Voltage-Sensitive Ion Channels: Biophysics of Molecular Excitability [1 ed.] 1402055242, 9781402055249, 9781402055256

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VOLTAGE-SENSITIVE ION CHANNELS

Voltage-Sensitive Ion Channels Biophysics of Molecular Excitability

by

H. RICHARD LEUCHTAG

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-5524-9 (HB) ISBN 978-1-4020-5525-6 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2008 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

To Alice, Clyde, Penny, Jeremy, Joshua and Ilana Leuchtag, and to the memory of my parents, Käthe (Wagner) Leuchtag and Rudolf Wilhelm Leuchtag, who with United States citizenship became Kathe and Rudolph Leuchtag Light

Contents Preface

xxiii

Ch. 1 EXPLORING EXCITABILITY 1. NERVE IMPULSES AND THE BRAIN 1.1. Molecular excitability 1.2. Point-to-point communication 1.3. Propagation of an impulse 1.4. Sodium and potassium channels 1.5. The action potential 1.6. What is a voltage-sensitive ion channel? 2. SEAMLESS NATURE, FRAGMENTED SCIENCE 2.1. Physics 2.2. Chemistry 2.3. Biology 3. THE INTERDISCIPLINARY CHALLENGE 3.1. Worlds apart 3.2. Complex systems 3.3. Interdisciplinary sciences bridge the gap

1 1 1 2 3 5 5 7 9 9 11 14 18 18 18 19

Ch. 2 INFORMATION IN THE LIVING BODY 1. HOW BACTERIA SWIM TOWARD A FOOD SOURCE 2. INFORMATION AND ENTROPY 3. INFORMATION TRANSFER AT ORGAN LEVEL 3.1. Sensory organs 3.2. Effectors: Muscles, glands, electroplax 3.3. Using the brain 3.4. Analyzing the brain 4. INFORMATION TRANSFER AT TISSUE LEVEL 5. INFORMATION TRANSFER AT CELL LEVEL 5.1. The cell 5.2. Cells of the nervous system 5.3. The neuron 5.4. Crossing the synapse 5.5. The “psychic” neuron 5.6. Two-state model "neurons" 5.7. Sensory cells 5.8. Effector cells 6. INFORMATION TRANSFER AT MEMBRANE LEVEL 6.1. Membrane structure 6.2. G proteins and second messengers 7. INFORMATION TRANSFER AT MOLECULAR LEVEL 7.1. Chirality 7.2. Carbohydrates vii

21 21 24 25 25 26 27 28 29 30 30 31 31 34 35 35 36 38 39 39 39 40 40 42

viii

CONTENTS 7.3. Lipids 7.4. Nucleic acids and genetic information 7.5. Proteins 8. INFORMATION FLOW AND ORDER 8.1. Information flow and time scales 8.2. The emergence of order

42 43 43 44 45 45

Ch. 3 ANIMAL ELECTRICITY 1. DO ANIMALS PRODUCE ELECTRICITY? 1.1. Galvani’s “animal electricity” 1.2. Volta’s battery 1.3. Du Bois-Reymond’s "negative variation" 2. THE NERVE IMPULSE 2.1. Helmholtz and conduction speed 2.2. Pflüger evokes nerve conduction 2.3. Larger fibers conduct faster – but not always 2.4. Refractory period and abolition of action potential 2.5. Solitary rider, solitary wave 3. BIOELECTRICITY AND REGENERATION 3.1. Regeneration and the injury current 3.2. Bone healing and electrical stimulation 3.3. Neuron healing 4. MEMBRANES AND ELECTRICITY 4.1. Bernstein's membrane theory 4.2. Quantitative models 4.3. The colloid chemical theory 4.4. Membrane impedance studies 4.5. Liquid crystals and membranes 5. ION CURRENTS TO ACTION POTENTIALS 5.1. The role of sodium 5.2. Isotope tracer studies 5.3. Hodgkin and Huxley model the action potential 5.4. Membrane noise 5.5. The patch clamp and single-channel pulses 6. GENETICS REVEALS CHANNEL STRUCTURE 6.1. Channel isolation 6.2. Genetic techniques 6.3. Modeling channel structure 7. HOW DOES A CHANNEL FUNCTION? 7.1. The hypothesis of movable gates 7.2. The phase-transition hypothesis 7.3. Electrodiffusion reconsidered 7.4. Ferroelectric liquid crystals as channel models

47 47 48 48 49 49 49 49 50 50 50 51 51 52 52 53 53 53 54 54 55 56 56 57 57 58 59 59 59 59 60 60 60 60 60 61

Ch. 4 ELECTROPHYSIOLOGY OF THE AXON 1. EXCITABLE CELL PREPARATIONS

63 63

CONTENTS

ix

1.1. A squid giant axon experiment 1.2. Node of Ranvier 1.3. Molluscan neuron 2. TECHNIQUES AND MEASUREMENTS 2.1. Space clamp 2.2. Current clamp 2.3. Voltage clamp 2.4. Internal perfusion 3. RESPONSES TO VOLTAGE STEPS 3.1. The current–voltage curves 3.2. Step clamps and ramp clamps 3.3. Repetitive firing 3.4. The geometry of the nerve impulse 4. VARYING THE ION CONCENTRATIONS 4.1. The early current 4.2. The delayed current 4.3. Divalent ions 4.4. Hydrogen ions 4.5. Varying the ionic environments 5. MOLECULAR TOOLS 5.1. The trouble with fugu 5.2. Lipid-soluble alkaloids 5.3. Quaternary ammonium ions 5.4. Peptide toxins 6. THERMAL PROPERTIES 6.1. Effect of temperature on electrical activity 6.2. Effect of temperature on conduction speed 6.3. Excitation threshold, temperature and accommodation 6.4. Stability and thermal hysteresis 6.5. Temperature effects on current–voltage characteristics 6.6. Temperature pulses modify ion currents 6.7. Temperature and membrane capacitance 6.8. Heat generation during an impulse 7. OPTICAL PROPERTIES 7.1. Membrane birefringence 7.2. Ultraviolet effects 8. MECHANICAL PROPERTIES 8.1. Membrane swelling 8.2. Mechanoreception

64 65 66 66 67 68 68 68 69 69 69 70 72 73 73 74 74 74 75 76 76 77 77 78 79 79 80 80 80 81 83 84 84 84 84 85 85 85 86

Ch. 5 ASPECTS OF CONDENSED MATTER 1. THE LANGUAGE OF PHYSICS 1.1. The Schrödinger equation 1.2. The Uncertainty Principle

89 89 89 90

x

CONTENTS 1.3. Spin and the hydrogen atom 1.4. Identical particles—why matter exists 1.5. Tunneling 1.6. Quantum mechanics and classical mechanics 1.7. Quantum mechanics and ion channels 2. CONDENSED MATTER 2.1. Liquids and solids 2.2. Polymorphism 2.3. Quasicrystals 2.4. Phonons 2.5. Liquid crystals 3. REVIEW OF THERMODYNAMICS 3.1. Laws of thermodynamics 3.2. Characteristic functions 4. PHASE TRANSITIONS 4.1. Phase transitions in thermodynamics 4.2. Transitions of first order 4.3. Chemical potentials, metastability and phase diagrams 4.4. Transitions of second order 4.5. Qualitative aspects of phase transitions 5. FROM STATISTICS TO THERMODYNAMICS 5.1. Phase space 5.2. The canonical distribution 5.3. Open systems 5.4. Thermodynamics of quantum systems 5.5. Phase transitions in statistical mechanics 5.6. Structural transitions in ion channels

Ch. 6 IONS IN THE ELECTRIC FIELD 1. REVIEW OF ELECTROSTATICS 1.1. Forces, fields and media 1.2. The laws of electrostatics 2. MOVEMENT OF IONS IN AN ELECTRIC FIELD 2.1. Current 2.2. Ohm's law 2.3. Capacitance and inductance 2.4. Circuits and membrane models 3. CABLE THEORY 3.1. The cable equations 3.2. Application to the squid axon 4. THERMODYNAMICS OF DIELECTRICS 4.1. Electrochemical potential 4.2. The Nernst-Planck equation 4.3. Thermodynamics of electric displacement and field 4.4. Electrets

91 92 93 93 94 95 95 96 97 97 98 99 99 102 103 103 104 105 106 108 108 108 110 110 111 112 113 115 115 115 117 119 119 119 121 122 123 123 125 126 126 126 127 128

CONTENTS 5. MOTIONS OF CELLS IN ELECTRIC FIELDS 5.1. Dielectrophoresis 5.2. Electrorotation 6. MOVEMENT OF IONS THROUGH MATTER 6.1. Movement of ions through liquid solutions 6.2. Surface effects 6.3. Movement of ions through solids 6.4. Ionic switches 6.5. Ionic polarons and excitons 7. SUPERIONIC CONDUCTION 7.1. Sodium-ion conductors 7.2. Superionic conduction in polymers and elastomers 7.3. Are ion channels superionic conductors?

xi 129 129 130 130 130 131 131 132 132 132 133 136 137

Ch. 7 IONS DRIFT AND DIFFUSE 1. THE ELECTRODIFFUSION MODEL 1.1. The postulates of the model 1.2. A mathematical membrane 1.3. Boundary conditions 2. ONE ION SPECIES, STEADY STATE 2.1. The Nernst-Planck equation 2.2. Electrical equilibrium 3. THE CONSTANT FIELD APPROXIMATION 3.1. Linearizing the equations 3.2. The current-voltage relationship 3.3. Comparison with data 4. AN EXACT SOLUTION 4.1. One-ion steady-state electrodiffusion 4.2. Finite current 4.3. Reclaiming the dimensions 4.4. Electrical equilibrium 4.5. Applying the boundary conditions 4.6. Equal potassium concentrations

139 139 140 141 141 142 142 144 146 147 148 149 151 151 152 156 156 159 161

Ch. 8 MULTI-ION AND TRANSIENT ELECTRODIFFUSION 1. MULTIPLE SPECIES OF PERMEANT IONS 1.1. Ions of the same charge 1.2. Ions of different charges 1.3. The Goldman–Hodgkin–Katz equation 2. TIME-DEPENDENT ELECTRODIFFUSION 2.1. Scaling of variables 2.2. The Burgers equation 2.3. A simple case 3. CRITIQUE OF THE CLASSICAL MODEL

163 163 163 164 165 166 168 168 169 170

xii

CONTENTS 173 173 174 174 175 176 176

Ch. 9 MODELS OF MEMBRANE EXCITABILITY 1. THE MODEL OF HODGKIN AND HUXLEY 1.1. Ion-current separation and ion conductances 1.2. The current equation 1.3. The independence principle 1.4. Linear kinetic functions 1.5. Activation and inactivation 1.6. The partial differential equation of Hodgkin and Huxley 1.7. Closing the circle 2. EXTENSIONS AND INTERPRETATIONS 2.1. The gating current 2.2. Probability interpretation of the conductance functions 2.3. The Cole–Moore shift 2.4. Mathematical extensions of the Hodgkin-Huxley equations 2.5. The propagated action potential is a soliton 2.6. Action potential as a vortex pair 2.7. Catastrophe theory version of the model 2.8. Beyond the squid axon 3. EVALUATION OF THE HODGKIN-HUXLEY MODEL 3.1. Current separation 3.2. Voltage dependence of the conductances 3.3. Time variation of the conductances 3.4. The separation of ion kinetics 3.5. We’re not out of the woods yet 4. THE CONCEPT OF AN ION CHANNEL 4.1. Pore or carrier – or what? 4.2. "Pore" and "channel": Shifting meanings 4.3. Limitations of the phenomenological approach

182 182 183 183 185 186 187 187 189 189 190 190 190 192 192

Ch. 10 ADMITTANCE TO THE SEMICIRCLE 1. OSCILLATIONS, NORMAL MODES AND WAVES 1.1. Simple pendulum 1.2. Normal modes 1.3. The wave equation 1.4. Fourier series 1.5. The Fourier transform of a vibrating string 2. MEMBRANE IMPEDANCE AND ADMITTANCE 2.1. Impedance decreases during an impulse 2.2. Inductive reactance 2.3. A simple circuit model 3. TIME DOMAIN AND FREQUENCY DOMAIN 3.1. Fourier analysis 3.2. The complex admittance 3.3. Constant-phase-angle capacitance

195 195 195 197 197 198 198 199 199 201 202 203 203 205 206

178 178 179 179 180 180

CONTENTS

xiii

4. DIELECTRIC RELAXATION 4.1. The origin of electric polarization 4.2. Local fields affect permittivity 4.3. Dielectric relaxation and loss 4.4. Cole–Cole analysis 5. FREQUENCY-DOMAIN MEASUREMENTS 5.1. Linearizing the model of Hodgkin and Huxley 5.2. Frequency response of the axonal impedance 5.3. Pararesonance 5.4. Impedance of the Hodgkin–Huxley axon membrane 5.5. Generation of harmonics ?5.6. Data fits to squid-axon sodium system 5.7. Admittance under suppressed ion conduction

207 207 208 209 211 212 213 214 216 216 217 217 217

Ch. 11 WHAT’S THAT NOISE? 1. STOCHASTIC PROCESSES AND STATISTICAL LAWS 1.1. Stochastic processes 1.2. Stationarity and ergodicity 1.3. Markov processes 2. NOISE MEASUREMENT AND ANALYSIS TECHNIQUES 2.1. Application of Fourier analysis to noise problems 2.2. Spectral density and autocorrelation 2.3. White noise 3. EFFECTS OF NOISE ON NONLINEAR DYNAMICS 3.1. An aperiodic fluctuation 3.2. The Langevin equation 4. NOISE IN EXCITABLE MEMBRANES 4.1. A nuisance becomes a technique 4.2. Fluctuation phenomena in membranes 4.3. 1/f noise 4.4. Lorentzian spectra 4.5. Multiple Lorentzians 4.6. Nonstationary noise 4.7. Light scattering spectra 5. IS THE SODIUM CHANNEL A LINEAR SYSTEM? 5.1. Sodium-current characteristics 5.2. Admittance and noise 6. MINIMIZING MEASUREMENT AREA 6.1. Patch clamping 6.2. Elementary stochastic fluctuations in ion channels

221 221 222 224 224 225 225 226 227 228 228 228 230 230 230 231 231 233 234 235 235 235 238 239 240 240

Ch. 12 ION CHANNELS, PROTEINS AND TRANSITIONS 1. THE NICOTINIC ACETYLCHOLINE RECEPTOR 2. CULTURED CELLS AND LIPOSOMES 2.1. Sealing the pipette to the membrane 2.2. Reconstitution of channels in bilayers 2.3. Reconstitution of sodium channels

243 244 245 245 246 247

xiv

CONTENTS 3. SINGLE-CHANNEL CURRENTS 3.1. Unitary potassium currents 3.2. Unitary sodium currents 4. MACROSCOPIC CURRENTS FROM CHANNEL TRANSITIONS 4.1. The two-state model 4.2. Ohmic one-ion channels 4.3. Time dependence 4.4. Critique of the methodology 5. PROTEIN STRUCTURES 5.1. Amino acids: Building blocks of proteins 5.2. Primary structure 5.3. Levels of structural organization 5.4. The alpha helix 5.5. The beta sheet 5.6. Domains and loop regions 5.7. Structure classifications and representations 5.8. Alpha-domain structures 5.9. Alpha/beta structures 5.10. Antiparallel beta structures: jelly rolls and barrels 6. METALLOPROTEINS 6.1. Metalloproteins in physiology and toxicology 6.2. Voltage-sensitive ion channels as metalloproteins 7. MEMBRANE PROTEINS 7.1. Membrane-spanning protein molecules 7.2. Crystallization of membrane proteins 7.2. Biosynthesis of membrane proteins 8. TRANSITIONS IN PROTEINS 8.1. Vibrations and conformational transitions 8.2. Allosteric transitions in myoglobin and hemoglobin 8.3. Allostery in ion channels

Ch. 13 DIVERSITY AND STRUCTURES OF ION CHANNELS 1. THE ROLE OF STRUCTURE 2. FAMILIES OF ION CHANNELS 2.1 Molecular biology 2.2. Evolution of voltage-sensitive ion channels 3. MOLECULAR BIOLOGY PROBES CHANNEL STRUCTURE 3.1. Genetic engineering of ion channels 3.2. Obtaining the primary structure 3.3. Hydropathy analysis 3.4. Site-directed mutagenesis 4. CLASSIFICATION OF ION CHANNELS 4.1. Nomenclature 4.2. Classification criteria 4.3. Toxins and pharmacology 4.4. Voltage-sensitive ion channels and disease

248 249 249 249 250 250 252 253 253 253 255 256 257 259 259 260 260 262 263 263 264 265 265 265 266 267 267 268 268 268 271 271 272 272 272 273 273 273 273 274 275 275 275 276 276

CONTENTS 5. POTASSIUM CHANNELS: A LARGE FAMILY 5.1. Shaker and related mutations of Drosophila 5.2. Diversity of potassium channels 5.3. Three groups of K channels 5.4. Voltage-sensitive potassium channels 5.5. Auxiliary subunits 5.6. Inward rectifiers 5.7. Potassium channels and disease 6. VOLTAGE-SENSITIVE SODIUM CHANNELS: FAST ON THE TRIGGER 6.1. Neurotoxins of VLG Na channels 6.2. Types of VLG Na channels 6.3. Positively charged membrane-spanning segments 6.4. Proton access to channel residues 6.5. Mutations in sodium channels 7. CALCIUM CHANNELS: LONG-LASTING CURRENTS 7.1. Function of VLG Ca channels 7.2. Structure of VLG Ca channels 7.3. Types of VLG Ca channels 7.4. Calcium-channel diseases 8. H+-GATED CATION CHANNELS: THE ACID TEST 9. CHLORIDE CHANNELS: ACCENTUATE THE NEGATIVE 9.1. Structure and function of chloride channels 9.2. Chloride-channel diseases 10. HYPERPOLARIZATION-ACTIVATED CHANNELS: IT’S TIME 11. CYCLIC NUCLEOTIDE GATED CHANNELS 12. MITOCHONDRIAL CHANNELS 13. FUNGAL ION CHANNELS–ALAMETHICIN 14. THE STRUCTURE OF A BACTERIAL POTASSIUM CHANNEL Ch. 14 MICROSCOPIC MODELS OF CHANNEL FUNCTION 1. GATED STRUCTURAL PORE MODELS 1.1. Structural gated pores 1.2. Selectivity filter and selectivity sequences 1.3. Independence of ion fluxes 1.4. Gates 1.5. A "paradox" of ion channels 1.6. Bacterial model pores and porins 1.7. Water through the voltage-sensitive ion channel? 1.8. Molecular dynamics simulations 2. MODELS OF ACTIVATION AND INACTIVATION 2.1. Armstrong model 2.2. Barrier-and-well models of the channel 2.3. The inactivation gate 2.4. Beyond the gated pore

xv 277 277 277 279 279 282 282 284 284 285 285 285 287 288 288 289 290 291 291 292 293 293 294 294 295 295 297 298 301 301 301 304 304 305 306 306 307 307 308 309 309 311 312

xvi

CONTENTS 3. ORGANOMETALLIC CHEMISTRY 3.1. Types of intermolecular interactions 3.2. Organometallic receptors 3.3. Supramolecular self-assembly by % interactions 4. PLANAR ORGANIC CONDUCTORS 5. ALTERNATIVE GATING MODELS 5.1. The theories of Onsager and Holland 5.2. Ion exchange models 5.3. Hydrogen dissociation and hydrogen exchange 5.4. Dipolar gating mechanisms 5.5. A global transition with two stable states 5.6. Aggregation models 5.7. Condensed state models 5.8. Coherent excitation models 5.9. Liquid crystal models 6. REEXAMINATION OF ELECTRODIFFUSION 6.1. Classical electrodiffusion – what went wrong? 6.2. Are the "constants" constant? 7. ORDER FROM DISORDER?

