-tRNA ligase (Brown, Brick and Blow, 1987) indicate that many enzymes actually tolerate substitutions within the active site without evident changes in the protein’s overall structure. The same cannot be said for mutations occurring at sites possessing functional groups essential for coordination with an active-site cofactor (e.g., a carboxylate or imidazolium ion that coordinates to an essential metal ion or 3-amino groups that bind pyridoxal 5-phosphate). Another example is the essential 3-amino groups that allow biotin or lipoamide to become covalently fixed by means of ATP-dependent post-translational enzyme modification reactions. Site-specific substitution of residues outside the active site can also affect the folding and stability of an enzyme, especially when mutations occur at sites needed to maintain critical domain-domain contacts as well as those at subunit-subunit interfaces. Enzyme stability is often decreased when residues deep within a protein are

466

substituted by residues that introduce charge (especially the introduction of arginine), which alters hydrogen-bonding networks or retains water molecules in regions that are usually buried. Mutations create large voids by removal of bulky side chains of hydrophobic groups needed for protein folding (e.g., replacement of an indole substituent by a methyl group in a Trp-to-Ala mutation). The lack of a detectable structural change within an active site can be deceptive, chiefly because the active site is that region within an enzyme that is likely to be most sensitive to even subtle structural dislocations. A substrate must be lodged with high precision, and the movement of key acid/base groups or hydrogen bonds may not be tolerated. Ideally, structures of the substrate-free and substrate ˚ resolution, or better, should be bound enzyme at 2-A obtained. In many crystallographic studies, the physiologic substrate is replaced by a slow substrate or a competitive inhibitor. Even normal substrates may be used, especially because they offer the advantage that catalytically important reaction intermediates may accumulate within the active sites of mutant enzymes, whereas they are most often turned over by the wild-type enzyme. In the case of phosphotransferases and other enzymes that break b-g or a-b phosphoanhydride bonds of nucleoside-5 triphosphates, the imido- or methylene analogues can be used to block b-g or a-b bond scission. For example, when ATP is the substrate, p(NH)ppA, p(CH2)ppA, p(CF2)ppA, pp(NH)pA, pp(CH2)pA, as well as certain phosphorothioate-containing nucleotides like [g-S] !-ATP and [b-S]-ATP may be used. While enzyme mutagenesis studies can be executed without the benefit of high-resolution structural data, most investigators (and virtually every research grant proposal reviewer) considers an adequate X-ray or NMR structural model to be the sine qua non for reliable interpretation of any observed changes in catalytic properties. Ideally, the high-resolution structure of the mutant enzyme should be determined in the absence and presence of a competitive inhibitor (or one of the substrates in a multi-substrate enzymic reaction) to eliminate the possibility that the mutant enzyme has undergone some structural change that precludes substrate binding. At an absolute minimum, the mutant enzyme should display the same circular dichroism spectrum and thermal stability as the wild-type enzyme. If the CD spectra (taken over the 190–235 nm range) of mutant and wild-type enzyme are the same, one may infer that the secondary structure has been maintained. Plots of DEllipticity versus temperature, DAbsorbance versus temperature, or DFluorescence versus temperature for an enzyme often over a 5–60 C range generally provides a way to estimate the melting temperature Tm for thermally induced unfolding. The Tm corresponds to the temperature at the mid-point in the unfolding curve. Such curves should show evidence of a sharp, cooperative transition from EnzFolded # EnzUnfolded. The current standard for demonstrating the presence of a well-folded enzyme is differential

Enzyme Kinetics

scanning calorimetry (see Table 4.1), because a change in heat capacity usually attends the transition from folded to unfolded states and vice versa. This method yields a thermochemical Tm value, which may or may not match that obtained by CD, UV/visible or fluorescence spectroscopy as temperature passes through the transition zone. The amplitude of the thermal melting curve is also proportional to the fraction of well-folded protein in the sample. Recommendation-3: Identify a candidate residue for substitution. Enzyme chemists often seek to define the role(s) of active-site residues in substrate binding and catalysis. Ideally, the choice of which residue to mutate should be supported by: (a) high-resolution structural data from X-ray crystallography or multi-dimensional NMR spectroscopy; (b) evidence of invariant conservation, as obtained from alignment and analysis of polypeptide sequences for the same enzyme from numerous organisms; and (c) a hypothesis that is based on some predictable role of a residue (or even a few residues) in the catalytic mechanism. Such a residue may be essential for substrate or cofactor binding or it might have a role in catalysis, perhaps as a key nucleophile or electrophile. Mechanistic inferences based on the known reaction mechanism of structurally similar enzymes can also be especially helpful, but we usually prefer to base a hypothesis on some aspect of a structural motif or a residue’s acid/base properties, polarity, or hydrophobic character. There is considerable latitude in the choice of amino acids to be inserted in place of the natural residue, but substitutions introduced through evolution (see Table 7.27) suggests that interchanges tend to be those that are relatively conservative (Plapp, 1982). These data apply to natural point mutations occurring throughout the protein structures, rather than those occurring within enzyme active sites. Bordo and Argos (1991) exploited the conserved topological structure in various protein families to establish an index for ‘‘structural equivalencing’’ of residues in homologous structures. They carried out a search for equivalent residue pairs in various topological families that were buried in protein cores or exposed at the protein surface and that, when mutated, still maintained the environment of the unmutated residue. They used the amino acid residues with atoms in contact with the mutated residue pairs to define the local environment. Bordo and Argos (1991) then constructed matrices of preferred amino acid exchanges that allowed them to deduce so-called preferred and avoided substitutions. Given the conserved atomic neighborhoods, such natural in vivo substitutions become a useful guide for studying the effects of point mutations deliberately introduced by site-directed mutagenesis. Figure 7.31 portrays their exchange matrix, thus providing ‘‘guidelines’’ for recommending ‘‘safe’’ amino acid substitutions (i.e., those substitutions that are least likely to disturb the protein

Chapter j 7 Factors Influencing Enzyme Activity

467

TABLE 7.27 Relatively Frequent Evolutionarily Accepted Point Mutations Residue

Accepted Substitution(s)

Isostere Gly

Ala

Ser, Gly, Thr Pro

Arg

Lys, His, Trp, Gln

Asn

Asp, Ser, His, Lys, Gln, Glu

Asp

Asp

Glu, Asn, Gln, His, Gly

Asn

Cys

Ser

Ser

Gln

Glu, His, Asp

Glu

Glu

Asp, Gln Asn, His

Gln

Gly

Ala, Ser

His

Asn, Gln, Asp, Glu, Arg

Iso

Val, Leu, Met

Leu

Met, Ile, Val, Phe

Lys

Arg, Gln, Asn

Met

Leu, Ile, Val

Phe

Tyr, Leu, Ile

Pro

Ala, Ser

Asn, Gln

Ser

Ala, Thr, Asn, Gly, Pro

Cys

Thr

Ser, Ala, Val

Val

Try

Phe, Tyr

Tyr

Phe, His, Trp

Val

Ile, Leu, Met

Phe

Source: Plapp (1995), based on the mutation data matrix of Dayhoff, Barker and Hunt (1983).

structure, either locally or in its overall folding pathway). Their guidelines, as portrayed in Fig. 7.29, indicate the residue substitutions that are most likely to allow us to probe the structural and functional significance of the substituted site, while minimizing destabilizing effects of steric clashes, changes in polarity, etc. Ng and Henikoff (2003) described how the SIFT (Sorting Intolerant From Tolerant) software program can be employed to predict whether an amino acid substitution is likely to affect protein function. Their approach permits users to prioritize substitutions for further study. They demonstrated that SIFT can distinguish between functionally neutral and deleterious amino acid changes in mutagenesis studies. SIFT is available at: http://blocks.fhcrc.org/ sift/SIFT.html. A related consideration is that a change in side chain charge may alter hydrogen-bonding interactions. For example, the hydrogen-bonding network in the active site of rat trypsin is substantially rearranged upon substitution of Aspartate-102 to Asparagine (Sprang et al., 1987). For other enzymes, activity depends on extended chains of hydrogenbonded water molecules. In some cases, the directionality of Grothuss chains may undergo rearrangement when residues with oppositely charged side chains are substituted for each other. (-)-Charge Residue R

H

O

δ

neutral Pro

(+)-Charge Residue R

Gly

Cys

Ser Ala Thr

Asp

Val Leu

Glu

lipophilic

acidic

Gln Phe

aromatic

Lys Tyr

Try

δ

H

δ

H

H Oδ

O

H

H

H

δ

Oδ

δ

H

H

H

Oδ

H O

H O H

δ

H O

δ

H

Asn

IIe Met

O

H

H

H

H

H O

basic

His Arg

FIGURE 7.31 Suggested amino-acid substitutions in site-directed mutagenesis experiments. Based on the structural analysis by Bordo and Argos (1997), each group of related amino acids is grouped as indicated by the term within the box. Amino acid residues connected by a dotted line may be substituted with 95% confidence that such a change is likely to be tolerated. (Note also that the extended alkyl side chain of lysine is known to participate in hydrophobic interactions in certain coiled-coiled structures, suggesting that a polar functional group need not be the only factor affecting the stability of an enzyme undergoing site-specific mutation.)

Several enzymes, including carbonic anhydrase, superoxide dismutase, and acetylcholine esterase, speed up their respective reaction to the point that proton transfer becomes the rate-determining step. These enzymes must therefore optimize the kinetics of proton transfer, which is strongly influenced by factors affecting the stability of hydrogenbonded molecular complexes (e.g., deviation from optimal colinearity of donor and acceptor atoms, deviations from the ˚ distance between atoms sharing the optimal 2.9–3.3 A bonding hydrogen, and the electrostatic properties of those atoms). Carbonic anhydrase presents an instructive case where the catalytic efficiency is so great (kcat > 106 s1) that proton transfer is rate-limiting (see Sections 2.5.8.c and 7.4.3). The rate was found to depend on the concentration of the protonated form of buffers in the solution. Indeed, Silverman and Tu (1975) adduced the first convincing evidence for the role of buffer in carbonic anhydrase catalysis through their observation of an imidazole buffer

468

dependent enhancement in equilibrium exchanges of an oxygen isotope between carbon dioxide and water. The effect is strictly on kcat, and kcat/Vmax is unaffected, because the latter is described by a rate law that does not include proton-transfer steps. Kresge and Silverman (1999) presented a thorough analysis of the role of hydrogen bonding in carbonic anhydrase catalysis, including results of sitedirected mutagenesis experiments. Recommendation-4: Conduct the required site-directed mutagenesis and isolation. Introduce the mutation, confirm the altered nucleotide sequence in the cDNA, incorporate the cDNA into an expression vector, grow the plasmid-carrying bacteria, lyse the cells, and isolate the expressed mutant enzyme. The most frequently used method for protein expression in E. coli is use of the pBR322 vector, which is an autonomously replicated plasmid that is maintained at high copy number by virtue of its ability to confer resistance to tetracycline and ampicillin. A common problem is the tendency of many bacterially expressed proteins to become incorporated into insoluble inclusion bodies. The bacterial protein secretory mechanisms can also be subverted to transfer the expressed protein to the periplasmic space and/or culture medium. For those enzymes requiring some essential post-translational modification available only in eukaryotes, the Bacculovirus and yeast expression systems are highly effective. Recommendation-5: Use one or several biophysical techniques to evaluate the mutant enzyme’s structural integrity and stability. The resulting protein requires structural characterization, beginning with, but obviously not limited to, the use of matrix-assisted laser desorption ionization time-of-flight (or MALDI-TOF) mass spectrometry to confirm that the expressed protein has the correct molecular mass. Many investigators use the expressed protein’s circular dichroism spectrum at several different temperatures to assess a mutant enzyme’s susceptibility to thermal denaturation. Most globular proteins are stable over a given temperature range (typically 5 to 45 C), but rapidly unfold over a narrow temperature range that brackets the melting temperature Tm. Analytical ultracentrifugation, especially the sedimentation equilibrium technique, also offers an independent way for determining the monomeric molecular weight as well as any tendency to oligomerize. Fluorescence spectra can also be a helpful indicator of a conformationally well-formed protein. Active enzyme titration (see Section 4.13.2) is another useful criterion of enzyme purity and activity. Ideally, the atomic structure should be re-determined by high-resolution X-ray and NMR structural techniques. Recommendation-6: Conduct steady-state kinetic experiments to determine Vm, Km, Vm/Km (or Ki for an

Enzyme Kinetics

inhibitor). The pH-rate profile may also reveal changes in catalytic properties. For studies on the roles of functional side chains, Plapp (1995) stressed the value of using those substitutions that are isosteric, or nearly so, with respect to the wild-type residue. Such an approach reduces the likelihood of significant structural changes attending site-directed substitution. Substitutions can also loosen or stiffen so-called hinge and loop regions, with resultant changes in catalytic activity (Ahrweiler and Frieden, 1991; Bullerjahn and Freisheim, 1992; Mas, Bailey and Resplendor, 1988). Any change resulting in the removal of a nucleophilic group participating in covalent catalysis should eliminate enzyme activity. Any mutation that replaces such an acid or base group with a water molecule and still retains some catalytic activity has probably caused a change in the chemical mechanism. Therefore, removal of an essential catalytic acid or base group through mutagenesis should reduce the value(s) of Vm/Km and/or Vm by five orders of magnitude or greatly increase the value of Km. Smaller changes in rate are of an indeterminate nature. Some parameters require highly accurate determinations of the concentration of active enzyme, not simply the total amount of UV-absorbing polypeptide or the correct molecular mass. This distinction recognizes that a significant fraction of many expressed proteins are incompletely folded. These properties can be analyzed with an activeenzyme assay, such as burst kinetics (see Section 4.4.4) or mechanism-based irreversible inhibition (see Section 6.3.2). Recommendation-7: To investigate more subtle aspects of enzyme catalysis, consider the use of kinetic isotope effect measurements, fast-reaction techniques, and/or isotope-exchange experiments. Because steady-state parameters like Vm/Km and Vm are often composites of many rate constants, a mutation that lowers a steady-state parameter by 50–100 may actually result in a much greater reduction in the rate of an elementary reaction. Such a situation can arise if the affected elementary reaction is substantially faster than another elementary reaction in a multi-step reaction mechanism. Fast reaction techniques are therefore superior to steady-state methods for determining the effects of site-directed mutations on catalysis. Other approaches such as kinetic isotope effect measurements, isotope exchange studies, positional exchange experiments, and kinetic isotope effects allow us to obtain other valuable mechanistic clues. Researchers may also investigate substrate exchange kinetics by solution NMR methods as well as properties of bound substrates by magic angle solid-state NMR. Many of these techniques are described in Chapters 9 and 10. Recommendation-8: Determine whether a mutation influences the action of an active-site directed irreversible inhibitor and a mechanism-based inhibitor. As already

Chapter j 7 Factors Influencing Enzyme Activity

discussed in Section 8.9.1, an active-site directed irreversible inhibitor (or affinity label) reacts with an essential catalytic functional group. Likewise, as discussed in Section 8.11, a mechanism-based inhibitor requires a catalysis to generate an activated electrophile to bring about irreversible enzyme inhibition. The electrophilic species is then intercepted by reaction with an active-site nucleophile. Recommendation-9: Investigate other structural consequences of the site-directed mutation. In some cases, mutations may be introduced to increase protein stability, to improve solubility, or to alter an enzyme’s tendency to oligomerize. Although many lower molecular weight enzymes rely on the presence of one or more disulfide cross links, larger proteins appear to have sufficient stability in the absence of –S–S– bridges. Because the cytoplasm and nucleus of most eukaryotic cells is a strongly reducing environment, disulfide bond formation is unlikely. Indeed, should any disulfide-containing protein be formed adventitiously the high molar excess of reduced glutathione (G-SH) with the cell would favor spontaneous reduction to the thiol form: Reaction-1: Enz-S–S-Enz + G-SH

Enz-S–S-G + Enz-SH

Reaction-2: Enz-S–S-G + G-SH

G-S–S-G + Enz-SH

Reaction-3: G-S–S-G + NADPH

2 G-SH + NADP+

Scheme 7.36 The third reaction, which is catalyzed by glutathione reductase, regenerates reduced glutathione. By contrast, many extracellular proteins are actually stabilized by disulfides formed spontaneously by neighboring thiols exposed to that oxidizing environment. Not surprisingly, commercial demands for highly stable enzymes has led to efforts to introduce disulfide linkages in ways that increase long-term stability of enzymes in bioreactors, especially when elevated temperature or the presence of organic solvents is necessary. Introducing a stabilizing disulfide linkage requires careful consideration of the enzyme’s three-dimensional structure. Obviously, cross-link formation requires that the two newly introduced residues must be nearby each other (distance z ˚ ), and the protein’s structure must be geometrically 2A compatible with linkage formation. An excellent example is T4 lysozyme, where Matsmura and Matthews (1989) introduced a disulfide linkage that spanned the enzyme’s active-site cleft. Because Cys-54 and Cys-97 might react with the later introduction of disulfide, the authors mutated these residues to Thr-54 and Ala-97. They then replaced Thr-21 and Thr-42 by cysteine residues, which under oxidative conditions formed the desired disulfide. Other approaches for increasing enzyme stability include substitutions that reduce the entropy of the unfolded protein (Matsmura and Matthews, 1989; Matthews, Nicholson and

469

Becktel, 1987; Thornton, 1981), or that eliminate destabilizing hydrophobic groups on an enzyme’s surface (Matsmura, Becktel and Matthews, 1989). The latter approach is also likely to reduce undesirable aggregation. Finally, because enzyme activity is often dramatically reduced in polar organic solvents, even under conditions where the folded structures are stable, Chen and Arnold (1991) used PCR-mediated random mutagenesis and screening in the presence of dimethylformamide (DMF) to probe for mutations that enhance activity in polar organic solvents. They found two amino acid substitutions (Gln103-Arg and Asp-60 Asn) that enhance Subtilisin E activity in the presence of DMF. These substitutions are located near the substrate binding pocket or in the active site, and they both affect kcat/Km for hydrolysis of a suitable peptide substrate. Notably, the effects of D60N are apparent only in the presence of DMF, highlighting the importance of screening in the organic solvent. They also observed that the [Asp-60-Asn/Gln-103-Arg/Asn-218-Ser] triple-mutant is 38 times more active than wild-type Subtilisin E in 85% DMF. This study illustrates the promise of mutagenesis in improving enzyme activity in polar organic media for ultimate use in chemical synthesis. Recommendation-10: Assess whether the energetic effects of multiple mutations are additive with respect to the Gibbs free energy changes for individual mutations. The Additivity Principle states that different properties of a molecular substance often contribute separately and additively to an overall thermodynamic property of a substance. In 1840, Hess introduced this concept in the form of the Law of Constant Heat Summation. This relationship allows us to estimate the heat of a reaction from collected measurements of seemingly different reactions, as long as the summation of a series of reactions yields the same overall chemical reaction as the one of interest. Thermodynamic additivity is path-independent, such that the energy change attending the conversion of reactant A to product P may be measured as a single-step or multi-step process: A # B, DGAB; with A # Q, DGAQ; Q # R, DGQR; R # S, DGRS; and S # B, DGSB, such that, DGAB ¼ DGAQ þ DGQR þ DGRS þ DGSB. Thermodynamic additivity requires that if two components, A and B, contribute independently to some process, then the total change in free energy (or enthalpy or entropy) is the sum of components, DG ¼ DGA þ DGB. Awt Bmut

G3

Amut Bmut G4

G1 Awt Bwt

G2

Amut Bwt

Scheme 7.37

Enzyme Kinetics

470

Consider two illustrative cases involving single and double point-mutations in carboxypeptidase A and triose-P isomerase. As discussed by Knowles (1987), the zincdependent exopeptidase known as carboxypeptidase A (CpA) possesses two residues Tyr-248 (Y248) and Tyr-248 (Y248) thought to be essential for the enzyme-catalyzed cleavage of peptide substrates. Replacement of either and both tyrosines by phenylalanine resulted in changes in kcat/ Km for selected substrates (Hilvert et al., 1986). Similarly, changes in kcat/Km as a result of the single and double point mutations in triose-P isomerase have been determined (Knowles, 1987). Changes in rate constant for each mutation could then be converted to free energy using the expression DDGz ¼ RT ln{(kcat/Km)State-2/(kcat/Km)State-1. This analysis allows us to conclude that these indicated substitutions in carboxypeptidase A are additive (left-hand thermodynamic cycle in Fig. 7.32), indicating that the mutations are independent. This is not the case for the sitespecific substitutions in triosephosphate isomerase (righthand thermodynamic cycle in Fig. 7.32). The latter are said to be interdependent, because thermodynamic additivity is not evident. Horovitz and Fersht (1990) also employed thermodynamic cycles, such as the one in Scheme 7.37, to dissect complex interactions amongst multiple residues. In this case, the pertinent parameter is the difference in the free energy changes (i.e., DDG ¼ DGmut – DGwt ¼ DGc) where DGc > 0 indicates reduced stability, and DGc < 0 indicates increased stability. Double-mutant cycles have been widely used in the field of protein engineering to measure intermolecular and intramolecular interactions. Ideally, there

Carboxypeptidase A

Y198F Y248F

Triose-P Isomerase

2.0

Y248F

E165D 3.5

1.5

E165D S96P 2.5

G kcal/mol 2.2

0.4 wild type

Y198F Independent & Additive Mutations

wild type

S96P Interdependent & Nonadditive Mutations

FIGURE 7.32 Independent and interdependent mutations in carboxypeptidase A and triose-P isomerase. The effects of the indicated substitutions were determined as changes in the apparent bimolecular rate constant for substrate (estimated as kcat/Km). Plotted on the vertical axis are DG values (expressed in kcal/mol). Redrawn from Knowles (1987) with permission of the publisher (Science).

should be no structural rearrangements other than the two single mutations. Structural perturbations elsewhere do not preclude the use of this method, as long as the sum of the changes in the single mutants equals the change in the double mutant. As pointed out by Kraut, Carroll and Herschlag (2003), however, there is no fundamental expectation of quantitative energetic additivity in chemical systems. In fact, additivity holds as an approximation only in special cases in which local factors dominate. Coulombic interactions between charges on protein surfaces contribute to protein stability, but their contribution has been difficult to assess quantitatively, in part because a mutation of one residue that is a partner in an ion pair, alters the energetics of many nearby groups beyond the coulombic energy between the ion-pair components. To approach this problem, Serrano et al. (1990) carried out an important series of experiments on barnase (acronym for Bacillus amyloliquefaciens ribonuclease), an enzyme comprised of a single polypeptide chain (Mol. Wt. ¼ 12,382 g) with no disulfide cross-links. The enzyme undergoes reversible unfolding induced either thermally or in the presence of organic solvents, and offers the advantage that protein structural changes by spectroscopic methods as well as by enzyme activity assays can be assessed. Such measurements of the energetics of unfolding of wild-type and mutant barnase may be used to quantify energies of interaction between side chains (Kellis, Nyberg and Fersht, 1988; Kellis et al., 1989) or long-range electrostatic interactions (Sali, Bycroft and Fersht, 1988; Serrano et al., 1990). Serrano et al. (1990) estimated the interaction energy between two charged residues, Asp-12 and Arg-16, in an a-helix located on the surface of a barnase mutant by invoking a double-mutant cycle involving wild-type Asp12:Thr-16, the two single mutants Asp-12:Thr-16-Arg and Asp-12-Ala:Thr-16, and the double-mutant Asp-12 Ala:Thr-16-Arg. The changes in free energy of unfolding of the single mutants were found to be non-additive, presumably as a consequence of the coulombic interaction energy. However, additivity was restored at high ionic strength, where nearby electrostatic interactions were screened by the added salt. The geometry of the ion-pair in the mutant was assumed to be the same as that in the highly homologous Bacillus intermedius RNase having Asp-12 and Arg-16 in the native enzyme. The ion-pair does not form a hydrogen˚. bonded salt bridge, but the charges are separated by 5–6 A The mutant barnase containing the ion-pair Asp-12:Arg-16 was determined to be more stable than wild type by 0.5 kcal/ mol, but only a part of the increased stability is attributable to the electrostatic interaction. Serrano et al. (1990) presented a formal analysis of how double-mutant cycles can be used to measure the energetics of pair-wise interactions. The reader will appreciate that coulombic interactions are long-range phenomena (see Section 2.2.1: Electrostatic

Chapter j 7 Factors Influencing Enzyme Activity

Interactions), and mutations in charged residues need not exhibit quantitative thermodynamic additivity. In view of its broad application in characterizing chemical and physical properties, additivity has been called the ‘‘fourth’’ law of thermodynamics (Benson, 1976). Thermodynamic additivity has been utilized to assess intrinsic sub-site binding energies (Allen, 1979) for enzymes that have substrate binding sub-sites and exo-sites topologically distant from the reactive site (e.g., polysaccharide depolymerases and proteinases). Dill (1997) recently discussed the merits and limitations of models that assume thermodynamic additivity and independence (of energy types, of neighbor interactions, of conformational freedom, of monomer contact pairing frequencies, etc.). He states that biological molecules may achieve stability in the face of thermal uncertainty, as polymers do, by compounding many small interactions; this summing can confound modelers, because application of the Additivity Principle leads to accumulated error. Finally, although the Fersht approach focuses on linear free-energy Brønsted relationships to analyze site-directed mutagenesis effects on catalysis, and treats the Brønsted exponent as being linearly correlated with the reaction coordinate at the transition state, Straub and Karplus (1990) suggested that, when the mutants differ solely through the formation or deletion of a hydrogen bond away from the reaction center, a linear free energy relation is to be expected only in limiting cases for which the Brønsted exponent is 0, 1 or N. They instead suggest that the experimental data may be better correlated with the conformational coordinate. Recommendation-11: Consider the possibility that residues that are outside the active site may modify catalysis or enzyme regulation. Although many pH kinetic studies and spectroscopic studies indicate that the acid/base groups within an enzyme’s active site are relatively isolated from other residues, slight changes in the polarity of a residue outside the catalytic center have been found to have discernible effects. For example, kinetic studies of E. coli aspartate aminotransferase by Gloss, Spencer and Kirsch (1996) revealed that several site-specific substitutions for Cys-191, a conserved residue situated outside the enzyme’s active site, nonetheless resulted in an 0.6–0.8 pH-unit alkaline shift in the pKa of the internal aldimine (Schiff’s base) joining the coenzyme pyridoxal phosphate (PLP) to Lys-258. The change in the pKa affected the pH dependence of the kcat/Km,Aspartate for the mutant enzymes. To understand the structure-function implications of these observations, Jeffery et al. (2000) determined the crystal structures of five maleate-bound forms (Cys-191-Ser, Cys-191 Phe, Cys-191-Tyr and Cys-191-Trp), as well as the maleate-free Cys-191-Ser ˚ resolution in the presence of PLP. transaminase, at ~2 A

471

The overall three-dimensional fold of each mutant enzyme was the same as the wild-type protein, except for a rotation of the mutated side chain around its Ca–Cb bond. This side chain rotation results in an altered pattern of hydrogen bonds connecting the mutant residue and the protonated Schiff’s base of the cofactor, which could account for the altered pKa of the Schiff’s base imine nitrogen. Such findings demonstrate how residues outside the active site can be important in determining subtle active-site geometries and in explaining how mutations outside the active site may affect catalysis. In a now classic series of experiments, the group led by British enzyme chemist Alan Fersht investigated the impact of binding energy, chiefly in the form of hydrogen bonding interactions, in the reaction catalyzed by tyrosine–tRNATyr ligase:

Tyr + ATP + tRNATyr

Tyr–tRNATyr + AMP + PPi

Scheme 7.38 Because expulsion of an oxygen atom as a hydroxide ion from the amino acid’s a-carboxylate group is highly disfavored at neutral pH, amino acyl–tRNA synthetases exploit a two-stage reaction mechanism:

Stage-1:

Tyr + ATP

Tyr-AMP + PPi

Stage-2: Tyr-AMP + tRNATyr

Tyr-tRNATyr + AMP

Scheme 7.39 In Stage-1, the enzyme catalyzes the formation of an amino acyl–adenylate intermediate, followed by Stage-2, in which the activated amino acyl group is transferred to the cognate tRNA. Amino acyl–adenylate formation in Stage-1 is unfavorable in solution (Keq,external z 10–7.5), but is highly favorable (Keq,internal z 2) within the synthetase’s active site. To achieve such stabilization, the enzyme employs a set of hydrogen bonds to achieve the internal stabilization effect amounting to a free energy change of ~9 kcal/mol (~37 kJ/mol). Xin, Lin and First (2000) examined variants at each position of the ‘‘Lys-Met-Ser-Lys-Ser’’ motif at positions 230–235 of tyrosyl-tRNA synthetase to test the hypothesis that this motif is involved in the catalysis of the second stage of the aminoacylation reaction (i.e., the transfer of tyrosine from the enzyme-bound tyrosyl-AMP intermediate to the tRNATyr substrate). Pre-steady-state kinetic studies showed that, while the rate constants for tyrosine transfer are similar to the wild-type value for all of the mobile-loop variants, the Lys-230-Ala and Lys-233A variants have increased

Enzyme Kinetics

472

dissociation constants (KdtRNA ¼ 2.4 and 1.7 mM), respectively) relative to the wild-type enzyme (KdtRNA ¼ 0.39 mM). In contrast, the KdtRNA values for the Phe-231 Leu, Gly-232-Ala, and Thr-234-Ala variants resemble that of the wild-type enzyme. The KdtRNA value for a loop deletion variant D227–234 is similar to that for the Lys-230 Ala/Lys-233-Ala double mutant (3.4 and 3.0 mM, respectively). Double-mutant free energy cycle analysis indicates there is a synergistic interaction between the side chains of Lys-230 and Lys-233 during the initial binding of tRNATyr (DDGint ¼ –0.74 kcal/mol). Xin, Lin and First (2000) suggested that while the KMSKS motif is important for the initial tRNATyr binding to the synthetase, this motif does not play a catalytic role in the second stage of the catalytic reaction cycle.

enhancement that is some 109 greater than catalysis by acetate ion (Hall and Knowles, 1975; Richard, 1984). The overall reaction equilibrium constant (Keq ¼ [DHAP]/[GAP] z 21) puts the DGreaction at approximately –2 kcal/mol. Some time ago, Rose (1962) proposed a chemical mechanism (Scheme 7.40) for the likely involvement of an enzyme-bound ene-diol intermediate in TIM catalysis, and he later demonstrated that (R)-[2-3H]-glyceraldehyde 3-P and (R)-[1-3H] hydroxyacetone-P largely exchange their tritium label with solvent as protons during TIM catalysis (Rose, 1975).

Recommendation-12: Treat every enzyme mutant as though it is an entirely new enzyme. An often-invalid assumption is that a seemingly straightforward point mutation only introduces a small change in polarity or ionic charge. Numerous published and anecdotal accounts, however, suggest that each mutant enzyme must be regarded as a ‘‘new’’ enzyme, one requiring detailed structural characterization before proceeding to mechanistic experiments. While many enzymes tolerate modest structural modifications, others often fail to fold, oligomerize, precipitate, etc. There are simply no reliable rules for predicting the impact of mutations on a particular enzyme’s structure or activity. The best way to proceed with site-specific mutagenesis is to recognize that a seemingly minor mutation may have unanticipated consequences. Therefore, the fullest possible characterization of all mutant enzymes should be considered (e.g., CD spectroscopy, calorimetry, sedimentation equilibrium ultracentrifugation, kinetics, and even highresolution NMR or X-ray crystallography). An experienced colleague once remarked that, after settling on a mutation of interest, it is now routine to prepare the mutant cDNA in its vector, bacterially express and purify that mutant, carry out the indicated series detailed isotope exchange and kinetic isotope effect experiments, analyze the rate data, and obtain the X-ray crystal structure – all within the span of one month. Because few labs can master all of these techniques, a durable collaboration with a crystallographer is the key to success.

7.15.5. Triose-Phosphate Isomerase: a Case Study in Directed Mutagenesis The deceptive simplicity of triosephosphate isomerase (TIM) action has inspired many highly inventive kinetic studies. This glycolytic enzyme catalyzes the reversible interconversion of (R)-glyceraldehyde 3-phosphate (GAP) and dihydroxyacetone phosphate (DHAP), exhibiting a catalytic rate

Scheme 7.40 In the above scheme, Glu-165 as the likely catalytic base B2 responsible for proton abstraction from carbon and His95 is thought to play multiple roles (see below). Numerous kinetic studies have confirmed that triose-P isomerase is a remarkably efficient enzyme. With a kcat of 3,500 s1 and a Km of 0.5 mM for DHAP, the apparent bimolecular association rate constant kcat/Km is 107 M1$s1), a value that is at or very near the diffusion limit. Based on the chemical reaction pathway, the minimal kinetic reaction mechanism for TIM catalysis may be depicted as:

E+GAP

E·GAP

E ·Z

E·DHAP

E + DHAP

Scheme 7.41 where E is a free enzyme, E$GAP is the substrate Michaelis complex, E$Z is the enzyme bound ene-diol intermediate, and E$DHAP is the product Michaelis constant. There are thus two chemical transition states, corresponding to: (1) conversion E$GAP to the E$Z intermediate; and (2) transformation of E$Z to E$DHAP, and two additional transition states controlling the binding/release rates for substrate and product. British chemists John Albery and Jeremy Knowles devised sets of experiments that allowed for the evaluation

Chapter j 7 Factors Influencing Enzyme Activity

of all eight rate constants, thereby establishing the energetics of the catalytic reaction cycle (see Fig. 5.10). As noted earlier, the design of informative site-directed mutagenesis experiments requires high-resolution structures, from which we may infer likely role(s) for residues in the substrate’s vicinity. The X-ray crystal structures of the ˚ (Albery and Knowles, yeast TIM were determined to 1.9 A ˚ 1986), and there is a 1.6-A structure for TIM with the competitive inhibitor phospho-glycolohydroxamate (Davenport et al., 1991). Interactions of the enzyme and its substrates have also been obtained from NMR spectroscopy (Browne et al., 1976; Webb et al., 1977) as well as Fourier Transform infrared spectroscopy (Belasco and Knowles, 1980). Given this abundance of structural data and the considerable literature on the kinetic and chemical data for triose-P isomerase, the Knowles group focused on TIM as a paradigm for systematic site-directed mutagenesis studies of enzyme catalysis. The following sub-sections highlight important conclusions derived from their work and related investigations.

7.15.5a. Role of Conserved Residue Glutamate-165 In the light of the alkylation of Glu-165 by active-sitedirected irreversible inhibitors, Strauss et al. (1985) and Raines et al. (1986) prepared wild-type TIM (wt-TIM) directly from chicken muscle as well as wild-type and its Glu-165-Asp point mutant after expression in E. coli. The

FIGURE 7.33 Three-dimensional structure of chicken muscle trioseP isomerase. Shown are the general structural features of the TIM dimer, including the well-known TIM barrel consisting of eight strands of parallel b-pleated sheet connected mainly by a-helical segments. Knowles (1990) identified the three main structural features of triose-P isomerase that give rise to highly efficient catalysis: (1) the substrate binding site consisting of four main-chain –NH– hydrogen bonds, of which two are situated at the end of a short a-helix in a manner making favorable electrostatic interactions with the three peripheral oxygen atoms of the phosphate group; (2) a catalytic constellation consisting of a catalytic base (Glu-165 anion), suitable for proton abstraction, as well as a neutral imidazole (with a perturbed pKa < 4.5); and (3) a flexible loop overlying the active site and holding reaction intermediates in place during catalysis.

473

specific catalytic activities of the wild-type isomerases were identical, attesting to the fact that bacterial expression is a valid way to study chicken TIM catalysis. They found, however, that the specific activity of the mutant enzyme is reduced by a factor of about 1,000. Raines et al. (1986) also found that such observed kinetic differences do not reflect a change in catalytic mechanism, such as the mutant enzyme aspartate acting as a general base through an ˚ intervening H2O molecule. In their analysis of the 2-A resolution structure of the Glu-165-Asp TIM mutant, Joseph-McCarthy et al. (1994) found that the catalytic base is oriented so as to use the carboxylate’s anti orbital for proton abstraction, whereas in the wild-type enzyme, the syn orbital is the likely base. They suggest that the 1,000 lower catalytic activity of this mutant is due either to the use of the less basic anti-orbital for proton transfer or to the greater distance between the base and the substrate. Raines et al. (1986) defined the complete reaction energetics for the Glu-165 Asp mutant, leading to the conclusion that only the free energies of the transition states for the two enolization steps were seriously affected. Each of the proton abstraction steps is about 1,000-fold slower in the mutant enzyme. Evidently, the excision of a methylene group from the side chain of the essential glutamate has little effect on the free energies of the intermediate states but dramatically reduces the stabilities of the transition states for the chemical steps in the catalyzed reaction. Finally, Blacklow et al. (1988) used the Glu-165-Asp TIM, which is known to be limited in rate by an enolization step in the catalytic mechanism, to interpret the behavior of the wild-type enzyme. Plots of the relative values of kcat/Km for catalysis by the wild-type enzyme (normalized with the corresponding data for the mutant enzyme) versus the relative viscosity have slopes close to unity, as predicted by the Stokes-Einstein equation for a diffusive process. In the presence of polymeric viscosogenic additives such as poly(ethylene)glycol, polyacrylamide, or Ficoll, no effect on kcat/Km is seen for wild-type TIM, consistent with the expectation that molecular diffusion rates are unaffected by the macroviscosity and are only slowed by the presence of smaller agents that raise the microviscosity. These results show that the reaction catalyzed by wild-type TIM is limited by the rate at which GAP adds to or leaves the active site.

7.15.5b. Role of Conserved Residue Histidine-95 As discussed by Lodi and Knowles (1991), there are at least three functions that could be fulfilled by His-95: (1) polarization of the substrate carbonyl, thereby facilitating pro-R hydrogen abstraction at Carbon-1 of DHAP by Glu-165; (2) stabilization of the reaction intermediate; and (3) facilitation of proton transfer to and from Oxygen-1 and Oxygen-2 atoms of the enediol(ate) intermediate prior to

Enzyme Kinetics

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G (kcal/mol)

20

Glu-165-Asp Mutant

10

0

ES

EZ

EP

E+P

E+S

Reaction Progress FIGURE 7.34 Reaction progress diagram for triose-phosphate isomerase. Wild-type TIM is shown as a solid line, and its Glu-165-Asp mutant form as a dotted line. Note that, although binding and release steps for substrate and product are unaffected by this site-specific substitution, there are higher transition states for the two enolization steps leading to the formation and conversion of the enediol intermediate. To construct the mutant profile, Raines and Knowles (1986) determined the following rate constants (with corresponding DG values) for the mutant enzyme: k1 ¼ 107 M1 s1 (14.1 kcal/mol); k1 ¼ 1.1  104 s1 (12.2 kcal/mol); k2 ¼ 2.2 s1 (17.3 kcal/mol); k-2 ¼ 12 s1 (16.3 kcal/mol); k3 ¼ 1.3  102 s1 (14.8 kcal/mol); k3 ¼ 33 s1 (15.6 kcal/mol); k4 ¼ 1.9  103 s1 (13.2 kcal/mol); and k4 ¼ 3.7  108 M1 s1 (12.0 kcal/mol). Rate constants for wild-type TIM were determined by Albery and Knowles (1976). The equilibrium constant value of 21.5 for the reaction catalyzed Glu-165-Asp mutant closely matches the Keq value of 22 previously reported for the wild-type enzyme. This is the expected result, because an enzyme cannot alter the equilibrium poise of its catalyzed reaction.

collapsing to form product. The stereospecificity of proton removal and the low level of proton transfer from DHAP to GAP suggested that a single enzymic base mediates the proton transfer process between carbon centers (Herlihy et al., 1976; Rose, 1962). Nickbarg et al. (1988) considered the possibility that catalysis by a single base might be inadequate to account for the remarkable catalytic efficiency of the native enzyme. Direct observation of enzymebound DHAP by Fourier transform IR spectroscopy revealed that the C¼O stretching frequency decreases by 19 cm1 relative to solution-phase substrate (Belasco and Knowles, 1980), suggesting the isomerase polarizes the substrate carbonyl by interacting with an active-site electrophile. Significantly, crystallographic data suggest the existence of a hydrogen bond between the d-nitrogen of His95 and the N-H group of the main-chain peptide bond of Glu-97, suggesting that the unprotonated His-95 imidazole ring might catalyze substrate enolization via electrostatic stabilization (Alber et al., 1981). Nickbarg et al. (1988) first reported that the His-95-Gln yeast TIM mutant has a 400 times lower specific catalytic activity than wild type. They also found that the substrate analogues 2-phosphoglycolate and phosphoglycolohydroxamate respectively bind 8 and 35 times more weakly to the

mutant isomerase. The enzyme mutant shows the same stereospecificity of proton transfer as the wild type, Even so, this TIM mutant failed to catalyze any appreciable exchange between protons of the remaining substrate and those of the solvent; however, the specific radioactivity of GAP formed in the forward reaction is 31% that of the solvent, while that of DHAP formed in the reverse reaction is 24% that of the solvent. The deuterium kinetic isotope effects observed with the mutant isomerase using (R)-2H]dihydroxyacetone phosphate and [2-2H]glyceraldehyde 3-phosphate are 2.15  0.04 and 2.4  0.1, respectively, indicating that substitution of Gln for His-95 greatly impairs the ability of the enzyme by stabilizing the reaction intermediate to such an extent that there is a change in the proton transfer pathways. Lodi and Knowles (1991) investigated the role of His-95 using I3C and 15N NMR to monitor pH-dependent changes in key residues. They studied both the wild-type enzyme and a mutant isomerase in which the single remaining active-site histidine was isotopically enriched in its imidazole ring. Their 15N NMR results unambiguously demonstrated that the imidazole ring of histidine-95 remains uncharged over the entire pH range of isomerase activity (i.e., from pH 5 to 9.9), effectively ruling out the traditional view that the positively charged imidazolium cation of His-95 donates a proton to the developing charge on the substrate’s carbonyl oxygen. Their results require that the first pKa of His-95 must be below 4.5, and later evidence suggested that the dipole of the N-terminal a-helix may perturb the histidine ionization. In later work, these investigators again relied on 15N NMR titration studies of the native enzyme and 13C NMR titration studies of the denatured enzyme to show that, while the pKa of histidine-95 is lowered by a least 2 units in folded versus unfolded states, the pKa of histidine103 is raised by about 0.6 units upon protein folding (Lodi and Knowles, 1991). Such complementary effects on the pKa, values of histidine-95 and histidine-103 suggest that dipolar effects associated with the short N-terminal a-helix underlie these perturbed pKa values. Other important work by Komives et al. (1991) also clarified the role of His-95. As noted above, the C¼O stretching frequency of DHAP bound to wild-type enzyme is lower than unbound DHAP, a property ascribed to an enzymic electrophile that polarizes the substrate carbonyl group toward the transition state for enolization. IR spectra of substrate bound to His-95-Gln and His-95-Asn mutants of yeast TIM have unperturbed carbonyl stretching frequencies between 1,732 and 1,742 cm1, and the lack of carbonyl polarization upon removal of His-95 suggested that His-95 is indeed the catalytic electrophile for dihydroxyacetone phosphate. They also found that Glu-165 is recruited to perform the proton transfers involving the substrate oxygens during catalysis by the His-95-Gln enzyme, strongly suggesting that His-95 fulfills this role in catalysis by the wild-type enzyme. Therefore, it seems likely that His-95 plays a dual role as an electrophile and as

Chapter j 7 Factors Influencing Enzyme Activity

a general acid/base in the catalysis of triose phosphate isomerization.

7.15.5c. Role of Conserved Residue Lysine-12 Comparison of TIM primary structures indicates that Lys-12 is a highly conserved residue, and its active-site location has invited speculation about potential role(s) in enhancing catalysis. The likely role of this residue in yeast TIM was elucidated by Lodi et al. (1994), who applied a combination of site-directed mutagenesis, Fourier transform infrared spectroscopy, enzyme kinetics, and X-ray crystallography. The Lys-12-Met mutant enzyme lacked detectable catalytic activity, and infrared experiments show no evidence of DHAP binding to the active site. Crystals of the enzyme grown in the presence of phosphoglycolohydroxamate, a potent reaction intermediate analogue, revealed an open active site with no bound inhibitor. Mutation of Lys-12 to arginine produces a protein with a 22 higher Michaelis constant and a 180 drop in maximal velocity. Moreover, mutation of Lys-12 to histidine yielded an enzyme showing virtually no catalytic activity at neutral pH. However, because the Lys-12-His enzyme is active below pH 6.1, Lodi et al. (1994) suggested that protonation of the histidine in this mutant is required for activity. Together with the structural results, it appears that the positive charge of Lys-12 is required for substrate binding.

7.15.5d. Role of Conserved Residue Cysteine-126 Cys-126 is a conserved residue that in native TIM is situated near the catalytic residue Glu-165. Gonza´les-Mondrago´n et al. (2004) examined the catalytic and stability properties of mutants Cys-126-Ala and Cys-126-Ser of wild-type TIM. Neither replacement induced significant changes in protein folding, as indicated by spectroscopic studies. Cys-126 Ala and Cys-126-Ser have Km and kcat values that are concomitantly reduced by only 1.5-fold and 4-fold, respectively, compared to those of wild-type TIM; in either case, however, the catalytic efficiency (kcat/Km) of the enzyme is barely affected. The affinity of TIM mutants for the competitive inhibitor 2-phosphoglycolate also increased slightly. In contrast, greater susceptibility to thermal denaturation resulted from mutation of Cys-126, especially when changed to Ser. By using values of the rate constants for unfolding and refolding, Gonza´les-Mondrago´n et al. (2004) estimated that, at 25 C, Cys-126-Ala and Cys-126-Ser are less stable than wild-type TIM by about 5.0 and 9.0 kcal/ mol, respectively. Moreover, either of these mutations slows down the folding rate by a factor of 10 and decreases the recovery of the active enzyme after thermal unfolding. Thus, Cys-126 is required for the proper stability and efficient folding of TIM rather than for enzymatic catalysis.

475

7.15.5e. Role of Conserved Active-Site V-Loop As discussed in Section 5.11, the induced-fit hypothesis of Koshland (1958) provided an explanation on how catalysis may be linked to conformational changes that reposition catalytic groups for: (a) improved substrate recognition; (b) removal of otherwise deactivating water molecules; and/or (c) greater catalytic efficiency. In principle, such changes may result in subtle rearrangements of the active site or larger scale changes involving segmental motion of subunits or the opening/closing of a flap or loop. Triose P isomerase possesses a mobile 10-residue loop (located at 166-Pro-Val-Trp-Ala-Ile-Gly-Thr-Gly-Leu-Ala-176), wherein the first and last three residues act as the N- and C-terminal hinges. This so-called V-loop has been shown via crystallography to leave the active site open to solvent in the unliganded form of the enzyme, and to encapsulate substrate analogues at the active site in the liganded form of the protein (Lolis and Petsko, 1990; Lolis et al., 1990; Noble et al., 1991; Wierenga et al., 1991). Interestingly, the structure of the loop itself remains unchanged in the loop-open and loop-closed states (Wierenga et al., 1992), indicating all loop residues appear to move as a rigid entity, creating a well-articulated active-site lid. The reader will also note that, because the catalytic residue Glu-165 lies only one residue away from this loop, changes in the loop conformation are apt to affect catalysis directly. Solid-state NMR experiments indicate that substrateindependent loop opening and closing motions occur on a catalytic time-scale (Williams and McDermott, 1995), suggesting further that the loop motions rely on a highly flexible hinge. To discover the protein structural elements needed for a loop proper closure, Sampson and Knowles (1992) selectively destabilized the open and closed forms of the enzyme with respect to each other. The mutant Tyr-164-Phe isomerase was prepared to evaluate the importance of the structure of the open form, and the mutant Glu-129-Gln, Tyr-208-Phe, and Ser-211-Ala enzymes allowed investigation of the closed form. The integrity of the loop itself was destabilized by making the Thr-172-Ala isomerase. Sampson and Knowles (1992) found that only those mutations that destabilize the closed form of the enzyme significantly perturb the catalytic properties of the isomerase. The apparent second-order rate constants (kcat/Km) of the Ser-211-Ala and Glu-129-Gln enzymes are reduced 30-fold, and that of the mutant Tyr-208-Phe enzyme is reduced 2,000-fold, from the level of wild-type TIM. The dramatic drop in activity of the Tyr-208 Phe enzyme is accompanied by a 200 increase in the Kd for the intermediate analogue phosphoglycolohydroxamate. They suggested that the most important property of the mobile loop of triosephosphate isomerase resides in the stability of the system when the active site contains ligand and the loop is closed.

476

FIGURE 7.35 Ribbon diagrams of open and closed V-loop conformations of triose-P isomerase. The O in the drawing indicates the Open-loop conformation (magenta), and the C indicates the Closed-loop conformation (green). The substrate analogue glycerol 3-phosphate (shown as a CPK model) is docked within the active site. Drawing from Jiang, Jung and Sampson (2004) with permission of the authors and publisher (American Chemical Society).

As noted above, Tyr-208 is essential for proper closure of the active-site loop, with its phenolic hydroxyl group forming a hydrogen bond with the amide nitrogen of the loop residue Ala-176. Both residues are conserved, and Tyr/Phe mutagenesis is attended by a 2000-times drop in the kcat/Km (Sampson and Knowles, 1992). The reaction catalyzed by the mutant enzyme shows a viscosity dependence using glycerol as the viscosogen and is consistent with the rate-limiting motion between the open and the closed loop conformations. A large primary isotope effect is observed with [l-2H]DHAP as substrate (kcat/Km)H/(kcat/ Km)D ¼ 6  13. Comparison of the energetics of the wildtype and mutant enzymes indicated that only the transition states flanking the enediol intermediate are affected substantially, suggesting either that loop closure and deprotonation are coupled and occur in the same ratelimiting step, or that these two processes happen sequentially, but interdependently. This finding is consistent with structural information that indicates that, upon loop closure, ˚ toward the substrate. the catalytic base Glu-165 moves 2 A The motion of TIM’s mobile loop is therefore linked to substrate deprotonation by the catalytic base Glu-165. Sun and Sampson (1998) determined the sequence requirements for TIM’s active-site loop by replacing the loop’s C-terminal hinge (residues 174–176) coding sequence with a genetic library consisting of all possible 8,000 amino acid combinations (i.e., 203) in the loop’s hinge. The most active of these were selected using in vivo complementation of the aforementioned TIM-deficient E. coli strain DF502. It was found that about 3% of the mutants complemented DF502 with an activity >70% of

Enzyme Kinetics

wild-type TIM activity. The sequences of these hinge mutants revealed that hinge flexibility could be restored in many ways. Finally, the active-site loop does not change conformation during opening and closing; instead, as noted by Xiang, Jung and Sampson (2004), loop motions are attended by local conformational changes in two hinges at the ends of the loop. Moreover, they found that glycine is never observed in hinge sequences from a wide variety of triose-P isomerases. They therefore tested the hypothesis that limited access to conformational space is required for protein hinges involved in catalysis. They replaced Val-167, and Try-168 in the N-terminal Pro-Val-Try hinge of chicken TIM by glycine residues to obtain ‘‘PGG’’-TIM and likewise mutated the C-terminal hinge K-174, Thr 175, and Ala-176 to obtain ‘‘GGG’’-TIM. The single-hinge mutants PGG and GGG had kcat values that were around 200 times lower than that of the wild-type TIM, with approximately 10 higher values for DHAP Michaelis constants. For the double-hinge mutant ‘‘PGG/GGG’’-TIM, the resulting kcat was reduced 2,500-fold, with a ten times higher Km value. By applying a combination of primary kinetic isotope effect measurements, isothermal calorimetric measurements, and 31 P-NMR spectroscopic titrations using the inhibitor 2-phosphoglycolate, Xiang, Jung and Sampson (2004) found that the mutants have a different ligand-binding mode than that of the wild-type enzyme. The predominant conformations of the mutants even in the presence of the inhibitor are loop-open conformations. They therefore concluded that mutation of the hinge residues to glycine probably allowed the loop to sample many more conformations, thereby reducing the population of the activeclosed conformations.

7.15.5f. Improving Catalytic Potency via Random Mutagenesis The convenience of conducting enzyme mutagenesis also invited the question: ‘‘How easy is it to improve the catalytic power of an enzyme?’’ Hermes, Blaklow and Knowles (1990) introduced random mutations in the DNA encoding the Glu-165-Asp mutant of yeast triose-P isomerase. As noted above, this mutant enzyme is sluggish as a consequence of significant changes in quantifiable kinetic parameters. After random mutagenesis over its whole length with ‘‘spiked’’ oligonucleotide primers, transformation of an isomerase-minus strain of E. coli was followed by selection of those colonies harboring an enzyme of higher catalytic potency. At least one million transformants from each of the ten spiked primers were subjected to selection on glycerol plates. As shown in Table 7.28, six amino acid substitutions in the sluggish Glu-165-Asp mutant modestly improve enzyme specific activity (i.e., by factors of 1.3 to 19). These suppressor sites are scattered across the sequence

Chapter j 7 Factors Influencing Enzyme Activity

477

(at positions 10, 96, 97, 167, and 233), but each of them lies reasonably close to the active site. As a result of these clever experiments, we now know that: (a) relatively few single amino acid changes can increase the catalytic potency of the Glu-165-Asp enzyme; and (b) all improvements involve alterations situated in, or very near, the active site. After subjecting mutant cDNA constructs for the expression of two sluggish isomerase mutants (Glu-165Asp-TIM and His-95-Asn-TIM) to the random mutagenesis with chemical reagents, Blacklow and Knowles (1990) selected for pseudo-revertant isomerases with increased catalytic potency. Remarkably, the same second-site mutation Ser-96-Pro improves the catalytic efficiency of these sluggish enzymes. Glu-165-Asp/Ser-96-Pro-TIM is a 20-times better catalyst than the Glu-165-Asp-TIM, and His-95-Asn/Ser-96-Pro-TIM is about 60 times more active than its parent His-95-Asn TIM. The change results in stabilization of the enediol(ate) intermediate and its flanking transition states. Blacklow, Liu and Knowles (1991) extended this work to investigate whether multiple second-site mutations would further enhance the catalytic action of single second-site pseudo-revertants of Glu-165Asp-TIM and His-95-Asn-TIM. They concluded that since the catalytic potency of each of the six second-site suppressor mutants can be further improved by the introduction of (at least) one of the other five changes, it is evident that none of the double mutants lies at a local catalytic maximum.

7.15.5g. Unresolved Aspects of TIM Catalysis Lest the foregoing sections leave the reader with the conclusion that TIM catalysis has been fully elucidated, it

seems appropriate to consider issues about the longstanding view of TIM catalysis. In that proton transfer mechanism for TIM catalysis, the carboxylate anion side chain of Glu-165 functions as a Brønsted base that abstracts a proton from the a carbonyl carbon of bound substrate, and the developing negative charge at the carbonyl carbon is stabilized by H-bonding to the neutral imidazole side chain of His-95; isomerization reaction is completed by reprotonation at the adjacent carbon of the enediol(ate) intermediate. Amyes, O’Donoghue and Richard (2001) found that ~80% of the enzymatic rate acceleration for TIM-catalyzed GAP isomerization can be attributed to interactions between the enzyme and the substrate’s remote, nonreacting phospho-dianion group, which stabilizes the transition state for deprotonation of a carbonyl carbon by 14 kcal/mol. That work showed that 1H-NMR spectroscopy can be used to monitor not only the velocity of TIM turnover of the ‘‘unnatural’’ substrate (R) glyceraldehyde in D2O, but also provides information about: (a) the velocity of formation of the isomerization product dihydroxyacetone that proceeds with intramolecular hydrogen transfer and with incorporation of solvent deuterium at C-1; and (b) the velocity of formation of (R)-glyceraldehyde deuterated at C-2. Building on the success of those studies, O’Donoghue, Amyes and Richard (2005) initially explored TIM-catalyzed isomerization of the physiological substrate (R)-glyceraldehyde 3 phosphate in D2O, to determine the effect of the substrate’s phospho-dianion group on the partitioning of the enediol(ate) intermediate in D2O between formation of H- and D-labeled products. However, their results have greater mechanistic importance. They found that the TIMcatalyzed isomerization of GAP in D2O resulted in a 49% yield of DHAP formed by intramolecular hydrogen transfer

TABLE 7.28 Summary of Kinetic Parameters for Wild-Type and Glu-165-Asp TIMs, and Pseudorevertants Derived by Random Mutagenesis of Glu 165-Asp TIM* Enzyme

kcat,forward (s1)

KDHAP (mM)

kcat,reverse (s1)

KGAP (mM)

kcat,f/KDHAP relative to Glu-175-Asp

Wild type

600

0.65

8300

0.42

270

Glu-165-Asp

4.1

1.2

4.2

0.078

1.0

Glu-165-Asp, Gly-10-Ser

4.7

1.1

14

0.18

1.3

Glu-165-Asp, Ser-96-Pro

3.4

0.053

68

0.066

19

Glu-165-Asp, Ser-96-Thr

9.5

1.3

17

0.10

2.1

Glu-165-Asp, Glu-97-Asp

5.8

0.93

10

0.079

1.8

Glu-165-Asp, Val-167-Asp

8.4

1.7

15

0.17

1.4

Glu-165-Asp, Gly-233-Arg

6.0

0.33

8.4

0.029

1.5

*From Hermes, Blacklow and Knowles (1990). Note: Selection is based on earlier observation that, while the isomerase-minus host strain will grow on glycerol (from which DHAP is made) containing lactate (which provides GAP), it cannot grow on either glycerol alone or lactate alone. Plasmid-mediated expression of wild-type isomerase from the trc promoter allows DF502 to grow on either glycerol or lactate as the sole source of carbon. When the mutant Glu-165-Asp isomerase is expressed from the same promoter, however, the transformants grow on lactate but not on glycerol. Colonies appearing on glycerol plates have more isomerase activity than the Glu-165-Asp mutant, and these were used to isolate and assay specific catalytic activity.

Enzyme Kinetics

478

from substrate to product, with smaller yields of [1(R)-2H]DHAP (d-DHAP) and [2(R)-2H]-GAP (d-GAP) labeled with solvent deuterium at C-1 and C-2, respectively. TIM Catalysis in Deuterium Oxide: H H

O

O(D)

H

OD

OD

OPO32–

OPO32– Intermediate

GAP

D2 O

H

H

H

O(D) D +

O OPO32– 49% intramolecular

O(D) O

H +

OPO32– 31% from solvent

O

D

OD OPO32-

21% from solvent

Scheme 7.42

Such results contrast sharply with earlier observations from the Knowles group that the reaction of tritium-labeled [1(R)-3H]-DHAP in H2O proceeds with up to only ~6% intramolecular transfer of the tritium label to the GAP product. The findings of O’Donoghue, Amyes and Richard (2005) clearly require re-evaluation of earlier conclusions about the dynamics of exchange of the hydron at the

carboxylic acid side chain of Glu-165 at the TIM-enediol(ate) complex with those of bulk solvent. More recently, O’Donoghue, Amyes and Richard (2008) labeled the catalytic with hydrogen (H) by abstraction of a proton from substrate GAP to form an enzyme-bound enediol(ate) in D2O solvent. Subsequent partitioning of this labeled enzyme between intramolecular H-transfer to form DHAP, and irreversible D exchange from solvent was examined by determining the yields of H- and D-labeled products. The products of the TIM-catalyzed reactions of GAP in D2O at a pD of 7.9 (I ¼ 0.15, NaCl) in solutions buffered by 0.021 and 0.083 M imidazole were determined by the same 1H-NMR analysis. The yield of hydrogenlabeled product DHAP remained constant as the concentration of the basic form of imidazole buffer is increased from 0.014 to 0.56 M. This finding shows that the active site of substrate-free TIM, which would be an open conformation needed for substrate binding, adopts a closed conformation at the enediolate complex intermediate, where the catalytic side chain is blocked from interacting with the imidazole dissolved in D2O. Figure 7.36 illustrates the working model proposed by O’Donoghue, Amyes and Richard (2008) for the reaction of hydrogen-labeled reactant in D2O that is consistent with known relevant data. In Pathway-A, isomerization involves hydron transfer between the two enediolate oxygens by consecutive proton transfer reactions of Glu-165. This pathway leads to formation of the deuterium-labeled product without deuterium incorporation into unreacted substrate. In Pathway-B, isomerization involves His-95 promoted deuterium transfer from O-2 to O-1 of the enediol(ate) intermediate. This pathway is thought to be the major isomerization

A

O O P

E•DHAP

O

Glu

Lys

O

H1

O HO1

O2 H N 3

C1 O1 H

N

N

His-95

Lys-12

2

1

C

His

His

H C2

Glu

B

Glu-165 O

Glu

His

E•I2

E •I1 A

B

Glu

Glu

His

Lys

Lys

His

Lys

E•GAP C Glu Lys

Lys His

FIGURE 7.36 Working model for hydrogen exchange reactions catalyzed by triose-P isomerase. See text for details. Drawing redrawn from O’Donoghue, Amyes and Richard (2008) with permission of the authors and publisher (Royal Society of Chemistry).

Chapter j 7 Factors Influencing Enzyme Activity

pathway, with intramolecular hydrogen label transfer from reactant to product. In Pathway-C, proton exchange occurs between Glu-165 and His-95 via the enediol(ate) intermediate. The vertical arrows indicate that reactants in one pathway can inter-convert to reactants in another. As pointed out by Rozovsky and McDermott (2007), the Albery-Knowles free energy profile (see Figs. 5.10 and 7.37) suggests: (a) that barriers of comparable heights for successive transition states lead to a condition where the steps are partially rate-limiting; and (b) that the free energies several of the intermediates are comparable to that of (Efree þ Sfree) or (Efree þ Pfree), implying that the E$S # E$P equilibrium is closer to unity than that of the bulk-phase Sfree # Pfree equilibrium. However, when 13C-enriched DHAP was bound to the enzyme and characterized at steady state over a range of sample conditions (chosen to minimize covalent modification of TIM by any methylglyoxal byproduct), DHAP was found by solution-state NMR to be the major species over the range from –60 to þ15 C (Rozovsky and McDermott, 2007). There was thus no indication that the enzyme preferentially stabilizes the reactive intermediate or the product. They conclude that the predominance of DHAP on the enzyme supports a mechanism, in which the initial proton abstraction in the reaction from DHAP to GAP is significantly slower than subsequent chemical steps. They further suggest that reprotonation, when forming GAP from DHAP, is likely to occur simultaneously with the active-site loop opening, with attendant product release. Physical coupling between the proton transfer and loop motion/ product release can be invoked by a partial or full proton transfer to the substrate’s phosphate triggering loop opening. As noted in earlier sections, active-site loop opening occurs on the time-scale as enzymatic turnover rate and is also sensitive to the chemical details of the bound ligand. They thus suggest that triggered release may assist in minimizing any back conversion of the newly formed GAP to DHAP. Finally, it is worth noting that the wealth of thermodynamic, spectroscopic, kinetic, and structural studies on TIM catalysis has made this enzyme fertile ground for the theoretical exploration of the reaction cycle and attendant structural changes. Guallar et al. (2004), for example, applied a combined Quantum Mechanical/Molecular Mechanical (QM/MM) treatment, based on protein structure modeling techniques, to provide a comprehensive, and quantitatively reliable, description of the TIM catalytic cycle. Their QM/MM methods enabled routine treatment of quantum regions of the active-site as large as 150–200 atoms, for which geometry optimizations can be carried out with sufficiently large basis sets to guarantee reasonably accurate structures. Without detailing all methods used, it suffices that they investigated the catalytic loop motion in TIM using methods for loop and side chain prediction, based on a protein molecular mechanics force-field and a continuum solvation model. Their goal was to model and compare the three reaction pathways shown in Fig. 7.37.

479

Key new findings of this work are: (a) identification of the mono anionic form of the substrate phosphate group as the principal channel for abstraction of the initial proton from DHAP; (b) an energy-based description of the opening and closing of the catalytic loop in the presence and in the absence of substrate, which is consistent with several experimental observations; and (c) detailed elucidation of all of the steps in the catalytic cycle in a fashion that can readily explain the isotope washout data reported by Knowles and co-workers. The fact that accurate free energies are obtained for a variety of barriers and intermediates, while preserving fidelity to the available crystallographic data, constitutes strong confirmation that the picture presented here is correct in its detail as well as in broad outline.

7.15.6. Chemical Rescue is a Method for Restoring Activity in some Mutant Enzymes The phenomenon of chemical rescue, first coined by Toney and Kirsch (1989), refers to the ability of an added acidic/ basic solute to reactivate (i.e., rescue) catalysis by an enzyme whose activity was lost by removal of an essential active-site functional group, usually as a result of random mutation or site-directed mutagenesis. In their seminal work on the chemical rescue of mutant aspartate aminotransferase, these investigators used site-directed mutagenesis to replace Lys-258, the endogenous general base, with an alanine residue. Although the resulting mutant was inactive, catalytic activity could then be restored by providing exogenous amines. By characterizing the effect of eleven amines, Toney and Kirsch (1989) were able to generate a Brønsted correlation (b ¼ 0.4) for the transamination of cysteine sulfinate. They concluded that localized mutagenesis should allow us to apply classical Brønsted analysis of transition-state structure to enzyme-catalyzed reactions. Virtually contemporaneous work from David Silverman’s laboratory is described in Section 7.4.3 dealing with the ability of certain buffer solutes, particularly methylimidazoles, to rescue carbonic anhydrase catalysis after removal of its active-site histidine residue by site-directed mutagenesis. Another good example of chemical rescue is the use of exogenous amines of a site-directed mutant of ribulose 1,5bisphosphate carboxylase/oxygenase, also lacking a key lysine residue (Harpel et al., 1994). These investigators demonstrated that ethylamine enhances the carboxylation rate of Lys-329-Ala by about 80 and strengthens complexation of 2-carboxyarabinitol 1,5-bisphosphate. Rescue of Lys-329 Ala follows an apparent Brønsted relationship with b ¼ 1, suggesting complete protonation of amine in the rescued transition state. Rate saturation with respect to amine concentration and the different steric preferences for amines between Lys-329 Ala and Lys-329-Cys

Enzyme Kinetics

480

enz

A H PiO

PiO

H OD

enz

CO2H H

Pi O

O

DO

O

H

enz

H

H

D PiO

H OD

O

H O

D

OH

N N

N

enz

H OH

H

PiO

OD

O

N

C

O

D N N

PiO

OD

D

CO2H

CO2H

Pi O

Pi O

OH

O

N

DO

H

D N

N

enz

CO2D

PiO

OH

DO

D N

B

enz

CO2

enz

CO2D

PiO

H

PiO

O

DO

H OD

DO

H

CO2D

PiO

H

N

OD

O H N

H N

N N

enz

CO2

Pi O

D O

H OH

N

FIGURE 7.37 Catalytic pathway for conversion of DHAP to GAP by triose-P isomerase. The first step, shown as occurring in the E$DHAP complex, involves the transfer of a proton from C1 of DHAP to the catalytic base, Glu-165. The different alternative paths between the two enzyme-bound intermediates E$I1 and E$I2 are shown in the lower-right square panel. A ball-and-stick representation is used for the atoms involved directly in each step and curved arrows indicate the primary proton transfers. In Path A and Path C, an enediol intermediate forms by transfer of a proton from His-95 (Path A) or Glu-165 (Path C) to the substrate O-2 atom. The enediol intermediate then forms enediolate by the transfer of a proton from O-1 to His-95 (Path A) or Glu165 (Path C). Path A is the classical and widely accepted mechanism first proposed by Albery and Knowles (1976). Path C is supported by NMR experiments (Harris, Abeygunawardana and Mildvan, 1997) and by earlier ab initio active-site theoretical models. Path B forms an enediolate intermediate directly by internal proton transfer from O-1 to O-2. The final catalytic step, common to all paths, involves proton transfer from Glu-165 to C-2 to form GAP. Proposed steps for the reaction should account for central experimental evidence: the substantial isotope washout when 2H and 3H substrates are used. From Guallar et al. (2004) with permission of the authors and publisher (Elsevier).

suggest that the amines bind to the enzyme in the position voided by the mutation.

7.15.7. Site-Directed Mutagenesis Suffers Significant Limitations While site-directed mutagenesis has become a cornerstone for enzyme and protein engineering, this powerful and farranging technique is not without limitations for the systematic investigation of enzyme catalysis. In most cases, even substrate binding is controlled by a constellation of active-site residues, thereby restricting the ability to create new enzymes by remodeling active sites to accommodate novel substrates. Because each enzyme tends to have its

own distinctive properties, mutagenesis has not provided many generalizable principles concerning enzyme action. If anything, enzyme chemists quickly learned that mutagenesis is much more useful in imparting greater stability to an enzyme of interest than as a tool for improving catalytic efficiency. The reviews of Admiraal et al. (2001) and Peracchi (2001) are valuable readings for those considering the inherent limitations of using site-specific mutagenesis to assess interactions between specific side chain groups in enzymes. The observation that the positions of so-called backbone ˚ or less is atoms within an enzyme are dislocated by 0.5 A frequently offered as evidence that site-directed mutagenesis of active-site residues is without significant effect on an enzyme’s overall structure. However, because

Chapter j 7 Factors Influencing Enzyme Activity

crystallographic determinations of protein structure are increasingly carried out at low temperature, it is wise to confirm that the mutant enzyme’s structure is unaffected at much higher temperatures where the actual rate studies are conducted. Seemingly innocuous mutations are known to greatly alter an enzyme’s structural integrity and stability. Two useful techniques for evaluating temperature-dependent changes in an enzyme’s structure are differential scanning calorimetry or circular dichroism. The basic goal is to evaluate the temperature range over which a structural change occurs, where the so-called melting temperature Tm is defined as the temperature corresponding to the mid-point in an observable structural transition. Mutant enzymes displaying different Tm values may be assumed to have altered tertiary structures. Using current practice, enzyme engineering fails to provide a convenient means to introduce gradual changes in enzyme structure. Given the exquisite precision with which various amino acid side chain groups participate in catalysis, the availability of only 19 other natural substitutions makes directed mutagenesis a rather clumsy approach. Even with the few naturally occurring one-carbon homologs (e.g., glutamate versus aspartate, and glutamine versus aspara˚ change in a side chain length gine), an approximate 1-A (upon adding or removing a methylene group) can hardly be regarded as fine-tuning of active-site geometry. One solution is to incorporate non-coded amino acid residues (i.e., unnatural amino acids for which Nature has no codon or corresponding amino acid-specific tRNA molecules). Unnatural amino acids can, however, be inserted by two promising methods: (a) in cell-free translation systems by the so-called amber suppression technique (Cornish et al., 1995); and (b) via the more efficient frame-shift suppression technique, by which otherwise non-coded amino acids are incorporated into proteins. In some instances, imperfect folding or lowered thermal stability of a bacterially expressed protein may drastically alter the kinetic properties of both wild-type and mutant enzymes. Such problems are most likely to arise when folding kinetics are slow, leading to the formation of metastable structures and/or insoluble inclusion bodies in bacteria used for expression. Low bacterial expression can be reduced by replacing the eukaryotic codons in a coding sequence by corresponding bacterial codons, for which there is an ample supply of cognate aminoacyl-tRNA ligases. Because prokaryotes and eukaryotes employ a different complement of chaperonins and heat shock proteins, some eukaryotic enzymes may also fail to adopt their catalytically active conformation. Mutants may also fail to form oligomers. Bacteria most often fail to introduce certain post-translational modifications (e.g., acetylation, methylation, N-linked glycosylation, phosphorylation ADP ribosylation, etc.). When necessary, expression experiments may be conducted in yeast and Bacculovirus systems, thereby avoiding or minimizing these effects.

481

Because hydrogen bonding is highly sensitive to local dipolar interactions, certain mutations can greatly reorganize the water structure within the active site, especially if a charged side-chain group is replaced by one of opposite charge. Enzymes like carbonic anhydrase require deprotonation of Enz$Zn[OH2] to regenerate the active enzyme in the form of Enz$Zn[OH]:

CO2 + Enz· Zn[OH-] Enz· Zn[OC(=O)OH]Enz· Zn[OH2]

Enz· Zn[OC(=O)OH]Enz· Zn[OH2] + HCO3Enz· Zn[OH]- + H

Scheme 7.43 While the overall polarity of the active site has not been changed substantially, reorientation of water dipoles is likely to affect catalysis markedly. It is reasonable to anticipate that even subtle changes in ionic charge or electronegativity may reorient hydrogen-bonding networks that serve as ‘‘proton wires’’ that shuttle protons in or out of an active site. Another limitation in the application of directed mutagenesis relates to the assumption that there is a straightforward thermodynamic basis for interpreting interactions by mutation experiments. To gauge the contribution of a particular molecular interaction between an active-site side chain group within an enzyme, a residue thought to be an interaction participant is replaced by another amino acid unable to interact in a like manner. The effect of this mutation is evaluated by measuring a relevant thermodynamic or kinetic parameter for both the wild-type protein and the mutant. Such measurements are converted to free energies (e.g., if the rate constants for some property of the wild-type and mutant enzymes are k and k9, respectively, then DG ¼ –RT ln(k9/k), and the difference in free energies between the two proteins is taken to be the energetic contribution of the interaction of interest (Fersht, 1998). Even so, the derived value for the change in the Gibbs free energy need not fully account for the effects from removal of the specific interaction, because the experimenter cannot control the mutation’s effects on the surrounding protein environment and/or rearrangements of the mutant enzyme’s conformation. If the free energies from double mutant cycles are additive, the observed energetic effects may simply be coincidental. Likewise, Gibbs free energy changes from double mutations sometimes yield DDG values that do not correspond to the sum of the free-energy changes from the individual single mutations (see also the last paragraph of Section 7.3.2: Enzymes Often Display pHdependent Changes in Activity). Perhaps the greatest limitation of site-directed mutagenesis is the assumption that the chemical mechanism of catalysis remains unchanged (Peracchi, 2001). Such an

Enzyme Kinetics

482

assumption is reasonable for minor changes in an active site, but removal of a key catalytic residue may cause the enzyme to adopt another reaction pathway, not necessarily of the same reaction type. What becomes clear is that an enzyme may have a range of mechanistic options, much like a More O’Ferrall-Jencks diagram (see Fig. 3.25), and conformational energy and active-site functional group availability may give rise to competing pathways for catalysis. In terms of perhaps the simplest form of catalysis (i.e., acid/base catalysis), the shortening of the side chain by a single methylene unit in a Glu-to-Asp mutant may change the nature of catalysis from specific base catalysis to general base catalysis (Scheme 7.44), and unless aided by a highresolution structure, such a change might prove to be nonobvious kinetically.

ENZ–Glu

ENZ–Asp

CH2 H2C

C

CH2

O

O

H

C O

O

H

R

O R X

X O

R

O

R

Scheme 7.44 It would not at all be surprising if enzymes possess their own ‘‘homeostatic’’ mechanisms that minimize the impact of mutations on the structure of their active sites. Such behavior is best explained by the as yet unproven idea that present day enzymes were once feeble catalysts that evolved into highly proficient catalysts through the accretion of seemingly minuscule advantages. If each incremental improvement boosted an enzyme’s turnover number, then the loss of a residue by directed mutagenesis may result in minor effects on catalysis. Moreover, because so many enzymes are conformationally supple, they may be able to tolerate mutation without substantial loss of catalytic activity, making it nearly impossible to predict the effects of certain mutations on catalysis. Combinatorial cassette mutagenesis, in principle, allows us to randomize two or three positions by oligonucleotide cassette mutagenesis, select for functional protein, and then sequence to determine the spectrum of allowable substitutions at each position. Repeated application of this

FIGURE 7.38 Crystallographic structures identifying critical residues determining whether NADD or NADPD is used as a redox coenzyme. Panel-A, Structural alignment of Escherichia coli isopropylmalate dehydrogenase, abbreviated IMDH (Wallon et al., 1997), with brown main-chain, and labels designating the amino acid residue followed by the site number and Thermus thermophilus IMDH (Hurley and Dean, 1984) (blue main-chain) showing the double H-bond (pink lines) critical to NADþ use. Only key residues are shown, with gray ¼ carbon; red ¼ oxygen; blue ¼ nitrogen; yellow ¼ phosphorus. Panel-B, Structural alignment of E. coli isopropylmalate dehydrogenase and E. coli isocitrate dehydrogenase, abbreviated IDH (green main chain) with NADPþ bound (Hurley et al., 1991) showing IDH residues (following the IMDH site numbering scheme) H-bonding to the 29-phosphate (2’P) of bound NADPþ (H bonds from the disordered Lys-289 not shown). Without delving into each specificity conferring interaction, note that the b-carboxylate of Asp-236 in the NADþ-dependent enzyme (Panel-A) would clash electrostatically with the C9-2-phoshoryl of NADPþ shown in Panel-B. However, as shown in Panel-B, Arg-236 is well aimed to stabilize the additional phosphate ester in NADPþ. From Lunzer et al. (2005) with permission of the authors and publisher (Science).

Chapter j 7 Factors Influencing Enzyme Activity

technique can reveal the number and type of substitutions allowed at each position. One example of directed enzyme evolution with respect to coenzyme specificity was recently provided by Lunzer et al. (2005). Based on the identification of six amino acid residues as being determinative with respect to NADþ or NADPþ usage by various structurally related dehydrogenases (Fig. 7.38), theses investigators developed a directed evolution approach using recombinant DNA techniques to switch the coenzyme specificity of E. coli isopropylmalate dehydrogenase from NADþ to NADPþ. Unlike most IMDHs, the E. coli IMDH already has one such residue (Arg-341) common to all NADPþ-dependent and structurally related isocitrate dehydrogenases. The remaining five replacements (Asp-236-Arg, Asp-289-Lys, Ile-290Tyr, Ala-296-Val and Gly-337-Tyr) were introduced into the coenzyme-binding pocket of E. coli IMDH by site-directed mutagenesis. They evaluated catalytic performance of the five-site mutant, finding that values of (kcat/Km)NAD ¼ 82,000 M1 s1 and (kcat/Km)NADP ¼ 84 M1 s1 for wildtype IMDH were changed dramatically to values of (kcat/ Km)NAD ¼ 180 M1 s1 and (kcat/Km)NADP ¼ 37,000 M1 s1 for the five-site mutant. Such results show an overall 20,000-fold change in kcat/Km for wild-type and mutant enzymes acting on NADþ or NADPþ. Significantly, the success of their efforts is indicated by the fact that recombinantly engineered ‘‘RKYVYR’’-IMDH is both as active and as specific toward NADPþ as is wild-type IMDH acting on NADþ. In cases where one or more of the sitespecific mutations was not introduced into the wild-type IMDH, Lunzer et al. (2005) observed that the resulting intermediates between the above two phenotypic extremes show that each amino acid contributes additively to enzyme function and that NADþ use is a global optimum. For other implications of this study for directed evolution, the reader should consult this illuminating report. Taken together, we are drawn to conclude that, although site-directed mutagenesis is not the ‘‘be-all and end-all’’ of enzyme science, judicious application of the technique can provide valuable insights into the nature of enzyme catalysis.

7.16. CONCLUDING REMARKS Considering that enzymes have evolved such complex catalytic and regulatory strategies to fulfill their roles within living systems, one should not be surprised by the wide range of physicochemical factors influencing their structures and kinetics. Depending upon their interests, enzyme kineticists routinely employ the rate relationships described in this chapter to investigate factors affecting enzyme structure, stability and/or catalysis. However, it should not be expected that the rapid-equilibrium and steady-state treatments described in this chapter will suffice in all cases.

483

The reader should also consult Chapter 9 for experiments exploiting isotope exchange and kinetic isotope effects as well as Chapter 10 for other fast-reaction kinetic approaches. Moreover, while this chapter focuses on many of the factors affecting enzyme catalysis, the reader will appreciate the inherent complexity of systems in which the simultaneous effect of two or more such factors is of interest. Treating all such factors even in a pair-wise manner would clearly be beyond the scope of this reference. For example, one may be interested in determine how pH and a specific metal ion simultaneously affect the kinetic properties of a particular enzyme. The reader would consider chemically reasonable kinetic schemes for pHdependent metal ion activation and then set about to derive the appropriate rate laws governing an enzyme’s interactions with metal ions and pH. Knowledge of the particular system as well as concepts already developed in this chapter should inform the investigator of likely mechanisms. Additional information on what is generally known about structural features of metal ion interactions with enzymes, such as that described in Chapter 2, would also help one to formulate an effective research plan. There is also the likelihood that, for many enzymes, metal ion binding would itself be attended by deprotonation of acidic groups in the metal ion binding site. Therefore, it may be anticipated that a metal ion binding would be strongly pH-dependent. For such problems, this chapter may thus be regarded as a starting-off point for developing a research strategy rather than an impenetrable and intellectually vapid compendium that merely lists rate equation after rate equation for every known interaction, however unlikely or unrealistic. Finally, efforts to analyze and understand the factors affecting biological catalysts have rewarded enzyme chemists with deeper insight into likely kinetic mechanisms and ways to study them. Enzyme chemists will someday (hopefully soon) learn to fashion new enzymes to suit their fancy and/or to meet their needs. When they do, lessons learned about the systematic characterization of factors affecting enzyme catalysis should allow them to improve and perfect their new creations.

Authoritative Readings from Methods in Enzymology Roles of Metal Ions Practical Design of Initial Rate Enzyme Assays, 63, Chapter 1. Kinetics of Metal-Activated Enzymes, 63, Chapter 11. Stability Constants for Metal-Ligand Complexes, 63, Chapter 12. Cr(III) and Co(III) Nucleotides as Chirality Probes and Inhibitors, 87, Chapter 11. NMR and ESR Studies on Cr(III) and Co(III) Nucleotides, 87, Chapter 12.

Enzyme Kinetics

484

Job Plots and the Method of Continuous Variation, 87, Chapter 27. Kinetics of Metal Ion-Nucleotide Requiring Systems, 249, Chapter 7.

pH Kinetics pH Kinetics in Enzyme Catalysis, 63, Chapter 9. pH Kinetics of Multisubstrate Enzymes, 87, Chapter 22. Buffers and Constant Ionic Strength in pH Studies, 87, Chapter 23. Two-Protonic-State Electrophiles as Mechanistic Probes, 87, Chapter 24.

Temperature Effects Temperature Effects in Enzyme Kinetics, 63, Chapter 10. Cryoenzymology: Enzyme Catalysis at Sub-zero Temperatures, 63, Chapter 13. Detecting Peroxodiferric Intermediates by Freeze-Quench Mo¨ssbauer, Resonance Raman, and XAFS Spectroscopies, 354, Chapter 32.

Catalysis Lipolysis Kinetics, 64, Chapter 14. Catalysis by Phospholipase A2, 249, Chapter 21.

Chapter 8

Kinetic Behavior of Enzyme Inhibitors Enzyme inhibitors are of immense importance. Every living organism relies on inhibitors as feedback signals that coordinate metabolic pathway activity: by reducing the catalytic activity of their respective enzyme targets, inhibitors throttle the flux of intermediates through a pathway. Inhibitors have also proven to be invaluable tools in countless efforts to discover, understand, and/or manipulate enzymes or their respective metabolic pathways. Biochemists long ago realized that genetic mutations and specific inhibitors are functionally equivalent tools for investigating the order of reactions within metabolic pathways, especially when a highly potent inhibitor causes pathway intermediates to accumulate to levels where they can be chemically or spectroscopically analyzed. Inhibitors actually offer the added advantage that intermediate degrees of enzyme activity are often attained by adjusting inhibitor concentration, and, unlike a mutation, a reversible inhibitor’s effect can be easily reversed by dilution. Reversible and irreversible inhibitors have also provided endless opportunities to explore the subtle aspects of enzyme catalysis and regulation. This chapter describes the modes of action of various reversible and irreversible enzyme inhibitors as well as the approaches for rationally designing high-affinity inhibitors, based on the inferred structures and properties of catalytic reaction cycle intermediates. While no single chapter can address all aspects of enzyme inhibition, this chapter considers inhibitor types, their distinguishing chemical features, their predicted and observed kinetic behavior, and other practical aspects of interest to those conducting experiments with enzyme inhibitors. Presentation of these topics within a single chapter also provides the reader with a convenient opportunity to compare and contrast their various modes of inhibitor action. Numerous examples also illustrate the pivotal role of enzyme kinetics in defining and improving the effectiveness of enzyme inhibitors. Allosteric inhibition and enzyme-catalyzed post-translational modification of target enzymes are considered separately in Chapter 12.

8.1. SCOPE AND SIGNIFICANCE OF ENZYME INHIBITION Most enzyme inhibitors bind within the active site, but regulatory inhibitors often bind at topologically remote Enzyme Kinetics Copyright Ó 2010, by Elsevier Inc. All rights of reproduction in any form reserved.

sites. Most are reversible, but many form covalent bonds with the enzyme. In all cases, inhibitors trap the enzyme into one or more catalytically inactive form(s), thereby reducing the concentration of active catalyst. Depending on the specific inhibitory mode, some inhibitor-containing complexes may also contain substrate molecule(s). Others use chemically reactive substrate analogues to achieve mechanism-based (or suicide) inhibition.

8.1.1. Distinguishing Reversible and Irreversible Enzyme Inhibitors Enzyme complexes with reversible inhibitors are stabilized mainly by non-covalent forces, although a few are known to form fully reversible covalent adducts (i.e., particularly imines, thioacetals, etc.). Most are characterized by Kd values which fall in the sub-micromolar to submillimolar range, indicative of rapid one-step reactions defined by the simple thermodynamic relation: Kd ¼ [E][I]/[EI] ¼ k2/k1 or Kd ¼ [ES][I]/[ESI] ¼ k2/k1. The value of the bimolecular rate constant k1 often falls within the 107–109 M1 s1 range, and the magnitude of k2 ranges from 104 s1 to 106 s1. In principle, dialysis or gel filtration should be sufficient to remove a reversible inhibitor from the enzyme, thereby restoring catalytic activity. When Kd values fall in the sub-nanomolar to femtomolar range, the most likely explanation is that the inhibitor interacts through a multi-step, ligand-induced isomerization (i.e., E þ I # E$I1 # E$I2 # E$I3, $$$), where the subscripts indicate slowly inter-converting species. In these cases, dialysis may prove completely ineffective, especially when inhibitors require days or weeks to dissociate. Reversible and irreversible inhibition can often be distinguished simply by diluting the enzyme-bound inhibitor into a solution containing a saturating level of substrate (Fig. 8.1). If the enzyme-inhibitor interaction is characterized by rapid association/dissociation (Kd ¼ k2/k1), a k2 value of 104 s1 to 106 s1 should result in complete dissociation of the inhibitor on the microsecond to millisecond time-scale. After dilution, the remaining low concentration of the reversible inhibitor will reduce enzyme activity only slightly relative to control samples assayed in the absence of inhibitor. When Kd values fall in the 1010 to 1015 M range, the off-rate constant often 485

Enzyme Kinetics

486

Product Formed

C R

SR I Time after Dilution FIGURE 8.1 Hypothetical time-courses for activity recovery after rapid dilution of enzyme-bound inhibitor into a solution containing saturating substrate concentration. C, Control sample having no inhibitor present; R, Reversible inhibitor bound as E$I complex at zero-time; SR, Slowly-reversible inhibitor bound as E$In complex at zero-time, but must isomerize through one or more intermediate states to form E$I complex, which then releases inhibitor; I, Irreversible inhibitor bound as E–I complex at zero-time. The actual time-scale for inhibitor release will depend to the transit time for isomerization of enzyme-inhibitor complexes to that enzyme species from which inhibitor is released.

falls in the range of 102 to 108 s1, such that complete inhibitor dissociation and activity recovery may require minutes to months.1 Finally, if the inhibitor is irreversibly fixed onto the enzyme, there will be no recovery of enzymatic activity.

The identification and characterization of new enzyme inhibitors is a major intellectual enterprise for enzyme chemists, pharmacologists, and medicinal chemists. The overarching goal is to develop systematic ways for improving the effectiveness (i.e., potency and specificity) of enzyme inhibitors. This quest began with Paul Ehrlich’s therapia magna sterilisans hypothesis, asserting that the cure of an infectious protozoal disease may be achieved by an agent administered as a single dose of sufficient potency to sterilize all the infected tissue, destroying the microbial pathogen contained therein. His systematic organochemical synthesis and search for the magic bullet to treat the syphilis-causing microbe Treponema pallidum was rewarded by discovery of diamidodioxyarsenobenzol, a highly effective trivalent arsenical also known as Preparation 606 or Salvarsan:

As HO

OH

As NH2 H2N OH HO

As OH H2N

Salvarsan and Its Likely Bioactive Form

8.1.2. Enzyme Inhibitors in Biomedicine Over the past century, biochemists have documented countless naturally occurring and synthetic compounds as well as simple inorganic ions that inhibit enzyme catalysis.

1 When inhibitor binding is extremely favorable, the DG for complexation is highly negative. Experience has shown that, in such instances, a single E$I complex is rarely formed. The initial complex E$I1 typically undergoes a sequence of isomerizations (i.e., E þ I # E$I1 # E$I2 # E$I3, $$$), such that DGTotal ¼ SDGi > k, then the distinction between partial and complete substrate inhibition will depend entirely on the quality of the experimental rate data.

1.0

1/

507

1/[Substrate] FIGURE 8.17 Predicted double-reciprocal plot (1/v versus 1/[S]) for substrate inhibition. If substrate inhibition is not too strong, extrapolation of a tangent line drawn for the linear region of Fig. 8.16 yields a reasonable estimate of Km and Vm.

Unfortunate contamination of a substrate by a noncompetitive inhibitor will also give the appearance of substrate inhibition. Equation 8.20 is readily obtained from the Eqn. 8.14 for a noncompetitive inhibition by replacing each inhibitor concentration term by a[S], where the fraction of a noncompetitive inhibitor within a substrate is: a ¼ [I]/[S].     1 1 a½S 1 Km a½S þ 1þ 1þ ¼ v Vm Kii Vm ½S Ki   1 a½S 1 Km a Km þ 8.20 ¼ 1þ þ Vm Kii Vm ½S Vm Ki   1 1 a½S aKm Km 1 þ 1þ þ ¼ Ki Vm ½S v Vm Kii

8.20a

Note that both the numerator and the denominator contain a substrate concentration term. When 1/v is plotted as a function of 1/[Substrate], the resulting curve will have the same concave-upward shape as that shown in Fig. 8.13, and the inhibition will be complete when the concentration of contaminated substrate is extrapolated to the (1/v)-axis. The observed degree of substrate inhibition behavior of an enzyme will depend on the value of a as well as the values of KS, Ki, and Kii.

.8 30

/Vm

.6 10 .4

,

1/V m Ki/Km = 1

1/

.2

0

0

5

10

[Substrate]/Km FIGURE 8.16 Plot of the fractional saturation of enzyme active sites (v/Vmax) versus the reduced substrate concentration [S]/Ks where Ki/ Ks [ 1, 10, 30, and infinity. Notice that there is an optimum in the rate dependence on substrate concentration.

1/V m

1/[Substrate] FIGURE 8.18 Double-reciprocal plot (1/v versus 1/[S]) showing partial inhibition. Note that the extrapolated curve intersects the 1/v axis.

Enzyme Kinetics

508

8.3.2. Fromm’s Alternative Substrate Inhibition Method Distinguishes Rival Multi-Substrate Kinetic Mechanisms As already mentioned in Section 3.5, when an enzyme is presented with a substrate S and an alternative substrate S9, the combined rate vobs will be vS þ vS9. However, if the rate assay is sensitive only to d[S]/dt or d[P]/dt, the presence of S9 will inhibit the conversion of S into P. Likewise, if the rate assay only detects d[S9]/dt or d[P9]/dt, the presence of S will bring about inhibition of vS9. Such effects are to be anticipated whenever S and S9 compete for the same site, because the enzyme is by definition the limiting component. While Haldane (1930) treated this case for a one-substrate enzyme, Wong and Hanes (1962) were the first to suggest that alternative substrate inhibition might provide clues about substrate binding order. It remained, however, for Fromm’s laboratory to be the first to exploit alternative substrate inhibition as a means for discriminating among rival multisubstrate kinetic mechanisms (Fromm, 1964; Rudolph and Fromm, 1971; Wong, 1975; Zewe, Fromm and Fabiano, 1964). The alternative substrate inhibition method requires suitable alternative substrates A9 and B9 that compete with substrates A and B, respectively. Furthermore, there is an implicit assumption that the enzyme mechanism is the same for both substrates, even though at least one rate constant for A and A9 or B and B9 must be different. The investigator studies the changes that occur in the double-reciprocal plots of initial rate data obtained in the absence and presence of one of the alternative substrates. In multi-substrate and multi-product enzymic reactions, the investigator can estimate the rate of the reaction by measuring: (a) the increase in concentration of a common product (i.e., a product formed by both the normal substrate and the alternative substrate); or (b) the increase in concentration of a specific product that is not directly formed by the alternative substrate. Consider, for example, hexokinase’s action on glucose in the presence of the alternative substrate mannose. In this case, the common product is ADP, which represents the sum of glucose 6-P and mannose 6-P production, whereas the specific product assay would only detect glucose 6-P production. Each procedure requires its own set of rate expressions and will produce distinctly different patterns in double-reciprocal plots for data collected at different, constant concentrations of the alternative substrate. Although at times useful, the common-product approach often produces nonlinear doublereciprocal plots that are more difficult to interpret unambiguously (Fromm, 1964; Rudolph and Fromm, 1971). Huang (1979) greatly simplified the otherwise cumbersome rate equations by a constant ratio approach, in which the concentrations of one substrate and its corresponding alternative substrate are maintained in a constant ratio. As we shall see, his approach linearizes most double-reciprocal plots, while still retaining the tell-tale non-hyperbolic nature

of 1/v versus 1/[B] plots in an ordered mechanism, when substrate A and its alternative substrate A9 are present in a constant concentration ratio. Case-1. Ordered Bi Bi Mechanism – Consider the following scheme showing how alternative substrate A9 interacts with an enzyme operating by an Ordered Bi Bi mechanism. k3[B] EA

k1[A]

k4

EAB EPQ

k2

k5 EQ k6[P]

k7

k8[Q] E

E k8'[Q']

k2 k1[A']

k3'[B'] EA' k4'

k5' EA'B EPQ'

EQ'

k7'

k6'[P]

Scheme 8.21 The initial velocity equation describing this scheme is:   1 1 Kb 1 1 Ka Kia Ka þ ¼ þ þ v V1 V1 ½B V1 ½A V1 ½B   ½A9  9 1þ 8.21 ðKb D½BÞ Kia 9 Kb 9 þ Ka 9 ½B As shown in Fig 8.19, this equation indicates that the presence A9 will behave as a competitive inhibitor relative to substrate A in plots of 1/v versus 1/[A]. The effect of A9 on plots of 1/v versus 1/[B] is to produce a series of concave-upward lines. When alternative substrate B9 is the alternative substrate, the following branched reaction pathway applies. EAB EPQ

k3[B] k4

k1[A] E

EA k2

k4' k3'[B']

k5 k7

k6[P] EQ

EAB' EP'Q

E k8[Q]

k6'[P'] k5'

Scheme 8.22 The initial-velocity equation accounting for the presence of alternative substrate B9 on d[P]/dt is: " ## " " 1 1 Ka Kb k3 9 ½B k5 9 1þ 1þ 1þ þ ¼ k7 ½A ½B v V1 ðk4 9 þ k5 9 Þ # Kia Kb h k3 9 k5 9 ½B i þ 8.22 1þ ½A ½B ðk2 9 k4 9 þ k5 9 Þ As shown in Fig. 8.20, this equation requires that the presence of alternative substrate B9 will generate noncompetitive inhibition patterns in a plot of 1/v versus 1/[A] as well as in a plot of 1/v versus 1/[B].

509

[A'] = 3x

[B'] = 3x

2x

2x

x

1/v

1/v

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

x

0

0

1 / [Substrate B]

1 / [Substrate A]

[B'] = 3x

[A'] = 3x

2x

1/v

1/v

2x

x

x

0

0

1 / [Substrate B] FIGURE 8.19 Initial-rate plots showing the expected effects of an alternative substrate A9 on an enzyme operating by an Ordered Ternary Complex Bi Bi kinetic mechanism. NOTE: The four lines through the data points in both plots are obtained by holding the concentrations of alternative substrate A9 at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for alternative substrate A9. The respective inhibition constants may be evaluated from secondary plot of slope versus inhibitor concentration for the data in left plots showing competitive inhibition patterns. See text for details.

Case-2. Rapid-Equilibrium Random Bi Bi Mechanism – Scheme 8.23 shows how alternative substrate A9 interacts with an enzyme operating by a Rapid Equilibrium Bi Bi mechanism. EA

B

A

EAB A

E

B

E+P+Q

EB A'

A' EA'

k

slowest step

B

k' EA'BslowestE + P + Q' step

Scheme 8.23

1 / [Substrate A] FIGURE 8.20 Initial-rate plots showing the expected effects of an alternative substrate B9 on an enzyme operating by an Ordered Ternary Complex Bi Bi kinetic mechanism. NOTE: The four lines through the data points in both plots are obtained by holding the concentrations of alternative substrate B9 at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for alternative substrate B9. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the left plot showing competitive inhibition patterns. See text for details.

where K1 ¼ [E][A]/[E$A], K2 ¼ [E$A][B]/[E$A$B], K3 ¼ [E][B]/[E$B], K4 ¼ [E$B][A]/[E$A$B], K19 ¼ [E][A9]/ [E$A9], K29 ¼ [E$A9][B]/[E$A9$B], K39 ¼ [E][B]/[E$B], K49 ¼ [E$B][A9]/[E$A9$B]. The initial-velocity equation for the Random Bi Bi mechanism in the presence of alternative substrate A9 can be written as: " " # # 1 1 Ka ½A9  Kb Kia Kb ½A9  1þ 9 þ 1þ 9 þ þ ¼ V1 ½B V1 ½A½B v V1 V1 ½A ka ka 8.23 This equation indicates that the presence A9 will behave as a competitive inhibitor relative to substrate A in plots of

Enzyme Kinetics

510

A

k

EAB

E+P+Q

slowest B step

B E B' EB'

A

EA B'

A

slowest step

EAB'

k'

E+P'+Q

Scheme 8.24 The initial-velocity equation for the Random Bi Bi mechanism in the presence of alternative substrate B9 can be written as: " " # # 1 1 Ka Kb ½B9  Kia Kb ½B9  1þ 9 þ 1þ 9 þ þ ¼ V1 ½A½B v V1 V1 ½A V1 ½B kb kb 8.24

Step-2: Prepare four stock solutions. All contain the same [Y] but different [Y9], such that b0 ¼ [Y9 ¼ 0]/[Y ¼ y]; b1 ¼ [Y9 ¼ y9]/[Y ¼ y]; b2 ¼ [Y9 ¼ 2–3 y9]/[Y ¼ y]; and b3 ¼ [Y9 ¼ 4 6 y9]/[Y ¼ y]. [A'] = 3x

[A'] = 3x

2x

2x

x

x

0

0

1 / [Substrate A]

1 / [Substrate B]

[B'] = 3x

[B'] = 3x

2x

2x

x

1/

FIGURE 8.21 Initial-rate plots showing the expected effects of alternative substrate A9 or B9 on an enzyme operating by a Rapid-Equilibrium Random Bi Bi kinetic mechanism. NOTE: The four lines through the data points in the upper two plots are obtained by holding the concentrations of alternative inhibitor A9 at zero (bottom line), X, 2X, and 3X, where X is the approximate dissociation constant for alternative substrate A9. The four lines through the data points in the lower two plots are obtained by holding the concentrations of alternative inhibitor B9 at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for alternative substrate B9. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the plots showing competitive inhibition patterns. See text for details.

Step-1: Determine the Km and V for Y9, so that the appropriate range of Y and Y9 concentrations can be estimated. If, for example, the respective Michaelis constants for Y and Y9 are 0.5 mM and 0.1 mM, then a b value (b ¼ [Y9]/[Y]) should make Y and Y9 roughly equivalent in their occupancy of substrate binding sites. However, if Y9 has a maximal velocity value that is twice as great as V for Y, a b value of 2.5 should make Y9 just about as effective a substrate as Y. The proper values can therefore be adjusted in terms if the relative values of kcat/Km for Y and Y9.

1/

As shown in Fig. 8.21, this equation indicates that the presence B9 will behave as a competitive inhibitor relative to substrate B in plots of 1/v versus 1/[B] and as a noncompetitive inhibitor in plots of 1/v versus 1/[A]. Finally, it should be noted that alternative substrates can be used to analyze three-substrate enzyme kinetic mechanisms. The interested reader should consult Table V-3 of Initial Rate Enzyme Kinetics by Fromm (1975).

Huang and Kaufman (1973) introduced a constant-ratio approach, in which the substrate and the alternative substrate are varied in a constant ratio. In this procedure, the [Substrate]-to-[Alternative Substrate] ratio is kept constant, such that their absolute concentrations can be increased simultaneously. The resultant graphical patterns of 1/v versus 1/[A] and 1/v versus 1/[A] plots and their characteristic replots are unique for each bisubstrate kinetic mechanism. Huang (1979) illustrated the protocol for studying the effect of Y9 on the graphical patterns of the two substrates (also written here as X and Y to emphasize that prior to securing all the necessary rate data, we cannot say a priori whether X is substrate A or substrate B).

1/

EB

8.3.3. Huang’s Constant-Ratio Alternative Substrate Inhibition Method Distinguishes Multi-Substrate Kinetic Mechanisms

1/

1/v versus 1/[A] and as a noncompetitive inhibitor in plots of 1/v versus 1/[B]. Now consider Scheme 8.24 showing how alternative substrate B9 interacts with an enzyme operating by a RapidEquilibrium Bi Bi mechanism.

x

0

1 / [Substrate B]

0

1 / [Substrate A]

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

511

Step-3: Obtain data for a plot of [ETotal]/v versus 1/[Y]. Carry out separate rate experiments in which each stock solution is varied (by addition of different amounts to a set total volume) in the presence of a fixed, nonsaturating concentration of substrate A. Because a common product is measured in all experiments, product formation may be greater when b is increased, thereby requiring some adjustment if stopped-time assays are used. The resulting data will generate four separate lines in a plot of [ETotal]/v versus 1/[Y]. (Note: In plotting these rate data, use 1/[Y] and not 1/{[Y] þ [Y9]}.) Step-4: Obtain data for a plot of [ETotal]/v versus 1/[X]. Carry out separate experiments in which the X concentration is varied, with each of the b0, b1, b2, and b3 stock solutions held at a non-saturating level at the same Y concentration. The resulting data will generate four separate lines in a plot of [ETotal]/v versus 1/[X]. Step-5: Compare the patterns to those indicated in Table 8.5 to distinguish the reaction mechanism and whether Y is A or B. In summary, this alternative substrate inhibition approach offers the distinct advantage that linear double-reciprocal plots can be more reliably analyzed than the curvilinear plots observed in the previous section (Huang, 1977; 1979).

8.3.4. Induced Substrate Inhibition is a Type of Abortive Complex Inhibition Binding of an inhibitor that mimics the first substrate in an ordered substrate binding mechanism may still allow the second substrate to bind. Provided that the first substrate’s addition is not of the rapid equilibrium type, the inhibitor’s binding will ‘‘induce’’ substrate inhibition by the second substrate.

Take, for example, the ordered kinetic mechanism (Scheme 8.25), in which an inhibitor I competes with substrate A and where substrate B binds to EI form an EIB ternary complex: A

EA

E

P

B

(EAB/EPQ)

El l

Q

EQ

E

ElB B

Scheme 8.25 In the absence of I, no inhibition occurs at any concentration of substrate B; however, when I is present, inhibition is observed at higher concentrations of substrate B. As pointed out by Cleland (1979), the rate equation for this situation is readily obtained by multiplying the terms corresponding to the free energy (namely Kia, Kb, and Ka[B]) in the denominator of the rate equation by {1 þ ([I]/Ki)[1þ ([B]/KIb)]}, where Ki is the dissociation constant for I from the E$I complex, and KIb is the dissociation constant for B from E$I$B. When expressed in double-reciprocal form, the resulting equation is:    1 1 1 Ka Kia Kb ½A 1þ þ ¼ þ v V1 V1 ½A ½A½B Ki 8.25 Kia Kb ½I Ka ½I½B þ þ V1 Kib Ki ½A V1 Kib Ki ½A The last term having [I] and [B] in the numerator is responsible for the induced substrate inhibition. If [I] ¼ 0,

TABLE 8.5 Graphical Patterns Predicted by the Common-Product, Constant-Ratio Alternative Substrate Inhibition Method for Bisubstrate Enzyme-Catalyzed Reactions Substrate A Varied Mechanism Theorell-Chance Ordered Ternary Complex Rapid-Equilibrium Random Ping Pong

Substrate Pair

1/[A] Plot

Intercept Plot

Substrate B Varied 1/[B] Plot

Intercept Plot

A and A9

N

N-nonlinear

N-nonlinear

N-nonlinear

B and B9

N

C

C

none

A and A9

N

N-nonlinear

N-nonlinear

N-nonlinear

B and B9

N

N

N

U

A and A9

N

N

N

N

B and B9

N

N

N

N

A and A9

N

U

U

N

B and B9

U

N

N

U

Symbols: C, competitive; N, noncompetitive; U, uncompetitive. Intercept Plot: Refers to a replot of the ordinate intercepts (obtained at different constant levels of the non-varied substrate) versus the reciprocal of the concentration of the nonvaried substrate. Source: Huang (1979).

512

this term drops out, and no inhibition can be observed. The presence of [A] in the denominator of this term indicates that the [I]/[A] ratio affects the observed extent of induced inhibition. An example of induced substrate inhibition was provided by Danenberg and Danenberg (1978) in their studies of the thymidylate synthase reaction. They found that the second substrate methylene tetrahydrofolate becomes a substrate inhibitor only in the presence of the inhibitor 5-bromo-deoxyuridine 59-monophosphate binding in place of dUMP.

8.4. PRODUCT INHIBITION This phenomenon, known simply as product inhibition, occurs when a fraction of the active catalyst molecule is diverted by the presence of a product into nonproductive complexes. In the classical Michaelis-Menten treatment of a one-substrate enzyme, the E$X complex breaks down irreversibly (i.e., E þ S # E$X / E þ P), such that no E$P complex accumulates, even in the presence of P. Later treatments, such as the classical and important report of Harmon and Niemann (1949) and Schwert and Eisenberg (1949) on trypsin-catalyzed hydrolysis of N-benzoylL-arginine amide, provided strong theoretical and practical evidence that reaction products are not inert. We now know that most enzymes operate by multi-step Uni Uni mechanisms (i.e., E þ S # E$S # E$X # E$P # E þ P), where the enzyme is inhibited by reaction product. In this case, P combines with the free enzyme species E to form the E$P complex, meaning that P directly competes with S for access to vacant active sites. In many respects, product P behaves in a manner akin to the action of a simple competitive inhibitor, except P is not a dead-end inhibitor (i.e., E$P can react to form E$S in the reverse direction). In multi-step Iso Uni Uni mechanisms (i.e., E9 # E; E þ S # E$S # E$X # E$P / E þ P or E þ S # E$S # E$X # E$P / E9 þ P, followed by E9 # E), the product may or may not compete with the substrate, depending on whether that product binds to the same enzyme form that combines with the substrate. As discussed below, the same is true for Uni Bi mechanisms, such as the above reaction catalyzed by trypsin. The important point is that, under appropriate experimental conditions, product inhibitors can provide valuable information about enzyme interactions of an enzyme with its substrate(s) and product(s). Nowhere is this more evident than in the catalysis of multisubstrate enzyme-catalyzed reactions. Furthermore, enzymatic reaction products are apt to accumulate when its respective enzyme experiences the ebb and flow of intermediary metabolism, and product inhibition is likely to be physiologically important within cells.

Enzyme Kinetics

8.4.1. The Alberty/Fromm Strategy Uses Product Inhibition Patterns to Distinguish Rival Multi-Substrate Kinetic Mechanisms Alberty (1958) first suggested that product interactions with a multisubstrate enzyme could provide valuable mechanistic insights. He also indicated that the order of substrate binding or product release might be inferred, if multisubstrate reactions are carried out in the presence of only one reaction product at a time, such that no net reverse reaction can take place. As first established experimentally by Fromm and Nelson (1962), this property greatly simplifies the mathematical treatment of product inhibition. Moreover, the order of substrate addition and product release determines the patterns of inhibition observed in plots of 1/v versus 1/[A] and 1/v versus 1/[B] (i.e., competitive pattern, meaning that lines in a double-reciprocal plot converge on the 1/v-axis); noncompetitive pattern, meaning that lines in a doublereciprocal plot converge to the left of the 1/v-axis, or uncompetitive pattern, meaning that lines in a doublereciprocal plot are parallel with respect to each other. Consider a bisubstrate reaction that is arbitrarily written as: M þ N # X þ Y, as if the order of substrate X and Y addition were unknown prior to the experiment. There are four possible product inhibition experiments: Step-1: Determine the dependence of initial velocity v on substrate M in the absence and presence of several constant concentrations of product X. In this initial velocity experiment, the concentrations of M should bracket the concentration range around KiM, while holding substrate N at an appreciable, but non-saturating concentration, say 3–5  KN. The product is then held at three or four constant concentrations, such that the experimenter collects data for 1/v versus 1/[M] at [X1], then as 1/v versus 1/[M] at [X2], etc. The data are then plotted as 1/v versus 1/[M], yielding a separate line for each concentration of product X. Step-2: Determine the dependence of initial velocity v on substrate N in the absence and presence of several constant concentrations of product X. In this initial velocity experiment, the concentrations of N should bracket the concentration range around KiN, while holding substrate M at an appreciable, but non-saturating concentration, say 3–5  KM. The product is then held at three or four constant concentrations, such that the experimenter collects data for 1/v versus 1/[N] at [X1], then as 1/v versus 1/[N] at [X2], etc. The data are then plotted as 1/v versus 1/[N], yielding a separate line for each concentration of product X. Step-3: Determine the dependence of initial velocity v on substrate M in the absence and presence of several constant concentrations of product Y. In this initial velocity experiment, the concentrations of M should

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

513

bracket the concentration range around KiM, while holding substrate N at an appreciable, but non-saturating concentration, say 3–5  KN. The product is then held at three or four constant concentrations, such that the experimenter collects data for 1/v versus 1/[M] at [Y1], then as 1/v versus 1/[M] at [Y2], etc. The data are then plotted as 1/v versus 1/[M], yielding a separate line for each concentration of product Y. Step-4: Determine the dependence of initial velocity v on substrate N in the absence and presence of several constant concentrations of product Y. In this initial velocity experiment, the concentrations of N should bracket the concentration range around KiN, while holding substrate M at an appreciable, but non-saturating concentration, say 3–5  KM. The product is then held at three or four constant concentrations, such that the experimenter collects data for 1/v versus 1/[N] at [Y1], then as 1/v versus 1/[N] at [Y2], etc. The data are then plotted as 1/v versus 1/[N], yielding a separate line for each concentration of product Y.

The initial-velocity equation for an enzyme operating by the above kinetic scheme is as follows:   1 1 Ka Kb Kia Kb ½Q 1þ þ þ þ ¼ 8.26 v V1 V1 ½A V1 ½B V1 ½A½B Kiq Notice that the factor (1 þ [Q]/Kiq) only affects the AB term in Eqn. 8.26, indicating that, in a Rapid-Equilibrium Random Bi Bi kinetic mechanism, the presence of product Q should give rise to a competitive inhibition pattern in plots of 1/v versus 1/[A] and 1/v versus 1/[B] (see Fig. 8.22). Case-2: Reaction carried out with substrates A and B, in the absence and presence of several fixed concentrations of product P (with [Q] ¼ 0). A

An instructive exercise is to examine the rate equations for product inhibitor for the most commonly encountered bisubstrate enzyme reactions.

8.4.2a. Rapid Equilibrium Random Bi Bi Kinetic Reaction Mechanism Because two different products P and Q are formed, we consider the two inhibition experiments. Case-1: Reaction carried out with substrates A and B, in the absence and presence of several fixed concentrations of product Q (with [P] ¼ 0). [Q] = 0, X, 2X & 3X A

P=0

B

EQ

EA E

EAB

EB

B

Q

EPQ

EP

Q

A

Scheme 8.26

B

EQ

EA E

EAB

EB

As we shall see later, a similar approach can in principle also be employed to characterize the nature of product inhibition in three substrate enzyme-catalyzed reactions.

8.4.2. Product Inhibition Equations for Various Two-Substrate Kinetic Mechanisms Indicate Potentially Unique Inhibition Patterns

[P] = 0, X, 2X & 3X P Q=0

EPQ

A

B

EP

Q=0

P

Scheme 8.27 The initial-velocity equation for an enzyme operating by the above kinetic scheme is as follows:   1 1 Ka Kb Kia Kb ½P þ 8.27 þ þ ¼ 1þ v V1 V1 ½A V1 ½B V1 ½A½B Kip The factor (1 þ [P]/Kip) only affects the AB term in Eqn. 8.27, indicating that, in a Rapid-Equilibrium Random Bi Bi kinetic mechanism, the presence of product P should give rise to a competitive inhibition pattern in plots of 1/v versus 1/[A] and 1/v versus 1/[B] (see Fig. 8.22).

8.4.2b. Ordered Sequential Bi Bi Kinetic Reaction Mechanism Because two different products P and Q are formed, we consider the two inhibition experiments. Case-1: Reaction carried out with substrates A and B, in the absence and presence of several fixed concentrations of product Q (with [P] ¼ 0). [Q] = 0, X, 2X & 3X

E

P=0

A

E

P=0

B

EA

EAB

EPQ

Scheme 8.28

EQ

Q

E

Enzyme Kinetics

514

[P] = 3x

[P] = 3x

2x

2x 1/

1/

x

x

0

0

1 / [Substrate A]

1 / [Substrate B] [Q] = 3x

2x

2x

x

x

1/

[Q] = 3x

1/

FIGURE 8.22 Predicted product inhibition patterns for the Rapid-Equilibrium Random Bi Bi kinetic mechanism in the absence of abortive complex formation. NOTE: The four lines through the data points in the upper two plots are obtained by holding the concentrations of product inhibitor P at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for P. The four lines through the data points in the lower two plots are obtained by holding the concentrations of product inhibitor Q at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for Q. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the plots showing competitive inhibition patterns. See text for details.

0

0

1 / [Substrate B]

1 / [Substrate A]

2x

2x 1/

[P] = 3x

1/

[P] = 3x

x

x 0

0

1 / [Substrate B]

1 / [Substrate A]

[Q] = 3x

2x

2x

x

x

1/

[Q] = 3x

1/

FIGURE 8.23 Predicted product inhibition patterns for the steady-state Ordered Ternary Complex Bi Bi kinetic mechanism in the absence of abortive complex formation. NOTE: The four lines through the data points in the upper two plots are obtained by holding the concentrations of product inhibitor P at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for P. The four lines through the data points in the lower two plots are obtained by holding the concentrations of product inhibitor Q at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for Q. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the plots showing competitive inhibition patterns. See text for details.

0

0

1 / [Substrate B]

1 / [Substrate A]

The initial-velocity equation for an enzyme operating by the above kinetic scheme is as follows:     1 1 Ka ½Q Kb Kia Kb ½Q þ ¼ 1þ 1þ þ þ V1 ½B V1 ½A½B v V1 V1 ½A Kiq Kiq

Case-2: Reaction carried out with substrates A and B, in the absence and presence of several fixed concentrations of product P (with [Q] ¼ 0). [P] = 0, X, 2X & 3X

8.28

Notice that the (1 þ [Q]/Kiq) affects both the A and AB terms in Eqn. 8.28, indicating that, in the Ordered Bi Bi Ternary Complex kinetic mechanism, the presence of product Q should give rise to a competitive inhibition pattern in a 1/v versus 1/[A] plot and a noncompetitive inhibition pattern in a 1/v versus 1/[B] plot (see Fig. 8.23).

E

P

B

A

EA

EAB

EPQ

Scheme 8.29

Q=0

E

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

515

The initial-velocity equation for an enzyme operating by the above kinetic scheme is as follows:     Kq ½P 1 1 ½P Ka Kb 1þ þ ¼ þ 1þ Kip Kiq V1 ½A V1 ½B v V1 Kip   Kq ½P Kia Kb 8.29 þ 1þ Kip Kiq V1 ½A½B The factor (1 þ [P]/Kip) only affects the B, AB, and 1/V1 terms in Eqn. 8.29, indicating that, in an Ordered Bi Bi Ternary Complex kinetic mechanism, the presence of product P should give rise to a noncompetitive inhibition pattern in plots of 1/v versus 1/[A] and 1/v versus 1/[B] (see Fig. 8.23).

    1 1 Ka ½Q Kb Kia Kb ½Q 1þ 1þ þ þ ¼ þ V1 ½B V1 ½A½B v V1 V1 ½A Kiq Kiq

8.30

Notice that the (1 þ [Q]/Kiq) affects the A, B, and ABterms in Eqn. 8.30, indicating that, in a Theorell-Chance Bi Bi kinetic mechanism, the presence of product Q should give rise to a competitive inhibition pattern in a 1/v versus 1/[A] plot and a noncompetitive inhibition pattern in a 1/v versus 1/[B] plot (see Fig. 8.24). Case-2: Reaction carried out with substrates A and B, in the absence and presence of several fixed concentrations of product P (with Q ¼ 0).

8.4.2c. Theorell-Chance Bi Bi Kinetic Reaction Mechanism

[P] = 0, X, 2X & 3X A

Because two different products P and Q are formed, we consider the two inhibition experiments.

E

Case-1: Reaction carried out with substrates A and B, in the absence and presence of several fixed concentrations of product Q (with [P] ¼ 0).

E

P=0

EQ

EA

EA

Q=0

P

EQ

E

Scheme 8.31

The initial-velocity equation for an enzyme operating by the above kinetic scheme is as follows:     1 1 Ka Kb ½P Kia Kb ½P ¼ þ 1þ 1þ þ þ v V1 V1 ½A V1 ½B Kip Kip V1 ½A½B

[Q] = 0, X, 2X & 3X B

A

B

Q

E

8.31

Scheme 8.30 The initial-velocity equation for an enzyme operating by the above kinetic scheme is as follows:

The factor (1 þ [P]/Kip) only affects the B and AB terms in Eqn. 8.31, indicating that, in a Theorell-Chance Bi Bi kinetic mechanism, the presence of product P should give rise to a competitive inhibition pattern in 1/v versus 1/[B]

2x

2x

x

1/

[P] = 3x

1/

[P] = 3x

x 0

0

1 / [Substrate B]

1 / [Substrate A]

2x

2x

x

x

1/

[Q] = 3x

1/

[Q] = 3x

0

0

1 / [Substrate A]

1 / [Substrate B]

FIGURE 8.24 Predicted product inhibition patterns for the Theorell-Chance Bi Bi kinetic mechanism in the absence of abortive complex formation. NOTE: The four lines through the data points in the upper two plots are obtained by holding the concentrations of product inhibitor P at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for P. The four lines through the data points in the lower two plots are obtained by holding the concentrations of product inhibitor Q at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for Q. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the plots showing competitive inhibition patterns. See text for details.

Enzyme Kinetics

516

plot and a noncompetitive inhibition pattern in a 1/v versus 1/[A] plot (see Fig. 8.24).

8.4.2d. Ping Pong Bi Bi Kinetic Reaction Mechanism

Case-2: Reaction carried out with substrates A and B, in the absence and presence of several fixed concentrations of product P (with Q ¼ 0). [P] = 0, X, 2X & 3X A

Because two different products P and Q are formed, we consider the two separate inhibition experiments. Case-1: Reaction carried out with substrates A and B, in the absence and presence of several fixed concentrations of product Q (with [P] ¼ 0).

E

P

EA

B

F

FP

Q=0

FB

EQ

E

Scheme 8.33

[Q] = 0, X, 2X & 3X

EA

FP

F

The initial-velocity equation for an enzyme operating by the above kinetic scheme is as follows:   1 1 Ka Kb ½P 8.33 1þ þ þ ¼ v V1 V1 ½A V1 ½B Kip

Q

FB

EQ

E

Scheme 8.32

The initial-velocity equation for an enzyme operating by the above kinetic scheme is as follows:   1 1 Ka ½Q Kb þ þ 1þ 8.32 ¼ V1 ½B v V1 V1 ½A Kiq Notice that the (1 þ [Q]/Kiq) only affects the A term in Eqn. 8.32, indicating that, in a Ping-Pong Bi Bi kinetic mechanism, the presence of product Q should give rise to a competitive inhibition pattern in a 1/v versus 1/[A] plot and an uncompetitive inhibition pattern in a 1/v versus 1/[B] plot (see Fig. 8.25).

8.4.3. Abortive Complex Formation Alters Idealized Product Inhibition Patterns for Two-Substrate Kinetic Mechanisms In what was the first product inhibition study to test the Alberty treatment, Fromm and Nelson (1961) examined the kinetics of the Klebsiella pneumoniae ribitol dehydrogenase [P] = 3x

[P] = 3x

2x 1/

x

2x x

0

0

1 / [Substrate A]

1 / [Substrate B]

[Q] = 3x

[Q] = 3x 2x

2x 1/

FIGURE 8.25 Predicted product inhibition patterns for the Ping Pong Bi Bi kinetic mechanism in the absence of abortive complex formation. NOTE: The four lines through the data points in the upper two plots are obtained by holding the concentrations of product inhibitor P at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for P. The four lines through the data points in the lower two plots are obtained by holding the concentrations of product inhibitor Q at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for Q. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the plots showing competitive inhibition patterns. See text for details.

The factor (1 þ [P]/Kip) only affects the B term in Eqn. 8.33, indicating that, in a Ping-Pong Bi Bi kinetic mechanism, the presence of product P should give rise to a competitive inhibition pattern in a 1/v versus 1/[B] plot and an uncompetitive inhibition pattern in a 1/v versus 1/[A] plot (see Fig. 8.25).

1/

E

B

x

x

1/

P=0

A

0

0

1 / [Substrate A]

1 / [Substrate A]

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

517

(Reaction: Ribitol þ NADþ # Ribulose þ NADH þ Hþ). Their experimental results for ribitol dehydrogenase kinetics in the direction of ribulose reduction (Fig. 8.26) could not be explained by any of the above product inhibition equations, suggesting that some unanticipated phenomenon was occurring. They soon recognized that the only reasonable explanation was that the effect of the oxidized coenzyme was potentiated by the formation of an additional enzyme species, namely the abortive ternary complex E$NADþ$Ribulose. Based on their work, Fromm and Nelson (1961) proposed that ribitol dehydrogenases operated by a mechanism (Scheme 8.34) accounting for the formation of two nonproductive, or abortive, complexes. NAD+

E

Ribulose

Ribitol

E-NAD+

[E-NAD+-Ribitol & E-NADH-Ribulose]

E-NAD+-Ribulose Abortive Complex

8.4.3a. Rapid Equilibrium Random Bi Bi Kinetic Reaction Mechanism (with Abortive Complexes) EAP B A P Kia

E

EAB EB

Kib B

Q

E

E-NADH-Ribitol Abortive Complex

Kp

Kb

EA

Kiq

EQ

E

EPQ

Ka

EP

Kq Q

A

A

Kip P

EAP

EBQ

Scheme 8.35

NADH

E-NADH

EBQ P B Q

Formation of the E$A$P and E$Q$B abortive complexes is indicated in Scheme 8.35 for the Rapid-Equilibrium Bi Bi kinetic mechanism. Case-1: Reacting components are A and B, in the absence and presence of several fixed concentrations of product Q.     1 1 Ka ½Q Kb Kia Kb ½Q þ 1þ 1þ þ þ ¼ V1 ½B V1 ½A½B v V1 V1 ½A Kiq Kiq

Scheme 8.34 The frequent occurrence of such nonproductive complexes in enzymology, especially for NADþ- and NADPþ-dependent dehydrogenases, suggests that enzyme active sites are pliable to the extent that they readily accommodate mismatched substrate-product complexes to form within their active sites. Because abortive complex formation has proved to be such a general observation, anyone interested in examining product inhibition must be cognizant of the effects of abortive complexes. For this reason, it is important to consider the effects of abortive complexes on the initial-rate equations for product inhibition of bisubstrate enzyme kinetics.

8.34

Case-2: Reacting components are A and B, in the absence and presence of several fixed concentrations of product P.     1 1 Ka Kb ½P Kia Kb ½P 1þ 1þ þ þ ¼ þ V1 ½A½B v V1 V1 ½A V1 ½B Kip Kip 8.35

These equations predict that an enzyme operating by a Rapid-Equilibrium Random Bi Bi kinetic mechanism will have the inhibition patterns shown in Fig. 8.27.

8.4.3b. Ordered Bi Bi Kinetic Reaction Mechanism (with Abortive Complexes) Formation of the E$A$P and E$Q$B abortive complexes are indicated in Scheme 8.33 for the ordered Bi Bi kinetic mechanism.

(+)NAD+ 1/

(–)NAD+

EA

E

EAB

E

B EQB

EAP

FIGURE 8.26 Effect of abortive complex formation on the initialrate behavior of bacterial ribitol dehydrogenase. Redrawn from Nelson and Fromm (1961) with permission of the authors and the American Society for Biochemistry and Molecular Biology.

EQ

EPQ

P

1/[Ribulose]

Q

P

B

A

Scheme 8.36 Case-1: Reacting components are A and B, in the absence and presence of several fixed concentrations of product Q.

Enzyme Kinetics

518

2x

2x

x

1/

[P] = 3x

1/

[P] = 3x

x

0

0

1 / [Substrate A]

1 / [Substrate B] [Q] = 3x

2x

2x

x

1/

[Q] = 3x

1/

FIGURE 8.27 Predicted product inhibition patterns for the Rapid-Equilibrium Random Bi Bi kinetic mechanism in the presence of abortive complex formation. NOTE: The four lines through the data points in the upper two plots are obtained by holding the concentrations of product inhibitor P at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for P. The four lines through the data points in the lower two plots are obtained by holding the concentrations of product inhibitor Q at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for Q. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the plots showing competitive inhibition patterns. See text for details.

x

0

0

1 / [Substrate B]

     Kq ½P 1 1 ½P Ka Kb ½P 1þ þ ¼ 1þ þ 1þ Kip Kiq V1 ½A V1 ½B v V1 Kip KIp   Kq ½P Kia Kb þ 1þ Kip Kiq V1 ½A½B

   1 1 ½B Ka ½Q ½B Kia Kb 1þ 1þ þ þ þ ¼ v V1 V1 KIb V1 ½A Kiq VKib Ka KIb   Kb Kia Kb ½Q 1þ þ þ V1 ½B V1 ½A½B Kiq

8.37

8.36 Case-2: Reacting components are A and B, in the absence and presence of several fixed concentrations of product P.

These equations predict an enzyme operating by an Ordered Bi Bi kinetic mechanism will have the inhibition patterns shown in Fig. 8.28.

2x x

2x

[P] 1/

1/

[P]

[P] = 3x

int

int

[P] = 3x

x 0

0

1 / [Substrate B]

1 / [Substrate A] [Q] = 3x

[Q] = 3x

int

int

2x 1/

x

2x

[Q]

[Q] 1/

FIGURE 8.28 Predicted product inhibition patterns for the steady-state Ordered Ternary Complex Bi Bi kinetic mechanism in the presence of abortive complex formation. NOTE: The four lines through the data points in the upper two plots are obtained by holding the concentrations of product inhibitor P at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for P. The four lines through the data points in the lower two plots are obtained by holding the concentrations of product inhibitor Q at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for Q. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the plots showing competitive inhibition patterns. The inset shows that a replot of intercept values versus product P concentration will be nonlinear.

1 / [Substrate A]

x

0

1 / [Substrate B]

0

1 / [Substrate A]

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

519

8.4.3c. Theorell-Chance Bi Bi Kinetic Reaction Mechanism (with Abortive Complexes) Formation of the E$A$P and E$Q$B abortive complexes are indicated in Scheme 8.34 for the Theorell-Chance Bi Bi kinetic mechanism. A k1 E

B k2

P

Q

k4 k5

k3 EA

EQ

P

B

EAP

EQB

k6 E

Scheme 8.37 Case-1: Reacting components are A and B, in the absence and presence of several fixed concentrations of product Q. " #    1 1 ½B Ka ½Q ½B Kia Kb 1þ þ 1þ 1þ þ ¼ V1 ½A v V1 KIb Kiq VKib Ka KIb   Kb Kia Kb ½Q þ 1þ þ V1 ½B V1 ½A½B Kiq

8.38

Case-2: Reacting components are A and B, in the absence and presence of several fixed concentrations of product P.    1 1 Ka Kb ½P ½P 1þ 1þ þ þ ¼ v V1 V1 ½A V1 ½B Kip KIp   Kq ½P Kia Kb þ 1þ Kip Kiq V1 ½A½B

8.39

These equations predict that an enzyme operating by a Theorell-Chance Bi Bi kinetic mechanism will have the inhibition patterns shown in Fig. 8.29. Table 8.6 illustrates the distinctive kinetic patterns observed with bisubstrate enzymes in the absence or presence of abortive complex formation. It should also be noted that the random mechanisms in this table (and in similar tables in other texts) are usually for rapid-equilibrium random mechanism schemes. Steady-state random mechanisms will contain squared terms in the product concentrations in the overall rate expression. The presence of these terms would predict non-linearity in product inhibition studies. This non-linearity might not be obvious under standard initial-rate protocols, but products that would be competitive in rapid-equilibrium systems might appear to be noncompetitive in steady-state random schemes, depending on the relative magnitude of those squared terms. Product inhibition of dehydrogenases has been especially well documented. For nearly all dehydrogenases, product inhibition studies support the idea that catalysis proceeds by an ordered ternary complex kinetic mechanism. Examples include ribitol dehydrogenase (Fromm and Nelson, 1962), lactate dehydrogenases (Zewe and Fromm, 1962), alcohol dehydrogenases (Wratten and Cleland, 1963), malate dehydrogenases (Raval and Wolfe, 1962), formate dehydrogenase (Peacock and Boulter, 1970), glyceraldehyde-3-P dehydrogenase (Engel, 1968), and malic enzyme (Hsu, Lardy and Cleland, 1967). The involvement of oxidized and reduced abortive ternary complexes and the nature of specific inhibitory effects with natural substrates and alternative substrates are indicated in the cited references. With glutamate dehydrogenase, product inhibition experiments (Engel, 1968) support other evidence indicating that GDH operates by

[Q] = 3x

2x

2x

1/

x

1/

[Q] = 3x

x

0

0

1 / [Substrate B]

1 / [Substrate A] [P] = 3x

[P] = 3x

x

2x 1/

1/

2x

x 0

0

1 / [Substrate B]

1 / [Substrate A]

FIGURE 8.29 Predicted product inhibition patterns for the Theorell-Chance Bi Bi kinetic mechanism in the presence of abortive complex formation. The inset shows that a replot of slope values versus product P concentration will be nonlinear. NOTE: The four lines through the data points in the upper two plots are obtained by holding the concentrations of product inhibitor Q at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for Q. The four lines through the data points in the lower two plots are obtained by holding the concentrations of product inhibitor P at zero (bottom line), X, 2X, and 3X, where X is the approximate inhibitor dissociation constant for P. The respective inhibition constants may be evaluated from secondary plots of slope versus inhibitor concentration for the data in the plots showing competitive inhibition patterns. See text for details.

Enzyme Kinetics

520

TABLE 8.6 Predicted Product Inhibition Patterns for Bisubstrate Kinetic Mechanismsa Product Inhibitor

1/[A] Plot no Abortives

1/[B] Plot no Abortives

1/[A] Plot with Abortives

1/[B] Plot with Abortives

Theorell-Chance and Iso-Theorell-Chance

P

Nb

C

Nc

Cd

Q

C

N

C

N

Ordered Bi Bi and Iso-Ordered Bi Bi

P

N

N

Nc

Nd

Q

C

N

C

N

Rapid Equilibrium Random Bi Bi

P

C

C

N

C

Q

C

C

C

N

Mechanism

Ping Pong Bi Bi

d

P

N

C

N

NLd

Q

C

N

C

NLd

a

The degree of saturation of substrate binding sites affects the tendency to form abortive complexes (such as EAP and EBQ in ordered mechanism), and the concentration range of the nonvaried substrate can therefore determine the resulting inhibition patterns (indicated as C ¼ Competitive; N ¼ Noncompetitive, and NL ¼ Nonlinear). In columns 3 and 4, the predicted patterns in the absence of abortive complex formation are shown. In columns 5 and 6, the predicted patterns in the presence of abortive complex formation are shown. b Re-plot of Intercept Values versus [I] will be parabolic (concave-up). c Re-plot of Slope Values versus [I] will be parabolic (concave-up). d Hyperbolic (concave-up), assuming abortive complexes form.

means of a random substrate addition pathway (Engel and Dalziel, 1970; Silverstein and Sulebele, 1973). Finally, product inhibition data suggest a random mechanism for NADP-linked isocitrate dehydrogenase (Londesborough and Dalziel, 1970; Wicken, Chung and Franzen, 1972). Although competitive inhibition and alternative substrate inhibition are the most valuable for determining steady-state kinetic mechanisms, product inhibition experiments provide valuable corroborating evidence. For example, Table 8.7 shows the predicted product inhibition patterns for an enzyme catalyzing a Rapid-Equilibrium Random Ter Ter mechanism, along with results determined experimentally for adenylosuccinate synthetase (Reaction: IMP þ LAspartate þ GTP # Adenylosuccinate þ GDP þ Pi), which is known on the basis of other experiments to operate by this mechanism. The predicted patterns in parentheses take into account the formation of abortive complexes. Fromm (1975) and Rudolph (1979) discussed the impact of abortive complex formation and other practical limitations on interpreting product inhibition experiments.

8.4.4. A Foster-Neimann Plot Permits the Analysis of Progress Curves for Enzyme in the Presence of Product Inhibition In their review of progress-curve analysis methods, Orsi and Tipton (1979) summarized how researchers can investigate aspects of product inhibition using a plot of [P]/t versus (1/t) ln{[S0]/[St]}). This plot was first suggested by Foster and Neimann (1953), for whom it is named. When product inhibition is competitive, the integrated equation for an enzyme operating by Uni Uni mechanism is: Vmax;f ½Ki  1 ½S0  ðKa  K1 Þ½Pt  ln ¼ þ t ½St  Ka ðKi þ S0 Þt Ka ðKi þ ½S0 Þ

8.40

By holding [S0] constant at various values of total enzyme concentration [ET], the plot yields straight lines, and rate parameters can be estimated without the need to evaluate the initial velocities separately. Foster-Niemann plots will be curved if: (a) there is an error in zero-time velocity data (Taraszka, 1962), a fact that

TABLE 8.7 Predicted and Observed Product Inhibition Patterns for Adenylosuccinate Synthetasea Product Present in Experiment Substrate Varied in Experiment

AMP-Succinate (P) Predicted/Observed

IMP (A)

GDP (Q) Predicted/Observed

Pi (R) Predicted/Observed

N/N

N/N

Aspartate (B)

N/N

N/N

N/N

GTP (C)

N/N

C/C

N/N

a

Cooper and Rudolph (1995). Effect of abortive complex formation on observed inhibition pattern.

b

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

requires the investigator to be precise with respect to the starting time and conditions of the reaction; or (b) the nature of the product inhibition is noncompetitive, as is the case with isomerization mechanisms (Rebholz and Northrop, 1994).

8.4.5. Product Inhibitors Often Provide Valuable Clues About Multisubstrate ‘‘Iso’’ Mechanisms As noted in Chapter 3, while central complex isomerization has no effect on the form of the initial-rate equation, isomerization of stable enzyme forms (i.e., those where no chemical reaction occurs until another substrate is added) introduces additional terms into the rate equation. A great many enzymes form ‘‘stable’’ enzyme species that isomerize. Peller and Alberty (1959) demonstrated that the magnitude of V1/[ETotal] can never exceed the magnitude of any unimolecular rate constant describing any reaction step in the direction of substrate going to product. Thus, isomerization mechanisms (or iso-mechanisms) are indicated: (a) if calculation of the rate constant for a mechanism yields a negative value; (b) if the value of a rate constant is less than V1/[ETotal]; or (c) if abnormal F values or product inhibition patterns are obtained. The following schemes illustrate several bisubstrate isomechanisms that have been observed experimentally. The first example is the Di-Iso Theorell-Chance mechanism, in which E$A and E$Q isomerize to E$A9 and E$Q9, respectively: A

EA'

EA

E

P

B

Q

EQ

EQ'

E

Scheme 8.38 The second example is known as the Mono-Iso Ping Pong Bi Bi kinetic mechanism: A

E

P

EA

B

F

Q

(FB/GQ)

G

E

Scheme 8.39 Note that an additional enzyme form G must isomerize to regenerate E at the end of the catalytic reaction cycle. In the absence of products, isomerizations of stable enzyme forms have no noticeable effect on initial rate equations. The isomerization steps of this type introduce additional rate constants to the mechanism, and they affect the Dalziel F parameters (see Chapter 5), as dictated by their Haldane relations. These quantitative effects, which may not be obvious to the inexperienced worker, are summarized in Table 8.8.

521

Of course, the formation of one or more abortive complexes increases the kinetic complexity and kinetic ambiguity, making the discrimination of iso-mechanisms much more difficult. The interested reader should also consult Fromm (1975), Rudolph (1979), or Rebholz and Northrop (1994; 1995) for a detailed discussion of iso-mechanisms.

8.4.6. The Metabolic Significance of Product Inhibition Merits Greater Consideration Although the use of product inhibition to analyze substrate binding order is fraught with many limitations, a good deal more attention should be given to the roles of product inhibition in metabolic regulation, especially when the experimenter seeks to understand the kinetic properties of an entire pathway. While allosteric feedback regulation (see Chapter 10) doubtlessly plays a major role in controlling the first committed step in a pathway, other non-allosteric enzymes are regulated by their immediate reaction products. In assessing the metabolic implications of product inhibition and abortive complex formation, Purich and Fromm (1972) carried out kinetic simulations to learn whether enzymes operating a particular kinetic mechanism have special regulatory advantages. The basic idea was that many isozymes are known to have different kinetic mechanisms and to differ with respect to their susceptibility to product inhibition and substrate-induced inhibition via abortive complex formation. Their simulations suggested that product inhibition is likely to be a physiologically significant feature of enzyme catalysis. Studies on yeast and brain hexokinase illustrate how enzymes catalyzing the same reaction can behave so differently. Gatt and Racker (1959) attempted to reconstitute glycolysis using highly purified mammalian enzymes. At that time, a good preparation of brain hexokinase was unavailable, and yeast hexokinase was used instead. In the presence of limiting ADP and orthophosphate, addition of mitochondria altered the kinetics of the reconstituted glycolytic system, leading to reduced lactate production when excess mitochondria was added. They also found that respiration was inhibited whenever the concentration of glycolytic enzymes was increased. Although these results resembled the well known Pasteur and Crabtree effects observed with intact cells, inhibition of lactate production in these reconstituted systems was attended by an unusually high accumulation of fructose 1,6-bisphosphate, a behavior that was clearly different from that observed with intact cells. Later work using mouse hexokinase in place of the yeast enzyme successfully reproduced all of the cellular glycolytic kinetics (Uyeda and Racker, 1965). The basis of these disparate behaviors relates to the 500 greater susceptibility of mammalian hexokinases to product inhibition by glucose-6-P (Crane, 1962; Zewe, Fromm and Fabiano, 1964). The potential significance of product inhibition is also illustrated by the pyruvate dehydrogenase reaction. The

Enzyme Kinetics

522

TABLE 8.8 Product Inhibition Patterns for Selected Iso-Mechanisms for Two- and Three-Substrate Enzyme-Catalyzed Reactions (No Abortive Complexes) Product Inhibition Patterns Varied Substrates Mechanism

Additional Terms in the Rate Equation

Product

A

B

C

Mono-Iso Theorell-Chance

ABQ, APQ

P

N

C

–

Q

N

N

–

P

N

C

–

Q

C

N

–

P

N

N

–

Q

N

N

–

P

N

N

–

(Scheme 5.48) Di-Iso Theorell-Chance

None

(Scheme 5.48) Mono-Iso Ordered Bi Bi

ABQ, APQ, ABPQ

Di-Iso Ordered Bi Bi

None

Q

C

N

–

Mono-Iso Ping Pong Bi Bi

AQ, ABQ, APQ

P

N

C

–

Q

N

N

–

AQ, BP, ABP, ABQ, APQ, BPQ

P

N

N

–

Q

N

N

–

P

N

N

N

Q

U

N

N

(Scheme 5.51) Di-Iso Ping Pong Bi Bi (Scheme 5.50) Mono-Iso Uni Uni Bi Bi Ping Pong

BCP, ABCP, BCPR, BPQR, BCPQR

(Scheme 5.52)

R

C

U

U

Di-Iso Bi Uni Uni Bi Ping Pong

CP, ACP, CPR, ABCP, ABCR, ABQR, APQR,

P

N

N

N

(Scheme 5.53)

CPQR, ABCQR, ABPQR

Q

U

U

N

R

N

N

U

ABR, ACQ, AQR, BCP, BPR, CPQ, ABCP,

P

N

N

U

ABCQ, APQR, BCPR, BPQR, CPQR

Q

U

N

N

R

N

U

N

Tri-Iso Hexa Uni Ping Pong

activity of mammalian PDH complex is inhibited by the immediate products of pyruvate oxidation, namely acetylCoA and NADH, and these inhibitory effects are reversed by CoA and NADþ, respectively (Bremer, 1969; Garland and Randle, 1964; Wieland et al., 1969). Bacterial PDH behaved in a like manner, with more sensitivity toward NADH than acetyl-CoA. NADH appears to bind to the flavoprotein component in the PDH complex, whereas acetyl-CoA is competitive with pyruvate and is reversed by AMP. With the mammalian enzyme, acetyl-CoA behaves as an uncompetitive inhibitor relative to pyruvate. Such distinctive differences are apt to alter the overall kinetics of pyruvate oxidation in a manner that is best adapted to the needs of the organism in which they are found. Finally, the reader should be wary of overly simplistic regulatory models based, in part, on product inhibition effects. A notable example is the adenylate energy charge model, a hypothesis that attempted to explain the metabolic basis for control of ATP utilization and regeneration (Atkinson, 1968; 1977). The cell’s energy charge was

estimated by substituting measured ATP, ADP, and AMP concentrations into the following expression: Energy Change ¼

½ATP þ 0:5½ADP ½AMP þ ½ADP þ ½ATP

8.41

The adenine nucleotide system is said to be ‘‘fully charged’’ at a limiting EC value of 1.00 (i.e., when only ATP is present) and ‘‘fully discharged’’ at an EC value of 0 (i.e., when only AMP is present). At intermediate values, the adenine nucleotides are interconverted by adenylate kinase, and their equilibrium concentrations are constrained by the adenylate kinase mass action ratio: Keq ¼ [ADP]2/ [ATP][AMP]. In the adenylate energy charge model, ATPutilizing enzymes (called U-systems) and ATP-regenerating enzymes (called R-systems) constitute opposing components of an overall homeostatic mechanism that buffers the energy charge in a range of 0.75–0.85. Accordingly, a drop in the cellular energy charge would be expected to reduce the velocity of ATP-utilizing reactions, while simultaneously stimulating ATP-replenishing processes.

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

8.5. MULTI-SUBSTRATE GEOMETRIC INHIBITORS Many enzymes operate by sequential-type mechanisms wherein two or more substrates reside simultaneously within an active site. Under favorable circumstances, two substrates can be joined together to obtain a new chemical compound possessing binding determinants that span a larger region of the active site. Such agents hold promise as enzyme-specific inhibitors, as opposed to analogues mimicking the binding determinants within a single substrate. Ideally, binding of an ideal multi-substrate geometric analogue would also release an amount of binding energy (DGAB) equivalent to the sum of the separate binding energies of the individual substrates (i.e., DGSubstrate-A þ DGSubstrate-B). Were such perfection realized, the overall equilibrium constant KAB ¼ KA  KB; in practice, however, such extremely high affinity has not been observed. It is likewise important to stress that multisubstrate geometric analogues are not transition-state analogues, because they do not typically mimic the bond

order and/or the geometry of those atoms or functional groups comprising the reaction center. HO N N

N H2N

OH O

N

NH2

N OO P O O

N

OO P O P O O O-

O

N

N

n

n = 4 or 5 HO

OH

Ap4A and Ap5A The utility of multi-substrate geometric analogues as high-affinity inhibitors and as diagnostic tools of substrate binding order was first assessed by Purich and

A

1/

180 M

90 M 0 AppppA 1 / [MgATP2-]

B 150 M 100 M

1/

While numerous investigations from Atkinson’s laboratory offered evidence of these predicted properties, later studies by Purich and Fromm (1972; 1973) demonstrated that the energy charge model is overly simplistic and that its principles are unlikely to constitute a useful model for quantitatively analyzing the control of energy metabolism. Chiefly, the energy charge model failed to account for the fact that U- and R-enzymes frequently have other phosphoryl-acceptor and -donor substrates, and fluctuations in these cosubstrate concentrations can greatly influence the so-called responsiveness of enzymes. In almost all of the previous studies, cosubstrates were maintained at a saturating concentration – far from the intracellular levels. Moreover, the concentration of orthophosphate was wholly omitted from their model, despite the fact that key regulatory steps in ATP utilization and regeneration are exquisitely sensitive to changes in orthophosphate concentration. The energy-charge model also failed to account for magnesium ion complexation by nucleotides, the nature of phosphoryl acceptor substrate, or the effects of adenine nucleotide compartmentation in living cells. Moreover, many signal transduction processes involve U-systems (e.g., ATPdependent protein kinases) to stimulate enzymes in other socalled R-systems by downstream activation of ATPregenerating enzymes. This fact would suggest that the initial hormone- or effector-stimulated kinase is itself unlikely to obey the predicted energy charge behavior of a U-system. In view of these considerations, as well as other technical limitations, the energy charge model is an inherently flawed attempt to rationalize the intracellular behavior of regulatory enzymes utilizing or replenishing ATP.

523

50 M 0 AppppA 1 / [AMP] FIGURE 8.30 Inhibition of rabbit muscle adenylate kinase by P1, P4-di-(adenosine-59) tetraphosphate. A, Plot of the reciprocal of the initial velocity v of the rabbit muscle adenylate kinase reaction versus the reciprocal of the millimolar ATP concentration in the absence and presence of AppppA. AMP was maintained at 0.2 mM, and ATP ranged from 0.11 to 1.0 mM. B, Plot of the reciprocal of the initial velocity v versus the reciprocal of the millimolar ATP concentration in the absence and presence of AppppA. ATP was maintained at 0.15 mM, and AMP ranged from 0.11 to 1.0 mM. Under these conditions, the formation constant (KF ¼ [MgATP2]/[ATP4]Free[Mg2þ]Free) equals 20,000 M1. Therefore, at an uncomplexed (or ‘‘free’’) magnesium ion concentration of 1 mM, >95% of the ATP was present as MgATP2 (i.e., Kstab  [Mg2þ]Free ¼ 20,000 M1  0.001 M ¼ 20 ¼ [MgATP2]/[Mg2þ]Free). Adenylate kinase activity was monitored continuously as decrease in NADH’s 340nm absorbance, using a coupled assay containing optimized concentrations pyruvate kinase, lactate dehydrogenase, P-enol pyruvate, potassium ion, and NADH.

Enzyme Kinetics

524

Fromm (1972), who studied the inhibition of adenylate kinase (Reaction: MgATP2 þ AMP # MgADP þ ADP) by the naturally occurring dinucleotide P1,P4-di-(adenosine-59)-tetraphosphate (Ap4A). The apparent inhibition constant for this compound proved to be only 10–15 times lower than the Michaelis constants for MgATP2 and AMP, far short of the ideal affinity of a multisubstrate geometric inhibitor. Even so, Ap4A was found to be a competitive inhibitor with respect to MgATP2 and AMP (Fig. 8.30), a finding that is consistent with the idea that rabbit muscle adenylate kinase operates by a random kinetic mechanism (Purich and Fromm, 1972). Later work by Lienhard and Secemski (1973) indicated that the compound P1,P5-di-(adenosine-59)-pentaphosphate (Ap5A) bound with higher affinity (KI z 10–15 mM). Ap5A is a valuable commercially available agent for suppressing the catalytic activity of the seemingly omnipresent adenylate kinase. There are several reasons why Ap4A and Ap5A might not exhibit the binding affinity anticipated on the basis of the individual binding energies for Enz$AMP and Enz$MgATP2 formation. First, as polyanions, Ap4A and Ap5A may form several metal ions to form complexes that the enzyme cannot recognize. Second, the extended P–O and O–P bonds of the ADP$$$PO 3 $$$AMP transition state are not duplicated by the multi-substrate geometric inhibitors. Third, stereochemical constraints may prevent forma˚ crystal tion of a high-affinity complex. Notably, the 2.6-A structure of the Enz$Ap5A complex revealed that both adenosine moieties of Ap5A cannot be bound simultaneously at yeast adenylate kinase’s ATP and AMP subsites (Egner, Tomasselli and Schulz, 1987) (see also Section 5.12: Induced-Fit Mechanism). Intriguingly, the dinucleoside polyphosphates Ap3A, Ap4A, Gp3A, and Gp4A are found in all cells in concentrations ranging from 0.05 to 100 mM. Gp3G and Gp4G have also been detected at sub-mM concentrations in E. coli and Saccharomyces cerevisiae, and reach even higher levels in the encysted embryos of Artemia-species brine shrimp. The yolk platelets of the latter contain a GTP:GTP guanylyltransferase (Reaction: 2 GTP # Gp4G þ PPi) that will also form the homo-dinucleotides G(59)p5(59)G, G(59)p3(59)G, dG(59)p4(59)dG, and G(59)pp(NH)pp(59)G as well as the hetero-dinucleotides G(59)p4(59)X, G(59)p4(59)I, and G(59)p4(59)A (Liu and McLenan, 1994). Delaney, Blackburn and Geiger (1997) reported that the activity of adenosine kinase (Reaction: Adenosine þ MgATP2 # MgADP þ 59-AMP) was unaffected by A(59)p2(59)A and A(59)p3(59)A, but was A(59)p5(59)A, and inhibited by A(59)p4(59)A, A(59)p6(59)A, where apparent IC50 values were 5.0, 3.3 and 500 mM, respectively. Inhibition of adenosine kinase activity by A(59)p4(59)A and four of its stable phosphonate analogues of A(59)p4(59)A tested was uncompetitive.

Sorci et al. (2006) examined the kinetic properties of structurally multi-substrate geometric NADþ analogues: P1di(adenosine-59-)-P3-(nicotinamide-riboside-59-)-triphosphate (or, Np3A); P1-di(adenosine 59-)-P4-(nicotinamide-riboside59-)-tetraphosphate (or, Np4A); and P1-di(adenosine 59-)P4-(nicotinate-riboside-59-)-tetraphosphate (or, Nap4A) with nicotinamide mononucleotide (NMN) adenylyltransferase (Reaction: NMN þ MgATP2 # NADþ þ PPi). The observed inhibition patterns in 1/v versus 1/[NMN] and 1/v versus 1/[MgATP2] plots also confirm that NMNAT operates by an ordered ternary complex kinetic mechanism (Magni et al., 1999). The organellespecific isoforms differ in the order of substrate addition. ATP binds first in the case of NMNAT1 and NMNAT2, but NMN is the first substrate to bind to NMNAT3. In all cases, the products are released sequentially, with PPi first, followed by NADþ. Another illuminating example was provided in the studies of Pope et al. (1998a,b,c) and Brown et al. (2000) who examined the action of the leucine-containing (red boxes) agent SB-234764. HO H3C

O

O

H3C

C C H2

C C

S N H

H

H

H2 C

H

C C

CH

CH C H2

O O

H2 C

H

HO

O

H C

OH C

C CH3

O

C H2

NH2

SB-234764 SB-234764 resembles the antibiotic mupirocin, itself a bisubstrate geometrical analogue of isoleucyl-adenylyl moiety at the 39-CCA terminus of tRNAile: H2N tRNAile N

N O

O P O

O

CH3

H3C H2C

N

H2C

CH

N

O

OH

O C

CH

O

NH2

Isoleucyl-Adenylyl Moiety of Amino Acylated tRNAile

Pope et al. (1998a,b,c) and Brown et al. (2000) demonstrated that SB-234764 is an extraordinarily potent inhibitor

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

(Ki z 1015 M) of bacterial isoleucyl-tRNA synthetase (Reaction: Ile þ MgATP2 þ tRNAIle # Ile–tRNAIle þ AMP þ PPi). Given such tight binding, it is not at all surprising that SB-234764 does not bind in a single-step process, but instead undergoes a series of isomerizations, as would be expected for tight-binding inhibitors. Williams and Northrop (1979) synthesized a tightbinding, multi-substrate analogue for gentamicin acetyltransferase (Reaction: Gentamicin þ CoA–S–COCH2 # CH3CO–N3–gentamicin þ CoA–SH). ClCH2CO–N3– -gentamicin was first formed enzymatically from gentamicin and the alternative substrate CoA–S–COCH2Cl, after which the inhibitory multi-substrate isoster CoA-–S–CH2CO–N3– gentamicin by reaction of ClCH2CO–N3–gentamicin with CoA–SH. Kinetic analysis of a time-dependent onset and reversal of inhibition of gentamicin acetyltransferase I by the purified multi-substrate analogue yielded a Ki z 109 M. The effectiveness of the multi-substrate analogue demonstrates that inhibitors of antibiotic resistance might be designed and prepared by specific enzymatic synthesis. Cullis et al. (1982) also observed the inhibition of histone acetylation by N-[2-(S coenzyme A)-acetyl] spermidine amide, a multi-substrate geometric analogue formed by joining coenzyme A with spermidine through an acetic acid linkage. This agent is a strong inhibitor (Ki < 108 M) of the acetylation of spermidine and histones by calf thymus histone acetylase. With isolated nuclei, the analogue inhibited acetylation of histones H2a and H2b much more strongly than acetylation of histones H3 and H4. Finally, noting the absence of a theoretical underpinning for the inhibition patterns for multi-substrate geometric inhibitors, Yu et al. (2006) derived the appertaining rate equations for likely sequential bisubstrate mechanisms.

525

Bernhard and Orgel (1959) also considered the possibility that naturally occurring high-affinity enzyme inhibitors might bear some structural analogy to the reaction’s transition state. Jencks (1966) and Wolfenden (1969) extended this concept by presenting other cogent examples. While noting that no inhibitor can be identical to a reaction’s transition state, they recognized that such inhibitors might well mimic certain strained shapes and electronic features that are typically only achieved in the transition state. Although exploitation of the transition-state inhibitor concept reflects the creativity of a great many investigators, Richard Wolfenden and Vern Schramm remain the most persuasive proponents of the rational design of transitionstate inhibitors (see also Section 8.12.4: Schramm’s Approach for Rational Drug Design is Based on Kinetic Assessment of Reaction Transition States).

8.6.1. The Energetics of Transition-State Stabilization Explains the Considerable Inhibitory Potency of Substances Resembling the Transition State As pointed out by Wolfenden (1995), the classical Haldane-Pauling model of a transition state stabilization model for enzyme catalysis offers a means to rationalize the extraordinarily high affinities achieved by some substrate analogues. The basic idea is that, because the active site has evolved to bind the substrate in its transition-state configuration, the enzyme sacrifices little energy typically needed to distort the substrate’s molecular shape and charge to resemble the transition state. The scheme shown in Fig. 8.31 permits the experimenter to

8.6. TRANSITION-STATE INHIBITORS Many highly potent natural and synthetic enzyme inhibitors bear scant resemblance to the structure of the substrate or product of the target enzyme reaction; they instead appear to resemble predicted catalytic reaction intermediates. An explanation for such behavior finds its roots in predictions by Linus Pauling (1946) about the essence of enzyme catalysis and the action of transitionstate inhibitors: If the enzyme were completely complementary in structure to the substrate, then no other molecule would be expected to compete successfully with the substrate in combination with the enzyme, which in this respect would be similar in behavior to antibodies; but an enzyme complementary to a strained substrate molecule would attract more strongly to itself a molecule resembling the strained substrate molecule than it would the substrate molecule.

x‡

Transition State Stabilization Energy

DGstabilization

(E·X)‡ E·P E+S

S

E·S

CATALYZED

DGreaction

UNCATALYZED E+P

P

FIGURE 8.31 Energetics of enzymatic and uncatalyzed reactions. Conversion of substrate S directly to the transition state in the absence of enzyme (proceeding directly to the transition state Xz) is energetically less favorable than the corresponding conversion on the enzyme surface (proceeding directly to the transition state (E$X)z).