313 313 314 316 317 318 318 320 321 322 322 322 323 323 324 324 325 325 326

Ch. 15 ORDER FROM DISORDER 1. COMPLEXITY AND CRITICALITY 1.1. The emergence of complexity 1.2. Power laws and scaling in physical statistics 1.3. Universality 1.4. Emergent phenomena 2. FRACTALS 2.1. Self-similarity 2.2. Scaling and fractal dimension 2.3. Fractals in time: 1/f noise 2.4. Fractal transport in superionic conductors 2.5. Self-organized criticality 3. ORDER, DISORDER AND COOPERATIVE BEHAVIOR 3.1. Temperature and entropy 3.2. The perfect spin gas 3.3. Thermodynamic functions of a spin gas 3.4. Spontaneous order in a real spin gas 4. FLUCTUATIONS, STABILITY, MACROSCOPIC TRANSITIONS 4.1. Fluctuations and instabilities 4.2. Convective and electrohydrodynamic instabilities 4.3. Spin waves and quasiparticles 4.4. The phonon gas 4.5. The spontaneous ordering of matter 5. PHASE TRANSITIONS 5.1. Order variables and parameters 5.2. Mean field theories 5.3. Critical slowing down and vortex unbinding

329 329 330 331 332 333 334 334 334 335 336 337 338 338 339 341 342 343 343 344 345 346 347 347 348 349 350

CONTENTS 6. DISSIPATIVE STRUCTURES 6.1. Thermodynamics of irreversible processes 6.2. Evolution of order 6.3. Synergetics 6.4. A model of membrane excitability Ch. 16 POLAR PHASES 1. ORIENTATIONAL POLAR STATES IN CRYSTALS 1.1. Piezoelectricity 1.2. Pyroelectricity 1.3. The strange behavior of Rochelle salt 1.4. Transition temperature and Curie-Weiss law 1.5. Hysteresis 1.6. Ferroic effects 2. THERMODYNAMICS OF FERROELECTRICS 2.1. A nonlinear dielectric equation of state 2.2. Second order transitions 2.3. Field and pressure effects 2.4. Chirality and self-bias 2.5. Admittance and noise in ferroelectrics 3. STRUCTURAL PHASE TRANSITIONS IN FERROELECTRICS 3.1. Order-disorder and displacive transitions 3.2. Spontaneous electrical pulses 3.3. Soft lattice modes 3.4. Hydrogen-bonded ferroelectrics 4. FERROELECTRIC PHASE TRANSITIONS AND CONDUCTION 4.1. Tris-sarcosine calcium chloride 4.2. Betaine calcium chloride dihydrate 4.3. Dielectric relaxation in structural transitions 4.4. Cole-Cole dispersion; critical slowing down 4.5. From ferroelectric order to superionic conduction 4.6. Ferroelectric semiconductors 5. PIEZO- AND PYROELECTRICITY IN BIOLOGICAL TISSUES 5.1. Pyroelectric properties of biological tissues 5.2. Piezoelectricity in biological materials 6. PROPOSED FERROELECTRIC CHANNEL UNIT IN MEMBRANES 6.1. Early ferroelectric proposals for membrane excitability 6.2. The ferroelectric–superionic transition model 6.3. Field-induced birefringence in axonal membranes 6.4. Membrane capacitance versus temperature 6.5. Surface charge 6.6. Field effect and the function of the resting potential 6.7. Phase pinning and the action of tetrodotoxin 7. THE CHANNEL IS NOT CRYSTALLINE

xvii 351 351 351 352 352 355 355 356 356 357 357 358 358 359 360 361 362 363 364 365 365 366 366 367 369 369 369 370 371 371 373 374 374 375 375 375 378 380 380 380 381 382 383

xviii

CONTENTS 387 387 387 388 388 391 392 393 394

Ch. 17 DELICATE PHASES AND THEIR TRANSITIONS 1. MESOPHASES: PHASES BETWEEN LIQUID AND CRYSTAL 1.1. Nematics and smectics 1.2. Calamitic and discotic liquid crystals 1.3. Helical structures: Cholesterics and blue phases 1.4. Columnar liquid crystals 2. STATES AND PHASE TRANSITIONS OF LIQUID CRYSTALS 2.1. Correlation functions in liquid crystals 2.2. Symmetry, molecular orientation and order parameter 2.3. Free energy of the inhomogeneous orientational structure 2.4. Modulated orientational structure 2.5. Free energy of a smectic liquid crystal of type A 2.6. Stability of the smectic phase 2.7. Phase transitions between smectic forms 2.8. Inversions in chiral liquid crystals 3. ORDER PARAMETERS UNDER EQUILIBRIUM CONDITIONS 3.1. Biaxial smectics 3.2. The role of fluctuations 3.3. Effect of impurities 4. FIELD-INDUCED PHASE TRANSFORMATIONS 4.1. Dielectric permittivity of liquid crystals 4.2. Unwinding the helix 4.3. The Fredericks transition 5. POLARIZED STATES IN LIQUID CRYSTALS 5.1. Flexoelectric effects in nematics and type-A smectics 5.2. Flexoelectric deformations 5.3. The flexoelectric effect in cholesterics 5.4. Polarization and piezoelectric effects in chiral smectics 5.5. The electroclinic effect 5.6. The electrochiral effect 6. THE FERROELECTRIC STATE OF A CHIRAL SMECTIC 6.1. Behavior of a liquid ferroelectric in an external field 6.2. Polarization and orientational perturbation 6.3. Surface-stabilized ferroelectric liquid crystals

394 395 396 397 398 399 399 399 400 401 401 402 402 403 404 405 405 406 407 408 408 408 409 411 413

Ch. 18 PROPAGATION AND PERCOLATION IN A CHANNEL 1. SOLITONS IN LIQUID CRYSTALS 1.1. Water waves to nerve impulses 1.2. Korteweg-deVries equation 1.3. Nonlinear Schrödinger equation 1.4. The sine-Gordon equation 1.5. Three-dimensional solitons

415 415 416 417 418 419 420

CONTENTS 1.6. Localized instabilities in nematic liquid crystals 1.7. Electric-field-induced solitons 1.8. Solitons in smectic liquid crystals 2. SELF-ORGANIZED WAVES 2.1. The broken symmetries of life 2.2. Autowaves 2.3. Catastrophe theory model based on a ferroelectric channe 2.4. The action potential as a polarization soliton 3. BILAYER AND CHANNELS FORM A HOST–GUEST PHASE 3.1. Protein distribution by molecular shape 3.2. Flexoelectric responses in hair cells 4. PERCOLATION THEORY 4.1. Cutting bonds 4.2. Site percolation and bond percolation 4.3. Two conductors 4.4. Directed percolation 4.5. Percolation in ion channels 5. MOVEMENT OF IONS THROUGH LIQUID CRYSTALS 5.1. Chiral smectic C elastomers 5.2. Metallomesogens 5.3. Ionomers 5.4. Protons, H bonds and cooperative phenomena Ch. 19 SCREWS AND HELICES 1. THE SCREW-HELICAL GATING HYPOTHESIS 2. ORDER AND ION CHANNELS 2.1. Threshold responses in biological membranes 2.2. Mean field theories of excitable membranes 2.3. Constant phase capacitance obeys a power law 2.4. The open channel is an open system 2.5. Self-similarity in currents through ion channels 3. FERROELECTRIC BEHAVIOR IN MODEL SYSTEMS 3.1. Ferroelectricity in Langmuir-Blodgett films 3.2. Observations in bacteriorhodopsin 3.3. Ferroelectricity in microtubules 4. SIZING UP THE CHANNEL MOLECULE 4.1. The size problem in crystalline ferroelectrics 4.2. Size is a parameter 5. THE DIPOLAR ALPHA HELIX 5.1. Structure of the  helix 5.2. Helix–coil transition 5.3. Dipole moment of the  helix 5.4. -Helix solitons in protein 5.5. Temperature effects in Davydov solitons

xix 421 422 422 422 422 424 425 427 429 429 430 430 431 433 434 435 436 437 437 438 439 439 443 443 445 445 446 447 447 447 448 448 449 450 451 452 452 453 453 454 454 454 457

xx

CONTENTS 6. ALPHA HELICES IN VOLTAGE-SENSITIVE ION CHANNELS 6.1. The -helical framework of ion channels 6.2. Channel gating as a transition in an  helix 6.3. Water in the channel––again?

459 459 460

Ch. 20 VOLTAGE-INDUCED GATING OF ION CHANNELS 1. ION CHANNEL: A FERROELECTRIC LIQUID CRYSTAL? 1.1. Electroelastic model of channel gating 1.2. Cole-Cole curves in a ferroelectric liquid crystal 1.3 A voltage-sensitive transition in a liquid crystal 2. ELECTRIC CONDUCTION ALONG THE ALPHA HELIX 2.1. Electron transfer by solitons 2.2. Proton conduction in hydrogen-bonded networks 2.3. Dynamics of the alpha helix 3. ION EXCHANGE MODEL OF CONDUCTION 3.1. Expansion of H bonds and ion replacement 3.2. Can sodium ions travel across an alpha helix? 3.3. Relay mechanism 3.4. Metal ions can replace protons in H bonds of ion channels 4. GATELESS GATING 4.1. How does a depolarization change an ion conductance? 4.2. Enzymatic dehydration of ions 4.3. Hopping conduction 5. INACTIVATION AND RESTORATION OF EXCITABILITY 5.1. Inactivation as a surface interaction 5.2. Restoration of excitability

465 465 465 466 467 468 468 469 469 470 471 471 472

Ch. 21 BRANCHING OUT 1. FERROELECTRIC LIQUID CRYSTALS WITH AMINO ACIDS 1.1. Amino acids with branched sidechains 1.2. Relaxation of linear electroclinic coupling 1.3. Electrical switching near the SmA*–SmC* phase transition 1.4. Two-dimensional smectic C* films 2. FORCES BETWEEN CHARGED RESIDUES WIDEN H BONDS 2.1. Electrostatics and the stability of S4 segments 2.2. Changes in bond length and ion percolation 2.3. Replacement of charged residues with neutrals 3. MICROSCOPIC CHANNEL FUNCTION 3.1. Tilted segments in voltage-sensitive channels 3.2. Segment tilt and channel activation 3.3. Chirality and bend 4. CRITICAL ROLES OF PROLINE AND BRANCHED SIDECHAINS 4.1. The role of proline

461

475 477 477 477 478 478 479 480 483 484 484 486 486 487 488 489 491 492 492 492 494 495 496 496

CONTENTS 4.2. The role of branched nonpolar amino acids 4.3. Substitution leads to loss of voltage sensitivity 4.4. Whole channel experiments 5. NEW DATA NEW MODELS

xxi 497 498 500

5.1. Amino acids dissociate from the helix 5.2. A twisted pathway in a resting channel 5.3. A prokaryotic voltage-sensitive sodium channel 5.4. Interactions with bilayer charges

501 503 503 503

6. TOWARD A THEORY OF VOLTAGE-SENSITIVE ION CHANNELS 6.1. The hierarchy of excitability 6.2. Block polymers 6.3. Coupling the S4 segments to the electric field 6.4. A new picture is emerging

504 505 506 506 506

INDEX

509

PREFACE The goal of this book is to explore the complexity of a microscopic bit of matter that exists in a myriad of copies within our bodies, the voltage-sensitive ion channel. We seek to investigate the way in which these macromolecules make it possible for the long fibers of our nerve and muscle cells to conduct impulses. These integral components of cell membranes are marvels of nature's evolutionary adaptation. To understand them we must probe the boundaries of physics and chemistry. Since function is intimately related to structure, we examine the molecular structure of channels, focusing on physical principles that govern all matter. With the application of genetic methods, our knowledge of ion channels has broadened and deepened. In the hope that research can help ameliorate suffering, we discuss the diseases that arise from channel malfunctions due to genetic mutations. This book is intended for students and scientists who are willing to travel into uncharted waters of an interdisciplinary science. We approach the subject of voltagesensitive ion channels from various points of view. This book seeks to give voice to the viewpoints of the physical and the biological scientist, and to bridge gaps in terminology and background. Readers may find this book to have both elementary and advanced aspects: For the reader trained in the biological sciences, it reviews background in physics and chemistry; for the reader trained in the physical sciences, it reviews background in physiology and biochemistry. Beyond the introductory chapters, we follow up concepts that may be as new and challenging to you, the reader, as at first they were to me. Ten years or so ago at a Biophysical Society meeting I was talking to a fellow channel scientist, one considerably younger than I. I happened to mention that, in my opinion, voltage-sensitive ion channels will eventually have to be investigated by quantum mechanical methods. “It’ll take a hundred years before that happens,” was his response before dashing off. This book is, in a sense, directed to that scientist. He and I are older now, and while I have learned that many things take longer than we expect, I would like him to consider that some things may take less long. While his estimate may well be right for a completely worked out solution to the problem of molecular excitability, there is no better time to begin working toward that goal than now. This book refers to results condensed-state physicists have obtained in materials that exhibit structural and behavioral properties similar to those of membranes containing voltage-sensitive ion channels. I hope that this book, by bringing together molecular excitability and condensed state physics, will confirm that biology and physics are parts of the same world. For this work I am indebted to many people. At UCLA, my professors Robert Finkelstein, David Saxon, Marcel Verzeano and Jean Bath stand out. James Swihart, my graduate adviser at the Indiana University Physics Department, taught me to sail the choppy seas of research; while in Europe, he discussed my thesis with Alan Hodgkin. Other influential professors at Indiana included Alfred Strickholm, Ludvik Bass (then a visiting professor from the University of Queensland, Australia) and

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Walter J. Moore. Helpful during my postdoctoral work at the New York University Physics Department were Morris Shamos, Robert Rinaldi, Abraham Liboff and Charles Swenberg, as well as Rodolfo Llinas and Charles Nicholson at the New York University Medical Center. By convincing me that classical electrodiffusion is inadequate as a mathematical model of excitable membrane currents, Fred Dodge and James Cooley prodded me into looking for the reason for that inadequacy. Harvey Fishman was my mentor and collaborator at the University of Texas Medical Branch in Galveston and the Marine Biological Laboratory at Woods Hole; he remains my friend. At Woods Hole I met and was inspired by Kenneth S. “Casey” Cole. At Texas Southern University, Floyd Banks, Sunday Fadulu, Debabrata Ghosh, Oscar Criner and Mahmoud Saleh were research collaborators. Discussions with Fred Cummings, Rita Guttman, Lee Moore, Tobias “Toby” Schwartz, Gabor Szabo, David Landowne, Malcolm Brodwick, Susan Hamilton, Arthur “Buzz” Brown, Richard “Spike” Horn, Tony Lacerda, Sidney Lang, Georg Zundel and others helped keep me focused. Donald Chang was instrumental in turning my focus from the membrane to the channel. Ichiji Tasaki has been a friend and colleague. They, together with William J. Adelman Jr., collaborated with me in organizing a conference and editing a book on structure and function in excitable cells, a precursor to this volume. Stewart Kurtz, Robert Newnham and other members of the Materials Research Laboratory of Pennsylvania State University provided valuable insights into ferroelectricity. I was fortunate in meeting Vladimir Fridkin, as our discussions have been fruitful. Vladimir Bystrov, my collaborator and friend, has applied his knowledge of physics and his boundless energy to research, writing, translating and organizing conferences. Hervé Duclohier invited me to his lab to put predictions of my channel model to an experimental test with his collaborators. Said Bendahhou and his colleagues extended the test from parts of channels to whole channels. Michael Green, a friend and colleague, and Fishman have read parts of this book and provided valuable criticism; any remaining errors are of course my own. I thank the many scientists on whose work I have depended, both those I have cited and—with sincere apologies—those I have not. My special gratitude goes to the authors whose illustrations provide figures in this volume, as well as to the permissions staffs of publishing houses and the Copyright Clearance Center. Jane Richardson kindly provided me with an updated version of a figure. The librarians who supplied me with research materials, particularly at the Butt–Holdsworth Memorial Library in Kerrville, Texas, deserve special mention. I appreciate the skill, patience and thoroughness of Springer editors Peter Butler, Tanja van Gaans and André Tournois, and typesetters Bhawna Narang and Nidhi Waddon. My wife and intellectual companion, Alice Leuchtag, has been a constant source of support and encouragement throughout the writing of this book. It is my hope that scientists will maintain an awareness of the outcomes of their research, applying science only to the building of a more just and peaceful world, in harmony with our planet. H. R. L.

CHAPTER 1

EXPLORING EXCITABILITY

Voltage-sensitive ion channels are macromolecules that act as electrical components in the membranes of living organisms. While we know that these molecules carry out important physiological functions in many different types of cells, scientists first became aware of them in the study of the impulses that carry information along nerve and muscle fibers. 1. NERVE IMPULSES AND THE BRAIN Our species, Homo sapiens, is unique among animals in its abilities to manipulate symbols, having developed languages and conceptualized space, time, matter, life, ethics and our place in the universe. These abilities are localized in the brain, about 1.4 kilograms of pink-gray organ. The complexity of the brain extends from macroscopic to microscopic—from its highly convoluted surface, through a labyrinth of lobes, tracts, nuclei and other anatomic structures, through a dense tissue of interconnected cells, through a rich mosaic of membranes, to the large molecules that make up those membranes and the membrane-spanning helical strands within them. It is remarkable that, despite the vast differences in human behavior from even that of our closest primate relatives, the molecular structures in our brains differ only in minor details from those of other mammals. Even more remarkable is the fact that such seemingly primitive forms as bacteria possess complex molecules that are shedding light on the details of related molecules in our brains. 1.1. Molecular excitability The human body, like the bodies of other living organisms, is a tumult of electrical activity. Just as an electrocardiogram shows us that the heart is a powerful generator of electric currents emanating from the coordinated action of its nerve and muscle cells, so an electroencephalogram demonstrates that the brain likewise generates electricity. The cells of the heart, brain and other organs produce electric currents in the form of transient ion flows across the membranes that cover them. The membranes are mosaic sheets of lipid and protein molecules. While the lipids form effective electrical

1

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insulators, proteins of a particular class are capable of dispatching pulses of rapid ion conduction. These switchable protein macromolecules are called ion channels.1 Ion channels of one type, ligand-gated ion channels, recognize and react to specific molecules in their environment. When these ligand molecules attach, the ion channel changes its conformation and starts (or stops) conducting ions. Examples of ligand-gated ion channels include receptors for tastes and odors, and the macromolecules that receive elementary messages from other cells in the form of chemical messenger molecules. Among these we find hormones, such as thyroxine and insulin, and neurotransmitters, such as dopamine and acetylcholine. Ion channels of another type switch their conductivity in response to a change of the voltage across the membrane. These channels make it possible for impulses to travel along nerve and muscle fibers; it is to these voltage-sensitive ion channels that this book is devoted. Hybrid channels exhibit both voltage and ligand sensitivity. The problem of the way ion channels respond to changes in membrane voltage—the problem of molecular excitability—has not been solved, although much progress has been made in this direction. This book will report on the background, history and ongoing efforts being made toward a solution of this problem. We will approach the problem from different directions, concerning ourselves not only with recent results, but also with earlier data and concepts. 1.2. Point-to-point communication To make large, multicellular organisms, evolution has had to solve the important problem of communication within the body. For stationary organisms such as plants, that problem was essentially solved by sending signal molecules, hormones, in the fluids that move up and down the body. Hormones also play an important part in communication within animals, but the amount and speed of the information that can be sent by this endocrine system is limited in specificity by the number of different hormones that can be synthesized and recognized, and in speed by the circulatory system that transports them. To generate a system of communication capable of controlling the muscles of the body, producing visual images and other sophisticated tasks in fast-moving organisms, the “blind watchmaker,” evolution, had to do better.2 A rapid point-to-point communication system was required. The solution, which appeared early during the evolution of such invertebrates as jellyfish, was for certain specialized cells to grow fibers of great length and to send waves of electrical and chemical energy along them. These nerve impulses are complex examples of solitary waves. They travel along a vast network of nerve fibers, the nervous system. Data about the external environment and the status of the body are fed into this point-to-point communications network from sense receptors. In vertebrates, the sense data are processed into responses and memories in the central nervous system, which consists of the brain and the spinal cord. Their outputs signal muscles to contract by way of neuromuscular junctions, and stimulate the endocrine system by activating glands. Nerve impulses are wonderful and mysterious. Every perceived sound, sight, smell and taste reaches our brains, and our consciousness, by nerve impulses. Every muscular movement, whether an eyeblink, a uterine contraction or a heartbeat, is

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controlled by nerve impulses. Even the release of chemical messengers such as adrenaline or testosterone is stimulated by nerve impulses. The nerve impulse is an integral part of what we mean by cellular excitability, the ability of living cells to respond to their environment. Nerve impulses are waves that move along axons, the long tubular fibers of neurons. One convenient way to study them is to record their electrical signatures, called action potentials. Action potentials can also be recorded from muscle, gland and other cells. Vast numbers of experiments on a great variety of animal and plant cells have been carried out by biological scientists to study the electrical responses of excitable membranes; see Figure 1.1.3 It is only by information from such experiments that our ideas regarding the underlying basis of excitability can be tested.

Figure 1.1. Dedication of a book on voltage-sensitive ion channels by Susumu Hagiwara. From Hagiwara, 1983.

1.3. Propagation of an impulse The question What is the scientific basis of excitability? has intrigued scientists for centuries. Isaac Newton evidently had a strong interest in this question as he posed these two Queries in his book Opticks4:

4

CHAPTER 1 Qu. 23. Is not Vision perform'd chiefly by the Vibrations of [the Aether], excited in the bottom of the Eye by the Rays of Light, and propagated through the solid, pellucid and uniform Capillamenta of the optick Nerves into the place of Sensation? And is not Hearing perform'd by the Vibrations either of this or some other Medium, excited in the auditory Nerves by the Tremors of the Air, and propagated by the solid, pellucid and uniform Capillamenta of those Nerves into the place of Sensation? And so of the other Senses. Qu. 24. Is not Animal Motion perform'd by the Vibrations of this Medium, excited in the Brain by the power of the Will, and propagated from thence through the solid, pellucid and uniform Capillamenta of the Nerves into the Muscles, for contracting and dilating them? ...