Enzyme Kinetics

526

estimate the maximal affinity that should be achievable if a compound were to approximate closely the electronic and stereochemical configuration of the enzyme and substrate in the transition state. An accurate estimation of KTX (or KTS) requires certain knowledge that the uncatalyzed reference reaction follows the same mechanism as the enzyme-catalyzed process. As shown in Fig. 8.31, the energetics of enzymecatalyzed reactions and corresponding uncatalyzed reference reactions can be understood by the cyclic path that accounts for substrate conversion to product by the uncatalyzed and enzymatic routes. The thermodynamic cycle depicted in Scheme 8.40 illustrates the energetic relationship between transition-state stabilization and catalysis, as originated with Kurz (1963)5 and later applied to explain the inhibitory power of transition-state analogues in enzyme-catalyzed processes (Lienhard, 1972; 1973; Wolfenden, 1969; 1972). Note that the uncatalyzed reaction is characterized by a transition state that is far less stable than its enzymatic counterpart. Note also that the initial and final conditions are the same for either route, an absolute requirement for any catalyzed process (i.e., the enzyme has no effect on the overall equilibrium constant). Kuncat E +''S''

E+S

configuration and the substrate in its short-lived transition state. Reaction Step-2: Proceeding from Species-(Sz) to Species(ESz), with product-over reactant defined by [E$Sz]/ [Sz][E], or 1/KTS, where KTS is the dissociation constant for the enzyme-bound transition state. Reaction Step-3: Proceeding from Species-(EþS) to Species-(ES), with product-over reactant defined by [E$S]/[E][S], or 1/KS, where KTS is the dissociation constant for the enzyme-substrate complex. Reaction Step-4: Proceeding from Species-(ES) to Species(ESz), with product-over reactant defined by [E$Sz]/ [E$S], or Kcat z, which is the quasi-equilibrium constant between the enzyme-bound substrate in its ground-state configuration and the enzyme bound substrate in its short-lived transition state. The Species-(EþS) / Species-(E$Sz) path by way of Species-(Sz) is represented by Kuncat z  1/KTS, equal to Kuncat  1/KTS, and the Species-(EþS) / Species-(ESz) path by way of Species-(ES) is represented by 1/KS  Kcat z , equal to Kcat z /KS. Therefore: Kuncat z Kcat z ¼ KTS KS

8.42

E+P

After cross-multiplication, we get: KTS

KS

(E''S'')

ES

E+P

Kcat

Scheme 8.40 The key parameter is the dissociation constant (KTS ¼ [E][Sz]/[E$Sz]) for the enzyme-bound transition state. To evaluate this parameter, we may analyze this thermodynamic cycle in terms of the pathways from Species-(EþS) to Species-(ESz). Reaction Step-1: Proceeding from Reactants-(EþS) to Species-(Sz), with product-over reactant defined by [Sz][E]/[S][E], or Kuncatz, which is the quasi-equilibrium constant between the substrate in its ground-state 5

Without specifically mentioning enzyme catalysis, Kurz (1963) explicitly showed how to calculate the virtual equilibrium constant for the dissociation of the catalyst from the transition state, based on experimental data, thereby setting the stage for estimating the affinity of transition-state enzyme inhibitors. His approach becomes quantitatively useful whenever one can measure the rates of reaction depending only on the catalyst’s presence (i.e., both processes operate by strictly analogous mechanisms). The same caveat applies to the enzymatic and reference reactions used to analyze the energetics of transition-state inhibitor action.

KTS ¼

KS Kuncat z Kcat z

8.43

Based on Absolute Rate Theory (see Section 3.8: Transition State Theory), the rate constant kuncat equals kn Kuncatz, where k is the transition-state transmission coefficient, and n is the vibration frequency of the scissile bond. Likewise, the rate constant kcat equals kn Kcatz. Therefore, Eqn. 8.43 may be rewritten in terms of the two rate constants: KTS ¼ KS 

kuncat kcat

8.44

If KS is nominally 104 M, and if kcat is 1012  kuncat, then KTS will be ~1016 M, suggesting that a perfect transition-state analogue would bind to the enzyme with a sub-femtomolar dissociation constant. Moreover, if the catalyzed rate is even greater, then the transition-state analogue would bind even more tightly. As illustrated in Fig. 8.32, most transition-state inhibitors bind in a multi-step reaction reflecting conformational rearrangements. While tight-binding transition-state analogues most often bind within the substrate binding sites, their high affinity or slow binding characteristics most often result in 1/v versus 1/[S] plots that do not resemble those expected for typical competitive inhibitors showing

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

fast

E..Iloose

E+I

slow

EItight

SLOW

FAST

E+I

G

E..Iloose

∆Gbinding

EItight

527

analogues, and many have proven to be powerful enzyme inhibitors. Several pertinent examples are presented in Table 8.9. The adenosine deaminase mechanism shown below illustrates the key feature in this aromatic nucleophilic substitution reaction. This mechanism suggests what agents might successfully mimic the stereochemical and/or electrostatic arrangement of atoms in the transition state. The tight binding of transition-state analogues relates to the fact that such analogues are complementary to the transitionstate configuration of the enzyme, whereas substrates in the ground state are not. Wolfenden (1972; 1995) also stressed that, while a substrate must necessarily suffer some loss in binding energy as it is contorted along its path to reach the transition state, the same is not true for transition-state analogues.

Progress of Binding Interaction FIGURE 8.32 Energetics of slow-binding transition-state inhibitors. As indicated in the earlier discussion, a genuine transition-state analogue should bind extremely tightly to an enzyme active site. If KS is nominally 105 M, and if kcat is 1010  kuncat, then KTS will be ~1015 M. While the binding process is likely to exhibit multiple conformational rearrangements, the system will behave as a two-step scheme shown here if one of the post-binding isomerizations is slow relative to the other steps.

H2O

O :NH3

NH2 N

H2O

N

N

N

K1 KS1

Kbi

E + S1S2

E + S1S2

ES1 + S2

E+P

KTS

KS2 ES1S2

ES1S2

E+P

Kcat

Scheme 8.41 With two reactants in multi-substrate enzyme-catalyzed reactions, the corresponding scheme must be amended (Scheme 8.41) to account for the fact that the two substrates react at a rate determined by the rate constant k describing the formation of a transition-state complex. Enzymatic rate enhancement in this situation results from the entropic advantage of bringing the two molecules together in the active site, in addition to the stabilization of transition state over ground state once they are bound.

8.6.2. There are Numerous Examples of Naturally Occurring and Synthetic Transition-State Analogues Enzyme chemists have already established an impressive record of success in developing numerous transition-state

N

NH

N

NH N

Rib

Transition State

Rib

rapid-equilibrium binding (see Section 8.7.2: Slow-Binding Inhibitors and Slow, Tight-Binding Inhibitors).

E + S1 + S2

NH2

Scheme 8.42 Returning to the adenosine deaminase reaction (Scheme 8.42), we may consider the action of two additional compounds. NDHPR has a Ki value of 105 M, whereas coformycin exhibits a Ki of approximately 1011 M. Both contain sp3 hybridization in the analogous position where water attacks during adenosine deaminase catalysis. Tight binding of coformycin is presumably the consequence of the structural complementarity of the enzyme and transitionstate analogue. H OH H C H N NH N

HO

N

H N

H

NH N

HO

N

O

O HO

OH H

OH

NDHPR

HO

OH

Coformycin

Table 8.10 lists several examples of highly effective transition-state mimics along with other multi-substrate geometric inhibitors that appear to share one or more transition-state features anticipated on the basis of the particular reaction mechanism.

Enzyme Kinetics

528

TABLE 8.9 Rate and Inhibitor Parameters for Selected Enzyme-Catalyzed and Uncatalyzed Reference Reactions Enzyme, Inhibitor, (Ki)

kun (s1)

Orotate decarboxylase Barbituric acid ribotide Ki ¼ 1  1011 M

2.8  1018

Adenosine deaminase 29-Deoxycoformycin Ki ¼ 2.5  1012 M

1.8  1010

Acetycholinesterase Methyl(trimethylammonio)ethyl borinate Ki ¼ 3.0  10–8 M

1.1  10–8

Cytidine deaminase Phosphapyrimidine riboside Ki ¼ 9.0  1010 M

3.2  1010

Triose-P isomerase Phosphoglycollo-NHOH Ki ¼ 4.0  10–8 M

Kcat (s1)

4.3  10–6

Km (M)

kcat/Km (M1s1)

KTX (M)

39

5.6  107

1.4  1017

5.0  1024

370

1.4  107

2.1  1012

1.3  1017

15,000

1.6  108

1.4  1012

7.0  1017

300

2.9  108

1.2  1012

1.1  1018

4,300

2.4  108

1.0  108

1.8  1014

Sources: Rudzicka and Wolfenden (1995) and Mader and Bartlett (1997).

Finally, some putative transition-state analogues have equilibrium dissociation constant values in the 10–100 pM range. A bimolecular rate constant kon for a protein with a low-molecular-weight ligand typically cannot exceed 108 M1 s1. If binding occurred as a one-step reaction, Kd would equal koff/kon, and koff would be very low, in the neighborhood of 102–103 s1 (putting t1/2 z 1–10 min). A more likely case is a two-step binding process, where Enz$Analogue1 isomerizes to Enz$Analogue2, allowing Kd to equal kþ1kþ2/k1k2.

8.6.3. High-Affinity Binding of Certain ‘‘Pro-Transition-State Analogues’’ is Triggered by Some Enzymes Given the wide spectrum of catalytic strategies adopted by enzymes, the classification of some transition-state inhibitors is problematical. While many agents are fully formed transition-state analogues, others require partial catalytic intervention by the target enzyme to generate the highaffinity transition-state analogue. An outstanding example of multi-step, pro-inhibitor activation is fluoroacetate, a powerful rodenticide (LD50 z 0.5 mg per 100 g body-weight). During fluoroacetate poisoning, citrate accumulates in rat brain to 1–2 mmol/g tissue, rather than its normal level of ~0.15 mmol/g tissue; by contrast, the levels of acetate, pyruvate, and a-ketoglutarate did not change (Buffa, Peters and Wakelin, 1951; Peters, 1957). Although this antimetabolite is itself a relatively poor inhibitor, fluoroacetate is converted to fluoroacetyl-CoA by acetyl-CoA synthetase (Reaction: FCH2COO þ CoA-SH þ ATP # Fluoroacetyl-CoA þ

AMP þ PPi) and onward to 2-fluorocitrate through the action of citrate synthase (Reaction: Fluoroacetyl-CoA þ Oxaloacetate # 2-Fluorocitrate þ Coenzyme A). The ()-erythro diastereomer of 2-fluorocitrate is a mechanismbased inhibitor that is first converted to fluoro-cis-aconitate by aconitase [EC 4.2.1.3], the [4Fe-4S]-containing enzyme (Reaction: Fluorocitrate # Fluoro-cis-aconitate þ H2O), followed by the addition of hydroxide ion and attendant loss of fluoride ion to form 4-hydroxy-trans aconitate (Lauble et. al. 1996). The latter remains bound to aconitase with extreme affinity. This behavior is also illustrated in the case of D-alanine: 2 D-alanine ligase reaction (i.e., D-Ala þ D-Ala þ MgATP # D-Ala-D-Ala þ MgADP þ Pi), which is essential for growth of certain enterococci (Walsh, 1993). Failure to synthesize this dipeptide prevents the cross-linking of the underlying peptidoglycan structure of the bacterial cell wall. The resulting aminoacyl-D-Ala-D-Ala strand is the target of vancomycin binding, such that this antibiotic arrests peptidoglycan cross-linking and blocks cell-wall synthesis. The kinetic reaction mechanism appears to be random, and for the reaction to proceed, all substrates must reside as an Enzyme$D-Ala$D-Ala$MgATP quaternary complex. Except for its activation of an a-carboxylate to form a peptide bond, the mechanism appears to be completely analogous to that catalyzed by glutamine synthetase, which forms a g-glutamyl-phosphate intermediate. There is strong evidence for the occurrence of an acyl-phosphate intermediate and subsequent attack by the amino group of a second D-Ala molecule. Indeed, a phosphinate dipeptide analogue is converted by enzymatic phosphorylation from a low-affinity inhibitor to an extremely tightly bound analogue of the ligase’s reaction

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

529

TABLE 8.10 Selected Examples of Transition-State Inhibitors Enzyme

Inhibitor

Reference

Adenosine deaminase

1,6-Dihydro-6-hydroxymethylpurine Ribonucleoside

Evans and Wolfenden (1970)

Coformycin

Sawa et al. (1967)

1,6-Dihydroinosine

Jones et al. (1989)

[1(S)-Aminoethyl][2-carboxy-2(R)- methyl1-ethyl]phosphinate

Duncan and Walsh (1988)

Alanine dehydrogenase

Oxalylethyl-NADH

Kapmeyer et al. (1983)

AMP nucleosidase

Formycin 59-phosphate

De Wolf, Fullin and Schramm (1979)

a-Amylase

D-malto-Bionolactone

Laszlo et al. (1978)

D-Alanine:D-alanine

ligase

a-N-Arabinofuranosidase

L-Arabino-1,4-lactone

Fielding et al. (1981)

Aspartate ammonia-lyase

3-Nitro-2-aminopropionate

Porter and Bright (1980)

Cytidine deaminase

Tetrahydrouridine

Cohen and Wolfenden (1971)

Phosphapyrimidine

Ashley and Bartlett (1984)

1,3-Diazepin-2-ol ribonucleoside

Marquez et al. (1980)

3,4 Dihydrouridine

Frick et al. (1989)

Dihydrofolate reductase

Methotrexate

Werkheiser (1976)

Dihydroorotase

Borocambamylethyl aspartate

Kinder et al. (1990)

Fumarate hydratase

3-Nitro-2-hydroxypropionate

Porter and Bright (1980);

S-2,3-Dicarboxyaziridine

Greenhut et al. (1985)

Methionine sulfoximine (ATP dependent)

Rowe et al. (1969)

Glutamine synthetase

Phosphinotricin (ATP-dependent)

Logusch et al. (1989)

Serine-borate complex

Tate and Meister (1978)

3b-Hydroxy-D5-steroid dehydrogenase

4-Aza-4-methyl-5-pregnane3,20-dione

Bertics, Edman and Karavolas (1984)

b-Lactamase

4-Aza-4-methyl-5-pregnane3,20-dione

Bertics, Edman and Karavolas (1984)

Ornithine transcarbamoylase

Nd-(N9-Sulfodiaminophosphinyl)-L ornithine

Langley et al. (2000)

Pepsin A

Pepstatin

Umesawa et al. (1970)

Phosphinic acid dipeptide

Bartlett and Keyer (1984)

Proline racemase

Pyrrole

Cardinale and Abeles (1968)

Pyrroline 2-carboxylates

Keenan and Alworth (1974)

Purine nucleoside phosphorylase

Immucillin H (systematic name ¼ (1S)-1(9-deazahypoxanthin-9-yl)-1,4 dideoxyl-1,4imino-D-ribitol)

Schramm (2002)

Pancreatic ribonuclease

Uridine-vanadate

Lindquist et al. (1973)

Triose-phosphate isomerase

2-Phosphoglycollate

Wolfenden (1970)

2-Phosphoglycolohydroxamate

Collins (1974)

Uracil-DNA-glycosylase

6-(p-n-Octylanilino)uracil

Focher et al. (1993)

intermediates. 1(S)-Aminoethyl-[2-carboxy-2(R)-methyl1-ethyl]-phosphinate is an ATP-dependent, slow-binding inhibitor of the D-Ala:D-Ala ligase from Salmonella typhimurium, and the enzyme-inhibitor complex (after ATP-dependent phosphorylation) has a half-life of 17 days at 37 C (Duncan and Walsh, 1988). Later solid-state NMR experiments (McDermott et al., 1990) and X-ray crystallographic work (Fan et al., 1994) confirmed that the inhibitor was phosphorylated in a manner that produces

a tightly bound transition-state analogue. Inhibitor phosphorylation generates a tight-binding inhibitor (i.e., E þ I # E$Iloose; ATP þ E$I # ATP$E$Iloose; ATP$E$Iloose / ADP$E$I-Ptight). Such behavior is analogous to the classical case of ATPdependent inhibition of glutamine synthetase (Reaction: 2 L-Glutamate þ MgATP þ NH3 # L-Glutamine þ MgADP þ Pi) by methionine sulfoximine (MSOX) and phosphinothricin.

Enzyme Kinetics

530

O

O

C O

C H2

C CH

O

CH3

C H2

C CH

P HO

C H2

OH

CH3

O

P

C O

O P

O

OH

Phosphinothricin-Phosphate

C CH

NH3

O

O H2 C

NH3

O

O

Methionine Sulfoximine CH3

O

Methionine Sulfoximine-Phosphate

NH3

O

C

O O

H2 C

HN

S

P

O

Glutamate

S

O

N

NH3

O

CH3 O

H2 C

O

O

O NH3

C

C O

Phosphinothricin O MSOX is a toxic bi-product formed from free and protein-bound methionine during the chemical bleaching of wheat flour to prepare white flour, whereas phosphinothricin is a natural product of the common soil microorganism Streptomyces hygroscopicus. The L isomer of the latter (developed by Hoechst and marketed as BASTAÔ) is the active ingredient of the herbicide glufosinate ammonium. Neocortical issues from cats treated with methionine sulfoximine can no longer maintain normal levels of glutamate, glutamine, and g-aminobutyrate when incubated in vitro with perfusion fluids (Peters and Tower, 1959). Later work by Ronzio and Meister (1968) demonstrated that incubation of ovine brain glutamine synthetase with MSOX and MgATP2 results in total loss of catalytic activity. The octameric enzyme stoichiometrically accumulates 8 mol methionine sulfoximine-phosphate and 8 mol ADP. These products are bound so tightly that the Kd for MSOX-P release approaches 1012 to 1013 M. In fact, MSOX-P and ADP are only released upon acid- or heat-induced denaturation. The convulsive conditions evoked by methionine sulfoximine were observed in individuals consuming toxic MSOX levels (~2 mM) in wheat flour that has been excessively bleached (Tower, 1969).

:NH3

P

O OH

NH3

O Glutamine Synthetase Transition-State

The tight binding of MSOX-P, as well as the analogous phosphorylated adduct of phosphinothricin (Berlicki and Kafarski, 2006; Gill and Eisenberg, 2001), may be rationalized by considering its resemblance to the likely catalytic transition state, wherein ammonia attacks a covalent g-glutamyl-phosphate intermediate (Meister, 1964; Todhunter and Purich, 1975). At times, this type of high-affinity inhibition strains the definition of reversible inhibitor. Avid non-covalent binding of the phosphorylated inhibitor (Kd z 1013 to 1015 M), which rivals that of avidin-biotin interactions, reminds us that covalent bonds between enzyme and inhibitor are not always needed to achieve extraordinary affinity. Work on 5-enolpyruvylshikimate-3-phosphate synthase (EPSPS) illustrates that inhibitor design is not an entirely predictable enterprise. This enzyme, which catalyzes the penultimate step of the shikimate biosynthesis pathway, is the plant target for the broad-spectrum herbicide

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

531

8.7. TIGHT-BINDING REVERSIBLE INHIBITORS A long-recognized fact is that not all inhibitors rapidly attain thermodynamic equilibrium between the free and enzyme-bound forms. Indeed, Peters (1955) reported that a period of approximately 15 minutes is required to achieve the full inhibitory effect of 2-fluorocitrate on aconitase (Reaction: Citrate # cis-Aconitate þ H2O). Moreover, although the inhibitor binds in place of citrate within the enzyme’s active site, the fully developed inhibition is noncompetitive, a feature that is consistent with a depleting effect on the total concentration of active enzyme. For some high-affinity multi-substrate geometric inhibitors, transition-state inhibitors, and other high-affinity inhibitors, the Kd value is often at or near the total enzyme concentration. In this situation, the concentration of enzyme-bound inhibitor represents a significant fraction of the total inhibitor concentration (Morrison and Walsh, 1988; Williams and Morrison, 1979). Easson and Stedman (1936) were probably the first to report this phenomenon when they described the inhibitory action of physostigmine on acetylcholine esterase (Reaction: Acetyl-Choline þ H2O # Choline þ Acetate).

8.7.1. Reversible Tight-Binding Inhibitors Undergo Slow Inhibitor-Induced Enzyme Conformational Changes

FIGURE 8.33 Crystallography of the phosphonate analogues bound to Escherichia coli EPSPS. Displayed are the electron densities derived from Fourier syntheses after the last refinement cycle, contoured for the ˚ resolution (left) and the (R)-phosphonate at (S)-phosphonate at 1.5-A ˚ resolution (right). From Priestman et al. (2005) with permission. 1.9-A

glyphosate. With the idea that the corresponding bacterial synthase may be a druggable target for novel antibiotic development, Priestman et al. (2005) analyzed the E. coli EPSPS kinetically and determined crystal structures for its complexes with (R)- and (S)-phosphonate analogues of the reaction’s tetrahedral intermediate (Fig. 8.33). Both diastereomers are competitive inhibitors with respect to the substrates shikimate-3-P and phosphoenolpyruvate. The (S)-form (Ki ¼ 750 nM), which corresponds structurally to the genuine tetrahedral intermediate, was found to bind about 50 more weakly than the (R)-analogue (Ki ¼ 16 nM). The latter induces conformational changes of Arg124 and Glu-341, the strictly conserved residues within the enzyme’s active site, giving rise to substantial alterations in the N-terminal domain. By contrast, the enzyme’s structure is unchanged upon binding of the (S)-phosphonate.

When the substrate has a negligible effect on the formation of Enzyme$Inhibitor complex, the net result is depletion (i.e., the removal of enzyme by the inhibitor from the reaction). The observed kinetic pattern is identical to the simple non competitive inhibition case; the substrate and the inhibitor do not affect each other’s binding, because only Vm is changed due to reduced enzyme concentration, while Km remains unaltered. The rate equation to be used for analyzing enzyme depletion is that for simple non competitive inhibition. An example of this enzyme depletion phenomenon is the ribonuclease inhibitor isolated from human placenta by Blackburn, Wilson and Moore (1977). This protein forms a one-to-one complex (Ki ¼ 3  1010 M) with bovine pancreatic RNase A, acting as a noncompetitive inhibitor for this enzyme. The interested reader should examine the related report by Newton et al. (1998), who investigated the mode of action of onconase, a cytotoxic ribonuclease with potent antitumor properties. One way to analyze tight-binding inhibitor action is to determine how v/[ET] depends on the total concentration of the inhibitor (Fig. 8.34). This approach also requires computer-assisted data fitting to extract the inhibition constant. It also suffers from the fact that small errors in velocity determinations can greatly affect the line shape at high total inhibitor concentrations.

Enzyme Kinetics

532

[ET]

[InhibitorT] FIGURE 8.34 Plot of initial velocity v versus the total concentration of inhibitor. See text for details.

Recognizing the restrictions imposed when dealing with tight-binding inhibitors, biochemists struggled to develop ways to analyze tight-binding inhibitor action, and four of the earliest equations stand out. Equation 8.45 was offered by Easson and Stedman (1936) and Strauss and Goldstein (1943) for enzyme inhibition to be a tight-binding noncompetitive inhibitor.  i 8.45 It ¼ It i þ K i  1i In this and the following equations, the degree of inhibition i ¼ (1  vi/v0), where vi is the inhibited velocity and v0 is the enzyme’s velocity in the absence of the inhibitor. Equations 8.46 and 8.47 were developed by Krupka and Laidler (1959) and Goldstein (1944), respectively, for a tight binding competitive inhibitor. In the latter equation, the parameter a in this equation equals vi/Vm.    v0 v0 At 8.46 It þ  1 Et þ K i 1 1þ vi vi Ka 

Henderson (1972) rearranged the general rate law for tight-binding inhibition, thereby obtaining Eqn. 8.49a for very high-affinity competitive inhibition, Eqn. 8.49b for very high-affinity uncompetitive inhibition, 8.49c for very high-affinity noncompetitive inhibition, and Eqn. 8.49d for very high-affinity mixed-type noncompetitive inhibition. !  v0 1 At þ Ka v0 It  1  ¼ Et þ Ki vi Ka vi

8.49a

!# " ,  v0 1 K a A t v0 It  1  ¼ Et þ ðAt þ Ka Þ þ vi Kis Ka vi 8.49b In these equations, v0 is the steady-state velocity in the absence of the inhibitor and equals N/D. Each equation predicts that a plot of [It]/(1  v/v0) versus v0/v will be linear (Fig. 8.35). The slope of the plotted line equals the apparent Ki and the intercept provides an independent check on any experimentally determined value total enzyme Et. The slope of the line is the apparent dissociation constant for the inhibitor. Secondary plots (from repeating the inhibition experiment at different substrate concentrations) will yield the Ki value. The vertical intercept equals Et. Hence,

8.47

In these equations, At is the total substrate concentration. The fourth (Eqn. 8.48), developed by Morrison (1969), represented a major advance in the analysis of tight-binding, dead-end inhibitors. Morrison treated the initial steady-state velocity of an enzyme reaction to be represented by the general expression: v0 ¼ (NEt/D), where the numerator term N contains the rate constants (which determine the maximal velocity Vm) as well as substrate concentrations, Et is the total enzyme concentration, and the denominator term D corresponds to the sum of terms representing the distribution of the enzyme in a particular form, which depends on the mechanism of inhibition. At is the total substrate concentration, and It is the total inhibitor concentration,

[I] / [1 – ( / 0)]

  At  aEt 1a 1 Ka a    Ka þ Et 1  a 1 þ At  aEt

It ¼ K i

such that It ¼ Ifree þ S(EIi). For the general cases where an inhibitor binds to several different enzyme forms to produce dead-end complexes with different inhibition constants Ki, Morrison (1969) obtained: vi ¼ NEt{D þ It SNi/Ki}, where Ni is each denominator term representing an inhibitorbinding enzyme form. " , # ! n X Ni It  Et 2 1 vi þ Nvi þ K D i¼1 i , ! n X Ni 2 ¼ 0 8.48  N Et D K i¼1 i

/

0

FIGURE 8.35 Henderson plot for the analysis of tight-binding inhibition. See text for details.

Chapter j 8 Kinetic Behavior of Enzyme Inhibitors

533

repeating the experiment at a different concentration of enzyme will produce a parallel line. If the Henderson equation or similar types are not employed, keep in mind that the inhibitor concentration [I] is the free inhibitor concentration. Determination of Ki may not be feasible if the rate assay is insensitive and requires an enzyme concentration much greater than Ki. Williams and Morrison (1979) derived the general rate equation for tight-binding inhibition as: # ! " , n X Ni It  Et 2 Nv þ 1 v þ K D i¼1 i , ! n X Ni 2  N Et D 8.50 N2 ¼ 0 K i i¼1 where the N term contains the rate constants determining the maximal velocity along with the substrate concentration(s): Ni denotes the denominator term(s) that represent enzyme form(s) reacting with the enzyme, and Ki terms are the respective dissociation constants for each enzyme$inhibitor complex. Other items in Eqn. 8.50 are [Et], concentration of total enzyme; [IT], concentration of total inhibitor; and D, representing the denominator of the rate equation in the absence of inhibitor. Note that: (a) when [ITotal] ¼ 0, v ¼ N[Et]/D; whereas (b) if [ET] > s, the transient term drops out, yielding the following oscillating function: D½XðtÞ ¼

D½XðtÞ D½XðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðeif Þ 1 þ iut 1 þ u2 t 2

10.45

where tanf ¼ us. In fact, although both oscillate with identical frequencies, the former lags behind the latter, yielding two sinusoidal waves displaced from each other by the phase angle f. A similar treatment explains the effects of periodic excitation of nuclei on relaxation processes observed in nuclear magnetic resonance experiments.

10.3.3. The Temperature-Jump Method is Probably the Most Versatile Chemical Relaxation Technique

where V0 is the initial voltage across the capacitor, R is the resistance of the sample solution between the discharge electrodes, t is the time after discharge, and C is the discharge capacitance (Hammes, 1974). If all other impedance losses are small compared to that occurring in the solution, the rate of temperature rise: dT i2 R ¼ dt 4:18CP rV

10.46

where the reaction must have a non-zero enthalpy change (i.e., DHo s 0). In such cases, the constant-pressure equilibrium constant KP will be perturbed by a sudden increase in temperature, and the change in the equilibrium position will depend on the magnitude of DHo/RT. (In principle, reactions exhibiting little or no temperature

10.48

where CP and r are the specific heat and density of the solution, and V is the volume of solution between the discharge electrodes. Upon combination of Eqns. 10.46 and 10.47 and integration from t ¼ 0 to t ¼ t at constant CP, r, V, and R, one obtains: dTðtÞ ¼

The equilibrium position of most biochemical reactions is temperature-dependent, such that: v ln KP DH o ¼  vT RT

dependence can often be linked to the formation of one reactant or product to another highly temperature-dependent process. For example, a reaction that produces or consumes a proton can be carried out in the presence of a buffer having a strongly temperature-dependent pKa value.) As noted earlier, Joule heating is accomplished by discharging a capacitor through a reacting system (at equilibrium) in an aqueous electrolyte solution within the reaction cuvette. This compartment is much like a cuvette used in absorption or fluorescence spectrophotometry, insofar as it contains optically transparent windows separated by a calibrated distance, typically 1 cm. However, as shown below, the temperature-jump cell must also contain the high-voltage electrode and the grounding electrode, suitably positioned to maximize the heating within a short period. Moreover, the electrodes must be free of any sharp edges that might permit unwanted sparking of the applied high voltage. Upon discharge of a capacitor, the current delivered to a reaction sample and supporting electrolyte within a temperature-jump apparatus can be expressed as:  V0 expðt=RCÞ 10.47 i ¼  R

CðV0 Þ2 R ½1  expð2t=RC 8:36CP rV

¼ dTN ½1  expð2t=RC

10.49

For experiments with quantities of precious biochemical components, the reaction volume can be kept to as little as 0.1–0.2 mL, and voltages of about 10 kV are required to obtain a 10 C temperature change in about 10 ms. Under such conditions, a 5–10% displacement of a 1 mM reactant, then the corresponding 50–100 mM change in reactant concentration, can easily be detected by absorption of fluorescence spectroscopy. If we consider an isomerization reaction with k1 and k1 each equal to 500 s1, then s will be about 1 ms, a relaxation period that would be well separated

Chapter j 10 Probing Fast Enzyme Processes

from the 3–10 ms period needed for Joule heating. It is usually feasible to continue the collection of rate data until the onset of thermal cooling to ambient temperature, typically 0.5–1 seconds. The schematic shown in Fig. 10.17 illustrates the general components that make up the temperature-jump apparatus. The temperature change is experienced by the reaction system as a change in rate constants, such that the system undergoes chemical relaxation. A micro-scale temperaturejump apparatus is shown in Fig. 10.18. Changes in sample absorbance or fluorescence are recorded by means of a rapid data acquisition device, and typical data are shown in Fig. 10.19. These can be fitted to a first-order decay curve and analyzed to obtain values for the rate constants. Other optical techniques, such as polarimetry and light scattering, have been employed. The temperature-jump technique can also be combined with rapid-mixing devices to increase the types of chemical processes that can be observed. Temperature-jump devices typically provide satisfactory signal-to-noise characteristics by employing bright light sources such as xenon or mercury vapor sources as well as quartz halide lamps. The choice of photomultiplier depends on the method of detection, with fluorescence measurements requiring a higher quality PM

CUVETTE Monochromator Lamp Power Supply

Spark Gap

hn

667

crosssection

A B E

A D

C

A

top view D

D E B

FIGURE 10.18 Micro-scale temperature-jump cuvette for fluorescence or light-scattering measurements. The body (A) is milled from Plexigas, Kel-F, or a similar material, with stainless steel inserts (B and C) acting as the ground electrode and high-voltage electrode, respectively. The light beam is focused onto the 0.2-mL sample by means of a conical quartz lens (D), and the mirrors (E) are positioned to double the intensity of the excitation (incident) light beam and the reflected (emitted) light beam. Drawing based on Hammes (1974).

PM Tube

H.V. capacitor

First-Order Reaction Kinetics

FIGURE 10.17 Block diagram of a temperature-jump apparatus. A spark-gap triggers the discharge of electrical energy stored within a capacitor, and discharge imparts a substantial electric field on the sample. As the principal charge carriers, small ions present in the reaction solution begin to migrate rapidly toward an oppositely charged pole; the acceleration of these ions must necessarily be balanced by frictional forces of the solvent, leading to the consequential release of energy as heat. Because the ions are present throughout the sample, this sudden heating occurs relatively uniformly throughout the sample. The rate at which the capacitor discharges is often the major factor limiting the time required for Joule heating. To assure rapid and uniform heating, large chemically inert, gold-plated electrodes are placed in the sample compartment and make direct and intimate contact with the reaction solution.

Signal Intensity

COMPUTER Highvoltage Source ground

Stable Baseline

Time (µsec) FIGURE 10.19 Typical tracing of a kinetic process observed in a temperature-jump experiment. Note that the amplitude of the noise envelope around each tracing is similar, a result indicating that the observed signal-to-noise ratio is not affected by the concentration of reactants present.

Enzyme Kinetics

668

tube than necessary for detecting changes in UV-visible absorbance. It is also worth noting that temperature-jump instruments can be advantageously fitted with a rapid-mixing device (Fig. 10.20), such that one can conduct relaxation kinetic studies after rapid mixing. One advantage of the stopped-flow temperature-jump technique is that relaxation kinetics can be applied at steady state, and not only for reacting systems near or at equilibrium. Under many conditions a chemical reaction can reach steady state in a millisecond and can persist in this time-invariant state for many seconds, thereby providing a wide window of opportunity for extending the utility of relaxation measurements. An extraordinary range of chemical and biochemical processes has been extensively analyzed by using temperature-jump and stopped-flow temperature-jump methods (Table 10.12). In most cases, the reaction is attended by a detectable change in the electronic spectrum using photomultiplier tubes or diode array detectors sensitive to UV or visible light. Although many reactions produce no such spectral change, the production or consumption of a proton

Microswitch Time Delay Stopping Syringe

Computer

Cuvette Lens

Monochromator

Lamp Power Supply

hν

Driving Syringe

A

B

Photomultiplier

Switching Circuit

Spark Gap

Spark Gap Trigger H.V. resistor

Pneumatic Driver

H.V. capacitor

High-voltage Power supply

ground

FIGURE 10.20 Block diagram of a stopped-flow temperature-jump apparatus. There are numerous applications for such combined instrumentation, especially for the investigation of very short-lived intermediates that would not otherwise be detected or time-resolved by separate application of stopped-flow or temperature-jump methods. When relaxation techniques are applied directly after rapid mixing of reactants, kinetic features of the short-lived intermediates and their elementary reactions are frequently far more evident. For additional details on instrument design and use, please consult Hammes (1974).

TABLE 10.12 Recommended Reports and Reviews for Selected Processes Investigated by the Temperature-Jump Kinetic Techniques Process/Reaction

Recommended Reviews

General Theory and Practice

Eigen (1964); Eigen et al. (1964); Hammes (1974); Eigen and De Maeyer (1974); Ilgenfritz (1977).

Acid/Base Reactions

Eigen (1964); Eigen et al. (1964); Rose and Stuehr (1971); Inskeep et al. (1968); Bernasconi (1976); Schuster, Wolschann and Tortschanoff (1977); Crooks (1983).

Proton Transfer

Schuster, Wolschann and Tortschanoff (1977).

Contractile Systems

Davis (1998); Davis and Harrington (1993); Davis (1998); Tinoco, Li and Bustamante (2006).

Electron Transfer

Diebler, Eigen and Matthies (1962); Hurwitz and Kustin (1964); Brandt et al. (1966); Medda et al. (1999); Narasimhulu (2007)

Enzyme Catalysis

Eigen and Hammes (1963); del Rosario and Hammes (1971); Hammes (1968a,b); Holler, Rupley and Hess (1970); Hammes and Schimmel (1970); Hammes and Wu (1971); Kirschner et al. (1971); Johnson (1992); Hammes (2002).

Enzyme Regulation

Hammes and Wu (1971); Kirschner et al. (1971); Kirschner (1971); Thusius (1977); Hammes (2002).

Hydration/Aquation

Eigen, Kustin and Munson (1962); Kustin and Lieberman (1964); Ahrens et al. (1964); Russo, Hura and Copley (2007).

Intramolecular Proton Transfer

Eigen (1964); Eigen et al. (1964); Bernasconi (1976).

Metal Ion Complexation

Hammes and Steinfeld (1970); Eigen and Wilkins (1965); Pasternack et al. (1972); Davies, Kustin and Pasternack (1969); Makinen, Pearlmutter and Stuehr (1969); Pasternack and Sigel (1970); Hague (1977).

Nucleic Acid Structure and Interactions

Craig, Crothers and Doty (1971); Pohl (1977); Po¨rschke (1977); Rigler, Ehrenberg and Wintermeyer (1977); Jovin and Striker (1977); Jose and Po¨rschke (2004); Orden and Jung (2008); Tinoco, Li and Bustamante (2006); Ma et al. (2007).

Protein Folding

Pohl (1977); Gruebele, Sabelko and Ballew (1998); Callender et al. (1998); Dyer, Gai and Woodruff (1998); Mizutani, Yamamoto and Kitagawa (2002); Hofrichter (2001); Callender and Dyer (2006); Mun˜oz (2007); Baldwin (2008); Hart et al. (2009).

Transmembrane Ion Transport

Grell and Oberba¨umer (1977); Krishnamoorthy (1986).

Chapter j 10 Probing Fast Enzyme Processes

during a reaction can be detected by using an indicator dye that produces a spectral change during its protonation or deprotonation. A dye is typically chosen with an appropriate pKa value and UV/visible spectrum. It is also necessary to work in a minimally buffered solution to ensure that the change in proton concentration is sensed by the indicator dye. Fluorescence may also be employed to follow relaxation kinetics of a chemical process. As mentioned in Chapter 4, many naturally occurring biochemical substances are highly fluorescent substances. For example, NADH has a strong fluorescence that is lost upon conversion to NADþ. Moreover, fluorescent analogues of otherwise non-fluorescent biochemicals are also widely available, making it possible to investigate those reactions that do not produce fluorescence change. Another common practice is to synthesize fluorescently tagged reactants, thereby optimizing their suitability for measurements of fluorescence enhancement, fluorescence quenching, fluorescence lifetimes, and/or fluorescence anisotropy/polarization. The key is to identify conditions where a change in some fluorescent property can be demonstrated to be directly proportional to a change in reactant or product concentration. The 90 angle between light source and detector in fluorescence spectrometers can also be used to study changes in light scattering, particularly for protein oligomerization or polymerization reactions. Finally, the laser-induced temperature jump technique is by far the most versatile and simplest of fast-initiation approaches for temperature-jump instruments (Ballew, Sabelko and Gruebele, 1996; Callender and Dyer, 2006). The dead-time of classical temperature-jump technique is limited to the ms time-scale. A laser-induced temperaturejump system, on the other hand, is not limited by the kinetics of capacitor discharge and the heat rise generated by the ensuing sudden movement of ions. The heating time is instead determined by the power and the emissive time of the pulsed IR light source as well as the lifetime (~1012 sec) of water’s vibrational/rotational excited state. The theoretical sample heating time is roughly 1012 sec. After the laser-induced jump in temperature, the thermal energy within the heated volume quickly dissipates, principally by diffusion through the rest of the sample volume as well as heat transfer to the still-cold surfaces of the cuvette. The kinetics of these events imposes an upper limit on the measurement time-scale. Rapid heating also has the effect of expanding the sample volume so suddenly that shock wave is generated, with the latter passing through the sample volume at the speed of sound (Dyer, Gai and Woodruff, 1998). A physically more realistic limit is therefore closer to 1010 to 109 sec. There are a number of laser optics configurations for inducing the temperature jump, some using back-surface reflection to make two passes of the light beam to maximize the heating effect,

669

whereas others use two lasers. Ballew, Sabelko and Gruebele (1996), for example, used two pulsed infrared lasers (wavelength ¼ 1.54 mm; pulse-time ¼ 1 ns) to illuminate two opposing sides of a T-jump cuvette, thereby developing a heating rate of ~5  109 K$s1. As configured, the dead-time for this configuration is less than 20 nanoseconds!

10.4. STOPPED-FLOW AND TEMPERATUREJUMP TECHNIQUES PROVIDE POWERFUL INSIGHTS INTO ENZYME CATALYSIS The systematic investigation of biochemical reaction rates has provided powerful insights regarding enzyme catalysis and metabolic regulation. Although a detailed account of flow and temperature-jump techniques is beyond the scope of this reference, it is still useful to consider a few additional illustrative cases.

10.4.1. Ribonuclease This enzyme (EC 3.1.27.5) facilitates the general acid/base catalyzed endonucleolytic cleavage to form 39-phosphomononucleotides ending in Cp or Up (Shen and Schlessinger, 1982; Williams, 1993). During catalysis, nucleoside-29,39-cyclic phosphodiesters accumulate, indicating the two-step reaction: X

X

X U

O

O O O P

O

O OH P O Y

G

O

HO

P

O OH O P O

G

OH

O P O OH

H2O

O

O

Y

O

O

O

O O

U

O

O

OH

O

O

U

O

HO

O

G

O OH O P O Y

In the first phase, facilitated attack of the 29-hydroxyl of RNA leads to the formation of the cyclic intermediate. During the second step, cyclic phosphodiester hydrolysis yields the 39-monoester. Classical studies by Nobelists Stanford Moore and William Stein established that His-12 and His-119 play critical catalytic roles, the former serving in the first step as a general base, and the latter acting as a general acid that protonates the leaving group. The second step is essentially the reverse of the first, with the enzyme returning to its original protonation state upon completing the reaction cycle.

Enzyme Kinetics

670

Note the formation of the 29,39-cyclic phosphodiester intermediate and subsequent attack of water to yield the 29-phosphomonoester. Although shown with cytosine, the same reaction mechanism applies to sites containing uracil. It is believed that the water molecule must await pseudorotation (see Section 9.1.2a) of the cyclic phosphodiester prior to in-line nucleophilic attack – not indicated here. Because the RNase reaction is unattended by a UV/visible spectral change, French and Hammes (1965) used a pH indicator dye to measure the proton released in their temperature-jump experiments. They found that the reaction kinetics were most consistent with a simple twostep isomerization/deprotonation reaction process. In this scheme, the second deprotonation step is rather fast, and the reaction kinetics are described by the equation: 1 kþ1 þ k1 ¼  KA s1 1þ þ ½H 

10.50

For RNase, kþ1 ¼ 780 s1, k1 ¼ 2470 s1, and KA ¼ 6.1 M. The most likely explanation for this two-step 10 sequence is that there is an opening and closing of the substrate groove on RNase, such that the pKA of 6.1 can be assigned to the active-site histidine. Interestingly, this process is absent from subtilisin-treated RNase or RNase that was carboxymethylated at His-119. It was also absent with unmodified RNase to which cytidine-29-P, cytidine39-P, pyrophosphate, or cytidine are bound. The specificity of these effects suggested to French and Hammes that their assignment of the pKA of 6.1 to His-119 was most likely correct. A large D2O solvent isotope effect was taken as evidence for the formation of a specific active-site

hydrogen bond (most likely one involving a carboxylate and protonated His-119) is involved in the rate-limiting step(s).

E’H

Transfer of amino groups between oxo- (or keto-) acid metabolites is one of the most fundamental reactions in intermediary metabolism. Figure 10.21 illustrates the chemical transformations corresponding to the elementary reactions in the mechanism catalyzed by alanine: oxaloacetate transaminase. Temperature-jump kinetics and spectroscopic data have been gainfully employed to determine each rate constant for the elementary reactions in the mechanism (Scheme 10.10) for mitochondrial aspartate:a-ketoglutarate transaminase, a homodimer of 45 kDa subunits (Hammes, 2002). The enzyme catalyzes amino group transfer from aspartate (Asp) to a-ketoglutarate (a-KG) to produce oxaloacetate (OAA) and glutamate (Glu).

Asp E εLys-PLP

E Asp-PLP Aldimine

E Aldimine

Asp-PLP

Asp-PLP E Quinoid

E Quinoid

Asp-PLP

OAA-PMP E Ketimine

E Ketimine

OAA-PMP

OAA E PMP

E PMP

OAA

E PMP

E + H+

10.4.2. Aminotransferase Catalysis: A Case Study in Temperature-Jump Kinetics

E εLys-PLP

E Ketimine

kA

In a second series of experiments aimed at characterizing interactions during and after substrate binding, Hammes and Walz (1969) established that the kinetics fit a substrate-induced conformational change (Scheme 10.9).

Asp E εLys-PLP

E Quinoid

EH

Scheme 10.9

Asp + E εLys-PLP

E Aldimine

k+1 k-1

Glu

E εLys-PLP + Glu

Glu-PLP

E εLys-PLP

Glu-PLP

E Aldimine

αKG-PMP

E Quinoid

αKG

E Ketimine

E PMP + OAA E PMP + αKG

Glu

Glu-PLP

Glu-PLP

αKG -PMP αKG

E PMP

2st HALF REACTION

1st HALF REACTION

Scheme 10.10

Chapter j 10 Probing Fast Enzyme Processes H2 H C C

OOC

COO

B:

NH3

[1]

671

H2 C OOC

C

COO

OOC

[2]

[3]

NH B

NH

C

OOC

COO

C

slow step

NH

H2 C

H2 C

H

NH

H OH

OH

OH O3PO

O3PO

O3PO

OH

COO

O3PO N H

CH3

N H

N H

CH3

H2 C C

OOC

COO

H H3C C

[8]

[4] COO

O

H3N

C

NH3

B

H H3 C

H H3C

COO

C

H 3C C

COO

[7]

NH

[6]

NH

H

OH

COO O3PO

[5]

NH

B: OH O3PO

OH O3PO

N H

CH3

CH3

N H

CH3

N H

CH3

OH O3PO

N H OOC

CH3

CH3 C

N H

CH3

O

FIGURE 10.21 Multi-step aminotransferase mechanism. See discussion in text as well as Scheme 10.11 for additional details.

The essential coenzyme pyridoxal-5 phosphate (PLP) alternates between its pyridoxal and pyridoxamine (PMP) forms in a Ping Pong mechanism. As mentioned in Chapter 4, Ping Pong mechanisms afford the experimenter the opportunity to study a half-reaction in the absence of the other substrate, resulting in the determination of every rate constant in the following scheme. Chemically, the second half-reaction traces the reverse of the first half-reaction to regenerate the PLP form along with the amino acid product. Without listing here the magnitude of each rate constant and analyzing the chemical impact of these constants, it is sufficient to say that transient kinetics and spectroscopic measurements demonstrated that catalysis proceeds via formation of an aldimine between the amino group of the amino acid and the aldehyde group of the coenzyme. The aldimine converts to a quinoid structure in the pyridine ring that subsequently decomposes into a transient ketimine that gives rise to the ketoacid and pyridoxamine-5-P. The coenzyme actually resides at the bottom of the active-site pocket as an imine linked to an active-site lysine. Binding of the amino acid substrate leads to a conformational change that is attended by transaldimination to yield the amino acid coenzyme aldimine intermediate that rearranges and subsequently decomposes into PMP and keto acid. Temperature-jump kinetic studies using the substrate

analogue b-erythro-aspartic acid yielded the rate constants for the formation and transformation of all intermediates in the transamination reaction (Hammes and Haslam, 1969). Structural studies have confirmed the mechanistic features noted above and identified key active-site residues (Christen, 1999; Kirsch et al., 1984). The active site is located at a subunit-subunit interface, such that they show C2 rotational symmetry relative to each other. The coenzyme lies at the bottom of the active site and is tethered to the enzyme by means of an imine with lysine258. Upon substrate binding, a conformational change allows the enzyme to close around the substrate, a step that may facilitate nucleophilic attack by dehydrating the active site.

10.4.3. Dihydrofolate Reductase: Another Outstanding Example This enzyme (abbreviated DHFR), which catalyzes the reaction of NADPH and 7,8-dihydrofolate (H2F) to produce NADPþ and 5,6,7,8-tetrahydrofolate (H4F), provides essential methyl-donor cofactors in the formation of several amino acids, purines, and thymidylate. The kinetics and mechanism of E. coli DHFR has been examined extensively through the application of fast reaction methods (Fierke, Johnson and Benkovic, 1987).

Enzyme Kinetics

672

E

5 M-1s-1 H2F

1.7 s-1

40 M-1s-1 20 s-1

E

950 s-1

E NADPH H2F

40 M-1s-1

20 M-1s-1 3.5 s-1

0.6 s-1

5 M-1s-1 200 s-1 13 M-1s-1 300 s-1

40 s-1

E NADPH

2.4 s-1 + + NADP E NADP E -1 -1 H4F 25 M s

2 M-1s-1 12.5 s-1

85 s-1 NADPH E E H4F 8 M-1s-1 H4F

1.4 s-1 25 M-1s-1

E

Scheme 10.11

Scheme 10.11 summarizes some of the rate constants for reactions of various enzyme species with reaction substrates and products, measured in the presence of excess ligand, as monitored by quenching of intrinsic fluorescence or a change in the efficiency of fluorescence resonance energy transfer between NADH and enzyme. These and other isomerization reactions of dihydrofolate reductase are discussed in detail by Fierke and Hammes (1995). The energetics of the DHFR catalytic reaction cycle is shown in Fig. 10.22.

TS1

15

TS2

TS3

∆G (kcal/mol) 0

E NADPH + H2F

NADPH EH F 2

+ H+

H+ NADPH EH F 2

ENADP HF

+

4

TS6

TS5

TS4

EH F 4

+

+ NADP + NADPH

NADPH

EH F 4

E NADPH + H4F

Reaction Progress FIGURE 10.22 Energetics of the Escherichia coli dihydrofolate reductase. The Gibbs free energy change is plotted on the vertical axis, expressed in units of kcal/mol. Calculated for 1.0 mM NADPH, 1.5 mM NADPþ, 0.3 mM dihydrofolate, 13 mM tetrahydrofolate, in 0.1 M NaCl (pH 7) at 298 K. Reprinted from Fierke, Johnson and Benkovic (1987) with permission of the American Chemical Society.

10.5. OTHER RELAXATION TECHNIQUES Although the temperature-jump technique is extraordinarily versatile, especially when combined with rapid-mixing methods, other techniques have provided additional ways to probe fast rate processes in complex multi-step biochemical mechanisms.

10.5.1. Pressure-Jump Methods Increase the Versatility of Relaxation Studies In this rapid kinetic technique for detecting fast-relaxation processes, a sudden change in externally applied pressure results in a change in the equilibrium constant for a particular system. The equilibrium position of many biochemical reactions is pressure-dependent, such that v ln KP DV o ¼  vP RT

10.51

where DV, the change in volume in the reaction, must be non-zero. Volume changes in the order of 20 cm3/mol are known to occur during charge neutralization, because water is packed more closely around two ions Hþ and A than around HA. Other reactions (such as HBþ # Hþ þ B) typically have values around 1 cm3/mol. Release of water from hydrophobic interactions is also attended by changes in DV. Because protein-ligand and protein-protein interactions involve ionic and hydrophobic interactions, pressure jump is well suited to their characterization. Protein folding equilibria are especially sensitive to changes in pressure, and not surprisingly, the packing of phospholipids in membrane bilayers results in significant volume changes. In any event, the relatively small DV values for most chemical reactions necessitate use of moderately high pressures (10–150 atm). A rule of thumb is that, under an applied pressure of 150 atm, the equilibrium constant for a reaction having a DV of 25 cm3/mol will change by around 10%. In a pressure-jump experiment, the reacting chemical system is placed in a device fitted with: (a) an observation port for UV/visible or fluorescence spectral measurements; (b) a housing for mounting a metal foil diaphragm; (c) a hydraulic pump for reaching desired pressures; (d) an accurate pressure gauge; and (e) an electronic triggering mechanism for simultaneously rupturing the diaphragm and initiating data acquisition (Fig. 10.23). Upon sudden depressurization, the pressure falls almost instantaneously to atmospheric pressure, as shock wave

Chapter j 10 Probing Fast Enzyme Processes

pressure sensor

673

sapphire windows

outlet valve h

50 L sample volume piezo piston

approach of the system to the new equilibrium position. The use of shock tubes also facilitates application of a periodic perturbation as a sinusoidal forcing function. This condition can result in a steady-state displacement of reactants from thermodynamic equilibrium, and this process may be analyzed by forcing function treatments (Ilgenfritz, 1977). Table 10.13 lists some of the rate processes investigated by means of pressure-jump relaxation methods.

10.5.2. Concentration Analysis (CCA)

FIGURE 10.23 Schematic drawing of a micro-volume pressure-jump apparatus. The 50-mL reaction volume requires 80 mL of sample for convenience of operation. Pressure is applied by means of a piezoelectric crystal stack separated from the sample by a thin membrane. A positive pressure jump of up to 100 bar can be achieved within 10 milliseconds. From Clegg and Maxwell (1976).

moves through the sample at the speed of sound. In water, the speed of sound is approximately 1.5  106 cm/ s, indicating that > k2b, or when the ratio of oxidized P1 to oxidized P2 is very high, predominant formation of reduced P1 will occur. Moreover, under typical laser flash conditions, the concentration of FlavinH* generated by the first reaction is less than 1 mM, and at protein concentrations above 10 mM, pseudo-first-order conditions will apply for the last three reactions in Scheme 10.12. An added complication is that the flavin quinone can undergo disproportionation, and transfer to P1 must be fast enough to avoid this complication. In any case, the reader will appreciate the value of laser flash photolysis as a technique for producing FlavinH*. Photolabile derivatives of ATP, GTP, 39,59-cyclic AMP, 39,59-cyclic GMP, and other nucleotides as well as peptides and neurotransmitters are now available for kinetic investigation of biochemical processes that utilize these substances or are allosterically activated/inhibited by these metabolites (Adams and Tsien, 1993; Corrie and Trentham, 1993; Givens et al., 1997; Kaplan, Forbush and Hoffman, 1978; Marriott, 1998; Pelliccioli and Wirz, 2002). Adenosine-59-triphospho1-(2-nitrophenyl)-ethanol, known widely as ‘‘caged ATP,’’ is

Enzyme Kinetics

676

an ATP derivative that is incapable of participating in ATPdependent hydrolase and phosphotransferase reactions (Kaplan, Forbush and Hoffmann, 1978). The acetophenone group can be removed photochemically by excitation with UV light (l ¼ 360 nm), promptly liberating ATP for kinetic experiments on ATP utilization that have 1–2 msec timescales. Noting that one einstein equals an Avogadro’s number of photons, a flash lamp producing 1016 quanta/ flash would produce ~15  109 einstein. If a given photoactivation reaction has a quantum yield of 0.1, then one flash would produce ~1.5 nanomol of uncaged metabolite. NH2 O

O

CH3 CH

P O O

O

P O

N

O P O

O

NO2

N

O

O OH

N N

OH

Photo-caged ATP (Adenosine-5´-triphospho1-(2-nitrophenyl)-ethanol)

OCH3

OCH3 O O

OCH3

O

S

O

hν +

H2O

OH

O

O

S

O

+ H+

O

OCH3

Photolysis of Photocaged Proton (2-Methoxy-5-Nitrophenyl Sulfate)

N,N,N9,N9-tetrakis[(oxycarbonyl)-methyl]-1,2-ethanediamine forms high-affinity complexes with calcium and magnesium (Kd ¼ 5 nM and 2.5 mM, respectively), and the metal ion is promptly liberated upon photolysis with 347nm light. Although the release of the free substrate from its photocaged precursor is generally fairly fast, there is a mistaken impression that photochemical reactions are inevitably extremely rapid and limited only by photon absorption (the latter occurring on the femtosecond timescale). In practice, the photochemical rate is almost always limited by the rate of dark reactions (i.e., chemical bond rearrangements occurring after photon absorption). These rearrangements are typically much slower, such that release of the substrate from its photo-cage typically occurs in the microsecond to submillisecond range. Walker et al. (1988) followed the rates of laser flash photolysis of Compound-1 by examining all three reaction products: ATP (measured enzymatically); 2-nitrosoacetophenone (measured as DAbs740nm), and Hþ (measured as DAbs with an indicator dye), and they also determined the decay kinetics of the aci-nitro intermediate. Two steps were kinetically resolved in aqueous solution at neutral pH: Step-1, a proton is released by rapid ionization of aci-Compound-1 to Compound-1 (pKa z 3.7); and Step-2, a slower reaction, in which free nucleotide and the side product 2-nitrosoacetophenone are formed at a rate matching the rate of decay of Compound2 monoanion. Walker et al. (1988) proposed the reaction mechanism in Scheme 10.13. CH3

O

CH

ADP

P O

Other photolabile ATP derivatives have been synthesized and characterized by Trenthan, Corrie and Reid (1992). As with any photolytic process, the efficiency of ATP liberation depends on the molar extinction coefficient of the photolabile agent as well as its quantum yield. The choice of reagent is also affected by ease of synthesis and stability of stock solutions maintained in the dark. Also shown above is a photo-caged proton, which upon photolysis rapidly yields a proton. This reagent has been employed in the rapid kinetic analysis of the sarcoplasmic reticulum calcium ATPase (Fibich, Janko and Apell, 2007). Another application of ‘‘caged’’ substrates exploits the photolability of N-(7a-carboxy-2-nitrobenzyl)carbamo ylcholine upon exposure to photons of 328-nm wavelength. Because this biochemically inactive substance can be converted to carbamoylcholine on the msec time-scale, Matsubara, Billington and Hess (1992) were able to examine the sub-millisecond rate of acetylcholine receptor channel opening. Photolabile chelators have been synthesized to achieve the rapid photo-release of divalent cations (Kaplan and Ellis-Davies, 1988). 1-(2-Nitro-4,5-dimethoixyphenyl)-

CH3 O

O

C

O

ADP

P O

h

O O

H2O

NO2

NO2

Fast O CH3 C O

+ ADP + H

RateDetermining Step +

N

H3C

O

ADP

P O O

C O N

OH

O

Scheme 10.13 One must not assume that the photo-caged nucleotide will be lodged within an enzyme’s active site, and photogenerated ATP may therefore take time to bind to the enzyme. Likewise, most ATP-dependent enzymes require MeATP2-, where the bound metal ion complexed to the b- and g-phosphoryls. The details of scheme remind us then that photo-caged ATP will not have its metal ion

Chapter j 10 Probing Fast Enzyme Processes

677

attached in its catalytically active form, such that the photo-generated divalent MeATP2 must subsequently rearrange to form the active metal-nucleotide complex. The kinetics of this metal ion-nucleotide interaction must therefore be considered when analyzing data any fast reaction experiments using photo-caged nucleotides. The properties in Scheme 10.13 also suggests new ways for generating ATP complexes with oxygen-exchange inert cations such as Cr(III) and Co(III). Another major limitation of this technique is that some biological substances are not sufficiently stable to endure exposure to so many photons. In particular, tryptophan residues are especially susceptible to photo-oxidation, as are many coenzymes and metabolites. Despite such limitations, photo-caged substrates can offer substantial advantages over rapid-mixing techniques. For example, a photo-caged substrate can be loaded into the active sites of a crystalline enzyme, and photolysis can then be employed to release the active substrate in time-resolved Laue crystallographic experiments (see Section 9.2.4).

10.6.2. Pulsed Radiolysis To excite and ionize target chemical substances by means of electron impact, the rapid reaction kinetic technique of pulsed radiolysis (time-scale ¼ 10–1,000 ps) typically uses a 3-MeV van de Graff accelerator (Fig. 10.25) or a microwave linear electron accelerator to generate the needed

pulse of electrons. Upon impact, susceptible chemical compounds rearrange, and their ensuing chemical reactions are then monitored by infrared, visible, or ultraviolet adsorption spectroscopy or by fluorescence spectroscopy (Dorfman, 1974). An excellent example of the application of pulsed radiolysis is provided by Hsu et al. (1996), who examined the action of manganese superoxide dismutase (MnSOD) on superoxide anion generated by pulse radiolysis (Fig. 10.26). The reaction was observed spectrally by measuring superoxide absorbance at 250–280 nm. Catalysis showed an initial burst of activity lasting ~1 ms, followed by the onset of an inhibited rate. The turnover number for human MnSOD at pH 9.4 and 20 C was kcat ¼ 4  104 s1 and kcat/Km ¼ 8  108 M1 s1. These catalytic properties of human MnSOD are qualitatively similar to those reported by Bull et al. (1991) for Thermus thermophilus MnSOD. Other applications of pulse radiolysis include: (a) kinetic studies on cytochrome c-d1 nitrite reductase from Thiosphaera pantotropha that provided crucial evidence for a fast intramolecular electron transfer from the c-heme to d1-heme (Kobayashi et al., 1997); (b) determination of the rate constant for hydroxyl and oxide radicals reacting with cysteine in aqueous solution (Mezyk, 1996); (c) direct demonstration of the catalytic action of monodehydroascorbate reductase (Kobayashi et al., 1995); (d) investigation of the generation and reactions of the

Metallic Sphero

Charging Belt

e e

Discharge Sphere

e e e

e

e

triggering device

e

e

e

e

data acquisition and display

e Motor

Van de Graaf Generator

PM tube

monochromator

Sample

Light Source

The ground serves as the source of electrons

FIGURE 10.25 Schematic diagram of a pulsed radiolysis device. This electrostatic generator uses one or more motor-driven dielectric belts to pick up electrons from an electroconductive comb at the electrical ground and then transfer them to a large hollow metal sphere, where they accumulate within the metal, often attaining very high voltages in excess of 1 MeV. Upon discharge, a research-grade van de Graaf generator, such as the one available at the Brookhaven National Laboratory, allows the stored electrons to be virtually instantaneously transferred to a cuvette containing a redox-active enzyme and its substrates. A 3-MeV van de Graaf generator typically discharges a 1.5-amp electron pulse that endures for 3–4 nsec. Such a pulse contains approximately 30–40 billion electrons!) In the case of superoxide dismutase, dioxygen molecules absorb a high-energy electron to produce superoxide anion. Because the released electrons often have energies in excess of 3 MeV, they are radioactive b-particles, and these pulsed radiolysis experiments must be conducted in a suitably shielded room to minimize exposure to the b radiation. (For comparison, phosphorus-32 nuclei emit electrons with a maximal energy of 1.709 MeV and an average energy of 0.69 MeV.)

Enzyme Kinetics

678

P-Mn(III):O-2

⌬Absorbance 250nm x103

O-2

40

20

k2 O 2

k-1

P-Mn(II)

P-Mn(III) H2O2

30

k1

k-4 k4

k-3

k3 O2

P-Mn(III):O-2

k-5 k5 X

O2-

k7

O2+H2O2

10

0

4 6 2 Elaspsed Time after Pulse, millisec

8

FIGURE 10.26 Time-course of human Mn SOD after generation of superoxide anion by pulsed radiolysis. The pulse radiolysis experiments were performed at Brookhaven National Laboratory using a 2 MeV van de Graaff accelerator. A path length of either 2.0 or 6.1 cm was used for all experiments. All UV-visible spectra were measured on a Cary 210 spectrophotometer at 25 C. Dosimetry was defined with the potassium thiocyanide dosimeter. The measured rate constants are based on the manganese concentration from atomic absorption spectroscopy and not on the protein concentration. All of the metal present is presumed to be bound in the active site and functioning independently of the other metal centers. Superoxide radicals (1–30 mM) were generated by exposing aqueous, air-saturated solutions to a high-dose electron pulse. The experiment demonstrates decrease in absorbance at 250 nm (E ¼ 2,000 M1 cm1) for a solution containing 0.5 mM enzyme, 50 mM EDTA, 10 mM sodium formate, and 2 mM sodium pyrophosphate at pH 9.4 and 20 C. The starting concentrations of superoxide were 11.6 mM (upper curve), 6.5 mM (middle curve), and 3.4 mM (lower curve). The calculated progress curves shown as solid lines were obtained by using the KINSIM software and the model of Bull et al. (1991). Reproduced with the permission of Hsu et al. (1996) and the American Chemical Society.

disulfide radical anion derived from metallothionein (Fang et al., 1995); (e) kinetic studies of NO$ reactions with HO2 (Goldstein and Czapski, 1995); and (f) the kinetics of enhanced intramolecular electron transfer in the ascorbate oxidase reaction in the presence of oxygen (Farver, Wherland and Pecht, 1994).

10.7. DATA ANALYSIS Unlike most chemical reaction mechanisms, enzymatic processes are almost inevitably characterized by multi-step mechanisms, some that are extremely fast and others that are often considerably slower. This simple fact emphasizes the need for devising the simplest possible experiments and for conducting robust statistical analysis. Because fast

reaction techniques allow one to discern many more kinetic features – and a correspondingly greater number of rate constants, statistical analysis is absolutely essential. Although aspects of enzyme rate analysis were presented in Section 4.8, especially global statistical analysis of enzyme rate data (Section 4.8.3), additional comments are provided here. Except under the simplest circumstances, nearly all observed relaxation processes are complex spectra that must, whenever practicable, be resolved into a set of decaying exponentials. For a system of n components, there are n relaxation times, an equal number of amplitudes, and one time-invariant baseline value, the latter being the final (t ¼ 8) value after the last exponential dies away. In most fast reaction studies, one obtains a plot of signal amplitude versus the elapsed time after mixing and/ or perturbation, usually by recording the absorbance or fluorescence change upon conversion of each species Xi to its successor. There are two basic ways of analyzing the resulting data: (a) iterative fitting of the data to the time evolution obtained for a set of differential equations, starting with initial estimates of rate constants; or (b) nonlinear least-squares fitting to a multi-exponential function. Calculation times for iterative fitting to differential rate equations, such as those obtained through the use of the Runge-Kutta method or Gear’s predictor-corrector protocol (see also Sections 3.4.2 and 4.73), are 10–100 longer than those required for nonlinear least-squares fitting to analytical functions of the following form: FðtÞ ¼ A0 þ A1 expðk1 tÞ þ A2 expðk2 tÞ þ A3 expðk3 tÞ þ $$$ þ B1 t

10.52

The first term A0 is a time-independent baseline correction, and the last term B1t is an optional time-linear component accounting for drift in the baseline, sample photobleaching, etc. Because biochemists are interested in detecting as many of the naturally occurring exponential processes, they typically push a technique to the limiting sensitivity of an instrument. In these cases, the observed signal-to-noise ratio is strongly affected by various types of noise (e.g., shot noise, Johnson (or thermal) noise, noise introduced by external electric fields from unshielded motors and other electrical devices, etc.). For this reason, the time-evolution of the observed signal amplitude must be analyzed statistically, typically through the use of a nonlinear least-squares fitting software (e.g., KINSIM-FITSIM, DynaFit, etc.). The ubiquity of high-powered personal computers has greatly facilitated this effort. The kinetic studies of Hazzard, Rong and Tollin (1991) on electron transfer from mitochondrial cytochrome c to cytochrome c oxidase, initiated by laser flash photolysis of

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TIME (sec) FIGURE 10.27 Time-course for electron transfer from mitochondrial cytochrome c to cytochrome c oxidase, initiated by laser flash photolysis of 5-deazariboflavin. (a) Single-exponential fit to the 605-nm data with a simultaneous record of the residuals; (b) singleexponential of the 505-nm tracing, showing noticeable deviations in the fitted curve and experimental data as well as the apparent deviations of the t ¼ 0 data; and (c) double-exponential of the 505-nm tracing, showing agreement in the fitted curve and experimental data as well as the lack of any deviations apparent at the t ¼ 0. The authors attributed the small deviation at the very beginning of each trace to contributions from cytochrome c reduction. With a two-exponential equation, the agreement between the observed and theoretical curve is markedly better. They conclude that the fast component of the kinetic process corresponds to direct reduction of the heme a component of cytochrome c oxidase by reduced cytochrome c. Reproduced with the permission of Hazzard, Rong and Tollin (1991) and the American Chemical Society.

5-deazariboflavin illustrate: (a) the virtue of analyzing a reaction at two different wavelengths; and (b) the value of plotting the calculated residuals for a particular fit. As shown in Fig. 10.27, at 604 nm, pseudo-first-order rate constants (kobs ¼ 409  7 s1) were obtained either by manual plotting of log(DSignal) versus time or by computer assisted fitting to a single exponential by the SI-FIT software from OLIS, Inc. Their data yielded linear log(DSignal) versus time plots for a period corresponding to over four half-lives. The 550-nm decay data, on the other hand, were

biphasic and required computer-assisted fitting to a double exponential using the SI-FIT software. The authors obtained kobs values of 416  24 s1 and 51  5 s1 for the twoexponential case, with respective amplitudes equal to 70% and 30% of the total amplitude measured in the experiment. Notably, the values for the main decay curve at 605 nm and 505 nm were statistically indistinguishable. They interpreted these results to indicate that the fast phase was the result of direct reduction of the heme a component of cytochrome c oxidase by reduced cytochrome c. Their results confirm and extend the experimental findings and conclusions of Antalis and Palmer (1982). It is also advisable to apply global analysis (see also 4.73), a strategy that entails the simultaneous analysis of a complete W  M array Y of absorbances A(l,t), where W is the number of wavelengths studied, and M is the number of measurements at delay time t for each kinetic trace (Bonneau, Wirz and Zuberbu¨hler, 1997). Multi-wavelength data provide the opportunity to determine of the relevant rate constants more accurately. Global analysis is then used to determine the absorption spectra of the individual species participating in a reaction, thus providing a means for discriminating between different mechanistic models. The elimination of linear parameters becomes a must for multi-wavelength data; otherwise, it becomes impossible to treat all unknown molar absorption coefficients at the different wavelengths as the result of both independent and unknown variables. With global analysis, the number of iteratively refined parameters remains equal to the number of rate constants, and is independent of the number of wavelengths used. The residuals between the model-specific function and the experimental data from least-squares analysis are given in matrix notation: R ¼ ðCt CÞ1 Ct Y  Y

10.53

For single wavelength data, the M  W matrix of absorbances simplifies to a vector of dimension M. The concentration matrix C is uniquely defined by the set of rate constants and is independent of the number of wavelengths monitored. An excellent example of how global analysis can be used to analyze a battery of spectral and kinetic data is the study of the pyridoxal phosphate-dependent enzyme dialkylglycine decarboxylase. This unique pyridoxal phosphate-dependent enzyme catalyzes both decarboxylation and transamination in its normal catalytic cycle. Steadystate kinetics of these reactions using alternate substrates were examined by Sun, Zabinski and Toney (1998), who validated an active-site model in which a single binding subsite mediates all bond-making-breaking in decarboxylation and transamination half-reactions. The unexpectedly low reactivity of several of these substrates prompted the study of their pre-steady-state kinetics to determine

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the nature of the rate-determining steps of the reactions (Sun, Bagdassarian and Toney, 1998). The substrate analogue 2-xmethyl-2-aminomalonate (or MAM) was particularly interesting, because it reacts 104-times slower than does aminoisobutyrate (or AIB) with dialkylglycine decarboxylase, yet it reacts about 105-times faster with free coenzyme in solution. As pointed out by the authors, such a dramatic reversal in reactivity on going from the bulk solution to the dialkylglycine decarboxylase active site was difficult to explain on purely mechanistic grounds. Experimental evidence suggested a slow, ratedetermining protein conformational change as a prerequisite to decarboxylation of MAM, thus explaining the enormous reactivity reversal. The reactions of several substrates, including MAM or 1-aminocyclopentane-1carboxylate, exhibit complex kinetics. Analytical expressions for observed rate constants for schemes describing their reactions were used in curve-fitting procedures to obtain estimates for the microscopic rate constants for the minimal mechanisms. By exploiting the fact that pyridoxal phosphate (PLP), pyridoxaamine phosphate (PMP), and various covalent substrate-coenzyme intermediates have characteristic UV/visible absorption bands above 300 nm, Sun, Bagdassarian and Toney (1998) also used rapid-scanning stopped-flow kinetic studies. Stopped-flow spectrophotometer absorbance scans (a total of 500) from 300 to 550 nm were collected on a logarithmic time-scale, with a UVenhanced diode array detector (2.6-ms per scan) and were analyzed globally using either SPECFIT (Spectrum Software Associates) or GLINT (Applied Photophysics). Observed rate constants were obtained by fitting the data to either a one-step or two-step, irreversible serial firstorder mechanism. These data and related fluorescence and circular dichroism spectral data, as well as results on solvent isotope effects, were used to obtain a unifying view of dialkylglycine decarboxylase catalysis. Sun, Bagdassarian and Toney (1998) observed that substrates fall into two groups: (a) those exhibiting simple, monophasic kinetics; and (b) those exhibiting biphasic kinetics. The rate of the 1-aminocyclo-propane-1-carboxylate (or AIB) half-reaction is likely limited by the decarboxylation step based on the simple kinetics and spectra obtained from global analysis. The spectra for the first species in the transamination half-reactions of L-alanine and L-aminobutyrate show long-wavelength absorption characteristics of a carbanionic quinonoid intermediate. This observation demonstrated that formation of the external aldimine intermediates and abstraction of the CR protons from them are rapid. The reactions of the slower substrates L-phenylglycine and 1-aminocyclohexane-1-carboxylate may have external aldimine formation as the ratedetermining step. The biphasic reactions of 2-methyl-2aminomalonate, 1-aminocyclopentane-1-carboxylate, isopropylamine, and glycine all have external aldimine

formation as the rapid observable step, based on the spectral changes observed in absorption and circular dichroism measurements. 2-Methyl-2-aminomalonate reacts 104-fold slower than does AIB with dialkylglycine decarboxylase, compared to 105-fold faster with coenzyme in solution. It was proposed that this radical reactivity reversal results from a slow protein conformational change that is a prerequisite to decarboxylation of MAM, which occurs rapidly thereafter. Circular dichroism measurements on active-site bound coenzyme provided evidence in support of their proposal. The binding of the non-covalent inhibitors pyruvate or lactate, or the covalently binding inhibitor 1aminocyclopropane-1-carboxylate, all induce a slow change in coenzyme circular dichroism that quantitatively parallels the slow decarboxylation of 2-methyl-2-aminomalonate. Fast circular dichroism changes are observed in the mixing time of these measurements for both 1-aminocyclopropane-1-carboxylate and 2-methyl-2-aminomalonate, indicating rapid external aldimine formation on this longer time-scale.

10.8. CONCLUDING REMARKS This chapter has focused on a few rapid-mixing and fastreaction techniques commonly employed to analyze enzyme catalysis. Space considerations prevent any detailed consideration of nuclear magnetic resonance, which is quintessentially a relaxation technique. Although only briefly considered in this reference, few techniques can rival the versatility of NMR spectroscopy in differentiating enzyme-bound ligands from free solution ligands, especially when conducted on proteins with molecular masses less than 25 kDa. One can often obtain a direct report on the rate and extent of substrate or inhibitor exchange, disclosing the binding mechanism, rate and equilibrium constants, conformational changes of enzyme-bound ligands, and pH-dependent changes in these processes. One can also directly evaluate the conformational dynamics of free and bound ligands. NMR parameters that provide such information include: (a) chemical shift data, revealing the dependence of nuclear magnetic energy levels on the electronic environment in a molecular complex; (b) relaxation phenomena, especially spin-spin relaxation time (T2) data; (c) Nuclear Overhauser Effects (NOEs), especially when conducted with appropriate isotopic labeling of probe ligands; and (d) magnetization transfer, wherein transfer efficiency reveals the local electromagnetic environment of the enzyme-bound ligand. The technique of time-resolved solid-state NMR spectroscopy, for example, permits the direct detection of transient enzyme intermediates, as first demonstrated with 5-enolpyruvylshikimate-3-phosphate (EPSP) synthase (Appleyard and Evans, 1993; Appleyard, Shuttleworth and Evans, 1994). This time-resolved method involves rapid

Chapter j 10 Probing Fast Enzyme Processes

freeze-quenching of enzyme-substrate mixtures at discrete time intervals, enabling the pre-steady-state kinetic trapping of transient species as a function of time, followed by low-temperature solid-state NMR analysis at each time point. Enzymatic reactions, which have either single or multiple intermediates, in principle, can be stopped along the reaction coordinate. When coupled with rotational echo double resonance (REDOR) solid-state NMR, active-site distances for discrete ground-state species as a function of time. (For additional consideration of the EPSP synthase reaction, see also Section 10.2.5a: Rapid Mix/Quench Mass Spectrometry). The interested reader should consult Craik and Wilce (1997) for a more complete discussion of the utility of NMR in investigating enzyme-substrate and enzyme-inhibitor interactions. The topics described in this chapter may leave the reader with the impression that rapid flow and relaxation methods are the ultimate kinetic tools, both fail-safe and utterly reliable. Experienced kineticists know that each technique has its own set of strengths and weaknesses. Moreover, the complexity of many enzyme reaction mechanisms often renders the determination of all relaxation times and amplitudes infeasible. Some relaxations, for example, are attended by extremely small signal amplitudes. This situation can arise if intermediate states are present at low concentration or if their associated spectral signal is just too faint. When the signal-to-noise ratio is low, the task of discerning multi-exponential processes is often difficult, if not impossible. Another frequently encountered feature of biochemical reaction is the presence of parallel or branched reaction pathways. Such conditions often yield a composite of several overlapping exponentials that only appear to be a single exponential decay. For example, closed integral forms of the differential kinetic equations have been published for nearly all reaction schemes examined in fast kinetic studies. Even so, deriving elementary rate constants and concentration profiles are not so easily obtained from the model parameters, as determined fitting procedures. As pointed out by Bonneau, Wirz and Zuberbu¨hler (1997), systems conforming to a sum of exponentials (Eqn. 10.21) fit equally well to a series of parallel first-order reactions or to any combination containing parallel, consecutive and reversible monomolecular elementary steps. Likewise, if kAB ¼ kCD for two parallel reactions (e.g., A / B and C / D), one will observe a single exponential, thereby obscuring the existence of two individual steps, not to mention the relative amplitudes of each process. When such mathematical ambiguity occurs, one must rely on additional information or chemical intuition regarding the reasonableness of individual molar absorption coefficients and/or rate constants. As an alternative to deconvoluting an observed relaxation time-course in terms of n relaxation times with n amplitudes, one can evaluate

681

well-defined averages of multiple relaxation times (Ilgenfritz, 1977). When one obtains a relaxation time that appears to be an average, it is useful to re-determine the relaxation time over a range of wavelengths. If the average relaxation time changes with wavelength, it is reasonable to infer that the process is multi-exponential and that certain intermediate species contributing to the signal at one wavelength are different from those observed at another wavelength. This test is useful because the average relaxation time contains the specific signal changes of the observed quantity (Ilgenfritz, 1977). Another useful test is to carry out computer-based simulations to determine whether it is reasonable to suspect the occurrence of an average relaxation time. The interested reader should consult the paper by Dyson and Isenberg (1971) that treated the problem of exponential curve analysis by a method of moments. As noted above, many enzymatic reaction steps produce weak or virtually undetectable spectral changes. In certain cases, a chromogenic substrate may be used in place of the natural substrate. This approach began over 50 years ago with the use of p-nitrophenylacetate to facilitate study of chymotrypsin catalysis. Even so, alternative substrates may not reveal all of the subtleties of enzymatic action on the natural substrate, raising serious questions as to whether the natural and chromogenic substrate operate by the same chemical mechanism. For example, p-nitrophenylacetate is an unlikely analogue of naturally occurring proteins, and despite all that has been learned about chymotrypsin, the enzyme’s action on protein substrates remains sketchy. Another approach is to use an indicator dye such as cresol red, to quantify the production/ consumption of protons. Finally, there is a continuous fluorimetric assay for reactions producing/consuming inorganic phosphate, employing the Ala-197-Cys mutant of E. coli phosphate binding protein labeled with N-[2(1-maleimidyl)ethyl]-7-(diethylamino)coumarin-3-carboxamide, with Pi binding causing a 5 enhancement in fluorescence emission at 474 nm (Brune et al., 1994). These investigators demonstrated their ingenious technique in single-turnover experiments on the hydrolysis of ATP bound to actomyosin (see Section 4.5.6: Continuous Fluorescence Assays are Now Available for Pi- and PPiProducing Reactions). Temperature-jump and other relaxation techniques may inevitably alter reactant concentrations other than those in the reaction of primary interest. In many cases, rapid change in temperature or pressure is likely to perturb enzyme protonation states, solution pH, or even the concentrations of free and complexed metal ions. Sample inhomogeneity can also limit fast reaction studies. While reactant purity is always a concern, especially with unstable enzymes, many artifacts can be traced directly to ineffective sample mixing and/or incomplete thermal equilibration. The resulting concentration and thermal gradients can distort the shape of

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an observed kinetic trace. Likewise, sudden stopping in stopped-flow devices often causes cavitation, wherein the resulting vapor gives rise to an inhomogeneous sample refractive index that severely compromises spectral measurements. Poor thermal equilibration of a sample prior to rapid mixing or temperature-jump can also artificially create the appearance of a multi-exponential process, even for an otherwise single-exponential process. Finally, many single-turnover reactions are conducted under conditions that lead to enzyme self-association, affecting the ability to reconcile fast reaction and steady-state kinetic data. Despite such limitations, fast reaction techniques constitute the single-most powerful set of tools for dissecting complex multi-step reaction mechanisms. As collaborations among chemists, physicists, and enzymologists extend the power and scope of fast reaction approaches, enzyme chemistry is apt to remain a fertile field for discovering new properties and principles of biological catalysis and control. Finally, Laue X-ray crystallography may well become the ultimate time-resolved kinetic technique for detecting and structurally analyzing enzyme reaction cycle intermediates (Bartunik et al., 1992; Moffat, 2001; Schlichting, 2000). The Laue method, which employs a stationary crystal placed in an intense multi-wavelength X-ray beam, allows sampling of a large fraction of reciprocal space (and thus of the diffraction data to be collected) in a single exposure. As described by Schlichting (2000), one can rationalize such behavior by inspecting Bragg’s Law (2d sin q ¼ nl) describing a constructive interference of X-rays of wavelength l reflected at angle q by the crystal’s parallel lattice planes (separated by distance d), and n is an integer. In the Laue geometry, a polychromatic X-ray beam allows the Bragg equation to be satisfied for many d-values simultaneously (i.e., one obtains solutions to numerous Bragg equations: 2d1 sin q ¼ nl1, 2d2 sin q ¼ nl2, 2d3 sin q ¼ nl3, $$$, 2di sin q ¼ nli, thereby providing a ‘‘snapshot’’ of the protein’s structure at the instant the crystal is irradiated). The popularity of the Laue method grew with the availability of powerful synchrotron X-ray sources (Moffat, 1997; Moffat, Szebenyi and Bilderback, 1984). Synchrotrons are high-energy electron (or positron) storage rings, into which the particles are injected in bunches and kept in orbit magnetically. Discontinuous X-ray emission occurs in 150-ps pulses, and depending on the number and separation of particle bunches injected in the storage ring, the emitted X-rays can occur as 150-ps flashes or as a quasi-continuous beam. Notably, the short flashes can be separated by a few microseconds, thereby allowing the experimenter to obtain multiple structural snapshots. The 150-ps pulse-time establishes a sub-nanosecond time-resolution limit on the technique. Notably, time-resolved Laue X-ray crystallography provides a way to obtain structures of the enzyme species on an ultra fast time-scale and, based on the intensity of the diffraction pattern, an estimate of the concentration of these species. When combined with molecular

dynamics calculations, the power of the Laue technique is greatly enhanced. With the availability of laser flash photolysis (Section 10.6.1) to initiate enzyme catalysis using ‘‘photo-caged’’ substrates already situated in the crystal, time-resolved Laue X-ray crystallography provides a method for directly analyzing enzyme catalysis. Using the photoprotein PYP as a model system, Schmidt et al. (2004) demonstrated how the singular value deconvolution (SVD) analysis separates the complete, time- and real spacedependent data matrix A into left singular vectors (lSVs) in matrix U, each of which contains time-independent structural information; right singular vectors (rSVs) in matrix VT, each of which contains the time dependence of the corresponding lSV; and the SVs in the diagonal matrix S, which serve as weighting factors, according to the equation: A ¼ USVT. This approach allowed them to evaluate rival chemical kinetic mechanism and to arrive at a self-consistent reaction kinetic mechanism (see Section 1.6: Prospects for Enzyme Science). Despite the great promise of using Laue X-ray crystallography as a time-resolved kinetic technique, there are several obvious limitations. First, standard monochromatic light sources tend to yield higher quality structural data than multi-wavelength (‘‘white light’’) sources, resulting in superior signal-to-noise ratios and more reliable scaling of structure factors (Bartnunik et al., 1992). Second, non-simultaneous or incomplete initiation of the enzyme-catalyzed reaction results in crystal mosaicism, resulting in increased complexity of the diffraction data. This limitation also includes any difficulties in loading the active sites of a crystalline enzyme with sufficient photo-activatable substrate. Third, there is always concern as to the effects of crystallization on enzyme kinetics and dynamics, because the crystal lattice may act to restrict or modify conformational changes essential for catalysis. Fourth, the cryosolvent itself may modify enzyme and/or substrate reactivity. Finally, the efficiency of data acquisition and analysis can present additional obstacles to obtaining a cleanly interpretable result.

FURTHER READING Chance, B. (2009). The Stopped-flow method and chemical intermediates in enzyme reactions – a personal essay. Photosyn Res., 80, 387. Eigen, M. (1968). New looks and outlooks on physical enzymology. Quart. Rev. Biophys. 1, 3. Fierke, C. A., & Hammes, G. G. (1995). Transient kinetic approaches to enzyme mechanisms. Meth. Enzymol. 249, 3. Hammes, G. G. (Ed.), (1973). Investigation of rates and mechanisms of reactions. Part II, vol. 6. in Techniques of Chemistry. WileyInterscience: New York. Johnson, K. A. (2003). Kinetic analysis of macromolecules: a practical approach. Oxford University Press, USA. Johnson, K. A. (2003). Fitting enzyme kinetic data with KinTek global explorer. Meth. Enzymol. 467, 601.

Chapter j 10 Probing Fast Enzyme Processes Nolting, B. (1999). Protein folding kinetics: biophysical methods. SpringerVerlag, New York. Pecht, I., & Rigler, R. (Eds.), (1977). Chemical relaxation in molecular biology. Springer-Verlag, Berlin.

Other Authoritative Readings from the Enzyme Kinetics Volumes of Methods in Enzymology The Temperature-Jump Method, 16, Chapter 1. The Pressure-Jump Method, 16, Chapter 2. Ultrasonic Methods, Frieder, 16, Chapter 3. Electric Field Methods, 16, Chapter 4. Polarographic Methods, 16, Chapter 5. Rapid Mixing: Stopped Flow, 16, Chapter 6. Rapid Mixing: Continuous Flow, 16, Chapter 7. Photochemical Reaction in Nucleic Acids, 16, Chapter 8.

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Fast Reactions Measured by Single and Repetitive Pulse Excitation, 16, Chapter 9. Fluorescence Methods in Kinetic Studies, 16, Chapter 10. Transient Kinetic Approaches to Enzyme Mechanisms, 249, Chapter 1. Rapid Quench Kinetic Analysis of Polymerases, Adenosine Triphosphatases, and Enzyme Intermediates, 249, Chapter 2. Rapid Mix-Quench MALDI TOF Detection of Intermediates, 354, Chapter 2. On-Line Rapid-Mixing Electrospray Mass Spectrometry, 354, Chapter 3. Detecting Peroxodiferric Intermediates by Freeze-Quench Mo¨ssbauer, Resonance Raman, and XAS Spectroscopies, 354, Chapter 32.

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Chapter 11

Regulatory Behavior of Enzymes Those familiar with living systems cannot but marvel at the ability of each cell to grow, to move, to sense, to communicate, to cooperate, to proliferate, to die and, even then to yield opportunity to its successors. We now recognize that metabolic regulation endows cells with adaptive mechanisms so resilient that early observers mistook such constancy as proof that organisms are animated by some life-sustaining vital force. Claude Bernard coined the term homeostasis to describe an organism’s internal stability, a property that arises from inherent feedback mechanisms that allow living systems to compensate automatically for environmental changes. Early on, the collective behavior of the whole appeared to exceed the sum of its component parts. It is worth recalling that so great a scientist as Pasteur was an adherent to the tenets of Vitalism, the belief that the vitality of the biotic world lies beyond chemistry and physics. Experience has shown, however, that hierarchically complex, large-scale networks can give rise to emergent properties (i.e., properties of an integrated system that cannot be predicted from the study of individual component enzymes and receptors). Emergent properties of living systems are evident in the inherent responsiveness of intermediary metabolism, in the far-reaching aspects of signal transduction, in the chromatinassociated processes governing gene expression, and most so in the behavior of highly robust networks of interconnected neurons, the latter giving rise to consciousness.1 We now know that enzymes (including many enzymelinked receptors) are the primary mediators of cellular regulation. They operate within a hierarchy of metabolic pathways – some enzymes producing or degrading the basic building blocks for cell growth, others transducing and amplifying sensory inputs, others mediating DNA replication, transcription, and protein synthesis, and still others generating the forces needed for cell motility, chromatin remodeling, cell division, and chemotaxis. Because enzymes are such powerful catalysts, they must be highly regulated at every level. By minimizing side-reactions and preventing the

formation of potentially toxic by-products, enzyme specificity/selectivity constitutes a first level of regulation. The ability of every enzyme to sense and respond to changes in the concentration of its substrates, cofactors, and reaction products represents a second level of regulation. But the most identifiable properties of metabolic regulation are enzyme cooperativity, allostery, interconversion by enzyme-catalyzed covalent modification, effector molecule-stimulated signal amplification, and substrate channeling. This chapter begins with a brief overview of enzyme regulation, followed immediately by a quantitative treatment of enzyme-ligand binding interactions, including the early models of Hill, Adair, and Pauling to emphasize the origins of cooperativity. These models are followed by thermodynamic treatments of the Monod-Wyman-Changeux and Koshland-Ne´methy-Filmer models for cooperativity. The focus of this chapter, however, is the kinetic characterization of regulatory enzymes, mainly as isolated components, but also as coordinated units within metabolic pathways. Later sections deal with the kinetic behavior of oligomerizing enzymes, multi-cyclic cascades of covalently interconvertible enzymes, substrate channeling, and Metabolic Control Analysis. Additional details on the treatment of enzyme activators are presented in Section 7.1.

11.1. OVERVIEW OF ENZYME REGULATION A shared goal of all modern molecular life scientists is to understand the physicochemical mechanisms and interactions responsible for homeostasis. At the cellular and intercellular levels, homeostasis arises from the ability of enzymes, membrane-bound and mobile receptors, as well as key regulatory proteins to sense changes in the concentration of substrates, coenzymes, activators as well as inhibitors,2 to integrate these input signals, and to make 2

1

Francis Crick’s 1994 book, entitled The Astonishing Hypothesis: The Scientific Search for the Soul, directly challenges Vitalism and fundamental religious concepts by asserting that physicochemical processes within the brain are entirely responsible for our consciousness and for what we call the mind. That we tend to ascribe to the mind many properties that appear to be beyond chemistry and physics, he would argue, is merely the consequence of our still primitive ability to examine the emergent properties of the brain’s neural networking. Enzyme Kinetics Copyright Ó 2010, by Elsevier Inc. All rights of reproduction in any form reserved.

In the IUPAC nomenclature, the terms effector and modifier are used in a more restrictive manner to describe substances that increase (activate) or decrease (inhibit) biological activity. Activators and inhibitors are said to be effectors, only if they have been shown to produce a physiologically significant effect. Inhibitors and activators are said to be modifiers, if they are substances artificially added to an enzyme system being studied in vitro. In many respects, this distinction overlooks the fact that some synthetic inhibitors become drugs, and all drugs produce a physiologically significant effect.

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appropriate responses that modify metabolic activity. Of particular significance are: (a) those enzymes catalyzing the first reaction (or committed step) within a metabolic pathway; (b) other regulatory (or pacemaker) enzymes situated at branch-points (or nodes) linking pathways; (c) kinase-containing receptors that detect and automatically modulate their own sensitivity to incoming signal molecules; and (d) so-called G-coupled receptors, where receptor occupancy is transduced by a physically coupled GTP-regulatory protein. By processing a battery of regulatory inputs, all four types of enzymes control metabolic throughput (or flux) and the concentrations of pathway intermediates. Other Michaelis-Menten-type enzymes within a pathway simply respond to changes in the availability of pathway intermediates. Because most metabolic pathways are interconnected at branch-points (or nodes), enzymes catalyzing the first steps of pathways must be closely regulated, often in a hierarchical manner. Fast signaling on the millisecond-to-second time-scale requires rapid binding of release of regulatory molecules to and from specific binding sites on target enzymes. On the time-scale of seconds to minutes, cellular regulation is most often achieved by hormone- and/or effector-controlled enzymes catalyzing post-translational modifications (e.g., phosphorylation, nucleotidylation, ADPRibosylation, acetylation, methylation, ubiquitinylation, proteolysis, etc.) of target enzymes, thereby altering the latter’s catalytic performance. Subsequent removal of the transferred group by certain phosphatases, phosphodiesterases, glycosylases, and hydrolases restores the target enzyme’s original activity state. Regulatory processes taking place on the minutes-todays time-scale often reflect changes in the rates of biosynthesis or degradation of regulatory enzymes. Certain signaling enzymes have very short biological half-lives (e.g., 12 min for ornithine decarboxylase), whereas other socalled house-keeping enzymes (like lactate dehydrogenase and cytochromes) are highly stable and persist for days, weeks, or even longer (Schimke, 1969). Finally, diurnal and longer-scale regulation is achieved through the expression of special genes, such CLOCK and PERIOD. Effector-mediated regulation allows enzymes catalyzing committed steps to be: (a) activated by signals arriving from other pathways; and/or (b) inhibited as pathway endproducts accumulation (Fig. 11.1). In his definitive review of the foundations and strategies of enzyme-mediated metabolic regulation, Stadtman (1966) credited the first reported case of feedback inhibition to Dische (1940), who described the inhibition of erythrocyte hexokinase by glycolytic end-products that are structurally unrelated to hexokinase’s substrates or products. Feedback inhibitors most often bind at discrete regulatory sites that are topologically remote from the active site. These inhibitors possess the ability to modify catalytic activity indirectly by first inducing conformational changes elsewhere in the target enzyme and in a manner that ultimately distorts the

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FIGURE 11.1 Representative mechanisms of feedback inhibition. Upper Scheme, Feedback inhibition of the first committed step in a linear multi-step metabolic pathway. Dotted-lines and minus-signs indicate allosteric feedback inhibition. Other enzymes (shown as light blue diamonds) in the pathway need not exhibit special regulatory properties and typically obey Michaelis-Menten kinetics. Lower Scheme, Coordinate feedback inhibition for a pathway that branches into two new pathways. Note that accumulation of end-product from each branched pathway only inhibits the first committed step in its own pathway (e.g., end-product M inhibits allosteric Enzyme-2, but not inhibit allosteric Enzyme-3), such that endproducts M and Z will not interfere with each other’s biosynthesis. However, accumulation of M will have the effect of diverting more of D into the lower pathway, whereas accumulate of Z will have the effect of diverting more of D into the upper pathway. Only when both endproducts, M and Z, accumulate, will the concentration of end-product D increase, with the desired effect of inhibiting allosteric Enzyme-1.

arrangement of active-site residues. Inhibitors have the effect of raising barriers to substrate binding and/or catalysis, whereas activators have the opposite effect. It should also be stressed that feedback inhibition at pathway branch-points presents the opportunity for highly coordinated metabolic regulation. Consider the case where a primary (or main) pathway branches into two subsidiary pathways. Such pathways are frequently regulated by the two generalized schemes that are shown in Panels (b) and (c) of Fig. 11.1. The simplest case is sequential feedback inhibition, which operates much like the linear pathway shown in Panel (a). The end product of one pathway (say that forming S6) does not directly influence the flux of the primary pathway supplying an essential substrate for the other pathway (say that forming S8). In the case of nested feedback inhibition, the ultimate end-product of each branch also inhibits enzymes catalyzing the first step of the primary pathway. The most common mode of enzyme regulation results from conformational changes that are either spontaneous or ligand-induced. In the case of a spontaneous conformational change, a protein may exist in two states X and Y that are in thermodynamic equilibrium with each other (i.e., X # Y), such that DG ¼ –RT ln([Y]/[X]). The equilibrium concentrations of X and Y are determined by the relative stabilities of each conformation. In other cases, a ligand molecule selectively alters the stability of one (or more) conformational state(s) of the enzyme. In either case, structural alterations may arise from short-range effects (i.e., at distances much less than a nanometer) or they can occur

Chapter j 11 Regulatory Behavior of Enzymes

from long-range changes at the quaternary structural level (i.e., through rearrangements in subunit-subunit interactions). The term allostery is derived from the Greek prefix allo, meaning ‘‘another,’’ and the suffix steros, meaning ‘‘shape’’ or ‘‘form.’’ (Note: Some authors erroneously infer that allostery implies the presence of ‘‘another site,’’ resulting in the incorrect inference that canonical noncompetitive inhibition and allosteric inhibition are somehow equivalent.) Allosteric interactions are most often associated with cooperativity (Koshland, Ne´methy and Filmer, 1966; Monod, Wyman and Changeux, 1965; Stadtman, 1966). Koshland and Hamadani (2002) categorized the main type of allosteric cooperativity as I3-cooperativity, a condition that applies whenever a protein consisting of initially identical subunits undergoes a ligandinduced conformational change that is inherently intramolecular (i.e., they occur within the subunits of a multisubunit enzyme). Ultimately, however, allosteric activation increases enzyme reaction rates by improving substrate saturation and/or stabilizing reaction transition states. The opposite is true of allosteric inhibition. Hemoglobin has served as the prototype for a cooperative ligand-binding protein for nearly a century (Perutz,

687

1990), and the enzymes listed in Table 11.1 may likewise be regarded as the prototypical allosteric enzymes. While most are oligomers of identical subunits (also often called protomers), some are made up of non-identical subunits, and still others contain regulatory subunits. A good example of the latter is Protein Kinase A (or PKA), which exists as a catalytically inactive R2C2 heterotetramer (R ¼ regulatory subunit; C ¼ catalytic subunit). Upon binding of 39,59cyclicAMP, the R2C2 complex rapidly dissociates into the R2$(cAMP)2 regulatory dimer and two catalytic subunits. Another example is aspartate carbamoyltransferse (Reaction: L-Aspartate þ Carbamoyl-Phosphate # N-CarbamoylL-Aspartate þ Pi), which catalyzes the first committed step in pyrimidine nucleotide biosynthesis. ATCase is a heterosubunit enzyme (R3)2(C2)3 made up of three catalytic C2 dimers flanked by two regulatory subunit trimers R3. This enzyme binds L-aspartate cooperatively, resulting in a so-called S-shaped (or sigmoidal) rate-saturation behavior (shown by the middle curve in Fig. 11.2). In the presence of the feedback inhibitor CTP, the enzyme’s activity is reduced, and the substrate concentration needs to achieve half-maximal saturation of the substrate L-aspartate (often symbolized as [S]0.5) to be shifted toward higher

TABLE 11.1 Several Classical Examples of Allosteric Enzymes Enzyme

Substrate(s)

Effector

References

Threonine Deaminase

L-Threonine

L-Isoleucine (þ)

Umbarger and Brown (1957; 1958)

Aspartate Transcarbamoylase

L-Aspartate

CTP ()

Gerhart and Pardee (1962; 1963; 1964)

Carbamyl-P

ATP (þ)

DCMP

dTTP ()

dCMP Aminohydrolase

dCTP (þ)

Scarano et al. (1963; 1964); Maley and Maley (1963; 1964) Okazaki and Kornberg (1964)

Deoxythymidine Kinase

Deoxythymidine

dTTP ()

Phosphofructokinase

Fructose-6-P

ATP ()

ATP

3’,5’-cAMP (þ)

Phosphorylase b

Glucose-1-P

ATP ()

Lysine-regulated Aspartokinase

L-Aspartate

dCDP (þ)

Glycogen þ Pi

Passonneau and Lowry (1962) Helmreich and Cori (1962); Madsen (1962)

5’-AMP (þ) L-Lysine (þ)

Stadtman et al. (1961)

L-Threonine ()

Stadtman et al. (1961)

L-Histidine

Martin (1962) Tomkins et al. (1963)

ATP Threonine-regulated Aspartokinase

L-Aspartate

ATP ATP-PPRP Pyrophosphorylase

PRPP ATP

Glutamate Dehydrogenase Acetyl-CoA Carboxylase

L-Glutamate

ATP or GTP ()

NAD

ADP (þ)

Acetyl-CoA

Citrate (þ)

ATP Bicarbonate

Martin and Vagelos (1962)

Enzyme Kinetics

688

Initial Reaction Velocity

Plus

ATP

Plus

CTP

[L-Aspartate] FIGURE 11.2 Plot of the initial velocity versus aspartate in the absence and presence of CTP feedback inhibition. The reaction mixture contained a fixed concentration of the nonvaried substrate carbamoylphosphate, and the concentration of L-aspartate was varied over a range sufficient to achieve rate saturation. Escherichia coli ATCase was maintained at 9.0 mg/mL (~3  108 M) in all measurements. This experiment was repeated in the presence of a fixed concentration of the activator ATP (upper curve) or the feedback inhibitor CTP (lower curve).

L-aspartate concentration. In the presence of the allosteric activator ATP, the enzyme’s activity is actually enhanced, and the mid-point (corresponding to [L-aspartate]0.5) in the ligand saturation curve is shifted toward lower L-aspartate concentration. In view of widespread interest in metabolic regulation, it is customary to consider aspects of cooperativity in modern enzyme kinetics books. Unlike enzymes exhibiting Michaelis-Menten kinetics, allosteric enzymes possess multiple interacting subunits (or protomers), each with its own active site and one or more allosteric sites. Cooperativity is the phenomenon wherein effector molecule binding on one subunit of an oligomeric protein results in local structural changes that are communicated to and from sites on other subunits. These binding interactions alter catalytic activity, often over narrow ranges of substrate and/or effector concentrations. The need for such responsiveness becomes clear when one realizes that most enzymes are not particularly responsive to changes in the concentration of input signals. Enzymes obeying the Michaelis-Menten kinetics can only undergo a 10 increase or decrease in activity over a 100 increase or decrease in substrate concentration. By contrast, cooperative enzymes can change their catalytic activity over much narrower concentration ranges, often experiencing a 10 increase in rate over a 3 increase in substrate concentration. Cooperativity in multi-subunit enzymes is most often analyzed in terms of the subunit binding interactions of Monod-Wyman-Changeux and Adair-Koshland models described later in this chapter. Weber (1972) suggested that even monomeric enzymes might also display cooperativity, if the polypeptide chain has two or more stable

conformations, of which at least one is stabilized by ligand binding. Reinhart (2004) also discussed the advantages of a model-independent approach for quantitative analysis of enzyme cooperativity and allosteric behavior. The underlying principles of thermodynamic linkage allow the investigator to characterize the apparent coupling parameters between pairs of ligands regardless of the oligomeric nature of the enzyme. Finally, it is also clear that DNA frequently undergoes transitions between two or more conformational states, with attendant changes in the affinity of its binding interactions with low- and high-molecular-weight metabolites. In this respect, DNA is allosteric (Chaires, 2008), and greater attention should be given to developing quantitative models like the MWC and KNF treatments to define DNA allostery. The regulatory analogy goes further, when one considers that chromatin is subject to ATP-dependent enzymecatalyzed conformational remodeling, much as proteins and enzymes are regulated by ATP-dependent enzymecatalyzed phosphorylation and nucleotidylation. While the likely complexity of a two-state protein interacting with a two-state DNA is daunting, such processes doubtlessly underlie the robustness of genetic control.

11.2. GENERAL STRATEGIES FOR MEASURING LIGAND BINDING Enzyme chemists must routinely determine the stoichiometry and affinity of ligand binding interactions with enzymes and other proteins. The tacit assumption is that the measurement technique has no effect on the tendency of the protein in question to establish a thermodynamic equilibrium (P þ L # P$L) between its free and bound forms: Kd ¼

½P½L ½P$L

11.1

where Kd is the dissociation constant, P is uncomplexed protein, L is the free or uncomplexed ligand concentration, and P$L is the protein$ligand complex. Under favorable circumstances, one can even characterize equilibrium binding interactions of enzymes with substrates, provided that one can omit some component essential for catalysis (e.g., metal ion, coenzyme, or essential activator). For example, one can study D-glucose binding to hexokinase as long as ATP and/or Mg2þ are absent. The parameter v is defined as the fractional saturation of a ligand binding site: v ¼

½P$L ½P$L ¼ ½PTotal  ½P þ ½P$L

11.2

which varies from 0 to 1, and the free ligand concentration at v ¼ 0.5 corresponds to the ligand dissociation constant Kd.

Chapter j 11 Regulatory Behavior of Enzymes

689

1.0

0.5

Kd

0

[Ligand]free

FIGURE 11.3 Saturation curve for simple noncoooperative ligand binding. According to Eqn. 11.2, the binding curve asymptotically approaches 1.0 at high ligand concentrations.

A more realistic situation occurs when the enzyme binds ligand both specifically and nonspecifically, as defined by the following binding equation: v ¼

½P$L þ a½L þ b½L2 ½P þ ½P$L

11.3

where a and b (with units of M1 and M2, respectively) are curve-fitting parameters. If b ¼ 0, the observed values can actually exceed unity, and the saturation curve continues to increase linearly as shown in Fig. 11.4. The curve shown in Fig. 11.3 will only be obtained if the ligand binding in Fig. 11.4 is corrected for nonspecific binding (corresponding to the dotted line). If b s 0, the last term in the saturation function Eqn. 11.3 will be nonlinear, and computer modeling then becomes the only practical means for abstracting the specific binding curve. Ligand binding experiments must be conducted under conditions assuring thermodynamic equilibrium, where the concentrations of each free and bound species rigorously obey their respective equilibrium constants. It also bears repeating that the observed curve is only as reliable as the

1.0

0.5 Nonspecific Binding

0

[Ligand] FIGURE 11.4 Saturation curve for ligand binding according to Eqn. 11.4. Notice that, when b ¼ 0, the binding curve asymptotically approaches a straight line with the slope a.

accuracy and precision of experimentally determined values of [P] and [P$L]. Shown in Table 11.2 are some of the most widely employed techniques for quantifying binding interactions. These techniques have been universally applied in the systematic investigation of enzyme interactions with substrates, cofactors, and metabolic regulatory molecules. Usually, equilibration can be readily achieved with most proteins that are reasonably stable (e.g., those remaining fully active for days or more). The same methods comprise the indispensable armamentarium of pharmacologists seeking to unravel receptor interactions with agonists and antagonists. Most rapidly equilibrating chemical reactions are complete within a few milliseconds; others involving slow conformational changes may require many minutes to reach equilibrium. In the case of equilibrium dialysis, the transfer of low-molecular-weight ligands across the dialysis membrane is slow (time-scale of hours). In spectroscopic assays, failure to account for spectral differences between free and bound ligand is a pervasive source of experimental error in equilibrium binding experiments. Although there is no a priori approach for predicting the magnitude of changes in the spectrum of ligands upon binding, a practice strongly recommended by Winzor and Sawyer (1995) is to titrate a fixed concentration of ligand with its macromolecular acceptor. The resulting spectral data are usually plotted as 1/y, where y is the measured spectroscopic parameter, versus 1/ Ci , where C i is total molar concentration of Species-i in the liquid phase. One must also correct for any dilution incurred during the titration, and it is advisable to use a nonlinear least-squares program to estimate the ordinate intercept, which represents the reciprocal of the spectral parameter of bound ligand. Finally, one cannot assume that the spectrum of bound ligand is the same for all sites in an oligomeric protein. The technique known as surface plasmon resonance (SPR) allows researchers to monitor biochemical interactions without the use of isotopic labels or absorption/ fluorescence reporter groups (Cush et al., 1993; Lo¨fa¨s and Johnsson, 1990). SPR is an optical phenomenon arising when surface plasmon waves are excited at a metal/liquid interface. Gold is an excellent conductor because its electrons are so easily promoted to higher energy orbitals, behaving in many respects like a gaseous cloud of ions akin to plasma (i.e., the highly ionized gases formed at extremely high temperatures). These free electrons in the detector’s gold foil surface are remarkably sensitive to changes in the electrostatic interactions occurring in the nearby liquid phase as a result of biospecific binding interactions. Moreover, the free electrons in this extremely thin foil determine the refractive index near the foil’s surface. In the BIACOREÒ instrument (LKB-Pharmacia, Inc.), SPR is detected using the gold film at the sensor probe tip. Incident light from the probe’s optical fiber undergoes total internal reflection at the surface interfaces,

Enzyme Kinetics

690

TABLE 11.2 Techniques for Measuring Ligand Binding Interactions with Enzymes Technique

Comments

Affinity Chromatography

This technique uses a matrix-bound ligand (or protein) as the stationary phase to measure binding interactions with a soluble protein (or ligand). The soluble component is placed in a buffer and passed through the column. Interaction with its matrix-immobilized binding partner retards its progress relative to that of some non-interacting reference solute (Andrews et al., 1973; Dunn and Chaiken, 1974; Nichol et al., 1974).

Affinity Electrophoresis

This technique relies on the ability of an ionic macromolecule to bind and transport a nonionic ligand through a gel under the influence of a moderate-to-high voltage electric field. An excellent example is the examination of glycogen binding to rabbit muscle phosphorylase (Winzor and de Jersey, 1989).

Equilibrium Dialysis

This technique uses a semipermeable membrane that freely permits passage of low molecular-weight ligands but not the enzyme or protein. Accordingly, the excess accumulation of a radioactively labeled ligand in the enzyme/protein-containing compartment depends on the concentration of enzyme as well as the number and affinity of ligand binding sites. A now classical example of the power of equilibrium dialysis is the determination by Englund et al. (1969) that DNA polymerase binds all four deoxy-nucleoside 59-triphosphate substrates at the same subsite in the catalytic center.

ESR Spectroscopy

This technique allows the experimenter to detect and characterize molecules with unpaired electrons (i.e., free radicals). The method also provides a way to analyze the binding of paramagnetic metal ions, such as hexaaquo Mn(II) ion, by quantifying the amount of unbound metal ion in equilibrium with an enzyme.

Fluorescence

This technique relies on an enhancement or quenching in intrinsic protein fluorescence (usually tryptophanyl and tyrosyl residues) upon reversible ligand binding (Lakowicz, 1999). Alternatively, interactions between proteins can be analyzed by changes in the fluorescence of cofactors (e.g., NADH, FADH, and N1,N6-ethenoATP) or extrinsic probes (e.g., dansyl chloride, ANS, pyrene, ethidium bromide, or proflavine monosemicarbazide). Protein–protein and protein–nucleic acid interactions may also be probed by changes in fluorescence anisotropy, a polarization parameter that may be evaluated either by steady-state or life-time measurements. When combined with the specificity of immunology or protein engineering of chimeric proteins containing green fluorescent protein, fluorescence has become a remarkably powerful and far-reaching technique – one that has fundamentally altered data acquisition in the molecular life sciences, particularly cell biology (see Section 4.5: Basic Fluorescence Spectroscopy).

Gel Filtration

This technique (also known as solute exclusion gel chromatography and molecular sieve chromatography) takes advantage of the fact that molecule-sized pores in the stationary medium bind and retard small solute molecules while excluding macromolecular solutes. For spherically shaped molecules, the fastest migrating solutes typically have the highest molecular weights. The most widely applied gel filtration method was developed by Hummel and Dreyer (1962).

Gel Retardation

This technique takes advantage of the observation that the electrophoretic mobility of DNA molecules and oligonucleotides is decreased when complexed to a protein (Fried and Crothers, 1981; Garner and Revzin, 1981). Two or more bands in an electrophoretic pattern are typically observed, the fastest corresponding to the free DNA and the others containing one or multiple proteins bound to the DNA. Cann (1989) provided a detailed treatment of this binding method.

Ion-Selective Electrodes

These electrochemical devices rapidly, reliably and selectively sense the concentration of an uncomplexed (or free) solute. The equilibrium potential developed after the electrode is immersed into a solution is directly related to log[Analyte]. The electrode is initially calibrated with suitable standardizing solutions of known analyte concentration, and the data are fitted by a linear leastsquares fitting protocol. The resultant working curve of Electrode Potential versus analyte concentration is then used to interpolate analyte concentration in experimental samples.

Isothermal Calorimetry

This method exploits the fact that most binding reactions are attended by the uptake or release of heat. An isothermal calorimeter measures DT ¼ (Tsample-cell  Treference-cell), when the sample cell contains the macromolecular receptor and an aliquot of ligand and the reference cell only contains the receptor and a ligand-free aliquot. The instrument records the amount of heat added to one of the cells to keep it isothermal with respect to the other cell.

Nitrocellulose Filters

This technique rests on the high-affinity binding of protein-bound DNA, but not DNA, to nitrocellulose membrane filters. DNA-binding to proteins can be rapidly and quantitatively analyzed by incubating [32P]-labeled DNA with a protein of interest (Jones and Berg, 1966; Riggs, Suzuki and Beecham, 1970). As little as 10–11 M DNA can be detected with high precision. The technique is most effective if nitrocellulose binding does not alter the protein’s affinity for DNA.

Chapter j 11 Regulatory Behavior of Enzymes

691

TABLE 11.2 Techniques for Measuring Ligand Binding Interactions with Enzymes – cont’d Technique

Comments

NMR Spectroscopy

This technique reports on changes in the local magnetic and electronic environment of a bound ligand or its macromolecular receptor by detecting changes in chemical shifts, ligand exchange kinetics, or other relaxation properties of the bound ligand.

pH Electrodes

This technique directly measures the free proton concentration in aqueous and certain mixed solvents. Some pH electrodes have sub-millisecond response times, extending their use to fast kinetic studies.

Plasmon Resonance

This highly sensitive technique capitalizes on the sensitivity of laser light impinging on the upper surface of a gold foil, the lower surface of which having covalently cross-linked to a ligand that interacts biospecifically with one or more proteins passing through a flow cell in which the foil is a component. A plasmon can be defined as an oscillating quantum mechanical wave of highly mobile electrons at or near the surface of metallic conductors (silver and gold foil). Binding of a protein (a polyelectrolyte) alters the distribution of these electrons to the effect that the oscillating wave changes the angle of reflectance of a incident laser light beam. The incident light rays create a evanescent wave that penetrates the foil ~100–150 nm before being ‘‘re-emitted’’ as reflected light rays. The evanescent wave is highly sensitive to protein interactions with the foil-bound ligand.

RIA and ELISA Methods

These antibody-based techniques (abbreviation for: Radio-Immuno Assay and Enzyme Linked ImmunoSorption Assay) are competitive ligand (antigen) binding techniques used primarily in clinical chemistry to analyze medically important biomarkers in human health and disease. The methods capitalize on the highly biospecific interaction of antibody (Ab) and antigen (Ag). In the RIA method, this attribute is combined with radiospecific activity measurements to achieve very high sensitivity for various ligands. In ELISA, the antibody is covalently attached to a support medium to facilitate separation of ligand free in solution and ligand bound to the Ab-matrix phase (Reaction: Ag þ Ab-matrix ¼ Ag$Ab-matrix). The interested reader should consult critical reviews of the technique (Hogg and Winzor, 1987; Hogg et al., 1987).

Ultracentrifugation

This technique represents a powerful means for analyzing the dependence of observed changes in sedimentation coefficients of complexes formed by self-association or by binding of dissimilar proteins to each other. In sedimentation velocity mode, a solution containing protein and ligand at thermodynamic equilibrium (Peq þ Leq ¼ P$Leq) is subjected to a centrifugal field of sufficient strength so as to pellet the protein (both as Peq and PLeq). The concentration of ligand in the region near the airwater meniscus is quickly depleted until it reaches Leq, the free ligand concentration. In sedimentation equilibrium mode (Van Holde and Baldwin, 1958), the centrifugal field is adjusted so as to establish an equilibrium gradient in the protein concentration, highest near the bottom and dropping off exponentially as one approaches the air-water meniscus. Sedimentation equilibrium is a well-defined process that obeys equilibrium hermodynamic, which allows one to determine the affinity and stoichiometry of protein-ligand and protein-protein binding interactions.

See also Table 4.1: Analytical Methods Employed in Rate Measurements.

but the SPR phenomenon reduces the reflected light intensity at certain wavelengths by altering the field strength of the evanescent wave. Binding interactions alter the refractive index close to the surface, such that the wavelength of the reflected light intensity minimum shifts. The BIACORE SPR probe allows the instrument to sense the spectral output of the reflected light. Because nearly all proteins have very similar specific refractive indices, SPR is a mass detector that does not depend on the nature of the interacting species. The surface of a sensor probe is coated with a carboxymethylated-dextran matrix that permits covalent attachment of a detecting molecule. An intrinsic advantage of this methodology is that extremely small volumes of material are needed because all of the binding interactions occur in an ultrathin layer. A disadvantage is that the output signal need not be a linear measure of the degree of protein-ligand complexation.

Some of these techniques described in Table 11.2 can be used to determine the actual kinetics of binding reactions. The use of fast reaction techniques, such as stopped-flow and temperature-jump, were already discussed in Chapter 10. As noted throughout this book, however, kinetic measurements are rarely as accurate as equilibrium measurements. A good rule-of-thumb is that, if you want to accurately determine a thermodynamic quantity, such as a dissociation constant, then apply an equilibrium measurement.

11.3. THE HILL EQUATION Recognizing that cooperative ligand binding is likely to be complicated, the English biophysicist A. V. Hill developed a simple ‘‘All-or-None’’ model to investigate the S-shaped oxygenation curve (Fig. 11.5) of hemoglobin.

Enzyme Kinetics

692

4 3

Peripheral Tissues

Y 1–Y

2

Lung

50

log

% of Total Sites Oxygenated

100

1 0 -1 -2 -3 -4

0

30-40

100

pO2 (mm Hg) FIGURE 11.5 Oxygen saturation curve for adult human hemoglobin in the presence of a physiologic bisphosphoglycerate (2 mM). Oxygen concentration is expressed as a partial pressure (units ¼ mm Hg). Lung O2 concentration z 100 mm Hg; peripheral tissue O2 concentration z 30 mm Hg.

In this model, a macromolecule with n fully saturable ligand sites is treated as though there are only two states: the uncomplexed macromolecule state M, and the fully bound state M$Ln. Consider the following equilibrium reaction: M þ nL ¼ MLn

11.4

for which the equilibrium dissociation constant is: K ¼

½M½Ln ½P$Ln 

11.5

Substituting for [M] in terms of [M$Ln], and writing the moles of bound ligand per total sites as a saturation function (i.e., Y L ¼ [M$Ln]/([M] þ [M$Ln]), where 0  Y L  1), we get Y L (1  Y L ) ¼ [L]n/K. Taking the logarithm of both sides, we get the Hill Equation:  YL log ¼ n log½L  log K 11.6 1  YL The linear region of the Hill Plot (Fig. 11.6) has a slope of nH, corresponding to the Hill coefficient. If the experimentally determined nH were found to equal the actual number n of ligand binding sites, the system would exhibit ‘‘infinite cooperativity’’ with respect to that ligand. No protein is known to satisfy fully the dictates of Hill’s infinite cooperativity treatment, and nH is invariably less than n. With hemoglobin, for example, oxygenation yields a Hill coefficient of about 2.8. Because hemoglobin has four subunits, each containing a heme group capable of undergoing reversible oxygenation, we could say that hemoglobin ‘‘behaves as though it has 2.8 infinitely cooperative sites.’’ The more rigorous statement would be: ‘‘While hemoglobin’s nH value of 2.8 indicates substantial subunit-subunit cooperativity, hemoglobin oxygenation is not ‘All-or-None,’ and intermediate states of oxygenation (i.e., Hb(O2), Hb(O2)2 and Hb(O2)3) must be present.’’

log [Ligand]

FIGURE 11.6 Hill plot describing the cooperative binding of n like ligand molecules. The dotted line is the theoretical binding curve (slope ¼ nH), as predicted by the Hill Equation. The plotted points represent the experimentally observed binding behavior. Notice that the points trace a linear dependence near the half-saturation point, allowing a reasonable estimate of nH.

Monod et al. (1963) showed that it is possible to convert the Hill Equation (see Section 11.2) into a corresponding rate equation involving kinetic constants in place of thermodynamic equilibrium constants. By replacing [E$Sn]/ [ETotal] by v/Vmax, they converted the fractional saturation expression to a corresponding velocity expression, yielding the expression:  v ¼ nH log½S  log K 11.7 log Vmax  v where v is the initial reaction velocity. The assumption of infinite cooperativity of the Hill treatment casts doubt on whether any enzyme kinetic model can ever completely satisfy this requirement. The Michaelis-Menten equation exhibits rapid pre-equilibrium binding of substrate, Purich and Fromm (1972b) demonstrated that Eqn. 11.7 can under such conditions apply to one-substrate enzyme reactions, such that v/Vm z Y A . If kcat is very large, however, rapid pre-equilibrium binding of substrate cannot occur, and v/Vm s Y A . For multisubstrate enzymes, use of the kinetic form of the Hill equation is dubious. Even if the second substrate is held at a saturating concentration, the relationship between [E$Sn]/[ETotal] and v/Vmax does not hold for ordered kinetic schemes. The situation is exacerbated when substrate binding is not purely of the rapid equilibrium type. There is also no a priori justification for assuming that the same rate is constant for conversion of substrate to product, especially with enzymes containing interacting active sites. Moreover, if product release is slow, as is the case for many enzymes, v/Vm s Y A . Finally, the Hill coefficient nH has been used as a rough index of enzyme responsiveness to changes in substrate concentration. For an enzyme obeying the simple MichaelisMenten equation, nH ¼ 1.0, and the sensitivity parameter [S]0.9/[S]0.1 ¼ 81, where [S]0.9 is the substrate concentration at which v/Vm ¼ 0.9, and [S]0.1 is the corresponding

Chapter j 11 Regulatory Behavior of Enzymes

corrected for nonspecific binding

_ n / [L]Free

substrate concentration at which v/Vm ¼ 0.1. As nH increases, this sensitivity parameter changes dramatically: for nH ¼ 2.0, [S]0.9/[S]0.1 ¼ 9; for nH ¼ 4.0, [S]0.9/[S]0.1 ¼ 3; for nH ¼ 8.0, [S]0.9/[S]0.1 ¼ 1.73; and for nH ¼ 16.0, [S]0.9/ [S]0.1 z 1.3. Therefore, highly cooperative enzymes exhibit significant changes in activity over a very narrow range of substrate concentration, making them ever-sharper ‘‘offon’’ switches as the Hill coefficient rises. In summary, although the Hill treatment played a historically notable role in the development of theories for enzyme cooperativity, it is inadvisable to employ the Hill treatment to analyze cooperative substrate binding as well as the initial-rate kinetics of cooperative enzymes.

Amount Bound

693

uncorrected data corrected data

[Ligand]

Slope = -1/Kd

Bmax

_ n

11.4. THE SCATCHARD EQUATION The well-known Scatchard Equation, which defines ligand binding in a manner permitting graphical estimation of ligand stoichiometry and their dissociation constant(s), may be written as: v n v ¼  ½L Kd K d

FIGURE 11.8 Saturation curve and the corresponding linear Scatchard plot (inset) obtained after correcting observed saturation curve for weak, nonspecific ligand binding. Scatchard plot for the corrected saturation data. Inset, Plot of the amount of bound ligand versus the concentration of labeled ligand, showing apparent lack of saturation in the top curve. Note that the curve becomes linear at high ligand concentration, thus defining the linear curve for weak nonspecific binding. Translating this same linear region of the top curve to the origin creates a reference curve for nonspecific binding. The dashed line is obtained by subtracting

11.8

ν /[L]F

where v is the fractional saturation of n fully saturable binding sites, [L] is the concentration of free (i.e., uncombined) ligand L, and Kd is the dissociation constant (Scatchard, 1949). In practice, most binding experiments are affected to various extents by nonspecific ligand binding. Such behavior occurs as a consequence of weak hydrophobic or electrostatic interactions between a protein and ligand. Nonspecific binding necessitates corrections that are usually made by subtracting the increment due to nonspecific binding from the total bound ligand to obtain the corrected saturation curve (Fig. 11.7).

n ν FIGURE 11.7 Scatchard plot for the independent binding of ligand X at n sites. The slope equals Kformation or 1/KD, and the horizontalaxis intercept yields the stoichiometry of ligand binding (i.e., mol ligand bound per mol oligomeric protein).

11.4.1. A Modified Scatchard Equation Accounts for Steric Hindrance Amongst Sites There are many known cases where the binding of ligand at one site on a protein is found to hinder subsequent ligand binding at other sites on the macromolecule. This situation arises when the distances between binding sites is not much greater than the molecular size of the ligand or when ligand binding disturbs the structure of its binding partner. Nonideal binding is particularly likely with enzymes having macromolecular substrates, such as protein kinases, polysaccharide-modifying enzymes, and certain phospholipases. Another famous example is the intercalation of acridine orange, ethidium bromide, and other hydrophobic substances like the carcinogen benz[a]pyrene into the voids between successive base-pairs in double-stranded DNA. In such cases, the standard Scatchard equation is inadequate, and the following treatment is more appropriate. Letting CF, C0, and bap, respectively, represent the free ligand concentration, the ideal site concentration at saturation, and the fraction of sites actually bound at saturation, we can define bapCo as the concentration of total sites bound, or CS. This approach allows us to write the equilibrium expression: Kap ¼ CF(CS  CB)/CB ¼ CF(bapC0  CB)/CB. Dividing the right-hand numerator and denominator by C0, and letting r equal CB/C0, the above equation becomes: Kap ¼ CF(bap – r)/r, which can be rearranged to yield: bap r r ¼  CF Kap Kap

11.9

Enzyme Kinetics

694

This equation defines the intrinsic dissociation constant for nonspecific binding of dyes to double-stranded DNA. (Note: Eqn. 11.10 reduces to the classical Scatchard equation when m ¼ 1.) This probabalistic model assumes that natural polymeric acceptors are sufficiently long to be treated as if they are infinite, and ligand binding to oligonucleotides cannot be analyzed.

11.4.2. The Scatchard Analysis may be Extended to Deal with Two Classes of Binding Interactions – One Strong and One Weak To treat this case, we will use two separate saturation functions, one for each class of sites: Class-1:

r1 b1 r1 ¼  ½LFree  K1 K1

11.11

Class-2:

r2 b2 r2 ¼  ½LFree  K2 K2

11.12

where r1 and r2 are defined as the fraction of ligand bound to sites 1 and 2, respectively, and b1 and b2 are the fractions of bound sites that are Type-1 and Type-2. We can define rT to be the sum of r1 and r2, and the analytical technique generally allows us to measure total ligand bound, or rT[LTotal]. Thus, rT/[L] ¼ {(b1  r1)/K1} þ {(b2  r2)/K2}, or, upon substituting (rT  r2) for r1, rT/[L] ¼ {[b1  (rT  r2)]/K1} þ {(b2  r2)/K2}. A plot of (rT/[L]) versus rT begins on the vertical axis at the intercept (equal to b1/K1 þ b2/K2) and has an initial slope (i.e., at rT » 0) equal to 1/K1; this initial linear region extrapolates to a horizontal-axis value of b1. At even higher ligand concentrations, the curve bows to the right and eventually reaches the horizontal intercept rT. Thus, on the basis of these experimentally derived values of K1 and b1, we can iteratively fit the entire experimental binding curve. Scatchard, Coleman and Shen (1957) first applied this treatment to investigate thiocyanate binding to human

two sites of different affinity

Slope = -1/K1

/ [L]Free

For this modified Scatchard equation, a plot of r/CF versus r will have a slope of 1/Kap and a horizontal-axis intercept of bap. An important example of the use of Scatchard analysis of steric hindrance is the intercalation of polycyclic aromatic dyes between successive base pairs in DNA (Bauer and Vinograd, 1970; Bresloff and Crothers, 1975). The limiting value of bap is about 0.25, indicating that about one in every four base-pairs can accommodate one acridine molecule at saturation. Random binding of a ligand to multiple m-residue sequences in an indefinite acceptor lattice has been treated by McGhee and von Hippel (1974), who obtained the following expression: !m1 r ð1  mrÞ 1  mr 11.10 ¼ CF K 1  ðm  1Þr

Slope = -1/K2 n1

n2

FIGURE 11.9 Scatchard analysis for strong and weak ligand binding. Observed curvilinear binding curve corresponds to the sum of two linear regions, in this case representing one weak and one strong binding site. The analysis works best when the respective binding constants have significantly different values. (Note that additional weak sites would have the effect of increasing the moles of ligand bound at higher ligand concentrations.)

serum albumin. They observed ten binding sites for this anion, each having a dissociation constant of 1–2 mM, as well as another 30 sites, each having an average dissociation constant w40 mM. Winzor and Sawyer (1995) have argued that the Scatchard method can be misleading with respect to the graphically evaluated parameters. A better practice is to employ computer programs allowing for analysis of binding data without using any transformations based solely on a Scatchard plot. If the experimenter still wishes, the final fitted data can still be plotted as a Scatchard plot, but the actual theory line should be based on the results of the computer fit, and not a ‘‘best fit’’ based on least-squares treatment for a line through plotted data points. Such programs also provide ways to correct for nonspecific binding as well as experimental error.

11.5. WYMAN’S LINKED FUNCTION ANALYSIS Ever since the cooperative heme-heme interactions of hemoglobin were first reported by Bohr (1903), this remarkable ‘‘information-transfer’’ reaction has excited the imagination of biologists, chemists, and physicists. When a macromolecule engages in two different association reactions with ligands X and Y at nearby sites, it is likely that the binding functions for these ligands will interact. Such sites are said to be ‘‘linked,’’ and the special saturation function describing their interactions is called a linked function. The following treatment of linked function behavior closely follows that developed by Wyman (1948; 1964). For macromolecule P having q sites, each able to

Chapter j 11 Regulatory Behavior of Enzymes

695

combine with one molecule of ligand X, one can use X to denote the total concentration of ligand bound by the macromolecule: q X X ¼ ½Po  iKi xi 11.13 i¼0

where [P0] is the concentration of the uncomplexed macromolecule, x is the activity of the ligand, and Ki is the apparent macroscopic association constant for the reaction: P þ iX ¼ PXi

11.14

Note also that K0 ¼ 1. The total concentration of macromolecule in all forms is: PT ¼ ½P1  þ ½P2  þ ½P3  þ $$$ þ ½Pq  q X iKi xi P ¼ ½Po 

11.15 11.16

i¼0

Using X to designate the amount of ligand bound per mole of macromolecule, we get: . 11.17 X ¼ S iKi xi S Ki xi i

i

If the Ki’s are independent of the total extent of the association reaction, then the above expression for X can be written in the following form: P v ln Ki xi 11.18 X ¼ v ln x The fractional saturation of the macromolecule (written here as x) is then: !, P X v ln Ki xi q 11.19 x ¼ ¼ v ln x q In the case of a second ligand binding to another nearby site, then the total concentration of the macromolecule in all its forms will be: q X r X P ¼ ½P0  Kij xi yj 11.20 i¼0 j¼0

where Kij is the apparent equilibrium constant for the reaction: P þ iX þ jY ¼ PXi Yj

11.21

We can now write separate expressions for the amounts of ligands X and Y bound per mole of macromolecule: #, " q X r X i j v ln x 11.22 X ¼ v ln Kij x y i¼0 j¼0

Y ¼

"

v ln

q X r X i¼0 j¼0

i j

Kij x y

#,

v ln y

11.23

PP PP For the double-sum , v ln is a perfect differential, called a linked function, ! ! vX vY ¼ 11.24 v ln y v ln x x

y

Note that, if the binding of X alters the binding of Y, the binding of Y must likewise alter the binding of ligand X. A linked function provides a quantitative expression that accounts for how ligands binding at interactive sites will influence each other. The concept of linkage lies at the heart of all ligand binding cooperativity.

11.6. THE MONOD-WYMAN-CHANGEUX MODEL The MWC model (named in honor of its formulators Nobelist Jacques Monod, Jeffries Wyman, and Jean-Pierre Changeux) seeks to explain the positive ligand-binding cooperativity observed with many allosteric proteins/enzymes. Contrary to overly simplified descriptions given in many elementary biochemistry textbooks, cooperativity in the MWC model need not result from an increase in ligand binding affinity as progress is made toward a higher degree of site occupancy. Indeed, the MWC model in its exclusive binding form requires that the intrinsic binding constants are the same for each binding site for like ligands within one conformational form. As will become evident from the discussion below, cooperativity in the MWC model actually results from the allosteric interactions controlling the availability of sites in two conformational forms, called the R and T states, that exist in equilibrium with each other, even in the absence of any ligand.

11.6.1. Several Key Properties of Allosteric Systems Suggested the Symmetry-Conserving MWC Model Monod, Wyman and Changeux (1965) based their model on an extensive analysis available in the extant literature on allosteric systems, particularly the seminal observations of Umbarger and Brown (1957; 1958) on threonine deaminase, Yates and Pardee (1956) on aspartate transcarbamoylase (ATCase), and Cohen et al. (1952) on the aspartokinases. In their original paper, they offered the following properties and definitions. Property-1. Homotropic effects are defined as cooperative binding interactions between sites that bind ligands that are identical in structure. For a hypothetical twosubstrate enzyme catalyzing the reaction: A þ B / Products, we would say that homotropic interactions refer to how substrate A molecules already occupying their binding sites influence the binding of additional substrate A molecules at similar sites on the oligomeric enzyme. Although rarely seen, the enzyme may also

Enzyme Kinetics

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exhibit homotropic effects with respect to the binding of substrate B molecules at sites that bind B. Property-2. Heterotropic interactions are defined as interactions between unlike ligand binding sites. In the above example, the binding of ligand A at its sites may interact heterotropically with ligand B binding at its sites on the protein, and vice versa. If a heterotropic interaction occurs, we say that the sites for A and B are linked. Property-3. Ligand binding cooperativity depends on an All or None (i.e., concerted) transition in the protein’s quaternary structure. This concerted transition, which occurs in the absence or presence of bound ligand, has the effect of simultaneously altering all sites for the same ligand. Property-4. Homotropic interactions are mainly responsible for the cooperativity of allosteric proteins. Cooperativity in a heterotropic ligand binding interaction stems from its linkage to homotropic cooperative ligand binding, and the MWC model treats them in such a manner. Any conditions, treatments, or mutations that change homotropic interactions will likewise alter heterotropic interactions. That an enzyme’s oligomeric structure is essential for cooperativity would be demonstrated by the observation that treatment of the enzyme with a reagent resulting in subunit dissociation should abolish all cooperativity.

11.6.2. MWC Ligand Saturation Functions are Simple Polynomials Accounting for Ligand Binding to One or Two Conformationally Distinct States of the Enzyme

constants KR ¼ [F][Ri–1]/[Ri] and KT ¼ [F][Ti–1]/[Ti]. In the exclusive binding model, KT is extremely large, such that ligand F only binds to R0, R1, R2, and R3. Assumption-5. There is reciprocal thermodynamic linkage among the binding sites, such that the relative concentrations of R- and T-states can be altered by changes in ligand concentration. This property is merely an example of the application of Le Chatelier’s Principle to the law of mass action. Assumption-6. All subunits within any single molecule are constrained by symmetry, such that no oligomer ever contains both R- and T-state subunits. In the context of the scheme for dimers, this statement excludes the possibility of R0T0 or R1T0 for a dimer. Assumption-7. Allosteric transitions between R0 and T0 are symmetry-conserving. There are therefore no mixedstate forms, such as R1T3, R2T2, or R3T1. Assumption-8. All binding sites with the R-state are equivalent and remain so at all degrees of saturation. Likewise, if ligand X binds to the T-state, then all binding sites for X to the T-state are equivalent and remain so at all degrees of saturation. Assumption-9. Ligand binding is rapid, such that equilibrium expressions with thermodynamic equilibrium constants can be used. This requirement excludes the need to determine the time-dependence of binding interactions. Two cases can be analyzed, based on whether the T-state binds ligand (Fig. 11.10). In the exclusive-binding case, oligomers in the T-state cannot bind ligand F and only

The MWC saturation function can be derived by applying the following assumptions. Assumption-1. The protein is assumed to be an oligomer containing a finite number n of identical subunits (called protomers). Therefore, one is dealing with equilibria among a definite number of components, defined by polynomial functions with exponents of degrees n and n–1. Assumption-2. The protein is assumed to exist only in two different, rapidly inter-converting conformational states. These two states are referred to as the R- and T-states, standing for ‘‘relaxed’’ and ‘‘taut’’ conformations. Assumption-3. There are always finite concentrations of ligand-free species R0 and T0, as defined by the allosteric constant L (i.e., L ¼ [T0]/[R0]). Notably, the T0 and R0 states always remain in rapid thermodynamic equilibrium with each other, even in the absence of ligand F or allosteric effectors. Assumption-4. The R- and T-states have different affinities for the ligand, and by convention, the R-state exhibits the higher affinity. Generally, each ligand F binding interaction is defined by the corresponding dissociation

FIGURE 11.10 Two limiting cases for a tetrameric protein operating by the Monod-Wyman-Changeux cooperativity model. Left, Exclusive Binding Scheme, wherein ligand F cannot bind to T0, and no T1, T2, T3 or T4 states can be formed. Right, Non-exclusive Binding Model, in which ligand F can combine with R0, R1, R2, and R3 as well as with T0, T1, T2, and T3, albeit with different affinities for R- and T-states. See text for additional comments.

Chapter j 11 Regulatory Behavior of Enzymes

697

site on R1. Note that the ‘‘2’’ in the denominator indicates that two sites on R2 can undergo dissociation:

11:27

Substituting the earlier expression for equation yields:

] = α2[

[

FIGURE 11.11 Free energy diagram showing the effect of the relative stability of R0- and T0-states on the shape of the ligand saturation function. Note that the T0-state serves as a buffer controlling the availability of subunits that are supplied in an all-or-nothing manner. The Gibbs free energy difference between successive R-states must be equivalent for the KF values to be equivalent.

serves to buffer the R0 concentration, as the ligand F concentration is varied. In the nonexclusive-binding case, oligomers existing in the T-state do bind ligand F, albeit more weakly than the R-state. In this case, the T-states no longer strictly serve a reservoir controlling the R0 concentration as the concentration of ligand F is varied. The case of a dimer illustrates the steps taken in the derivation of the MWC saturation function for the model: T0

R0

R1

R2

Scheme 11.1

into the last

]

11:28

The saturation function Y F accounts for the moles of bound ligand F divided by the total moles of protein binding sites, in this case requiring the ‘‘2’’ in the denominator to convert each dimer concentration into the concentration of sites: 11:29 Substituting from the earlier expressions into the last equation yields:

a þ a2 YF ¼ ¼ 2 L þ 1 þ 2a þ 2a

11:30a að1 þ aÞ

L

þ ð1 þ aÞ2

11.30b

The value of the allosteric constant L is a measure of the relative abundance of T0- and R0-states, which in turn strongly affects the value of a needed to give half-maximal saturation of the total binding sites. The vertical dotted lines in each of the ligand saturation plots in Fig. 11.12 are

where there is a pre-existing equilibrium between ligandfree states T0 and R0:

]/[

]

Positive Cooperativity

11:25

To simplify the subsequent expressions, the concentration of ligand F can be normalized with respect to its dissociation constant KF, such that: a ¼ [F]/KF. Ligand F binds to either of two open sites on R0, requiring the ‘‘2’’ in the numerator to convert the R0 concentration into the concentration of available binding sites:

nYF / [L]Free

= [

No Cooperativity

Negative Cooperativity

11:26 nYF

An equivalent statement is that there are two ways for ligand F to bind to the R0 dimer, but only one way for F to dissociate from R1. Ligand F can also bind to the one open

FIGURE 11.12 Scatchard plot of ligand binding exhibiting negatively cooperative, independent, and positively cooperative binding. See text for comments.

Enzyme Kinetics

698

provided to indicate how the a change in the value of L results in a shift in the position of the mid-point of the saturation curves. The actual value of L for a particular allosteric protein is determined by differences in the stability subunits in the R and T conformations. These structural features are determined by the protein’s amino acid sequence, and mutations affecting the relative stability of the T0- and R0-states would certainly be expected to alter the ligand concentration needed for half-maximal saturation.

11.6.3. The Saturation Function may be Generalized to Explain Ligand Binding to an Oligomer with n Symmetry-Conserved Sites The nonexclusive binding of ligand F to the R- and T-states is defined by the following set of linked reactions: R0 + F R1 + F R2 + F R3 + F

R0

T0

R1 R2

T0 + F

R3 R4

T1 + F T2 + F T3 + F

T1 T2

½R1  ¼ n½R0 

½F KR

ðn  1Þ ½F ½R1  2 KR « 1 ½F ½Rn  ¼ ½Rn1  n KR

½R2  ¼

½F KT ðn  1Þ ½F ½T2  ¼ ½T1  2 KT « 1 ½F ½Tn  ¼ ½Tn1  n KT ½T1  ¼ n½T0 

11.33

The ligand saturation function Y F accounts for the fraction of total sites occupied by ligand F:

YF

 1½R1  þ 2½R2  þ . þ n½Rn  þ1½T1  þ 2½T2  þ . þ n½Tn   ¼  ð½R0  þ ½R1  þ . þ ½Rn Þ n þð½T0  þ ½T1  þ . þ ½Tn Þ 

11.34

where the integers 1, 2, 3, n indicate the number moles of ligand present on species R1 and T1, R2 and T2, R3 and T3, ., Rn and Tn, respectively. By defining the reduced concentrations of ligand (i.e., a ¼ [F]/KR; likewise, if c ¼ KR/KT, then ca ¼ [F]/KT), the saturation function Y F for nonexclusive binding becomes:

T3 T4

Scheme 11.2 For the case depicted in Scheme 11.2, the protein is distributed into ten species, such that: ½RTotal  ¼ ½R0  þ ½R1  þ ½R2  þ ½R3  þ ½R4 

11.31a

½TTotal  ¼ ½T0  þ ½T1  þ ½T2  þ ½T3  þ ½T4 

11.31b

The ligand-free forms are in thermodynamic equilibrium, as indicated by: ½T0  ¼ L½R0 

stoichiometric coefficients are shown in the following equilibrium relationships:

11.32

In the MWC model, ligand F interactions with an oligomeric protein are governed by a single microscopic dissociation constant KF. For an oligomeric protein Pn, however, the concentration of binding sites is not merely [R0]; instead, the concentration of unoccupied sites is n[R0], meaning that, for the first ligand binding reaction (R0 þ F / R1), F can react with any one of the n empty binding sites. For a tetramer, the concentration of ligand-free subunits is thus 4[R0], or 4[R0]. In the reverse reaction (R1/ R0 þ F), the concentration of occupied sites is 1[R1]. These stoichiometric coefficients are also known as statistical factors that account for the multiple ways that a ligand can be added to or lost from. For a tetramer, these statistical factors are 4, 3/2, 2/3, and 1/4. These

Y FðNONEXCLUSIVEÞ ¼

Lcað1þcaÞn—1 það1þaÞn—1 Lcað1þcaÞn það1þaÞn

11.35

Notice also that the saturation function for exclusive binding to the R-state species is a limiting case that can be readily obtained by eliminating all ligand-bound Tstates: Y FðEXCLUSIVEÞ ¼

að1 þ aÞn—1 Lþað1þaÞn

11.36

Note also that allosteric proteins that are composed of numerous protomers (i.e., n is large) will be defined by correspondingly higher-order polynomial functions, and all other parameters being equal, they will exhibit sigmoidal curves that rise more sharply than those having fewer subunits. For any system having an n value of 1, there is no cooperativity. In this case, the saturation function becomes: Y FðEXCLUSIVEÞ ¼

að1 þ aÞ0 Lþað1þaÞ1

¼

a Lþað1þaÞ

11.37

This equation lacks higher-order polynomial terms and cannot give rise to a sigmoidal (or S-shaped) saturation curve. Note also that, when n ¼ 0 and L ¼ [T0]/[R0] ¼ 0, the

Chapter j 11 Regulatory Behavior of Enzymes

699

enzyme is a monomer that exists entirely in the R-state, and the above equation reduces to: YF ¼

½R1  ½R0 ð½F=KF Þ a ¼ ¼ 11.38 ½R0  þ ½R1  ½R0 ð1 þ ð½F=KF ÞÞ 1þa

The latter equation is formally equivalent to the Michaelis-Menten equation.

11.6.4. The MWC Model Also Accounts for the Effects of Positive and Negative Allosteric Modifiers Interactions of the protein with activators and inhibitors finetune the magnitude of the apparent allosteric constant L9:

L0 ¼

n X

i¼1 n X i¼1

¼

½Ti  ½Ri 

¼

½T0  ½R0 

½I ½T0  1 þ KI ½R0  1 þ

!n

n X ½Ti  i¼1 n X

½T0 

½Ri  ½R  i¼1 0

11.39

ð1 þ bÞn !n ¼ L ð1 þ gÞn ½A

DGswitch-over

KA

Therefore, the following equation accounts for the case where ligand F binds exclusively to the R-state in the presence of activator and inhibitor: YFðEXCLUSIVEÞ ¼ ¼

að1 þ aÞn—1 L0 það1þaÞn

að1 þ aÞn—1 ð1 þ bÞn L þ að1 þ aÞn ð1 þ gÞn

value of the apparent allosteric constant will be determined by the relationship: L9 ¼ L /(1 þ g)n, where g is the activator’s reduced concentration (g ¼ [A]/Ka), and n is the number of activator binding sites. Because a smaller L9 value decreases the overall stability of the T-state, less energy is required to shift the equilibrium toward the Rstate. The concentration of ligand F is the only source of Gibbs free energy to shift the T-to-R equilibrium, such that the saturation curve will be shifted to the left (i.e., toward lower [F]) in the presence of activator. It seems doubtful that an activator and inhibitor for any given enzyme are simultaneously present in the same cellular compartment. Were it so, the enzyme would be subjected to a molecular-level ‘‘tug of war,’’ where the magnitude of L9 would be determined by the ratio (1 þ b)n/(1 þ g)n. If the value of this ratio exceeds unity, the inhibitor’s effects would dominate, and the mid-point for half-maximal binding of ligand F would shift to the right. If the value of this ratio is less than unity, the inhibitor’s effects would dominate, and the mid-point for half-maximal binding of ligand F would shift to the left. The so-called switch-over point is the reduced ligand F concentration aswitch, at which the apparent allosteric constant L9 ¼ 1 (Wyman and Gill, 1990).

11.40

Inhibitor I binds to and stabilizes the T0-state of an allosteric protein obeying the MWC exclusive binding model. In the absence of any activator (i.e., g ¼ 0) the value of the apparent allosteric constant will be determined by the relationship: L9 ¼ L(1 þ b)n, where b is the inhibitor’s reduced concentration (b ¼ [I]/Ki), and n is the number of inhibitor binding sites. At higher L9 values, the overall stability of the T-state increases, a higher concentration of ligand F is required to drive the equilibrium toward the R-state. Because the concentration of ligand F is the only source of Gibbs free energy to shift the T-to-R equilibrium, it follows that the saturation curve is shifted to the right (i.e., toward higher [F]) in the presence of inhibitor. In just the opposite manner, binding of activator A stabilizes the R-state. In the absence of any inhibitor (i.e., b ¼ 0), the

  n ¼ RT ln Lð1þbÞ ¼ RT ð1þgÞn

ln L0 ¼ 0; at aswitch-over

11.41

When a is at the switch-over point, the energy difference between R- and T-states is zero. Finally, while the Monod-Wyman-Changeux model accounts for key aspects of positive cooperativity, the model cannot account for negative cooperativity. Negative cooperativity requires there to be changes in the individual binding constants for ligand binding. Any experimental evidence demonstrating negative cooperativity immediately excludes the MWC model as a candidate mechanism. That said, Viratelle and Seydoux (1975) advanced a pseudoconservative transition model that explains both positive and negative cooperativity in terms of a two-state model. At least one of the states must contain two sites of differing affinity for the same ligand, such that the T-state is in equilibrium with an R-state, itself having n / 2 high-affinity sites and n/2 low-affinity sites.

11.7. THE KOSHLAND-NE´METHY-FILMER MODEL This ligand-induced conformational change model introduced by Koshland, Ne´methy and Filmer (1966) explains allosteric cooperativity in terms of sequential changes in quaternary structure. Unlike the MWC concerted-transition

Enzyme Kinetics

700

model, the fraction of total sites occupied in the KNF model exactly matches the fraction of total subunits capable of binding ligands. Furthermore, the KNF model allows for the possibility of positive cooperativity, independent binding, and negative cooperativity by assigning different dissociation constants (e.g., K1 ¼ [P0][F]/[PF1], K2¼ [PF1][F]/[PF2], K3 ¼ [PF2][F]/[PF3],$$$, Kn ¼ [PFn-1][F]/[PFn]) for n sites that can bind ligand F. Such binding behavior is presented as a Scatchard plot in Fig. 11.12. Cooperativity in the KNF model is actually the result of the allosteric interactions that control the affinity of sites for ligand.

11.7.1. The KNF Model is Rooted in Adair’s Treatment of Polyvalent Ligand Binding Interactions

hexaphosphate. Singular value decomposition of the resulting difference spectra at different pO2 values revealed two distinct optical transitions, indicating that the optical response to O2 binding is nonlinear. The degree of nonlinearity was evaluated by fitting the data to the Adair equation, while accounting for two optical parameters: D31 representing the change upon binding the first and second O2 molecules, and D32 (w0.8  D31) for the change upon binding the third and fourth O2 molecules. The equilibrium constants were: b1 ¼ 0.081 Torr1, b2 ¼ 2.53  103 Torr2, b3 ¼ 1.25  103 Torr3, and b4 ¼ 1.77  106 Torr4, where the Adair equation is written as: YS ¼

b1 pO2 þ b2 ðpO2 Þ2 þ b3 ðpO2 Þ3 þ b4 ðpO2 Þ4

1 þ b1 pO2 þ b2 ðpO2 Þ2 þ b3 ðpO2 Þ3 þ b4 ðpO2 Þ4 11.43

For a detailed discussion of the structural ramifications and cooperativity of O2 binding to hemoglobin, the reader should consult Perutz et al. (1998). The general Adair equation Y S for the binding of a ligand X to a multi-site protein where Ki represents the macroscopic association constant for the ith site and n is the total number of sites is  i n  Y X 1 i½Xi i K j¼1 i i¼1 nY s ¼ 11.44   i n X Y 1 i½Xi  1þ K j¼1 i i¼1

In 1925, Gilbert Adair developed a saturation function describing the binding of oxygen to hemoglobin. Adair anticipated the possibility that the microscopic dissociation constants can change when a ligand binds to a multi-subunit protein, and the KNF model has been occasionally referred to as the Adair-Koshland-Ne´methy-Filmer model. Likewise, as mentioned in Section 11.6.2, Pauling (1935) developed a heme-heme interaction model that has led some to refer to the KNF model as the Pauling-Koshland-Ne´methy-Filmer model. Let k1 be the microscopic dissociation constant for O2 binding to Hb and k2, k3 and k4 likewise for O2 binding to the second, third, and fourth sites. If Y S represents the fractional saturation of the ligand binding (i.e., fraction of occupied sites divided by total concentration of all sites), then:  ½O2  3½O2 2 3½O2 3 ½O2 4 þ þ þ K1 K2 K1 K2 K3 K1 K2 K3 K4 K1 YS ¼  4½O2  6½O2 2 4½O2 3 ½O2 4 1þ þ þ þ K1 K2 K1 K2 K3 K1 K2 K3 K4 K1 11.42

11.7.2. The KNF Model Incorporates Elements of Pauling’s Site Interaction Model

There is a progressive increase in oxygen binding affinity upon oxygenation of hemoglobin, with K1 z 300 K4 in the absence of bisphosphoglycerate (BPG) and K1 z 1,000 K4 in the presence of 2 mM BPG. In both cases, oxygen binding by hemoglobin exhibits positive cooperativity (Tyuma, Imai and Shimizu, 1973). Analogous Adair-type equations can be derived for the binding of other ligands to multi-site proteins. If the microscopic dissociation constants are not equal, then the Adair equation predicts a non-hyperbolic (i.e., cooperative) curve. Values for K1 and K4 can be obtained by studying ligand binding at very low and very high ligand concentrations. Values for K2 and K3 can be obtained from computer fitting protocols or by using a Hill plot. Ownby and Gill (1990) measured changes in the Soret band (400–440 nm) spectrum of hemoglobin A0 at different O2 partial pressures (pO2) in the presence of 30 mM inositol

In the mid-1930s, Linus Pauling developed a heme-heme interaction model to explain what was the most accurate and extensive data (Ferry and Green, 1929) on the oxygen binding equilibrium for horse hemoglobin in phosphate and borate buffers. Their experimental data points for the ratio y (defined as mol O2 absorbed per mol O2 bound at saturation) versus p (defined as the oxygen pressure, expressed in units of mm mercury) at different pH values fell on or near a smooth curve produced by the Hill equation: y ¼ Kpn/(1 þ Kpn), where K is a proportionality constant, and n ¼ 2.6. As noted in Section 11.2, the Adair treatment requires four formation constants K1, K2, K3, and K4 to fit data sets plotted as the fractional saturation y versus the partial pressure p of the reversible ligand O2. In the absence of any detailed subunit structural data, Pauling (1935) offered a two-parameter cooperativity model, based on the following postulates:

While the Adair treatment represents a useful method for analyzing ligand binding phenomena, it provides no explanation for why initially identical sites would behave as though they have different dissociation (or association) constants.

Chapter j 11 Regulatory Behavior of Enzymes

701

(a) hemoglobin molecule contains four hemes, each having the equivalent affinity for O2; (b) each heme interacts with k neighboring hemes, and such connections occur in pairs; and (c) the interactions for connected heme-pairs are likewise equivalent. Pauling (1935) distinguished: the k ¼ 0 case, wherein no interaction occurs between hemes, as defined by the independent binding equation: y ¼ Kp/(1 þ Kp), with n ¼ 1 and K has inverse pressure units of p–1; the k ¼ 1 case, in which heme-heme interaction occurs, leading to the equation: y ¼ Kpn/(1 þ Kpn), where 1  n  2; and the k ¼ 2 case, where hemes are located at the corners of a square, and interactions occur along the sides of the square, as shown in the following interaction diagram. Hb

Hb

Hb

Hb

Hb

Hb

Hb

Hb

Hb

Hb

k=0

k=1

Hb Hb k=2

To obtain an equation showing a higher-order dependence of y on O2 partial pressure on the basis of the square arrangement, we note that, at the oxygen pressure p, the relative amounts of the six different molecular species present are as follows: 1

4α K ap2p 2

4 K ap p

Hb

Hb

Hb

HbO2

Hb

HbO2

Hb

Hb

Hb

Hb

Hb

HbO2

Hb

HbO2

Hb

HbO2

O2Hb

HbO2

O2Hb

HbO2

O2Hb

HbO2

O2Hb

Hb

2K ap2p 2

2

4α K ap3p3

α4K ap4p4

Values of the constants K9 and a define: (a) DGoxy, representing the free energy change for oxygen addition to heme, (e.g., DGoxy ¼ RT ln K9); and (b) DGstab, representing the stabilization free energy per pair-wise interaction, shown above as heavy red lines (e.g., DGstab ¼ RT ln a). The resulting equation is: y ¼ [K9p þ (2a þ 1)K29p2 þ 3a3K39p3 þ a4K492p4]/[1 þ 4K9p þ (4a þ 2) K29p2 þ 4a3K39p3 þ a4K492p4]. This equation provided a good fit to the oxygenation data of Ferry & Green (1929), when K9 ¼ 0.033 and a ¼ 12. Gill et al. (1986) analyzed cooperative free energies for nested allosteric models as applied to ligand binding to human hemoglobin, based on the detailed cooperative free energies for each of the ten different ligated cyanomet species (Smith and Ackers, 1985). Their approach is an extension of the general principle of hierarchical levels of allosteric control, or ‘‘nesting’’ (Wyman, 1972), based on the simple two-state MWC model. Cooperative binding in the T-state is

described using simple interaction rules first devised by Pauling (1935) and requires only three parameters to describe the cooperative free energies of the 10 ligated species of hemoglobin. Hellmann and Decker (2002) also applied a nested MWC cooperativity model to analyze the mechanism of GroEL-associated ATPase activity, based on its existence in different protein-folding cycle conformations.

11.7.3. The KNF Model Accounts for Both Positive and Negative Cooperativity The Koshland-Ne´methy-Filmer model is based on the following assumptions. First, in the absence of ligand (or substrate), the oligomeric protein exists in only one conformational form. Second, ligand binding first induces a conformational change within the subunit to which it alone is bound (i.e., unlike the MWC model, there is no requirement for maintaining oligomer symmetry). Third, that induced conformational change alters the subunit-subunit interface, allowing for the possibility that one or more adjacent subunits will alter its vacant sites will exhibit increased affinity (positive cooperativity), reduced affinity (negative cooperativity), or no change in ligand affinity (independent binding). Fourth, all binding interactions are defined thermodynamically in terms of dissociation equilibrium constants. To illustrate the steps taken in the derivation of the KNF saturation function, consider Scheme 11.3 describing ligand binding for a dimer.

P0

P1

P2

Scheme 11.3

The first molecule of ligand F binds to either of two open sites on P0, requiring the ‘‘2’’ in the numerator to convert the P0 dimer concentration into the concentration of available binding sites on that dimer:

11:45

Ligand F can also bind to the one open site on P1. Note that the ‘‘2’’ in the denominator indicates that two sites on P2 can dissociate:

11:46

Enzyme Kinetics

702

The saturation function accounts for the moles of bound ligand divided by the total moles of protein binding sites, in this case requiring the ‘‘2’’ in the denominator to convert each dimer concentration into the concentration of sites: 11:47 Substitution for each protein-ligand species yields:

11:48a

  ½F ½F 1þ K1 K2 YF ¼ 2½F ½F2 1þ þ K1 K2 K1

11.48b

Equations for the general case of n subunits are presented in the original paper by Koshland, Ne´methy and Filmer (1966).