Much has been learned since Newton's time (including the facts that the ether doesn't exist and that nerve impulses can make muscles contract but not dilate), yet the core of the question of excitability remains unresolved; a fundamental understanding of the molecular basis of the action potential still eludes us. The neuron's long nerve fiber, the axon, is a long cylinder, which in some neurons of vertebrates has a fatty covering of myelin over it. The myelin sheath speeds up the action potential. For simplicity let us begin by considering an unmyelinated axon. The axon is bounded by a membrane called the axolemma, and it contains a watery, fibrous gel called the axoplasm. The axon is bathed in a body fluid that is essentially blood plasma, an aqueous solution rich in sodium ions, like seawater. The axoplasm has a much lower concentration of sodium ions, but a much higher concentration of potassium ions than the exterior solution. The anions, cations and neutral molecules present in the two solutions are distributed so as to make the solutions electrically neutral and at the same osmotic pressure. The sodium and potassium concentration differences represent two independent sources of energy. A voltage measurement can tell us that a healthy axon, ready for a nerve impulse, is electrically charged. As a result of surface charges on the membrane, the axoplasm is negative relative to the external solution. The internal potential relative to the external solution in the inactive cell is called the resting potential. In a typical nerve axon, the potential of the axoplasm relative to the external medium (which serves as ground potential) is about -70 mV. Sending a message requires a source of energy. Supplying energy only at the transmitting point would be inadequate, because the message would then diminish and be lost in the background noise. Energy sources therefore must be distributed all along the communication line. In this way, as in a line of carefully placed upright dominoes, there is no limit to the length of the path. However, a line of dominoes can send only one message—until energy is provided to set the dominoes up again. Thus for ongoing communication two sources of energy are needed: one to transmit the message (an impulse sufficient to knock the dominoes down) and another to restore the metastable order of the system (work to set them up again). Such a strategy is used in the body to propagate nerve and muscle impulses. The nerve or muscle fiber is maintained in a high-energy state far from thermal equilibrium, known as the resting state. This term is a misnomer because the “resting” membrane is highly charged by a strong electric field. The resting voltage across a nerve membrane is usually about 70 mV, with the inside negative. Combining this with

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5

the membrane thickness L of about 5 nm (1 nm = 10-9 m = 10 '), we see that the average resting electric field, E = V/L, is of the order of 107 V/m, a very high field. So the “rest” of a resting membrane is a tense one indeed! In this state, only a small stimulus is needed to initiate a wave in which the fiber rapidly falls to a state of lower energy. Part of the energy made available in this process must be passed along to a neighboring section to carry the wave on. This is the fast system, often called the sodium system for the current of sodium often ions involved in it. After the impulse has passed, the high-energy resting state—perhaps better called the excitable state—is restored to ready the system for the next impulse. This is accomplished by the slow or delayed system, often called the potassium system. 1.4. Sodium and potassium channels The high concentration of sodium ions on the outside relative to that inside the cell would tend to drive them in. In addition, their positive charge attracts them toward the negative interior of the axon. For these two reasons, the external sodium ions are at a high electrochemical potential energy relative to the axoplasm, which would drive them across the membrane through any available pathway. Macromolecules called sodium channels within the axonal membrane provide such pathways under certain conditions. When these conditions are met, the channels are said to be open; otherwise, they are closed. The terms “open” and “closed” are convenient labels but, as we shall see in the following chapters, should not be taken too literally. The voltage-sensitive sodium channels are pathways for sodium ions only when the membrane is partially depolarized. It takes only a rather modest depolarization (lowering of the absolute value of the voltage from resting) to reach the threshold at which the probability becomes high for the sodium channel to remodel itself into a different configuration, in which it becomes a selective ion conductor. Not all the sodium channels in an axon open to allow sodium ions to enter the axoplasm, however, and within a brief period of about 0.7 s most of them close again, even while the depolarization is maintained. Now the sodium system is said to be inactivated. Restoring the excitable state—setting the dominoes upright again—is the job of the potassium channels. Like the sodium channels, these are glycoprotein molecules embedded in the fatty membrane. The probability that the axonal potassium channels will open increases upon depolarization, but only after a brief delay. We emphasize an important point: The opening and closing of voltagesensitive ion channels is not rigidly controlled by the membrane voltage. These are stochastic events, so that only their probability is voltage-dependent, as we will explore in Chapter 11. 1.5. The action potential Now we can begin to see how a nerve impulse travels: Suppose a group of sodium channels in a region of an axon were to open. Then external sodium ions there would quickly enter the axon, pushed by the concentration difference and pulled by the

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electrostatic force. As they carry their positive charges into the axoplasm, they drive the internal voltage toward zero and beyond, to positivity. As this depolarization spreads out within a local region surrounding the group of channels, neighboring Na channels sense it, respond and stochastically open, carrying the action forward. Like the line of dominoes, the array of sodium channels exists in a metastable situation; its destabilization spreads by local interactions. Thus the signal is carried from channel to channel and down the axon to its terminal. Because of inactivation, sodium channels close after a brief opening. After a delay the potassium ions flow outward, driven by their electrochemical potential difference. Because the K+ concentration is higher inside the cell, this current is oppositely directed to that of the sodium ions. The outward current restores the resting potential difference. It will take a little longer for that patch of axon to become excitable again; this refractory period is due to inactivation. The voltage-sensitive channels are restored to their excitable configurations by a shift of their molecular configurations, and the axon is ready to conduct another impulse.

Figure 1.2. The action potential rises from the level of the resting potential to a positive peak, then drops at a slower rate to the resting potential. It may “undershoot” the resting level and approach it from below. The time marker, 500 Hz, shows that the action potential is complete in about 2 ms. This figure, published by Hodgkin and Huxley in 1939, is one of the first pictures of a complete action potential. From Smith, 1996. Reprinted by permission from MacMillan Publishers Ltd: Nature 144:710-711 copyright 1939.

An action potential, then, is a traveling electric wave normally initiated by a threshold depolarization, a sufficiently large lessening of the resting potential. (Alternately, it may be initiated by heating or injuring the axon or muscle fiber.) The entire action potential at a given point is completed in two to three milliseconds; it

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7

propagates along the axon, which may be as short as a millimeter or as long, for example, as a giraffe's leg. Figure 1.2 shows the time course of an action potential.5 The action potential is not a localized phenomenon. As the inward current flows, the ions spread in both directions, depolarizing adjacent regions. This activates neighboring sodium channels, moving the impulse ahead. If the depolarization has been applied, experimentally, to an excitable region of an excised axon, the action potential may start off in either direction, depending on electrode placement. However, once the impulse has started, it will only continue in the forward direction, since the sodium channels in the backward direction are inactivated. In the living organism the anatomy of the cell ensures that the impulse travels in only one direction, from cell body to axon terminal. We have seen that a useful way to look at a nerve axon is that it is a system of metastable units extended along a line, like a row of dominoes. The only thing that keeps the sodium ions from flowing in and the potassium ions from flowing out until electrical and diffusional equilibrium is attained is the impermeability of the membrane. Any breach in that impermeability will initiate an ion current. Evolution has found a way to harness that metastability by breaking the membrane's impermeability with two separate sets of molecules, permeable to different ions, thereby creating an efficient and adaptable system of information transfer. One type of ion channel is necessary to permit the signaling current to flow, and another to carry an opposing current to restore the membrane to its excitable condition. In many nerve and muscle membranes, the sodium channel plays the first, and the potassium channel the second role. This is not always the case; for example, calcium channels take the place of sodium channels at the axon terminal; the calcium ions they import into the cell trigger transmission of the signal across the synapse. Necessary as well is the energy-requiring job of maintaining the different ion concentrations inside and outside the cell; this job is carried out by metabolically driven membrane molecules called ion pumps. This brief (and incomplete) description shows us in general terms how an action potential works and what a voltage-sensitive ion channel does. What is missing from this simple picture is an understanding of the way the ion channels themselves work. That is the riddle of molecular excitability. Here begins the trail that we will seek to follow in this book. 1.6. What is a voltage-sensitive ion channel? The electric currents that are measured in experiments on axons are due to the movement of positive ions across the axolemma. The major part of the membrane area is impermeable to ions; it is occupied by a double layer of lipid molecules. Lipids are amphiphilic molecules, arranged with their polar, hydrophilic heads facing outward to the aqueous phases. Because the inner regions of the membranes are composed of the hydrophobic tails, ions lack the energy to enter, much less traverse them. It is by way of the ion channels, glycoprotein molecules that extend through the lipid bilayer, that ions may, under certain conditions, pass. The relationship between the bilayer and the protein molecules intrinsically embedded within it has been explored by the

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freeze–fracture technique; see Figure 1.3.6 The carbohydrate chains of the glycoproteins are seen extending outward into the extracellular phase.

Figure 1.3. Schematic sketch of a cell membrane, showing the relation of intrinsic protein molecules to the lipid bilayer. From C. U. M. Smith, 1996, after B. Safir, 1975.

Among the various types of membrane proteins we shall focus on the ones directly involved in excitability. We have already mentioned the sodium channel and the calcium channel, rapidly switching conductors of their respective ions, and the slower potassium channel. These glycoprotein molecules are called voltage-sensitive (or voltage-dependent)7 ion channels, because it is the voltage across the membrane that controls their ion conductance. Because the ion concentrations inside the axon are different from those outside, the concentration differences act, along with the potential difference, to move the ions. The voltage plays two roles: Its decrease impels a change in the conformation of the molecules in their ionic environment, and it helps to drive the ions across. In later chapters of this book we will review how these channels behave in various circumstances, that is, their function, and how they are put together, their structure. We will seek to answer the questions: ! ! !

How do the ions pass so rapidly through the voltage-sensitive ion channel? How does the channel manage to select specific types of ions to carry? What transformations does the conformation of the channel undergo that

EXPLORING EXCITABILITY ! !

9

convert it from nonconducting to conducting and back? How are the opening and closing transformations coupled to the electric field? How does the structure of channels determine their function?

These are difficult questions and, although various models have been proposed, the full answers to them are not yet known. We can expect the answers to be rather subtle, and that it will require a great deal of fundamental knowledge to understand them. For this reason let us now take a brief tour through some aspects of the sciences of physics, chemistry and biology and their interdisciplinary combinations. 2. SEAMLESS NATURE, FRAGMENTED SCIENCE One of the fundamental tenets of science is that nature is a seamless unity. Yet a survey of science as it is actually carried on shows that, in practice, science is divided into disciplines represented by departments with little communication between them. This division into physics, chemistry, biology and other branches, historically necessary though it was, has resulted in a fragmented science. 2.1. Physics Physics is a set of general concepts that deal with such concepts as space, time, force, motion, electricity, magnetism, sound, light and the fundamental structure of matter. These concepts are as important to living as to nonliving things, to “the trees and the stones and the fish in the tide.”8 Newton's mechanics is the flagship theory of classical physics. Classical mechanics allows us to isolate a problem from its environment. Newton's three laws are sufficient for many applications but fail in two realms: the fast-moving and the microscopic. The two revolutions that dealt with these realms are relativity and quantum mechanics. In solving a mechanical problem, the direct application of Newton's laws is usually not the easiest way to proceed. Instead of analyzing forces, the concept of energy gives us a more convenient approach, because of the important law that energy is conserved. The concept of energy conservation extends far beyond mechanics, because energy takes many forms, including heat, electrical, magnetic, elastic and chemical—even mass, as relativity shows, is a form of energy. Energy is not necessarily associated only with particles, but can be found in space, in the form of fields—electric, magnetic and gravitational. One branch of physics directly pertinent to voltage-sensitive ion channels is electrodynamics, which deals with electricity and magnetism. While mechanics describes a world of three independent dimensions, length, time and mass, nature provides another dimension: electric charge. This dimension adds some interesting phenomena: Resting charges produce electrostatic attractions and repulsions; when charges move, they also produce magnetic fields, perpendicular to the velocity or current. Electric and magnetic fields in space produce electromagnetic waves. Thus

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electrodynamics includes optics, a subject that deals not only with light, but with the entire electromagnetic spectrum, from gamma rays to radio waves. Mechanics and electrodynamics contain the basic laws governing the behavior of individual particles and their interactions with each other, but because matter consists of enormously large numbers of particles, new behavior emerges from their aggregations. Laws such as the gas laws describe matter in bulk. Studies of heat engines led to the concept of entropy, a measure of the disorder of a system, and to the formulation of the laws of thermodynamics. The kinetic theory of gases uses statistical techniques to sum the effects of many individual collisions into statements about pressure, volume and temperature. An ion channel is composed of tens of thousands of atoms; the number of electrons runs into the millions. How can we expect to see any sort of ordered behavior from such a vast assemblage of particles? Since the channel does exhibit regular responses to electric field changes, the particles must clearly be acting together in some sort of collective behavior. The branch of physics and chemistry that has been developed to apply the laws of mechanics to large numbers of atoms and molecules is statistical mechanics. Statistical mechanics allows us to understand phase transitions, collective changes in molecular ordering such as the melting of ice. The loss, when heated, of magnetic polarization in a ferromagnet and of electric polarization in a ferroelectric material are other examples of phase transitions. A closed system is one in which neither energy nor matter is allowed to enter or leave. Such a system is subject to the first law of thermodynamics, that the energy of the system remains constant, and the second law, that the entropy of the system can not decrease. These laws cannot be simply applied to open systems, which exchange energy and matter with their environment. Living organisms are open systems. The growth of a tree from a seed is an example of a system in which entropy decreases. However, if the tree together with the air, water and soil surrounding it and the light source illuminating it are isolated, the entropy of the entire closed system will increase. The decrease of the tree's entropy is more than compensated by the increase in entropy of the other components of the system. The quantum revolution originated in some seemingly minor phenomena that could not be explained by classical physics. One of these was the distribution of wavelengths of light given off by heated bodies, such as the resistance wire in a toaster or light bulb. With increasing temperature a heated object glows red, then orange, then white. The frequency of the light most strongly emitted rises with temperature in a way that classical physics was unable to explain. Max Planck took the bold step of postulating that radiant energy could be given off only in discrete packets of a magnitude that was directly proportional to the frequency of the emitted light. With this unprecedented assumption he was able to obtain a perfect fit to the data for wavelength distribution of energy emitted from a black body. Planck named these packets of energy quanta, and we call his proportionality constant h Planck's constant. The energy of a quantum is E = hv, where v is the frequency of the radiant wave emitted. Planck's constant h has dimensions of action, energy times time, or momentum times displacement. Its value is 6.63 × 10-34 joule second. We shall also use the constant ħ = h/2% = 1.054 × 10-34 J s. In terms of

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the angular frequency 7 = 2%v, the energy of a quantum is written ħ

(2.1)

Planck's discovery led to the resolution of other riddles that classical physics was unable to solve, but also raised new questions. Einstein used Planck's quanta to explain the way electrons are emitted by metal surfaces illuminated by ultraviolet light, but his successful theory of the photoelectric effect required light to be composed of particles, photons. This seemed either to contradict numerous experiments showing the wave nature of light, or to require light to have a dual nature, both wave and particle. Since we cannot deny experimental data, we must accept the duality of light. To account for the stability of the atom, Niels Bohr postulated that only a discrete set of orbits could be permitted and that electrons had to jump from orbit to orbit to emit or absorb quanta of light. The energy of a photon would be Planck's constant times the difference between the frequencies of the orbits. Louis de Broglie explained the discrete patterns of the spectral lines of atoms by assuming that the electron acts as a standing wave wrapped around the nucleus. The number of nodes in the standing wave became the principal quantum number n, giving the electronic structure an energy En = nhv, where v is the orbital frequency. Erwin Schrödinger wrote the equation governing this wave. It became clear that light is not the only thing with a dual nature: Electrons, and indeed all objects, have both particle and wave properties. The predictions of quantum electrodynamics have been shown to be accurate to more than 12 significant figures. Bohr came to the realization that quantum mechanics would have profound implications for biology, requiring us to renounce a completely deterministic account of life processes in favor of a probabilistic description. These concerns were later taken up by Schrödinger, Max Delbrück and others.9 2.2. Chemistry Chemistry is the study of matter and its interactions. It emerged from its alchemical beginnings when it began to study the qualitative and quantitative properties of pure substances, which could be separated into elements and compounds. Compounds can be broken down into their elements, which they contain in definite proportions by weight. These facts validated the Greek concept of atoms, which may be seen as bonding together to form molecules. One of the major goals of chemistry was to understand the nature of the bonds that connect atoms into molecules. Early pictures of this chemical bond showed atoms as balls with hooks that could engage one another. The pictures served to express in symbolic language the idea of the connection of two atoms to each other. The hook picture was discarded when the chemical bond was finally understood in the 1920s as a consequence of electromagnetism and quantum mechanics. The study of reactions among elements, with analyses of weights, volumes, temperatures and other quantities, revealed remarkable periodicities, the exploration of which led to the Periodic Table. A proper alignment of the columns of the Table

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required the concept of atomic number, which turned out to be the number of protons in the atom's nucleus. The number of neutrons varies; isotopes have the same atomic number but a different atomic mass number, the number of protons and neutrons. The rows are numbered by periods, or principal quantum numbers, while the columns, numbered by Roman numerals, indicate the number of electrons in the outer, valence, shell. The atomic weight is the average mass number of the isotopes present, weighted by their relative abundance on Earth. Molecular weight is the sum of the atomic weights in a molecule. Its unit is the dalton (Da); for macromolecules, the kilodalton is a more practical unit. For example, the molecular weight of the blood protein hemoglobin is 64.65 kDa and that of the sodium channel about 250 kDa. In the study of ion channels we will focus on the atoms that form the structures of living organisms and the metal ions that course through the channels. The element carbon is the core component of living matter. The importance of carbon derives from its four valence electrons, which form a tetrahedral bond pattern. The boundaries between organic chemistry—the study of complex carbon-containing molecules—and inorganic chemistry are not sharp, and the study of ion channels requires attention to both. Other molecules prevalent in living organisms are hydrogen, oxygen, nitrogen, phosphorus and sulfur. These form the compounds that form the bodies of organisms and underlie their metabolism, reproduction and other life activities. Still other elements, including calcium, potassium, sodium, chlorine, magnesium, iron, iodine, cobalt, zinc, molybdenum, copper, manganese and selenium, are involved in such vital protein activities as oxygen transport, information processing and enzymatic reactions. Alkali metals, the elements in Group I, are important as ions: starting from the top of the Table, we have lithium, sodium, potassium, rubidium, cesium and francium. Above them lies hydrogen, which has unique properties. Sodium, in the form of ions, is highly concentrated in seawater and blood. As we saw, it plays an important role in nerve conduction, mediated by sodium channels, which preferentially but not exclusively conduct sodium and lithium ions. Potassium is a larger, heavier atom than sodium, with chemical properties similar to those of sodium but sufficiently different that the two elements can be easily separated. Potassium channels play important physiological roles including the restoration of the resting state by the outward current. The alkaline earths of Group II include beryllium, magnesium, calcium and strontium. Calcium and magnesium both have important physiological functions, including their role in ion channels, particularly the ubiquitous calcium channels. The nonmetals in Group VII, the halogens fluorine, chlorine, bromine and iodine, form negative ions. Chlorine plays the role of counterion to cations such as sodium, potassium and calcium. Chloride channels, like potassium channels, help to restore the resting potential after a depolarization. The rare gases, helium, neon, etc., have no direct relevance to living organisms because of their inertness. However, their stable configuration of filled outer shells makes elements of Groups VI and VII electron acceptors, and elements of Groups I and II electron donors. The readiness of these elements to ionize, together with the electrostatic attraction of the ions, accounts for ionic bonding.

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Closed outer shells can be formed also by the sharing of electrons, which accounts for covalent bonds. The nature of these bonds was explained quantum mechanically by constructive interference of the electron waves of the valence electrons; a pair of electrons of opposite spin from the two bonded atoms is shared between them. As a result of the overlapping of orbitals, the electron density between the nuclei increases, leading to a net attraction. The carbon atom, for example, may share one, two or three pairs of electrons with another atom, thereby forming single, double or triple bonds, respectively. The forces between atoms to form molecules, and between molecules, are electrical in nature. A measure of the attraction an atom has for the electrons it shares with another atom is its electronegativity. In each row of the periodic table, electronegativity increases from left to right; the alkali metals are the most electropositive, followed by the alkaline earths. Within any column, electronegativity decreases downward, with increasing atomic number. Covalent bonds are formed between atoms of equal or nearly equal electronegativities. Unequal values of electronegativity lead to ionic bonds. Covalent bonds with unequally shared electrons are said to have a partial ionic character, producing a polar covalent bond. This type of bond is seen in the water molecule, H2O, in which the oxygen atom, with eight protons, attracts electrons more strongly than the lone proton of the hydrogens. This leaves the oxygen atom with a partial negative charge and the hydrogens with partial positive charges. As a consequence, water has a high electric dipole moment, which accounts for its many unusual properties. Water is a good solvent for ionic compounds, but nonpolar compounds, such as oils, do not dissolve in water; they are hydrophobic. The electrostatic repulsion between covalent bonds determines the structure of the molecule formed by them. Thus the methane molecule, CH4, in which each of the four valence electrons of the carbon atom pairs covalently with the single electron of a hydrogen atom, forms a tetrahedron; each C-H bond angle is 109.5°. A third important type of bond is the hydrogen bond, a relatively small attraction between a hydrogen atom and an electronegative atom such as oxygen or nitrogen. The hydrogen bond is weak but when present in large numbers can determine the structure of a molecule. Water, the compound essential to terrestrial life, is highly polar because of the strong electronegativity of the oxygen atom; hydrogen bonds of type OH###O are prevalent in liquid water and ice. Of primary importance is the dipole moment of the water molecule and the hydrogen bonds linking adjacent molecules together. Strong electrolytes in water solution break up into ions, which become hydrated with a water shell. Living organisms have evolved in water, and organisms that live on land retain water within their cells and body fluids. While the properties of water are extremely important to the processes of living, they do not necessarily determine the properties of the membranes that separate aqueous compartments. Chemical reactions are reversible, although in practice the backward reaction may be too small to be detectible. In a redox reaction, one molecule is oxidized, losing electrons, while another is reduced, gaining them. The speeds of reactions are greatly increased by catalysts, substances that work by contact or by intermediate reactions. Practically all biological catalysts, enzymes, are proteins, although a class of nucleic acids, RNA, also has enzymatic properties.

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The large and complex molecules of living organisms are polymers of functional groups, such as methyl (-CH3), amino (-NH3), carboxyl (-COOH), sulfhydryl (-SH) and phosphate (-PO4). The polymers may have linear or branched structures. Four major types of biochemical compounds have evolved: Carbohydrates, sugars and their polymers; lipids, including oils, fats and waxes; nucleic acids, such as deoxyribonucleic acid (DNA), ribonucleic acid (RNA) and adenosine triphosphate (ATP); and proteins, which serve as structural components, enzymes, or membrane proteins—such as ion channels. 2.3. Biology Biology studies the way in which living things develop, grow, adapt, reproduce, change their environment, and die. Life is a systemically maintained nonequilibrium state. The living system can maintain itself far from equilibrium for only a limited time, returning to equilibrium in death. The ordered chemical pathways of life, its metabolism, can only exist at temperatures low enough that their functional order is not destroyed, but not so low that fluids crystallize. Life on Earth is adapted to conditions on our planet, which themselves are modulated by living processes. The maintenance of an ordered system in a far-from-equilibrium state requires a steady input of energy into the system, which therefore must be an open system. Because the second law of thermodynamics does not apply directly to open systems, the growth of order in organisms undergoing respiration in no way contradicts this law. Living systems are composed of cells, which maintain their essential components within a membrane that allows for a controlled exchange of matter and information with the environment. The functions of cells are based on the chemistry and physics of structures composed of nucleic acids, proteins, lipids and carbohydrates. Living organisms contain a genetic code chemically expressed in deoxyribonucleic acid (DNA), by which they reproduce, allowing the species to survive the death of the individual. Single-celled organisms without formed nuclei, prokaryotes, reproduce by cell division. The bacteria, the only living organisms in existence for three billion of the 3.8 billion years of life on Earth, established many of the life-favoring conditions of the Earth, including the composition of the atmosphere: photosynthetic bacteria generated the oxygen of the air. Mitochondria and chloroplasts, descendants of bacteria, live within the cells of eukaryotic organisms, which have complex cells with defined nuclei, and maintain a symbiotic relationship with them. The prokaryotic bacteria and the eukaryotic protists are single-celled; the other eukaryotes—fungi, plants and animals—are multicellular. To maintain homeostasis, all organisms, prokaryotic and eukaryotic, must exchange not only matter and energy but also information with their environment. The single cell of a unicellular organism contains all the structures necessary for independent existence: DNA; the complex machinery to divide the cell and synthesize biomolecules; organelles containing enzymes for energy conversion and other functions; receptors and effectors to interact with the environment, and many other systems to maintain a stable existence in a changing environment. Because the

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surface-to-volume ratio shrinks as size increases, and enough surface area is required for the efficient exchange of nutrients and waste products with the environment, cells are necessarily small. This limitation was overcome in multicellular organisms, but at a price: New mechanisms were required for communication within the body in order to maintain homeostasis. In plants, a system of hormones suffices. In animals, the demands of mobility requires, in addition to the endocrine system, a system of point-topoint communication—the nervous system. The study of structure, anatomy, is a prerequisite to the search for explanation of the functions of the parts of the body, physiology. Clearly, physiology also requires a knowledge of physics—indeed, the two names are derived from the same root. General physiology seeks to elucidate the basic principles underlying the behavior of many biological systems. One of these pervasive phenomena is cellular excitability, the ability of cells to respond to stimuli. The detailed examination of conduction in nerve cells and associated cells is the task of neurophysiology. Neurobiology traces is history far back to the ancient Egyptians and Incas, who carried out brain operations on humans, as shown by archeological studies of skulls and documented in Egyptian papyri; knowledge of the electrical nature of nerve and muscle phenomena is much more recent. Charles Darwin, on a trip around the world as scientific officer of the Beagle, saw finchlike birds that differ from finches in their beaks, claws and lifestyle; tortoises in the Galapagos; octopi changing color. A vast array of species—why so many? Where do they all come from? They appear to be related.10 Organisms have many offspring, but many of these die before reproducing. Which ones survive? That is mostly decided by accident, but the offspring are not entirely the same. They possess a variety of traits, and the ones with favorable traits—faster, stronger, more adaptable or more resilient—have a better chance to survive by escaping starvation, predation, disease or other dangers. And, surviving, to reproduce and to pass the characteristics of their heredity, the information we now call their genes, to their offspring. The variety of living species as well as the traces of bygone forms, the fossils, give us a map to past development. That the continuing development of new species has indeed been possible is due to the fact that the genetic code of DNA is, while remarkably stable, subject nevertheless to occasional chance errors induced by chemical alteration or radiation. These mutations lead to offspring that are in most cases less suited to survive and so frequently die without progeny. However, in occasional, rare, cases, a mutation favors survival. Organisms with such favorable attributes have a higher probability of surviving and passing these attributes—now known to be due to their mutated DNA—to their offspring, increasing their representation in the gene pool. Competition and isolation, along with further favorable mutations, contribute to the development of new species. If a population with a new mutation becomes isolated from its parent population, an entirely new species may arise, no longer able to breed with the parent species. That, Darwin concluded, must be the answer to the question of the variability of species. Remarkably, another man had this idea at the same time; the theory of

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evolution by organic adaptation is the product of Darwin and Alfred Russel Wallace jointly. A new science was born, but many details remained to be filled in by others. To focus on one detail, the production of new species is not, as Darwin thought, a steady process. The fossil record shows that it proceeds in spurts more accurately described as a series of punctuated equilibria, as discovered by Stephen J. Gould and Niles Eldridge.11 The ability of living organisms to give rise to new forms, such as feathers from scales, is due to chance alterations of the DNA molecule, coupled with natural selection. These mutations arise from attacks on the DNA by free radicals, causing errors to creep into the genetic code. Most mutations reduce the individual's ability to survive; some are neutral. The likelihood of favorable changes, such as those that converted scales into feathers, is minuscule in a single generation, but accumulates with time, because they provide a higher probability of survival to the individual lucky enough to inherit them. The advantages that an organism derives from a mutation can transfer: Feathers served as heat insulators before enabling flight. This benefit is enjoyed by subsequent heirs of the new gene. Fossil studies allow the process of evolution to be traced back to the first living organisms, the bacteria. In these one-celled organisms, 1-2 m in diameter, we see the first examples of metabolism, photosynthesis, locomotion and reaction to external stimuli. The biochemical and biophysical capabilities of present-day protists, fungi, animals and plants have their origins in bacterial evolution. An evolutionary “tree” can be drawn that connects living and fossil forms. In it, branches not only diverge (as when bilaterally symmetrical animals split away from those with radial symmetry), but they also merge. An interesting example of this is the formation of protists from bacteria. The protists and other eukaryotes possess organelles that are modified bacteria, with their own DNA. Strictly speaking, the principles of evolution apply only to the entire organism, since parts of organisms, such as organs and cells, must die when the organism dies. Nevertheless, it is possible to speak of the evolution of proteins. Evolutionary trees have been established for cytochrome c and many other proteins. The oxygen-carrying molecules hemoglobin and myoglobin evolved into their many present forms from the divergence of a single ancestral globin gene.12 In the same way evolutionary trees have been established for voltage-sensitive ion channels, as we will see in Chapter 13. The way in which offspring inherit specific characteristics from each of their parents was a riddle to Gregor Mendel, the founder of genetics. The properties of hybridized plants in crossing experiments showed that traits were segregated in alternate forms, now called alleles. Mendel’s quantitative investigations on plants showed the influence of two “factors,” today called genes, one from each parent. The genes combine randomly and (in most cases, as we now know) independently of other genes. Often the effect of one gene on the organism will be masked by another, dominant, gene. If the dominant allele is present, it will be expressed in the organism regardless of whether the DNA contains one or two copies of it. The recessive gene will be expressed in the phenotype—the structure, physiology and behavior of the organism—only if it is present in both genes. Mendel's rules provided statistical laws by which the probabilities of the outcomes of hybrid crossings could be calculated.

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The genes are present in the DNA of the chromosomes. During meiosis, the production of haploid gametes, the two alleles of the diploid parent cell segregate from each other; each gamete produced has the same probability of owning one or the other member of the pair of alleles. In experiments in which he crossed plants that were hybridized for two different characteristics, such as seed shape and seed color, Mendel found that the results could be interpreted by his law of independent assortment. Since the genes that led to this result were on different chromosomes, we can state that genes located on different chromosomes assort independently of each other. Genes on the same chromosome, however, do not assort independently, because of the phenomenon of crossing over, whereby the parent chromosomes swap parts of each other. Since nearby genes on a chromosome will tend to cross over together while genes farther apart will have a greater probability of becoming separated, data from cross-over experiments can be used to construct a genetic map. Such a map shows the location of the identified genes on the linear chromosome. While reproduction may be a simple cell division, it frequently requires an exchange of the genetic material, DNA, between organisms, as in sexual reproduction, the fusion of two gametes. The exchange of DNA requires complex cooperative behavior, which presupposes the exchange of information between the individuals. Social behavior has evolved as an important feature of animals. In humans, this has led to the development of formal languages, science, trade and government. The way the simple DNA molecule, with only four different nucleotide components, is able to code for polypeptide chains composed of 20 amino acids was resolved by the realization that a group of three nucleotides was required to code for one amino acid residue. This is the triplet code. X-ray diffraction studies by Rosalind Franklin in the laboratory of Maurice Wilkins, and modeling by Francis Crick and James Watson, yielded the structure of DNA: an antiparallel double helix with complementary bases paired and connected by hydrogen bonds. Separation of the two strands allows enzymes to replicate the DNA, an essential precursor to cell division. A more indirect approach is required to read the code and convert it into polypeptide chains. This process requires an intermediate nucleic acid, RNA, which serves in messenger and transfer roles and in the synthesis of the polypeptide. The code is transferred to messenger RNA by a process called transcription. The triplets, transcribed to complementary RNA, are called codons. The genetic code is a “dictionary” between the 64 codons and the 20 amino acids. The cellular process of translation builds the polypeptide chain according to the code in the messenger RNA. Genetic methods are highly effective ways of studying biological systems. A study can focus on a particular phenotype, such as a cell shape or protein structure. It is remarkable that many of the genes and proteins in neurons have already been identified in such simple organisms as the nematode Caenorhabditis elegans and the fruitfly Drosophila melanogaster. The Human Genome Project is sketching the entire gene map of humans. Ecology tells us that life is a vast multidimensional hierarchical organization, from molecule to supramolecular structure such as a membrane, to organelle, cell, tissue, organ, organ system and organism. Organisms form populations, which combine

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with populations of other species to form communities, which in turn interact with their environment to establish an ecosystem, a part of the Earth's biosphere. 3. THE INTERDISCIPLINARY CHALLENGE The above review of the elements of their sciences facts suggests that physicists, chemists and biologists traverse orbits in fairly well separated universes. A vast gulf still remains between physics and physiology. 3.1. Worlds apart A need for repeatability and precision, and a practical desire to solve the simplest problems first, caused physics to separate from its beginnings, in which time was measured by one’s pulse, temperature calibrated by human body temperature and electricity measured by the strength of shocks. While biology was exploring and describing the vast complexity of the living world, physics discovered the power of mathematics as a language to express its reasoning and laws. At least on the experimental side, biology is close to physics. It has long used physical instruments, beginning with the microscope and the centrifuge, to delve into the world of the cell. Electronic, optical, acoustical and nuclear instruments are routine sights in biological laboratories, along with isotopes, ultracold temperatures and other experimental techniques developed in physical and chemical laboratories. In the realm of theory, on the other hand, biology and physics remain worlds apart. The division of science into discrete compartments, once a convenient simplification, is now a barrier to progress. 3.2. Complex systems In recent years science has taken upon itself the task of dealing with complex systems. The laws of physics are simple, so why is the world, particularly the biological world, so complicated? In clouds, sand dunes, ocean waves and biological molecules we see a tendency of nature to form structures. But the outcomes of physical processes are also sensitively dependent on initial conditions, a dependence called chaos. Organization and chaos coexist in complex systems, which are characterized by forces between particles, conservation of quantities such as particle number and angular momentum, and symmetries. Problems become simpler when nature provides us with well separated scales of space, time or energy. In biological systems, however, such convenient scale separations are lacking; organisms are characterized by order at many levels. The computer modeling of complex systems in many cases develops into a pattern in which the behavior is dominated by abrupt jumps. We see such intermittency in the sudden onset of stormy weather, ice ages and plagues. Such emergent properties are also seen in phase transitions such as melting, boiling and sublimation.13 The competition between order and complexity may also govern biological systems, such

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as the opening and closing of ion channels.14 Chapter 15 deals with these concepts of critical phenomena as they bear upon the problems of cellular and molecular excitability. These new concepts are helping to bridge the culture gap between the physical and biological sciences. 3.3. Interdisciplinary sciences bridge the gap The insights and perspectives of either one of the classical disciplines alone will not be enough to solve the problems of molecular excitability. The emergence of interdisciplinary sciences—biophysics, biological physics, biochemistry and biophysical chemistry—has been helpful in bridging the chasm between physics and biology. New journals and new departments have appeared, providing intermediate territories. But problems remain. Different branches of science speak languages with different vocabularies and grammars and so view the world differently. Jargons separate the sciences. Scientists know that they must communicate their ideas precisely, so they must invent new words and define the meaning of existing words more precisely. In bringing together two fields of endeavor with overlapping vocabularies, we must watch for ambiguities in meaning. The process of combining the perspectives of physics and chemistry, or physics and biology, is sometimes referred to a “reduction,” and the term “reductionism” is often used in a derogatory sense. This unfortunate association hides an important fact: When it is discovered that certain chemical phenomena can be explained by quantum and statistical physics, the chemical content is not lost. There is a tightening up as pieces of the puzzle fit together, but the chemical and the physical insights are still there (to the extent that they were correct in the first place) when they are merged into a single picture. We can expect similar simplifications from the joining of physical and biological concepts. We can portray biophysics as an attempt to bridge two ways of thinking, two languages, two literatures and two styles of research. The reconciliation of these differences can be very productive, as the history of interdisciplinary science demonstrates. However, the attempt to bridge two disciplines as different as physics and biology requires us to stretch between widely separated conceptual bases. NOTES AND REFERENCES 1. Bertil Hille, Ion Channels of Excitable Membranes, Third Edition, Sinauer Associates, Sunderland, MA, 2002. 2. R. Dawkins, The Blind Watchmaker, Penguin, New York, 1988. 3. S. Hagiwara, Membrane Potential-Dependent Ion Channels in Cell Membrane: Phylogenetic and Developmental Approaches, Raven, New York, 1983. Reprinted by permission from Wolters Kluwer Health. 4. Isaac Newton, Opticks, Dover Publications, New York, 1952. 5. C. U. M. Smith, Elements of Molecular Neurobiology, Second Edition, John Wiley, Chichester, 1996, 276; A. L. Hodgkin and A. F. Huxley, Nature 144:710-711, 1939. 6. Smith, 130; B. Safir, Scientific American. 234(4):29-37, 1975.

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7. The alternative term “voltage-dependent ion channels,” although commonly used, is not precise. It is not the channels but their conformation that is dependent on voltage. These molecules are sensitive to, but not dependent on, the potential difference across the membrane. 8. From Poem in October by Dylan Thomas. 9. Erwin Schrödinger, What is Life? , Cambridge University, Cambridge, 1955; Max Delbrück, Mind from Matter? An Essay on Evolutionary Epistemology, Blackwell Scientific, Palo Alto, 1986. 10. Charles Darwin, The Origin of Species by means of Natural Selection of the Preservation of Favored Races in the Struggle for Life, Avenel Books, New York, 1979. 11. Stephen J. Gould, Wonderful Life, Norton, New York, 1989. 12. Richard E. Dickerson and Irving Geis, Hemoglobin: Structure, Evolution, and Pathology, Benjamin/Cummings, Menlo Park, California, 1983. 13. Nigel Goldenfeld and Leo P. Kadanoff, Science 284:87-89, 1999; J. A. Krumhansl, in Nonlinear Excitations in Biomolecules, edited by M. Peyrard, Springer, Berlin, and Les Editions de Physique, Les Ulis, 1995, 1-9. 14. S. A. Kauffman, The Origin of Order, Oxford University, Oxford, 1993; ___, At Home in the Universe, Oxford University, 1995.

CHAPTER 2

INFORMATION IN THE LIVING BODY

Information streams from our surroundings, enters our sense organs and converges on the brain. There it is processed, with information previously stored, to issue an outgoing stream of commands to our muscles and glands. In touch with our surroundings, we use information to carry out our life activities. To promote their survival, organisms need information: from which direction the sun is shining, where food and water can be found, how to avoid predators. The negative entropy of the organism's metabolism can be applied to make that information available. The problem, then, is to convert this information into survival-enhancing responses to environmental stimuli. This requires information processing, which occurs at all the hierachical levels of the organism. The detection of information from the outside of a cell is carried out by receptor molecules embedded in the bounding plasma membranes. The response to the data received must be by effectors that either react mechanically or emit light, electric fields or chemical substances into the environment. Mechanical reactions may propel the organism or a part of it through space or emit an acoustical wave. In bacteria, mechanical reactions include motions of flagella or cilia. Information processing also takes place at time scales much longer than the lifetime of an individual organism. The species itself adapts, by organic evolution, to long-term changes in the environment. Because of the relationships between species, cooperative as well as competitive and predational, changes in one species often lead to changes in other species. New species arise when habitats become separated. Species become extinct from loss of habitat, catastrophes or other causes. In this chapter we will examine some of the systems of biological information processing and their basis in informational macromolecules. 1. HOW BACTERIA SWIM TOWARD A FOOD SOURCE The manner in which organisms extract information from their environment and process it to promote their survival is illustrated by chemosensitivity in bacteria. Motile bacteria are sensitive to chemical substances in their environment, swimming up gradients of attractants, and down gradients of repellents. While bacteria are known to have existed for more than 2.5 billion years, this process, bacterial chemotaxis, is hardly simple. 21

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The molecular biology of bacterial transduction has been studied in the rod-shaped cell Escherichia coli, which resides in our guts.1 These bacteria are propelled by the rotation of 8-10 flagella, a filament of which consists of a single array of subunits of a protein called flagellin. The tubular array, 0.02 m in diameter by 5-20 m long, is twisted into a helix. Although rotary motion is uncommon as a physiological adaptation, the flagellum is rotated at about 100 rev/sec by a cellular mechanism energized by a transmembrane H+ gradient. When the flagella all rotate in a counterclockwise direction, the flagella form a single bundle that propels the bacterium forward smoothly. Clockwise rotation, on the other hand, pulls the flagella outward and the bacterium tumbles irregularly; see Figure 2.1.

Figure 2.1. Rotary motion in bacterial flagella. A. Flagellar motion on counterclockwise rotation. B. Flagellar motion on clockwise rotation. From C. U. M. Smith, 1996. Copyright John Wiley & Sons Limited. Reproduced with permission.

The motion of the cell resulting from alternation of the flagellar rotation is a three-dimensional random walk consisting of runs of smooth swimming interspersed with periods of chaotic tumbling. When the cell is placed into a gradient of a chemical attractant, its swimming runs are found to be longer when directed toward the source than in any other direction. Thus the bacterium homes in on a source of attraction and, conversely, away from a source of repulsion. This system features sensory adaptation: a prolonged immersion in an attractant or repellent leads to desensitization. The attractant or repellent molecules are sensed by transmembrane molecules called receptor–transducer (R-T) proteins. Depending on the particular type of chemical, the interaction is either by a direct binding or indirectly, by receptor molecules. The genes for the R-T proteins have been isolated and their DNA codes determined. From this sequence, their membrane-spanning protein segments are identified. Each R-T protein responds to a specific set of molecules. One of these, Tsr, is sensitive to the amino acids serine, an attractant, and leucine, a repellent. The site that binds the attractant or repellent molecule is in a part of the sequence that projects out of the membrane. Inside the membrane, domains to which signaling groups attach are in the cytoplasm. The arrival of a repellent or attractor molecule generates a

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chemical signal that modulates the direction of rotation of the flagellar motor, as in Figure 2.2.

Figure 2.2. Interaction of attractant molecules with receptor and receptor-transducer molecules in a bacterial membrane. From C. U. M. Smith, 1996. Copyright John Wiley & Sons Limited. Reproduced with permission.

The signaling pathway that connects the flagellum to the R-T protein has been traced by genetic analysis to involve four proteins, called CheA, CheW, CheY and CheZ. When the Tsr R-T protein accepts a serine repellent, it undergoes a conformational transition, which travels to the signaling domain, where CheW and CheA are activated. CheA accepts a phosphate group from an energy carrier, adenosine triphosphate (see Section 7.4) and passes it on to CheY, which diffuses through the cytoplasm to the flagellar motor. There the phosphorylated CheY induces clockwise rotation with subsequent cell tumbling. The role of CheZ is to desensitize the system by dephosphorylating CheY, terminating its influence.2 The binding of the attractant leucine induces a conformational change in Tsr that inactivates CheA and CheW. The flagellar motor then resumes its counterclockwise rotation and the cell swims forward. The ability of biological systems to process information was established during the evolution of the bacteria, in the first billion years after the formation of the Earth.