11.7.4. While not an Enzyme, Hemoglobin Provided Many Clues About Allostery Perutz (1965) put the MWC allosteric model for hemoglobin oxygenation on a stereochemical basis by invoking an equilibrium between a low-affinity, tense-state (or T-state), itself constrained by salt-bridges between the four C-termini, and a high-affinity, relaxed-state (or R-state) lacking these salt-bridges. He proposed that the tension exerted by the salt-bridges in the T-state is transmitted to the heme-linked histidines, thus restraining iron-atom movements into the porphyrin plane as required for oxygen binding. In this way, the equilibrium was thought to be governed primarily by an out-of-plane iron atom in fivecoordinated (high-spin state) deoxy-Hb and an in-plane hexa-coordinated iron atom (low-spin state) oxy-Hb. At the b-hemes, the distal valine and histidine block the oxygencombining site in the T-state where tension was thought to strengthen that blockage. Perutz et al. (1998) attributed the linear proton release with early oxygen uptake to the sequential rupture of T-state salt-bridges and an accompanying drop in the pKa of the weak bases participating in these linkages. The basic quaternary motions in the hemoglobin molecule were viewed as a rotation of two ab-dimers, each behaving as a nearly rigid unit. In an earlier description, Muirhead and Perutz (1963) suggested that the molecular dimensions of hemoglobin ˚  change during oxygenation, putting rough values of 50 A ˚ ˚ ˚ ˚ ˚ 55 A  64 A and 50 A  55 A  69 A for deoxy- and oxyHb. The idea that hemoglobin undergoes a substantial change shape was given further credence by the finding of Goers and Schumaker (1970) that hemoglobin undergoes a large 4% increase in its sedimentation coefficient s upon

oxygenation. As pointed out by Sanders, Purich and Cannell (1981), however, the good agreement between that 4% increase and the aforementioned change in molecular ˚ dimensions was fortuitous on three counts: (a) a 5 A increase in one dimension, without compensating decreases in the other dimensions, is incompatible with a rotation of one ab-dimer relative to the other; (b) the value of the dimer-tetramer equilibrium constant was not then known for the sedimentation experiments; and (c) the dependence of s–1 on the concentration of oxy- and deoxy-Hb was incorrectly assumed to be identical. In fact, the quasi-elastic laser light experiments indicated that the diffusion coefficient of oxy-Hb tetramers was 0.8% smaller than that of deoxy-Hb, and that, in the limit of zero concentration, the oxy-Hb diffusion coefficient was found to be 1.5  1% smaller than that of oxy-Hb (Sanders, Purich and Cannell, 1980). These results agreed well with ellipsoids of revolution modeled on the basis of X-ray coordinates for atoms defining the molecular surface of oxy- and deoxy-Hb. Numerous spectroscopic measurements (including UVvisible, fluorescence, NMR, and ESR techniques) probed the dependence of Y F and R for hemoglobin on the oxygenation. In the MWC model, Y F  R, because there is the pre-existing T0-R0 equilibrium requires that there must be some R0 even when O2 is zero. In the KNF model, Y F ¼ R, because, in the absence of a pre-existing T0-R0 equilibrium, both Y F and R must be zero when O2 is zero. A key limitation in such studies, however, is evident in the original MWC paper, where the R0 concentration in the absence of O2 is taken to be no more than 1% of the T0 concentration. Unambiguous assessment of small differences in Y F and R is technically challenging, especially when physiologically meaningful measurements should be conducted at or near hemoglobin’s extremely high concentration (~370 mg/mL) in red blood cells. Molecular crowding as well as nonideal protein-protein and protein-ligand interactions affect measurements at such high protein concentrations. Based on numerous oxygenation and spectroscopic experiments conducted in laboratories worldwide, Ackers (1998) concluded that hemoglobin obeys a hybrid model that resembles features of sequential KNF-like models, with oxygenation attended by quaternary structure switching when two O2 molecules are bound. His own work on the energetics of cooperativity of these intermediates, each with assigned quaternary structures, revealed quaternary switching from form T to form R, whenever heme-site binding creates a tetramer with at least one ligated subunit on each dimeric half-molecule. This ‘‘symmetry rule’’ translates the configurational isomers of heme-site ligation into six observed quaternary transition switch-points. Cooperativity arises from both ‘‘concerted’’ quaternary switching and ‘‘sequential’’ modulation of binding within each quaternary form, T and R. Binding affinity is regulated through a hierarchical code of tertiary-quaternary coupling that includes the classical allosteric models as limiting

Chapter j 11 Regulatory Behavior of Enzymes

703

cases. Perutz et al. (1998) also suggested that hemoglobin oxygenation involves elements of both MWC and KNF models. Those interested in the details of hemoglobin cooperativity should also consult the outstanding review by Eaton et al. (1999), who framed the historical development of this problem in the context of the Monod-WymanChangeux site-availability model versus the ‘‘Pauling’’Koshland Ne´methy-Filmer site-affinity model. They concluded that the weight of evidence clearly favors the MWC model. First, Shulman, Hopfield and Ogawa (1975) showed how the vast array of equilibrium, spectroscopic, and complex kinetic behavior is consistent with the twostate MWC allosteric model. Second, Szabo and Karplus (1972) clarified the relation between Perutz’s stereochemical mechanism (Perutz, 1970; 1989) and the MWC model. Third, time-resolved spectroscopy from the picosecond to the millisecond regime, together with an improved understanding of protein physics from studies on myoglobin, led to an interpretation of hemoglobin’s complex kinetics in terms of the MWC model (Eaton et al., 1999). Finally, while the precise distribution of ligation intermediates along the oxygen-binding curve would inform us about the underlying mechanism, determining this distribution has been problematic. Except for affinity differences, O2 and CO bind to hemoglobin in a similar manner. Using a clever cryogenic technique, Perrella and Di Cera (1999) directly determined the CO ligation intermediate distribution, finding that at low saturation, CO has slightly higher affinity to b-chains and that pair-wise interactions are more pronounced between a-chains. At high saturation, the two chains tend to behave identically. The detailed distribution of CO-ligated states cannot be reconciled with either the MWC or KNF models, suggesting a more subtle mechanism for hemoglobin cooperativity. One caveat is that a much

No Cooperativity

Kd-1 =

[P0][L] [P1]

Positive Cooperativity P0

P0

∆G1

∆G1 P1

higher affinity of CO interactions with hemoglobin may alter the distribution of ligation intermediates to such an extent that we cannot be sure that inferences can be made as to the precise distribution of O2-ligated states along the oxygen binding curve.

11.7.5. Negative Cooperativity Distinguishes KNF Models from MWC Models The term negative cooperativity describes the condition wherein binding of a ligand molecule decreases a protein’s affinity for subsequent molecules of ligand (Levitsky and Koshland, 1969). Negative cooperativity cannot occur in the Monod-Wyman-Changeux allosteric transition model, because a single dissociation constant defines ligation at all equivalent sites, both for the exclusive and non-exclusive cases. Therefore, any experimental observation of negative cooperativity in ligand binding must be regarded as prima facie evidence against an enzyme’s operation by an MWCtype model. That said, the occurrence of negative cooperativity per se is insufficient proof that the Koshland Ne´methy-Filmer treatment describes the enzyme’s behavior. Other models may be found to define the ligation state distribution along the ligand saturation curve more accurately. Phosphofructokinase (Reaction: Fructose 6-P þ MgATP2 # Fructose 1,6 Bisphosphate) catalyzes the first committed step in glycolysis. Hill and Hammes (1975) conducted equilibrium binding experiments on the interaction of rabbit muscle phosphofructokinase (PFK) with fructose 6-phosphate (F6P) and fructose 1,6-bisphosphate (FBP). The binding isotherms for both F6P and FBP exhibited negative cooperativity at pH 7.0 and 8.0 in the presence of 1–10 mM potassium phosphate at PFK concentrations where it exists as a mixture of dimers and

Nagative Cooperativity P0

∆G1

P1 [P1][L] [P2]

∆G2

P1 ∆G2

P2

P2

∆G2

K1 = K2 [L]

Amount Bound

Amount Bound

P2 binding gets tighter

K1>K2 [L]

Amount Bound

Kd-2 =

binding gets looser

K1

K2 [L]

FIGURE 11.13 Effect of changes in the magnitude of ligand dissociation constants on the ligand saturation curves for a dimeric protein obeying the sequential transition model. In the KNF model, positive cooperativity is the consequence of successive incremental increases in binding energy (i.e., DG1 < DG2 < DG3 $$$ < DGn); negative cooperativity is the consequence of successive incremental decreases in binding energy (i.e., DG1 > DG2 > DG3 $$$ > DGn), and independent binding, where DG1 ¼ DG2 ¼ DG3 $$$ ¼ DGn. Note: The double-humped rate-saturation curve shown for the case of negative cooperativity is deliberately exaggerated. One more often observes a non-hyperbolic curve that extends over a wider than typical substrate concentration range. When plotted as v/[S] versus v (the velocity equivalent of a Scatchard plot), negative cooperativity is evidenced by the curve shown in Fig. 11.12.

704

tetramers (pH 7.0), or as tetramers (pH 8.0) and at pH 7.0 in the presence of 5 mM citrate (where the enzyme exists primarily as dimers). The enzyme binds 1 mol F6P or FBP per mol 80-kDa monomer. At aggregation states smaller than the tetramer, the saturation of the enzyme with either ligand is attended by tetramer formation, increased enzymatic activity, and quenching of the PFK’s intrinsic fluorescence. At protein concentrations where aggregates higher than the tetramer predominate, the FBP binding isotherms are hyperbolic. The authors quantitatively analyzed these data in terms of a model in which: (a) the dimer exhibits extreme negative cooperativity in binding the F6P or FBP; (b) the tetramer shows less negative cooperativity; and (c) aggregates larger than the tetramer have little or no binding cooperativity. Phosphate is a competitive inhibitor of the fructose phosphate sites at both pH 7.0 and 8.0, while citrate inhibits a complex, noncompetitive behavior. In the presence of p(NH)ppA, the F6P binding isotherm is sigmoidal at pH 7.0, but hyperbolic at pH 8.0. The sigmoidal initial velocity v versus [F6P] curves at pH 7.0 were attributed to a heterotropic interaction between ATP and F6P binding sites, thereby altering the homotropic interactions between F6P binding sites. Homotropic interactions between fructose 6-phosphate binding sites are thought to give rise to positive, negative, or no cooperativity in a manner that depends on pH, aggregation state, and metabolic effectors. Probably the most abundant enzyme in Nature, ribulose 1,5-diphosphate carboxylase (EC 4.1.1.39), also known as ‘‘Rubisco,’’ catalyzes the reaction of D-ribulose 1,5-bisphosphate with CO2 to produce two molecules of 3-phospho-D-glycerate. The enzyme can also utilize O2 in the absence of carbon dioxide, in this case producing 3phospho-D-glycerate and 2-phosphoglycolate. When CO2/ Mg2þ-activated spinach leaf rubisco is incubated with the transition-state analog 2-carboxyarabinitol 1,5 bisphosphate, an essentially irreversible complex is formed. The extreme stability of this quaternary complex permitted Johal, Partridge and Chollet (1985) to use native analytical isoelectric focusing, anion-exchange chromatography, and non-denaturing gel electrophoresis to probe the mechanism of the binding process and the effects of ligand tight-binding on the structure of the protein molecule. Changes in the chromatographic and electrophoretic properties of the enzyme upon tight binding of the inhibitor reveal that the ligand induces a conformational reorganization which extends to the surface of the protein molecule and, at saturation, results in a 16% decrease in apparent molecular weight. Their analysis of ligand binding by isoelectric focusing demonstrated that incubating the protein with a stoichiometric molar concentration of ligand (expressed on a mol site basis) results in an apparently charged homogeneous enzyme population with an isoelectric point of 4.9. Moreover, substoichiometric levels of ligand produce differential effects on each of the charge microheterogeneous native

Enzyme Kinetics

enzyme forms. These stoichiometry-dependent changes in electrofocusing band patterns were employed as a probe of cooperativity in the ligand tight-binding process. The tightbinding reaction was shown to be a case of negative cooperativity. Aspartate carbamoyltransferase (EC 2.1.3.2), also known as aspartate transcarbamylase and carbamylaspartotranskinase, catalyzes the first committed step (i.e., the transfer of a carbamoyl group from carbamoyl phosphate to the a-amino group of L-aspartate to form N-carbamoyl-Laspartate with the liberation of orthophosphate) in pyrimidine biosynthesis (Kantrowitz and Lipscomb, 1988; Lipscomb, 1994). The Escherichia coli enzyme (molecular weight 310,000) is the prototypical allosteric regulatory protein. This enzyme (ATCase) can be dissociated by mercurials into two catalytic trimers and three regulatory dimers. ATCase is feedback-inhibited by the pyrimidine nucleotide biosynthetic end-product CTP and is activated by ATP. To explore the subunit-subunit ‘‘cross-talk’’ underlying enzyme cooperativity, Eisenstein et al. (1992) constructed a comprehensive set of hybrid molecules consisting of wild-type and mutationally altered catalytic subunits. The mutant enzymes that were virtually devoid of activity had Gly-128 replaced by Asp or Arg. The kinetic properties of these hybrid enzyme-like molecules were analyzed to evaluate the basis for the unusual quaternary constraint demonstrated by an inter-subunit hybrid containing one wild-type catalytic subunit, one inactive mutant subunit (containing the Gly to Asp replacement), and three wildtype regulatory subunits. A similar inter-subunit hybrid was constructed from the wild-type catalytic subunit and the mutant in which Gly-128 was replaced by Arg, and it too demonstrated a pronounced decrease in activity relative to that expected for a hybrid containing three active sites. Moreover, neither of these hybrid holoenzymes exhibited the cooperativity with respect to aspartate that is characteristic of wild-type ATCase. In contrast, hybrid holoenzymes containing at least one wild-type chain in each catalytic subunit showed cooperativity. Also, hybrid enzymes containing different arrangements of five, four, three, or two wild-type catalytic chains with an appropriate complement of mutant chains had specific activities proportional to the number of wild-type chains in the holoenzymes. Exceptions were observed only in hybrids in which one of the two subunits in the holoenzyme was composed completely of mutant catalytic chains. For these hybrids, the negative complementation was manifested as a much lower enzyme activity than expected from the number of wild-type chains in the enzyme and the loss of cooperativity. Thus, the activity and allosteric properties of these hybrids are dependent on the arrangement of catalytic chains in the holoenzyme, in contrast to results obtained for hybrids containing native and chemically modified catalytic chains. Intra-subunit hybrid catalytic trimers containing one or two wild-type chains exhibited one-third and two-thirds

Chapter j 11 Regulatory Behavior of Enzymes

the activity of the intact wild-type catalytic subunit, respectively, indicating the dominant negative effect that was seen in inter-subunit hybrid holoenzymes absent within trimers. To gain an insight into the origin of apparent negative cooperativity of the thermophilic F1-ATPase (an a3b3g complex), Ono et al. (2003) compared Km values from ATPase activity (Reaction: MgATP2 # MgADP þ Pi) and MgATP2 measurements with Kd values for MgATP2 binding to mutant sub-complexes of F1-ATPase, a(Trp-463Phe)3b(Tyr-341-Trp)3g and a(Lys-175-Ala/Thr-176-Ala/ Trp-463-Phe)3b(Tyr-341-Trp)3g. For a(Trp-463-Phe)3b(Tyr341 Trp)3g, apparent Km values (4.0 and 233 mM) did not agree with apparent Kd values in the order of 0.016 and 13 mM obtained from fluorescence quenching of the added tryptophan residue. On the other hand, in the case of a(Lys175-Ala/Thr-176-Ala/Trp-463-Phe) 3 b(Tyr-341-Trp) 3 g, which lacks noncatalytic nucleotide binding sites, the apparent Km of ATPase activity (10 mM) roughly agreed with the highest Km of fluorescence measurements (27 mM). The results indicate that in case of a(Trp-463-Phe)3b(Tyr341 Trp)3g, the activating effect of ATP binding to noncatalytic sites dominates overall ATPase kinetics and the highest apparent Km of ATPase activity does not represent the ATP binding to a catalytic site. In the case of a(Lys-175Ala/Thr-176-Ala/Trp-463 Phe)3b(Tyr-341-Trp)3g, the Km of ATPase activity reflects the ATP binding to a catalytic site due to the lack of noncatalytic sites. The Eadie-Hofstee plot of ATPase reaction by a(Lys-175-Ala/Thr-176-Ala/Trp463-Phe)3b(Tyr-341-Trp)3g was rather linear compared with that of a(Trp-463-Phe)3b(Tyr-341-Trp)3g, if not perfectly straight, indicating that the apparent negative cooperativity observed for wild-type F1-ATPase is due to the ATP binding to catalytic sites and noncatalytic sites. Thus, the frequently observed Km9s of 100–300 mM and 1–30 mM range for wild-type F1-ATPase correspond to ATP binding to a noncatalytic site and catalytic site, respectively. Despite considerable effort, relatively few enzymes have been unambiguously proven to exhibit negative cooperativity. This observation should not be surprising, given the fact that most spectroscopic methods suffer the disadvantage that there is no definitive way to demonstrate that an observed incremental change in some spectral property is a direct linear measure of moles of bound ligand. This problem is compounded when substrate binding is attended by catalysis, and it goes without saying that kinetic instability is a contributing complexity. Moreover, because substrate analogs often fail to reproduce the kinetic properties of natural substrate, their use in place of their natural counterparts is questionable. It is also true that analogs often exhibit different on- and off-rate constants, even when their respective dissociation constants appear to be quite similar. Finally, a confounding problem in efforts to define an enzyme’s regulatory mechanism using binding data is that

705

binding behavior is not predictive of kinetic behavior. An interesting example of the decoupling of substrate binding and enzyme activity is evident for the Bacillus subtilis CTP:glycerol-3-phosphate cytidylyltransferase. This enzyme catalyzes the reaction of CTP with glycerol-3-phosphate to form CDP-glycerol; the latter is a precursor of poly-(glycerol phosphate), a teichoic acid found in the cell walls of many Gram-positive bacteria. In examining the substrate binding interactions through the quenching of intrinsic tryptophan fluorescence, Sanker, Campbell and Kent (2001) found that the Kd values for initial binding of each substrate were 1,000-fold lower than the Km values from activity measurements. They also found that substrate binding exhibited negative cooperativity, such that subsequent substrate molecules occurred with markedly decreasing affinities. Interestingly, this negative binding cooperativity is not observed in any of the kinetic measurements, suggesting that catalysis cannot occur unless the enzyme is ‘‘fully loaded’’ (i.e., all four substrate molecules are bound to the enzyme dimer).

11.7.6. Fraction-of-the-Sites Behavior: The Case of Escherichia coli Alkaline Phosphatase An extreme example of negative cooperativity is fraction-of the-sites activity, a phenomenon observed when only a fraction of subunits bind substrate or when some substratebound subunits are inactive. The classical case is called half-of-the site behavior, where activity is attributable to only half of the subunits within an even numbered oligomer (see below). In the binding change mechanism for the FOF1ATP synthase exhibits what may be called two-thirds-ofthe-sites activity, because one of the three sites always remains unoccupied (see Section 13.10: ATP Synthase: Boyer’s Binding Change Mechanism). While fraction-ofthe-sites behavior seems inherently wasteful, some enzymes appear to gain advantage by ‘‘borrowing’’ ligand binding energy in one or more sites to destabilize binding at another site, such that the resulting activity versus substrate concentration curve can fulfill some special metabolic requirement. This narrative about bacterial alkaline phosphatase (Reaction: Phosphomonoester þ H2O # Alcohol þ Phosphate) illustrates the value of rigorous experimentation in the characterization of negative cooperativity. During the mid-1970s, the nature of strong negative cooperativity exhibited by alkaline phosphatase presented the intriguing possibility that substrate binding energy could be employed to facilitate catalysis by this dimeric protein. To explain how an idled subunit might play a role in overall catalysis, Lazdunski et al. (1971) proposed the ‘‘flip-flop’’ model, where dephosphorylation of the phosphoryl-enzyme intermediate at one active site is activated by the binding of substrate to the second subunit. In each catalytic cycle, one

Enzyme Kinetics

706

subunit ‘‘flips’’ from ES to SE, just as the other ‘‘flops’’ between SE and ES (Lazdunski, 1972). One can make the analogy to a two-cylinder internal combustion engine, where adding fuel to the idle cylinder somehow triggers ignition in the other cylinder.

II OH

OH

E +A P

E A P 1

E OH

E 2

E

6 O P A-OH 3

E

OH

O P E

E A P

OH

I

O P

E 5

E

+ A-OH

OH

In the Lazdunski scheme, A–P represents the phosphomonoester substrate, E$A–P is the Michaelis complex, and E–O–P stands for the covalent phosphoryl-enzyme. The mechanism proposed may be described as follows: Step-1, substrate binds (and, in doing so, blocks additional substrate binding; Step-2, serine hydroxyl phosphorylation promotes release of the alcohol A–OH portion of the phosphomonoester; Step-3, phosphoryl-enzyme next allows the binding of a second molecule of substrate; and Step-4, the resulting hybrid species (containing both a Michaelis complex; and Step-5, phosphoryl-enzyme) may liberate Pi or form the diphosphorylated enzyme intermediate plus the alcohol product. In the first instance, substrate binding is coupled to enzyme dephosphorylation. Chappelet-Tordo et al. (1974) referred to this reciprocating mechanism as a flip-flop mechanism. Alternatively, enzyme phosphorylation (Step-5) may be coupled to dephosphorylation (Step-6). In their analysis of the flip-flop model, Bale, Chock and Huang (1980a) considered the following reaction scheme for the flip-flop behavior, where substrate S represents a phosphomonoester ROP, and E–P is the phosphorylated enzyme intermediate.

Pi k5

E-P

k-1 ROH

ES Pi

k2 k4 k3[S] k-3

Note that the flip-flop mechanism gives rise to [S] and [S]2 terms that cancel to yield the following Michaelis-type expression: v k2 k4 ½S=ðk2 þ k4 Þ ¼ k—1 k3 k4 þ k1 k2 ðk—3 þ k4 Þ ½ET  þ ½S k1 k3 ðk2 þ k4 Þ

4 + A-OH

k1[S]

v k1 k2 k3 k4 ½S2 ¼ ½ET  fk—1 k3 k4 þ k1 k2 ðk—3 þ k4 Þg½S þ k1 k3 ðk2 þ k4 Þ½S2 11.49

O P

Scheme 11.4

E

E$S. They showed that the initial rate equation for Scheme 11.6 is given by:

SE-P

Scheme 11.5 Binding of S to E–P facilitates the dephosphorylation and results in the formation of S$E which is kinetically indistinguishable from E$S. (The direct hydrolysis of E$P to E (dashed arrow) is tacitly assumed to be negligible compared with the substrate-facilitated pathway of S$E–P þ

11.50

Without describing all details, it is sufficient to say that, when Bale, Chock and Huang (1980a) derived corresponding rate equations accounting for the presence of an alternative substrate S9, they found that the additional reaction pathways generate second-power terms that do not cancel. Therefore, non-hyperbolic kinetic behavior is required, if flip-flop catalysis occurs in the presence of S and S9, because such a model predicts that, in the presence of fixed S9 concentrations (say [S9]1, [S9]2, [S9]3, etc.), plots of 1/v versus 1/[S] should be nonlinear. In experiments (Fig. 11.14) with several different phosphomonoesters acting as either S or S9 kinetic data, however, they only obtained linear competitive inhibition patterns. Moreover, when the alternative substrate S9 is present at constant ratios to substrate S, linear intersecting doublereciprocal plots are predicted. Instead, parallel plots were obtained for three different S–S9 pairs. Nonlinear, noncompetitive inhibition by the product P1 is predicted for the flip-flop model, but linear competitive inhibition was observed. In addition, the kcat determined at pH 8, 25 C in 0.1 M Tris-HC1 is 27 s1, which agreed very well with NMR measurements of the off-rate of ~25 s1 for inorganic phosphate under identical conditions (Hull et al., 1976). All the kinetic experiments by Bale, Chock and Huang (1980a) are consistent with a mechanism in which an idled subunit plays no discernible role, indicating that the flip-flop mechanism is not operative in alkaline phosphatase catalysis. Bale, Chock and Huang (1980b) also examined alkaline phosphatase with respect to its fast reaction kinetic behavior. Results of earlier transient-phase kinetic measurements showed that at acidic pH, the enzyme catalyzes phosphate ester hydrolysis with an initial burst of alcohol production followed by a steady-state rate. Because alcohol production is accompanied by the phosphorylation of the enzyme, it was suggested that the dephosphorylation reaction is rate-limiting. However, at alkaline pH values, the initial burst was not observed when an organophosphate was used as the substrate, leading to the suggestion that at an

Kapp, µM

Chapter j 11 Regulatory Behavior of Enzymes

6 4 2 0

5 [CMP], µM

10

1/

10

5

0

0.5 1.0 1 / [NASBIP], µM

FIGURE 11.14 Double-reciprocal plots for the hydrolysis of 6bromo-2-hydroxy-3-naphthoyl-O-anisidine-phosphate by alkaline phosphate in the presence of alternative substrate CMP. Velocity is expressed in arbitrary fluorescence units. Inset, Kapp as a function of [CMP]. From Bale, Chock and Huang (1980a) with permission of the authors and the American Society for Biochemistry and Molecular Biology.

alkaline pH, the rate-limiting step involves a conformational change occurring prior to or during the phosphorylation of the enzyme (Bloch and Schlesinger, 1973). In contrast, utilizing another substrate analogue, O-p-phenylazophenyl phosphorothioate, Chlebowski and Coleman (1972; 1974) reported that alcohol production proceeds with an initial burst and that the amplitude of the burst corresponds to the formation of 2 mol product per dimer at pH 8.5. When Bale, Chock and Huang (1980b) conducted rapid kinetic experiments on alkaline phosphatase, they were able to demonstrate that catalysis proceeds with an initial burst of alcohol production, followed by a steady-state rate when an organophosphate is used as the substrate. Kinetic analysis of the burst-phase revealed that the equilibrium of the initial binding of the substrate, 4-methyl-umbelliferyl-P, to the enzyme was rapid and that the 4-methyl-umbelliferone formation was fast. (The choice of this substrate is based on the fact that esters of 4-methyl-umbelliferone do not fluoresce unless the ester linkage is hydrolyzed to form the fluorophore.) Analysis of the steady-state phase of the reaction yielded a Pi-release rate constant that agreed with the kcat determined by initial-rate studies. Dephosphorylation of the phosphoryl enzyme prepared at pH 5.7 was studied by the pH-jump technique, using a three-syringe stopped flow apparatus. The data showed that dephosphorylation is not rate-limiting in the catalytic cycle and that the presence of substrates or inhibitor has no effect on this step. The lack of effect of substrates on the rate of dephosphorylation and on the rate of phosphate dissociation indicates that the flip-flop mechanism, in which the

707

product release is supposedly facilitated by the binding of a second substrate molecule, is not valid for alkaline phosphatase. One serious limitation of the above study is that 4-methyl-umbelliferyl-P is an unnatural substrate chosen solely for its fluorogenic properties. One must give serious consideration to the possibility that this bicyclic fluorophore may be lodged in the active site in such a manner that the kinetic mechanism is fundamentally altered. This situation reminds us of the vital importance of working with natural substrates! Finally, Huang, Rhee and Chock (1982) also presented a cogent analysis of the kinetics of subunit-subunit interactions and enzyme cooperativity. They also describe some of the pitfalls in various treatments of enzyme cooperativity.

11.8. OTHER COOPERATIVITY MODELS That the aforementioned cooperativity models are not the only explanations for cooperativity is illustrated by the following brief accounts of other models.

11.8.1. Hybrid Cooperativity Models Eigen (1967) described a generalized ligand-binding scheme (Fig. 11.15), in which the MWC and KNF models represent extreme cases of a manifold of cooperative ligand

L

L

L

L

L

L

L

Eigen’s General Model

L

L

KNF Induced-Fit Model

L

L

MWC Exclusive Model

L

L

L

L

MWC Nonexclusive Model

FIGURE 11.15 Four schemes for cooperative ligand binding for a dimer possessing one ligand binding site per subunit. (Red ¼ ligand-free subunits; Blue ¼ ligand containing subunits.) For simplicity, not all microscopic species are represented.

Enzyme Kinetics

708

11.8.2. The Duke, Le Nove`re and Bray Conformational Spread Model The Conformational Spread (or CS) model describes cooperative behavior and ligand binding properties of oligomeric proteins assembled into closed-ring structures (Duke, Le Nove`re and Bray, 2001). Each functional subunit (or protomer) within a ring can only exist in either of two conformational states, designated as (þ) for active and (–) for inactive, that inter-convert rapidly and stochastically. Ligand affinity is taken to be higher for protomers in the (þ) conformational state, and the nonzero coupling energy EJ also increases the tendency of a protomer to adopt the same conformation as its immediate neighbor(s). Under such constraints, thermodynamic linkage dictates that increased ligand binding raises the likelihood of a protomer adopting the (þ)-conformational state. At any given ligand concentration, the activity of protomers in the ring fluctuates over time, due to: (a) individual stochastic flipping; (b) random site occupation; and (c) dissociation of ligand molecules. An inherent feature of this treatment is that the kinetics and thermodynamics can be tuned to attain the circumstance where the ring spends the majority of the time in one or the other of two extreme states, wherein the ring behaves as a two-state switch. Sensitivity to a change in ligand concentration is determined by the number of protomers in the ring structure. The CS model reproduces the kinetics and sensitivity of the 34-membered FliM protein ring that

A ∆G

binding behaviors. The MWC case gives rise to cooperativity by controlling binding site availability by means of concerted, symmetry-conserving changes in oligomer structure. The KNF model gives rise to cooperativity by controlling affinity by means of sequential ligand-induced conformational changes. In its simplicity, conciseness, and geometry, the Monod treatment is an exceptionally elegant model, with fewest adjustable parameters. On the other hand, the Adair-Koshland Ne´methy-Filmer treatment offers the inherent advantage of generality. As argued by Haber and Koshland (1967), the MWC case may even be regarded as a special case of the most general KNF model. Aspartate transcarbamoylase is also thought to involve ligand-induced rearrangements in its local inter-subunit structure, while also displaying more global, concerted changes in quaternary structure. Stevens and Lipscomb (1992) suggested that CTP binding to the T-state or ATP binding to the R-state reorients several key residues, resulting in a CTP-induced decrease (or an ATP-induced increase) in the size of the nucleotide binding site and a related CTP-induced decrease (or ATP-induced increase) in the extension of the outer parts of the dimer of the regulatory chains, R1 and R6. As a result, CTP pinches the ˚ in the R-state; ATP regulatory dimers together by 0.3 A ˚ in the T-state. pushes the regulatory dimers apart by 0.3 A These changes influence key residues in the R1–C1 interface of the R-state and the R1–C1 and R1–C4 interfaces of the Tstate, such that the separation of catalytic trimers (c3$$$c3) ˚ by CTP in the R-state and increased is decreased by 0.5 A ˚ by 0.4 A by ATP in the T-state. Smaller effects on c3$$$c3 are observed when CTP binds to the sterically crowded Tstate (or when ATP binds to the elongated R-state). These changes reorient key residues in the active site (e.g., catalytic chain residue Arg-229, a residue involved in aspartate binding). This pattern for action of CTP and ATP in perturbing the regulatory dimer, and consequently both the structure and flexibility in critical parts of the T-state or R-state, was termed the nucleotide perturbation mechanism. In other instances, cooperativity was found to depend on delicate readjustments in subunit-subunit interactions, bringing about major changes in ligand binding. For example, pyruvate kinase’s non-allosteric M1 isozyme can be converted into an allosteric enzyme simply by changing a single amino acid residue located at the intersubunit contact (Ikeda, Tanaka and Noguchi, 1997). An Ala-398-Arg substitution resulted in pronounced cooperativity, with nH and [S]0.5 values of 2.7 and 0.41 mM, respectively, compared with values of 1.0 and 0.049 mM for the wild-type enzyme. Such results indicated that Ala398 is a critical residue allowing the enzyme to prefer the R-state. Kolodziej, Tan and Koshland (1996) likewise demonstrated that a mutation at Ser-68 in the binding pocket of Salmonella typhimurium aspartate receptor results in a change from positive cooperativity to negative cooperativity.

B

EA –EL + ÉA

EL – EA –EL

Coupling energy = –EL

Coupling energy = 0

C

At Low Ligand Concentration inactive = protomer

At Medium Ligand Concentration = ligand or effector

At High Ligand Concentration = active protomer

FIGURE 11.16 States of a protomer within a protein ring that is susceptible to conformational spread. (a) Transitions between inactive and active conformations are accompanied by an energy change. Association with an effector molecule also selectively stabilizes the active conformation. Shown is the symmetric case in which active / inactive transition involves a free energy change þ EA in the absence of ligand and – EA when the ligand is bound; (b) Coupling energy of conformation-dependent interactions between neighboring protomers favor juxtapositioning of like conformations (þ)/(þ) or ()/() and disfavor juxtaposing of (þ)/() states. Redrawn from Duke, Le Nove`re and Bray (2001) with permission.

Chapter j 11 Regulatory Behavior of Enzymes

controls the clockwise and counterclockwise motions of the bacterial flagellar motor. Smaller oligomeric proteins, even the a2b2 hemoglobin structure, can be analyzed by the conformational spread model. In fact, the canonical MonodWyman-Changeux and Koshland-Ne´methy-Filmer models are limiting cases of their conformation spread model. This treatment may also be applicable to acetylcholine binding to the acetylcholine receptor, the a2bgd subunit ring structure of which has quasi-five-fold symmetry.

11.8.3. V-Type Allosteric Systems

Vm

Although most allosteric enzymes involve K-type interactions (where the K signifies changes in ligand affinity), Monod, Wyman and Changeux (1965) anticipated that other enzymes might modulate their activity through changes in catalytic efficiency. They used the descriptor ‘‘Vmax-type’’ or ‘‘V-type’’ to describe this less frequently observed form of allosteric behavior. An excellent example of a V-type allosteric enzyme is E. coli phosphoglycerate dehydrogenase (PGDH), a homotetramer catalyzing the NADþ-dependent formation of {sc}D{/sc}-3-phosphohydroxypyruvate from D-phosphoglycerate. This enzyme is regulated by serine, which weakly affects the enzyme’s Km but alters Vmax substantially (Sugimoto and Pizer, 1968a,b). Kinetic studies indicate a minimum of two serine-binding sites, although the crystal structure of PGDH with bound serine as well as direct serine-binding studies revealed four serine-binding sites (Grant, Schuller and Banazak, 1996; Schuller, Grant and Banaszak, 1995). The serine-binding sites reside at the interface between regulatory domains of adjacent subunits. Two serine molecules bind at each of the two regulatory domain interfaces in the enzyme. When all four serine binding sites are occupied, one observes nearly total inhibition; however, with only one bound serine at each interface 85% inhibition is observed. Tethering of the regulatory domains by multiple hydrogen bonds between each subunit appears to prevent structural changes required in the catalytic cycle. Part of the conformational change may

709

involve a hinge, where two antiparallel beta-sheets are joined within the regulatory domain). Serine binding may prevent the enzyme from closing the cleft between the substratebinding domain and the nucleotide-binding domain. If this conformational change is necessary for catalysis, then a plausible model emerges for Vmax-type allosteric control that arises from serine-regulated movement of rigid domains about flexible hinges.

11.9. OLIGOMERIZATION-DEPENDENT CHANGES IN ENZYME ACTIVITY For some enzymes, catalytic activity depends on the state of enzyme oligomerization. If a ligand (e.g., substrate, activator, or inhibitor) binds preferentially to one oligomerization state, changes in the concentration of that ligand will inevitably cause the enzyme to undergo association or dissociation. This linkage of oligomerization to protein concentration and/or ligand concentration is a molecular level example of La Chatelier’s principle of reciprocal action. Likewise, if biosynthesis or degradation of enzyme results in a change in total enzyme concentration, the state of oligomerization and catalytic activity may be altered. An early indication that enzyme oligomerization might play a regulatory role was obtained by Olsen and Anfinsen (1952), who demonstrated that glutamate dehydrogenase (GDH) undergoes reversible concentration-dependent association-dissociation. Later work by Frieden (1963) and Tomkins et al. (1965) demonstrated that, in the presence of its coenzyme, the GDH aggregation and activity are regulated by GTP and ADP. Thusius (1977) described the application of right-angle light scattering in stopped-flow and temperature-jump modes to investigate GDH polymerization (see Section 4.4.10). Escherichia coli deoxythymidine kinase (Reaction: dTMP þ MgATP2 # dTDP þ MgADP) is subject to feedback inhibition by the end-product dTTP and activation by dCDP, dCTP, and various other deoxynucleoside 59-diand tri-phosphates (Okazaki and Kornberg, 1964). As shown in Fig. 11.18, those enzymes that activate or inhibit the dT kinase produce a marked change in the apparent sedimentation coefficient, whereas little or no change in sedimentation rate was detected with nucleotides showing no activating or inhibitory effect (Iwatsuki and Okazaki, 1967). These investigators concluded that the degree of dT kinase dimerization correlates with the effect on enzymatic activity.

Vm≠ 0, when [A]=0

[Activator] FIGURE 11.17 Plot of the catalytic activity of a V-type allosteric enzyme versus the concentration of a metabolic activator. Note: To discern changes that are strictly associated with changes in maximal velocity, rate measurements are carried out in the presence of saturating, but noninhibitory, concentration(s) of substrate(s).

11.9.1. Enzyme Self-Association can Alter Catalytic Activity A classical example of oligomerization-dependent changes in enzyme activity is rabbit muscle phosphorylase a (Reaction: Glycogenn þ Pi # Glucose 1-P þ Glycogenn-1), the

Enzyme Kinetics

Percentage of Control Activity

710

[ET,Total]). Substitution of [ED,Total] ¼ ([ETotal]  [ET,Total]) and [ET,Total] ¼ ([ETotal]  [ED,Total]) into Eqn. 10.31 yields

+dCDP 700 500 300

+dADP 100

+dTTP 3

4

5

Svedberg value FIGURE 11.18 Relationships between observed changes in enzyme activity and an increase in the sedimentation rate for deoxythymidine kinase. Various effector concentrations that bring about the indicated changes in catalytic activity. Reproduced from Iwatsuki and Okazaki (1967) with permission.

phosphorylated form that is generated post-translationally by the action of phosphorylase kinase (Reaction: Phosphorylase b þ ATP # Phosphorylase a þ ADP). Numerous studies had demonstrated that phosphorylase a is a 195-kDa dimer at low concentration, but is a 390-kDa tetramer at high concentrations. In seeking to assess the relative catalytic efficiency of the dimer and tetramer, Huang and Graves (1970) recognized the important principle that, unless an enzyme undergoes an association-dissociation reaction that is attended by a change in catalytic efficiency, the specific catalytic activity of an enzyme (i.e., v/[ETotal]) cannot change as a function of enzyme concentration. They therefore used light scattering measurements to quantify the enzyme’s dimer-tetramer equilibrium, and simultaneously measured the catalytic activity of phosphorylase. Their experiments revealed a dilution-dependent increase in the enzyme’s specific activity that could be accounted for quantitatively by a decrease in molecular weight. Their approach begins with the following association-dissociation reaction: 11.51

This reaction is defined by the equilibrium dissociation constant Kd ¼ [ED]2/[ET], where ED and ET are the dimeric and tetrameric species, respectively. The observed specific activity (written as the activity per unit of protein) is the weight-average value of the specific activities of the dimer fD and the tetramer fT:

f ¼

v ½ETotal 

¼

vD þ vT ½ETotal 

½ED;Total  ½ET;Total  vD vT þ   11.52 ½ED;Total  ½ET;Total  ½ETotal  ½ETotal  f ¼

11.53

½ET;Total  ¼

½ETotal ðfD  fÞ fD  fT

11.54

½ED;Total fD þ ½ET;Total fT ½ETotal 

where v is the observed initial velocity and ETotal is the total weight concentration of enzyme (i.e., [ETotal] ¼ [ED,Total] þ

½ETotal ðf  fT Þ2 g  ðfD  fT ÞðfD  fÞ L

11.55

2½ETotal ðf  fT Þ2 mol  L MD ðfD  fT ÞðfD  fÞ

11.56

Kd ¼ or: Kd ¼

f ¼

½ETotal ðf  fT Þ fD  fT

Substitution of these expressions into the original equation explicitly defines the equilibrium constant in terms of specific activities:

6

ETetramer # 2EDimer

½ED;Total  ¼

Rearranging this expression: 1 ½ETotal ðf  fT Þ 1 ¼ þ 2  fT Þ ðf ðf  fT Þ Kd ðfD  fT Þ D

11.57

A plot of 1=ðf  fT Þ versus ðf  fT Þ yields 1/(fD – fT), where fT s fT, as the ordinate-intercept and –Kd/ (fD – fT) as the abscissa-intercept. Therefore, if specific activity fT is known independently, one can evaluate both fD and Kd. Moreover, in the limit of infinite protein concentration (i.e., as 1/[ETotal] / 0), f approaches fT, such that a plot of f versus 1/[ETotal] allows one to estimate fT from the vertical-axis intercept. The concept of a more active phosphorylase a dimer was reinforced by the findings that the dimeric form has a lower activation energy of 11.3 kcal/mol and a greater affinity for glycogen; whereas the tetrameric form has a considerably higher activation energy of 22.6 kcal/mol and a lower affinity for glycogen (Huang and Graves, 1970). The alternative hypothesis that the tetramer per se is inactive was also disproved, because a change in specific activity would not have correlated with the change in molecular weight. Importantly, because the light scattering technique yields a weight-averaged molecular weight, it was necessary for Huang and Graves (1970) to correct the scattering data to obtain an accurate estimate of the fraction of protein present as dimer and tetramer. Light scattering measurements of the type are also extremely tedious, owing to the strong light scattering by sub-microscopic particles. The quality of the data presented in this report attests to their painstaking efforts to obtain accurate data. Neal, Purich and Cannell (1984) describe centrifuging a modified light scattering cuvette to ‘‘de-dust’’ protein solutions by sedimenting and trapping sub-microscopic particles in a 1-mm pore Teflon plug situated at the bottom of the cuvette. Given the power of modern nanosecond fluorescence spectroscopy as well as the utility of fluorescence

Chapter j 11 Regulatory Behavior of Enzymes

711

anisotropy measurements, the theory developed by Huang and Graves (1970) should enjoy widespread use in characterizing enzymes subject to self-association.

binds rapidly, reversibly, and preferentially to one state. In this case, such that:

11.9.2. Some Substrates Alter Enzyme Oligomerization and Catalytic Activity Substrates and/or regulatory effectors often alter an enzyme’s state of oligomerization, depending on the differential stabilization of the dissociated and associated states. Wyman and Gill (1990) present numerous examples of ligand-induced protein oligomerization. A ligand may also have the net effect of promoting oligomerization, and such processes frequently operate by mechanisms where oligomerization first requires a ligand-induced conformational change. In the case where ligand-induced oligomerization is kinetically facile, the resulting oligomer may or may not exhibit an enhanced or diminished capacity for dimerization, again depending on the dissociated and associated states. Consider the simplest two-state case of a monomer P1 existing in rapid equilibrium with dimer P2, where only the monomer binds ligand X. Letting KE ¼ [P2]/[P1]n, K ¼ [P1][X]/[P1$X]; and a ¼ [X]/K, the saturation function is: Y ¼

að1 þ aÞ

ð1 þ aÞ2 þ 2KE ½E

Km9 ¼

p  ði  1Þ ½Pm ½X H i ½Pm X

11.59

Kn9 ¼

p  ði  1Þ ½Pn ½X H i ½Pn Xi 

11.60

For this system, the saturation function is: Y ¼

pað1 þ aÞp1 þ qcaKE ½Pm nm1 ð1 þ caÞq1 ð1 þ aÞp þ nKE ½Pm nm1 ð1 þ caÞq

11.61

where [Pm] and [Pn] are the concentrations of protomer and oligomer, a ¼ [X]/Km9, and c ¼ [Pn]/[Pm]. A more general scheme for the monomer-dimer case can be written in the following manner: KE 2E + 2X

2X + E2 K2'

K1'

E2X K2' E2X

11.58

2EX KEX

Scheme 11.6

This equation has a form that is similar in some respects to the saturation function for exclusive ligand binding in the Monod-Wyman-Changeux cooperativity model. In this model, a sigmoidal curve for binding of ligand X is observed. Frieden (1967) and Nichol et al. (1967) offered general treatments for ligand-induced protein oligomerization. The basic assumptions are that: (a) a protein exists in either of two states (i.e., as a protomer Pm consisting of m subunits possessing q equivalent ligand sites, and as an oligomer Pn having n subunits with p equivalent ligand sites); (b) Pm and Pn are in rapid equilibrium with each other, defined by the equilibrium constant KE ¼ [Pn]/[Pm](nm); and (c) the ligand

where E is the monomer, E2 is the dimer. The saturation function Y ¼ v=nVmax can then be written in terms of initial velocity v, total enzyme concentration [ET], the rate constants k1 and k2 for the breakdown of ES1 and ES2, yielding: k1 að1 þ aÞn1 þ k1 Keq ½ET dað1 þ daÞ2n1 v ¼ nVmax ð1 þ aÞn þ Keq ½ET dað1 þ daÞ2n

10.62

While the quadratic nature of ligand-induced enzyme dimerization conspires to make this expression somewhat

TABLE 11.3 Examples of Effector-Induced Changes in Enzyme Self-Association Type of Effect

Enzyme

Effector

Reference

Inhibitor-induced

Homoserine Dehydrogenase

L-Threonine

Datta, Gest and Segal (1964)

Activator-induced

Isocitrate Dehydrogenase

ATP, NADH

Chen, Brown and Plaut (1964)

Acetyl-CoA Carboxylase

Citrate

Vagelos, Alberts and Martin (1963)

Threonine Deaminase

AMP, ADP

Phillips and Wood (1964); Hirata et al. (1965); Whiteley (1966)

Inhibitor-induced

Deoxythymidine Kinase

DTTP

Iwatsuki and Okazaki (1967)

Activator-induced

dCMP Deaminase

DTTP

Maley and Maley (1964)

Enzyme Kinetics

712

complicated, computer-assisted data fitting allows one to extract the relevant constants, provided that measurements were conducted over an adequate range of protein and ligand concentrations. In principle, oligomerization will also be strongly dependent on the total enzyme concentration, even in the absence of the ligand. The important conclusion is that, except for an additional term defining enzyme oligomerization, the general form of these saturation functions accounting for cooperative ligand binding have the same general polynomial form as those for the MWC and KNF models. Therefore, the success of efforts to experimentally discern differences among the three models will depend almost entirely on the strength with which oligomerization is linked to changes in ligand concentration.

11.10. HYSTERESIS

Product Formed

Product Formed

There is no a priori requirement that regulatory enzymes undergo rapid conformational isomerization, and given the fact that slow isomerizations allow for more robust interactions of enzymes and ligands, one can readily imagine that there will be circumstances where effective cellular control might require a slow or retarded response to an input signal. A succession of slow enzyme conformational changes can result in a timing mechanism that allows a metabolic process to undergo a gradual time-dependent change in activity. These structural isomerizations often require many seconds before the full extent of change in catalytic activity can occur. In other cases, the enzyme may exhibit ‘‘memory’’ (i.e., it may remain in a ligand-induced conformational state for a period of time after the ligand has already dissociated). Any slow change in catalytic activity is considered to be a form of hysteresis, so named because a curve depicting the time evolution of a change in activity of a hysteretic enzyme (Fig. 11.19) resembles the shape of

final

initial= 0

initial≠ 0

initial final

Time FIGURE 11.19 Product formation by an enzyme undergoing a timedependent change in activity. Top, Transition from an initial velocity vinitial of zero to a faster final constant velocity vfinal. Bottom, Transition from an initial non-zero velocity vinitial (right panel) to a faster final constant velocity vfinal.

an electronic hysteresis loop. Hysteresis may provide a living system with a broader spectrum of responses than those occurring promptly upon ligand adsorption or desorption. Several mechanisms can, in principle, generate a slow response of enzyme activity to a change in some solution variable (e.g., concentration of substrate, product, activator, inhibitor, hydrogen ion, etc.). In one case, the substrate binds and induces a slow conformational change from a totally inactive form E$S to a more active form E9$S: S

E

slow step

E' S

ES

Scheme 11.7 In the second case, the free (or uncomplexed) form of enzyme E isomerizes slowly to a form E9 that can bind substrate. slow step

E

E'

S E' S

Scheme 11.8 If two conformational states differ in catalytic activity, slow isomerizations can lead to hysteresis or lag-phase kinetics. The following equation describes the time dependent evolution of a first-order transition from v1 to v2: ðv2  vt Þ ¼ ðv2  v1 Þet=t lnðv2  vt Þ ¼ 

 t lnðv2  v1 Þ t

11.63

11.64

where t is the relaxation time (equal to 1/k) for the transition. A good example of a hysteretic enzyme is bovine liver phosphofructokinase, for which t is in the order of 0.3–0.5 min (Frieden, 1968), roughly the same time-scale thought to be relevant for observed oscillations in the glycolytic rate. The case of nickel ion-dependent activation of calmodulin-dependent phosphoprotein phosphatase (CPP) illustrates how time-dependent conformational changes attend the binding of metal ions (King and Huang, 1983). This phenomenon is most conveniently examined using 10 mM p-nitrophenyl-phosphate (colorless), which is hydrolyzed to yield orthophosphate and p-nitrophenol (bright yellow), in the presence of saturating concentrations of calmodulin (~0.1 mM), 0.2 mM calcium ion, 50 mM potassium chloride, and 20 mM EDTA in 50 mM HEPES buffer (pH 7.6). After CPP (4.6 nM) was added and allowed to totally inactivate, Ni2þ was added, and the time-dependent rise in the product’s 405-nm absorbance was followed

Chapter j 11 Regulatory Behavior of Enzymes 10

6

20

tracing from continuous enzyme assay

A405nm x 1000

A405nm x 100

8

Ni2+ = 0.68 mM

4 2

10

5

2

τ = 1.15 min = 1/kobs

τ

713

1/kobs = 1.15 min

0

1

2

Time (min) 1

0

2

3

4

5

6

Time (min) FIGURE 11.20 Time-dependent reactivation of calmodulin-dependent phosphoprotein phosphatase upon addition of nickel ions. The left panel shows the resulting lag phase reactivation process, and the extrapolated line (dotted) allows for the determination of t, the apparent reactivation lag-time. (The lag-time t was determined, based on the duration from the zero-time point to the extrapolated line, as indicated in the figure.) The right panel shows the first-order kinetics of CPP reactivation. Redrawn from King and Huang (1983) with permission of the authors and publisher.

continuously in a spectrophotometer. Typical data are shown in Figs. 11.20 and 11.21. The authors successfully demonstrated that the enzyme was activated by the binding of one equivalent of divalent cation (determined by K1). Higher metal ion concentrations resulted in deactivation by binding at a second lower affinity metal ion site (K2 z 10  K1). Table 11.4 summarizes the types of hysteretic behavior of various enzymes. The metabolic significance of hysteresis is still inadequately documented. In this respect, one cannot discount the possibility that the observed hysteretic behavior is merely an artifact of enzyme purification and/or

(min)

3

low-temperature storage. Such considerations underscore the need for rapid purification of enzymes under conditions that minimize loss of cofactors (e.g., ligands, metal ions, etc.) that are essential for maintenance of an enzyme’s structural integrity. There are other causes of lag-phase kinetics during enzyme rate measurements. For example, Rudolph, Purich and Fromm (1968) found that bacterial coenzyme A-linked aldehyde dehydrogenase (Reaction: Acetaldehyde þ CoASH þ NADþ # S-Acetyl-Coenzyme A þ NADH þ Hþ) has an essential active-site thiol that must be reduced by dithiothreitol or 2-mercaptoethanol prior to rate measurements. No lag in enzyme rate is observed in this case, whereas a slow time-course of activation occurs when thiol and substrates are combined simultaneously with the bacterial dehydrogenase (see Fig. 4.8). The need to be wary of apparent hysteresis is illustrated in the case of hexokinase, where the lag-time t was reported to be 3 min (Shill and Neet, 1975). At pH 6.5, the enzyme exhibited negative cooperativity with respect to MgATP2, with a slow burst. Intriguingly, both negative cooperativity and burst are lost (a) when kinetic measurements are carried out at pH 8, (b) if citrate is added to the assay at pH 6.5, or (c) provided that MgGTP2 is used in place of MgATP2 at pH 6.5. Morrison (1979) has suggested that the observed hysteresis in the hexokinase reaction is most likely the consequence of slow, tightbinding inhibition (see Section 6.2.7) by AIATP1 complex present in certain commercial ATP samples contaminated with aluminum ion. In fact, an overwhelming body of kinetic evidence indicates that hexokinase is not a hysteretic enzyme (Purich et al., 1973). Finally, lags are frequently observed artifacts in coupled enzyme assays, especially when the auxiliary enzyme has a much higher Michaelis constant than the primary enzyme, or when the auxiliary enzyme concentration is insufficient for rapid attainment of the initial-rate condition.

11.11. ENZYME AMPLIFICATION CASCADES

2

K2 = 4.03 min–1 K1 = 2.08 mM

1

–1/K1

1/K2 0

2

4

6

1 / [ Ni2+ ], mM FIGURE 11.21 Lag-time (t) in the reactivation of calmodulin-dependent phosphoprotein phosphatase versus 1/[Ni2D]. Values for K1 and k2 were obtained by least-squares fitting to the equation: t ¼ (1/k2){1 þ (K1/[Ni2þ])}. Size of data points were chosen to indicate the experimental uncertainty. Redrawn from King and Huang (1983) with permission of the authors and publisher.

The early discoveries of Cori and Green (1943) and Krebs and Fischer (1956) showing that the a and b forms of glycogen phosphorylase can be inter-converted by phosphatases and kinases, led to the broader realization that cells contain linear and multi-cyclic enzyme cascades arranged so that each enzyme in a step or cycle is itself a substrate for a subsequent enzyme-catalyzed covalent modification reaction. Such behavior results in metabolic signalamplifying cascade, in which an initially small input signal results in a massive output signal (Wald, 1965). Examples of the cascades include: (a) retinal photoreception, resulting in the production of about 50,000 molecules of 39,59-cyclicGMP per photon absorbed by rhodopsin; (b) blood clotting,

Enzyme Kinetics

714

TABLE 11.4 Selected Examples of Hysteretic Enzyme Kinetics Enzyme

Observation

Rate Constant

Reference

Phosphofructokinase

Interconversion of active tetramer to inactive dimer

0.023 s1

Frieden (1968)

Chorismate Synthase

Slow activation of dimer

0.0011 s1

Gaertner and Cole (1973)

Glutamate Dehydrogenase

Activation upon slow GTP-induced dissociation

0.35 s1

Frieden and Colman (1967)

Pyruvate Kinase

Activation of tetramer

0.023 s1

Badway and Westhead (1976)

Pyruvate Carboxylase

Activation of tetramer

0.035 s1

Bais and Keech (1972)

Adenylyl Cyclase

Slow activation by ITP and IDP

0.023 s1

Rodbell et al. (1971)

Threonine Deaminase

Hysteretic and cooperative dimer

0.01 s1

Hatfield (1977)

in which countless molecules of fibrin are produced by thrombin-catalyzed proteolysis of fibrinogen; and (b) receptor-induced protein phosphorylation cascades that control all eukaryotic cells. The systematic investigation of the kinetics of these cascade systems promises to provide insights about the design features accounting for the signal amplification and information processing achieved by such hierarchies of enzymes with other enzymes serving as their substrates. Covalent interconversion of enzymes is well established as a fundamental theme in metabolic regulation (Wald, 1965; Krebs, 1994; Stadtman, 2001). Prototypic enzymecatalyzed reversible interconversion reactions include: (a) ATP-dependent phosphorylation of glycogen phosphorylase and glycogen phosphorylase kinase; (b) ATPdependent adenylylation of bacterial glutamine synthetase; and (c) NAD-dependent ADPribosylation of various membrane-associated proteins. Developments over the last few decades confirm that most cascades regulated by protein kinase, kinase-kinase, and kinase-kinase-kinase reactions arranged in highly branched pathways appear to act competitively and simultaneously on a variety of targets. In the case of bacterial nitrogen metabolism, an important observation was that ammonium ion at an elevated concentration can suffice for glutamine in the reactions catalyzed by the various glutamine-dependent enzymes leading to the synthesis of key nitrogen end-products (e.g., ATP, histidine, carbamoyl-P, CTP, glucosamine, and tryptophan). There is thus little need for glutamine synthetase when ammonium ion is freely available in high concentrations to bacterial cells. To accommodate fluctuations in NH3 and glutamine, E. coli and other gram-negative bacilli evolved an elaborate mechanism for rapidly modulating the activity of glutamine synthetase, thereby avoiding wasteful expense of ATP in the net synthesis of glutamine, whenever

the latter is not needed. In explaining the remarkable and versatile regulation of bacterial nitrogen metabolism via the glutamine synthetase nucleotidylation cascade (Adler, Purich and Stadtman, 1975; Chock and Stadtman, 1977, 1980; Stadtman and Ginsburg, 1974), the late Earl Stadtman suggested that a metabolic system is probably most robust

Glutamine Synthetase Cascade HO

Gln Syn is a Dodecamer

PPi

ATP

ATase AMP-O

OH

HO

O-AMP

PII-A OH

UR

UT

AMP-O

O-AMP

PII-D HO

AMP-O

OH

O-AMP

Active With Mn2+

Active With Mg2+ ATase AMP

H2O

FIGURE 11.22 The Escherichia coli glutamine synthetase (GS) cascade. Metabolic regulation of glutamine synthesis by multiple effector molecules that directly reflect the status of nitrogen metabolism in the bacterium. In this scheme, GS is the metabolically active form of glutamine synthetase, and GS-AMP is the metabolically inactive adenylylated enzyme. (For clarity, only three subunits in the GS dodecamer are shown.) The same converting enzyme, designated ATase for adenylyltransferase, catalyzes adenylylation and deadenylylation to form GS-AMP and GS, respectively. As shown in the diagram, however, those metabolites serving as positive effectors for adenylylation behave as negative effectors for deadenylylation, and vice versa. Even more striking is the involvement of a bi-functional enzyme UR/UT, which serves to release UMP from PII UMP or to uridylate tyrosinoyl residues on PII. The dotted lines from PII and PII-UMP indicate the ability of these forms of the regulatory protein to direct the ATase’s action on glutamine synthetase. From Adler, Purich and Stadtman (1975) with permission of the American Society for Biochemistry and Molecular Biology.

Chapter j 11 Regulatory Behavior of Enzymes

when multiple enzymes within a pathway are regulated by a common set of input parameters (Fig. 11.22). His systematic investigations on the adenylylation and deadenylylation of glutamine synthetase indicated that the common inputs are glutamine (Gln), ammonia (NH3), a-ketoglutarate (a-KG) and ATP, which act in an opposing manner (e.g., Gln and high NH3 favor adenylylation, whereas low NH3, a-KG and ATP favor deadenylylation). Significantly, these same metabolites bind to and regulate both glutamine synthetase itself and the other enzymes that control its state of adenylylation. By simultaneously acting on several target enzymes multiple-cycle cascades, activators or inhibitors can reinforce their potency. Stadtman believed that similar multiple-input strategies would be found to regulate protein kinase cascades as well as the blood-clotting protease cascade. Unidirectional activation of the intrinsic and extrinsic pathways of the blood clotting cascade exemplify a linear sequence of proteolysis steps that amplify an initial signal into a massive, well-controlled biological response. In this respect, enzyme cascades are not merely amplification systems; the multiplicity of enzymes involved in a cascade allows for a multiplicity of effector binding sites, thereby enhancing a cascade’s responsiveness to changes in a regulatory input signal. This behavior is illustrated by the numerous calcium ion binding sites in the blood clotting cascade (Fig. 11.23), and a small change in the concentration of this divalent cation elicits a major change in the rapidity with which fibrin is generated in the last step of this sequence of converting proteolytic zymogens into their catalytically active forms.

Intrinsic Pathway

BLOOD-CLOTHING PROTEASE CASCADE

Damaged Surface

XIIinact

XIIact XIinact

Extrinsic Pathway

XIact

Trauma

IXinact

VIIact

IXact

Xinact pro-Thrombin

VIIact

Xinact

Xact

Thrombin

Fibrinogen

FIBRIN

FIGURE 11.23 Schematic representation of the intrinsic and extrinsic pathways for activation of proteases in the blood-clotting cascade, culminating in the conversion of fibrinogen to the major clotting constituent fibrin. (Asterisks indicate calcium-activated proteases that must be converted from their inactive (black type) to their active (gray) forms.) These proteases contain g-carboxy-glutamate residues that potentiate the affinity for calcium ion that is present at a high plasma concentration.

715

e1 Multi-Cycle Enzyme Cascade

ON

OFF

ENZ1

ENZ1

OFF

X + ENZ2

ON

ENZ2 + Y

OFF

X + ENZ3

ON

ENZ3 + Y

OFF

X + ENZ4

ON

ENZ4 + Y

FIGURE 11.24 Cascade of Enzyme-Catalyzed Interconversion Reaction. A multi-step enzyme interconversion cascade illustrating the sequential modification and conversion of target enzymes from one state of activity to another. Although written here as a cascade of increasing catalytic activity, enzyme-catalyzed covalent modification can either activate or inhibit target enzymes, depending on the particular system under study.

Likewise, deficiency in vitamin K or coumarin analogs can greatly reduce the susceptibility of these proteases to calcium ion as a consequence of failure to form g-carboxyglutamyl residues. A schematic representation of a generalized multi-cyclic cascade is shown in Fig. 11.24, where ‘‘o’’ and ‘‘m’’ respectively designate the original unmodified and the modified enzyme forms in cycles marked 1, 2, and n. Accordingly, the kinetic properties of cascade ‘‘converter’’ enzymes are of paramount significance in the operation of covalently inter-converting enzyme systems. Stadtman and Chock (1977a,b) and Chock and Stadtman (1980) led the way in developing kinetic models for these cascades, and based on extensive simulations, they offered the following concepts regarding the kinetic properties and responsiveness of enzyme cascades: Concept-1. Covalent interconversion of an enzyme can be a highly dynamic process, and the specific activity of the target enzyme is determined by coupling of two opposing cascades, resulting in a steady-state distribution of active and inactive forms of an interconvertible enzyme. The target enzyme’s activity smoothly and continuously responds to feedback signals that alter the efficiency of the enzymes catalyzing the inter-conversion steps as well as the ability of the target enzyme to be an effective substrate for the inter-converting enzymes. Concept-2. A minimum of three enzymes are typically involved in each interconvertible enzyme cycle that can integrate many allosteric effectors into a single output, the specific activity of the target enzyme in the cascades.

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This arrangement also allows for multiple signal inputs as a consequence of effector binding at separate sites on cascade enzymes. Concept-3. Cyclic cascades can achieve great signal amplification, and interconvertible enzymes can respond to effectors at concentrations considerably below their respective dissociation constants. Concept-4. Cyclic cascades can propagate a sigmoidal dose-response curve showing how an increasing concentration of allosteric effector manifests itself in activity changes of the target enzyme. Salient features of the Stadtman-Chock treatment were verified in the elegant studies of Mura, Chock and Stadtman (1981) on factors affecting the state of adenylylation of glutamine synthetase in Lubrol W2C-permeabilized E. coli cells (Mura and Stadtman, 1981). These cells were permeable to small molecules, and all components of the bicyclic covalent interversion system (i.e., uridylylation/deuridylylation and adenylylation/ deadenylylation activities) are retained with cells cultured with 10 mM glutamine as the sole nitrogen source. The protein components of the adenylylation/deadenylylation cascade are retained with permeabilized cells grown in the presence of 5 mM glutamine. Under the experimental conditions employed, only the monocyclic cascade was investigated, because the bifunctional uridylyltransferaseuridylyl-removing enzyme (or UTase/UR Enz) that catalyzes the uridylylation and deuridylylation requires the use of the Pn regulatory protein, and Pn is selectively inactivated by the Lubrol treatment. Mura, Chock and Stadtman (1981) demonstrated that nSS (i.e., the steady-state extent of glutamine synthetase adenylylation, corresponding to the average number of covalently bound AMP molecules per dodecamer) in the permeabilized E. coli cells is determined by the concentration of various substrates and/or metabolic effectors added to the suspending buffer. In both permeabilized cells grown in the presence of 5 and 10 mM glutamine, the nSS value depends upon the concentration of substrates (ATP, Pi) and allosteric effectors (ATP, glutamine, and a-ketoglutarate) of adenylyltransferase, whereas only in cells grown in 10 mM glutamine is the value of nSS affected by the UTase/UR-Enz substrate UTP and its allosteric effectors CMP and Mn2þ. Importantly, the 5and 10-mM glutamine-grown permeabilized cells exhibited characteristics predicted by the theoretical treatment of mono- and bicyclic cascade systems (Stadtman and Chock, 1978), respectively. The value of nSS in both cell preparations increased as the [glutamine]/[a-ketoglutarate] concentration ratio in the suspending buffer was increased, and both preparations exhibited high signal amplification (i.e., the concentration of glutamine required to sustain an nSS value of 6.0 was much lower

than the Michaelis constant of adenylyltransferase for glutamine). Moreover, the signal amplification factor for 10 mM glutamine-grown cells was much greater than that for 5 mM glutamine-grown cells, and only the former could elicit a cooperative type of response of nSS to increasing glutamine concentration. Because the theoretical treatment relied on kinetic parameters determined by earlier in vitro studies, Mura, Chock and Stadtman (1981) concluded that regulation of glutamine synthetase adenylylation in the permeabilized cells occurs by the same mechanism as established with purified cascade enzyme components and that the principles of monocyclic and bicyclic cascade models operate in situ in the permeabilized cells. Ligand-induced changes in a ‘‘target’’ enzyme’s conformation or oligomerization state can also influence their performance as macromolecular substrates for converter enzymes. This is to be anticipated because target enzymes often possess a battery of binding sites for substrates and/or effectors and because target enzymes are known to have their own allosteric control mechanisms. In principle, the Km and/or kcat of a converter enzyme may be altered by the target enzyme’s conformation or oligomerization. Stadtman and Ginsburg (1974) and Adler, Purich and Stadtman (1975) pointed out how the informationprocessing capacity of the bacterial glutamine synthetase cascade is likely to be increased by the binding of regulatory molecules at sites on target and converter enzymes. A more complete analysis of cascade dynamics remains to be achieved, even for the most rigorously investigated cascade prototypes (i.e., glycogen phosphorylase, glutamine synthetase, or blood clotting). In the simplest of these, namely the glycogen phosphorylase cascade, the number of input parameters can be staggering (e.g., epinephrine, 39,59cyclicAMP, ATP, ADP, AMP, Pi, and glucose – not to mention the confounding effects of having target and modifying enzymes embedded within the physically constraining environment of the glycogen particle). The daunting task of analyzing the dynamic behavior of covalent interconversion cascades has required the use of restrictive assumptions to reduce kinetic complexity without losing a sense of the system’s robustness. A serious limitation of the Stadtman-Chock treatment of multi-cyclic cascades is an implicit reliance on the rapid-equilibrium treatment to simplify their rate equations. As we already have observed with the much simpler bisubstrate random substrate kinetic mechanisms, the rapid equilibrium equation is incredibly simpler than the steady-state equation (compare, for example, Eqs. 6.75 and 6.72). One may anticipate that more realistic steady-state treatments of multi-cyclic cascades would yield rate laws that are too complex for meaningful analysis. If such is the case, one may entertain reservations about both the physical realism and generality of the Stadtman-Chock treatment.

Chapter j 11 Regulatory Behavior of Enzymes

In seeking greater understanding of the kinetic constraints on covalent interconversion cascades, Goldbeter and Koshland (1981) advanced the novel concept known as zero-order ultrasensitivity. By modeling the dependence of the steadystate rates of modifying reactions on the concentrations of their macromolecular substrates (i.e., unmodified target proteins) and products (i.e., covalently modified targets), these investigators observed that a target’s fractional modification should be quite sensitive to the ratio of the limiting rates of the modifying enzymes. Sensitivity was particularly strong when the kinetics are zero-order with respect to its target (i.e., [Unmodified Protein Substrate] >> Km). Their findings suggest that a multi-enzyme cascade with reversible converter enzymes has three potential ways for enhancing its sensitivity beyond that expected from Michaelis-Menten kinetics: (1) conventional cooperativity that could occur for any enzyme with a Hill coefficient exceeding unity; (2) multistep ultrasensitivity, resulting from an ability of a given ligand (or its messenger) to act in more than one step; and/or (3) zero-order ultrasensitivity, wherein the converter enzymes operate under saturating conditions to amplify the response to a signal. A given pathway or cascade can, in principle, use any or all such mechanisms to enhance signal sensitivity. Because glycogen phosphorylase (Reaction: Glycogenn þ Pi # Glycogennþ1 þ Glucose-1-P, where n indicates the number of glucosyl units) is so abundant in muscle, Meinke, Bishop and Edstrom (1986) undertook in vitro kinetic experiments on the covalent modification of phosphorylase by phosphorylase kinase (Reaction: Phosphorylase b þ MgATP2 # Phosphorylase a þ MgADP, where the b and a indicate the unphosphorylated and phosphorylated forms) and phosphoprotein phosphatase (Reaction: Phosphorylase a þ H2O # Phosphorylase b þ Pi). Employing protein concentrations found in skeletal muscle, Meinke, Bishop and Edstrom (1986) observed the predicted zero-order ultrasensitivity phenomenon. An observed Hill coefficient nH of 2.3 indicated the apparently cooperative nature of the system with respect to its macromolecular substrate. The work of Huang and Ferrell (1996) on the mitogen-activated protein kinase cascade, found that this cascade shows a sharp stimulus-response curve that may be fundamentally important in the function of those molecular switches controlling such high-level processes as mitogenesis. Protein kinases often exhibit selectivity, rather than absolute specificity, for target proteins. Such behavior is reminiscent of the ability of certain blood clotting enzymes to substitute for each other, albeit with reduced efficiency. Morishita, Kobayashi and Aihara (2006) conducted computer simulations to investigate the time evolution of signal amplification cascades composed of M signaling molecular species whose numbers are assumed to be constant and equal to N for all steps of the cascade (Fig. 11.25). Each signaling molecule has only two states: inactive and active. Inactive molecules in the (iþ1)th step

717

initial input

1st 1 step 1

2nd 2 step 2

= Inactive = Active

Mth M step M

FIGURE 11.25 Schematic diagram of the signaling cascade. The input signal activates the inactive signaling molecules of the first step. The activated signaling molecules of the ith step catalytically activate the inactive signaling molecules of the (i þ1)th step. The activated molecules of each step spontaneously become inactive.

are catalytically activated by active ones in the ith step, and the molecules in the first step are activated by the input to the cascade. The activated molecules at each step then become spontaneously inactive. They examined the performance of signal transduction in a signaling cascade by focusing on the influence of three parameters: the number of signaling molecules N, the input intensity tp, and the cascade step-i. They defined four characteristics of the transient response: (a) the signal integral: Z N IðN; i; tÞ ¼ Xi ðtÞdt 11.65 i¼0

defined in terms of N, i, t, and t; (b) signal amplitude A(N,i,t), equal to the product of maximal response Maxi for ith step and Xi(t); (c) signaling time T(N,i,t), defined as the time at which Xi reaches its maximum, and is defined only when the signal amplitude is greater than zero; and (d) signal duration D(N,i,t), defined as the signal integral divided by the signal amplitude D(N,i,t) ¼ I(N,i,t)/A(N,i,t), and is again defined only when the signal amplitude is greater than zero. For numerical calculations of the time series of Xi(t) and the statistics of the characteristics defined above, they used Gillespie’s algorithm, a Monte Carlo method, to numerically calculate sample paths obeying the chemical master equation (Gillespie, 1976). The statistics of the characteristics were calculated from 10,000 independent samples. Morishita, Kobayashi and Aihara (2006) found that each step of the signaling cascade has an optimal number of

718

molecules at which the average signal amplitude of the response is maximized for i 2. This optimal number was shown to be a consequence of the balance between the failure of signal propagation by signal loss and the signal amplification by fluctuations in stochastic reactions. Morishita, Kobayashi and Aihara (2006) showed that a small number of signaling molecules endows the cascade with an ability to actively discriminate true signals and error signals. A trade-off relation is found between reliable signal transduction with low false-negative errors and specific signal transduction with low false-positive errors. In addition, they found that the specificity and the reliability are balanced for an optimal N even if a biased requirement for either the specificity or the reliability is imposed. Furthermore, several properties of the cascade (such as the signal duration and the signaling time) are strongly influenced by the stochasticity originating from the small number of signaling molecules. Finally, the factors governing these highly dynamic cascades are also apt to respond to a complex hierarchy of regulatory effectors beyond the confines of any one pathway. Future kinetic work on the time-evolution of cascades, rather than a steady-state rate treatment, is likely to reveal how time-dependent changes in the responsiveness are related to the kinetic properties of individual components.

11.12. SUBSTRATE CHANNELING Observing that the intracellular concentrations of enzymes are in many cases surprisingly high, far above those used in most kinetic experiments, Srere (1985) suggested that functionally related enzymes are apt to interact and form noncovalent, multi-enzyme complexes that he termed metabolons. He further predicted that these supramolecular complexes may be endowed with special properties, particularly the ability to increase metabolic flux by mechanisms allowing scarce pathway intermediates to be immediately available to the next enzyme within a metabolic pathway. Although many reaction intermediates are covalently tethered and transferred within such multienzyme complexes as the keto-acid dehydrogenase and fatty acid synthase complexes, Srere’s ideas gave rise to the idea that non-covalently bound intermediates are channeled by favorably arranged subunit-subunit interactions. However, recognizing that some have assumed that two or more pools of the same low-molecular-weight metabolite can coexist within the cytosol or another such compartment, Barros and Martinez (2007) recently used Brownian diffusion and enzyme/transporter turnover numbers to analyze the behavior of metabolites generated at different sites within the same compartment. They described a metabolite domain in terms of: (a) its strength or amplitude (i.e., the ratio between metabolite concentration at the source and the

Enzyme Kinetics

average concentration in the compartment); and (b) its physical size or extension (i.e., an arbitrary parameter defined as the distance between source and the point at which the concentration falls below twice the compartment average). The amplitude ranges from unity in a perfectly homogeneous compartment, where there is no flux, to a value of infinity for the strongest domain. Their analysis, which can be applied to any other metabolite, if its rate of production, average concentration, and diffusion coefficient are known, indicates that ATP, glucose, pyruvate, lactate, and glutamate cannot be concentrated at their sources of production/transport to an extent that would appreciably affect their downstream targets. The amplitudes for the above metabolites varied from 1.0008 to 1.01, and their extension values consistently equaled zero. For these and other metabolites produced by slow enzymes or transporters and present at mM or higher concentrations, Barros and Martinez (2007) concluded that the cytosol behaves as a well-mixed, homogeneous compartment. X-ray crystallographic analysis of certain multi-subunit enzyme complexes has already demonstrated that two or more subunits often provide separate active sites for catalyzing the overall reaction (Raushel, Thoden and Holden, 2003). While such a division of labor had been anticipated on the basis of the activities of isolated subunits as well as sequence-based predictions of binding motifs and affinity labeling experiments, enzyme crystallography unequivocally established that the active sites responsible for sequential transformations of metabolites are frequently remote from each other, often situated at ˚ from each other. In several distances of 25–100 A instances, it has been possible to identify the actual channel that serves as a conduit for essential intermediates between two or more active sites. The phenomenon of substrate channeling refers to the direct physical transfer of non-covalently bound intermediates(s) from the active site of the enzyme producing the intermediate to the active site of an enzyme catalyzing a next step in a metabolic pathway. An implied feature of substrate channeling is that there must either be direct contact between the two active sites, or a channel or tunnel that facilitates the intermediate’s transfer while limiting its escape. From a kinetic perspective, channeling requires the transferred intermediate to remain enzyme-bound without loss to the bulk solvent. Given the substantial genetic burden for maintaining the structurally constrained subunit-subunit interfaces comprising a tunnel, substrate channeling must offer compelling advantages with respect to catalysis. As shown in Table 11.4, ammonia, indole, and carbon monoxide are three intermediates now known to be shuttled by means of tunnels. Were ammonia permitted to diffuse freely away from enzymes using glutamine’s g-amide to generate nascent ammonia, >99% would almost instantaneously be

Chapter j 11 Regulatory Behavior of Enzymes

protonated to form ammonium ion at neutral pH, thus quenching its nucleophilicity. The free base form of indole per se is not particularly reactive; nor is it, for that matter, all that toxic. Even so, its limited water solubility would almost assuredly result in nonspecific capture by other proteins, thus reducing the efficient synthesis of tryptophan. Likewise, given the well known affinity of heme proteins for carbon monoxide, there can be little doubt that this diatomic molecule would be promptly intercepted by the abundant cytochromes within bacteria. Aside from minimizing side reactions, tunnels may play a role in choreographing the timing of catalysis at active sites that would otherwise be too remote from each other to operate efficiently. Given the likelihood that the intermediate’s motions are restricted within the lumen of the tunnel, the dimensionality of diffusion is effectively reduced to one dimension. Adam and Delbru¨ck (1968) considered how the reduced dimensionality of diffusion can: (a) lower the time required for a metabolite or particle originating at point P to reach point Q; and (b) increase the capture or catch of molecules by other molecules localized in the vicinity of target point Q. They also pointed out how the

719

ability to manage the dimensionality of diffusion represents a selective advantage in evolution, suggesting further that fully developed, internally compartmentalized cells enjoy a much greater range of options for controlling metabolic processes. Even so, there is no a priori reason why transit through a tunnel is necessarily a diffusionlimited process. For tunnels of sufficient length and volume, more than one molecule of the intermediate may be entrained, such transit through the tunnel would not be rate-determining. One view is that the enzyme has an H-bonding network (or proton wire) that conducts outbound protons from the carbonic anhydrase active site to the bulk solvent (Tu et al., 1989); one can envisage a tunnel filled with an extended chain of interacting ammonia molecules. Although hydrogen bonding in liquid ammonia is by no means as pervasive as in water, enzyme systems have provided us with countless surprises. Finally, given the division of labor of multiple protein and RNA components in ribosomes (Wilson and Noller, 1998; Yusupov et al., 2001), substrate channeling is apt to contribute in large measure to high-fidelity proofreading during protein biosynthesis.

TABLE 11.5 Examples of Enzymes Employing Tunnels to Transfer Reactive Intermediates Between Active Sites Enzyme

Comments

Tryptophan Synthase

The a-subunit catalyzes the conversion of indole-3-glycerol phosphate to the indole intermediate and glyceraldehyde-3-P, and the PLP-containing b-subunit catalyzes the condensation of the indole intermediate with L-serine to yield L-tryptophan. The enzyme possesses a hydrophobic tunnel (length ~25 A˚) for passing indole between these active sites (Rhee et al., 1997). Single turnover kinetic experiments verify that the concentration of the indole intermediate is extremely low, consistent with the direct and rapid transfer of the indole intermediate (Anderson, Miles and Johnson, 1991).

Carbamoyl-P Synthetase

The small subunit catalyzes glutamine hydrolysis to form glutamate and unprotonated ammonia, which is used by the large subunit to form carbamoyl-P via carboxy-phosphate and carbamate intermediates. Synchronization of the active sites within CPS is facilitated by a 96 A˚ long tunnel (Thoden et al., 1997). Phosphorylation of bicarbonate to carboxy-phosphate appears to be the trigger for the hydrolysis of glutamine and release of ammonia into the tunnel (Raushel, Thoden and Holden, 2003).

Glutamine PRPP Amidotransferase

The enzyme catalyzes glutamine hydrolysis to form glutamate and unprotonated ammonia, the latter of which ˚ to react with PRPP to yield phospho-ribosylamine (Krahn et al., 1997). traverses ~20 A

Asparagine Synthetase

This dimeric enzyme catalyzes reaction of ammonia (again generated by glutamine hydrolysis), ATP, and aspartate to form asparagine via an a-aspartyl-AMP intermediate. A molecular tunnel (~19 A˚ length), mainly formed by backbone atoms and hydrophobic side chains, connects the two active sites (Larsen et al., 1999).

Glutamate Synthase

The NADPH-dependent, 3Fe-4S enzyme catalyzes the reductive incorporation of ammonia (produced by ˚ tunnel) to a second active site to react with the hydrolysis of L-glutamine and transferred by means of a 30-A C-2 carbon of 2-oxoglutarate, yielding two L-glutamate molecules (van den Heuvel et al., 2002).

Glucosamine 6-P Synthase This enzyme catalyzes the glutamine-dependent synthesis of fructosimine 6-P which subsequently isomerizes to glucosamine 6-phosphate. The hydrophobic tunnel connecting the glutaminase and isomerase domains appears to be blocked by Trp 74, and glutamine binding is thought to trigger the movement of the indole ring of Trp 74 to permit ammonia transfer (Teplyakov et al., 2001). Carbon Monoxide Dehydrogenase/ Acetyl-CoA Synthase

This multi-functional enzyme, which contains a [Fe4S4]-cubane bridged to a copper-nickel binuclear site, ˚ long conduit for transferring CO generated at the active site of one subunit for possesses a 138-A incorporation into acetyl-CoA at the active site on the other subunit. This physical mechanism undoubtedly limits reaction of CO with numerous heme-containing redox enzymes (Doukov et al., 2002).

Enzyme Kinetics

720

11.12.1. Several Criteria Define Substrate Channeling Kinetic treatments of substrate channeling (Anderson, 1999; Wu et al., 1991) provide kinetic criteria for assessing whether an enzyme’s catalytic mechanism is likely to employ substrate channeling. Scheme 11.9 describes two mechanisms by which a substrate can be converted to a metabolic product via an obligate intermediate. In the dissociative (or diffusional) case, enzyme E1 converts substrate A to intermediate B, which is then released to the medium (indicated by the blue arrow); B is free to diffuse away from the E1$E2 dimer or may bind to the active site of enzyme E2 on the original or another E1$E2 dimer complex, allowing B’s conversion to C. In the non-dissociative (or direct channeling) case, B is instead transferred between E1 and E2 while confined within a tunnel. escape

A

tunnel

C

B E1

B

A E1

E2

dissociative

C E2

nondissociative

Scheme 11.9 There are several approaches for evaluating whether substrate channeling occurs. In the transient-time approach, the basic experiments shown in Fig. 11.26 are conducted to determine the immediacy of B conversion to metabolite C. In Experiment-1, if substrate B can bind rapidly to Enzyme2, B will be converted to C promptly at a steady rate without any delay. In the second experiment, we measure the timeevolution of A-to-C conversion, which eventually reaches the same limiting rate as that observed in Experiment-1. The basic assumptions are that: (a) binding of B to the first

enzyme does not alter the rate observed in Experiment-1; and (b) the reverse reaction has no effect on the transienttime. To analyze for channeling, one carries out the two separate rate measurements shown in Fig. 10.23. If the observed rate of A-to-C conversion is faster than expected on the basis of a rate equation that takes into account the ratio of Vmax and Km (dashed line). B

B

A E1

C

B

A E1

C channelling

E2

Final Product C has high specific radioactivity

Scheme 11.10 We can also use an isotope dilution approach to evaluate the likelihood of channeling. The asterisk on B in the diffusion model indicates that radiolabeled B will suffer dilution in its radiospecific activity if A) is reacted with enzyme E1E2 in the presence of various concentrations of unlabeled B. In the absence of added B, the specific activity will be {cpm B)/ (mol B))}, where cpm stands for counts/min determined in a liquid scintillation counter and mol stands for the amount of substance B, usually determined enzymatically, spectrophotometrically, or with fluorescence intensity. If the measurements are repeated in the presence of x mol B and y mol B, the specific activity will be decreased, according to the ratios {cpm B)/(x mol B þ mol B))} and {cpm B)/(y mol B þ mol B))}. The transient-phase kinetics of substrate-exchange can be represented by release of ligand B from the first enzyme E1, followed by its binding to a second enzyme E2, as shown in the following scheme:

E1B Product C Formed

Final Product C has low specific radioactivity

escape & exchange

E2

Experiment-1 Enz-2 C B

B + E2

k1 k-1 k2 k-2

E1 + B E2B

Scheme 11.11

τ

Experiment-2 Enz-1 Enz-2 A B C Time

Transient Phase = 1/kobs FIGURE 11.26 Transient-time analysis for substrate channeling. See text for explanation of lag-phase.

The composite transient kinetic rate constants for the above mechanism can be written as: l1 xk1 þ k1 ½E1  þ k2 ½E2  þ k2 l2 x

k1 ðk2 þ k2 ½E2 Þ þ k1 k2 ½E1  l1

11.66 11.67

where the rate constants match those presented in the two-step scheme. This approach allows us to evaluate

Chapter j 11 Regulatory Behavior of Enzymes

721

the individual rate constants in the kinetic pathway for substrate exchange between successive active sites. Anderson (1999) describes how we may use the rapid mixquench technique under single-turnover conditions (see scheme below). The enzyme is treated as a reactant whose concentration exceeds the concentration of the limiting radiolabeled substrate. B E1

C entry & catalysis

E2 B B

A

no entry

Final Product C has zero specific radioactivity

Scheme 11.12

11.12.2. Tryptophan Synthase is an Outstanding Example of Substrate Channeling In this reaction, L-serine and indole-3-glycerol phosphate react to form tryptophan and glyceraldehyde 3-P in two separate reactions (Scheme 11.13). 2-O

3PO

H H

OH α-Subunit

2-O

3PO

OH

H

CHO + OH

N H

N H

CO2 HO H C H

CO2 H NH3

200 s–1

β-Subunit

H2C

+ N H

H + H O 2 NH3

N H

Scheme 11.13

The E. coli enzyme is an a2b2 tetramer. The a-subunit catalyzes a retro aldol condensation, whereas the b-subunit mediates a PLP-dependent b-elimination. A detailed kinetic analysis by Anderson, Miles and Johnson (1991) of wildtype tryptophan synthase resulted in the kinetic scheme shown in Fig. 11.27 showing the communication between a and b subunits.

E + Serine

10 µM–1s–1

20 s–1

Indole-Glycerol 3-P • E 8 s–1

0.16 s–1

24 s–1

Indole-Glycerald 3-P • E* 2 µM–1 s–1

20 s–1

0.2 µM–1 s–1

200 s–1

Indole + Glycerald 3-P + E

0.14 µM–1s–1

E • Serine

Activate

Indole-Glycerol 3-P • E*

Indole + Glycerald 3-P • E

In Experiment-1 (upper path in Scheme 11.12), the reaction rate of B) (red) from solution at E2 (conversion of B) / C)) in a single turnover is compared to results from Experiment-2 (lower path in Scheme 11.12) measuring the rate of A) / B) / C) in a single enzyme turnover. If C) formation is faster, B) must have transferred between sites without dissociation.

β-Subunit

Indole-Glycerol 3-P + E

11 s–1

C E2

E1

Final Product C has high specific radioactivity

α-Subunit

8 s–1

0.16 s–1

E~Amino Acid Indole 2 µM–1s–1 Channel E* ~ Amino Acid . Indole 0.1 s–1

1000 s–1

E* • Tryptophan 0.5 µM–1s–1

8 s–1

E + Tryptophan

FIGURE 11.27 Reaction mechanism and rate constants for catalytic events at separate active sites of tryptophan synthase. See text for additional details. Adapted from Anderson (1999).

Formation of the serine adduct with the pyridoxal 5-P (or PLP) cofactor within the b-subunit activates events at the a-subunit. Upon transfer through a channel lined with hydrophobic residues to facilitate its movement, indole arrives at the next active site and reacts with the serine-PLP adduct to form tryptophan. Finally, channeling of charged substrates between the active sites of bi-functional enzymes or bi-enzyme complexes appear to be significantly enhanced by favorable interactions with the electrostatic field of the enzymes. Transfer efficiencies are a measure of the probability that substrate leaving the active site of the first enzyme will reach the active site of the second enzyme before escaping out into bulk solution. Experimental indicators of channeling, on the other hand, are factors that decrease in the transient (lag) time for appearance of the final product of the coupled enzyme reaction or a decrease in the susceptibility of the overall reaction rate to the presence of competing enzymes or competitive inhibitors. Elcock, Huber and McCammon (1997) analyzed the connection between simulated transfer efficiencies and experimental observables, by extending previously reported analytical approaches to combine the simulated transfer efficiency with the Michaelis-Menten kinetic parameters Km and Vmax of the enzymes involved. They derived expressions allowing both transient times and steady-state rates to be calculated, and they applied this approach to two systems that have been studied both theoretically and

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experimentally. The first was the bifunctional enzyme dihydrofolate reductase-thymidylate synthase (DHFR-TS), which exhibits an experimentally observed decrease in transient times that is consistent with a transfer efficiency of around 80%. In the second case of the citrate synthasemalate dehydrogenase fusion protein, a transfer efficiency of 73% was found to be consistent with the experimental transient time measurements. Independent analysis of the effects of adding the competing enzyme aspartate aminotransferase yields a transfer efficiency of 69%, in excellent agreement with the transient time results. In both cases, the experimental transfer efficiencies are in good agreement with those obtained from simulations that include electrostatic interactions. They also analyzed the discrepancy between simulation and experiment, in that a competitive inhibitor in the DHFR-TS system yields qualitatively different results from those predicted on the theoretical grounds.

11.12.3. The Once-Confusing Story of NADD Transfer Between Dehydrogenases Illustrates the Need for Careful Studies on Substrate Channeling The following brief account underscores the folly of proposing channeling based on limited experimental findings and inadequate kinetic theory.

Nitrobenzaldehyde + NADH

ADH

NAD+ + Nitrobenzyl alcohol

GPDH Glyceraldehyde-3P + NAD+ + Pi 1,3-Bisphosphoglycerate + NADH

Scheme 11.14

Srivastava and Bernhard (1984; 1985; 1986) claimed that reduction of benzaldehyde and p-nitrobenzaldehyde by NADH using horse liver alcohol dehydrogenase (ADH) was faster if NADH was bound to glyceraldehyde-3-P dehydrogenase (GPDH). The rate of reduction of aldehyde substrate with GPDH$NADH appeared to follow a Michaelis-type concentration dependence; even so, the apparent Km for GPDH$NADH complex was higher than that for free NADH. Reaction velocity was independent of GPDH concentration when [GPDH] > [NADH]Total. These investigators suggested: (a) that NADH was channeled from GPDH to ADH through the initial formation of a GPDH$NADH$ADH complex; and (b) that glycolytic enzymes form multi-enzyme complexes for channeling of intermediate metabolites. Chock and Gutfreund (1988) re-examined the kinetics of NADH transfer and concluded that coenzyme transfer between GPDH and ADH proceeds by a dissociative

mechanism, not by direct transfer through a ternary complex. While their conclusions were disputed (Srivastava et al., 1989). Brooks and Storey (1991) reported that they could reproduce the results of Srivastava et al. (1989), but that a mathematical solution of the direct-transfer-mechanism equations of Srivastava et al. (1989) did not adequately describe the experimental properties of the reaction rate at increasing ADH concentrations. Subsequent studies by Wu et al. (1991) yielded compelling evidence for a dissociative mechanism as well as a clear set of kinetic criteria for testing whether an enzyme exhibits substrate channeling.

11.12.4. Substrate Hydration may also Affect Channeling Measurements In assessing the potential for substrate channeling during glycolysis, Ova´di and Keleti (1978) adduced evidence suggesting that glyceraldehyde 3-P formed in the aldolase is channeled to glyceraldehyde-3-P dehydrogenase (GPDH): Aldolase Fructose 1,6-Bisphosphate DHAP + Glyceraldehyde-3P Glyceraldehyde-3P + NAD++Pi

GPDH 1,3-Bisphosphoglycerate + NADH

Scheme 11.15 These investigators found that the Michaelis constant of the intermediate product, namely glyceraldehyde-3-P, produced by aldolase could be determined with the GPDH coupled reaction, as shown in Scheme 11.12. The observed Km value corresponded to that of the aldehyde (active) form of glyceraldehyde-3-P, although under their conditions, the aldehyde is hydrated to its diol faster than the enzymic reaction catalyzed by GPDH. They suggested that above a certain concentration of the enzymes the glyceraldehyde-3-P formed by aldolase is channeled to GPDH. Pettersson and Pettersson (1999) reanalyzed this coupled enzyme system and convincingly demonstrated the crucial effect of nonenzymic hydration (i.e., that glyceraldehyde 3-P is hydrated, forming a gem-diol species that is unreactive in the subsequent GPDH reaction) on the interpretation of observed transient reaction rates and isotope dilution measurements for the aldolase–GPDH coupled reactions. They investigated a model (Scheme 11.13) in which the substrate S is converted irreversibly to product P by an enzyme operating under pseudo-first-order conditions determined by the rate constant k. Substrate S was assumed to be in equilibrium with Shyd which cannot interact with the enzyme. In this scheme, the catalytically reactive form S is supplied at a constant rate v, which may or may not equal zero.

Chapter j 11 Regulatory Behavior of Enzymes

S

k

P

k1 k-1 Shyd Scheme 11.16 Based on their kinetic treatment, Pettersson and Pettersson (1999) experimentally tested the system and found that under the experimental conditions employed by Ova´di and Keleti (1978), the kinetics of the dehydrogenase-catalyzed oxidation of glyceraldehyde 3-P are governed by two exponential transients, with the faster phase reflecting interaction of the dehydrogenase with the substrate’s aldehyde form, and the slower phase the nonenzymic dehydration of the substrate’s diol form. The rapid transient is dominant in the coupled two-enzyme reaction, because the aldolase produces the substrate in its aldehyde form, thus explaining why the lag-time constant observed by Ova´di and Keleti (1978) corresponds to the Km of the aldehyde form. Pettersson and Pettersson (1999) also found that the aldehyde–diol interconversion has no readily detectable effect on the transient-state kinetics of the coupled reaction, simply because diol dehydration is very slow (life-time z 140 s) compared with that of the rapid transient (where k z 20–100 s1). The converse holds when the dehydrogenase acts upon glyceraldehyde 3-P added to (and equilibrated with) solution. That the main transient governing the latter reaction is much slower than the transient observed in the coupled reaction was taken as further evidence for the channeling of glyceraldehyde 3-P in the coupled reaction (Orosz and Ova´di, 1987). Nonetheless, Pettersson and Pettersson (1999) found this rate discrepancy is anticipated precisely in the free-diffusion mechanism. Likewise, their analysis lends no support to the conclusion of Orosz and Ova´di (1987) that the experimentally observed isotope dilution is lower than expected for a free-diffusion mechanism, indicating the likely presence of a leaky channel for glyceraldehyde 3-phosphate transfer. Pettersson and Pettersson (1999) found instead that the observed isotope dilution factor is in excellent agreement with that expected for the free-diffusion mechanism, and inevitably justifies the conclusion that the isotope dilution experiment of Orosz and Ova´di (1987) provides exceptionally convincing evidence for the absence of significant contributions from channeling.

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may even behave entirely differently than inferences drawn by analysis of individual components. Thus, while treating enzymes as pathway building blocks, the central focus of MCA is that the integrated metabolic network operates within its cellular context. One seeks to understand largescale dynamics of metabolic and physiological systems through modeling, simulation, and experimentation cast in terms of the sensitivity or responsiveness of metabolic flux to input signals. Engineers use the term robustness to describe the degree to which a system or process can still function effectively in the face of stresses and/or even invalid inputs. In statistics, the same term applies to a test or property that is not seriously disturbed by violations of the assumptions on which it is based. And in medicine, ‘‘robust’’ intones a condition of enhanced stability and vigor. That metabolic pathways are likewise robust probably attests to the ability of their component enzymes to adopt new steady-state conditions and their ability to respond to multiple inputs from activators and inhibitors, as well as environmental factors (e.g., temperature, pH, pMg2þ, etc.). The key ideas of Metabolic Control Analysis are expertly set out in the book Understanding the Control of Metabolism by Fell (1997) as well as the recent review article ‘‘Signaling Control Strength’’ by Westerhoff (2008). Both focus on the large-scale integration of cellular processes that are hierarchically arranged and dynamically interactive. MCA emphasizes the need to focus on metabolic flux control, eschewing the classical notion of a rate-determining enzyme reaction and looking instead at the cumulative and synergistic effects of pathway organization. Shown in Fig. 11.28 is a simplified schematic that Westerhoff (2008) offered to illustrate the hierarchical organization of intracellular processes. Although multiple metabolic pathways are not readily represented in such a simplified diagram, the diagram indicates that the robustness of metabolism is doubtlessly linked to the presence of multiple steady states that ‘‘buffer’’ metabolic fluxes against changes in most of their component enzymes. Pathways are highly organized multi-step systems designed to maintain a steady-state metabolic throughput.

J S X1 RDS

1f 1r

X2

2f 2r

X3

3f

⋅ ⋅ ⋅X i 3r

if ir

⋅⋅⋅ Xi+1

(n-1)f

Xn

(n-1)r

J

P

Scheme 11.17

11.13. METABOLIC CONTROL ANALYSIS Enzyme kineticists have been highly successful reductionists who have dissected complex catalytic and regulatory mechanisms, often defining their elementary steps. Even so, the emerging discipline, known by the rubric Metabolic Control Analysis (or MCA), offers the perspective that the whole is much greater than the sum of its parts. The whole

where the flux J is the difference (vf  vr) for the forward and reverse steps in each successive reaction. Most steps are reversible enzyme-catalyzed reactions that operate at or near equilibrium, with the direction of flow depending on what is essentially a one-way supply of substrate and one-way removal of product P. In most pathways, the first step is thermodynamically favorable, so much so that its substrate S

Enzyme Kinetics

724

where a change in substrate D[A] results both in a change in velocity Dv and a change in velocity DJ, such that: DJ Dvi ¼ ðvf  vr Þ J

FIGURE 11.28 Hierarchical subdivision of cellular biochemistry. Depicted here is a schematic of the membrane and other eschelons of metabolic organization with a prokaryotic cell. Each level is illustrated in terms of DNA metabolism, mRNA metabolism, and intermediary metabolism of proteins, enzymes, and metabolites that can also enter or leave the cell. Arrows indicate multiple signal inputs, actions, and fluxes operating in parallel and/or series. As suggested by Westerhoff (2008), gene expression networks can be viewed differently than traditional metabolic pathway in that there is only a flow of information, not mass. DNA replication, mRNA metabolism, protein metabolism and intermediary metabolism are described as separate metabolic networks (levels) connected only by the concentrations of substances in one level influencing reactions rates in the other levels. Not shown is the structural metabolism that governs the assembly and turnover of macromolecular complexes, molecular machines, and replisomes. Eukaryotic cells are, of course, necessarily more complex, with additional processes such as vesicle trafficking, nuclear-cytoplasmic exchange, chromatin structure and remodeling, and various membranocytoskeletal components required for cell-cell integration of metabolic processes and information transfer. Diagram from Westerhoff (2008) with permission of the author and publisher (Academic Press).

is irreversibly committed to the pathway. Under steady-state conditions, the dissipation of free energy is minimized by a nearly constant flow or flux from X1 to X2 to X3, eventually forming end-product Xn, which in turn is irreversibly committed to another pathway. This mathematical approach allows one to interpret observable changes in metabolic flux for a pathway made up of multiple steps, each catalyzed by an enzyme. The rate of metabolite flow, or flux Ji, through the i th enzyme-catalyzed reaction in a metabolic pathway is the net reaction rate (units ¼ DM/Dt), Ji ¼ vf i  vr i A much simpler representation is:

J S X1 RDS

1f 1r

J X2 P

Scheme 11.18

11.68

11.69

where the fractional change in flux through the rate-limiting step is DJ/J, and the fractional change in reaction velocity is Dvf relative to (vf – vr). One can define Ji under different conditions that: (a) pathway is in a steady-state condition, where the flux through each reaction is identical; (b) system is at equilibrium, where Ji ¼ 0; and (c) reaction is far from equilibrium. The integrated kinetic properties of metabolic pathways are considered at the level of flux control coefficients, the fractional changes in pathway flux attending infinitesimally fractional changes in the activity of a particular enzyme operating within one pathway. Consider the following hypothetical pathway scheme, starting with the first ‘‘EXase’’-catalyzed reaction that converts component X0 to its immediate product X1, and proceeding through ‘‘Ydh’’catalyzed reaction that acts on pathway intermediate Y, and terminating at some sink.

X0

EXase JXase

X1

Y

EYdh JYdh

Sink

Scheme 11.19

For any one enzyme in the pathway, one can evaluate its flux control coefficient, vJydh/v[EXase] from a plot of flux JYdh as a function of the concentration of enzyme EXase (Fig. 11.29A). Alternatively, one may determine v ln JYdh/ v ln[EXase] from a plot of ln(JYdh) as a function of the natural logarithm of EXase concentration (Fig. 11.29B). A flux control coefficient value of zero indicates that the metabolic flux is completely insensitive to a change in the activity of that enzyme, whereas a value of one would mean that the pathway is wholly dominated by the activity of one particular step. Without providing the proof here, suffice it to say that MCA’s summation theorem requires confirmation of an intuition that all flux control coefficients sum to unity. Fell (1997) pointed out several important conclusions reached on the basis of MCA. Contrary to traditional studies of metabolic regulation, few pathways have a single ratedetermining step. For example, while phosphofructokinase (PFK) is generally considered to regulate the rate of glycolysis in many cells, a 3.59 increase in PFK concentration was without effect on the anaerobic glycolytic rate in yeast (Heinisch, 1986). These observations suggest that there are more rate-contributing steps. Furthermore, while traditional approaches focus regulatory effects exerted at an early pathway step controlling metabolite supply, MCA

Chapter j 11 Regulatory Behavior of Enzymes

Flux, Jydn

A

∂Jydh/∂JXase

j

e

[EXase], µM

In(Flux), In(Jydn)

B

∂ In(Jydh)/∂ In(JXase)

Inj

In e

In[EXase] FIGURE 11.29 MCA analysis of metabolic flux. A, Plot of metabolic flux versus the concentration of enzyme Xase; B, Plot of natural logarithm of metabolic flux versus the natural logarithm of enzyme Xase concentration.

theory suggests that regulation is more effective when later steps control demand for key metabolic intermediates. Although the flux control coefficient vJYdh/v[EXase] is written as a dependence on enzyme concentration, the determining factor is enzyme activity, not necessarily enzyme concentration. For enzymes whose activity depends

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linearly on its concentration, one can evaluate the flux control coefficient by methods indicated in Table 11.6. Metabolic flux is measured before perturbing the enzyme concentration (activity) and after the system readjusts to its new steady state. In experiments based on the use of an inhibitor, it is imperative to be certain that the inhibitor is specific for the target enzyme. Moreover, the use of extremely tightly bound, or better still covalently bound, inhibitors assures that a rise in substrate concentration, as would be anticipated as the pathway adjusts to a new steady state, will not reduce the potency of the inhibitor. In principle, cells may be treated with a specific irreversible inhibitor that is radioactively labeled, and a sample of these cells could be lysed at the same time that a flux measurement is made. Subsequent electrophoresis and radiometric analysis should permit an estimation of the ratio of unmodified to covalently modified enzyme (r ¼ Eunmod/Emod) which would permit an accurate estimate of active enzyme at each inhibitor concentration (i.e., [Eactive] ¼ [Eactive]  r). Another promising technique would entail micro-injection of a photo-activatable form of the enzyme of interest (see Table 11.6 and Section 9.3.1). The application of metabolic control analysis also deals with elasticity coefficients, which are measures of the fractional change in metabolic flux caused by an infinitesimal change in the concentration of a positive or negative effector. This metabolic control parameter, symbolized by e, known as elasticity (3 h (v ln v/v ln x) h (xvv/vvx) provides a measure of the direction and magnitude of a change in enzyme (or pathway) velocity as a function of a change in the concentration x of substance X acting as a substrate or effector. When 3 equals zero, v shows no functional dependence on the concentration of X. If 3 is positive,

TABLE 11.6 Experimental Approaches for Altering the Activity of a Target Intracellular Enzyme Method

Comment

DNA Transfection

While enzyme over-expression by transfection allows an experimenter to increase the concentration of an enzyme of interest, cells are so inherently homeostatic that the concentrations of other enzymes, regulatory proteins, and low-molecular-weight metabolites may be altered. The cell is very likely to make compensating adjustments, to the effect that one cannot be certain the sole effect is that of the enzyme of interest.

Microinjection

This approach offers a potential method for abruptly increasing an enzyme’s concentration within a cell. Given the 1–2 msec period for a 100 kDa protein to diffuse 10–20 mm, the cytoplasm should be thoroughly mixed almost immediately after micro-injection. Compartmentation of a target enzyme may also limit the effectiveness of both over-expression or micro-injection. Some organelles are also thought to expel proteins whose concentrations exceed certain intra-organellar thresholds.

Titration with Inhibitor

The general availability of highly specific enzyme inhibitors, especially those of the mechanism-based and transition-state type, allow one to reduce the effective concentration of an enzyme of interest.

Photoactivation

A highly promising approach is micro-injection of an enzyme temporarily inhibited by a photolabile ligand, such that flux measurements can be made before and after flash photolysis.

Micropinocytosis

This process permits highly efficient, lipid raft-mediated transfer of fusion proteins containing a membranepenetrating sequence (e.g., RQIKIWFQNRRMKWKK in Drosophila antennapedia and RRRQRRKKR in HIV TAT protein) directly across the peripheral membrane and into the cytoplasm of eukaryotic cells within 15–30 min.

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v increases as the concentration X increases; if 3 is negative, v decreases as X increases. The connectivity theorem indicates that the control of a system can be solved by determining the flux control coefficients and elasticity coefficients for each component step in a pathway. Delgado and Liao (1995) recently demonstrated that time-scale separation is an effective way to localize metabolic control to only a few enzymes. They considered model pathways in which the eigenvalues of the Jacobian of the system are widely separated (i.e., systems with time-scale separation). Their treatment assumes the system possesses a unique, asymptotically stable steady state and that the reaction steps of the system under analysis are well represented by linear kinetics around the steady state. While cells may produce many enzymes at far greater concentrations than needed to maintain a certain steady state, they can achieve time-scale separation by controlling the expression of only a few enzymes. The over-expressed enzymes catalyze what are termed ‘‘fast’’ reactions and lead to small response times of the system. For these ‘‘fast’’ reactions, metabolite control coefficients are small compared with ‘‘slow’’ reactions and do not effectively control overall flux. Nonetheless, at pathway branch points, ‘‘fast’’ reactions may be mutually competitive and result in significant control coefficients for fluxes to the branches. Metabolic control analysis treats each enzyme within a metabolic pathway as an elementary or irreducible unit. Enzymes have not evolved to optimize the catalysis and regulation of individual catalyzed reactions; instead enzymes are merely the ‘‘microprocessors’’ operating in the setting of a larger self-adaptive ‘‘computing’’ network that is far more robust and ‘‘intelligent’’ than any individual gene product. The following may be a helpful analogy from enzymology. Today, those overly enamored with fast reaction kinetics tend to look down upon steady-state approaches, even saying such silly things as: ‘‘It’s the dawn of the new enzymology, don’t get left in the steady state.’’ As noted in the previous chapter, many rapid reaction measurements are conducted as single-turnover reactions, where the concentrations of reacting species are present at aberrantly different concentrations present in the steady state. Only upon reconciling the fast reaction kinetic data with those obtained in steady-state experiments can we be confident that we are not errantly delving into a virtual reality of disintegrated facts. After all, the principles of nonequilibrium thermodynamics instruct us that the steadystate is an inherently more stable condition – one that may still offer conceptual and experimental opportunities to the new map-makers venturing into the frontiers of metabolic homeostasis. There is growing recognition that new mathematical approaches must be developed to study metabolic regulation at the cellular and subcellular levels. Measurements based on the behavior of populations of cells reflect a composite or average kinetic behavior that obscures the metabolism

Enzyme Kinetics

occurring within cells and subcellular compartments. A new technique makes possible the measurement of these processes in individual cells, it is becoming essential to consider the relationship between classical (deterministic) and stochastic (probabilistic) biochemical kinetics, especially as they apply to the kinetic properties of small systems (i.e., systems that have few component molecules that they are subject to significant statistical fluctuations). Earlier work indicated that the mean concentration dynamics are governed by differential equations that are similar to those in classical chemical kinetics, when expressed in terms of the stoichiometry matrix and timedependent fluxes (Leonard and Reichl 1990; McQuarrie, Jachimowski and Russell, 1964; Samoilov, Plyasunov and Arkin, 2005; Thakur, Rescigno and DeLisi, 1978; Zheng and Ross, 1991). As discussed by Goutsias (2007), each flux can be decomposed into a macroscopic term accounting for the effect of mean reactant concentrations on the rate of product synthesis and a mesoscopic term accounting for the effect of statistical correlations among interacting reactions. He developed a mathematical framework for probing fundamental relationships between classical and stochastic behavior of biochemical reaction systems consisting of irreversible uni- or bi-molecular elementary reactions. Because computation of fluxes and mean concentration dynamics often requires intensive Monte Carlo simulation, Goutsias (2007) derived differential equations that can be solved by standard numerical techniques to obtain more accurate approximations of fluxes and mean concentration dynamics than those obtained with the classical approach. Although far more work is essential for broader application of these principles to metabolic control analysis of a complex manifold of biochemical reactions arranged in both linear and branched reaction pathways, such approaches are also likely to provide important clues about single-molecule enzyme kinetics. Investigating regulatory processes at the level of enzymes, compartments, cells and beyond thus remains a central focus of modern molecular life sciences. Efforts to understand homeostasis gives all of us a unity of purpose. Although much has been gleaned from studies of individual elements (e.g., enzymes, substrates, cofactors, products, activators, and inhibitors), we now know there is so much more to learn about metabolic integration. Such efforts must not minimize the work of enzymologists and pharmacologists to delve into the chemical mechanisms of individual enzymes and enzyme-coupled receptors, because those chemical studies are doubtlessly the best way for chemists to create new drugs and biological control agents. Recent discoveries in chromatin remodeling and epigenetics also emphasize that we need to be open to the discovery of new metabolic processes. It is noteworthy that Metabolic Control Analysis finds its roots in the work of Kacser and Burns (1973), who, in part, were motivated by their deep interest in genetics to explain why most null mutations tend

Chapter j 11 Regulatory Behavior of Enzymes

to be recessive. They assumed, by extension of Beadle’s and Tatum’s ‘‘one-gene/one enzyme’’ rule, that there was likely to be a one-to-one correspondence between gene dosage and enzyme concentration. Such assumptions led them to consider why the flux through an overall pathway might not be affected by a doubling in the concentration of an individual enzyme. We now know, however, that epigenetic silencing through site specific, enzyme-catalyzed DNA methylation is a highly effective mechanism (viz. X chromosome inactivation in females) for leveling gene dose and likewise the expression of individual genes. This observation does not minimize MCA’s significance, and although this textbook was written mainly to inspire students to apply kinetic principles to the systematic investigation of individual enzyme reactions, Metabolic Control Analysis represents a most opportune field for further inquiry – one that doubtlessly will both challenge and reward the cadre of young biochemists and biophysicists who will soon succeed us.

11.14. CONCLUDING REMARKS Although much of enzyme regulation has focused on the development and testing of equilibrium models for subunit cooperativity and allostery, it must be stressed that enzymes most often operate away from thermodynamic equilibrium, and almost certainly so within cells. That this must be so is evident in the fact that steady states are the inherently nonequilibrium, energy-dissipating processes so essential for Life. Indeed, while force generation and locomotion (Chapter 13) are the most macroscopically apparent signs that an organism is alive, the orderly management opensystem energy flows remains of utmost importance for Life. In this respect, steady states are features more fundamental than any other ascribable attribute of any living organism. By minimizing the rate of entropy increase and by stabilizing energy flow, steady states give rise to the solute gradients, morphogenesis, chemotaxis, etc., needed to attain the cellular and intercellular structures and the biochemical processes required for organisms to grow, to move, to reproduce, and to sense the world around them. So robust is the overall stability of nonequilibrium steadystates – especially those that are nested within large-scale, multi-node networks, that considerable energy is required to displace, even transiently, a target process from one steady-state condition to another. At the physiologic level, homeostasis is a manifestation of the feedback mechanisms that maintain steady-state poise at the organ and inter-organ level. Given their catalytic roles in mediating the energyconsuming and -producing reactions and metabolic pathways within all organisms, enzymes provide the underlying physicochemical mechanisms for such orderly flow of metabolic energy. As discussed in Section 5.2, the great

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English enzymologist-turned-geneticist/evolutionist J. B. S. Haldane introduced the concept that enzyme catalysis is often best considered as a steady-state process. The isotope exchange experiments of Albery and Knowles (Section 5.4.6) confirmed Haldane’s principle in terms of the fluxes and internal energetics of triose-P isomerase and proline racemase catalysis. Subsequent investigations attest to their principle that enzymes bind and transform their substrates by means of a manifold of intermediates that attain the steady-state condition such that they exhibit little tendency toward a single rate-determining step. The pathways of intermediary metabolism, respiration, as well as oxidativeand photo-phosphorylation are likewise characterized by steady-state fluxes, and the same appears to hold true for many aspects of gene expression, cellular motility, and ion/ solute transport. It seems clear then that the resilience of living cells and entire organisms is derived, at least part, from hierarchically arranged steady states. Metabolic steady-states are inevitably created and stabilized by enzymes operating within their local metabolic pathways and subcellular compartments, such that the rate constants of individual enzymes (and their isozymes) are tuned to satisfy the metabolic needs of cells. Subunit cooperativity and allostery doubtlessly play key roles in adjusting pathway flux, as do higher echelons of metabolic regulation, such as those achieved principally by signal transduction pathways driven by nucleoside triphosphate-dependent enzymes (e.g., protein kinases and phosphoprotein phosphatases) and mechanoenzymes (e.g., GTP-regulatory proteins, active transporters, and chromatin-remodeling components). Even higher levels of control are manifested through the timely expression of tissue-specific transcription factors, which are the developmentally choreographed master switches that globally control chromatin remodeling and dictate the pathways used by differentiated cells, even specifying the isozyme to be formed within a certain cell type. While the concepts presented in this chapter provide a basis for such efforts, the research literature presents countless examples of how catalytic and regulatory mechanisms converge. When viewed from this perspective, the systematic investigation of the kinetics of enzyme regulation remains ripe for unprecedented discovery, whether pursued as isolated catalysts or as complete metabolic pathways.

FURTHER READING Boyer, P. D., Gresser, M., Vinkler, C. et al. (1997) in Structure and Function of Energy-Transducing Membranes (van Dam, K., and van Gelder, B.F., eds., pp. 261–274, Elsevier, Amsterdam. (A classic paper dealing with the binding-change mechanism for ATP synthase as well as negative cooperativity.) Fell, D. (1997). Understanding the Control of Metabolism. London: Portland Press.

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Huang, C. Y., Rhee, S. G., and Chock, P. B. (1982). (A lucid review of multi-site cooperativity, with many illustrative examples.) Ann. Rev. Biochem., 51, 935. Mulquiney, P. J., and Kuchel, P. W. (2003). Modeling Metabolism with Mathematica. Boca Raton: CRC Press. (A hands-on approach for simulating enzyme kinetics and metabolic fluxes.) Schachman, H. K. (1988). Can a simple model account for the allosteric transition of aspartic transcarbamoylase? J. Biol. Chem., 263, 18583. Westerhoff, H. V. (1987). Thermodynamics and Control of Biological Free-Energy Transduction. Holland: Elsevier. Westerhoff, H. V., and van Dam, K. (1987). Thermodynamics and Control of Biological Free-Energy Transduction. Holland: Elsevier.

Enzyme Kinetics

OTHER AUTHORITATIVE READINGS FROM METHODS IN ENZYMOLOGY Cooperativity in Enzyme Function: Equilibrium and Kinetic Aspects, 64, Chapter 7. Hysteretic Enzymes, 64, Chapter 8. Enzyme Cascade Kinetics and Control, 64, Chapter 12. Alternative Substrate Studies of Enzyme-Catalyzed Chemical Modification, 64, Chapter 13. Cooperativity in Enzyme Function: Equilibrium and Kinetic Aspects, 249, Chapter 20. Substrate Channeling Mechanisms, 308, Chapter 6.

Chapter 12

Single-Molecule Enzyme Kinetics Enzyme kinetic experiments are normally carried out under conditions where countless enzyme molecules react with an even greater number of substrate molecules. A 1-nL aliquot of a 1-mM enzyme solution, for example, contains ~108 enzyme molecules. While the idea of observing the action of enzyme molecules in singulo was a biochemist’s pipe dream only a decade ago, such experiments are now feasible as a consequence of advances in protein science, optics, fluorescence and solid-state electronics. Reaction trajectories can now be reliably determined for individual enzyme molecules that are physically isolated from each other by attachment to solid surfaces or supramolecular structures, during confinement within a gel or polymer matrix, or as they operate freely within an extremely small volume element. Site-directed mutagenesis also affords reliable routes for inserting fluorophores at precise positions, for remodeling active sites to achieve desired catalytic properties, for tethering enzymes to surfaces, and for improving enzyme stability, especially for investigation at extreme dilution. There have also been significant advances in the detection of reaction rates, particularly through the use of synthetic, photostable fluorophores that can be inventively attached to enzymes and substrates. Other intensely fluorescent polypeptides, such as green fluorescent protein (and its colored variants) as well as luciferase, can be attached to enzymes or proteins by recombinant DNA methods. As will become clear in this chapter, breakthroughs in enzyme chemistry, materials science, chemical physics, high-speed digital computers, as well as kinetic theory have spurred the development of single-molecule kinetics.

12.1. GENERAL COMMENTS ON SINGLEMOLECULE ENZYME KINETICS Enzyme chemists and statistical physicists are similarly intrigued by the stochastics of enzyme catalysis and cooperativity (e.g., activity fluctuations, pausing, waiting-time distributions, static disorder, fluctuating reactant concentrations, etc.). Such information affords the opportunity to compare individual and ensemble-averaged properties, thereby bridging chemistry’s microscopic and macroscopic worlds. The Ergodic Hypothesis asserts that the timeaverage of a physical quantity or chemical configuration along a time trajectory for an individual member within Enzyme Kinetics Copyright Ó 2010, by Elsevier Inc. All rights of reproduction in any form reserved.

a homogeneous ensemble is equivalent to the ensembleaveraged value of that same quantity or configuration at any given time (Gillespie, 1992; Norris, 1997; van Kampen, 1992). In other words, statistical ensemble averages are truly representative to time averages of the system, albeit with statistical variation. A powerful justification for conducting single-molecule observations has been to test the Ergodic Hypothesis by examining whether individual members are indeed representative of the overall population of molecules (Xie and Trautman, 1998). Beyond the aforementioned issues, there are other compelling reasons why enzyme chemists are motivated to conduct single-molecule experiments on enzyme catalysis. First, there is a longstanding desire to observe a catalyst operating in real time. Although chemistry is a powerfully inferential science, viewing enzymes directly doubtlessly changes the way we think about catalysis. Simply put: ‘‘Seeing always trumps believing.’’ Second, synthetic catalysts lack highly defined structures on the subnanometer length-scale, whereas well-structured enzyme molecules are apt to divulge intimate details of catalysis that cannot be glimpsed with synthetic catalysts. Third, direct observation may reveal conformational changes and protein-protein interactions that may not be so easily detected in crystallography or magnetic resonance spectroscopy. X-ray methods require co-crystallization of an enzyme with its macromolecular substrate, and NMR can rarely be practiced on protein complexes having a molecular mass greater than 30–40 kDa. Such considerations are especially relevant to mode-of-action studies on energase-type reactions (e.g., motors, protein folding machines, GTP-regulatory proteins, and ribosomal elongation machinery) that almost always have macromolecular binding partners. Fourth, examination of temperature effects on the kinetics of individual enzyme molecules can be used to characterize the statistical thermodynamic behavior of a rate-determining reaction that is nested amidst a series of elementary steps within a catalytic cycle. Fifth, single molecule observations promise to reveal topological constraints on enzymes acting on polymeric substrates (e.g., topoisomerases mediating changes in DNA super-coiling or chromatin compaction). Enzyme chemists and cell biologists are likewise interested in the detailed motions and processivity of molecular motors as they crawl on the surfaces of cytoskeletal fibers. Fifth, there is a compelling need to discover single-molecule

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DNA sequencing approaches that avoid errors typically introduced by polymerase chain reaction methods. Finally, nanoscience will soon make demands on single-enzyme molecule kinetics in the development of devices consisting of natural and synthetic components working together as an integrated unit. Direct single-molecule enzyme kinetic observations promise to accelerate the design and fabrication of novel nano-scale biosensors and other mechanical devices (e.g., switches, pumps, and actuators), with presently unimaginable applications. Finally, it is worthwhile briefly noting some of the forces acting on single-enzyme molecules. Table 12.1 lists the major physical and chemical forces acting on single-enzyme molecules in aqueous solutions. Note that the magnitude of electrostatic forces will depend on many factors, including the sites and density of ionic charges as well as the effects of ionic strength on these interactions. Other effects, such as centrifugal, gravitational, and laser optical-trapping forces, are comparatively weaker. For additional comments on electrostatic interactions between an enzyme and its

substrate, see Section 2.2: Forces Affecting Enzyme Structural Stability and Interactions. Those interested in a detailed and elegant account of the mechanical forces affecting biological molecules should consult Chapters 2 to 6 from Mechanics of Motor Proteins and the Cytoskeleton by Howard (2001).

12.2. DEMONSTRATION OF SINGLEMOLECULE REACTION RATES The activity of single lactate dehydrogenase molecules (Reaction: Lactate þ NADþ # Pyruvate þ NADH) was first investigated by Xue and Yeung (1995), using a very highly diluted enzyme sample that was placed into capillary tubes to reduce turbulent flow and convective currents that might otherwise disperse the enzymatically produced NADH molecules. Because the net concentration of NADH in any zone was very low, Fick’s Law dictates that diffusive broadening of the zone would likewise be manageable.

TABLE 12.1 Forces Influencing the Behavior of Enzymes at the Single-Molecule Level Force

Comment

Brownian Forces

Brownian forces are associated with momentum transfers to and from other colliding solvent and solute molecules. The magnitude of the Brownian force FB is given by the equation: FB ¼ (2kBTz)1/2W(t), where kB is the Boltzmann constant, T is the absolute temperature, z is the coefficient of viscous drag, and W(t) is a Wiener function, a continuous-time stochastic process for models involving Brownian dynamics. At room temperature, a 100-kDa globular protein (with a molecular mass m of 1.66  10–19 g, a density r of 1.35 g/cm3, a volume of ~120 nm3, a spherical particle radius of ~3 nm, and a translational diffusional coefficient D of ~70 mm2/s), would have an average speed Dx/Dt of ~9 m/s. The average velocity is stochastic, and the protein will change direction often. The number n of random moves per time increment Dt is proportional to the D, such that n/Dt ¼ 2D/(Dx)2, where Dx is the displacement, such that Dt will range from 5 to 20 ns. Other details are discussed in Chapter 13.

Viscous Forces

Like Brownian forces, drag forces arise from momentum transfers to and from other colliding solvent and solute molecules as the object particle moves in a stationary liquid. The viscous drag force FV is directly proportional to the particle’s velocity, as given by the equation: FV ¼ z Dx/Dt, where z is the coefficient of viscous drag. Average viscous drag on a 100 kDa protein is ~60 pN$s/m, resulting in viscous forces of ~400–500 pN. Other details are discussed in Chapter 13.

Electrostatic Forces

Caused by Coulombic interactions between ions. When placed at a 1-nm distance, two monovalent ions exert an attractive or repulsive force of ~2 pN, depending on the signs of their charges. Such forces are strongly attenuated by ionic screening, especially at the ionic strength values typically found in cells. Surface-tethered proteins will also experience the strong local effects of surface charge. See details on electrostatic potential in Section 2.2.1, ionic strength effects in Section 7.5, and surface double-layer behavior in Section 7.12.1.

Noncovalent Forces

Associated with reversible binding between biological macromolecules. For a one-step binding interaction having a Kd z 10–7 M, the noncovalent interaction energy is typically less than 10–30 pN-nm. Noncovalent forces depend on distance over which the force is exerted. A force of 4–5 pN will rupture a single hydrogen bond, and hydrogen bond-stabilized protein domains rarely resist rupture forces above ~200 pN, regardless of the large number of hydrogen bonds. Other details are discussed in Chapter 13.

Covalent Forces

Shared electron-pair bonds are extremely strong (e.g., rupture of a C–C single bond requires ~2,000 pN). Covalent surface-tethered enzymes will remain tightly associated with the surface in most single-molecule enzyme kinetic experiments. Other details are discussed in Chapter 13.

Adhesive Forces

Proteins adhere to common and borosilicate glass primarily by electrostatic interactions and to polystyrene and unmodified quartz (pure silica) by hydrophobic interactions. Adhesion requires numerous near-surface encounters, resulting in presently unpredictable time-scales for adhesion. Constantly applied forces of 100–300 pN are typically needed to dislodge a protein from glass, suggesting that surface desorption is slow.

Chapter j 12 Single-Molecule Enzyme Kinetics

731

Indeed, NADH fluorescence was found to accumulate within weakly fluorescent zones, suggesting that NADH had been produced enzymatically at sites corresponding to spatially isolated enzyme molecules. The relative activity of each enzyme molecule was then determined by the intensity of fluorescence emission (at 460 nm) within each zone, which was sampled by electrophoretically transporting the NADH molecules through the capillary and past a microfluorimetric detector. The latter consisted of an excitation laser and fluorescence detector (Fig. 12.1a). The resulting histogram (Fig. 12.1b) documented the frequency of zones exhibiting a particular relative activity. The distribution for the 79 individual LDH molecules sampled in this manner was highly asymmetric, indicating that the rate of the most active enzyme molecules exceeding the slowest by a factor of ~15. Surprisingly, the observed dispersion in single-molecule LDH activity was remarkably persistent, as evidenced by the finding that each isolated enzyme molecule produced a zone of characteristic intensity, even during an extended assay period. Although Xue and Yeung (1995) attributed such persistent behavior to different enzyme conformations, each separated by energy barriers of varying heights, the

A Relative Fluorescence

2800

2600

2400

2200

2000 0

1

2

3

4

5

6

Time (minutes)

B Frequency

15

10

5

0 0

500

1000

1500

Relative Activity FIGURE 12.1 Enzymatic activity of single LDH molecules. A, Fluorescent zones of NADH molecules formed by spatially isolated lactate dehydrogenase molecules; B, Histogram showing variation in observed activity of individual enzyme molecules. Extremely dilute enzyme (7.6  1017 M) was placed into capillary tubes (diameter ¼ 20 mm, corresponding to a 3-pL volume per 1-cm length) in the presence of 1 mM NADþ, 3 mM lactate, and 20 mM Tris buffer (pH 9.1). Reproduced from Xue and Yeung (1995) with permission of the authors and Macmillan Magazines, Ltd.

basis of such conformations remains uncertain. A more likely explanation is that the microenvironment of the capillary’s interior surface was responsible for changes in enzyme activity arising from differential LDH adsorption. (Partial denaturation or oligomer dissociation at extreme dilution are other less likely possibilities.) Nonspecific heterogeneous adsorption of the enzyme might orient some active sites of the LDH tetramer nearer to the surface, resulting in reduced substrate access to those enzymes whose active sites facing the glass surface versus those active sites with unobstructed access to substrate. Another possibility is that other adsorbed substances or variation in the electrostatic potential of the glass itself could influence substrate access. Whatever the explanation, the significance of Xue and Yeung (1995) was that activity measurements had been made for the first time under conditions allowing the interrogation of the catalytic behavior of spatially isolated single-enzyme molecules. In related work, Uhl, Pilarczyk and Greulich (1998) microinjected highly diluted LDH-1 into a drop of substrate solution, forming a bubble of enzyme within the drop of substrate. Cloud-like reaction zones of NADH molecules appeared around the spatially isolated enzyme molecules (Fig. 12.2). The kinetics of the NADH formation in every fluorescent zone as well as the size of each zone could be described by zero-order NADH production and diffusioncontrolled loss of NADH molecules from the reaction zone. Based on the extent of enzyme dilution and statistical analysis, they concluded that only few enzyme molecules at the center of the fluorescent reaction zones were responsible for catalyzing NADH formation. Craig et al. (1999) also examined the single-molecule kinetic behavior of calf intestinal mucosal alkaline phosphatase (Reaction: Phosphomonoester þ H2O # Alcohol þ Pi) by mixing a highly dilute enzyme solution with another containing fluorogenic substrate at high concentration. They loaded the reacting solution into a capillary (10 mm internal diameter), such that the diluted enzyme molecules could convert its weakly fluorescent substrate 29-[2-benzthiazole]69-hydroxybenzthiazole-phosphate into a pool of intensely fluorescent product 29-[2-benzthiazole]-69-hydroxybenzthiazole. An electric field (400 V/cm) was then applied for 1 min across the capillary, and the local pools of fluorescent product were swept past a high-sensitivity fluorescence detector. With this experimental setup, Craig et al. (1999) observed a series of fluorescent peaks, each corresponding to those product molecules that had accumulated around a single-enzyme molecule. In a single-molecule experiment using highly purified alkaline phosphatase, a set of eight peaks was detected (Fig. 12.3), albeit with poor resolution of the fourth and fifth peaks. Enzyme activity was estimated from the integrated area beneath each peak using nonlinear regression analysis. Absolute activity was estimated by calibrating the instrument’s signal strength at known concentrations of

Enzyme Kinetics

732

FIGURE 12.2 Time-dependent increase in fluorescence intensity of NADH produced within discrete reaction zones. Each zone of variable size is statistically distributed at positions where substrate solutions make contact with the highly diluted LDH-1. Careful inspection of this series of time-lapse images confirms (a) the progressive timedependent increase in image intensity and (b) the spherelike puffs of localized NADH fluorescence signal intensity. The time-dependent rise in intensity is consistent with the expected time-course of LDH catalysis. Moreover, the ‘‘puffs’’ appear to be local domains that are predicted for Brownian motion: each NADH molecule diffuses haphazardly without significant net migration from its point of origin (i.e., its original site of enzymatic synthesis). The frame-to-frame steadiness of these images also confirms the absence of bulk-phase convective currents. Consult the text for additional details. Reproduced from Uhl, Pilarczyk and Greulich (1998) with permission of the authors and the publisher.

a

b

c

d

e

f

g

h

i

fluorescent product. The turnover rate constant was 30  5 s1. Craig et al. (1999) ascribed the observed long-term, molecule-to-molecule differences in activity and apparent activation energy to enzyme micro-heterogeneity. They also concluded that there is no need to invoke the energy landscape model to explain molecule-to-molecule variation in enzyme activity. The above examples represent multiple-turnover assays of what were inferred to be single-enzyme molecules. This condition obscures unambiguous observation of a singleenzyme molecule as it passes through a catalytic cycle, thereby revealing the stochastics of forming catalytic intermediates and their interconversion. The promise of single-molecule enzyme kinetics in establishing reaction trajectories was realized in the experiments of Lu et al.

(1998). These investigators determined trajectories over the course of more than 600 individual turnovers of a single cholesterol oxidase molecule. This FAD-dependent oxidoreductase (EC 1.1.3.6) catalyzes the reaction of cholesterol with dioxygen to form cholest-4-en-3-one and hydrogen peroxide: H3C H3C

CH3

H

H3C H

H3C H

HO

O2

Signal (arbitrary units)

H2O2

H3C H3C H

H3C H

CH3 H3C

H

O

Scheme 12.1 0

40

80

120

160

200

Migration Time (sec) FIGURE 12.3 Single-molecule enzyme measurement on highly purified Esherichia coli alkaline phosphatase. The reaction mixture was incubated for 30 min at 40 C. Only the portion of the electropherogram corresponding to the single-molecule peaks is shown. Reproduced from Polakowski et al. (2000) with permission of the authors and the American Chemical Society.

Cholesterol oxidase operates by a typical Ping Pong mechanism: in the first half-reaction, the naturally fluorescent Enz$FAD is reduced by cholesterol to generate a nonfluorescent Enz$FADH, and in the second half-reaction, Enz$FADH is reoxidized upon reaction with molecular oxygen to form H2O2 (Kass and Sampson, 1995; MacLachlan et al., 2000).

Chapter j 12 Single-Molecule Enzyme Kinetics

H2O2

k-1

BLINK OFF

BLINK ON

Microstate-4 Enz-FADH2 O2 (nonfluorescent)

Microstate-2 (fluorescent) Enz-FAD S

k'-1 k'+1[O2]

A Counts

k+1[S]

16 8 0

P

Microstate-3 Enz-FADH2 (nonfluorescent)

Scheme 12.2

B Occurrence

Microstate-1 (fluorescent) Enz-FAD

733

12.3. KINETIC TREATMENT OF SINGLEMOLECULE ENZYME BEHAVIOR Whether pursued by steady-state or rapid-reaction techniques, the chemical kinetic reaction velocity rises smoothly with reactant concentration, simply because so many enzyme molecules are involved. By contrast, the kinetic behavior of single molecules is inevitably stochastic (Delbru¨ck, 1940;

1000

1500

Time (msec) 40

20

0 0

Lu et al. (1998) physically isolated individual cholesterol oxidase molecules from each other within an agarose gel (99% water) under conditions that permitted the enzyme to tumble freely. When supplied with cholesterol and dioxygen, catalysis was observed as single enzyme molecules appeared to ‘‘blink’’ on and off, alternating between the enzyme-bound FAD and FADH species. Each on-/off-cycle corresponds to one enzymatic turnover. The single-molecule trajectory shown in Fig. 12.4A has several notable features. First, the emission-on and emissionoff times respectively indicate the duration of the reductive and oxidative phases of the catalytic reaction cycle. Second, the relatively uniform amplitude of the fluorescence change is consistent with the behavior of single enzyme molecule. Third, the duration of each fluorescent state is stochastic, such that the waiting-time of each state cannot be predicted from the duration of a previous on- or off-phase. Finally, because the oxidase catalyzes D5-ene-3b-hydroxysteroid oxidation to a D5-3-ketosteroid, followed by isomerization to the D4-3-ketosteroid, the Ping Pong reaction has at least six steady-state rate constants, the observation of two fluorescent states obviously cannot provide sufficient data to evaluate all six rate constants unambiguously. Based on the catalytic reaction cycle shown in Scheme 10.1, the observable reaction cycle comprises a fluorescent phase (lumping together Microstates-1 and -2) and a non-fluorescent phase (lumping together Microstates-3 and -4). This fact is evident by the non-exponential distribution of emission on-time (Fig. 12.4B), indicating that the forward reaction involves more than one elementary reaction.

500

500

1000

1500

2000

2500

On-Time (msec) FIGURE 12.4 Enzymatic cycle of cholesterol oxidase and real-time observation of enzymatic turnovers of a single cholesterol oxidase molecule. A, Each on-off cycle in the emission intensity trajectory corresponds to an enzymatic turnover. ct, count; ch, channel. B, Distribution of emission ontime derived from the intensity trajectory. The non-exponential distribution reflects the fact that the forward reaction involves more than one elementary reaction. The solid line represents a simulated two-exponential process (e.g., E þ S / ES, k1[S] ¼ 2.5 s1; and ES / E þ S, k2 ¼ 15.3 s1). Reproduced from Xie and Lu (1999) with permission of the authors and the American Society for Biochemistry and Molecular Biology.

Kramers, 1940), meaning that the behavior observed at any instant cannot be predicted on the basis of a molecule’s past history. Each elementary step in a multi-step enzyme mechanism is defined by a rate constant that on the macroscopic scale is averaged in classical chemical kinetics. In singlemolecule observations, however, the time needed for the same elementary step fluctuates, depending on the Boltzmanndistributed energy of the molecule at the instant that the reaction occurs. These fluctuations are not the consequence of undesirable noise: they are instead valuable data, from which single-molecule kinetic information may be extracted. As pointed out by Qian and Elson (2002), the kinetics of reactant concentration changes cannot be applied, and must instead be defined by the probability of change by considering the likelihood that an enzyme molecule will be in a particular conformation at time t. This approach has roots in stochastic kinetic treatments (Ball and Rice, 1992; Conti, 1984) of ion channel trajectories (i.e., records of the opening and closing of single ion channels), determined potentiometrically using patch-clamp techniques (Sakmann and Neher, 1995). In making single-molecule enzyme kinetic measurements, there are three basic goals: (a) observing individual trajectories, which are simply accurate records of the stochastic conformational dynamics and/or chemical kinetics of the enzyme during the catalytic cycle; (b) determining the number of substrate or product molecules consumed or produced by an enzyme molecule; and (c) quantifying time-dependent fluctuations of single enzyme molecules. Unlike typical macroscopic rate equations,

Enzyme Kinetics

734

which are continuous and differentiable at every point, many single-molecule phenomena are often characterized by substrate-‘‘on’’ and substrate-‘‘off’’ states that can be treated as Markov processes (Norris, 1997). Any signal that lumps together multiple kinetic states will be intrinsically nonMarkovian, except when one of the steps within that lumped manifold of steps is so abundant so as to dominate the kinetic behavior of that ‘‘lumped’’ signal. It is also important to keep in mind that, during the course of most single-molecule kinetic experiments, the macroscopic substrate and product concentrations are essentially constant. The now classical example of single-molecule biochemical kinetics is the opening and closing of gated ion channels, which while technically not enzymes, do facilitate transfer of solute or solvent molecules across a membrane. As discussed by Nobelists Sakmann and Neher (1995), these events may be observed by their ingenious patch-clamp methodologies. Each small area of membrane contains one or at most a few channel molecules, and when successfully captured by a micropipette, individual ion crossing events of channels located within the membrane patch may be detected potentiometrically as discrete changes in ion currents. Gate-opening and -closing of these channels can be viewed as a simple two-state model resembling Michaelis-Menten behavior (i.e., State-1 ¼ Free Enzyme, and State-2 ¼ ‘‘E$S’’ complex).

E+S

k+1 k–1

EX

k+2 k-2

E+P

or

E

α β

EX

Scheme 12.3 In the latter representation, a ¼ {kþ1[S] þ k2[P]} and b ¼ (k1 þ kþ2), and the kinetics obey a single exponential term, exp[(a þ b)t]. Assuming that the fluorescence signals for each state are fA and fB, such that the expectation is K ¼ a/b. Qian and Elson (2002) developed a nonequilibrium steady-state enzyme model for a threeconformational state system such as that obeying a twointermediate enzyme kinetic mechanism.

E+S

k+1 k–1

ES

k+2 k–2

EP

k+3 k–3

E+P

Scheme 12.4 The kinetics of the single enzyme can be treated by a three-state Markov process:      PE   kþ1 ½S  k3 ½P  k1 kþ3     d    ðtÞ ¼ kþ1 ½S k1  kþ2 k2 PES     dt  k3 ½P kþ2 k2  kþ3  PEP     PE      PES   PEP  12.1

where PE(t), PES(t), and PEP(t) are the probabilities of the enzyme being in the E, ES, and EP states at time t. The kinetics are characterized by a time-independent term l0 ¼ 0 and two exponential terms exp[–l1t] and exp[–l2t]. The l’s can be real or imaginary. When imaginary, the kinetics are described by a transcendental function that can oscillate. If all the constants as well as [S] and [P] are specified, each l value can be computed from the eigenvalues of the above matrix, such that:  pffiffiffiffi l1;2 ¼ 0:5 kþ1 ½S þ k1 þ kþ2 þ k2 þ kþ3 þ k3 ½P  D 12.2

where: D ¼ ½kþ1 ½S  k1  kþ2  k2  kþ3 þ k3 ½P2  4ðkþ2  k3 ½PÞðkþ3  k1 Þ

12.3

Oscillatory behavior becomes evident when the substrate concentration within the open intervalpffiffiffiffiffiffiffi from pffiffiffiffiffiffiffi lies pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½k  k Þ þ ð k þ k =k to ½k þ ð k þ3 1 þ1 2 þ2 þ p2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ22 kþ3 þ k1 Þ =kþ1 . This finding appears to explain experimentally observed oscillations (Edman and Rigler, 2000). Again, according to Qian and Elson (2002), the trajectory of a single-enzyme molecule is a stochastic process, wherein the enzyme jumps among the three states seemingly at random. For the three states, the following steady-state probabilities are given: PSS 1 ¼

kþ2 kþ3 þ kþ3 k1 þ k1 k2 l1 l2

12.4

PSS 2 ¼

kþ1 ½Sðkþ3 þ k2 Þ þ k2 k3 l1 l2

12.5

kþ1 kþ2 ½S þ ðkþ2 þ k1 Þk3 ½P l1 l2

12.6

PSS 3 ¼

The treatment of Qian and Elson (2002) makes it possible to deduce the transition probabilities Pij(t, t þ t) for the enzyme being in state-j at time t. In the steady-state condition, the transition probabilities are independent of t. Among the nine such functions of form Pij(t), where subscripts i,j ¼ 1, 2, and 3, are the following:  k3 þ l2 PSS 3 P13 ðtÞ ¼ expðl1 tÞ l 1  l2  k3 þ l1 PSS 3 expðl2 tÞ þ PSS þ 12.7 3 l2  l1  l2 ð1  PSS 1 Þ þ kþ1 ½S þ k3 ½P expðl1 tÞ P11 ðtÞ ¼ l2  l1  kþ1 ½S þ k3 ½P þ l1 ð1  PSS 1 Þ expðl2 tÞ þ l1  l2 þ PSS 1

12.8

Chapter j 12 Single-Molecule Enzyme Kinetics

735

l2 ð1  PSS 3 Þ þ kþ3 þ k2 P33 ðtÞ ¼ expðl1 tÞ l2  l1  kþ3 þ k2 þ l1 ð1  PSS 3 Þ þ expðl2 tÞ þ PSS 3 l1  l2 12.9 

While state probabilities and transition probabilities are independent of the signal used in kinetic measurements, system-specific assumptions about the signal-generating states must also be considered. In view of work carried out by Edman and Rigler (2000) on the horseradish peroxidase reaction, Qian and Elson (2002) considered the case where only the EP complex (State-3 in the above treatment) produces a significant fluorescence (f) signal (i.e., f(1) ¼ f(2) ¼ 0, and f(3) ¼ 1). The mean fluorescence signal can be written as: hf i ¼

3 X i¼1

SS f ðiÞPSS i ¼ P3

12.10

The corresponding time-correlation function is: C2 ðtÞ ¼ Df ð0ÞDf ðtÞ ¼ ¼

P33 ðtÞPSS 3

3 X

i;j ¼ 1

2  ðPSS 3 Þ

2 f ðiÞf ðjÞPSS i Pi;j ðtÞ  hf i

12.11

Note that the presence of the P33(t) term introduces two exponentials (i.e., A exp(l1t) þ B exp(l2t)). Without going into greater detail, suffice it to say that higher-order correlation functions may be used if the experimental rate data are of high enough quality. Most single-molecule studies are conducted under conditions where the substrate is held at a constant concentration. This situation is always favored when an extremely small amount of enzyme is present in the experiment. When changes in substrate concentration are likely, then the situation becomes more complicated, and focus must be on the joint probability of an enzyme being an ion of a particular conformation and of n molecules of substrate being available for reaction with that enzyme conformation. Those interested in learning how such situations are treated should consult Qian and Elson (2002). English et al. (2005) tested the Michaelis-Menten equation at the single-molecule level by monitoring extended time traces of enzymatic turnovers for individual b-galactosidase molecules by a fluorescent detection system (see Fig. 12.5) allowing them to observe the release of product, one molecule at a time. They found evidence of a molecular memory phenomenon, characterized by clusters of turnover events separated by periods of low activity. This behavior arises at high substrate concentrations and persists over the msec-to-sec time range. They attributed this molecular memory phenomenon to the presence of

interconverting conformers with broadly distributed lifetimes. While they proved that the Michaelis-Menten equation still applies to even a fluctuating single enzyme, they suggested that the interconverting conformers exhibit different times for catalytic turnover. Min et al. (2005) have reviewed the single-molecule enzyme kinetic measurements exhibiting large temporal fluctuations of the turnover rate constant over a broad time-scale (from 1 ms to 100 s). The rate constant fluctuations, which they attributed to dynamic disorder, are associated with fluctuations of the protein conformations known to occur on the same timescales. They discussed the unique information extractable from these experiments and how these results may be reconciled with the ensemble-averaged Michaelis-Menten equation. They also suggest that theoretical models based on the generalized Langevin equation treatment of Kramers’ barrier crossing problem for chemical reactions can account for the observation of dynamic disorder and highly dispersed kinetics. As noted above, single-molecule enzyme turnover experiments most often measure parameters relating to the probability density f(t) for the stochastic waiting time t for individual turnovers. Kou et al. (2005) point out that the probability density is best reconciled with ensemble kinetics, and thereby provides information on dynamic disorder associated with fluctuations in the catalytic rates arising from enzyme conformational isomerizations of the type shown in Scheme 12.5.

E1

k11[S]

a12 a21 E2

En

k21

b12 b21

k12[S] k-12

ES2

k13[S] k-13

k1n[S] k-1n

ESi

ESn

P + E01

k31

E1

g12 g21 k22

b23 b32

a23 a32 Ei

ES1

k-11

P + E02

k32

E2

b23 b32 k23

P + E0i

k33

k2n

P + E0n

k3n

Ei

En

Scheme 12.5 where additional rate constants are required to account for these conformational isomerizations. The ‘‘vertical’’ transitions correspond to conformational changes that may or may not increase the catalytic efficiency, whereas ‘‘horizontal’’ transitions correspond to parallel paths for a onesubstrate enzyme operating by a Briggs-Haldane-type steady-state scheme. In principle, an enzyme may meander its way through such an array of conformation and reaction

736

Enzyme Kinetics

FIGURE 12.5 Single-molecule b-galactosidase kinetics. A, Schematic representation of the immobilization of single b-galactosidase molecule to a streptavidin-coated polystyrene bead by means of a flexible PEG linker. The bead binds to the hydrophilic biotin-PEG surface of the glass coverslip. As the chromogenic resorufin-b-D-galactopyranoside (RGP, Compound 1) substrate in buffer solution is converted to a fluorescent resorufin (R, Compound 2) product by the single-enzyme molecule, each product molecule is detected before it rapidly diffuses out of the confocal detection volume; B, Schematic representation of the photobleaching and detection beams. A lens focuses a 550-mW, 560-nm photobleaching beam to a 200-nm diameter spot surrounding the bead. The beam is coupled into the 100-nm-thick flow cell by a prism atop a quartz slide. A water immersion (WI) objective is used to focus a 1-mW detection beam (l ¼ 560-nm) onto a diffraction-limited spot around the bead and to collect the emission for detection with a photon-counting avalanche photodiode detector; C, Turnover time trace of a single b-galactosidase molecule at 20 nM RGP. Left, fluorescence intensity as a function of time for a b-galactosidase molecule undergoing enzymatic turnovers, each giving a fluorescence burst. Middle, data for the same enzyme molecule after addition of 200 mM phenylethylÒ D-thiogalactopyranoside inhibitor. Right, data for a bead without enzyme (no inhibitor). All time traces are obtained with 0.5-ms time bins; D, Turnover time traces of a single b-galactosidase molecule at 100 nM RGP. Dashed line represents the threshold used to determine waiting times between two adjacent bursts. The intensity histogram of the enzymatic time trace is shown at right. The time trace has 0.5-ms time bins. Reproduced from English et al. (2005) with permission of the authors and Macmillan Magazines, Ltd.

stages, proceeding along the lowest-energy course. Each interconversion affects certain intrinsic catalytic rate constants, and this dynamic disorder causes f(t) to exhibit highly stretched multi-exponential decay kinetics at high substrate concentrations and apparent mono exponential decay kinetics at low substrate concentrations. Kou et al. (2005) derived a single-molecule Michaelis-Menten-like equation for the reciprocal of the first moment of f(t), which, like the Michaelis-Menten equation, shows a hyperbolic dependence on the substrate concentration. While this single-molecule Michaelis-Menten-like equation applies

when the enzyme-conformer inter-conversion rates are slower than the catalytic rate, the apparent Michaelis constant is a complicated function of the catalytic rate constants for individual conformer reactions. One problem with complicated reaction mechanisms of the sort shown in Scheme 12.5 is that highly stretched multiexponential decay kinetics may be nothing more than an artifact. In particular, the likelihood that in singulo enzyme kinetics may well reflect the various ways in which an immobilized enzyme interacts with that surface must always be considered. Brief and long-lived encounters of

Chapter j 12 Single-Molecule Enzyme Kinetics

a surface-tethered enzyme molecule with its nearby surface would be tantamount to false ‘‘conformers’’ of the enzyme, the interconversions of which would be completely irrelevant to the solution-phase behavior of the enzyme. With the exception of enzymes tethered via natural biospecific interactions (such as the use of a surface-tethered enzymebinding protein), the same may be said for data obtained in any technique that physically confines an enzyme for purposes of convenient single-molecule measurement. Because substrate molecules might alter enzyme-surface interactions, we cannot confidently predict how a tethered enzyme may behave as a function of substrate concentration.

12.4. VIDEO MICROSCOPY Video microscopy has flourished since the advent of highspeed cameras capable of acquiring very weak fluorescent images of biological molecules, combined with fast computing methods for digitizing, processing, and analyzing acquired images. While video microscopy finds widest applications in the life sciences, chemists, engineers, and biophysicists are also employing video microscopy with increasing frequency. Cell biologists perfected video microscopy to observe cell motility. The discovery of cytoskeletal molecular motors would have been impossible were it not for the inventive efforts of Robert Allen and Shinya Inoue´. Allen Video-Enhanced Contrast DifferentialInterference Contrast (AVEC-DIC) microscopy achieves l/10 spatial resolution (as compared to the standard l/2 resolution for typical optical microscopy) and enjoys the capacity to digitally subtract the background or a previous image to achieve higher contrast and updated images, respectively. Allen, Travis and Allen (1981) and Inouye´ (1986) actually pioneered this technique to investigate cell motility and organelle trafficking. Whether employing DIC optics or dark-field and fluorescence methods, video enhancement has changed the way we observe small particles. Inoue´’s efforts also revolutionized the theory of video microscopy. In DIC (or Nomarski) optics, two beams of a polarized light source are allowed to take slightly different paths before passing through a refractive specimen, such that they interfere with each other when once again recombined. The resulting image has a distorted three-dimensional appearance that strongly enhances the line and edge features of a magnified object. DIC yields sharper and more distinctive boundaries among objects that have different refractive indices at the illuminating wavelength l. Another major advantage of video microscopy is that the subtle grayness of a diffracted image can be enhanced to a degree that allows shapes and lengths to be discerned more effectively. (The author found that one may obtain a similar visual effect by processing a black-and-white, phase-contrast microscope

737

image with the edge-enhancing ‘‘Emboss’’ image filter feature available in Adobe PhotoshopÔ.) Whether employing DIC optics or dark-field and fluorescence methods, video enhancement greatly enhances our ability to observe and track the movements of macromolecular structures, cells, and other small particles. To illustrate this point, consider the so-called ‘‘sliding tubule’’ assay, one of the most widely used assays of microtubule motors. Motor molecules are first tethered to the surface of a microscope coverslip, and a sample of assembled microtubules is then placed atop the motor-covered surface. This assay detects functioning motors by the sliding movements of individual microtubules (Fig. 12.6). Because microtubules are assembled in a head-to-tail fashion from ab-heterodimers, they exhibit a polarity, with the faster growing end called the (þ)-end and the other designated as the ()-end (Purich and Kristofferson, 1984). Any motor locomoting toward the (þ)-end will therefore cause the microtubule to ‘‘slide’’ toward the ()-end, with the opposite true for ()-end-directed motors. Beyond the single-molecule work described in this chapter, video microscopy has been used to make real-time observations on the growth of 65-nm sperm acrosomal processes (Tilney and Inoue´, 1982), the assembly/disassembly of 25-nm diameter microtubules (Allen et al., 1981; Koonce and Schliwa, 1986; Schnapp et al., 1985), and the sliding of 10-nm actin filaments (Toyoshima et al., 1987; Yanagida et al., 1984).

12.4.1. Kinesin Takes One 8-nm Step per ATP Molecule Hydrolyzed The molecular motor kinesin is an energase that transduces the Gibbs free energy of ATP hydrolysis into step-wise translocations along microtubules (MTs). Head-to-tail

(+)

(-) t = t1

(+)

(-) t = t2

(+)

(-) t = t3

FIGURE 12.6 Cartoon representation of ‘‘gliding’’ microtubule assay for kinesin motility. As a point of reference, a single kinesin molecule pushing a microtubule (radius z 15 nm; length < 20 mm) at a 1-mm/s rate exerts a force of ~5 pN (Hunt et al., 1994).

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a Kinesin:Bead Ratio = 5.8

Number

12 8 4 0

b

0

1

2

3

4

5

6

7

8

FIGURE 12.7 Movement along microtubules of 200-nm beads coated with heterotetrameric (a2b2) kinesin. (a) DIC images indicating the progress of one bead (marked with an asterisk) along a microtubule; (b) Histograms of the bead run-lengths at one different kinesin/bead ratio. Black bars indicate that beads detached before reaching MT ends. Light bars correspond to the other beads, including those that could no longer be observed because they moved beyond the field of view. Reproduced from Coy, Wagenbach and Howard (1999) with permission of the authors and the American Society for Biochemistry and Molecular Biology.

ATPase Rate per Molecule (S-1)

polymerization of ab-tubulin heterodimers during microtubule self-assembly creates the 8-nm repeats observed in each protofilament of the resulting MT, and the fact that kinesin exhibits an 8-nm step-size indicates that this molecular motor has a stereospecific interaction with only one type of tubulin subunit. When saturated with ATP, kinesin can move at rates of 0.8 mm/s, corresponding to about one hundred 8-nm steps/s. Coy, Wagenbach and Howard (1999) measured the stoichiometry of kinesin by comparing its stepping rate vmotility to its ATP hydrolysis rate vATPase. When a single kinesin molecule is moving on the microtubule surface, the ratio vmotility/vATPase yields what the authors call the fuel economy of the motor, namely the number of ATP molecules hydrolyzed per step. Likewise, division of vmotility/ vATPase by the step size (d ¼ 8 nm) yields the number of steps per ATP hydrolyzed. They examined the speed of kinesin molecules adsorbed to 200-nm casein-coated silica beads. Flow cells were constructed from coverslips treated with 3-aminopropyl triethoxysilane, which covalently labels the surface with positively charged groups and facilitates the adsorption of negatively charged microtubules. (These microtubules were taxol stabilized to arrest their assembly/disassembly dynamics). After introducing kinesin-coated beads in the presence of 1 mM ATP, motility was observed directly by DIC microscopy using an inverted microscope fitted with a high-sensitivity CCD camera. The steady-state rate of ATP hydrolysis was also determined in the presence of 1 mM ATP and a variable concentration of taxol-stabilized MTs. Typical rate data are shown in Fig. 12.7 for beads having 0.088, 0.88, and 5.8 kinesin molecules per bead. The ATP hydrolysis rate data shown in Fig. 12.8 were fitted to the Michaelis-Menten equation to estimate the apparent maximal rate constant kcat for MT-stimulated ATP hydrolysis. Coy, Wagenbach and Howard (1999) also obtained values for the apparent Michaelis constant (Km), corresponding to the microtubule concentration (expressed in terms of tubulin dimers) needed to reach 0.5 Vmax. The authors analyzed their data to determine whether the motors on those beads having more than one bound motor were fully activated in the hydrolysis assay. They also corrected their data for the effect of ATP depletion and accumulation of ADP and orthophosphate on the observed hydrolysis rate. The self-consistent and convincing finding of this investigation is that kinesin translocates at 9.4  1.3 per ATP hydrolyzed for the hetero-tetrameric motor and 8.7  0.7 nm/ATP for homo-dimeric kinesin. After correcting for an 8.1 nm step-size and a 3% effect of ATP depletion and of ADP/Pi accumulation on the hydrolysis, Coy, Wagenbach and Howard (1999) obtained a corrected value of 1.08  0.09 steps per ATP hydrolyzed. These data on kinesin stepping ATP hydrolysis are only consistent with models predicting a tight coupling between the chemical and mechanical cycles of the motor.

100 80 60 40 20 0

0

20

40

60

Tubulin Concentration (mM) FIGURE 12.8 Steady-state kinetics of microtubule-stimulated ATP hydrolysis by heterotetrameric (open circles) and homodimeric (closed circles) kinesin-coated beads. The estimated kcat values are 94 and 88 s1, respectively, for the heterotetrameric and homodimeric motors; likewise, the estimated Km values are 11 and 7.4 mM, respectively. Reproduced from Coy, Wagenbach and Howard (1999) with permission of the authors and the American Society for Biochemistry and Molecular Biology.

12.4.2. Dark-Field Microscopy Affords Direct Observation of Microtubule Assembly/ Disassembly Dynamics Hotani and Horio (1986) used dark-field microscopy to observe stochastic microtubule (MT) growth and shortening, but technically speaking, the scattering behavior giving rise to the dark-field images required the gain/loss of

Chapter j 12 Single-Molecule Enzyme Kinetics

739

d

Optical Path in Dark-Field Microscopy condenser lens

objective lens

ocular lens

a

f e

light rays

g specimen

Dark Field

b c

annular stop FIGURE 12.9 Illumination of microscopic objects by dark-field microscopy. Yellow lines are ray tracings. Most of the incident light rays are blocked by central region (black), the annular stop. Arrangement of lens permits only oblique, side-on illumination of the specimen.

multiple tubulin dimers before observing a perceptible change in microtubule length. Thus, while a single microtubule could be imaged, the kinetics of individual elongation steps could not be recorded. This imaging technique creates contrast between side-illuminated object and surrounding dark field or background. In the basic optics set-up for the light path (shown in Fig. 12.9), the dark-field condenser produces an annular cone of illumination, and its aperture is greater than the objective’s aperture. Light is scattered by the object to the objective, whereas little if any of the unscattered light reaches the objective. Therefore, in the absence large-molecule light scattering, the field of view is totally black, or dark. Another advantage is that there is no need to introduce an extrinsic fluorescence reporter group. A simple truth is that dark-field microscopy is extraordinarily sensitive to the presence of macromolecular aggregates and large floating particles. Keeping a sample sufficiently free of adventitious scatter centers (dirt particles) becomes a major hurdle in nearly all cases. By visualizing the dynamic behavior of individual microtubules in vitro by dark-field microscopy, Hotani and Horio (1986) demonstrated that growing and shortening populations of microtubules coexist in steady-state conditions. Because there are approximately 1,600 tubulin dimers per mm of MT length, it should be obvious that dark-field microscopy is unable to report on the gain or loss of individual tubulin dimers. Furthermore, the time-scale of dimer addition or dissociation from individual microtubules takes place on a 5–10 msec time-scale (Karr, Kristofferson and Purich, 1980; Purich and Kristofferson, 1984), whereas observations on MT lengthening and shortening were made on a much longer time-scale (>5 min). Even so, the observations of Hotani and Horio (1986) played a major role in establishing the dynamicity of assembled microtubules. The real-time video recordings (Fig. 12.10) of Horio and Hotani (1986) revealed that both ends of a microtubule exist

FIGURE 12.10 Instability of microtubules visualized as paired darkfield microscopic images. Several microtubules were attached to a glass surface to observe and record the trajectories (‘‘life histories’’) of their endwise growing and/or shortening. Dynamic images were recorded on videotape (b). To demonstrate their length changes, two images differing in time by roughly five and one-half min were photographed as a double exposure to place corresponding tubules next to each other with a slight positional offset. (The image shown here was enhanced with PhotoshopÔ for presentation in this chapter.) Pairwise comparisons of these images unambiguously demonstrated the shortening and lengthening of certain microtubules. Methods: Tubulin was prepared from bovine brain by three assembly/disassembly cycles in the warm and cold, respectively, in the presence of GTP, and microtubule-associated proteins were removed from tubulin by passage of the microtubule proteins through a phosphocellulose column. Microtubule assembly was carried out 37 C with 1.7 mg/mL tubulin in 90 mM PIPES buffer (pH 6.9) containing 1.8 mM EGTA, 2.7 mM GTP, and 0.9 mM magnesium sulfate. After a 2-min incubation, a 4-mL sample was transferred onto a glass slide, mounted with a coverslip, and observed under dark-field illumination. (The scale bar in the upper right of the composite micrograph ¼ 10 mm.) Reproduced from Horio and Hotani (1986) with permission of the authors and Macmillan Magazines, Ltd.

in either the growing or the shortening phase, and alternate quite frequently between the two phases in a stochastic manner. Moreover, growing and shortening ends can coexist on a single microtubule, one end continuing to grow simultaneously with shortening at the other end. They observed no correlation in the phase conversion either among individual microtubules or between the two ends of a single microtubule. The two ends of any given microtubule have remarkably different characteristics; the active end grows faster, alternates in phase more frequently and fluctuates in length to a greater extent than the inactive end. Microtubule-associated proteins, which are thought to bind simultaneously to neighboring tubulin dimers situated within a microtubule lattice, suppress the phase conversion and stabilize MTs in the growing phase.

12.5. OPTICAL TWEEZERS This technique, also known as ‘‘laser trapping’’ microscopy, allows an experimenter to impart 5–50 piconewton forces on small objects as a consequence of momentum transferred

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to a microscopic object from photons in a laser beam (Ashkin and Dziedzic, 1985; Ashkin et al., 1986). The microscope’s objective lens is used to focus a laser beam onto a very small, diffraction-limited spot, and the objective has a high numerical aperture that creates a steeply converging beam. In operation, it is important to establish a steep three-dimensional gradient of intensity around the focal point, such that a particle of higher refractive index than the surrounding medium (e.g., polystyrene beads in water) is effectively pushed toward the region of highest intensity. Under these conditions, the bead is said to be ‘‘trapped’’ by the gradient of forces acting on the object. Ultimately, photons impart a trapping force. The socalled radiation pressure (i.e., the force per area exerted on an object’s surface) depends on the photon flux, defined as the number of photons passing through a given area per unit time. The force on a dielectric object depends on the change in momentum of light induced due to refraction of the light by the object. The total force exerted is determined by the difference between the momentum fluxes entering and exiting the object. In the case of optical tweezers, the radiation pressure is provided by a laser microbeam (at a non-absorbing wavelength) on very small objects such as a bacterium or a micron-sized polystyrene sphere. Consider two rays of light, denoted a and b, that are focused by a lens (Fig. 12.11), where each ray is refracted at the surface of the bead so that its direction of propagation changes. A ray of light impinging on the interface of the sphere has its direction of propagation, and therefore its momentum, changed by the interaction of the light with the sphere. Since the sphere changes the momentum of the light, an equal and opposite change in momentum of the bead occurs. The resultant force of the light on the sphere due to refraction is always in the direction of the focus of the light (shown as the intersection of two yellow lines). In short, optical trapping works because laser light refracts through transparent objects in such a way that there is always more light pressure pushing the object towards the focal point than there is pushing it away from the focal point. Radiation forces due to refraction can also be used to pull a transparent object. With the aid of a beam-splitter and additional optics required to focus two beams, laser-tweezers experiments can also be arranged to trap several objects, simultaneously. The resulting double laser-trap allows a fascinating range of biophysical experiments to be carried out, and, as shown in Fig. 12.12, be applied and controlled by the tensile forces on biopolymers with beads attached to their two ends. The successful experimenter will exercise due care to compensate for errors arising from drifts in the relative alignment of the laser beams. For optimal results, the laser beams must be aligned to within less than the diameter of the bead. Alternatively, ‘‘two-bead’’ experiments with a single laser trap can be carried out by the use of a micropipette while holding one bead. A good example is the use of optical tweezers to assess how tension on duplex DNA alters the polymerase

Optical “Tweezers” Technique b

a

F

Lens Fb

Fa

Bead b

a

FIGURE 12.11 A tracing of light-rays illustrating the interaction of light with a particle in a laser optical trapping experiment. Parallel rays enter a small, refractile sphere from above and are bent because the sphere acts like a lens. Before entering, the rays travel vertically with zero horizontal momentum; upon deflection, they pick up horizontal momentum. Because momentum must be conserved, an equal and opposite momentum change is imparted on the sphere. If the beam were perfectly uniform, the reaction forces would cancel, resulting in no net sideways component. In a gradient, however, the asymmetry in the light gives rise to an imbalance in the reaction forces and the object is pulled towards the brighter side.

A

Myosin bead

bead Actin Filament

B

Hairpin bead

bead RNA strand

FIGURE 12.12 Double-focusing laser tweezers optical trapping experiments. A, Scheme for imparting tension to stretch an actin filament for studies on myosin motility; B, Scheme for measuring force-dependent unfolding of partially folded RNA strand.

and exonuclease activities of T7 phage DNA polymerase (Wuite et al., 2000).

12.5.1. Optical Tweezers Facilitated SingleMolecule Studies on RNA Polymerase Messenger synthesis by RNA polymerase is a processive reaction that is punctuated by pauses that allow transcription factors to bind and modify subsequent mRNA synthesis (Nudler, 1999; von Hippel, 1998). Such action is obscured when studying a population of RNA polymerases (RNAPs), because the collective asynchronous action of individual RNAPs results in ensemble-averaged kinetics. Real-time observation on single RNAP molecules reveals the interplay

Chapter j 12 Single-Molecule Enzyme Kinetics

A

optical trap streptavitin-coated microsphere DNA

RNA

LDNA HA-tagged RNAP

coverslip coverslip is moved to maintain a constant 4 pN force

B Nucleotides transcribed

between active elongation and nonproductive states. An individual RNAP trajectory also exposes variations in the behavior of a single RNAP molecule. While transcriptional pausing is known to induce dispersion of the population, the debate centers on whether or not the stochastic behavior of a structurally homogeneous population is sufficient to generate the observed levels of asynchrony, or whether the occurrence of stable RNAP conformations must also be invoked. A study of single RNAP molecules suggests that an elongating RNAP population is composed of RNAPs in distinct states that elongate at different intrinsic rates and are more or less likely to pause (Davenport et al., 2000). However, the average elongation rates reported in the previous work were significantly slower than solution rates, a condition that complicates interpretation of the fast and slow elongation modes observed (Davenport et al., 2000). Adelman et al. (2002) examined elongation by single RNAP molecules on a template lacking known regulatory pause sites. An important observation of these single-molecule enzyme kinetic studies was the occurrence of transient pauses lasting 1–30 sec. The average elongation rates in these experiments were identical to those obtained from bulk solution assays of transcription, indicating that the RNAP molecules studied were fully active. Individual RNAPs exhibited homogeneous elongation dynamics, with differences among RNAPs arising from random switching between a single active elongation mode and the paused state. Adelman et al. (2002) developed a method for specific, flexible immobilization of a hemagglutinin (HA) epitopetagged RNAP on a coverslip surface coated with antibody against this tag. The HA tag was fused to the a-subunit’s ˚ radius of motion with respect C-terminus, resulting in a 70-A to the RNAP’s central core. RNAP elongation complexes immobilized in this way (Fig. 12.13) are highly active, and their behavior on the surface is nearly identical to that observed in bulk solution. Transcript elongation by ten individual RNAP molecules is shown in Fig. 12.13B. The average elongation rate for the 30 individual wild-type RNAPs studied was 12 nucleotides transcribed per second (nt/s). In vitro transcription assays were performed under identical conditions (22 C and 1 mM ribonucleoside substrates) on the same template, and the elongation rate was 12–13 nt/s. These single-molecule experiments revealed no sites at which a large proportion of RNAPs paused, and the observed transient pauses occurred with low probability throughout the gene in the absence of dominant sequence effects. By ascertaining that the single RNAP molecules investigated behaved like typical bulk solution-phase RNA polymerase, Adelman et al. (2002) determined the fundamental kinetic features of transcription elongation under conditions that allow direct comparison to population studies. Their examination of single wild-type RNAP molecules revealed that the asynchrony of transcription elongation does not result from distinct modes of RNAP

741

2000 1800 1600 1400 1200 1000 800 600 400 200 0 0

20

40

60

80

100

120

140

Time (seconds) FIGURE 12.13 Single molecule observations on RNA polymerase. A, A stalled ternary elongation complex composed of Escherichia coli RNAP (bearing an hemagglutinin HA epitope tag, DNA template, and a short RNA) was specifically immobilized on a coverslip surface through interaction with a nonspecifically adsorbed anti-HA antibody. A streptavidin-coated microsphere was attached to a biotin located on the downstream end of the DNA, and the microsphere was held at a fixed position throughout elongation by an optical trap. During transcription, the template DNA was pulled through the RNAP, leading to a decrease in length of downstream DNA (LDNA). Feedback control moved the coverslip toward the optical trap so that a constant force of 4 pN was maintained on the RNAP. Force of this magnitude has been shown to have no detectable effect on elongation rate or pausing (Davenport et al., 2000; Wang et al., 1998); B, Elongation trajectories of ten single RNAP molecules plotted as nucleotides transcribed versus time. Reproduced from Adelman et al. (2002) with permission of the authors and the Proceedings of the National Academy of Sciences USA.

movement. Rather, the observed heterogeneity in elongation rates on the rpoB gene results from statistical variations caused by the stochastic switching of RNAPs from a single active elongation mode to a transiently paused state. This conclusion contrasts with the results of a single-molecule study by Davenport et al. (2000), who reported that a transcribing RNAP population is heterogeneous, with distinct fast and slow elongation modes. Adelman et al. (2002) noted that differences may have resulted from the use of distinct DNA templates and/or alterations in the activities of immobilized RNAPs. These possibilities were difficult to assess, because the earlier study did not provide a comparative solution analysis of elongation under their conditions.

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Adelman et al. (2002) also pointed out that Davenport et al. (2000) reported a slower average overall elongation rate: 8.0  3 nt/s versus 12.0  2.1 nt/s from the newer work. Discrepancies may have arisen from the way that the instantaneous velocity was calculated and the use of a ‘‘peak’’ velocity distribution. To calculate an instantaneous velocity, Davenport et al. (2000) averaged data over 15 sec, which is a significantly longer time-averaging window. Adelman et al. (2002) measured the half-life of active elongation between pauses for wild-type RNAPs to be 12.7 sec, with the pausing half-life of 1.6 sec.

12.5.2. Optical Trapping Facilitates SingleMolecule Studies of RecA Polymerization on Double-Stranded DNA The 38.8 kDa RecA protein, an essential participant both in recombination and DNA repair, binds cooperatively to DNA, increasing the latter’s length by 50% and altering the twist relative to double-stranded DNA (dsDNA). During DNA strand exchange during recombination, a singlestranded DNA (ssDNA) strand must replace a homologous strand on a dsDNA molecule. A mechanistic clue about RecA’s role is that the protein self-assembles in a manner akin to actin and tubulin, where depolymerization appears to be linked to nucleoside-59 triphosphate hydrolysis. To probe the mechanism underlying RecA’s role in recombination, Shivashankar et al. (1999) directly measured the RecA’s polymerization kinetics and evaluated changes in the entropic elasticity of DNA using single-molecule optical tweezers approaches (Fig. 12.14). In a typical experiment, the force required to extend the DNA coil by a distance z, where z ¼ (r  r0), with r representing the position of the DNA attachment to the bead and r0 standing for the end attached to the coverslip. The force FDNA(z) is measured by the bead’s displacement d from the center of the optical trap, where the force-distance relationship is determined by the effective voltage V(d) which exhibits Hookean behavior (i.e., a parabolic dependence of FDNA(z) on displacement) near the trap’s center: VðdÞ ¼

ad2 2

Force-Extension Measuring Device: optical trap 3-mm bead with tethered l-DNA molecule (red)

RecA

lens

Piezoelectric drive

quadrant photon detector FIGURE 12.14 Force-extension measurements on RecA interactions with a tethered l-DNA molecule using back scattering and optical tweezers techniques. Light from a near-infrared laser diode (power ¼ 150 mW, wavelength ¼ 830 nm) was focused by use of an infinitycorrected, oil-immersion Zeiss Neofluor 100 objective (NA ¼ 1.3) to construct the laser trap (indicated as a dotted-line well). Co-linear with the IR laser beam, light from a second laser (power ¼ 8 mW; wavelength ¼ 633 nm) was used to measure scattering from a polystyrene bead with a covalently tethered l-phage DNA molecule. The backscattered light from the particle was focused onto a quadrant detector. Force-extension measurements were recorded by moving the piezoelectrically displaceable stage (dynamic range ¼ 30 mm; step-size ¼ 10 nm), while simultaneously recording the displacement of the tethered bead. The trapping stiffness was ~0.05  0.01 pN/nm for a 100-mW laser. For other details, see Shivashankar and Libchaber (1998). Reproduced from Shivashankar et al. (1999) with permission of the authors and the National Academy of Sciences.

optical trap. Such trap-induced DNA extension apparently serves to lower DEact for RecA binding to DNA, to the effect that RecA polymerization is greatly accelerated. Finally, in the presence of ATP, RecA undergoes single site nucleation and linear elongation at the nuclei on individual dsDNA molecules, the latter proceeding at a rate of ~12 monomers/s (Fig. 12.16a), with a depolymerization rate of ~2 monomers/s. In the presence of ATPgS, multiple nucleation sites are formed, resulting in a sigmoidal growth curve (Fig. 12.16b). The latter finding reminds us that lasertrap observations are direct and unambiguous.

12.12

where the stiffness a is estimated from the range of thermal fluctuations of an untethered bead. During RecA polymerization, the molecule is kept under a 6-nN tension by the trap, a force that is sufficient to reduce entropic conformations of DNA. Shown in Fig. 12.15 are plots of FDNA(z) versus DNA extension for RecA polymerization on DNA in the presence of ATP (Panel A) and ATPgS (Panel A). To reduce the entropically controlled conformations of DNA during the RecA polymerization, the DNA molecule was maintained under a constant tension of 6 pN with the

12.5.3. Actin-Based Listeria Motility Exhibits Monomer-Sized Stepping Although the mechanoenzymatic basis of actin-based motility is considered at length in the last section of Chapter 13, it is appropriate to describe here the particle-tracking experiments of Kuo and McGrath (2000) on the motility of the untracellular pathogen Listeria monocytogenes within the peripheral cytoplasm of Cos7 host cells (Fig. 12.17). Their particle-tracking microscope consisted of a highsensitivity quadrant diode detector and a nanometer-scale repositioning piezoelectric stage, relying on a low-power

Chapter j 12 Single-Molecule Enzyme Kinetics

FIGURE 12.15 The force exerted by DNA as a function of piezoelectrically induced extension at various stages of RecA assembly on DNA. A, In the presence of ATP; and B, in the presence of ATPgS. Curves for naked DNA (circles) and DNA covered fully by RecA (diamonds). In the top panel, a force-extension curve is also included for partially covered DNA (15 min) and another force-extension curve for a sample after depolymerization (300 min). Reproduced from Shivashankar et al. (1999) with permission of the authors and the Proceedings of the National Academy of Sciences USA.

A

743

laser (1 mm/sec maximal rates of actin-based motility. They also could not explain the basis of regular 5.4-nm stepping; nor could their ideas explain why motility was unaffected by (þ)-end capping proteins that would be expected to block monomer incorporation into free filaments (see Section 13.12: Actoclampin Molecular Motors).

12.6. ATOMIC FORCE MICROSCOPY This force-measuring microprobe technique, introduced by Binnig, Quate and Gerber (1986), employs an electromechanically positioned probe, most often consisting of monolithic silicon moving over the surface of a sample to measure contour dimensions as well as the energies of probe interactions with a surface. AFM probes must be sharp enough to reveal the detailed structure of macromolecules (e.g., proteins, polysaccharides, DNA, etc.), molecular assemblies (e.g., membranes, cell walls, amyloid, etc.), as well as cells and tissues, all of which are positioned on synthetic and natural surfaces. They are most often fabricated from single-crystal silicon (radius of curvature: sharp tip z 10 nm; super-sharp tip < 2-nm radius) that can be

Atomic Force Microscopy Laser

quadrant detector

θ

Tip cantilever specimen FIGURE 12.18 Schematic design of atomic force microscope, showing displacement of cantilever arm during operation. Note that displacements of the anvil are determined optically with great accuracy. See text for additional explanation of the mechano-optical interface. See text for additional comments.

doped with chemical functional groups suitable for biomolecule coupling, diamond-coated for extended wear, or magnetically coated. The probe (Fig. 12.18) is positioned at the tip of a free-ended cantilever that deflects in response to the force between tip and sample. A laser beam is reflected onto positions A and B on a quadrant photo-diode detector. AFM instruments typically use a raster-scanning mode that records a two-dimensional scan of the specimen’s surface. For small displacements, the difference signal (i.e., D ¼ cpmA – cpmB) changes as the cantilever bends in a manner predicted by Hooke’s Law, allowing the interaction force between the tip and sample to be determined. Movement of the tip over the sample is executed by an extremely ˚ ngstrom precise, piezoelectric ceramic device with a sub-A resolution along x-, y- and z-axes. There are two commonly applied measurement methods: (a) contact mode, in which the probe maintains contact with the surface, with more or less force applied as the tip is ‘‘dragged’’ over the specimen; and (b) tapping mode, where the probe at the end of an oscillating cantilever (usually at its resonant frequency) taps the surface only for a very small fraction of its oscillation period. In either case, AFM measurements tend to be dominated by short-range, van der Waals forces between oscillating dipoles. Alternatively, a probe protein or ligand can be cross-linked to the tip by covalent modification, such that the force effects on the formation and rupture of biospecific bonding interactions can be examined. The so-called non-contact mode may be employed for AFM imaging. Here, the cantilever must be made to oscillate above the surface of the sample at a distance that lies outside the repulsive regime of typical intermolecular force curves. A thin intervening layer of water then forms a forcesensing capillary bridge between probe and sample, thereby allowing the instrument to sense structural features of the probed surface. The spatial resolution of the AFM technique depends strongly on the nature of the probe. Integrated AFM cantilever-tip assemblies are typically fabricated from silicon and are pyramidal in shape (cone angles ¼ 20–30 ; radius of curvature ~5–10 nm). Variation in performance from one tip to another can be quite large, making it difficult to control at the scale relevant to high-resolution structural imaging. In addition, tips undergo wear during scanning, making it difficult to obtain a quantitative assessment of the tip’s contributions to image broadening. Owing to increased pressures at the tip–sample interface, sharper tips exhibit the greatest tendency to undergo wear. In view of their mechanical properties as well as structural integrity and resilience, carbon nanotubes are quickly becoming the tip of choice in AFM measurements. Nanotubes possess a honeycomb network, called a graphene sheet, with its constituent sp2-hybridized carbon atoms forming a seamless cylinder that can be micrometers long. When an oscillating nanotube tip is brought into contact with a sample, the cantilever

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oscillation amplitude dampens strongly, but then recovers partially as the nanotube buckles. The nanotube can be viewed by dark-field microscopy (500–1,000 magnification), a property used to guide their attachment to standard AFM tips by means of a micromanipulator. The nanotube is then attached, after transferring adhesive to the AFM tip by briefly touching the tip surface to commercial adhesive tape. Nanotube deflection at the elastic buckling point corresponds to the Euler buckling force FB, FB ¼

p2 EI L2

12.13

where E is the Young’s modulus, I is the moment of inertia (I ¼ pr4), L is the nanotube tip-length, and r is the nanotube radius. Figure 12.19 illustrates the high resolution that can be accomplished by state-of-the-art AFM using carbon nanotube probes. Acetylcholinesterase (Reaction: Acetylcholine þ H2O # Acetate þ Choline) plays an absolutely essential role in neurotransmission by depleting the neurotransmitter after its release into the synaptic gap. When present in even slight excess over the concentration of acetylcholinesterase, nerve toxins that totally inhibit this enzyme bring about death with startling promptness. Yingge et al. (1999) used atomic force microscopy to investigate the force spectrum between cholinesterase and its substrate as well as the influence of its inhibitors and re-activators on enzyme-substrate

complexation. Acetylcholinesterase was immobilized covalently on the surface of gold-coated mica, and the substrate was likewise attached to the silicon nitride tip of the AFM probe. After scanning the surface to locate enzyme molecules, the probe was centered over individual enzyme molecules to record the force-distance spectrum for each before and after addition of substrate, inhibitors, and reactivators. Without delving into the specific findings, suffice it to say that the approach demonstrated subtle binding features that could not be obtained from standard enzyme kinetics. Finally, Ratcliff and Erie (2001) developed an accurate three-step AFM method for determining the molecular volume of medium-sized proteins: Step-1: image plane fitting to be certain that the background is flat; Step-2: image analysis using the NIH Image SXM freeware; and Step-3: volume calculation, achieved without making any a priori assumption about the molecule’s shape. The volume is then calculated using the formula, Vi ¼ Ai(Mi  S), where Vi is the volume of the ith particle, Ai is the area of the image, Mi is the total height, and S is the height of the surface plane. Their approach was sufficiently sensitive to determine the dimer dissociation constant (Kd ¼ 1.4 mM) for a DNA helicase by counting the number of monomers and dimers in a sample. Their value agreed with values based on hydrodynamic data. Because the technique requires very small sample sizes (i.e., measurements on only 300–1,000 molecules), this approach promises to be especially valuable whenever a protein is especially scarce.

12.7. NEAR-FIELD OPTICAL MICROSCOPY

FIGURE 12.19 GroES molecular chaperonin imaged by AFM with a nanotube tip. Large scan area in (a) shows both ‘‘dome’’ and ‘‘pore’’ conformations, representing the two sides of GroES facing up. A higher resolution image (b) of the pore side shows the heptameric symmetry, which matches well with the crystal structure. The corresponding rotationally reconstructed image (c) is consistent with these data. Reproduced from Cheung, Hafner and Lieber (2000) with permission of the authors and the Proceedings of the National Academy of Sciences USA.

This microscopic technique differs from typical far-field optical microscopes that use lenses or mirrors to form an image from light that has propagated over a distance very much greater than the wavelength of the light illuminating the specimen. For green light (l z 532 nm), far-field microscopes have an imaging distance in the millimeter range, and, even with the best design, the smallest object or feature d that can be resolved in far-field mode is ~1 mm. This phenomenon, called diffraction-limited resolution, gives rise to the l/2 Abbe´ diffraction limit. As first described by Synge (1928), however, the light field near the surface of an object also contains spatial frequencies that are higher than those that can be imaged by far-field optics. These higher spatial frequencies (which are a manifestation of the evanescent waves associated with surface-reflected light of various wavelengths) constitute what is called the illuminated object’s ‘‘near-field.’’ The higher frequency light of the evanescent wave provides important information about a specimen’s fine structure. However, the intensity of the evanescent wave decays exponentially over a distance that is less than the wavelength of incident light (see Section 12.7.3), meaning that light revealing fine-structure details

746

never reaches the detector in commonly used far-field optical microscopes. In Near-field Scanning Optical Microscopy (NSOM), an excitation beam, consisting of sub-wavelength light, is guided to the specimen by means of a small aperture formed by the tapered end of an optical fiber (Fig. 12.20) that is coated with aluminum to prevent light from leaking elsewhere along the fiber shaft. Any resulting fluorescence is then collected with a microscope fitted with a high numerical aperture objective lens (Fig. 12.21). The specimen is then raster-scanned at a height of a few nanometers above its surface, allowing the experimenter to obtain optical images at ~50-nm resolution (de Lange et al., 2001; Trautman and Ambrose, 1997; Xie, 1996). Although near-field microscopy permits spectroscopic imaging at resolutions substantially better than the Abbe´ diffraction limit, the 50-nm resolution cannot provide structural information about enzymes or even structures of the size of ribosomes. Even so, the NSOM technique can be used to obtain a fluorescence signal from individual molecules, particularly when enzyme molecules are spaced distances that allow resolution of images that are at least a few pixels apart in the CCD camera. Another major advantage is that the tip of the NSOM probe can be functionalized and covalently modified to attach fluorescently labeled proteins for FRET measurements on the kinetics of protein-ligand binding interactions. NSOM is thus ideally FIGURE 12.20 The near-field optical probe. A, An optical fiber is pulled to a final diameter of 20–120 nm and subsequently coated with aluminum. This coating serves to confine the light to the tip region. A subsequent etching step results in a flat and circular endpoint and aperture. The aperture functions as a miniature light source, and its diameter primarily determines the optical resolution of the microscope; B, The principle of surface-specific excitation. The optical near-field generated at the aperture has significant intensity only in a layer of 104 M1$cm1 and high quantum yields. Recombinant DNA techniques also afford efficient routes for expressing fusion proteins containing green fluorescent protein (GFP) or its numerous colormodified derivatives.

12.8.1. Epifluorescence Permits Uniform Sample Illumination In standard fluorescence photometry, the photomultiplier tube detects the emission signal after excitation is achieved

Chapter j 12 Single-Molecule Enzyme Kinetics

Objective

Optical fiber

Laser

Distance detection

Sample

y x Scanners

Polarizers

Feedback

Objective Long-pass filter

Detector 2

z

747

FIGURE 12.22 Schematic representation of the role of the chromatic beam-splitting filter in the illumination and observation of a specimen by epifluorescence microscopy. Although shown separately, both illumination and observation are carried out simultaneously.

Detector 1 FIGURE 12.21 Schematic layout of a near-field scanning optical microscope. The NSOM probe is a tapered optical fiber. Laser light is passed into the fiber and is used to excite fluorophores as the probe scans the sample surface. The probe-sample distance is maintained constant at t), where t) is the correlation time which will be on the same time-scale as the average period of fluctuations of the force F(t). Using this formalism that t) is a direct measure of the relaxation time needed for the degrees of freedom responsible for the force to return to equilibrium after being suddenly disturbed, one can ask for a certain time t1 what the probability is that the force F(t) assumes a value between F(t1) and F(t1) þ dF(t1). One can evaluate the joint probability that at a time t1 the force lies between F(t1) and F(t1) þ dF(t1) and that at t2 the force lies between F(t2) and F(t2) þ dF(t2). The approach is non-invasive, applies to molecular dynamic studies on small numbers of molecules, and yields chemical reaction rate constants and/or diffusion coefficients.

Chapter j 12 Single-Molecule Enzyme Kinetics

microscope. Such a volume element within a solution containing 1 nM enzyme should on average contain one (or just a few) enzyme molecule(s) that diffuses as it catalyzes a reaction, interacts with a regulatory protein, or self-associates or dissociates. Laser-induced fluorescence from that volume element fluctuates as fluorescently tagged molecules are rarified or concentrated in that volume element. The major breakthrough was achieved by combining the FCS technique with confocal detection. Autocorrelation is used to provide a measure for the self-similarity of a time signal, such that fluctuations in the fluorescence signal are quantified by temporally autocorrelating the recorded intensity signal.

12.9.2. One- and Two-Photon FCS Provides a Highly Versatile Enzyme Probe One-Color FCS (Fig. 12.30, left) is a sensitive probe of the concentrations and mobilities of biochemical substances (Arago´n and Pecora, 1976; Elson and Magde, 1974; Rigler et al., 1998). The method has been gainfully exploited to examine ligand binding/unbinding to macromolecules (Kinjo and Rigler, 1995; Schwille, Meyer-Almes and Rigler, 1997) as well as the kinetics of self-assembly, polymerization, and aggregation (Palmer and Thompson, 1987). Another FCS application concerns the intramolecular fluctuations associated with intermittent emission

753

behavior, termed ‘‘blinking’’ or ‘‘flickering,’’ which typically occurs on the msec time-scale (Garcia Parajo et al., 2000; Veerman et al., 1999; Weber et al., 1999; Zumbusch and Jung, 2000). When such phenomena take place on faster time-scales than residence times of molecules within an illuminated volume (Magde, Elson and Webb, 1972), the frequency of ‘‘blinking’’ or ‘‘flickering’’ reveals important dynamic features of the microenvironment (Haupts et al., 1998; Widengren et al., 1995; Widengren and Rigler, 1998). Blinking or flickering can arise from: (a) the intersystem crossing from a singlet to a triplet state; (b) conversion to a different protonation state; (c) quenching upon binding or release of a ligand (i.e., substrate, coenzymes, metal ions, regulatory effectors, etc.); or (d) quenching upon isomerization of one or more enzyme bound intermediates. Sudden irreversible loss of fluorescence from single molecules may also result from photobleaching. Due to its inherent measurement selectivity, Two-Color Cross-Correlation Analysis (Eigen and Rigler, 1994; Rarbach et al., 2001; Schwille, 2001b; Schwille, Bieschke and Oehlenschla¨ger, 1997) is far superior to conventional FCS as a probe of interactions between two different molecular species. The basic strategy (Fig. 12.30, right) is to label both species with spectrally distinct dyes and to record the spontaneous, yet coordinated, fluctuations for each in the two detection channels. Coordinated fluctuations in both probes unambiguously prove the existence of physical or chemical linkage between the two fluorophores. As pointed

FIGURE 12.30 Design features of single-color and dual-color fluorescence correlation spectrometers. See text and cited article for details. Reproduced from Haustein and Schwille (2000) with permission of the authors and publisher.

Enzyme Kinetics

754

out by Katrin et al. (2002), two-color cross-correlation analysis is, in many respects, a dynamic analogue of colocalization techniques frequently used in fluorescence imaging (Fig. 12.30). However, the dual-color crosscorrelation technique offers the added advantage of minimizing false-positive signals associated with the extremely low probability of coordinated fluctuations of two fully independent measurement parameters. Two-color cross-correlation analysis has been used to probe kinetics of irreversible association processes (Schwille, Bieschke and Oehlenschla¨ger, 1997), DNA polymerization by means of the polymerase chain reaction (or PCR) method (Rigler et al., 1998), the early aggregation events in prion protein rod formation (Bieschke and Schwille, 1997; Bieschke et al., 2000), binding of doublestranded DNA to transcription activator proteins (Rippe, 2000). The method has proven to be particularly valuable in characterizing enzyme kinetics, such as the endonucleolytic cleavage of a double-labeled substrate by EcoRI (Kettling et al., 1998) and other nucleases (Koltermann et al., 1998). Cross-correlation analysis requires two appropriately labeled molecular species as well as stable optical instrumentation permitting maximum overlap of two illumination beams in a confocal fashion (Heinze et al., 2000). Joint excitation of spectrally distinct fluorophores by a single infra-red (IR) illumination source relies on the fact that twophoton excitation spectra have different selection rules (i.e., a quantum chemistry rule or tendency stating when a particular transition is likely or very unlikely (or in the parlance of spectroscopists allowed or forbidden, respectively) than one-photon counterparts (Denk, Strickler and Webb, 1990; Xu, Shear and Webb, 1997). Two-photon cross-correlation analysis has been demonstrated with a large number of dye combinations. A conceptual difference between one- and two-color FCS analysis concerns how measurement parameters are evaluated. One-color autocorrelation analysis mainly focuses on characteristic time-scales underlying the fluorescence fluctuations, allowing extraction of such rate data as decay times for particle mobility and internal dynamics. As pointed out by Katrin et al. (2002), the most important parameter in two-color FCS analysis is the amplitude of the cross-correlation function, a parameter containing all of the required information concerning the relative concentration of doubly labeled particles (Schwille, 2001b; Schwille, Bieschke and Oehlenschla¨ger, 1997). In principle, the amplitude reflects temporal coincidences of fluctuations in the two simultaneously recorded signals. The greater the number of joint fluctuations in both channels relative to the number of fluctuations in each single separate channel, the higher the relative amplitude of the cross-correlation function with respect to the amplitudes of the autocorrelation curves (Heinze et al., 2000). Therefore, rather than recording the full temporal correlation function containing

all dynamic information about the fluctuation decay, a simplified mode of data analysis can be obtained from the cross-correlation amplitudes (Winkler et al., 1999). This fact greatly shortens measurement times and likewise simplifies data processing. Dual-color FCS is thus especially well suited for studies on living systems, where the illumination-dose should ideally be kept to a minimum. The authoritative paper by Katrin et al. (2002) presents the crucial measurement parameters.

12.9.3. Basic Kinetic Theory The following brief account of cross-correlation analysis is based on the elegant treatment of Kohl et al. (2002). The generalized fluorescence correlation function is: Gij ðtÞ ¼

hdFi ðtÞdFj ðt þ tÞi hFi ihFj i

12.18

where Fi(j)(t) are the fluorescence signals. For i ¼ j, this equation is the autocorrelation function for one molecular species, as observed in a single detection channel. If i s j, signal fluctuations of different fluorescent species are recorded simultaneously and cross-correlated, yielding the cross-correlation function (designated as Gij or Gx). Under ideal conditions (i.e., when there is minimal spectral interaction between the reporter dyes, including Fo¨rster resonance energy transfer), Gij(0) is directly proportional to the relative concentration of double-labeled species (i.e., those molecules simultaneously bearing both reporter groups), because such dyes are effectively co-localized and must virtually simultaneously ‘‘witness and testify’’ about similar kinetic environments. The absolute concentration of double-labeled molecules is obtained from the corresponding auto-correlation amplitudes, as given by: hCij i ¼

Gx ð0Þ Veff Gi ð0ÞGj ð0Þ

12.19

where Veff is the effective detection volume. (Obviously, if no double-labeled molecules actually exist, Gx(0) ¼ 0, and so does CCijD.) The temporal decay of the cross-correlation curves can be evaluated under ideal conditions by a three-parameter model for one diffusing species: !1=2  r02 t t 1 1þ 2 GðtÞ ¼ Gð0Þ, 1 þ tD z0 t D ¼ Gij ð0ÞDiff ij

12.20

where tD is defined as the average lateral diffusion time for a single double-labeled molecule, r0 and z0 are lateral and axial dimensions of Veff ¼ (p/2)3/2r02z0. If Veff is known, then r0 and z0 may be estimated. Alternatively, by using a pure dye of known diffusivity, Veff can be estimated for a particular instrument’s optical configuration. Equation 11.20

Chapter j 12 Single-Molecule Enzyme Kinetics

755

accounts for the correlation function and the occupation times for the translational diffusion for a single fluorescent species. As noted by Kohl et al. (2002), when FRET also takes place, the approach for evaluating the auto- and crosscorrelation amplitudes must be modified. In this case, another parameter known as molecular brightness (h), measured in units of photons$molecule1$s1, must be introduced. For a particular dye, h is determined in FCS experiments by multiplying its respective autocorrelation amplitude by the average fluorescence count rate. If hi,F and hj,F represent the FRET-unaffected brightness values (i.e., those obtained with non-interacting single-labeled protein molecules), the auto-correlated amplitude changes to:

GiðjÞ ðtÞ ¼

hCiðjÞ ih2iðjÞ þ hCij ih2iðjÞ;F

ðVeff hCiðjÞ ihiðjÞ þ hCij ihiðjÞ;F Þ2

12.21

Likewise, the cross-correlated amplitude becomes: G*ij ðtÞ ¼

hCij ihi;F hj;F Veff ðhCi ihi þ hCij ihi;F Þ  ðhCj ihj þ hCij ihj;F Þ

12.22

To evaluate the impact of FRET on the observed correlation amplitudes, the parameters fi ¼ hi,F/hi and fj ¼ hj,F/hj must be considered. Then the relative autocorrelation amplitudes in the presence and absence of FRET will be:  Cij 2 Cij 1 þ fiðjÞ GiðjÞ;FRET ð0Þ C0 C0 ¼

 C C GiðjÞ;NoFRET ð0Þ ij ij 2 þ fiðjÞ 1 C0 C0

12.23

Similarly, relative cross-correlation amplitudes in the presence and absence of FRET will be: GCij =C0 ;FRET ð0Þ

GCij =C0 ;NoFRET ð0Þ

ff i j ¼  Cij Cij Cij Cij þ fi þ fj 1  1 C0 C0 C0 C0 12.24

where Cij/C0 is the actual relative fraction of crosscorrelating molecules relative to the initial concentration of double-labeled molecules C0. This notation also allows us to correct for time-dependent substrate if C0 represents the initial concentration of labeled substrate molecules in an enzyme-catalyzed reaction. Those seeking a detailed discussion of the theoretical underpinnings of cross-correlation analysis should also consult Schwille, Meyer-Almes and Rigler (1997), Koltermann et al. (1998), and Schwille (2001a). Several papers cited in Section 10.3 provide useful practical advice concerning the design and application of FCS instrumentation for one- and two-photon work.

12.9.4. Proteolytic Cleavage may be Fruitfully Investigated by FCS Kohl et al. (2002) employed recombinant DNA methods to join green fluorescent protein GFP and the red fluorescent protein DsRed by a 32-residue polypeptide linker that possessed a tobacco etch virus (or TEV) protease cleavage site. This substrate allowed them to carry out both FRET and dual-color cross-correlation analysis without the need for extrinsic fluorescent labeling. As illustrated in Fig. 12.31, proteolysis cleaves the substrate into two enzyme bound fragments GFP–P1 and P2–DsRed, and subsequent release from the enzyme results in the concomitant loss of FRET and the cross-correlation signal.

FIGURE 12.31 Schematic representaion of the STEV-ST and the fluorescent protease assay. Left: domain structure of STEV-ST. Composition and linear arrangement of STEV-ST are illustrated (not drawn to scale). The rsGFP and DsRed domains of STEV-ST are joined by a 32-aa protein linker containing the protease recognition sequence from the TEV protease. A C-terminal Strep-Tag II was added for protein purification. The vertical numbers indicate the amino acid position within the protein. Cleavage in the peptide linker region of the reporter construct by TEV protease terminates both FRET and cross-correlation because of the separation of the fluorescent proteins; Right: autocorrelation functions and photon counts per molecule in kHz (Inset) for rsGFP and DsRed were determined in parallel during an incubation of 110 nM STEV-ST with 197 nM rTEV. Changes in fluorescence intensity of the FRET donor (rsGFP) and acceptor (DsRed) were detected immediately after enzyme addition, whereas autocorrelation G(0) values and fluorescent particle numbers remained constant. The monitored increase of rsGFP fluorescence corresponds to approximately 35% FRET in the intact substrate. Reproduced from Kohl et al. (2002) with permission of the authors and the Proceedings of the National Academy of Sciences USA.

Enzyme Kinetics

756

0.03

0.02

0

G(τ)

40 0.01

0 0.001

160

0.01

0.1

1.0

τ (msec)

10

100

FIGURE 12.32 Experimental and fitted cross-correlation curves measured during proteolytic cleavage. The curves were averaged from two successive 20-s measurements each. During the course of the reaction, the amplitudes of G(0) gradually decreased whereas the corresponding diffusion times remained constant, assuring the identity of the substrate. Although the original figure contains many more tracings, those shown here are illustrative of the protease’s time-dependent behavior. Reproduced from Kohl et al. (2002) with permission of the authors and the Proceedings of the National Academy of Sciences USA.

0 min 12 min

0.04 22 min

G (τ)

This fluorescence-based protease assay was examined by FCS on a single-molecule scale. In a reaction mix containing 236 nM rTEV and 100 nM STEV-ST, the protease digest was monitored twice per minute over a 10-min interval by continuously recording photon count rates for rsGFP and DsRed emission; the resulting autocorrelation computations for signals from both fluorescence detection channels was also computed. Despite a large increase in the green fluorescence count rate and a decrease in the red fluorescence red count rate, the corresponding autocorrelation functions remained constant. The average molecular brightness per molecule at any given time is easily determined by multiplying the autocorrelation amplitude by the photon count rate, and FCS allows for very sensitive monitoring of the emission properties in the presence and absence of FRET. As plotted in Fig. 12.32, the photon emission rate per single molecule increases for rsGFP by more than 50% from 4.6 kHz (GFP, with FRET) at the beginning of the enzymatic reaction to 7 kHz (GFP, no FRET) at the end. This increase corresponds to approximately 35% of energy transfer in the intact substrate, which agreed reasonably well with the value of 40% determined from the emission spectra analysis at higher substrate and enzyme concentrations. The consistency of the computed autocorrelation function amplitudes throughout the proteolytic cleavage demonstrated that there was essentially no loss of fluorescent molecules from adsorption or photo-degradation. Cleavage of STEV-ST results in the dissociation of the two fluorophores (Fig. 12.32) and, consequently, a decay of the cross-correlation amplitude. Figure 12.33 shows the time-course of cross-correlation curves recorded over 0, 40,

54 min 95 min

0.02

165 min

0.00 1E-3

0.01

0.1

1

10

100

τ [ms] FIGURE 12.33 Time-dependent loss in the amplitude of crosscorrelation curves during the cleavage of a double-labeled DNA substrate by EcoR1 restriction endonuclease. Labeled DNA (10 nM) and unlabeled DNA (80 nM) were reacted with 1.6 nM of the nuclease, and the observed fluorescence signals were processed in the cross-correlation mode and fitted to Eqn. 12.11. Reproduced from Kettling et al. (1998) with permission of the authors and the Proceedings of the National Academy of Sciences USA.

and 190 s after combining 110 nM STEV-ST and 437 nM rTEV. Green and red fluorescence were measured continuously, and cross-correlation analysis was carried out in intervals of 40 s (low enzyme concentrations) and 20 s (high enzyme concentrations). The decay of cross-correlation amplitudes Gx(0) closely resembles the decrease of the intact substrate concentration Cij. However, in the presence of FRET, there exists no simple linear relationship between Gx(0) and Cij. Rather, the determination of absolute substrate concentrations requires both the autocorrelation amplitudes and fluorescence count rates. Kohl et al. (2002) commented that their approach using an all-protein fluorescent substrate assay for monitoring proteolytic degradation results in an extremely sensitive and selective real-time method for following dynamic processes under physiological conditions. Furthermore, this type of assay has readily adaptable cellular applications. The limits imposed by alternative techniques, such as standard FRET, with regard to the distances between the fluorophores, can be overcome by the use of dual-color cross-correlation, which takes into account the dynamic concomitant movement of dye molecules independent of their size and proximity. The combination of the fluorescent proteins, DsRed and rsGFP, appears to be well suited for simultaneous excitation with TPE. DsRed, despite its slow maturation and tendency to aggregate as a native protein, worked well within the fusion construct being linked to protein fragments at both its N- and C-terminal ends. In conclusion, this study provides proof in principle for the extension of twophoton cross-correlation spectroscopy applications to protein-based reporter molecules in diagnostic screening and cell biological research.

Chapter j 12 Single-Molecule Enzyme Kinetics

757

12.9.5. Endonucleolytic Cleavage has also been Examined by FCS

12.10. ‘‘ZERO-MODE’’ WAVEGUIDES FOR SINGLE-MOLECULE ANALYSIS

In their examination of EcoR1 endonuclease, Kettling et al. (1998) carried out a cross-correlation analysis at different time points during the endonucleolytic cleavage of a 66-base-pair DNA substrate containing a palindromic GAATTC cleavage sequence, and doubly labeled at opposing ends with the fluorescent dyes Rhodamine Green and Cy5 (red). The first probe was excited at 480 nm with an argon-ion laser, and the second was excited at 633 nm with a helium-neon laser. As noted above, dual-color crosscorrelation analysis, unlike traditional FRET analysis, monitors the concomitant movement of two distinct fluorescence reporter groups in a manner that is not subject to ˚ FRET distance limit. Hydrolysis releases the 50- to 90-A two fragments, resulting in the time-dependent decrease in the amplitude of the cross-correlation signal (Fig. 12.34). These investigators determined the reaction rate over a 1–160 nM substrate concentration range, and as shown in Fig. 12.34, the data showed excellent agreement with the Michaelis-Menten equation. The Michaelis constant was 14  1 nM, and maximal velocity was 0.74  0.03 nM/min. Other methods for evaluating kinetic parameters for this endonuclease relied on data obtained over narrow substrate concentration ranges. In this respect, the dual-color FCS experiments allow the experimenter the opportunity to make kinetic measurements over a broad substrate concentration range.

To isolate individual molecules, most single-molecule optical approaches typically require pM-to-nM fluorescently tagged molecules, thereby precluding the investigation of biochemical reactions where the enzyme must be present at concentrations greater than 100 nM. In fact, oligomeric enzymes frequently undergo dilution-dependent dissociation and inactivation, and many proteinnucleic acid interactions are likewise of intermediate affinity. Levene et al. (2003) demonstrated a novel device for analyzing single-molecule enzyme kinetics at high fluorophore concentration. They constructed metal-clad waveguides that exhibit a cut-off wavelength, above which no propagating modes exist inside the waveguide. This cut-off wavelength is related to the size and shape of the guide. Because no propagating modes exist, Levene et al. (2003) referred to these guides as ‘‘zero-mode waveguides.’’ The rapid decay of illumination incident to the entrance of such guides effectively creates zeptoliter sampling volumes (1 zL ¼ 1021 L) within the guide apertures, thus allowing direct observation of chemical reactions. At enzyme concentrations of 10–5 M, these ‘‘zepto cuvettes’’ would on average hold single-enzyme molecules. Their zero-mode waveguides are optically efficient, because the illuminating light is used near the entrance of the guide, before substantial attenuation occurs; by contrast, NSOM uses typically 0.1% of the incident light as a consequence of attenuation through the aperture. Implementation of zero-mode waveguides was achieved using small holes in a metal film deposited on a microscope coverslip (Fig. 12.35). The metal film acts as the cladding, and the contents of the hole compose the core of the waveguide. Millions of such holes can be constructed on a single coverslip, providing numerous observation chambers for massive parallel observation of single-molecule reaction kinetics. For direct observation of single-molecule enzymatic activity, enzymes are adsorbed onto the bottom of the waveguides in the presence of a solution containing the fluorescently tagged ligand/substrate molecules. The coverslip is illuminated through a microscope objective from below, and the fluorescence is collected back through the same objective (Fig. 12.36). In this case, the observed signal consists of fluorescence from the ligand molecule that is in the active site of the enzyme. This signal must be distinguished from a background of freely diffusing fluorescent ligand within the waveguide’s observation volume. The limited dimensions of the observation volume reduce the number of observable diffusing molecules and the correlation time of the resulting fluorescence fluctuations, allowing for facile discrimination of the signal from background.

0.50

0,8

v [nM min-1]

v [nM min-1]

0.75

0.25

0,6 0,4 0,2 0,0 0,000

0.00 0

50

0,015

0,030

v / S [min-1]

100

0,045

150

S [nM] FIGURE 12.34 Michaelis-Menten and Eadie-Hofstee plots for EcoR1 endonuclease. Labeled DNA at a final concentration of 0.8 nM was mixed with different amounts (0–130 nM) of unlabeled DNA and incubated with 160 pM EcoRI in the reaction buffer at 27 C. The reactions were monitored on-line and the initial rates v were derived by linear regression of data points of the first 5–20 min. (Inset) Calculations from an Eadie Hofstee plot lead to a KM value of 14  1 nM and Vmax of 0.74  0.03 nM min1. Reproduced from Kettling et al. (1998) with permission of the authors and the Proceedings of the National Academy of Sciences USA.

12.11. PROSPECTS

Because the optical intensity and the output coupling efficiency at the entrance of the waveguide are close to those for free space (Levene et al., 2003), zero-mode waveguides offer an optically efficient, highly parallel, and relatively simple platform for performing a wide variety of biochemical assays.

The history of enzyme kinetics is punctuated by the invention of new approaches for glimpsing previously unobservable aspects of catalysis. The opportunities presented by singlemolecule kinetic measurements appear to be the ultimate level of observation, allowing researchers to record the trajectories of individual enzyme molecules as they bind, interconvert, and release substrates/products. Despite the appeal of these approaches, it is unlikely that they will approach the currency of fast reaction methods and isotopic probes, much less initial-rate techniques. Of all the singlemolecule kinetic approaches now at the enzyme chemist’s disposal, three are especially promising. Although the potential of fluorescence correlation spectroscopy was recognized some time ago (Magde et al., 1972), more up-todate developments in biophotonics, especially with respect to the two-photon strategies, are propelling the investigation of single-molecule reaction dynamics. One can justifiably anticipate that many insightful FCS experiments on singleenzyme molecules will be conducted in the future. A second highly promising branch of single-molecule enzyme kinetics is concerned with how force is generated in energase-type reactions (see Chapter 12). The third approach is the use of zero-mode waveguides, which in principle should be suitable for high-concentration FCS and cross-correlation experiments. As enzyme chemists carry out more single-molecule experiments, it will always be necessary to distinguish the

Chapter j 12 Single-Molecule Enzyme Kinetics

observed behavior governed by reaction stochastics versus experimental artifacts. There is a clear need for side-by-side comparisons of real-time measurements of single-molecule events with those fixed-time assays of accumulated product formed by single enzyme molecules. A detailed examination of the properties of covalently tethered enzymes may also influence the design of large-scale bioreactors, perhaps providing insight for improving bioreactor throughput, specificity, and stability. Because so many enzymes are oligomers of catalytically active individual subunits, their suitability for use in single-molecule enzyme kinetic experiments is compromised by their likely simultaneous action. One solution is to create hybrid oligomers consisting of only a single catalytically active subunit (shown below in red).

759

chemical equilibria are well understood. One thing is certain: the single-molecule approach is one of the most direct ways for exploring the role of enzyme dynamics in catalysis. It remains to be seen whether the thermodynamics of small systems (Hill and Hammes, 1975) can be extended to deal with conditions imposed in single-molecule kinetic experiments conducted in extremely small volumes. Rapid advances in our ability to observe single-molecule phenomena have already stimulated efforts to develop the theoretical framework for single-molecule kinetics, and further work in this field is likely to provide powerful and persuasive ideas about the origins of enzyme catalytic efficiency.

active subunit tether inactive subunit surface

An advantage of this approach would be that one or more of the catalytically inactive subunits (shown in blue) may be modified by site-directed mutagenesis, or otherwise, to facilitate surface immobilization. The experimenter could thus arrange to have the active site of catalytically active subunit free of direct interactions with the immobilization surface. Finally, our intuition about chemical reactivity is schooled by the detailed examination of rate processes under conditions where the rules governing thermodynamics and

FURTHER READING Selvin, P. R., and Ha, T. (2008). In Single-Molecule Techniques: A Laboratory Manual. Woodbury, New York: Cold Spring Harbor Laboratory Press. Marriott, G., and Parker, I., eds (2003), Biophotonics, Part A (vol 360) and Part B (vol 361), Methods in Enzymology, Academic Press, San Diego. A valuable compilation of best-practice techniques in detection of single-molecule enzyme systems. Bustamante, C. (2008). In singulo Biochemistry: When Less Is More. Annu. Rev. Biochem., 77, 45. Herbert, K. M., Greenleaf, W. J., and Block, S. M. (2008). Single-Molecule Studies of RNA Polymerase: Motoring Along. Annu. Rev. Biochem., 77, 149. Xie, X. S., Choi, P. J., Li, G.-W., et al. (2008). Single-Molecule Approach to Molecular Biology in Living Bacterial Cells. Annu. Rev. Biochem., 37, 417.

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Chapter 13

Mechanoenzymes: Catalysis, Force Generation and Kinetics Force is vital. While many organisms can live without oxygen, none can survive without generating force. And were we to search for life on some far-off planet totally devoid of carbon, we would surely find that, more than any other biotic signature, force generation is the common thread linking all self-perpetuating life forms. Mechanoenzymes power cellular locomotion, proliferation, and irritability – the telltale signs of life. ATP-dependent cytoskeletal motors generate locomotive forces. The actomyosins generate the contractile forces needed for muscle contraction and leading edge retraction, whereas the actoclampins generate the expansile forces for vesicle propulsion and cell crawling. Some ATP-dependent motors (e.g., PICH, or Plk1-interacting checkpoint ‘‘helicase’’) drive chromatin compaction; other motors (e.g., SWI-SNF mechanoenzyme complex) promote nucleosome shifting for transcription factor access; and the microtubule motors (e.g., dyneins, kinesins, and possibly microtubule endtracking motors) power mitosis in a highly choreographed manner. Even viruses and bacteriophages rely on ATPdependent force-generating motors to stuff RNA or DNA into what otherwise would be empty capsids. The discovery of Naþ,Kþ-ATPases and the later demonstration of their role as ion pumps by the Danish Nobelist Jens Skou explained how cells maintain the transmembrane cation gradients (i.e., [Naþ]out > [Naþ]in and [Kþ]in > [Kþ]out) needed for generating and propagating nerve impulses. In fact, all ion pumps generate electrical potentials, corresponding to the work per unit charge needed to generate the desired transmembrane ion balance. Add to that the bioenergetic roles of the ATP synthases and electron transport systems in oxidative phosphorylation and photophosphorylation, and one quickly realizes that the significance of mechanoenzymes to life can never be overstated. As noted in Chapter 1, protein conformational changes are likely to be essential for enzyme catalysis, and we may conjecture that all biological catalysis involves force-gated cues that trigger conformational changes associated with one or more segments along the catalytic trajectory. The present chapter focuses on force-associated conformational changes in mechanoenzymatic (or energase-type) reactions. Central concepts include conformational effects on noncovalent binding interactions, the mechanical energy yield Enzyme Kinetics Copyright Ó 2010, by Elsevier Inc. All rights of reproduction in any form reserved.

of ATP hydrolysis, the nature of force-dependent rate constants, as well as the stall force that prevents motor advance. Also discussed is the pivotally important concept that the action of motors is virtually unidirectionally, even when operating at energies exceeding the thermal energy (1 kBT) by factors of only 10 to 15. We likewise consider the unifying ideas concerning the actions of molecular motors and how they are often subject to large fluctuations in conformational and chemical states. Also considered in this chapter are brief descriptions of some mechanisms, by which energases harness the Gibbs energy of different stages in ATP hydrolysis (e.g., P–O bond-scission, energized conformational change(s), and/or the release of ADP, Pi, or Hþ). These principles and relevant kinetic properties are illustrated by considering several prototypical molecular motors. Additional single-molecule enzyme approaches for analyzing mechanoenzyme kinetics were already considered in Chapter 12.

13.1. BRIEF OVERVIEW OF ENERGASE-TYPE REACTIONS That ATP is the universal store of metabolic energy was popularized by the German-American Nobelist Fritz Lipmann, who first considered phosphoryl-transfer potential as the principle underlying bioenergetics. ATP is continually produced, utilized, and regenerated in living organisms. The main idea is that oxidation of C–H and C–C bonds to water and carbon dioxide is the principal energy source. Inasmuch as all C–H and C–C bond formation is ultimately the product of photosynthesis, the sun is the energy source for nearly all life forms on earth. The approximate chemical free energy captured via photophosphorylation (when attended by non-cyclic electron transport) is approximately 1.25 P–O–P bonds (as ATP) per absorbed photon. ATP formation, hydrolysis and resynthesis are also key to solute transport, anabolic and catabolic phases of metabolism, excitable membranes polarization/depolarization, genetic coding and decoding of information, and nearly all mechanical work achieved by living systems. 761

Enzyme Kinetics

762

Creatine∼P C H + O2 H2O + CO2

METABOLIC DYNAMO

STORE

UTILIZATION

Transport Metabolism

ATP

Bio-electricity

ADP+ Pi

Information Work

Scheme 13.1 Lipmann’s metabolic dynamo (Scheme 13.1) illustrates how oxidative metabolism generates excess ATP that then drives various cell processes (Lipmann, 1941). This diagram depicts oxidative metabolism of C–H and C–C bonds and the ensuing steps in respiration as a hypothetical mechanoenzymatic process that generates ATP. In addition to its role in substrate-level phosphorylation reactions, ATP is hydrolyzed within the active sites of mechanoenzymes to drive the force-generating processes shown on the right. In mammals, excess ATP is stored in the form of creatine-P, which is formed reversibly by creatine kinase (Reaction: MgADP þ Creatine-P # MgATP2 þ Creatine). In this metabolic dynamo, energy generated through metabolic respiration is stored mainly as phosphorylated compounds (ATP, ADP, creatine-P, etc.), and phosphoryl group transfer (e.g., MgATP2 þ X # X–P þ MgADP) is a universal mechanism for retaining group-transfer potential during catalysis. Hydrolysis of the P–O–P bonds liberates the Gibbs free energy needed to drive energy-requiring cellular functions, either directly or indirectly. Some enzymes utilize catalytic mechanisms that chemically couple the hydrolysis of phosphorylated metabolites to biosynthetic reactions, whereas others use the liberated energy to power work-requiring reactions. Little could the biochemists of the 1940s have imagined that the Gibbs energy liberated during ATP hydrolysis energizes more than a thousand different protein-sized molecular machines, each having its own affinity-modulated mechanism to accomplish mechanical work. A major breakthrough was the post-World War II development of biological electron microscopy, which almost immediately revealed the ultrastructure of striated muscle and biomembranes. Such information, coupled with the discovery of myosin’s intrinsic ATPase activity (Engelhardt and Lyubinova, 1939) led to the actomyosin cross-bridge cycle model (Huxley, 1954), and Skou (1957) discovered the sodium, potassium ATPase responsible for ‘‘bioelectricity.’’ Others soon discovered the proteins that formed the membrane-associated motors responsible for active transport of various other ions and solutes, and a theme emerged that transport is most often coupled to an energy-yielding reaction ATP hydrolysis. The subsequent discovery of solute gradient-driven counter-transport explained how the processes (Reaction: Xout þ Yin # Xin þ Yout) that appeared to be independent of ATP hydrolysis ultimately

required the latter in a gradient-forming process (Reaction: Yout þ ATP þ H2O # Yin þ ADP þ Pi þ Hþ). Likewise, when a protein kinase-mediated phosphorylation results in a conformational change, that process too is intrinsically mechanochemical! We now know that Lipmann was absolutely correct about ATP’s pivotal role in cellular metabolism, as were his ideas about the vital importance of P–O bond cleavage in driving otherwise unfavorable chemical reactions. Importantly, a substantial fraction of the ATP generated by oxidative phosphorylation and photophosphorylation is not needed to drive the biosynthetic reactions. It is instead needed to power the mechanochemical reactions responsible for ion/ solute transport, regulation of intermediary metabolism, bioelectricity, information storage and transfer, and mechanical work. Some of these mechanoenzymatic reactions are listed in Table 13.1. Drawing on the architectural concept of tensegrity that defines the push/pull force balance within geodesic domes, Ingber (1997, 1998) advanced the concept of cellular tensegrity to explain how cells might convert mechanical forces of gravity, hemodynamic shear, and motility into chemical responses and vice versa. He predicted that regulatory processes within cells are hard-wired for immediate response to mechanical stresses transmitted from the extracellular matrix via cell surface receptors that are coupled to the internal cytoskeleton. Mechanical signals are integrated with other environmental signals and then transduced into a biochemical response through forcedependent changes in the molecular and/or mechanical properties of membranocytoskeletal processes. His ideas are compatible with the assertion here that cytoskeleton is responsible for dynamic cellular entasis, meaning that, by means of surface tethering, cytoskeletal filaments and their ATP-dependent motors are inherently able to generate both contractile and/or expansile forces, allowing cells to attain and sustain dynamic tension. Likewise, literally hundreds of mechanoenzymes (or energases) are continually grasping, twisting, rotating, stretching, and/or translocating other subcellular components, rearranging their structures and interactions through pushing, pulling, twisting, and writhing forces. Among the cellular components that are targets of energase-mediated structural rearrangement or translocation are countless macromolecular structures, including membranes, vesicles, chromatin, and various unfolded and folded proteins – even those elongating polypeptide chains just emerging from ribosomes. Indeed, the ribosome is the quintessential mechanochemical device that exploits force for the vectorial synthesis and disposition of nascent polypeptides. As previously discussed in Section 1.4.3, most mechanoenzymatic reactions are currently misclassified as energydissipating ATPases (Reaction: ATP þ H2O # ADP þ Pi þ Hþ) and GTPases (Reaction: GTP þ H2O # GDP þ Pi þ Hþ). Recognizing mechanoenzymes transduce chemical

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

763

TABLE 13.1 A Survey of Energase Functions and Reactions Catalyzed Cellular Process

Reaction

Bioenergetics Ca2þ-Transporting ATPase (Jencks, 1980)

Ca2þin þ ATP # Ca2þout þ ADP þ Pu

F1Fo ATP Synthase (Boyer and Kohlbrenner, 1981)

GradientState-1 þ ATP # GradientState-2 þ ADP þ Pi

CF1CFo ATP Synthase (Penefsky, 1974)

GradientState-1 þ ATP # GradientState-2 þ ADP þ Pi

Transport PEP:Protein Phosphotransferase (Postma and Roseman, 1976)

Hexoseout þ 2-P-enolpyruvate # Hexose-6-Pin þ Pyruvate

Hþ/Kþ-Exchanging ATPase (Sachs et al., 1968; 1992)

Protoninþ Kþout þ ATP # Protonout þ Kþin þ ADP þ Pi

Steroid-Transporting ATPase (Nagao et al., 1995)

Steroidinþ ATP # Steroidout þ ADP þ Pi

Vitamin B12-Transporting ATPase (Friedrich, de Veaux and Kadner, 1986)

Cobalaminout þ ATP # Cobalaminin þ ADP þ Pi

Xenobiotic-Transporting ATPase (Loe, Deeley and Cole, 1998)

Druginþ ATP # Drugout þ ADP þ Pi

Intermediary Metabolism Endopeptidase La (Goldberg, 1992; Goldberg et al., 1994)

Bound Polypeptide þ 2ATP # Released Fragments þ 2ADP þ 2Pi

GTP-Regulatory Proteins (Rodbell, 1992)

Interaction State1 þ GTP # Interaction State2 þ GDP þ Pi

Proteasome ‘‘ATPase’’ (Rivett et al., 1997; Mason et al., 1998)

Bound Polypeptide þ ATP # Released Fragments þ ADP þ Pi

Enzyme Catalysis Nicotinate Phosphoribosyl-transferase (Gross, Rajavel and Grubmeyer, 1998)

Enz-Activity-State1 þ ATP # Enz-Activity State2 þ ADP þ Pi

Sulfate Adenylyltransferase (Peck, 1974; Gavel et al., 1998)

Enz-Activity State1 þ ATP # Enzyme Activity State2 þ ADP þ Pi

Bio-Electricity Choline-Transporting ‘‘ATPase’’ (Saier, 2000)

Cholineout þ ATP þ H2O # Cholinein þ ADP þ Pi

Naþ-Exchanging ATPase (or Na/K-ATPase) (Skou, 1957; 1960; Anner, 1985)

Naþin þ Kþout þ ATP # Naþout þ Kþin þ ADP þ Pi

DNA Remodeling Deoxyribonuclease ATPase (site-specific) DNA Topoisomerases ‘‘ATPase’’ (Wasserman and Cozzarelli, 1986; Wang, 1998)

DNA þ ATP # Fragmented DNA þ ADP þ Pi

DNA-Winding-State1 þ ATP # DNA Winding-State2 þ ADP þ Pi

Exodeoxyribonuclease V (Muskavitch and Linn, 1981; Lehman, 1971)

DNA þ ATP # Fragmented DNA þ ADP þ Pi

Nucleoplasmin ‘‘ATPase’’ (Ito et al., 1996; Laskey et al., 1993)

Compaction-State1 þ GTP # Compaction-State2 þ GDP þ Pi

Protein Biosynthesis/Folding Initiation Factors (Kurzchalia et al., 1984)

Interaction-State1 þ GTP # InteractionState2 þ GDP þ Pi

Elongation Factors (Rodnina et al., 1997; Clark and Nyborg, 1997)

Position1 þ GTP # Position2 þ GDP þ Pi

Chaperonin ‘‘ATPases’’ (Hemmingsen et al., 1988)

Folding-State1 þ ATP # Folding-State2 þ ADP þ Pi

(Continued)

Enzyme Kinetics

764

TABLE 13.1 A Survey of Energase Functions and Reactions Catalyzed – cont’d Cellular Process

Reaction

Cell Motility, Traffic and Division Actoclampin Motor (Dickinson and Purich, 2002)

Clamp-locked State1 þ ATP # Clamp-released State2 þ ADP þ Pi

Actomyosin Motor (Lymn and Taylor, 1971)

Contractile-State1 þ ATP # Contractile-State2 þ ADP þ Pi

Dynein Motor (Gibbons, 1988; Khan and Sheetz, 1997)

MT-Position1 þ ATP # MT-Position2 þ ADP þ Pi

Kinesin Motor (Khan and Sheetz, 1997; McIntosh and Porter, 1989; Sablin et al., 1998)

MT-Position1 þ ATP # MT-Position2 þ ADP þ Pi

Membrane Organelle Formation Dynamin ‘‘GTPase’’ (Warnock and Schmid, 1996; McClure and Robinson, 1996)

Membrane-Fission-State1 þ GTP # Fission-State2 þ GDP þ Pi

Peroxisome-Assembly ‘‘ATPase’’ (Lee and Wickner, 1992; Tsukamoto et al., 1995)

Membrane-State1 þ ATP # Membrane-State2 þ ADP þ Pi

Phospholipid-Flipping ATPase (Morris et al., 1993; Vermeulen, Briede and Rolofsen, 1996)

Phospholipid-State1 þ GTP # Phospholipid-State2 þ GDP þ Pi

Chloroplast Protein ATPase (Cline, Ettinger and Theg, 1992)

Pre-Proteinout þ ATP # Pre-Proteinin þ ADP þ Pi

bond energy into mechanical work and not heat, Purich (2001) asserted: (a) that enzyme classification based solely in terms of covalent-bond chemistry ignores the essential substrate- and product-like conformational states of mechanoenzymes; (b) that the Enzyme Commission and its successor, the IUPAC-IUBMB Joint Commission on Biochemical Nomenclature (JCBN), break their own rule against naming enzymes on the basis of partial reactions and have thereby perpetuated such pedagogically indefensible terms as ‘‘ATPase’’ and ‘‘GTPase’’; (c) that such errors can be cured by building on Pauling’s assertion that stable, longlived chemically distinct interactions may be regarded as chemical bonds,1 allowing the more encompassing definition of an enzyme as a catalyst for making and/or breaking chemical bonds, whether covalent and noncovalent; (d) that, unlike covalent bonds with bond energies on the order of

400 kJ/mol, the DG for transformation of noncovalent substrate and product states is typically 20 to 30 kJ/mol, low enough that ATP hydrolysis (DG ¼ 40 kJ/mol) can readily drive mechanoenzymatic reactions; (e) that, if anything, mechanoenzymatic reactions (e.g., NoncovalentState1 þ ATP þ H2O # ADP þ Pi þ Noncovalent-State2 þ Hþ) more closely resemble synthases, because both mechanistically link ATP hydrolysis to chemical bond transformations (e.g., compare Keq ¼ [ADP][Pi][State2]/ [ATP][State1] to Keq ¼ [Gln][ADP][Pi]/[ATP][Glu][NHþ 4 ], the latter corresponding to the equilibrium constant for the glutamine synthetase reaction); (f) that the thousands of mechanochemical reactions constitute a unique seventh class of enzyme-catalyzed reactions; and (g) that the name energase emphasizes their central role in transducing chemical and mechanical energy.2 Purich (2001) also

1

2

In his classical book, The Nature of the Chemical Bond, Linus Pauling stated: ‘‘We shall say that there is a chemical bond between two atoms or groups of atoms in case that the forces acting between them are such as to lead to the formation of an aggregate with sufficient stability to make it convenient for the chemist to consider it as an independent molecular species.’’ His prescient use of the words ‘‘sufficient stability’’ allows one to consider long-lived noncovalent interactions in much the same way as we treat coordinate covalent bonds present in many transition metal ion complexes with reversibly bound ligands. Reversible protein-ligand and protein-protein complexes are also of sufficient stability to be considered an independent molecular species, and the same may be said of solutes and ions separated by an impermeable membrane (e.g., Solutein and Soluteout).

In proposing the energase classification, Purich (2001) took issue with the suggestion of Tanford (1983) that ‘‘ATPases’’ may be treated as two groups: those using the DG of ATP hydrolysis to drive molecular discrimination, and those catalyzing biosynthetic reactions. Taking glutamine synthetase as an example, the extent of enzyme-bound g-glutamyl-P formation (Todhunter & Purich, 1975) indicates the internal equilibrium (Enz$ATP$Glutamate Enz$ADP$g-glutamyl-P) is w10,000x more favorable than solution-phase acyl-P formation. Such findings demonstrate unambiguously that biosynthetic enzymes also exploit molecular discrimination. While Tanford also suggested that only biosynthetic ‘‘ATPases’’ form phosphorylated intermediates, it is also clear that mechanoenzymes (e.g., SR calcium pump and PEP phosphotransferase) are transiently phosphorylated during catalysis.

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

argued that low-activity ‘‘ATPases’’ result from uncoupling, the result of some nonphysiologic treatment that disrupts the linkage between mechanical and chemical steps, resulting in futile energy dissipation. For example, in the absence of its actin filament partner, myosin is a feeble ATPase that produces heat and no work; however, the complete actomyosin molecular motor, consisting of myosin and its actin filament partner, hydrolyzes ATP at a 60–200 greater rate and in a manner coupled to myosin’s advance on the filament. Likewise, treatment of mitochondria with nitrophenol disrupts physiologic coupling of Mitchell’s chemiosmotic proton gradient and Boyer’s ATP synthase, unbridling the latter’s latent ‘‘ATPase’’ activity. The advent of laser tweezers techniques, atomic force microscopy, deformable cell culture substrata, and a host of other force-sensing/measuring techniques (see Chapter 12) provided the capacity to quantify the forces generated by cytoskeletal motors as well as those needed to disrupt noncovalent bonding interactions. Some examples are listed in Fig. 13.1. In most cases, ATP and GTP are the immediate sources of P–O–P bond energy, with other systems utilizing acetyl-P, P-enoyl-pyruvate, and ion gradients. However, because phosphoryl group transfer from ATP drives the biosynthesis of GTP and other phosphorylated metabolites, nearly all energase-type reactions are driven, either directly or indirectly, by the Gibbs energy of ATP hydrolysis (Purich, 2001). Moreover, ATP is itself produced primarily by energase-type reactions in photophosphorylation and/or oxidative phosphorylation, involving ATP synthases coupled to proton gradients generated by electron-transport chain mechanoenzymes lodged within the membranes of photoplasts and mitochondria. Figure 13.1 also emphasizes the nexus of enzyme catalysis and force generation and further suggests that there are numerous instances where force is likely to alter the kinetic properties of enzymes. Living systems rely on a variety of affinity-modulated binding interactions that are driven by the free energy derived from the hydrolysis of nucleoside 59-triphosphate, acetylphosphate, phosphoenolpyruvate, etc. Representative reactions of active transporters and GTP-regulatory proteins may be written as follows: ATP + H2O Solutein

G-Protein Target (Substrate-like State)

1pN

10 actomyosin MT-dynein MT-kinesin actoclampin

100

bacterial flagellum λ-phage DNA B–S transition

Ig/Fn β-domain disassembly biotinavidin binding

HIV RTase

protein domain movements

10

leading edge motility

Listeria rocket-tail propulsion actin filament tensile strength

29 Phage DNA packing motor

1pN

cell-tosubstratum linkages

antibodyantigen interactions

ATP synthase

conjectural Hill-type Brownian ratchet in actin motility

1000

protein adsorbed onto glass

100

1000

FIGURE 13.1 Force-dependent reactions and cellular processes and the forces they generate or require. The four cytoskeletal molecular motors (box at upper left) operate by harnessing the Gibbs free energy of ATP hydrolysis to generate an equilibrium force F, equal to DGhydrolysis/d, where d is the operating distance, or step size. Minimal step sizes are 5.4 and 8 nm, corresponding to the respective monomer repeat distance of actin filaments and microtubules (MTs). Muscle actomyosins appear to employ a larger step size (~11 nm), putting the maximal force generated at 1.5–3 pN (Guilford et al., 1997). Under reversible work conditions and prompt ATP hydrolysis, actoclampin molecular motors would be expected to exhibit stall forces of ~8–9 pN over a 5.4-nm step. ATP hydrolysis-dependent MT-dynein and MT-kinesin molecular motors would likewise be expected to have stall forces of ~5–6 pN. Because irreversible work involves kinetic effects, the realistic stall forces are likely to be less.

ATP + H2O Myosin Filament (Filament Position-1)

ADP + Pi Myosin Filament (Filament Position-2)

13.3

ADP + Pi Soluteout

13.1 ATP + H2O

765

ADP + Pi

Because all of the above reactions are characterized by noncovalent substrate-like or product-like states, they may be represented by the following generic reaction: ATP + H2O Noncovalent Substrate-like State

G-Protein Target (Product-like State)

13.2

ADP + Pi Noncovalent Product-like State

13.4

Enzyme Kinetics

766

In its simplest form, we may write the following: ATP + H2O

ADP + Pi

StateS

StateP

13.5 where ‘‘S’’ and ‘‘P’’ are substrate- and product-like noncovalent bonding interactions. Standard enzymes (i.e., those catalyzing the covalent bond transformations of intermediary metabolism) also rely on their conformational flexibility to catalyze covalent bondmaking-breaking (see Section 1.5.10 and Fig. 1.8), whereas mechanoenzymes harness energy derived from making/ breaking of covalent bonds to drive changes in conformation to generate force. This mechanoenzymatic behavior is shown in Scheme 13.1, where conformation-coordinate is on the horizontal axis, and the covalent chemistry coordinate on the vertical axis. Conformational Coordinate Covalent Chemistry Coordinate

E1 S

E2 S

En

Ei S

S

ES1

ES2

ESi

EnS

E1P

E2P

EiP

EnP

P

P

E1

E2

P Ei

P En

Scheme 13.2 The vertically arranged reaction steps in each column of Scheme 13.2 correspond to an enzyme operating by a steady-state one-substrate (Briggs-Haldane-type) scheme, whereas the horizontally arranged transitions indicate conformational states that the enzyme adopts as it progresses along the enzyme’s conformational coordinate. As is required for any scheme that is a composite of adjoining thermodynamic cycles, the net free energy change around any cycle must be zero (see Section 3.5.7). Not all pathways are of equivalent potential energy or catalytic efficiency, and the enzyme is thought to meander within this array of protein conformations and catalytic-cycle stages, such that it takes the most efficient route for catalysis. That route corresponds to the path of lowest energy. The threedimensional reaction coordinate diagram representing Scheme 13.1 is shown in Fig. 13.2, with the conformationcoordinate on the x-axis, DG on the y-axis, with the covalent chemistry coordinate on the z-axis. Because each ATP- or GTP-dependent energase has its own intrinsic rate constant for nucleotide hydrolysis, and because the force-dependent structural rearrangement of each energase target often

FIGURE 13.2 Three-dimensional reaction coordinate diagram for mechanoenzymes. These enzymes harness the Gibbs free energy of ATP or GTP hydrolysis to drive force-generating conformational changes indicated on the horizontal axis. A hypothetical reaction trajectory is indicated by the filled red circles and black line. Notice also that the roughened surface in this three-dimensional potential energy diagram indicates that the actual reaction trajectory depends both on the energetics of the mechanoenzyme conformational changes as well as the energetics of the hydrolysis reaction.

depends on noncovalent interactions, the duration of a particular affinity-modulated binding interaction can be fine-tuned for optimal effectiveness.

13.2. THE DRIVING FORCE FOR AFFINITYMODULATED MOLECULAR MOTORS A system is said to be at thermodynamic equilibrium when its energy equals the thermal noise, the latter corresponding to RT on a per-mole basis or to 1 kBT on a permolecule basis. The same system is considerably far from equilibrium when its energy is >10 RT/mol or >10 kBT/ molecule. Under most circumstances, living organisms operate in the reversible-work range of ~3 RT/mol (or ~3 kBT/molecule), where most processes are still reversible, or nearly so. The biotic trick is for molecular motors to operate efficiently, while maintaining good order. The second law of thermodynamics dictates that the energy associated with thermal noise cannot be harnessed to perform work. So, most molecular motors are instead driven by the DG of ATP or GTP hydrolysis, corresponding to 14–15 kBT/molecule under physiologic conditions (e.g., 3–10 mM ATP, ~0.5 mM ADP, 3–5 mM Pi, pH ¼ 7.2, and pMg2þ ¼ 2–3). An important difference between molecular motors and the macro-scale motors is that molecular motors are optimized to operate in a thermal bath, a microscopic environment wherein thermal noise is significant relative to the motor’s energy consumption (Gabry, Pesz and Bartkiewicz, 2004). Although operating at only a modest distance from thermal equilibrium, molecular motors successfully manage noise and space-time asymmetry to generate the forces

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

needed for such biologically useful functions such as cellular motility, nerve sensation, solute and organelle transport, signal transduction, as well as the separation/ segregation of subcellular particles. All energase-type processes therefore generate force by coupling noncovalent protein-protein or protein-ligand binding interactions to critically important protein conformational changes that are most often driven by the hydrolytic cleavage of thermodynamically unstable anhydride bonds of ATP, GTP, acetylphosphate, and phospho-enolpyruvate. Any mechanochemical reaction may be parsed into the mechanical reaction component (e.g., StateS # StateP) and the ATP hydrolysis component (e.g., ATP þ H2O # ADP þ Pi), and both are factors in the overall equilibrium constant: StateProduct ½ADP½Pi   13.6 Keq ¼ StateSubstrate ½ATP The overall equilibrium constant Keq may be parsed into two terms: Kmech ¼ [StateP]eq/[StateS]eq, and Khyd ¼ [ADP]eq[Pi]eq/[ATP]eq, such that Keq ¼ Kmech  Khyd. Likewise, DG9rxn ¼ RT ln Keq ¼ RT ln{Kmech  Khyd} ¼ DG9mech þ DG9hyd. It is DG9hyd that indicates the Gibbs energy available to drive the mechanical reaction component (i.e., StateS # StateP). As described in Section 3.11.5, DG9hyd using the disequilibrium ratio (r ¼ G/Keq), where G is the mass action ratio [StateP]obs[ADP]obs[Pi]obs/ [StateS]obs[ATP]obs, can be estimated. Note further that r ¼ rmech  rhyd. The key factor rhyd depends on Ghyd ¼ [ADP]obs[Pi]obs/[ATP]obs, and KATPase ¼ [SADP][SPi]/ [SATP], the latter having an approximate value of 3  105 M. Each summation is made for the concentrations of all ‘‘isomeric’’ molecular species, and, in the case of an Mg2þdependent mechanoenzyme, the relevant summation for ATP would be: [SATP] ¼ [MgATP2] þ [MgHATP1] þ [KATP3] þ [NaATP3] þ [ATP4] þ [HATP3], which is usually taken at pH ¼ 7, pMg2þ ¼ 2, and physiologic levels of sodium and potassium ions. For a Ca2þ-dependent process, the relevant summation for ATP would be: [SATP] ¼ [CaATP2] þ [CaHATP1] þ [KATP3] þ [NaATP3] þ [ATP4] þ [HATP3], which is usually taken at pH ¼ 7,

767

pCa2þ z 5.5, and physiologic levels of sodium and potassium ions. The values for [SADP] and [SPi] can be similarly defined. In practice, as long as substantial concentrations of other metal ions are not present, the values for [SATP], [SADP], and [SPi] are approximated by [ATP]total, [ADP]total, and [Pi]total, respectively. Various G values for MgATP hydrolysis and their corresponding DG9hyd values are presented in Table 13.2. (Given the pivotal importance of mitochondrial ATP synthase as well as ATP-dependent cytoplasmic processes, the mitochondrial value of G9/K9eq and DG9hyd is also provided.) The DG9hyd values in Table 13.3 specify the quantity of reversible work that can be accomplished by a system whose mechanical and chemical processes are tightly coupled. The simplest definition of work w is that amount of energy required to displace an object over a distance Dx while experiencing a constant force F. Although biochemists typically express energy in units of kcal/mol and kJ/mol, each molecular motor harnesses the energy of individual ATP molecules, one at a time. In a sense, molecular motors are quantized, meaning that the cleavage of a single P–O–P bond in ATP releases a packet of energy equal to approximately 38 kJoules/N0, where N0 is Avogadro’s number. Because one Joule equals one Newton$meter, a simple calculation puts the per-molecule, reversible-work equivalent of ATP hydrolysis at ~100 pN-nm. To achieve directed motion, a molecular motor transduces chemical energy into mechanical energy by relying on protein conformational change to generate the forces needed to drive a mechanochemical process. For each molecular motor, there is an energetically unfavorable (or ‘‘uphill’’) phase of the catalytic reaction cycle, involving a protein conformational change that must be coupled mechanically to a downhill-phase driven by the Gibbs free energy released upon hydrolysis of nucleoside 59-triphosphate, most often ATP or GTP. In other words, the DG associated with NTP binding, phosphoanhydride bond hydrolysis, and/or dissociation of NDP and/or Pi is the ‘‘downhill’’ phase which can, in principle, be harnessed to drive mechanoenzyme catalysis, with a concomitant generation of force. A typical

TABLE 13.2 Dependence of the DG for ATP Hydrolysis Under Various Conditions P P G9 ¼ [ ADP][Pi]/[ ATP]a 5

K9/G9

[ATP]/[ADP]

10 M

1

0

0.0000001

103 M

102

0.00001

1.0 M

105

11 34

0.1

57

1000

M

106

103 M

108

105 M

1010

10

a

DG (kJ/mol)

1

Calculated for Pi ¼ 10 mM.

28 46

Relevant Condition Thermodynamic equilibrium

0.01 10

Mitochondrial matrix Cytoplasm

Enzyme Kinetics

768

TABLE 13.3 Dependence of Reversible Work (Expressed as Force Exerted Over a Distance) on the DG for ATP Hydrolysis Under Various Conditions at 37 C DG kJ/mol 0

DG/kBT

pN exerted over 0.1 nm

0

0

pN exerted over 1 nm

pN exerted over 5.4 nma

pN exerted over 8 nmb

pN exerted over 10 nm

0

0

0

0

~4

183

18.3

3.4

2.3

1.83

28

~11

465

46.5

8.6

5.8

4.65

34

~13

564

56.4

10.4

7.1

5.64

46

~18

764

76.4

14.1

9.5

7.64

57

~22

946

94.6

17.5

11.8

9.46

11

a

Longitudinal monomer repeat distance in an actin filament. Longitudinal dimer repeat distance in a microtubule protofilament.

b

molecular motor consuming ~100 ATP molecules/sec operates at ~1017 watts (i.e., 1 ATP/s yields ~100 pN$nm/s; 100 ATP/s yields ~104 pN$nm/s, or ~104  1012  109 N$m/s ¼ ~1017 N$m/s). This power of a molecular motor may be compared to a thermal-noise power of ~108 watts continually ‘‘washing’’ over a molecular motor in aqueous medium. The 108 W value represents a motor exchanging ~4  1021 joules (i.e., kB  300 K O 1013 s; 1.38  1023 J$K1  300 K O 1013 s; ~4  108 J/s) with its environment over a relaxation time of 1013 s, the latter corresponding to the shortest time for heat dissipation (i.e., the time-scale of vibrational stretching of a chemical bond). Amazingly, the power associated with thermal energy is therefore some 108–109 times the power derived from ATP hydrolysis for directed motion of a molecular motor. Thermal noise is so large that nanoscale motions of a protein are best described as a random walk, and this may be compared to riding a bicycle in a hurricane. The viscous drag on a molecular motor is also so large that a motor protein immediately attains mechanical equilibrium. Moreover, within the low-Reynolds number regime in which the system operates, viscous force effects for molecular motors are much greater than any effect of inertial forces. Therefore, molecular motors do not glide inertially; instead, the resulting dynamics are such that motor velocity is proportional to force. Despite such limitations, molecular motors are highly efficient in managing their affinity-modulated interactions with their respective binding partner (e.g., myosin binding to an actin filament). In most cases, a regular array of nearby binding sites helps to assure motor processivity, even in the face of high thermal noise. The energy comes from chemical reaction (P–O bond hydrolysis), but any required momentum is borrowed from the thermal bath. Each mechanoenzyme possesses its own molecular discrimination mechanism comprised of a succession of conformational transitions between affinity-modulated interaction states between its initial, substrate-like ‘‘S’’ state and final,

product-like ‘‘P’’ state. For the example given in Scheme 13.3, the term ‘‘Condition’’ designates a position, conformation, solute concentration, or component abundance, and the asterisk indicates its conformationally energized state (i.e., the stored conformational energy that can be harnessed during phosphate release). Conditioninitial + Enz·ATP (Conditioninitial)·Enz·ATP + H2O (Conditioninitial)*·Enz·ADP·Pi (Conditionfinal)·Enz·ADP + ATP Conditioninitial + ATP + H2O

(Conditioninitial)·Enz·ATP (Conditioninitial)*·Enz·ADP·Pi + H+ (Conditionfinal)·Enz·ADP + Pi Conditionfinal + Enz·ATP + ADP Conditionfinal + ADP + Pi + H+

Scheme 13.3 There are, of course, many other possibilities, among them: (a) mechanisms wherein ATP binding triggers the key conformational transition; (b) schemes wherein ADP release drives the work step; and (c) the intriguing possibility that the proton liberated upon ATP hydrolysis is temporarily stored within an energized hydrogen-bond network. The operant mechanism can only be divulged through detailed kinetic analysis of the ATP hydrolysis mechanism and attendant conformational changes. In some cases, once freed from its mechanoenzyme partner, the final conformational state (Conditionfinal) of the macromolecular target is stable. An example is the position of a dynein molecule advancing on the surface of a microtubule. In other cases, once freed from its mechanoenzyme partner, Conditionfinal may quickly isomerize from Conditioninitial to Conditionfinal. Examples of the latter are the macromolecular targets of most G-proteins. Although the b,g P–O–P bond in ATP and GTP are the primary source for the Gibbs energy in most energase reactions, hydrolysis of P–O–P bonds in acetyl-P, phosphoenol-pyruvate, and pyrophosphate also serve to drive other

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

motors (Table 13.4). Consider, for example, P-enolpyruvatedependent glucose phosphotransferase (PTS) system from Escherichia coli. Its P-enolpyruvate-dependent phosphotransferase (EC 2.7.3.9) catalyzes the reaction of P-enolpyruvate with an histidyl residue in a 9-kDa heatstable protein (HPr) to produce pyruvate and phosphorylHPr. Protein-Np phosphohistidine:sugar phosphotransferase (EC 2.7.1.69), the second component of this phosphotransferase system, completes the transport process by phosphorylating hexose. Therefore, the Gibbs energy of P–O bond scission in P-enolpyruvate drives PTS mediated hexose transport (Reaction: Hexoseout þ 2-P-enolpyruvate # Hexose-6-Pin þ Pyruvate). Finally, the very substantial change in the Gibbs energy for NADH (or NADPH) oxidation drives the mitochondrial electron transport system, which is a transmembrane Hþ gradient-forming energase. These reactions are attended by the net phosphorylation of three molecules of ADP per NADH molecule. In this respect, any energase directly powered by NAD(P)H oxidation would be exceedingly powerful on a per-molecule basis. Many ATP- and GTP-hydrolyzing mechanoenzymes are oligomers of structurally related subunits that form short arc- or ring-structures that writhe and/or contract by means of bond rearrangements among neighboring subunits. Aside from generating force upon hydrolysis of the b-g P–O bond,

TABLE 13.4 Energy Source for Various AffinityModulated Mechanoenzymes Acting as Clamps, Latches and Switches Reaction

Energy Source

Interaction Partner

Active Transporters

ATP PEP

Ligand on low-concentration side of gradient

ATP-forming

pH

abg subunit Interface

pH Gradientforming

ATP

abg subunit Interface

ATPase Futile Cycle

ATP

None

ATP Synthase

Myosin

ATP

Actin filament

Electron Transport Chain

emf

Proton on high pH side of gradient

Dynein and Kinesin

ATP

Microtubule

GTP-Regulatory Proteins

GTP

Target protein

DNA Topoisomerase

ATP

Dimer interface

ATP Sulfurylase

GTP

Active site

Actoclampin

ATP

Actin filament end-tracking motor protein

Abbreviations: PEP = P-enol-pyruvate; pH = pH Gradient; and emf = electromotive force.

769

TABLE 13.5 Other Energy Sources for Energases Metabolite Acetyl-phosphate Phospho-enol-pyruvate Pyrophosphate NADH

DGo Hydrolysis/Oxidation 43.1 kJoule/mol

62.0 kJoule/mol

33.5 kJoule/mol

218 kJoule/mol

these mechanoenzymes possess an oligomer-organizing mechanism that coordinates force generation and subunit rearrangements. Other energase reactions operate in the complete absence of P–O bond cleavage. For example, ketoacid decarboxylation provides the substantial energy needed to drive certain sodium ion transporters (Reaction: Oxaloacetate þ Naþin # Naþout þ Pyruvate þ CO2). Indeed, Dimroth (1981) first demonstrated that the Naþ-pumping oxaloacetate decarboxylase of Klebsiella aerogenes could be reconstituted upon incorporation into lecithin liposome membranes using the detergent dilution method with octylglucoside. A steady-state [Naþ]internal/[Naþ]external ratio of ~30 could be established by oxaloacetate decarboxylation. Oxaloacetate-dependent Naþ transport was almost completely abolished by the Naþ ionophores nigericin or trinactin, but the uncoupler carbonylcyanide-p-trifluoromethoxyphenylhydrazone (CCCP) was without effect. Dimroth’s work demonstrated that oxaloacetate decarboxylase functions as a primary active Naþ pump by converting the energy of the decarboxylation reaction into an Naþ gradient. Finally, proton-translocating electron transport systems are specialized redox energase reactions, wherein electron transfer per se drives a protein conformational change that alters the pKa value(s) of one or more acid/base groups, resulting in transmembrane proton extrusion. NADH, the main metabolic reductant formed from the oxidation of foodstuffs, initiates the process by reducing electron transport proteins (ETP) situated in the inner mitochondrial membrane. Before culminating in the reduction of molecular oxygen, electron transport via ETP Complexes I, III, and IV results in the accumulation of protons within the intermembrane space. Exactly how the electron-motive force of electrochemical gradient generates the proton-motive force remains an area of intense investigation. The current view is that electron transport drives a conformational change that lowers the pKa values of ETP carboxyl groups that upon exposure to the inter-membrane space release protons. When viewed in this manner, the multi-protein electron transport complex is a highly specialized affinity modulated (or molecular discrimination) mechanism. The resulting proton gradient is itself an important source of Gibbs free energy for ATP synthesis (see Section 13.8).

Enzyme Kinetics

770

The affinity-modulated binding partners of energases are mainly proteins, nucleic acids, and membranes, although low-molecular-weight solutes are handled by pumps and transporters. All energases hydrolyze ATP at some point in their catalytic reaction cycle, but not necessarily during the force-generating step. The reaction cycle as an initial binding step can be envisaged (e.g., E þ S # E$S, where E is the energase and S is its affinity-modulated ligand), followed by a series of conformational transitions and product release (e.g., E þ S # (E$S1) # (E$S2) # $$$ # (E$Sn) # (E$Pn) # $$$ # (E$P2) # (E$P1) # E þ P), of which one step is energetically linked to ATP hydrolysis. (In fact, mechanoenzymes operate by bisubstrate kinetic mechanisms, wherein the target ligand and ATP may react much the same way as enzymes that catalyze covalent bond transformations using two substrates.) A detailed understanding of the nature of the attendant conformational changes in mechanoenzyme and its target protein or ligand remains of paramount importance in understanding the mechanisms of energase action. Because these mechanoenzymes have structures that react with and may accommodate mechanical forces in complicated ways, it is relevant to consider the qualitative features of force-induced bond rupture of single noncovalent bonds and ensembles involving only several noncovalent bonds.

13.3.1. Bond Energetics may be Described as Potential Energy Functions To appreciate the behavior of force effects on bond stability and rupture, we first consider a single bonding interaction between two atoms joined together by covalent bonds. Depending on the nature of the atoms connected by a bond and the number of electrons linking them together, a bond is constantly stretching and/or rotating about some equilibrium bond length or angle. Covalent bonds typically exhibit a potential energy well (Fig. 13.3) that may be defined by potential energy functions. One such function is the Lennard-Jones potential, which has the following form:    s 12 s 6 13.7  UðrÞ ¼ 43 r r Here, the interaction energy is the maximal energy of attraction (i.e., corresponding to the depth of the energy well), and s is one of the adjustable parameters with the same units as r. The r12 term is known as the Pauli repulsion term, observed at very short distances due to unfavorable overlapping of electron orbitals, and the r6 term defines the attractive interaction observed over longer

distances. Unfortunately, there are no corresponding potential energy functions for noncovalent bonding interactions. Smoluchowski (1916) showed that at steady-state a particle undergoing Brownian motion on a one-dimensional potential energy landscape U(r) has a probability P(r) of being at a location r, given by solution of the differential equation:  d dP P dU 13.8 þ 0 ¼  dr dr kB T dr with appropriately assigned boundary conditions. If the bonds behave as simple springs operating by Hooke’s Law (e.g., force f ¼ kDr, where k is a linear force constant (units ¼ Newton/m or dyne/cm), and Dr is the spring’s displacement from its relaxed equilibrium length req to its extended position rx (i.e., Dr ¼ rx  req). The potential energy is U(r) ¼ kDr2/2. Note in Fig. 13.3 that thermal energy has the effect of causing the bond to vibrate with greater amplitude at higher energy, thereby increasing the inter-atomic distance of the A–B bond along the r-axis. If the bond is stretched (weakened) to its transition-state configuration [A$$$$$B]z, then the bond has an equal probability of reforming or breaking completely. Likewise, if a force is applied in a manner having the effect of increasing the inter-atomic distance A–B from its equilibrium bond distance A–Beq to successively more distant interactions (e.g., A$$B, A$$$$B, A$$$$$B, etc.) until disruption of [A$$$$$B]z results in separate entities A and B.

13.3.2. Kramers Developed an Insightful Bond Rupture Model The kinetics of force-induced bond disruption was treated by Kramers (1940). For a far-from-equilibrium system of

Energy

13.3. QUALITATIVE FEATURES OF FORCE-INDUCED NONCOVALENT BOND RUPTURE

[A

B]‡

A–B

Interatomic Distance, r FIGURE 13.3 Bond-rupture energetics for a diatomic molecule A–B on a classical energy landscape. The horizontal line represents the equilibrium bond distance between atoms A and B. See text for details.

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

bonded states confined locally within a deep energy well, the escape from the bound state can be modeled as a constant diffusive flux of thermal states along its preferred path along the bond energy landscape (i.e., proceeding over the saddle point defining the transition state). The following Arrhenius-type rate law describes the force-dependent rate constant for disruption of a bond treated as a Hookean spring: k ¼

D exp½DEb ðf Þ=kB T lc  lTS

13.9

Energy

where D is a one-dimensional diffusion coefficient, lc (equal to !dr$exp[DEb(r)/kBT]) is the length (or distance) over which the bonded states are spread thermally, lTS (equal to !dx$exp[DETS(r)/kBT]) is the length to the transition state, and DEb(r) z kc(r  rc)2, with the force constant kc ¼ (v2E/vr2), and DETS(x) z kTS(r  rc)2, and kTS ¼ (v2E/vr2), such that lc ¼ (2pkBT/kc)2, and lTS ¼ (2pkBT/kTS)2. An applied force increases the likelihood of surmounting the saddle-point in the energy landscape for bond disruption. For a simple two-dimensional reaction trajectory (Fig. 13.4), a force F acting over a distance Dr has the effect of increasing both the inter-atomic bond distance r and the potential energy of the now stretched A–B bond by DEF. Such a force reduces the energy required for bond rupture at the transition state [A$$$$$B]z. If the bond-breaking rate follows an Arrheniustype rate law (k ¼ k0e(DE  DEF)/RT), an applied force will generally tend to increase the rate of bond dissociation. Note also that F is the average force, whereas the slope DE/Dr gives the instantaneous force Fr at inter-atomic distance r. A corollary is that observation of a force-insensitive bond is

[A

B]‡

A–BF ∆EF = F∆X

A–B ∆r

Interatomic Distance, r FIGURE 13.4 Force effects on simple bonds. Effect of an applied force on the rupture of a simple A–B bond, as depicted on a classical bond energy landscape. In the case of classical covalent and electrostatic bonds, the inter-atomic distance is determined by the energetics and orientation of interacting orbitals. For many non-classical bonds observed in biology, however, the A–B inter-atomic distance also depends on macromolecular conformation, inasmuch as biopolymers supply the backbone structure holding interacting atoms sufficiently near each other for a bond to form.

771

consistent with an inability of an applied force to be effectively coupled to the vibrational mode of the bond(s) responsible for a noncovalent binding interaction. Multi-stage chemical reactions, wherein bond rupture is an elementary reaction, reflect random walks on a free energy landscape in the space of molecular configurations. The simplified reaction-coordinate diagrams that we use to depict the one-dimensional trajectory along an energy landscape are the lowest energy passages, such that other higher energy shortcuts over a mountainous landscape are rarely, if ever, taken. The highest-lying accessible pass on the trajectory passes through the transition state Xz. In energasetype reactions, the energy landscape is remodeled by the Gibbs free energy liberated by ATP hydrolysis to create new, more expeditious pathways rather than awaiting improbable transitions driven merely by thermal motions. An important, but not necessarily obvious conclusion, is that proteins often behave as though they exist in just a few conformational states, despite the fact that they are made up of thousands, and occasionally even tens of thousands, of atoms.

13.3.3. Noncovalent Bonding Interactions are Inherent to Mechanoenzyme Action The force-dependent reactions of energases and other mechanochemical proteins involve binding interactions comprised of a network of noncovalent bonding interactions. Indeed, macromolecular bonds are far more complex, involving more widely distributed atomic-scale interactions – some of an electrostatic character (e.g., salt bridges and hydrogen bonds), others of a hydrophobic nature, and still others dominated by very short-range van der Waals forces. Force-dependent disruption of such a network is apt to exhibit a tortuous trajectory in configurational space, and we cannot easily deal quantitatively with such interactions. Even so, the lowest lying reaction path is often represented as reaction progress along a scalar reaction coordinate r. The resulting mountainous energy landscape leads to a spectrum of bond rupture behaviors and rates under an applied force. In reality, rupture of a complex molecular bond is likely to involve passage over a cascade of activation barriers (Evans, 2001). We may now consider the simplified bonding diagram shown in Fig. 13.5 for force-dependent rupture of a subunitsubunit or protein-ligand interaction comprised principally of two bonds. The bond shown in red is already taut, whereas a second much less strained bond is indicated by the comparatively more relaxed blue curve. Because the taut-most bond is already at or near the transition state for rupture, a relatively small force might be expected to bring about its rupture. Even so, the overall subunit-subunit or protein-ligand interaction would not be lost upon rupture of that bond, because these subunit-subunit or protein-ligand interactions would be maintained through the action of the second bond, which remains intact at low-to-intermediate force. Only at high force would the latter interaction be lost.

Enzyme Kinetics

772

Associated

Fully ruptured

High Force

Energy

equilibrium bond-length

Associated

Medium Force

Low Stability

High Force

Low Force unstretched or disengaged bonding interaction

vacant binding sites

Low Force

Moderate Stability

Very High Force

Dissociated

FIGURE 13.5 Diagram for force-induced dissociation of a multiplebond system joining two proteins or protein segments. Two proteins or protein segments, shown here in light blue and pink, are joined by straight line (indicating equilibrium bond energy) and curved line (indicating a weak bonding interaction), representing two noncovalent binding interactions formed by bond-acceptor sites (shown as small white circles) and bond-donor sites (shown in bright red and dark blue). With two (or more) bonding interactions, there is an opportunity to achieve intermediate states, of which some may prove to be exceptionally stable. Such a multiple-bond system may require a much higher force for complete rupture. The same principles apply to multiple non-covalent bonding interactions responsible for ligand binding to a protein.

Now consider the more complicated manifold of step-wise force-dependent interactions shown in Fig. 13.6 and the bond-rupture diagram in Fig. 13.7. The reader will appreciate that these diagrams are highly simplified and conjectural. Clearly, we cannot so easily represent the collective atomic level interactions responsible for protein-protein bonds involving multiple noncovalent interactions. Even so, these diagrams do point out the general principles likely to explain how force might lead to sequential rupture of structures stabilized by a set of nearby bonds. The structural stability of proteins under the influence of mechanical force is of paramount importance in forcegenerating enzyme reactions, especially those associated with the cytoskeletal fibers responsible for the contractile and expansile actions of muscle and nonmuscle cells. Unlike the rupture of simple bonds, the structural complexity of proteins allows for more intermediate states to become populated over an intermediate force regime, corresponding to stability energies of 5–15 kBT (3–10 kcal). Such complexity also gives rise to multi-exponential kinetics for force-induced rearrangements in protein structure and motions. There is presently limited quantitative information on how an ensemble of non-covalent bonding interactions contributes to protein stability, and much less is known about how proteins. What is clear is that these interactions are rarely the consequence of a single non-covalent bond. They are instead a composite of numerous weak interactions, each contributing only a few kJ/mol of bond energy to the overall protein bonding interaction. Although the individual effects of these contributing interactions tend to be of short duration, their collective action may result in complex kinetic behaviors. Weakly associated species tend to

High Stability

FIGURE 13.6 Force-interaction diagram for force effects on stability of multiple-bond systems. Two proteins or protein segments, shown here in light blue and pink, are joined by two noncovalent binding interactions formed by bond-acceptor sites (shown as small white circles) and bonddonor sites (shown in bright red and dark blue). With two (or more) non-covalent bonding interactions, a system can achieve states of greater or lesser stability, depending on the strengths of individual bonding interactions. The relative stability of these noncovalent bonding interactions likewise determines in large measure the kinetics of force-induced changes in bonding and the stability of protein-protein and protein-ligand interactions.

Energy

vacant site after bond-rupture

equilibrium bond-length

[A

B]‡

A–BF A–B

Interatomic Distance, r FIGURE 13.7 Energy landscape for the force-dependent bond rupture. Shown is a scheme by which single-step bond rupture (or multiple-step bond rupture) leads to a force-stabilized intermediate (indicated by dashed line). In principle, the intermediate state can be more, less, or the same stability than the bonding interaction in the absence of an imposed force.

dissociate when exposed to a pulling force, especially if the applied force is exerted for a sufficient period relative to the lifetime (toff ¼ 1/koff) for spontaneous dissociation. Even so, if each isolated bonding interaction experiences a bonddisrupting force applied for a period shorter than toff, the bond may resist detachment. Therefore, the key to understanding measurements of bond strength lies in the relation between the lifetimes of the applied force and that of the underlying bonding interactions. It is worth noting that in many cases no force may be required for bond rupture to occur. A good example is a reversible binding interaction, where the researcher need only allow sufficient time for

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

ASSOCIATED

ASSOCIATED

bonds at their equilibrium bond-length

[1]

one ruptured bond

DISSOCIATED

[3]

k–

k’–

k+

k’+

Abruptly applied high force

+

DISSOCIATED

[2]

Constantly applied low force

two ruptured bonds

+ NO force is required if sufficient time is provided for spontaneous dissociation.

+ Force

Energy

dissociation to occur. In such cases, application of a modest applied force (Pathway-[2] in Figure 13.8) is often sufficient to maintain reactants in the dissociated state. Ligand binding is an important factor that, depending on the detailed energetics, can promote or retard the rupture of noncovalent bonding interactions. Figure 13.9 illustrates how binding of ligand L might result in an additional bonding interaction that cannot occur in the ligand’s absence. The degree of stabilization is determined by the magnitude of the DG for the protein-ligand binding interaction. Although relatively few studies have unambiguously examined the effect of force on a chemical reaction rate constant, Wiita et al. (2006) studied the effect of force on the reductive cleavage of a protein -S–S- bond by reduced dithiothreitol. They used a recombinant protein held under constant tension exerted by a laser optical trap (see Section 12.5: Optical Tweezers). Their single-molecule ‘‘forceclamp’’ consisted of two beads, each attached to each end of a recombinant protein corresponding to cardiac titin’s immunoglobulin-like domain-27, an 89-residue b-sandwich protein with well characterized mechanical properties. They used mutagenesis to introduce a disulfide bond in the I27-domain situated between residues 32 and 75, which on the basis of NMR data are positioned near each other. Reduction of disulfide bonds through the thiol disulfide exchange (Chemical Reaction: R-S–S-R þ R9–SH # R-S–

773

Fully ruptured

Low Stability k+1 [L] High Stability

k-1

L

Progress of Bond Dissociation FIGURE 13.9 Force-interaction diagram showing how ligand binding may alter force-induced bond rupture. Two proteins or protein segments, shown here in light blue and pink, are joined by two noncovalent binding interactions formed by bond-acceptor sites (shown as small white circles) and bond-donor sites (shown in bright red and dark blue). The binding of ligand L to a binding site (large white circle) results in a highly stabilized protein-ligand complex that resists force-induced dissociation. The binding energy DGbinding for ligand F will therefore increase the apparent DGbond-rupture. See text for details.

S-R- þ R–SH) is crucial in regulating protein function and is known to occur in mechanically stressed proteins. By applying a constant stretching force to the single disulfide bond and measuring their rate of reduction by DTT, Wiita et al. (2006) observed that the reduction rate was linearly dependent on DTT concentration (as expected for a bimolecular reaction) but was exponentially dependent on the applied force (increasing 10 over a 300-pN force range). This result predicts that the disulfide bond lengthens by 0.034 nm at the transition state in the thiol disulfide exchange reaction. Their findings suggest that mechanical force can play a role in disulfide reduction in vivo, especially in mechanochemically stressed proteins such as molecular motors and proteins under mechanical tension. For sake of comparison, 300-pN force acting over a distance of 0.034 nm corresponds to about one-tenth the Gibbs energy liberated during ATP hydrolysis.

+ DISSOCIATED

FIGURE 13.8 Force-interaction diagram showing how time-underforce alters multiple-bond dissociation. Two proteins or protein segments, shown here in light blue and pink, are joined by two noncovalent binding interactions formed by bond-acceptor sites (shown as small white circles) and bond-donor sites (shown in bright red and dark blue). The fully bonded associated state can undergo either force-dependent and force-independent rupture of a pair of non-covalent bonds. [1] For the fully associated state, an abruptly applied force of high magnitude must be applied to rupture both bonds to bring about dissociation. [2] Constant application of a force of lower magnitude will also bring about dissociation, provided there is sufficient time for partial bond rupture. [3] With the passage of time on the scale of 1/k9, the interaction dissociates spontaneously and no force is required for complete dissociation.

13.3.4. Noncovalent Bonds may be Classified as Ideal, Slip, and Catch Bonds Although typically overlooked in general enzymology, almost all enzymes generate, exert, or transmit forces in various ways and to various degrees during catalytic reaction cycles. This is to be expected because covalent bond rearrangements and reactant-induced conformational changes are inevitably associated with force constants for bending and/or stretching. It must therefore be true that enzyme catalysis is likewise regulated in one or another way by force. Recognizing that enzyme catalysis is a complex, multi-step process, biophysicists have focused on ligand binding to cell surface receptors, particularly those involved

Enzyme Kinetics

774

in cell adhesion to other cells or bacterial pathogens. In certain instances, force has the effect of lowering the likelihood for bond rupture (Figs. 13.10 and 13.11). To rationalize how leukocytes actually manage to tether and roll on vascular surfaces and to explain the increased strength of such connections as shear forces build at higher flow rates, Dembo et al. (1988) offered the following operational definitions of noncovalent bonding interactions: Ideal Bond – Any noncovalent bonding interaction with a lifetime that remains essentially unchanged as an applied force is increased (i.e., koff is force-independent). Catch Bond – Any noncovalent bonding interaction with a lifetime that is prolonged as an applied force is increased, thereby converting short-lived tethers into longer-lived tethers (i.e., koff is inversely related to an increase in the applied force). Slip Bond – Any noncovalent bonding interaction with a lifetime that is shortened as an applied force is increased (i.e., koff increases as the applied force increases). As discussed below, these definitions emphasize the significance of force effects on the kinetics of noncovalent protein-ligand and protein-protein interactions. Treating such interactions as a spring, Dembo et al. (1988) assumed the bound-state spring and the transition-state spring either have the same stiffness or the same resting length (i.e., only one of the two parameters of the two springs is assumed to differ, resulting in either slip bonds or catch bonds). In the most general case, both the elastic constant and the resting length can differ for the transition-state spring and the bound-state spring (Zhu, Lou and McEver, 2005), with the Force-Induced Increase in the Activation Energy for Bond Rupture

∆G ∆EL

distance

distance F F

Low-Force

ASSOCIATED STATES Low Force

Groups are initially at their equilibrium bond length, where only the weak bond (red) offers any stability.

DISSOCIATED STATE Higher Force

As the weak bond ruptures, the much stronger catch-bond (blue) forms and then stabilizes associated state.

At high force, both bonds eventually rupture, leading to dissociation.

FIGURE 13.11 Diagram for force-induced conversion of an ideal noncovalent bond into a catch bond. Two proteins or protein segments, shown here in light blue and pink, are joined by two noncovalent binding interactions formed by bond-acceptor sites (shown as small white circles) and bond-donor sites (shown in bright red and dark blue). Some noncovalent bonds exhibit increased lifetimes (i.e., greater stability) when stretched by mechanical forces. In each case, the catch-bond eventually transitions into an ordinary slip-bond that becomes increasingly shorterlived as the tensile force on the bond is further increased (see Pereverzev et al., 2005 and Pereverzev and Prezhdo, 2006 for a theoretical treatment of transitions of catch-bonds into force-sensitive slip-bonds).

force-dependent off-rate constant predicted to depend exponentially on a quadratic function of force: 0 koff ðf Þ ¼ koff exp½dl  f =kB T

High-Force

FIGURE 13.10 Diagram for force-induced protein structural reorganization, with increased inter-subunit stability. The hypothetical protein structure is reorganized in a force dependent manner, such that the increased complementarity of subunit binding sites is characterized by a more stable subunit-subunit interface and consequently a higher activation energy for bond rupture (i.e., DELz > DEH z).

13.10

where k0off is the zero-force off-rate constant, kB is the Boltzmann constant, T is the absolute temperature, and bondlength difference dl ¼ lTransition-state  lBound-state. If the transition-state and bound-state happen to have the same bond-length but different spring constants, such that dk ¼ (kTransition-state  kBound-state), then another slip bond is obtained: 0 exp½dkf 2 =2k2 kB T koff ð f Þ ¼ koff

∆EH ∆G

Force-Sensitive Transition to a Catch-Bonded State

13.11

In Dembo’s treatment, the off-rate constant is an exponentially increasing function of the strain energy applied to that spring-like bond. In the most general case, both the resting length and the elastic constant are apt to differ for the transition-state spring and the bound-state spring (Piper, 1997), and the off-rate constant depend exponentially on a quadratic function of force. As discussed later in this chapter, muscle contraction is driven by cyclic affinity-modulated noncovalent binding interactions between actin filaments and the motor enzyme myosin. Conformational changes in the actin-myosin binding interface occur in concert with the binding of ATP, binding to actin, and loss of hydrolytic by-products (namely ADP, orthophosphate, and an Hþ), but the nature of the conformational changes and the bond strength of the actomyosin complex are still unclear. The force-dependent kinetics of the actomyosin bond may be particularly

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

important at high loads, where myosin may detach from actin before achieving its full power stroke. Guo and Guilford (2006) demonstrated that over a physiological range of rapidly applied loads, actomyosin behaves as a catch bond, with an increasing lifetime with loads increasing up to a maximum at 6 pN. They made the surprising observation that the myosin-ADP bond has longer lifetimes under load than rigor bonds, although the load at which bond lifetime is maximal remains unchanged. Guo and Guilford (2006) also found that the actomyosin bond lifetime is ultimately dependent not only on load, but also the loading history, meaning that the order and duration of the applied force affects the dynamics of protein-protein interactions. Their data also suggest a complex relationship between the rate of actomyosin dissociation and muscle force and shortening velocity. The 6-pN load for achieving the maximum bond lifetime is at or near the force generated by a single myosin molecule during isometric contraction. Such findings suggest that catch bonds between loadbearing molecules are tuned mechano-kinetically to best advantage for their physiological environment.

13.3.5. Green Fluorescent Protein Unfolding/ Refolding is Force-Dependent Although there are high-resolution X-ray structures for many molecular motors, there is a paucity of unambiguous structural data on the detailed structural changes as force is generated. The same is unfortunately true for all enzymes. For this reason, we must look to other nonenzymatic systems to glean information about how proteins respond to mechanical force. Recognizing that the intrinsic fluorescence of green fluorescent protein (GFP) depends on its intact structure, Dietz and Rief (2004) studied the force-induced unfolding of green fluorescent protein GFP by means of atomic force microscopy (see Section 13.5). AFM is an extremely sensitive biophysical technique that can generate reasonably accurate free energy landscapes for individual protein molecules. In principle, rate constants can be obtained for both protein unfolding and refolding, which can be used to characterize the energetics of the unfolding-refolding equilibrium. Another advantage is that there may be one or several local unfolding-refolding intermediates on the trajectory between the extreme, fully folded and fully unfolded conformational states. Such intermediates are indicated by the arching red arrows shown (a) in Fig. 13.12, where the local minima on the trajectory are shown as depressions in the free energy landscape, and (b) in Fig. 13.13, where the trajectory is written as a reaction-coordinate diagram, annotated with cartoons of likely GFP structures. The first transition from GFPDa to GFPDaDb occurs close to equilibrium. From their data, Dietz and Rief (2004) extracted a DG of 22  4 kBT between GFP and GFPDa (the latter meaning the helix deleted form). The contour length increases upon detachment of seven amino acids to

775

FIGURE 13.12 Free energy landscape for force-induced unfolding of green fluorescent protein. Arrows indicate the low-energy trajectory of the mechanical unfolding pathway. Starting from the completely folded structure in the global minimum, the pathway is biased along the Nterminal-to-C-terminal direction. Upon unfolding of the N-terminal a-helix, the protein arrives at a local minimum (GFPDa). The direction of the pathway then changes, because the protein responds to the detachment of the a-helix by turning and reorienting along the new N-terminal-to-C-terminal direction, as determined by the line connecting between the first structured residue at the N-terminus and the last structured residue at the C terminus of the folded portion of the intermediate. The new bias induces detachment of a b-strand leading to the second local minimum at GFPDaDb. The protein reorients, with the protein transitioning directly into the completely unfolded state, at least at the temporal resolution of their experiment. EXPERIMENTAL: Measurements were conducted on a custom-built atomic force microscope fitted with goldcoated cantilevers with the following 6 and 30-pN$nm spring-constants and resonance frequencies of 1.5 and 8.5 kHz, respectively. For quantitative analysis of contour lengths and unfolding forces, the force versus extension traces were fitted to an interpolation formula (Bustamante et al., 1994) for a worm-like chain model: F(x) ¼ (kBT/p)[0.25(1  x/ L)2  0.25 x/L], where L is the contour length of the stretched protein, p is the persistence length, and x is the distance between attachment points of the protein. A p value of 0.5 nm provided the best fit over the 20–150 pN force regime and was held fixed at this value for all fits. From Dietz and Rief (2004) with permission of the authors and the National Academy of Sciences.

DL ¼ 3.2  1 nm. Their data did not allow them to directly determine the position or height of the transition barrier. However, based on their < 1-ms estimated refolding-time for the a-helix, they estimated an upper-limit of 14 kBT for this barrier height. The position of the barrier for the transition from GFP to GFPDa is 0.28  0.03 nm from the minimum, with an activation barrier of 23 kBT. Detachment of the b-strand from the barrel leads to an additional contour length increase of 6.8  0.6 nm. From the GFPDaDb state, GFP unfolds into the completely denatured state leading to a contour-length increase of 69.8 nm. The associated barrier height is 20–25 kBT at a position of 0.55 nm.

Enzyme Kinetics

776

Free Energy [kBT]

GFP

GFP∆α∆β ∆α∆β

GFP∆α ∆α

23

20-25

3.7 22

0.55 3.2

0.28

69.3

6.5 End-to-end Distance [nm]

FIGURE 13.13 Free energy landscape for the force-induced GFP unfolding. Although the unfolding pathway for GFP meanders along a threedimensional landscape, this pathway may nonetheless be depicted as progress along a one-dimensional reaction coordinate. For simplicity, the distances contain the directly measured increases in contour length upon detachment of amino acids, and these numbers are likely to be smaller at small forces, at which the polymer is incompletely stretched. From Dietz and Rief (2004) with permission of the authors and the National Academy of Sciences.

13.4. KELLER-BUSTAMANTE TREATMENT OF MOLECULAR MOTOR BEHAVIOR In their elegant exposition on molecular motor action, Keller and Bustamante (2000) focused on the significance of microscopic fluctuations, which often are directly observable in single-molecule experiments, and which largely disappear in the long-term ensemble averages typically observed in bulk experiments. Their physical picture of a molecular motor is that of a microscopic fluctuating machine, the operation of which corresponds to a random walk or diffusion process on the potential energy surface of the system. The diffusional fluxes resulting from the random walk define the motor’s chemical reaction rate and mechanical velocity. These motors harness the energy of ATP, GTP, and related molecules of high group-transfer potential to bias the random walk, thereby achieving directed motion. The processive action of tethered motors is another significant feature of motor action. Although there are by now many treatments of molecular motor action, the theory developed by Keller and Bustamante (2000) is especially appealing, and is closely paraphrased here for accuracy. In their treatment, a molecular motor is a force- and motion-generating mechanoenzyme, usually acting: (a) when associated with an underlying polymeric track, such as an actin filament, a microtubule, or a nucleic acid molecule; (b) when localized in a membrane or nuclear pore, as in the case of active transporters, membrane-fusion proteins, and nuclear-cytoplasmic sorters

within the nuclear pore; or (c) when combined with another protein on a membrane or in free solution, as is the case with GTP-regulatory proteins. Their large-scale, ensembleaveraged properties are what we most often observe macroscopically as rate constants for the hydrolysis of ‘‘fuel’’ molecules like ATP, GTP, P-enolpyruvate, etc. Although our macroscopic view of a molecular motor is that of a seemingly simple and well-behaved process, the microscopic view is that of a force- and motion-generating cycle, wherein a succession of conformational states of the motor and its binding partner are driven by: (a) chemical events (e.g., ATP binding, P{sb}O bond-breaking, and/or ADP or Pi release); and (b) rapid, incessant and seemingly haphazard thermal fluctuations within and between the motor and its binding partners. The detailed behavior of motors are accounted for in terms of system variables, which can be divided into two classes: (a) the chemical variable, which measures chemical reaction progress in terms of the chemical change per se as well as its associated change in Gibbs energy; and (b) mechanical variables, including all conformational and positional variables. The conformational variable describes changes in the motor’s shape and its detailed interactions with its binding partner (i.e., microtubules for dynein and kinesin, actin filament for actomyosin and actoclampin, or DNA molecules for helicases and topoisomerases). The positional variable defines the motor’s location on its track (i.e., sites on subunits within microtubules, actin filament, or DNA molecules) or its rotational angle, as in the case of

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

Position

Coordina te

d

Free Energy

DV 2D

Reaction Coordinate

B

1.05 0.9 0.75 0.6 0.45 0.3 0.15

0

20

40

60

80

100

time (msec)

Coordina te

C

Position

The system variables define an n-dimensional state space for the motor, with each point representing a unique conformation of the motor molecule. Associated with each conformation x1, x2, ., xn is a free energy V(x1, x2, ., xn) called the potential of mean force (McQuarrie, 1976), which has the property that its derivatives with respect to x1, x2, ., xn are the (time or ensemble) average forces, CFiD equal to [vV(x1, x2,., xn)]/vx1], where xj s xi. As described by the potential of mean force that can in principle be calculated by integrating the Boltzmann factor, exp[2U(x1, x2, ., xn, y1, y2, ., ym)/kBT], over the bath variables, y1, y2, ., ym, while holding the system variables constant, where U(x1, x2, ., xn, y1, y2, ., ym) is the full potential for all degrees of freedom in the system, including protein, solvent, and other solution variables. Both entropic and enthalpic contributions to the free energy are included in the potential of mean force, so both entropic and mechanical forces are accounted for. Because the potential of mean force is an equilibrium quantity, all bath variables (which do not appear in V) are implicitly assumed to be at equilibrium. If, in the simplest case, a motor can be described by only two system variables, the potential of mean force V(x1, x2) defines a two-dimensional potential energy surface on which the molecular motor moves (Panel A, Fig. 13.14). Along a line parallel to x1, the chemical variable, this surface looks like a typical reaction free energy diagram. The local minima on this curve represent stable or metastable species separated by free energy barriers determining the rates of chemical reactions, which is proportional to the probability of transitions from one local minimum to another. After each chemical reaction cycle, the enzyme returns to its initial state, and the free energy must decrease by a fixed amount that is closely related to the macroscopic free energy for the chemical reaction. Therefore, the free energy surface is periodic in the chemical variable, except for a linear term that accounts for the free energy of reaction. Along a line parallel to x2, the position variable, the surface gives the local free energy changes associated with movement of the motor along its track. Inasmuch as the track is periodic, the potential must also be periodic, and in the absence of external forces the overall free energy change in one full step d along the track is zero. For example, for actomyosin, x2 would be the position of myosin along an actin filament, and the free energy surface along x2 may have a periodic series of minima representing the stable

DX1

Position (nm)

13.4.1. Motor Molecule Motions are Analyzed in Terms of State Space and the Potential of Mean Force

A

Free Energy

rotary motors (e.g., the bacterial flagellar motor or the F1F0 ATP synthase). Here, x1 is the chemical variable, x2 is the position variable, and x3 to xn are mechanical variables describing internal motions within the motor protein.

777

Reaction Coordinate FIGURE 13.14 Stochastic action of a molecular motor. A, Hypothetical potential energy surface (potential of mean force) for a simple motor with two system variables. The surface is periodic, with four unit cells shown. The trajectory in the lower right shows the path of a hypothetical system point executing a random walk on the surface. B, Simulated run of position versus time data, calculated using the Langevin equations for a two-dimensional system with the potential surface in Panel-A. C, Kinetic scheme overlaid on the potential energy surface in (A). The fine lines show the boundaries of the regions corresponding to each macroscopic intermediate species. Each macroscopic species is identified with a minimum of the potential, and transitions between species are associated with low energy pathways between minima. From Keller and Bustamante (2000) with permission of the authors and the Biophysical Journal.

778

binding sites for myosin on the filament. Altogether, the potential must satisfy the key condition (Magnasco, 1994) that V(x1 þ Dx1, x2, .) ¼ V(x1, x2, .) þ DV, and V(x1, x2 þ d, .) ¼ V(x1, x2, .), where DV is a constant, Dx1 is the period along x1, and d is the period along x2 (i.e., the step size for the motor). In a molecular motor the mechanical and chemical variables must be coupled in some way so that progress along the chemical reaction leads to movement. The nature of this coupling is contained in the contours of V(x1, x2, ., xn). Therefore, all the important features of a molecular motor are determined by the potential of mean force, and the choice of V(x1, x2,., xn) defines the mechanism and properties of the motor.

13.4.2. Molecular Motors Operate Stochastically As described by Keller and Bustamante (2000), the chemical and mechanical operation of a molecular motor can be described by a potential energy function V, and the movements of the motor are treated as a point moving on an n-dimensional potential energy surface. While motion at the macroscopic level is governed by classical equations of motion that predict smooth trajectories, the situation is different at the microscopic level. Here, the system’s interactions with so-called bath variables (representing the solvent and all degrees of freedom not explicitly accounted for in the system variables) becomes significant, and at a given temperature T, each of the bath variables has energy of the order of kBT. This energy is significant relative to other features of the potential energy surface and is usually much larger than the motor’s kinetic energy. The bath variables may therefore exert significant effects on the motion of the system variables, and it is assumed that these effects are random. This physical picture can be well described by a collection of classical Langevin equations: g1dx1/dt ¼  vV/vx1 þ F1(t) þ dF1(t), g2dx2/dt ¼  vV/vx2 þ F2(t) þ dF2(t), .., gndxn/dt ¼  vV/vxn þ Fn(t) þ dFn(t), where g1, g2, ., gn are damping constants dF1, dF2, ., and dFn are random bath forces, and F1(t), F2(t), ., Fn(t) are external forces which may include, for example, a load force opposing the motion of the motor. While these external forces may depend on time, they are assumed be independent of the system variables. The classical inertial forces, mixi, are neglected in the last set of equations, indicating that all motions are over-damped. This is a good approximation for the relatively slow timescale of much of the experimental data on molecular motors. Any very fast motions (e.g., vibrations of parts within the motor with megahertz or higher frequencies) that show significant inertial behavior in proteins are averaged out in the 0.1-msec to min time range. The damping terms, gidxi/dt, are simple frictions, and do not allow for any ‘‘memory’’ (i.e., forces caused by reaction of the bath at a later time due to motions in x at an earlier time) on the experimental time

Enzyme Kinetics

scale. The effects of the bath variables appear in three ways in the Langevin equations: (a) in the damping terms on the left hand side gidxi/dt; (b) in the stochastic forces on the right-hand-side, dF1(t), dF2(t), etc.; and (c) in the potential of mean force V. The stochastic forces are defined to have zero mean (i.e., any force that does not average to zero is included in the ‘‘external’’ forces F1, F2, etc.), such that CdF1(t)D ¼ 0 for all i. In addition, the fact that the damping terms are written as simple frictions requires that the stochastic forces have d-function time correlation, meaning that a force fluctuation at time t is completely uncorrelated with another force fluctuation an infinitesimal time later. The value of the force at any one time is taken to have a Gaussian distribution, which is consistent with a physical situation in which the actual forces are much faster than the time between experimental observations, so the apparent force is the sum of many small impulses. As expected for a bath at equilibrium, the statistical properties of the bath forces depend only on time intervals and not on the absolute value of time. Finally, the bath forces acting on different variables i and j are uncorrelated at all times. The nature of the stochastic forces is evident in the spectral density of fluctuations, which is the Fourier transform of the correlation P function, where CjdF1(v)j2D ¼ CdFi(t)dFi(tþt)Dexp(ivt)dt (over the indefinite interval from –N to þN) equals 2gikBT. Note that from the last term the intensity of fluctuations for d-function for correlated forces is independent of frequency and is accordingly called white noise.

13.4.3. Motors ‘‘Walk’’ on the Potential Energy Surface During Chemical and Positional Transitions The treatment presented so far defines the behavior of a system moving on a potential energy surface V(x1, x2, ., xn), subjected to white noise of intensity 2gikBT at all frequencies. The presence of random forces causes the trajectory of the system point, [x1(t), x2(t), ., xn(t)] to be random as well. Individual trajectories therefore have little significance by themselves. The important quantities are those that describe the statistics of many trajectories, and the proper solution to the Langevin equation (see Glossary) is a probability distribution of trajectories. The approximations made in the previous section – the fact that the bath forces lose all correlation after an infinitesimal time, and the neglect of inertial forces so that the equations of motion are first order in time – mean that the system loses all memory of previous positions after each step. The motion of x1(t), x2(t), etc., is therefore a Markov walk or diffusion process, described by a probability density, w(x1, x2, ., xn; t), for observing the walker at location x1, x2, ., xn at time t, given that it had distribution w0(x1, x2, ., xn) at the initial time, t0. Because probability is conserved, w mustP obey the continuity equation: vw/vt þ =J ¼ vw/vt þ (vJi/vxi), integrated from 1 to n, where the n-dimensional gradient

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

= equals (v/vx1, v/vx2, .., v/vxn), and the n-component probability current density J equals (J1, J2, .., Jn). Here, Jn equals kBT vw /gnvx1 þ fiw /gI, where fi , equal to – (vV/vx1) þ Fi(t), is the force acting along the ith dimension. Substituting the latter two equations into the continuity equation yields the Smoluchowski equation: n  vx X kB T v2 w 1   2 þ ð f ; wÞ ¼ 0 13.12 þ vt i ¼ 1 gi gi vxi which, for the one-dimensional case, becomes: vw kB T v2 w 1 v  2þ   vt gi vx gi vxi





 vV þ FðtÞ w ¼ 0 vx 13.13

The one-dimensional Smoluchowski equation bearing the name of its innovator is actually a special case of the Fokker-Planck equation, and was first derived by 1906. A second-order partial differential equation, the Smoluchowski equation is one that can be solved for w(x1, x2, . . . , xn; t) at any time t, provided that the distribution w0 is known at the initial time t0. Shown in Panel A of Fig. 13.14 is a hypothetical twodimensional potential energy surface plotting V(x1, x2) for a molecular motor having only two degrees of freedom (i.e., one chemical variable and one mechanical variable). According to the stochastic theory, the operation of a single motor during a single cycle is a random walk on this surface, which is periodic along the chemical axis (except for its uniform tilt) and along the position axis, as is required for periodic mechanochemical action – catalytic turnover of ATP hydrolysis and periodic movement along its track. Four unit cells are shown, each of which contains three potential energy minima (labeled A, B, and C in the unit cell in the upper right). Each minimum can be reached from the neighboring minima by low-energy ‘‘passes’’ between them. Together these passes define a low-energy path through the conformational space of the motor. The low-energy path, in turn, defines the most probable sequence of conformational changes as the motor goes through one mechanochemical cycle. During a cycle the diffusing system point will tend to stay near the minima of the deep wells, but will occasionally make transitions between wells through the passes. Hence the wells correspond to the stable states that would be found in kinetics experiments, and the low-energy passes between the wells define the reaction coordinates for transitions between kinetic intermediates. The entire surface has a uniform tilt along the chemical axis. The drop in energy in one unit cell is the constant energy, DV. The tilt represents the thermodynamic driving force for the chemical reaction, and biases the diffusion process toward the products of the chemical reaction and away from reactants. At a given instant of time the system

779

point may step in any direction, but over many steps the system will, on average, drift in the direction of the tilt. The long trough in the center of Fig. 13.14 (Panel A) is the crucial region where chemistry is coupled to mechanical motion. As long as the low-energy path is parallel to the chemical variable (as it is for transitions between the three closely spaced wells) no net change in position takes place. Experimentally, the motor would be seen to fluctuate about a fixed location on its track while purely chemical processes take place; but in the trough region the tilt of the potential in the chemical direction drives movement along the mechanical direction, and chemical energy is transduced into mechanical motion. Panel B of Fig. 13.14 is a run of simulated single molecule data (motor position versus time) for a motor with the potential surface in Panel A. The simulation was carried out by numerically integrating the Langevin equations for the chemical and position variables, x1 and x2, with V given by the surface in Panel A of Fig. 13.14, zero external forces, Fi(t), and a stochastic force, dFi(t). Only the intrinsic fluctuations of the system itself are shown; no attempt has been made to add the instrumental noise present in experimental data. While the motor goes through the purely chemical part of its cycle, its position fluctuates rapidly, but the average velocity is zero. As the system enters the trough region a rapid stepping motion is observed with a large positive velocity. A second, smaller step occurs as the system falls from the left-hand well (labeled A in Panel C, Fig. 13.14) to the lower well (labeled C in the same panel). After these steps the average position again becomes constant and the average velocity drops to zero. Though this is a purely hypothetical example, the qualitative behavior – rapid steps separated by relatively long pauses – is similar to that observed in real motors (for examples, see Coppin et al., 1996, 1997; Hua et al., 1997; Schnitzer and Block, 1997). Finally, the interested reader should consult Keller and Bustamante (2000) for their further treatment of the energetics of ATP hydrolysis, the nature of force generation, stalling forces (including predicted plots of Motor Translocation Velocity versus Load), as well as predicted hydrolysis kinetics and translocation velocities for several model cases. Another excellent review dealing with multiscale motility of molecular motors is that presented by Lipowsky and Klumpp (2004).

13.5. CALCIUM ION PUMP: CHEMICAL SPECIFICITY VERSUS VECTORIAL SPECIFICITY Calcium ion is a major regulatory signal in nearly all cells, and its metabolic impact is further enhanced through its binding to calmodulin, the multivalent calcium ion signaltransducer (see Section 7.1.9: Activation of 39,59-cyclic AMP Phosphodiesterase by Calcium Ion-calmodulin Complexes).

Enzyme Kinetics

780

In muscle, the sarcoplasmic reticulum (SR) is a subcellular tubulovescular compartment that physically surrounds each actomyosin-rich myofibril. Release of calcium ion into the sarcoplasm initiates muscle contraction; during relaxation, Ca2þ is transported back into the lumen of the SR through the action of the calcium ion pump concentrated in the membrane of the sarcoplasmic reticulum. ATP-dependent pumps also play a major role in controlling the steady-state calcium ion concentration in other cellular compartments. Most enzymes only have a single specificity for its substrate, which is then transformed catalytically to product, followed by product dissociation to restart the catalytic cycle. This mechanoenzymatic reaction, on the other hand, displays two topologically controlled chemical specificities reflecting the physical accessibility of a mechanoenzyme for the target ligand’s substrate-like and product-like forms, and there is no net covalent transformation of the ligated molecule in the affinity-modulated reaction. The SR calcium pump distinguishes the cytoplasmic calcium ion (Ca2þCyto) and the SR lumen calcium ion (Ca2þSR). In his studies on sarcoplasmic reticulum ATPase, William Jencks (1980; 1989) was among the first to recognize that, to comprehend the nature of mechanochemical coupling, two fundamental questions must be answered. First, how does energetic coupling for ATP binding and transported-solute binding to various enzyme intermediates allow catalytic turnover to proceed at a biologically useful rate? In other words, how does the transporter avoid forming reaction-cycle intermediates that are either too stable or too unstable. Second, what is the underlying mechanism for the stoichiometric coupling between the chemical reaction of ATP hydrolysis and the physical reaction such as the active transport of ions? In short, highly effective coupling should make certain that neither the chemical reaction nor physical reaction would occur unless both can occur. He distinguished the chemical specificity phase from the vectorial specificity phase of the pump’s mechanoenzyme action. Chemical specificity can be thought of as being governed by an enzyme’s affinity, as reflected in the value of the dissociation constant KiS for its substrate. By analogy to the half-reactions found in redox reactions, the catalytic reaction cycles of affinity-modulated energases may be separated into:

as the phosphoanhydride bond in ATP); or (b) the energetically coupled dissipation of a transmembranal chemiosmotic (or electrochemical) gradient. The sarcoplasmic reticulum calcium ATPase is an energase that actively transports calcium ions against an otherwise unfavorable metal ion concentration gradient by coupling the vectorial movement of two calcium ions per molecule of ATP hydrolyzed: ATP + H2O Ca2+Cyto

Ca2+SR

13.14 The calcium ion gradient within most cells is such that the transport of a single calcium ion between the cytoplasmic and lumenal compartments requires only 6–8 kBT of energy, as compared to 21 kBT/mol available from ATP hydrolysis. Therefore, this mechanoenzyme maximizes efficiency by simultaneously transporting two calcium ions per molecule of ATP hydrolyzed. Jencks (1980) analyzed the action of the SR calcium pump by means of the catalytic reaction cycle shown in Fig. 13.15. The model shows two principal energase states, based on the intuitive mechanism (Fig. 13.16) of Makinose (1973), who first suggested how phosphorylation-dependent transport of calcium ions might take place. In both schemes, the specificities of the pump for catalysis change in the two enzyme states. Jencks (1980) pointed out that coupling is determined by: (a) the chemical specificity in phosphoryl transfer; and (b) vectorial specificity for ion binding. While most enzymes exhibit only a single specificity for catalysis, energases have different specificities, as indicated by affinity, depending on what is bound to the enzyme. To define the pump’s mechanism of action, Jencks (1989) proposed two rules governing the chemical specificity needed by the pump to achieve active calcium ion transport (i.e., its susceptibility toward phosphorylation and hydrolysis at various stages in the reaction cycle).

H2O

Pi

E

E P

Chemical Phase – referring to the ensemble of reaction steps wherein a change in the conformation of the ligandbound energase and/or its target ligand is driven energetically by a chemical reaction. The latter phase harnesses the Gibbs free energy provided: (a) by the cleavage of a weak chemical bond (such

2Ca2+ Cytoplasm ECa

2

ATP 2 Ca2+ SR Lumen ECaP2

al ic ity em ific Ch ec Sp

Ligand-binding Phase – referring to the ensemble of reaction steps wherein the target ligand binds to the mechanoenzyme and undergoes ligand-induced conformational isomerizations.

ADP + Pi

ADP

l ia or city t c ifi Ve ec Sp

FIGURE 13.15 Catalytic cycle for the ATP-dependent sarcoplasmic reticulum calcium ion pump. The various enzyme forms of this transmembrane mechanoenzyme are shown in a manner akin to that of any soluble-phase enzyme, except that the steps affecting chemical specificity and vectorial specificity are indicated. See text for additional details.

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

SR Lumen

ATP + 2 Ca2+

Cytoplasm ATP

2 Ca2+

G (kcal/mol)

Cytoplasm

SR Lumen

Pi

ADP

Cytoplasm ~P Ca2+

2 SR Lumen

781

ECa2

Pi

+3

1 0

ECa 2 + Pi –5

2 1

4 -16

–10

2

-5

E+Pi

3

ECaP 2

-14

E P

Cytoplasm ~P

2 Ca2+ SR Lumen

FIGURE 13.16 Reaction cycle for the ATP-dependent sarcoplasmic reticulum calcium ion pump. See text for additional details.

Chemical Specificity Rule-1. Only the calcium ion-bound species E$(Ca2þ)2 can react with ATP to yield P~E$(Ca2þ)2. On the other hand, for the reaction to operate efficiently in the reverse direction, only P~E$(Ca2þ)2 can react with ADP. Chemical Specificity Rule-2. Only the free enzyme E reacts with Pi to give phosphorylated enzyme E~P. Likewise, for the reaction to operate efficiently in the reverse direction, only the phosphorylated enzyme E~P is permitted to react with water to undergo hydrolysis. In reckoning how the calcium pump exploits autophosphorylation to achieve active transport, Jencks (1989) offered the following rules defining the pump’s vectorial specificity for calcium ion binding: Vectorial Specificity Rule-1. Only the free enzyme E binds and releases Ca2þ on the cytoplasmic side of the membrane (i.e., the outside of the sarcoplasmic reticulum). Vectorial Specificity Rule-2. Only the phosphorylated enzyme E~P binds Ca2þ on the inside of the vesicle (i.e., lumenal side of the sarcoplasmic reticulum). These rules allow the pump to be both an efficient and vectorial active transporter. Were the enzyme to take a path that did not conform to either chemical specificity rules, ATP would be hydrolyzed without the transport of calcium ions, and the system would no longer be productively coupled. Were the enzyme to violate either rule for vectorial specificity, calcium ion would leak out of the vesicle, and the system would no longer be productively coupled. This energase must bind calcium ions with high affinity on the low calcium ion concentration side (i.e., the cytoplasm), and with low affinity on the high calcium ion concentration side (i.e., the lumen of the sarcoplasmic reticulum). The enzyme relies in the energy of ATP

–15 FIGURE 13.17 Energetics of Ca2D and Pi binding to ATP-dependent sarcoplasmic reticulum calcium pump. Reaction cycle for calcium pump showing free energy changes (kcal/mol) for the affinity-modulated reaction cycle, with reaction steps indicated by numbers in black circles. The diagram shows that, although the binding of calcium ions and phosphate ion is favorable, the presence of both is attended by an interaction energy of þ7.6 kcal/mol (indicated by the red bar). The numbers in the lower left refer to the energy change, as expressed in kcal/mol.

hydrolysis to accomplish the mechanical work of switching the enzyme from its high-affinity to its low-affinity states. Jencks (1980) analyzed this behavior in terms of the binding site interaction energy (see Fig. 13.17). Phosphoenzyme formation from orthophosphate ion is slightly favorable, and the binding of calcium ion to the free enzyme appears to have a sub-micromolar dissociation constant. Even so, calcium ion binding to the phosphorylated pump is weaker by 7.6 kcal/mol. This mutual destabilization explains the escaping tendency of calcium ion, even in a region of high calcium ion concentration. Another indication that conformational changes are likely to account for changes in ion affinity was provided by Keillor and Jencks (1996), who examined the kinetic properties of the ovine renal Naþ,Kþ ATPase. This enzyme was pre-incubated with 120 mM sodium and 3 mM magnesium and then allowed to react with ATP (concentration range ¼ 0.01–2.00 mM) to form a covalent phosphoenzyme intermediate (E–P). These investigators observed that the first-order rate constant for phosphorylation increases hyperbolically with ATP concentration to a maximum value of 4.6  102 s1 and K0.5 ¼ 75 mM. If phosphoryl-transfer is assumed to be rate-limiting, then the approach to equilibrium that yields 50% E–P in the presence of ADP obeys the following rate law: kobserved ¼ kf þ kr þ 9.2  102 s1. However, for E–P formation from E$Na3 with 1.0 mM ATP and 2.0 mM ADP, kobserved ¼ 4.2  102 s1, indicating that phosphoryl transfer from bound ATP to the enzyme is not rate-limiting for E–P formation from E$Na3. Their results suggest that there is likely to be a rate-limiting conformational change of the E$Na3$ATP intermediate, followed by rapid phosphoryl transfer, with kcat 3,000 s1. While similar mechanisms are likely to underlie the action of other ATP-driven pumps such as the Naþ,Kþ ATPase and the Hþ-pumping ATPase, there is no a priori basis for believing that the mechanisms need be identical.

Enzyme Kinetics

782

In principle, the degree of ATP-dependent affinity modulation may be tuned to the physiologically requisite [Solute]in/[Solute]out ratio, with due consideration for the role of electrostatic charge effects. If the [Solute]in/[Solute]out ratio is relatively low, then a smaller fraction of the Gibbs energy for ATP hydrolysis would be required to achieve and maintain that ratio. However, if the [Solute]in/[Solute]out ratio is high, such as the [Hþ]in/[Hþ]out ratio needed to reach a gastric pH of ~1, then a far greater increment of the ATP hydrolysis energy would be needed.

13.6. ACTOMYOSIN MECHANISM The sliding-filament hypothesis (Huxley and Hanson, 1954; Huxley and Niedergerke, 1954) advanced muscle biophysics by overturning earlier speculations that contractile proteins somehow underwent extraordinarily great force-generating changes in overall length. Instead, myosin molecules localized in so-called thick filaments of striated muscle were proposed to slide on actin-rich thin filaments. The idea of myosin forming a transient, force-generating cross-bridge with an actin filament was proposed by Huxley (1957) to explain contractile forces in terms of repetitive cross-bridge cycles allowing the thick filaments to make larger scale excursions than predicted by a simple cross-bridging interaction. Keilley and Meyerhof (1948) first demonstrated that myosin catalyzed ATP hydrolysis, and later work showed that its hydrolytic activity was greatly stimulated by the presence of intact actin filaments. Based on fast reaction kinetic studies of interactions of myosin with actin filaments in the presence of MgATP2, Lymn and Taylor (1971) advanced the Lymn-Taylor cross-bridge cycle, in which ATP binding brought about the dissociation of the globular myosin head from its actin filament partner in the actomyosin molecular motor complex prior to the P–O–P bond hydrolysis step and that rebinding of the actin filament to myosin activates the ATPase activity. In the absence of an actin filament, myosin could catalyze rapid ATP hydrolysis, but the ADP and Pi remained bound for many seconds. Relying on changes in protein fluorescence, Bagshaw et al. (1972) demonstrated that ATP binding to myosin was attended by a conformational isomerization (asterisk) prior to the hydrolysis step and that isomerizations also limited A.Moc.D A.Moo

1

A.Moo.T 2 A.Mco.T Mco. T

A.Moc.T 3a

A.Mcc.T

return stroke

Mcc.T

3b

A.Moo.T

working stroke

A.Moo strong

A.Moc.D.P

A.Moo.D.P binding

A.Mcc.D.P

A.Mco.D.P binding

hydrolysis

Mcc.D.P

Scheme 13.4

state

weak state

futile cycle

Mco.D.P

the rate of Pi and ADP release (Bagshaw and Trentham, 1974; Bagshaw et al., 1974). As described by Zeng et al. (2004), the structural and kinetic events associated with ATP binding to myosin and the mechanism of actomyosin dissociation can now be explained as described in Scheme 13.4. Here, A ¼ Actin; M ¼ Myosin; T ¼ ATP, D ¼ ADP, and P ¼ Orthophosphate. The letters ‘‘o’’ and ‘‘c’’ refer to an open or closed switch state, respectively, with Switch-1 preceding Switch-2. This model assumes that the primary effect of actin is on the rate and position of the Switch-1 closed-open transition (i.e., the Moc$D$P state is insignificant and omitted from the pathway). However, actin is assumed to have no significant effect on the equilibrium of the Switch-2 open-to-closed transition compared with the equivalent reaction in the absence of actin. The enhanced fluorescence state (denoted by an asterisk in the text) is assumed to be related to Switch-2 closing and the lever arm in the pre-power stroke position. Switch-2 opening is coupled to the lever arm swing and the working stroke. Switch-1 closing results in a weak actin binding state. The dominant route that minimizes futile cycling is highlighted in gray. The mechanism shows how mechanochemical coupling can be achieved without a direct influence of actin on Switch-2 or lever arm position; rather, actin has an indirect effect via activation of Pi release. One limitation of such models is that they do not reveal the detailed nature of the affinity modulation mechanism for myosin’s interactions with an actin filament. Finally, Kojima, Ishijima and Yanagida (1994) determined that a pulling force of ~400 pN exerted along the filament’s longitudinal axis is sufficient to break an actin filament. By analogy to linked chains, this value would be expected to decrease at greater filament lengths, as reflected in the greater probability of linkage failure. Much greater contractile forces are, of course, generated in the trailing pole (or uropodium) of migrating cells. If such strong forces were focused on a filament, the filament would likely break, thereby limiting the magnitude of contractile forces that could be generated.

13.7. GTP-REGULATORY PROTEINS GTP hydrolysis-dependent mechanoenzymes participate in polymerization of cytoskeletal proteins tubulin and septin, in polypeptide elongation/translocation on ribosomes, and in the trafficking of membranes and vesicles. Even so, the largest group of GTP-regulatory proteins participate in signal transduction pathways. They include the small GTPases (so named to indicate their low molecular weight) and the receptor-associated heterotrimeric G-proteins. In response to specific input signals or receptor occupancy, these signal-transducing mechanoenzymes control metabolism by mediating the activation or inhibition of their respective target. All G-proteins exist in at least two

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

conformational states, one characterized by the presence of bound GTP, and the other by GDP. Their interconversion is accomplished by an intrinsic ‘‘GTPase’’ activity as well as the participation of GTPase-activating proterins (GAPs) and nucleotide-exchange factors (NEFs). Bound GTP is converted to bound-GDP by the GAP-activated GTPase, whereas bound GDP is converted to bound GTP by release of GDP and binding/exchange of GTP in the presence of NEFs. The rate of interconversion between active and inactive states is limited by GTPase hydrolysis, and GTPase-activating proteins (GAPs) can accelerate hydrolysis, as can G-protein association with a conformational state of its effector. In this respect, certain G-proteins are clocks, which are controlled by their intrinsic rate constants for GTP hydrolysis, Pi release, and nucleotide exchange. G-proteins behave as switches, adapters, latches, and/or sensors. The small GTPases have a central six-stranded b-pleated sheet surrounded by a helices. Five polypeptide loops form the guanine nucleotide binding site that is highly conserved among the numerous members of this protein superfamily. Specific examples of GTP-regulatory proteins are: (a) transducin, the retinal regulatory protein that, in its bound-GTP state, stimulates cyclic GMP phosphodiesterase; (b) elongation factor Tu, a ribosomal protein that in its GTPbound complex (EF-Tu$GTP) moves an aminoacyl-tRNA into the ribosome-mRNA complex A-site for aminoacylation of a growing polypeptide; and (c) Ras, another G-protein that, in its GTP-containing form, activates its downstream tyrosine kinase targets. b-g-Phosphoanhydride bond cleavage of bound GTP9 allows each of these regulatory proteins to undergo a conformational transition from its stimulatory state to its quiescent state. In the classical reaction cycle for GTP-regulated proteins (Fig. 13.18), a G protein-linked receptor possesses a stereospecific ligand-receptor site on the outer cell surface, and a Ga$Gb$Gg-heterotrimer binding site lodged on the inside surface. Binding of the physiologic ligand at its specific receptor binding site activates the bound Ga$Gb$ Gg-heterotrimer, causing the Ga-subunit to release a GDP molecule and to bind a GTP molecule. This nucleotide exchange reaction allows the GTP$Ga subunit to be released from its Gb$Gg partner. The newly dissociated Ga subunit then initiates signal transduction events by activating (or inhibiting) the activity of its target enzyme. With time, the Ga subunit catalyzed the hydrolysis of its bound GTP, returning the Ga subunit to its inactive GDP$Ga form that then recombines with Gb$Gg, thereby re-forming a deactivated Ga$Gb$Gg-heterotrimer. These mechanoenzymes catalyze in-line, SN2 GTP hydrolysis, wherein the g-phosphoryl undergoes inversion of configuration as it is transferred directly to water. This consistent finding excludes the occurrence of a phosphorylenzyme intermediate. The kinetics of GTP hydrolysis and guanine nucleotide exchange determine the time-dependence

783

G.GDP R + A = A.R* A.R*.G.GDP

Pi

NEP

A.R*.G.

GAP

A.R*.G.GTP *G.GTP FIGURE 13.18 Canonical catalytic reaction cycle for GTP-regulatory proteins. Step-1, Agonist-A starts the process by activating receptor-R; Step-2, activated receptor R* binds to G$GDP; Step-3, GDP dissociates from *R$G$GDP complex; Step-4, *R$G binds GTP and activates G-protein as *R$G$GTPU; Step-5, activated receptor R* departs, leaving behind the activated *G$$GTP$ complex to stimulate ‘‘downstream’’ metabolic processes; and Step-6, *G$$GTP$ hydrolyzes, thereby deactivating G-protein and terminating the activated metabolic condition. In this model, the activated receptor *R plays a catalytic role by activating more than one G-protein. This is evident in the reaction cycle. Note also that the signal transduction efficiency Etrans of G-proteins is measured by the ratio of active to inactive enzyme (i.e., Etrans ¼ ‘‘GTPase’’$GTP/ ‘‘GTPase’’$GDP). Etrans is proportional to the ratio of the dissociation rate constant (kdissociation) for the ‘‘GTPase’’$GDP complex to the hydrolysis rate constant (khydrolysis) for the ‘‘GTPase’’$GTP complex. Therefore, nucleotide exchange proteins (NEPs) and GTPase-activating proteins (GAPs) play an important role in determining the duration of ‘‘GTPase’’$ GTP$Target Protein and ‘‘GTPase’’$GDP$Target-Protein complexes.

of activated or inhibited states of the particular G-protein. Gia1, for example, has a GTP hydrolysis turnover number (kcat) of ~3 min1; Ras has a kcat of 0.3 min1; and EF-Tu has a value of 0.003 min1. Tetracoordinate AlF4– ion activates Ga subunits by mimicking GTP’s g-phosphoryl group as it undergoes hydrolysis and assumes a pentavalent geometry. Depending on the specific G-protein, GTPgS binding is either slowly hydrolyzed or nonhydrolyzable, thereby sustaining the activated state of GTP regulatory proteins.

13.8. AAAD MECHANOENZYMES The AAAþ (standing for ATPases Associated with various cellular Activities) mechanoenzyme family is large and functionally diverse. These enzymes have the capacity to alter the conformation of specific target proteins and they play important roles in various cellular processes, including proteolysis, membrane fusion, cytoskeletal regulation, protein folding, and DNA replication (see Table 13.6). These energases are united by the presence of similar structural elements in their ATP binding/hydrolysis domain, the latter creating and transmitting mechanical forces (Iyer

Enzyme Kinetics

784

TABLE 13.6. Some Examples of AAAþ Mechanoenzymes Name

Target

Cellular Function

Katanin

Microtubules

Severs microtubule cytoskeleton

NSF and p97

Pex1p and Pex6p

Drives the asymmetric priming of two Peroxisomal membrane fusion partners

p97/Cdc48p

Uncertain

Cytosolic chaperone required for endoplasmic reticulum-associated protein degradation

Clamp loader

DNA polymerase processivity clamp

Loads processivity clamp onto doublestranded DNA

Replicative helicase

Double-stranded DNA

Alters DNA structure during replication

NVL2

Ribosomal protein L5

Regulates ribosome biogenesis

Dynein

Microtubule

Intracellular, flagellar, and ciliary motility

et al., 2004; Ogura and Wilkinson, 2001; Patel and Latterich, 1998; Vale, 2000). AAAþ family members have been identified in a wide range of organisms, ranging from bacteria to mammals. Their broad involvement in living systems attests to how evolution has crafted interfaces between an integral force-generating domain and a target protein (substrate) recognition domain. While Arabidopsis thaliana appears to possess the most (~140), most eukaryotes contain 50–80 different AAAþ proteins (e.g., budding yeast, for example, has ~50). In eukaryotes, for example, DNA is folded and wound into nucleosomal arrays, which then undergo further compaction to form diffuse chromatin and dense chromosome structures. Compaction has a major impact on the efficient control of nuclear processes requiring enzyme access to the DNA sequences during recombination, replication, transcription, and repair. Chromatin remodeling is accomplished at two levels: first by enzymes that covalently modify nucleosomal histones (e.g., acetylation-deacetylation, phosphorylation-dephosphorylation, methylation-demethylation, as well as ADP-ribosylation); and second by multi-subunit AAAþ mechanoenzyme complexes that disrupt histone-DNA interactions. The latter enzymes forcibly pry open chromatin in reactions that often limit the rates of DNA replication and transcription. AAAþ mechanoenzymes harness the Gibbs energy of ATP binding and/or hydrolysis to transmit conformational forces to their macromolecular targets (substrates), thereby inducing conformational changes to their respective

substrate proteins. Their ATP-binding/hydrolysis domain is the engine that generates a conformational change, and an attached interaction domain serves as the transmission that mechanically deforms the macromolecular AAAþ mechanoenzyme’s substrate (target). By opening or closing otherwise closed or open conformations of the target, these work proteins exert inhibitory and activating effects that control the target’s role in cellular physiology. Therefore, just as the so-called G-proteins are misnamed GTPases, the AAAþ enzymes are not true ATPases; both catalyze chemical reactions that are doubtlessly of the energase type.

13.8.1. The AAAD ATPases Possess Common Structural Elements All AAAþ ATPases contain structurally conserved, 200- to 250-residue ATP-binding domains containing both wellknown Walker A and Walker B motifs, phosphoryl sensor elements, as well as several other characteristic structural features. As described in the excellent review of Hanson and Whiteheart (2005), the AAAþ domain (Fig. 13.19) consists of two sub-domains. The wedge-shaped N-terminal Rossman Fold consisting of a b-sheet of parallel strands arranged in a b5–b1–b4–b3–b2 sequence at its core. This sequence, along with the insertion of b4 between b1 and b3, as well as the lack of other strands adjacent to b2, distinguishes AAAþ domains from other nucleotide-binding domains. Beyond these shared structural features, AAAþ domain diversity arises from the number and position of the a-helices connecting strands of the central b-sheet and

FIGURE 13.19 Typical structural features of AAAD domains. The crystal structure of the second AAAþ domain of NSF103 (Protein Data Bank accession code 1D2N) is presented as a model AAAþ domain. Approximate positions of the key structural elements are highlighted, and the bound nucleotide analog p(NH)ppG (coordinated by Mg2þ) is shown in stick representation. From Hanson and Whiteheart (2005) with permission of the authors and publisher.

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

from insertions between strands (Hanson and Whiteheart, 2005). The C-terminal sub-domain is composed of several b-helices (Beyer, 1997; Frickey and Lupas, 2004; Sauer, et al., 2004) and is an important distinguishing feature of the AAAþ family compared with other nucleotide-binding proteins. This sub-domain lies above the wide end of the N-terminal wedge and forms a partial lid over the nucleotide-binding site. Evidence of this variation-on-a-theme nature of AAAþ mechanoenzyme evolution are the many observations that their substrate interactions can be reliably manipulated by introducing mutations at key positions in their nucleotide binding/hydrolysis domain and their macromolecular substrate recognition/deformation regions. Hanson and Whiteheart (2005) tabulated these mutations for manipulating ATP and substrate interactions (Table 13.7). Some of the site-directed mutagenesis observations suggest strategies by which putative macromolecular substrates for a given AAAþ protein can be readily identified by noncovalently trapping the AAAþ Enz$S complex as a highaffinity noncovalent complex.