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The process of bacterial chemotaxis is a complete sensory–motor system in a single cell. In multicellular organisms, sensory and motor systems become separated, and in animals a central nervous system intervenes between them. It is interesting to note, however, that evolutionary traces of bacterial systems can be detected in mammalian macromolecules. 2. INFORMATION AND ENTROPY Bacterial chemotaxis is an example of the importance of information processing in the everyday (and every-instant) activities of living organisms. But what is information? If a system can occupy a discrete number N of states, we may not know which of these states it will occupy at a given time. The probability of a particular state or choice, on the assumption they are equally likely, is 1/N. For a tossed coin, N = 2, so the probability of a "tail" is ½; for a rolled die, N = 6, so the probability of a [ : ] is 1/6. The probability of random appearance of a particular character is the reciprocal of the number of choices. We can use the simplest case, N = 2, as our canonical example. If we know that the coin has landed heads up, we have one bit of information. When n coins are tossed, the number of possible outcomes is P = 2n. The information is the number of digits when written in binary. In bits, it is defined as the log to base 2 of the number of possibilities.3 (2.1) The probability that today is your birthday is, to someone who doesn't know it, 1/365, since, neglecting leap years, it is one of 365 (= 101,101,101 in binary) possibilities. If you tell me your birthday, you are giving me log2 365 = 8.51 bits of information. By the properties of the logarithm function, information relating to independent events is an additive property. This basic unit, the bit, measures information of all kinds, from the information on this page to the information transmitted by a bee in its dance. Entropy can be roughly described as the disorder to which closed systems tend; for a more precise definition see Chapter 5. Information is negative entropy. Every living organism is an island of information in a sea of growing entropy. This information is processed within the organism at the organ, tissue, cell and molecular levels. It is moved from organism to organism by means ranging from the transfer of plasmids to the publication of journal articles. Nonliving objects undergoing irreversible processes, such as marble statues in acid rain, gain entropy and lose their ordered forms. Living organisms are different; they are open systems. They assimilate energy from their environment—from photons or from food—and lose entropy in discarding waste matter and heat. During their lifetimes they grow and become highly ordered. Information transfer from the environment requires signal transduction, the conversion of the energy carried by an incoming signal (sound, light, odor molecules, etc.) to a form useful to the organism. There information is processed generally in a way to help the organism adapt by acting upon its environment in some way.

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In single-celled organisms, this information is converted fairly directly into a response, as we saw in the case of bacterial chemotaxis. In multicellular organisms, however, a great deal of coding and processing is necessary. While in plants this signaling is limited to the transport of hormones (with some exceptions, such as the sensitive plant Mimosa pudica and the Venus fly trap), in animals it involves also the propagation of nerve impulses and the release of neurotransmitters. The signal received as, for example, an image on the retina, is converted by receptor molecules into nerve impulses that travel to specific regions of the brain. It is then sent via efferent neurons to the appropriate muscular or glandular effectors. This may result in a muscular movement or secretion, requiring a transduction of the train of impulses into a contraction or biochemical synthesis. These processes thus form a sequential system, from sensory transduction to the coding and inward conduction of impulses, to processing and memory in ganglion or brain, and then outward conduction to muscle or gland. 3. INFORMATION TRANSFER AT ORGAN LEVEL While the laws that govern natural processes are assumed to be the same whether they refer to stars, horses or atomic nuclei, their application is quite different. These are all ordered structures, but the level of complexity and their environment is quite different. Living things exhibit properties not generally seen in nonliving things, such as metabolism, genetic reproduction and organic evolution. Nonliving matter, such as a crystal, can often be described by two length scales, macroscopic (crystal dimensions) and microscopic (size of the molecular unit cell). On the other hand, living matter is ordered at many hierarchical levels, each with its own length scale: ORGANISM — ORGAN — CELL — MEMBRANE — MOLECULE e.g., animal brain neuron axolemma ion channel Levels higher than organismic and lower than molecular, as well as the tissue level between organ and cell, have been omitted for simplicity. The nervous system is closely related to all the other systems; it, along with the endocrine system, transfers information throughout the body. The central nervous system receives information from every organ and sends information to every organ. The command and control structure that is embodied in this communication network is responsible for unifying the body into a coordinated system. 3.1. Sensory organs Information comes to us through our external senses: vision, hearing, the chemical senses of taste and smell and the skin senses of heat, cold and touch. There are also

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internal senses that tell us the orientation of the head relative to the gravitational field, and the locations, orientations and stresses of body parts. The light-receiving part of the eye is the retina; the rest of the structure essentially serves to position and focus the image on it. The light-absorbing cells in the retina, the photoreceptors, are the highly sensitive rods and the color-receiving cones. In these cells, information from incident photons stimulates the formation of nerve impulses. The retina is in a sense part of the brain, to which it is attached by the optic nerve; neurons in the retina begin the processing of the information that becomes the representation of the image in the brain proper. The organs of audition and balance are embedded in the solid skull. Sound, received at the eardrum and impedance-matched by the bones of the middle ear from air to water acoustics, forms standing waves in a spiral structure, the cochlea. A delicate resonating membrane, the basilar membrane, fringed by hair cells, divides the aqueous cochlea into two pathways, traversed by the sound so that incoming and outgoing waves interfere to form standing waves. As different locations along the basilar membrane resonate at different frequencies, the cilia of the hair cells are agitated and the cells convert acoustic information into nerve impulses, which travel to the brain via the acoustic nerve. Balance information is recorded by the vestibular organ of the ear, consisting of three semicircular canals lined with hair cells that detect acceleration, and two sacs, each containing calcium carbonate crystals that respond to the gravitational field. The hair cells are examples of mechanoreceptors. Smell and taste are closely related senses, which depend on the recognition of molecules by sensory systems based on chemoreceptors. The vomeronasal organ, present in almost all terrestrial vertebrates, sends neural information to brain structures controlling the reproductive behavior.4 Sensations of heat, cold, sharp or blunt, heavy or light contact are subserved by special sensory cells in the skin. Additional senses include the ability to detect electric fields (by electroreceptors in the lateral organs of fish) and infrared radiation (by photoreceptors in the pit organs of certain snakes). In addition to these external senses, internal senses monitor the body. Muscle tension is signaled by sensory neurons whose terminals are wrapped around muscle fibers. They help control muscular responses by a feedback system, exemplified by the knee-jerk reflex. Reflexes are mediated by synapses in the spinal cord. The reflex arc consists of an afferent neuron that carries sensory information to the spinal cord and a motor neuron with which it synapses, directly or by way of an interneuron. The efferent motor neuron processes both excitatory and inhibitory stimuli to form responses that promote a contraction of one muscle while deterring that of another. Pain receptors are nerve endings located throughout the body that respond to cellular disturbance or injury. 3.2. Effectors: Muscles, glands, electroplax Muscle cells, myocytes, are contractile fibers of three types: Skeletal muscles carry out voluntary movements; smooth muscles line the arteries, uterus and digestive, urinary and other tracts; cardiac cells form the highly coordinated tissues of the atria and

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ventricles of the heart. All myocytes are under the control of nerve fibers that synapse with them. Muscle cells are also excitable cells in their own right, conducting impulses along their length. This excitation is coupled to the contractile mechanism by specialized membranous structures. Glands are organs that contain secretory cells. Exocrine glands, such as salivary, mammary and sweat glands, secrete fluids through ducts. Endocrine glands controlled by the nervous system, such as the adrenal medulla, posterior pituitary and pineal gland, secrete their hormones into the bloodstream, which carries them to target cells elsewhere in the body. Some organs, such as the pancreas, ovary and testes, have both endocrine and exocrine functions. 3.3. Using the brain The brain is neither a hydraulic device (as the philosopher René Descartes speculated) nor a computer (as some contemporary writers assert), even though moving fluids (blood and cerebrospinal fluid) and information-processing capabilities are both important aspects of it. The brain is an organ of the body, playing an essential role in the maintenance and survival of the organism. Like other organs, it has a metabolism, requiring a steady supply of nutrients and oxygen as well as the removal of wastes and excess heat. Unlike a computer, the brain is not assembled from parts; it grows by cell division. The brain–computer analogy also breaks down on a deeper level: Unlike a computer, the brain programs itself, by conditioning and learning. The brain is not a fixed structure. Although a computer may be modified by a replacement of components, its configuration—the hardware—normally remains constant, while the software and the contents of its memory are variable. Not so for the brain, which is fully functional in childhood, a period during which it grows rapidly. The brain's structure is changeable, within limits. If a part is injured, another part often takes over its functions. Learning, a requirement for mental growth, proceeds incrementally, new knowledge being integrated into previous knowledge. When the new knowledge conflicts with earlier, emotions may be aroused. Despite the fact that brains and digital computers are quite different, they are similar in one respect: They contain components that can exist in two or more discrete states, between which they switch in response to an input. In neuronal membranes these are the ion channels. The brain, along with the spinal cord, is part of the body's rapid informationprocessing system, the central nervous system. Its inputs are our senses and its outputs are mediated by our muscles and glands. The information acquired from the external environment is subject to further processing in the brain. Learned experiences allow us to recognize seen objects, heard voices, the touch of a familiar hand, the movement of an elevator in which we are standing. This interpreted sensation is perception. Our perception can be fooled; the fooling of our visual perception by optical illusions is as common as seeing a movie. An image may be ambiguous, capable of more than one interpretation. A black-and-white figure may be seen as a vase or two faces. In processing such an ambiguous image, the mind can select one interpretation

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or the other, but not both simultaneously—even when we know that both exist. We can be misled not only by optical illusions, but also by auditory ones and those of other modalities. Since understanding is analogous to perception in their dependence on brain processes, we may expect that cognition itself can be fooled.5 3.4. Analyzing the brain Let us now look at the brain as the object of our analyses. A human brain, prepared for dissection, has roughly the size and shape of a large cauliflower. In its natural state in the living animal, however, the brain is a soft, throbbing mass enclosed in the skull and covered by its glistening protective meninges. Of course, it is not the human brain that is mainly used in neurophysiological research. If "curiosity killed the cat," it is often human, not feline, curiosity that did it. We owe a great debt of gratitude to the animals—mammals, birds, amphibians, reptiles, fish and others—that involuntarily sacrifice their lives for scientific research. The brain receives the lion's share of the body's blood supply. A complex distribution of arteries and veins provides the blood on which the brain depends. However, the blood is filtered by the structural complex known as the blood–brain barrier, so that its cells, except in the case of trauma, only come into contact with the cerebrospinal fluid. Pools of cerebrospinal fluid occupy the central canal of the spinal cord and the ventricles of the brain. The distribution of blood supply shifts to brain areas that are currently active—a property used in imaging techniques for locating a particular brain function and for diagnosis. The study of the human brain in all its complexity would be impossible without some system. We have to pick up the thread somewhere, and traditionally this is done by starting with an embryo and following its development, or ontogeny. This not only gives us a systematic scheme to understand brain structure but, remarkably, also helps us helps us understand how the brain developed in the evolution of the species, or phylogeny. The parallelism between phylogenetic evolution and ontogenetic development, albeit imperfect, allows us to compare the human brain to the brains of other animals, living or extinct: The development of the embryonic brain follows many of the stages that the brain followed in its billion-year evolution. Brain and skin derive from the same embryonic layer, the outer cell layer called ectoderm. A strip of ectoderm curls into a gutter, which closes to form the neural tube. Starting as three bumps on one end of the hollow neural tube in the embryo, the brain grows in complexity and size. In invertebrates such as worms, the front bump, the forebrain, plays a role in olfaction; the midbrain, in the detection of vibration and pressure, and the hindbrain, in vision. In mammals, things are far more complex; the forebrain and hindbrain each further divide into two additional parts. The embryonic mammalian brain is divided into the telencephalon and diencephalon, which make up the forebrain; the mesencephalon, the midbrain; and the metencephalon and rhombencephalon, the hindbrain. The forebrain grows out of all proportion to the rest to develop into the cerebral hemispheres, which cover the brain except for the front part of the hindbrain, from which emerges the cerebellum, important in the coordination of bodily motions. Each hemisphere communicates with the

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opposite side of the body, and the two hemispheres send messages to each other over a great bridge, the corpus callosum. The rest of the neural tube becomes the spinal cord, which together with the brain makes up the central nervous system. From it, branching bundles of nerve fibers, the cranial and spinal nerves, extend outward to the skin and all other organs of the body. When our ancestors, more than 3 million years ago, walked upright, the brain had to adapt and make a 90-degree bend: The bipedal posture that freed their hands imposed a new anatomical structure upon their brains. The study of the brain and its many functions—regulation, sensation, perception, control of muscles and glands, memory, emotion, cognition—is too deep to engage in here, except for a few generalizations. One of these is the dynamicism of the brain. Like the respiratory, circulatory and female reproductive systems, the nervous system operates on an intrinsic rhythm, the sleep–wakefulness cycle. Constantly active even during sleep, the brain is a busy organ indeed during the waking hours. The structure responsible for controlling the sleep-wakefulness cycle is the ascending reticular activating system, which originates in the brainstem, where it receives information from virtually all sensory pathways. Acting on the cerebral cortex by way of groups of cells called nuclei in the thalamus, it results in the maintenance of sleep or consciousness.6 The brain’s functions are highly localized. The brainstem contains important centers in which breathing, heartbeat and other vital functions are performed. From the floor of the thalamus extends the hypothalamus, which connects with the pituitary, the master gland that controls the endocrine system. Special locations have been mapped for emotions, long-term memory, and the serial processing of information from the eyes and ears. One area of the human cerebrum is devoted to understanding language, another to forming words. Within the convoluted surface of the cerebral hemispheres are motor and sensory domains, with distorted maps of the human body neatly laid out on them of the bodily regions they subserve. The brain is subject to injuries and diseases, both functional and structural. Brain injuries have given scientists many clues to the relation between brain structure and function. The case of Phineas Gage, a worker much of whose prefrontal cortex was destroyed in an industrial accident, is famous for the insight it provides into that structure.7 Probing the brains of patients with brain tumors has given us functional maps of the cerebral hemispheres. Drugs have provided insights into the function of receptors and ion channels. Some pathologies have been traced to their mutations; these channelopathies will be discussed in Chapter 13. 4. INFORMATION TRANSFER AT TISSUE LEVEL If we look at a thin slice, suitably stained, of any organ through the optical microscope, we see an arrangement of cells. Remarkably, in the vicinity of any cell, we find a blood capillary and a nerve fiber. In this way, both of the great information systems, hormonal and neural, reach into the neighborhood of every living cell of the body. These systems connect the body into a unified system.

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Because of its proximity to a capillary, every cell receives a blood supply. Thus it has access to the gases, nutrients and hormones the blood contains at any moment. Because of its proximity to a neuron, each cell is connected to the brain or spinal cord. Muscle and skin tissue are innervated by excitatory and inhibitory motor fibers, as well as by sensory fibers. Other tissues may only contain sensory nerve fibers. Every living tissue of the body, except in the brain itself, contains pain fibers. 5. INFORMATION TRANSFER AT CELL LEVEL Cells are basic units of living organisms. In one-celled organisms, they carry out all the functions of the organism. Some of this autonomy remains even in the cells of multicellular organisms, as these cells can often be grown in artificial media as cell cultures. 5.1. The cell The earliest known cells, Archaebacteria, were free-living and contained within their outer membrane all the functions necessary to life: respiration, digestion, excretion, reproduction. Many new forms evolved from these. Some developed the ability to photosynthesize, splitting water molecules and releasing oxygen molecules as a byproduct. In these cells we find photoreceptors that transduce sunlight into the main source of biological energy. The oxygen that entered the environment from photosynthesis, although lethal to many organisms, became the basis for a new form of respiration, aerobic, far more efficient than the earlier, anaerobic, form. Eventually cells developed a peculiar form of cooperation: Cells were infected by predatory cells but tamed them, using their special talents for the host's own benefit. Cells that could respire aerobically became subunits of other cells in the process of endosymbiosis. These subunits are organelles called mitochondria. The endosymbiotic organelles that conferred the ability to photosysnthesize are chloroplasts. Other organelles also are believed to have endosymbiotic origins.8 Endosymbiosis changed cells in a fundamental way. The new cells, eukaryotes, look quite different under the microscope from the earlier prokaryotes. They are larger (5-30 m, as compared to the bacteria's 1-10 m) and far more complex. The characteristic for which they are named (eukaryote = "good nucleus") is the nucleus, clearly demarcated by an envelope consisting of a porous double membrane. The matrix material of which a cell is composed is a gel called the cytoplasm. Gels have interesting properties associated with the sol–gel transition. In the transition from sol to gel, new bonds form between long polymer chains to form a threedimensional structure. An example of the importance of sol–gel transformations is the method of movement of the ameba. Ameboid locomotion occurs in many types of cells, including our white blood cells, which destroy pathogens. Enclosed in the eukaryotic cell's bounding plasma membrane are the nucleus and other organelles, including the mitochondria, which oxidize food molecules to produce molecular energy packages such as adenosine triphosphate (ATP). The ribosomes are organelles that carry out protein synthesis. Other organelles include the

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endoplasmic reticulum, Golgi body, chloroplast (in algae and plants), vesicles, vacuoles, cilia and flagella. The study of ion channels in organellar membranes has opened new avenues of research. The cytoskeleton, a network of protein fibers in the cytoplasm, helps determine the shape of the cell. The types of fibers in the cytoskeleton of a neuron are microfilaments, neurofilaments and microtubules (see Chapter 19, Section 3.3). The neurofilaments, intermediate in diameter between actin filaments (about 5 nm) and microtubules (about 20 nm) are also called intermediate filaments. In Alzheimer's disease, neurofilaments form chaotic tangles in the brain. The maintenance of cell shape by the cytoskeleton has been studied in great detail in red blood cells, partly because of its relation to the genetic disease sickle-cell anemia. The cytoskeleton is anchored to the cell membrane by specific protein molecules. The cell is enclosed by a fluid two-dimensional structure, the plasma membrane, 6-10 nm thick. In addition to the plasma membrane, some cells, including the cells of algae, plants and fungi, are also surrounded by a cell wall, which provides rigidity to the cell. Composed of a phospholipid bilayer with embedded proteins, the plasma membrane maintains the integrity of the cell by controlling inflows and outflows of materials. The cell membrane normally supports an electric potential difference and is traversed by ionic currents. 5.2. Cells of the nervous system Nervous tissue contains two types of cells, neurons and glia. Neurons carry messages along their fibers and transmit them from their terminals to other cells, and so are involved in information processing. Glial cells of various types maintain the mechanical, chemical and electrical conditions necessary for the neurons' functioning. The glia, of various types and sizes, outnumber the neurons ten to one and play important supporting roles in the nervous system. 5.3. The neuron Neurons are the primary information-processing cells of the nervous system. Figure 2.3 shows several types of neurons.9 Motor neurons send excitatory and inhibitory commands from the spinal cord to muscles. The mitral cell is a sensory neuron that decodes an odor message and transmits it to the brain. The pyramidal cell undergoes modifications at its synapses; see Section 5.5. Purkinje cells are highly branched neurons of the cerebellum. The human brain contains some 1011 neurons, each making roughly one thousand synaptic connections on average. Thus the estimated number of synapses in the brain is the astronomical figure of 1014. A simple motor neuron, which sends information from the central nervous system to a muscle cell, consists of a perikaryon or soma, the cell body that contains the nucleus and many organelles, a number of branching dendrites, and a single axon, which may divide into branches. Each branch ends in a terminal bouton or terminal containing fluid-filled vesicles. The terminal

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Figure 2.3. Types of neurons: motor, sensory, pyramidal and Purkinje. From Nicholls, Martin and Wallace, 1992.

communicates with another cell, called the postsynaptic or subsynaptic cell. This may be another neuron or a muscle or gland cell. The region of apposition of the two cells is called the synapse. Much information processing takes place at the synapse, where information is transmitted from one cell to another. Although there are simple electrical synapses, called gap junctions, most synapses are chemical. In these, information is passed by the transfer of special substances called neurotransmitters. At chemical synapses the presynaptic terminal emits this transmitter substance, which diffuses across the synaptic cleft to bind to chemoreceptor molecules in the postsynaptic membrane. Synapses are classified and named by the neurotransmitter they release; a neuron that releases acetylcholine is cholinergic, one that releases adrenaline is adrenergic. Synapses exhibit a great deal of variation; a simple form is shown in Figure 2.4.

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Figure 2.4. Schematic chemical synapse. From Nicolls, Martin and Wallace, 1992.