13.8.2. DNA Processivity Clamp Loader: An AAAD Mechanoenzyme The DNA polymerase clamp, which was first identified as a protein factor that greatly enhances the efficiency of DNA replication by E. coli DNA polymerase (Hurwitz and Wickner, 1974; Wickner, 1976), is required by both prokaryotic and eukaryotic DNA polymerases (O’Donnell, 1987; 2006; O’Donnell and Kuriyan, 2006; Maki and Kornberg, 1988). When bound to a sliding clamp, DNA

785

polymerase processivity (i.e., its repetitive catalytic action without dissociating from its polymeric substrate) increases from tens of nucleotides incorporated per DNA binding event to thousands. The increased processivity permits polymerases to accomplish the complete replication of an entire genome. The clamp’s closed multi-subunit ring structure (the right-handed, flattened spiral of which resembles a ‘‘lockwasher’’) accommodates a molecule of duplex DNA in a manner that allows it to slide freely. Given the clamp’s propensity to adopt closed-ring conformation, uncatalyzed clamp loading onto DNA is highly inefficient. A specialized AAAþ mechanoenzyme known as a clamp loader is therefore essential for rapid ATP-dependent loading of the clamp onto DNA. Once the clamp is loaded, its loader dissociates, and the polymerase binds at a nearby site, allowing the polymerase and DNA to remain associated as the DNA template is copied. On the leading strand, where DNA synthesis is continuous, one clamp is required at each replication fork. On the lagging strand, where DNA is synthesized discontinuously, a clamp is needed for each Okazaki fragment synthesized. In E. coli, where the replication fork progresses at rates of at least 500 nucleotides/ sec, clamp loading on the lagging strand must be rapid. The clamp-bound polymerase proceeds along the DNA at rates limited chiefly by the rate of nucleotide incorporation. Sliding clamps are also known to interact with a number of other protein factors and enzymes needed for different aspects of DNA metabolism. The simple elegance of a sliding torroidal clamp in confining polymerases on their nucleic acid templates is thus a splendid example of how, in Nature, form follows function and vice versa. As shown in Fig. 13.20, the

TABLE 13.7 Molecular Toolbox for Manipulating Interactions of AAAþ Proteins* Domain/Motif

Key Residue(s)

Typical Mutation

Effects

Walker A

K in GXXXXGK[T/S]

Lys / Ala

Inhibits nucleotide binding

Walker B

E in hhhhDE

Glu / Gln

Impairs ATP hydrolysis, functions as a substrate trap

Sensor 1

A polar residue at the end of b-strand-4

N/The / Ala

Impairs ATP hydrolysis

Sensor 2

R in the sequence GAR near the N end of b-strand-7

Arg / Ala/Met

Impairs ATP hydrolysis, sometimes also ATP binding

Arginine fingers

R residues in the SRH at the end of a-helix 4

Arg / Ala

Impairs ATP hydrolysis in most AAAþ proteins

Arg / Glu

Impairs ATP hydrolysis and oligomerization

Ty / Xaa

Mutations impair macromolecular substrate binding and processing, with little or no effect on ATP hydrolysis or oligomerization.

Pore loop

YVG in the 22 loop

Val / Xaa Gly / Xaa Abbreviations used are: h, hydrophobic residue; SRH, second region of homology; Xaa, any amino acid. *The table is based mainly on the AAAþ protein review by Hanson and Whiteheart (2005).

Enzyme Kinetics

786

Eukaryotic Clamp (PCNA)

Escherichia coli Clamp (β)

FIGURE 13.20 Ribbon diagrams of eukaryotic clamp PCNA (left) and the E. coli b-clamp (right). Peptide loops on the outer edge of the rings covalently link domains to form individual subunits. A single PCNA (name originally given as: Proliferating Cell Nuclear Antigen) subunit is formed by covalently linking two domains and a-b subunit by linking three domains (Bowman, O’Donnell and Kuriyan, 2004). Thus, PCNA contains three inter-domain interfaces, and b contains two, that interact only through noncovalent bonds and must be opened to allow DNA to pass into the ring’s aperture. The diagrams presented below illustrate the structural organization of the clamps into six domains of similar fold (circles) arranged in a ring. Figure and legend courtesy of Professor Linda Bloom.

eukaryotic sliding clamp PCNA has three identical subunits whereas the E. coli sliding clamp has only two; however, the eukaryotic subunit is a pseudo-dimer and the E. coli monomer is pseudo-trimeric. Once assembled into a ring, their subunits are arranged in a head-to-tail fashion, giving the clamps a pseudo-hexameric symmetry relative to a rotational axis extending through the center and of their front and back faces. DNA polymerase and the clamp loader are known to interact with the same clamp face in a mutually exclusive manner. In fact, the a-subunit of DNA polymerase III interacts with the b-clamp via its extreme seven C-terminal residues, and the Pol III interaction site is the same site as that of the d-subunit of the clamp loader, thereby providing a structural basis for switching between the clamp loading machinery and the polymerase itself. This constraint means the clamp loader must disengage from the clamp before the polymerase can bind. Clamp loaders have a functional core consisting of five subunits, each of which contains three structurally homologous domains joined by flexible linkers (Panels A–C, Fig. 13.21). As shown in Panel A, the N-terminal domains (I and II) share homology with other members of the AAAþ family, but the C-terminal domain (III) is unique to clamp loaders. The loader’s five subunits are also arranged in a ring via tight noncovalent interactions between the C-terminal domains (Panels B and C). The N-terminal domains I and II are loosely packed, as if suspended from a ‘‘collar’’ formed by the C-terminal domains, and a large opening exists on the N-terminal face. Beyond the clamp loader’s conserved five-subunit core, the RFC-A subunit contains N-terminal and C-terminal

extensions (absent in other subunits) of unknown function. The N-terminal region is not required directly for clamp loading or yeast viability and was deleted from RFC-A in the crystal structure (Panel D, Fig. 12.21). Three alternative RFC complexes are formed in which RFC-A is replaced by other proteins. In one such complex, S. cerevisiae Rad24 or human Rad17 substitutes for RFC A to form a clamp loader required for a checkpoint response during S-phase. Although the molecular details are still unclear, Rad24/ Rad17-RFC, along with the checkpoint clamp, is recruited to replication forks stalled at sites of DNA damage, where it plays a role in mediating a checkpoint response.

13.8.2a. ATP-dependent Clamp Loading Ultimately, ATP binding and hydrolysis promote conformational changes in the clamp loader that modulate its affinity for the clamp and DNA. In its simplest terms, ATP binding promotes conformational changes that give the clamp loader a high affinity for the clamp and DNA, with ATP hydrolysis having the opposite effect and reducing clamp loader affinity for the clamp and DNA. While the E. coli clamp loader was thought to load a clamp on the leading strand in the absence of ATP hydrolysis (Glover and McHenry, 2001; Johanson and McHenry, 1984), this result is a likely artifact of weaker Loader$Clamp interactions in assays carried out in the presence of ATPgS (Anderson et al., 2007). Although the five core clamp loader subunits share homology with AAAþ proteins, only four subunits (A, B, C, and D) in RFC and three subunits (B, C, and D) in the E. coli clamp loader, are functionally active ATP hydrolyzing units. ATP binding sites are located at the interface of domain I and domain II within a subunit, and each contains conserved Walker A and Walker B sequence motifs. Arrangement of the subunits in a ring places ATP binding sites at the interface of adjacent subunits (Panels A–C, Fig. 13.21). Conserved Arg fingers extend from one subunit to interact with ATP bound to the neighboring subunit. This arrangement promotes dynamic coupling of ATP binding and hydrolysis to conformational changes in the complex that modulate its affinity for the clamp and DNA. Biochemical studies with RFC suggest that ATP sites fill sequentially, such that binding two molecules of ATPgS promotes binding of either PCNA or DNA, and binding of PCNA or DNA promotes binding of a third molecule of ATPgS, and formation of a ternary RFC$PCNA$DNA complex promotes binding a fourth ATPgS molecule (Gomes, Schmidt and Burgers, 2001). In contrast, all three sites in the E. coli g complex bind ATP in the absence of the clamp or DNA (Johnson and O’Donnell, 2003).

13.8.2b. Clamp and DNA Binding Reactions Structural and biochemical studies support a model in which the clamp loader binds the clamp via contacts made between

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

787

FIGURE 13.21 Structural features of Saccharomyces cerevisiae RFC and its interaction with PCNA. The color code for RFC (abbreviation for: Replication Factor C) corresponds to S. cerevisiae/ human subunits: Rfc1/p140 (purple); Rfc4/p40 (blue); Rfc3/p36 (red); Rfc2/p37 (green) and Rfc5/p38 (yellow). A, Ribbondiagram of the scRfc2 subunit bound to ATPgS (white). B, Diagram illustrating the arrangement of RFC subunits in a ring with Arg finger (wedge-like protrusion) extending from one subunit to the ATP site of an adjacent subunit. C, View of scRFC, with downward view of the ‘‘collar’’ formed by the C-terminal domains. Notice that subunits spiral around an axis through the center of the ring, and the N-terminal domain of one subunit approaches the ATP binding site (ATPgS in white) of the adjacent subunit. D, Ribbon diagram of scRFC bound to PCNA. RFC-A, RFC-B, and RFC-C contact the face of PCNA, but RFC-D and RFC-E do not. The clamp is closed, even though the A-, B-, C-, and D-subunits contain bound ATPgS (white), perhaps due to mutation of Arg fingers to Ala and/or substitution of ATPgS for ATP. In the open RFC$PCNA complex, the cyan and magenta PCNA subunits are anticipated to ‘‘swing up’’ and bind the N-terminal face of RFC (see panel F below). E, Diagram illustrating the anticipated ‘‘footprint’’ (black circles) for RFC binding to PCNA. F, The clamp-loading reaction can be divided into two phases, based on ATP requirements: Stage-1, formation of a ternary Loader$Clamp$DNA complex promoted by ATP binding, and Stage-2, release of the clamp on DNA promoted by ATP hydrolysis. The diagram illustrates the structures of PCNA and RFC with individual protein domains represented by spheres or ovals; note that RFC-A contains an extra C-terminal domain that lies in the gap between RFC-A and RFC-E. In the ternary complex, the RFC$PCNA complex adopts a conformation that spirals like DNA. The clamp is opened out of plane. Duplex DNA enters the protein complex through the large opening on the N-terminal face of RFC, and the single-stranded template overhang likely exits through the gap between RFC-A and RFC-E. Figure and legend courtesy of Professor Linda Bloom.

the N-terminal domain (I) of each clamp loader subunit and one face of the clamp (Panels D–F, Fig. 13.21). Hydrophobic residues in a conserved sequence motif in the A-subunit bind a hydrophobic pocket on the clamp that is located near the interface of neighboring domains and the covalent linker peptide. The RFC$PCNA structure in Panel D of Fig. 13.21 shows that the B- and C-subunits make similar contacts, but to a lesser extent. In this structure the clamp is not opened, and it is likely that productive formation of an open clamp Loader$Clamp complex requires each of the five clamp loader subunits to make similar contacts with the clamp. Given the six-domain structure of the clamp, five of the six inter-domain regions could interact with individual clamp loader subunits and the sixth would be free to open (Panels E and F, Fig. 13.21). Subunits within the clamp loader adopt a spiral conformation relative to an axis through the center of the ring, and this probably facilitates the opening of the clamp in an outof-plane conformation. The duplex portion of a primed template enters the Loader$Clamp complex via a large opening present on the N-terminal face of the clamp loader and extends up towards the collar (see Panel F, Fig. 13.21). The clamp loader fits on DNA in a manner similar to a ‘‘screw cap’’ with the subunits and clamp spiraling around

the duplex with the same pitch as the helix. In one model, the clamp loader would bind DNA in a manner in which the 39-primer-end would run into the collar formed by the C-terminal domains and the single-stranded template overhang would exit in the gap between N terminal domains of the A- and E-subunits.

13.8.2c. Dynamic Interactions in Clamp Loading onto DNA Structural studies only provide a fairly static view of clamp loading. To load clamps, a succession of conformational changes between loader, clamp, and DNA must occur in an appropriate time ordered sequence. Although the modes of action are apt to be quite similar for prokaryotic and eukaryotic clamp loading, the kinetics of the E. coli clamp loading reactions have advanced at a steady pace by taking advantage of: (a) fluorescence anisotropy to directly examine the association/dissociation of loader (enzyme) and clamp (macromolecular substrate) in the presence of DNA and ATP; (b) stopped-flow burst-phase kinetics to separate pre-steady-state and steady-state phases; (c) stopped-flow measurements using a continuous fluorescence assay of Pi release (see Section 4.5.7) to establish

Enzyme Kinetics

788

13.9. GRADIENT-DRIVEN MECHANOENZYMATIC PROCESSES With the arguable exception of the ribosome, the ATP synthase and the bacterial flagellar motor are amongst the oldest and most elaborate mechanoenzyme complexes invented by living systems. Because their structures and general modes of operation are well described in most graduate-level textbooks (Alberts et al., 1994; Voet and Voet, 2003), the topic is treated cursorily here. A shared feature of ATP synthase and the bacterial flagellar motor is Rotary Catalysis, meaning that these mechanoenzymes must generate torque fT. As a measure of how hard an object is rotated, torque depends on the applied force F, the length of the lever arm connecting axis of rotation and point of that applied force, as well as the angle of rotation. In the case of the ATP synthase, the torque presumably is used to force the substrates together in a manner that chemically promotes extrusion of the elements of water from ADP and orthophosphate that are juxtaposed within each active site. (Alternatively, rotation may facilitate the conversion of a tightly bound product state into a loosely bound product state.) In its overall synthetic action, the synthase takes up a proton as it produces ATP. The bacterial rotary flagellar motor is likewise a torque-generating mechanoenzyme that propels the microorganism. Unlike the eukaryotic flagellum, which generates a true whipping motion, the bacterial flagellum rotates in a corkscrew like motion.

intermembrane 2H+ removal of protons space (outside)

lowers pH

Oxidoreductase

Aox

BH2

Cox

C

∇

the rates and stoichiometry of ATP hydrolysis; (d) radiometric assays using subunits loaded with radiolabeled ATP; and (e) fluorescence resonance energy transfer (FRET) to probe near distance interactions (on the ~1–6 nm lengthscale) between neighboring subunits. Bloom (2006) describes how these approaches have been employed to provide important insights about clamp loading. Clamp loader interactions with its binding partners must be dynamic, so that each binding (or hydrolysis) event promotes a conformational change in the clamp loader that facilitates the next step in the reaction. On a very basic level, the clamp loader must initially have a high affinity for the clamp and DNA to bring these macromolecules together, but then must have a low affinity to release the clamp on DNA for use by a DNA polymerase. This affinity modulation is accomplished in part by ATP binding and hydrolysis. In an ATP-bound conformational state, the clamp loader has a high affinity for the clamp and DNA, and on hydrolysis of ATP, the affinity is decreased and the clamp loader releases the clamp on DNA. However, ATP binding and hydrolysis alone cannot provide a sufficient mechanism for ordered affinity modulation that supports an efficient clamp loading reaction. Some additional level of regulation of ATP binding and hydrolysis is likely to exist.

Ared

Oxidoreductase

pH

Cred

B

A

matrix space (outside)

2H+

expelled protons raises pH

FIGURE 13.22 Essential features of Mitchell’s Chemiosmotic Theory for oxidative phosphorylation. Shown is the inner-mitochondrial membrane with its embedded proton pump that extrudes protons when driven by the electron transport system. In Mitchell’s scheme, protontranslocating pumps are a pair of oxidoreductases ‘‘A’’ and ‘‘B’’ that link proton transport to electron transport by means of electron carriers such as ubiquinones and flavins, each accepting two protons as well as two electrons. See text for additional details.

As illustrated in Fig. 13.22, we now know that rotary catalysis of ATP synthase is powered by energy derived from mitochondrial transmembrane proton gradients. Oxidative phosphorylation is thus a process that harnesses the free energy of NADH oxidation (Reaction: NADH þ Hþ þ 0.5 O2 # NADþ þ H2O) to drive net ATP synthesis (Reaction: ADP þ Pi þ Hþ # ATP þ H2O). The DG for NADH oxidation is obtained by the following simple calculation:    9 ¼ 0:82 V  ð0:32 VÞ ¼ 1:14 V, such DE 9 DEox9  DEred that the corresponding change in Gibbs energy (given by:   DG 9 ¼ nFDE 9 ) equals – 220 kJ/mol. While this calculation verifies that NADH oxidation yields sufficient Gibbs free energy for such purposes, intensive research on mitochondria failed to demonstrate the formation of a telltale covalent intermediate in oxidative phosphorylation, leaving the underlying mechanism in doubt for decades. Then, in 1961, British biochemist Peter Mitchell provided the crucial link for understanding bioenergetics of oxidative phosphorylation. His Chemiosmotic Principle (Fig. 13.22) revealed how energy can be stored by a transmembrane proton gradient. He suggested that the inner mitochondrial membrane is a closed proton-impermeant membrane containing proton pumps that catalyze the reaction: H+INNER

H+MATRIX

ETS

ETS

Energized

De-Energized

Scheme 13.5

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

driven by the flow of NADH-derived electrons via the electron transport system (ETS). The fundamental principle is that the ATP synthase and the bacterial rotary flagellum (Fig. 13.23) utilize the Gibbs free energy difference of the proton gradient to generate force. As described in the next section, the ATP synthase operates by Boyer’s Binding-Change Mechanism or some variant thereof. Because the physiologic DG for ATP synthesis is þ34.5 kJ$mol1, at least two protons are required (more if the efficiency of energetic coupling is low). Therefore, the central question becomes: What is the DG of Hþ transport against a Hþ gradient? Starting with the relevant experimental data (e.g., DpH ¼ pHinside – pHoutside z 0.75, and the transmembrane electric potential DJ ¼ þ150 mV), we seek the corresponding DG value, given by:

    DG ¼ DG 9 þ RT ln Hþ P = Hþ N þ ZF DJ 13.15 

where F is Faraday’s constant. For DG 9 ¼ 0, DG ¼ RT ln([Hþ]P/[Hþ]N) þ ZFDJ, such that: DG ¼ 2.303 RT log([Hþ]P/[Hþ]N) þ ZF DJ ¼ 2.303 RT (pHN – pHP) þ ZF DJ ¼ 2.303 RT DpH þ ZF DJ ¼ 2.303 RT (0.75) þ (þ1) F(0.75) z þ20 kJ/mol Hþ. (Note: The total protonic potential difference Dp in volts, may also be expressed as: Dp ¼ emf – 2.303  RT/F  DpH). Based on this estimate, the DG for NADH oxidation is sufficient to drive 10–11 protons against the electrochemical gradient. While this

789

quantity of available energy could in principle drive the synthesis of 6–7 ATP molecules from ADP and orthophosphate, the actual number is 2–3. Because molecular motors harness the Gibbs free energy attending the dissipation of a proton gradient, such that fT ¼ DGgradient/150d, where d is the motor’s elementary displacement. If machinery-dependent parameters remain invariant, the generated torque fT should be proportional to the potential applied in accordance with the energy balance equation: fT ¼

n3qe Dp d

13.16

where n is the number of protons translocated, 3 is the efficiency, qe is the proton’s electrical charge, and Dp is the unitary proton potential. Finally, as discussed by Khan and Sheetz (1997), proton fluxes should vary linearly with force if the coupling process is close to equilibrium. At high loads that near the equilibrium stall force, the torque scales roughly linearly with the applied potential, and the force sustained is the same, whether the potential is supplied as an electrical or chemical potential. The structural and biophysical data attesting to the aforementioned design and operation features are clearly beyond the scope of this kinetics textbook.

FIGURE 13.23 Chemiosmotic generation of torque by rotary motors. A, Bacterial flagella consist of: (a) a reversible rotary motor and drive shaft extending from the cytoplasm, through the wall and ending at the outer membrane; (b) a short proximal hook, which serves as a flexible coupling to; (c) a long helical filament acting as a propeller. Torque is generated between a stator (itself connected to the rigid peptidoglycan cell wall) and a rotor connected to the flagellar filament. Motile bacteria actively respond to external stimuli by modulating the direction of rotation of their flagella. The presence of a chemoattractant, such as aspartate, or removal of a chemorepellent, such as leucine, enhances counterclockwise (CCW) rotation, causing cells to extend actively motile runs that carry them in a favorable direction, often at speeds >25 mm/s. Such behavior is manifested through the action of the protein kinase CheA, which phosphorylates a key effector molecule CheY. When bound to FliM (a component at the base of the flagellar motor), phosphorylated CheY promotes clockwise rotation. B, FOF1 ATPase consists of two structurally and functionally distinct parts: the membrane-integrated ion-translocating FO complex and the peripheral F1 complex, which carries the catalytic sites for ATP synthesis and hydrolysis. The enzyme catalyzes the synthesis of ATP from ADP and inorganic phosphate utilizing the energy of an electrochemical ion gradient. ATP synthases can also hydrolyze ATP, thereby generating a transmembrane proton gradient. Proton and stator structures are shaded light and dark gray, respectively. The curved arrows indicate the probable proton pathways.

Enzyme Kinetics

790

13.10. ATP SYNTHASE: BOYER’S BINDING CHANGE MECHANISM A triumph for enzyme kinetics is Nobel Laureate Paul Boyer’s formulation of the Rotary Catalysis Model for the ATP synthase reaction. Known widely by the unassuming name ‘‘Binding Change Mechanism’’, this model accounts for ATP synthesis in terms of energy-driven extreme changes in nucleotide binding affinity, by explaining how ATP synthase can reversibly transduce a proton-motive force into the net synthesis of ATP and H2O from ADP and orthophosphate (Hutton and Boyer, 1979; Boyer, 2003; Boyer, Cross and Momsen, 1973). The clues about extreme, cooperative subunit-subunit interactions that are the essence of the binding change mechanism were inferred almost entirely from kinetic data. As elucidated in Boyer’s laboratory. The Binding Change Mechanism suggests that at any instant nucleotide binding is different within each of the synthase’s three ab-heterodimeric units, such that: (a) in one state, called the Loose- (or L-) Conformation, ADP and Pi are loosely bound; (b) in

Boyer’s Binding Change Mechanism Loose

Loose ADP + Pi

1

S

ADP Pi S ATP

ATP Open

Tight

Open

Tight

2 Tight

Tight

ATP

ADP Pi S

S 3 Loose

Open

ATP + H2O

Energy Input

ATP Loose Open

FIGURE 13.24 Affinity-modulated ATP synthesis by ATP synthase. In the Binding Change Mechanism, each of the three ab-dimers adopts a conformationally distinct form that is constrained by its rotational position in the synthase’s trimeric (ab)3 complex. As explained in the text, energy-driven rotation of the stator simultaneously is thought to change the conformations of all three ab-dimers, resulting in extreme positive cooperativity in the tight-binding conformation as well as extreme negative cooperativity in the loose-binding conformation. Note: Positive and negative cooperativity represent another way of implying the occurrence of affinity-modulated interactions. See text for additional details.

a second state, called the Tight- (or T-) Conformation, water is extruded from ADP and Pi to form ATP; and (c) in the third state, called the Open- (or O-) Conformation, ATP is readily released from its binding site. As shown in Fig. 12.24, transition between these states depends on the position of the Stator, the rotational position of which is determined by mechanically coupled transmembrane proton-translocating (see Fig. 13.23). Translocation of four protons down the proton gradient drives conformational changes within each ab-unit, as they are forced to rotate past the stator. Rotation of the g-subunit, which is located deep within the pseudo-three-fold axis between the (ab)3 complex, is believed to deform the surrounding catalytic subunits. In the binding change mechanism, the major energy-requiring step is not ATP synthesis; rather, energy is required to drive simultaneous and highly cooperative binding of its substrates (ADP, Pi, and Hþ) and to release its products (ATP and H2O). The rotation is actuated by the energy released during proton translocation. At maximal rates of catalysis, the ATP synthase rotates at ~8,800 rpm, and with three molecules of ATP synthesized per turn, kcat is approximately 400 s1. Kinetic studies using an oxygen18 tracer indicate that this turnover rate is limited by product release (Boyer, 2002). Elucidation of this energy-transducing mechanism was the crowning achievement of Paul Boyer’s life-long pursuit of understanding enzyme catalysis by innovating numerous kinetics techniques, including isotope exchange at equilibrium and oxygen-18 tracer methods for analyzing phosphotransferase, phosphohydrolase, and phosphodiesterase mechanisms. Although the Binding Change Mechanism provides the broad outlines for rotary catalysis, more up-to-date work has provided a fuller picture of active-site nucleotide occupancy and the linkage of the synthase’s chemical and mechanical reactions. More up-to-date work suggests that, beyond the usual substrate binding energy and transition-state stabilization in single-site, non-cooperative enzymes, positive catalytic cooperativity between the three catalytic sites plays a major role of rate acceleration by F 1 (Senior, Nadanaciva and Weber, 2000). Indeed, exclusive filling of catalytic Site-1 with MgATP2 (in what has been termed uni-site catalysis) is attended by extremely low net hydrolysis rate of ~0.001 s1 (Penefsky and Cross, 1991). Simultaneous filling of two sites (so-called ‘‘bi-site catalysis’’ model) results in a substantially greater rate acceleration, amounting to 2–3% of Vmax. However, physiological rates of ~100 s1 require filling of all three sites (‘‘tri-site’’ or multi-site catalysis model) with MgATP2 (Lo¨bau, Weber and Senior, 1998; Weber et al., 1993; Weber, Dunn and Senior, 1999). Under rapid steady-state conditions, multi-site hydrolysis occurs, and two of the catalytic sites contain bound MgADP while the third contains bound MgATP2 (Weber, Bowman and Senior, 1996).

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

791

(2007) also suggest that the 80 sub-step is driven by ATP binding, whereas the 40 sub-step is driven by release of ADP or Pi (Yasuda et al., 2001). After a 80 sub-step, g dwells at the 80 angle for ~2 ms, during which two 1 ms reactions take place (Yasuda et al., 2001), with one identified as ATP hydrolysis (Shimabukuro et al., 2003). Nishizaka et al. (2004) established that the ATP hydrolyzed there is the molecule that had been bound 200 previously (Fig. 13.25B) and an ATP molecule that is bound at 0 is destined to be hydrolyzed after g rotates through another 120 þ 80 angle. Adachi et al. (2007) used high-speed imaging of g-rotation to show that the other of the two 1-ms reactions at 80 is Pi release, and that the Pi release drives the last 40 sub-step. They also demonstrated that ADP is released at 240 after it is bound as ATP at 0 , by direct observation of the binding and release of a fluorescent ATP analogue 20-O-Cy3-EDAATP (Oiwa et al., 2003) in a single molecule of F1ATPase. With the possible alternative in Fig. 13.25C, the findings of Adachi et al. (2007) essentially verify the basic scheme as shown in Fig. 13.25B, which had been first proposed by Weber and Senior (2000).

Although direct observation of ATP synthase rotary catalysis was first accomplished by Noji et al. (1997), another important goal has been to establish the actual coupling scheme between the g-subunit rotation and chemical reactions. Adachi et al. (2007) proposed the scheme in Fig. 13.25 for ATP-driven rotation in the direction of ATP hydrolysis, noting that, in principle, ATP synthesis by reverse rotation would follow the same scheme by tracing individual steps in reverse. The same group previously demonstrated that rotation of F1-ATPase occurs in steps of 120 , each driven by hydrolysis of one ATP molecule (Adachi et al., 2000; Yasuda et al., 1998). At sub-millisec time resolution, the 120 step can be further resolved into 80–90 and 40–30 sub-steps. Although they proposed initial sub-step amplitudes as 90 and 30 (Yasuda et al., 2001), subsequent work by HironoHara et al. (2001), Nishizaka et al. (2004), and Shimabukuro et al. (2003) indicated they are closer to 80 and 40 . Although experimental precision does not allow absolute distinction and a possibility remains that a third small sub-step may exist between the two (Kinosita et al., 2004), they presently favor the latter values. Adachi et al.

-A-

ADP Release

ATP ATP binding binding awaiting ATP binding

120° 80°

T

T

0°

iv

T

DP ii

T

P iv

T

DP T ii’

T

T

T

P T

T

T

T

D DP iv”

ii” P

i’

T iii”

T i” T

DP

DP P

iii

T T

D

ii’ T

DP

DP T iv’

T

T

T i

iii’

D

ii

-C-

T

T i’

D

T

T

T

T

T iii

T

D

DP

T

D i

240°

200°

hydrolysis awaiting awaiting ATP pi release binding

-B-

360° 320°

Pi release in (B) Pi release in (B)

iii’

DP T iv’

D

P

T i”

T iii”

T

P

DP iv”

T

T ii”

FIGURE 13.25 Scheme for mechanochemical coupling of ATP synthase. (A) Schematized time-course of stepping rotation, wherein the vertical axis measures the rotary angle g, and the horizontal axis represents elapsed time. (Note: Colored events occurring in the catalytic site are shown in the same color in (B) or (C).) (B) Corresponding states for nucleotides situated in the three synthase’s catalytic sites. Each of the three circles represents one of three b subunits, each hosting a hydrolase/synthase catalytic site. The central gray ellipsoid represents the g-subunit, the thick arrow showing its orientation, with the ‘‘twelve o’clock’’ position in (i) corresponding to 0  in (A). Molecular species derived from the same ATP molecule are shown in the same color. Small arrows show the progress in this major reaction pathway; the configurations (ii), (ii), and (ii9) shown below the major path represent the instant immediately after ATP binding (i.e., the start of an 80 sub-step). (C) An alternative scheme in which Pi release lags behind ADP release. Figure and legend from Adachi et al. (2007) with permission of the authors and publisher (Elsevier).

Enzyme Kinetics

792

We may safely anticipate that the kinetic properties of each proton-driven ATP synthase and ATP hydrolysis-driven proton pump will prove to be systemdependent, meaning that adaptations are optimized for oxidative phosphorylation, photophosphorylation, and vesicle acidification. Learning how each energase works and how different organisms exploit rotary catalysis will be a significant research enterprise for many years to come.

13.11. ROLE OF ATP IN PROTEIN FOLDING The American enzyme chemist Christian Anfinsen was awarded the Nobel Prize in 1972 (see Table 1.2), for his demonstration in the early 1960s that the 124-residue polypeptide chain of pancreatic RNase folds spontaneously to form a native, compact enzyme possessing full catalytic activity. Even so, spontaneous folding of longer polypeptides cannot occur on biologically relevant timescales (Gething and Sambrook, 1992). As pointed out by Ranson, White and Saibal (1998), cellular conditions of high protein concentration, elevated temperature, physiologic ionic strength, etc. often conspire to favor aggregation as opposed to correct folding. From the rates of protein synthesis in rapidly dividing microorganism (e.g., 20–30 min for E. coli), it is evident that proteins fold into their specific three-dimensional conformations on a millisecond-to-second time-scale, far faster than would be possible if the molecule were to search the entire conformation space to find its lowest possible energy state. It now seems clear that proteins must carry out massive, parallel searches of protein conformation space, aided by the stabilizing effects of local minima allowing partially folded structures to persist as they transit along the folding trajectory in search of their final well-folded conformation. In fact, cells produce helper proteins, or chaperonins, that bind to unfolded polypeptide chains and facilitate their rapid and correct folding. Among these are the so-called heat shock proteins, of which many are ATPand GTP-dependent mechanoenzymes induced at elevated temperatures where the greater tendency for proteins to unfold is apt to require additional mechanisms to facilitate their refolding.

13.11.1. GroEL/GroES is a Model ‘‘Foldase’’ System The Escherichia coli chaperonin GroEL, also designated as GroEL14 to indicate its tetradecameric subunit structure, is arranged as two barrel-shaped heptamer rings consisting of fourteen identical 58-kDa subunits. Its cochaperonin GroES (or GroES7) is itself a ring composed

of seven identical 10-kDa subunits. In the presence of MgATP2, GroES7 forms stable complexes that cap the apical regions of the GroEL14 barrel. Together, GroEL and GroES7 mediate the refolding of over half of E. coli proteins (Viitanen, Gatenby and Lorimer, 1992). GroEL14 binds to nascent polypeptides, a process that in most cases depends on its interaction with GroES (Lissin, 1995). It was known for some time that GroEL possesses potassium ion-dependent ATPase activity (Viitanen et al., 1990), and several studies have shown that the apical domains of the GroEL swing outwards upon ATP and GroES binding (Chen et al., 1994; Saibil et al., 1993), indicating that the exposure of the hydrophobic substratebinding domains is triggered by ligand-induced conformational changes (Gibbons et al., 1996). Through the use of sedimentation-velocity ultracentrifugation and dynamic light-scattering to follow the protein folding process, Walters et al. (2002) demonstrated that ATP hydrolysis is required for the correct interactions of GroES7 and GroEL14.

13.11.2. GroEL/GroES Mediates an Annealing/Folding Cycle Although many details of chaperonin-mediated protein folding remain to be elucidated, Corrales and Fersht (1996) proposed a mechanism for the coordinated action of GroEL and GroES (Fig. 13.27) that: (a) reconcile the seemingly opposing views that the chaperone as a folding cage and an annealing machine; (b) accounts for the formation of different types of complexes of GroEL and GroES; and (c) defines a likely role for ATP hydrolysis. GroEL’s resting intracellular form in the absence of denatured proteins is thought to be the very fast folding form (Species-1). Fast-folding proteins that fold within a second or so, when bound to Species-1, will become native before appreciable quantities of ATP are hydrolyzed by GroEL, since its ATPase activity is low, with kcat z 0.04 s1 in the absence of denatured protein and 0.16–0.8 s1 in the presence of denatured protein. Thus, fastfolding proteins are quickly ejected by the GroES$GroEL$Nucleotide Complex before ATP hydrolysis, whereas slow-folding proteins enter the next cycle via the ATP hydrolase activity. Corrales and Fersht (1996) postulated that the ATPase activity of GroES$GroEL$Nucleotide Complex is a gatekeeping function that allows only slow folding proteins to be subjected to later chaperoning events. The choice between Form-2 / Form-3 and Form-2 / Form-4 paths in Fig. 13.26 may be described as a kinetic partitioning step. They also suggested that ATP binds in the ring of GroEL that is opposite to GroES (i.e., acting in trans).

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

DENATURED STATES OR MISFOLDED INTERMEDIATES

ATP

1

ADP

NATIVE PROTEIN OR PARTLY FOLDED PROTEIN

ES

ATP

ATP ADP

2

FAST FOLDERS

ADP

3

ES

SLOW FOLDERS

GATEKEEPING ATPase Pi + ADP +

“ANNEALING”

5

4

ES

CYCLING ATPase

ATP

ES

7

ES

ADP ATP

6

ATP ATP

ATP ATP

8

ES

9

ADP ATP

NATIVE PROTEIN OR PARTLY FOLDED PROTEIN

FIGURE 13.26 Minimal mechanism for GroE-mediated protein folding. Square-cornered rectangles represent the T-state, having strong affinity for peptides and weak affinity for GroES and ATP. Squarecornered rectangles represent the R-state, having weak affinity for peptides and strong affinity for GroES and ATP. A denatured state binds to Species-1, the GroEL$GroES$ATP$ADP complex. Fast folders, such as barnase, rapidly fold to give Species-3 and release folded protein. (Fast folding parts of larger proteins may also fold rapidly to give partly folded proteins on Species-3, where they may translocate or dissociate.) For slow folders, GroES dissociates from Species-2 on ATP hydrolysis to give Species-4, the T-state. There is an equilibrium with the R-state Species-6, induced by the binding of ATP. GroES does not directly give Species-2 on binding to Species-6. The T-state Species-4 has the potential for unfolding to a compact denatured state, as evidenced from the catalysis of 1H 4 2H exchange in barnase. The top cycle (Species-1 / Species-2 / Species-3) is a selection step that uses ATP hydrolysis as a gate-keeping action to allow access of slow folders to the bottom cycle which encompasses rounds of annealing, if necessary, and any folding or translocation. This cycle uses ATP hydrolysis to pump the conversion of R-state to T-state. The experiments on barnase have shown that it folds in the GroEL complexes Species-2 and Species-6 and in GroEL$GroES complexes that are likely to be Species-2 and Species-7, and that annealing processes occur (Species-4 / Species-5) and the complex Species-8 exists. All the illustrated allosteric changes are observed in these reactions. Other states, involving additional conformations of proteins, different combinations of ATP and ADP, or binding of denatured proteins to different rings, may be involved. Reproduced from Corrales and Fersht (1996) with permission of the author and publisher.

793

Peptides initially bind to the trans ring, but GroES dissociates on the hydrolysis of ATP and rebinds to the opposite end of GroEL so that the peptide ends up in the cavity cis to GroES (i.e., in the cavity contiguous with GroES). ATP hydrolysis in the trans ring expels GroES from the complex. Apart from the 20–30% of the protein that folds slowly because its peptidyl-prolyl bonds are in the wrong conformation, barnase folds too rapidly to enter the chaperoning cycle directly (Species-4 / Species-9 in Fig. 13.26). We have reconstructed events in the cycle by measuring the individual steps directly by mixing the reagents in a suitable order. The ATPase reaction (Species-2 / Species-4) causes the concomitant dissociation of the GroEL$GroES complex, a process that is also promoted by bound peptides. GroES will recombine with the GroEL$Nucleotide$ Proteindenatured complex after exchange of nucleotide. We mimicked this step by adding GroES to a preformed GroEL$BarnaseDenatured complex in the presence of physiological concentrations of ADP (0.38 mM) and ATP (2.7 mM). This gave a slower folding form of the GroEL*$GroES*$Nucleotide$Proteindenatured complex (kf ¼ 0.5 s1). Proteins that fold much more slowly than barnase will enter the chaperoning cycle after the first round of ATPase activity. As those proteins slowly fold, there will be many rounds of ATP hydrolysis that cause the expulsion of GroES, which is followed by its rebinding. We have studied directly the GroEL$Barnasedenatured complexes that should be present when GroES is not bound. Shown here are the allosteric conversions that occur on the binding of ATP to the GroEL$Barnasedenatured complexes. Annealing is most efficient when denatured barnase is bound to the T-state, which binds the denatured state most tightly (i.e., free GroEL or GroEL$ADP complexes: the binding of denatured proteins enhances the dissociation of nucleotides (37)). Folding takes place most rapidly from the R-states (Species-6 and Species-7) and slowly from Species-4. There will be a pulse of annealing every time ATP is hydrolyzed and causes the transient release of GroES and formation of the T-state. Thus ATP hydrolysis pumps GroEL from the weaker-binding/better-folding R-state, which is the predominant species, to the stronger binding/stronger annealing T-state. The protein can leave in a partly folded state from the cycle.

13.12. ACTOCLAMPIN MOLECULAR MOTORS Actin-based motility is indispensible for all cell crawling, intracellular organelle propulsion, cell-cell communication, morphogenesis, as well as synaptogenesis within neural networks. Eukaryotic chemotaxis requires exquisite spatiotemporal coordination of pathways for

Enzyme Kinetics

794

13.12.1. Hill-Type Mechanisms for Force Generation by Polymerizing Free-ended Filaments Three decades of fruitful investigation on the in vitro polymerization of actin monomers gave rise to a natural inclination to interpret all aspects of cellular motility in terms of free-filament nucleation and elongation. The small-system thermodynamist Terrell Hill first considered the possibility that elongation of free-ended filaments might generate force (Hill, 1981; Hill and Kirschner, 1982). Hill suggested that the maximal force Fmax that can be generated is: Fmax ¼

DG ½ActinFree ¼ RT ln ½ActinðþÞcrit d

13.17

where F is the force, d is the distance over which the force is exerted, R is the universal gas constant, T is the absolute temperature, and [M] is the actin monomer concentration. The propulsive force Fmax ¼ DG/d ¼ 0, when the free monomer concentration equals the critical concentration, whereas Fmax z 1–2 pN when [M]Free equals [Actin](–)crit. The value of [M] is confined by the minus-end critical concentration, simply because any higher monomer concentration would result in ()-end polymerization – an unlikely circumstance in that ()-end disassembly is needed to supply monomers for sustained (þ)-end elongation.

ensemble of ~80 filaments

bacterium Motile Surface

chemattractant signal detection as well as for rapid assembly, operation, and turnover of the force-generating machinery responsible for cell migration. Actin motility also propels endosomes, exosomes, phagosomes, and even the intracellular pathogens such as Listeria monocytogenes, Shigella flexneri, Rickettsia, Vaccinia and Variola. The actin motile apparatus must exert and maintain an appropriate push-pull force balance on the membrane, thereby allowing for rapid extension and retraction of lamellipodia and filipodia. The evanescent nature of actin filament formation and turnover at the cell’s leading-edge makes the task of uncovering the underlying mechanisms for expansile force generation all the more daunting. Unlike the stable thick and thin filament ultrastructure of striated muscle, the EM images of which directly suggested how a cross-bridge cycle might generate contractile forces, actin filaments at the cell’s leading edge are formed, remodeled, and disassembled within seconds as the cell continues to advance at speeds exceeding 1 mm/s. Given the enormous number of actin cytoskeleton regulatory proteins in most cells, it is not surprising that the nature of cell crawling stubbornly remained so elusive until very recently.

Scheme 13.6 Such ideas were the genesis of so-called Brownian Ratchet mechanisms, wherein the (þ)-ends of an ensemble of elongating filaments (see Scheme 13.6) randomly flex away from the surface of a motile object (hereafter called the motile surface) to create sufficient space (called the intercalation distance) for 5.4-nm monomers to insert between the filament ends and the motile surface (Mogilner and Oster, 1996; Peskin, Odell and Oster, 1993). Later, as evidence indicated that filaments were most likely tethered to the motile surface, and in view of the emerging role of Arp2/3 complex in filament nucleation, Mogilner and Oster (2003) offered the Elastic Tethered Ratchet Model. They proposed that nascent filaments are formed by and are transiently associated with the Arp2/3 nucleation complex residing on the motile surface. Some filaments dissociate to become the ‘‘working filaments’’ that elongate freely until capped and lose contact with the surface. During motility, the attached filaments would be expected to be in tension and resist the forward progress of the bacterium/bead, whereas the dissociated filaments would be expected to be in compression and would generate the propulsive force. Even so, the maximal force that can be generated by the Tethered Ratchet Model would be the same as the aforementioned 1–2-nN limit.

13.12.2. The Actoclampin Hypothesis: Concerning the Existence and Action of Cytoskeletal Filament End-Tracking Motors To solve the longstanding riddle of how actin polymerization might generate forces needed for cell crawling and other actin-based motile processes, Dickinson and Purich (2002) broke with the traditional assumption that free-ended filament elongation powers actin-based motility. They instead proposed the Actoclampin Molecular Motor Hypothesis asserting that all actin-based motility arises from forces generated by filament end-tracking motors. They argued that such tracking proteins mediate processive insertional polymerization, meaning that actin filaments

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

issue into end-tracked filaments without ever releasing the filament’s (þ)-end during active elongation. The also showed for the first time how cytoskeletal filament endtracking can operate in the presence or absence of ATP hydrolysis (Dickinson and Purich, 2002; Caro, Dickinson and Purich, 2004). An important consequence of the Actoclampin Hypothesis is that various filament end-tracking proteins can form (þ)-end molecular motors, thereby allowing multiple actin-based motile processes to operate within the cytoplasm of the same cell. Such motor versatility also implies regulatory versatility, whereby motility within different subcellular regions can respond to their own unique cues from their associated signal-transduction pathways. In their 2002 paper, Dickinson and Purich also proposed the Lock, Load and Fire Model, a subsidiary hypothesis asserting that affinity-modulated tracking mechanism involves ATP hydrolysis-dependent attenuation of tracker affinity for its filament partner. Such models generate significantly greater force than possible with any Brownian Ratchet (BR) model (Dickinson and Purich, 2002; Caro, Dickinson and Purich, 2004). This novel molecular motor model was the first to predict that force is generated through the processive action of a membrane-bound filament endtracking protein (or clampin), whose release from its highaffinity interaction with a filament (þ)-end can be driven by the Gibbs free energy of filament-bound ATP hydrolysis: Fmax ¼

DGATP-hydrolysis d

13.18

where d ¼ 5.4 nm. Given that, under physiologic conditions, a DGATP-hydrolysis value of ~20 kBT corresponds to ~50 pN$nm of reversible work, we can estimate that Fmax would be 18 pN. Of course, the fast rate of assembly may impose kinetic constraints that reduce this value. In the mechanoenzymatic cycle shown in Fig. 13.27, Locking refers to the initial state, in which the clampin is tightly bound onto an ATP-containing subunit situated at the (þ)-end of its filament partner. Loading refers to the direct transfer of a Profilin$Actin$ATP monomer to the elongation site on the (þ)-ends, an event that triggers ATP hydrolysis (Firing) on an adjacent clamped subunit. The energy of ATP hydrolysis is predicted to drive a conformational change that is attended by a lowering of clampin affinity for its now ADP-actin partner affinity, thereby facilitating its translocation and re-locking to newly added ATP-containing subunit at or near the filament terminus. Unlike the classical cytoskeletal motors myosin, dynein, and kinesin, which generate pulling (or contractile) forces as they processively track along the sides of an actin filament or microtubule, the actoclampins generate pushing (or expansile) forces by permitting surface-tethered filaments to elongate, even in the face of substantial opposing force (Dickinson and Purich, 2002).

795

In contrast to inherently contractile actions of myosin, dynein, and kinesin exerted on a preformed actin filament or microtubule, the forward force generated by clamped-filament motors is an inherently expansile action. To name these (þ)-end filament-tracking motors, Dickinson and Purich (2002) suggested the name actoclampin, a composite of two root words, ‘‘acto-’’ from actin (as in actomyosin) and ‘‘clamp’’ (meaning a clasping device used for strengthening flexible moving objects and for securely fastening two or more components), followed by the suffix ‘‘in’’ to indicate its protein origin. The clampins include VASP, neuronal Wiscott-Aldrich Syndrome protein (N-WASP/WAVE), and formin. Evidence for processive filament end-tracking motors includes: (a) shared structural features and cellular locations of VASP, formins, and other WASP/WAVE like proteins; (b) the role of profilin during actin assembly; and (c) insensitivity of motility to capping protein. Work on formin-based motility (Kovar and Pollard, 2004; Romero et al., 2004; Vavylonis et al. 2006) has demonstrated that formins processively incorporate monomers onto filament ends and generate force in end-tracking mechanisms similar to that originally proposed by Dickinson and Purich (2002). Likely structural features of the ActA-VASP end-tracking motors are summarized in Fig. 13.28.

13.12.3. Processive Single-Filament End-Tracking by ActA-VASP Complex As the gold standard for in vitro actin-based particle propulsion, Listeria locomotion occurs solely by means of its ActA surface protein, which initiates formation and elongation of actin filament-rich ‘‘rocket-tails.’’ ActA-based motility has been widely exploited to unravel the underlying force-generating mechanism in the propulsion of Listeria as well as ActA-coated particles and vesicles. In the Tethered Ratchet Model, ‘‘working’’ free-ended filaments elongate freely until their (þ)-ends become capped. During motility, the Arp2/3-attached filaments are under tension, impeding forward progress of the bacterium/bead, whereas the working filaments become compressed, allowing them to generate the propulsive force. In the Actoclampin Model, however, single type filaments is formed, and these filaments undergo insertional (þ)-end monomer-addition. Their elongating ends remain persistently bound to the motile object – not by means of a ()-end association with Arp2/3 (which would be the wrong direction for propulsion), but by means of affinity-modulated interactions of the filaments’ (þ)-ends with an end-tracking protein. In this respect, all working filaments remain attached to the surface by means of surface-associated proteins that track the elongating (þ)-ends of their filament partners by simple monomer-addition cycles.

Enzyme Kinetics

796

New Actin• ATP monomers load onto Filament.

[1]

Next Cycle Begins.

New Energy Well (unoccupied) at new (+)-End of Filament

Remnant Energy Well

Clamp in Energy Well at Terminal-Subunits in of an Actin Filament

[2]

5.4-nm shift

Clamp in penultimate energy Well

ATP hydrolysis weakens clamp binding affinity.

[3]

Clamp Occupies New Terminal Energy Well

Clamp Shifts & Re-locks onto Actin.ATP at new terminus.

New Energy Well (empty)

Clamp is “Lifted” into Now Shallow Energy Well

FIGURE 13.27 ‘‘Lock, Load & Fire’’ model for an affinity-modulated filament end-tracking motor. The reaction begins with a clamped filament, the energy status of which is schematically represented by a green circle in the deep potential energy well situated immediately below the terminal Actin$ATP (red subunits). Other shallow energy wells (located at 5.4-nm intervals along the filament) correspond to the greatly attenuated clamp affinity for Actin$ADP$Pi or Actin$ADP (both shown as blue subunits) in the filament. Each filament growth cycle includes: Locking – wherein each filament’s (þ)-end is tightly bound to its ‘‘clampin’’ partner (such as VASP, N-WASP, formin, etc.); Loading – wherein new ATP-containing monomers add onto the filament end (represented here by the two additional red monomers and the unoccupied deep potential energy well positioned immediately below them), and Firing – wherein ATP hydrolysis attenuates clamp-binding affinity, as indicated by the conversion of a deep energy well to a shallow energy well. After shifting and re-locking (treated as the diffusive translocation and subsequent binding of the new filament end to the clamp, as shown as the green circle in the deep, terminal potential energy well). Profilin (small light blue circles) facilitates monomer addition by directly transferring Actin$ATP complex from the loading zone to the elongating (þ)-end (Dickinson, Southwick and Purich, 2002).

FIGURE 13.28 Key structural features of N-terminal EVH1 surface-tethering domain – Binds Ena/VASP proteins in actin filament endto FPPPP-containing adapter protein on the tracking motors. The modular structure of surface of the motile object. VASP possesses: (a) a Ena/VASP homology domain 2 (EVH2) having two actin-binding domains (G-actin-binding domain, GAB and Factin-binding domain FAB); (b) repetitive GPPPPP domains that bind profilin-actin and presumably supply monomers for efficient VASP-facilitated elongation (Barzik et al., D 2005; Kang et al., 1997); and (c) an EVH1 domain that links VASP to the motile surface (i.e., membrane or bacterial surface) via the DFPPPP sites in zyxin/vinculin, the DLPPPP Affinity-modulated GPPPPP Sites– sites in migfilin, or the FEFPPPP sites in ActA. clamp – Binds to Bind profilin Moreover, VASP shares these same key Action-ATP at (+)-end structural elements with WASP and WAVE/ GPPPPP Binding SCARE proteins, with close sequence site Tetramerization domain – homology between WASP/WAVE’s WH2 motif coiled-coil region and VASP’s GAB motif, between WASP’s CA motif of the VCA domain and VASP’s FAB maintains tetramer motif (Chereau and Dominguez, 2006). Formins structure Profilin-Actin have three functionally analogous domains: (a) an actin monomer binding FH2 domain, which binds to (þ)-end terminal filament subunits; (b) a proline-rich profilin-binding domain, which is believed to capture Profilin$Actin and transfer new subunits on the filament (þ)-ends; and (c) a surface-anchoring domain linking the formin and elongating filament end to cell membrane sites of polymerization. These proteins all have surface-anchoring domains and proline-rich regions that are spatially situated relative to the WH2/GAB and C/FAB motifs in a manner consistent with the Dickinson-Purich end-tracking hypothesis (Chereau and Dominguez, 2006; Chereau et al., 2005). These structural similarities, and the localization of these proteins at filament (þ)-ends during rapid polymerization and force generation, strongly suggest their common function in actin assembly.

T

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

An inherent limitation on experiments on many-filament ensembles, such as those propelling Listeria or similarly sized particles or vesicles, is that the processivity of individual filaments could not be demonstrated unambiguously. Therefore, to establish whether elongating (þ)-ends detach or remain associated with ActA during many monomeraddition cycles, actin filament elongation from Act A-immobilized nanospheres was examined under conditions favoring the spatial isolation and direct imaging of individual filaments. This was possible by using 50-nm beads that were sparsely cross-linked to Listeria ActA protein, from which it is possible to nucleate and elongate mainly one, or at most a few, individual filament(s). To minimize confounding thermal and/or convective motions, the ActA-coated nanospheres were immobilized by glutaraldehyde cross-linking to amino-modified microscope coverslips. The bead-laden coverslips were then assembled into simple flow chambers permitting facile replacement of motility solutions consisting of concentrated actin-rich cell extract, supplemented with fluorescently labeled actin, and an ATP-generation system.

To determine whether filaments elongated from 50-nm ActA-nanospheres had their (þ)- or ()-ends bound to the bead surface, we sequentially used green- and red-colored fluorescently labeled actin as a ‘‘time-stamp’’ to indicate the location and extent of early and late monomer incorporation. Elongation was first carried out for 2–4 min with Oregon-green actin, which was then chased by extract containing rhodamine actin. Subsequent glutaraldehyde fixation facilitated survey of thousands of beads for twocolor filaments (i.e., those containing both Oregon greenand rhodamine-labeled actin). Although comprising only ~8% of all bead-bound filaments, those filaments with adjoining segments that exclusively contained Oregon green- or rhodamine-labeled actin (Fig. 13.29) demonstrated that single filaments elongate while tethered. Scans of each fluorophore’s intensity along the length of these two-color filaments also showed a sharp transition from the Oregon green-region to the rhodamine-region, attesting to the efficiency and relative rapidity of the exchange of dyecontaining motility solutions. Significantly, that segment containing the most recently added fluorophore (rhodamine)

B Number Observed

A

797

120

1 minute

100

2 minute

80

5 minute

60

10 minute

40 20 0

1

2

3

4

Number of filaments attached to nanoparticle

D Normalized Intnesity

C

FIGURE 13.29 Single-filament elongation of filaments on 50-nm ActA-immobilized beads. A, Examples of single actin filaments terminating at the surface of 50-nm beads. The beads were amino-silanized by treatment with 3-aminopropyltriethoxysilane (APES), then modified with glutaraldehyde (2% vol/vol) to generate free surface-linked mono-aldehydes, with which ActA protein (0.3 mg/mL) was permitted to react for 30 min. Beads were then immobilized on an APES-treated microscope coverslip. After assembly of a flow cell, the immobilized ActA-coated beads were exposed to a VASP- and Mena-rich brain cytosol extract containing ~5 mM unlabeled rabbit muscle actin for various times in 20 mM Hepes (pH 7.7), 50 mM KCl, 2 mM MgCl2, 5 mM EGTA, 1 mM EDTA, 0.5 mM ATP, 0.5 mM DTT, 100 mM sucrose. The sample was then prepared for electron microscopy by critical-point drying and coating with platinum/carbon, and imaging was done in a transmission electron microscope. B, Histogram showing the timecourse of filament formation on 25-nm Acta-beads, with filaments clearly terminating at the bead surface. C, Examples of two filaments elongating in the two-color change assay, in which Oregon-green-labeled actin (green) was first followed by transfer to Rhodamine-labeled actin (red). Actin filament endtracking was accomplished with 50-nm ActA-coated silica beads that were prepared and processed as described above and imaged under a NikonTE2000E total internal reflectance fluorescence microscope (TIRFM). D, Plot of normalized green and red fluorescence signals for single bead-bound, end-tracked actin filaments. The observed fluorescence color-change pattern in Panels C and D unequivocally demonstrates filament polarity, with the last-added actin incorporated nearest the ActA-coated bead (C. Sturm, J. Phillips, K. Interliggi, W. Zeile, R. B. Dickinson, and D. L. Purich, unpublished findings).

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invariably appeared nearest the bead, whereas the segment containing the first-added fluorophore (Oregon-green) always farthest from the bead. Such results demonstrate that monomers added exclusively to individual filaments while their (þ)-ends remained at the bead surface. In separate experiments, we reversed the order of additions of fluorophore-labeled actin, and the resulting banding patterns of individual filaments were likewise reversed (data not shown). These observations of elongating single filaments with their (þ)-ends bound to ActA beads strongly argue for processive monomer addition. Had individual filaments detached during each monomer-addition cycle, detaching filaments would have quickly diffused away from the surface within the time required to add a new monomer, even at relatively slow elongation rates of a few monomers per second. This assertion is based on the expected large displacement of a diffusing filament away from the bead surface, with pffiffiffiffiffiffiffiffi an estimated root-mean-square value xrms ðtÞ ¼ 2tD, where D ¼ kB T lnð2L=dÞ=2phL is the longitudinal diffusivity of the filament, L is the filament’s length, t is the time over which the displacement occurs, kBT is Boltzmann’s constant  absolute temperature, h is the fluid viscosity, and d is the filament diameter (~7 nm). Even by conservatively estimating the extract viscosity as ~10 cp (or 10-times the viscosity of water) a 100-nm actin filament would diffuse (on average) more than 200 nm away from the surface (i.e., more than twice the filament’s length) within the time needed to capture a monomer from a 5-mM G-actin solution (i.e., t ¼ (kon[G-actin])1 z 20 ms for an on-rate constant kon ~ 10 mM1 s1). Any detaching filament would have been convectively swept away from the immobilized ActA-beads whenever new motility medium passed through the flow chamber. Therefore, it was highly unlikely that the filaments underwent an attachment/detachment cycle during each monomer-addition/tracker-translocation cycle or even a single detachment over the course of thousands of monomer-addition/tracker-translocation cycles. Collectively, these findings unambiguously demonstrate that the (þ)-ends of individual filaments elongate by processive insertional polymerization for hundreds of monomeraddition cycles, while their (þ)-ends remain persistently tethered to the ActA-bead surface. While clearly at odds with all previously proposed models for actin-based motility requiring unattached ends for monomer addition, processive insertional polymerization from ActA-immobilized nanospheres is consistent with the actoclampin end-tracking molecular motor hypothesis.

13.12.4. Properties of Actoclampin Motors The actoclampin model provides a mechanistic basis for the earlier work of Lindberg, Hoglund and Karlsson (1981), who inferred the presence of processive actoclampin-like

Enzyme Kinetics

motors from their ultrastructural analysis based on electron micrographs of the leading edge of motile glial cells. They also proposed that Profilin$Actin complex is the immediate precursor for filament elongation at membrane-associated ‘‘organizing units’’ that incorporate actin monomers while the elongating filament ends remain tethered to the membrane. It is essential to emphasize that the actoclampin filament end-tracking model must not be confused with nonprocessive models, such as the insertional polymerization model of Loisel et al. (1999) involving a monovalent linkage between VASP and a growing filament. Because that linkage is broken in each monomer-addition cycle in that model, any resulting push-pull force balance would always be lost after each monomer-addition cycle. Dickinson, Caro and Purich (2004) stressed that the essential properties of an active (i.e., ATP-driven) processive filament end-tracking motor are ATP hydrolysis on near-terminal subunits, triggered by monomer addition, ATP hydrolysis to reduce affinity and release the tracking protein, and processivity, maintained by multivalent interactions with the two sub-filaments. They also identified four highly advantageous properties of end-tracking motors: (i) maintenance of a strong continuous possession of the filament end to the propelled object during cycles of monomer addition; (ii) an ability to harness the energy of filament-bound ATP hydrolysis to yield substantially more energy for work than provided by the free energy of monomer addition alone; (iii) a reliable mechanical scheme for coupling elongation and force generation, even when the filament is oriented perpendicular to the surface; and (iv) the ability to promote force-independent polymerization under moderate compressive or tensile forces. These and other properties are briefly considered below.

13.12.4a. Role of Profilin$Actin$ATP Complex in Clamped-Filament Elongation Lindberg, Hoglund and Karlsson (1981) first proposed that the Profilin$Actin complex is the ‘‘substrate’’ for filament elongation, as catalyzed by membrane-tethered endtracking motors. Zeile, Purich and Southwick (1996) and Kang et al. (1997) observed that microinjection of a peptide analogue of VASP’s GPPPPP registers promptly arrested intracellular VASP motility. Dickinson, Southwick and Purich (2002) first advanced the concept that motility enhancement by VASP-mediated accumulation of Profilin$Actin$ATP complex on a motile surface should require direct transfer of monomers from VASP to filament ends that are persistently associated with the VASP motor. In their consideration of cytoskeletal filament-end tracking, Dickinson, Caro and Purich (2004) proposed a cofactor-assisted end-tracking model (Fig. 13.30) accounting for profilin’s central role in actin-based motility. This concept also fits with profilin’s action in formin-based

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

Profilin-Actin loading 1 High-affinity tether 2

4 High-affinity tether

Direct transfer of Profilin-Actin to (+)-end

799

If Arp2 and Arp3 mimic the two terminal actin subunits of a filament (þ)-end, binding of a monomer coupled to N-WASP can likewise trigger hydrolysis on the terminal subunit on the adjacent sub-filament. These ideas are compatible with a mechanism (shown in Fig. 13.31), in which the binding of Arp2/3 complex, VASP, and monomeric actin culminate in ATP hydrolysis-driven assembly of an active end-tracking motor unit.

13.12.4c. Processivity 3

Ready to load Profilin-Actin

FIGURE 13.30 Hypothetical action of a cofactor-assisted, dimeric end-tracking motor. Step-1: Soluble cofactor and monomer combine and bind to the end-tracking unit. Step-2: Cofactor-Monomer add to the filament end. Step-3: Other tracking unit and cofactor dissociate from the adjacent protofilament. (This step can be facilitated by ATP hydrolysis energy to modulate the affinity of the cofactor and/or the tracking unit for the filament.) Step-4: Reloading, wherein the cofactor may return immediately to solution, or remain transiently bound to the tracking unit or to the filament. The motile object always maintains a firm purchase on the propelling rocket tail (Dickinson, Caro and Purich, 2004).

end-tracking motors, as first proposed by Dickinson, Caro and Purich (2004) and experimentally demonstrated by Romero et al. (2004). Profilin’s role in direct-transfer of monomers by VASP is also supported by the results of Grenklo et al. (2003), who found that introduction of profilin covalently cross-linked to actin markedly reduced Listeria motility and brought about tail detachment, in a manner that required profilin’s polyproline-binding motif.

13.12.4b. ATP Hydrolysis In the Lock, Load and Fire mechanism, release and/or translocation of the tracking protein requires prompt penultimate ATP hydrolysis. This idea finds precedence in the studies of Karr, Podrasky and Purich (1979), who sought to explain how a single boundary layer of Tubulin$GTP might continually persist at the (þ)-ends of microtubules undergoing rapid elongation. They proposed the penultimate nucleotide hydrolysis pathway, in which each incoming Tubulin$GTP activates immediate GTP hydrolysis within the Tubulin$GTP already situated at the microtubule’s end. Triggering of ATP hydrolysis on the penultimate subunit upon addition of a new actin monomer is also suggested by the results of Dayel and Mullins (2004). They found that that binding of a new actin-monomer together with the VCA peptide (with WH2 domain binding the actin monomer and C binding to Arp3) triggered rapid hydrolysis of ATP on Arp2.

As discussed in Chapters 5 and 7, many enzymes that bind to and act on polymeric substrates do so in a processive manner. Dickinson, Caro and Purich (2004) suggested that processivity is an essential feature of actoclampin filament end-tracking motors, one that requires a multivalent tracker to maintain its binding interactions with actin filament. Such a property can be achieved by oligomeric proteins (e.g., VASP is a tetramer and formin a dimer) or by the binding of multiple end-tracking units to a multivalent adaptor protein (e.g. trivalent Nck binding to N-WASP by way of three SH-3 domains (Rivera et al., 2004; Rohatgi et al., 2001), or by binding interactions with ActA’s four EFPPPP sequences to VASP (Machner et al., 2001), or simply by the close proximity of end-tracking molecules imparted by random immobilization on a surface (e.g., by surface-adsorbed N-WASP in biomimetic particle propulsion assays (Wiesner et al., 2003).

13.12.4d. Protection from Capping Proteins Because polymerization of free-ended filaments is potently inhibited by capping proteins, protection from capping proteins is a signature property anticipated by the actoclampin filament end-tracking hypothesis. Formin molecules engaged in processive filament end-tracking are known to inhibit the binding of capping proteins (Kovar, Wu and Pollard, 2005; Romero et al., 2004; Zigmond et al., 2003). Notably, protection of a capping protein had been previously demonstrated for VASP (Bear et al., 2002), a property that can be explained if VASP similarly engages in processive end-tracking. With the consistent role of poly-proline registers, GAB and FAB motifs described above, Barzik et al. (2005) found that VASP’s EVH2 domain was sufficient to protect (þ)-ends from capping protein, and FAB and GAB motifs were required. Moreover, profilin enhanced the ability of VASP to protect (þ)-ends from capping protein, in a manner that required interactions of profilin with G-actin and VASP.

13.12.4e. Propulsion of Hard and Soft Particles Several published studies of propulsion of 0.25–10 mm particles, such as polystyrene microspheres, oil droplets, and vesicles, under relatively well-defined conditions

Enzyme Kinetics

800

End-Tracker ActA

ActA carries VASP to Arp2/3 nucleus

Profilin site Actin site

Profilin

2 End-Tracking Motor is now assembled

Actin ATP

ADP

ActA binds & activates Arp2/3 complex

1

Arp2/3 docked on Mother Filament

Hydrolysis

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ActA arm is released from Arp2/3 complex

Rapid End-Tracking Begins

FIGURE 13.31 Potential roles for ActA, Arp2/3 complex, and Ena/VASP proteins in the assembly and operation of Listeria (D)-end-tracking motors. Listeria ActA protein binds to Arp2/3 complex already docked on a mother filament (blue-colored subunits shown in partial profile), and the bound Arp2/3 complex interacts with one monomer-loading/filament-binding arm of a multivalent end-tracker in a manner facilitating ATP hydrolysisdependent motor assembly. After initiation, the two arms of the end-tracker (shown here as a dimer, but more likely a tetramer in the case of VASP) alternately supply Profilin$Actin$MgATP2 via numerous rounds of processive ATP hydrolysis-dependent elongation. As first proposed by (Dickinson and Purich, 2002), actin filament (þ)-end-tracking motors engage multiple arms in ‘‘loading’’ and ‘‘locking’’ interactions that deliver new monomers while maintaining a persistent purchase on its elongating filament partner (i.e., each motor is processive). Essential components and interactions: Listeria surface protein ActA (yellow) with Arp2/3 filament docking site (yellow plus-sign) and end-tracker protein docking site (yellow disk); endtracker protein (dark gray disk with extended arms), with profilin docking site (lavender circle) and actin monomer docking site (green square); Arp2/3 complex, with Arp2 (medium blue), Arp3 (rose), and other subunits (light gray); partial subunit lattice of ‘‘mother’’ filament (dark blue); and ATP (red) and ADP (blue) within Actin, Arp2, and/or Arp3.

in vitro, have provided further insight into the mechanism of actin-based motility. Dickinson and Purich (2006) also demonstrated that diffusion-limited, force-insensitive elongation of end-tracked filaments explains the signature properties of hard (rigid) and soft (deformable) particles undergoing actin-based motility. First, the observed particle speed dependence on particle size should arise from the greater characteristic diffusion length of larger particles, and predicted speeds were found to be in good agreement with published data (Bernheim-Groswasser et al., 2002; Wiesner et al., 2003). Second, the model predicts that the concentration gradient arising from monomer consumption should result in faster filament growth and compressive stress build-up on outer filaments of the rocket tail, in balance with tensile stresses on the slower filaments at the tail center. For rigid particles, the faster elongation rates must be accommodated by changes in filament orientation, resulting in a local increase in F-actin concentration. This stress differential between slower and faster filaments should increase until the stress on the tense filaments in the tail center causes their detachment from the particle surface, allowing the particle to proceed forward at a faster rate, thereby initiating a cycle of saltatory motion.

Chemical processes exhibit two types of rate-limiting action. When we say an enzyme-catalyzed reaction is diffusion-limited, we mean that the ensuing chemical step(s) is/are so rapid that the reaction occurs as soon as the reactant and enzyme encounter each other. When we say an enzyme-catalyzed reaction is reaction-limited, we mean that the ensuing chemical step(s) is/are so slow that the reaction occurs relatively slowly after reactant and enzyme combine with each other. An important prediction of the Dickinson and Purich (2006) treatment of actin-based vesicle propulsion is that, when the propulsion speed is diffusion-limited, any experimental conditions that reduce the filament-end density at the surface should increase the monomer concentration at the surface and thereby enhance particle speed. As noted in Chapter 5, Albery and Knowles (1976) suggested that evolution produces a nearly perfect catalyst, whenever the physics of substrate diffusion to and the release of product from the active site limit the efficiency of catalysis. While this concept is normally applied to the diffusion of low-molecular-weight substrates, it is now obvious that the action of actoclampin motors may be perfected to such a degree that motility is limited only by the rate at which the polymerization substrate

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

Profilin$Actin$ATP complex can arrive at the end of a filament undergoing clamped-filament elongation.

13.12.4f. Generality of Cytoskeletal Filament End-tracking Motors Although the end-tracking motor concept was originally advanced to explain actin-based motility, the concept is likely to apply to any stepwise advance of a motor unit along a linear polymer. Dickinson, Caro and Purich (2004), for example, suggested that GTP hydrolysis-driven endtracking motors probably account for microtubule-based processes. Kinetochores bind specifically to GTP-containing MT (þ)-ends (Severin et al., 1997), and several proteins have been identified in the kinetochore to bind microtubule (þ)-ends and to participate in force generation (Schuyler and Pellman, 2001). Of particular interest is the kinetochore-associated protein EB1, which concentrates at elongating microtubule (þ)-ends and promotes microtubule assembly and stabilization (Bu and Su, 2001; Tirnauer and Bierer, 2000; Tirnauer et al., 2002a). EB1 readily associates with the elongating GTP-rich (þ)-ends (Schroer, 2000), but dissociates uniformly along the length of assembled MTs (Tirnauer et al., 2002b). How GTP hydrolysis microtubules regulates EB1 localization remains unclear; co-polymerization of EB1-tubulin-GTP complexes from solution to (þ)-ends appears unlikely given the much lower (~100) intracellular concentration of EB1 relative to tubulin (Tirnauer et al., 2002a,b). EB1 may be recruited to (þ)-ends by adenomatous polyposis coli (APC) protein (Fodde et al., 2001a; Mimori-Kiyosue and Tsukita, 2001) (other kinetochore proteins). Notably, APC is a multimeric protein (Joslyn et al., 1993) that binds both to EB1 (MimoriKiyosue, Shiina and Tsukita, 2001) and to microtubule (þ)-ends (Kaplan et al., 2001). Because both APC and EB1 bind to MT ends (Mimori-Kiyosue, Shiina and Tsukita, 2001) and apparently to each other (Fodde et al., 2001b) in the kinetochore, an attractive possibility is that EB1 and APC (or another kinetochore protein) bind to GTPcontaining protofilament subunits in a ternary complex, which is subsequently disrupted by the GTP hydrolysis. If tracking units operated on all or several of the 13 microtubule protofilaments simultaneously, a cofactor-assisted model akin to that proposed for profilin in formin and ActAVASP motors would account for how the kinetochore facilitates rapid monomer addition and force generation during tight possession of the GTP-rich filament end. Another fascinating case is nematode sperm cell motility, wherein filament assembly from the major sperm protein (MSP) dimer results in dynamic membrane protrusions in a manner that closely resembles actin-based motility in other eukaryotic cells (Roberts and Stewart, 1997; Stewart and Roberts, 2005). Whereas actin-based motility is driven by addition of ATP-bound actin subunits onto actin filament plus-ends located at the cell membrane, MSP dimers

801

assemble from solution into nonpolar filaments that lack a nucleotide binding site. Filament polarity and on-filament ATP hydrolysis, although essential for actin-based motility, thus appear to be unnecessary for membrane protrusions by MSP. What powers the steady-state dimer flux arising from near-membrane MSP filament assembly and distal diassembly further back in the cytoplasm. As a potential resolution to this paradox, Dickinson and Purich (2006) proposed a model for MSP filament assembly and force generation by means of MSP filament-associated endtracking proteins. In this model, ATP hydrolysis drives affinity-modulated, processive interactions between membrane-associated proteins and elongating filament ends. However, in contrast to the actoclampin model for actin filament end-tracking motors, tracker-bound ATP hydrolysis is proposed to activate MSP dimer insertion and tracker translocation. The MSP end-tracking model predicts properties (i.e., ATP dependence, Michaelis-Menten ratesaturation behavior, and the need for other soluble non-MSP components within the cytoplasm) that are consistent with several key observations of MSP-based motility: (a) need for added ATP; (b) persistent membrane attachment of elongating filaments; (c) polymerization of filament ends at the membrane with distal depolymerization of free-filament ends away from the membrane; as well as (d) the observed rate-saturation behavior with respect to the concentration of non-MSP soluble cytoplasmic components.

13.12.4g. End-tracking Motors: Summary A hallmark of any successful theoretical treatment is its ability to explain wide-ranging and unanticipated properties without any need to alter the basic underlying mechanism, and despite apparent differences in the propulsion of hard (rigid) and soft (deformable) particles, the model readily accounts for the push-pull force balance established when an ensemble of surface tethered end-trackers operates stochastically under diffusion- or reaction-limited modes and is restricted by the advance of the slowest advancing end-tracker. The end-tracking motor hypothesis therefore represents a true paradigm shift for the entire field of actinbased motility and essentially reverses many widely held assumptions about filament assembly and force generation. Rather than filaments nucleating within the cytoplasm – away from the motile surface – and then growing until their free ends encounter and push against the surface, filaments nucleate and elongate by insertional assembly at the membrane itself. Rather than requiring free ends to elongate and push, filaments can instead push most effectively while persistently attached to the membrane with end-tracking proteins. Rather than growing with the force-sensitive kinetics of a thermal ratchet, end-tracking proteins allow the typically slower step of monomer binding to be insensitive to forces. And rather than the delayed ATP hydrolysis observed during elongation of free-ended filaments, rapid

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hydrolysis would be required for rapid advance of tightbinding end-tracking motors. One must entertain the likelihood that end-tracking motors are functionally diverse with respect to the nature of each tracker’s interactions with the (+)-end of a cytoskeletal filament and with regard to the role(s) of ATP hydrolysis in triggering and/or energizing tracker translocation during each monomer-incorporation/filament-elongation cycle. There is no basis to assume a priori that the affinitymodulated clamping domains of all end-tracking proteins need be identical, either structurally or energetically. Actin filament end-tracking proteins, for example, belong to several distinct protein families (e.g., Ena/VASP family, formin/mDia family, and WASP/N-WASP family), and such structural diversity offers the possibility that actin-filament end-tracking motors are uniquely adapted to fulfill specific force-requiring tasks. Each task may impose a different set of force and energetic requirements as well as reaction cycle kinetics, and the affinity of the tracking domain in each clampin may be suited to support such division of labor. Low force-requiring motors may even be able to advance in the absence of ATP hydrolysis, whereas high force-generating motors would require ATP hydrolysis for sufficient attenuation of tracker affinity to achieve rapid elongation in the face of greater loads. In this respect, even though the very same actin filament may be part of two different endtracking motors, each of the latter would be able to display different force-generating properties. The heart of this argument is that each end-tracking protein has its own set of noncovalent binding interactions with its filament partner and that these interactions are shaped by Natural Selection. End-tracking motor diversity is also suggested by the fact that only a subset of known end-tracking motors requires the active participation of Arp2/3 filament-initiation complex for the assembly of an active end-tracking motor. Even greater versatility may, of course, be achieved by the expression of various muscle and non-muscle actins. Within a single vertebrate, there are up to six forms of actin, each differing with respect to its amino acid sequence, antigenic sites, as well as tissue-specific expression. Such diversity in end-tracking motor function is also indicated by the fact that bacterial actin (MreB) possesses at most, only 17% sequence homology relative to mammalian actin, with even less conservation in the structure of MreC, the microbial homodimeric membrane-spanning protein that is likely to serve as an end-tracking protein. Finally, several investigators have wrongly conflated the generalizable features of end trackers, as first defined in the Actoclampin Hypothesis (Dickinson and Purich, 2002; Dickinson, Caro and Purich, 2004) with the specific actions predicted in its subsidiary hypothesis (i.e., the Lock, Load & Fire Model). Failure to observe an exact match of ATP hydrolysis kinetics and monomer incorporation kinetics does NOT refute the validity of the Actoclampin Hypothesis. Such a finding only suggests that a particular motor

Enzyme Kinetics

may not engage rapid stoichiometricv ATP hydrolysis under the imposed experimental conditions or that ATP is unnecessary, given the nature of the tracker’s interactions with its filament partner. Another intriguing possibility is that ATP hydrolysis may only be activated whenever elongation stalls, much like the role of ATP hydrolysis in protein folding reactions of chaperonins. GTP hydrolysis likewise enhances ATP sulfurylase catalysis: little or no GTP hydrolysis occurs at high sulfate concentration, whereas stoichiometric GTP hydrolysis (relative to AMP-sulfate synthesis) enhances reaction rate by providing an additional thermodynamic impetus to the catalytic reaction cycle (Leyh, 1999). Such a contingent hydrolysis mechanism would greatly enhance the catalytic options of end-tracking motors by allowing elongation to proceed, when elongation is rapid and facile, and by engaging ATP hydrolysis to drive the mechanochemical reaction cycle, whenever elongation is retarded by load or otherwise. In summary, the above considerations suggest that endtracking motors are apt to display a broad spectrum of mechanochemical reaction cycles that have been crafted by means of Natural Selection to satisfy specific cellular requirements. In this sense, each particular motor can be described by the Actoclampin Molecular Motor Model, whether or not it engages the Lock, Load & Fire mechanism using ATP hydrolysis to facilitate rapid end-trackng against substantial loads, and the Actoclampin Model successfully captures all the essential features of actinbased motility.

13.13. CONCLUDING REMARKS AND PROSPECTS This chapter’s focus on biological force generation represents a fitting way to conclude this chapter and, for that matter, the entire book. We have observed that the definition of an enzyme as any biological catalyst that enhances reactivity through the facilitated making/breaking of chemical bonds unifies the action of traditional enzymes that catalyze covalent bond transformations and mechanoenzymes that instead catalyze noncovalent bond transformations. There is a satisfying complementarity in that traditional enzymes use transient noncovalent interactions with substrates to facilitate the making/breaking of covalent bonds, and mechanoenzymes harness the energy liberated upon cleavage of covalent bonds in molecules like MgATP2 to generate force associated with the making/ breaking of noncovalent bonds. Nowhere is this more evident than with structurally related enzymes hexokinase and actin: the former phosphorylates glucose to fuel glycolysis, and the latter ‘‘phosphorylates’’ water to generate mechanical forces. Another unifying concept is that all forms of catalysis result from the well-managed, sequential application of

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

force. When we say that an intermediate is stabilized or destabilized during a catalytic reaction cycle, we are implicitly asserting that enzyme-associated forces are responsible for a stabilizing or destabilizing effect. For example, when some MgATP2-dependent mechanoenzymes operate against a load (i.e., an opposing force F) over some distance d, the force-dependent, one-substrate Michaelis-Menten equation takes the form: v ¼ [ETot]kcat  exp{Fd/kBT]/{1 þ Km/[MgATP2]. This equation indicates that the reaction’s activation energy DEact increased by an increment DEmechanical, equal to Fd when a sufficient load or stalling force is applied. By analogy, it may be postulated that, during catalysis, traditional enzymes likewise undergo a series of conformational/chemical changes that commensurately exert appropriately compensatory mechanical forces to stabilize or destabilize enzyme-bound reactants and enzyme-bound intermediates. Consider the following sequence for a one-substrate reaction: E þ S # (E$Sz); (E$S)z # (E$S); (E$S) # (E$X); (E$X) # (E$X)z; (E$X)z # E$P; E$P # (E$P)z; (E$P)z # E þ P, where the parentheses indicate all enzyme-bound reactants, with (E$S)z, (E$X)z and (E$P)z representing successive transition states. Whether induced by substrates, other ligands, or thermally, all enzyme conformational changes (e.g., (E$S) # (E$S1)z; (E$S1)z # (E$S1); (E$S1) # (E$S2)z; (E$S2)z # (E$S3); $$$$; (E$Si1) # (E$Si)z; (E$Si)z # (E$Siþ1); etc.) require the movement of atoms, as influenced by the force constants associated with each covalent bond or noncovalent interaction. In this context, the enzyme may be regarded as a force-generating and/or force-sensing template that adjusts to each successive stage in a catalytic reaction. Conformational flexibility, especially that required to appropriately stabilize a succession of transition states in multi-stage enzyme reaction cycles, is a special quality, one that most polypeptides are unable to achieve. As discussed in Chapter 1, too much stabilization can freeze catalysis, whereas too little stabilization can likewise result in feeble catalysis. Rate enhancement is a cardinal feature of enzyme catalysis, and molecular motors operating at 400–500 catalytic cycles per second are ten to one thousand times slower than most enzymes catalyzing covalent bond transformations. At least three factors account for the slower rates of mechanoenzymes. First, mechanoenzymatic reactions involving the diffusion and conformational rearrangement of macromolecular substrates (e.g., proteins, nucleic acids, membrane vesicles, etc.) are intrinsically slower, especially when mechanochemical forces must be generated. Second, cellular processes apparently do not require faster action. Muscle contraction and cell crawling, for example, are fast enough to match the pace of the cell cycle, and there is no need for driving the evolution of faster mechanoenzymes. And third, when operating far away from their reversible work regime, mechanoenzymes are likely to pay added kinetic and thermodynamic ‘‘penalties,’’ the latter limiting

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the maximum forces they can generate. In any case, kcat values of 400–500 s1 are quite respectable, corresponding to catalytic rate enhancements likely to exceed 109 to 1010. Consideration of molecular motor fuel economy (i.e., the number of ATP molecules hydrolyzed per step) by Coy, Wagenbach and Howard (1999) in Section 12.4.1 raises the general issue of how to express molecular motor efficiency. While physicists have traditionally evaluated the efficiency of macroscopic engines and motors in terms of the ratio of work output to energy input, biological molecular motors must be evaluated in terms of the energy required to achieve a certainty of outcome. Processes like mitosis require extraordinarily high fidelity in their action in sorting, disjoining and moving chromosomes to daughter cells, such that there is less concern with the amount of energy expended than in achieving that mitosis with adequate fidelity and in a timely manner. Even so, the number of ATP molecules hydrolyzed to initiate motor docking, to generate needed propulsive forces, to ensure adequate motor processivity, and to load and eventually to release its cargo is truly miniscule, when compared to the enormous number of ATP molecules formed and hydrolyzed during intermediary metabolism. When viewed in the fuller context of cellular trafficking and structural remodeling, certainty of outcome is one of the ‘‘biotic tricks’’ needed to support life, as we know it. Finally, this chapter concludes with cautionary comments regarding studies focusing on nucleoside 5’triphosphate (NTP) analogues. The action of almost every mechanoenzyme M on its target T depends directly on binding and/or hydrolysis of the metal-nucleotide complex or the release of its hydrolysis products. Structural biologists therefore seek the fullest repertoire of the mechanoenzyme structures, ideally including all the relevant target-free forms M$(MeNTP2–), M$(Pi$MeNDP), and M$(MeNDP), as well as the corresponding target-bound species. Because conformational changes lie at the heart of the force-generating/work-achieving steps, key objectives are (a) to identify changes in how the nucleotide is posed within the active site, and (b) to observe corresponding conformational changes in the mechanoenzyme and/or its target ligand. A crucial limitation in obtaining such structural information is that the catalytic cycles of mechanoenzyme are short-lived (occurring in the 10 msec to 1-sec range), and fleeting intermediates often fail to accumulate in sufficient abundance for structural analysis. The active sites of ATP- and GTP-dependent mechanoenzymes have therefore been probed using one or more synthetic compounds assumed to mimic crucial intermediates. For ATP-requiring mechanoenzymes, the following partial list of ATP analogues offers the uninitiated great optimism that meaningful insights will be forthcoming. Probes like N1,N5-ethenoATP, 29(39)-O-NBD-ATP, and 29(39)-O-Mant-ATP permit ligand binding measurements and distance surveys, the latter by Fo¨rster resonance energy

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transfer. Other probes like TEMPO-ATP and VO2+-ATP provide details on the local environment, using EPR spectroscopy. The imidodiphosphonates p(NH)ppA and pp(NH)ppA as well as the methylene diphosphonates p(CH2)ppA and pp(CH2)ppA serve as a nonhydrolyzable and/or slowly hydrolyzing ATP isosteres. Photoaffinity reagents like 8-azidoATP, Bz2-ADP, and vanadyl-ADP allow one to lodge nucleotides (and even nucleotide analogues) as covalent adducts within active sites. The phosphorothioates ATP[aS], ATP[bS] and ATP[gS] are often alternative substrates, competitive inhibitors, and/ or stereochemical probes of phosphoryl transfer. Disulfide linkage of phosphorothioates to mechanoenzymes is also feasible, especially when a key active-site serine residue can be mutated to cysteine. Stable exchange-inert metal ionATP complexes, such as Cr(III)ATP and Cr(III)ADP, facilitate an analysis of metal-nucleotide stereochemistry. Binary complexes of ADP-orthovanadate and ADP-AlF41– are frequently employed as analogues of pentavalent transition states formed during ATP hydrolysis. Finally, caged compounds, such as adenosine-59-triphospho-1-(2-nitrophenyl)-ethanol, may be used to generate ATP by photolysis within a few milliseconds. While this armamentarium of nucleotide compounds is impressive, each of the above-mentioned compounds has its unique set of structural, acid-base, metal ion-binding, and hydrogen- bonding properties that inevitably compromise, rather than commend, its use as an analogue of MgATP2–, CaATP2–, or their likely hydrolysis transition-states. For example, while p(NH)ppA is touted as virtually isosteric ˚ ) and P–N–P with ATP (e.g., its P–N bond-length (1.68 A  ˚ and bond-angle (127.2 ) compare favorably with 1.63 A  128.7 value for ATP), the bridging –NH– can act both as a electron-pair acceptor (N–H) and donor (:N–H). As a consequence, p(NH)ppA binds Mg2+ by means of one (or both) of its two b-phosphoryl oxygens as well as the P–N–P nitrogen itself. The latter interaction departs radically from the metal nucleotide interactions observed with MgATP2– or CaATP2–, so much so that its Mg2+ complex is fundamentally different, remaining intact under both basic and acidic pH values and inducing a 31P chemical shift not observed with MgATP2–. So, while ATP and p(NH)ppA are virtually isosteric, their metal ion complexes are not. Likewise, the ˚) P–C–P bond angle (~117 ) and P–C bond length (1.79 A within p(CH2)ppA and pp(CH2)ppA, as well as higher phosphoryl pKa values, make the methylene diphosphonate ‘‘analogues’’ relatively poor ATP mimics. Even the manganous ion complexes are problematic: MnATP2– and MnADP complexes have stability constants that are at least 4-8x greater than MgATP2– and MgADP, with measurable effects on the pKa values of the b and g phosphoryls. Mn2+ also prefers nitrogen atom-containing ligands, and its coordination number, geometry, and bond-lengths are all substantially different than observed in MgATP2– or CaATP2–. The tervalent metal ions in Cr(III)ATP1–,

Enzyme Kinetics

Co(III)ATP1–, and Rh(III)ATP1– also greatly modify the acid/base and oxygen exchange behavior, so much so that these exchange-inert metal ions are hardly likely to have much in common with MgATP2– or CaATP2–. In some respects, Cr(III)ATP1–, Co(III)ATP1–, and Rh(III)ATP1– more closely resemble catalytically inactive MgHATP1– and MgHGTP1– complexes. Likewise, the precise geometries, bond-lengths, and acid/base properties of ADP-vanadate and ADP-AlF41– binary complexes do not faithfully reproduce the stereoelectronic nature of the actual pentavalent transition states; nor are they likely to mimic the precise nucleotide racking within the active site that is energetically linked to force generation or attenuation of binding interactions between the mechanoenzyme and its binding partners. Finally, even photo-caged analogues suffer the disadvantage that, once liberated, the enzyme-bound ATP must bind its catalytically required metal ion and undergo conformational adjustments that are likely to introduce kinetic steps completely unrelated to pathways for mechanochemically coupled hydrolysis of MgATP2– or CaATP2–. Unlike NTP-dependent kinases and synthases, which undergo many hundreds to thousands of turnovers per second and bind MeNTP2– in the 0.01–3 mM range, mechanoenzyme-mediated NTP hydrolysis is far slower, and MeNTP2– complexes are bound far more tightly (often in the 0.01–1 mM range). With kinases and synthases, analogues like p(NH)ppA, p(CH2)ppA, and Cr(III)ATP most often form thermodynamic complexes used mainly as competitive inhibitors to kinetically determine substrate binding order or as stereochemical probes of metalnucleotide binding. In mechanoenzymes, however, ATP binding gives rise to the all-important g-phosphoryl sensor, within which the nucleotide, its metal ion, and the g-phosphoryl are posed in a fundamentally different manner. Actin, for example, binds ATP at least 1000x more tightly than p(NH)ppA, casting doubt on any assertion that p(NH)ppA may be a useful probe of actin-MgATP2– interactions. Moreover, as addressed in Section 7.1.8, the concept of agonist efficacy rationalizes the behavior of structurally related ligands interacting at the same receptor site. Indeed, pharmacologists successfully distinguish between (a) affinity, as measured by site occupancy parameters, such as ligand dissociation constants Kd and binding stoichiometry Y, and (b) action, as measured by the receptor’s ability to couple site occupancy to its signaling function. A key finding is that site occupancy does not necessarily imply that two receptor-bound ligands A and A9 will adopt identical poses within the binding site. In many cases, two ligands merely compete for an overlapping part of the agonist site, such that ligand A9 fails to evoke the full effect of the natural ligand A, even when their reduced concentrations [A]/KA and [A9]/KA9 are identical. Therefore, while MgATP2– and p(NH)ppA might compete for the very same active site within a mechanoenzyme, there is no justification for assuming that they reproduce each other’s

Chapter j 13 Mechanoenzymes: Catalysis, Force Generation and Kinetics

detailed sub-site interactions. A related issue is that enzymes are conformationally supple, so much so that many ATP-like compounds are likely to adopt conformations that are irrelevant to the actual configurations achieved with MgATP2– and CaATP2–. After decades of working with various ATP- and GTPlike compounds to study of phosphotransferases, synthases, as well as actin and tubulin polymerization, this author has become wary of their utility in the analysis of mechanoenzymes. There is a tendency to over-interpret the observed behavior of these analogues, particularly when inferring how forces might be generated and transmitted or how ATP hydrolysis may bring about changes in a mechanoenzyme’s affinity for its natural binding partner(s). The caveats noted in the last few paragraphs raise serious questions about whether any of above ATP ‘‘analogues’’ can ever inform us unambiguously about how ATP and its g-phosphoryl are realistically posed as reaction intermediates. When we use such compounds to analyze the reaction pathways of mechanoenzymes, we are denying the obvious – namely that the entire forcegenerating apparatus focuses atomic-level forces on and around the g-phosphoryl group, such that binding, force generation (power stroke) and/or movement (translocation) depend exquisitely on protein conformational changes linked to ATP binding, b-g P–O bond scission, ADP and/or orthophosphate release, and possibly even proton release. And when we publish our findings with these so-called analogues, we ask others to join in this deception, such that we fool each other into believing it is possible to glean useful information. Therefore, except for their use to seize molecular motors mid-course in a catalytic cycle, thereby blocking their function in motility, translocation, elongation, transport, etc., detailed structural studies with so-called ATP analogues are rarely as insightful as was once believed. In truth, they are just as likely to mislead us. While there is a need for new ways to explore mechanoenzyme catalysis, the best current approach is to use a combination of techniques, namely (a) stopped-flow fluorescence anisotropy to measure the

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rates and extent of macromolecular association; (b) stopped-flow FRET measurements to detect conformational changes in mechanoenzymes and/or their target; (c) stopped-flow fluorescence for the continuous detection of released Pi using MDCC-labeled phosphate-binding protein; and (d) rapid mixing/quenching experiments to evaluate ATP hydrolysis rate constants. Moreover, because the ATP-dependent action of many mechanoenzymes on their affinity-modulated binding partners can now be directly probed by single-molecule techniques, there is good reason to believe their use will provide fundamental insights into the physics of biological force generation. Given advances in fluorescence spectroscopy, video microscopy, and Laue diffraction methods, we may confidently anticipate great gains in our understanding of mechanoenzyme catalysis in the not-so-distant future.

FURTHER READING Boyer, P. D. (2002). Catalytic Site Occupancy During ATP Synthase Catalysis. FEBS Lett., 512, 29–32. Dickinson, R. B., Caro, L., & Purich, D. L. (2004). Force Generation by Cytoskeletal End Tracking Proteins. Biophys J., 87, 2838–2854. Howard, J. (2001). Mechanics of Motor Proteins and the Cytoskeleton. Sunderland, MA: Sinauer. 384. Ingber, D. E., & Tensegrity, I. (2003). Cell Structure and Hierarchical Systems Biology. J. Cell Sci., 116, 1157–1173, Tensegrity II. How Structural Networks Influence Cellular Information Processing Networks, J. Cell Sci. 116, 1397–1408. Keller, D., & Bustamante, C. (2000). The Mechanochemistry of Molecular Motors. Biophys. J., 78, 541. Khan, S., & Sheetz, M. P. (1997). Force Effects on Biochemical Kinetics. Annu. Rev. Biochem., 66, 785–805. Schliwa, M. (Ed.). (2001). Molecular Motors. Berlin: Wiley-VCH. 684. Tamanoi, F., and Hackney, D.D. (2004) Energy Coupling and Molecular Motors, (3rd ed.). Vol. 23, Enzymes, Academic Press, Orlando. 468. Zeng, W., Conibear, P. B., Dickens, J. L., Cowie, R. A., Wakelin, S., Ma´lna´si-Csizmadia, A., & Bagshaw, C. R. (2004). Dynamics of Actomyosin in Relation to the Cross-Bridge Cycle. Philos. Trans. Royal Soc. London, Part B Biol. Sci., 359, 1843–1855.

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References

Abbott, S. J., Jones, S. R., Weinman, S. A., et al. (1979). J. Am. Chem. Soc., 101, 4323. Abdullah, M., French, D., & Robyt, J. F. (1966). Arch. Biochem. Biophys., 114, 595. Abeles, R. H., & Dolphin, D. (1976). Acc. Chem. Res., 9, 114. Abeles, R. H., & Maycock, A. L. (1976). Acc. Chem. Res., 9, 313. Abeles, R. H., Frey, P. A., & Jencks, W. P. (1992). Biochemistry. p. 838 Boston: Jones and Bartlett. Abrahams, J. P., Leslie, A. G., Lutter, R., & Walker, J. E. (1994). Nature, 370, 621. Abu-Soud, H. M., Wang, J., Rousseau, D. L., & Stuehr, D. J. (1999). Biochemistry, 38, 12446. Ackers, G. K. (1998). Adv. Prot. Chem., 51, 185. Adachi, K., Oiwa, K., Nishizaka, T., Furuike, S., et al. (2007). Cell, 130, 309. Adachi, K., Yasuda, R., Noji, H., Itoh, H., et al. (2000). Proc. Natl. Acad. Sci. USA, 97, 7243. Adair, G. S. (1925). J. Biol. Chem., 63, 529. Adam, G., & Delbru¨ck, M. (1968). In A. Rich, & N. Davidson (Eds.), Structural Chemistry and Biology (pp. 198–215). San Francisco: Freeman. Adams, E. (1972). The Enzymes (3rd ed.), 6, 479. Adams, E. (1976). Adv. Enzymol., 44, 69. Adams, S. R., & Tsien, R. Y. (1993). Annu. Rev. Physiol., 55, 755. Adelberg, E. (1955). J. Biol. Chem., 216, 431. Adelman, K., LaPorta, A., Santangelo, T. J., et al. (2002). Proc. Natl. Acad. Sci. USA, 99, 13538. Adler, S. P., Purich, D. L., & Stadtman, E. R. (1975). J. Biol. Chem., 250, 6264. Admiraal, S. J., Meyer, P., Schneider, B., et al. (2001). Biochemistry, 40, 403. Agarwal, R. P., Spector, T., & Parks, R. E. (1977). Biochem. Pharmacol., 26, 359. Agmon, N., & Edelstein, A. L. (1997). Biophys. J., 72, 1582. Agrawal, N., Hong, B. Y., Mihai, C., & Kohen, A. (2004). Biochemistry, 43, 1998. Ahrens, M. L., Maass, G., Schuster, P., & Winkler, H. (1964). J. Am. Chem. Soc., 92, 6134. Ahrweiler, P. M., & Frieden, C. (1991). Biochemistry, 30, 7801. Akazawa, T., Takabe, T., & Kobayashi, H. (1984). Trends Biochem Sci., 9, 380. Alber, T., Banner, D. W., Bloomer, A. C., et al. (1981). Philos. Trans. R. Soc. London, B, 293, 159. Alber, T., Petsko, G. A., & Tsemoglou, D. (1976). Nature (London), 263, 297. Alberts, A. W., & Vagelos, P. R. (1972). The Enzymes (3rd ed.), 6, 37. Alberts, B., Bray, D., Lewis, J., Raff, M., et al. (1994). Molecular Biology of the Cell (3rd ed.). New York: Garland Publishing. Alberty, R. A. (2006). Biophys. Chem., 124, 11.

Alberty, R. A. (1956). Adv. Enzymol., 17, 1. Alberty, R. A. (1958). J. Am. Chem. Soc., 80, 1777. Alberty, R. A. (1961). The Enzymes (2nd ed.), 5, 531. Alberty, R. A. (1969). J. Biol. Chem., 244, 3290. Alberty, R. A. (1983). IandEC Fund., 22, 218. Alberty, R. A. (1992a). Biophys. Chem., 42, 117. Alberty, R. A. (1992b). J. Chem. Educ., 69, 493. Alberty, R. A. (1994). Chem. Rev., 94, 1457. Alberty, R. A. (2003). Thermodynamics of Biochemical Reactions (p. 397). Hoboken, New Jersey: Wiley-Interscience. Alberty, R. A. (2004). J. Biol. Chem., 279, 27831. Alberty, R. A. (2008). Biophys. Chem., 132, 114. Alberty, R. A., & Hammes, G. G. (1958). J. Phys. Chem., 62, 154. Alberty, R. A., & Massey, V. (1954). Biochim. Biophys. Acta, 13, 347. Albery, W. J. (1975). In E. Caldin, & V. Gold (Eds.), Proton Transfer Reactions (pp. 263–316). London: Chapman and Hall. Albery, W. J. (1993). Adv. Phys. Org. Chem., 28, 139. Albery, W. J., & Davies, M. H. (1969). Trans. Faraday Soc., 65, 1059. Albery, W. J., & Davies, M. H. (1972). J. Chem. Soc. Faraday Trans., I, 167. Albery, W. J., & Knowles, J. R. (1976). Biochemistry, 15, 5627 & 5631. Albery, W. J., & Knowles, J. R. (1986). Biochemistry, 25, 2572. Albery, W. J., & Knowles, J. R. (1987). J. Theor. Biol., 124, 137. Albery, W. J., & Knowles, J. R. (1977). Angew. Chem. Int. Ed. English, 16, 285. Albright, C. F., Giddings, B. W., Liu, J., et al. (1993). EMBO J., 12, 339. Alkema, W. B., de Vries, E., Floris, R., & Janssen, D. B. (2003). Eur. J. Biochem., 270, 3675. Allart, B., Gatel, M., Guillerm, D., & Guillerm, G. (1998). Eur. J. Biochem., 256, 155. Allen, J. D. (1979). Meth. Enzymol., 64, 248. Allen, R. D., Travis, J. L., & Allen, N. S. (1981). Cell Motil., 1, 275. Allingham, J. S., Smith, R., & Rayment, I. (2005). Nature Struct. Molec. Biol., 12, 378. Allison, R. D. (1985). Meth. Enzymol., 113, 419. Allison, R. D., & Purich, D. L. (1979). Meth. Enzymol., 63, 1. Allison, R. D., & Purich, D. L. (1980). Meth. Enzymol., 64, 3. Allison, R. D., & Purich, D. L. (2002). Meth. Enzymol., 354, 455. Allison, R. D., Todhunter, J. A., & Purich, D. L. (1976). J. Biol. Chem., 252, 6046. Altman, S. R. (1993). The RNA World. Cold Spring Harbor: Cold Spring Harbor Press. p. 239. Amad, M. H., Cech, N. B., Jackson, G. S., & Enke, C. G. (2000). J. Mass Spectrom., 35, 784. 807

808

Amici, A., Emanuelli, M., Ruggieri, S., et al. (2002). Meth. Enzymol, 354, 149. Amyes, T. L., O’Donoghue, A. C., & Richard, J. P. (2001). J. Am. Chem. Soc., 123, 11325. Anderson, C. M., Zucker, F. H., & Steitz, T. A. (1979). Science, 204, 375. Anderson, J. B. (1995). Adv. Chem. Phys., 91, 381. Anderson, J. M. (1976). J. Chem. Educ., 53, 561. Anderson, K. S. (1999). Meth. Enzymol., 308, 111. Anderson, K. S. (2003). In K. A. Johnson (Ed.), Kinetic Analysis of Macromolecules (p. 19). New York: Oxford University Press. Anderson, K. S., & Johnson, K. A. (1990). J. Biol. Chem., 265, 5567. Anderson, K. S., Miles, E. W., & Johnson, K. A. (1991). J. Biol. Chem., 266, 8020. Anderson, K. S., Sikorski, J. A., & Johnson, K. A. (1988). Biochemistry, 27, 7395. Anderson, N. W., & Thompson, J. F. (1979). Phytochemistry, 18, 1953. Anderson, S. G., Williams, C. R., O’Donnell, M., & Bloom, L. B. (2007). J. Biol. Chem., 282, 7035. Anderson, S. R., Anderson, V. E., & Knowles, J. R. (1994). Biochemistry, 33, 10545. Anderson, V. E. (1998). Comprehensive Biochemical Catalysis, 2, 115. Anderson, V. E. (2005). Arch. Biochem. Biophys., 433, 27. Andrews, P., Kitchen, B. J., & Winzor, D. J. (1973). Biochem. J., 135, 897. Anner, B. M. (1985). Biochim. Biophys. Acta, 832, 335. Antalis, T. M., & Palmer, G. (1982). J. Biol. Chem., 257, 6194. Anthony, C. (1996). Biochem. J., 320, 697. Anthony, C. (1998). Comprehensive Biochemical Catalysis, 3, 155. Anthony, R. S., & Spector, L. B. (1970). J. Biol. Chem., 245, 6739. Anthony, R. S., & Spector, L. B. (1971). J. Biol. Chem., 246, 6129. Anthony, R. S., & Spector, L. B. (1972). J. Biol. Chem., 247, 2120. Antoniou, D., Caratzoulis, S., Kalyanaraman, C., et al. (2002). Eur. J. Biochem., 269, 3103. Antosiewicz, J., McCammon, J. A., & Gilson, M. (1994). J. Mol. Biol., 238, 415. Appleyard, R. J., & Evans, J. N. S. (1993). J. Magn. Reson., Ser. B, 102, 245. Appleyard, R. J., Shuttleworth, W. A., & Evans, J. N. S. (1994). Biochemistry, 33, 6812. Arago´n, S. R., & Pecora, R. (1976). J. Chem. Phys., 64, 1791. Archontis, G., & Simonson, T. (2005). Biophys. J., 88, 3888. Arfin, S., & Umbarger, H. (1969). J. Biol. Chem., 244, 1118. Arias, I. M., Doyle, D., & Schimke, R. T. (1969). J. Biol. Chem., 244, 3303. Armstrong, F. A. (1999). Biochem. Soc. Trans., 27, 206. Armstrong, F. A. (2008). Philos. Trans. Roy. Soc. Lond., Series B Biol. Sci., 363, 1263. Arnett, E. M., & Reich, R. (1980). J. Am. Chem. Soc., 102, 5892. Arnold, F. H., & Georgiou, G. (2003). Directed Enzyme Evolution: Screening and Selection Methods. Totowa, New Jersey: Humana Press.

References

Arnold, J. P., Cheng, M.-S., Cullis, P. M., & Lowe, G. (1986). J. Biol. Chem., 261, 1985. Ashkin, A., Dziedzic, J. M., Bjorkholm, J. E., & Chu, S. (1986). Optics Lett., 11, 288. Ashley, G. W., & Bartlett, P. A. (1984). J. Biol. Chem., 259, 13621. Astumian, R. D., & Bier, M. (1995). Biophys. J., 68, 219. Atkinson, D. E. (1968). Biochemistry, 11, 4030. Atkinson, D. E. (1977). Trends Biochem. Sci., 2, 198. Ator, M. A. & Ortiz de Montellano, P. (1990). The Enzymes (3rd ed.), 19, 214. auf der Mauer, A., O’Hara, B. P., Wood, S. P., & Mann, N. H. (1993). Curr. Biol., 3, 265. Auld, D. S., Galdes, A., Geoghegan, K. F., et al. (1984). Proc. Natl. Acad. Sci. USA, 82, 5041. Avnir, D., Braun, S., Lev, O., & Ottolenghi, M. (1994). Chem. Mater., 6, 1605. Axley, M. J., Bock, A., & Stadtman, T. C. (1991). Proc. Natl. Acad. Sci. USA, 88, 8450. Babior, B. M. (1975). Acc. Chem. Res., 8, 376. Bachelard, H. S. (1971). Biochem. J., 125, 249. Badarau, A., & Page, M. I. (2006). Biochemistry, 45, 10654. Badway, J. A., & Westhead, E. W. (1976). J. Biol. Chem., 251, 5600. Baerga-Ortiz, A., Rezaie, A. R., & Komives, E. A. (2000). J. Mol. Biol., 296, 651. Bahnson, B. J., & Anderson, V. E. (1989). Biochemistry, 28, 4173. Bahnson, B. J., & Klinman, J. P. (1995). Meth. Enzymol., 249, 373. Bahnson, B. J., Park, D., Kim, K., et al. (1993). Biochemistry, 32, 5503. Bais, R., & Keech, B. (1972). J. Biol. Chem., 247, 3255. Baker, B. M., & Murphy, K. P. (1998). Meth. Enzymol., 295, 294. Baker, B. R. (1967). Design of Active-Site-Directed Irreversible Inhibitors: The Organic Chemistry of the Enzyme Active-Site. New York: J. Wiley. Baker, J. E., Krementsova, E. B., Kennedy, G. G., et al. (2004). Proc. Natl. Acad. Sci. USA, 101, 5542. Baldwin, A. N., & Berg, P. (1966). J. Biol. Chem., 241, 839. Baldwin, R. L. (2008). Annu. Rev. Biophys., 37, 1. Bale, J. R., Chock, P. B., & Huang, C. Y. (1980a). J Biol. Chem., 255, 8424. Bale, J. R., Chock, P. B., & Huang, C. Y. (1980b). J Biol. Chem., 255, 8431. Ball, F. G., & Rice, J. A. (1992). Math. Biosci., 112, 189. Ballew, R. M., Sabelko, J., & Grubele, M. (1996). Proc. Natl. Acad. Sci. USA, 93, 5759. Ballou, D. (1971). Instrumentation for the Study of Rapid Biological Oxidation-Reduction Reactions by EPR and Optical Spectroscopy, Ph. D. Dissertation, University of Michigan, Ann Arbor. High Wycombe, England: Available through University Microfilms, Ltd. Balsera, M. A., Wriggers, W., Oono, Y., & Schulten, K. (1996). J. Phys. Chem., 100, 2567. Bambara, R. A., Uyemura, D., & Choi, T. (1978). J. Biol. Chem., 253, 413. Barbas, C. F., III, Heine, A., Zhong, G., et al. (1997). Science, 278, 2085.

References

Bard, A. J., & Faulkner, L. R. (2000). Electrochemical Methods: Fundamentals and Applications (2nd ed.). New York: John Wiley and Sons. Barfield, M., & Karplus, M. (1969). J. Amer. Chem. Soc., 91, 1. Barlow, D. J., & Thornton, J. M. (1983). J. Mol. Biol., 168, 867. Barman, T. E., & Gutfreund, H. (1964). In B. Chance, R. Eisenhardt, Q. H. Gibson, & K. K. Lonberg-Holm (Eds.), Rapid Mixing and Sampling Techniques in Biochemistry (p. 339). New York: Academic Press. Barnett, R., Hogenkamp, H. P. C., & Abeles, R. H. (1966). J. Biol. Chem., 241, 1483. Barros, L. F., & Martinez, C. (2007). Biophys. J., 92, 3878. Barsacchi, D., Cappiello, M., Tozzi, M. G., et al. (1992). Biochim. Biophys. Acta, 160, 163. Barshop, B. A., Wrenn, R. F., & Frieden, C. (1983). Anal. Biochem., 130, 134. Bar-Tana, J., & Cleland, W. W. (1974). J. Biol. Chem., 249, 1263. Bartlett, G. J., Porter, C. T., Borkakoti, N., & Thornton, J. M. (2002). J. Mol. Biol., 324, 105. Bartlett, P. A., & Marlowe, C. K. (1987). Biochemistry, 26, 8553. Bartlett, P. H., & Keyer, W. B. (1984). J. Am. Chem. Soc., 106, 4282. Bartunik, H. D., Bartunik, L. J., Viehmann, H., et al. (1992). Phil. Trans. Roy. Soc. London A, 340, 209. Barzik, M., Kotova, T. I., Higgs, H. N., et al. (2005). J. Biol. Chem., 280, 28653. Bashford, D., & Karplus, M. (1990). Biochemistry, 29, 10219. Basolo, F., & Pearson, R. G. (1967). Mechanisms of Inorganic Reactions (2nd ed.). New York: Wiley. Basran, J., Sutcliff, M. J., & Scrutton, N. S. (1999). Biochemistry, 38, 3218. Basran, J., Sutcliff, M. J., & Scrutton, N. S. (2001). J. Biol. Chem., 276, 24581. Bates, D. M., & Watts, D. G. (1988). Nonlinear Regression Analysis and its Applications. New York: John Wiley. Batzold, F. H., Benson, A. M., Covey, D. F., et al. (1976). Adv. Enz. Regul., 14, 243. Bauer, H., Fritz-Wolf, K., Winzer, A., et al. (2006). J. Am. Chem. Soc., 128, 10784. Bauer, W. L., & Vinograd, J. (1970). J. Mol. Biol., 47, 419. Baum, C. L., Reschly, E. J., Gayen, A. K., et al. (1997). J. Biol. Chem., 272, 6490. Baxter, N. J., Olguin, L. F., Gilicnik, M., et al. (2006). Proc. Natl. Acad. Sci. USA, 103, 14732. Baykov, A. A., Cooperman, B. S., Goldman, A., & Lahti, R. (1999). Prog. Mol. Subcell. Biol., 23, 127. Bayley, H., & Knowles, J. R. (1977). Meth. Enzymol., 46, 69. Bazemore, L. R., Takahashi, M., & Radding, C. M. (1997). J. Biol. Chem., 272, 14672. Bear, J. E., Svitkina, T. M., Krause, M., et al. (2002). Cell, 109, 509. Beard, T. M., & Turner, N. J. (2002). Chem. Comm., 246. Bearne, S. L., & Wolfenden, R. (1997). Biochemistry, 36, 1646. Beastall, G. H., Rees, H. H., & Goodwin, T. W. (1971). FEBS Lett., 18, 175. Beattie, J. A., & Oppenheim, I. (1979). Principles of Thermodynamics. Amsterdam: Elsevier.

809

Begley, T. P., & Ealick, S. E. (2004). Curr. Opin Chem. Biol., 8, 508. Beinert, H. (2000). J. Biol. Inorg. Chem., 5, 2. Belasco, J. G., & Knowles, J. R. (1980). Biochemistry, 19, 472. Belasco, J. G., & Knowles, J. R. (1983). Biochemistry, 22, 122. Belasco, J. G., Albery, W. J., & Knowles, J. R. (1986). Biochemistry, 25, 2552. Belasco, J. G., Bruice, T. W., Albery, W. J., & Knowles, J. R. (1986a). Biochemistry, 25, 2558. Belasco, J. G., Bruice, T. W., Fisher, W. J., et al. (1986b). Biochemistry, 25, 2564. Bell, A. T. (2003). Science, 299, 1688. Bell, R. M., & Koshland, D. E., Jr. (1970). Biochem. Biophys. Res. Commun, 38, 539. Bell, R. P. (1978). In N. B. Chapman, & J. Shorter (Eds.), Correlation Analysis in Chemistry: Recent Advances (pp. 55–84). New York: Plenum Press. Bell, R. P. (1980). The Tunnel Effect in Chemistry. New York: Chapman and Hall. Belle, V., Fournel, A., Woudstra, M., et al. (2007). Biochemistry, 46, 2205. Bender, M. L. (1960). Chem. Rev., 60, 53. Bender, M. L. (1971). The Mechanism of Homogeneous Catalysis from Proton to Protein. New York: Wiley. Bender, M. L., & Komiyama, M. (1978). Cyclodextrin Chemistry. New York: Springer. Benkovic, P. A., Bullard, W. P., De Maine, M., et al. (1974). J. Biol. Chem., 249, 930. Benkovic, S. J. (1979). Meth. Enzymol., 63, 370. Benkovic, S. J., & Shray, K. J. (1976). Adv. Enzymol., 44, 139. Benkovic, S. J., & Shray, K. J. (1979). Meth. Enzymol., 63, 139. Benkovic, S. J., Hammes, G. G., & Hammes-Schiffer, S. (2008). Biochemistry, 47, 3317. Benner, S. A. (1982). Experientia, 38, 633. Benson, S. W. (1976). Thermochemical Kinetics: Methods for Estimation of Thermochemical Data and Rate Parameters (2nd ed.). New York: Wiley. Berg, O. G. & Jain, M. K. (2002). Interfacial Enzyme Kinetics (p. 328). Hoboken: Wiley. Berg, O. G., Gelb, M. H., Tsai, M.-D., & Jain, M. K. (2001). Chem. Rev., 101, 2613. Berger, R. L., Balko, B., & Chapman, H. F. (1968). Rev. Sci. Instrum., 39, 493. Berger, R. L., Balko, B., Borcherdt, W., & Friauf, W. (1968). Rev. Sci. Instrum., 39, 486. Bergmann, M., & Fruton, J. S. (1937). J. Biol. Chem., 117, 193. Bergmann, M., & Fruton, J. S. (1941). J. Biol. Chem., 138 (124), 321. Bergmann, M., Zervas, L., & Fruton, J. S. (1936). J. Biol. Chem., 115, 893. Bergmann, M., Zervas, L., & Fruton, J. S. (1935). J. Biol. Chem. 109–325. Bergmeyer, H. U. (1986). Methods of Enzymatic Analysis, Vols. 1–12 (3rd ed.). New York: J. Wiley and Sons. Berlicki, L., & Kafarski, P. (2006). Bioorg. Med. Chem., 14, 4578. Berlin, C. M., & Schimke, R. T. (1965). Mol. Pharmacol, 1, 149. Bernasconi, C. F. (1976). Relaxation Kinetics. New York: Academic Press.

810

Bernhard, S. A., & Orgel, L. E. (1959). Science, 130, 625. Bernhardt, P. V. (2006). Austr. J. Chem., 59, 233. Bernheim-Groswasser, A., Wiesner, S., Golsteyn, R. M., et al. (2002). Nature, 417, 308. Berry, H. (2002). Biophys. J., 83, 1891. Berry, R. S. (1960). J. Chem. Phys., 32, 933. Berthelot, M. (1862). Ann. Chim. Phys., 366, 110. Berthelot, M., & Pe´an de Saint-Gilles, L. (1862). Ann. Chim. Phys., Ser., 111 (66), 5. Berti, P. R. (1999). Meth. Enzymol, 308, 355. Bertics, P. J., Edman, C. F., & Karavolas, H. J. (1984). J. Biol. Chem., 251, 2863. Bessman, M. J., Frick, D. N., & O’Handley, S. F. (1996). J. Biol. Chem., 271, 25059. Beyer, A. (1997). Protein Sci., 6, 2043. Bieschke, J., & Schwille, P. (1997). In J. Slavik (Ed.), Fluorescent Microscopy and Fluorescent Probes, Vol. 2 (p. 81). New York: Plenum Press. Bieschke, J., Giese, A., Schulz-Schaeffer, W., et al. (2000). Proc. Natl. Acad. Sci. USA, 97, 5468. Bigeleisen, J. (1955). J. Chem. Phys., 23, 2264. Bigeleisen, J. (1962). Tritium in the Physical and Biological Sciences, International Atomic Energy Agency, Vol. 1, 161. Bijvoet, J. M., Peerdeman, A. F., & van Bommel, A. J. (1951). Nature, 168, 271. Binnig, G., Quate, C. F., & Gerber, C. (1986). Phys. Rev. Lett., 56, 930. Birchmeier, W., Wilson, K. J., & Christen, P. (1973). J. Biol. Chem., 248, 1751. Black, S. D., & Coon, M. J. (1987). Adv. Enzymol., 60, 35. Blackburn, P., Wilson, G., & Moore, S. (1977). J. Biol. Chem., 252, 5904. Blacklow, S. C., & Knowles, J. R. (1990). Biochemistry, 29, 4099. Blacklow, S. C., Liu, K. D., & Knowles, J. R. (1991). Biochemistry, 30, 8470. Blacklow, S. C., Raines, R. T., Lim, W. A., et al. (1988). Biochemistry, 27, 1158. Blanchard, J. S., & Cleland, W. W. (1980). Biochemistry, 19, 3543. Bla¨tter, W. A., & Knowles, J. R. (1979). Biochemistry, 18, 3927. Bla¨tter, W. A., & Knowles, J. R. (1980). Biochemistry, 19, 738. Bloch, W., & Schlesinger, M. J. (1973). J. Biol. Chem., 248, 5794. Bloom, L. B. (2006). Crit. Rev. Biochem. Mol. Biol., 41, 179. Bloomberg, V. A., Crothers, D. M., & Tinoco, I., Jr. (1974). Physical Chemistry of Nucleic Acids. New York: Harper and Row. Bloomfield, V., Peller, L., & Alberty, R. A. (1962). J. Am. Chem. Soc., 84, 4375. Blount, K. F., & Uhlenbeck, O. C. (2005). Annu. Rev. Biophys. Biomol. Struct., 34, 415. Blume, A., & Hillmann, M. (1986). Eur. Biophys. J., 13, 343. Bock, C. W., Katz, A. K., Markham, G. D., & Glusker, J. P. (1999). J. Amer. Chem. Soc., 121, 7360. Bock, C. W., Kaufman, A., & Glusker, J. P. (1994). Inorg. Chem., 33, 419. Bohr, C. (1903). Zentr. Physiol., 17, 682. Bolesch, D. G., & Keasling, J. D. (2000). J. Biol. Chem., 275, 33814. Bonneau, R., Wirtz, J., & Zuberbu¨hler, A. D. (1997). Pure and Appl. Chem., 69, 1.

References

Bonnet, G., Krichevsky, O., & Libchaber, A. (1998). Proc. Natl. Acad. Sci. USA, 95, 8602. Bonvin, M. J. J. (2006). Curr. Opinion. Struct. Biol., 16, 194. Bordo, D., & Argos, P. (1991). J. Mol. Biol., 217, 721. Bordo, D., Djinovic-Caarugo, K., & Bolognesi, M. (1994). J. Mol. Biol., 238, 366. Borel, J. F., & Kis, Z. L. (1991). Transplantation Proceedings, 23, 1867. Borsting, P., & Nielsen, P. (2002). Chem. Commun., 18, 2140. Bosco, D. A., Eisenmesser, E. Z., & Kern, D. (2002). Proc. Natl. Acad. Sci. USA., 99, 5247. Bott, A. W. (2004). Current Separat., 21, 1. Bouhss, A., Dementin, S., van Heijenoort, J., et al. (2002). Meth. Enzymol., 354, 189. Bourne, Y., Radic, Z., Sulzenbacher, G., et al. (2006). J. Biol. Chem., 281, 29256. Bowman, G. D., O’Donnell, M., & Kuriyan, J. (2004). Nature, 429, 724. Boyd, R. K. (1977). Chem. Rev., 77, 93. Boyer, P. D. (1959). Arch. Biochem. Biophys., 82, 387. Boyer, P. D. (1960). Ann. Rev. Biochem., 29, 1544. Boyer, P. D. (1997). Ann. Rev. Biochem., 88, 717. Boyer, P. D. (2002). J. Biol. Chem., 277, 39045. Boyer, P. D. (2003). In I. Grenthe (Ed.), Nobel Lectures, Chemistry 1996–2000. Singapore: World Scientific Publishing Co. Boyer, P. D., Cross, R. L., & Momsen, W. (1973). Proc. Natl. Acad. Sci. USA, 70, 2837. Boyer, P. D., Koeppe, J. O., & Luchsinger, W. W. (1956). J. Am. Chem. Soc., 78, 356. Boyer, P. D., & Kohlbrenner, W. E. (1981). In B. R. Selman, & S. Selman-Reimer (Eds.), Energy Coupling in Photosynthesis (p. 231). Amsterdam: Elsevier. Bradford, M. M. (1976). Anal. Biochem., 72, 248. Branchaud, B. I., & Turner, B. E. (2002). Meth. Enzymol., 354, 415. Brand, L., & Johnson, M. L. (Eds.) (1992). Numerical Computing Methods, Part A, Meth. Enzymol Vol. 210. San Diego: Academic Press. Brand, L., & Johnson, M. L. (Eds.) (2004). Numerical Computing Methods, Part D, Meth. Enzymol Vol. 210. San Diego: Academic Press. Brandt, K. G., Parks, P. C., Czerlinski, G. H., & Hess, G. P. (1966). J. Biol. Chem., 241, 4180. Brashear, W. T., & Parsons, S. M. (1975). J. Biol. Chem., 250, 6885. Braun, M., Lu¨nsdorf, H., & Bu¨ckmann, A. F. (1991). Eur. J. Biochem., 196, 439. Braunheim, B. B., & Schwartz, S. D. (1999). Meth. Enzymol, 308, 398. Braunstein, A. E., & Goryachenkova, E. V. (1984). Adv. Enzymol., 56, 1. Bray, R. C. (1961). Biochem. J., 81, 189. Bray, R. C., & Patterson, R. (1961). Biochem. J., 81, 194. Breaker, R. R., & Li, Y. (1994). Chem. Biol., 1, 223. Bremer, J. (1969). Eur. J. Biochem., 8, 535. Bresloff, J. L., & Crothers, D. M. (1975). J. Mol. Biol., 95, 103. Breslow, R. (1958). J. Am. Chem. Soc., 80, 3719. Breslow, R. (2005). Artificial Enzymes (pp. 194). Hoboken: Wiley.

References

Brevet, A., Roustan, C., Desvages, G., et al. (1974). Eur. J. Biochem, 39, 141. Bridger, W. A., & Cohen, L. H. (1968). J. Biol. Chem., 243, 644. Bridger, W. A., Millen, W. A., & Boyer, P. D. (1968). Biochemistry, 7, 3608. Briggs, G. E., & Haldane, J. B. S. (1925). Biochem. J., 19, 338. Britton, H. G. (1966). Arch. Biochem. Biophys., 117, 167. Britton, H. G. (1973). Biochem. J, 133, 255. Britton, H. G. (1977). Biochem. J, 161, 517. Britton, H. G. (1994). Biochem. J., 302, 965. Britton, H. G. (1997). Biochem. J., 321, 187. Brocklehurst, K. (1982). Meth. Enzymol., 87, 427. Brocklehurst, K. (2002). In R. Eisenthal, & M. Danson (Eds.), Enzyme Assays: A Practical Approach (pp. 157–170). London: Oxford University Press. Brocklehurst, K., & Malthouse, J. P. (1981). Biochem. J., 193, 819. Brønsted, J. N. (1928). Chem. Rev., 5, 322. Brønsted, J. N., & Pedersen, K. (1924). Zeit. Phys. Chem., A108, 185. Brooks, S. P., & Storey, K. B. (1991). Biochem. J., 278 (Part 3), 875. Brown, A. J. (1892). J. Chem. Soc., 61, 369. Brown, A. J. (1902). J. Chem. Soc., 81, 373. Brown, G. K., Fox, R. M., & O’Sullivan, W. J. (1975). J. Biol. Chem., 250, 7352. Brown, K. A., Brick, P., & Blow, D. M. (1987). Nature, 326, 416. Brown, M. J. B., Mensah, L. M., Doyle, M. L., et al. (1998b). J. Biol. Chem., 273, 31691. Brown, M. J., Mensah, L. M., Doyle, M. L., et al. (2000). Biochemistry, 39, 6003. Browne, C. A., Campbell, I. D., Keiner, P. A., et al. (1976). J. Mol. Biol., 100, 319. Bruice, T. C. (1972). Cold Spring Harbor Symp. Quant. Biol., 36, 21. Bruice, T. C. (2002). Acc. Chem. Res., 35, 139. Brune, M., Hunter, J. L., Corrie, J. E. T., & Webb, M. R. (1994). Biochemistry, 33, 8262. Bruns, N., & Tiller, J. C. (2005). Nano Lett., 5, 45. Brush, E. J., & Kozarich, J. W. (1992). The Enzymes, 20, 317, 3rd ed. Bryant, F. R., Benkovic, S. J., Sammons, D., & Frey, P. A. (1981). J. Biol. Chem., 256, 5965. Buchholz, K., Kasche, V., & Bornscheuer, U. T. (2005). Biocatalysts and Enzyme Technology (p. 465). Weinheim, Germany: Wiley-VCH. Buchwald, S. L., & Knowles, J. R. (1980). J. Am. Chem. Soc., 102, 6601. Buchwald, S. L., Friedman, J. M., & Knowles, J. R. (1984). J. Am. Chem. Soc., 106, 4911. Buddenbaum, W. E., & Shiner, V. J., Jr. (1977). Isotope Effects on Enzyme-Catalyzed Reactions. In W. W. Cleland, M. H. O’Leary, & D. B. Northrop (Eds.). Baltimore: University Park Press. Bu¨chner, E. (1897). Ber. deut. Chem. Ges., 30, 117. Bu¨chner, E. (1897). Ber. deut. Chem. Ges., 31, 568. Buffa, P., Peters, R. A., & Wakelin, R. W. (1951). Biochem. J., 48, 467.

811

Bull, C., Niederhoffer, E. C., Yoshida, T., & Fee, J. A. (1991). J. Am. Chem. Soc., 113, 4069. Bull, H. G., Garcia-Calvo, M., Andersson, S., et al. (1996). J. Am. Chem. Soc., 118, 2359. Bullerjahn, A. M. E., & Freisheim, J. H. (1992). J. Biol. Chem., 267, 864. Bulos, B., & Handler, P. (1965). J. Biol. Chem., 242, 3283. Bunton, C. A. (1969). Nucleophilic Substitution at the Saturated Carbon Atom. New York: McGraw-Hill. Burbaum, J. J., & Knowles, J. R. (1989). Biochemistry, 28, 9306. Burbaum, J. J., Raines, R. T., Albery, W. J., & Knowles, J. R. (1989). Biochemistry, 28, 9293. Burger, R. M., & Lowenstein, J. M. (1975). Biochemistry, 14, 2362. Burman, J. D., Harris, R. L., Hauton, K. A., et al. (2004). FEBS Lett., 576, 442. Burnell, J. N., & Whatley, F. R. (1977). Biochim. Biophys. Acta, 481, 246. Burton, K. (1959). Biochem. J., 71, 388. Burton, K., & Krebs, H. A. (1953). Biochem. J., 54, 94. Bustamante, C., Marko, J. F., Siggia, E. D., & Smith, S. (1994). Science, 265, 1599. Butcher, E. C., & Lowry, O. H. (1976). Anal. Biochem., 76, 502. Butler, A. (1998). In M. Sinnot (Ed.), Comprehensive Biological Catalysis Vol. 3 (p. 427). London: Academic Press. Cabodevilla, J. F., Odriozola, L., Santiago, E., & MartinezIrujo, J. J. (2001). Eur. J. Biochem., 268, 1163. Cahn, R. S., Ingold, C. K., & Prelog, V. (1966). Angew. Chem. Int. Ed. Engl., 5, 385. Caldwell, S. R., Newcomb, J. R., Schlecht, K. A., & Raushel, F. M. (1991). Biochemistry, 30, 7438. Callaghan, P. T. (1991). Principles of Nuclear Magnetic Resonance Microscopy. Oxford: Clarendon Press. Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatics. New York: John Wiley. Callender, R. H., Dyer, R. B., Gilmanshin, R., & Woodruff, W. H. (1998). Annu. Rev. Phys. Chem., 49, 173. Campbell, D. P., Carper, W. R., & Thompson, R. E. (1982). Arch. Biochem. Biophys., 215, 289. Cane, D. E., Yang, G., Coates, R. M., et al. (1992). J. Org. Chem., 57, 3454. Cann, J. R. (1989). J. Biol. Chem., 264, 17032. Cantor, C. R., & Schimmel, P. R. (1980). Biophysical Chemistry. Vol. 1, pp. 281–288. San Francisco: Freeman. Cao, L. (2006). Carrier-bound Immobilized Enzymes: Principles, Application and Design. Berlin: Wiley-VCH. p. 578. Capson, T. L., Peliska, J. A., Kaboord, B. F., et al. (1992). Biochemistry, 31, 10984. Carafoli, E., & Krebs, J. (Eds.) (2000). Topics in Biol. Inorg. Chem, Vol. 3. Berlin: Springer-Verlag. Caratzoulas, S., Mincer, J., & Schwartz, S. D. (2002). J. Am. Chem. Soc., 124, 3270. Cardinale, G. J., & Abeles, R. H. (1968). Biochemistry, 7, 3970. Cardosi, M. F. (1990). Scand. J. Clin. Lab. Invest. Suppl., 202, 27. Careaga, C. L., & Falke, J. J. (1992). J. Mol. Biol., 226, 1219. Carey, P. R., Carriere, R. G., Lynn, K. R., & Schneider, H. (1976). Biochemistry, 15, 2387. Carey, P. R., Carriere, R. G., Phelps, D. J., & Schneider, H. (1978). Biochemistry, 17, 1081.

812

Carper, R., Chang, K. W., Thorpe, W. G., et al. (1974). Biochim. Biophys. Acta, 358, 49. Carreras, C. W., & Santi, D. V. (1995). Annu. Rev. Biochem, 64, 721. Carroll, L. J., Mehl, A. F., & Dunaway-Mariano, D. (1989). J. Am.. Chem. Soc., 111, 5965. Carter, P., & Wells, J. A. (1987). Science, 237, 384. Carter, P., & Wells, J. A. (1988). Nature, 332, 564. Cascio, P., Call, M., Petre, B. M., et al. (2002). EMBO J., 21, 2636. Caserta, G., & Cervigni, T. (1974). Proc. Nat. Acad. Sci. USA, 71, 4421. Casey, E. J., & Laidler, K. J. (1951). J. Am. Chem. Soc., 73, 1455. Cassidy, C. S., Lin, J., & Frey, P. A. (1997). Biochemistry, 36, 4576. Castellan, G. W. (1963). Ber. Bunsenges. Physik. Chem., 67, 898. Castillo, E., & Lopez-Munguia, A. (2004). J. Biotechnol., 114, 209. Castro, G. R., & Knubovets, T. (2003). Crit. Rev. Biotechnol., 23, 195. Cautrecasas, P., Wilcek, M., & Anfinsen, C. B. (1968). Proc. Natl. Acad. Sci. USA, 61, 636. Cech, T. R. (1993). The RNA World. Cold Spring Harbor: Cold Spring Harbor Press. p. 239. Celio, M. R., Pauls, T., & Schwaller, B. (Eds.) (1996). Guidebook to the Calcium-Binding Proteins. Oxford: Oxford University Press. Cha, S. (1968). J. Biol. Chem., 243, 820. Cha, S., Agarwal, R. P., & Parks, R. E. (1975). Biochem. Pharmacol., 24, 2187. Cha, Y., Murray, C. J., & Klinman, J. P. (1989). Science, 243, 1325. Chaires, J. B. (2008). ACS Chem. Biol., 3, 207. Chance, B. (1943). J. Biol. Chem., 151, 553. Chance, B. (1949a). Arch. Biochem., 22, 224. Chance, B. (1949b). Arch. Biochem., 24, 410. Chance, B. (1950). J. Biol. Chem., 182, 649. Chance, B. (1951). In J. T. Edsall (Ed.), Enzymes and Enzyme Systems. Cambridge: Harvard University Press. Chance, B. (1955). Disc. Faraday Soc., 20, 205. Chance, B. (1974). Investigation of Rates and Mechanisms of Reactions, Part II. In Techniques of Chemistry, Vol. 6 (p. 5). New York: Wiley-Interscience. Chang, C. H., Ballinger, M. D., Reed, G. H., & Frey, P. A. (1996). Biochemistry, 35, 11081. Chang, K.-C., & Hammer, D. A. (1999). Biophys J., 76, 1280. Chang, L. M. S. (1975). J. Mol. Biol., 93, 219. Changeux, J. P., & Edelstein, S. J. (2005). Science, 308, 1424. Chaplin, M., & Bucke, C. (1990). Enzyme Technology. Cambridge, England: Cambridge University Press. Chapman, J. M., & Ayrey, G. (1981). The Uses of Radioactive Isotopes in the Life Sciences, George. London: Allen and Unwin. Chappelet-Tordo, D., Fosset, M., Iwatsubo, M., et al. (1974). Biochemistry, 13, 1788. Cheesman, M. R., Ankel-Fuchs, D., Thauer, R. K., & Thompson, A. J. (1989). Biochem. J., 260, 613. Chelsky, D., & Parsons, S. M. (1975). J. Biol. Chem., 250, 5669. Chen, C. S., Fujimoto, G., Girdaukis, G., & Shi, C. J. (1982). J. Am. Chem. Soc., 104, 7294.

References

Chen, J.-L., & Greider, C. W. (2003). EMBO J., 22, 304. Chen, K., & Arnold, F. H. (1991). Bio/Technology, 9, 1073. Chen, R. F., Brown, D. M., & Plaut, G. W. E. (1964). Biochemistry, 3, 552. Chen, Y. (1973). J. Chem. Phys., 59, 5810. Chen, Y., & Hill, T. L. (1973). Biophys. J., 13, 1276. Chen, E., Wen, Y., Lewis, J. W., et al. (2005). Rev. Sci. Instrum., 76, 831. Cheng, M. C., & Marsh, E. N. (2005). Biochemistry, 44, 2686. Cheng, Y., Suen, J. K., Zhang, D., et al. (2007). Biophys. J., 92, 3397. Chereau, D., & Dominguez, R. (2006). J. Struct. Biol. E-pub. Chereau, D., Kerff, F., Graceffa, P., et al. (2005). Proc. Natl. Acad. Sci. USA, 102, 16644. Cheung, C. L., Hafner, J. H., & Lieber, C. M. (2000). Proc. Natl. Acad. Sci. USA, 97, 3809. Chiba, S. (1997). Biosci. Biotechnol. Biochem., 61, 1233. Chin, J., La¨ngst, G. G., Becker, P. B., & Widom, J. (2004). Meth. Enzymol., 376, 1. Chipman, D. M., & Sharon, N. (1969). Science, 165, 459. Chlebowski, J. F., & Coleman, J. E. (1972). J. Biol. Chem., 247, 6007. Chlebowski, J. F., & Coleman, J. E. (1974). J. Biol. Chem., 249, 7192. Cho, H., Krishnaraj, R., Kitas, E., et al. (1992). J. Am. Chem. Soc., 114, 7296. Cho, Y.-K., & Northrop, D. B. (1999). Biochemistry, 38, 7470. Chock, P. B., & Gutfreund, H. (1988). Proc. Natl. Acad. Sci. USA, 85, 8870. Chock, P. B., & Stadtman, E. R. (1977). Proc. Natl. Acad. Sci. USA, 74, 2761–2766. Chock, P. B., & Stadtman, E. R. (1980). Meth. Enzymol., 64, 297. Chollet, R., & Anderson, L. L. (1976). Arch. Biochem. Biophys., 176, 344. Chollet, R., & Anderson, L. L. (1977). Biochim. Biophys. Acta., 482, 228. Chorghade, M. S. (2006). Drug Discovery and Development, Drug Discovery. (p. 456) New York: Wiley Interscience. Chow, C. K., & Palecek, S. P. (2004). Biotech. Prog., 20, 449. Chowdhry, V., & Westheimer, F. H. (1979). Annu. Rev. Biochem, 48, 293. Christen, P. (1970). Esperentia, 26, 337. Christen, P. (1999). Adv. Enzymol., 74, 255. Christensen, U. (2000). Biochem. J., 349, 623. Christiansen, I. A. (1949). Acta Chem. Scand, 3, 493. Christianson, D. W. (1991). Adv. Protein Chem., 42, 281. Chunduru, S. K., Mrachko, G. T., & Calvo, K. C. (1989). Biochemistry, 28, 486. Ciriolo, M. R., Battistoni, A., Falconi, M., et al. (2001). Eur. J. Biochem., 268, 737. Clark, B. F., & Nyborg, J. (1997). Curr. Opin. Struct. Biol., 7, 110. Clark, J. B. (2002). In R. Eisenthal, & M. Danson (Eds.), Enzyme Assays: A Practical Approach (pp. 141–148). London: Oxford University Press. Clark, W. M. (1960). Oxidation-Reduction Potentials of Organic Systems. Williams and Wilkins. Clausen, T., Laber, B., & Messerschmidt, A. (1997). Biol. Chem., 378, 321. Clegg, B. W., & Maxwell, R. M. (1976). Rev. Sci. Inst., 47, 1383.

References

Cleland, W. W. (1963). Biochim. Biophys. Acta, 67, 104. Cleland, W. W. (1967). Adv. Enzymol., 29, 1. Cleland, W. W. (1970). The Enzymes (3rd ed.). Vol. II, 1, (P. D. Boyer. Cleland, W. W. (1973). J. Biol. Chem., 248, 8353. Cleland, W. W. (1975). Biochemistry, 14, 3220. Cleland, W. W. (1977a). Adv. Enzymol., 29, 1. Cleland, W. W. (1977b). Adv. Enzymol., 45, 273. Cleland, W. W. (1977c). Isotope Effects on Enzyme-Catalyzed Reactions. In W. W. Cleland, M. H. O’Leary, & D. B. Northrop (Eds.). Baltimore: University Park Press. Cleland, W. W. (1979). Meth. in Enzymol., 63, 500. Cleland, W. W. (1980). Meth. in Enzymol., 64, 104. Cleland, W. W. (1982a). CRC Crit. Rev. Biochem., 13, 385. Cleland, W. W. (1982b). Meth. Enzymol., 87, 159. Cleland, W. W. (1982c). Meth. Enzymol., 87, 366. Cleland, W. W. (1986). In C. Berlasconi (Ed.), Investigations of Rates and Mechanisms of Reactions, Vol. 6 (p. 791). New York: Wiley. Cleland, W. W. (1990). Faseb J., 4, 2899. Cleland, W. W. (1995). Meth. Enzymol., 249, 341. Cleland, W. W. (2005). Arch. Biochem. Biophys., 433, 2. Cleland, W. W., & Kreevoy, M. M. (1994). Science, 264, 1887. Cleland, W. W., Frey, P. A., & Gerlt, J. A. (1998). J. Biol. Chem., 273, 25529. Cleland, W. W., O’Leary, M. H., & Northrop, D. B. (Eds.) (1977). Isotope Effects on Enzyme Catalyzed Reactions. Baltimore: University Park Press. Cline, K., Ettinger, N. F., & Theg, S. M. (1992). J. Biol. Chem., 267, 2688. Clouet-d’Orval, B., & Uhlenbeck, O. C. (1996). RNA, 2, 483. Clouet-d’Orval, B., & Uhlenbeck, O. C. (1997). Biochemistry, 36, 9087. Cohen, R. M., & Wolfenden, R. (1971). J. Biol. Chem., 246, 7561. Cohn, M. (1963). Biochemistry, 2, 623. Cohn, M., & Hu, A. (1978). Proc. Natl. Acad. Sci. USA, 75, 200. Cohn, M., & Leigh, J. S., Jr. (1962). Nature, 193, 1037. Cole, R. B. (2000). J. Mass Spectrom., 35, 763. Cole, R. B. (1997). Electrospray Ionization Mass Spectrometry: Fundamentals, Instrumentation and Applications. New York: John Wiley and Sons. Coleman, J. E. (1998). Curr. Opinion Chem. and Biol., 2, 222. Coleman, J. E. (1992). Annu. Rev. Biochem., 61, 897. Coll, R. J., Compton, P. D., & Fink, A. L. (1982). Meth. Enzymol., 87, 66. Collins, K. D. (1974). J. Biol. Chem., 249, 136. Collins, K. D. (1997). Biophys. J., 72, 65. Collins, K. D., & Washabaugh, M. W. (1985). Q. Rev. Biophys., 18, 323. Colman, R. F. (1990). The Enzymes, 3rd edn. 19, 283. Connors, K. A. (1987). Binding Constants: The Measurement of Molecular Complex Stability (p. 411). New York: J. Wiley. Conti, F. (1984). Curr. Top. Membr. Transp., 22, 371. Contillo, L. G. (1994). In J. W. Crabb (Ed.), Techniques in Protein Chemistry, Vol. V (p. 493). New York: Academic Press. Cook, P. F., & Cleland, W. W. (1981). Biochemistry, 21, 1797. Cook, P. F., & Cleland, W. W. (2007). Enzyme Kinetics and Mechanism (p. 416). New York: Garland Science.

813

Coon, M. J., & Koop, D. R. (1983). The Enzymes (3rd ed.), 16, 645. Cooper, B. F., & Rudolph, F. B. (1995). Meth. Enzymol., 249, 188. Cooperman, B. S. (1982). Meth. Enzymol., 87, 526. Copeland, R. A. (2000). Enzymes: A Practical Introduction to Structure, Mechanism, and Data Analysis (2nd ed.). New York: Wiley. Copeland, R. A. (2005). Evaluation of Enzyme Inhibitors in Drug Discovery: A Guide for Medicinal Chemists and Pharmacologists. Hoboken: Wiley-Interscience. Copley, M., & Fromm, H. J. (1968). Biochemistry, 6, 3503. Corey, D. R., Willet, W. S., Coombs, G. S., & Craik, C. S. (1995). Biochemistry, 34, 11531. Corey, E. J., Ortiz de Montellano, P. R., Lin, K., & Dean, P. D. G. (1967). J. Am. Chem. Soc., 89, 2797. Cori, G. T., & Green, A. A. (1943). J. Biol. Chem., 151, 31. Corless, R. M., Gonnet, G. H., Hare, D. E. G., & Jeffrey, D. J. (1993). Technical Report Cs-93-03, Department of Computer Science. Canada: University of Waterloo. Cornelius, R. D., & Cleland, W. W. (1978). Biochemistry, 17, 3279. Cornforth, J. W. (1973). Chem. Soc. Rev., 2, 1. Cornish, V. W., Mendel, D., & Schultz, P. G. (1995). Angew. Chem. Int. Ed. Engl, 34, 621. Cornish-Bowden, A. (1979). Principles of Enzyme Kinetics. London: Butterworth. Cornish-Bowden, A. (1984). J. Theor. Biol., 108, 451. Cornish-Bowden, A. (1994). Biochem. J., 301, 621. Cornish-Bowden, A. (1995). Analysis of Enzyme Kinetic Data. Oxford: Oxford University Press. Cornish-Bowden, A. (2001). Methods, 24, 181. Cornish-Bowden, A. (2002). In R. Eisenthal, & M. Danson (Eds.), Enzyme Assays: A Practical Approach (pp. 249–268). London: Oxford University Press. Corrales, F. J., & Fersht, A. R. (1996). Proc. Natl. Acad. Sci., 93, 4509. Corrie, J. E. T., & Trentham, D. R. (1993). In H. Morrison (Ed.), Biological Application of Photochemical Switches, vols. 1–2. New York: Wiley. Cotter, R. (1997). Time-of-Flight Mass Spectrometry: Instrumentation and Applications in Biological Research. Columbus: American Chemical Society. Cotton, F. A. (1990). The Crystal Field Theory: Chemical Applications of Group Theory (3rd ed.). New York: J. Wiley. Cotton, F. A., Wilkinson, G., & Gaus, P. (1995). In Basic Inorganic Chemistry (3rd ed.). New York: J. Wiley. Coulson, A. F. W., Knowles, J. R., & Offord, R. E. (1970). J. Chem. Soc. Chem. Commun., 1, 7. Coulson, A. F. W., Knowles, J. R., Priddle, J. D., & Offord, R. E. (1970). Nature, 227, 180. Cowan, J. A. (1992). Comm. Inorg. Chem., 13, 293. Craig, D. B., Arriaga, E. A., Wong, J. C. Y., et al. (1996). J. Am. Chem. Soc., 118, 5245. Craig, M. E., Crothers, D. M., & Doty, P. (1971). J. Mol. Biol., 10, 1949. Craig, S. P., III, & Eakin, A. E. (2000). J. Biol. Chem., 275, 20231. Craik, D. J., & Wilce, J. A. (1997). Methods in Molecular Biology. New York: Humana Press. In Protein NMR Techniques, Vol. 60, p. 195.

814

Crandall, W. V., & Lowe, M. E. (2001). J. Biol. Chem., 276, 12505. Crane, R. K. (1962). In P. D. Boyer, H. A. Lardy, & K. Myrback (Eds.) (3rd ed.) The Enzymes, 6 (p. 47). Crane, R. K., & Dorando, F. C. (1980). Ann. N.Y. Acad. Sci., 339, 46. Creighton, D. J., & Murthy, N. S. R. K. (1990). The Enzymes, 19, 323, 3rd edn. Crooks, J. E. (1983). J. Phys. E: Sci. Instrum., 16, 1142. Cruz, J. A., Sacksteder, C. A., Kanazawa, A., & Kramer, D. M. (2001). Biochemistry, 40, 1226. Csa´sza´r, A. G., Czako´, G., Furtenbacher, T., et al. (2005). J. Chem. Phys., 122, 214305. Cullis, P. M., & Lowe, G. (1978). J. Chem. Soc. Chem. Commun., 12, 512. Cullis, P. M., & Lowe, G. (1981). J. Chem. Soc. Perkin Trans., 1, 2317. Cullis, P. M., Wolfenden, R., Cousens, L. S., & Alberts, B. M. (1982). J. Biol. Chem., 257, 12165. Cunningham, B. C., & Wells, J. A. (1989). Science, 244, 1081. Cush, R., Cronin, J. M., Stewart, W. J., et al. (1993). Biosensors Bioelectronics, 8, 347. Czerlinski, G. H. (1966). Chemical Relaxation: An Introduction to Theory and Application of Stepwise Perturbation (p. 309). New York: Marcel Dekker. D’Amico, S., Gerday, C., & Feller, G. (2002). J. Biol. Chem., 277, 46110. D’Amico, S., Marx, J.-C., Gerday, C., & Feller, G. (2003). J. Biol. Chem., 278, 7891. Dabiri, G. A., Sanger, J. M., Portnoy, D. A., & Southwick, F. S. (1990). Proc. Natl. Acad. Sci. USA, 87, 6068. Dafforn and Koshland (1971). Proc. Natl. Acad. Sci. USA, 68, 2463. Dafforn and Koshland (1973). Biochem. Biophys. Res. Communs., 52, 779. Dahlquist, F. W., & Purich, D. L. (1975). Biochemistry, 14, 1980. Dahlquist, F. W., Rand-Meir, T., & Raftery, M. A. (1969). Biochemistry, 8, 4214. Dahm, S. C., & Uhlenbeck, O. C. (1990). Biochimie, 72, 819. Dai, Y., Hedstrom, L., & Abeles, R. H. (2000). Biochemistry, 39, 6498. Dalziel, K. (1957). Acta Chem. Scand, 11, 1706. Dalziel, K. (1963). J. Biol. Chem., 238, 2850. Danenberg, P. V., & Danenberg, K. D. (1978). Biochemistry, 17, 4018. Daniel, R. M., Finney, J. L., Re´at, V., et al. (1999). Biophys. J., 77, 2184. Daniel, R. M., Smith, J. C., Ferrand, M., et al. (1998). Biophys. J., 75, 2504. Daniel, R. M., Danson, M. J., & Eisenthal, R. (2001). Trends Biochem. Sci., 26, 223. Dantas, G., Kuhlman, B., Callender, D., et al. (2003). J. Mol. Biol., 332, 449. Darbre, A. (1986). In A. Darbre (Ed.), Practical Protein Chemistry: A Handbook. Chichester: John Wiley and Son. Darnault, C., Volbeda, A., Kim, E. J., et al. (2003). Nature Struct. Biol., 10, 271. Darvey, I. G. (1976). Mol. Cell. Biochem., 11, 3.

References

Darvey, I. G., Shrager, R., & Kohn, L. D. (1975). J. Biol. Chem., 250, 4696. Datta, P., Gest, H., & Segal, H. L. (1964). Proc. Natl. Acad. Sci. USA, 51, 125. Davenport, R. C., Bash, P. A., Seaton, B. A., et al. (1991). Biochemistry, 30, 5821. Davenport, R. J., Wuite, G. J., Landick, R., & Bustamante, C. (2000). Science, 287, 2497. Davidson, V. L. (1999). Acct. Chem. Res., 33, 87. Davies, D. R. (1990). Annu. Rev. Biophys. Biophys. Chem., 19, 189. Davies, G. J., Wilson, K. S., & Henrissat, B. (1997). Biochem. J., 321, 557. Davies, G., Kustin, K., & Pasternack, R. F. (1969). Inorg. Chem., 8, 1535. Davis, J. S. (1998). Adv. Exper. Med. Biol., 453, 343. Davis, J. S., & Harrington, W. F. (1993). Adv. Exp. Med. Biol., 332, 513. Davis, L., & Metzler, D. E. (1972). The Enzymes, 7, 33, 3rd edn. Davis, M. E., & McCammon, J. A. (1990). Chem. Rev., 90, 509. Davis, R. P. (1961). The Enzymes, 5, 454, 2nd edn. Dawson, R. M. C., Elliott, D. C., Elliott, W. H., & Jones, K. M. (1986). Data for Biochemical Research (3rd ed.). Oxford: Clarendon Press. Day, L. A., Singer, S. J., & Sturtevant, J. M. (1962). J. Am. Chem. Soc., 84, 3769. Dayel, M. J., & Mullins, R. D. (2004). PLoS Biol., 2, E91. Dayhoff, M. O., Barker, W. C., & Hunt, L. T. (1983). Meth. Enzymol., 91, 524. De Freitas, D. M., Castro, M. C. A., & Geraldes, C. F. G. C. (2006). Acct. Chem. Res., 39, 283. Deans, N. L., Allison, R. D., & Purich, D. L. (1993). Biochem. J., 286, 243. Decicco, C. P., Nelson, D. J., Luo, Y., et al. (2001). Biorg. Med. Chem. Lett., 11, 2561. Deems, R. A., Eaton, B. R., & Dennis, E. A. (1975). J. Biol. Chem., 250, 9013. Deisenhofer, J., Epp, O., Miki, K., et al. (1984). J. Mol. Biol., 180, 385. Deisenhofer, J., Epp, O., Miki, K., et al. (1985). Nature (London), 318, 618. DeKock, R. L., & Gray, H. B. (1989). Chemical Structure and Bonding. Mill Valley, California: University Science Books. de Lange, F., Cambi, A., Huijbens, R., de Bakker, B., Rensen, W., Garcia-Parajo, M., Rensen, W., Wvan Hulst, N., & Figdor, C. G. (2001). J. Cell Sci., 114, 4153. del Rosario, E. J., & Hammes, G. G. (1971). Biochemistry, 10, 716. Delaney, S. M., Blackburn, G. M., & Geiger, J. D. (1997). Eur. J. Pharmacol., 332, 35. Delbru¨ck, M. (1940). J. Chem. Phys., 8, 120. Delgado, J., & Liao, J. C. (1995). Biosystems, 36, 55. della Ragione, F., Porcelli, M., Carteni-Farina, M., et al. (1985). Biochem. J., 232, 335. DeMartino, G. N., & Slaughter, C. A. (1999). J. Biol. Chem., 274, 22123. Dembo, M., Tourney, D. C., Saxman, K., & Hammer, D. (1988). Proc. Royal Soc. London, 234, 5583. Demchuk, E., & Wade, R. (1996). J. Phys. Chem., 100, 17373.

References

Denbigh, K. (1966). The Priciples of Chemical Equilibrium: with Applications in Chemistry and Chemical Engineering. Cambridge University Press. Denk, W., Strickler, J. H., & Webb, W. W. (1990). Science, 248, 73. Dennis, E. A. (1973). Arch. Biochem. Biophys, 158, 485. Denu, J. M., & Dixon, J. E. (1995). Proc. Natl. Acad. Sci. USA, 92, 5910. Deperthes, D. (2002). Biol. Chem., 383, 1107. DeRose, V. J. (2002). Chem. Biol., 9, 961. Desai, G., Panick, G., Zein, M., et al. (1999). J. Mol. Biol., 288, 461. Despa, F., & Berry, R. S. (2007). Biophys. J., 92, 373. Despa, F., Ferna´ndez, A., & Berry, R. S. (2004). Phys. Rev. Lett., 93, 228104. Deutscher, M. P. (1990). Guide to Protein Purification (p. 894), Vol. 182, Methods in Enzymology, New York: Academic Press. DeWolf, W. E., Jr., Fullin, F. A., & Schramm, V. L. (1979). J. Biol. Chem., 254, 10868. Dewolf, W., Jr., & Segel, I. H. (2000). J. Enzyme Inhib., 15, 311. Di Cera, E. (2006). J. Biol. Chem., 281, 1305. Di Sabato, G., & Jencks, W. P. (1961). J. Am. Chem. Soc., 83, 4400. Dice, J. F., & Goldberg, A. L. (1975). Arch. Biochem. Biophys., 170, 213. Dick, T. P., Ruppert, T., Groettrup, M., et al. (1996). Cell, 86, 253. Dickerson, R. E., & Geis, I. (1983). Hemoglobin: Structure, Function, and Pathology (p. 4245). Menlo Park: Benjamin/ Cummings Publishing. Dickinson, R. B., & Purich, D. L. (2002). Biophys. J., 82, 605. Dickinson, R. B., & Purich, D. L. (2006). Biophys. J., 91, 1548. Dickinson, R. B., Caro, L., & Purich, D. L. (2004). Biophys. J., 87, 2838. Dickinson, R. B., Southwick, F. S., & Purich, D. L. (2002). Arch. Biochem. Biophys., 406, 296. Diebler, H., Eigen, M., & Matthies, P. (1962). Z. Naturforsch, 166, 6146. Dietz, H., & Rief, M. (2004). Proc. Natl. Acad. Sci. USA, 101, 16192. Dill, K. A. (1997). J. Biol. Chem., 272, 701. Dill, K.A., & Bromberg, S. (2003). Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, (p. 666). New York: Garland Science. Dimcheva, N., Horozova, E., & Jordanova, Z. (2002). Z. Naturforsch., 57, 883. Dimroth, P. (1981). J. Biol. Chem., 256, 11974. Dimroth, P. K. G. (2000). Biochim. Biophys. Acta, 1458, 374. Dimroth, P. K. G., & Matthey, U. (1998). Biochim. Biophys. Acta, 1365, 87. Dische, Z. (1940). Bull. Soc. Chim. Biol., 23, 1140. Dixon, M. (1953). Biochem. J., 55, 170. Dixon, M., & Webb, E. C. (1979). Enzymes (3rd ed.). New York: Academic Press. Dixon, N. E., Gazzola, T. C., Blakeley, R. L., et al. (1975). J. Am. Chem. Soc., 97, 4131. Dobson, C. M., & Ferch, A. R. (1995). Protein Folding. Cambridge, England: Cambridge University Press.

815

Dorfman, L. M. (1974). In G. G. Hammes (Ed.) (3rd ed.), Investigation of Rates and Mechanisms of Reactions, Part II, vol. VI (p. 463ff). New York: Wiley-Interscience. Doudna, J., & Cech, T. (2002). Nature, 418, 228. Doukov, T. I., Iverson, T. M., Seravalli, J., et al. (2002). Science, 298, 567. Douzou, P. (1977). Adv. Enzymol., 45, 157. Douzou, P., & Balny, C. (1977). Proc. Natl. Acad. Sci. USA, 74, 2297. Dowd, J. E., & Riggs, D. S. (1965). J. Biol. Chem., 240, 863. Dowd, P., Ham, S., & Hershline, R. (1992). J. Am. Chem. Soc., 114, 7613. Dowers, T. S., Rock, D. A., & Jones, J. P. (2004). J. Am. Chem. Soc., 126, 8868. Downward, J. (1992). Curr. Biol., 2, 329. Drauz, K., & Waldmann, H. (2002). Enzyme Catalysis in Organic Synthesis: A Comprehensive Handbook, 2 vols. (p. 1500). Berlin: Wiley-VCH. Drott, J., Lindstro¨m, K., Rosengren, L., & Laurell, T. (1997). J. Micromech. Microeng., 7, 14. Ducruix, A., & Giege, R. (1982). Crystallization of Nucleic Acids and Proteins: A Practical Approach. Oxford, UK: IRL Press. Duffield, R. B., & Calvin, M. (1946). J. Am. Chem. Soc., 68, 557. Duggleby, R. G., & Morrison, J. F. (1977). Biochim. Biophys. Acta, 481, 297. Duggleby, R. G., & Morrison, J. F. (1978). Biochim. Biophys. Acta, 526, 398. Duggleby, R. G. (1985). Biochem. J., 228, 55. Duggleby, R. G. (1986a). Biochem. J., 235, 55. Duggleby, R. G. (1986b). J. Theor. Biol., 123, 67. Duggleby, R. G. (1994). Biochim. Biophys. Acta, 1205, 268. Duggleby, R. G. (1995). Meth. Enzymol., 249, 61. Duggleby, R. G. (2001). Methods, 24, 168. Duggleby, R. G., & Clarke, R. B. (1991). Biochim. Biophys. Acta, 1080, 231. Duggleby, R. G., & Nash, J. C. (1989). Biochem. J., 257, 57. Duggleby, R. G., & Northrop, D. B. (1989). Bioorg. Chem., 17, 177. Duggleby, R. G., & Wood, C. (1990). Biochem. J., 270, 843. Duine, J. A., Frank, J., & Jongejan, J. A. (1987). Adv. Enzymol., 59, 169. Duke, T. A., Le Nove`re, N., & Bray, D. (2001). J. Mol. Biol., 308, 541. Dunaway-Mariano, D., & Cleland, W. W. (1980). Biochemistry, 19, 1496, 1506. Duncan, K., & Walsh, C. T. (1988). Biochemistry, 27, 3709. Dunford, H. B. (1998). In M. Sinnot (Ed.), Comprehensive Biological Catalysis, vol. 3 (p. 195). London: Academic Press. Dunford, H. B. (1999). Heme Peroxidases. New York: WileyVCH. Dunn, B. M., & Chaiken, I. M. (1974). Proc. Natl. Acad. Sci. USA, 71, 2382. Dwivedi, C. M., Ragin, R. C., & Uren, J. R. (1982). Biochemistry, 21, 3064. Dyer, R. B., Gai, F., & Woodruff, W. H. (1998). Accounts Chem. Res., 31, 706. Eady, R. R., Robson, R. L., Richardson, T. H., et al. (1987). Biochem. J., 244, 197.

816

Earnhardt, J. N., & Silverman, D. N. (1998). In M. Sinnot (Ed.), Comprehensive Biological Catalysis, vol. 1 (p. 483). London: Academic Press. Easson, L. H., & Stedman, E. (1936). Proc. Roy. Soc. London, Series B, 121, 142. Easterby, J. S. (1981). Biochem. J., 199, 155. Eaton, W. A., Henry, E. R., Hofrichter, J., & Mozzarelli, A. (1999). Nat. Struct. Biol., 6, 351. Eckstein, F., Romaniuk, P. J., & Connolly, B. A. (1982). Meth. Enzymol, 87, 197. Edelson, D. (1976). Computers in Chem., 1, 29. Ederer, M., & Gilles, E. D. (2007). Biophys. J., 92, 1846. Edman, L., & Rigler, R. (2000). Proc. Natl. Acad. Sci. USA, 97, 8266. Eger, B. T., & Benkovic, S. J. (1992). Biochemistry, 31, 9227. Eger, B. T., Kuchta, R. D., Carroll, S. S., et al. (1991). Biochemistry, 30, 1441. Egner, U., Tomasselli, A. G., & Schulz, G. E. (1987). J. Mol. Biol., 195, 649. Ehlert, F. J. (1988). Molec. Pharmacol., 33, 187. Eigen, M., & Wilkins, R. G. (1965). Adv. Chem. Ser., 49, 55. Eigen, M. (1954). Z. Phys. Chem (Leipzig) NF1 176. Eigen, M. (1964). Angew. Chem. Int., Ed, 3, 1. Eigen, M. (1968). Q. Rev. Biophys., 1, 3. Eigen, M., & de Maeyer, L. (1955). Int. Z. Elektrochemie, 59, 986. Eigen, M., & de Maeyer, L. (1963). Tech. Org. Chem.. 8 (Part II), 895. Eigen, M., & de Maeyer, L. (1974). Investigation of Rates and Mechanisms of Reactions, Part II. In Techniques of Chemistry, Vol 6. (p. 63). New York: Wiley-Interscience. Eigen, M., & Hammes, G. G. (1963). Adv. Enzymol, 25, 1. Eigen, M., & Rigler, R. (1994). Proc. Natl. Acad. Sci. USA, 91, 5740. Eigen, M., & Schoen, J. (1955). Z. Elektrochem, 59, 483. Eigen, M., & Tamm, K. (1962). Z. Elektrochem, 66, 93, and 107. Eigen, M., Kruse, W., Maass, G., & De Maeyer, L. (1964b). Prog. React. Kinet., 2, 285. Eigen, M., Kustin, K., & Munson, R. (1962). Z. Phys. Chem. N.F., 31, 140. Einspahr, H., & Bugg, C. E. (1980). Acta Cryst., B36, 264. Einstein, A. (1920). Sitzber. Preuss. Akad. Wiss., Phys. Math. KI, 380. Eisenmesser, E. Z., Bosco, D. A., Akke, M., & Kern, D. (2002). Science, 295, 1520. Eisenstein, E., Han, M. S., Woo, T. S., et al. (1992). J. Biol. Chem., 267, 22148. Eisenthal, R., & Cornish-Bowden, A. (1974). Biochem. J., 139, 715. Eisenthal, R., Danson, M. J., & Hough, D. W. (2007). Trends in Biotechnol, 25, 247. Elcock, A. H., Huber, G. A., & McCammon, J. A. (1997). Biochemistry, 36, 16049. Elion, G. B., Burgi, E., & Hitchings, G. H. (1952). J. Am. Chem. Soc., 74, 411. Ellis, J. J., & Morrison, J. F. (1982). Meth. Enzymol., 87, 405. Ellis, R. J. (2001). Trends Biochem. Sci., 26, 597. Ellis, R. J., & Minton, A. P. (2003). Nature, 425, 27. Elson, E. L., & Magde, D. (1974). Biopolymers, 13, 1.

References

Emerson, M. T., Grunwald, E., & Kromhout, R. A. (1960). J. Chem. Phys., 33, 547. Emilsson, G. M., Nakamura, S., Roth, A., & Breaker, R. R. (2003). RNA, 9, 907. Engel, P. C. (1968). PhD Dissertation. Oxford University. Engel, P. C., & Dalziel, K. (1970). Biochem. J., 118, 409. Engelhardt, V. A., & Ljubinova, M. N. (1942). Nature, 144, 668. Engers, H. D., Bridger, W. A., & Madsen, N. B. (1969). J. Biol. Chem., 244, 5936. Englander, S. W., & Kallenbach, N. R. (1983). Quarterly Rev. Biophys., 16, 521. English, B. P., Min, W., van Oijen, A. M., et al. (2005). Nature Chem. Biol., 2, 87. Englund, P. T., Huberman, J. A., Jovin, T. M., & Kornberg, A. (1969). J. Biol. Chem., 244, 3038. Eppler, R. K., Komor, R. S., Huynh, J., et al. (2006). Proc. Natl. Acad. Sci. USA, 103, 5706. Epstein, D. M., Benkovic, S. J., & Wright, P. E. (1995). Biochemistry, 34, 11037. Erez, T., Gdalevsky, G. Y., Torchinsky, Y. M., et al. (1998). Biochim. Biophys. Acta, 1384, 365. Erijmann, L. E., & Klegg, R. M. (1998). Biophys. J., 75, 453. Erion, M. D., Stoeckler, J. D., Guida, W. C., et al. (1997). Biochemistry, 36, 11735. Ermak, D. L., & McCammon, J. A. (1978). J. Chem. Phys., 69, 1352. Escobar, W. A., Tan, A. K., Lewis, E. R., & Fink, A. L. (1994). Biochemistry, 33, 7619. Esson, W. (1866). Phil. Trans. Roy. Soc. London Ser. A., 156, 220. Esteban, J. A., Banerjee, A. R., & Burke, J. M. (1997). J. Biol. Chem., 272, 13629. Eu, B. C. (1994). Kinetic Theory and Irreversible Thermodynamics. New York: Wiley. Euler, H. V., Josephson, K., & Myrba¨ck, K. (1924). HoppeSeyler’s Z. Physiol. Chem., 134, 39. Euler, L. (1779) ‘‘De serie Lambertina plurimisque eius insignibus proprietatibus’’. Originally published in Acta Academiae Scientarum Imperialis Petropolitinae, 1779, 1783, 29–51 Also in Opera Omnia Series 1, Volume 6 (pp. 350–369). Evans, B., & Wolfenden, R. (1970). J. Am. Chem. Soc., 92, 4751. Evans, E. (2001). Annu. Rev. Biochem., 30, 105. Evans, J. A., & Deutscher, M. P. (1978). J. Biol. Chem., 253, 7276. Evans, M. G., & Polanyi, M. (1935). Trans. Faraday Soc., 31, 875. Evans, M. G., & Polanyi, M. (1936). Trans. Faraday Soc., 32, 1333. Everett, R. R., Kanofsky, J. R., & Butler, A. (1990). J. Biol. Chem., 265, 4908. Everts, S. (2008). Chem. and Engin. News, 29, 15. Eyring, H. (1935a). Chem. Rev., 17, 65. Eyring, H. (1935b). J. Chem. Phys., 3, 107. Eyring, H., & Stern, A. E. (1939). Chem. Rev., 24, 2. Falke, J. J., & Koshland, D. E., Jr. (1987). Science, 237, 1596. Fan, C., Moews, P. C., Walsh, C. T., & Knox, J. R. (1994). Science, 266, 439. Fang, T.-Y., Alechina, O., Aleshin, A. E., et al. (1998). J. Biol. Chem., 273, 19553. Fang, X.-W., Thiyagarajan, P., Sosnick, T. R., & Pan, T. (2002). Proc. Natl. Acad. Sci. USA, 99, 8518.

References

Fang, X., Wu, J., Wei, G., et al. (1995). Int. J. Radiat. Biol., 68, 459. Fara, J., & Urquhart, J. (1984). Trends Pharmacol. Sci., 5, 21. Farver, O., Wherland, S., & Pecht, I. (1994). J. Biol. Chem., 269, 22933. Fasella, P., & Hammes, G. G. (1967). Biochemistry, 6, 1798. Fauzi, H., Kawakami, J., Nishikawa, F., & Nishikawa, S. (1997). Nucl. Acids Res., 25, 3124. Fedor, M. J. (2000). J. Mol. Biol., 297, 269. Fedor, M. J., & Uhlenbeck, O. C. (1990). Proc. Natl Acad. Sci. USA, 87, 1668. Fedor, M. J., & Uhlenbeck, O. C. (1992). Biochemistry, 31, 12042. Feher, G., & Weissman, M. (1973). Proc. Natl. Acad. Sci. USA, 70, 870. ˚ qvist, J. (2002). Theor. Chem. Acc., 108, 7184. Feierberg, I., & A Feldman, P. L., Griffith, O. W., Hong, H., & Stuehr, D. J. (1993). J. Med. Chem., 36, 491. Fell, D. (1997). Understanding the Control of Metabolism. London: Portland Press. Ferguson, W. J., & Good, N. E. (1980). Anal. Biochem., 104, 300. Fernley, H. N. (1974). Eur. J. Biochem., 43, 377. Ferrer, J. L., Jez, J. M., Bowman, M. E., et al. (1999). Nat. Struct. Biol., 6, 775. Ferry, R. M., & Green, A. A. (1929). J. Biol. Chem., 81, 175. Fersht, A. R. (1972). J. Mol. Biol., 64, 497. Fersht, A. R. (1974). Proc. Roy. Soc. London, Ser. B, 187, 397. Fersht, A. R. (1977). Enzyme Structure and Mechanism. San Francisco: W.H. Freeman (p. 371). Fersht, A. R. (1985). Enzyme Structure and Mechanism (2nd ed.). (p. 475). New York: W.H. Freeman and Co. Fersht, A. R. (1987). Trends Biochem. Sci., 12, 301. Fersht, A. R. (1998). Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding (p. 650). New York: W.H. Freeman. Fersht, A. R., Ashford, J. S., Bruton, C. J., et al. (1975). Biochemistry, 14, 1. Fersht, A. R., Mulvey, R. S., & Koch, G. L. E. (1975). Biochemistry, 14, 13. Fibich, A., Janko, K., & Apell, H.-J. (2007). Biophys. J., 3, 3092. Fielding, A. H. (1981). J. Chem. Soc., Perkin Trans., 1, 1013. Fierke, C. A., & Hammes, G. G. (1995). Meth. Enzymol., 249, 1. Fierke, C. A., Johnson, K. A., & Benkovic, S. J. (1987). Biochemistry, 26, 4085. Fink, A. L. (1973). Biochemistry, 12, 1736. Fink, A. L. (1974). J. Biol. Chem., 249, 5027. Fink, A. L. (1976). Biochemistry, 15, 1580. Fink, A. L. (1977). Acc. Chem. Res., 10, 233. Fink, A. L., & Ahmed, A. I. (1976). Nature (London), 263, 294. Fink, A. L., & Geeves, M. A. (1979). Meth. Enzymol. 63, 336. Fink, A. L., & Meehan, P. (1979). Proc. Natl. Acad. Sci. USA, 76, 1566. Finley, T. H., Valinsky, J., Sato, K., & Abeles, R. H. (1972). J. Biol. Chem., 247, 4197. First, E. A. (1998). In M. Sinnott (Ed.), Comprehensive Biological Catalysis: A Mechanistic Reference, vol. I (p. 596). London: Academic Press. Fischer, E. (1894). Ber. Deutsches Chem. Ges., 27, 2985. Fischer, G., Tradler, T., & Zarnt, T. (1998). FEBS Lett., 426, 17. Fischer, S., & Gra¨ber, P. (1999). FEBS Lett., 457, 327.

817

Fisher, L. M., Albery, W. J., & Knowles, J. R. (1976). Biochemistry, 15, 5621. Fisher, J., Charnas, R. L., & Knowles, J. R. (1978). Biochemistry, 17, 2180. Fisher, H. F., Conn, E. E., Vennesland, B., & Westheimer, F. H. (1953). J. Biol. Chem., 202, 687. Fisher, L. M., Albery, W. J., & Knowles, J. R. (1986a). Biochemistry, 25, 2529. Fisher, L. M., Albery, W. J., & Knowles, J. R. (1986b). Biochemistry, 25, 2538. Fisher, L. M., Belasco, J. G., Bruice, T. W., et al. (1986). Biochemistry, 25, 2543. Fletcher, S. J., Herlihy, J. M., Albery, W. J., & Knowles, J. R. (1976). Biochemistry, 15, 5612. Fletterick, R. J., Bates, D. J., & Steitz, T. A. (1975). Proc. Natl. Acad. Sci. USA, 72, 38. Flinterman, M., Farzaneh, F., Habib, N., et al. (2009). Mol Ther., 17, 334. Floss, H. G. (1982). Method. Enzymol. 87, 126. Floss, H. G., & Tsai, M.-D. (1979). Adv. Enzymol. 50, 243. Focher, F., Verri, A., Spadari, S., et al. (1993). Biochem. J., 292, 883. Folk, J. E. (1982). Meth. Enzymol. 87, 36. Folk, J. E. (1983). Adv. Enzymol. 54, 1. Fontecave, M., Nordlund, P., Eklund, H., & Reichard, P. (1992). Adv. Enzymol., 65, 147. Forchhammer, K., Boesmiller, K., & Bock, A. (1991). Biochimie, 73, 1481. Forster, A., & Hill, C. P. (2003). Trends Cell Biol., 13, 550. Fo¨rster, T. (1948). Ann. Phys. (Leipzig), 2, 55. Foster, R. J., & Niemann, C. (1953). Proc. Natl. Acad. Sci. USA, 39, 999. Fox, D. K., Meadow, N. D., & Roseman, S. (1986). J. Biol. Chem., 261, 13498. Fox, B. G. (1998). In M. Sinnott (Ed.), Comprehensive Biological Catalysis, vol. 3 (p. 261). London: Academic Press. Francesconi, M. A., Donella-Deana, A., Furlametto, V., et al. (1989). Biochim. Biophys. Acta, 999, 163. Francisco, W. A., Knapp, M. J., Blackburn, N. J., & Klinman, J. P. (2002). J. Am. Chem. Soc., 124, 8194. Franck, J., & Rabinowitch, E. (1934). Trans. Faraday Soc., 30, 120. Francklyn, C. S. (2008). Biochemistry, 47, 11695. Frandsen, T. P., & Svensson, B. (1998). Plant Mol. Biol., 37, 1. Frankel, B. A., Kruger, R. G., Robinson, D. E., et al. (2005). Biochemistry, 44, 11188. Frauenfelder, H., Parak, F., & Young, R. D. (1988). Annu. Rev. Biophys. Biophys. Chem., 17, 451. Frau´sto da Silva, J. J. R., & Williams, R. J. P. (2001). The Biological Chemistry of the Elements (2nd ed.). Oxford: Clarendon Press. Freire, E. (2002). Nature Biotechnol., 20, 15. French, D. (1957). Carbohydrate Chem., 12, 189. French, T. C., & Hammes, G. G. (1965). J. Am. Chem. Soc., 87, 4669. French, T. C., & Hammes, G. G. (1969). J. Am. Chem. Soc., 91, 1278. French, D., Levine, M. L., Pazur, J. H., & Norberg, E. (1950). J. Am. Chem. Soc., 72, 1746.

818

Frey, P. A. (1987). In D. Dolphin, R. Poulson, & O. Avramovic (Eds.), Pyridine Nucleotide Coenzyme: Chemical, Biochemical, and Medical Aspects, vol. 2B (p. 462). New York: Academic Press. Frey, P. A. (1992). The Enzymes, 20, 147. Frey, P. A. (2001). Ann. Rev. Biochem., 70, 121. Frey, P. A., & Abeles, R. H. (1966). J. Biol. Chem., 242, 5369. Frey, P. A., & Hegeman, A. D. (2007). Enzymatic Reaction Mechanisms (p. 768). New York: Oxford University Press. Frey, P. A., Eisenberg, M. K., & Abeles, R. H. (1967). J. Biol. Chem., 241, 2732. Frey, P. A., Kokesh, F. C., & Westheimer, F. H. (1971). J. Am. Chem. Soc., 93, 7266. Frey, P. A., Whitt, S. A., & Tobin, J. B. (1994). Science, 264, 1927. Frick, L., Yang, C., Marquez, V. E., & Wolfenden, R. (1989). Biochemistry, 28, 9423. Frickey, T., & Lupas, A. N. (2004). J. Struct. Biol., 146, 2. Fried, M. G., & Crothers, D. M. (1981). Nucleic Acids Res., 9, 6505. Frieden, C. (1959). J. Biol. Chem., 234, 2891. Frieden, C. (1963). J. Biol. Chem., 238, 3299. Frieden, C. (1967). J. Biol. Chem., 242, 4045. Frieden, C. (1968). In E. Kramme, & A. Pihl (Eds.), The Regulation of Enzyme Activity and Allosteric Interactions (p. 59). New York: Academic Press. Frieden, C. (1997). Trends in Biochem. Sci., 22, 317. Frieden, C., & Colman, R. F. (1967). J. Biol. Chem., 242, 1705. Friedlander, G., Kennedy, J. W., Macias, E. S., & Miller, J. M. (1981). Nuclear and Radiochemistry (chap. 9) (3rd ed.). New York: Wiley. Friedrich, M. J., de Veaux, L. C., & Kadner, R. J. (1986). J. Bacteriol., 167, 928. Fritsch, F. N., Shafer, R. E., & Crowley, W. P. (1973). Comm. ACM, 16, 123. Fromm, H. J. (1961). Biochim. Biophys. Acta, 52, 199. Fromm, H. J. (1964). Biochim. Biophys. Acta, 81, 413. Fromm, H. J. (1967). Biochim. Biophys. Acta, 139, 221. Fromm, H. J. (1969). Eur. J. Biochem., 7, 385. Fromm, H. J. (1970). Biochem. Biophys. Res. Communs., 40, 692. Fromm, H. J. (1975). Initial Rate Enzyme Kinetics. New York: Springer-Verlag. Fromm, H. J., & Fromm, S. J. (1999). Biochem. Biophys. Res. Commun, 265, 448. Fromm, H. J., & Nelson, D. R. (1962). J. Biol. Chem., 237, 215. Fromm, H. J., & Zewe, V. (1962a). J. Biol. Chem., 237, 1661. Fromm, H. J., & Zewe, V. (1962b). J. Biol. Chem., 237, 3027. Fromm, H. J., Zewe, V., & Fabiano, R. (1964). J. Biol. Chem., 239, 1625. Frost, A. A., & Pearson, R. G. (1961). Kinetics and Mechanism (2nd ed.). New York: Wiley. Funatsu, T., Harada, Y., Higuchi, H., et al. (1997). Biophys. Chem., 68, 63. Furakawa, Y., Yamada, R., & Iwashima, A. (1981). Acta Vitaminol. Enzymol., 3, 145. Fyfe, J. A., Miller, R. L., & Krenitsky, T. A. (1973). J. Biol. Chem., 248, 3801. Gabdoulline, R. R., & Wade, R. C. (1998). Methods, 14, 329.

References

Gabry, B. J., Pesz, K., & Bartkiewicz, S. J. (2004). Physica A: Stat. Theoret. Phys., 336, 112. Gaertner, F. H., & Cole, K. W. (1973). J. Biol. Chem., 248, 4602. Gaitonde, M. K., & Richter, D. (1956). Proc. Roy. Soc. B, 145, 83. Galburt, E. A., & Stoddard, B. L. (2001). Physics Today, 54, 33. Galdes, A., Auld, D. S., & Vallee, B. L. (1983). Biochemistry, 22, 1888. Gallo, K. A., & Knowles, J. R. (1993). Biochemistry, 32, 3981. Gallo, K. A., Tanner, M. E., & Knowles, J. R. (1993). Biochemistry, 32, 3991. Gamero-Castano, M., & Fernandez de la Mora, J. (2000). J. Mass Spectrom., 35, 790. Gandour, R. D., & Schowen, R. L. (1978). Transition States of Biochemical Processes. New York: Plenum Press. Ganzhorn, A. J., & Plapp, B. V. (1988). J. Biol. Chem., 263, 5446. Ganzhorn, A. J., Green, D. W., Hershey, A. D., et al. (1987). J. Biol. Chem., 262, 3754. Gao, J., & Truhlar, D. G. (2002). Ann. Rev. Phys. Chem., 53, 467. Garces, E., & Cleland, W. W. (1969). Biochemistry, 8, 633. Garcia-Parajo, M. F., Segers-Nolten, G. M. J., Veerman, J. A., et al. (2000). Proc. Natl. Acad. Sci. USA, 97, 7237. Garcı´a-Viloca, M., Gao, J., Karplus, M., & Truhlar, D. G. (2004). Science, 303, 186. Garfinkel, D., Garfinkel, L., Pring, M., et al. (1970). Ann. Rev. Biochem., 39, 473. Garland, P. B., & Randle, P. J. (1964). Biochem. J., 91, 6, c-7c. Garner, M. M., & Revzin, A. (1981). Nucleic Acids Res., 9, 3047. Garrard, L. J., Bui, Q. T. N., Nygaard, N., & Rauschel, F. M. (1985). J. Biol. Chem., 260, 5548. Garrido-del Solo, C., Garcia-Canovas, F., Havsteen, B. H., & Varon-Castellanos, R. (1993). Gass, J. D., & Meister, A. (1970). Biochemistry, 9, 1380. Gatt, S., & Racker, E. (1959). J. Biol. Chem., 234, 1024. Gavel, O. Y., Bursakov, S. A., Calvete, J. J., et al. (1998). Biochemistry, 37, 16225. Gawlita, E., Caldwell, W. S., O’Leary, M. H., et al. (1995). Biochemistry, 34, 2577. Gear, C. W. (1971). Numerical Initial-Value Problems in Ordinary Differential Equations. Englewood Cliffs, New Jersey: Prentice-Hall. Gehring, H., Rando, R. R., & Christen, P. (1977). Biochemistry, 16, 4832. Geoghegan, K. F., Emery, M. J., Martin, W. H., et al. (1993). Bioconjug. Chem., 4, 537. Gerhart, J. C., & Pardee, A. B. (1962). J. Biol. Chem., 237, 891. Gerhart, J. C., & Pardee, A. B. (1963). Cold Spring Harbor Symp. Quant. Biol., 28, 505. Gerhart, J. C., & Pardee, A. B. (1964). Federation Proc., 20, 727. Gerlt, J. A. (1992). Enzymes, 20, 95. Gerstein, M., & Chothia, C. (1996). Proc. Natl. Acad. Sci. USA, 93, 10167. Gething, M. J., & Sambrook, J. (1992). Nature, 355, 33. Getzoff, E. D., Cabelli, D. E., Fisher, C. L., Parge, H. E., et al. (1992). Nature, 358, 347. Getzoff, E. D., Tainer, J. A., Stempien, M. M., et al. (1989). Proteins Struct. Funct. Genet., 5, 322. Ghisla, S., & Massey, V. (1980). J. Biol. Chem., 255, 5688. Gibbons, D. L., Hixson, J. D., Hay, N., et al. (1996). J. Biol. Chem., 271, 31989.

References

Gibbons, I. R., Cosson, M. P., Evans, J. A., et al. (1978). Proc. Natl. Acad. Sci. USA, 75, 2220. Gibbons., I. R. (1988). J. Biol. Chem., 263, 15837. Gibbs, C. S., & Zoller, M. J. (1991). J. Biol. Chem., 266, 8923. Gibbs, J. W. (1948). The Collected Works of J. Willard Gibbs, Vol. 1. New Haven, CT: Yale University Press. Gibson, Q. H., Swoboda, B. E. P., & Massey, V. (1964). J. Biol. Chem., 239, 3927. Gilbert, S. P., & Johnson, K. A. (1994). Biochemistry, 33, 19510. Gill, H. S., & Eisenberg, D. (2001). Biochemistry, 40, 1903. Gill, I., & Ballesteros, A. (2002). Trends Biotech., 18, 282 & 469. Gill, S. J., Robert, C. H., Coletta, M., et al. (1986). Biophys. J., 50, 747. Gillespie, D. T. (1976). J. Comp. Phys., 22, 403. Gillespie, D. T. (1977a). J. Phys. Chem., 81, 2340. Gillespie, D. T. (1977b). J. Stat. Phys., 16, 311. Gillespie, D. T. (1980). J. Chem. Phys., 72, 5363. Gillespie, D. T. (1981). J. Chem. Phys., 75, 704. Gillespie, D. T. (1992). Markov Processes: An Introduction for Physical Scientists. San Diego: Academic Press. Gillespie, D. T. (2007). Annu. Rev. Phys. Chem., 58, 35. Gilliland, G., Perrin, S., Blanchard, K., & Bunn, H. F. (1990). Proc. Natl. Acad. Sci. USA, 87, 2725. Gimona, M. (2006). Nature Rev. Mol. Cell Biol., 7, 68. Giranda, V. L., Berman, H. M., & Schramm, V. L. (1988). Biochemistry, 27, 5813. Givens, R. S., Jung, A., Park, C.-H., et al. (1997). J. Am. Chem. Soc., 119, 8369. Glaser, L. (1972). The Enzymes, 6, 355. Glasstone, S., Laidler, K. J., & Eyring, H. (1941). The Theory of Rate Processes. New York: McGraw Hill. Glickman, M. H., & Klinman, J. P. (1996). Biochemistry, 35, 12882. Gloss, L. M., Spencer, D. E., & Kirsch, J. F. (1996). Protein Struct. Funct. Genet., 24, 195. Glover, B. P., & McHenry, C. S. (2001). Cell, 105, 925. Glusker, J. P. (1971). The Enzymes, 5, 413, 3rd edn. Glusker, J. P., Katz, A. K., & Bock, C. W. (1999). Rigaku J., 16, 8. Glusker, J. R. (1991). Adv. Protein Chem., 42, 1. Gobry, V., Bouchard, G., Carrupt, P. A., et al. (2000). Helvet. Chim. Acta, 83, 1465. Goers, J. W., & Schumaker, V. N. (1970). J. Mol. Biol., 28, 125. Gold, V. (1960). Trans. Faraday Soc., 56, 255. Goldberg, A. L. (1992). Eur. J. Biochem., 203, 9. Goldberg, A. L., Marshall, R. P., Chung, C. H., & Maurizi, M. R. (1994). Meth. Enzymol., 244, 350. Goldbeter, A., & Koshland, D. E., Jr. (1981). Proc. Natl. Acad. Sci. USA, 78, 6840. Golding, B. T., & Buckel, W. (1998). In M. Sinnott (Ed.), Comprehensive Biological Catalysis, vol. 3 (p. 239). London: Academic Press. Goldstein, S., & Czapski, G. (1995). Free Radic. Biol. Med., 19, 505. Gomes, X. V., Schmidt, S. L., & Burgers, P. M. (2001). J. Biol. Chem., 276, 34776. Gomori, G. (1943). J. Biol. Chem., 148, 139. Gonza´les-Mondrago´n, E., Zubillaga, R. A., Saavedra, E., et al. (2004). Biochemistry, 43, 3255.

819

Gonzalez, M. A., Webb, M. R., Welsh, K. M., & Cooperman, B. S. (1984). Biochemistry, 23, 797. Good, N. E., & Izawa, S. (1972). Meth. Enzymol., 24B, 53. Good, N. E., Winget, G. D., Winter, W., et al. (1966). Biochemistry, 5, 467. Goudar, C. T., Sonnad, J. R., & Duggleby, R. G. (1999). Biochim. Biophys. Acta, 1429, 377. Gould, R. M., & Plapp, B. V. (1990). Biochemistry, 29, 5463. Gould, S. M., & Tawfik, D. S. (2005). Biochemistry, 44, 5444. Goutsias, J. (2007). Biophys. J., 92, 2350. Grabowski, R., & Buckel, W. (1991). Eur. J. Biochem., 199, 89. Grant, G. A., Schuller, D. J., & Banaszak, L. J. (1996). Protein Sci., 5, 34. Grant, K. L., & Klinman, J. P. (1989). Biochemistry, 28, 6597. Gra¨slund, A. (2002). Meth. Enzymol., 354, 117. Graves, D. J., & Wang, J. H. (1972). The Enzymes, 7, 435, (3rd ed.). Graves, K. L., & Hardy, L. W. (1994). Biochemistry, 33, 13049. Green, D. W., Sun, H. W., & Plapp, B. V. (1993). J. Biol. Chem., 268, 7792. Greenhut, J., Umezawa, H., & Rudolph, F. B. (1985). J. Biol. Chem., 260, 6684. Grell, E., & Oberba¨umer, I. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 371). Berlin: Springer-Verlag. Grenklo, S., Geese, M., Lindberg, U., et al. (2003). EMBO Rep., 4, 523. Griffiths, A. D., & Tawfik, D. S. (2003). EMBO J., 22, 24. Griffiths, W., Jonsson, A. P., Liu, S., et al. (2001). Biochem. J., 355, 545. Grima, R., & Schnell, S. (2006). Biophys. Chem., 124, 1. Grimsby, J., Sarabu, R., Corbett, W. L., et al. (2003). Science, 301, 370. Groettrup, M., Soza, A., Eggers, M., et al. (1996). Nature, 381, 166. Groll, M., & Huber, R. (2003). Int. J. Biochem. Cell Biol., 35, 606. Groll, M., Ditzel, L., Lowe, J., et al. (1997). Nature, 386, 463. Gross, J. W., Hegeman, A. D., Vestling, M. M., & Frey, P. A. (2000). Biochemistry, 39, 13633. Gross, J. W., Rajavel, M., & Grubmeyer, C. (1998). Biochemistry, 37, 4189. Gruebele, M., Sabelko, J., Ballew, R., & Ervin, J. (1998). Accounts Chem. Res., 31, 699. Grunwald, E., Loewenstein, A., & Meiboom, S. (1957). J. Chem. Phys., 27, 630. Guallar, V., Jacobson, M., McDermott, A., & Friesner, R. A. (2004). J. Mol. Biol., 337, 227. Guengerich, F. P. (1991). J. Biol. Chem., 266, 10019. Guengerich, F. P., Miller, G. P., Hanna, I. H., et al. (2002). J. Biol. Chem., 277, 33711. Guerrier-Takada, C., Gardiner, K., Maresh, T., et al. (1983). Cell, 35, 849. Guigas, G., & Weiss, M. (2008). Biophys. J., 94, 90. Guilford, W. H., Dupuis, D. E., Kennedy, G., et al. (1997). Biophys. J., 72, 1006. Gulbinsky, J. S., & Cleland, W. W. (1968). Biochemistry, 7, 566. Guldberg, C. M., & Waage, P. (1867). E´tudes sur les Affinite´s Chimiques, Brøgger and Christie. Christiania (Oslo). Guldberg, C. M., & Waage, P. (1879). J. Prakt. Chem., 2 (19), 69.

820

Gulnik, S. V., Suvorov, L. I., Majer, P., et al. (1997). FEBS Lett, 413, 379. Gunsalus, I. C., & Sligar, S. G. (1978). Adv. Enzymol., 47, 1. Guo, B., & Guilford, W. H. (2006). Proc. Natl. Acad. Sci. USA, 103, 9844. Guo, H. C., & Collins, R. A. (1995). EMBO J., 14, 368. Guo, H., Cui, Q., Lipscomb, W. N., & Karplus, M. (2001). Proc. Natl. Acad. Sci. USA, 98, 9032. Gutfreund, H. (1955). Faraday Soc. Disc., 20, 1. Gutfreund, H. (1972). Enzymes: Physical Principles. London: Wiley-Interscience. Gutfreund, H. (1987). Biophys. J., 26, 605. Gutfreund, H., & Sturtevant, J. M. (1959). Biochem. J., 73, 1. Ha, T., Enderle, T., Chemla, D. S., et al. (1996a). Phys. Rev. Lett., 77, 3979. Ha, T., Enderle, T., Ogletree, D. F., et al. (1996b). Proc. Natl. Acad. Sci. USA, 93, 6264. Ha, T., Glass, J., Enderle, T., et al. (1998). Phys. Rev. Lett., 80, 2093. Ha, T., Rasnik, I., Cheng, W., et al. (2002). Nature, 419, 638. Ha, T., Ting, A. Y., Liang, J., et al. (1999). Proc. Natl. Acad. Sci. USA, 96, 893. Haber, J., & Koshland, D. E., Jr. (1967). Proc. Natl. Acad. Sci. USA, 58, 2087. Hackney, D. D. (1990). The Enzymes, 19, 1 (3rd ed.). Hackney, D. D., Stempel, P. D., & Boyer, P. D. (1980). Meth. Enzymol., 64, 60. Hagedoorn, P. L., Chen, T., Schroder, I., et al. (2005). J. Biol. Inorg. Chem., 10, 259. Hagner-McWhirter, A., Hannesson, H. H., Campbell, P., et al. (2000). Glycobiology, 10, 159. Hagner-Mcwhirter, A., Lindahl, U., & Li, J. P. (2000). Biochem. J., 347, 69. Hague, D. N. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 84). Berlin: SpringerVerlag. Haiker, H., Lengsfeld, H., Hadvary, P., & Carriere, F. (2004). Biochim. Biophys. Acta, 1682, 72. Hajduk, P. J., & Greer, J. (2007). Nat. Rev. Drug. Discov., 6, 211. Hajduk, P. J. (2006). J. Med. Chem., 49, 6972. Hajduk, P. J., Sheppard, G., Nettesheim, D. G., et al. (1997). J. Am. Chem. Soc., 119, 5818. Hakala, M. T., Glaid, A. J., & Schwert, G. W. (1956). J. Biol. Chem., 221, 191. Haldane, J. B. S. (1930). Enzymes. London: Longmans, Green and Co. Haldane, J. B. S., & Stern, K. G. (1932). Allgemeine Chemie der Enzyme. Dresden: Verlag von Steinkopff. Hall, A., & Knowles, J. R. (1975). Bochemistry, 14, 4348. Hall, D., & Minton, A. P. (2003). Biochim. Biophys. Acta, 1649, 127. Hall, N. F., & Alexander, O. R. (1940). J. Am. Chem. Soc., 93, 230. Halpern, J. (1985). Science, 227, 869. Halpern, J. (1986). Pure Appl. Chem., 58, 575. Hamdan, S., Bulloch, E. M., Thompson, P. R., et al. (2002). Biochemistry, 41, 5266. Hamilton, G. A. (1985). Adv. Enzymol., 57, 85. Hammes, G. G. (1968a). Adv. Prot. Chem., 23, 1. Hammes, G. G. (1968b). Acc. Chem. Res., 1, 321.

References

Hammes, G. G. (1974). In Techniques of Chemistry. Investigation of Rates and Mechanisms of Reactions, Part II, vol. 6 (p. 147). New York: Wiley-Interscience. Hammes, G. G. (2002). Biochemistry, 41, 8221. Hammes, G. G., & Eigen, M. (1963). Adv. Enzymol., 25, 1. Hammes, G. G., & Fasella, P. (1964). J. Am. Chem. Soc., 85, 3929. Hammes, G. G., & Haslam, J. L. (1969). Biochemistry, 8, 1591. Hammes, G. G., & Kochavi, D. (1962). J. Am. Chem. Soc. 84, 2069, 2073, 2076. Hammes, G. G., & Schimmel, P. R. (1966). J. Phys. Chem., 70, 2319. Hammes, G. G., & Schimmel, P. R. (1967). J. Phys. Chem., 71, 917. Hammes, G. G., & Schimmel, P. R. (1970). The Enzymes, 2, 67, (3rd ed). Hammes, G. G., & Steinfeld, J. I. (1962). J. Am. Chem. Soc., 84, 4639. Hammes, G. G., & Walz, F. G. (1969). J. Am. Chem. Soc., 91, 1278. Hammes, G. G., & Wu, C.-W. (1971). Science, 172, 1205. Hammes-Schiffer, S. (2002). Biochemistry, 41, 13335. Hammond, G. S. (1955). J. Am. Chem. Soc., 77, 334. Hanes, J. W., & Johnson, K. A. (2007). Analyt. Biochem., 372, 125. Hanna, R., & Doudna, J. A. (2000). Curr. Opinion Chem. Biol., 4, 166. Hanozet, G. M., Simonetta, M., Barisio, D., & Guerritore, A. (1979). Arch. Biochem. Biophys., 196, 46. Hansen, D. H., & Knowles, J. R. (1981). J. Biol. Chem., 256, 5967. Hanson, J. A., Duderstadt, K., Watkins, L. P., et al. (2007). Proc. Natl. Acad. Sci. USA, 104, 18055. Hanson, P. I., & Whiteheart, S. W. (2005). Nature Rev. Molec. Cell Biol., 6, 519. Hanson, R. L. (1979). J. Biol. Chem., 254, 888. Hanson, R. L., Rudolph, F. B., & Lardy, H. A. (1973). J. Biol. Chem., 248, 7852. Hanson, T. L., & Fromm, H. J. (1965). J. Biol. Chem., 240, 4133. Harcourt, A. V. (1867). J. Chem. Soc., 20, 460. Harcourt, A. V., & Esson, W. (1865). Proc. Roy. Soc., 14, 470. Harmon, K. M., & Niemann, C. (1949). J. Biol. Chem., 178, 743. Harpel, M. R., & Hartman, F. C. (1994). Biochemistry, 33, 5553. Harris, C. M., & Massey, V. (1997). J. Biol. Chem., 272, 28335. Harris, G. S., & Kosarich, J. W. (1996). Curr. Opinion Chem. Biol., 1, 259. Harris, T. K., Abeygunawardana, C., & Mildvan, A. S. (1997). Biochemistry, 36, 14661. Harris, T. K., & Mildvan, A. S. (1999). Proteins, 35, 275. Harris, T. K., Zhou, Q., & Mildvan, A. S. (2000). J. Molec. Struct., 615, 163. Hart, T., Hosszu, L. L. P., Trevitt, C. R., et al. (2009). Proc. Natl. Acad. Sci. USA, 106, 5651. Hartley, B. S., & Kilby, B. A. (1954). Biochem. J., 56, 288. Hartman, F. C., & Harpel, M. R. (1993). Adv. Enzymol., 67, 1. Hartrampf, G., & Buckel, W. (1986). Eur. J. Biochem., 156, 301. Hartridge, H., & Roughton, F. J. W. (1926). Proc. Camb. Phil. Soc., 23, 450. Hartridge, H., & Roughton, F. J. W. (1923). Proc. R. Soc., A104, 376.

References

Hartshorn, S. R., & Shiner, V. J., Jr. (1972). J. Am. Chem. Soc., 94, 9002. Hashimoto, T., & Yamada, Y. (1994). Ann. Rev. Plant Physiol. Plant Mol. Biol., 45, 257. Hasnain, S. S. (1994). Pure and Appl. Chem., 66, 51. Hass, L. F., & Byrne, W. L. (1960). J. Am. Chem. Soc., 82, 947. Hatefi, Y., & Stiggall, D. L. (1976). The Enzymes, 13, 175 (3rd ed.). Hatfield, G. W., Ray, W. J., Jr., & Umbarger, H. E. (1970). J. Biol. Chem., 245, 1748. Haupts, U., Maiti, S., Schwille, P., & Webb, W. W. (1998). Proc. Natl. Acad. Sci. USA, 95, 13573. Hauser, H., & Semenza, G. (1983). Crit. Rev. Biochem., 14, 319. Haustein, E., & Schwille, P. (2007). Annu. Rev. Biophys. Biomol. Struct., 36, 151. Havir, E. A., & Hanson, K. R. (1973). The Enzymes, 7, 75. Havir, E. A., & Hanson, K. R. (1975). Biochemistry, 14, 1620. Hayashi, H., Tanase, S., & Snell, E. E. (1986). J. Biol. Chem., 261, 11003. Haynes, R., & Walsh, K. A. (1969). Biochem. Biophys. Res. Commun., 36, 235. Hayward, S., Kitaom, A., & Go, N. (1994). Protein Sci., 3, 936. Hazzard, J. T., Rong, S., & Tollin, G. (1991). Biochemistry, 30, 213. He, Z. H., Chillingworth, R. K., Brune, M., et al. (1997). J. Physiol., 501, 125. Hearon, J. Z., Bernhard, S. A., Friess, S. L., et al. (1959). The Enzymes, 1, 49, (2nd ed). Heberle, J. (2004). Biophys. J., 87, 2105. Hegeman, A. D., Gross, J. W., & Frey, P. A. (2002). Biochemistry, 41, 2797. Hegg, L. A., & Fedor, M. J. (1995). Biochemistry, 34, 15813. Heinemann, U., & Saenger, W. (1982). Nature, 299, 27. Heinisch, J. (1986). Mol. Gen. Genet., 202, 75. Heinze, K. G., Rarbach, M., Jahnz, M., & Schwille, P. (2002). Biophys. J., 83, 1671. Heller, A. (1990). Accts. Chem. Res., 23, 128. Hellmann, N., & Decker, H. (2002). Biochim Biophys Acta, 1599, 45. Helmreich, E., & Cori, C. F. (1964). Proc. Natl. Acad. Sci. USA, 51, 131. Hemmingsen, S., Woolford, M. C., van der Vies, S. M., et al. (1988). Nature, 333, 330. Hendee, W. R. (1973). Radioisotopes in Biological Research. New York: Wiley. Henderson, P. J. F. (1972). Biochem. J., 127, 321. Henot, F., & Pollack, R. M. (2000). Biochemistry, 39, 3351. Henri, V. (1903). Lois Ge´ne´rales de L’Action des Diastase. Paris: Herman. Hentzler-Wildman, K., & Kern, D. (2007). Nature, 450, 964. Herlihy, J. M., Maister, S. G., Albery, W. J., & Knowles, J. R. (1976). Biochemistry, 15, 5601. Hermes, J. D., Blacklow, S. C., & Knowles, J. R. (1990). Proc. Natl. Acad. Sci. USA, 87, 696. Hermes, J. D., Roeske, C. A., O’Leary, M. H., & Cleland, W. W. (1982). Biochemistry, 21, 5106. Herschlag, D. (2001). RNA, 7, 535. Herschlag, D., & Jencks, W. P. (1986). J. Am. Chem. Soc., 108, 7938.

821

Herschlag, D., & Jencks, W. P. (1990). Biochemistry, 29, 5172. Hersh, L. B. (1974). J. Biol. Chem., 249, 5208. Hersh, L. B. (1982). J. Biol. Chem., 257, 12820. Hersh, L. B., & Jencks, W. P. (1968). J. Biol. Chem., 242, 3468. Hertel, K. J., Peracchi, A., Uhlenbeck, O. C., & Herschlag, D. (1997). Proc. Natl. Acad. Sci. USA, 94, 8497. Hertel, K. J., & Uhlenbeck, O. C. (1995). Biochemistry, 34, 1744. Hertel, K. J., Hershlag, D., & Uhlenbeck, O. C. (1994). Biochemistry, 33, 3374. Highbarger, L. A., Gerlt, J. A., & Kenyon, G. L. (1996). Biochemistry, 35, 41. Highes, K. T. (2002). In R. Eisenthal, & M. Danson (Eds.), Enzyme Assays: A Practical Approach (pp. 79–102). London: Oxford University Press. Hilderbrandt, R. L. (1969). J. Chem. Phys., 51, 1654. Hill, D. E., & Hammes, G. G. (1975). Biochemistry, 14, 203. Hill, T. L., & Kirschner, M. W. (1982). Int. Rev. Cytol., 78, 1. Hill, R. L., & Teipel, J. W. (1971). The Enzymes, 5, 539, 3rd edn. Hille, R. (2002). Trends Biochem. Soc., 27, 360. Hilvert, D., Gardell, S. J., Rutter, W. J., & Kaiser, E. T. (1986). J. Am. Chem. Soc., 108, 5298. Himes, R. H., & Harmony, A. K. (1973). CRC Crit. Rev. Biochem., 1, 501. Hindmarsh, A. C. (1983). In R. S. Stepleman (Ed.), Scientific Computing (pp. 55–64). Amsterdam: North Holland. Hirata, M., Tokushige, M., Inagaki, A., & Hayaishi, O. (1965). J. Biol. Chem., 240, 1171. Hirohara, H., Bender, M. L., & Stark, R. S. (1974). Proc. Natl. Acad. Sci. USA, 71, 1643. Hirono-Hara, Y., Noji, H., Nishiura, M., et al. (2001). Proc. Natl. Acad. Sci. USA, 98, 13649. Hoa, G. H., Hamel, G., Else, A., et al. (1990). Anal. Biochem, 187, 258. Hoare, J. P. (1985). In A. J. Bard, R. Parsons, & J. Jordan (Eds.), Standard Potentials in Aqueous Solution (pp. 49–66). New York: IUPAC/Marcel Dekker. Hodsdon, J. M., Brown, G. M., Sieker, L. C., & Jensen, L. H. (1990). Acta Crystallogr. B., 46, 54. Hoffman, B. M. (1991). Acc. Chem. Res., 24, 164. Hoffman, B. M. (2002). Acc. Chem. Res., 36, 522. Hofmeister, F. (1888). Naunyn-Schmiedebergs Archiv fu¨r Experimentelle Pathologie und Pharmakologie. Leipzig, 24, 247. Hofrichter, J. (2001). Methods Mol. Biol., 168, 159. Hogg, P. J., & Winzor, D. J. (1987). Arch. Biochem. Biophys., 254, 92. Hogg, P. J., Johnston, S. C., Bowles, M. R., et al. (1987). Mol. Immunol., 24, 797. Hol, W. G. (1985). J. Prog. Biophys. Molec. Biol., 46, 149. Holler, E., Rupley, J. A., & Hess, G. P. (1970). Biochem. Biophys. Res. Commun., 40, 166. Hollfelder, F., & Herschlag, D. (1995). Biochemistry, 34, 12255. Hollfelder, F., Kirby, A. J., & Tawfik, T. S. (2001). Nature, 383, 60. Holliday, G. L., Almonacid, D. E., Bartlett, G. J., et al. (2006). Nucl. Acids Research, 35 (Database issue), D515. Holliday, G. L., Bartlett, G. J., Almonacid, D. E., et al. (2005). Bioinformatics, 21, 4315. Holzhutter, H. G., & Kloetzel, P. M. (2000). Biophys. J., 79, 1196.

822

Hong, L., & Tang, J. (2004). Biochemistry, 43, 4689. Honig, B., & Nichols, A. (1995). Science, 268, 1144. Hopfield, J. J. (1974). Proc. Natl. Acad. Sci. USA, 71, 4135. Hopkins, A. L., & Groom, C. R. (2002). Nature. Rev. Drug. Discov., 1, 727. Horenstein, B. A., & Schramm, V. L. (1993a). Biochemistry, 32, 7089. Horenstein, B. A., & Schramm, V. L. (1993b). Biochemistry, 32, 9917. Horio, T., & Hotani, H. (1986). Nature, 321, 605. Horovitz, A., & Fersht, A. R. (1990). J. Mol. Biol., 214, 613. Horovitz, A., & Fersht, A. R. (1992). J. Mol. Biol., 224, 733. Howard, J. (2001). Mechanics of Motor Proteins and the Cytoskeleton. Sunderland, MA: Sinauer. pp. 367. Howarth, J. V., Millar, N. C., & Gutfreund, H. (1987). Biochem. J., 248, 677. Hsu, J. L., Hsieh, Y., Tu, C., et al. (1996). J. Biol. Chem., 271, 17687. Hsu, R. Y., Lardy, H. A., & Cleland, W. W. (1967). J. Biol. Chem., 242, 5207, and 5215. Huang, C. Y. (1977). Arch. Biochem. Biophys., 184, 488. Huang, C. Y. (1979a). Meth. Enzymol., 63, 54. Huang, C. Y. (1979b). Meth. Enzymol., 63, 486. Huang, C. Y. (1982). Meth. Enzymol., 87, 509. Huang, C. Y., & Graves, D. J. (1970). Biochemistry, 9, 660. Huang, C. Y., & Kaufman, S. (1973). J. Biol. Chem., 248, 4242. Huang, C. Y., Chau, V., Chock, P. B., et al. (1981). Proc. Natl. Acad. Sci. USA, 78, 871. Huang, C. Y., Rhee, S. G., & Chock, P. B. (1982). Ann. Rev. Biochem., 51, 935. Huang, C. Y., Zhou, R., Yang, D. C. H., & Chock, P. B. (2003). Biophysical Chemistry, 100, 143. Huang, C.-Y. F., & Ferrell, J. E. (1996). Proc. Natl. Acad. Sci. USA, 93, 10078. Huber, R. E., & Hurlburt, K. L. (1986). Arch. Biochem. Biophys., 246, 411. Huffman, D. L., & O’Halloran, T. V. (2001). Annu. Rev. Biochemistry, 70, 677. Huheey, J. E., Keiter, E. A., & Keiter, R. L. (1993). Inorganic Chemistry: Principles of Structure and Reactivity (4th ed.). Reading, MA: Addison-Wesley. Hull, W. E., Halford, S. E., Gutfreund, H., & Sykes, B. D. (1976). Biochemistry, 15, 1547. Hummel, J. P., & Dreyer, W. J. (1962). Biochim. Biophys. Acta, 46, 358. Hunkapiller, M. W., Forgac, M. D., & Richards, J. H. (1976). Biochemistry, 15, 5581. Hunt, A. J., Gittes, F., & Howard, J. (1994). Biophys. J., 67, 766. Hurley, J. H., & Dean, A. M. (1984). Structure, 2, 1007. Hurley, J. H., Dean, A. M., Koshland, D. E., Jr., & Stroud, R. M. (1991). Biochemistry, 30, 8671. Hurst, R. D., & Clark, J. B. (2002). In R. Eisenthal, & M. Danson (Eds.), Enzyme Assays: A Practical Approach (pp. 149–156). London: Oxford University Press. Hurwitz, J., & Wickner, S. (1974). Proc. Natl. Acad. Sci. USA, 71, 610. Hurwitz, R., & Kustin, K. (1964). Inorg. Chem., 3, 823. Hutton, R. L., & Boyer, P. D. (1979). J. Biol. Chem., 254, 9990.

References

Huyghues-Despointes, B. M., Pace, C. N., Englander, S. W., & Scholtz, J. M. (2001). Methods Mol. Biol., 168, 69. Hvidt, A., & Nielsen, S. O. (1966). Adv. Protein Chem., 21, 287. Hwang, J. K., & Warshel, A. (1988). Nature, 334, 270. Ikeda, Y., Tanaka, T., & Noguchi, T. (1997). J. Biol. Chem., 272, 20495. Ilgenfritz, G. (1977). Chemical Relaxation in Molecular Biology. In I. Pecht, & I. Rigler (Eds.). Berlin: Springer-Verlag. p. 1. Imoto, T., Johnson, L. N., North, A. C. T., et al. (1972). The Enzymes, 7, 842 (3rd ed.). Ingber, D. E. (1997). Annu. Rev. Physiol., 59, 575. Ingber, D. E. (1998). Sci. Amer., 278 (1), 48. Ingram, L., Conway, T., & Alterthum, F. (1991). Ethanol production by Escherichia coli strains co-expressing Zymomonas PDC and ADH genes, US Patent 5,000,000, issued March 19, 1991. Washington, DC: United States Patent and Trademark Office. Inouye´, S. (1986). Video Microscopy. New York: Plenum. Inskeep, W. H., Jones, D. L., Silfvast, W. T., & Eyring, E. M. (1968). Proc. Natl. Acad. Sci. USA, 59, 1027. Irimescu, R., & Kato, K. (2004). Tetrahedron Lett., 45, 523. Ito, T., Tyler, J. K., Bulger, M., et al. (1996). J. Biol. Chem., 271, 25041. Iwata, S., Ostermeier, C., Ludwig, B., & Michel, H. (1995). Nature, 269, 660. Iwatsuki, N., & Okazaki, R. (1967). J. Mol. Biol., 29, 1967. Iyer, L. M., Leipe, D. D., Koonin, E. V., & Aravind, L. (2004). J. Struct. Biol., 146, 11. Jabri, E., Carr, M. B., Hausinger, R. P., & Karplus, P. A. (1995). Science, 268, 998. Jack, W. E., Terry, B. J., & Modrich, P. (1982). Proc. Natl. Acad. Sci. USA, 79, 4010. Jacob, C. A., & Ortiz de Montellano, P. R. (1986). Biochemistry, 25, 4705. Jacob, F., Joris, B., & Frere, J. M. (1991). Biochem. J., 277, 647. Jacob, M., Holtermann, G., Perl, D., et al. (1999). Biochemistry, 38, 2882. Jaeger, K. E., & Eggert, T. (2004). Curr. Opin. Biotechnol., 15, 305. Jaenicke, R., Schurig, H., Beaucamp, N., & Ostendorp., R. (1996). Adv. Protein Chem., 48, 181. Jain, M. K., Gelb, M. H., Rogers, J., & Berg, O. G. (1995). Meth. Enzymol, 249, 567. Janes, S. M., Mu, D., Wemmer, D., et al. (1990). Science, 248, 981. Janes, S. M., Palcic, M. M., Scaman, C. H., et al. (1992). Biochemistry, 31, 12147. Janin, J., & Deville-Bonne, D. (2002). Meth. Enzymol., 354, 118. Janmey, P. A. (1995). Structure and Dynamics of Membranes. Amsterdam: Elsevier. p. 805. Janmey, P. A. (1998). Physiol. Rev., 78, 763. Janson, C. A., & Cleland, W. W. (1974). J. Biol. Chem. 249, 2562 (2567), 2572. Jarvest, R. L., & Lowe, G. (1981). Biochem. J., 199, 273. Jarvest, R. L., Lowe, G., & Potter, B. V. L. (1981). Biochem. J., 199, 427. Jarvis, T. C., Ring, D. M., Daube, S. S., & von Hippel, P. H. (1990). J. Biol. Chem., 265, 15160.

References

Jeffery, C. J., Gloss, L. M., Petsko, G. A., & Ringe, D. (2000). Protein Engin., 13, 105. Jencks, W. P. (1963). Annu. Rev. Biochem., 32, 659. Jencks, W. P. (1966). In E. P. Kennedy (Ed.), Current Aspects of Biochemical Energetics (p. 273). New York: Academic Press. Jencks, W. P. (1969). Catalysis in Chemistry and Enzymology (p. 288). New York: McGraw-Hill, New York (also available from Dover Publications.). Jencks, W. P. (1972). Chem. Rev., 72, 705. Jencks, W. P. (1975). Adv. Enzymol., 43, 219. Jencks, W. P. (1980). Adv. Enzymol., 51, 75. Jencks, W. P. (1981). Proc. Natl. Acad. Sci. USA, 78, 4046. Jencks, W. P. (1989). J. Biol. Chem., 264, 18858. Jencks, W. P. (1997). Annu. Rev. Biochem., 6 (6), 1. Jencks, W. P., & Knowles, J. R. (1976). Crit. Rev. in Biochem., 4, 165. Jencks, W. P., & Page, M. I. (1974). Biochem. Biophys. Res. Communs., 57, 887. Jen-Jacobson, L., Kurpiewski, M., Lesser, D., et al. (1983). J. Biol. Chem., 258, 14638. Jez, J. M., & Penning, T. (1998). Biochemistry, 57, 9695. Jez, J. M., Bowman, M. E., & Noel, J. P. (2001). Biochemistry, 40, 14829. Jiang, L., Althoff, E. A., Clemente, F. R., et al. (2008). Science, 319, 1387. Jiang, Y. L., & Stivers, J. T. (2002). Biochemistry, 41, 11236. Joesten, M. D., & Schaad, L. J. (1974). Hydrogen Bonding. New York: Marcel Dekker. Johal, S., Partridge, B. E., & Chollet, R. (1985). J. Biol. Chem., 260, 9894. Johanson, K. O., & McHenry, C. S. (1984). J. Biol. Chem., 259, 4589. Johnson, A., & O’Donnell, M. (2003). J. Biol. Chem., 278, 14406. Johnson, K. A. (1992). The Enzymes, 20, 1. Johnson, K. A. (1995). Meth. Enzymol, 249, 38. Johnson, K. A. (2003). In K. A. Johnson (Ed.), Kinetic Analysis of Macromolecules (p. 1). New York: Oxford University Press. Johnson, K. A. (2008). J. Biol. Chem., 283, 26297. Johnson, K. A., & Benkovic, S. J. (1990). The Enzymes, 19, 159. Johnson, K. A., Simpson, Z. B., & Blom, T. (2009a). Analyt. Biochem., 387, 30. Johnson, K. A., Simpson, Z. B., & Blom, T. (2009b). Analyt. Biochem., 387, 20. Johnson, L. N., & Barford, D. (1990). J. Biol. Chem., 265, 2409. Johnson, P. E., Abbott, S. J., Orr, G. A., et al. (1976). Biochemistry, 15, 2893. Johnston, M., Raines, R., Walsh, C. T., Jr., & Firestone, R. A. (1980). J. Am. Chem. Soc., 102, 4241. Jones, J. B., & Desantis, G. (1998). Acct. Chem. Res., 32, 103. Jones, O. W., & Berg, P. (1966). J. Mol. Biol., 22, 199. Jones, S. R., Kindman, L. A., & Knowles, J. R. (1978). Nature, 275, 564. Jones, W., Kurz, L. C., & Wolfenden, R. (1989). Biochemistry, 28, 1242. Jonsson, T., Edmondson, D. E., & Klinman, J. P. (1994). Biochemistry, 33, 14871. Jonsson, T., Glickman, M. H., Sun, S. J., & Klinman, J. P. (1996). J. Am. Chem. Soc., 118, 10319.

823

Jorpes, J. E. (1966). Jac Berzelius, His Life and Work. Stockholm: Almqvist and Wiksell. p. 112. Jose, D., & Porschke, D. (2004). Nucl. Acids Res., 32, 2251. Joseph-McCarthy, D., Lolis, E., Komives, E. A., & Petsko, G. A. (1994). Biochemistry, 33, 2815. Joseph-McCarthy, D., Rost, L. E., Komives, E. A., & Petsko, G. A. (1994). Biochemistry, 33, 2824. Josephson, L., & Cantley, L. C., Jr. (1977). Biochemistry, 16, 4572. Josephson, N. S., Kuntz, K. W., Snapper, M. L., & Hoveyda, A. H. (2001). J. Am. Chem. Soc., 123, 11594. Joslyn, G., Richardson, D. S., White, R., & Alber, T. (1993). Proc. Natl. Acad. Sci. USA, 90, 11109. Jou, D., & Llebot, J. E. (1990). Introduction to the Thermodynamics Processes of Biological Processes. New York: Prentice Hall. Jovin, T. M., & Striker, G. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 244). Berlin: Springer-Verlag. Joyce, B. K., & Himes, R. H. (1966). J. Biol. Chem., 241, 5716. Jyothirmayi, N., & Ramadoss, C. S. (1991). Biochim. Biophys. Acta, 1083, 193. Kacser, H., & Burns, J. A. (1973). Symp. Soc. Exp. Biol., 27, 65 (reprinted in: Biochem. Soc. Transactions (1995) 23, 341). Kaiser, E. T., & Lawrence, D. S. (1984). Science, 226, 505. Kalousek, F., Darigo, M. D., & Rosenberg, L. E. (1980). J. Biol. Chem., 255, 60. Kang, F., Laine, R. O., Bubb, M. R., et al. (1997). Biochemistry, 36, 8384. Kantrowitz, E. R., & Lipscomb, W. N. (1988). Science, 241, 669. Kanyo, Z. F., Scolnick, L. R., Ash, D. E., & Christianson, D. W. (1996). Nature, 383, 554. Kaplan, K. B., Burds, A. A., Swedlow, J. R., Bekir, S. S., Sorger, P. K., & Na¨thke, I. S. (2001). Nat. Cell Biol., 3, 429. Kaplan, J. H., & Ellis-Davies, G. C. R. (1988). Proc. Natl. Acad. Sci. USA, 85, 6571. Kaplan, J. H., Forbush, B., III, & Hoffmann, J. F. (1978). Biochemistry, 17, 1929. Kapmeyer, H., Pfleiderer, G., & Trommer, W. E. (1976). Biochemistry, 15, 5024. Karlsson, A., Parales, J. V., Parales, R. E., et al. (2000). J. Inorg. Biochem., 78, 83. Karlsson, J. (1986). Adv. Cardiol., 35, 35. Karplus, P. A. (1997). Prot. Sci., 6, 1302. Karr, T. L., Kristofferson, D., & Purich, D. L. (1980). J. Biol. Chem., 255, 8560. Kaseda, K., Yokota, H., Ishii, Y., et al. (2000). J. Bact., 182, 1162. Kashiwaya, Y., Sato, K., Tsuchiya, N., et al. (1994). J. Biol. Chem., 269, 25502. Kass, I. J., & Sampson, N. S. (1995). Biochem. Biophys. Res. Commun., 206, 688. Katrin, G. H., Rarbach, M., Michael, J., & Schwille, P. (2002). Biophys. J., 83, 1671. Katsumata, M. (1969). J. Theor. Biol., 24, 294. Katz, A. K., Glusker, J. P., Beebe, S. A., & Bock, C. W. (1996). J. Amer. Chem. Soc., 118, 5752. Katz, B., & Kossiokof, A. (1986). J. Biol. Chem., 261, 15480. Kauzmann, W. (1959). Adv. Prot. Chem., 14, 1. Kayne, F. J. (1973). The Enzymes, 8, 353 (3rd ed.).

824

Kebarle, J. (2000). J. Mass Spectrom., 35, 804. Keck, J. L., Goedken, E. R., & Marqusee, S. (1998). J. Biol. Chem., 273, 34128. Keenan, M. V., & Alworth, W. L. (1974). Biochem. Biophys. Res. Commun., 165, 118. Keillor, J. W., & Jencks, W. P. (1996). Biochemistry, 35, 2750. Keller, D., & Bustamante, C. (2000). Biophys. J., 78, 541. Kellis, J. T., Jr., Nyberg, K., & Fersht, A. R. (1989). Biochemistry, 28, 4914. Kellis, J. T., Jr., Nyberg, K., Sali, D., & Fersht, A. R. (1988). Nature, 333, 784. Kennelly, P. J., & Krebs, E. G. (1991). J. Biol. Chem., 266, 15555. Kennes, C., Pries, F., Krooshof, G. H., et al. (1995). Eur. J. Biochem., 228, 403. Kenyon, G. L., & Reed, G. H. (1983). Adv. Enzymol., 54, 367. Kerfelec, B., Allouche, M., Colin, D., et al. (2008). Proteins, 73, 828. Kermse, W. (1971). Carbene Chemistry (2nd ed.). New York: Academic Press. Kettling, U., Koltermann, A., Schwille, P., & Eigen, M. (1998). Proc. Natl. Acad. Sci. USA, 95, 1416. Khan, S., & Sheetz, M. P. (1997). Annu. Rev. Biochem., 66, 785. Khangulov, S. V., Sossong, T. M., Jr., Ash, D. E., & Dismukes, G. C. (1998). Biochemistry, 37, 8539. Khedouri, E., Wellner, V., & Meister, A. (1964). Biochemistry, 3, 824. Khmelnitsky, Y. L., Budde, C., Arnold, J. M., et al. (1997). J. Am. Chem. Soc., 119, 11554. Kicska, G. A., Tyler, P. C., Evans, G. B., et al. (2002a). J. Biol. Chem., 277, 3219. Kicska, G. A., Tyler, P. C., Evans, G. B., et al. (2002b). J. Biol. Chem., 277, 3226. Kiefer, J. R., Mao, C., Braman, J. C., & Beese, L. S. (1998). Nature, 391, 304. Kim, B. K., Cha, S., & Parks, R. E., Jr. (1968). J. Biol. Chem., 243, 1771. Kim, E.-J., Kim, H.-P., Hah, Y. C., & Roe, J. H. (1996). Eur. J. Biochem., 241, 1781. Kim, I. Y., Veres, Z., & Stadtman, T. C. (1992). J. Biol. Chem., 267, 19650. Kim, I. Y., Veres, Z., & Stadtman, T. C. (1993). J. Biol. Chem., 268, 12555. Kim, I.-C., Cha, J.-H., Kim, J.-R., et al. (1992). J. Biol. Chem., 267, 2210. Kim, S. C., & Raushel, F. M. (1986). Biochemistry, 25, 4744. Kim, T.-J., Nguyen, V. D., Lee, H.-S., et al. (2001). Biochemistry, 40, 14182. Kim, T. J., Nguyen, V. D., Lee, H. S., et al. (2001). Biochemistry, 40, 14182. Kinder, D. H., Frank, S. K., & Ames, M. M. (1990). J. Med. Chem., 33, 819. King, E. L., & Altman, C. (1956). J. Phys. Chem., 60, 1375. King, M. M., & Huang, C. Y. (1983). Biochem. Biophys. Res. Commun., 114, 955. Kinjo, M., & Rigler, R. (1995). Nucleic Acids Res., 23, 1795. Kintses, B., Simon, Z., Gyimesi, M., et al. (2006). Biophys. J., 91, 4605. Kinzie, S. D., Thevis, M., Ngo, K., et al. (2003). J. Am. Chem. Soc., 125, 4710.

References

Kiremer, M. L. (1989). Eur. J. Biochem., 185, 651. Kirsch, J. F. (1977). In W. W. Cleland, M. H. O’Leary, & D. B. Northrop (Eds.), Isotope Effects on Enzyme-Catalyzed Reactions. Baltimore: University Park Press. Kirsch, J. F., Eichele, G., Ford, G. C., et al. (1984). J. Mol. Biol., 174, 497. Kirschner, K. (1971). Curr. Top. Cell Regul., 4, 167. Kirschner, K., Gallego, E., Schuster, I., & Goodall, D. (1971). J. Mol. Biol., 58, 29. Kiss, R., Bozoky, Z., Kova´cs, D., et al. (2008). FEBS Lett., 582, 2149. Kisselev, A. F., Akopian, T. N., & Goldberg, A. L. (1998). J. Biol. Chem., 273, 1982. Kisselev, A. F., Kaganovich, D., & Goldberg, A. L. (2002). J. Biol. Chem., 277, 22260. Kitaoka, M., Sasaki, T., & Taniguchi, H. J. (1992). J. Biochem., 112, 40. Kitchen, D. B., Decornez, H., Furr, J. R., & Bajorath, J. (2004). Nature Revs., 3, 935. Kitson, F. G., Larsen, B. S., & McEwen, C. N. (1996). Gas Chromatography and Mass Spectrometry: A Practical Guide. New York: Academic Press. Kleeman, J. E., & Parsons, S. M. (1976). Arch. Biochem. Biophys., 175, 687. Klibanov, A. M. (2001). Nature, 409, 241. Klibanov, A. M. (2003). Curr. Opinion Biotechnol., 14, 427. Kline, P. C., & Schramm, V. L. (1993). Biochemistry, 32, 13212. Klink, B. U., Goody, R. S., & Schleidig, A. J. (2006). Biophys. J., 91, 981. Klinman, J. P. (1977). In W. W. Cleland, M. H. O’Leary, & D. B. Northrop (Eds.), Isotope Effects on Enzyme-Catalyzed Reactions (p. 176). Baltimore: Univ. Park Press. Klinman, J. P. (2003). Biochim. Biophys. Acta, 1674, 131. Klinman, J. P. (2006a). Biochim. Biophys. Acta, 1757, 981. Klinman, J. P. (2006b). Philos. Trans. Royal Soc., London B, Biol. Sci., 361, 1323. Klinman, J. P., & Mu, D. (1994). Ann. Rev. Biochem., 63, 299. Kloetzel, P. M. (2004). Nature Immunol., 5, 661. Klotz, I. M., Walker, F. M., & Pivan, R. B. (1946). Biochem. J., 68, 1486. Kluger, R. (1992). The Enzymes, 20, 271 (3rd ed.). Klumpp, G. W. (1982). Reactivity in Organic Chemistry. New York: Wiley. pp. 167–179. Knapp, J. K., & Klinman, J. P. (2002). Eur. J. Biochem., 269, 3113. Knight, K., & Scrutton, N. S. (2002). Biochem. J., 367, 19. Knight, W. B., & Cleland, W. W. (1989). Biochemistry, 28, 5728. Knowles, J. R. (1980). Ann. Rev. Biochem., 49, 877. Knowles, J. R. (1987). Science, 236, 1252. Knowles, J. R. (1990). Phil. Transact: Biolog. Sci., 332, 115. Knowles, J. R. (1991). Nature, 350, 121. Knowles, J. R. (2002). Science, 299, 2067. Knox, W. E., Auerbach, V. H., & Lin, E. E. E. (1956). Physiol. Rev., 36, 164. Kobayashi, C., & Takada, S. (2006). Biophys J., 90, 3051. Kobayashi, K., Koppenhofer, A., Ferguson, S. J., & Tagawa, S. (1997). Biochem., 36, 13611. Kobayashi, K., Tagawa, S., Sano, S., & Asada, K. (1995). J. Biol. Chem., 270, 27551.

References

Kobayashi, M., & Shimizu, S. (1999). Europ. J. Biochem., 261, 1. Kobayashi, T., & Adachi, S. (2004). Biotechnol. Lett., 26, 1461. Kobayashi, T., & Laidler, K. J. (1974). Biotechnol. Bioeng., 16, 99. Kohen, A. (2006). In A. Kohen, & H.-H. Limbach (Eds.), Isotope Effects in Chemistry and Biology (p. 744). Boca Raton, Florida: Taylor and Francis. Kohen, A., & Klinman, J. P. (1999). Chem. Biol., 6, R191. Kohen, A., Cannio, R., Bartolucci, S., & Klinman, J. P. (1999). Nature, 399, 496. Kohl, T., Heinze, K. G., Kuhlemann, R., et al. (2002). Proc. Natl. Acad. Sci. USA, 99, 12161. Kohler, A., Cascio, P., Leggett, D. S., et al. (2001). Mol. Cell., 7, 1143. Kojima, H., Ishijima, A., & Yanagida, T. (1994). Proc. Natl. Acad. Sci. USA, 91, 12962. Kokesh, F. C., & Westheimer, F. H. (1971). J. Am. Chem. Soc., 93, 7270. Kolodziej, A. F., Tan, T., & Koshland, D. E., Jr. (1996). Biochemistry, 35, 14782. Koltermann, A., Kettling, U., Bieschke, J., et al. (1998). Proc. Natl. Acad. Sci. USA, 95, 1421. Komives, E. A., Chang, L. C., Lolis, E., et al. (1991). Biochemistry, 30, 3011. Kondepudi, D. K., & Prigogine, I. (1998). Modern Thermodynamics: From Heat Engines to Dissipative Structures. New York: Wiley. Kono, N., & Uyeda, K. (1973). J. Biol. Chem., 248, 8603. Konopka, J. M., Lardy, H. A., & Frey, P. A. (1986). Biochemistry, 25, 5571. Koonce, M. P., & Schliwa, M. (1986). J. Cell Biol., 103, 605. Kopelman, R. (1986). J. Stat. Phys., 42, 185. Kopelman, R. (1988). Science, 241, 1620. Kornberg, A. (1991). For the Love of Enzymes: Odyssey of a Biochemist. Cambridge, MA: Harvard University Press. Kornberg, A., & Khorana, H. B. (1961). Biochem. Prep., 8, 110. Kornberg, R., & McConnell, H. M. (1971). Biochemistry, 10, 1111. Koshkin, V., & Dunford, H. B. (1999). Biochim. Biophys. Acta, 1431, 47. Koshland, D. E., Jr. (1952). J. Am. Chem. Soc., 74, 2286. Koshland, D. E., Jr. (1958). Proc. Natl. Acad. Sci. USA, 44, 98. Koshland, D. E., Jr. (1959a). J. Cell. Comp. Physiol., 54, 245. Koshland, D. E., Jr. (1959b). The Enzymes, 1, 98 (2nd ed.). Koshland, D. E., Jr. (1963a). Science, 142, 1533. Koshland, D. E., Jr. (1963b). Cold Spring Harbor Symp. Quant. Biol., 28, 473. Koshland, D. E., Jr. (1993). Protein Sci., 2, 1364. Koshland, D. E., Jr., & Hamadani, K. (2002). J. Biol. Chem., 277, 46841. Koshland, D. E., Jr., Ne´methy, G., & Filmer, D. (1966). Biochemistry, 5, 365. Kosow, D. P., & Rose, I. A. (1971). J. Biol. Chem., 246, 2618. Kou, S. C., Cherayil, B. J., Min, W., et al. (2005). J. Phys. Chem., 109, 10968. Kovar, D. R., & Pollard, T. D. (2004). Proc. Natl. Acad. Sci. USA, 101, 14725. Kovar, D. R., Wu, J. Q., & Pollard, T. D. (2005). Mol. Biol. Cell, 16, 2313.

825

Kovaleva, E. G., & Plapp, B. V. (2005). Biochemistry, 44, 1279. Kowalski, A., Wyttenbach, C., Langer, L., & Koshland, D. E., Jr. (1956). J. Biol. Chem., 219, 719. Koya, V., Lu, S., Sun, Y. P., et al. (2008). Diabetes, 57, 757. Krahn, J. M., Kim, J. H., Burns, M. R., et al. (1997). Biochemistry, 36, 11061. Kramer, D. M., Sacksteder, & Cruz, J. A. (1999). Photosynth. Res., 60, 151. Kramers, H. A. (1940). Physica-Utrecht, 4, 284. Krantz, A., Copp, L. J., Coles, P. J., et al. (1991). Biochemistry, 30, 4678. Krauss, G., Ro¨mer, D., Riesner, D., & Maass, G. (1973). FEBS Letts, 30, 6. Kraut, D. A., Carroll, K. S., & Herschlag, D. (2003). Annu. Rev. Biochem., 72, 517. Krebs, E. G. (1994). Trends Biochem. Sci., 19, 439. Krebs, E. G., & Fischer, E. H. (1956). Biochim. Biophys. Acta, 20, 157. Kreevoy, M. M., & Truhlar, D. G. (1986). In C. F. Berlasconi (Ed.), in Techniques in Chemistry (A. Weissberger. Investigation of Rates and Mechanisms of Reaction, vol. 6 (pp. 13–95). New York: Wiley, series ed.), Part 1. Kresge, A. J. (1964). Pure Appl. Chem., 8, 243. Kresge, A. J. (1977). In W. W. Cleland, M. H. O’Leary, & D. B. Northrop (Eds.), Isotope Effects on Enzyme Catalyzed Reactions (p. 37). Baltimore: University Park Press. Kresge, A. J., & Silverman, D. N. (1999). Meth. Enzymol, 308, 276. Kreuzer, H. J. (1981). Nonequilibrium Thermodynamics and Its Statistical Foundations. London: Oxford University Press. Kreyszig, E. (1993). Advanced Engineering Mathematics (p. 929) (7th ed.). New York: John Wiley. Krishna, R. (2003). Applications of Pharmacokinetic Principles in Drug Development (p. 556). Berlin: Springer. Krishnamoorthy, G. (1986). Biochemistry, 25, 6666. Krishnaswamy, P. R., Pamiljans, V., & Meister, M. (1960). J. Biol. Chem., 235, PC39–40. Krishnaswamy, P. R., Pamiljans, V., & Meister, A. (1962). J. Biol. Chem., 237, 2932. Kroll, J. S., Langford, P. R., & Loynds, B. M. (1991). J. Bacteriol., 173, 7449. Krupka, R. M. (1965a). Biochemistry, 4, 429. Krupka, R. M. (1965b). Biochemistry, 5 (1983), 1988. Kuby, S. A., & Noltmann, E. A. (1962). The Enzymes, 6, 515, (2nd ed.). Kuchar, J., & Hausinger, R. P. (2004). Chem. Rev., 104, 509. Kuki, A., & Wolynes, P. G. (1987). Science, 236, 1647. Kuliopulos, A., Mildvan, A. S., Shortle, D., & Talalay, P. (1989). Biochemistry, 28, 149. Kumar, S., & Nussinov, R. (1999). J. Mol. Biol., 293, 1241. Kumar, S., & Nussinov, R. (2002). Biophys. J., 83, 1595. Kung, H. F., & Stadtman, T. C. (1971). J. Biol. Chem., 246, 3378. Kung, H. F., & Tsai, L. (1971). J. Biol. Chem., 246, 6436. Kuo, S. C., & McGrath, J. L. (2000). Nature, 407, 1026. Kuper, J., Llamas, A., Hecht, H. J., et al. (2004). Nature, 430, 803. Kurota, A., Kamata, Y., & Yamauchi, F. (1990). Agric. Biol. Chem., 54, 1557. Kurth, T., Grahn, S., Thormann, M., et al. (1998). Biochemistry, 37, 11434.

826

Kurz, J. L. (1963). J. Am. Chem. Soc., 85, 987. Kurzchalia, T. V., Bommer, U. A., Babkina, G. T., & Karpova, G. G. (1984). FEBS Lett., 175, 313. Kustin, K., & Lieberman, E. C. (1964). J. Phys. Chem., 68, 3869. Kuzmic, P. (1996). Anal. Biochem, 237, 260. Kuzmic, P. (1999). Anal. Biochem., 267, 17. Labute, P., & Santavy, M. (2006). Online. http://www.chemcomp. com/journal/pboltz.htm. Lahiri, S. D., Zhang, G., Dunaway-Mariano, D., & Allen, K. N. (2003). Science, 299, 2067. Lai, H. Y., & Axelrod, B. (1975). Biochim. Biophys. Acta., 391, 121. Laidler, K. J. (1987). Chemical Kinetics (p. 531) (3rd ed.). New York: Harper and Row. Laidler, K. J., & Bunting, P. S. (1974). Meth. Enzymol., 64, 227. Laidler, K. J., & Bunting, P. S. (1980). Meth. Enzymol., 64, 227. Lajtha, A. (1959). J. Neurochem., 3, 358. Lajtha, A. (1964). Internat. Rev. Neurobio., 6, 1. Lakowicz, J. R. (2006). Principles of Fluorescence Spectroscopy (p. 954) (3rd ed.). New York: Springer. Langley, D. B., Templeton, M. D., Fields, B. A., et al. (2000). J. Biol. Chem., 275, 20012. Larsen, T. M., Boehlein, S. K., Schuster, S. M., et al. (1999). Biochemistry, 38, 16146. Laskey, R. A., Mills, A. D., Philpott, A., et al. (1993). Phil. Trans. Roy. Soc. London B Biol. Sci., 339, 263. Laszlo, E., Hollo, J., Hoschke, A., & Savosi, G. (1978). Carbohydr. Res., 61, 387. Lauble, H., Kennedy, M. C., Emptage, M. H., et al. (1996). Proc. Natl. Acad. Sci. USA, 93, 13699. Lauffer, M. A. (1975). Entropy-Driven Processes in Biology. Berlin: Springer- Verlag. Laurence, D. J. R. (1957). Meth. Enzymol., 4, 174. Laurents, D. V., & Baldwin, R. M. (1998). Biophys. J, 75, 428. Lawson, J. W. R., & Veech, R. L. (1979). J. Biol. Chem., 254, 6528. Layne, E. (1957). Meth. Enzymol., 3, 447. Lazdunski, M. (1972). In B. L. Horecker, & E. R. Stadtman (Eds.), Current Topics in Cellular Regulation, vol. 6 (p. 267). New York: Academic Press. Lazdunski, M., Petitclerc, C., Chappelet, D., & Lazdunski, C. (1971). Eur. J. Biochem., 20, 124. Leadlay, P. F., Albery, W. J., & Knowles, J. R. (1976). Biochemistry, 15, 5617. Leatherbarrow, R. J., Fersht, A. R., & Winter, G. (1985). Proc. Natl. Acad. Sci. USA, 82, 7840. Lebioda, L., Stec, B., Brewer, J., & Tykarska, E. (1991). Biochemistry, 30, 2823. Leboida, L., & Stec, B. (1991). Biochemistry, 30, 2817. Lederer, F. (1991). In B. Curti, S. Ronchi, & G. Zanetti (Eds.), Flavins and Flavoproteins 1990, Proceedings of the Tenth International Symposium (p. 773). New York: Walter de Gruyter. Lee, L. V., Vu, M. V., & Cleland, W. W. (2000). Biochemistry, 39, 4808. Lee, M. H., Gautam-Basak, M., Wooley, C., & Sander, E. G. (1988). Biochemistry, 27, 1367. Lee, Y. J., & Wickner, R. B. (1992). Yeast, 8, 787. Lee, Y.-F., Tawfik, D. S., & Griffiths, A. D. (2002). Nucleic Acids Res., 30, 4937.

References

Lehman, I. R. (1971). The Enzymes, 4, 251 (3rd ed.). Lehmann, H. (1936). Biochem. Z., 286, 3367. Lei, C., Shin, Y., Liu, J., & Ackerman, E. J. (2002). J. Am. Chem. Soc., 124, 11242. Leitner, D. M. (2008). Ann. Rev. Phys. Chem., 59, 233. Leonard, D., & Reichl, L. E. (1990). J. Chem. Phys., 92, 6004. Lesser, D. R., Kurpiewski, M. R., & Jen-Jacobson, L. (1990). Science, 250, 776. Leung, D. W., Chen, E., & Goeddel, D. V. (1989). Technique (Philadelphia), 1, 1115. Levene, M. J., Korlach, J., Turner, S. W., et al. (2003). Science, 299, 682. Leveque, V. J., Vance, C. K., Nick, H. S., & Silverman, D. N. (2001). Biochemistry, 40, 10586. Levin, M. K., Wang, Y. H., & Patel, S. S. (2004). J. Biol. Chem., 279, 26005. Levins, R. (1966). Am. Scientist, 54, 421. Levinthow, L., & Meister, A. (1954). J. Biol. Chem., 209, 265. Levitsky, A., & Koshland, D. E., Jr. (1969). Proc. Natl. Acad. Sci. USA, 62, 1121. Levy, M., Griswold, K. E., & Ellington, A. D. (2005). RNA, 11, 1555. Lewandowicz, A., Rudzinski, J., Tronstad, L., et al. (2001). J. Am. Chem. Soc., 123, 4550. Lewis, P. S. (1926). Biochem. J., 24, 978. Leyh, T. S. (1999). Meth. Enzymol., 249, 48. Li, N., Batzer, A., Daly, R., et al. (1993). Nature, 363, 85. Li, S., Furie, B. C., Furie, B., & Walsh, C. T. (1997). Biochemistry, 36, 6384. Li, Y., & Breaker, R. R. (1999a). Curr. Opin. Struct. Biol., 9, 315. Li, Y., & Breaker, R. R. (1999b). J. Am. Chem. Soc., 121, 5364. Li, Z.-H., Bulychev, A., Kotra, L. P., et al. (1998). J. Am. Chem. Soc., 120, 13003. Li, Z., Sau, A. K., Furdui, C. M., & Anderson, K. S. (2005). Anal. Biochem., 343, 35. Liang, Z. X., & Klinman, J. P. (2004). Curr. Opin. Struct. Biol., 14, 648. Lieberman, I., Kornberg, A., & Simms, E. S. (1955). J. Biol. Chem., 215, 403. Lienhard, G. E. (1972). Ann. Rev. Med. Chem., 7, 249. Lienhard, G. E. (1973). Science, 180, 149. Lienhard, G. E., & Secemski, I. I. (1973). J. Biol. Chem., 248, 1121. Lin, J., Westler, W. M., Cleland, W. W., et al. (1998). Proc. Natl. Acad. Sci. USA, 95, 14664. Lindahl, P. A. (2002). Meth. Enzymol., 354, 296. Lindberg, U., Hoglund, A. S., & Karlsson, R. (1981). Biochimie, 63, 307. Lindquist, R. N., Lynn, J. L., Jr., & Lienhard, G. E. (1973). J. Am. Chem. Soc., 95, 8762. Lindqvist, Y., Schneider, G., & Wihko, P. (1994). Eur. J. Biochem., 221, 139. Lindsay, J. P., Clark, D. S., & Dordick, J. S. (2004). Biotechnol. Bioeng., 85, 553. Lindskog, S. (1997). Pharmacol. Therapeut., 74, 1. Lindskog, S., Henderson, L. E., Kannan, K. K., et al. (1971). The Enzymes, 5, 587 (3rd ed.). Lineweaver, H., & Burk, D. (1934). J. Amer. Chem. Soc., 56, 658. Liou, Y.-C., Sun, A., Ryo, A., et al. (2003). Nature, 424, 556.

References

Lipinski, C. A. (2000). J. Pharmacol. Toxicol. Meth., 44, 235. Lipmann, F. (1941). Adv. Enzymol., 1, 99. Lipmann, F. (1971). Wanderings of a Biochemist, Krieger Publ. Malabar: Florida. Lipowsky, R., & Klumpp, S. (2005). Physica A: Statistical Mechanics and its Applications, 352, 53. Lippard, S. J., & Berg, J. M. (1994). Principles of Bioinorganic Chemistry. Mill Valley, California: University Science Books. Lipscomb, W. N. (1994). Adv. Enzymol., 68, 67. Lissin, N. M. (1995). FEBS Lett., 361, 55. Little, D. I., Poat, P. C., & Giles, I. G. (1982). Eur. J. Biochem., 124, 499. Liu, A., Po¨tsch, S., Davydov, A., et al. (1998). Biochemistry, 37, 16369. Liu, J. (2005). Biophys. Chem., 120, 207. Liu, J. J., & McLennan, J. G. (1994). J. Biol. Chem., 269, 11687. Llamas, A., Mendel, R. R., & Schwarz, G. (2004). J. Biol. Chem., 279, 55241. Lloyd, N. C., Morgan, H. W., Nicholson, B. K., & Ronimus, R. S. (2005). Angew. Chem. Int. Ed. Engl., 44, 941. Lo¨bau, S., Weber, J., & Senior, A. E. (1998). Biochemistry, 37, 10846. Lockwood, T. D., Minassian, I. A., & Roux, L. (1982). Biochem. J., 206, 239. Lodeiro, S., Schulz-Gasch, T., & Matsuda, S. P. (2005). J. Am. Chem. Soc., 127, 14132. Lodi, P. J., & Knowles, J. R. (1991). Biochemistry, 30, 6948. Lodi, P. J., & Knowles, J. R. (1993). Biochemistry, 32, 4338. Lodi, P. J., Chang, L. C., Knowles, J. R., & Komives, E. A. (1994). Biochemistry, 33, 2809. Loe, D. W., Deeley, R. G., & Cole, S. P. (1998). Cancer Res., 58, 5130. Loewenstein, A., & Szo¨ke, A. (1962). J. Am. Chem. Soc., 84, 1151. Loewus, F. F., Ofner, P., Fisher, H. F., et al. (1953). J. Biol. Chem., 202, 699. Lo¨fa¨s, S., & Johnsson, B. (1990). J. Chem. Soc. Chem. Commun.. 1526. Logusch, E. W., Walker, D. M., McDonald, J. F., & Franz, J. E. (1989). Biochemistry, 28, 3043. Loisel, T. P., Boujemaa, R., Pantaloni, D., & Carlier, M. F. (1999). Nature, 401, 613. Lolis, E., Alber, T., Davenport, R. C., et al. (1990). Biochemistry, 29, 6609. Lolis, E., & Petsko, G. A. (1990). Biochemistry, 29, 6619. Londesborough, J. C., & Dalziel, K. (1970). In H. E. Sund (Ed.), Pyridine Nucleotide-Dependent Dehydrogenases (p. 315). Berlin: Springer-Verlag. Longstaff, C., Campbell, A. F., & Fersht, A. R. (1990). Biochemistry, 29, 7339. Los, J. M., Simpson, L. B., & Wiesner, K. (1956). J. Am. Chem. Soc., 78, 1564. Losel, R. M., Wise, J. G., & Vogel, P. D. (1997). Biochemistry, 36, 1188. Lowe, C. R. (1989). Philos. Trans. Roy. Soc. London Series B, Biol. Sci., 324, 487. Lowe, G., & Potter, B. V. L. (1982). Biochem. J., 201, 665. Lowe, G., & Sproat, B. S. (1978). J. Chem. Soc., Chem. Commun. 565.

827

Lowe, G., Cullis, P. M., Jarvest, J. L., et al. (1981). Philos. Trans. Roy. Soc. London Series B, 293, 75. Lowe, J., Stock, D., Jap, B., et al. (1995). Science, 268, 533. Lowry, O. H., & Passonneau, J. V. (1972). A Flexible System of Enzymatic Analysis. New York: Academic Press. Lowry, O. H., Passanneau, J. V., Schulz, D. W., & Rock, M. K. (1961). J. Biol.Chem., 238, 2756. Lowry, O. H., Rosbrough, N. J., Farr, A. L., & Randall, R. J. (1951). J. Biol. Chem., 193, 265. Lu, W.-Y., Starovasnik, M. A., Dwyer, J. J., et al. (2000). Biochemistry, 39, 3575. Lu, H. P., Xun, L., & Xie, X. S. (1998). Science, 282, 1877. Luciano, F., Kesmir, C., Mishto, M., et al. (2005). Biophys. J., 88, 2422. Lueck, J. D., & Fromm, H. J. (1973). FEBS Lett., 32, 184. Lueck, J. D., & Fromm, H. J. (1974). J. Biol. Chem., 249, 1341. Lueck, J. D., Ellison, W. R., & Fromm, H. J. (1973). FEBS Lett., 30, 321. Lumry, R. (1959). The Enzymes (2nd ed.), vol. 1 (p. 157). New York: Academic Press. Lumry, R., Smith, E. L., & Glantz, R. R. (1951). J. Am. Chem. Soc., 73, 4330. Lundblad, R. L. (1994). Techniques in Protein Modification (p. 296). Boca Raton, Florida: CRC Press. Lundblad, R. L. (2004). Chemical Reagents for Protein Modification (pp. 352) (3rd ed.). Boca Raton, Florida: CRC Press. Lunzer, M., Miller, S. P., Felsheim, R., & Dean, A. M. (2005). Science, 310, 499, 1. Luo, F., Yang, Y., Chen, C. F., et al. (2007). Bioinformatics, 23, 207 (Erratum:23, 916). Luo, J., & Bruice, T. C. (2004). Proc. Natl. Acad. Sci. USA, 101, 13152. Lutz, O., Nolle, A., & Staschewski, D. (1978). Z. Naturforsch. A: Phys. Chem. Kosmophys., A33, 380. Lymn, R. W., & Taylor, E. W. (1971). Biochemistry, 10, 4617. Ma, Y. Z., & Taylor, E. W. (1995). Biochemistry, 34, 13233. Ma, H., Wan, C., Wu, A., & Zewail, A. H. (2007). Proc. Natl. Acad. Sci. USA, 104, 712. Macgregor, R. B., Jr., Clegg, R. M., & Jovin, T. M. (1985). Biochemistry, 24, 5503. Machner, M. P., Urbanke, C., Barzik, M., et al. (2001). J. Biol. Chem., 276, 40096–40103. MacLachlan, J., Wotherspoon, A. T., Ansell, R. O., & Brooks, C. J. (2000). J. Steroid Biochem. Mol. Biol., 72, 169. MacNeal, R. K., & Purich, D. L. (1978). J. Biol. Chem., 253, 4683. MacNeal, R. K., Webb, B. C., & Purich, D. L. (1977). Biochem. Biophys. Res. Commun, 74, 440. Madsen, N. B. (1964). Biochem. Biophys. Res. Communs, 15, 390. Madsen, N. D. (1986). The Enzymes, 17, 365, (3rd ed.). Magde, D. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 2). Berlin: SpringerVerlag. Magde, D., Elson, E. L., & Webb, W. W. (1972). Phys. Rev. Lett., 29, 705. Maglia, G., & Allemann, R. K. (2003). J. Am. Chem. Soc., 125, 13372. Magnani, M., Serafini, G., Bianchi, M., et al. (1991). J. Biol. Chem., 266, 502.

828

Magni, G., Amici, A., Emanuelli, M., et al. (1999). Advan. Enzymol., 73, 135. Maier, V. P., Tappel, A. L., & Vollman, D. H. (1955). J. Amer. Chem. Soc., 77, 1278. Maister, S. G., Pett, C. P., Albery, W. J., & Knowles, J. R. (1976). Biochemistry, 15, 5607. Majerus, P. W., Alberts, A. W., & Vagelos, P. R. (1965). Proc. Natl. Acad. Sci. USA, 53, 410. Makara, G. M. (2007). J. Med. Chem., 50, 3214. Maki, S., & Kornberg, K. (1988). J. Biol. Chem., 263, 6555. Makinen, W. B., Pearlmutter, A. F., & Stuehr, J. F. (1969). J. Am. Chem. Soc., 91, 4083. Makinose, M. (1973). FEBS Lett., 37, 140. Maley, F., & Maley, G. F. (1963). Science, 141, 1278. Maley, G. F., & Maley, F. (1964). J. Biol. Chem., 239, 1168. Malmstro¨m, A. (1984). J. Biol. Chem., 259, 161. Malmstro¨m, A., & Aberg, L. (1982). Biochem. J., 201, 489. Malmstro¨m, A., Roden, L., Feingold, D. S., et al. (1980). J. Biol. Chem., 255, 3878. Malmstro¨m, B. G. (1961). The Enzymes, 5, 471 (2nd ed.). Malmstro¨m, B. G., Andre´asson, L.-E., & Reinhammer, B. (1975). The Enzymes, 12, 507 (3rd ed.). Malmstro¨m, B. G., & Rosenberg, A. (1959). Adv. Enzymol., 21, 131. Malthouse, J. P. G., & Brockelhurst, K. (1978). Biochem. Soc. Trans., 6, 250. Manjunath, P., Shenoy, B. C., & Roa, M. R. R. (1983). J. Appl. Biochem., 5, 235. Mannervik, B. (1982). Meth. Enzymol., 87, 370. Manstein, D. J., Massey, V., Ghisla, S., & Pai, E. F. (1988). Biochemistry, 27, 2300. Manstein, D. J., Pai, E. F., Schopfer, L. M., & Massey, V. (1986). Biochemistry, 25, 6807. March, J. (1992). Advanced Organic Chemistry (4th ed.). New York: Wiley. Marcinkeviciene, J., Luo, Y., Graciani, N. R., et al. (2001). J. Biol. Chem., 276, 23790. Marcus, R. A. (1956). J. Chem. Phys., 24. 966 & 979. Marcus, R. A. (1965). J. Chem. Phys., 43, 679. Marcus, R. A. (1968). J. Phys. Chem., 72, 891. Marcus, R. A. (2006). J. Chem. Phys., 125, 194504. Marcus, R. A., & Sutin, N. (1985). Biochim. Biophys. Acta, 811, 265. Maresca, A., Temperini, C., Vu, H., et al. (2009). J. Am. Chem. Soc., 131, 3057. Markley, J. L. (1975). Biochemistry, 14, 3546. Markley, J. L., Northrop, D. B., & Royer, C. A. (Eds.) (1996). High Pressure Effects in Molecular Biophysics and Enzymology. New York: Oxford University Press. Marohnic, C. C., Crowley, L. J., Davis, C. A., et al. (2005). Biochemistry, 44, 2449. Marquez, V. E., Liu, P. S., Kelley, J. A., et al. (1980). J. Med. Chem., 23, 713. Marriott, G. (Ed.) (1998). Caged Compounds; Method Enzymology (p. 529), vol. 291. New York: Academic Press. Marsh, E. N., & Ballou, D. P. (1998). Biochemistry, 37, 11864. Marshall, M., & Cohen, P. P. (1966). J. Biol. Chem., 241, 4197. Martensen, T. M., & Graves, D. J. (1980). Meth. Enzymol., 64, 325.

References

Martin, A. M., Horton, N. C., Lusetti, S., et al. (1999a). Biochemistry, 38, 8430. Martin, A. M., Sam, M. D., Reich, N. O., & Perona, J. J. (1999b). Nature Struct. Biol., 6, 269. Martin, D. B., & Vagelos, P. R. (1962). J. Biol. Chem., 237, 1787. Martin, R. G. (1963). J. Biol. Chem., 238, 257. Martin, S. L., & Kantrowitz, E. R. (1999). Biochemistry, 38, 2842. Mas, M. T., Bailey, J. M., & Resplendor, Z. E. (1988). Biochemistry, 27, 1168. Mascaro, J., Horhammer, R., Eisenstein, S., et al. (1977). 99, 273. Mason, G. G., Murray, R. Z., Pappin, D., & Rivett, A. J. (1998). FEBS Lett., 430, 269. Massey, V. (1953). Biochem. J., 53, 72. Massey, V., & Hemmerich, P. (1975). The Enzymes, 12, 191 (3rd ed.). Massey, V., Ghisla, S., & Kieschke, K. (1980). J. Biol. Chem., 255, 2796. Massey, V., Palmer, G., & Ballou, D. P. (1973). In T. E. King, H. S. Mason, & M. Morrison (Eds.), Oxidases and Related Systems (p. 25). Baltimore: University Park Press. Masson, P., & Balny, C. (2005). Biochim. Biophys. Acta, 1724, 440. Matayoshi, E. D., Wang, G. T., Krafft, G. A., & Erickson, J. (1990). Science, 247, 954. Matsubara, N., Billington, A. P., & Hess, G. P. (1992). Biochemistry, 31, 5507. Matsumura, M., & Matthews, B. W. (1989). Nature, 334, 406. Matsumura, M., Becktel, W. J., & Matthews, B. W. (1989). Science, 243, 792. Matsuzawa, T. (1978). Seikagaku, 50, 1073. Matthews, B. W. (1986). Harvey Lectures, 81, 33. Matthews, B. W., Nicholson, H., & Becktel, W. J. (1987). Proc. Natl. Acad. Sci. USA, 84, 6663. Matthews, D. J., & Wells, J. A. (1993). Science, 260, 1113. Maxwell, J. C. (1868). Phil. Mag. (IV), 35, 134. McCammon, J. A. (1987). Science, 238, 486. McCammon, J. A., & Northrup, S. H. (1981). Nature, 293, 316. McClure, S. J., & Robinson, P. J. (1996). Mol. Membr. Biol., 13, 189. McClure, W. R. (1969). Biochemistry, 8, 2782. McClure, W. R., & Jovin, T. M. (1975). J. Biol. Chem., 250, 4073. McClure, W. R., & Chow, Y. (1980). Meth. Enzymol., 64, 284. McDermott, A. E., Creuzet, F., Griffin, R. G., et al. (1990). Biochemistry, 29, 5767. McGhee, J. D., & von Hippel, P. H. (1974). J. Mol. Biol., 86, 449. McGillivray, I. H. (1930). Biochem. J., 24, 891. McIntosh, J. R., & Porter, M. E. (1989). J. Biol. Chem., 264, 6001. McMaster, M. C., & McMaster, C. (1998). GC/MS: A Practical User’s Guide. New York: John Wiley and Sons. McPherson, A. (1999). Crystallization of Biological Macromolecules. New York: Cold Spring Harbor Laboratory Press. McPherson, M. J., Stevens, C., Baron, A. J., et al. (1993). Biochem. Soc. Trans., 21, 752. McQuarrie, D. A. (1976). Statistical Mechanics. New York: Harper Collins. McQuarrie, D. A., Jachimowski, C. J., & Russell, M. E. (1964). J. Chem. Phys., 40, 2914. Meadows, D. H., Jardetsky, O., Epand, R. M., et al. (1968). Proc. Natl. Acad. Sci. USA, 60, 766.

References

Meadows, D. H., Markley, J. L., Cohen, J. S., & Jardetsky, O. (1967). Proc. Natl. Acad. Sci. USA, 58, 1307. Medda, R., Padiglia, A., Bellelli, A., et al. (1999). FEBS Lett., 453, 1. Medda, R., Padiglia, A., Pedersen, J. Z., et al. (1995). Biochemistry, 34, 16375. Meiboom, S. (1961). J. Chem. Phys., 34, 375. Meiler, J., & Baker, D. (2006). Proteins, 65, 538. Meinke, M. H., Bishop, J. S., & Edstrom, R. D. (1986). Proc. Natl. Acad. Sci. USA, 83, 2865. Meister, A. (1962). The Enzymes, 6, 443 (2nd ed.). Meister, A. (1973). The Enzymes, 10, 699 (3rd ed.). Melander, L., & Saunders, W. H. (1980). Reaction Rates of Isotopic Molecules. New York: Wiley Interscience. Mendel, R. R., & Bittner, F. (2006). Biochim. Biophys. Acta, 1763, 621. Mentch, F., Parkin, D. W., & Schramm, V. L. (1987). Biochemistry, 26, 921. Menzo, S., Monachetti, A., Balotta, C., et al. (2003). AIDS, 17, 663. Merritt, E. A., Sundaralingam, M., Cornelius, R. D., & Cleland, W. W. (1978). Biochemistry, 17, 3274. Mesecar, A. D., Stoddard, B. L., & Koshland, D. E., Jr. (1997). Science, 277, 202. Messerschmidt, A. (1998). In M. Sinnot (Ed.), Comprehensive Biological Catalysis, Vol. 3 (p. 401). London: Academic Press. Messerschmidt, A., & Weverdagger, R. (1996). Proc. Natl. Acad. Sci. USA, 93, 392. Messing, R. A. (1975). Immobilized Enzymes for Industrial Reactors. New York: Academic Press. Metzler, D. E. (2004). Biochemistry: The Chemical Reaction of Living Cells. New York: Academic Press. Meyer, K. H. (1911). Liebigs Ann. Chem., 380, 211. Mezyk, S. P. (1996). Radiat. Res., 145, 102. Michaelis, L. (1952). The Enzymes, 1, 1 (2nd ed.). Michaelis, L., & Davidsohn, H. (1911). Biochem. Z., 35, 386. Michaelis, L., & Menten, M. (1913). Biochem. Z., 49, 333. Michel, H. (1998). Proc. Natl. Acad. Sci. USA, 95, 12819. Michel, H., Behr, J., Harrenga, A., & Kannt, A. (1998). Ann. Rev. Biophys. Biomol. Struct., 27, 329. Midelfort, C. F., Gupta, R. K., & Rose, I. A. (1976). Biochemistry, 15, 2178. Mieyal, J. J., & Abeles, R. H. (1972). The Enzymes, 7, 515 (3rd ed.). Mildvan, A. S. (1970). The Enzymes, 2, 445 (3rd ed.). Mildvan, A. S. (1979). Adv. Enzymol., 49, 103. Mildvan, A. S., Harris, T. K., & Abeygunawardana, C. (1999). Meth. Enzymol., 308, 219. Mildvan, A. S., & Scrutton, M. C. (1967). Biochemistry, 6, 2978. Mildvan, A. S., Massiah, M. A., Harris, T. K., et al. (2002). J. Molec. Struct., 615, 163. Mildvan, A. S., Xia, Z., Azurmendi, H. F., et al. (2005). Arch. Biochem. Biophys., 433, 129. Millar, N. C., Howarth, J. V., & Gutfreund, H. (1987). Biochem. J., 248, 683. Miller, B. G., & Raines, R. T. (2004). Biochemistry, 43, 6387. Miller, G. P., & Benkovic, S. J. (1998). Biochemistry, 37, 6327. Mills, G. A., & Urey, H. C. (1940). J. Am. Chem. Soc., 62, 1019.

829

Milman, H. A., Cooney, D. A., & Huang, C. Y. (1980). J. Biol. Chem., 255, 1862. Milner, Y., Michaels, G., & Wood, H. G. (1978). J. Biol. Chem., 263, 87. Mimori-Kiyosue, Y., & Tsukita, S. (2001). J. Cell Biol., 154, 1105. Mimori-Kiyosue, Y., Shiina, N., & Tsukita, S. (2001). Curr. Biol., 10, 865. Min, W., English, B. P., Luo, G., et al. (2005). Acc. Chem. Res., 38, 923. Minton, A. P. (1980). Biophys Chem., 12, 271. Minton, A. P. (1983). Biophys Chem., 18, 139. Minton, A. P. (2005). Biophys J, 88, 971. Mitchell, P. (1961). Nature, 191, 144. Miziorko, H. M., & Lorimer, G. H. (1983). Ann. Rev. Biochem., 52, 507. Miziorko, H., & Eckstein, F. (1984). J. Biol. Chem, 259, 13037. Mizuguchi, H., Cook, P. F., Tai, C. H., et al. (1999). J. Biol. Chem., 274, 2166. Mizutani, Y., Yamamoto, K., & Kitagawa, T. (2002). Tanpakushitsu Kakusan Koso, 47, 670. Moffat, K. (2001). Chem. Rev., 101, 1569. Moffat, K., Szebenyi, D., & Bilderback, D. (1984). Science, 223, 1423. Mogilner, A., & Oster, G. (1996). Biophys. J., 71, 3030. Mogilner, A., & Oster, G. (2003). Biophys. J, 84, 1591. Mohr, L. H., Smith, L. E. H., & Raftery, M. A. (1973). Arch. Biochem. Biophys., 159, 505. Moncrief, M. B. C., Hom, L. G., Jabri, E., et al. (1995). Protein Sci., 4, 2234. Mongan, J. (2004). Mol. Design, 18, 433. Monod, J., Changeux, J. P., & Jacob, F. (1963). J. Mol. Biol., 6, 306. Monod, J., Wyman, J., & Changeux, J.-P. (1965). J. Mol. Biol., 12, 88. Montague, C., Rhee, K. W., & Carlson, F. D. (1983). J. Muscle Res. Cell Mot., 4, 95. Montero, C., & Lorente, P. (1991). Biochem. J., 275, 327. Montgomery, R. K., Sybicki, M. A., Forcier, A. G., & Grand, R. J. (1981). Biochim. Biophys. Acta, 661, 346. Mor, A., Reboud-Ravaux, M., Mazaleyrat, J. P., & Wakselman, M. (1988). Thromb. Res., 35 (Suppl. 8). Morales, M. F., & Goldman, D. E. (1955). J. Am. Chem. Soc., 77, 6069. More O’Ferrall, R. A. (1970). J. Chem. Soc. B, 274. Morehouse, K. M., & Mason, R. P. (1988). J. Biol. Chem., 263, 1204. Morehouse, K. M., Sipe, H. J., Jr., & Mason, R. P. (1989). Arch. Biochem. Biophys., 273, 158. Morild, E. (1981). Adv. Protein Chem., 34, 93. Morishita, Y., Kobayashi, T. J., & Aihara, K. (2006). Biophys. J., 91, 2072. Moriyami, Y., & Nelson, N. (1989). J. Biol. Chem., 264, 3577. Morrison, J. F. (1979). Meth. Enzymol., 63, 257. Morrison, J. F., & Cleland, W. W. (1966). J. Biol. Chem., 241, 673. Morrison, J. F., & Cleland, W. W. (1980). Biochemistry, 14, 3127. Morrison, J. F., & James, E. (1965). Biochem. J., 97, 37. Morrison, J. F., & Walsh, C. T. (1988). Adv. Enzymol., 61, 201.

830

Moscovitz, Y., & Srebnik, S. (2005). Biophys. J., 89, 22. Moss, J., & Lane, M. D. (1971). Adv. Enzymol., 35, 1. Mourad, N., & Parks, R. E., Jr. (1966). J. Biol. Chem., 241, 271. Mrachko, G. T., Chunduru, S. K., & Calvo, K. C. (1992). Arch. Biochem. Biophys., 294, 446. Mudd, C. P., & Berger, R. L. (1988). J. Biochem. Biophys. Meth., 17, 171. Mudd, S. H. (1973). The Enzymes, 8, 121 (3rd ed.). Mudd, S. H., & Mann, J. D. (1973). J. Biol. Chem., 238, 2164. Muller-Steffner, H. M., Augustin, A., & Schuber, F. (1996). J. Biol. Chem., 271, 23967. Mullins, L. S., & Raushel, F. M. (1995). Meth. Enzymol., 249, 398. Mun˜oz, V. (2007). Annu. Rev. Biophys., 36, 395. Mura, U., & Stadtman, E. R. (1981). 256, 13014. Mura, U., Chock, P. B., & Stadtman, E. R. (1981). 256, 13022. Musafia, B., Buchner, V., & Arad, D. (1995). J. Mol. Biol., 254, 761. Muskavitch, K. M. T., & Linn, S. (1981). The Enzymes, 14, 233 (3rd ed.). Musser, S. M., Stowell, M. H., & Chan, S. I. (1995). Adv. Enzymol., 71, 79. Muzyczka, N., Poland, R. L., & Bessman, M. J. (1972). J. Biol. Chem., 247, 7116. Nagao, K., Taguchi, Y., Arioka, M., et al. (1995). J. Bacteriol., 177, 1536. Nagy, M., & Ribet, A.-M. (1977). Eur. J. Biochem., 77, 77. Nakajima, N., Tanizawa, K., Tanaka, H., & Soda, K. (1986). Agric. Biol. Chem., 50, 2823. Nakano, S., Proctor, D. J., & Bevilacqua, P. C. (2001). Biochemistry, 40, 12022. Nakayama, C., Wataya, Y., Santi, D. V., et al. (1981). J. Med. Chem., 24, 1161. Nambiar, K. P., Stauffer, D. M., Kolodziej, P. A., & Benner, S. A. (1983). J. Am. Chem. Soc., 105, 5886. Narasimhulu, S. (2007). Biochim. Biophys. Acta, 1770, 360. Nayal, M., Hitz, B. C., & Honig, B. (1999). Prot. Sci., 8, 676. Nazi, I., & Wright, G. D. (2005). Biochemistry, 44, 13560. Neal, D. G., Purich, D. L., & Cannell, D. S. (1984). J. Chem. Phys., 80, 3469. Neet, K. E., & Koshland, D. E., Jr. (1966). Proc. Natl. Acad. Sci., USA, 56, 1606. Neet, K. E., Nanci, A., & Koshland, D. E., Jr. (1968). J. Biol. Chem., 243, 6392. Negelein, E., & Haas, E. (1935). Biochem. Z., 282, 206. Nelson, M. J., & Cowling, R. A. (1990). J. Am. Chem. Soc., 112, 2820. Nelson, M. J., Cowling, R. A., & Seitz, S. P. (1994). Biochemistry, 33, 4966. Nelson, M. J., Seitz, S. P., & Cowling, R. A. (1990). Biochemistry, 29, 6897. Newman, J., Peat, T. S., Richard, R., et al. (1999). Biochemistry, 38, 16105. Newton, D. L., Boque, L., Wlodawer, A., et al. (1998). Biochemistry, 37, 5173. Ng, P. C., & Henikoff, S. (2003). Nucleic Acids Res., 31, 3812. Ng, R. (2004). Drugs-From Discovery to Approval (p. 368). New York: Wiley-Liss.

References

Nichol, L. W., Jackson, W. J. H., & Winzor., D. J. (1967). Biochemistry, 6, 2449. Nichol, L. W., Ogston, A. G., Winzor, D. J., & Sawyer, W. H. (1967). Biochem. J., 143, 435. Nicholls, A., & Honig, B. (1991). J. Comp. Chem., 12, 435. Nicholls, D. G., & Ferguson, S. J. (2002). Bioenergetics (p. 297) (3rd ed.). London: Academic Press. Nicholson, C., & Sykova, E. (1998). Trends Neurolog. Sci., 21, 207. Nicholson, C., & Tao, L. (1993). Biophys. J., 65, 2277. Nicholson, H., Tronrud, D. E., Becktel, W. J., & Matthews, B. W. (1992). Biopolymers, 32, 1431. Nickbarg, E. B., Davenport, R. C., Petsko, G. A., & Knowles, J. R. (1988). Biochemistry, 27, 5948. Nielsen, J. E., & McCammon, J. A. (2003). Protein Science, 12, 1894. Nikolic-Hughes, I., Rees, C. D., & Herschlag, D. (2004). J. Am. Chem. Soc., 126, 11814. Ning, J., & Fromm, H. J. (1968). Biochem. Biophys. Res. Commun., 32, 672. Ning, J., Purich, D. L., & Fromm, H. J. (1969). J. Biol. Chem., 244, 3840. Nishikawa, S., Morioka, H., Kim, H., et al. (1987). Biochemistry, 26, 8620. Nishizaka, T., Oiwa, K., Noji, H., et al. (2004). Nat. Struct. Mol. Biol., 11, 142. Noble, M. E. M., Wierenga, R. K., Lambeir, A.-M., et al. (1991). Proteins: Struct., Funct., Genet., 10, 5049. Noji, H., Yasuda, R., Yoshida, M., & Kinosita, K., Jr. (1997). Nature, 386, 299. Nordlie, R. C. (1982). Meth. Enzymol., 87, 319. Nordlie, R. C., & Fromm, H. J. (1959). J. Biol. Chem., 234, 2523. Nordlund, P., & Eklund, H. (1990). Nature, 345, 593. Nordlund, P., Sjo¨berg, B.-M., & Eklund, H. (1990). 345, 593. Norris, J. R. (1997). Markov Chains. Cambridge: Cambridge University Press. Northrop, D. B. (1969). J. Biol. Chem., 244, 5808. Northrop, D. B. (1975). Biochemistry, 14, 2644. Northrop, D. B. (1977a). Biochemistry, 14, 2644. Northrop, D. B. (1981). Ann. Rev. Biochem., 50, 103. Northrop, D. B. (1982). Meth. Enzymol., 87, 607. Northrop, D. B. (1991). In P. F. Cook (Ed.), Enzyme Mechanism from Isotope Effects. Boca Raton, Florida: CRC Press. Northrop, D. B. (1994). Biochem. J., 301, 623. Northrop, D. B. (1999). J. Am. Chem. Soc., 121, 3521. Northrop, D. B. (2002). Biochim. Biophys. Acta, 1595, 71. Northrop, D. B., & Wood, H. G. (1969). J. Biol. Chem., 244, 5820. Northrop, D. B. (1977b). In W. W. Cleland, M. H. O’Leary, & D. B. Northrop (Eds.), Isotope Effects on Enzyme-Catalyzed Reactions. Baltimore: University Park Press. Northrup, S. H., Allison, S. A., & McCammon, J. A. (1984). J. Chem. Phys., 80, 1517. Novoa, W. B., Winer, A. D., Glaid, A. J., & Schwert, G. W. (1959). J. Biol. Chem., 234, 1143. Nudler, E. (1999). J. Mol. Biol., 288, 1. Nun˜ez, S., Antoniou, D., Schramm, V. L., & Schwartz, S. D. (2004). J. Am. Chem. Soc., 126, 15720. Nygaard, A. P. (1963). The Enzymes, 7, 557 (2nd ed.).

References

O’Brien, M., Scurria, M., Rashka, K., et al. (2005). Promega Notes. No, 91, 6. O’Brien, P. J., & Herschlag, D. (1999). Chemistry and Biology, 6, R91. O’Donnell, M. E. (1987). J. Biol. Chem., 262, 16558. O’Leary, M. H. (1977). In W. W. Cleland, M. H. O’Leary, & D. B. Northrop (Eds.), Isotope Effects on Enzyme-Catalyzed Reactions. Baltimore: University Park Press. O’Leary, M. H. (1992a). Meth. Enzymol., 64, 83. O’Leary, M. H. (1992b). The Enzymes, 20, 235 (3rd ed.). O’Rear, J. L., Wang, S. L., Feig, A. L., et al. (1890). J. Chem. Soc., 57, 834931. O’Sullivan, W. J., & Smithers, G. W. (1979). Meth. Enzymol., 63, 294. O’Donnell, M. E. (2006). J. Biol. Chem., 281, 10653. O’Donnell, M., & Kuriyan, J. (2006a). Curr. Opin. Struct. Biol., 16, 35. O’Donoghue, A. C., Amyes, T. L., & Richard, J. P. (2005). Biochemistry, 44, 2610. O’Donoghue, A. C., Amyes, T. L., & Richard, J. P. (2008). Org. Biomolec. Chem, 6, 391. Ogston, A. G. (1948). Nature, 162, 963. Ogura, T., & Wilkinson, A. J. (2001). Cells, 6, 575. Ogura, Y. (1955). Arch. Biochem. Biophys., 57, 288. O’Hara, P., Goodwin, P., & Stoddard, B. L. (1995). J. Appl. Cryst., 28, 829. Ohtaka, H., & Freire, E. (2005). Prog. Biophys. Molec. Biol., 88, 193. Oiwa, K., Jameson, D. M., Croney, J. C., et al. (2003). Biophys. J., 84, 634. Okazaki, R., & Kornberg, A. (1964). J. Biol. Chem., 239, 269. Okuda, J., Miwa, I., Inagaki, K., et al. (1978). J. Biochem., 84, 993. O’Leary, M. H., Yamada, H., & Yapp, C. J. (1981). Biochemistry, 20, 1476. Olsen, J. A., & Anfinsen, C. B. (1952). J. Biol. Chem., 197, 67. Ono, S., Hara, K. Y., Hirao, J., et al. (2003). Biochim. Biophys Acta., 1607, 35. Oppenheimer, N. J. (1984). J. Am. Chem. Soc., 156, 3032. Oppenheimer, N. J., & Handlon, A. L. (1992). The Enzymes, 20, 453 (3rd ed.). Orden, A. V., & Jung, J. (2008). Biopolymers, 89, 1. Orgel, L. E. (1960). An Introduction to Transition-Metal Chemistry: Ligand Field Theory. New York: John Wiley and Sons. Orosz, F., & Ova´di, J. (1987). Biochim. Biophys. Acta, 915, 53. Orr, G. A., Simmons, J., Jones, S. R., et al. (1978). Proc. Natl. Acad. Sci. USA, 75, 2230. Orsi, B. A., & Tipton, K. F. (1979). Meth. Enzymol, 63, 159. Ortiz de Montellano, P. R. (Ed.) (1986). Cytochrome P-450: Structure, Mechanism, and Biochemistry. New York: Plenum. Ost, T. W., & Daff, S. (2005). J. Biol. Chem., 280, 965. Osterman, H. L., & Walz, F. G., Jr. (1976). Biochemistry, 17, 4124. Ostwald, W. (1984). J. Prakt. Chem., 2 (29), 385. Ottesen, M. (1971). In D. Glick (Ed.), Meth. Biochem. Anal., vol. 20 (p. 135). New York: Wiley. Otto, M. R., Lillo, M. P., & Beechem, J. M. (1994). Biophys. J., 67, 2511. Ouellet, L., Laidler, K. J., & Morales, M. F. (1952). Arch. Biochem. Biophys., 39, 37.

831

Ova´di, J., & Keleti, T. (1978). Eur. J. Biochem., 85, 157. Ownby, D. W., & Gill, S. J. (1990). Biophys Chem., 37, 395. Ozols, J. (1990). Meth. Enzymol., 182, 587. Pace, J. (1930). Biochem. J., 24, 606. Page, M. I., & Jencks, W. P. (1971). Proc. Natl. Acad. Sci. USA, 68, 1678. Pai, E. F., & Schulz, G. E. (1983). J. Biol. Chem., 258, 1752. Paiva, A. A., Tilton, R. F., Jr., Crooks, G. P., et al. (1997). Biochemistry, 36, 15472. Paiva, A. M., Vanderwall, D. E., Blanchard, J. S., et al. (2001). Biochim. Biophys. Acta, 1545, 67. Palfey, B. A. (2003). Kinetic Analysis of Macromolecules. In K. A. Johnson (Ed.) (p. 203). New York: Oxford University Press. Palfey, B. A., & Massey, V. (1998). In M. Sinnot (Ed.), Comprehensive Biological Catalysis, vol. 3 (p. 83). London: Academic Press. Palm, D. (1968). Eur. J. Biochem., 5, 270. Palmer, D. R., Garrett, J. B., Sharma, V., et al. (1999). Biochemistry, 38, 4252. Palmer, A. G., 3rd, & Thompson, N. L. (1987). Biophys. J., 52, 257. Palmore, G. T., Bertschy, H., Bergens, S. H., & Whitesides, G. M. (1998). J. Electroanal. Chem., 443, 155. Pandey, A., Nigam, P., Soccol, C. R., et al. (2000). Biotechnol. Appl. Biochem., 31, 135. Parkin, D. W., Mentch, F., Banks, G. A., et al. (1991). Biochemistry, 30, 4586. Passonneau, J. V., & Lowry, O. H. (1962). Biochem. Biophys. Res. Comm., 7, 10. Pasternack, R. F., & Sigel, H. (1970). J. Am. Chem. Soc., 92, 6146. Pasternack, R. F., Angwin, M., Gipp, I., & Reingold, R. (1972). J. Inorg. Nucf. Chem., 34, 2329. Pasteur, L. (1858). Compt. Rend., 46, 615. Patel, S., & Latterich, M. (1998). Trends Cell Biol., 8, 6571. Pauling, L. (1935). Proc. Natl. Acad. Sci. USA, 21, 186. Pauling, L. (1945). The Nature of the Chemical Bond. Ithaca, New York: Cornell University Press. p. 3. Pauling, L. (1946). Chem. Engin. News, 24, 1375. Pauling, L. (1947). General Chemistry (p. 568). San Francisco: W.H. Freeman. Pauling, L. (1960). The Nature of the Chemical Bond. Ithaca, New York: Cornell University Press. Pauling, L. (1970). General Chemistry. New York: Dover Publications. p. 329. Payen, A., & Persoz, J. (1833). Annal. Chem. Phys., 2me Se´rie, 53, 73. Peacock, D., & Boulter, D. (1970). Biochem. J., 120, 763. Pearson, R. B., & Kemp, B. E. (1991). Meth. Enzymol., 200, 62. Pecht, I., & Lancet, D. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 306). Berlin: Springer-Verlag. Peck, H. D. (1974). The Enzymes, 10, 651, 3rd edn. Peller, L., & Alberty, R. A. (1959). J. Am. Chem. Soc., 81, 5907. Pelliccioli, A. P., & Wirz, J. (2002). Photochem. Photobiol. Sci., 1, 441. Penefsky, H. S. (1974). The Enzymes, 10, 375 (3rd ed.). Penefsky, H. S., & Cross, R. L. (1991). Adv. Enzymol, 64, 173. Penner, M. H., & Frieden, C. (1987). J. Biol. Chem., 262, 15908.

832

Penner-Hahn, J. E. (1998). In M. Sinnott (Ed.), Comprehensive Biological Catalysis, vol. 3 (p. 439). London: Academic Press. Peracchi, A. (2001). Trends Biochem. Sci., 26, 497. Pereverzev, Y. V., & Prezhdo, O. V. (2006). Biophys. J., 91, L19. Pereverzev, Y. V., Prezhdo, O. V., Forero, M., et al. (2005). Biophys. J., 89, 1446. Perrin, D., & Dempsey, B. (1974). Buffers for pH and Metal Ion Control. London: Chapman and Hall. Pershad, H. R., Hirst, J., Cochran, B., et al. (1999). Biochim. Biophys. Acta, 1412, 262. Perutz, M. F. (1964). Sci. Amer., 211, 64. Perutz, M. F. (1970). Nature, 228, 726. Perutz, M. F. (1990). Annu. Rev. Physiol., 52, 1. Perutz, M. F. (1991). Mechanisms of Cooperativity and Allosteric Regulation in Proteins. New York: Cambridge University Press. Perutz, M. F., Wilkinson, A. J., Paoli, M., & Dodson, G. G. (1998). Ann. Rev. Biophys. Biomol. Struct., 27, 1. Perutz, M. F. (1989). Quart. Rev. Biophys, 22, 139. Peskin, C. S., Odell, G. M., & Oster, G. S. (1993). Biophys. J., 65, 316. Peters, B., Janek, K., Kuckelkorn, U., & Holzhutter, H. G. (2002). J. Mol. Biol., 318, 847. Peters, E. L., & Tower, D. B. (1959). J. Neurochem., 5, 80. Peters, R. A. (1955). Disc. Faraday. Soc., 20, 189. Peters, R. A. (1957). Adv. Enzymol., 18, 113. Peterson, G. L. (1983). Meth. Enzymol., 91, 95. Petrev, D., & Honig, B. (2003). Meth. Enzymol., 374, 492. Pettersson, H., & Pettersson, G. (1999). Eur. J. Biochem., 262, 371. Pfister, K., Ka¨gi, J. H. R., & Christen, P. (1978). Proc. Natl. Acad. Sci. USA, 75, 145. Philipp, M., Tsai, I. H., & Bender, M. L. (1979). Biochemistry, 18, 3769. Phillips, A. T., & Wood, W. A. (1964). Biochem. Biophys. Res. Communs, 15, 530. Phillips, D. C. (1964). Proc. Natl. Acad. Sci. USA, 57, 484. Phillips, D. C. (1967). Sci. Am., 215 (November), 78. Phillips, R. C., George, P., & Rutman, R. J. (1969). J. Biol. Chem., 244, 3330. Pieklik., J., & Guynn, R. W. (1975). J. Biol. Chem., 250, 4445. Pike, S., & Duggleby, R. G. (1987). Biochem. J., 244, 781. Pimentel, G. C., & McClellan, A. L. (1960). The Hydrogen Bond. San Francisco: Freeman. Pingoud, A., Riesner, D., Boehme, D., & Maass, G. (1973). FEBS Letts, 30, 1. Piper, J. W. (1997). Force Dependence of Cell Bound E-Selectin/ Carbohydrate Ligand Binding Characteristics, PhD Dissertation. Atlanta: School of Mechanical Engineering, Georgia Institute of Technology. 1997. Plapp, B. V. (1970). J. Biol. Chem., 245, 1725. Plapp, B. V. (1982). Meth. Enzymol., 87, 469. Plapp, B. V. (1995). Meth. Enzymol., 249, 91. Plapp, B. V. (2009). Arch. Biochem. Biophys. (e-pub ahead of press, July 2009). Pluria, D. H., Schomberg, H., Richard, J. P., et al. (1980). Biochemistry, 19, 325. Pochet, R. (Ed.) (2000). Calcium: The Molecular Basis of Calcium Ion. Dordrecht: Kluwer Academic Publishers.

References

Pocker, Y., & Sarkanen, S. (1978). Adv. Enzymol., 47, 149. Pohl, F. M. (1968). Eur. J. Biochem., 4, 373. Pohl, F. M. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 282). Berlin: SpringerVerlag. Polakis, E. S., & Bartley, W. (1966). Biochem. J., 99, 521. Polakowski, R., Craig, D. B., Skelley, A., & Dovichi, N. J. (2000). J. Am. Chem. Soc., 122, 4853. Polanyi, J. C., & Schreiber, J. L. (1974). In H. Eyring, W. Yost, & D. Henderson (Eds.), Kinetics of Gas Reactions: Physical Chemistry – An Advanced Treatise, vol. 6A (p. 383). New York: Academic Press. Polgar, L., & Bender, M. L. (1966). J. Am. Chem. Soc., 88, 3153. Pollack, S. J., Jacobs, J. W., & Schultz, P. G. (1986). Science, 234, 1570. Pollegioni, L., Falbo, A., & Pilone, M. S. (1992). Biochim. Biophys. Acta, 1120, 11. Pontremoli, S., Luppis, B., Wood, W. A., et al. (1965). J. Biol. Chem., 240, 3464. Pontremoli, S., Melloni, E., De Flora, A., & Horecker, B. L. (1973). Proc. Natl. Acad. Sci. USA, 70, 661. Pontremoli, S., Melloni, E., Michetti, M., et al. (1982). Proc. Natl. Acad. Sci. USA, 79, 2451. Pope, A. J., Lapointe, J., Mensah, L., et al. (1998a). J. Biol. Chem., 273, 31680, 31691. Pope, A. J., McVey, M., Fantom, K., & Moore, K. J. (1998b). J. Biol. Chem., 273, 31702. Pope, A. J., Moore, K. J., McVey, M., et al. (1998c). J. Biol. Chem., 273, 31691. Popja`k, G. (1970). The Enzymes (3rd ed.), 2, 115. Po¨rschke, D. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 191). Berlin: SpringerVerlag. Porter, D. J. T., & Bright, H. J. (1980). J. Biol. Chem., 255, 4772. Po¨rschke, D. (1998). Biophys. J., 75, 528. Porter, D. J. T., & Bright, H. J. (1980). J. Biol. Chem., 255, 4772. Porterfield, W. W. (1993). Inorganic Chemistry: A Unified Approach. San Diego: Academic Press. Postma, P. W., & Roseman, S. (1976). Biochim. Biophys. Acta, 457, 213. Potier, N., Rogniaux, H., Chevreux, G., & Van Dorsselaer, A. (2005). Meth. Enzymol., 402, 361. Prabhu, K. S., & Ramadoss, C. S. (2000). Indian J. Biochem. Biophys., 37, 6. Pratt, J. M. (1985). Chem. Soc. Rev., 14, 161. Press, W. H., Vetterling, W. T., & Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing. New York: Cambridge University Press. Pre´vost, G. P., Lonchampt, M. O., Holbeck, S., et al. (2006). Cancer Res., 66, 9227. Priestman, M. A., Healy, M. L., Becker, A., et al. (2005). Biochemistry, 44, 3241. Prusiner, S., Davis, J., & Stadtman, E. R. (1976). J. Biol. Chem., 251, 3447. Purich, D. L. (1979). Enzyme Kinetics and Mechanism Subseries, Part A: Initial Rate and Inhibitor Methods. In Methods in Enzymology, vol. 63. San Diego: Academic Press. Purich, D. L. (1980). Enzyme Kinetics and Mechanism Subseries, Part B: Isotopic Probes and Complex Enzyme Systems.

References

In Methods in Enzymology, vol. 64. San Diego, CA: Academic Press. Purich, D. L. (1982). Enzyme Kinetics and Mechanism Subseries, Part C: Intermediates, Stereochemistry, and Rate Studies. In Methods in Enzymology, vol. 87. San Diego, CA: Academic Press. Purich, D. L. (1995). Enzyme Kinetics and Mechanism Subseries, Part D: Developments in Enzyme Dynamics. In Methods in Enzymology, vol. 249. San Diego, CA: Academic Press. Purich, D. L. (1996). Contemporary Enzyme Kinetics and Mechanism. New York: Academic Press. Purich, D. L. (1998). Adv. Enzymol., 72, 9. Purich, D. L. (2001). Trends Biochem. Sci., 26, 417. Purich, D. L. (2002). Meth. Enzymol., 354, 168. Purich, D. L. (2005). In Encyclopedia of Molecular Cell Biology and Molecular Medicine, vol. 11 (p. 27). Weinheim, Germany: Wiley-VCH. Purich, D. L., & Allison, R. D. (1980). Meth. Enzymol., 64, 3. Purich, D. L., & Allison, R. D. (2000). Handbook of Biochemical Kinetics. New York: Academic Press. Purich, D. L., & Allison, R. D. (2002). The Enzyme Reference. New York: Academic Press. Purich, D. L., & Fromm, H. J. (1971). J. Biol. Chem., 246, 3456. Purich, D. L., & Fromm, H. J. (1972a). Arch. Biochem. Biophys., 149, 307. Purich, D. L., & Fromm, H. J. (1972b). Curr. Top Cell. Reg., 6, 131. Purich, D. L., & Fromm, H. J. (1972c). Biochem. J., 130, 63. Purich, D. L., & Fromm, H. J. (1972d). Biochem. Biophys. Res. Communs, 47, 916. Purich, D. L., & Fromm, H. J. (1972e). Biochim. Biophys. Acta, 268, 1. Purich, D. L., & Fromm, H. J. (1972f). Biochim. Biophys. Acta, 276, 563. Purich, D. L., & Fromm, H. J. (1972g). J. Biol. Chem., 247, 249. Purich, D. L., & Fromm, H. J. (1973). J. Biol. Chem., 248, 461. Purich, D. L., & Southwick, F. S. (1996). Biochem. Biophys. Res. Communs., 231, 686. Purich, D. L., & Southwick, F. S. (1999). Meth. Enzymol., 308, 93. Purich, D. L., Fromm, H. J., & Rudolph, F. B. (1973). Adv. Enzymol., 39, 249. Putman, S. J., Coulson, A. F. W., Farley, I. R. T., et al. (1972). Biochem. J., 129, 301. Pyle, A. M. (2002). J. Biol. Inorg. Chem., 7, 679. Purich, D. L., & Kristofferson, D. (1984). Adv. Protein Chem., 36, 133. Qasba, P. K., Ramakrishnan, B. E., & Boeggeman, E. (2005). Trends Biochem. Sci., 30, 53. Qian, H., & Elson, E. L. (2002). Biophys. Chem., 101, 565. Quastel, J. H., & Woolbridge, W. R. (1928). Biochem. J., 22, 689. Quastel, J. H., & Woolf, B. (1926). Biochem. J., 20, 545. Quemeneur, E., Moutiez, M., Charbonnier, J.-B., & Menez, A. (1998). Nature, 391, 301. Racker, E. (1976). A New Look at Mechanisms in Bioenergetics. New York: Academic Press. Racker, E., & Krimsky, I. (1952). J. Biol. Chem., 198, 731. Rader, J. I., & Huennekens, F. M. (1973). The Enzymes, 9, 197, (3rd ed.).

833

Radisky, E. S., & Koshland, D. E., Jr. (2002). Proc. Natl. Acad. Sci. USA, 99, 10316. Radisky, E. S., Lee, J. M., Lu, C.-J. K., & Koshland, D. E., Jr. (2006). Proc. Natl. Acad. Sci. USA, 103, 6835. Radzicka, A., & Wolfenden, R. (1995). Science, 267, 90. Raines, R. T. (1998). Chem. Rev., 98, 1045. Raines, R. T., & Knowles, J. R. (1986). Ann. N.Y. Acad. Sci., 471, 266. Raines, R. T., & Knowles, J. R. (1987). Biochemistry, 26, 7014. Raines, R. T., Sutton, E. L., Straus, D. R., et al. (1986). Biochemistry, 25, 7142. Raison, J. K. (1973). Symposia of the Soc. for Exper. Biol., 27, 485. Rajagopalan, P. T., Zhang, Z., McCourt, L., et al. (2002). Proc. Natl. Acad. Sci. USA, 99, 13481. Ralph, E. C., & Fitzpatrick, P. F. (2005). Biochemistry, 44, 3074. Ralph, E. K., & Grunwald, E. (1969). J. Am. Chem. Soc., 91, 2422. Ramachandra Shastry, M. C., & Roder, H. (1998). Nature Structural Biology, 5, 385. Ramachandra Shastry, M. C., Luck, S. D., & Roder, H. (1998). Biophys. J., 74, 2714. Ramaswamy, S., Eklund, H., & Plapp, B. V. (1994). Biochemistry, 33, 5230. Rando, R. (1974). Biochemistry, 13, 3859. Ranson, N. A., White, H. E., & Saibil, H. R. (1998). Biochem. J., 333, 233. Rao, J. K., Erickson, J. W., & Wlodawer, A. (1991). Biochemistry, 30, 4663. Rao, K. S., & Lederer, F. (1998). Protein Sci., 7, 1531. Rarbach, M., Kettling, U., Kolterman, A., & Eigen, M. (2001). Methods, 24, 104. Ratcliff, G. C., & Erie, D. A. (2001). J. Am. Chem. Soc., 123, 5632. Raushel, F. M., & Cleland, W. W. (1973). J. Biol. Chem., 248, 8174. Raushel, F. M., & Cleland, W. W. (1977). Biochemistry, 16, 2176. Raushel, F. M., Thoden, J. B., & Holden, H. M. (2003). Acc. Chem. Res., 36, 539. Raval, D. N., & Wolfe, R. G. (1962). Biochemistry, 1, 1112. Ray, W. J., Jr., & Hatfield, G. W. (1970). J. Biol. Chem., 245, 1753. Ray, W. J., Jr. (1983). Biochemistry, 22, 4625. Rebholz, K. L., & Northrop, D. B. (1993). Biochem. J., 296, 355. Rebholz, K. L., & Northrop, D. B. (1994). Analyt. Biochem., 216, 285. Rebholz, K. L., & Northrop, D. B. (1995). Meth. Enzymol., 249, 211. Rechsteiner, M., Realini, C., & Ustrell, V. (2000). Biochem. J., 345, 115. Record, M. T., Jr., Zhang, W., & Anderson, C. F. (1998). Adv. Prot. Chem., 51, 281. Rees, D. C. (1985). Proc. Natl. Acad. Sci. USA, 82, 3082. Rees, D. C., & Farrelly, D. (1990). The Enzymes, 19, 38 (3rd ed.). Rees, D. C., Lewis, M., Honzatko, R. B., et al. (1981). Proc. Natl. Acad. Sci. USA, 78, 3408. Rees, H. H., Goad, L. J., & Goodwin, T. W. (1969). Biochim. Biophys. Acta, 176, 892. Reetz, M. T. (2004). Proc. Natl. Acad. Sci. USA, 101, 5716.

834

Reetz, M. T., Zonta, A., Schimossek, K., et al. (1997). Angew. Chem. Int. Ed. Engl., 36, 2830. Regenfuss, P., Clegg, R. M., Fulwyler, M. J., et al. (1985). Rev. Sci. Instrum., 56, 283. Reich, N. O., & Mashhoon, N. J. (1993). J. Biol. Chem., 268, 9191. Reichard, P. (1993). Science, 260, 1773. Reinhart, G. D. (2004). Meth Enzymol., 380, 187. Renard, M., & Fersht, A. R. (1973). Biochemistry, 12, 4713. Rendina, A. R., & Orme-Johnson, W. H. (1978). Biochemistry, 17, 5388. Revel, J. P., & Ball, E. G. (1959). J. Biol. Chem., 234, 577. Reynard, A. M., Hass, L. F., & Boyer, P. D. (1988). J. Biol. Chem., 236, 2277. Reynolds, L. J., Garcia, G. A., Kozarich, J. W., & Kenyon, G. L. (1988). Biochemistry, 27, 5530. Rhee, S., Parris, K. D., Hyde, C. C., et al. (1997). Biochemistry, 36, 7664. Rhoads, D. G., & Lowenstein, J. M. (1968). J. Biol. Chem., 243, 3963. Rich, D. H., & Sun, E. T. O. (1980). Biochem. Pharmacol., 29, 2205. Richard, J. P. (1984). J. Am. Chem. Soc., 106, 4926. Richard, J. P., & Frey, P. A. (1978). J. Am. Chem. Soc., 100, 7757. Richard, J. P., Carr, M. C., Ives, M. H., & Frey, P. A. (1980). Biochem. Biophys. Res. Comm., 94, 1052. Richard, J. P., Prasher, D. C., Ives, M. H., & Frey, P. A. (1979). J. Biol. Chem., 254, 4339. Richards, F. H. (1977). Annu. Rev. Biophys. Bioeng., 82, 151. Richards, F. M., Eisenberg, D. S., Kim, P. S., & Matthews, C. R. (2000). Protein Folding Mechanisms, vol. 53. New York: Advances in Protein Chemistry, Academic Press. Rickert, K. W., & Klinman, J. P. (1999). Biochemistry, 38, 12218. Rieder, S. V., & Rose, I. A. (1959). J. Biol. Chem., 234, 1007. Riggs, A. D., Suzuki, H., & Beecham, J. M. (1970). J. Mol. Biol., 48, 67. Rigler, R., Ehrenberg, M., & Wintermeyer. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 219). Berlin: Springer-Verlag. Rigler, R., & Elson, E. (2001). Fluorescence Correlation Spectroscopy: Theory and Applications. Berlin: Springer. Rigler, R., Fo¨ldes-Papp, Z., Meyer-Almes, F.-J., et al. (1998). J. Biotechnol., 63, 97. Rippe, K. (2000). Biochemistry, 39, 2131. Rishavy, M. A., & Cleland, W. W. (2000). Biochemistry, 39, 4569. Rivera, G. M., Briceno, C. A., Takeshima, F., et al. (2004). Curr. Biol., 14, 11. Rivett, A. J., Mason, G. G., Murray, R. Z., & Reidlinger, J. (1997). Mol. Biol. Rep., 24, 99. Roach, P. J. (1991). J. Biol. Chem., 266, 14139. Robertson, J. G. (2005). Biochemistry, 44, 5561. Robyt, J. F. (1998). Essentials of Carbohydrate Chemistry. New York: Springer. Roca, J., & Wang, J. C. (1994). Cell, 77, 609. Rocchia, W., Alexov, E., & Honig, B. (2001). J. Phys. Chem. B, 105, 6507. Rodbell, M. (1992). Curr. Top. Cellular Reg., 32, 1. Rodbell, M., Birnbaumer, L., Pohl, S. L., & Kraus, H. M. J. (1971). J. Biol. Chem., 246, 1877.

References

Rodgers, J., Femec, D. A., & Schowen, R. L. (1982). J. Amer. Chem. Soc., 104, 3263. Rodnina, M. V., Savelsberg, A., Katunin, V. I., & Wintermeyer, W. (1997). Nature, 385, 37. Roe, S. (2001). Protein Purification Techniques: A Practical Approach (2nd ed.). New York: Oxford University Press. pp. 280. Roepstorff, P. (2000). In P. Jolle`s, & H. Jo¨rnvall (Eds.), Proteomics in Functional Genomics: Protein Structure Analysis (p. 81). Basel: Birkha¨user Verlag. Rohatgi, R., Nollau, P., Ho, H. Y., et al. (2001). J. Biol. Chem., 276, 26448. Romero, S., Le Clainche, C., Didry, D., et al. (2004). Cell, 119, 419. Ronzio, R. A., & Meister, A. (1968). Proc. Natl. Acad. Sci. USA, 59, 164. Roques, B. P., Noble, F., Crine, P., & Fournie´-Zaluski, M.-C. (1995). Meth. Enzymol., 248, 263. Rosanoff, M. A. (1906). J. Am. Chem. Soc., 28, 114. Rose, I. A. (1962). Brookhaven Symp. Biol., 15, 293. Rose, I. A. (1975). Adv. Enzymol., 43, 491. Rose, I. A. (1982). Meth. Enzymol., 87, 84. Rose, I. A. (1995). Meth. Enzymol, 249, 315. Rose, I. A., & Rose, Z. B. (1969). Comprehensive Biochemistry, 17, 126. Rose, I. A., Grunberg-Manago, M., Korey, S. R., & Ochoa, S. (1954). J. Biol. Chem., 211, 737. Rose, I. A., O’Connell, E. L., Litwin, S., & Bar-Tana, J. (1974). J. Biol. Chem., 249, 5163. Rose, M. C., & Stuehr, J. (1971). J. Am. Chem. Soc., 93, 4350. Rosenberry, T. I. (1975). Adv. Enzymol., 43, 103. Rossotti, F. J. C., & Rossotti, H. (1961). The Determination of Stability Constants. New York: McGraw-Hill. Roughton, F. J. W. (1934). Proc. Roy. Soc., B115, 475, 495. Roughton, F. J. W., & Chance, B. (1953). In S. L. Friess, E. S. Lewis, & A. Weissberger (Eds.), Rates and Mechanisms of Reactions (1st ed.). Technique of Organic Chemistry, vol. VIII (p. 669). New York: Interscience Publishers. Roustan, C., Brevet, A., Pradel, L.-A., & van Thoai, N. (1973). Eur. J. Biochem, 37, 248. Roustan, C., Fattoum, A., & Pradel, L. A. (1976). Biochemistry, 15, 2172. Rowe, W. B., Ronzio, R. A., & Meister, A. (1969). Biochemistry, 8, 2674. Roy, J. R., & Abraham, T. E. (2003). Chem. Rev., 104, 3505. Roy, J. R., & Abraham, T. E. (2006). J. Molec. Catalysis B: Enzymatic, 38, 31. Rozovsky, S., & McDermott, A. E. (2007). Proc. Natl. Acad. Sci. USA, 104, 2080. Ruan, K., & Weber, G. (1988). Biochemistry, 27, 3295. Rudnick, G., & Abeles, R. H. (1975). Biochemistry, 14, 4515. Rudolph, F. B. (1979). Meth. Enzymol., 63, 411. Rudolph, F. B., & Fromm, H. J. (1971a). Arch. Biochem. Biophys., 147, 515. Rudolph, F. B., & Fromm, H. J. (1971b). J. Biol. Chem., 246, 2104. Rudolph, F. B., & Fromm, H. J. (1971c). J. Biol. Chem., 246, 6611. Rudolph, F. B., & Fromm, H. J. (1979). Meth. Enzymol., 63, 138.

References

Rudolph, F. B., Baugher, B. W., & Beissner, R. S. (1979). Meth. Enzymol., 63, 22. Rudolph, F. B., Purich, D. L., & Fromm, H. J. (1968). J. Biol. Chem., 243, 5539. Ruiz-Gayo, M., Baamonde, A., Turcaud, S., et al. (1992). Brain Res., 571, 306. Rupert, P. B., Massey, A. P., Sigurdsson, S. T., & FerreD’Amare, A. R. (2002). Science, 298, 1421. Russo, D., Hura, G. L., & Copley, J. R. D. (2007). Phys. Rev. E, 75, 409. Russell, J. M. (2000). Magnesium and the Cell. New York: Academic Press. Ruszczycky, M. W., & Anderson, V. E. (2006). J. Theoret. Biol., 243, 328. Ryan, O., Smyth, M. R., & Fagain, C. O. (1994). Essays Biochem., 28, 129. Ryves, W. J., & Harwood, A. J. (2001). Biochem. Biophys. Res. Commun., 26, 280. Sablin, E. P., Case, R. B., Dai, S. C., et al. (1998). Nature, 395, 813. Sachs, G., Besane`on, M., Shin, J. M., et al. (1992). J. Bioenerg. Biomembr., 24, 301. Sachs, G., Collier, R. H., Shoemaker, R. L., & Hirschowitz, B. I. (1968). Biochim. Biophys. Acta, 162, 210. Saenger, W. (1984). In J. L. Atwood, J. E. Davies, & D. D. MacNicol (Eds.), Inclusion Compounds, vol. 2 (pp. 231– 259). New York: Academic Press. Saibil, H. R., Zheng, D., Roseman, A. M., et al. (1981). J. Biol. Chem., 256, 10456. Saier, M. H., Jr. (2000). Microbiology, 146, 1775. Sakmann, B., & Neher, E. (1995). Single-Channel Recording (2nd ed.). New York: Plenum Press. Sakurai, M., Cook, P. F., Haseman, C. A., & Uyeda, K. (2000). Biochemistry, 39, 16238. Sali, D., Bycroft, M., & Fersht, A. R. (1988). Nature, 335, 741. Salowe, S. P., & Hermes, J. D. (1998). Arch. Biochem. Biophys., 255, 165. Samoilov, M., Plyasunov, S., & Arkin, A. P. (2005). Proc. Natl. Acad. Sci. USA, 102, 2310. Sampson, N. S., & Knowles, J. R. (1992). Biochemistry, 31, 8488. Sanders, A., Purich, D. L., & Cannell, D. S. (1981). J. Mol. Biol., 147, 583. Sanker, S., Campbell, H. A., & Kent, C. (2001). J. Biol. Chem., 276, 37922. Santoro, S. W., & Joyce, G. F. (1997). Proc. Natl. Acad. Sci. USA, 94, 4262. Saric, T., Graef, C. I., & Goldberg, A. L. (2004). J. Biol. Chem., 279, 46723. Sartorius, C., Gerber, M., Zeppezauer, M., & Dunn, M. (1987). Biochemistry, 26, 871. Sasahara, K., McPhie, P., & Minton, A. P. (2003). J. Mol. Biol., 326, 1227. Sauer, R. T., Bolon, D. N., Burton, B. M., et al. (2004). Cell, 119, 9. Sauve, A. A., Munshi, C., Lee, H. C., & Schramm, V. L. (1998). Biochemistry, 37, 13239. Sawa, T., Fukagawa, Y., Homma, I., et al. (1967). J. Antibiot., 204, 227. Scapin, G., Reddy, S. G., Zheng, R., & Blanchard, J. S. (1997). Biochemistry, 36, 15081.

835

Scarano, E., Geraci, G., Polzella, A., & Campanile, E. (1963). J. Biol. Chem., 238, 1556. Scarano, E., Geraci, G., & Rossi, M. (1964). Biochem. Biophys. Res. Comm., 16, 239. Scatchard, G. (1949). Ann. N.Y. Acad. Sci., 51, 660. Scatchard, G., Coleman, J. S., & Shen, A. L. (1957). J. Am. Chem. Soc., 79, 12. Schachman, H. K. (1988). J. Biol. Chem., 263, 18583. Schaeffer, M., van Vlijmen, H. W. T., & Karplus, M. (1998). Adv. Prot. Chem., 51, 1. Schardinger, F. (1905). Centr. Bakt. Parasitenk., 14, 772. Scharschmidt, M., Fisher, M. A., & Cleland, W. W. (1984). Biochemistry, 23, 5471. Schechter, I., & Berger, A. (1967). Biochem. Biophys. Res. Commun., 27, 157. Scheuring, J., & Schramm, V. I. (1997). Biochemistry, 36, 8215. Schimandle, C. M., Mole, L. A., & Sherman, I. W. (1987). Mol. Biochem. Parasitol., 23, 39. Schimerlik, M. I., & Cleland, W. W. (1973). J. Biol. Chem., 248, 8418. Schimerlik, M. I., Rife, J. E., & Cleland, W. W. (1975). Biochemistry, 14, 5347. Schimke, R. T. (1964). J. Biol. Chem., 239, 3808. Schimke, R. T. (1969). Curr. Topics Cell. Reg., 1, 77. Schimke, R. T., Sweeney, E. W., & Berlin, C. M. (1965). J. Biol. Chem., 240, 322 & 4609. Schlichting, I. (2000). Acc. Chem. Res., 33, 532. Schloss, J. V., & Hixon, M. S. (1998). In M. Sinnott (Ed.), Comprehensive Biochemical Catalysis, 2 (p. 43). Schloss, J. V., Porter, D. J. T., Bright, H. J., & Cleland, W. W. (1980). Biochemistry, 19, 2358. Schmid, R. W., & Reilley, C. N. (1957). Analyt. Chem., 29, 264. Schmidt, D. E., Jr., & Westheimer, F. H. (1971). Biochemistry, 10, 1249. Schmidt, D. M. Z., Mundorff, E. C., Dojka, M., et al. (2003). Biochemistry, 42, 8387. Schnapp, B. J., Vale, R. D., Sheetz, M. P., & Reese, T. S. (1985). Cell, 40, 455. Schneider, B., Xu, Y. W., Sellam, O., et al. (1998). J. Biol. Chem., 273, 11491. Schnell, S., & Maini, P. K. (2000). Bull. Math Biol., 62, 483. Schnell, S., & Mendoza, C. (2000). J. Theor. Biol., 187, 207. Schnitzer, M. J., & Block, S. M. (1997). Nature, 388, 386. Schobert, B. (1998). Eur. J. Biochem., 254, 363. Schoenheimer, R. (1942). The Dynamic State of Body Constituents. Cambridge, Massachusetts: Harvard University Press. Schonbaum, G. R., & Chance, B. (1976). The Enzymes, 8, 363 (3rd ed.). Schowen, R. (1977). In W. W. Cleland, M. H. O’Leary, & D. B. Northrop (Eds.), Isotope Effects on Enzyme-Catalyzed Reactions. Baltimore: University Park Press. Schowen, R. (1988). In J. F. Liebmann, & A. Greenburg (Eds.), Mechanistic Principles of Enzyme Activity (p. 119). New York: VCH. Schowen, R., & Schowen, K. (1982). Meth. Enzymol., 87, 551. Schramm, V. L. (1998). Biochim. Biophys. Acta, 1587, 107. Schramm, V. L. (2002). Biochim. Biophys. Acta, 1587, 107. Schramm, V. L. (2003). Acct. Chem. Res., 36, 588. Schramm, V. L. (2005). Curr. Opin. Struct. Biol., 15, 604.

836

Schramm, V. L., & Purich, D. L. (Eds.) (1999). Enzyme Kinetics and Mechanism, Part E: Energetics of Enzyme Catalysis, vol. 249. San Diego, CA: Academic Press. Methods in Enzymology, vol. 249. Schramm, V. L., & Wedler, F. C. (Eds.) (1986). Manganese in Metabolism and Enzyme Activation. Orlando: Academic Press. Schreiber, S. L. (1991). Science, 251, 283. Schroeder, G. K., Lad, C., Wyman, P., et al. (2006). Proc. Natl. Acad. Sci. USA, 103, 4052. Schroer, T. A. (2000). Traffic, 1, 3. Schuller, D. J., Grant, G. A., & Banaszak, L. J. (1995). Nature Struct. Biol., 2, 69. Schulz, G. E. (1992). Faraday Discuss., 93, 85. Schulz, E., Jaenicke, R., & Knoche, W. (1976). Biophys. Chem., 4, 253. Schuster, P., Wolschann, P., & Tortschanoff, K. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 107). Berlin: Springer-Verlag. Schuyler, S. C., & Pellman, D. (2001). Cell, 105, 421. Schwarz, G., & Mendel, R. R. (2006). Annu. Rev. Plant Biol., 57, 623. Schwert, G. W., & Eisenberg, M. A. (1949). J. Biol. Chem., 179, 665. Schwille, P. (2001a). Cell. Biochem. Biophys., 34, 383. Schwille, P. (2001b). In E. L. Elson, & R. Rigler (Eds.), Fluorescence Correlation Spectroscopy: Theory and Applications (p. 360). Berlin: Springer. Schwille, P., Bieschke, J., & Oehlenschla¨ger, F. (1997). Biophys. Chem., 66, 211. Schwille, P., & Kettling, U. (2001). Curr. Opin. Biotech., 12, 382. Schwille, P., Meyer-Almes, F.-J., & Rigler, R. (1997). Biophys. J., 72, 1878. Scopes, R. K. (1993). Protein Purification: Principles and Practice (p. 380) (3rd ed.). Berlin: Springer. Scott, W. G. (1999). Quart. Rev. Biophys., 32, 241. Scott, W. G. (2001). J. Mol. Biol., 311, 989. Seber, G. A. F., & Wild, C. J. (1989). Nonlinear Regression (p. 624). New York: Wiley. Seebeck, F. P., & Hilvert, D. (2003). J. Am. Chem. Soc., 125, 10158. Segal, H. L., & Kim, Y. S. (1963). Proc. Nat. Acad. Sci. USA, 50, 912. Segal, H. L., Kachmar, J. F., & Boyer, P. D. (1952). Enzymologia, 15, 187. Segel, L. A. (1988). Bull. Math. Biol., 50, 579. Segel, L. A., & Slemrod, M. (1989). SIAM Rev., 31, 446. Selden, L. A., Kinosian, H. J., Estes, J. E., & Gershman, L. C. (1999). Biochemistry, 38, 2769. Senior, A. E., Nadanaciva, S., & Weber, J. (2000). J. Exper. Biol., 203, 35. Serrano, L., & Fersht, A. R. (1989). Nature, 342, 296. Serrano, L., Horovitz, A., Avron, B., et al. (1990). Biochemistry, 29, 9343. Serrano, R., Gancedo, J. M., & Gancedo, C. (1973). Europ. J. Biochem., 34, 479. Severin, F. F., Sorger, P. K., & Hyman, A. A. (1997). Nature, 388, 888. Seymour, S. L., & Klinman, J. P. (2002). Biochemistry, 41, 8747. Sha¨fer, G., & Kardinahl, S. (2003). Biochem. Soc. Transact., 31, 1330.

References

Sham, Y. Y., Muegge, I., & Warshel, A. (1998). Biophys. J., 74, 1744. Shan, S. O., & Herschlag, D. (1999). Meth. Enzymol., 308, 246. Shan, S. O., Loh, S., & Herschlag, D. (1996). Science, 272, 97. Sharrocks, A. D., Yang, S. H., & Galanis, A. (2000). Trends Biochem. Sci., 25, 448. Shaw, D. J. (1980). Introduction to Colloid and Surface Chemistry (3rd ed.). London: Butterworth and Co. Shaw, E. (1970). The Enzymes, 1, 91 (3rd ed.). Shechter, I., Sweat, F. W., & Bloch, K. (1970). Biochim. Biophys. Acta, 220, 463. Shen, V., Imamoto, F., Schlessinger, D., & Schlessinger. (1982). J. Bacteriol., 150, 1489. Sherman, H. C., Thomas, A. W., & Baldwin, M. E. (1919). J. Am. Chem. Soc., 41, 231. Sheu, K.-F., Ho, H.-T., Nolan, L. D., et al. (1984). Biochemistry, 23, 1779. Sheu, K.-F., Richard, J. P., & Frey, P. A. (1979). Biochemistry, 18, 5548. Shih, I. H., & Been, M. D. (2001). Proc. Natl. Acad. Sci. USA, 98, 1489. Shill, J. P., & Neet, K. E. (1975). J. Biol. Chem., 250, 2259. Shim, H., Hong, S.-B., & Raushel, F. M. (1998). J. Biol. Chem., 279, 17445. Shimabukuro, K., Yasuda, R., Muneyuki, E., et al. (2003). Proc. Natl. Acad. Sci. USA, 100, 14731. Shimizu, T., & Johnson, K. A. (1983). J. Biol. Chem., 258, 13833. Shimojo, N., Naka, K., Uenoyama, H., et al. (1993). Clin. Chem., 39, 11. Shindell., D. M., Magagnosc, C., & Purich, D. L. (1978). J. Chem. Educ., 55, 708. Shinnzawha-Itoh, K., Nakashima, R., Yaono, R., & Yoshikawa, S. (1995). Science, 269, 1069. Shintaku, M., Matsuura, K., Yoshioka, S., et al. (2005). J. Biol, Chem., 280, 40934. Shivashankar, G. V., Feingold, M., Krichevsky, O., & Libchaber, A. (1999). Proc. Natl. Acad. Sci. USA, 96, 7916. Shivashankar, G. V., & Libchaber, A. (1998). Appl. Phys. Lett., 73, 417. Shoaf, W. T., & Jones, M. E. (1973). Biochemistry, 12, 4039. Shoichet, B. K., McGovern, S. L., Wei, B., & Irwin, J. J. (2002). Current Opinion in Chem. Biol., 6, 439. Shuker, S. B., Hajduk, P. J., Meadows, R. P., & Fesik, S. W. (1996). Science, 274, 1531. Shulman, R. G., Hopfield, J. J., & Ogawa, S. (1975). Quart. Rev. Biophys, 8, 325. Shurki, A., Strajbl, M., Villa`, J., & Warshel, A. (2002). J. Am. Chem. Soc., 124, 4097. Sideraki, V., Wilson, D., Kurz, L., Quiocho, F., & Rudolph, F. (1996). Biochemistry, 35, 15019. Siegel, M. I., Wishnick, M., & Lane, M. D. (1972). The Enzymes, 6, 169, (3rd ed.). Sigel, H., & Sigel, A. (1998). Metal Ions in Biological Systems, vol. 26. New York: Marcel Dekker, Inc. Sigel, H., & Sigel, A. (Eds.) (1995). Vanadium and Its Role in Life. In Metal Ions in Biological Systems, vol. 31. New York: Marcel Dekker. Sigel, R., Vaidya, A., & Pyle, A. M. (2000). Nature Struct. Biol., 7, 1111.