Information enters the neuron at the dendrites and soma, which typically are covered with hundreds of synapses, some at spines. Some are excitatory, driving the cell voltage in a positive direction and so promoting the formation of an impulse at the subsynaptic axon, and some are inhibitory, making the cell more negative and thereby resisting the formation of an action potential. The pattern of action potentials, or spikes, forms as a result of an integration of these impulses at the axon hillock that joins the axon to the soma. These spikes travel as electrochemical waves along the axons and its branches, sending coded messages to the subsynaptic cells. Since the spike can be considered an all-or-nothing event, a neuron carrying, say, eight spikes per second is transmitting information at a rate of 8 bits per second. According to Donald Hebb’s rule, if neuron A repeatedly contributes to the firing of neuron B, the efficiency of A in firing B increases. This forms the neural basis of Pavlovian conditioning and associative learning.10 Since the axonal terminals may be more than one meter away from the cell body, the question arises as to how proteins and other materials are transported to the terminal. Axonal transport has been studied by radioactive and fluorescent tracers and by video microscopy. These studies show that vesicles containing proteins and other materials are driven along microtubules by vesicle-associated proteins, both outward from the soma and back toward it (orthograde and retrograde transport). The vesicles move by a rapid process at several hundred millimeters a day, and by slow processes at 1-10 mm per day.11 Neurons are classified as either myelinated or unmyelinated, according to whether or not their axons are wrapped in periodically interrupted insulating regions of

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myelin. The tiny gaps between the regions of the axon covered with myelin are called nodes of Ranvier, or simply nodes. Because the transmembrane flow of ions is limited to the nodes, conduction of the action potential along myelinated axons is called saltatory, i.e., jumping, conduction. When axons of the same diameter are compared, conduction speed is much greater in myelinated than unmyelinated axon. To put it another way, an organism can gain impulse speed without sacrificing compactness by choosing myelination. Myelination is only seen in vertebrate animals. Neurons function in receiving information at various sites on the dendrites and soma, integrating excitatory and inhibitory inputs to form a train of action potentials, conducting the impulses to one or more target locations, and releasing neurotransmitters at the terminals to transmit the information to postsynaptic cells. In addition, some neurons have feedback loops in their structure; others release neurohormones into the interstitial fluid. 5.4. Crossing the synapse Briefly, synaptic transmission works like this: When an action potential arrives, it spreads over the terminal. Because the terminal has ion channels that are different from those in the rest of the axon, the inward current is not of sodium but of calcium ions. When the calcium ions reach the vesicles in the terminal, they stimulate them to discharge their neurotransmitters into the synaptic cleft in the process of exocytosis; see Section 5.8. After diffusing across, the neurotransmitter molecules bind to receptors on the postsynaptic membrane, which are ligand-gated channels. In addition to this ionotropic action, neurotransmitters may also act in a metabotropic response, as discussed in Section 6 of this chapter. Neurotransmitters provide a new dimension in brain complexity, far from the outworn telephone-exchange metaphor of the brain, in which synapses are viewed as mere switches. While peripheral nervous system neurons all secrete the same transmitter, acetylcholine, central nervous system neurons paint with a rich palette of neurotransmitters. There are some fifty different neurotransmitters (and neuromodulators, which exert a regulatory effect over a larger distance), of different chemical families. In addition to acetylcholine, there are amino acids (glutamate, aspartate, glycine), amino acid derivatives (serotonin, dopamine, noradrenaline, aminobutyric acid), nucleic acids (adenosine and its phosphates) and peptides (endorphin, oxytocin, substance P, bombesin). Even as small a molecule as nitric oxide (NO) acts as a neuroactive molecule, although it is not sequestered into vesicles; it diffuses, affecting neurons as far as 100 m away from its site of synthesis.12 The existence of a variety of neurotransmitters in organs greatly increases the rate of information transmission. Synapses provide two very important functions in information processing: plasticity, the ability to change the structure of the nervous response by the growth of new synapses and the decay of unused ones, and memory, to which they contribute.

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5.5. The “psychic” neuron One of the most remarkable neurons is the pyramidal cell of the cerebral cortex, dubbed by neuroanatomist Santiago Ramon y Cajal the “psychic” neuron of the brain. This name is apt, and not only because of its impressive pyramid shape, with a long axon, an extended dendrite studded with spines at the neuron’s apex, and a bushy dendritic arbor at its base; see Figure 2.3. It connects its own region of the cortex with distant cortical regions as well as with subcortical effectors. It integrates thousands of afferent inputs; through its efferent connections it regulates skeletal and smooth muscles. Unlike that of other cortical neurons, the pyramidal cell’s information output is transmitted by way of the neurotransmitter glutamate. Its reactions are modulated by a number of other neurotransmitters. The prefrontal cortex of primates carries out higher functions, such as the “working memory,” by which an item of information can be held “in mind” for several seconds and be updated moment by moment. Prefrontal pyramidal neurons exhibit tonic activity triggered by the brief presentation of a stimulus, giving a neural basis for the observations of “out of sight—out of mind” behavior in patients with prefrontal lesions. Specific neurons are coded to items of information in object space, such as a person’s face. Pyramidal neurons are compartmentalized, in the sense that some regions possess dopamine receptors and other regions have serotonin receptors. The receptors have been shown to modulate memory fields and cognitive processes. Pyramidal cells represent vast stretches of the brain’s “information highway,” being privy to the current cortical environment as well as repositories of stored knowledge.13 5.6. Two-state model “neurons” Now that we have seen something of the marvelous complexity of neurons, we must acknowledge that the word “neuron” has been adopted by a group of engineers, computer scientists and mathematicians for another, much simpler, structure. These model neurons are modeled as simple on–off components, comparable to transistors. As we have seen, biological neurons are highly evolved structures, possessing an enormous number of inputs at the dendritic tree and perikaryon and multiple outputs due to branching of the axon at the terminals. The rich variety of synaptic types and neurotransmitters make the neuron a structure of great versatility and power, far more complex than a simple two-state device. The confusion between real and binary neurons is compounded when it is used to assess the complexity of the brain. According to some model calculations, the brain is comparable to a computer consisting of 1011 transistors, one per neuron.14 By underestimating the variety and complexity of the human brain by many orders of magnitude, such calculations embolden unrealistic speculations of technological “brains.” The most important application of binary neurons is their assembly into neural nets. Although far from describing biological brains, such neural networks have been used to solve difficult problems, becoming a growing trend in computer science.

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Furthermore, the insights they provide into the way changing brain connections underlie learning in simple models may give us useful clues to the organizational principles of the brains of living organisms. For example, Per Bak and Dimitris Stassinopoulos have studied a “toy model” of a brain, the computer results of which suggest that brains operate in a self-organized critical state.15 The concept of selforganized criticality will be discussed in Chapter 15 of this book. 5.7. Sensory cells Sensory cells function in detecting events in the body and in the external environment, and transducing this information into the language of action potentials. Mechanoreceptors, cells that detect mechanical pressure, include stretch receptors in muscle, hair cells in the inner ear and touch receptors in the skin. Hair cells are present in both the basilar membrane, where they react to sounds of specific sound frequencies, and in the vestibular apparatus, where they react to the position of the otoliths. Taste and smell depend on chemoreceptors, which sensitively react to molecular structures. We have already seen an example of bacterial chemoreception in the first section of this chapter. A particular substance may have drastically different effects when transduced by different detectors. For example, there are two types of receptors sensitive to acetylcholine: the nicotinic acetylcholine receptor, a ligand-gated ion channel found in skeletal muscles, where it reacts to acetylcholine emitted by motor neurons, and the muscarinic acetylcholine receptor, which is found on smooth and cardiac muscle. Many chemoreptors utilize systems of interacting proteins, including the G proteins discussed in Section 6.2. Figure 2.5 illustrates the olfactory transduction pathway. An odorant molecule is carried through the mucus layer and binds to the receptor of a G protein, which releases an alpha subunit. This stimulates adenylyl cyclase to produce elevated levels of cAMP, opening cyclic nucleotide gated channels. The ionic current leads to a generator potential, which may result in the formation of an action potential in the axon.16 The lateral organs of certain fish, such as rays and skates, are highly sensitive to electric fields. They contain electroreceptors known as ampullae of Lorenzini, which respond to stimuli as small as 3 V.17 Some bacteria, insects, birds and whales exhibit sensitivity to magnetic fields.18 While magnetic fields produced by electric currents in the human heart, brain and skeletal muscle have been measured,19 the question, “How do organisms detect magnetic fields?” is still controversial. However, the use of the mineral magnetite by magnetotactic bacteria for orientation provides a biophysical mechanism for magnetoreception and suggests a role for this sense modality in early evolution.20 Migrating birds and fish navigate by following the Earth’s magnetic lines of force.21

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Figure 2.5. An olfactory receptor neuron. The transduction of the binding of an odorant molecule by an odor receptor to second messenger system occurs in a fine dendrite called a cilium. The depolarization of the ciliary membrane spreads to the membrane of the soma, where it produces action potentials that travel along the axon to the olfactory bulb. From Broillet and Firestein, 1996.

Electromagnetic radiation arriving at the retina is detected by the light receptor structures, rods and cones. In the embryo, these appear first as cilia, which specialize to form stacks of disks containing visual pigment, rhodopsin in the case of rods. Rod cells have been shown to be sensitive to single photons. Ion channels play an important role in rod photoreception: In the dark, there is a current of sodium ions between outer and inner segments of the rod; illumination stops this current. The pit organs of rattlesnakes are lined with receptors sensitive to infrared radiation. Noxious thermal, mechanical and chemical stimuli are detected by sensory neurons known as nociceptors, located on pain fibers. They are also characterized by their sensitivity to capsaicin, the main pungent ingredient of “hot” chili peppers.22

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5.8. Effector cells The information received by receptor cells is shunted to various parts of the brain, where it is processed by interneurons receiving information from many sources. For example, information from the retinal rods and cones may lead to changes in the ciliary muscles controlling the tension of the lens, and hence focus; the external recti controlling eye rotation; and, by way of a series of intermediate structures, radiate to the visual projection area in the occipital lobes of the cerebral hemispheres. Further processing, involving also the cerebellum, may for example lead to coordinated motions such as the movements of head, arms and legs to swing a racket in a return of a served tennis ball. The pupillary light reflex requires the flow of information through the optic nerves and tracts, and synapses in various ganglia and nuclei. Eventually responses are generated by the effectors, muscles and glands. Excitability is by no means limited to the cells of the nervous system. Although the primary function of muscle cells, or myocytes, is to contract, they also conduct impulses. Many of the types of ion channels we will encounter in this book reside in the membranes of myocytes as well as neurons. Myocytes that effect the movement of one bone relative to another are skeletal muscle cells, known, because of their striped appearance in the microscope, as striated cells. Striated muscles are under conscious control of the nervous system. Another form of muscle cells are the smooth muscle cells, which raise hairs, contract arteries, transport food through the digestive tract, and perform many other autonomic functions in the body. The third type of myocytes are the cardiac cells, which are highly adapted to their function of lifelong coordinated rhythmic contraction. An interesting modification of muscle cells are the electric organs found in electric eels and other electric fish, which deliver discharges of as much as 600 volts to repel predators and stun prey. The electroplax of these organs contains nicotinic acetylcholine receptors and voltage-sensitive sodium channels; see Chapter 13. Glandular cells have the ability to secrete materials to the outside. This process, exocytosis, is the opposite of swallowing an object or bit of fluid, called endocytosis. The material to be secreted, such as milk, a hormone or a neurotransmitter, is produced at the endoplasmic reticulum and gathered into a vesicle at the Golgi apparatus. The vesicle drifts to the plasma membrane, the membranes fuse together to form an opening, and the vesicle contents are expelled. While the ability to secrete is the specialty of glandular cells, it is present in many other types of cells, particularly the neuron. In fact, the brain itself can be viewed as an immense gland.23 Protein secretions are synthesized according to RNA coding at ribosomes attached to the endoplasmic reticulum. There the polypeptide may enter the lumen or—if destined to be a membrane protein—thread itself across the membrane. Processing occurs at the Golgi body, where secretory vesicles bud off. The vesicles may be transported down the length of the axon to the terminal, where they are ready to be released when an action potential arrives.

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6. INFORMATION TRANSFER AT MEMBRANE LEVEL The cell membrane is the bounding structure of the cell, separating and distinguishing it from its environment. For the cell to remain alive, it must be an open system, so the membrane must be “leaky.” Leakiness is, however, not a very appropriate term, since the requirement for survival is a highly selective permeability: Water must equilibrate, oxygen and nutrients must enter freely, carbon dioxide and other metabolic products must leave freely, but toxic materials must be denied entry. Furthermore, the excitable membrane must be electrically insulating when not conducting an impulse, but carry substantial currents when it is. 6.1. Membrane structure These requirements are elegantly met by the biological membrane, which not only covers the cell, but also forms intracellular organelles. It consists of a fluid lipid bilayer with intrinsically located proteins, including ion channels, pumps and receptors. The lipid molecules form two sheets, with their hydrophobic tails facing inward and half of their hydrophobic heads facing the cell's aqueous environment and the other half facing toward the cell's aqueous interior. As we saw in Figure 1.3, this bilayer is pierced by protein structures, which may be limited to the inner or outer leaflet of the bilayer, or span the membrane between the aqueous media. 6.2. G proteins and second messengers The excitability of membranes in heart, smooth muscle and secretory cells, and the somata and dendrites of neurons, is subject to modulation. Their electrical properties constantly adjust to the varying needs of the organism. The complexity of these adaptations is well illustrated by the function of receptors, such as chemoreceptors. This process utilizes regulatory membrane proteins called G proteins. These are members of a class of guanine-binding proteins. G proteins function as switches capable of turning on or off the activity of other molecules, and they do so for specific time durations. The G proteins operate by a process called collision coupling. The function of G proteins is to transmit messages from receptors to effectors within the cell membrane, such as enzymes and ion channels. Some enzymes send out second messengers, which trigger biochemical changes elsewhere in the cell; this is called the metabotropic response. Figure 2.6 shows a schematic diagram of the induction of a second messenger, inositol 1,4,5-triphosphate (IP3), from an incoming first messenger, not shown, which binds with the G-protein-coupled receptor R. The activated receptor splits off two subunits, G and G. The G subunit, activated by a guanosine triphosphate (GTP) energy carrier, migrates along the membrane to dock with and activate the effector phospholipase (PLC). The PLC then catalyzes a reaction that yields diacylglycerol (DAG) and the IP3 second messenger.24

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Figure 2.6. Schematic diagram of the induction of a second messenger, IP3, from an incoming first messenger that activates the G-protein-coupled receptor R. This causes the G subunit to migrate along the membrane to dock with and activate the effector phospholipase (PLC). From C.U.M. Smith, 1992. Copyright John Wiley & Sons Limited. Reproduced with permission.

The multiplicity of steps in this process has the valuable consequence of amplifying the signal. The production of a different second messenger, in this case cyclic adenosine 3',5'-monophosphate (cAMP), is catalyzed by the enzyme adenylyl cyclase. 7. INFORMATION TRANSFER AT MOLECULAR LEVEL Descending the hierarchy of size, we arrive at the molecular level. Here quantum effects become important, although classical physics remains adequate to explain many phenomena. Broken symmetries, exemplified at the organism level by the positions of the heart and liver, which break the bilateral symmetry of the body, become even more important in molecules. As we shall see in Chapter 18, broken symmetry may be considered a fundamental characteristic of life. 7.1. Chirality If you raise your left hand in front of a mirror, your image's right hand will go up, and it will have the shape of your right hand turned the other way, not that of your left hand. The property of handedness is called chirality. Lord Kelvin defined this concept in 1884: “I call any geometrical figure or group of points chiral and say it has chirality if its image in a plane mirror ideally realized, can not be brought to coincide with itself.”25 A helical screw is an example of a chiral object: The mirror image of a righthand screw is a left-hand screw. Highly symmetrical molecules such as O2, H2O and

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benzene are nonchiral, while others, such as alanine and all other amino acids except glycine, are chiral, see Figure 2.7. Alanine comes in two forms, L-alanine and D-alanine, where L and D stand for left-handed (levo) and right-handed (dextro) respectively. Every solid made of chiral molecules—unless it is an equal mix of right-handed and lefthanded ones, called a racemic mixture—is chiral. A chiral crystal may be made of nonchiral molecules; e.g., the SiO2 molecules of -quartz are helically ordered.26

Figure 2.7. Chiral objects: quartz crystals, snails, screws, S- and R-alanine, hands and a spinning particle. From Kitzerow and Bahr, 2001.

Some living organism possess (at least approximate) mirror symmetry, but not all. In Bourgogne, France, where snails are cultivated for food, a million right-handed snails are found for every left-handed one. Alice, in Through the Looking-Glass, said, “Perhaps looking-glass milk isn't good to drink.” It isn't; protein molecules are composed of left-handed amino acids. Many organic molecules are chiral because they contain a carbon atom with four different ligands. The ligands are not in the same plane but are situated at the corners of a tetrahedron surrounding the central carbon atom. The absolute configuration of the molecule is designated rectus (R) or sinister (S).27 For molecular chirality in amino acids and carbohydrates, the configuration is designated by the prefix “L” or “D” instead of the absolute configuration, R or S. We observe two facts: 1. The two configurations are not equally distributed; biological nature has a preference in handedness even at the molecular level. L-amino acids occur much more

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frequently than D-amino acids, and proteins are made entirely of L-amino acids. Similarly, there is a natural preference for D-sugars. 2. Chirality has important physiological effects, as the taste, odor and drug activities of the enantiomers can differ greatly. While the amino acid (S)-asparagine has a bitter taste, the R enantiomer has a sweet taste. A disastrous example of the importance of chiral differences was seen in the drug thalidomide, prescribed in the 1960s as an antinausea agent for pregnant women. While the S enantiomer had the desired properties, the R enantiomer, also present in the racemic mixture, acted as a teratogen, causing serious birth defects in the babies. Since then, the chirality of drugs has been carefully controlled. A question that arises from these observations is whether the origin of the natural preference in chirality of biomolecules arose as a consequence of an accidental fluctuation that was amplified by evolutionary processes, or whether it arose from a systematic chiral perturbation at a more fundamental level.28 In the 1950s physicists demonstrated that left-handed and right-handed structures are not energetically equivalent in the nuclear weak force. Because of this so-called parity violation, neutrons are left-handed, while antineutrons are righthanded. The electrons of an atom may be left- or right-handed, and the symmetry of a stable atom may be broken by the absorption of light. The coupling due to electromagnetic forces between the electron's spin and its orbital motion causes a preference in the handedness of helical molecules. Quantum mechanical calculations show that the L-enantiomers of naturally occurring amino acids and the D-enantiomers of sugars do have slightly lower energy than their unpreferred enantiomers. Kinetic studies of model systems in which fluctuations and an external chiral influence (such as circularly polarized light) are present show that a period of 15,000 years is sufficient for a systematic chiral influence to determine which enantiomer will dominate. The enantiomeric homogeneity in nature thus may well have been caused by the asymmetry of the weak interactions, but this hypothesis cannot be proved within a time scale of human societies. 7.2 Carbohydrates Carbohydrates, sugars and their polymers, consist of carbon, hydrogen and oxygen. Simple sugar units, monosaccharides, often assume ring configurations. In the polymerization process, the rings may form linear chains or branched structures called polysaccharides. These units are frequently found to be attached to proteins, forming glycoproteins, discussed in Section 7.5 below. 7.3 Lipids Lipids are a great variety of molecules containing carbon, hydrogen, oxygen, nitrogen and phosphorus atoms. Simple constituents of lipids are the fatty acids, which are chains of hydrocarbons with a carboxyl group at one end. Typical of all lipids, they are nonpolar at the hydrocarbon end and polar at the carboxyl terminal; thus their long

INFORMATION IN THE LIVING BODY

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hydrocarbon tails are at home in a nonpolar environment, while the polar carboxyl groups find their lowest energy position in a polar environment, such as a water surface: The polar heads of the lipids are hydrophilic, which the nonpolar tails are hydrophobic. This amphipathic property of lipids determines many of the structures they form, including membranes. The lipids of a membrane form its matrix, to which proteins and carbohydrates confer specific properties. Three groups of lipids are of major importance to biological membranes: phospholipids, glycolipids and steroids. Phospholipids consist of two long fatty acid chains (nonpolar tails) attached via glycerol and a phosphate group to a hydrophilic characterizing group (polar head) of choline, ethanolamine, serine or inositol. Glycolipids, such as sphingomyelin and gangliosides, form the second group of membrane lipids. Steroids differ from the others by their characteristic flat ring structure. Biomembrane structures are based on the separation of two aqueous phases by a lamellar bilayer consisting of a mixture of lipids containing various types of protein molecules. The bilayer is fluid, allowing both lipid and protein molecules to migrate within the membrane. The unusual combination of order and fluidity exhibited by lipids is an example of liquid crystals; see Chapter 17.29 7.4. Nucleic acids and genetic information Nucleic acids carry out essential functions in reproduction and energy transformation. Deoxyribonucleic acid (DNA) encodes genetic information in a readable form; while it is highly stable in the cell nucleus, its relatively rare changes constitute the important mutations that allow organisms, and species, to adapt to changing conditions. Its form is the famous double helix. Ribonucleic acid (RNA) is a highly adaptable molecule that carries the genetic message, forms part of the ribosomes that build proteins, shuttles amino acids and even has catalytic capabilities. Nucleic acids are polymers of nucleotides, which consist of a purine or pyrimidine base, a pentose sugar (ribose for RNA and deoxyribose for DNA) and a phosphate unit. In DNA, the purine bases are the two-ring adenine and guanine, abbreviated A and G; the pyrimidines are the single-ring cytosine and thymine: C and T. Since the set {A, G, T, C} can be written {00, 01, 10, 11}, each nucleotide can store and transmit two bits of information. In RNA, T is replaced by U. Adenosine triphosphate (ATP) is an important nucleotide molecule in metabolic transformations; it functions as an energy carrier because of the repulsions of its negatively charged phosphate groups. Energy from ATP is coupled to other reactions by the transfer of one or two phosphate groups. Analogous compounds, such as the GTP discussed in Section 6.2, have similar properties. 7.5. Proteins Proteins are polymers of amino acids, attached end to end. To make an amino acid, we would start with a central (alpha) carbon and attach to its four bonds an amino group,

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a hydrogen atom, a carboxyl group and one additional group, called the side group or R group, that determines the type of the amino acid. For example, if R is a methyl group (the nonpolar side group CH3) the amino acid is alanine. For the simplest amino acid, glycine, the R group is another hydrogen and the molecule is nonchiral, as already mentioned. For any other choice of R, there are two possible structures, D and L. Biological amino acids have the L configuration. The polymerization of amino acids to form a polypeptide is due to a connection by covalent bonds called peptide bonds. Twenty different amino acids are produced from DNA code and translated from a sequence of three RNA molecules, known as a codon. Since each RNA nucleotide contains one of four bases, U, C, A and G, there are 43 = 64 triplet codons, enough for two or more representations of each of the 20 amino acids. Other amino acids are produced from these 20 by enzymatic action. Proteins are formed from one or more polypeptides by posttranslational processing, which involves folding and formation of intramolecular and intermolecular bonds. We discuss the hierarchical structure of proteins in Chapter 12, Section 5. Proteins serve many functions: as structural building blocks, as enzymes that catalyze metabolic reactions, and as informational molecules such as the ion channels that constitute the theme of this book. Proteins may be found dissolved in blood or other body fluids or, as in the case of ion channels, embedded in membranes. Short proteins, called peptides, serve as messenger molecules. Proteins are often found in combination with lipids, forming lipoproteins. Glycoproteins possess carbohydrate side chains. The attachment of these oligosaccharide units, generally anionic, occurs at specific locations of a few amino acids. This process, called glycosylation, is carried out in the lumen of the endoplasmic reticulum and the Golgi body. Carbohydrate groups extending outward from a cell's membrane play an important role in the recognition of other cells. 8. INFORMATION FLOW AND ORDER Information flows rapidly through the body by the propagation of impulses along axons and nerve fibers. At synapses, information is transmitted by formation of vesicles containing a variety of neurotransmitters and their release into the synaptic cleft. More slowly, the convection of chemical messengers conveys hormonal information. Axonal transfer moves materials, and with them information, in both directions, away from and toward the perikaryon. Information also flows by the growth of nerve and muscle fibers and other structures. Within the cell, information is copied from its DNA archive into various forms of RNA, to be used in the synthesis of proteins. These in turn have many informational jobs, including the speeding of chemical reactions as enzymes, and so helping produce other substances in the body. Membrane proteins control the traffic across the boundaries of organelles and cells, where they direct energetic and informational processes. Voltage-sensitive ion channels play key roles in the control of bodily processes, thought and consciousness.

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8.1. Information flow and time scales Information flows at many levels and time scales. Interaction with the environment yields information in the hard language of survival. The probability of survival is enhanced by adaptation. Survival patterns help determine favorable genes transmitted to offspring, thus shaping changes in the gene pool of the DNA of a species. The information flow in evolution spans many generations. It sometimes results in the origin of new species and the extinction of others.30 In the growth of the organism, information flows from the stored genetic code of DNA to RNA, particularly messenger RNA, which provides the template for the synthesis of polypeptides. These, singly or in combination, become processed into proteins. Among these proteins are the enzymes that speed the synthesis of carbohydrates and lipids, enabling the biosynthesis of the body. Thus information flows from nucleic acids to proteins, and from there to lipids and carbohydrates. A third way in which information flows is by signaling within the body. The flow of information from the environment via sense organs to the nervous system, and thence to the body's effectors, proceeds at a much higher speed than body growth. This form of information flow is a consequence of cellular excitability. Faster yet is the passage of information from molecule to molecule, or even within a single macromolecule. The explanation of this molecular excitability, as displayed in the voltage-sensitive ion channels, is a major goal of biophysics. 8.2. The emergence of order In the ordered structures of life, from the molecular level to the organismic level, we see again and again the emergence of new kinds of order from increasing complexity. From the patterns of atoms arise molecules; from the interactions of molecules come supramolecular aggregates such as membranes and organelles; from the coordination of these emerge cells; and from them, tissues, organs and organisms. Each level appears to be understandable within its proper concepts, at its own scale, but is of no help in the understanding of the next “lower” level. Understanding the way a reflex arc works is of no help in understanding the basis of the action potential, and that knowledge in turn is of not much use in trying to understand the way a voltage-sensitive ion channel works. On the other hand, the understanding of a more fundamental level can be of considerable help in making sense of the next higher level. For example, knowledge of atomic structure helps us understand the formation of molecules and crystals. The way properties at a more complex level of organization can arise from the properties of its simpler components is called emergence; the new properties are emergent properties. Dealing with emergence is not simple or direct; special mathematical techniques such as those of statistical mechanics are required.

46

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1. 2. 3. 4.

5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

30.

C. U. M. Smith, Elements of Molecular Neurobiology, Second Edition, Wiley, 1996, 252-256. R. B. Bourrett, K. A. Borkovich and M. I. Simon, Ann. Rev. Biochem. 60: 401-441, 1991. Leon Brillouin, Science and Information Theory, Academic, 1962. A. Cavaggioni, Carla Mucignat-Caretta, G. Sartor and R. Trindelli, in Neurobiology: Ionic Channels, Neurons and the Brain, edited by Vincent Torre and Franco Conti, Plenum, New York, 1996, 165-173. See, e. g., Piattelli-Palmarini, Inevitable Illusions: How Mistakes of Reason Rule Our Minds, John Wiley & Sons, New York, 1994. John Nolte, The Human Brain: An Introduction to its Functional Anatomy, Mosby-Year Book, St. Louis, 1993, 386f. Nolte, 382. Lynn Margulis and Dorion Sagan, Microcosmos: Four Billion Years of Evolution from Our Microbial Ancestors, Summit Books, New York, 127-136. John G. Nicolls, A. Robert Martin and Bruce G. Wallace, From Neuron to Brain: A Cellular and Molecular Approach to the Function of the Nervous System, Third Edition, Sinauer Associates, 1992. J. A. Scott Kelso, Dynamic Patterns. The Self-Organization of Brain and Behavior, MIT Press, Cambridge, MA, 1995. Irwin B. Levitan and Leonard K. Kaczmarek, The Neuron: Cell and Molecular Biology, Oxford University, New York, 1991, 16-19. See e.g. Smith, 318-349. Patricia S. Goldman-Rakic, Ann. N. Y. Acad. Sci. 868:13-26, 1999. Roger Penrose, The Emperor's New Mind, Oxford University, New York, 1989. Per Bak, How Nature Works: The Science of Self-Organized Criticality, Springer, New York, 1996, 175-182. Marie-Christine Broillet and Stuart Firestein, in Neurobiology: Ionic Channels, Neurons, and the Brain, edited by Vincent Torre and Franco Conti, Plenum, New York, 1996, 155-164. With kind permission of Springer Science and Business Media. Jin Lu and Harvey M. Fishman, Biophys. J. 67:1525-1533,1994. Ulrich Warnke, in Bioelectrodynamics and Biocommunication, edited by Mae-Wan Ho, Fritz-Albert Popp and Ulrich Warnke, World Scientific, Singapore, 1994, 365-386. B. N. Cuffin and D. Cohen, J. Appl. Physics 48:3971-3980, 1977. R. P. Blakemore, Science 190:377-379, 1975. J. L. Kirschvink, M. M. Walker and C. E. Diebel, Current Opinion in Neurobiology 11:462-467, 2001. M. Tominaga and D. Julius, in Control and Diseases of Sodium Dependent Transport Proteins and Ion Channels, edited by Y. Suketa E. Carafoli, M. Lazdunski, K. Mikoshiba, Y. Okada and E.M. Wright, Elsevier Science, Amsterdam, 2000, 119-122. Smith, 289-317. Smith, 158. Kelvin, Baltimore Lectures. H. Kitzerow and C. Bahr, in Chirality in Liquid Crystals, edited by H. Kitzerow and C. Bahr, Springer, 2001, 1-27. With kind permission of Springer Science and Business Media. Kitzerow and Bahr, 3 Kitzerow and Bahr, 5ff. D. Chapman, Biological Membranes, Academic Press, 1968; Liquid Crystals & Plastic Crystals, Vol. 1: Physico-Chemical Properties and Methods of Investigation, edited by G. W. Gray and P. A. Winsor, Ellis Horwood Limited, Chichester 1974, 288-307; in Liquid Crystals: Applications and Uses, vol. 2, edited by B. Bahadur, World Scientific, Singapore, 1991. Charles Darwin, The Origin of Species, Avenel Books, New York, 1979.

CHAPTER 3

ANIMAL ELECTRICITY

It is a fundamental tenet of science that the laws of nature are universal. Thus the task of biophysicists is to account for the workings of living organisms in terms of physical laws. This book seeks to trace this endeavor in the area of voltage-sensitive ion channels. Like all subjects, electrophysiology has a history. In this chapter we will briefly review the early history of bioelectricity.1 The common view of science as an edifice that is built up steadily, with no retreats, is not correct, as pointed out by Thomas Kuhn.2 Sometimes a “brick” inserted into the structure doesn't fit right, and the parts built on top of it become ragged and uneven. Experimental results become puzzling and inexplicable. Eventually the problem becomes so obvious that something has to be done about it. Part of the building has to be dismantled, the misfit brick removed and replaced, and rebuilding started from there. So the history of science, in addition to its phases of steady growth, also has its revolutionary changes, which Kuhn called paradigm shifts. These shifts require a readjustment in our concepts, which is difficult at best. The field of cellular and molecular excitability has had its share of both successful and proposed paradigm shifts. Revolutionary changes have often begun with phenomena that could not be explained by existing theories. Experimental data may suggest new theories, but to be successful a theory must be consistent with all observed phenomena. 1. DO ANIMALS PRODUCE ELECTRICITY? The frog's leg, dissected from the animal, twitched. Luigi Galvani and his wife, Lucia, were at the dissecting table while an electrostatic machine was charging nearby. With the scalpel touching the frog’s leg at the instant a spark leaped across the terminals of the machine, the leg muscles contracted. The intimate relation between electricity and living processes has been demonstrated.3 The existence of bioelectricity in electric catfish has been known since about 2600 BC, according to Egyptian records.4 Electric fish have electric organs they use to repel predators, stun their prey and attract mates. Today, molecular biologists use these electric organs to study the molecular basis of bioelectricity; see Chapter 12. 47

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Electricity received its name from the Greeks, who knew that rubbing a piece of amber (elektron) gave it the property of picking up pieces of lint and other small objects. Like lodestone's attraction for iron, it mysteriously acted at a distance. The science of electricity developed over many centuries. The Leyden jar, invented in the 1740s, consists of a glass jar covered inside and out with metal foils. An electrode passing through a rubber stopper connects to the inner foil with a chain. The jar was charged with an electric machine operated by hand to produce a separation of charge by frictional electricity. The amount of charge stored in the Leyden jar was estimated by the strength of the electric shock it gave the experimenter—the first electric measuring instrument was the human body! Benjamin Franklin used a Leyden jar in his famous, and dangerous, kite experiment to demonstrate the electric nature of lightning in 1751. 1.1. Galvani’s “animal electricity” Galvani carried out an important series of experiments, the results of which he published in 1791. He used a frog’s hind leg, attached to its sciatic nerve, a bundle of nerve fibers sheathed in connective tissue that connects to the spinal cord. Galvani discovered a new property of electricity, electric current, which became known as galvanic electricity. When he hung a frog leg on a hook on his balcony, he saw the leg twitch when it touched the metal railing. Since he could see no external source of electricity, Galvani declared that it came from the frog itself. This, he asserted, demonstrated the existence of animal electricity, electric charge generated by the organism itself. Although we know today that such bioelectric charge separation exists, his experiment had not demonstrated it.5 1.2. Volta’s battery Alessandro Volta found a fallacy in Galvani's argument: Noting that the hook and railing were made of dissimilar metals, brass and iron, Volta showed in 1800 that the junction of unequal metals was sufficient to generate electricity. His voltaic pile of dissimilar metals and cloth soaked in salt solution served as a continuous source of electric current, opening a new era in chemistry and physics. Humphrey Davy used it to isolate a number of elements, including sodium, potassium, calcium and magnesium, by electrolysis in 1808. Hans Christian Oersted discovered in 1820 that a steady electric current produced a magnetic field that rotated a magnetized needle. The development of the galvanometer, a sensitive device for measuring current, followed shortly thereafter. Michael Faraday expressed the chemical effect of an electric current in his laws of electrolysis, published in 1834. Among his many other contributions, Faraday described an unusual silver compound, which today is recognized as the first known example of a superionic conductor; see Chapter 6, Section 7.

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1.3. Du Bois-Reymond’s “negative variation” Galvani's hypothesis of animal electricity received new support when it was shown, by Galvani and Alexander von Humboldt, that a nerve–muscle preparation could be stimulated by contact with a freshly cut muscle. In the 1830s Carlo Matteuci proved in a series of experiments that a wound gives rise to a current of injury at its surface, vindicating Galvani’s belief in animal electricity. In 1843 Emil du Bois-Reymond found with a galvanometer that a steady current flowed from the intact portion to the cut end of a frog nerve. This current decreased when the nerve was excited. This “negative variation,” as du Bois-Reymond called it, was later called the action current. In 1848, du Bois-Reymond generated a theory of nerve excitation on the basis of his observations on stimulation of a frog nerve–muscle preparation. He proposed that the excitatory effect is a function only of the rate of rise of the applied current. After several decades of influence, his theory was contradicted by new data on the amplitude of the threshold pulse (the lowest current strength that excited the nerve) and was abandoned.6 2. THE NERVE IMPULSE New measurements, based on new techniques, shed light on the nature of the nerve impulse, preparing the ground for quantitative models. 2.1 Helmholtz and conduction speed Hermann von Helmholtz in 1850 made the first measurement of the speed of nervous conduction, overthrowing the notion that nerve conduction was so rapid that its propagation velocity could never be measured. Helmholtz dissected a frog's leg, leaving several centimeters of sciatic nerve intact. In one of his experiments, he attached the free end of the muscle to a lever that made a mark on a rotating cylindrical surface. The time from stimulation to the muscle twitch was less when the nerve was stimulated near its attachment to the muscle than when the stimulating electrode was farther back on the nerve. The length of nerve between the two electrode positions divided by the time difference gave a mean conduction speed of 27.3 meters per second. We now know that the sciatic nerve is a bundle of many nerve fibers, with conduction speeds ranging from 10 to 100 m/s. 2.2. Pflüger evokes nerve conduction Edouard F. W. Pflüger in 1859 made a series of studies on the stimulation of nerve–muscle preparations with steady currents. He reported that nerve propagation can be evoked with weak currents by stimulation with a negative electrode (cathode), and that, with larger currents, a nerve impulse can be initiated by either starting or stopping the current; stimulation with a positive electrode (anode) can block the conduction of the impulse.

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The development of artificial solutions such as Ringer's solution allowed cells to remain active for longer periods of time. This, along with more sophisticated circuits and microelectrodes, provided the basis for new experiments. 2.3. Larger fibers conduct faster—but not always An important milestone was passed when experiments were first carried out on single fibers, rather than bundles of nerve fibers. This allowed the relation between fiber diameter and conduction rate of simple axons to be studied.7 An important discovery was that some axons are covered with a myelin sheath interrupted by periodic gaps about every 2 mm. These gaps, nodes of Ranvier, speed the nerve impulse by the jumping of action currents from node to node, a process called saltatory conduction. A nerve such as the optic or sciatic may contain thousands of motor and sensory fibers, both myelinated and unmyelinated. 2.4. Refractory period and abolition of action potential In muscle or nerve, a shock given briefly after the end of an action potential will fail to evoke a response. This absolute refractory period was first found in cardiac muscle and nerve trunks. It is markedly prolonged by lowering the temperature. E. D. Adrian and Keith Lucas found that a subnormal response could be evoked by a strong second shock during a recovery time called the relative refractory period. The absolute refractory period coincides with the duration of the action potential. The action potential can be terminated by a brief pulse of inward current. After a weak pulse the action potential resumes its course, but after a sufficiently strong pulse the membrane potential falls to its resting value; the action potential is abolished in a roughly all-or-none manner.8 2.5. Solitary rider, solitary wave In 1834, John Scott Russell, while riding on a horse, studied the motion of a small boat in a canal. He observed that a lump of water formed in front of the boat when it was suddenly stopped. This lump, the first recorded example of a solitary wave or soliton, moved forward with constant speed and shape. Experimentation in a tank of water showed that solitons in shallow water could be generated by the movement of a piston or the insertion of a block into the water. In 1895, Diederik Korteweg and G. de Vries derived a partial differential equation that described the shallow-water waves Russell had observed. Two features of this equation are a dispersive term and a nonlinear term. The effect of dispersion to spread the energy spectrum of the pulse is balanced by the effect of nonlinearity to distort its shape. From this dynamic balance the solitary wave emerges as an independent entity. In another type of solitary wave, dissipative effects are in a dynamic balance with nonlinear effects. A burning candle is a common example of nonlinear diffusion. The power radiated by the flame is equal to the downward speed of the candle’s surface times the rate at which energy is released by the burning vaporized wax.

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In 1900, Wilhelm Ostwald described experiments on metal wires in acids. When the oxide layer of the metal was disturbed by scratching, a reaction propagated along the wire at a speed of about 1 m/s. This phenomenon was interpreted as a model of nerve propagation.9 The word soliton was coined in 1965 by Norman J. Zabusky and M. D. Kruskal, who studied their decomposition, collision and particle-like properties by computation. The concept of soliton propagation was applied by Alexander S. Davydov to molecular systems such as the  helical structures in proteins. In Chapters 18 and 19 we will apply this concept to both action potentials and the surge of ions across the membrane through the ion channel. 3. BIOELECTRICITY AND REGENERATION The story of electricity and life is intimately connected with studies of regeneration.10 The fact that plants and primitive animals possess the ability to regrow lost parts is illustrated by the study of regeneration, such as that of planarian flatworms, in the laboratory. 3.1 Regeneration and the injury current The scientific study of regeneration began in the 18th century. Crayfish, lobsters and crabs were shown to regrow legs and claws that had been amputated. Hydra, a small animal living in ponds, showed remarkable powers of regrowth. Lazzaro Spallanzani discovered the ability of the salamander to regrow its limbs, tail, jaw, and the lenses of its eyes. After the discovery of the cell, scientists concerned themselves with the process of differentiation, by which a cell becomes specialized. A fertilized egg divides by mitosis and proliferates. The growing embryo separates into three tissue layers: the endoderm, whose cells make glands and viscera; the mesoderm, which gives rise to muscle, bone and the circulatory system, and the ectoderm, which grows into skin, sense organs and the nervous system. Once a cell has differentiated, it appears incapable of reverting to the neo-embryonic form that can convert to a different type of cell. If such dedifferentiation were impossible, there could be no regeneration of one type of cell—bone, muscle, skin—from another. In a salamander that had a limb cut off, regeneration begins with a proliferation of epidermal cells to form an apex over the stump. In about a week, this apical cap encloses a ball of undifferentiated cells. The so-called “totipotent cells of this blastema then differentiate into bone, cartilage, blood vessel and skin cells. Grafting experiments show that, as the new limb forms, its shape appears from a pattern of organization that depends on its location on the body surface. Information therefore must pass from the body to the blastema, which also has information in the DNA of its cell nuclei. Where do the undifferentiated cells of the blastema come from? Experiments showed that the formation of the blastema depended on the arrival of regrowing nerve “

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fibers. These branch and form terminal buds that make synaptic connections with the epidermal cells. The interactions at these neuroepidermal junctions do not depend on action potentials or acetylcholine. The clue to the question of what passes across the gap came from A. M. Sinyukhin, who related the regeneration of tomato plants to the current of injury at the wound. This electric current initially flows inward but reverses to outward after the callus is formed. As the outward current increased, cells more than doubled their metabolic rate, became more acidic and produced more vitamin C. Externally applied currents in the same direction augmented these effects. 3.2. Bone healing and electrical stimulation Robert O. Becker sought to solve the problem of nonhealing bone fractures. He compared injury currents in salamanders, which regenerated limbs, with those in frogs, which did not regenerate. While the salamander’s injury current reversed during regeneration, the frog’s only returned gradually to its preamputation baseline. The passage of electric currents through the aquarium water accelerated regeneration in larval salamanders. In the 1920s, Elmer J. Lund found that the polarity of regeneration could be controlled by small currents. Electric potentials were found on the surfaces of slime molds, worms, hydras, salamanders and mammals, including humans. Did the electric field gradient control the distribution of growth inhibitors and stimulators? Studies of planarians showed that these flatworms possess an electric potential gradient that controls the regrowth of cut parts. The polarity of a planarian could be controlled by passing a current through it. When these experiments showed that external electric fields could be used to determine whether a head or a tail would grow from a cut piece, interest grew in the healing effects of direct currents. These findings, applied to the problem of nonhealing bone fractures in humans, led to the discovery of piezoelectricity in bone. Piezoelectricity, discussed in Chapter 16, transduces a mechanical stress into an electric polarization. This polarization plays an important role in the organization of bone cells along lines of maximum stress, allowing bone shapes to adapt to their function.11 Implantation of synthetic biopolymers that generate piezoelectric currents have been shown to induce the growth of bone.12 Recent studies deal with the exposure of humans to electromagnetic waves, ever more prevalent in our technological society. Studies suggest that exposure to extremely low frequency electromagnetic radiation increases stress and the incidence of disease.13 3.3. Neuron healing W. E. Burge found in 1939 that the voltage between the head and other body surfaces became more negative during physical activity, diminished in sleep and became positive under general anesthesia. This suggested that an analog system of information transfer is present in addition to the digital information transfer by action potentials. Since the

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threshold of an action potential depends on the resting potential, any change in external potential can affect the level of neural activity. In such a change in the vicinity of an injured cell, the activity of a neuron would be depressed as a result of the injury potential. Harvey M. Fishman and collaborators studied the sealing of injured axons. Their surprising finding was that the giant axons of squids, crayfishes and earthworms respond to cutting by endocytic formation of vesicles that fuse with each other to yield axosomes with diameters up to 5 m in the axoplasm of the injured region. The axosomes move to the cut ends and accumulate to form a temporary plug that restores a barrier sufficient to allow recovery of axonal electrical function until a permanent seal (a continuous plasma membrane) is restored.14 4. MEMBRANES AND ELECTRICITY In 1883 Svante Arrhenius showed that certain compounds, strong electrolytes, dissociate into charged fragments, ions, in water, a finding that prepared the way for the membrane hypothesis. 4.1. Bernstein's membrane theory The concept that the nerve impulse was due to ionic effects at the membrane bounding the nerve fiber, the axolemma, was developed by L. Hermann, M. Cremer, Julius Bernstein, Jacques Loeb and others.15 Bernstein's 1868 membrane theory stated that living cells were electrolytes contained within a poorly permeable membrane, which supports a potential difference with the inside negative, and that during activity the membrane becomes so permeable to ions as to abolish the potential. The theory was based on measurements with the newly developed capillary electrometer and string galvanometer, which allowed him to estimate cell phase boundary potentials of 0.02V up to 0.08 V. However, Bernstein's depolarization was only a decline toward zero potential; the discovery of the sign reversal of an action potential was to come later. 4.2. Quantitative models Ostwald in 1890 studied artificial precipitation membranes and compared them to the membranes that enclose living protoplasm, pointing out their similarity as semipermeable membranes. The quantitative basis for this hypothesis had already been begun for the special case of an electrically neutral membrane in 1889 by Walther Nernst and in 1890 by Max Planck, whose quantum theory we have already discussed. U. Behn extended these results in 1897, anticipating later developments by including the role of sodium. We will study the equation of Nernst and Planck in Chapters 7 and 8, which are devoted to electrodiffusion, a theory that has had a profound influence on membrane biophysics. The laying of a telegraph cable across the Atlantic Ocean required a careful analysis of current flows, which was carried out by William Thomson (Lord Kelvin). The application of his cable equation to nerve and muscle fibers led to a quantitative model of the axon, which will be discussed in Chapter 6, Section 3.