References

Sikorski, R. S., Wang, L., Markham, K. A., et al. (2004). J. Am. Chem. Soc., 126, 4778. Silman, I. M., & Katchalski, E. (1966). Annu. Rev. Biochem., 35, 873. Silverman, D. N. (1982). Meth. Enzymol., 87, 732. Silverman, D. N., & Lindskog, S. (1988). Acc. Chem. Res., 21, 30. Silverman, D. N., & Tu, C. K. (1975). J. Amer. Chem. Soc., 97, 2263. Silverman, D. N., & Vincent, S. H. (1983). Crit. Rev. Biochem., 14, 207. Silverman, R. B. (1988). Mechanism-Based Enzyme Activation: Chemistry and Enzymology (Vols. 1–2). Boca Raton: CRC Press. Silverman, R. B. (1995a). The Organic Chemistry of Drug Design and Discovery. New York: Academic Press. Silverman, R. B. (1995b). Meth. Enzymol., 249, 240. Silverman, R. B. (2000). The Organic Chemistry of EnzymeCatalyzed Reactions (2nd ed.). New York: Academic Press. Silverman, R. B. (2002). The Organic Chemistry of EnzymeCatalyzed Reactions (p. 717). San Diego: Academic Press. Silverman, R. B., Muztar, A. J., Levy, M. A., & Hirsch, J. D. (1983). Life Sci., 32, 2717. Silverman, S. K., & Cech, T. R. (2001). RNA, 7, 161. Silverstein, E., & Boyer, P. D. (1964). J. Biol. Chem., 239, 3901 & 3908. Silverstein, E., & Sulebele, G. (1973). Biochemistry, 12, 2164. Singer, S. J. (1967). Adv. Protein Chem., 22, 1. Singh, A. E., Thornton, E., & Westheimer, F. H. (1962). J. Biol. Chem., 237, PC3007. Sinnott, M. L., & Souchard, I. J. L. (1972). Biochem. J., 133, 89. Sinnott, M. L., & Viratelle, O. M. (1972). Biochem. J., 133, 81 (143), 751. Sinnott, M. L. (Ed.) (1998). Comprehensive Biological Catalysis: A Mechanistic Reference, vols. I–IV. San Diego: Academic Press. Sinnott, M. L., & Jencks, W. P. (1980). J. Am. Chem. Soc., 102, 2026. Sinnott, M. L., Garner, C. D., First, E., & Davies, G. (1998). The Lexicon of Terms and Concepts in Mechanistic Enzymology. In Comprehensive Biological Catalysis: A Mechanistic Reference, Vol. 4 (pp. 77–79). San Diego: Academic Press. Skarstedt, M. T., & Silverstein, E. (1976). J. Biol. Chem., 251, 6775. Skoog, D. A., Holler, F. J., & Nieman, T. A. (1998). Principles of Instrumental Analysis Saunders College Publishing (pp. 253– 271) (5th ed.). Philadelphia: Harcourt-Brace. Skou, J. C. (1957). Biochim. Biophys. Acta, 23, 394. Skou, J. C. (1960). Biochim. Biophys. Acta, 42, 6. Slama, J. T., Radziejewski, C., Oruganti, S. R., & Kaiser, E. T. (1984). J. Am. Chem. Soc., 106, 6778. Sleep, J., Herrmann, C., Barman, T., & Travers, F. (1994). Biochemistry, 33, 6038. Smith, C. A., & Rayment, I. (1996). Biochemistry, 35, 5404. Smith, E. L. (1951). Adv. Enzymol., 12, 191. Smith, E. L., Davis, N. C., Davis, E., & Spackman. (1954). The Mechanism of Enzyme Action. In W. D. McElroy & B. Glass (Eds.) (p. 291). Baltimore, Maryand: Johns Hopkins Press. Smith, F. R., & Ackers, G. K. (1985). Proc. Natl. Acad. Sci. USA, 50, 5347.

837

Smith, H. T., Staudenmayer, N., & Millett, F. (1977). Biochemistry, 16, 4971. Smith, L. E. H., Mohr, L. H., & Raftery, M. A. (1973). J. Am. Chem. Soc., 95, 7497. Smith, M. (1997). Synthetic DNA and biology. In B. G. Malmstro¨m (Ed.), Novel Lectures, Chemistry (pp. 1991– 1995). Singapore: World Scientific Publishing. Smoluchowski, M. (1916). Phys. Z., 17, 557. Snyder, E. E., Buoscio, B. W., & Falke, J. J. (1990). Biochemistry, 29, 3937. Sols, A., & Marco, R. (1970). Curr. Top. Cellular Reg., 2, 173. Sonesson, A. W., Elofsson, U. M., Brismar, H., & Callisen, T. H. (2006). Langmuir, 22, 5810. Song, S., Jahansouz, H., & Himes, R. H. (1993). FEBS Lett., 332, 150. Song, Y. H., Zhang, Y. J., Shen, T. Y., et al. (2004). Biophys. J., 86, 2017. Spector, L. B. (1980). Proc. Natl. Acad. Sci. USA, 77, 2626. Sprang, S., Standing, T., Fletterick, R. J., et al. (1987). Science, 237, 905. Srere, P. A. (1985). Trends Biochem. Sci., 10, 109. Srivastava, D. K., & Bernhard, S. A. (1984). Biochemistry, 23, 4538. Srivastava, D. K., & Bernhard, S. A. (1985). Biochemistry, 24, 623. Srivastava, D. K., & Bernhard, S. A. (1986). Science, 234, 1081. Srivastava, D. K., Smolen, P., Betts, G. F., et al. (1989). Proc. Natl. Acad. Sci. USA, 86, 6464. Stadtman, E. R. (1966). Adv. Enzymol., 28, 76. Stadtman, E. R. (2001). J. Biol. Chem., 276, 44357. Stadtman, E. R., & Chock, P. B. (1977a). Proc. Natl. Acad. Sci. USA, 74, 2761. Stadtman, E. R., & Chock, P. B. (1977b). Proc. Natl. Acad. Sci. USA, 74, 2766. Stadtman, E. R., & Chock, P. B. (1978). Cum. Top. Cell. Regul., 13, 53. Stadtman, E. R., & Ginsburg, A. (1974). The Enzymes, 10, 755, (3rd ed.). Stadtman, T. C., & Elliott, P. (1957). J. Biol. Chem., 228, 983. Stage-Zimmermann, T. K., & Uhlenbeck, O. C. (1998). RNA, 4, 875. Sta¨helin, H. F. (1996). Experientia, 52, 5. Stanford, N. P., Szczelkun, M. D., Marko, J. F., & Halford, S. E. (2000). EMBO J., 19, 6546. Staros, J. V., Bayley, H., Standring, D. N., & Knowles, J. R. (1978). Biochem. Biophys. Res. Commun., 80, 568. Stayton, M. M., & Fromm, H. J. (1979). J. Theor. Biol., 78, 309. Steinfeld, J. I., Francisco, J. S., & Hase, W. L. (1999). Chemical Kinetics and Dynamics (2nd ed.). Upper Saddle River, New York: Prentice-Hall. Steinfeld, J. I., & Hammes, G. G. (1963). J. Phys. Chem., 67, 528. Steinman, H. M. (1985). J. Bacteriol., 162, 1255. Stevens, L. (1992). In R. Eisenthal, & M. J. Danson (Eds.), Enzyme Assays: A Practical Approach. Oxford: IRL Press. Stevens, R. C., & Lipscomb, W. N. (1992). Proc. Natl. Acad. Sci. USA, 89, 5281. Stewart, M., & Roberts, T. M. (2005). Adv. Protein Chem., 71, 383. Steyaert, J., & Wyns, L. (1993). J. Mol. Biol., 229, 770.

838

Steyaert, J., Hallenga, K., Wyns, L., & Stanssens, P. (1990). Biochemistry, 29, 9064. Stiles, J. R., & Bartol, T. M. (2000). In E. D. Schutter (Ed.), Computational Neuroscience: Realistic Modeling for Experimentalists. New York: CRC Press. Stoddard, B. L. (2001). Methods, 24, 125. Stoddard, B. L., Dean, A., & Bash, P. A. (1996). Nat. Struct. Biol., 3, 590. Storer, A., & Cornish-Bowden, A. (1974). Biochem. J., 141, 205. Storm, D. R., & Koshland, D. E., Jr. (1970). Proc. Natl. Acad. Sci. USA, 66, 445. Stoscheck, C. M. (1990). Meth. Enzymol., 182, 50. Strassman, M., Thomas, A., & Weinhouse, S. (1953). J. Am. Chem. Soc., 75, 5135. Strassman, M., Thomas, A., & Weinhouse, S. (1955). J. Am. Chem. Soc., 77, 1261. Straub, J. E., & Karplus, M. (1990). Prot. Engineering, 3, 673. Straus, D., Raines, R., Kawashima, E., et al. (1985). Proc. Natl. Acad. Sci. USA, 82, 2272. Strauss, O. H., & Goldstein, A. (1943). J. Gen. Physiol., 26, 559. Strawser, L. D., & Larrabee, A. R. (1976). J. Biol. Chem., 251, 720. Strickland, S., Palmer, G., & Massey, V. (1975). J. Biol. Chem., 250, 4048. Strittmatter, P. (1964). J. Biol. Chem., 240, 4481. Stryer, L. (1978). Annu. Rev. Biochem., 47, 819. Stubbe, J. (1990a). Adv. Enzymol., 63, 349. Stubbe, J. (1990b). J. Biol. Chem., 265, 5329. Stubbe, J., Fish, S., & Abeles, R. H. (1980). J. Biol. Chem., 255, 236. Stuehr, J. (1974). Investigation of Rates and Mechanisms of Reactions, Part II. In Techniques of Chemistry, Vol. 6 (p. 237). New York: Wiley-Intercience. Sugimoto, E., & Pizer, L. I. (1968a). J. Biol. Chem., 243, 2081. Sugimoto, E., & Pizer, L. I. (1968b). J. Biol. Chem., 243, 2090. Sullivan, S. M., & Holyoak, T. (2008). Proc. Natl. Acad. Sci. USA, 105, 13829. Sumner, J. B. (1926). J. Biol. Chem., 69, 435. Sun, D. P., Liao, D. I., & Remington, S. J. (1989). Proc. Natl. Acad. Sci. USA, 86, 5361. Sun, J., & Sampson, N. S. (1998). Protein Sci., 7, 1495. Sun, S., Bagdassarian, C. K., & Toney, M. D. (1998). Biochemistry, 37, 3876. Sun, S., Zabinski, R. F., & Toney, M. D. (1998). Biochemistry, 37, 3869. Sundaram, P. V., Tweedale, A., & Laidler, K. J. (1970). Can. J. Chem., 48, 1498. Svensson, R., Greno, C., Johansson, A. S., et al. (2002). Anal. Biochem., 311, 171. Swain, C. G., Stivers, E. C., Reuwer, J. F., Jr., & Schaad, L. J. (1958). J. Am. Chem. Soc., 80, 5885. Switzer, R. L., & Simcox, P. D. (1974). J. Biol. Chem., 249, 5304. Synge, E. H. (1928). Phil. Mag., 6, 356. Szabo, A., & Karplus, M. (1972). J. Mol. Biol., 72, 163. Szabo, A., Shoup, D., Northrup, S. H., & McCammon, J. A. (1982). J. Phys. Chem., 77, 4484. Szedlacsek, S. E., & Duggleby, R. G. (1995). Meth. Enzymol., 249, 144.

References

Szedlacsek, S. E., Ostafe, V., Duggleby, R. G., et al. (1990). Biochem. J., 265, 653. Szent-Gyo¨rgyi, A. (1939). On Oxidation, Fermentation, Vitamins, Health and Disease. Baltimore, Florida: William and Wilkins. Szorenyi, E. T., Dvornkiova, P. D., & Degtyar, P. G. (1949). C.R. Acad. Sci. U.R.S.S., 67, 341. Tan, G. O., Ensign, S. A., Ciurli, S., et al. (1992). Proc. Natl. Acad. Sci. USA, 89, 4427. Tanford, C. (1962). J. Amer. Chem. Soc., 84, 4240. Tanford, C. (1980). The Hydrophobic Effect: Formation of Micelles and Biological Membranes (2nd ed.). New York: Wiley. Tanford, C., & Kirkwood, J. C. (1957). J. Am. Chem. Soc., 79, 5340. Tang, L., Lutje-Spelberg, J. H., Fraaije, M. W., & Janssen, D. B. (2003). Biochemistry, 42, 5378. Tanner, M. E., & Kenyon, G. L. (1998). Comprehensive Biochemical Catalysis, 2, 7. Tanner, K. G., Landry, J., Sternglanz, R., & Denu, J. M. (2000). Proc. Natl. Acad. Sci. USA, 97, 14178. Tanner, M. E., Gallo, K. A., & Knowles, J. R. (1993). Biochemistry, 32, 3998. Taoka, S., Tu, C., Kistler, K. A., & Silverman, D. N. (1994). J. Biol. Chem., 269, 17988. Taraszka, M. (1962). Ph.D. Dissertation. Madison: Univ. of Wisconsin. Tate, S. S., & Meister, A. (1978). Proc. Natl. Acad. Sci. USA, 75, 4806. Taube, H. (1959). J. Can. Chem., 37, 129. Tauber, H. (1943). The Chemistry and Technology of Enzymes. New York: J. Wiley. Tauler, A., Lin, K., & Pilkis, S. J. (1990). J. Biol. Chem., 265, 15617. Tawfik, D. S., & Griffiths, A. D. (1998). Nature Biotechnol., 16, 652. Tayhas, G., Palmore, R., & Whitesides, G. M. (1994). Enzymatic Conversion of Biomass for Fuels Production (271), ACS Symposium Series No. 566, Chapter 14. Taylor, J. D., & Halford, S. E. (1989). Biochemistry, 28, 6198. Taylor, J. D., Badcoe, I. G., Clarke, A. R., & Halford, S. E. (1991). Biochemistry, 30, 8743. Taylor, R. F., & Schultz, J. S. (1996). Handbook of Chemical and Biological Sensors (p. 604). Boca Raton, Florida: CRC Press. Taylor-Ringia, E. A., Garrett, J. B., Thoden, J. B., et al. (2004). Biochemistry, 43, 224. Tcherkez, G. G. B., Farquhar, G. D., & Andrews, T. J. (2006). Proc. Natl. Acad. Sci. USA, 103, 7346. Teague, W. E., Jr., & Dobson, G. P. (1992). J. Biol. Chem., 267, 14084. Teague, W. E., Jr., & Dobson, G. P. (1999). J. Biol. Chem., 274, 22459. Teilum, K., Maki, K., Kragelund, B. B., et al. (2002). Proc. Natl. Acad. Sci. USA, 99, 9807. Teipel, J. W., Hass, G. M., & Hill, R. L. (1999). J. Biol. Chem., 243, 5684. Teplyakov, A., Obmolova, G., Badet, B., & Badet-Denisot, M.-A. (2001). J. Mol. Biol., 313, 1093. Terry, B. J., Jack, W. E., & Modrich, P. (1985). J. Biol. Chem., 260, 13130. Tewari, Y. B., & Goldberg, R. N. (1988). Biophys. Chem., 29, 245.

References

Thakur, A. K., Rescigno, A., & DeLisi, C. (1978). J. Phys. Chem., 82, 552. Theeuwes, F., & Yum, S. I. (1976). Ann. Biomed. Eng., 4, 343. Theorell, H., & Chance, B. (1951). Acta Chem. Scand., 5, 1127. Theorell, H., & McKee, J. S. M. (1961). Acta Chem. Scand., 15, 1797, 1811 & 1824. Thielking, V., Alves, J., Fliess, A., et al. (1990). Biochemistry, 29, 4682. Thoma, J. A., & Koshland, D. E., Jr. (1960). J. Am. Chem. Soc., 82, 3329. Thomas, C. A., Jr. (1956). J. Amer. Chem. Soc., 78, 1861. Thompson, G. A., & Meister, A. (1977). J. Biol. Chem., 252, 6792. Thomson, J. B., Tuschi, T., & Eckstein, F. (1996). Nucleic Acids and Molecular Biology, 10, 173. Thomson, J. F. (1960). Arch. Biochem. Biophys., 90, 1. Thorn, S. N., Daniels, R. G., Auditor, M.-T. M., & Hilvert, D. (1995). Nature, 373, 228. Thornberry, N. A., Bull, H. G., Taub, D., et al. (1991). J. Biol. Chem., 266, 21657. Thornton, E. K., & Thornton, E. R. (1978). In R. D. Gandour, & R. L. Schowen (Eds.), Transition States of Biochemical Processes (pp. 3–76). New York: Plenum Press. Thornton, J. M. (1981). J. Mol. Biol., 151, 261. Thorslund, A., & Lindskog, S. (1967). Eur. J. Biochem., 3, 117. Thusius, D. (1977). In I. Pecht, & R. Rigler (Eds.), Chemical Relaxation in Molecular Biology (p. 339). Berlin: SpringerVerlag. Tilney, L. G., & Inoue´, S. (1982). J. Cell Biol., 93, 820. Timasheff, S. N. (1993). Annu. Rev. Biophys. Biomol. Struct., 22, 67. Timasheff, S. N. (1998). Adv. Prot. Chem., 51, 355. Timasheff, S. N. (2002). Proc. Natl. Acad. Sci. USA, 99, 9721. Tinoco, I., Jr., Li, P. T., & Bustamante, C. (2006). Quart. Rev. Biophys, 39, 325. Tipton, K. F., & Dixon, H. B. F. (1979). Meth. Enzymol., 63, 183. Tirnauer, J. S., & Bierer, B. E. (2000). J. Cell Biol., 149, 761. Tirnauer, J. S., Canman, J. C., Salmon, E. D., & Mitchison, T. J. (2002a). Mol. Biol. Cell, 13, 4308. Tirnauer, J. S., Grego, S., Salmon, E. D., & Mitchison, T. J. (2002b). Mol. Biol. Cell, 13, 3614. Tobe´, S., Heams, T., Vergne, J., Herve´1, G., & Maurel, M.-C. (2005). Nucleic Acids Res., 33, 2557. Todd, A. E., Orengo, C. A., & Thornton, J. M. (2002). Trends Biochem. Sci., 27, 419. Todd, M. J., & Gomez, J. (2001). Anal. Biochem., 296, 179. Todhunter, J. A., & Purich, D. L. (1975). J. Biol. Chem., 250, 3505. Todhunter, J. A., Reichel, K. B., & Purich, D. L. (1976). Arch. Biochem. Biophys., 174, 120. Tolbert, N. E. (1981). Ann. Rev. Biochem., 50, 133. Tollin, G., Meyer, T. E., & Cusanovich, M. A. (1986). Biochim. Biophys. Acta, 853, 29. Tomkins, G. M., Yielding, K. L., Talal, N., & Curran, J. F. (1963). Cold Spring Harbor Symp. Quant. Biol., 28, 461. Tomkins, G. M., Yielding, K. L., Curran, J. F., et al. (1965). J. Biol. Chem., 240, 3793. Toney, M. D., & Kirsch, J. F. (1989). Science, 243, 1485. Tormos, J. R., Wiley, K. L., Seravalli, J., et al. (2005). J. Am. Chem. Soc., 127, 14538.

839

Torres, R. A., Schiott, B. S., & Bruice, T. C. (1999). J. Am. Chem. Soc., 121, 8164. Torrey, H. C. (1955). Phys. Rev., 104, 563. Totrov, M., & Abagyan, R. (2008). Curr. Opin. Struct. Biol., 18, 178. Tower, D. B. (1969). In H. H. Jasper, A. A. Ward, & A. Pope (Eds.), Basic Mechanisms of the Epiilepsies (p. 611). Boston: Little and Brown. Toyoshima, Y. Y., Kron, S. J., McNally, E. M., et al. (1987). Nature, 328, 536. Toyoshima, C., Nakasako, M., Nomura, H., & Ogawa, H. (2000). Nature, 405, 647. Tramontano, A., Janda, K. D., & Lerner, R. A. (1986). Science, 234, 1566. Traniello, S., Melloni, E., Pontremoli, S., et al. (1972). Arch. Biochem. Biophys, 149, 222. Traniello, S., Pontremoli, S., Tashima, Y., & Horecker, B. L. (1971). Arch. Biochem. Biophys., 146, 161. Trautman, J. K., & Ambrose, W. P. (1997). In T. Basche, W. E. Moerner, M. Orrit, & U. P. Wild (Eds.), Single Optical Detection, Imaging, and Spectroscopy (p. 191). Weinheim: VCH. Trentham, D. R. (1977). Biochem. Soc. Trans., 5, 5. Trentham, D. R., Corrie, J. E. T., & Reid, G. P. (1992). Biophys. J., 61, A295. Truhlar, D. G., Garrett, B. C., & Klippenstein, S. J. (1996). J. Phys. Chem., 100, 12771. Tsai, C. S., Burgett, M. W., & Reed, L. J. (1973). J. Biol. Chem., 248, 8348. Tsai, M.-D., & Chang, T. T. (1980). J. Am. Chem. Soc., 102, 5416. Tsukamoto, T., Miura, S., Nakai, T., et al. (1995). Nat. Genet., 11, 395. Tsukihara, T., Aoyama, H., Yamashita, E., et al. (1996). Science, 272, 1136. Tsukihara, T., Shimokata, K., Katayama, Y., et al. (2003). Proc. Natl. Acad. Sci. USA, 100, 15304. Tu, C. K., Silverman, D. N., Forsman, C., et al. (1989). Biochemistry, 28, 7913. Turro, N. J. (1979). Modern Molecular Photochemistry. Menlo Park, California: Benjamin-Cummings Publ. Co. Tweto, J., Dehlinger, P., & Larrabee, A. R. (1972). Biochem. Biophys. Res. Commun., 251, 720. Tyuma, I., Imai, K., & Shimizu, K. (1973). Biochemistry, 12, 1491. Tzafriri, A. R., & Edelman, E. R. (2004). J. Theor. Biol., 226, 303. Udenfriend, S. (1962). Fluorescence Assay in Biology and Medicine, Chapter 7. New York: Academic Press. Udgaonkar, J. B., & Hess, G. P. (1987). Biophys. J., 52, 873. Uhl, V., Pilarczyk, G., & Greulich, K. O. (1998). Biol. Chem., 379, 1175. Uhlin, U., & Eklund, H. (1994). Nature, 370, 533. Ullrich, V., & Duppel, W. (1975). The Enzymes, 12, 253, (3rd ed.). Umbarger, H. E., & Brown, B. (1957). J. Bact., 73, 105. Umbarger, H. E., & Brown, B. (1958). J. Biol. Chem., 233, 415. Umezawa, H., Aoyagi, T., Morishima, H., et al. (1970). J. Antibiot., 23, 259. Unrau, P. J., & Bartel, D. P. (1998). Nature, 395, 260–263. Unrau, P. J., & Bartel, D. P. (2003). Proc. Natl. Acad. Sci. USA, 100, 15393.

840

Urban, P., & Lederer, F. (1984). Eur. J. Biochem., 144, 345. Urquhart, J., Fara, J., & Willis, K. L. (1984). Ann. Rev. Pharmacol. and Toxicol., 24, 99. Usher, D. A., Richardson, D. I., Jr., & Eckstein, F. (1970). Nature, 228, 663. Usher, K. C., Remington, S. J., Martin, D. P., & Drueckhammer, D. G. (1994). Biochemistry, 33, 7753. Utter, M. F., Barden, R. E., & Taylor, B. L. (1975). Adv. Enzymol., 42, 1. Uyeda, K., & Racker, E. (1965). J. Biol. Chem., 240, 4682. Uyemura, D., Bambara, R. A., & Lehman, I. R. (1975). J. Biol. Chem., 250, 8577. Vagelos, P. R., Alberts, A. W., & Martin, D. B. (1963). J. Biol. Chem., 238, 533. Vale, R. D. (2000). J. Cell Biol., 150, F13. Valente, A. P., Miyamoto, C. A., & Almeida, F. C. (2006). Curr. Med. Chem., 13, 3697. Vallee, B. L., & Falchuk, K. H. (1993). Phys. Rev., 73, 79. Vallee, B. L., & Galdes, A. (1984). Adv. Enzymol., 56, 283. Vallee, B. L., & Williams, R. J. P. (1968). Chem. in Britain, 4, 397. Van Beers, E. H., Bu¨ller, H. A., Grand, R. J., et al. (1995). Crit. Rev. Biochem. Mol. Biol., 30, 197. Van Berkel, G. J. (2000). J. Mass Spectrom., 35, 773. van den Broek, B., Noom, M. C., & Wuite, G. J. L. (2005). Nucleic Acids Res., 33, 2676. van den Burg, B. (2003). Curr. Opin. Microbiol., 6, 213. van den Heuvel, R. H. H., Ferrari, D., Bossi, R. T., et al. (2002). J. Biol. Chem., 277, 24579. Van Hall, T., Sijts, A., Camps, M., et al. (2000). J. Exp. Med., 192, 483. Van Holde, K. E., & Baldwin, R. L. (1958). J. Phys. Chem., 62, 734. van Kampen, N. G. (1992). Stochastic Processes in Physics and Chemistry. Amsterdam: Elsevier. van Kooten, O., Snel, J. F. H., & Vredenberg, W. J. (1986). Photosynth. Res., 9, 211. Van Slyke, D. D., & Zacharias, G. (1914). J. Biol. Chem., 19, 181. van Tilbeurgh, H., Sarda, L., Verger, R., & Cambillau, C. (1992). Nature, 359, 159. Vanhooke, J. L., Benning, M. M., Raushel, F. M., & Holden, H. M. (1996). Biochemistry, 35, 6020. Vas, M., Berni, R., Mozzarelli, A., et al. (1979). J. Biol. Chem., 254, 8480. Vasquez-Vivar, J., Martasek, P., Hogg, N., et al. (1999). Meth. Enzymol, 301, 169. Vavylonis, D., Kovar, D. R., O’Shaughnessy, B., & Pollard, T. D. (2006). Molec. Cell, 21, 455. Veech, R. L., Lawson, J. W., Cornell, N. W., & Krebs, H. A. (1979). J. Biol. Chem., 254, 6538. Veerman, J. A., Garcia-Parajo, M. F., Kuipers, L., & van Hulst, N. F. (1999). Phys. Rev. Lett., 20, 2155. Velazquez-Campoy, A., Muzammil, S., Ohtaka, H., et al. (2003). Curr. Drug Target – Infectious Disorders, 3, 311. Veres, Z., Kim, I. Y., Scholz, T. D., & Stadtman, T. C. (1994). J. Biol. Chem., 269, 10597. Verger, R., Rietsch, J., & Desnuelle, P. (1977). J. Biol. Chem., 252, 4319. Vermeulen, W. P., Briede, J. J., & Rolofsen, B. (1996). Mol. Membr. Biol., 13, 95.

References

Viadiu, H., & Aggarwal, A. K. (2000). Mol. Cell, 5, 889. Vick, J. E., Schmidt, D. M. Z., & Gerlt, J. A. (2005). Biochemistry, 44, 11722. Vihinen, M., & Mantsala, P. (1989). Crit. Rev. Biochem. Mol. Biol., 24, 329. Viitanen, V. P., Lubben, T. H., Reed, J., Goloubinoff, P., O’Keefe, P. D., & Lorimer, G. H. (1990). Biochemistry, 29, 5665. Viitanen, P. V., Lubben, T. H., Reed, J., et al. (1992). Protein Sci., 1, 363. Villa´, J., & Warshel, A. (2001). J. Phys. Chem. B, 33, 7887. Villa´, J., Strajbl, M., Glennon, T. M., et al. (2000). Proc. Natl. Acad. Sci. USA, 97, 11899. Villafranca, J. E., Howell, E. E., Oakley, S. J., et al. (1987). Biochemistry, 26, 2182. Villafranca, J. J. (1982). Meth. Enzymol., 82, 180. Villafranca, J. J. (1989). Meth. Enzymol., 177, 390. Villafranca, J. J., & Nowak, T. (1992). The Enzymes, 20, 63, (D. S. Sigman, ed.). Vilter, H. (1984). Phytochemistry, 23, 1387. Viola, R. E. (2000). Adv. Enzymol., 74, 295. Viola, R. E., & Cleland, W. W. (1982). Meth. Enzymol., 87, 353. Viola, R. E., Morrison, J. F., & Cleland, W. W. (1980). Biochemistry, 19, 3131. Viratelle, O. M., & Seydoux, F. J. (1975). J. Mol. Biol., 92, 193. Virtanen, J. A., Cheng, K. H., & Somerharju, P. (1998). Proc. Natl. Acad. Sci. USA, 95, 4964. Vocadlo, D. J., Davies, G. J., Laine, R., & Withers, S. G. (2001). Nature, 412, 835. Voet, D., & Voet, J. G. (2005). Biochemistry (3rd ed.). New York: J. Wiley. Volkenstein, M. V., & Goldstein, B. N. (1966). Biochim. Biophys. Acta, 115, 471. von Hippel, P. H. (1998). Science, 281, 660. Von Hungen, K., Mahler, H. R., & Moore, W. J. (1968). J. Biol. Chem., 243, 1415. Wade, R. C., Davis, M. E., Luty, B. A., et al. (1993). Biophys. J., 64, 915. Wagner, J., Lerner, R. L., & Barbas, C. F., 3rd (1995). Science, 270, 1797. Wald, G. (1965). Science, 150, 1028. Waley, S. G., Mill, J. C., Rose, I. A., & O’Connell, E. L. (1970). Nature, 227, 181. Walker, J. W., Reid, G. P., McCray, J. A., & Trentham, D. R. (1988). J. Am. Chem. Soc., 110, 7170. Wallon, G., Kryger, G., Lovett, S. T., et al. (1997). J. Mol. Biol., 266, 1016. Walsh, C. T. (1979). Enzymatic Reaction Mechanisms. San Francisco, CA: Freeman Publications. Walsh, C. T. (2006). Posttranslational Modifications of Proteins: Expanding Nature’s Inventory (p. 576). New York: Roberts and Co. Walsh, C. T., Jr. (1980). Accts. Chem. Res., 13, 148. Walsh, C. T., Jr. (1993). Science, 261. 308 and 262, 164. Walsh, C. T., Jr., & Spector, L. B. (1971). Arch. Biochem. Biophys., 145, 1. Walsh, J. D., & Miller, A. F. (2003). J. Molec. Struct. (Theor. Chem), 623, 185.

References

Walter, C. (1974). J. Theoret. Chem., 44, 1. Walters, C., Errington, N., Rowe, A. J., & Harding, S. E. (2002). Biochem. J., 364, 849. Wang, C. H., Willis, D. L., & Loveland, W. D. (1975). Radiotracer Methodology in the Biological, Environmental, and Physical Sciences. Englewood Cliffs, New Jersey: PrenticeHall. Wang, J. C. (1998). Q. Rev. Biophys., 31, 107. Wang, M. D., Schnitzer, M. J., Yin, H., et al. (1998). Science, 282, 902. Wang, R., Chait, B. T., Wolf, I., et al. (1999). Biochemistry, 38, 14573. Wang, S. X., Mure, M., Medzihradszky, K. F., et al. (1996a). Science, 273, 1078. Wang, W., Popov, V. V., Minikawa, N., et al. (1996b). Biochemistry, 35, 95. Wang, Y., Zhang, Y., & Ha, Y. (2006). Nature, 444, 179. Wang, Z. X. (1993). Analyt. Biochem., 213, 370. Warburg, O., & Christian, W. (1941). Biochem. Z., 310, 384. Ward, R. L. (1969). Biochemistry, 8, 1879. Warner, T. G., & Dennis, E. A. (1975). J. Biol. Chem., 250, 8004. Warnock, D. E., & Schmid, S. L. (1996). BioEssays, 18, 885. Warren, S., Zerner, B., & Westheimer, F. H. (1966). Biochemistry, 5, 817. Warshel, A. (1978). Proc. Natl. Acad. Sci. USA, 75, 5250. Warshel, A. (1981). Biochemistry, 20, 3167. Warshel, A. (1991). Computer Modeling of Chemical Reactions in Enzymes and Solutions. New York: J. Wiley. Warshel, A. (2003). Ann. Rev. Biophys. Biomol. Struct., 32, 425. Warshel, A., & Floria´n, J. (1998). Proc. Natl. Acad. Sci. USA, 95, 5950. Warshel, A., & Levitt, M. (1976). J. Mol. Biol., 193, 227. Warshel, A., & Papazyan, A. (1996). Proc. Natl. Acad. Sci. USA, 93, 13665. Warshel, A., & Russell, S. T. (1984). Q. Rev. Biophys., 17, 283. Warshel, A., Sharma, P., Mitsunori, K., et al. (2006). Chem. Rev., 106, 3210. Warwicker, J., & Watson, H. C. (1982). J. Mol. Biol., 157, 671. Wasserman, S. A., & Cozzarelli, N. R. (1986). Science, 232, 951. Watanabe, Y., & Groves, J. T. (1992). The Enzymes, 20, 405, (3rd ed.). Waters, M. J., & Duggleby, R. G. (1993). J. Recept. Res., 13, 1135. Watts, D. C. (1973). The Enzymes, 8, 384 (3rd ed.). Webb, M. R. (1992). Proc. Natl. Acad. Sci. USA., 89, 4884. Webb, M. R. (2003). In K. A. Johnson (Ed.), Kinetic Analysis of Macromolecules (p. 131). New York: Oxford University Press. Webb, M. R., & Trentham, D. R. (1980). J. Biol. Chem., 255, 8629. Webb, M. R., & Trentham, D. R. (1981). J. Biol. Chem., 256, 1775. Weber, G. (1957). Nature, 180, 1409. Weber, G. (1972). Biochemistry, 11, 864. Weber, J., Bowman, C., & Senior, A. E. (1996). J. Biol. Chem., 271, 18711. Weber, J., Dunn, S. D., & Senior, A. E. (1999). J. Biol. Chem., 274, 19124. Weber, J., & Senior, A. E. (2000). Biochim. Biophys. Acta, 1458, 300.

841

Weber, J., Wilke-Mounts, S., & Senior, A. E. (1994). J. Biol. Chem., 269, 20462. Weber, J., Wilke-Mounts, S., Hammond, S. T., & Senior, A. E. (1998). Biochemistry, 37, 12042. Weber, J., Wilke-Mounts, S., Lee, R. S. F., et al. (1993). J. Biol. Chem., 268, 20126. Weber, W., Helms, V., McCammos, J. A., & Langhoff, P. W. (1999). Proc. Natl. Acad. Sci. USA, 96, 6177. Wechselmann, W. (1913). The Pathogenesis of Salvarsan Fatalities. Saint Louis, USA: Fleming-Smith Publishers. (translated by C. Martin). Wedekind, J. E., Reed, G. H., & Rayment, I. (1995). Biochemistry, 34, 4325. Wei, Y., Lin, J., & Frey, P. A. (2001). Biochemistry, 40, 11279. Weikl, T. R., & von Deuster, C. (2009). Proteins, 75, 104. Weil, M. H., & Afifi, A. A. (1970). Circulation, 41, 989. Weiss, P. M., Garcia, G. A., Kenyon, G. L., et al. (1988). Biochemistry, 27, 2197. Weiss, S. (1999). Science, 283, 1676. Wenner, J. R., & Bloomfield, V. A. (1999). Biophys J, 77, 3234. Werkheiser, W. C. (1976). Biochemistry, 15, 5024. Werner, A. (1893). Classics in Coordination Chemistry. In: Z. Anorg. Chem., 3. New York: Dover Publications. 267 (Translated in Kauffman, G.B. (1968), Part 1, pp. 9–88. Werth, B. (1994). The Billion Dollar Molecule: One Company’s Quest for the Perfect Drug (p. 445). New York: Simon and Schuster. Westerhoff, H. V. (2008). J. Theoret. Biol., 252, 555. Westheimer, F. H. (1968). Accts. Chem Res., 1, 61. Westheimer, F. H. (1987). Science, 235, 1173. Whitby, F. G., Masters, E. I., Kramer, L., et al. (2000). Nature, 408, 115. Whiteley, H. R. (1966). J. Biol. Chem., 241, 4890. Whitley, P. L., & Tanford, C. (1962). J. Biol. Chem., 237, 1735. Whittaker, J. W. (1999). Essays Biochem., 34, 155. Wicken, J. S., Chung, A. E., & Franzen, J. S. (1972). Biochemistry, 11, 4766. Wickner, S. (1976). Proc. Natl. Acad. Sci. USA, 73, 3511. Widengren, J., Mets, U., & Rigler, R. (1995). J. Phys. Chem., 99, 13368. Widengren, J., & Rigler, R. (1998). Cell Mol. Biol., 44, 857. Wieland, O., von Jagow-Westermann, B., & Stukowski, B. (1969). Hoppe-Seyler’s Z. Physiol. Chem., 350, 329. Wierenga, R. K., Noble, M. E. M., & Davenport, R. C. (1992). J. Mol. Biol., 224, 1115. Wierenga, R. K., Noble, M. E. M., Vriend, G., et al. (1991). J. Mol. Biol., 220, 995. Wiesner, S., Helfer, E., Didry, D., et al. (2003). J. Cell Biol., 160, 387. Wiita, A. P., Ainavarapu, S. R., Huang, H. H., & Fernandez, J. M. (2006). Proc. Natl. Acad. Sci. USA, 103, 7222. Wikstro¨m, M. (1977). Nature (London), 266, 271. Wilhelmy, L. (1850). Ann. Physik u. Chem., Poggendorff’s, 81. 413 & 499. Wilkinson, G. N. (1961). Biochem. J., 80, 324. Wilkinson, K. D., & Rose, I. A. (1979). J. Biol. Chem., 254, 12567. Williams, J. C., & McDermott, A. E. (1995). Biochemistry, 34, 8309.

842

Williams, J. W., & Morrison, J. F. (1979). Meth. Enzymol., 63, 437. Williams, J. W., & Morrison, J. F. (1981). Biochemistry, 20, 6024. Williams, J. W., & Northrop, D. B. (1979). J. Antibiot. (Tokyo), 32, 1147. Williams, J. W., Morrison, J. F., & Duggleby, R. G. (1979). Biochemistry, 18, 2567. Williams, R. J. P. (1970). Quart. Rev. Chem. Soc. London, 24, 331. Williams, R. J. P. (1982). FEBS Lett., 140, 3. Williams, R. J. P. (1993). Trends. Biochem. Sci., 18, 115. Williams, R. J. P. (2003). Chem. Comm., 10, 1109. Wilson, D. J., & Konermann, L. (2003). Anal. Chem., 75, 6408. Wilson, D. J., & Konermann, L. (2004). Anal. Chem., 76, 2537. Wilson, K. S., & Noller, H. F. (1998). Cell, 92, 337. Wimalasena, D. S., Jayatillake, S. P., Haines, D. C., & Wimalasena, K. (2002). Biochem. J., 367, 77. Winer, A. D., & Schwert, G. W. (1959). J. Biol. Chem., 234, 1155. Winkler, F. K., Banner, D. W., Oefner, C., et al. (1993). EMBO J., 12, 1781. Winkler, T., Kettling, U., Koltermann, A., & Eigen., M. (1999). Proc. Natl. Acad. Sci. USA, 96, 1375. Winstein, S., Clippinger, E., Fainberg, A. H., et al. (1956). J. Amer. Chem. Soc., 78, 328. Winzor, D. J., & Sawyer, W. H. (1995). Quantitative Characterization of Ligand Binding (p. 168). New York: Wiley-Liss. Wirsching, P., Ashley, J. A., Lo, C. H., et al. (1995). Science, 270, 1775. Wissner, A., Overbeek, E., Reich, M. F., et al. (2003). J. Med. Chem., 44, 2719. Withers, S. G., Rupitz, K., & Street, I. P. (1988). J. Biol. Chem., 263, 7929. Woenckhaus, C., Jeck, R., & Jo¨rnvall, H. (1979). Europ. J. Biochem, 93, 65. Wold, F. (1971). The Enzymes, 5, 499, (3rd ed.). Wolf, D., & Williams, R. D. (1974). J. Chem. Educ., 51, 319. Wolfe, M. S., & Kopan, R. (2004). Science, 305, 1119. Wolfe, R. M. (1992). Radioactive and Stable Isotope Tracers in Biomedicine: Principles and Practice of Kinetic Analysis. New York: Wiley-Liss. Wolfenden, R. (1969). Nature (London), 223, 704. Wolfenden, R. (1970). Biochemistry, 9, 3404. Wolfenden, R. (1972). Acct. Chem. Res., 5, 10. Wolfenden, R. (1999). Bioorg. Med. Chem., 7, 647. Wolfenden, R., Snider, M., Ridgway, C., & Miller, B. (1999). J. Am. Chem. Soc., 121, 7419. Wolfsberg, M., & Stern, M. J. (1964). Pure Appl. Chem., 8, 225, & 8, 325. Womack, F. C., & Colowick, S. P. (1979). Proc. Natl. Acad. Sci. USA, 76, 5080. Wong, I., Patel, S. S., & Johnson, K. A. (1991). Biochemistry, 30, 526. Wong, J. T.-F. (1965). J. Am. Chem. Soc., 87, 1788. Wong, J. T.-F. (1975). Kinetics of Enzyme Mechanisms. New York: Academic Press. Wong, J. T.-F., & Hanes, C. S. (1962). Can. J. Biochem. Physiol., 40, 763. Wong, Y. H., & Sherman, W. R. (1985). J. Biol. Chem., 260, 11083. Wood, P. M. (1988). Biochem. J., 253, 287.

References

Woodard, R. W., Tsai, M.-D., Floss, H. G., et al. (1980). J. Biol. Chem., 255, 9124. Woodward, R. B., & Hoffmann, R. (1970). The Conservation of Orbital Symmetry. Weinheim, Germany: Verlag Chemie. Woska, D. C., Xie, Z.-L. D., Gridnev, A. A., et al. (1996). J. Am. Chem. Soc., 118, 9102. Woycechowsky, K. J., Vamvaca, K., & Hilvert, D. (2007). Adv. Enzymol., 75, 241. Wratten, C. C., & Cleland, W. W. (1963). Biochemistry, 2, 935. Wright, D. J., Jack, W. E., & Modrich, P. (1999). J. Biol. Chem., 274, 9820. Wright, D. J., King, K., & Modrich, P. (1989). J. Biol. Chem., 264, 11816. Wu, X. M., Gutfreund, H., Lakatos, S., & Chock, P. B. (1991). Proc Natl Acad Sci USA, 88, 497. Wuite, G. J. L., Smith, S. B., Young, M., et al. (2000). Nature, 404, 103. Wurtz, C. A. (1880). Compt. rendu., 91, 787. Wyman, J. (1948). Adv. Prot. Chem., 4, 407. Wyman, J. (1964). Adv. Prot. Chem., 19, 223. Wyman, J. (1972). Curr. Top. Cell. Reg., 6, 207. Wyman, J., & Gill, S. J. (1990). Binding and Linkage: Functional Chemistry of Biological Molecules. Mill Valley, California: University Science Books. Wyss, M., Smeitink, J., Wevers, R. A., & Wallimann, T. (1992). Biochim. Biophys. Acta, 1102, 119. Xiang, J., Jung, J.-Y., & Sampson, N. S. (2004). Biochemistry, 43, 11436. Xie, X. S. (1996). Acct. Chem. Res., 26, 598. Xie, X. S., & Lu, H. P. (1999). J. Biol. Chem., 274, 15967. Xie, X. S., & Trautman, J. K. (1998). Annu. Rev. Phys. Chem., 49, 441. Xin, Y., Li, W., & First, E. A. (2000a). J. Mol. Biol., 303, 299. Xin, Y., Li, W., & First, E. A. (2000b). Biochemistry, 39, 340. Xu, W., Dunn, C. A., Jones, C. R., et al. (2004). J. Biol. Chem., 279, 24861. Xu, Y., Eads, J., Sacchettini, J. C., & Grubmeyer, C. (1997). Biochemistry, 36, 3700. Xu, C., Shear, J. B., & Webb, W. W. (1997). Anal. Chem., 69, 1285. Xue, Q., & Yeung, E. S. (1995). Nature, 373, 681. Yagi, K. (1991). A Study of Enzymes, 2, 271. Yamamoto, S., Lin, K., & Bloch, K. (1969). Proc. Natl. Acad. Sci. USA, 63, 110. Yan, B. X., & Sun, Y. Q. (1996). J. Biol. Chem., 272, 3190. Yanagida, T., Nakase, M., Nishiyama, K., & Oosawa, F. (1984). Nature, 307, 58. Yang, A. S., Gunner, M. R., Sampogna, R., et al. (1993). Proteins, 15, 252. Yang, J., Bruno, W. J., Hlavack, W. S., & Pearson, J. E. (2006). Biophys J., 91, 1136. Yang, K., & Metcalf, W. W. (2004). Proc. Natl. Acad. Sci. USA, 101, 7919. Yang, S. Y., & Schulz, H. (1987). Biochemistry, 26, 5579. Yarus, M. (1992). Trends Biochem. Sci., 17, 130. Yasuda, R., Noji, H., Kinosita, K., Jr., & Yoshida, M. (1998). Cell, 93, 1117. Yasuda, R., Noji, H., Yoshida, M., et al. (2001). Nature, 410, 898. Yates, R. A., & Pardee, A. B. (1956). J. Biol. Chem., 221, 757. Yeo, Y., & Park, K. (2004). J. Control. Release, 100, 379.

References

Ygueribide, J., Epstein, H. F., & Stryer, L. (1970). J. Mol. Biol., 51, 573. Yingge, Z., Zhao, D., Bai, C., & Wang, C. (1999). Life Sciences, 65, 253. Yokoyama, K., & Gelb, M. H. (2001). In F. Tamanoi, & D. S. Sigman (Eds.), The Enzymes, Vol. XXI (p. 117). New York: Academic Press. Yokoyama, K., & Kayanuma, Y. (1998). Anal. Chem., 70, 3368. Yoon, T. P., & Jacobsen, E. N. (2003). Science, 299, 1691. Yorita, K., Misaki, H., Palfey, B. A., & Massey, V. (2000). Proc. Natl. Acad. Sci. USA, 97, 2480. Yoshino, M. (1987). Biochem. J., 248, 815. Youn, H.-D., Kim, E.-J., Roe, J.-H., et al. (1996). Biochem. J., 318, 889. Yu, M., Magalha˜es, L. B., Cook, P. F., & Blanchard, J. S. (2006). Biochemistry, 45, 14788. Yuan, L., Craig,., S. P., III, McKerrow, J. H., & Wang, C. C. (1992). Biochemistry, 31, 806. Yusopov, M. M., Yusopova, G. Z., Baucom, A., et al. (2001). Science, 292, 883. Zabinski, M., & Walz, F. G., Jr. (1976). Arch. Biochem. Biophys., 175, 558. Zaks, A., & Klibanov, A. M. (1985). Proc. Natl. Acad. Sci. USA, 82, 3192. Zaks, A., & Klibanov, A. M. (1986). J. Am. Chem. Soc., 108, 2767. Zaks, A., & Klibanov, A. M. (1988). J. Biol. Chem., 263, 8017. Zanghellini, A., Jiang, L., Wollacott, A. M., et al. (2006). Protein Sci., 15, 2785.

843

Zatman, L. J., Kaplan, N. O., & Colowick, S. P. (1953). J. Biol. Chem., 200, 197. Zaug, A. J., & Cech, T. R. (1986). Science, 231, 470. Zeche, D. L., Konermann, L., Withers, S. G., & Douglas, D. J. (1998). Biochemistry, 37, 7664. Zeile, W. L., Purich, D. L., & Southwick, F. S. (1996). J. Cell Biol., 133, 49. Zemal, H. (1987). J. Am. Chem Soc., 109, 1875. Zeng, W., Conibear, P. B., Dickens, J. L., et al. (2004). Philos. Trans. Roy. B, Biol. Sci., 359, 1843. Zerner, B. (1991). Bioorg. Chem., 19, 116. Zewe, V., & Fromm, H. J. (1965). Biochemistry, 4, 782. Zewe, V., Fromm, H. J., & Fabiano, R. (1964). J. Biol. Chem., 239, 1625. Zheng, Q., & Ross, J. (1991). J. Chem. Phys., 94, 36. Zhou, G., Ellington, R., & Chapman, M. S. (2000). Biophys. J., 78, 1541. Zhu, C., Lou, J., & McEver, R. P. (2005). Biorheology, 42, 443. Zhu, D.-M., Dustin, M. L., Cairo, C. W., & Golan, D. E. (2007). Biophys. J., 92, 1022. Zigmond, S. H., Evangelista, M., Boone, C., et al. (2003). Curr. Biol., 13, 1820. Zimmerle, C. T., & Frieden, C. (1989). Biochem. J., 258, 381. Zimmerman, S. B., & Minton, A. P. (1993). Annu. Rev. Biophys. Biomol. Struct., 22, 27. Zolotnitsky, G., Cogan, U., Adir, N., et al. (2004). Proc. Natl. Acad. Sci. USA, 101, 11275. Zumbusch, A., & Jung, G. (2000). Single Mol., 1, 261.

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Appendix

Abbreviations & Symbols ROMAN LETTERS AND SYMBOLS SI symbol for absorbance (unitless)

A

SI symbol for the pre-exponential term in DA A, B, C,.

Change in absorbancy Substrate A, B, C,.

Bq C

Becquerel (unit of radioactivity ¼ 1 disintegration per second) Coulomb

C or c c or Ci D

Molar concentration (M or mol/liter) Symbol for Curie (old unit of radioactivity) SI symbol for translational diffusion constant (meter2/second)

Da Drot

Dalton Rotational diffusion constant

e

Unit charge on an electron (1.6022  1019 coulomb)

Ea [ET], or Eo F G DG



DGz

General symbol for enzyme Activation energy (units ¼ kCal/mol or kJoule/mol) Total enzyme concentration Faraday (9.6485  104 coulomb$mol1)

Gibbs free energy (units ¼ kCal/mol or kJoule/mol) Standard Gibbs free energy change per mole Standard Gibbs free energy of activation

H DH h

Enthalpy (units ¼ kCal/mol or kJoule/mol) Standard enthalpy change per mole

J

Joule(107 erg or 1 volt$coulomb) Macroscopic equilibrium constant

K or Keq

Planck’s constant (6.626  1034 joule$second or 6.626  1027 erg$second)

Kap KD

Acid dissociation constant Dissociation constant for ligand A, B, C, . Apparent equilibrium constant Dissociation constant

KF

Dissociation constant for ligand F for an allosteric protein

Ki

Macroscopic inhibition constant Macroscopic ionization constant

Ka KA, KB, .

845

Appendix

846

Kia, Kib, .

Dissociation constants in enzyme kinetics

KR

Dissociation constant for ligand that binds to R-state of allosteric protein Equilibrium constant for dissociation of ES complex Dissociation constant for ligand that binds to T-state of an allosteric protein

KS KT K1, K2, K3 k or kB kcat kcat/Km kH/kD

Stepwise binding or dissociation constants for an oligomeric receptor Boltzmann constant (R/No) (1.3806  1023 joule$K1$molecule1) Catalytic constant; turnover number (units ¼ s1) Specificity constant (units ¼ M1$s1)

L

Kinetic isotope effect (unitless) Allosteric constant equal to [T0]/[R0]

n

Number of gram-molecular weights (or moles)

N N0 nH or nHill

Newton (unit of force) Avogadro’s number (6.0221  1023 mol1)

pKa

Hill coefficient –log10Ka

r

Radius of a molecule or particle

R

Universal gas constant (8.3144  107 erg$K–1$mol–1; 8.3144 joule$K1$mol–1; 1.9872 calorie $K1$mol1; 80.082057 liter$atm$K1$mol1) Standard entropy change

DS DSz t1/2 Veff Vm or Vmax V/K v vinit or v0

Standard entropy of activation Half-life Effective volume Maximal velocity (units ¼ M$s1) Ratio of Vmax to Km (units ¼ M1$s1)

Reaction velocity of enzyme-catalyzed reaction (units ¼ M$s1) Initial velocity of enzyme-catalyzed reaction (units ¼ M$s1)

GREEK LETTERS AND SYMBOLS b

Reduced concentration ([I]/KI) for allosteric protein

g

Reduced concentration ([A]/KA) for allosteric protein

h l mi mi

Viscosity Wavelength Chemical potential of ith species per mole Standard chemical potential per mole

n

Frequency

t

Relaxation time (t) Lag time (s)

Glossary

Acidity – The tendency for a Brønsted acid (e.g., A–H # A– + H+) or Lewis acid adduct (e.g., Men+(OH2) # Me(n1)+(OH) + H+) to act as a proton donor, expressed in terms of its dissociation constant in water. With reference to a solvent, this term is usually restricted to Brønsted acids. If the solvent is water, the pH value of the solution is a good measure of the protondonating ability of the solvent, provided that the concentration of the solute is not too high. For concentrated solutions or for mixtures of solvents, the acidity of the solvent is best indicated by use of an acidity function. Acid-Labile Sulfides – The bridging sulfur atoms in iron-sulfur proteins are often referred to as acid-labile sulfides, because treatment of such proteins with acids generates H2S. Alpha Effect – Enhanced nucleophilicity of an attacking nucleophile attributable to one or more unshared pairs of electrons on an atom that lies immediately adjacent to the nucleophilic group. The increased nucleophilicity of hydrazines and hydroxylamines is due to an alpha effect, as is the increased nucleophilicity of HO2– compared to HO– in solution. This effect on nucleophile reactivity can be significant for substitution reactions at a carbonyl group, at an unsaturated carbon, or in reactions of a nucleophile with a carbocation. The effect is largest when the transition state displays considerable bond formation. Substitution reactions at a saturated carbon generally tend to exhibit little or no alpha effect. Substances exhibiting alpha effects in their reactions consistently deviate from the anticipated structure-reactivity correlations known for simple nucleophiles. One explanation for the alpha effect is ground-state destabilization: repulsive electronic interactions between the alpha atom’s lone-pair and the nucleophile occur in the ground-state, and such destabilization is expected to be relieved as a covalent bond is forming in the transition state of a nucleophilic substitution reaction. Reduced solvation in molecules exhibiting the alpha effect may also play a role in the increased nucleophilicity. Another factor may be the ability of the alpha lone pair to stabilize any partially positive group formed in the transition state. Ambident Nucleophile – A term describing a chemical species containing two distinct, yet strongly interacting, reactive centers at which a covalent bond can be formed. These centers are positioned so that a reaction at one center either stops (or greatly retards) a reaction at the other center. The more basic group in the ambident anion preferentially reacts with the polarizable group under nucleophilic attack. Ambident anions consistently deviate from anticipated structure-activity correlations known for substances containing only a single reactive group. Examples of such molecules include conjugated

nucleophiles such as the enolate anion. Such nucleophiles have potentially two attacking atoms (in the case of the enolate anion, the oxygen or the a-carbon); reaction conditions affect which will be the more prevalent species. Other examples include the cyanide (NC–) and the nitrite (NO2–) ions. Anchimeric Assistance – The well-known facilitation of nucleophilic displacement reactions (particularly solvolytic reactions) resulting from a suitably positioned neighboring group. This functional group may help in directing the nucleophile to the electrophilic center or it may increase reactivity by stabilizing a reaction intermediate. A likely case is the acid catalyzed hydrolysis of phenylglycosides. Anchimeric assistance may also explain slight changes in enzyme mechanisms resulting from the presence of a neighboring group. Anchor Principle – A concept suggesting that the relatively large size of coenzymes and other substrates is likely to assist in the proper positioning of the reaction center on the enzyme (Jencks, 1975). Anchoring is attended by a corresponding loss of translational and rotational motion. The binding energy includes the energy required for achieving the proper orientation. The binding of well-anchored large substrates may initiate the conformational reorganization of the enzyme’s active site. If these factors are enhanced in the transition state, so too will they favorably contribute to catalysis. Anomeric Effect – An observed effect of certain substituents on the conformation and stability of glycosides. In the case of aand b-glucosides, when the substituent attached to the anomeric position of D-glucose is a nonpolar alkyl group, such a substituent will favor the equatorial position (i.e., as in the b form). However, if the substituent is polar, the a-position (i.e., axial) is favored. Thus with polar moieties, a-glucosides exhibit greater stability than b-glucosides. Antibonding Orbital – The combination of two atomic orbitals to form a bond between two elements actually produces two molecular orbitals having different energies. The molecular orbital with the lower energy is referred to as the bonding orbital; the other orbital tends to push atoms apart and is termed the antibonding orbital. Binary Complex – A noncovalent complex between two molecules. Binary complex often refers to an enzyme-substrate complex, designated ES in single-substrate reactions or as EA or EB in certain multisubstrate enzyme-catalyzed reactions. Bonding Orbital – A molecular orbital having a lower energy level than the atomic orbitals from which it is formed. Such an orbital can contain two electrons, and their presence results in a strong bond when the overlap of the atomic

847

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orbitals is large. Two overlapping atomic orbitals combine to yield one low-energy bonding orbital (designated s) and one high-energy antibonding orbital (designated s*). Two paired electrons are sufficient to fill the s orbital, and any additional electrons must occupy the high-energy s* orbital where, rather than stabilizing the bond, they lead to repulsion between the atoms. Bond Number – The number of electron-pair bonds shared by two atoms. In acetone, for example, the bond number for the carbon and oxygen is two, whereas the bond number is one for carbon bonded to hydrogen. Bond Order – The number of bonding electron pairs between two atoms. Single bonds have a bond order of 1, double bonds have a bond order of 2, and triple bonds have a bond order of 3. A fractional bond order is possible in reaction transition states as well as in certain resonance-stabilized molecules and ions. Bond order therefore is an index of the degree of bonding between two atoms in a molecule. In valence-bond theory, the bond order for a particular bond is calculated as the weighted sum of all the canonical forms of the molecule. For example, if benzene were completely represented by two Kekule´ resonance forms, then one would predict that a carbon-carbon bond in benzene has a bond order of 1.5. However, the three so-called Dewar structures also contribute to the structure of benzene. The carboncarbon bond order for benzene using valence-bond theory is thus determined to be 1.463. Hence, a carbon-carbon bond in benzene is not halfway between a single and a double bond, but actually somewhat less. Bond orders are calculated differently in molecular orbital theory. The order is determined from the weights of the atomic orbitals in each molecular orbital. With this method, the carbon-carbon bond order in benzene is 1.67. Bridging Ligand – A ligand that is capable of simultaneous binding to two or more central metal atoms within a transition metal ion complex, such that the bridged metal ions form a polynuclear coordination complex. Catalytic Antibody – Antibodies that catalyze chemical reactions by virtue of their complementary high-affinity interactions with transition-state analogs, against which they were raised. Catalytic Proficiency – A measure of an enzyme’s ability to lower the activation barrier for the reaction of a substrate in solution. Catalytic proficiency is a unit-less parameter that equals kcat divided by the rate constant (knon) for the uncatalyzed (or reference) reaction. In this book, the preferred term for kcat/knon is catalytic rate enhancement (symbol e). See Section 4.4: Additional Comments on Kinetic Parameters. Catalytic RNA – Naturally occurring and synthetic RNA molecules that catalyze sequence specific self-cleavage and other transesterification reactions. See Section 1.3.4: Ribozyme Catalysis; and Section 4.5: Ribozyme Kinetics. Chemical Shift – A term used in NMR spectroscopy to designate the displacement in the magnetic resonance frequency of a nucleus as a consequence of the electronic environment in which the nucleus resides. Because moving electrons generate their own magnetic fields, a nucleus surrounded by these electrons experiences an effective field, Heff, which is defined

by (1 – a)Ho, where a is the so-called screening constant and Ho is the applied magnetic field. A chemical shift is typically reported as a dimensionless displacement (units ¼ parts per million, or simply ppm) from a reference standard. If the magnetic field is varied while the radiofrequency v is held constant, then the chemical shift (ppm) equals {[Hsample – Hreference]/Hreference}  106. If the microwave frequency is varied at constant magnetic field, the chemical shift (ppm) equals {[vsample – vreference]/vreference}  106. Tetramethylsilane is frequently used as a proton chemical shift reference compound in organic chemistry, because it yields a single sharp NMR signal in a region well removed from the resonance lines of most other protons. See Nuclear Magnetic Resonance. Commitment to Catalysis – A quantitative measure of the tendency of the enzyme-substrate complex poised for catalysis to proceed onward to product formation, as opposed to its tendency to return into free enzyme and substrate. See Section 4.4: Additional Comments on Kinetic Parameters. Coordination Number – The integral number of molecules or ions that directly interact with a metal ion center in a transition metal ion complex. Corrin – The ring-contracted (i.e., missing a meso-position carbon-20) porphyrin-like carbon skeleton of various B12 vitamins, cofactors and derivatives. De-inhibition – Any ligand-induced process that nullifies the inhibitory effect of an enzyme-bound inhibitor. In such cases, the ligand is said to de-inhibit the system. Such behavior is distinct from activation, because increased activity is not observed when the deinhibiting ligand interacts with the enzyme in the absence of the inhibitor. Diffusion Time – The period of time t required for a molecule with a diffusion coefficient D (units of cm2 s–1) to diffuse a mean square distance x2, as given by the expression t ¼ x2/2D. For example, a molecule with a diffusivity of 10–5 cm2 s–1 requires about 0.5 seconds to diffuse about 10 micrometers, which corresponds to the dimension of many cells. Eigenfunction – Any solution to one or more differential equations, for which only certain parameters are allowed. Einstein – A unit used principally in photochemistry to designate an Avogadro’s number of quanta (i.e., 6.0221  1023 quanta).

Electroactive Species – Any chemical compound that can be oxidized or reduced in a manner that generates a flow of electrons from a solution to an electrode. Electron Sink – That region within a molecule having sufficient electrophilic character that it can capture or trap an electron during the course of a reaction. Electron Spin Resonance – A powerful spectroscopic technique for analyzing free radicals and spin-states of metal ions exhibiting paramagnetism. Like protons in NMR, an unpaired electron has a spin and magnetic moment, such that, when placed in a magnetic field, it has two possible orientations corresponding to quantum numbers equal to þ1/2 or 1/2. These two orientations define two energy states differing by an energy DE equal to (hg/2p)H, where g is the gyromagnetic ratio of the electron, and transitions between these states occur upon absorption of radiation if frequency v ¼ (g/2p)H. For

Glossary

free electrons, g is ~1000x larger than for protons, and the frequency for free electrons is ~10 gigahertz. To observe the ESR phenomenon, the sample must be placed in a resonant cavity connected by means of a microwave guide to a microwave generator (either a klystron or a Gunn diode). The sample and its surrounding resonant cavity is then mounted between the pole faces of a powerful electromagnet. The resonant absorption of the microwave energy is measured in a first derivative mode (i.e., DAbsorbance/DH, where H is the magnetic field), giving rise to a peak and trough pattern for each absorption peak. Under favorable conditions, ESR spectrometers can detect free radicals at sub-picomolar concentrations. ESR spectra are also obtained with paramagnetic substances like Mn2+, Cr3+, and ionic iron species. The interested reader should consult the outstanding brief description provided in volume IV of Comprehensive Biological Catalysis (Sinnott. 1999). Electrophile – A reagent or reactant that is preferentially attracted to a region of high electron density within a particular molecular entity (or, in the case of intramolecular processes, within the same molecular entity). Typically, an electrophile bonds by accepting both bonding electrons for the reaction partner (termed the nucleophile). The term electrophile is also used with respect to certain polar radicals exhibiting high reactivities with sites of high electron density. Electrophilic Catalysis – Enhanced chemical reactivity arising from the participation of a Lewis acid in catalyzed or uncatalyzed chemical reactions. Electrophilicity – The relative reactivity expressed by an electrophile, as measured by the relative rate constants for a particular reaction of different electrophiles for a common substrate. Electrostatic Surface Potential – A measure of the electric charge density and distribution along the surface of molecules. The surface potential is approximately determined by the laws of classical electrostatics or more rigorously by quantum mechanics. The magnitude of the attractive or repulsive energy (U) between two point charges (q1 and q2) separated at a distance r is equal to (kq1q2/Dr), where k is 9  109 joulemeter/coulomb2 and D is the dielectric constant of the medium. To estimate the electrostatic surface potential, calculations are made for the attractive or repulsive interactions experienced by a probe proton passing above the molecule’s surface at a fixed distance. Elementary Reaction – A reaction that takes place on a molecular scale in a single step following an individual collision or other elementary process and in which no stable intermediate need be postulated and no simpler reaction can be suggested.

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Encounter-Controlled Rate – A reaction velocity equal to the rate of encounter of reacting molecular entities (also known as diffusion-controlled rate). For a bimolecular reaction in aqueous solutions at 25 C, the corresponding second-order rate constant for the encounter controlled rate is typically about 1010 M–1 s–1. Endergonic Process – A change or reaction for which the Gibbs free energy (DG) becomes more positive. (The origin of this compound term is derived from the Greek prefix end- which is a contraction of endo and the Greek root ergon, standing for work.) Energase – An unofficial term for mechanochemical and affinitymodifying enzyme reactions (Purich, 2001). Rooted in the word ‘‘energy’’, the term reinforces the idea that these enzymes transduce the DG for the hydrolysis of nucleoside 5’triphosphate (or other similar sources of Gibbs free energy) into translations, rotations or twists, or transmembrane solute and/or electrogenic gradients. Under normal physiologic conditions, energases couple nucleotide hydrolysis to force production. Energases participate in what may be regarded as a long-lived catalytic cycle. Overt ATPase or GTPase activity only occurs when an energy-transducing step of an energase process is disrupted by nonphysiologic treatments or uncoupling agents. In this respect, the current Enzyme Commission designation of energase-type reactions as ATPases and GTPases is erroneous. Entering Group – The atom, ion, or group of a molecular entity that approaches and forms a covalent or coordinate covalent bond with the substrate during a reaction. Any attacking nucleophile is an entering group. See also Leaving Group. Enthalpy-Driven Reaction – Any reaction or process for which the enthalpy change is large and negative, such that the DH term greatly dominates the –TDS term in the Gibbs equation (DG ¼ DH –TDS), and acts thereby as the thermodynamic driving force for that reaction or process. Enzyme – Biological catalyst that accelerates the rate of making/ breaking of chemical bonds (Purich, 2001). While appearing to be no more encompassing than previous definitions focused on the making/breaking covalent bonds, the term chemical bond also encompasses other long-lived bonding interactions. Because many protein conformational states and numerous protein-ligand complexes are often sufficiently long-lived to exhibit chemically definable properties, their formation and/or transformation should be categorized as chemical reactions. This new definition of enzyme catalysis includes: (a) forcegenerating ‘‘ATPases’’; (b) Active and passive transporters; (c) Peptidyl translocases, (d) GTP-regulatory proteins, as well as (e) Ligand exchange factors. See Section 1.3.5: Redefining Biological Catalysis to Encompass Mechanochemical Processes.

Enantioselective Reaction – A reaction in which an optically inactive compound (or achiral center of an optically active molecule) is selectively converted to a specific enantiomer (or chiral center).

Epimerization – The stereochemical isomerization or structural rearrangement resulting in the interconversion of epimers.

Encounter Complex – Any weakly attractive, short-lived complex that is typically formed as an intermediate in a reaction mechanism. When there are only two such molecular entities engaged in the formation of a particular encounter complex, that complex is often called an encounter pair.

Equilibrium Isotope Effect (EIE) – The ratio of equilibrium constants Kl/Kh, where subscript l stands for the light (lowermass) isotopic molecule, and subscript-h represents the heavy (higher-mass) isotopic molecule. As with KIEs, there are three possibilities: (a) if Kl/Kh > 1.0, the EIE is said to be normal;

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(b) if Kl/Kh ¼ 0, the isotope effect is said to be absent or undetectable; and (c) if Kl/Kh < 0, the EIE is said to be an inverse isotope effect. Note that: Kl/Kh ¼ exp{–(DG l – DG h)/ RT}, which in principle reflects contributions from DH l, TDS l, DH h, and TDS h. See Section 9.8.3.: Measurement of Isotope Effects by the Equilibrium Perturbation Method. Excited State – A state of higher energy experienced by a molecular entity, relative to its ground state. In photic processes, the excited state can be achieved by absorption of photons, by heating, or by extreme pressurization. Exergonic – An adjective describing a process, change, or reaction in which the change in Gibbs free energy (DG) is a negative value. If a compound has a negative change in the standard Gibb’s free energy of formation, the substance is referred to as an exergonic compound. Exothermic – An adjective describing a process, change, or reaction that releases heat. If pressure is constant and no work other than PDV work is done, then DH ¼ qP where DH is the enthalpy and qP is the heat released at constant pressure. Thus, in such systems, exothermic systems will have negative DH values. Flux – A rate of transfer of entities, particles, fluids through a given point, surface, or pathway. For example, the different pathways for a particular enzyme-catalyzed reaction will have a different flux through each of those pathways. Footprinting – Any technique designed to characterize binding interactions by determining the accessibility of the backbone of macromolecules to cleavage or modification reactions. For nucleic acid interactions, footprinting was originally accomplished by changes in phosphodiester accessibility to DNase I, but numerous chemical and enzymatic methods continue to be elaborated. Force – A vector quantity that changes the state of rest or motion of matter. The force, F, is equal to the change in momentum, r, with respect to time (i.e., F ¼ dr/dt). If the mass, m, of the entity is constant, the force is that which produces an acceleration, a (also a vector quantity), of that entity (i.e., F ¼ ma). The SI unit for force is the newton. Formation Constant – An association constant (units ¼ M–1) for an equilibrium binding reaction of molecules (e.g., A + B # A–B), most frequently the binding interactions of a macromolecule ligand (which can be a proton, substrate, effector, anion, or cation such as a metal ion). Concentrations of the products of an association or formation reactions appear in the numerator of formation constants. Macromolecules and ligands may undergo pH-dependent changes in charge, and many binding reactions are often also characterized by changes in entropy and/or enthalpy. Thus, formation constants frequently depend on such variables as pH, temperature, and pressure. A stability constant is a special formation constant defining the extent of metal ion complexation with a ligand. Fractionation Factor – A ratio defining the equilibrium isotopic distribution of two isotopes between two different chemical species. If X, followed by a subscript, represents the mole fraction of an isotope (denoted by that same subscript), then the fractionation factor, often symbolized by f, with respect to chemical species A and B is (X1/X2)A/(X1/X2)B. Fractionation factors can also refer to different sites, A and B, within the same

chemical species. As an example, the deuterium solvent fractionation factor, used in studying solvent isotope effects, is f ¼ (XD/XH)solute/(XD/XH)solvent. Consider the fractionation of protium and deuterium in the following reaction: SH + ROD # SD + ROH. The equilibrium constant is: K ¼ [SD][ROH]/ [SH][ROD], and dividing the numerator and denominator by [SH][ROH] yields the fractionation factor: f ¼ ([SD]/[SH])/ ([ROD]/[ROH]). Fractionation factors are determined by the number and kinds of atoms directly attached to the carbon atom bearing the exchanging hydrogen. Buddenbaum & Shiner (1977) tabulated H/D fractionation factors for various substances relative to acetylene at 25 C. Other values are presented by Schowen (1977) and in Appendix A in Isotope Effects on Enzyme Catalyzed Reactions (Cleland, O’Leary & Northrop, 1977). See also Section 9.8.7: Solvent Isotope Effects. Franck-Condon Principle – The consistently observed spectroscopic behavior indicating that the most likely electronic transition will occur without changes in the positions of the nuclei (e.g., little change in the distance between atoms) in the molecular entity and its environment. Such a state is known as a Franck-Condon state, and the transition is referred to as a vertical transition. In such transitions, the intensity of the vibronic transition is proportional to the square of the overlap interval between the vibrational wavefunctions of the two states. General Acid Catalysis – Catalysis of a reaction by a Brønsted acid, including the protonated solvent S, forming the lyonium ion, SH+. The rate of a general acid-catalyzed reaction is increased by an increase in the lyonium ion concentration and/ or by an increase in the concentration of other acids, even when [SH+] is held constant. Experimentally, general acid catalysis can be distinguished from specific acid catalysis by analysis of the effect of buffer concentration on the overall reaction rate. General Base Catalysis – Catalysis of a reaction by a Brønsted base, including the deprotonated solvent SH or SH+, forming the lyate ion, S– or S, respectively. Experimentally, general base catalysis can be distinguished from specific base catalysis by analysis of the effect of buffer concentration on the overall reaction rate. Global Analysis – A systematic algorithm for data analysis that permits the experimenter to recover estimates of lifetimes (t) from a multi-exponential time-dependent process F(t). The process is described by the generalized P sum-of-exponentials expression, involving n steps: F(t) ¼ ai exp[ – t/ti], where the summation is taken from (i ¼ 1) to (i ¼ n), and ai is the amplitude associated with each of the individual steps. This algorithm simultaneously analyzes data from multiple experiments. This is accomplished applying an internally consistent set of fitting parameters to nonlinear least squares software adapted to perform model-dependent summations of these nonlinear least-squares equations. Haldane Relationship – A quantitative relation between the reaction equilibrium constant Keq and the Vmax,forward, Vmax,reverse, Km,forward, and Km,reverse for enzyme-catalyzed reactions. The Haldane Relationship shows how kinetic parameters are thermodynamically constrained by the reaction equilibrium constant. See Section 4.2.4: The Haldane Relationship and Section 4.3.4: Haldane Relationship for Two-Intermediate Case.

Glossary

Heavy Atom Isotope Effect – An isotope effect (on either kinetic or equilibrium process) resulting from substitution by isotopes other than those of hydrogen. Hertz – The SI unit for frequency (symbolized by Hz) equal to reciprocal seconds. The term hertz should only be used with respect to cycles per second and not for radial (circular) frequency or angular velocity. Heterotropic Effect – Cooperative ligand binding in which the bound ligand influences the binding or reaction of a different ligand to the same macromolecule. The influence of the ligand can result in either weakened binding (hence, negative heterotropic cooperativity) or in enhanced binding (hence, positive heterotropic cooperativity). High-spin State – The state having the largest number of unpaired electrons is one of two alternative electronic states (the other being the Low-spin state) arising when the separation between the highest occupied and the lowest unoccupied molecular orbitals is not large. Hit – A term used in high-throughput screening to indicate a library component (or element) that demonstrates enzyme inhibition in excess of a basal (or background) value. Homotropic Interaction – A term used in describing the interactions among like ligand binding sites in cooperative saturation. If an enzyme binds two ligands, say X and Y, then the effect of the binding of molecule X on the binding of other molecules of X is termed a homotropic interaction. Likewise, the effect of the binding of molecule Y on the binding of other molecules of Y is termed a homotropic interaction. Effects arising from the influence of ligand X binding on ligand Y binding, and vice versa, are termed heterotropic interactions. Host-Guest Interactions – Structural recognition and binding interactions involving a naturally occurring or synthetic ‘‘host’’ (or template) and a smaller ligand known as a ‘‘guest’’. Examples of host molecules include the cyclodextrans and the so-called crown ethers. Host-guest interactions have served as useful systems for modeling enzymic catalysis, resolving optical isomers of complicated organic molecules, making analogs for micellar catalysis, and for understanding the selective cation transport properties of valinomycin-like antibiotics. Hydrolases – A major class of enzymes that catalyze hydrolytic cleavage reactions. Examples include esterases, phosphatases, sulfatases, nucleases, glycosidases, peptidases, proteinases, and amidases. Although ATP- and GTP-hydrolyzing mechanoenzymes are classified as hydrolases, they are so misnamed for their catalysis of a partial reaction (ATP + H2O # ADP + Pi). See Energases. Hysteresis – Slow transitions produced by enzyme isomerizations. This behavior can lead to a type of cooperativity that is generally associated with ligand-induced conformational changes. A number of enzymes are also known to undergo slow oligomerization reactions, and these enzymes may display unusual kinetic properties. If a lag in product formation is observed, it is advisable to determine the time course of enzyme activation or inactivation following enzyme dilution.

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Inner Coordination Sphere – The region of a metal ion complex where ligands make direct binding interactions with the central metal ion. When the ligands do not completely neutralize the positive ionic charge of the central ion, other ions or electronrich substances will become loosely associated with the complex through so-called outer coordination sphere interactions. Inner-Filter Effect – An apparent decrease in the emission quantum yield and/or distortion of the band shape due to the reabsorption of emitted radiation. If such an effect is not corrected or compensated for, results of an emission experiment may prove to be incorrect. This is especially true in fluorescence quenching experiments conducted to evaluate the stoichiometry and affinity of ligand binding. Inner-Sphere Electron Transfer – Electron transfer between two metal centers sharing a ligand or atom in their respective coordination shells. Electron transfer between two centers in which the interaction between the donor and acceptor centers in the transition state is significant (i.e., greater than 20 kJ/mol). Interconvertible Enzyme – An enzyme that exists in two or more defined forms arising as a consequence of enzyme-catalyzed covalent modifications (e.g., phosphorylation, acetylation, ADPribosylation, etc.) at specific amino acid residues. An example is glycogen phosphorylase, which is the substrate for phosphorylase kinase (Reaction: Phosphorylase b (a relatively inactive dimer of 90 kDa subunits) + MgATP2– # MgADP + Phosphorylase a, a catalytically active tetramer) and glycogen phosphorylase phosphatase (Reaction: Phosphorylase a + H2O # Pi + Phosphorylase b). Zymogen activation by a protease is not an example of an interconvertible enzyme, because this hydrolysis reaction is not reversible. See Section 11.10: Enzyme Amplification Cascades. Intermediacy – A condition of kinetic competency pertaining to the conversion of substance A to another substance P by means of one or more intervening species, say I1 and I2, in a series first-order reaction: A / I1 / I2 / $ $ $ Ii / P (See Section 3.4.1: Series First-Order Processes). Intermediacy requires that the overall A-to-P transit time tAP equals 1/k1 + 1/k2 + $ $ $ + 1/ki, where k1, k2, ., ki, etc., are the rate constants for A / I1, I1 / I2, and Ii / P, respectively. While this equation establishes the kinetic criteria for establishing that a solution-phase compound is a true intermediate, additional binding steps are needed for binding of a putative intermediate compound to the enzyme. In the context of enzyme catalysis, intermediacy is depicted as the unidirectional flux through the reaction sequence: E + A / E$A / E$I1 / E$I2 / $ $ $ E$Ii / E$P / E$P. Each step in an enzyme-catalyzed reaction is typically reversible, and some interconversion steps may engage a second reactant (kx[B]), one must use net rate constants (e.g., k19, k29,., ki9, etc.) to obtain the flux through each step. See Section 5.2.3: Cleland’s Net Reaction Rate Method and Section 5.4.3: Additional Comments on Partial Exchange Reactions. Internal Conversion – A radiationless mode of energy loss (decay) of an already photochemically excited molecule, whereby some or all of the electronic energy of the absorbed photon is converted into kinetic energy, with concomitant

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changes in molecular motion (mainly vibration). This very rapid process releases energy thermally (i.e., without loss of a photon), resulting in conversion of the molecule to (a) a lower-lying excited state, (b) a long-lived triplet state, or (c) radiationless return to the ground state. (The lifetime for internal interconversion is less than 10–11 s, whereas fluorescence and phosphorescence have respective lifetimes that are 102–103 and 107–108 times longer.) In most molecules, the originally absorbed photon results in formation of the first excited state. In the case of chorophyll, however, absorption of blue light promptly generates the second excited state, followed by efficient internal conversion to its first excited state. In this manner chlorophyll, with its extensive network of conjugated double bonds, retains the ability to harvest a significant fraction of absorbed photic energy over a wider spectrum of incident light. Internal Equilibrium – A thermodynamic equilibrium established between two successive enzyme-bound reactant states E$Si and E$Si+1 (KESi-internal ¼ [E$Si+1]/[E$Si]), E$Ii and E$Ii+1 (KEI-internal ¼ [E$Ii+1]/[E$Ii]), or E$Pi or E$Pi+1 (KEPi-internal ¼ [E$Pi+1]/[E$Pi]), where S, I, or P are the substrate, intermediate, or product. Intramolecular Catalysis – Acceleration of a chemical reaction (at a particular site on a molecular entity or complex) by one or more functional groups located at other site(s) within the same molecular entity. Intramolecular catalysis can often be expressed by comparison to the rate of the reaction when the functional group(s) is(are) removed from the same molecular entity or by determination of the effective molarity of the catalytic group. Intramolecular Isotope Effect – An isotope effect (either kinetic or equilibrium) resulting from reactions in which the different isotopes occupy chemically equivalent alternative reactive sites within the same molecular entity. In such cases, isotopically distinct products are formed. Intramolecular Kinetic Isotope Effect – A kinetic isotope effect observed by a single reactant, having isotopic atoms at equivalent reactive positions, which reacts to produce isotopomeric products with a nonstatistical distribution. The pathway favored will be the one having lower force constants for the displacement of the isotopic nuclei in the transition state. Intrinsic Binding Energy – The standard free energy change (DGint) corresponding to the energy released when a protein binds to a compound (or a substituent group of that compound) without destabilization or energy losses attributable to translational, rotational, and internal entropy. As pointed out by Jencks (1975), the intrinsic binding energy will never be directly observable, because binding of one molecule to another is almost always attended by a net loss of translational and rotational entropy. In other words, the observed binding energy of protein-ligand bond formation typically reflects losses of translational and rotational degrees of freedom, often amounting to –40 entropy units. The intrinsic binding energy will also be underestimated by the net effect of the mutual freezing out of rotations and vibrations in the interacting components. Likewise, a protein or enzyme that destabilizes its ligand by distortion (i.e., transition-state stabilization as

opposed to ground-state stabilization) will also have an intrinsic binding energy that is better estimated through the binding of a so-called transition-state analog. Furthermore, Jencks (1975) has stressed that the free energy of binding is affected by induced-fit interactions as well as nonproductive substrate complexation; these interactions are likely to exercise control over specificity without contributing significantly to catalytic rate enhancement. Intrinsic Kinetic Isotope Effect – An effect of isotopic substitution within a reactant or substrate on a specific step in an enzyme-catalyzed reaction. The magnitude of an intrinsic isotope effect may not equal the magnitude of an isotope effect on collective rate parameters such as Vmax or Vmax/Km, unless the isotopically sensitive step is the rate-limiting or rate contributing step. Ionic Atmosphere – A term used in electrostatic descriptions of ions to denote the continuous electric charge density [r(r)] surrounding an ionic species. On average, an ion will be surrounded by a spherically symmetrical distribution of counter ions that form its ion atmosphere. Ionization Energy – The minimum amount of energy or work (i.e., enthalpy) required in the gaseous phase to remove an electron from an isolated molecular entity (typically in its vibrational ground state). Adiabatic ionization energy is that energy needed to produce a new molecular entity that is also in its vibrational ground state. If the product is not in this ground state and possesses the energy dictated by the Franck-Condon principle, the energy needed is referred to as the vertical ionization energy. Isoelectronic – Referring to two or more molecular entities having the same number and connectivity of atoms (although not necessarily the same elements) as well as the same number of valence electrons. Thus, CO, N2, and NO+ are isoelectronic, as are ketene (CH2¼C¼O) and diazomethane (CH2¼N¼N). Isomerases – A major class of enzymes that catalyze changes within one molecule. Examples include racemases, epimerases, mutases, and tautomerases. Isomerization – A chemical reaction in which the principal product is isomeric with respect to the principal reactant. Isomerization can occur in the absence of molecular rearrangements (for example, between conformational isomers). Isotope Exchange – A reaction in which the reactants and products exchange atoms or groups of atoms that are chemically identical and differ only in isotopic composition. For example, the exchange of deuterium in D2O with the labile hydrogens of glucose in solution represents this type of isotope exchange. The exchange of an isotope from one isotopically substituted substrate in a multi-substrate enzyme-catalyzed reaction, with at least one of the other substrates. Isotope Scrambling – A change in the distribution of isotopes within a specified set of atoms of a particular molecular entity or entities as a consequence of enzyme catalysis. Isotopically Sensitive Step – An elementary reaction or step (in a chemical process) for which the rate constants are altered by an isotopic substitution in substrate, product, or solvent.

Glossary

Isozyme – Enzymes differing in amino acid sequence, but catalyzing the same chemical reaction. Isozymes need not operate by the same catalytic mechanism. Joule – A unit of energy (symbolized by J) equal to 107 ergs. In the SI system, the joule replaces the calorie, which is equivalent to 4.184 J. The joule formally equals the heat generated by an ampere flowing through a 1-ohm resistor over a one second interval. Kinetic Resolution – Partial or complete resolution of the enantiomers in a racemate as a consequence of unequal reaction rates of enantiomers with an enzyme or another chiral agent. See Section 5.9: Enantiomeric Enrichment and Anomeric Specificity. Lag Time – 1. The pre-steady-state portion of progress curve, during which there is an exponential increase in the slope of the product concentration versus time plot. 2. The slow increase in the slope of the product concentration versus time progress curve often observed when using coupled-enzyme assay protocols. Investigators strive to minimize such lag times. 3. A slow increase in the slope of the product concentration versus time or progress curve. Such systems respond slowly to rapid changes in substrate concentrations and are often associated with reaction schemes containing a slow isomerization step. Langevin Equation – A stochastic differential equation of the form:

_ mvðtÞ ¼ gvðtÞ  vx UðxðtÞ; sðtÞÞ þ f ðtÞ þ jðtÞ

used in coarse-grained modeling of the stepwise motions of molecular motors. Included are: the particle’s mass m; accel_ eration vðtÞ; viscous coefficient g; nonlinear periodic potential UðxðtÞ; sðtÞÞ that is a function of particle’s position x(t) and state s(t); an external force term f(t); as well as a zero-mean white Gaussian noise term j(t) having a variance of 2gT, where T is the absolute temperature. A motor’s movements on its polymer track (e.g., actin filament and microtubule) may be treated as a particle undergoing one-dimensional diffusion along the reaction trajectory, while jostled haphazardly by the thermal motions of solvent molecules. For a spherical particle of radius r, the mass m is given by: m ¼ 4(prr3)/3, the drag coefficient z is given by: z ¼ 6pht, and the timescale t0 for inertial equilibration is given by: t0 ¼ m/z ¼ (2r/9h)r3, where r is the particle’s density, and h is viscosity. For molecular motors, the effects of viscous drag are so great that the motor attains mechanical equilibrium almost instantaneously For particles of 5–50 nm radius, t0 is in the range of 5–30 ps – extremely short relative to the ms time-scale for the catalytic cycle typical of most mechanoenzymes. Thermal noise is, in fact, so considerable that the nanoscale motions of molecular motors are best described as a random walk. The Langevin equation contains force terms accounting for solvent effects as well as random force effects, which, when combined with Newton’s second law, account approximately for the effects arising from otherwise neglected degrees of freedom. Langevin equations are inherently stochastic, and by accounting for frictional drag and random momentum transfers to and from the motor from thermal motions, they are well suited for describing how solvent molecules alter molecular motor kinetics by expediting transition-state barrier crossing (Astumian and Bier, 1995).

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Latent Activity – The amount of total enzymatic activity that becomes manifest only after disruption of membranous barriers between enzyme and substrate or upon removal of some otherwise inhibitory factor. Membrane disruption is often achieved by treatment with detergent to solubilize the enzyme. One example is the so-called microsomal glucose-6 phosphatase1, an enzymatic activity that is located in the lumen of the endoplasmic reticulum but becomes trapped as a latent activity in microsome vesicles upon mechanical disruption of cells. Lead – A term used in high-throughput screening experiments to indicate a library component that consistently scores as a hit. See Hit. Leaving Group – An atom, group, or moiety (either charged or neutral) that detaches from the main portion of the substrate or reactant in a given reaction. If the leaving group detaches and carries away the electron pair that formerly constituted the bond to the central atom, the leaving group is said to be nucleofugal. If the electron pair is not removed, the leaving group is said to be electrofugal. A leaving group may also be termed an exiphile. See also Entering Group. Lewis Acid – Any chemical species having a vacant orbital and thus acting as an electron-pair acceptor from a Lewis base (Lewis, 1923). Examples of Lewis acids include: BF3, Fe3+, Na+, Ca2+, and SO3, as well as Zn2+ in carbonic anhydrase. Lewis Base – Any chemical species that can donate a pair of electrons to a Lewis acid to form a Lewis adduct (Lewis, 1923). Examples of Lewis bases include: NH3, R–NH2, R–OH, CO, CO32–, and R–SH. Lifetime – The time (symbolized by t) needed for a concentration of a molecular entity to decrease, in a first-order decay process to e–1 of its initial value. In this case, the lifetime (sometimes called mean lifetime) is equal to the reciprocal of the sum of rate constants for all concurrent first-order decompositions. If the process is not first-order, the term apparent lifetime should be used, and the initial concentration of the molecular entity should be provided. Ligases – A major class of enzymes, also referred to as synthetases that catalyze the joining of two entities with the concomitant hydrolysis of molecules such as ATP or GTP. (Mechanoenzymes share many aspects in common with ligases. Although ATP- and GTP-hydrolyzing mechanoenzymes are classified as hydrolases, they are so misnamed for their catalysis of a partial reaction (ATP + H2O # ADP + Pi).) See Energases. Limit Dextran – An oligosaccharide that cannot be further degraded by a polysaccharide hydrolase or a polysaccharide phosphorylase. The types of limit dextrans formed by various amylases and phosphorylases can provide useful information about ligand binding at the active site, and the structure of a limit dextran is also useful for recognizing and identifying new enzyme activities. Linear Free-Energy Relation – This classical chemical kinetic approach for analyzing the structures of transition states is based on the chemist’s ability to introduce modest changes in the structure of a reactant and then to quantify how such changes perturb the kinetics (k) and equilibrium (Keq) of a reaction of interest. In favorable circumstances, one observes a linear relationship between the change in

Glossary

854 activation energy DGz and the change in equilibrium free energy DG0. In the Brønsted treatment, the so-called b value is a measure of extent of covalent bond-making or -breaking in the reaction, with b ¼ 0 implying no bond-making/breaking in the approach to the transition state, b ¼ 1 implying _ complete making/-breaking, and intermediate b values indicating partial bond-making/-breaking. A plot of the log k or log Keq for one series of reactions versus the log k or log Keq for a related series of reactions. (Recall that at constant temperature and pressure the logarithm of an equilibrium constant is proportional to DG , and the logarithm of a rate constant is proportional to DGz). An example of a linear free energy relation is the Hammett sr-equation. With equilibrium constants, this relationship is given by the expression: log (Kx/KH) ¼ sXr, where KH is the equilibrium constant where the substituent is H, Kx is the equilibrium constant where the substituent is X, sX is a constant characteristic of the substituent X, and r is a constant for a given reaction under a well-defined set of experimental conditions. Lone Pair – A pair of nonbonding electrons localized in the valence shell on a single atom. Examples are the lone electron pair on the nitrogen atom in ammonia and the two lone pairs on the oxygen atom in water. In these cases, the lone pairs participate in hydrogen bonding interactions. Loose Ion Pair – An ion pair in which the constituent ions are separated by one or more solvent (or other neutral) molecules. If X+ and Y– represent the constituent ions, a loose ion pair is usually symbolized by X+j jY–. The constituent ions of a loose ion pair can readily exchange with other ions in solution; this provides an experimental means for distinguishing loose ion pairs from tight ion pairs. In addition, there are at least two types of loose ion pairs: solvent-shared and solvent-separated. Low-Spin Complex – This spin state, which occurs when the highest-occupied molecular orbital is occupied by two paired electrons ([Y), is one of two alternative electronic states arising when the separation between the highest occupied and the lowest unoccupied molecular orbitals is not large. The lowspin state is the ground state if the one-electron energy required to promote an electron to the lowest-unoccupied molecular orbital is larger than the energy required to pair up two electrons in the highest energy occupied molecular orbital. In any coordination entity with a particular dn (1 < n < 9) configuration and a particular geometry, if the n electrons are distributed so that they occupy the lowest possible energy levels, the entity is a low-spin complex. If some of the higher energy d orbitals are occupied before all the lower energy ones are completely filled, then the entity is a high-spin complex. Detection of a low-spin state most often indicates a change in oxidation state of a metal ion. Mass-Action Ratio – The ratio of the product of the concentrations (or activities) of all the products of a reaction to the product of the concentration (or activities) of all the reactants (or substrates). This ratio, often symbolized by G, will change as the reaction progresses until equilibrium is reached (i.e., at t ¼ N), at which point G ¼ Keq.

Maximal Velocity (Vmax) – The limiting velocity of an enzymecatalyzed reaction (units ¼ DMolarity per second) Vmax must

not be confused with kcat (units ¼ s–1) or specific activity (units ¼ mmol product per second per milligram of enzyme).

Maxwell – A unit, symbolized by Mx, of magnetic flux through a square centimeter normal to a field of one gauss. One maxwell is equal to 10–8 weber. Mean Transit Time – The average period of time during which a metabolite (or drug) remains in the volume of distribution; also referred to as the mean residence time. Michaelis Constant – A composite kinetic parameter (units ¼ molarity) consisting of at least two rate constants and defining the substrate concentration at which v is 0.5 Vmax. See Section 4.4: Additional Comments on Kinetic Parameters. Michaelis-Menten Equation – A one-substrate rate equation based on the rapid-equilibrium formation of ES complex, which then decomposes slowly to liberate free enzyme and product P, thereby defining the dependence of initial reaction velocity v on the free substrate concentration [S] or total enzyme. See Section 5.1: Michaelis-Menten Kinetics. Microscopic Reversibility – A physical principle requiring that, under equilibrium conditions, any molecular process (or reaction) and the reverse of that process (or reaction) will occur with the same frequency. This principle was verified through the use of certain quantum mechanical expressions for transition probabilities. The Principle of Microscopic Reversibility and its large-scale consequence, known as the Principle of Detailed Balancing, enable investigators to understand the mechanism of the reverse reaction to the same level of accuracy as that achieved for the forward reaction. This principle has only limited application to reactions that are not at equilibrium. Furthermore, the Principle of Microscopic Reversibility does not apply to reactions commencing with photochemical excitation. Mixing Time – The period that elapses before two or more solutions are thoroughly mixed in a chemical kinetic experiment. In most manually controlled chemical kinetic studies, the mixing time is rarely a factor affecting accurate data acquisition; however, the mixing time can be significant in rapid kinetic processes studied by continuous and stopped-flow kinetic techniques. Modifier – A term applied to substances or agents that either depress or elevate the action of a particular catalyst. While often used synonymously with effector, the International Union of Biochemistry has suggested that the term modifier is more appropriate for those substances that are artificially added to in vitro systems. Other investigators restrict the use of the term modifier to those agents that covalently modify the structure of the catalyst, thereby altering the catalyst’s activity. Molecular Dynamics – Any of a series of computational methods for deducing molecular structure and reactivity by treating molecules or a system atoms comprising a segment of a macromolecule in terms of quantum and molecular mechanics. The basic goal is to explain molecular structure on the basis of the forces associated with and between bonded and non-bonded atoms. Quantum mechanical treatments become cumbersome when the number of atoms within the system is increased. For this reason one tends to use molecular mechanics, which is an empirical method, also called

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force-field calculations. These approximate methods allow one to estimate the relative energy differences among various molecular conformations by taking into account natural bond lengths and angles, influence of strain, torsional interactions, and attractive and/or repulsive electronic, dipolar, and van der Waals forces.

concentration of active catalyst. This technique usually allows one the opportunity to evaluate the catalytic rate constant kcat, which is the first-order decay rate constant for the rate-determining step for each cycle of catalysis, and one can evaluate the magnitude of other parameters such as the substrate’s dissociation constant or Michaelis constant.

Mole Fraction – A measure of the fractional composition of a solution containing one or more substances in amounts nA, nB, nC, . (where ni in the amount-of-substance i expressed in mol), such that for any component, say B, mole fraction ¼ XB ¼ nB/ [nA + nB + nC + .]. Technically, one includes solvent as one of the components when expressing mole fraction in chemical thermodynamics, but in describing dilute biological solutions, nsolvent is often omitted. With multi-component solutions, one may choose to analyze the fractional composition of any two (or more) substances while not including others held constant in the experiment (e.g., concentrations of buffer components, proton, supporting electrolyte(s), enzyme, etc.). For two components, XA ¼ nA/[nA + nB] and XB ¼ nB/[nA + nB]. In this case, XA + XB ¼ 1, and this formulation allows one to analyze binding interactions between two such components by the method of continuous variation.

n / p* Transition – An electronic transition in which an electron in a nonbonding (e.g., lone pair) orbital (also called an n-orbital) is promoted to a p-antibonding orbital. The excited state arising from such a promotion is often referred to as an n-p* state. An n-orbital electron typically interacts strongly with a polar solvent; this is less likely to be the case for an electron in a p* orbital. Therefore, the energy difference between n and p* orbital electrons will increase when a substance is placed in a more polar solvent; this is manifested as a shift to shorter wavelength (often called a blue shift) for light absorption.

Mo¨ssbauer Spectroscopy – The Mo¨ssbauer effect refers to the absorption and reradiation of a gamma ray by a non-recoiling nucleus. Gamma rays of extremely narrowly defined wavelength exhibit this effect with certain nuclei, especially 57Fe, allowing Mo¨ssbauer spectroscopy to detect very subtle changes in electron configuration, chemical bonding, and oxidation state during substrate binding and enzymic catalysis. The following are Mo¨ssbauer isomer shifts for several classes of iron complexes: Fe(II), high-spin, d ¼ ~1.3 mm/sec; Fe(III), _ high-spin, d ¼ ~ 0.5-0.7 mm/sec; Fe(II), low-spin, d ¼ _ ~0.1 mm/sec; and Fe(III), low-spin, d ¼ ~0 mm/sec (Lippard _ and Berg, 1994). The interested reader should consult the outstanding entry on Mo¨ssbauer spectroscopy in volume IV of Comprehensive Biological Catalysis (Sinnott, 1999). Multiple Dead-End Inhibition – Reduced catalytic activity that occurs whenever an inhibitor binds more than once to a single enzyme form (or forms). While standard double-reciprocal plots are usually linear, secondary replots of the data (i.e., plots of slopes and/or intercepts versus [I], the concentration of the inhibitor) will be nonlinear depending on the relative magnitude of the [I]2, [I]3, ., and [I]n terms in the rate expression. Cleland (1963) presented more complex examples of multiple dead-end inhibition in which the kinetic expression for the slope or intercept is a function containing polynomials of the inhibitor concentration in both the denominator and numerator. For example, if the slope is equal to a function having the form (a + b[I] + c[I]2)/(d + e[I]) in which a, b, c, d, and e are constants or collections of constants, then the slope is said to be a 2/1 function (the numbers representing the highest power of [I] in the numerator and denominator, respectively). The nonlinearity of the slope replot, in this case, is dependent on the relative magnitudes of the constants in the expression. Multiple-Turnover Conditions – Reaction conditions that permit a catalyst to operate for many catalytic cycles. Multipleturnover conditions are usually obtained by maintaining the substrate concentration in substantial excess over the

n / s* Delocalization – Delocalization of a free electron pair into an antibonding or s*-orbital. n / s* Transition – An electronic transition in which an electron in a nonbonding (e.g., lone pair) orbital is promoted to an antibonding or s*-orbital. Natural Abundance – A quantitative measure of isotope composition relative to the abundance of all isotopic forms found in nature. Values for those stable isotopes most commonly employed in biological tracer experiments are: 1H, 99.985%; 2H, 0.015%; 12C, 98.90%; 13C, 1.10%; 14N 99.63%; 15 N, 0.37%; 16O, 99.76%; 17O, 0.04%; and 18O, 0.20%. Negative Cooperativity – Ligand interactions with oligomeric or polymeric macromolecules, for which binding of the first (or preceding) ligand molecule decreases the likelihood for binding of the next (or subsequent) ligand molecule. In the Koshland-Ne´methy Filmer model, negative cooperativity occurs when the dissociation constant for ligand binding to the (i + 1)-site is greater than the dissociation constant for ligand binding to the ith site. Note that negative cooperativity cannot occur in the Monod-Wyman-Changeux allosteric transition model, because the dissociation constant is equivalent for all sites. Thus, positive cooperativity can only result in this binding mechanism as a consequence of the ‘‘recruitment’’ of binding sites from the T-state in an all-or-none transition to the R-state. Any occurrence of negative cooperativity can be regarded as prima facie evidence against the applicability of the MonodWyman-Changeux model to the system under investigation. Newton – The SI unit (symbolized by N) for force; equal to kilogram-meter-second2. The name of this classical mechanical term stems from Isaac Newton’s theory of classical mechanics, such that F ¼ ma, where acceleration a ¼ dv/dt ¼ d2r/dt2. NIH Shift – An intramolecular hydrogen migration observed in the hydroxylation of aromatic rings in certain enzymecatalyzed reactions as well as some chemical reactions. The rearrangement was first observed at the National Institutes of Health (hence the name ‘‘NIH’’) in studies of the synthesis of L-tyrosine from L-phenylalanine via phenylalanine hydroxylase. Observation of this shift requires site-specific deuteration of the aromatic substrate.

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Nitrene – A molecule containing a neutral nitrogen atom with four nonbonding electrons. While two of these nonbonding electrons are paired, each of the others may have parallel spins (thus, triplet state) or antiparallel spins (singlet state). Generated photochemically, nitrenes are the nitrogen analogs of carbenes. Nonlinear Inhibition – This term usually applies to reversible inhibition of an enzyme-catalyzed reaction in which nonlinearity is detected (a) in a double-reciprocal plot (i.e., 1/v versus 1/[S]) in the presence of different, constant concentrations of inhibitor or (b) in replots of slope or intercept values obtained from primary plots of 1/v versus 1/[S]). Nonlinearity replots can be attributable to a number of factors: cooperativity; multiple inhibition (for example, the formation of an EI2 complex); partial inhibition; substrate inhibition; inhibitor induced substrate inhibition; tight-binding enzyme inhibition; or in certain cases of product inhibition. Before proceeding to consider these more complicated explanations, one should first take measures to exclude trivial reasons for nonlinearity, such as lack of constant pH, temperature variability, instability of substrate and protein, etc. Nuclear Magnetic Resonance – The magnetic resonance phenomenon arises when applied magnetic field H0 interacts with the magnetic field generated by a nucleus (here, for simplicity, a proton with nuclear spin states of +1/2 and –1/2) having angular momentum as a consequence of its spinning motion. The spinning nucleus has a magnetic field strength, indicated by its magnetic moment m, which becomes direc_ tionally oriented with the applied magnetic field. Because the nucleus precesses, m is oriented at angles of u and (180 – u) relative to the lines of force in an applied field. At angle u, m is _ aligned with the direction of the applied field, whereas at angle  (180 – u), the opposite is true. For this reason, the respective potential energy of these two states is –mH0 sin u and +mH0 _ sin u, resulting in a quantized energy difference DE given by _ the expression: DE ¼ 2mH0 sin u ¼ hn, where h is Planck’s _ constant. Thus, electromagnetic radiation with a frequency n can be absorbed, thereby promoting the system to the higher energy state. Such energies correspond to photons in the radiofrequency range, and a radiofrequency generating coil is positioned around the sample within the cavity of the NMR spectrometer. NMR provides valuable information regarding molecular structure, and we can briefly consider two phenomena, namely chemical shift and spin-spin coupling. See Chemical Shift. One readily senses the great versatility and power of NMR as a structural probe by considering so-called splitting patterns (or, more formally, absorption band multiplicities). These splitting patterns define the positions of nuclei within a molecule. Splitting is a consequence of reciprocal magnetic interactions between spinning nuclei, and the interaction likewise depends on electronic structure. Spin-spin coupling creates multiplets from what would otherwise appear as sharp resonance lines, and the coupling strength J depends on the particular nuclei under examination. For example, in the case of n spin-coupled protons, the multiplet will have n + 1 lines. A single nearby proton will split the observed proton resonance into a doublet of equal or (1:1) intensity, two nearby electrons will split the observed proton resonance into a triplet with

Glossary

intensities of 1:2:1, and so on. The intensities are always coefficients of a binomial expansion. So-called multi-dimensional NMR techniques can provide important information about macromolecular conformation. In these cases, the sequence of a protein is already known, and establishing covalent connectivity between atoms is not the goal. Rather, one seeks through space information that can reveal the solution conformation of a protein or other macromolecule. Two- or three-dimensional techniques use pulses of radiation at different nuclear frequencies, and the response of the spin system is then recorded as a free-induction decay (FID). Techniques like COSY and NOESY allow one to deduce the structure of proteins with molecular weights less than 20,000–25,000). The following naturally occurring nuclides are of sufficient natural abundance and have nuclear spin properties that commend their use in NMR investigations of enzymic processes: 1H or protium (Natural Abundance ¼ 99.985%), Spin I ¼ ½; 2H or deuterium (Natural Abundance ¼ 0.015%), Spin I ¼ 1; 13C or carbon-13 (Natural Abundance ¼ 1.10%), Spin I ¼ ½; 15N or nitrogen-15 (Natural Abundance ¼ 0.366%), Spin I ¼ ½; 17O or oxygen-17 (Natural Abundance ¼ 0.003%), Spin I ¼ 5/2; and 31P or phosphorus-31 (Natural Abundance ¼ 100%), Spin I ¼ ½. Still others can be substituted into biological molecules to provide a diverse range of opportunities. For example, fluorine [19F (Natural Abundance ¼ 100%), Spin I ¼ ½] can be substituted for hydrogen in many cases. Nucleophile – A molecular entity preferentially attracted to an electrophilic region or some other site of low electron density in another molecule. A nucleophile typically brings a pair of electrons to the second molecular entity, thereby qualifying it as an electron-pair donor (or Lewis base). Outer-Sphere Electron Transfer Reaction – A well-established electron transfer path observed with redox-active transition metal ion complexes, especially when two reacting species require more time to exchange ligands than that required to undergo electron transfer. Each complex maintains its full complement of bound ligands during electron transfer, and the transferred electron must traverse both coordination shells. Oversaturation – A phenomenon associated with noncompetitive product inhibition, wherein the halftimes for approach to equilibrium divided by the initial substrate concentration are observed to increase with increasing substrate concentrations. As pointed out by Cleland (1990), this would not be the case for such time courses (normalized with respect to substrate concentration) if the product inhibition were competitive. In the case of proline racemase (Fisher, Albery and Knowles, 1986), the observation of oversaturation suggests that the enzyme operates by an iso mechanism (i.e., the form of free enzyme reacting with the L-isomer is different from the enzyme form reacting with the D-isomer). Pacemaker Enzyme – The slowest enzyme-catalyzed reaction within a metabolic pathway; sometimes referred to as the rate-limiting step (although that term is also used to describe a step in the reaction pathway of a single reaction). The pacemaker step may change, depending on availability of

Glossary

substrates and/or effectors as well as reaction conditions. For example, in red blood cells, hexokinase is thought to be the pacemaker for glycolysis, but hexokinase and phosphofructokinase both act as glycolytic pacemakers in many other cell types. Parallel Reactions – 1. A set of reactions in which a specific reactant can proceed to react by two or more pathways, each independent of the others. 2. A set of reactions in which a common product is produced from different reactants, each reaction being independent of the others. The term ‘‘parallel reactions’’ is not a synonym for ‘‘side reactions’’. Partial Competitive Inhibition – This type of inhibition differs from that exhibited by classical competitive inhibitors, because the substrate can still bind to the EI complex and the EIS complex can go on to form product (albeit at a slower rate) without the inhibitor being released from the binding site. While standard double-reciprocal plots of partial competitive inhibitors will be linear (except for some steady-state, i.e., nonrapid-equilibrium, cases), secondary slope replots will be nonlinear. Partial Inhibition – A form of reversible inhibition in which the inhibitor-substrate(s)-enzyme complex can still generate product(s), without the inhibitor dissociating (albeit at a slower rate when compared to the inhibitor-free system). Partial Pressure – For a mixture of gases in a specified volume, the partial pressure of a component of the mixture is the pressure exerted by that compound if it were the sole occupant of the volume. For example, the partial pressure of oxygen is around 152 mm of mercury or about 0.21 atmospheres at sea level and 298 K. Pascal – The SI unit (symbolized by Pa), equal to one newton per square meter, for pressure or stress. One pascal equals ten baryes, 10–5 bar, and 9.869233  10–6 atmosphere.

Patch Clamp – An electrophysiologic ion current-measuring device that is comprises (a) an excised membrane patch that has become sealed to the orifice of a glass micropipette and (b) a voltage clamp electronic circuit for measuring ion channel currents. The patch clamp relies on formation of a giga ohm seal between the recording pipette and the cell membrane. This approach allows the investigator to prepare (a) inside-out patches with their cytoplasmic membrane face exposed to the bath solution, and (b) outside-out patches with their extracellular membrane face exposed to the bath solution. The technique offers the advantage that solution composition on each side of the membrane patch can be readily manipulated. Electrophysiologic measurements are both accurate and precise.

Phospholipid Flip-Flop – The pair-wise exchange of phospholipid molecules across a membrane bilayer. The polar head groups of phospholipids resist entry into the apolar core of the membrane bilayer, and phospholipid flip-flop is both improbable and sluggish (i.e., kexchange z 10–5 s–1). Kornberg and McConnell (1971) used nitroxide spin-labeled phospholipid vesicles to follow the time course of flip-flop. In their clever experiments, ascorbic acid was added to the bulk solvent to reduce any nitroxide spin label on the exterior surface of the lipid. Then, the time course of flip-flop could be determined by

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the time course of the diminution in the amplitude of the ESR spectrum that measured the amount of spin-label remaining on the inner, and thus solvent-inaccessible membrane surface. Peptidomimetic – A synthetic compound containing non-peptidic structural elements allowing it to behave as an isosteric mimic of a naturally occurring parent peptide. Pharmacophore – A hypothetical ensemble of steric and electronic features believed to be essential for the biospecific action of a drug on its macromolecular target (e.g., enzyme, receptor, adaptor proteins, structural protein, DNA, RNA, etc.). Phosphorescence – 1. Long-lived luminescence. 2. Luminescence involving a change in spin multiplicity (e.g., triplet-to-singlet, singlet-to-triplet, quartet state-to-doublet state, etc.). Photolysis – A light-induced bond-cleavage reaction. Pi-to-Pi* (p/p*) Transition – An electronic transition in which a p-orbital electron is promoted to an antibonding p-orbital. The excited state arising from such a promotion is often referred to as a p–p* state. Pi-to-Sigma* (p/s*) Transition – An electronic transition in which a p-orbital electron is promoted to a s antibonding orbital. Planck’s Constant – The proportionality constant, symbolized by h, relating the frequency of electromagnetic radiation to its quanta of energy (i.e., E ¼ hn) : h ¼ 6.6260755  10–34 J$s (or 6.6260755  10–27 erg$seconds). The reduced quantity S, equal to h/2p, is used for expressing energy relationships for harmonic processes. Positive Cooperativity – Any set of ligand interactions with oligomeric or polymeric macromolecules, such that binding of the first (or preceding) ligand molecule increases the likelihood for binding of the next (or subsequent) ligand molecule. In the Monod-Wyman-Changeux allosteric transition model, the dissociation constant for ligand interactions is equivalent for all sites, but cooperativity results from the disproportionate ‘‘recruitment’’ of binding sites from the so-called T-state in an all-or-none transition to the R-state. In the Koshland-Ne´methyFilmer model, the dissociation constant for ligand binding to the PL(i) state is lower than the dissociation constant for ligand binding to the PL(i–1) state. Potential Energy Surface (or Potential Energy Landscape) – A diagram depicting the variation of the potential energy associated with the reactants and products of an elementary reaction in which the energetically easiest progress (for movement of a representation point of the reaction system) is plotted as a function of two or three coordinates (usually representing molecular geometries) of a multidimensional, potential energy hypersurface. Typically, the diagram is a depiction having isoenergetic contour lines and resembles a topographic map. In most two-coordinate systems, for elementary reactions, the coordinates chosen are related to two significant variables that change during the course of a reaction (e.g., two distinct interatomic distances). For more complicated reactions involving a number of elementary reactions, care should be exercised in selecting the coordinates in such systems. In some publications a third coordinate represents the standard Gibbs free energy instead of potential energy. The energetically

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easiest route on the contour map is referred to as the potential energy profile. Pre-equilibrium – A term (also referred to as ‘‘prior equilibrium’’) denoting any reversible step that precedes an irreversible step or the rate-limiting step in a multistage reaction mechanism. The so-described reaction step must have rate constants of sufficient magnitude to establish a rapid, fully reversible equilibrium between its reactants and products. The first association/dissociation equilibrium leading to the formation of EX complex from E and S in the MichaelisMenten treatment is an example of a pre-equilibrium. Pressure-Jump Method – A fast-reaction kinetic technique used to achieve a rapid change in external pressure that results in a sudden change in the equilibrium constant for a particular system. The investigator then analyzes the rate of approach of the system to the new equilibrium position. Pre-Steady-State Phase – The initial period of nonlinear product formation, commencing with the initiation of the reaction and ending when the system is at steady state. Typically, the presteady-state phase lasts from milliseconds to a few seconds after mixing reactants. The time-course of pre-steady-state rate processes often can be evaluated using stopped-flow, temperature-jump, and mix-quench methods. Primitive Change – A simple molecular change into which an elementary reaction can conceptually be further dissected. Primitive changes would include bond rupture, bond rotation, change in bond length, a redistribution of charge, etc. The concept of primitive change is useful when considering elementary reactions, but a primitive change does not itself represent a discrete chemical process. Prosthetic Group – A very tightly bound coenzyme. Examples of prosthetic groups include the hemes, cobalamin coenzymes, pyridoxal 5-phosphate, lipoic acid, and biotin. FAD and FMN are also very held by most flavin-dependent oxidoreductases. In many instances, distinctions are made purely on the basis of affinity and do not apply consistently. For example, while NAD+ and NADP+ binding affinities are relatively high, dialysis is frequently sufficient for resolving (removing) these coenzymes from their binding sites on NAD(P)+ dependent oxidoreductases. Quantitative Structure-Activity Relationship (QSAR) – Mathematical relationships that link a drug’s pharmacological activity to its chemical and/or structural features, usually by various statistical, pattern recognition, and structure-determination techniques. Quantum Yield – A value (symbolized by f) equal to the number of molecules transformed via a reaction per quantum of light absorbed. It is synonymous to quantum efficiency and is the reciprocal of the quantum requirement. A primary quantum yield is the fraction of light absorbing molecules that are converted in a particular process. Unfortunately, such values are often very difficult to measure. A product quantum yield is the ratio of the number of molecules of product formed per number of quanta absorbed by the reactant. That product formation is the rate-determining step is a strong possibility if the observed f value does not vary with experimental conditions. The differential quantum yield is given by f ¼ {d[x]/dt}/n

where d[x]/dt is the rate of change of the concentration or of the amount of some measurable quantity and n is the amount (e.g., moles) of photons absorbed per unit time. The integral quantum yield is simply defined as the number of events per number of photons absorbed. Quaternary Complexes – Complexes formed from the ordered or random combination of four distinct entities. Many threesubstrate enzymes proceed via the formation of a reversible E$A$B$C quaternary complex. Racemases – A class of enzymes that catalyze the interconversion of one enantiomer with its mirror image. Care must be exercised in applying this term. For example, the enzyme that interconverts D-methylmalonyl-CoA to L-methylmalonylCoA is not a racemase, but is instead an epimerase; the two coenzyme A derivatives are diastereoisomeric, and not enantiomeric, with respect to each other. Racemization – Any reaction, either chemical or biochemical, which results in the conversion of an optically active molecular entity to a optically inactive, or racemic, mixture. Radical (or, Free Radical) – 1. Any molecular entity possessing an unpaired electron. The modifier ‘‘unpaired’’ is preferred over ‘‘free’’ in this context. The term ‘‘free radical’’ is to be restricted to those radicals, which do not form parts of radical pairs. Further distinctions are often made, either by the nature of the central atom having the unpaired electron (or atom of highest electron spin density) such as a carbon radical (e.g., $CH3) or whether the unpaired electron is in an orbital having more s character (thus, s radicals) or more p character (hence, p radicals). Whenever presenting the radical molecular entity in a manuscript, the structure should always be written with a superscript dot or, preferably, a center-spaced bullet (e.g., $OH, $CH3, Cl$). 2. Any substituent or moiety bound to a molecular entity. IUPAC suggests that this older term should be abandoned, preferring instead usage of ‘‘groups’’, ‘‘moieties’’, or ‘‘substituents’’. Radical Ion – An electrically charged radical. Hence, a radical cation carries a positive charge and a radical anion carries a negative charge. The charge and the odd electron are often localized with the same atoms of the molecular entity. According to IUPAC recommendations, if the unpaired electron and the charge cannot be associated with specific atoms of the molecular entity (or they are both localized to the same atom), then a superscript (or center space) dot or bullet should precede the superscript charge (+ or –). Random Walk – A path resulting from a set of successive steps (or excursions), whose individual lengths and directions are only randomly related to each other. Thus, the preceding step has no influence at all on any subsequent step. Rapid-Start Complex – The complex that RNA polymerase forms at the promoter site just prior to initiation. Some bacterial promoters require high NTP concentrations to initiate efficient transcription, because this represents a ‘‘status report’’ on the stores of ATP, UTP, GTP, and CTP needed for RNA synthesis. Nature has evolved a kinetic control device: high initiating ATP and GTP concentrations must be present to stabilize an otherwise short-lived polymerase-promoter complex. The reader may also recall that bacterial translation is also tightly

Glossary

controlled, and amino acid starvation leads to ppGpp synthesis, the so-called stringent response agent that also potently inhibits RNA polymerase. Such kinetic control ensures that NTP and amino acid concentrations are adequate before transcription and translation occur. Rate Constant – The proportionality constant allowing one to equate reaction velocity (having units of M$s–1) to the molarity of reactant(s) involved in the reaction. For zero-orderprocesses, v ¼ k, where k has units of Ms–1; for unimolecular processes, v ¼ k[X], where k has units of s–1; for bimolecular processes, v ¼ k[X]2 or k[X][Y], where k has units of M–1 s–1; and for termolecular processes, v ¼ k[X]3 or k[X]2[Y] or k[X][Y]2 or k[X][Y][Z], where k has units of M–2 s–1. Rate-Contributing Step – Any reaction step having a rate constant whose magnitude is nearly that of the rate constant in the slowest step in a reaction mechanism. Such a step is said to be a contributing factor in slowing down the reaction rate. Rate-Controlling Step – Synonym for rate-determining step and for rate-limiting step. If a rate constant for an elementary reaction has a stronger influence on the overall rate than any other rate constant in the mechanism, then the step associated with that rate constant is referred to as the rate-controlling step. Note that the rate-controlling step may change with a corresponding change in reaction conditions. Redox Potential – The reduction-oxidation potential (typically expressed in volts) of a compound or molecular entity measured with an inert metallic electrode under standard conditions against a standard reference half-cell. Any oxidation-reduction reaction, or redox reaction, can be divided into two half-reactions, one in which a chemical species undergoes oxidation and one in which another chemical species undergoes reduction. In biological systems the standard redox potential is defined at pH 7.0 versus the hydrogen electrode and partial pressure of dihydrogen of 1 bar.

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Reference Reaction – A nonenzymic reaction used to evaluate the magnitude of the catalytic rate enhancement achieved by a corresponding enzyme-catalyzed reaction. For example, Radzicka and Wolfenden reported that orotic acid is decarboxylated with a t1/2 of 78 million years (or, about 2.5  1015 seconds) at room temperature in neutral aqueous solution, as suggested by the kinetics determined in sealed quartz tubes maintained at elevated temperatures. Thus, based on the maximal rate of the extremely proficient enzyme orotidine 59-phosphate decarboxylase, the rate enhancement is estimated to be somewhere around 1017. Based on a transition-state binding model, Radzicka and Wolfenden estimated the intrinsic binding energy of the altered substrate in its transition state corresponds to a dissociation constant lower than 5  10–24 M! There are limitations in the use of reference reactions for this purpose, and the chief concern relates to the possibility that two related reactions (or even the same reaction) may proceed by different mechanisms. Moreover, there can be changes in the molecularity of the nonenzymic and enzymic processes. If a covalent intermediate forms in the enzymic reaction, there may simply be no appropriate cognate nonenzymic reaction. Finally, the order of a multistep process may not be the same for both reactions, even though the reactions are otherwise fundamentally similar. Relaxation – The adjustment of a system of linked chemical reactions to a new state of equilibrium in response to the abrupt addition/removal of energy to/from the system. In temperaturejump relaxation methods, energy is added by electrical discharge of a capacitor, which results in Ohmic heating of ions suddenly moving toward their respective electrodes. In pressure-jump methods, the sudden rupture of a foil diaphragm causes depressurization, and the system responds to a loss of energy. In concentration correlation analysis, the chaotic movement of molecules leads to an increase or decrease in the statistically average number of molecules in a volume element, and the system relaxes in response to the perturbation from thermodynamic equilibrium.

Reduced Concentration – A normalization parameter used in treating ligand binding equilibria to convert two extensive variables, Kdissociation and substrate concentration, into a parameter whose value is related to the fractional saturation of ligand binding sites. For the simple Michaelis-Menten treatment, v ¼ Vmax/{1 + Km/[S]}, if R is the reduced concentration parameter (R ¼ [S]/Km), then v ¼ Vmax/{1 + R–1}. The extent of substrate saturation of two Michaelis-Menten enzymes with different Km values will be identical if their R values are the same. For example, if R ¼ 1, then v/Vmax ¼ 0.5 for both enzymes. Likewise, R values of 5 and 10 yield respective velocities of 0.833 Vmax and 0.909 Vmax. Reduced concentrations are also useful in the MonodWyman-Changeux cooperativity model, where a ¼ [F]/KR and ca ¼ [F]/KT. This makes polynomial functions simpler to handle. For example, if ligand F binds exclusively to the R-state, then the ligand F saturation function, YF, for an n-site protein equals (1 + a)n–1/{L + (1 + a)n}, where L is the allosteric constant. Similarly, b ¼ [I]/Ki and g ¼ [A]/Ka, where [I] and [A] are the inhibitor and activator concentrations, respectively.

Rieske Iron-Sulfur Protein – A mitochondrial respiratory chain iron-sulfur protein containing a [2Fe–2S] cluster with two coordinated cysteinyl sulfur ligands and two coordinated histidyl imidazole ligands. The term is also applied to similar proteins isolated from photosynthetic organisms and microorganisms that contain similarly coordinated [2Fe 2S] clusters.

Reduced Mass – The quantity m, equal to mAmB/(mA + mB), used to describe classical harmonic motion of bonded atoms of masses mA and mB.

Rotational Correlation Time – The period required for reduction of the fraction of some rotational correlated property to l/e (or 0.367) of its initial value.

Relaxation Time – The period of time that must elapse for an exponentially decreasing variable to fall from its original value Xinitial to Xinitial/e (or, approximately 0.3679 Xinitial). In 1866, Maxwell first used the term ‘‘time of relaxation’’ to define the period needed for the elastic force of a fluid to decay to X/e, where X is the initial value of the imposed force. Resonance Raman Spectroscopy – A vibrational spectroscopic technique used to characterize and assign vibrations that are directly connected with a chromophore, as a means of identifying metal-ligand interactions, particularly those that have visible electronic spectra. The interested reader should consult the outstanding brief description provided in volume IV of Comprehensive Biological Catalysis (Sinnott, 1999).

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Rotational Diffusion – The haphazard rotational motions of molecules or one or more segments of a molecule. This diffusional process strongly influences the mutual orientation of molecules (particularly large ones) as they encounter each other and proceed to form complexes. Rotational diffusion can be characterized by one or more relaxation times, ti, describing the motion of a molecule or segment of volume, V, in a medium of viscosity, h, as shown in the following equation: ti ¼ hV/(RT), where R and T are the universal gas constant and the absolute temperature, respectively. Because the rotating unit is rarely spherically symmetric in size and electronic charge, its rotational diffusional behavior is likely to be a composite of several tau values. Rod-like structures are especially likely to exhibit such behavior. The same is true for molecules having regions or segments that rotate relatively independently of each other. Under favorable conditions where the rotating unit is characterized by a fluorescence excited state lifetime of suitable magnitude, one can use time resolved fluorescence anisotropy measurements to infer the time-scale of rotational motions. Rule of the Geometric Mean (RGM) – A rule, originally proven by Bigeleisen (1955), asserting that isotopic disproportionation equilibrium constants are nearly identical with the classical value. The RGM has proved to be useful in determining whether certain double-proton transfer reactions are concerted or stepwise. Isotopic rate constants for concerted double-proton transfer reactions were found to be related by kHH/kHD ¼ kHD/kDD. Observation that double proton transfer reactions deviate from the rule of the geometric mean constitutes evidence in favor of hydrogen tunneling or a step-wise process via an intermediate. Rv – A ratio used to assess the degree of cooperativity exhibited by an enzyme. It is equal to the true Vmax value (typically extrapolated from the high-substrate-concentration end of a double-reciprocal plot) divided by the apparent Vmax value obtained from extrapolating the asymptote in the lowsubstrate-concentration portion of the double-reciprocal plot. For a noncooperative system, Rv will equal one; positively cooperative systems will have values greater than one; and negatively cooperative systems will have values less than one. This method requires good estimates of the asymptotes. S0.5 – A useful parameter for cooperative enzymes, corresponding to the concentration of substrate needed for one-half of the maximal saturation of the cooperative enzyme. S0.9/S0.1 (or, RS) – A kinetic parameter indicating the ratio of substrate concentrations needed to achieve reaction velocities equal to 0.1 Vmax and 0.9 Vmax. For an enzyme obeying the Michaelis-Menten equation, S0.9/S0.1 equals 81, indicating that such enzymes exhibit modest sensitivity of reaction rate relative to changes in the substrate concentration. Many positively cooperative enzymes have S0.9/S0.1 values between five and ten, indicating that they can be turned on or off over a relatively narrow substrate concentration range. Saddle Point – The point of minimal potential energy in the trajectory of reactants to products in a chemical reaction. A reaction’s saddle point indicates the geometry and energy of reactants as they approach and pass the transition state of a reaction. Scissile Bond – A covalent bond that is broken during the course of a chemical reaction.

Glossary

Second – The SI unit of time, corresponding to the duration of 9,192,631,770 periods for the radiative transition from the (F ¼ 4, mF ¼ 0) hyper-fine spectral energy level to the (F ¼ 3, mF ¼ 0) level of a ground-state Cesium-133 atom. Secondary Isotope Effect – A change in reaction rate that results from isotopic substitution adjacent to the site of bondbreaking/-making. The classical example in enzymology is the hydrolysis of phenylglucosides by lysozyme, for which the so-called deuterium isotope effect (symbolized as kH/kD) was found to be 1.14. This result was consistent with an expected change in electronic hybridization at positions C-1 and oxygen (within the glucopyranose ring) if a carbonium ion (or carboxonium) intermediate formed during the reaction cycle. Secondary Plot (or Replot) – A graphical method for simplifying the evaluation of kinetic parameters in multi-variable rate equations to examine the functional dependence of a kinetic property (e.g., enzyme activity, inhibition, activation, etc.) in terms of slopes, intercepts, and/or their curvilinear dependence of the property on that variable. For example, the rate equation (1/v ¼ 1/Vm + Km /Vm [S]{1+ [I]/Ki}) for a competitive inhibitor I of a one-substrate enzyme indicates that a convergent primary plot of 1/v versus 1/[S], with 1/v-Intercept ¼ 1/Vm and Slope ¼ (Km/Vm){1 + [I]/Ki}. Therefore, secondary plot of primary plot Slope versus [I] will yield a straight line, with a vertical-axis Intercept9 ¼ Km /Vm and Slope9 ¼ 1/Ki. Shifted Binding – A phenomenon observed with polymeric substrate interacting with subsites comprising an enzyme active site, wherein a second substrate molecule binds in a nonproductive mode within the subsites and sterically shifts the otherwise more energetically favorable binding of a productively bound substrate. In such cases, it may be necessary to keep the concentration of the polymeric substrate low enough to avoid such problems. Single-Turnover Conditions – Reaction conditions that only permit a catalyst to pass through a single round of catalysis. Single-turnover conditions are usually obtained by limiting the substrate concentration relative to the concentration of active catalyst. Occasionally, single turnover conditions can also be achieved by limiting the period of reaction. Siroheme – The heme-like prosthetic group in a sulfite reductase and nitrite reductase, which catalyze the six-electron reduction of sulfite and nitrite to sulfide and ammonia. Soft Acid – A relatively large electron pair-acceptor atom of low positive charge, high polarizability, and low electronegativity (Pearson, 1963; 1966). Examples: Cu+, Ag+, I2, Br2, and carbenes. Soft Base – A substance with loosely held valence electrons (examples: R–SH, R–S–, I–, and CN–). The electron-donor atoms are of low electronegativity and high polarizability (Pearson, 1963). Klopman (1968) suggested that soft Lewis bases bind to soft Lewis acids to give complexes dominated by their respective frontier molecular orbitals (i.e., the highest occupied molecular orbital and the lowest unoccupied molecular orbital). Solvation – A stabilizing interaction between a solute and the solvent, or between groups on an insoluble substance and the surrounding solvent.

Glossary

Solvent-Accessible Surface Area – The surface traced out by the center of a solvent probe molecule as it is rolled over the surface of a protein whose three-dimensional structure has been determined at the atomic level (Lee and Richards, 1971). Solvent-accessible surface areas can be calculated by various computer algorithms (Baker and Murphy, 1998), and differences in solvent-accessible areas can be used to characterize the energetics of surface hydration as a function of changes in protein conformation, oligomerization, and complexation. Solvolysis – Whenever a nucleophilic substitution reaction uses the solvent as the nucleophile, the reaction can be referred to as a solvolysis reaction (or simply as solvolysis). Examples of solvolysis include hydrolysis (reaction with water), ethanolysis (reaction with ethanol), acetolysis (reaction with acetic acid), and formolysis (reaction with formic acid). All solvolysis reactions exhibit first-order kinetics, because the concentration of the solvent does not change to any appreciable extent during the solvolytic reaction. This fact is often misinterpreted as an indication that the reaction type is SN1. To be sure, polar solvents such as water, alcohols, and amines should be regarded as good nucleophilic agents. A traditional method for distinguishing SN1 and SN2 reactions in physical organic chemistry is to add a low concentration of a substance that is a better nucleophile than the solvent. If the rate of substitution remains unchanged, then one can infer that the reaction is of the SN1 type; however, should the nucleophilic substitution rate increase, an SN2 type reaction is likely. Of course, the most direct means for distinguishing SN1 and SN2 reactions is by stereochemistry, when practicable. Specific Activity – 1. The intrinsic variable expressed usually as international units (U) of enzyme activity per milligram protein (or nucleic acid in the case of ribozymes). One U corresponds to the conversion of 1 micromole substrate into product per minute. A katal corresponds to the conversion of one mole substrate per second. Hence, 1 U corresponds to 16.67 nkat. As an experimental parameter, specific activity is especially useful during enzyme purification when one must develop strategies for maximizing specific activity while ensuring that adequate yield is maintained. Likewise, one may use specific activity as a measure of purity of any protein or other biomolecule that possesses some assayable response (such as drug binding or metabolite transport). 2. The intrinsic variable expressed as units of radioactivity (in becquerels or, more traditionally, curies) per mole of a substance. One Bq corresponds to one disintegration per second (dps) and one Ci to 3.70  1010 Bq. This parameter is especially useful in quantifying the amount of substance in biological samples. Specific Catalysis – The acceleration of a reaction by a unique catalyst rather than a family of related substances or materials. The term is commonly used with respect to H+ or OH– catalysis (thus, specific acid and specific base catalysis). Specificity – A measure and/or description of how specific an enzyme is toward a substrate or class of substrates or toward an effector or class of effectors. For effectors (or for ligands binding to macromolecules that are not enzymes), this specificity is readily measured by dissociation (or, association) constants. For enzymes, specificity is best quantitated by the Vmax/Km ratio. See Specificity Constant. It is crucial, in the

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complete characterization of an enzyme, that the specificity of the enzyme be known in detail. This includes stereospecificity and anomeric specificity when appropriate. It should also be recalled that biomolecules often exist in different stable conformations and in different states of ionization and/or metal chelation. For example, for ATP-dependent enzymes, it may be necessary to know the relative specificity of MgATP2– and MgHATP– or with MnATP2– and MnHATP–. If accurate values are known for the Vmax/Km ratios of a large set of substrates and alternative substrates, it may be possible to determine the topology of the enzyme’s active site and to design specific active-site inhibitors. Spin Label – A stable paramagnetic group that has been attached to a molecular entity. Experiments with ESR measurements can reveal aspects of the microenvironment of the spin label. If the paramagnetic group is not covalently attached to the molecular entity of interest, the term ‘‘spin probe’’ is usually applied. Stability Constant – An equilibrium association or formation constant (units expressed as molarity–1 or M–1) for ligand binding to a metal ion. See Section 2.8: Metal Ions in Enzyme Active Sites, Section 6.8: Working with ATP-Dependent Enzymes, Section 7.1.7: Activation of 3959-Cyclic AMP Phosphodiesterase by Calcium Ion-Calmoduliin Complexes, and Section 7.2: Metal-Nucleotide Complexes as Substrates. State Function – A function (such as energy) that is dependent only on the state of the system and not on the pathway as to how that state was reached. Examples of state functions include internal energy, enthalpy, entropy, Gibbs free energy, Helmholtz free energy, pressure, volume, temperature, etc. Work and heat are not state functions. Sticky Substrates – A designation given to a substrate (or ligand) that displays a strong tendency to remain bound to its binding site, and, in the case of enzymes, to undergo enzymatic catalysis with greater ease than to dissociate. Cleland (1982) showed that the stickiness of a substrate is a measure of the ratio of the net constant for reaction of the first collision complex through the first irreversible step to the rate constant for dissociation of that collision complex. The strength of enzyme-substrate complexation can be altered by solution variables (e.g., temperature, ionic strength, pH, solvent polarity, etc.) or factors such as proteolytic alteration of enzyme structure as in the case of yeast hexokinase’s interaction with D-glucose (Rose et al., 1974). Stiffness (or Stiffness Instability) – A term used in numerical integration of rate processes to describe computational difficulties arising whenever reaction steps within a mechanism are each characterized by widely disparate time constants. In such a case, the most rapid step must be integrated repetitively over such small time intervals that these intervals are far too small for efficient integration of the slower reaction process. As a result, excessively long periods for computation are required for accurate integration of both steps in a mechanism. Stoichiometric Number – A number (usually symbolized by v followed by a subscript denoting the species) equal to the coefficient of that species in a particular reaction. For example, for the reaction A + 2B # 3P, the stoichiometric number for P is 3. By convention, stoichiometric numbers are positive for products and negative for reactants.

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Stokes Shift – If an electronic transition results in both absorption and luminescence, then the Stokes shift is the difference (in either wavelength or frequency units) between the band maxima. If the luminescence occurs at a shorter wavelength, the difference is often referred to as an anti-Stokes shift. Substrate-Assisted Catalysis – The facilitation of enzymic catalysis achieved by the presence of a substituent located on the substrate at a site away from the position of bond-making/breaking. Substrate-assisted catalysis by serine proteases is well documented (Carter and Wells, 1987). Protease substrates are usually described as having sequences .P3–P2–P1–P19– P29–P39., where P1–P19 lie on each side of the scissile bond. Substrate Synergism – The cardinal feature of Ping Pong mechanisms is the ability of the enzyme to catalyze partial exchange reactions as a result of the independence of the substrate’s interactions with the enzyme. The second substrate is obliged to await the dissociation of the first product before it may bind to the enzyme. Multi-substrate enzymes frequently mediate such partial reactions, which may be related to important steps in catalysis. One enzyme proposed to be of this sort is succinyl-CoA synthetase, but the partial reactions are relatively slow, and the participation of such reactions in catalysis becomes difficult to assess. Indeed, slow partial exchanges have been interpreted as proof of contamination of a particular enzyme with another enzyme or a small amount of the second substrate, an indication that the mechanism is not Ping Pong, or that the presence of other substrates may markedly increase the rates of elementary reactions giving rise to the partial exchange. Bridger et al. (1968) proposed that the latter phenomenon be termed substrate synergism, and they examined this enzyme-substrate interaction by deriving appropriate rate laws for various exchanges. Their conclusion was that the rate of a partial exchange reaction must exceed the rate of the same exchange reaction in the presence of all the other substrates if the same catalytic steps and efficiencies are involved. If the opposite relation is observed, one must consider the possibility that synergism exists. Lueck and Fromm (1973) examined the significance of partial exchange rate comparisons and focused on often misleading comparisons of exchange rates made with respect to initial velocity data. Swain-Schaad Relationship – Quantitative relationships between magnitudes of deuterium and tritium primary kinetic isotope effects on chemical reactivity: kH/kT ¼ (kH/kD)1.442 and kH/kT ¼ (kD/kT)3.26–3.34.

Synergism Quotient (Qsyn) – A parameter used in isotope exchange experiments of enzyme-catalyzed reactions purported to have a Ping Pong mechanism. This parameter assesses whether substrate synergism is present and to what degree. For a Ping Pong Bi Bi mechanism, Qsyn ¼ {(Rmax, A-P)–1 + (Rmax, B-Q)–1}/(Vmax, f1 + Vmax, r1) where Vmax, f and Vmax, r are the maximum initial-rate velocities for the forward and reverse reaction, respectively, and Rmax,A-P and Rmax,B-Q are the maximum rates of exchange seen in the half-reactions. A synergism quotient of unity indicates a Ping Pong mechanism with no substrate synergism. For values significantly larger than one, either the mechanism is not a simple Ping Pong scheme or substrate synergism is present.

Glossary

Synergistic Inhibition – Synergistic inhibition is a term used to describe inhibition induced by a second inhibitor. A particular inhibitor (I1) may be a weak inhibitor for a certain enzyme; however, in the presence of a second inhibitor (I2), inhibition is greatly enhanced. An example of synergistic inhibition is the inhibitory effect of inorganic pyrophosphate on trichodiene synthase by certain aza analogs (Cane et al., 1992). Both enantiomers of serine are weak inhibitors of g-glutamyl transpeptidase (L-serine has a Ki value of 10.7 mM), but in the presence of borate ion, they are highly potent inhibitors (Revel and Ball, 1959; Tate and Meister, 1978; Thompson and Meister, 1977; Allison, 1985). System – A term used in thermodynamics to designate a region separated from the rest of the universe by definite boundaries. The system is considered to be isolated if any change in the surroundings (i.e., the portion of the universe outside of the boundaries of the system) does not cause any changes within the system. Tautomerism – A rapid and usually reversible internal isomerization between two or more structurally distinct compounds. The intramolecular process entails a heterolytic cleavage of a chemical bond followed by a recombination of the fragments, usually accompanied by the migration of a double bond or by ring opening and ring closing. The most common form of tautomerism involves the transfer of a proton (i.e., prototropy or prototropic rearrangement). An example of this form of tantomerism is keto-enol tautomerism between a carbonyl group having an a-hydrogen and its enol form. This reaction can be facilitated by both general acid and general base catalysis. In most cases, the equilibrium favors the keto form. Telomerase – An enzyme that uses an RNA template to add DNA to the ends of chromosomes. Telomerases are normally active only in stem cells and those cells giving rise to sperm and egg, but telomerase also undergoes activation when cells become cancerous. In the latter case, telomerase action allows transformed cells to replicate without a limit, a process termed ‘‘immortalization’’. Template Challenge Method – A procedure used to assess the processivity of nucleic acid polymerases. In this method, the investigator perturbs the polymerization reaction proceeding on one template by the addition of a new template (McClure and Chow, 1980). Tensegrity – The concept that cells are hard-wired to respond to externally applied mechanical stresses by means of pushing and pulling forces generated and transmitted by the membrano-cytoskeleton. See Section 13.1. Ternary Complex – A complex involving three species (such as the EAB complex formed in the ordered Bi Bi mechanism). Tetrahedral Intermediate – An intermediate, often transient, appearing in a chemical or enzymatic reaction in which a carbon atom, which had been double-bonded (i.e., in a trigonal structure) in a particular molecular entity, has been transformed to a carbon center having a tetrahedral arrangement of substituents. Thermodynamic Control – If a particular molecular entity(ies) participates in two or more parallel reactions and the proportion of the resulting products is determined by the relative equilibrium constants for the interconversion of reaction

Glossary

intermediates on or after the rate determining step(s), then the more prevalent product is said to be thermodynamically controlled (i.e., the more stable product will be the one formed in highest amounts). If the reactions are reversible and the system is allowed to go to equilibrium, the favored product is the thermodynamically controlled species. A synonymous term is equilibrium control. Thermodynamic pKa – The pKa value of an ionizable group extrapolated to zero ionic strength. Tight Ion Pair – An ion pair in which the constituent ions are not separated by a solvent or other intervening molecule. Tight ion pairs are also referred to as contact ion pairs. If X+ and Y– represent constituent ions, then a tight ion pair would be symbolized by X+Y–. An example of a tight ion pair would be the case in which an enzyme stabilizes a carbonium ion with juxtaposed negatively charged side chain groups. Transferase – Any enzymes that catalyzes the transfer of a group or moeity from one compound to another. The groups being transferred can be one-carbon units such as methyl, hydroxylmethyl, carbamoy-, or amidino moieties. Enzymes transferring aldehyde or ketonic groups such as transketolase are members of this class. Other examples include acyltransferases, glycosyltransferases, transaminases, phosphotransferases, and sulfotransferases. Transition State – The short-lived chemical configuration or intermediate species corresponding to the least-stable configuration (i.e., that form present at the highest point (saddle-point or coll) on the lowest-lying reaction path on the potential energy landscape/surface/trajectory connecting a reactant with its reaction product. Transition-state configurations often involve partial bonds as bonding orbitals make or lose contact/overlap with each other. See Section 1.2.2: Explaining the Magnitude of Enzyme Rate Enhancement; Section 3.6: Transition-State Theory; Section 8.6: Transition-State Inhibitors; Section 8.12.4: Rational Drug Design by Transition-State Assessment. Transmission Coefficient – A ratio used in transition-state theory (symbolized by k) to represent the probability that the activated complex will go on to form product(s) rather than return to reactants. In most cases, k is taken to be approximately one; however, if reactants do not obey the Boltzmann law or if the temperature is very high, then the coefficient can be less than one. Triple-Competitive Method – A method used to determine primary intrinsic isotope effects. In this procedure, three differently labeled substrates are used to react with a labeled co-substrate and the distribution of the labels in the products is measured (Northrop, 1982).

863

V Systems – Cooperative enzyme systems in which the presence of an allosteric effector results in an alteration of the Vmax value of the system (as opposed to changes in the Km value(s): K systems). Walden Inversion – The inversion of stereochemical configuration at a chiral center in a nucleophilic substitution reaction. The uncatalyzed reaction is bimolecular: i.e., it is an SN2 reaction (in fact the Walden inversion was noted before the criteria for SN2 reactions were presented). Wash-Out – The loss of an isotopic label from one or more sites of a labeled compound in the absence or presence of an enzyme. Nonenzymatic tritium and deuterium washout occurs whenever a carbanion or ylide is sufficiently stable so as to promote proton release. Enzymatic tritium and deuterium washout also occurs when a carbanion is formed and the protonated base can undergo exchange with protons from the solution. The carbon8 atom of many purines forms an ylide, and tritium or deuterium bound at this position can undergo facile exchange. For this reason, many tritium labeled nucleotides and nucleosides should be purified immediately prior to use. Another good example is proline racemase, which is thought to proceed by two-base catalysis through a transition state that has two thiols sandwiching a proline carbanion. Wedler-Boyer Isotope Exchange Protocol – A convenient useful protocol in equilibrium exchange studies, which removes the problem associated with the formation of abortive complexes and makes it more convenient to study the effects of modifiers on the exchange rates. In this procedure, rates of isotope exchange are measured as a function of the absolute concentration of any one of the substrates or products under conditions such that the concentrations of all substrates and products are being varied in a constant ratio (i.e., Keq). Womack-Colowick Continuous Dialysis Method – A ligand binding technique that relies on the ability to monitor the concentration of uncomplexed ligand in equilibrium with a macromolecule by determining the ligand’s rate of transport across a dialysis membrane (Womack and Colowick, 1973). The technique allows one to make successive additions of ligand to the upper chamber containing the protein, and one can typically obtain sufficient binding data within 20 min to construct a Scatchard plot for ligand binding at a single protein concentration. A miniaturized flow-dialysis cell has also been described (Brown and Reichard, 1969).

Turnover Number – A kinetic parameter, also known as kcat, indicating the number of substrate molecules converted to product per enzyme molecule per second. See Section 4.4: Additional Comments on Kinetic Parameters.

X-Ray Absorption for Fine Structure (XAFS) – A local environmentally sensitive structural probe based on the absorption of X-radiation by a central metal ion within a metal-ligand coordination complex. With suitable model compounds as reference standards, the XAFS technique can inform the experimenter about the nature and arrangement of ligands around a metal of interest. The technique can also sense changes ˚ . The in metal-ligand bond lengths to an accuracy of 0.03 A interested reader should consult the outstanding brief description provided in volume IV of Comprehensive Biological Catalysis (Sinnott, 1999).

Volt – The SI unit for electric potential and for electromotive force (symbolized by V) equivalent to one joule per coulomb. It is the difference in electric potential needed for a one ampere current to flow through a resistance of one ohm.

Zero-Order Reactions – Reactions in which the velocity (v) of the process is independent of the reactant concentration, following the rate law v ¼ k. Thus, the rate constant k has units of M$s1. An example of a zero-order reaction is a Michaelis-Menten

Triplet State – A state having a total electron spin quantum number of one.

Glossary

864

enzyme-catalyzed reaction in which the substrate concentration is much larger than the Michaelis constant. Under these conditions, if the substrate concentration is raised even further, no change in the velocity will be observed (since v ¼ Vmax). Thus, the reaction is zero-order with respect to the substrate. However, the reaction is still first order with respect to total enzyme concentration. When the substrate concentration is not saturating then the reaction ceases to be zero order with respect to substrate. Reactions that are zero-order in each reactant are exceedingly rare. Thus, zero-order reactions address a fundamental difference between order and molecularity. Reaction order is an empirical relationship. Hence, the term pseudo-zero order is actually redundant. All zero-order reactions cease being so when no single reactant is in excess concentration with respect to other reactants in the system.

Zwitterion – A molecular entity containing oppositely charged acid/base groups. An example is +H3NCH(CH3)COO–, which is by far the predominant species of alanine at neutral pH. The term ‘‘zwitterion’’ is a German word designating the presence of an ‘‘inner salt’’ or an internal ionic species within a molecule. Most biochemistry textbooks indicate that zwitterions are electrically neutral, lacking any overall net charge. This is only true for a substance containing one positively charged and one negatively charged ionizable group. Polyvalent molecules may have zwitterionic forms with overall net charge. For glutamic acid, one zwitterionic species (+H3NCH(CH2CH2COOH)COO–) is in fact neutral at low pH; the other is a monoanion present at higher pH (+H3NCH(CH2CH2COO–)COO–). Note: A zwitterion must not be confused with an ylide, which has opposite electric charges residing on adjacent atoms.

Zero-Point Energy – The residual, undampable energy (designated E0) of a harmonic oscillator in the ground state, as predicted by the Heisenberg Uncertainty Principle asserting that the position and momentum of a particle cannot both be known or determined with certainty.

Zymogen – An enzyme precursor or pro-enzyme, designated by the suffix ‘‘-ogen’’ to indicate that it can generate its catalytically active (e.g., trypsinogen is the precursor of trypsin, much as pro-elastase is the precursor of elastase), usually by means of enzyme-catalyzed proteolysis.

Index

A

AAA+ mechanoenzymes, 783–788 ATP binding/hydrolysis domain, 783, 784 cellular functions, 783, 784 site-directed mutagenesis, 785 structure, 784–785 see also DNA processivity clamp loader Abeles, R., 16 Abscisic acid, 106 Absolute stereochemical configuration, 9, 147 Cahn-Ingold-Prelog specification system, 148–149 Absorbance, 242 Absorbance correlation analysis, 674 Absorption spectroscopy, 221 electron orbital quantum states, 240 use of reference standards, 229 Abzymes (catalytic antibodies), 10–11 Acetate kinase, 275, 356 phosphotransfer stereochemistry, 582, 584, 585 Acetate thiokinase see Acetyl-coenzyme A synthetase Acetoacetate decarboxylase, 30, 126 perturbed pKa value, 404–405 Acetolactate synthase, 127 Acetone-butanol fermentation, 13 Acetone-cyanohydrin lyase, 126 Acetyl-coenzyme A, 126 Acetyl-coenzyme A synthetase, 528 metal ion coordination scheme, 112 Acetyl-phosphate, 184, 275, 355 energase reactions, 765, 768 Acetylcholine receptors, 388, 676, 709 Acetylcholinesterase, 70, 71, 305–306, 309, 467, 531 atomic force microscopy, 745 irreversible inhibition, 540 Acetylene hydratase, 107 Acetylenecarboxylate hydratase, 126 Acid-base active site residues, 57 Acid-base catalysis Brønsted theory, 409–412 efficiency of enzyme catalysis, 30 metal ion activities, 82, 83 Acidity, for definition see Glossary Acid-labile sulfides, for definition see Glossary Acidophilic bacteria, uses of enzymes, 431 Aconitase, 126, 528, 531 prochirality, 150 Acridone synthase, 71 Actin, 62, 776 filamant sliding, 737 filament elongation, diffusion-limited, 199 filament turnover, 794 myosin interaction, 774, 782

nucleotide exchange reaction, 24 role of profilin, 395 Actin-based motility actoclampin model see Actoclampin Brownian ratchet mechanism, 794 cellular functions, 793–794 Hill mechanism, 794 Listeria monocytogenes, 795, 797 optical tracking experiments, 742–744 Lock, Load and Fire mechanism, 795, 799 processive single-filament end-tracking, 795–798 ATP hydrolysis coupling, 798–799 propulsion of hard/soft particles, 799–800 Tethered Ratchet model, 794, 795 Activation energy, 25–26, 429 Activation overpotential, 157 Activators, 379–391, 686 allosteric, 382 applications, 380 binding interactions, 2, 382 continuous variation (Job Plot), 389–390 efficacy/affinity, 388 biphasic effects, 391 catalytic, 382 definitions, 381 degree of activation, 381–382 effectors, 2, 685 essential, 381 binding to otherwise unreactive substrate, 385 induced-fit mechanisms, 320 low-molecular-weight substances, 383 metabolic targets, 380 mixed, 382 modifiers, 2, 685 non-consumed substrate (pseudo-essential activators), 386–387 nonessential, 381 kinetic mechanisms, 387–388 post-translational modification, 383 rate-limiting rebinding following release, 385–386 redox, 383 reversible combination with enzyme after substrate binding (substrate-induced activation), 384 before substrate binding, 383–384 random binding to substrate, 384–385 specific, 382 time-dependent activation, 390–391 Active sites, 53–172 acid-base residues, 57 additional functional groups, 77–81 g-carboxyl-L-glutamate, 80–81 pyruvoyl residues, 81

selenocysteine, 57, 79–80 specialized amino-acid residues, 79–81 thiol adduct to tyrosyl, 81 topaquinone, 81 tryptoquinone, 81 tyrosyl quinone, 81 vitamin-based coenzymes, 77–78 affinity labeling agent interactions, 539, 540 binding energy, 60–61 catalytic residues, 54 functional classification, 57–59 identification techniques, 54–55 multi-domain/subunit, 59 tetheting to nearby structural elements, 59 chirality, 576 cofactor-activating residues, 58 cofactor/coenzyme binding, 55 competitive inhibitors as stereochemical probes, 499–500 conformation gating, 199 crystalline enzyme kinetics, 457 definition, 54 desolvation, 194 diversification, 70–77 structural aspects, 71–74 efficiency of enzyme catalysis, 26, 27 covalent intermediate formation, 30 electrostatic charge neutralization, 27–28 pre-organization, 30 irreversible inhibitor interactions, 539, 543 kinetic mechanism, 540–541 lids/flaps, 60–61, 321–322, 323, 325, 452 loops, 61, 322, 323, 468, 471, 475–476 low-barrier hydrogen bonds, 67–68 metal ions, 2, 55, 81–112 ribozymes, 22, 23 modified residues, 58 nucleophilic residues, 57, 206 pH effects ionizable catalytic groups, 397, 398, 400, 406 kinetic parameters, 399–400 local microenvironment, 404 photoaffinity labeling, 547–550 polymeric substrate-modifying enzymes, 112–116 post-translational modification, 55 primer-stabilizing residues, 58 radical-forming residues, 58 site-directed mutagenesis, 480, 481 substrate recognition subsites, 113–114 designation scheme, 115 signal-transducing protein kinases, 115 substrate-induced conformational change, 319, 320, 321, 322

865

Index

866

Active sites (Continued ) subunits (complex active sites) Ping Pong mechanisms for transfer of reactant moieties, 358 substrate channeling, 718–719 transition-state stabilizing residues, 57–58 volume, 59 water-activating residues, 58 Activity factors influencing, 379–483 site-directed mutagenesis studies, 468 units, 289, 290 Activity coefficient, 417 effect of ionic strength, 418, 419 Actoclampin, 63, 199, 397, 761, 776, 793–802 ATP hydrolysis coupling, 795, 798–799 force generation processes, 794–795 Lock, Load & Fire mechanism, 795, 799 molecular motor hypothesis, 794–795, 798–801 processivity, 329, 798 profilin actions, 798, 799 protection from capping proteins, 799 Actomyosin, 397, 761, 765, 776, 780 bond kinetics, 774–775 mechanism, 782 mechanochemical cycle, 48 sliding filament/cross-bridge cycle model, 762, 782 Acyclic diene metathesis polymerization, 8 Acyl carrier protein, 139 Acyl-enzyme intermediate amide/peptide hydrolysis, 135 chymotrypsin/serine proteases, 20, 143 ester hydrolysis, 134 formation at sub-zero temperatures, 436–437 Acyl-phosphate intermediate, 606 N-Acylamino acid racemase, 75 Acylation, 61 (Acyloxy)methyl ketones, 546 Adair equation, 700 Adair, G., 700 Adair-Koshland model see KoshlandNe´methy-Filmer cooperativity model Adaptive inhibition, drug design, 567 Adder-Mixer, 223 Addition reactions, 120 Additivity principle, 469, 471 reaction coordinate diagrams, 208 Adducts, covalent enzyme-substrate, 139 Adenine, 57 Adenomatous polyposis coli (APC) protein, 801 Adenosine deaminase, 86, 527 Adenosylcobalamin, 97, 100, 112 kinetic isotope effects, 629 S-Adenosylhomocysteine, 122 S-Adenosylmethionine decarboxylase, 126 Adenylate cyclase, 233, 388 Adenylate energy charge model, 522–523 Adenylate kinase, 340, 425, 522 inhibition, 524 substrate-induced conformational change, 321–322 Adenylosuccinate lyase, 126

Adenylosuccinate synthetase, 364, 520 Adenylyltransferase, 318 Adhesive forces, enzyme influences at singlemolecule level, 730 Adiabatic assumption, 201 Adjacent associative nucleophilic displacement reactions, 580 ADP-ribosylase, kinetic isotope effects, 628–629 Adverse drug effects, 486–487 Affinity agonist-receptor interactions, 388 Koshland-Ne´methy-Filmer cooperativity model, 700 Affinity chromatography, 690 Affinity electrophoresis, 690 Affinity labeling agents active site interactions, 539 experimental applications, 540–541 irreversible enzyme inhibition, 539 kinetic mechanism, 540–544 quiescent inactivators, 546 syncatalytic, 546–547 time-dependent deactivation, 545–546 Agonist-receptor interactions, 388 Alanine:oxaloacetate transaminase, 671 D-Alanine ligase, 528, 529 Alanine racemase, 76 Alanine scanning mutagenesis, 462–464 Alberty, R.A., 16, 210 formalism for biochemical thermodynamics, 211–212 Alberty/Fromm approach, multi-substrate kinetic mechanisms, 512–513 Albery, W.J., 18, 600 Alcohol dehydrogenase, 100, 118, 306, 338, 364, 519 active site-directed irreversible inhibition, 543, 545 high pressures effects, 439 kinetic isotope effects, 611, 628, 629 NAD+ sub-site interactions, 500–501 NAD+ transfer, 722 quantum mechanical hydrogen tuneling, 614 stereospecific hydride transfer, 576–577, 586 stopped-flow kinetics of NAD+ binding, 645–647 zinc ions, 645 Aldehyde decarboxylase, 97 Aldehyde dehydrogenase, 107 Aldehyde ferredoxin oxidoreductase, 94, 107 Aldehyde oxidase, 106 Aldehyde oxidoreductase, 364 Aldolase, 127, 335, 364, 587, 633 substrate channeling, 722 Aldose-ketose isomerization, cis-enediol mechanism, 138 Aldose-ketose reductases, 74 ALEXA(TM) dyes, 249, 250 Alkaline phosphatase, 76, 231, 356, 638 chromogenic substrates, 243 dissociative transition state, 411–412 ‘‘Flip-Flop’’ model, 705–706, 707 negative cooperativity, fraction-of-sites activity, 705–707 pH effects, 407–408

single-molecule activity, 731–732 thermal inactivation, 426 Alkaliphilic bacteria, uses of enzymes, 431 Allen video-enhanced contrast differentialinterference contrast (AVEC-DIC) microscopy, 737 Allosteric cooperativity, 687 activators, 382 heterotropic interactions, 696 homotropic effects, 695–696 I3, 687 key properties, 695–696 Koshland-Ne´methy-Filmer model, 699–707 Monod-Wyman-Changeux model, 695–699 nested models, 701 protomers (interacting subunits), 687, 688 V-type allosteric systems, 709 Allosteric enzymes, 2, 3, 20, 687 Allostery, 687 a effect, for definition see Glossary a-amylase extremophilic, activity-stability relationship, 431–433 ionic strength effects, 420 N-a-benzoyl-DL-arginine-p-nitroalanine, 243 a-cyclodextrins, 446, 447 a-glucosidase, 122 a-helix, 61, 62 3a-hydroxysteroid dehydrogenase, 74 a-lytic protease, 400 solvent kinetic isotope effects, 626 5a-reductase, inhibition by finasteride, 556 a,b-barrel, 74 a,b-hydrolase fold, 71 Alternative substrates, 492, 575 induced-fit mechanisms, 326 multi-substrate kinetic mechanism differentiation Fromm’s method, 508–510 Huang’s constant-ratio approach, 510–511 reaction mechanism elucidation, 19 Altman, S., 22 Alzheimer’s disease, 3, 536 Ambident nucleophile, for definition see Glossary Ambiguous reaction mechanisms see Kinetic ambiguity Amide hydrolysis, 135 acyl-enzyme intermediate, 135 carboxypeptidase-type general base mechanism, 136 carboxypeptidase-type nucleophilic mechanism, 135 Amine dehydrogenase, 128 Amine oxidase enzyme electrodes, 167 quantum mechanical hydrogen tuneling, 614 Amine-catalyzed b-keto acid decarboxylation, 138 D-Amino acid oxidase, 2, 128, 316, 317–318 L-Amino acid racemization, 55 Amino acids genetic code, 56, 57 intrinsic fluorescence, 247 pKa values, 56

Index

protein structure, 55–56, 61 residue molecular weights, 56 Aminoacyl-tRNA ligase, 57, 61 Aminoacyl-tRNA synthases, 2 kinetic proofreading, 455, 456, 457 ‘‘double-check’’ mechanism, 455 site-directed mutagenesis, 471–472 2-Amino-butenoate (vinyl-L-glycine), 550 4-(2-Amino-ethyl)benzenesulfonyl fluoride (AEBSF), 225, 226, 227 Amino-luciferin, 255 2-Amino-6-mercapto-7-methylpurine, 243 Aminopeptidases, 109 Aminopropyl triethoxysilane, 167 Aminotransferases, stopped-flow temperaturejump investigations, 671 Amount-of-substance determinations, 276 AMP nucleosidase, 122 Amplification cascades, 713–718 signaling, 718 Stadtman-Chock model, 715–716, 717 zero-order ultrasensitivity model, 717 Amylase enzyme electrodes, 167 multiple (repetitive) attack, 327–328 Amyloid precursor protein, 536 Amyotrophic lateral sclerosis (Lou Gehrig’s disease), 3 Analytical ultracentrifugation, site-directed mutagenesis studies, 468 Anchimeric assistance, 55, 330–331, 382 for definition see Glossary Anchor principle, for definition see Glossary Anfinsen, C., 18, 792 Angiotensin-converting enzyme, 488 inhibitors, 488 8-Anilino-1-naphthalenesulfonate, 249 Anomeric effect, for definition see Glossary Anomeric specificity, 317–318 Antibodies, catalytic (abzymes), 10–12 ‘‘bait-and-switch’’ approach, 10 immunogens, 10 limitations, 10–11 product release, 11 multi-stage catalysis, 11 protein carriers, 10 reactive immunization approach, 11 semi-synthetic enzymes (synzymes), 11 Antibodies, enzyme inhibitors in drug design, 572 Antibonding orbital, for definition see Glossary Antipain, 226 Apoenzymes, 12 metal ion acquisition, 84–85 Aprotinin, 225, 226 Aquation, metal-ligand, 89 Arabinose (fucose) dehydrogenase, 363 Arginase, 105–106 Arginine kinase equilibrium constant determination, 281–284 substrate-induced conformational change, 322 Argininosuccinate lyase, 128, 403 Arrhenius law, 201, 435

867

Arrhenius plots, 427–428 compensation effect, 428 nonlinear, 428–429 pure tuneling region, 615 tunnel correction, 615 Arrhenius, S., 6, 201 Arsenate, 387 Arsenical irreversible enzyme inhibitors, 539 Aryldialkylphosphatase, 84 Ascorbate oxidase, pulsed radiolysis, 678 Ascorbate-dependent enzymes, 57 Asparagine synthetase, 358 Aspartate:a-ketoglutarate transaminase, 671 Aspartate aminotransferase mechanism-based inhibition, 550 site-directed mutagenesis, 471 syncatalytic labeling, 546–547 Aspartate ammonia lyase, 128 Aspartate carbamoyltransferase, 687–688, 695 ligand binding cooperativity, 704–705, 708 Aspartate transcarbamoylase see Aspartate carbamoyltransferase Aspartic proteases, 60 Aspartokinases, 695 Association constants, 289 Associative reactions elimination, 125, 126 nucleophilic second-order substitution, 124 phosphate ester hydrolysis covalent catalysis, 137 general acid-base catalysis, 137 Asymmetric carbon atom, 147 Atom-percent, 256 Atom-percent excess, 256, 263 Atomic absorption spectroscopy, metal ion–enzyme/enzyme substrate complexes, 93 Atomic force microscopy, 744–745, 765, 775 contact mode, 744 tapping mode, 744 ATP, 761–762 generation, 109–110, 761, 762 hydrolysis, 212, 762, 764, 765 coupling to force-generation, 68–69, 766–769 hydrogen bonding network, 69 sarcoplasmic reticulum calcium ion pump, 780 substrate regeneration, 276 ATP phosphoribosyltransferase, 149, 355, 363, 364 ATP sulfurylase, 301, 310 ATP synthase, 23, 393, 765, 788 Boyer’s Binding Change Mechanism, 790–792 photophosphorylation, 420 rotatory catalysis, 748, 788, 791 transmembrane proton gradients, 788–789 ATP-dependent enzymes, 272–275 kinases, 104 metal ion/substrate complexes, 272–273, 393–394 MgATP2-/MgGTP2-, 273 stability constants, 273, 274, 275

phosphoryl acceptor solutions preparation, 273–274 ATPase, 429, 430, 762, 764–765 GroEL activity, 792, 793 reversible cold inactivation, 434 stopped-flow calorimetry, 644 Autocatalysis, 7, 22, 206–207 Michaelis complex formation, 207 Automatic pipettes/pipetting, 229–230 Auxiliary enzyme assays see Coupled enzyme assays Azadirachtin, 488 Azathioprine, 487 3’-Azido-2’ deoxythymidine (AZT), 490, 492 Azomethane decomposition, 176

B Bacitracin, 488 Backbone fluctuations, 36 Bacteria hydrogenases, 107 optimal temperature ranges, 430 peptidoglycan synthesis, 146 temperature-related growth rates, 430 Bacterial flagellar motor, 709, 788 rotatory catalysis, 788 transmemebrane proton gradients, 788–789 Bagshaw, 19 ‘‘Bait-and-switch’’ strategy, 60 Baker, B.R., 539 Balanced polymorphism, 64 Barnase, 470 GroEL-mediated folding, 793 Bases, DNA structure, 453, 454 Beer-Lambert absorbance law, 240–241, 644 Bell, R.P., 16, 615 Bender, M.L., 12 Benign prostatic hyperplasia, 556 Bentonite, 383 Benzoylformate decarboxylase, 128 Bequerel (Bq), 261–262 Bergmann, M., 18 Berzelius, 5 Bestatin, 225, 226, 227 b particles, 260 b-N-acetyl-glucosaminidase, 243 b-amylase, 115 b-barrel, 63 b-cyclodextrins, 446, 447 b-galactosidase, 95, 279, 447 single-molecule kinetics, 735 substrate recognition subsite designation scheme, 115 b-keto acids, amine-catalyzed decarboxylation, 138 b-lactamase, 385–386 alanine scanning mutagenesis, 463 b-lysine 5,6-aminomutase, 132, 133 b-pleated sheet, 61 b-secretase, 536 b-site amyloid precursor protein-cleaving enzyme (BACE), statine-based inhibitor, 536–537 Bi, in Cleland notation system, 336

Index

868

BIM-46174, 487 Binary complexes, 338 Binary inhibitors, 501 Binding activation, 382 Binding energy, 37, 211 active site, 60–61 metal ion affinity, 83 efficiency of enzyme catalysis, 28 hydrogen bonding, 67 Binding interactions allosteric, 2 modulators/effectors, 2 polymeric substrates, 332 quantification with fluorescence anisotropy, 250–252 see also Ligand binding Bioavailability, 561 Lupinski’s ‘‘rule-of-five index’’, 568 Biocytin, 54 Biofuels production, 47, 233 Biological catalysts, 12–15 efficiency, 25–34 Biological membranes, 447–448 interfacial double layer, 448–449 interfacial enzyme catalysis, 449 intramembrane proteolysis, 453 lipid rafts/caveolae, 453 molecular crowding effects, 453 phospholipid flip-flop, 448 units for tethered components, 453 Bioreactor/biosensor integrated devices, 440 Bioreactors, 440 immobilized enzymes, 439, 440 flow tube tethering, 443 methods of immobilization, 441 open systems, 440 Biotin, 335, 358, 464 enzyme immobilization techniques, 441 transcarboxylase multi-site catalysis, 31 Biotin (propionyl-coenzyme A-carboxylase) synthetase, 77, 132 Bis(oxazoline), 9 Bisubstrate kinetic mechanisms, 335–340 Cleland notation, 336–338 differentiation using competitive inhibitors (Fromm’s method), 494–498 examples, 338–340 graphical analysis, 359 Haldane relations, 359–360 kinetic, 360 thermodynamic, 359–360 iso mechanisms, 338, 521, 522 ordered binary complex sequential mechanism, 337 ordered ternary complex sequential mechanism, 336–337 Bi Bi, 336–337 Bi Uni, 336–337 pH effects, 408–409 Ping Pong mechanisms, 337–338 product inhibition patterns, 513–516, 521, 522 effects of abortive complex formation, 516–520 quantitative analysis, 360–366 crossover-point analysis, 363–364

Dalziel phi method, 360–362 Fromm’s point-of-convergence method, 362–363 random sequential mechanisms Bi Bi, 336 Bi Uni, 336 rapid equilibrium equations, 349–352 steady-state rate equations, 341–349 Cleland’s net reaction rate method, 345–347 Fromm’s systematic method, 341–343 random kinetic mechanism, 347–349 Theorell and Chance ordered binary complex mechanism, 347–348 two-step computer-assisted method, 343–345 ‘‘Blank’’ reaction rates, 219 ‘‘Blinking’’ (intramolecular fluctuations) fluorescence correlation spectroscopy, 753 single-molecule fluorescence, 750, 751 Blood clotting cascade, 713–714, 715 inhibitors, 488–489 intrinsic/extrinsic pathway activation, 715 Blow, D., 21 Body temperature homeotherms, 429–430 poikilotherms, 430 Boltzmann distribution, 35, 192–194 single-molecule kinetics, 733 Bond number, for definition see Glossary Bonding orbital, for definition see Glossary Boric acid, 412, 413 Bottlenecks, 349 rapid-mixing experiments, 638 zero-order (saturation) kinetics, 289 Bovine serum albumin, 228 catalytic rate enhancement, 11–12 cryoprotectant, 223 Boyer, P.D., 18, 339, 592, 790 binding change mechanism, 789, 790–792 isotope exchange at equilibrium, 586–603 Bragg’s law, 457 Branch points/nodes of metabolic pathways, 2 regulatory enzyme activities, 686 Branched pathways enzyme kinetic properties, 372–373 Ping Pong mechanism, 357, 372–373 Brande´n, C., 645, 647 Breslow, R., 12 Bridging ligand, for definition see Glossary Briggs-Haldane steady-state treatment, 293–297, 301 derivation, 294 reverse-reaction rate, 295 steady-state assumptions, 296–297 substrate concentration effects, 295 British Anti-Lewisite (dimercaprol), 383 Britton’s flux ratio method, 596–599 4-(3-Bromoacetylpyridino)butyl diphosphoadenosine, 543 Bromo-hydroxyacetone-phosphate, 542, 543 3-Bromo-2-ketoglutarate, 542 Bromoperoxidase, 97, 107–108 Brønsted theory of acid-base catalysis, 204, 409–412

Brown, A., 287, 289 Brownian motion, 196, 198 enzyme influences at single-molecule level, 730 molecular crowding effects, 446 Brownian ratchet mechanism, 743–744, 794 Bubble formation avoidance, 223 Bu¨chner, E., 13 Buffer catalysis, 206, 414–415 Buffer effects, 412–416 active participation in enzyme catalysis, 206, 414–415 contaminants, 412, 413 cryoenzymology, 435 maintenance of constant ionic strength, 415–416 three-component systems, 416 Buffer selection, 413–414 pKa values, 413, 414 temperature dependence, 414 Burst amplitude, enzyme concentration determination, 276, 278 Burst-phase kinetics, 356, 638 enzyme-bound intermediates detection, 657 stopped-flow techniques, 655–656 Butylcholinesterase, 70 1-Butyl-3-methylimidazolium cation, 424

C Cage effects of solvent molecules, 195 Caged ATP, photochemistry, 675–676 Cahn-Ingold-Prelog system, tartrate, 9 absolute stereochemical configuration specification, 148–149 prochirality specification, 149–150 Calcineurin inhibitors, 501 Calcium, 82 blood clotting cascade regulation, 715 EGTA complex formation, 97 enzyme activation, 96–97 essential activator of otherwise unreactive substrates, 385 nucleotide complexation, 395 ATP-dependent enzyme activity, 274, 393 Calcium ATPase see Calcium ion pump Calcium ion pump, 779–782 ATP hydrolysis coupling, 780 calcium ion binding, 781 chemical specificity phase of activity, 780, 781 vectorial phase of activity, 780, 781 Calcium-calmodulin, 97 39,59-cyclic AMP phosphodiesterase activation, 388–389 Calmodulin, 97, 388, 779 Calmodulin-dependent phosphoprotein phosphatase, 712 Calorimetry, 222 Calpain, 225, 572 Calpastatin, 572 Canonical structures, 119 Captopril, 488 Carbamoyl-serine-ammonia-lyase, 126 Carbanions, 119 intermediates, 125–130

Index

Carbenes, 119 photochemical reactions, 548, 549 Carbenium intermediates, 122–123 Carbon monoxide, cytochrome c oxidase binding, 162 Carbon monoxide dehydrogenase, 107, 151 Carbon, reactive forms, 119 Carbon–carbon bonds, 119 formation/cleavage reactions, 125, 126–127 Carbonic anhydrase, 1, 109, 126, 196, 467, 627 active site zinc, 111, 414–415 Lewis acid properties, 415 buffer effects, 415, 467–468 chemical rescue of mutant enzyme, 479 equilibrium exchange kinetics, 591–592 mechanism-based inhibition, 557 rate limiting deprotonation, 415, 627 site-directed mutagenesis, 467–468, 481 stopped-flow kinetic techniques, 642 substrate channeling, 719 Carbonium ions, 119, 121 Carboxycyclohexadienyl dehydratase, 126 Carboxypeptidase, 84, 109 temperature effects, 428 Carboxypeptidase A cryoenzymology and kinetics, 436, 437 general base catalysis, 136 peptide hydrolysis, nucleophilic mechanism, 135 site-directed mutagenesis, 470 Carrier-linked pro-drugs, 566 Catalase, 301, 302, 335 enzyme electrodes, 167 Catalysis, 5–12 activation energy reduction, 203, 204 autocatalysis, 7, 22, 206–207 buffer, 206, 414–415 chemical, 203–207 electrophilic, 206 heterogeneous, 205 historical aspects, 5–7 homogeneous, 205 interfacial, 205 nucleophilic, 205–206 rate enhancement, 1, 8, 67, 203–204, 205 reaction cycle see Catalytic cycle reaction mechanisms, 134–138, 637 time evolution of reaction, 204–205 transient covalent intermediate formation, 205 transition state stabilization, 204 Catalytic activation, 382 Catalytic antibodies, 10–11 for definition see Glossary Catalytic constant (kcat), 289 Catalytic converters, 6 Catalytic cycle, 53, 204, 309 internal equilibria, 309 time, 204–205, 303, 304 Catalytic proficiency, 1 evolution, 308–309 Catalytic promiscuity, 41, 74 diversification of enzymes, 74–77 genome-searching approaches to identification, 76–77 molecular moonlighting, 74

869

Catalytic rate enhancement, 1, 203–204, 205 active site low-barrier hydrogen bonds, 67–68 synthetic catalysts, 8 Catch bonds, 774, 775 Catechol-O-methyltransferase, 151 Cathepsin B, irreversible inhibition, 546 Cathepsin M, 225 Caveolae, 453 Cavitation, 682 Cech, T., 22 Cell permeabilization, 48, 49 Cell surface-bound enzymes, 443, 444 Cell survival, 49 Cell-free extracts, historical aspects, 13 Cellobiohydrolase, 115 Cellular tensegrity, 762 Central complexes, 338, 637 Ceruloplasmin, 100 Cha method, steady-state random Bi Bi mechanism, 349 Chalcone synthase, 71–72 Chance, B., 16, 18, 339, 643, 647 Changeux, J.P., 18, 695 Chaotropes, 421 Chaperonins, 25, 792 apoenzyme metal ion complex formation, 84–85 protein folding, role of ATP hydrolysis, 792–793 Charge-transfer complexes, 70 Chemical bonds, 764 Chemical dynamics, 4, 34 Chemical kinetics, 4, 34, 171–214 composite multi-stage mechanisms, 184–192 solution behavior of reacting molecules, 194–201 study methods, 171, 172 Chemical mechanism, 19, 637 Chemical potential, 417 Chemical rescue of mutant enzymes, 415, 479 Chemical shift, 680 for definition see Glossary Chemical space, exploration for drug design, 569 Chemical trapping, enzyme-substrate covalent intermediates, 145–146 Chemoluminescence, 254 enzyme rate assays, 254–255 instrumentation, 254 Chemostat, 177 Chi-square data fitting, 267 Chirality, 147 active site, 576 coenzymes, 576 enantioselective catalysts, 9–10 limitations, 10 enzymatic recognition, 576 substrates, 576 Chirality center, 147 Chitinases, 115 Chloramphenicol, 566 Chloroethanesulfonate, 413 Chloro-hydroxyacetone-phosphate, 542, 543 Chloroperoxidase, 86 activity in organic solvents, 424

Chlorophyll a, 105 Chlorophyll b, 105 Cholesterol oxidase, single-molecule enzyme cycle, 732–733 Choline acetyltransferase equilibrium constant determination, 280, 281 equilibrium exchange behavior, 595–596 Choline oxidase, enzyme electrodes, 167 Chromatin remodeling, 49, 688, 784 Chromium-nucleotide complexes, 395–396, 397 Chromogenic substrates, 243 Chymostatin, 226 Chymotrypsin, 71, 73, 309, 681 active site low-barrier hydrogen bonds, 67–68 activity in organic solvents, 424 acyl-enzyme intermediate, 142, 143 double-displacement mechanism, 142 formation at sub-zero temperatures, 436, 437 burst kinetics, 276, 278, 356 catalytic mechanism, 20–22, 32, 572 general base catalysis, 20 push–pull proton transfer, 20 catalytic rate enhancement, 21 rapid mix/quench experiments, 652 stopped-flow calorimetry, 644 substrate recognition subsites, 114 turnover number, 303–304 Chymotrypsinogen, 21–22 trypsin cleavage, 22 Circe Effect, 29 Circular dichroism, 221 site-directed mutagenesis studies, 468, 481 tervalent metal ion-nucleotide complex substrates, 396 cis-enediol mechanism, aldose-ketose isomerization, 138 cis-platin, 86 Citrate, blood-coagulation inhibition, 488–489 Citrate synthase, 528 substrate channeling, 722 Clampins, 795 Classification of enzymes/reactions, 23, 25, 118 Enzyme Commission (EC) scheme, 118 mechanoenzymes, 764 see also Databases development Clausius-Clapeyron equation, 438 Cleland, W.W., 18, 336, 338, 339 computer programs for pH kinetic analysis, 402 equilibrium perturbation, 612–613 exchange-inert metal-nucleotide complexes, 395–397 isotope trapping, kinetic treatment, 603–606 kinetic mechanisms notation, 336–338 low-barrier hydrogen bonds, catalytic role, 31–32 net reaction rate method for unbranched kinetic mechanisms, 345–347 rules for analysis of reversible dead-end inhibitors, 505 CLOCK gene, 686 Clp ATPase, 25

Index

870

Cobalamin (vitamin B12), 81, 97, 99 adenosyl free radical intermediates, 132 metal ion hexa-coordinate complex, 97 Cobalamin (vitamin B12)-dependent enzymes, 131, 132–133, 575–576 kinetic isotope effects, 629 Cobalt, 82 enzyme activation, 97–100 b-lactamase, 385–386 enzyme complex coordination schemes, 112 nucleotide complexes, 395, 397 Cobalt-dependent enzyme reactions, 98–99 Cocaine, 488 Codeine, 488 Codons, 57 Coenzyme A-linked aldehyde dehydrogenase, 227, 233 hysteretic behavior, 713 Coenzyme F430, 107 Coenzymes, 12, 77–78, 335 active site binding, 55 enzyme-bound covalent coenzyme–substrate intermediates formation, 30, 142, 144 induced-fit mechanisms, 320 intrinsic fluorescence, 248 irreversible binding, 78 metal ion interactions, 78 vitamin-based, 77–78 metabolic roles, 77 Cofactor-activating residues, 58 Cofactors, 335 active site interactions, 55, 58 multi-enzyme system enzyme-tethered, 31 Coformycin, 527 Cohn, M., 18 Coincidence circuitry, 261 Colchicine, 488, 559 Cold inactivation, 223 Colipase, 452 Collagenase, 109 Colls, 197, 202 Combinatorial cassette mutagenesis, 482 Commercial applications of enzyme technology, 14 Commercially available enzymes, 15, 232 Commitment to catalysis, 307–308, 723–724 for definition see Glossary regulatory enzymes, 686 Committed step of metabolic pathway, 2 Compartmentalization, in vitro directed enzyme evolution, 39–40 Compensation effect, 428 Competitive inhibitors, 489–501 active site stereochemical probes, 499–500 binary, 501 dead-end, 493 use in distinguishing multi-substrate mechanisms (Fromm’s method), 494–498 Dixon plots, 498–499 double inhibitor interactions, substrate subsite binding (Yonetani-Theorell treatment), 500–501 enzyme complex formation, 489–490 kinetic studies, 490

kinetic properties, 492–493 mutally exclusive inhibition, 490 partial, 387, 493–494, 504 product inhibition, 512 tight-binding reversible, 532 Complete (linear) inhibition, 504–505 Complex enzyme, 12 Component reactions see Elementary reactions Compounds, transient covalent enzymesubstrate, 141–142 Computationally designed enzymes, 46 design algorithm, 44 Computer-assisted steady-state rate equation derivation, 343–345 Concentration determinations, protein/ enzyme, 276–278, 673–674 active enzyme determination techniques, 276, 278 active-enzyme titration, 278 burst amplitudes, 276, 278 specific activity, 276 rate data, 292 total protein determination techniques, 276, 277, 278 Concentration overpotential, 157 Configuration, stereochemical, 147 see also Absolute stereochemical configuration Conformation gating, 199 Conformation, stereochemical, 147 Conformational change, force generating, 761, 767 AAA+ mechanoenzymes, 784 molecular motor directed motion, 767 Conformational change, ligand-induced, 325 activator release/rebinding, 385 crystalline enzyme kinetics, 457 desolvation facilitation, 324 enzyme regulation, 686 induced-fit mechanisms, 319, 320, 326 computer-based docking algorithms, 327 multi-substrate reactions, 335 mutually exclusive inhibition, 490 tight-binding reversible inhibitors, 531–533 transition state inhibitors, 526 Conformational change, spontaneous, 686 Conformational flexibility, 320 computer modeling, 327 lyophilized salt-enzyme preparations, 424 pH-dependence, 400 psychrophilic enzymes, 433 Conformational spread model, 708–709 Conformer, 147 Connectivity, 139 Consensus sites, 63 Conservation equation Briggs-Haldane steady-state treatment, 294 Michaelis-Menten kinetics, 288 Constant heat summation law, 469 Contamination avoidance, initial rate enzyme assays, 227–228 partial-exchange reactions, 355, 356 Continuous fluorimetric assays, 681

Continuous processes, 171 rate assays, 217, 218–219 Continuous variation analysis, activator binding site number/affinity determination, 389–390 Continuous-flow techniques, 16, 639 rapid-mixing methods, 172 reaction intermediates detection, 640, 643 Control factor, 188 Cooperativity, 685, 688 conformational spread model, 708–709 hemoglobin oxygenation model, 702–703 Hill equation, 691–693 hybrid models, 707–708 Koshland-Ne´methy-Filmer model, 699–707 Monod-Wyman-Changeux model, 695–699 negative, 700, 701–702, 703–705 fraction-of-sites activity, 705–707 positive, 700, 701–702 Scatchard equation, 693–694 Wyman’s linked function analysis, 694–695 see also Allosteric cooperativity Coordination number, 82, 85 metal ion–ligand complexation, 85–87 Copper, 82 chiral catalysts, 10 enzyme activation, 100–101 ligand complexation, 86 molybdenum metabolism relationship, 78 Copper zinc superoxide dismutase, 84 ionic strength effects, 420 Copper-albumin, 100 Copper-containing enzymes, 102 galactose oxidase, 81 Copper-containing proteins, 100–101 Corrins, 81, 97 for definition see Glossary Coulomb’s Law, 64 Coumarins, 489, 557 Counter-transport rate, 597 Coupled (auxiliary) enzyme assays, 217, 235, 237–240 lag phase, kinetic treatment, 238–239 Covalency, definition, 85 Covalent bonds, 118 definition of enzymes, 23, 24 influences at single-molecule level, 730 Covalent intermediates efficiency of enzyme catalysis, 26, 30–31 enzyme-tethered co-factors, 31 serine proteases, 20, 21 stereospecificity, 31 Cram, D.J., 12 Creatine kinase, 63, 115–116, 275, 300, 340, 762 equilibrium exchange behavior, 594–595 metal ion coordination scheme, 111 Creatine phosphate, 111, 762 Crick, F.H.C., 174, 454, 685 Critical micelle concentration, 447–448 Critical micelle temperature, 448 Critical surface pressure, 452 Cross-correlation analysis see Fluorescence correlation spectroscopy

Index

Cross-linked enzyme crystals, 459–460 productivity, 460 Cross-linking agents, enzyme immobilization, 440 Cross-metathesis, 8 Crossed molecular beam experiments, 4 Crotonase, 613 Cryoenzymology, 18, 435–438 acyl-enzyme formation, 436–437 buffer selection, 435 cryosolvent selection, 435–436 enzymes studies, 436 limitations, 437–438 practical considerations, 435–436 rationale, 435 Cryoprotectants, 223, 427 Cryosolvents, 435 Crystal field splitting, 87–88 Crystal field stabilization energy, 87–88 Crystal field theory, 87–88 Crystalline enzyme kinetics, 457–460 cross-linked crystals, 459–460 enzyme-substrate complexes preparation, 457 technical aspects, 458–459 time-resolved Laue X-ray crystallography, 459 video absorption spectroscopy, 459 CTP:glycerol-3-phosphate cytidyltransferase, 705 Curie (Ci), 261 Cuvettes, 241, 242 cleaning, 241–242 Cyanocobalamin (vitamin B12), 87 39,59-Cyclic AMP phosphodiesterase, calcium-calmodulin activation, 388–389 39,59-Cyclic AMP-dependent protein kinase, 388 alanine scanning mutagenesis, 462 39,59-Cyclic GMP phosphodiesterase, active site-directed inhibitors, 561–562 39,59-Cyclic GMP-dependent protein kinase, 162 Cyclic voltammetry, 160 Cycloamyloses see Cyclodextrins Cycloartenol synthase, 72, 122 Cyclodextrins, 446 host–guest interactions (inclusion complex formation), 446–447 Cyclomaltodextrin glucanotransferase, 446 Cyclophyllin, 74, 501 Cyclophyllin A (prolyl-peptidyl isomerase), 35–36 chemical shift mapping, 35 Cyclosporin, 501 Cystathione g-synthase, mechanism-based inhibition, 556 Cystathionine b-lyase, 128 Cysteine, 383 Cysteine-phosphate intermediate, phosphate ester hydrolysis, 138 Cytidine, 57 Cytochrome b5 reductase, kinetic isotope effects, 611 Cytochrome c, 92, 633 laser flash photolysis, 678–679 Cytochrome c oxidase, 101, 161–162 inhibitors, 162

871

Cytochrome c-d1 nitrate reductase, pulsed radiolysis, 677 Cytochrome P450 2D6, 269 Cytochrome P450 2E1, 111 Cytochrome P450s enzyme electrodes, 167 metal ion coordination scheme, 111 Cytochromes, 81 Cytoskeleton, 444 dynamic cellular entasis, 762 molecular motors, 761, 765, 772 chemical/mechanical cycle coupling, 738 video microscopy, 737

D Dalziel, K., 18, 339, 359 phi method, bisubstrate mechanisms quantitative analysis, 360–362 Damko¨hler number, 444 Dark-field microscopy, microtubule assembly/ disassembly, 738–739 Databases development, 46–47 Daughter ion, 257 Debye-Hu¨ckel limiting law, 417–418, 449 Definitions of enzymes, 23, 764 De-gassing solutions, 223 Degradation prevention, 224–227 Degree of inhibition, 537–538 De-inhibition, 382–383 for definition see Glossary Deionized water, 228 DelPhi software, 65, 66 d, 256 D5-3-ketosteroid isomerase, alanine scanning mutagenesis, 463 D4-3-ketosteroid-5-reductase, 74 Dennis, E., 18 Dental caries, 750 Deoxycytidylate hydroxymethyltransferase, 128 Deoxyhemoglobin, 87 3-Deoxy-D-manno-octulosonate 8-phosphate synthase, quench-flow experiments, 652 Deoxyribonuclease, random scission kinetics, 329 Deoxyribonucleic acid see DNA Deoxyribonucleosides, 57 Deoxyribozyme, 313 Deoxythymidine kinase, dimerization, 709 Depolymerases, 327, 328 active sites, 112, 113 nonproductive interactions in steady-state treatments, 332–333 random scission kinetics (endodepolymerases), 329–330 substrate-assisted catalysis, 330–331 DeSa monochromator, 648 Desnuelle, 18 Desolvation, 200–201 active sites, 194 conformational change facilitation, 324 electrostatic interactions, 64, 65 Detailed balance principle, 189–190, 209 thermodynamic cycle analysis, 190–192

Detoxification, 111 Deuterium oxide cryoprotectant, 223 Dextran, glucosyltransferase binding, single molecule fluorescence observations, 750–751 Diabatic electron-transfer process, 202 Dialkylglycine decarboxylase, 95 rapid-scanning stopped-flow kinetic studies, 679–680 Diaphorase, enzyme electrodes, 167 Diastase, 5, 12, 13 Diastereoisomers (distereomers), 147 Diethyl pyrocarbonate (ethoxyformic anhydride), 545 Differential scanning calorimetry mutant enzymes, 481 psychrophilic enzymes, 433 Differential-interference contrast (Nomarski) microscopy, 737 kinesin movement along microtubules, 738 Diffusion, 196–197, 220 limitation on chemical processes in water, 196–197 modeling, 199 molecular crowding effects, 445, 446 molecular size/shape influences, 198 Diffusion time, for definition see Glossary Diffusion-limited (diffusion-controlled) reactions, 198 dimensionality influence, 198–199 macromolecule self-assembly, 199 Smoluchowski equation, 197–198, 199 Diffusion-limited rate constants, 197–198 electrostatic effects, 199–200 Diffusional anisotropy, 444 Diffusional encounter, 197 Digital reaction simulator, 182–183 Dihydrodipicolinate reductase, 128 Dihydrofolate reductase, 36–37, 464 methotrexate interaction, 534, 535–536, 749–750 single-molecule kinetics, 749–750 stopped-flow temperature-jump investigations, 671–672 substrate channeling, 722 Dihydroorotate dehydrogenase, 579 Di-isopropyl-fluorophosphate, 539 Dimercaprol (British Anti-Lewisite), 383 Dimethylformamide, 223 Dimethylsulfoxide, 223, 229, 434 Dioldehydrase, 121, 132, 575–576 Dipole–dipole interactions charge-transfer complexes, 70 enzyme structural stability/interactions, 66 hydrophobic interactions, 69 van der Waals interactions, 70 Direct rate assays, 217 Direct-linear plot, 291 limitations, 291–292 Directed evolution, 39–40, 483 enantioselectivity optimization, 317 Direction of chemical reaction, 210 Disaccharidases, substrate recognition subsite designation scheme, 115 Dische, Z., 18

Index

872

Discontinuous rate assays, 217 see also Stopped-flow techniques Discrete processes, 171 Disequilibrium ratio, 213 Dissociation constants, 289 Dissociative reactions elimination, 125 nucleophilic first-order substitution, 121 phosphate ester hydrolysis covalent catalysis, 137 general acid-base catalysis, 136 Distributive mechanisms, depolymerases, 327–328 Disulfide linkages, site-directed mutagenesis studies, 469 Dithioerythreitol, 231, 383 Dithionite, 157 Dithiothreitol, 231, 383 Diversification of enzymes, 70–77 role of catalytic promiscuity, 74–77 role of pseudogenes, 77 structural aspects, 71–74 Dividing surface see Transition-state barrier Dixon plots, limitations in, 498–499 DNA, 61, 453, 454 allosteric behavior, 688 packing, 444 polycyclic aromatic dye binding, steric hindrance, 694 polyelectrolyte effects, 422 polymerization, two-color cross-correlation analysis, 753 RecA in recombination, optical trapping experiments, 742 single-molecule sequencing approaches, 730 transcription errors, 454 type II restriction endonuclease target sequence recognition/binding, 323 DNA methylases, 49 DNA methyltransferases, 40 DNA polymerase clamp, 785 AAA+ mechanoenzyme clamp loader see DNA processivity clamp loader structure, 786 DNA polymerases, 270, 426 crystalline kinetics, 458 kinetic proofreading, 454–455, 456 laser optical tweezer experiments, 740 processivity, 328, 329 clamp-dependence, 785 replicative accuracy, 453–454 substrate-induced conformational change, 325–326 see also DNA repair enzymes DNA processivity clamp loader, 785–788 DNA binding, 787–788 DNA polymerase clamp interactions, 785, 786–787 ATP binding/hydrolysis, 786, 788 DNA polymerase processivity dependence, 785 structure, 786 DNA repair enzymes, 3, 113, 454 Docking prediction, 560 Domains, protein structure, 62, 63 motifs, 63

Dopamine b-monooxygenase, 387 Double inhibitor interactions, substrate subsite binding (Yonetani-Theorell treatment), 500–501 Double-layer theory, 421–422, 449 Gouy Chapman layer, 421, 449 oriented diffusion, 421, 422, 449 Stern layer, 422, 449 Double-reciprocal plot see Lineweaver-Burk plot Douzou, P., 18 Drift tube, 259 Drug design, 488, 558–572 adaptive inhibition, 567–568 distal-site potentiator approach, 571 enzyme targets, 44, 553–554 high-throughput screening, 559–560, 569 inhibitory potency, 561–562 lead compounds, 44, 488, 559, 560 fragment-based design, 569–570 macromolecular inhibitors, 572 mechanism-based inhibitors, 563 metabolic control analysis, 572 pharmacophores, 561 pro-drugs, 566–567 RNA interference, 571 Schramm’s transition state-based strategy (molecular similarity approach), 563–566 screening approaches, 563 Lupinski’s ‘‘rule-of-five index’’, 568 target enzymes, 558–561 targets, 487 ‘‘druggable’’, 558–559 virtual (in silico) screening, 560 Duality principle, 164 Duplication, gene, 70, 460 DynaFit(TM), 269, 678 Dynamic mechanism, 637 Dynein, 397, 776 processivity, 329

E Eadie-Hofstee plot, 291 Eckstein, F., 18 Ectoenzymes, 443 EDTA, protease inhibition, 225, 226, 227 Efficacy, agonist-receptor interactions, 388 Efficiency of enzyme catalysis, 25–34 acid-base catalysis, 30 covalent catalysis, 26, 30–31 enzyme conformational flexibility, 32–33 evolutional aspects, 310 force-sensing/force-gated mechanisms, 26, 33–34 internal equilibrium constants, 309 intrinsic binding energy, 28 metal ion facilitation, 26, 32 reactant approximation/orientation, 26, 28–29 reactant-state (ground state) destabilization, 26, 29 transition state stabilization, 26–28, 29 low-barrier hydrogen bonds, 31–32 Efficiency function, 309–310 Ehrlich, P., 486

Eigen, M., 18, 639, 640, 675 Eigenfunction, for definition see Glossary Einstein, for definition see Glossary EKB-569, 550, 552 Eklund, H., 647 Elastase acyl-enzyme formation at sub-zero temperatures, 436, 437 solvent kinetic isotope effects, 626 stopped-flow kinetic techniques, 642 Elasticity coefficients, 725 Electric double layer, 421–422, 449 diffuse region, 421, 422, 449 Gouy-Chapman model, 421, 449 Stern layer, 422, 449 Electric field effects, metal ion complex color/magnetic properties, 87–89 Electroactive species, for definition see Glossary Electrokinetic potential (z), 422 Electron microscopy, 41, 762 Electron paramagnetic resonance spectroscopy, 654 freeze-quench experiments, 653–654, 655 lipase active site lid opening, 452 Electron sink, for definition see Glossary Electron spin resonance, for definition see Glossary Electron spin resonance spectroscopy, 172, 654 enzyme reaction intermediates detection, 654 free radical intermediates, 132 freeze-quench experiments, 654 ligand binding measurement, 690 metal ion–enzyme/enzyme substrate complexes, 94 metal-nucleotide complexes, 392 Electron transfer reactions, 152, 155–169 free radicals, 131 heme proteins/enzymes, 161 Marcus theory analysis, 162–163 oxidation potentials, 155 photosynthesis, 164 quantum mechanical tuneling, 155 redox active centers, 154, 158 redox enzyme mechanisms, 164–166 reduction potentials, 156–159 measurement, 158 respiratory chains, 160–162 simple kinetic models, 163–164 Electron-nuclear double resonance (ENDOR) spectroscopy, metal ion–enzyme/ enzyme substrate complexes, 94 Electronic transitions, 244 radiationless, 244 singlet state, 244 Electrophiles, 16, 119 for definition see Glossary hydration, 200 two-protonic state, 545 Electrophilic catalysis, for definition see Glossary Electrophilic reactions, 206 additions (heterolytic), 120 covalent intermediate formation, 30 substitutions, 120

Index

Electrophilicity, for definition see Glossary Electrospray ionization mass spectrometry, 259 rapid mix/quench experiments, 650–652 Electrostatic effects desolvation, 64, 65, 200–201 diffusion influences, 198, 199–200 enzyme influences at single-molecule level, 730 enzyme structure/interactions influence, 64–66 site-directed mutagenesis, 470 structural representation, 65–66 transition state stabilization Coulombic interactions, 27 efficiency of enzyme catalysis, 26, 27–28 Electrostatic field, 65 Electrostatic surface potential, 65 for definition see Glossary Elementary reactions, 16, 176, 637, 638 fast reaction techniques, 640 molecularity, 176 pH effects, 401 primitive changes, 637 rate constants, 303 reaction coordinates, 208 reaction mechanism elucidation, 19 relaxation amplitudes, 660 study methods development, 16 Elements essential for life, 82 Elimination reactions, 125, 126 Emergent properties of complex systems, 2, 685 Empirical rate equation, 172–174 Emulsin, 12 Enalapril, 488 Enantiomeric enrichment, 316–318 chirality-conferring step, 316 Enantiomeric excess, 147 Enantiomers, 147 Enantioselective reaction, for definition see Glossary Enantioselectivity, 316–317 directed evolution, 317 Encounter complex, 195, 197, 198, 419 diffusion-limited reactions, 198 dissociation constants, 198 Encounter contolled rate, for definition see Glossary Encounter distance, 198 Encounter-limited (encounter-controlled) reactions, 195, 198 Endergonic process, for definition see Glossary Endodepolymerases, 328 random scission kinetics, 329–330 Endoglycosidases, 115 Endonucleases, 113–114 Endopolymerases, 112–113, 328 Endoproteinases, 115 Energases, 25, 38, 47, 53, 729, 761–766 definition, 764–765 see also Glossary functions/reactions catalyzed, 763–764 Energy charge model, evidence against, 523 Energy flow during enzyme catalysis, 37 Enhancement factor, 7

873

Enolase, 75, 87, 129, 300 5-Enolpyruvoyl-shikimate-3-phosphate synthase, 271, 530–531, 573, 680–681 tetrahedral intermediate, ectrospray ionization ion-trap mass spectrometry, 651 Enoyl-coenzyme A hydratase, 239 isotope equilibrium perturbation effects, 613 Environmental awareness, 232 EnzFitterÒ, 269 EnzFit(TM), 269 Enzyme Commission (EC) classification scheme, 46–47, 118 Enzyme, for definition see Glossary Enzyme electrodes, 166–169 applications, 169 design features, 166–167 electron transfer mediators, 167 redox-active enzymes, 168 sensitivity/reproducibility, 169 Enzyme linked immunosorption assay (ELISA), 691 EnzymeKinetics Pro(TM), 269 Epidermal growth factor receptor, 552 Epigenetics, 49 Epimerization, for definition see Glossary 2,3-Epoxypropanol-phosphate, 542 Equilibrium constant, 7, 278, 288, 289 determination techniques, 278–284 internal, 300 temperature effects, 427–428 Equilibrium dialysis, 689, 690 Equilibrium exchange rate equations exchange of isotope labeled species, 589–592 equilibrium treatment, 589–590 steady-state treatment, 590 predicted patterns, 591 Equilibrium isotope effects, 586, 612 definition, 608 see also Glossary solvent isotope effects, 623 Equilibrium isotope perturbation method, 612–613 Equilibrium reactant concentration calculation, 419 Equilibrium thermodynamics, 210 Alberty formalism, 211–212 ErbB2, 552 Erectile dysfunction, phosphodiesterase inhibitors and, 561 Ergodic hypothesis, 45, 729 Error, experimental, 174 Error-prone polymerase chain reaction, in vitro evolution of enzymes, 39 Essential activators, 381 binding, 382 catalytic, 382 metal ions, 84, 385 mixed, 382 Esson, E., 185 Ester hydrolysis, 134 acyl-enzyme intermediate, 134 with no intermediate, 134 see also Phosphate ester hydrolysis

Ethanol-ammonia lyase, 121 Ethanolamine ammonia-lyase, 132 Ethoxyformic anhydride (diethyl pyrocarbonate), 545 Ethyleneimine, 184 N-Ethylmaleimide, 226 1-Ethyl-3-methylimidazolium cation, 424 Euler method, reaction rate analysis, 183 Eupergit(TM), 441 Evanescent wave phenomena, 749 Evolutionary aspects, 460, 482 catalytic proficiency, 308–309 efficiency of catalysis, 310 intermediates, 300 perfected enzymes, 310 Excited state, for definition see Glossary Exergonic, for definition see Glossary Exiphile, 119, 121 Exodepolymerases, 328 Exons, 61 Exopolymerases, 112, 113, 327, 328 Exothermic, for definition see Glossary Experimental error, 174 Explicit ion effects, 421 Expression vectors, site-directed mutagenesis studies, 468 Extent of chemical reaction, 210 Extremophilic bacteria, 430 Extremophilic enzymes applications, 431 structural stability, 430–433 activity relationship, 431–433 Eyring’s absolute rate theory, 203

F F0F1-ATP synthase fraction-of-sites activity (negative cooperativity), 705 rotatory catalytic mechanism, 748 F1-ATPase, 393, 395 negative cooperativity, 705 rotatory catalysis, 790, 791 Farnesyl diphosphate synthase, 123 Fast reaction techniques see Rapid reaction techniques Fast-atom bombardment mass spectrometry, 258 Fatty acid synthesis, 138–139 Feedback inhibition, 2, 686 Feedback regulation, 2 Fermentation science, 13 Ferricyanide, 157, 158 Ferrocene, 158 Ferrochelatase (protoheme ferrolyase), 83, 85, 92 Fersht, A.R., 18 Fick’s law, 212 Finasteride, 556 Fink, A., 18 First-order kinetics, 177–180 isotope exchange at equilibrium, 586–587 mechanism-based inhibitors (suicide substrates), 552, 555–556 Michaelis-Menten processes, 288

Index

874

First-order kinetics (Continued ) multi-stage mechanisms reversible, 186 series, 185 parallel first-order reactions diverging from common reagent, 179–180 leading to common product, 180 rate law derivation, 178 see also Pseudo first-order kinetics First-order reactions, 175, 176 Fischer, E., 13, 146, 319, 567 Fischer Projection, 9 Fischer-Kilani aldose synthesis, 256 Fischer-Rosanoff convention, 147 FITSIM, 18, 183, 268, 269, 537, 678 FitSpace ExplorerÒ, 271 FK506, 501 Flash photolysis, 638, 639, 658, 665, 675–677 apparatus, 675 data analysis, 678–679 Flavin coenzymes, 78 electron transfer reactions, 152 hydride transfer reactions, 578–579, 580 photochemistry, 675 stereochemistry, 579 Flavin transhydrogenases, 165–166 Flavodoxin, 66 Flickering clusters, 195 ‘‘Flip-flop’’ model, 705–706, 707 Flow techniques (rapid reaction techniques), 639, 640–653 Fluor (scintillator), 261, 262 Fluorescein, 249, 747 photobleaching, 250 Fluorescence, 240, 244 enhancement, 249 intrinsic/extrinsic reporter groups, 247–250 ligand binding measurement, 690 quantum yield, 244 quenching, 249 spectra, 244 temperature-jump techniques, 667, 669 Fluorescence anisotropy, 243, 268 binding interactions quantification, 250–252 instrumentation, 251 scattered light artifacts, 251 single molecule observations, 751 Fluorescence correlation spectroscopy, 244, 674, 751–756 applications, 753–754 kinetic theory, 753–754 observations on small volumes, 751, 752–753 one-color, 753 proteolytic cleavage investigations, 755–756 two-color cross-correlation analysis, 753–754 Fluorescence microscopy, 444, 746–751 direct observation of rotatory catalysis, 748 sample illumination, 746–747 see also Total internal reflection fluorescence microscopy Fluorescence microspectrophotometry, crystalline enzyme kinetics, 458–459

Fluorescence resonance energy transfer (FRET), 37, 243, 252–253, 747 acceptor, 252 adenylate kinase substrate-induced conformational change, 321 cross-correlation analysis, 755 proteolytic cleavage investigations, 755–756 DNA processivity clamp loader observations, 788 donor, 252 protease assays, 253 single molecule observations, 751 Fluorescence spectrometer (spectrophotofluorimeter), 244–246 coincidence phenomena, 245 inner-filter effects, 245–246 limitations, 245 luminometry, 254 shot noise, 245 Fluorescence spectroscopy, 172, 221, 243–255 enzyme assays, 248, 250 instrumentation see Fluorescence spectrometer (spectrophotofluorimeter) ligant binding titration experiments, 246 metabolite quantification, 246–247 phosphate/pyrophosphate-producing reactions, continuous assays, 253–254 reaction progress-curve quantitative analysis, 268–270 site-directed mutagenesis studies, 468 solutions standardization, 230 Fluoroacetate, 528 5-Fluorodeoxyuridylate, 552 Fluorophores, 244, 729, 746 extrinsic macromolecule labeling, 248–250 nucleic acids incorporation, 249 quenching, 244 Flux control coefficients, 724 Flux ratio method (Britton), isotope exchange behavior, 596–599 definition, 596 multi-substrate kinetic mechanisms, 598–599 ordered ternary complex mechanism, 598–599 random ternary complex mechanism, 599 one-substrate kinetic mechanisms, 597 reaction flux measurement, 596–597 Flux, reaction, 596 counter-transport rate, 597 isotopomer measurement, 597 metabolic control analysis, 723–724, 725, 726 regulatory enzymes, 686 transport rate, 597 Foldases, 25 Footprinting, for definition see Glossary Force, for definition see Glossary Force generation, 761, 773 coupling to ATP/GTP hydrolysis, 766–769 green fluorescent protein unfolding/refolding observations, 775 mechanoenzymes, 766 Force-gated processes, 37 conformational changes, 209 efficiency of enzyme catalysis, 26, 33–34

Force-induced noncovalent bond rupture, 769–775 energetics, 770 Kramers model, 770–771 mechanoenzyme catalysis, 771–773 Force-measuring microprobes, 744 Formaldehyde ferredoxin oxireductase, 107 Formate dehydrogenase, 519 kinetic isotope effects, 627–628 Formation constant, for definition see Glossary Formins, 795, 799 Formyltetrahydrofolate synthetase, 110 Fo¨rster radius, 252 Fo¨rster resonance energy transfer see Fluorescence resonance energy transfer (FRET) Foster-Neimann plots, 520–521 Fractal kinetics, 176 Fraction-of-sites activity, 705–707 Fractionation factors for definition see Glossary isotope exchange rates, 600 Fragment ion, 257 Fragment-based lead design, 569–570 Free radicals, 119, 131–134 active site residues formation, 58 heterolytic generation, 131 homolytic generation, 131 addition reactions, 120 paramagnetic properties, 131 radical trapping approaches, 133 reaction intermediates, 132 rearrangement reactions, 120–121 1,2-shift reaction, 120 substitution reactions, 120 Freeze-dried enzymes, 224 activity in organic solvents, 423 salt-enzyme preparations, 424 Freeze-quench experiments, 653–655 apparatus, 654 biochemical reactions probed, 654 Frequency factor, 201 Frieden, C., 18 three-substrate enzyme kinetic analysis protocol, 368–369 see also Hysteretic enzymes Fromm, H.J., 18, 339, 341 adenylosuccinate synthetase, 364, 369, 520 CoA-linked acetaldehyde dehydrogenase, 369 energy charge model, evidence against, 523 hexokinase catalysis and control, 225, 606 limitations on Dixon plots in analyzing reversible inhibition mode, 499 multi-substrate kinetic mechanisms alternative substrate inhibitor method, 508–510 competitive inhibitor method, distinguishing kinetic mechnsisms, 494–498 point-of-convergence method, 362–363 substrate-ratio protocol for distinguising three-substrate enzymes, 369–370

Index

systematic method for deriving rate equations, 341–343 Huang’s modification, 343 two-step computer-assisted derivation method, 630 Frozen storage, 224, 427 Fructokinase, isotope trapping experiments, 606 Fructose bisphosphatase, 224–225, 430 alanine scanning mutagenesis, 462–463 Fructose 1,6-bisphosphate aldolase, 225 Fructose-6-phosphate 2-kinase, alanine scanning mutagenesis, 462–463 Fruton, J., 18 Fumarase, 275, 429, 587 equilibrium isotope effects, 612 Fumarate hydratase, 129 Functional groups adaptive inhibition in drug design, 567 irreversible inhibitor interactions, 539 reagents modifying, 541

G G-proteins, receptor-associated, 686, 782 Galactose binding protein, 97 Galactose oxidase, 81 enzyme electrodes, 167 g particles, 260 g-carboxyl-L-glutamate, 80–81 g-cyclodextrins, 446 g-glutamyltransferase, 386–387, 505–506 g-secretase, 453, 536 Gauss-Newton iterative data fitting, 267 Gear’s predictor-corrector algorithm, reaction rate analysis, 183–184, 297, 678 Gel filtration, 690 Gel retardation, 690 Gelatinase, 109 Gelb, M., 18 Gene duplication, 70, 460 General acid catalysis, 204, 409, 410 chymotrypsin/serine proteases, 20 for definition see Glossary General base catalysis, 409, 410 carboxypeptidase amide/peptide hydrolysis, 136 chymotrypsin/serine proteases, 20 for definition see Glossary Genetic basis of disease, 49 Gentamicin acetyl transferase, inhibition, 525 Gibbs energy, 212 Gibbs equation, 210 Gibbs, J.W., 212 Glass cuvettes, 241 Glass surfaces passivation, 228 thermal equilibration, 220, 227 ultraviolet radiation absorption, 241 wall effects, 228 Glassware washing, 228 Global Kinetic ExplorerÒ, 270–271 Global statistical analysis (global analysis), 174 for definition see Glossary fluorescence anisotropy data, 251 rate data, 270–272

875

relaxation kinetics data, 679–680 software, 270–271 Glucan 1,4-a-glucosidase, 122 Glucanases, substrate recognition subsite designation scheme, 115 Glucoamylase, 403–404 substrate recognition subsite designation scheme, 115 Glucokinase, 115–116, 303, 633 activators, 40, 379, 381 Glucose 6-phosphatase, 357, 364, 373 isotope exchange processes, 569 Glucose dehydrogenase, 363 Glucose isomerase, 97 Glucose oxidase, 78, 160 enzyme electrodes, 167 Glucose phosphotransferase, 769 D-Glucose, stable isoptomers, 256 Glucosyltransferase, single molecule fluorescence observation of dextran binding, 750–751 Glufosinate ammonium, 530 Glutamate decarboxylase, solvent kinetic isotope effects, 626 Glutamate dehydrogenase, 35, 338, 519 isotope equilibrium perturbation effects, 613 polymerization, 709 Glutamate mutase, 100 adenosylcobalamin coenzyme kinetic isotope effects, 629 Glutamate oxidase, enzyme electrodes, 167 Glutamate racemase, 129, 603 Glutamate synthase, 358 Glutamine amidohydrolase, reversible cold inactivation, 434 Glutamine synthetase, 2, 48–49, 104, 106, 218, 228, 318, 319, 335, 364, 386, 391, 401, 714 acyl-phosphate intermediate, 144–145, 146, 606 adenylylation/deadenylylation cascade, 715, 716 competitive inhibitor studies, 499–500 covalent intermediates, 146 side-reactions, 142–145 g-glutamyl phosphate intermediate, 604, 606 isotope exchange at equilibrium, 587–588 transition state inhibitors, 529–530 Glutathione, 383 Glutathione reducatse, 579 Glutathione-S-transferase, 252 Glyceraldehyde 3-phosphate dehydrogenase, 117, 519 crystalline kinetics, 457 substrate channeling, 722 Glycerokinase, 363 Glycerol cryoprotectant, 223 reversible cold inactivation effects, 434 Glycerol-dehydrase, 121 Glycine reductase, 80 Glycogen, 445 Glycogen phosphorylase, 233, 387 a versus b forms, 233 amplification cascade, 713, 714, 717–718

Glycogen phosphorylase kinase, 714 Glycogen synthase kinase-3, 96 Glycosidases kinetic isotope effects, 619 substrate recognition subsite designation scheme, 115 Glycosyl hydrolases, substrate recognition subsite designation scheme, 115 Glycosylation, 61 Glycosyltransferases, 60 Glyphosate, 573 Gold nanoparticle catalysts, 8 Good’s buffers, 413 Gouy-Chapman (outer diffuse) layer, 421, 449 Grafted polymers, enzyme immobilization, 425 Gramisidin, 488 Granddaughter ion, 257 GRASP (Graphical Representation and Analysis of Structural Properties) software, 65–66 GRASP2, 66 ‘‘Green Chemistry’’, 9, 47 Green fluorescent protein force-dependent unfolding/refolding, 775 fusion protein expression, 746, 755–756 GroEL foldase, 95 GroEL/GroES, 25, 792 ATP hydrolysis-dependence, 792, 793 protein folding, 792–793 Ground-state destabilization, 37, 205, 333 efficiency of enzyme catalysis, 26, 29 Grubbs, R., 9 Grubbs Ruthenium Catalyst, 8 GTP hydrolysis, 782 coupling to force-generation, 766–769 substrate regeneration, 276 GTP-dependent enzymes, 393 kinases, 104 GTP-protein-coupled receptors, 487–488 GTP-regulatory proteins, 782–783 nucleotide exchange factors, 395 GTPase-activating proteins (GPAs), 395, 782 GTPases, 762, 764 Guanine, 57 Guanine nucleotide dissociation simulator, 395 Guldberg, C., 6 Gutfreund, H., 18

H Haber, H., 6 Haber process, 6 Haldane, J.B.S., 15, 16, 26, 294, 506, 507 Haldane relationship, 280, 296, 300–301 bisubstrate kinetic mechanisms, 359–360 kinetic, 360 thermodynamic, 359–360 for definition see Glossary equilibrium constant determination, 280 Half-life first-order processes, 178 protein degradation, 634 radioisotopes, 260, 261 signaling/housekeeping enzymes, 686 Half-sandwich complexes, 167

Index

876

Haloalkane dehalogenase, 71 kinetic isotope effects, 630 Halohydrin dehalogenase, 317 Haloperoxidases, 107, 426 Halophilic bacteria, uses of enzymes, 431 Hammes, G., 18, 32 Hansen, J., 13 Haptens, catalytic antibody design, 10 Hartley, B.S., 18 Heat exchangers, 220 Heat shock proteins, 792 Heavy atom isotope effect, for definition see Glossary Heisenberg uncertainty principle, 164, 203, 609, 614 Helicases, 776 Heme, 81, 101, 163, 335 cytochrome c oxidase, 161–162 cytochrome P450 coordination scheme, 111 factors affecting electron transfer, 161 synthesis, 85 Hemoglobin, 87, 319, 444, 640 conformational spread model, 709 oxygen binding (heme-heme cooperativity), 687, 694, 702–703 Adair model, 700 Pauling’ site interaction model, 700–701 oxygen saturation curve, 691–692 Henri, V., 287 Heparosan-N-sulfate-glucuronate 5-epimerase, 129 Heterobifunctional cross-linkers, enzyme immobilization, 440 Heterogeneous catalysis, 7, 205 Heterotopic interaction, for definition see Glossary Hexokinase, 1, 59, 232, 303, 331, 335, 360, 363, 429, 575, 633 active site substrate-induced conformational change, 320, 321 alternative substrate inhibition, 508 ATPase activity, 320, 321 coupled (auxiliary) enzyme assays, 235, 237 de-inhibition, 382–383 equilibrium exchange behavior, 594 isotope exchange, 589 feedback inhibition, 686 interactions with hexoses/pentoses, 320–321 isotope trapping experiments, 604, 605 kinetic mechanism, 225, 605–606 lag-phase kinetics, 713 limitations of intial-rate data, 364–365 MgATP2- effects, 393–394, 396–397 phosphoryl transfer reaction stereochemistry, 580 pressure effects, 439 product inhibition, 521 protease cleavage sites, 225 High performance liquid chromatography, 172 isoenzyme detection, 216 High-spin state, for definition see Glossary High-throughput screening directed evolution of enzymes, 39 drug design, 559–560, 569 substrate specificity determination, 41, 42

Hill, A.V., 691 Hill coefficient, or constant, 691 Hill equation, 691–693 Hirudin, 269 Histidine ammonia lyase, 206 Histone acetylase, inhibition, 525 Historical aspects, 12 enzyme science, 12 enzyme technology, 13 multi-substrate steady-state kinetics, 338 Hit, for definition see Glossary HIV protease inhibitors, 488 adaptive, drug design, 568 HIV reverse transcriptase, 391 39-azido-29 deoxythymidine (AZT) inhibition, 492 ionic strength effects, 420 Hofmeister series, 421 Homeostasis, 2, 213, 482, 685 Homeotherm body temperature, 429–430 Homobifunctional cross-linkers, enzyme immobilization, 440 Homoenzymes, 12 Homogeneous catalysis, 7, 205 Homoserine transacetylase, solvent kinetic isotope effects, 626 Homotropic interaction, for definition see Glossary Host-Guest interactions, for definition see Glossary Hoveyda Ruthenium Catalyst, 8 Huang, C.Y., 18 constant ratio alternative substrate inhibition method, 510–511 derivation of steady-state rate equations, 343 Human growth hormone, 462 Hydride transfer, 150 Hydrogen bonds, 66 active site catalytic residues, 59 enzyme structural stability, 66–69 Grotthu¨ss chains (proton-conducting wires), 67, 481 hydrophobic interactions, 69 Lewis acid–Lewis base interactions, 82 low-barrier, 67 active site, 67–68 efficiency of enzyme catalysis, 31–32 transition state stabilization, 31–32, 68 pH-dependent kinetic effects, 400 protein structure, 61, 62 proton transfer reactions, 67, 195–196 short strong, 68 site-directed mutagenesis effects, 467–468, 481 substrate binding energy, 67 water molecules, 194, 195–196, 197 solvent properties, 195 Hydrogenation, catalytic, 7–8 Hydrolases, 23, 71, 335 burst kinetics, 356–357 for definition see Glossary pH optima, 398 Hydronium ions, 195, 196 Hydrophobic interactions entropy-driven processes, 69 enzyme structural stability, 69

(S)-2-Hydroxyacid oxidase, 129 Hydroxylamine, 184 Hydroxylapatite, 383 p-Hydroxy-mercuribenzoate, 539 6-(1S-Hydroxy-3-methylbutyl)-7-methoxy2H-chromen-2-one, 557 3-Hydroxy-3-methylglutaryl-coenzyme A reductase, 335 inhibition, 573 reversible cold inactivation, 434 Hydroxynitrile lyase, 71 (4-Hydroxyphenol)pyruvate dioxygenase, mix/quench experiments, 652–653 Hydroxy-proline, 57 Hyperthermophilic bacteria, uses of enzymes, 431 Hypothesis-based research, 172–173 proof by exclusion, 174 Hypoxanthine:guanine phosphoribosyltransferase, 122 Hysteresis, for definition see Glossary Hysteretic enzymes, 324, 712–713 activator biding, 390 inhibitor binding, 535

I

I3-cooperativity, 687 IC50, 538 Ideal bond, 774 Imines (Schiff’s bases), 139, 140 Immobilized enzymes bioreactor applications, 440, 442 cross-linking agents, 440 kinetic behavior, 442 flow tube tethered enzymes, 442–443 relevance to cellular conditions, 443–444 stability/kinetics, 439–444 supporting media, 440 Immucillin, 564, 566 Immunosuppressive agents, 487, 501 IMP dehydrogenase, 6-mercaptopurine inhibition, 487 in silico drug design distal-site potentiator approach, 571 fragment-based lead design, 569–570 volume filling/interaction-matching, 570 virtual screening, 560 Inactivation, enzyme assay design, 223–224, 231 J-Plot, 231–232 see also Inhibitors, irreversible Induced substrate inhibition, 511 Induced-fit mechanism, 319–327 computer-based ligand-docking, 327 energetics, 324 enzyme specificity, 319, 324–326 generalized catalytic reaction cycle, 326 selected-fit mechanism comparison, 324–326 Inducers, 16, 383 Induction period, 185 Ingold, K., 16 Inhibitors, 2, 265, 485–573, 686 adaptive, 567 applications, 489 biomedical, 486–491 biphasic effects, 391

Index

competitive see Competitive inhibitors dead-end, 493 Cleland’s rules for analysis, 505 use in distinguishing multi-substrate mechanisms, 494–498 drug design targets/lead molecules, 44 effectors, 685 feedback inhibition, 686 irreversible, 485–486 active site-directed, 539, 540–541, 543 affinity labeling agents, 539–547 kinetic mechanism, 540–544 pH effects, 544–545 rational design, 539–540 substrate competition effects, 544 therapeutic see Drug design tight-binding reversible inhibitor differentiation, 544 linear (complete), 504–505 mechanism-based see Mechanism-based inhibitors (suicide substrates) modifiers, 685 multi-substrate geometric analogues, 523–524 noncompetitive, 494, 495, 501–502, 512, 531, 532 partial, 387, 489, 493–494, 504–505 pH-dependent effects, 407 product see Product inhibition production methods, 44–45 reversible, 485, 489–506 degree of inhibition, 537–538 IC50, 538 percentage inhibition, 537–538 potency measures, 537–538 tight-binding see Tight-binding reversible inhibitors site-directed mutagenesis studies, 468–469 substrate see Substrate inhibition synergistic, 505–506 total, 489, 493 transition-state see Transition-state inhibitors uncompetitive, 494, 495, 502–503, 512, 532 Initial velocity, 174, 289 versus Log[Substrate] plot, 292 Initial velocity (V0), 215 Initial-rate enzyme experiments, 3, 16, 53, 181–182, 217–284, 287 Briggs-Haldane treatment, 293–297 data analysis see Rate data, analysis design, 215–232 activity purity, 216–217 ‘‘blank’’ rates determination, 219 bubble formation avoidance, 223 contamination avoidance, 227–228 continuous rate measurements, 217, 218–219 discontinuous rate measurements, 217 enzyme degradation prevention, 224–227 enzyme inactivation avoidance/ management, 223–224, 231 experimenter alertness, 232 isoenzymes, 216 lag-phase kinetic behavior recognition/ minimization, 227 mixing uniformity, 220, 223

877

order-of-addition effects detection, 228–229 precision of measurements, 229–230 problem diagnosis, 231–232 randomizing measurement order, 230–231 reference standards, 229 single-point assays, associated errors, 218 special requirements, 220 substrate concentration, 216 thermal equilibration, 220, 227 working solutions assembly, 264–265 equilibrium constant determination, 280–281 Michaelis-Menton processes, 287–293 reaction rate dependence on active enzyme concentration, 292 multi-substrate reactions, 335–378 multiple-turnover processes, 16 one-substrate reactions, 287–335 rate parameters, 289–290 rate plots, 290–292 single-turnover processes, 16 techniques, 221–222 three-substrate enzymes, 368 Frieden protocol, 368–369 Fromm substrates-ratio protocol, 369–370 Initial-rate phase, definition, 215–216 Inner-coordination sphere, for definition see Glossary Inner-filter effects, 245–246 for definition see Glossary Inner-sphere electron transfer, for definition see Glossary Insecticides, 488 Instantaneous velocity (vt), 174 Instrumentation noise, 230 Interactions, enzyme-reactant, 64–70 electrostatic, 64–66 hydrogen bonding, 66–69 hydrophobic, 69 ion–dipole and dipole–dipole, 66 p-cation, 70 van der Waals, 70 Interconvertible enzyme, for definition see Glossary Interfacial catalysis, 205, 447–453 enzyme in scooting/hopping mode, 450, 451 processive turnover, 450 units for tethered components, 453 Intermediacy, for definition see Glossary Intermediates, 1, 117, 118 burst-phase kinetic investigations, 658 carbanion, 125–130 carbenium, 122–123 covalent, 140, 205, 352 chemical trapping, 145–146 detection, 138–146 enzyme-bound coenzyme–substrate compounds, 142, 144 enzyme-bound substrate–substrate compounds, 142, 143 enzyme-substrate adducts, 139–141 enzyme-substrate compounds, 141–142 isotope trapping, 603–604

side-reactions, 142–145 transfer within complex active sites, 358 cryoenzymology studies, 435, 436–437, 438 electron spin resonance detection, 654 electrophilic catalysis, 206 evolutionary aspects, 300 exchange reactions, 588 external, 300 group-transfer reactions, 124–125 internal equilibria, 309 equilibrium constants, 300 multiple, 299–300 Haldane relationship, 300–301 steady-state treatment, 298–301 nucleophilic catalysis, 124–125, 205 probes of enzyme catalysis, 38 rapid-mixing continuous-flow detection methods, 640, 643 series first-order processes, 185 substrate channeling, 718–719 time-resolved solid-state nuclear magnetic resonance spectroscopy, 680 Internal conversion, for definition see Glossary Internal equilibria, 309–310 equilibrium constants, 300, 309, 310 International Unit (IU), 290 Internuclear distance, 201 Intracellular enzyme kinetics, 48–49 Intramembrane proteolysis, 453 Intramolecular catalysis, for definition see Glossary Intramolecular Isotope Effect, for definition see Glossary Intramolecular kinetic Isotope Effect, for definition see Glossary Intrinsic binding energy for definition see Glossary efficiency of enzyme catalysis, 28 Introns, 61 Invertase, 5, 12, 15, 287, 289 Ion, 257 chaotropic, 421 dipole interactions, 66 explicit effects, 421 fragment, 257 Hofmeister series, 421 isotopologue, 257 isotopomeric, 257 kosmotropic, 421 m/z ratio, 257 metastable, 257 molecular, 257 molecular mass, 257 principal, 257 product, 257 Ion channels, 733, 734 Ion current intensity, 257 Ion pairs electrostatic interactions, 64–65 longer-range, 65 N–O bridges, 65 nucleophilic internal substitution reactions, 124 salt bridges, 65 transition metal complex reactions, 90

Index

878

Ion ratio mass spectrometry (IRMS), 259–260, 263 Ion-selective electrodes, 690 Ionic atmosphere, for definition see Glossary Ionic reactants electrostatic interactions, 199–200 nonaqueous liquids, 424–425 Ionic screening, 200, 419 Ionic strength activity coefficient effect, 418, 419 buffer system maintenance, 415–416 definition, 417 effects on enzyme kinetics, 416–422 equilibrium reactant concentration calculation, 419 explicit ion effects, 421 macromolecular polyelectrolytes, 419, 422 rate constant effects (primary salt effect), 418–419 thermodynamic activity, 419 Debye-Hu¨ckel limiting law, 417–418, 419 Ionization biomolecules for mass spectroscopy, 258 solute, 196 Ionization energy, for definition see Glossary Iron, 81, 82, 101, 103 enzyme complex coordination schemes, 111, 112 Iron-containing proteins, 57, 101, 103, 111 Iron-sulfur centers, 101 Iron-superoxide dismutase, 84 Iso mechanisms see Isomerization mechanisms Isocitrate dehydrogenase, 38 irreversible inhibition, 542 Isoelectronic, for definition see Glossary Isoenzyme, for definition see Glossary Isoenzymes (isozymes), 63 Haldane relationships, 360 initial-rate enzyme assays, 216 Isoforms, 64 Isoleucyl-tRNA synthetase, inhibition, 525 Isomer group thermodynamics, 211–212 Isomerases, 23 for definition see Glossary Isomerization mechanisms (iso-mechanisms), 300, 314–315 bisubstrate kinetic mechanisms, 338 Iso Uni Uni, 338 multisubstrate, kinetic schemes, 370–372 product inhibition, 521, 522 Isopentenyl pyrophosphate isomerase, 150 Isopropylmalate dehydrogenase, site-directed mutagenesis, 483 Isothermal calorimetry, 690 Isotope exchange at equilibrium, 586–603 back exchange processes, 596 Britton’s flux ratio method, 596–599 equilibrium exchange rate equations, 589–592 equilibrium treatment, 589–590 steady-state treatment, 590 experimental focus on isotopic atoms, 587–589

first-order kinetics, 586–587 fractionation factors, 600 intermediate exchange reactions, 588 mass action ratio, 588–589 medium exchange reactions, 588 ordered Bi Bi ternary complex mechanism, 592–593 predicted patterns for enzyme kinetic mechanisms, 591 random Bi Bi ternary complex mechanism, 594 Theorell-Chance ordered Bi Bi binary complex mechanism, 595–596 transition-state energetics, 599–603 Albery-Knowles kinetic treatment, 600 Isotope exchange, for definition see Glossary Isotope fractionation factors, 600 equilibrium perturbation, 612 solvent kinetic isotope effects, 624 Isotope labeling experiments, 575–632 enzyme stereochemistry definition, 576–585 equilibrium isotope exchanges, 586–603 protein turnover, 631–634 proximal labeling, 585 reaction-center labeling, 585 remote labeling, 585 site-directed mutagenesis, 468 substrate labeling, 585–586 see also Kinetic isotope effects Isotope scrambling, for definition see Glossary Isotope trapping method, enzyme-bound substrate partitioning kinetics, 603–604 Isotopic ratio, 257 ion ratio mass spectrometry (IRMS), 259–260 Isotopically sensitive step, for definition see Glossary Isotopologue ion, 257 Isotopomeric ion, 257 Isotopomers, 608 kinetic isotope effects, 608, 609 zero-point energy differences, 609–612 reaction flux measurement, 597 IUPAC Gold Book, 47

J J-Plot, 231–232 Jain, M., 18 Jencks, W.P., 10, 18, 637 Job Plot, 390 Johnson (thermal) noise, 230 Jo¨rnvall, H., 647 Joule, for definition see Glossary

K Kaiser, E.T., 12 Katal (kat), 290 Ketol-enol tautomerism, 201 Ketosteroid isomerase, 68 Ketyl-acid reductoisomerase, reaction intermediates, 121, 123 Kinase-containing receptors, 686 Kinases docking domains, 331 nucleotide substrate binding/discharge, 394 peptide microarray priming, 331

Kinesin, 397, 776 ATP hydrolysis-related step rate, 737–738 processivity, 329 Kinetic ambiguity, 173, 192 multi-substrate initial-rate data, 364–366 steady-state processes with multiple internal isomerizations, 299 Kinetic equivalence principle, 192 Kinetic isotope effects, 586, 607–628 absent (indeterminate), 609 applications, 608 Biegeleisen-Mayer treatment, 610 definitions, 608 equilibrium perturbation measurements, 612–613 influence of other reaction steps, 619–622 internal competition assessment method, 611 intrinsic, 609, 620, 621–622, 623 inverse, 609 normal, 609 notation, 608 observed, 609 influence of multiple isotopically sensitive steps, 622–623 pH dependence, 628 primary, 586, 608 isotopomer zero-point energy differences, 609–612 proton-transfer enzyme studies, 627–631 quantum mechanical hydrogen tuneling, 613–616 rate constants, 609 reaction equilibria, 609 remote labeling, 608–609, 628 secondary, 586, 608 nucleophilic SN1-/SN2-type mechanisms differentiation, 616–619 solvent isotope effects, 623–627 substrate binding effects, 629–630 temperature dependence, 615–616 Arrhenius rate plots, 615 Kinetics, enzyme, 19, 637 definition, 3 historical development, 15–19 models, 3–4 Kinetochore, microtubule interactions, 801 King and Altman method, rate equation derivation, 341 KINSIM, 18, 183, 268, 269, 537, 678 Kirchhoff, V., 5 Kirschner, K., 18 Klinman, J.P., 18, 647 Knowles, J.R., 18, 26, 582, 585, 600 Koshland, D.E.Jr, 18, 319, 320 Koshland-Ne´methy-Filmer model, 688, 699–707 negative cooperativity, 701–702, 703–704 oxygen binding to hemoglobin, 700–701, 702, 703 positive cooperativity, 701–702 saturation function, 701–702 Kosmotropes, 421 Kramers, H., 770 Ku¨hne, W., 13

Index

L Laccase, crystalline enzyme kinetics, 459–460 Lactate dehydrogenase, 66, 129, 264–265, 320, 335, 338, 519, 633 equilibrium exchange behavior, 593 single-molecule activity, 730–731 steady-state rate equation, 341 Lactate, enzyme electrodes, 167 Lactate oxidase, enzyme electrodes, 167 Lactose permease, 24 lacZ, 279 Lag time, for definition see Glossary Lag-phase kinetic behavior, 712, 713 coupled (auxiliary) enzyme assays, 238–239 recognition/minimization in initial rate enzyme assays, 227 see also Hysteretic enzymes Laidler, K.J., 18 Laminar mixing, role in solute dissolution, 220 Langmuir trough, 177, 451–452 Lanosterol synthase, 72 Laser trapping see Optical tweezers Laser-induced temperature-jump technique, 669 Latent activity, for definition see Glossary Laue X-ray crystallography, 38, 682 Lead compounds, 44, 488, 559 docking prediction, 560 fragment-based design, 569–570 high throughput screening, 559–560 inhibitory potency, 561–562 ‘‘off-target’’ effects prediction, 562–563 screening approaches, 563 virtual (in silico) screening, 560 Lead, for definition see Glossary Least-squares data fitting, 267 pKa determination, 402 relaxation kinetics, 678 Leaving group, for definition see Glossary Legendre transform, 212 Lemoine, G., 5 Lerner, R., 18 Leucine aminopeptidase, 94 Leukocyte tethering/rolling, 774 Leupeptin, 225, 226 Levansucrase, activity in organic solvents, 424 Lewis acids, 82–83 for definition see Glossary efficiency of enzyme catalysis, 26, 32 enzyme-bound metal ions, 56, 81, 82, 83 magnesium, 103 zinc, 108–109, 111 Lewis bases, 82 for definition see Glossary Lewis, G.N., 82 L’Hoˆpital’s rule, 289 Lid-gated active sites, 60–61, 452 induced-fit mechanism, 325 substrate-induced conformational change, 321–322 Lifetime, for definition see Glossary Ligand binding efficiency, 388, 570 Hill equation, 691–693 Koshland-Ne´methy-Filmer model, 699–707 measurement strategies, 688–691

879

metal ion complexation, 85–87 Monod-Wyman-Changeux model, 695–699 oligomerization induction, enzyme activity regulation, 711–712 Scatchard equation, 693–694 steric hindrance, 693 Wyman’s linked function analysis, 694–695 see also Binding interactions Ligand field stabilization energy, 89 active-site metal ion binding, 83 Ligand field theory, 88–89 Ligand/ligancy, definitions, 85, 86 Ligases, 23, 40 for definition see Glossary Light absorption, 240 concentration dependence (Beer’s law), 240–241 electronic transitions, 244 radiationless, 244 singlet states, 244 Light emission, chemoluminescence, 254 Light scattering, initial rate enzyme assays, 222 Limit dextran, for definition see Glossary Linderstro¨m-Lang, C., 61 Linear free-energy relation, for definition see Glossary Lineweaver-Burk plots, 264, 290 competitive inhibitor kinetics, 494 limitations, 290 noncompetitive inhibitor kinetics, 502 substrate inhibition, 507 two-substrate reactions, 336 uncompetitive inhibitors, 503 Linked function, 422, 695 Wyman’s analysis, 694–695 Lipases colipase interactions, 452 interfacial catalysis, 449, 451 hopping mode, 451 Langmuir trough experiments, 451–452 organic solvent effects, 422–423 surfactant effects, 452–453 Lipid rafts, 453 Lipmann, F., 761, 762 Lipoamide, 78, 464 Lipoic acid, 78, 358 Lipoxygenase, 447 quantum mechanical hydrogen tuneling, 614 Liquid scintillation counting, 261–262, 263, 268 coincidence circuitry, 261 counting efficiency, 262 counting period, 262 data analysis, 262–263 isotope-based rate measurements, 576 quenching, 262 sample preparation, 262 Lithium, 95–96 Lock-and-key model, 13, 319, 567 London dispersive forces (van der Waals interactions), 70 Lone pair, for definition see Glossary Loops, active site, 322, 323 site-directed mutagenesis, 468, 471, 475–476 substrate binding (acceptor-binding site), 61

substrate-induced conformational change, 322 Loose ion pair, for definition see Glossary Low-spin complex , for definition see Glossary Luciferase, 254–255 Luciferase-based assays, 255 Luciferyl-AMP (LAMP), 254 Luminometer, 254 Lupinski’s ‘‘rule-of-five index’’, drug design, 568 Lyases, 23 Lysine tyrosylquinone coenzyme, 81, 132 Lysine-2,3-aminomutase, 97, 133 D-Lysine-5,6-aminomutase, 133 Lysozyme, 59, 121 carbenium ion mechanism, 121 glycosidic linkage cleavage, 333 glycosyl-enzyme intermediate, 618 ground-state destabilization, 333 intermediates, 125 kinetic mechanism, computational analysis, 28 oxacarbenium ion ion-pair stabilization, 333 Phillips mechanism, 333 pKa value calculation, 406 secondary kinetic isotope effects, 616–617, 619 substrate binding, 333 substrate recognition subsite designation scheme, 115 substrate-assisted catalysis, 331

M MACiE database, 46–47 Magic bullet, 486 Magnesium, 82, 103–105 ATP/GTP complexes, 103–104, 109, 110, 111, 273, 385, 395 ATP-dependent enzyme requirement, 273–274, 393–394 hexokinase hydrolysis, 320, 321 stability constant, 273 enzyme activation, 96 enzyme complex coordination schemes, 109, 110, 111 ligand binding, 103 Magnetic stirring bars, 228 Major sperm protein, 801 Malaria, 565 Malate dehydrogenase, 519 isotope equilibrium perturbation effects, 612 Malic enzyme, 519 isotope equilibrium perturbation effects, 612 kinetic isotope effects, 628 Mandelate racemase, 309 Manganese, 82, 105–106 nucleotide complexation, ATP-dependent enzyme rate experiments, 274 Manganese superoxide dismutase, 84, 105, 165 pulsed radiolysis, 677 Manometry, 221 Marcus, R., 155

Index

880

Marcus theory, 155 electron transfer reactions analysis, 162–163 Markov processes, single-molecule enzyme kinetics, 734 Mass action law, 6, 174, 176 Mass action ratio, 213, 588–589 Mass spectrometers classification, 258 design features, 257–258 detector, 258, 259 high-vacuum chamber, 258 inlet port, 258 ion source, 258 mass analyzer, 258 ion ratio, 259–260 time-of-flight, 259 unit mass resolution, 257 Mass spectrometry, 172, 222 applications, 258 basic principle, 257 complex biochemical substances analysis, 258–259 definitions, 256–257 isotope-based rate measurements, 576 positional isotope exchange reactions, 607 rapid mix/quench experiments, 650–652 soft ionization methods, 258 stable isotopes detection, 255 see also Electrospray mass spectrometry; Fast-atom bombardment mass spectrometry; Ion ratio mass spectrometry; Matrix–assisted laser–desorption ionization mass spectrometry; Matrix–assisted laser–desorption ionization–time–of–flight mass spectrometry Mass-to-charge ratio, 257, 258 Massey, V., 18 MathematicaÒ, 343, 345 MatLabÒ, 183 Matrix-assisted laser-desorption ionization mass spectrometry (MALDI-MS), 258–259 Matrix-assisted laser-desorption ionization–time-of-flight mass spectrometry (MALDI-TOF MS), 259 irreversible active site-directed inhibitors, 542 partial proteolysis assessment, 227 rapid mix/quench experiments, 651 site-directed mutagenesis studies, 468 Matrix-immobilized enzymes see Immobilized enzymes Maximal possible enrichment (MPE), 234 Maximal velocity (Vmax), 289 for definition see Glossary Maxwell, for definition see Glossary Maxwell-Boltzmann distribution see Boltzmann distribution Mean, 267 Mean transit time, for definition see Glossary Mechanism, systematic characterization, 3 stages, 3–4

Mechanism-based inhibitors (suicide substrates), 550–552, 555–558 design strategies, 550 drug design, 563 kinetic mechanism, 552, 555–556 noncovalent, 557–558 versatility, 556 Mechanoenzymes, 53, 253, 388, 397, 761–805 affinity-modulated, 768, 769 chemical phase of reaction cycle, 780 ligand binding, 770, 780 ATP binding sites, 395 catalytic mechanism, 23–25 cellular functions, 762, 763–764 chemical-to-mechanical energy transduction (force generation), 25, 766 classification, 762, 764 energy sources, 768–769 gradient-driven processes, 788–789 metal-free ATP4-/GTP4- utilization, 394–395 molecular motors see Molecular motors noncovalent bonding interactions, 771–773 nucleotide exchange factors, 395 subunit rearrangements, 769 Medium exchange reactions, 588 Melting temperature mutant enzymes, 481 site-directed mutagenesis studies, 468 Membrane transporters, 24 Membrane-bound enzymes, 444, 447 Memory, molecular, 712, 735 pH, 423 Menaquinone synthesis, 75 Menten, M., 15, 288 2-Mercaptoethanol, 231, 383 6-Mercaptopurine, 487 Mercurial irreversible enzyme inhibitors, 539 Mesophilic bacteria, 430 Messenger RNA, 57, 61 transcriptional elongation kinetics, 740–741 translation errors, 454 Metabolic control analysis, 48, 723–726 connectivity theorem, 725 drug design, 572 elasticity coefficients, 725 experimental approaches, 725 flux control coefficients, 724 summation theorem, 724 Metabolites domains, 718 fluorescence spectroscopy quantification, 246–247 Metabolons, 718 Metal carbene catalysts, 8 Metal ions, 12, 56 activators biphasic effects, 391 essential, 84, 385 of otherwise unreactive substrates, 385 active site, 2, 55, 81–112 essential role in catalysis, 81–84 free radical formation induction, 81 properties of specific ions, 95–109 redox properties, 81, 83 structural/chemical features, 83–85

substrate-binding templates, 81, 83, 205 thermodynamic control, 83 ATP complexes, ATP-dependent enzyme requirement, 272–273, 393–394 coenzyme interactions, 77, 78 contaminants, 228 coordination complexes, 85–87 coordination number, 2, 86 coordination number, 3, 86 coordination number, 4, 86 coordination number, 5, 86–87 coordination number, 6, 87 crystal field stabilization energy, 87–88 d-orbital energy of ligands, 88 field effects influencing color/magnetic properties, 87–89 ligand field splitting (molecular orbitalbased approach), 88–89 schemes for specific enzymes systems, 109–112 cosubstrate binding, 78 electron spin resonance, 94 electrostatic transition state stabilization, 27 enzyme catalysis facilitation, 26, 32 enzyme/enzyme substrate complexes, 93–95 atomic absorption spectrometry, 93 binding strength assessment using chelators, 93–94 electron-nuclear double resonance (ENDOR) spectrometry, 94 equilibrium constant effects, 279 essential for life, 82 induced-fit mechanisms, 320 initial apoenzyme complex formation, 84–85 Lewis acids, 56, 81, 82–83 ligand complexation, 85–87 hydration atmosphere, 85, 89–90 low-affinity protein-binding sites, 83 precipitation/aggregation effects, 391 reversible enzyme binding, 84 solutions standardization, 230 structure-stabilizing role, 85 Metal-activated enzymes definition, 84 metal ion binding affinity, 84 see also Metal ions, active site Metal-nucleotide complex substrates, 391–397 analytical techniques, 392 exchange-inert mechanistic probes, 395–397 tervalent metal ions, stability, 395–396 Metallo-catalysts, 6 Metalloenzymes definition, 84 metal ion binding affinity, 84 redox reactions, 159–160 see also Metal ions, active site Metallopepetidases, 84 Metalloproteinases, 84 Metallothioneins, 100 Metastable ion, 257 Metathesis, 8–9 Methenyl-tetrahydrofolate cyclohydrolase, 110 Methionine adenosyltransferase, 151, 296 Methionine aminopeptidase, 97

Index

Methionine g-lyase, mechanism-based inhibition, 556 Methionine sulfoximine, 529–530 Methotrexate, 533, 534 dihydrofolate reductase interaction, 749–750 time-dependent reversible inhibition, 535–536 Methyl transfer reaction stereochemistry, 150–152, 153 Methylcobalamin, 97 Methyl-coenzyme M reductase, 107 Methylenetetrahydrofolate dehydrogenase, 110 Methylglyoxal synthase, 68 Methylitaconate D-isomerase, 122 Methylmalonyl coenzyme A carboxytransferase, 97 Methylmalonyl coenzyme A mutase, 132 N-Methyl-tryptophan oxidase, kinetic isotope effects, 628 R-Mevelonate, prochirality, 150 Michael acceptors, mechanism-based inhibitor design, 550, 552 Michaelis complexes, 338 zymogen activation, 207 Michaelis constant, 293 commitment to catalysis relationship, 307 for definition see Glossary implications for substrate affinity/specificity, 301–303 physiological substrate concentrations, 303 specificity constant relationship, 304–307 substrate capture by enzyme, 306 Michaelis, L., 15, 166, 288 Michaelis-Menten equation, 15–16 for definition see Glossary derivation, 288 single-molecule enzyme kinetics, 736–737 Michaelis-Menten kinetics conservation equation, 288 idealized versus realistic behavior, 291, 292–293 initial-rate kinetics, 287–293 nonproductive substrate binding, 293 rate data plotting methods, 290–292 reaction rate dependence on active enzyme concentration, 292 rectangular hyperbolic saturation curves, 289 simplifying assumptions, 288, 292 substrate concentrations, 288 first-order zone, 288 mixed-order zone, 288 zero-order zone, 288–289 Micro Electro-Mechanical Systems (MEMS), 440 Micro Total Analysis Systems, 440 Micro-pipetting, 229–230 Microarrays, peptide enzyme specificity, 331–332 priming, 331 Microbial contaminants, 228 Microdiffusion technique, 222 Microscopic reversibility, for definition see Glossary Microscopic reversibility principle, 189, 204, 209, 214, 580

881

Microtubules, 62, 776 assembly/disassembly, 737 dark-field microscopy, 738–739 GTP hydrolysis-driven end-tracking motors, 801 kinesin interaction, 738 sliding tubule assay, 737 Mildvan, A.S., 18 MIMIC Software, 184 Mitchell, P., 18, 788 Mitochondria electron transport system, 769, 789 iron-sulfur centers, 101, 103 transmemebrane proton gradients, 789–789 Mix/follow experiments, 638 Mix/quench experiments, 638 Mixed activation, 382 Mixed reaction order, 176 Mixed-type inhibition noncompetitive, tight-binding reversible inhibitors, 532 partial, 494 Mixing time, for definition see Glossary Mixing uniformity, 220, 223 cryoenzymology, 435 initial rate enzyme assays, 220, 223 Modified residues, active site, 58 Modules, protein structure, 63 Molar concentration, 289 determination, 276 Molarity, 6, 289 Mole fraction, for definition see Glossary Molecular chaperones see Chaperonins Molecular crowding, 176, 194, 196, 444–446, 453, 702 kinetic effects, 446 Molecular dynamics, 64 for definition see Glossary macromolecule conformational flexibility modeling, 327 simulations, 38 Molecular ion, 257 Molecular ion mass, 257 Molecular memory, 712, 735 pH, 423 Molecular moonlighting, 74 Molecular motors, 729 affinity-modulated interactions, 768 cellular functions, 767 directed motion, 767, 776 force generation coupling to ATP/GTP hydrolysis, 766–769 generality of filament end-tracking concept, 801 Keller-Bustmante treatment, 776–779 chemical variables, 776, 777 mechanical variables, 776, 777 position change on potential energy surface, 778–779 positional variables, 776, 777 state space analysis, 777–778 stochastic nature, 778 single molecule experiments, 776 thermal environment, 766, 768 transmembrane proton gradients, 789–789

Molecular orbital methods, 4 Molecular similarity drug design, 563 kinetic mechanism elucidation, 563–564 Molecularity, 176–177 Molybdenum, 82, 106 Molybdenum cofactor (MoCo), 78, 97 Monoamine oxidase B, quantum mechanical hydrogen tuneling, 614 Monod, J., 18, 695 Monod-Wyman-Changeux model, 688, 695–699, 707–708 hemoglobin oxygenation, 702, 703 saturation functions, 696–698 Monodehydroascorbate reductase, pulsed radiolysis, 677 Monodentate ligand, 90 complexation with aquated metal ions, 90–91 Moore, S., 18, 669 More O’Ferrall-Jenks diagram, 209 Morphine, 488 2-(N-Morpholino)ethane sulfonate, 413 Mo¨ssbauer spectroscopy, for definition see Glossary Motifs, protein structure, 63 Multi-drug resistance factor (MDR2), 448 Multi-stage (multi-step) mechanisms chemical kinetics, 184–192 computational design algorithm, 44 detailed balance principle, 189–190 thermodynamic cycle analysis, 190–192 first-order kinetics kinetically competent intermediates, 185 reversible, 186 series, 185 transit time, 185 induced-fit mechanisms, 319–320 catalytic reaction cycle, 326 kinetic equivalence principle, 192 mechanistic ambiguity, 192 Michaelis-Menten kinetics, 289 microscopic reversibility principle, 189 pH effects, 400–401 product inhibition, 512 rapid-equilibrium treatment, 186, 187, 188 rapid-mixing experiments, 637, 638 rate-contributing steps, 189 rate-controlling (rate-limiting) step, 188–189, 289 reaction coordinate diagrams, 207–210 reaction progress-curve analysis, 268–269 relaxation kinetics, 661–663 second-order kinetics, 186 steady-state treatment, 186, 187–188 temperature-dependent changes, 189 Multi-substrate enzymes, 374–378 Multi-substrate reactions alternative substrate inhibition Fromm’s method, 508–510 Huang’s constant-ratio approach, 510–511

Index

882

Multi-substrate reactions (Continued ) Cleland’s notation, 336–338 competitive inhibitor use in distinguishing mechanisms (Fromm’s method), 494–498 empirical rate equation, 173 historical aspects, 338 initial-rate kinetics, 335–378 isotope exchange behavior, flux ratio method (Britton), 598–599 order of substrate binding, 335 practical issues, 264–265 working solutions assembly, 264–265 product inhibition patterns Alberty/Fromm approach, 512–513 ‘‘iso’’ mechanisms, 521, 522 see also Bisubstrate kinetic mechanisms Multi-subunit enzymes, 62 allosteric regulation, 62 cooperativity, 688 inter-subunit cross-talk, 2 reversible cold inactivation, 433–434 subunit-subunit interactions, 62 Multi-tasking proteins, 74 Multidentate ligand, 91 transition metal complex interactions, 91–92 Multiple dead-end inhibition, for definition see Glossary Multiple (repetitive) attack, 327 Multiple-focusing mass spectrometers, 255 Multiple-turnover conditions, for definition see Glossary Multiple-turnover processes, 16 Muscarinic acetylcholine receptor, agonist binding, 388 Mutant enzymes, 3 chemical rescue, 479 Mutation, 70, 454, 460, 466, 467 adaptive inhibitors in drug design, 567–568 ‘‘over-perfection’’, 25 Mutation rate, 454 Myasthenia gravis, 540 Myosin actin interaction, 774, 782 forces exerted by single molecules, 749 intrinsic ATPase activity, 762, 765 processivity, 329, 330

N NAD (nicotine adenine dinucleotide), 78, 160 intrinsic fluorescence, 248–249 NAD-dependent histone/protein deacetylase, 49, 122 NAD-dependent poly-ADPR polymerases (PARPs), 49 NAD-linked dehydrogenases enzyme electrodes, 167 kinetic isotope effects, 611 NAD-NADP transhydrogenase, 363 NADH:cytochrome b5 reductase, 165 NADH-dependent hydride transfer, 578 isotope labeling, 577–578 stereochemistry, 576–579 NADH-dependent nitrate reductase, 106 NADP-dependent formate dehydrogenase, 107

NADP-linked isocitrate dehydrogenase, 520 kinetic isotope effects, 628 Nanoscience, 730 Natural abundance, for definition see Glossary Natural selection, 460 active site diversification, 70–71, 73–74 Near-Attack-Conformers (NACs), 29 Near-field optical microscopy, 745–746 Near-field scanning optical microscopy, 244, 746 Negative cooperativity, for definition see Glossary Nernst equation, 156 Nerve toxins, 539–540, 745 Neuronal Wiscott-Aldrich syndrome protein (WASP), 795, 799 Neutron crystallography, 42 New biological catalyst design, 42–44 computational design algorithm, 44 Newman Projection, 9 Newton, for definition see Glossary Nickel, 7, 8, 82, 106–107 enzyme complex coordination schemes, 112 ligand complexation, 86 Nickel iron hydrogenase, 159 Nickel-superoxide dismutase, 84, 107 Nicotinamide mononucleotide adenylyltransferases, 63 inhibition, 524 NIH shift, for definition see Glossary Nitrenes for definition see Glossary photochemical reactions, 548, 549 Nitric oxide, cytochrome c oxidase binding, 162 Nitric oxide synthase, 165, 253 ferrous heme complex, 165 stopped-flow kinetic techniques, 642 tetrahydrobiopterin cofactor, 165 Nitrile hydratase, 97 Nitrocellulose filters, 690 Nitrogen–oxygen acyl shifts, 61 Nitrogen–oxygen bridges, 65 Nitrogenases, 84, 107 p-Nitrophenyl-N-acetyl-b-D-glucosaminide, 243 p-Nitrophenylphosphate, 243 Nobel Prize Awards for enzyme science research, 17 Nodes of metabolic pathways, 2 regulatory enzyme actions, 686 Noise, electronic instrument, 230 Nojirimycin, 488 Nonadiabatic electron-transfer process, 202 Noncompetitive inhibitors, 501–502 partial, 494, 495 product inhibition, 512 tight-binding reversible, 531, 532 Noncovalent catalysis, 23, 24 Noncovalent forces bonding interactions, 773–775 enzyme influences at single-molecule level, 730 Nonessential activators, 381 binding, 382 catalytic, 382

kinetic mechanisms, 387–388 mixed, 382 Nonlinear inhibiton, for definition see Glossary Nonproductive interactions depolymerase kinetics, 332–333 substrate binding, 293 Norrish, G.W., 639, 675 Northrop, D.B., 18 Nuclear magnetic resonance, for definition see Glossary Nuclear magnetic resonance spectroscopy, 16, 20, 41, 172, 260, 400, 680 cyclophyllin A (prolyl-peptidyl isomerase) conformational exchange, 35–36 isoptomeric metabolites, 256 isotope-based studies, 576 ligand binding measurement, 691 metal ion coordination complexes, 94, 95 metal-nucleotide complexes, 392 nuclear Overhauser effects, 680 positional isotope exchange reactions, 607 protein tertiary structure, 62 radioactive isotope-labeled substrate, 255 relaxation data, 680 short strong hydrogen bonds, 68 site-directed mutagenesis, 466, 468, 474 structure-activity relationships, 570 Nuclear Overhauser effects, 680 Nuclease, single molecule fluorescence observations of conformational dynamics, 751 Nucleic acids, polyelectrolyte effects, 422 Nucleophiles, 16, 119 active site functional groups, 57, 206 for definition see Glossary group-transfer reactions, 124–125 hydration, 200 Nucleophilic catalysis, 124–125, 205–206 covalent intermediates, 30, 125 Ping Pong reactions, 206 ribozymes, 22 serine proteases, 20 Nucleophilic reactions, 119, 197 additions (heterolytic), 120 adjacent associative, 580 anchimeric assistance, 330 carboxypeptidase amide/peptide hydrolysis, 135 rearrangements, 120 substitutions, 120, 121, 618–619 first-order (SN1 reactions), 121, 123, 178–179, 616 internal mechanism, 124 second-order (SN2 reactions; inline mechanism), 124, 618 secondary kinetic isotope effects, 616–619 Nucleoside 59-diphosphate kinase, 141, 275, 340, 366, 373 double-displacement reactions, 141–142 isotope trapping experiments, 606 phosphoryl transfer reaction stereochemistry, 580 Ping Pong kinetic mechanism, 141, 352, 353–354, 356

Index

Nucleoside diphosphate phosphatase, phosphoryl transfer reaction stereochemistry, 580 Nucleoside hydrolases carbenium ion mechanism, 121 kinetic isotope effects, 631 Nucleoside monophosphate kinases, 366 Nucleoside phosphorylases, kinetic isotope effects, 631 Nucleoside triphosphate regeneration reactions, 275, 276 39-Nucleotidase, 109 59-Nucleotidase, partial competitive inhibition, 493 Nucleotide-dependent reactions, stereochemistry, 579–585 Nucleotide-dependent transphosphorylases, 103 Nucleotide-exchange factors, 395, 782 Nucleotidylation, 61 Nudix hydrolase, 104

O Occam’s razor, 174 Olefin metathesis, 8–9 Oligomerization enzyme activity regulation, 709–712 ionic strength effects, 420 pressures effects, 438, 439 substrate effects, 711–712 Oligovinyl sulfonates, 413 Opium, 488, 559 Optical silica cuvettes, 241 Optical tweezers, 740–744, 765 single-molecule enzyme observations, 740–741 Orbital steering, 18 efficiency of enzyme catalysis, 26, 28–29 Order-of-addition (order-of-mixing) effects, 228–229 Ordered sequential mechanism Bi Bi, product inhibition patterns, 513–515 abortive complex effects, 517–518 Bi Ter, 366 competitive inhibitor use for defining mechanism, 496 Haldane relationships, 359, 360 quantitative analysis, 359, 360, 361, 362, 364 rapid equilibrium treatment, 350–351 steady-state rate equations, 341–343 Ter Ter, 366–367 Theorell-Chance reactions, 347–348, 366 Ordered ternary complex mechanism Bi Bi mechanism alternative substrate inhibition, 508–509 equilibrium isotope exchanges, 592–593 flux ratio method (Britton), 598–599 Organic chemical reactions, 119 additions, 120 elimination reactions, 125 general reaction mechanisms, 120–121 oxidation–reduction, 121 rearrangements, 120–121 substitution reactions, 120

883

Organic solvent effects, 422–425 pH memory, 423 polarity influences, 423–424 site-directed mutagenesis studies, 469 Organomercurial lyase, 130 Orgel, L., 454 Oriented diffusion, 449 Orotidine monophosphate decarboxylase, 130 Orotidine-59-phosphate decarboxylase, 129, 308 Osmotic effects, 196 Osmotic pump, 177 Ostwald, W., 6, 13 Outer-sphere electron transfer reaction, for definition see Glossary Ovalbumin, 228 Over-inhibition, 573 ‘‘Over-perfection’’ mutations, 25 Overall catalytic mechanism, 19 Overall reaction order, 176 Overpotential, 157 Oversaturation, for definition see Glossary Oxacarbenium ions, 119 ion-pair stabilization, 333 Oxalate decarboxylase, 130 Oxaloacetate decarboxylase, 769 Oxidation–reduction reactions, 121, 152, 155 active site metal ion activities, 81, 83 complex metalloenzymes, 159–160 electrochemical analytic techniques, 159–160 half-reaction components, 16 redox centers in enzymes/redox proteins, 154 redox potentials, 156–159 measurement, 156, 157–158 variation, 158 Oxidative phosphorylation, 762, 765, 788 chemiosmotic principle, 788–789 Oxidoreductases, 23, 84 2-Oxoglutarate dehydrogenase, 78 Oxyhemoglobin, 87 Oxyphosphorane intermediate, ribozyme catalytic mechanism, 23

P Pacemaker enzymes, 686 for definition see Glossary Paclitaxel, 40 Palladium, 7, 8 Palm, D., 18 Papain, 15, 66 substrate recognition subsites, 114 Pardee, A., 18 Parent ion, 257 Parsimony principle, 174 Partial competitive inhibition, for definition see Glossary Partial inhibition, for definition see Glossary Partial pressure, for definition see Glossary Partial-exchange reactions erroneous exchange experiments, 354–356 experimental protocol, 356 Ping Pong Bi Bi mechanism, 353–354

Pascal, for definition see Glossary Patch-clamp methods, 733, 734 for definition see Glossary Patterns, protein structure, 63 Pauling, L., 24, 26, 27, 50, 525, 764 site interaction model, 700–701 Pauling-Koshland-Ne´methy-Filmer cooperativity model see KoshlandNe´methy-Filmer model Pdx1, 572 Peak intensity, mass spectrum, 257 Peak, mass spectrum, 257 Pe´clet number, 444 Pee Dee Belemnite isotope standard, 255 Penetratin sequences, 51 Penicillin, 486, 488, 559 Penicillin acylase, 447 solvent kinetic isotope effects, 626 Pepsin, 7, 13, 207 substrate recognition subsites, 114 Pepsinogen, 7 autocatalysis, 207 Pepstatin A, 225, 226, 227 Peptidases, 335 Peptide bonds, 55, 56 dipole moments, 66 planarity, 56 Peptide hydrolysis acyl-enzyme intermediate, 135 carboxypeptidase-type general base mechanism, 136 carboxypeptidase-type nucleophilic mechanism, 135 Peptidyl-(acyloxy)methyl ketones, 546 Percent-purity, 234 Percentage inhibition, 537–538 Percentage recovery, 234 PERIOD gene, 686 Peroxidase, 16, 101, 164–165, 735 activity in organic solvents, 424 enzyme electrodes, 167 time-resolved spectral analysis, 649 Perrin, J., 250, 251 Perutz, M.F., 702 pH effects, 30, 397–412 active site, 399–400 activity changes, 398–401 reversibility, 408 bisubstrate enzyme kinetics, 408–409 Brønsted theory of acid-base catalysis, 409–412 buffers, 177 selection, 413 enzyme stability during elevated temperature exposure, 425 equilibrium constant, 279 inhibition, 407, 544–545 initial rate enzyme assays, 221 kinetic isotope effects, 628 long-range electrostatic effects, 401 mechanism changes, 407–408 metal-ligand hydrolysis, 90 multiple active enzyme protonation states, 406–407

Index

884

pH effects (Continued ) pKa determination methods, 401–404 Alberty-Massey method, 401–402 experimental techniques, 405 nonlinear least-squares fitting, 402 protein structural calculations, 405–406 solution phase/intrisic pKa value comparisons, 404 substrate ionizable groups, 407 pH memory, 423 pH optimum, 397–398 pH Stat, 177, 221 Phage-display profiles, enzyme specificity, 331–332 Pharmacodynamics, 561 Pharmacokinetics, 561 Pharmacophore, 561 for definition see Glossary 1,10-Phenanthroline, 225, 226 Phenylalanine ammonia lyase, 206 Phenylalanine hydroxylase, 103 Phenylalanine-tRNA synthase, 335 Phosphate binding protein, 253 Phosphate ester hydrolysis covalent catalysis associative mechanism, 137 dissociative mechanism, 137 cysteine-phosphate intermediate mechanism, 138 general acid-base catalysis associative mechanism, 136 dissociative mechanism, 136–137 Phosphate-producing enzymes, continuous fluorescence assays, 253–254 Phosphatidylcholine flippase, 448 Phosphatidylserine decarboxylase, 451 Phosphinothricin, 529–530 Phosphite-dependent hydrogenase, 76 Phosphodiester hydrolases, 103 Phosphoenolpyruvate carboxykinase, 325 Phosphoenolpyruvate carboxylase, kinetic isotope effects, 628 Phosphoenolpyruvate, energase reactions, 765, 768, 769 Phosphofructokinase, 266, 430, 724 hysteretic behavior, 712 negative binding cooperativity, 703–704 reversible cold inactivation, 434 Phosphoglucomutase, 105 Phosphoglycerate dehydrogenase, 709 2-Phospho-D-glycerate hydrolase, 87 Phosphoglycerate kinase, 355 Phosphohydrolases, 575 Phospholipase A2, 449–450, 451 Phospholipase C, 451 Phospholipase D, 453 Phospholipases interfacial catalysis, 449, 450–451 processive turnover, 450 scooting/hopping mode, 450 Langmuir trough experiments, 451–452 surface-dilution approach, 451 Phospholipid flip-flop, 448 for definition see Glossary

Phospholipids, 447 biological membranes, 447 dynamics, 447–448 flip-flop, 448 Langmuir trough experiments, 451–452 micelles formation, 447–448 Phosphomonoester hydrolases, 104 Phosphopantothenoylcysteine decarboxylase, 130 Phosphoprotein phosphatase, 717 Phosphoramidon, 226 Phosphorescence, 240 for definition see Glossary Phosphoryl transfer reactions, 762, 765 coupling to force-generating conformational change, 767–768 secondary kinetic isotope effects, 618 stereochemistry, 579 definition with isotopically labeled nucleotides, 580–585 nucleoside 59-triphosphate-hydrolizing reactions, 585 pseudorotation, 580–581 Phosphorylase a, oligomerization-dependent activity changes, 709–710 Phosphorylase kinase, 233, 710, 717 Phosphorylases, 123, 253 Phosphorylation, 61 peptide microarray priming, 331 Phosphotransferases, 340, 575 MgATP2-/CaATP2- requirement, 394, 395 Phosphotriesterase, 40, 86, 412 Photoaffinity labeling active sites, 547–550 limitations, 549–550 types of agent, 548 Photo-caged substrates, 638, 675–676 Photochemical reactions, 177, 548–549 efficiency (quantum yield), 549 Photolability, fluorescent labels, 250 Photolysis, for definition see Glossary Photophosphorylation, 761, 762, 765 ionic strength effects, 420–421 Photosynthesis, 761 electron transfer reactions, 164 Physiologic temperature, 429–430 bacteria, 430 homeotherms, 429–430 poikilotherms, 430 Physostigmine, 531 p-cation interactions, 70 Pi-to-Pi* transition, for definition see Glossary Pi-to-Sigma* transition, for definition see Glossary Ping Pong kinetic mechanisms, 340 Bi Bi mechanism, 289, 352–358 burst kinetics, 356 competitive inhibitor experiments, 496–497 equilibrium exchange rate equation derivation, 589–590 partial-exchange reactions, 353–354 product inhibition patterns, 516 steady-state rate equation, 352–353 substrate inhibition, 357–358

bisubstrate kinetic mechanisms, 337–338 Bi Bi scheme, 337 ordered Uni Uni Bi Bi scheme, 337 branched pathways, 357, 372–373 isomerization reactions Di-Iso Bi Bi, 371 Di-Iso Bi Uni Uni Bi, 372 Mono-Iso Bi Bi, 371–372, 521 Mono-Iso Uni Uni Bi Bi, 372 isotope trapping experiments, 606 limitations of intial-rate data, 364 multi-site, 31, 54, 358 transfer of reactant moieties, 358 nucleophilic catalysis, 206 nucleoside 59-diphosphate kinase, 141 single-molecule enzyme experiments, 732–733 stopped-flow temperature-jump investigations, 671 three-substrate schemes Hexa Uni, 367 ordered sequential Bi Uni Uni Bi, 367 ordered sequential Uni Uni Bi Bi, 367–368 random Bi Uni Uni Bi substrate addition, 368 random Uni Uni Bi Bi substrate addition, 368 transient covalent intermediates, 141 Pipettes/pipetting, precision of measurements, 229–230 Planck’s constant, for definition see Glossary Plant secondary metabolites, 488, 559 Plapp, B.V., 18, 647 Plastic labware washing, 228 Plastic surfaces passivation, 228 thermal equilibration, 220 wall effects, 228 Platinum, 5, 7, 8 on charcoal, 8 ligand complexation, 86 Poikilotherm body temperature, 430 Point mutations, 3, 64 location specification, 64 Poisson-Boltzmann equation, 65, 66, 194, 405, 422 Polanyi, M., 6 Polarimetry, 12, 221 reaction progress-curves quantitative analysis, 268–270 Polyelectrolyte effect, 422 Polyketide synthases, 71–72 Polymerases, 112–113, 327, 328 active sites, 112–113 distributive mechanisms, 327–328 kinetic proofreading, 454–455 processive mechanisms, 327–329 sequence recognition subsites, 112–113, 114–115 Polymeric substrates, 327–333 nonproductive interactions in steady-state treatments, 332–333 shifted binding, 332 single-molecule enzyme experiments, 729

Index

Polymerization reactions, pressures effects, 438 Polymorphisms, 461 Polypyrroles, 158 Polysaccharidases, substrate recognition subsite designation scheme, 114–115 Porphyrins, 78, 81, 83 Porter, 639, 675 Positional isotope exchange, 606–607 Positive cooperativity for definition see Glossary see also Ligand binding Post-translational modification, 19, 20, 61, 233 active site residues, 55 enzyme activation, 383 Potassium, 82 enzyme activation, 95–96 Potential energy surface for definition see Glossary see also Reaction coordinate diagrams Precision of measurements, 229–230 Pre-equilibrium, for definition see Glossary Preferential interactions, 196 Preferential solvent effect, 223 Pressure effects, 438–439 Pressure-jump methods, 639, 658, 672–673 apparatus, 672 for definition see Glossary rate process investigations, 673 Pre-steady-state phase, for definition see Glossary Primary salt effect, 418–419 Primer-stabilizing active site residues, 58 Prime(TM), 269, 327 Priming, 331 Primitive change, for definition see Glossary Principal component analysis, 38 conformational flexibility modeling, 327 Principal ion, 257 Prion protein, 753 Processive mechanisms, 330 exopolymerases, 327–329 Prochirality, 149, 576 Cahn-Ingold-Prelog specification, 149–150 Prodrugs, 492, 566–567 Product, Cleland notation system, 336 Product inhibition, 512–523 metabolic significance, 521–523 multi-substrate kinetic mechanisms differentiation (Alberty/Fromm approach), 512–513 ‘‘iso’’ mechanisms, 521, 522 progress curve analysis with Foster-Neimann plots, 520–521 two-substrate kinetic mechanisms, 513–516 patterns with abortive complex formation, 516–520 Product ion, 257 Product release, 306 active site binding energy, 61 Productivity, cross-linked enzyme crystal catalysts, 460

885

Profilin, 395, 795 actin adenine nucleotide exchange reaction, 24, 25 actin complexes, 798, 799 Progress curve analysis, 182–183, 310–311 numerical integration, 183 product inhibition, Foster-Neimann plots, 520–521 quantitative, 268–270 software, 268–269 steady-state rate measurements, 268 transient-state rate measurements, 268 series first-order processes, 185 induction period, 185 tight-binding reversible inhibition, computer-assisted method, 535 Prolidase, 97 Proline racemase, 129 equilibrium isotope effects, 612 isotope rate measurements, 600, 601–603 transition-state energetics, 601–602 Prolyl hydroxylase, 57 Proof by exclusion, 174 Proofreading, kinetic, 453–456 Propionyl-coenzyme A carboxylase, 129 Prospects for enzyme science, 34 cell survival, 49 chromosomal remodeling, 49 database developments, 46–47 direct enzyme therapy, 50–51 energy flow during enzyme catalysys, 37 enzyme inhibitor production, 44–45 epigenetics, 49 genetic basis of disease, 49 in singulo enzyme catalysis, 45–46 intracellular enzyme kinetics, 48–49 methods of enzyme dynamics analysis, 34–37 new biological catalyst design, 42–44 probes of enzyme catalysis, 38 protein folding, 38–39 substrate specificity, 39–42 Prostaglandin endoperoxide synthase, 269 Prosthetic groups, 12, 78, 335 Proteases, 335 bacterial, 225 fluorescence correlation spectroscopy, 755–756 fluorescence resonance energy transfer (FRET) assays, 253 inhibitors, 225, 226, 227, 233, 488–489 see also HIV protease inhibitors intramembrane, 453 microarray/phage-display profiles, 331–332 purified enzyme degradation, 224–225 partial proteolysis assessment, 227 retroviral, 60 substrate recognition subsites, 114–115 zine-containing, 109 Proteasomes, 61, 313–314 19S cap complex, 313–314 20S core particle, 313 a subunits, 314 b-subunit protease activity, 313 Protective colloid effect, 223, 425

Protein concentration see Concentration determinations, protein/enzyme Protein DataBase (PDB), 41, 46, 66 Protein dynamics, 34, 209 ‘‘hot spots’’ during catalysis, 36 time-scale, 36 Protein film voltammetry, 160 Protein folding, 38–39, 61 chaperonin-mediated, role of ATP hydrolysis, 792–793 entropy-driven processes, 69 enzyme electrostatic interactions, 27 metal ion involvement, 84 Protein geranylgeranyl-transferase, 305 Protein kinase A, 233, 687 Protein kinases, 41, 575 consensus phosphorylation sites, 116 substrate recognition subsite phosphorylation, 115–116 Protein molecular volume, atomic force microscopy, 745 Protein phosphatases, substrate recognition subsite dephosphorylation, 115 Protein structure amino acid sequence, 55–56, 61 selenocysteine incorporation, 57 calculations, pKa value determination, 405–406 background energy, 406 born energy, 405 pair-wise interaction energy, 406 consensus sites, 63 domains, 62, 63 motifs, 63 hierarchical schemes, 61–62, 63 higher-order, 62–63 modules, 63 patterns/repeats, 63 post-translational modification, 57 primary, 61 quaternary, 62 secondary, 61–62 subunits, 62 tertiary, 62, 63 Protein synthesis, 57 noncognate amino acid incorporation, 454 proofreading, 455, 719 Protein-tyrosine phosphatases, stopped-flow kinetic techniques, 642 Proteome, 41–42 Protoheme ferrolyase (ferrochelatase), 83, 85, 92 Proton inventory experiments, solvent kinetic isotope effects, 626–627 Proton transfer reactions aqueous solutions, 195–196 hydrogen bond networks, 195–196 facilitated, 196 rate constants, 665 relaxation techniques, 663–665 Proton wires, 67, 481 carbonic anhydrase activity, 415 substrate channeling, 719 Pseudo first-order kinetics, 180–181

Index

886

Pseudo-essential activators (non-consumed substrate), 386–387 Pseudogenes, diversification of enzymes, 77 Pseudorotation, 148 phosphoryl transfer reaction stereochemistry, 580–581 Psychrophilic bacteria, 430 Psychrophilic enzymes activity-stability relationship, 431–433 uses, 431 Pterin-based redox cofactors, 78 Pulse-chase experiments protein turnover measurement, 632 see also Isotope trapping Pulsed radiolysis, 658, 677–678 Purich, D.L. actoclampin end-tracking molecular motors, 63, 199, 397, 793–802 CoA-linked acetaldehyde dehydrogenase, substrates ratio method, 369–370 Dixon plots, limitations in analyzing reversible inhibition mode, 499 energases, 25, 38, 47, 53, 729, 761–766 energy charge model, evidence against, 523 enzyme, new unifying definition, 24 glutamine synthetase g-glutamyl phosphate intermediate, 604, 606 isotope exchange at equilibrium, 587–588 Lock, Load & Fire mechanism, 795, 799 tubulin:tyrosine ligase, kinetic mechanism, 369 Purification, enzyme, 232–235 catalytic activity parameters, 234, 235 rationale, 232–234 techniques, 234–235 trouble-shooting, 234, 236–237 Purine nucleoside hydolase drug design, 564 kinetic mechanism, 563–564 Purine nucleoside phosphorylase, 123, 253, 363, 564 antimalarial drug design, 565 carbenium ion mechanism, 121 chromogenic substrates, 243 Push–pull proton transfer chymotrypsin catalytic mechanism, 20 efficiency of enzyme catalysis, 26 kinetic isotope effects, 622 PYP (Ectothiorhodospira halophila photoactive yellow protein), 38 Pyranopterins, 86 Pyrene, 249 Pyridine nucleotide transhydrogenase, 363 Pyridoxal kinase, substrate inhibition, 506 Pyridoxal-5-phosphate, 78, 335 Pyridoxal-5-phosphate-dependent enzymes, 57 Pyridyl disulfide, 545 Pyrophosphatase, 299 Pyrophosphate continuous fluorescence assays, 254 energase reactions, 768 Pyrrolo-quinoline quinone, 81 Pyruvate carboxylase, 95, 335

Pyruvate dehydrogenase, 78, 358 enzyme-tethered co-factors, 31 product inhibition, 522 Pyruvate formate-lyase, 132 Pyruvate kinase, 94, 275 kinetic isotope effects, 628 ligand-binding cooperativity, 708 metal ion coordination scheme, 109–110 Pyruvoyl groups, active site, 81

Q Q10 temperature quotient, 429 Quadrupole mass spectrometers, 255 Quantitative structure-activity relationship (QSAR), for definition see Glossary Quantum mechanical hydrogen tuneling, 34, 54, 92–83 electron transfer reactions, 155 enzymes exhibiting, 614 kinetic isotope effects, 613–616 temperature dependence, 614, 615 Quantum mechanics, 4, 20 tunnel correction, 16 Quantum yield, for definition see Glossary Quartz cuvettes, 241 Quasi-elastic light scattering, 674 Quasi-steady-state approximation, 188, 297 Quasi-thermodynamic transition state model, 202 Quaternary complexes, 338 Quenching, 244, 249 collisional, 244 liquid scintillation counting, 262 Quiescent inactivators, 546 Quinoproteins, active site, 81

R Racemases, for definition see Glossary Racemic mixtures, enantioselective enzyme resolution, 316 Racemization, 61 L-amino acids to D-amino acids, 55 for definition see Glossary Racker, E., 18, 117, 275, 486 Radiationless transitions, 244 Radical ion, for definition see Glossary Radical (or free radical), for definition see Glossary Radioactive decay, 178, 260–261 radioactivity quantification, 261–263 rates, 260–261 scinitllation counting data analysis, 262–263 Radioimmunoassay, 691 Radioisotope assays chromatographic separation, 262 equilibrium constant determination, 280 internal reactant species, 309 liquid scintillation counting, 261–262 measurement error, 262 reaction progress-curve quantitative analysis, 268 reaction rate processes, 221, 255, 260–264 specific activity, 263–264 amount of substance analysis, 263–264

Radioisotopes half-life, 260, 261 radioactivity quantification, 261–263 Raftery, M., 18 Random error, 266 Random kinetic mechanisms, 359, 362 alternative substrate inhibition, 509–510 bisubstrate Bi Bi, 336 Bi Uni, 336 rapid equilibrium assumptions, 349, 350 competitive inhibitor use for definition, 494–496 Haldane relationships, 359 product inhibition patterns, 513 abortive complex effects, 517 steady-state rate equations, 347–349 simplifying approach (Cha method), 349 Ter Ter, 367 ternary complex equilibrium isotope exchanges, 594 flux ratio method (Britton), 599 Random number table, 230 Random scission kinetics, endodepolymerases, 329–330 Random walk, for definition see Glossary Randomizing measurement order, 230–231 Raney nickel, 8 Rapid equilibrium assumptions, 186, 187, 188, 349 nonessential enzyme activators, 387–388 ordered sequential kinetic mechanism, 350–351 random kinetic mechanisms, 349, 350 Rapid flow methods, 638, 639, 640–653 Rapid mix/quench mass spectrometry, 650–652 Rapid mixing continuous-flow methods device design, 640, 643 dead time, 640 turbulence, 643 reaction intermediates detection, 640, 643 Rapid mixing devices, 638, 639, 640 temperature-jump apparatus combination, 668 Rapid mixing/quenching techniques, 172, 650–653 apparatus, 639, 649 burst-phase kinetics, 655–656 experimental findings, 650 Rapid reaction techniques, 16, 327, 637–682 data analysis, 678–679 flow methods, 639, 640–653 internal equilibrium constant determination, 310 limitations, 681 range, 638–640 rate data, 188 relaxation methodsRelaxation techniques time-scale, 638 Rapid scan devices for spectroscopic analysis, stopped-flow experiments, 647–648 Rapid-start complex, for definition see Glossary

Index

Rate constant, 174 bimolecular, electrostatic effects, 199–200 for definition see Glossary diffusion-limited, 197–198 temperature effects, 427–428 Rate data analysis, 265–272 global, 270–272 residuals, 267 weighting, 266–268 enzyme concentration detemination, 292 plotting methods, 290–292 reaction progress-curves, 268–270 reporting, 284 see also Reaction rate Rate equations, 173 competitive inhibitor kinetics, 492–493 empirical, 172–174 Fromm’s systematic method for deriving, 341–343 two-step computer-assisted method, 343–345 Rate saturation kinetics, 1, 177 Rate-contributing steps, 189, 724 for definition see Glossary Rate-controlling (rate-determining/ratelimiting) steps, 188–189, 198, 724 for definition see Glossary Michaelis-Menten mechanism, 289 multi-step enzymes, 289 Raw residuals, 267 RB-101, 566–567 Reactant orientation, efficiency of enzyme catalysis, 26, 28–29 Reaction center, isotopic substitution, 608 kinetic isotope effects, 608 remotely labeled reactant effects, 608–609 Reaction coordinate, 208 Reaction coordinate diagrams, 207–210 additive principle, 208 Reaction flux see Flux, reaction Reaction kinetics simulator, 182–183 Reaction mechanism concept, 19–25 Reaction order, 175–176 Reaction pathway, 197 Reaction rate, 4, 6, 171, 287 analytic strategies, 181–184 Euler method, 183 Gear’s predictor-corrector method, 183–184 initial rate methodInitial rate enzyme experiments progress curves see Progress curve analysis Runge-Kutta method, 183 definition, 174–175 empirical rate equation, 172–174 Michaelis-Menton processes, 292 multi-stage mechanismsMulti-stage (multistep) mechanisms simulations see Simulations of kinetic mechanisms

887

study methods, 171, 172 radioactive isotopes, 255 single-molecule enzyme experiments, 730–733 stable isotopes, 255–266 units, 174 Reaction trajectory see Trajectory of reaction Reaction velocity, 174–175, 289 Rearrangement reactions, 120–121 RecA single-molecule observations with optical tweezers, 742 stopped-flow kinetic techniques, 642 Recombinant DNA technology, 15 Recombinant enzymes, proteolytic susceptibility, 225 Rectangular hyperbolic rate curves, 289 Red fluorescent protein, fusion protein expression, 755–756 Redox activators, 383 Redox centers, 154, 155–156 complex metalloenzymes, 159 Redox enzymes, 196 electron transfer mechanisms, 164–166 enzyme electrodes, 155 see also Oxidation–reduction reactions Redox potentiometry, 157–158 Reduced substrate concentration (ALPHA), 290 Reference reaction, 1 for definition see Glossary Reference standards, 229 Refrigerator storage, 223, 224, 228, 427 Regression analysis, 266 Regulation, enzyme, 19, 637, 685–727 amplification cascades, 713–718 hysteresis, 712–713 oligomerization-dependent activity changes, 709–712 overview, 685–688 pacemaker enzymes, 686 substrate channeling, 718–723 see also Cooperativity Relative configuration, 147 Relative permitivity, 199 Relaxation, chemical, 639, 680 definition, 656–658 Relaxation kinetics, 172, 638, 639, 658–665, 681 average relaxation times, 663 data analysis, 678–679 elementary reactions, 660 multi-step reactions, 661–663 one-step reactions, 660–661 proton-transfer processes, 663–665 reactions unimolecular in forward direction and bimolecular in reverse direction, 659 single/periodic perturbation effects, 665–666 unimolecular isomerizations, 658–659 Relaxation time, for definition see Glossary Rennet, 13 Reorganization energy, 27 Repeats, protein structure, 63 Repetitive (multiple) attack, 327 Residual sum-of-squares values, 266

Residuals, 267 scatter plots, 267–268 Resistance correlation analysis, 674 Resolution, mass spectrum, 257 Resolving power (R), mass spectrometry, 257 Resonance energy transfer, 244 Resonance Raman spectroscopy, for definition see Glossary Respiratory chains, 160–162 Restriction endonucleases, 113–114, 290, 446 DNA target sequence binding, 323 induced-fit interactions, 322–324 ionic strength effects, 419 sequence recognition subsites, 114 two-color cross-correlation analysis, 753, 757 Resveratrol, 488 Retinal photoreception, 713 Retro-aldolase, new biological catalyst design, 42–44 Reverse pipetting, 229 Reverse transcriptase-mediated polymerase chain reaction, 233 Rhodamine, 249, 250, 251, 747 Rhodanese, 66 Rhodium, 8 Rhodopsin, 713 Ribitol dehydrogenase, 338 product inhibition, 516–517, 519 Ribonuclease, 228, 331 folding, 792 inhibitors, 531 oligovinyl sulfonate, 413 site-directed mutagenesis, 470 stopped-flow temperature-jump investigations, 669–670 Ribonuclease A, 113–114, 397 stereochemistry, 581 Ribonuclease H, 391 Ribonuclease T1, 403 site-directed mutagenesis, 464 Ribonucleic acid see RNA Ribonucleoside diphosphate reductase, metal ion coordination scheme, 112 Ribonucleotide reductase, 103, 132 free radical intermediates, 132, 133 freeze-quench experiments, 654–655 Ribosomes, 57, 61 Ribozymes catalytic mechanism, 22–23 hairpin, 311–312, 313 hydrostatic pressure effects, 438–439 hammerhead, 22, 311, 312, 313 kinetic isotope effects, 630 solvent, 626–627 kinetics, 311–313 metal ion activation, 95–96 orbital steering, 29 oxyphosphorane intermediate, 23 phosphoryl transfer, 22 stopped-flow kinetic techniques, 642 transition-state stabilization, 313 Ribulose,5-bisphosphate carboxylase/ oxygenase (Rubisco), 1, 40, 130, 479 negative binding cooperativity, 704 reversible cold inactivation, 434

Index

888

L-Ribulose-5-phosphate 4-epimerase, kinetic isotope effects, 612 Ricin, 177 kinetic isotope studies, 631 Rieske iron-sulfur proteins, 101 for definition see Glossary Ring-closing metathesis, 8 Ring-opening metathesis, 8 RNA catalytic, 23 see also Ribozymes polyelectrolyte effectsMessenger RNA; Transfer RNA, 422 RNA interference, drug design, 571 RNA polymerase, 61 hydrostatic pressure effects, 438 single-molecule observations with optical tweezers, 740–741 RNA-induced silencing complexes (RISCs), 571 RO-28-1675, 381 Robustness of metabolic pathways, 723 Rose, I.A., 18, 627 Rosetta Match algorithm, 42 Rotational correlation time, for definition see Glossary Rotational diffusion, for definition see Glossary Rotational echo double resonance (REDOR), 681 Rotatory catalysis, 788 Boyer’s binding change mechanism, 790–792 transmemebrane proton gradients, 788–789 Rubredoxins, 86, 101 ‘‘Rule-of-five index’’ (Lupinski), drug design, 568 Runge-Kutta method, reaction rate analysis, 183, 678

S S0.5, for definition see Glossary S0.9/S0.1 (or, Rs), for definition see Glossary Sabatier, P., 6 Sabatier Principle, 6 Saddle-points, 197, 202 for definition see Glossary Salen complex, 9 Salt bridges, 65 Salt effects see Ionic strength Salting-in, 224 Salvarsan, 486–487 Santi, D.V., 31 Sarcoplasmic reticulum, 780 calcium ATPase, 780 calcium ion binding, 781 catalytic reaction cycle, 780–781 chemical specificity, 780, 781 vectorial phase of activity, 780, 781 Sarin, 539 Saturation kinetics see Zero-order kinetics SB-234764, 524–525 Scale-limited precision of measurement, 229 Scatchard equation, 693–694 steric hindrance modification, 682 strong and weak ligand binding, 694 Scatter plots, 267 Scha¨rdinger dextrins see Cyclodextrins

Schellman, J., 61 Schramm, V., 19, 525 transition state-based drug design strategy, 563–566 Schrock, R., 9 Schwert, G., 339 Scintillation counting see Liquid scintillation counting Scintillation counting fluid, 261, 262 Scintillator (fluor), 261, 262 Scissile bond, 608 for definition see Glossary Scurvy, 57 Second, for definition see Glossary Second-order kinetics, 175, 180 reversible multi-stage mechanisms, 186 Secondary isotope effect, for definition see Glossary Secondary plot (or replot), for definition see Glossary Selected-fit mechanism, 324–326 Selectivity, 685 Selenocysteine, 57, 79–80 tRNA formation, 57 Selenocysteine synthase, 57 Selenophosphate synthetase, 145 Self-assembly processes, 196 diffusion limitation, 199 Self-association, 709–711 Self-defrosting refrigerators, 224 Semialdehyde hydrolase, 71 Septin, 62 Sequential mechanisms bisubstrate kinetic mechanisms ordered binary complex, 337 ordered ternary complex, 336–337 random, 336 isotope trapping experiments, 606 limitations of intial-rate data, 364 Sequestered substrates, 446–447 L-Serine dehydratase, metal ion coordination scheme, 112 Serine proteases, 71, 356, 364 active site low-barrier hydrogen bonds, 67–68 acyl-enzyme intermediate amide/peptide hydrolysis, 135 formation at sub-zero temperatures, 436 catalytic mechanism, 20 double-displacement reaction, 337–338 inhibitors, 488 solvent kinetic isotope effects, 626 substrate specificity, 302 substrate-assisted catalysis, 331 Shifted binding for definition see Glossary polymeric substrates, 332 Shock tubes, 639, 658 pressure-jump techniques, 673 Shock waves, 639 Shore, J., 647 Short-chain fatty acyl-coenzyme A synthetase see Acetyl-coenzyme A synthetase Shot noise, 230 fluorescence spectroscopy, 245

Side-reactions, 142–145 Sildenafil, 561–562 Lupinski’s ‘‘rule-of-five index’’, 568–569 Simple enzyme, 12 Simulations of kinetic mechanisms, 182–183 continuous simulation, 184 Gear’s algorithm, 184 stochastic simulation, 184 Simultaneous action of two enzymes on same substrate, 318–319 Single-molecule enzyme experiments, 16, 45–46, 729–759 artifacts of enzyme-surface interactions, 736–737 fluorescent microscopy, 746–751 forces influencing enzyme behavior, 730 future prospects, 758 kinetic treatment, 733–737 dynamic disorder considerations, 735–736 Markov processes, 734 stochastic approaches, 733–734 molecular motors, 776 optical tweezers, 740–741 rationale, 729 reaction rates, 730–733 video microscopy, 737–739 zero-mode waveguides, 757–758 Single-point initial rate assays, associated errors, 218 Single-turnover conditions, for definition see Glossary Single-turnover processes, 16, 637 Singlet states, 244 Singular value decomposition, 38 Siroheme, for definition see Glossary Site interaction model, 700–701 Site-directed mutagenesis, 233, 460–483, 729 AAA+ mechanoenzymes, 785 alanine scanning technique, 462–464 catalytic antibody production, 11 chemical rescue of mutant enzymes, 479–480 directed evolution of enzymes, 39, 40 early experiments on subtilisin, 461, 462 enzyme stability, 480 expression systems, 481 goals, 460 investigation strategies, 464–472 limitations, 480–483 protein stability effects, 460, 469 temperature-dependent changes evaluation, 481 Skou, J., 761, 762 Sliding tubule assay, 737 Slip bond, 774 Slip plane, 422 Slow, tight-binding inhibitors, 534 time-dependent behavior, 533–535 Slow-binding inhibitors, 534 binary inhibitors, 501 time-dependent behavior, 533–535 Small GTPases, 782, 783 Small interfering RNAs, 571 Smoluchowski diffusion equation, 197–198, 199

Index

Sodium, 82 enzyme activation, 95–96 ion transporters, 769 Sodium potassium ATPase, 781–782 Sodium-glucose symport system, 24–25 Soft acid, for definition see Glossary Soft base, for definition see Glossary Soft ionization methods, 258 Solubility product, 177 Solubility-enhancing agents, 446 order-of-addition (order-of-mixing) effects, 229 Solute ionization, 196 Solute-selective electrodes, 222 Solutions behavior of reacting molecules, 194–201 preparation, precision of measurements, 230 standardization, 230 Solvation, for definition see Glossary Solvent cage, 195, 197, 200 Solvent kinetic isotope effects, 623–627 fractionation factors, 624 inverse, 623 normal, 623 practical aspects of determination, 625–626 primary, 624 secondary, 624 simplified theory, 624–625 Solvent-accessible surface area, for definition see Glossary Solvolysis, for definition see Glossary Sortase transpeptidase (SrtA), solvent kinetic isotope effects, 626 SPECFIT software, 680 Specific acid catalysis, 204, 409, 410 Specific activation, 382 Specific activity, 234 definition, 276 see Glossary enzyme concentration determination, 276 radioisotope assays, 263–264 turnover number (kcat) relationship, 304 Specific base catalysis, 409, 410 Specificity, 685 for definition see Glossary Specificity constant, 304–307 Spectrally invisible reactions, 575 Spectrophotofluorimeter see Fluorescence spectrometer Spectrophotometers, use of reference standards, 229 Spin label, for definition see Glossary Splicing, 61 Stability constants for definition see Glossary metal ions complexes of ATP/ADP, 273, 274, 275 Stabilization forces, 64–70 electrostatic interactions, 64–66 hydrogen bonding, 66–69 hydrophobic, 69 ion–dipole and dipole–dipole, 66 p-cation, 70 van der Waals, 70 Stabilizing additives, 233

889

Stable isotopes commercial availability, 255 mass spectrometry, 255 metabolites position-specific labeling, 255–256 universal labeling, 256 natural abundance, 255 reaction rate measurement, 255–264 Stadtman, E.R., 19 Stagnant layer, 422 Stall force, 761 Standard deviation, 266 State function, for definition see Glossary Statine-based inhibitor, 536 Statistical analysis, 174, 266, 267 rate dataRate data, analysis Steady-state approximation, 186, 187–188 Steady-state processes, 293 Briggs-Haldane treatment, 293–297 homeostasis, 2 key features, 294, 296 approximate delay-time, 297 substrate/enzyme concentration ratio, 297 multiple internal isomerization effects, 298–299 site-directed mutagenesis studies, 468 two-intermediate treatment, 298–301 derivation, 298 reverse reaction scheme, 298 Steady-state rate equations, 341–349 Cleland’s net reaction rate method for unbranched kinetic mechanisms, 345–347 Fromm’s systematic method, 341–343 Ping Pong Bi Bi reaction, 352–353 random kinetic mechanism, 348 simplifying approach (Cha method), 349 Theorell and Chance ordered binary complex mechanism, 347–348 two-step computer-assisted method, 343–345 Stein, W., 18, 669 Stereochemistry, 146–152 Cahn-Ingold-Prelog specification of absolute stereochemical configuration, 148–149 definitions, 147–148 flavin enzymes, 579 isotopically labeled substrate experiments, 576–578 methyl transfer reactions, 150–152, 153 NADH-dependent hydride transfer, 576–579 nucelotide-dependent reactions, 579–585 prochirality specification, 149–150 Stereoelectronic effect, 148 efficiency of enzyme catalysis, 29 Stereoisomerism, 148 Stereospecificity, 2, 6, 576 alcohol dehydrogenase, 576–577 competitive inhibitors as active sites probes, 499–500 covalent intermediates formation, 31 ester synthesis, 15 flavin enzymes, 579 Steric hindrance, 693 modified Scatchard equation, 682

Stern layer, 422, 449 Debye-Hu¨ckel length, 449 Stickiness, substrate, 401 for definition see Glossary isotope trapping experiments, 603, 604, 605, 606 Stiffness (or stiffness instability), for definition see Glossary Stochastic behavior, 171 chemical rate processes simulation, 184 single-molecule kinetics, 733 Stoichiometric titration measurements, 159–160 Stokes shift, for definition see Glossary Stopped-flow calorimeter, 644 Stopped-flow instruments, 639, 643–644 cavitation artefacts, 682 cryoenzymology, 435 field-jump, 645 mixing devices, 172 Stopped-flow techniques, 643–645 amount-of-substance information, 644 apparatus see Stopped-flow instruments applications, 642 burst-phase kinetics, 655–656 dead-time, 644 rapid-scan devices for spectroscopic analysis, 647–648 reaction time-evolution information, 644 solvent kinetic isotope effects, 626 Stopped-flow temperature-jump technique, 668, 669–671 Stopped-time assays crystalline enzyme kinetics, 458 high pressures experiments, 438 radioactive isotope-labeled substrate, 255 Storage conditions, 223–224 STRENDA Commission checklist for enzyme rate data reporting, 284 Streptomycin, 488 Strictosidine synthase, 130 Stromelysin (membrane metalloproteinase 3), 109 inhibitor design, 570 Structural mechanism, 19, 637 Subdiffusion, 446 Substitution reactions, 120 nucleophilicNucleophilic reactions, substitutions Substrate Cleland notation system, 336 isotopic labeling, 585–586 Substrate binding, 302 conformational change, 319 kinetic proofreading, 454 differential, 302 induced-fit mechanisms, 319–324 nonproductive, 293 preferential transition-state, 302 uniform, 302 Substrate capture, 306 Substrate channeling, 718–723 bi-functional enzymes/bi-enzyme complexes, 721–722 kinetic criteria, 720–721, 722 proton wires, 719

Index

890

Substrate channeling (Continued ) substrate hydration effects, 722–723 transfer efficiencies, 721–722 tunnels, 718–719 Substrate inhibition, 289, 506–511, 544 induced, 511 nonlinear by excess substrate, 506–507 Ping Pong mechanisms, 357–358 Substrate specificity, 1–2, 6, 39–41 active site binding energy, 61 active site substrate recognition subsites, 112–113, 114–115 designation scheme, 114–115 signal-transducing protein kinases, 115 directed enzyme evolution, 39–40 induced-fit mechanisms, 319, 324–326 Michaelis constant relationship, 301–303 microarray/phage-display profiles, 331–332 simultaneous action on different substrates, 315 specificity constant, 304–307 stereochemistry, 146 substrate binding mechanisms, 302, 303 Substrate synergism, 355–356 for definition see Glossary Substrate-assisted activation, 382 depolymerases, 330–331 Substrate-assisted catalysis, for definition see Glossary Substrate-directed activation, 382 Substrate-induced activation, 384 [Substrate]/v versus [Substrate] plot, 291 Subtilisin, 28, 224 site-directed mutagenesis, 461, 462, 464 Succinate dehydrogenase, 160 malonate inhibition, 490 o-Succinyl-benzoate synthase, 74–75 Succinyl-coenzyme A synthetase, 355 Sucrose a-glucosidase, 123 Sucrose cryoprotectant, 223 Suicide substrates see Mechanism-based inhibitors Sulfite oxidase, 106 Sumner, J.B., 22 Superfamilies, enzyme, 70, 71 Superoxide dismutases, 84, 196, 467 Supporting media, enzyme immobilization, 440 Surface plasmon resonance, ligand binding assays, 689, 691 Surface-dilution approach, 451 Surfactants, lipase effects, 452–453 Swain relationships, for definition see Glossary Swain-Schaad relation, 600, 622 Syncatalytic affinity labeling agents, 546–547 Synergism coefficient, 356 Synergism quotient (Qsyn), for definition see Glossary Synergistic inhibition, 505–506 for definition see Glossary Synthetases, nucleotide substrate binding/ discharge, 394 Synthetic catalysts, 7–12 catalytic antibodies, 10–12 catalytic hydrogenation, 7–8

chiral catalysis, 9–10 high-throughput screening, 10 limitations, 7 metathesis, 8–9 synthetic enzymes, 12 Synthetic enzymes, 12 Syphilis, treatment with phenylarsine oxides, 486 System, for definition see Glossary

T T4 phage DNA replication, 446 T7 phage DNA polymerase, optical tweezer experiments, 740 Tabun, 539 Tadalafil, 561 TADDOLate ligand, 9 Takadiastase, 13 Takamine, J., 13 Talc, 383 Taq DNA polymerase, 426, 430 Tartrate, absolute stereochemical configuration, 9 Tautomerism, 16 for definition see Glossary Taxol, 488 Technology, enzyme commercial applications, 14, 15 commercially important enzymes, 15 historical aspects, 13 Telomerase, 49 for definition see Glossary Temperature effects, 425–438 activity, 425–426 reversibility, 428 Arrhenius plots, 427–428 compensation effect, 428 nonlinear, 428–429 bacterial growth rates, 430 equilibrium constant, 427–428 irreversible inactivation, 425 kinetics, 426–427 kinetic isotope effects, 615–616 quantum mechanical hydrogen tuneling, 614, 615 Q10 temperature quotient, 429 rate constant, 427–428 relevance to living organisms, 429–430 reversible cold inactivation in multi-subunit enzymes, 433–434 single-molecule enzyme experiments, 729 site-directed mutagenesis studies, 481 stability, 425–426, 429 Temperature, laboratory, 430 Temperature-jump method, 639, 658, 666–669, 681 apparatus, 667 micro-scale, 667 rapid-mixing devices, 668 temperature-jump cell, 666 dead-time, 669 reports/reviews, 668 stopped-flow technique, 668, 669–671 Ter, Cleland notation system, 336 Terminology, enzyme, 12 Ternary complexes, 338

tert-butylbromide solvolysis, 178–179 Tethered ratchet model, 794, 795 7,7’,,8’-Tetracyanoquinodimethane, 8, 158 Tetrahedral intermediate, for definition see Glossary Tetrahydrobiopterin cofactor, nitric oxide synthase, 165 TexasRed(TM), 249, 250 Theorell, H., 19, 647 Theorell-Chance kinetic mechanism, 337 Di-Iso, 371, 521 ordered Bi Bi, 348 abortive complex effects, 519–520 equilibrium exchange rate equation derivation, 590 equilibrium isotope exchanges, 595–596 product inhibition patterns, 515, 519 ordered sequential Ter Ter mechanism, 366–367 pH effects, 408–409 quantitative analysis, 360, 361, 362, 364 Therapeutic index, 487, 571 Therapy, enzyme, 42, 50–51 autologous versus heterologous enzymes, 50 replacement therapy, 50 Thermal energy of reacting molecules, Boltzmann distribution, 192–194 Thermal enzyme inactivation, 425–426 kinetics, 426–427 reversible cold inactivation in multi-subunit enzymes, 433–434 Thermal enzyme stability, 426 Thermal equilibration, initial rate enzyme assays, 220, 227 Thermal (Johnson) noise, 230 Thermal niches of living organisms, 429 Thermochemical data, equilibrium constant determination, 280 Thermodynamic activity ionic strength effects, 419 Debye-Hu¨ckel limiting law, 417–418, 419 site-directed mutagenesis, 469–471, 481 Thermodynamic control, for definition see Glossary Thermodynamic cycles detailed balance evaluation, 190–192 induced-fit mechanisms, 324 Thermodynamic mechanism, 637 Thermodynamic pKa, for definition see Glossary Thermodynamic principles, 210–214 direction/extent of chemical reaction, 210 equilibrium, 210–212 non-equilibrium, 212–214 Thermodynamic transition state model, 202 Thermodynamic-kinetic modeling, 192 Thermolysin, 109 Thermophilic bacteria, 426, 430 uses of enzymes, 431 Thiamin diphosphate, 127 Thimidylate synthase, 31 6-Thioguanine, 487 Thiohemiacetal adduct, 139 5-Thiopyrimidine, 184

Index

Third-order processes, 175 Thomson, J., 258 Three-substrate enzyme kinetics, 366–370 competitive inhibitors to differentiate mechanisms (Fromm’s method), 496, 498 initial-velocity experiments, 368 Frieden protocol, 368–369 Fromm substrates-ratio protocol, 369–370 kinetic schemes, 366–368 Threonine deaminase, 695 Thrombin, 71, 95, 714 ionic strength effects, 419–420 Thymidylate synthase, 151, 512 mechanism-based inhibition, 552 Tight ion pair, for definition see Glossary Tight-binding reversible inhibitors, 525, 531–537 computer-assisted progress curve analysis, 535 irreversible inhibitor differentiation, 544 rate equations, 531–533 slow inhibitor-induced enzyme conformational change, 531 time-dependent behavior, 533–535 Time evolution of catalytic process, 204–205 Time-clamping, rapid-mixing continuous-flow methods, 640 Time-of-flight mass spectrometers, 255, 259 Time-resolved electrospray ionization mass spectrometry, quench-flow experiments, 652 Time-resolved Laue X-ray crystallography, 459, 682 Time-resolved solid-state nuclear magnetic resonance spectroscopy, 680 Time-resolved spectral analysis, stopped-flow experiments, 648–649 Time-scale, 37 biological reactions, 641 chemical processes, 171–172, 641 enzymatic catalysis, 34, 35, 36 fast reaction techniques, 638 flash photolysis, 638 protein structural fluctuations, 35, 36 rapid-mixing experiments, 638, 640 regulatory enzyme activity, 686 relaxation methods, 638 Titanium complexes, 10 Titin, force-clamp experiment, 773 Titration, active enzyme concentration determination, 278 Topaquinone, 81, 132 Topaquinone copper amine oxidase, 101 Topoisomerases, 776 Total internal reflection fluorescence microscopy, 244, 749, 750–751 Tracee, 257 Tracer, 257 Trajectory of reaction chemical dynamics, 4, 32, 33 energase-catalysed reactions, 771 kinetic isotope effects, 608

891

single-molecule observations, 45, 729, 732–733, 734 fluorescent microscopy, 750 stochastic processes, 733 transition state theory, 201, 203 Transaminases, 335 Transcarboxylase, 31, 358 Transcription, 61 errors, 454 kinetic proofreading, 454–455 messenger RNA elongation kinetics, 740–741 punctuated pauses, 740 trans-epoxysuccinyl-L-leucylamido-(4guanidino)butane (E-64), 225, 226, 227 Transfer RNA, 57 Transfer RNA nucleotidyltransferase, 363 Transferases, 23 Ping Pong Bi Bi mechanism, 356–357 Transit time coupled (auxiliary) enzyme assays, 238–239 series first-order processes, 185 Transition metal catalysis di-/poly-nuclear metal cluster complexes, 8 hydrogenation, 7–8 maximization of active catalytic surface, 8 olefin metathesis, 8–9 surface properties, 8 Transition metal complexes, 85 color/crystal field splitting energy, 87–88 ligand field stabilization energy, 89 reaction mechanisms, 89–93 electron transfer reactions, 92–93 metal-ligand hydrolysis, 89–90 monodentate ligand complexation with aquated metal ions, 90–91 multidentate ligand interactions, 91–92 square planar, 86 tetrahedral, 86 Transition state, 197 for definition see Glossary isotope rate measurements analysis, 599–603 Albery-Knowles treatment, 600 kinetic isotope studies, 608, 631 preferential substrate binding, 302 reaction coordinate diagrams, 208, 209 solvation, 200, 201 thermodynamic reciprocity, 32 Transition state analogues catalytic antibody production, 10, 11 drug design strategy (Schramm’s molecular similarity approach), 563–566 Transition state barrier, 201, 203, 597 bimolecular rate constant relationship, 202 Transition state inhibitors, 525–531 energetics, 525–527 enzymatic pro-inhibitor activation, 528–531 enzyme-bound transition state dissociation constant, 526 naturally-occurring, 527 synthetic, 527 Transition state stabilization, 6–7, 37 active site residues, 57–58 catalytic antibody design, 10

efficiency of enzyme catalysis, 26–27, 29, 31–32, 204, 526 electrostatic, 26, 27 energetics, 27, 525 low-barrier hydrogen bonds, 31–32, 68 ribozymes, 313 Transition state theory, 201–204 equilibrium assumption, 203 molecularity, 176 product dividing surface, 202, 203 reactant dividing surface, 202, 203 reaction coordinate, 202, 208 saddle point (coll) passage, 202 simplifying assumptions, 201 solvation, 200 Translation, 61 errors, 454 kinetic proofreading, 455 Transport rate, 597 Trehalose cryoprotectant, 223 Trentham, 19 2,,5-Trihydroxyphenylanaline quinone, 4, 81 Triose-phosphate isomerase, 27, 63, 68, 296, 309, 364, 627 active site-directed irreversible inhibition, 542–543 isotope rate measurements, 599–600, 601 reaction energetics, 310, 601 site-directed mutagenesis, 470, 472–479 transient ene-diolate intermediate, 600, 601, 627 Triple-competitive method, for definition see Glossary Triplet state, 244 for definition see Glossary Trituration, 224 Trypsin, 70, 331, 467, 512 acyl-enzyme formation at sub-zero temperatures, 436, 437 chromogenic substrates, 243 chymotrypsinogen cleavage, 22 substrate recognition subsites, 114 substrate specificity, 302 Tryptophan fluorescence, 247–248 Tryptophan synthase, substrate channeling, 721–722 Tryptophan tryptophyloquinone, 81 Tryptophanase, reversible cold inactivation, 434 Tryptophyloquinone, 132 Tryptoquinone, 81 Tubulin:tyrosine ligase, 369 Tubulin, 63, 488, 498, 782 kinesin interaction, 738 Tungsten, 78, 82, 107 Tungsten pyranopterin, 107 Tunnel correction, 16 Tunnelling see Quantum mechanical hydrogen tuneling Turbidity correlation analysis, 674 Turbidity, initial rate enzyme assays, 222 Turnover number, 234, 303–304 for definition see Glossary specific activity relationship, 304

Index

892

Turnover, protein, 631–632 control mechanisms, 633 isotope labeling experiments, 631–634 pulse-chase labeling techniques, 632–633 steady-state concentrations, 631–632 Tyrosinase, enzyme electrodes, 167 Tyrosine fluorescence, 247–248 Tyrosine ligase, 498 Tyrosine phosphatase, rapid mix/quench experiments, 651 Tyrosine-tRNA ligase, 464 Tyrosyl quinone, 81

U Ubiquitination, 61 UDP-N-acetylglucosamine enoylpyruvyl transferase (MurA), 579 UDP-galactose 4-epimerase, 159, 335 UDP-glucose epimerase, 386 Ultracentrifugation, 691 Ultraviolet microphotometry, crystalline enzyme kinetics, 457–458 Ultraviolet/visible absorption spectroscopy, 172, 240–243 chromogenic substrates, 243 cuvettes, 242 cleaning, 241–242 elementary reaction relaxation amplitudes, 660 flash photolysis, 675 instruments, 241 stray light minimization, 242–243 metal-nucleotide complexes, 392 progress-curve quantitative analysis, 268–270 sensitivity, 242 solutions standardization, 230 temperature-jump techniques, 668–669 Ultraviolet/visible spectrophotomer, 241 Umbarger, H., 18 Uncompetitive inhibitors, 502–503 clinically useful, 504 partial, 494, 495 product inhibition, 512 tight-binding reversible, 532 Uni, Cleland notation system, 336 Unit mass resolution, 257 Universal Protease Substrate(TM), 227 Uracil, 57 Uracil DNA glycosylase, 269 Urease, 13, 22 crystalline kinetics, 458 nickel ion complexes, 84, 107 Usher, D., 18 UTP, regeneration from UDP, 275

V V systems, for definition see Glossary van der Waals interactions, 70 van der Waals radius, 70 Vanadium, 82, 107–108 Vancomycin, 488, 528 van9t Hoff, J.H., 6 Vardenafil, 561 Variance, 266–267 Variant enzymes, 64

VASP, 795, 799, 801 Velocity of reaction, 174–175, 289 Vennesland, B., 576 vent DNA polymerase, 426 Verger, R., 18 Vibrational energy, catalysis promotion, 34–35 Video absorption spectroscopy, crystalline enzyme kinetics, 459 Video microscopy, 737–739 Vinblastine, 488, 559 Vincristine, 488 Vinleursine, 488 Vinyl-L-glycine (2-amino-butenoate), 550 Viral vectors, enzyme therapy, 50–51 Viscosity effects, 411 enzyme influences at single-molecule level, 730 VisualEnzymics(TM), 269 Vitalism, 685 Vitamin B12Cobalamin; Cyanocobalamin Vitamin K, blood clotting cascade regulation, 715 Vitamin K-dependent carboxylase, 385 carbanion intermediate, 127, 130 Vitamins deficiency, 77 intrinsic fluorescence, 248 see also Coenzymes Volt, for definition see Glossary

W Waage, P., 6 Walden inversion, for definition see Glossary Wall effects, 228 Warburg, O., 117, 352 Warfarin, 489 Water, 194 active site residues activation, 58 cage effect, 195, 197 deionized for enzyme rate assays, 228 diffusion limitation on chemical processes, 196–197 dissociation, 195 electron transfer, 196 hydrogen bond networks, 194, 195, 200 hydrophobic interactions, 196 isotopomeric forms, 623 liquid structure, 195 osmotic effects, 196 preferential interactions, 196 proton transfer, 195–196, 663–665 solute ionization, 196 solvent interface (electric double layer), 421–422 solvent kinetic isotope effects, 623 solvent properties, 194–195, 200 Water bath, 220 Watson, J.D., 174 Webb, E., 18, 19, 585 Weizmann, C., 13 Werner, A., 86 Westheimer, F., 19, 30, 547, 548, 576 White noise, 230 Wilhemy, 5 Wolfenden, R., 19, 525

Womack-Colowick continuous dialysis method, for definition see Glossary Wong, J.T., 19 Working solutions assembly, multisubstrate kinetics, 264–265 Wyman, J., 18, 695 linked function analysis, 694–695

X X-Ray absorption for fine structure (XAFS), for definition see Glossary X-ray diffraction, 19, 41, 42, 457, 459 active site substrate binding, 60 carbonic anhydrase, 109, 111 cryoenzymology studies, 436 acyl-enzyme formation, 437 drug lead compound screening, 560 F1-ATPase, 395 hexokinase, 320 Laue method, 38, 682 lysozyme, 333 metal ion coordination complexes, 94 multi-subunits enzyme complexes with separate active sites, 718 protein tertiary structure, 62 site-directed mutagenesis, 466, 468, 472 stereochemistry of compounds, 147 time-resolved, 38, 459 Xanthine dehydrogenase, stopped-flow kinetic techniques, 642 Xanthine oxidase, 106 enzyme electrodes, 168–169 Xylanase substrate recognition subsite designation scheme, 115 time-resolved electrospray ionization mass spectrometry, 651

Y Yeast cell division control protein, 395

Z Zero-mode waveguides, 757–758 Zero-order kinetics, 175, 177 Michaelis-Menten processes, 288 Zero-order reactions, for definition see Glossary Zero-point energy, 609 for definition see Glossary isotopomer differences, 609–612 solvent isotope effects, 623 z-potential, 422 Zidovudine (AZT), 490, 492 Zinc, 108–109 enzyme activation, 100 enzyme complex coordination schemes, 111, 112 Lewis acid activity, 108–109 Zinc ion transporter, 109 Zinc-finger motifs, 109 Zirconium complexes, 10 Zwitterion, for definition see Glossary Zymogens, 7, 21, 64 activation, 207 for definition see Glossary