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4.3. The colloid chemical theory Loeb noted that salt solutions containing only sodium ions cannot maintain excitability, but that this poisonous effect can be antagonized by calcium ions. Since these ions exist partly in combination with proteins, he argued, the substitution of one ion for another changes the physical properties of the protein. Irritability depends on the presence of Na+, K+, Ca2+ and Mg2+ in the right proportions, and a change in these proportions can generate or inhibit activity. Loeb’s theory was amplified by R. Höber in 1905, who found that the effect of ions on excitability follows the lyotropic series, the series of anions discovered by Hofmeister as critical for “salting out” proteins in water. He also showed that dyes stain a nerve soaked in potassium-rich solution, but less so in a calcium-rich solution. He explained the parallelism between stainability and excitability by a loosening of the colloidal membrane in the former case and a compaction in the latter. An important link between membranes and bioelectrogenesis was made clear in 1911 by F. G. Donnan, who studied the equilibrium distribution of ions between two solutions separated by a semipermeable membrane when one of the solutions contained a colloidal suspension.16 Torsten Teorell in 1935 extended the Donnan theory to explain membrane permeability in terms of the partition effects due to differential solubility. K. H. Meyer and J. F. Sievers in 1936 published an independent version of this “fixed charge” theory. Teorell built and analyzed a porous membrane system that exhibited oscillations due to coupling between ion and water fluxes. 4.4. Membrane impedance studies Hugo Fricke in 1923 determined the capacitance of a red blood cell membrane as 0.81 F/cm2. With the additional assumption that the membrane might have the dielectric constant of an oil, 3, he estimated the thickness of the membrane to be about 3.3 nm—a molecular dimension! Kenneth S. Cole extended Fricke's work, carrying out a series of studies on the impedance of sea urchin eggs. The analyses required for the interpretation of these measurements led Cole, together with his brother Robert H. Cole, to develop a powerful new mathematical formulation of impedance problems. We will discuss this formulation, now a part of condensed-state physics theory, in Chapter 10. The rediscovery by J. Z. Young that squid have axons of unusually large diameter revolutionized the study of neurobiology. With these “giant” axons it was possible to insert electrodes into a single fiber. In 1939, K. S. Cole and Howard Curtis made impedance measurements that demonstrated the fall of membrane resistance during excitation.17 Their results on squid axon, suggested by their earlier experiments on the large cylindrical single cell of the alga Nitella, showed that an action potential was accompanied by a large increase in membrane conductance. Since Figure 3.1 shows that the voltage change comes first, it suggests that the membrane depolarization precedes the change in membrane conductance.

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Figure 3.1. Superposition of a band measuring membrane resistance decrease on a line showing the action potential during the passage of an impulse across a squid giant axon. The marks indicate 1 ms time intervals. From Cole and Curtis, 1939. Reproduced from The Journal of General Physiology, 1939, 22:649-670. Copyright 1939 The Rockefeller University Press.

The study of membrane impedance was an important step in the electrical modeling of excitable membranes; see Chapter 10. 4.5. Liquid crystals and membranes In 1854 R. Virchow, a biologist known for his fundamental contributions to cell theory, described myelin figures. These lipid–water systems were the first known representatives of a state of matter intermediate between liquid and solid, the liquid crystals.18 The first systematic description of liquid crystals was made in 1888 by the botanist Friedrich Reinitzer, who studied the phase transitions of cholesteryl benzoate. On heating the solid structure, Reinitzer observed the formation of a turbid liquid at 145.5°C, which became transparent on further heating to 178.5°C, thus showing three distinct phases, solid, liquid crystal and liquid.

The physicist Otto Lehmann, who coined the term liquid crystal, found that many organic compounds exhibited a liquid crystalline phase marked by the mechanical properties of a liquid but the optical properties of a crystalline solid.

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Since these phases are neither liquids nor crystals, G. Friedel in 1922 proposed the term mesomorphic states. He separated them into three classes, smectic, nematic and cholesteric; these are described in Chapter 17. Liquid crystals, sometimes called paracrystalline materials, have long been recognized as important aspects of biological structures and their functions. Joseph Needham wrote19 [L]iving systems actually are liquid crystals, or, it would be more correct to say, the paracrystalline state undoubtedly exists in living cells... The paracrystalline state seems the most suited to biological functions, as it combines the fluidity and diffusibility of liquids while preserving the possibilities of internal structure characteristic of crystalline solids.

F. Rinne, in 1933, and J. D. Bernal, in 1933 and 1951, pointed out the intimate connection between naturally occurring liquid crystals and life processes. Nowhere is this connection more compelling than in the structures and functions of biological membranes, which are a type of liquid crystal. 5. ION CURRENTS TO ACTION POTENTIALS Let us now turn our attention to the ions that cross the excitable membrane and the currents they carry. 5.1. The role of sodium E. Overton in 1902 showed that frog muscles became inexcitable when immersed in solutions with less than one-tenth the normal concentration of sodium chloride. Of the two ions of NaCl, the chloride ion was not the essential one; it could be replaced by nitrate, bromide and other anions without inducing a loss of excitability. That left the sodium ion; it could only be replaced without loss of excitability by its close relative, the lithium ion. In 1943, David Goldman published an analysis of electrodiffusion applied to nerve membranes.20 Here he simplified the mathematical model by what became known as the constant-field approximation. The Goldman equation was extended by Hodgkin and Bernhard Katz to provide an equation that gave a value for the equilibrium voltage across a membrane with multiple species of ions, positive and negative. In 1949 Hodgkin and Katz carried out experiments on squid giant axon, in which they replaced the external sea water with solutions deficient or enriched in

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Figure 3.2. Demonstration by Hodgkin and Katz that the height and rate of rise of the action potential depends on sodium-ion concentration. Record 1 shows an action potential (with respect to resting potential) in seawater. Records 2-8 show the slowing and diminution of the action potential 30,46, 62, 86, 102, 107 and 118 seconds after an isotonic dextrose solution without sodium was applied to the axon. Traces 9 and 10 show the return of the action potential 30 and 90-500 seconds after the reapplication of seawater. From C. Hammond, 1996.

sodium ions. They showed that both the height and the rate of rise of the action potential depended strongly on sodium-ion concentration; see Figure 3.2.21 Variation of the potassium-ion concentration resulted mainly in a change in the resting potential. These findings supported their sodium hypothesis: that the permeability of the axonal membrane in the active state was primarily to sodium ions, while its permeability in the resting state was mostly to potassium ions. 5.2. Isotope tracer studies Experiments on epithelial membranes, such as toad bladder and frog skin provided new techniques and data. Isotope tracer studies to study the fluxes of individual ion species were carried out by Hans Ussing, who also developed mathematical equations to analyze them. 5.3. Hodgkin and Huxley model the action potential During World War II many biophysicists were involved in military activities. At the end of the war, Cole established a Department of Biophysics at the University of

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Chicago. With George Marmont, an electrical engineer, he devised methods and circuits to “tame” the action potential. A long internal wire electrode created an isopotential region within the axon by greatly lowering the axon's internal resistance. This space clamp effectively prevented the action potential from propagating, allowing its time variation to be examined without spatial complications. To establish a firm electrical control of the membrane, Cole developed a feedback system, the voltage clamp. This This system allowed the experimenter to study the currents elicited by an imposed voltage stimulus. Joined by Andrew Huxley, Hodgkin and Katz then modified the Goldman equation to describe the electrical behavior of squid axon under voltage clamp. Hodgkin and Huxley conducted a series of incisive experiments on voltage-clamped squid axons. By replacing the sodium ions with impermeant ions, they were able to separate the membrane current into its different ionic components, and gained enough information from their experiments to describe these ion currents quantitatively. This work reached its completion in four papers published in 1952 by Hodgkin and Huxley, in which they generated a set of model equations on the basis of their measurements.22 Remarkably, these equations allowed them to reproduce a close approximation to the shape of an action potential. As the underlying mechanism of excitability, Hodgkin and Huxley proposed that the voltage dependence of the conductance change is due to movements of a charged or dipolar component of the membrane; see Chapter 9. An enormous expansion of electrophysiology occurred as these ideas and techniques were applied to many different types of cells of different species. One important new development was the internal perfusion of the axon. Another was the use of toxins such as the pufferfish toxin tetrodotoxin for ion separations. M. F. Schneider and W. Knox Chandler, and separately Clay Armstrong and Francisco Bezanilla, and R. D. Keynes and E. Rojas, were challenged by Hodgkin and Huxley's speculation that the opening of an ion channel was due to the response of an ionic or dipolar unit within it, a concept that suggested that the motion of this unit could be detected if ionic currents were suppressed. These groups observed gating currents, presumably corresponding to such movements.23 5.4. Membrane noise Suppose you insert a tiny electrode into the brain of a snail, goldfish or cat. As you push the probe deeper into the brain, you will observe regions where spikes are far apart in time and other places where they are more frequent. But one thing you will notice at all locations: The time between spikes is random; the patterns seldom repeat. This randomness is seen at the cell level and at the membrane level as well. In 1966, Hans E. Derksen and Alettus A. Verveen reported that the resting potential of nerve membranes is not steady but is subject to spontaneous fluctuations.24 The analysis of these spontaneous electrical fluctuations plays an important role both in the isolation of ion channels and our understanding of excitability. We will discuss noise measurements and their physical interpretation in Chapter 11.

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5.5. The patch clamp and single-channel pulses The need for accurate noise measurements spotlighted the importance of electrode area: Since random noise averaged out over the surface of large electrodes, its measurement required small, patch, electrodes. In the hands of Alfred Strickholm and Harvey Fishman, the patch clamp took form.25 It was brought to a high point of development by Bert Sakmann and Erwin Neher, whose technique made it possible to see currents from single channel molecules.26 Other studies showed that these molecules were proteins. These new techniques ushered in a new world of single-channel pulse studies that are helping decipher the structures and the functions of these ion channels. 6. GENETICS REVEALS CHANNEL STRUCTURE The investigation of the structure of ion channels required the isolation of the proteins and a means of characterizing them. 6.1. Channel isolation The nerve poison tetrodotoxin served as a powerful tool in setting apart the sodium channel. By incorporating proteins into bilayers in vesicles it became possible to study their electrical properties. Raimundo Villegas and colleagues were able to show that the structures responsible for excitability are single protein molecules, to isolate these channel molecules and estimate their molecular weight.27 After these accomplishments, the science of genetics was called into play. 6.2. Genetic techniques The discovery of the structure of DNA opened the door to a great variety of techniques for characterizing and altering genetic materials, known as genetic engineering. A group led by Shosaku Numa in the laboratory of Masaharu Noda used recombinant DNA techniques to clone the nicotinic acetylcholine receptor and, in 1983, the sodium channel from electric eel.28 From the complementary DNA they were able to deduce the primary structures of these molecules. From the primary structures and the known properties of the amino acids, channel structures were modeled. The Noda group found that the sodium-channel molecule has a mass of about 260 kDa and consists of four roughly equal repeats. A membrane-spanning segment in each homologous repeat, S4, was found to have a pattern of repeated, positively charged residues. The S4 segments were identified as voltage sensors of voltage-sensitive ion channels. Studies in fruit-fly genetics later provided a wedge by which the potassium channel could be approached. A mutation in these flies, Shaker, that led to tremors, was found to originate in a certain potassium channel. Many ion channels have been identified, and their structure is emerging. Research is also clarifying the relationship between channel malfunction and disease. The classification of ion channels will be discussed in Chapter 13.

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6.3. Modeling channel structure The property of hydrophobicity of the constituent amino acids gave an indication of their location in the protein relative to the lipid bilayer; amino acid residues in the bilayer tend to be more hydrophobic than those near the aqueous surfaces. This property made it possible to begin making models of the channel’s physical structure from its amino acid sequence. Recent successes in crystallizing certain bacterial channels has given renewed impetus to the modeling process. 7. HOW DOES A CHANNEL FUNCTION? As knowledge of the structure and function of ion channels grew, the question remained: What is the relation between structure and function? More simply put, how does an ion channel work? Experiments based on naive interpretations of the predictions of Hodgkin and Huxley continued to be useful, but the question of the structure–function relationship remained an open problem. 7.1. The hypothesis of movable gates The discovery of single-channel currents helped establish a scientific ideology in which the channel was viewed as a gated pore. The problem appeared to be one of finding a movable part of the channel that could occlude or clear the aqueous pathway over which ions traversed the channel. The arrival of molecular engineering seemed to promise that these structures would become apparent. That promise has not been fulfilled; molecules do not behave like macroscopic devices. 7.2. The phase-transition hypothesis As one alternative explanation, Ichiji Tasaki proposed a model of two stable states, with a phase transition between them.29 Other proposals included an analogy between channels and semiconductors, and the effects of dipolar mechanisms.30 Microscopic models of channel function are discussed in Chapter 14. 7.3. Electrodiffusion reconsidered In 1973 I reconsidered the electrodiffusion model, developing exact solutions to the equations and using these to generate current–voltage curves. Although the solutions to the nonlinear problem were qualitatively different from the linear solutions, the I–V curves were quite close to those obtained from the constant-field approximation. The exact solutions turned out to be no better descriptors of the behavior of excitable membranes than the conventional approximations.31 Thus the problem appeared to be not in the approximations but in the formulation and application of the model.

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Since some assumption or assumptions of the model must be wrong, I was drawn to the assumption that the parameters of the equations were taken to be constants. The parameter that stood out particularly was the dielectric permittivity, the factor by which the force of an electrostatic interaction is reduced by a particular medium. What if it or the ionic mobility, or both, depended on the electric field? That is not a far-fetched idea; after all, Hodgkin and Huxley had made their conductances depend on voltage. The problem of the electric-field dependence of the dielectric permittivity had already been worked out by condensed-state physicists studying materials called ferroelectrics. In 1920, crystals of sodium potassium tartrate tetrahydrate, commonly known as Rochelle salt, were found to have electrical properties unlike any material seen before.32 Rochelle salt is the first known example of this class of materials; others were found slowly. Since the phase transitions that these materials undergo in an electric field are strikingly analogous to the transitions of ferromagnetic materials such as iron in a magnetic field, these materials were named ferroelectrics. 7.4. Ferroelectric liquid crystals as channel models The understanding of ferroelectrics has grown rapidly, and many new ferroelectric materials, including polymers and liquid crystals, were discovered. It was in the 1960s and 1970s that several suggestions of possible ferroelectric behavior in nerve and muscle membranes were made. A. R. von Hippel wrote33 in 1969 that “True relations may exist between ferroelectricity, the formation of liquid crystals, and the generation of electric impulses in nerves and muscles.” A class of liquid crystals, smectic C*, exhibits phase transitions controlled by the electric field. These ferroelectric liquid crystals are made of elongated molecules that form layers in which they are tilted with respect to the layer perpendicular. Furthermore, they must be chiral. From the literature I, later joined by Vladimir Bystrov, noted a number of similarities between the behavior of ferroelectric liquid crystals and that of ion channels of excitable membranes.34 A certain class of amino acids present in ion channels, with branched nonpolar sidechains, was found to be particularly effective in these transitions35; see Chapter 21. NOTES AND REFERENCES Kenneth S. Cole, Membranes, Ions and Impulses, University of California, Berkeley, 1968, 1972; A. L. Hodgkin, Conduction of the Nervous Impulse, Charles C. Thomas, Springfield, 1964; William J. Adelman, Jr., in Structure and Function in Excitable Cells; edited by William J. Adelman, Jr., Van Nostrand Reinhold, New York, 1971, 274-319; Torsten Teorell, in Structure and Function in Excitable Cells, edited by Donald C. Chang, Ichiji Tasaki, William J. Adelman, Jr. and H. Richard Leuchtag, Plenum, New York, 1983, 321-334. 2. Thomas S. Kuhn, The Structure of Scientific Revolutions, University of Chicago, Chicago, 1962. 3. Robert O. Becker and Gary Selden, The Body Electric: Electromagnetism and the Foundation of Life, William Morrow, New York, 1985. 4. Cole, 1. 5. Mary A.B. Brazier, A History of Neurophysiology in the 19th Century, Raven, New York, 1988. 1.

62 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

CHAPTER 3 Ichiji Tasaki, Physiology and Electrochemistry of Nerve Fibers, Academic, 1982, 22ff. Tasaki, 87. Tasaki, 65-69. Alwyn Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structures, Second Edition, Oxford University, 2003, 1-6. Becker and Selden, 25-76. Becker and Selden, 118-149. Maurice V. Cattaneo, in Electrical and Optical Polymer Systems, edited by Donald L. Wise, Gary E. Wnek, Debra J. Trantolo, Thomas M. Cooper and Joseph D. Gresser, Marcel Dekker, Inc., New York, 1998, 1213-1222. Becker and Selden, 271-329. H. M. Fishman, K. P. Tewari and P. G. Stein, Biochim. Biophys. Acta 1023:421-435, 1990; H. M. Fishman and G. D. Bittner, NIPS 18:115-118, 2003. Cole, 54. W. J. Moore, Physical Chemistry, Third Edition, Prentice-Hall, Englewood Cliffs, N.J., 1962, 760. Kenneth S. Cole and Howard J. Curtis, J. Gen. Physiol. 22:649-670, 1939. Glenn H. Brown and Jerome Wolken, Liquid Crystals and Biological Structures, Academic, New York, 1979. Joseph Needham, Biochemistry and Morphogenesis, Cambridge, 1942, 661. D. E. Goldman, J. Gen. Physiol. 27:37-60, 1943. A. L. Hodgkin and B. Katz, J. Physiol. 108:37-77,1949. By permission of Wiley- Blackwell Publishing. Reprinted from Constance Hammond. Cellular and Molecular Neurobiology, Academic, San Diego, 1996, 119, with permission from Elsevier. A. L. Hodgkin and A. F. Huxley, J. Physiol. (Lond.) 116:449-472; 116: 473-496; 116: 497-506; 117: 500-544, 1952. M. F. Schneider and W. K. Chandler, Nature (Lond.) 242:244-246, 1973; C. M. Armstrong and F. Bezanilla, Nature (Lond.) 242:459-461, 1973;_ J. Gen. Physiol. 63:533-552, 1974; R. D. Keynes and E. Rojas, J. Physiol. (Lond.) 239:393-434, 1974. H. E. Derksen and A. A. Verveen, Science 151:1388-1389, 1966. H. M. Fishman, Proc. Natl. A cad. Sci. USA 70:876-879, 1973; H. M. Fishman, J. Membrane Biol. 24:265-277, 1975. O. P. Hamill, A. Marty, E. Neher, B. Sakmann and F. J. Sigworth, Pflügers Arch.391:85-100, 1981. Raimundo Villegas, Gloria M. Villegas, Zadila Suárez-Mata and Francisco Rodriguez, in Structure and Function in Excitable Cells, edited by D. C. Chang, I. Tasaki, W. J. Adelman Jr. and H. R. Leuchtag, Plenum, New York, 1983, 453-469. M. Noda,, S. Shimizu, T. Tanabe, T. Takai, T. Kayano, T.Ikeda, H. Takahashi, H. Nakayama, Y. Kanaoka, N. Minamino, K. Kangawa, H. Matsuo, M. A. Raftery, T.Hirose, S. Inayama, H. Hayashida, T. Miyata and S. Numa, Nature 312, 121-127, 1984. I. Tasaki, Physiology and Electrochemistry of Nerve Fibers, Academic, New York, 1982. L. Y. Wei, Bull. Math. Biophys. 31: 39-, 1969; Ann. N. Y. Acad. Sci. 227: 285-, 1974. H. Richard Leuchtag, thesis; H. R. Leuchtag and J. C. Swihart, Biophys. J. 17:27-46, 1977. M.E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon, Oxford, 1977, 1. A. R. von Hippel, J. Phys. Soc. Japan 28 (suppl.):1-6, 1970. H. R. Leuchtag, J. Theor. Biol. 127, 321-340, 1987; 341-359, 1987; H. R. Leuchtag and V. S. Bystrov, Ferroelectrics 220:157-204, 1999. O. Helluin, M. Beyermann, H. R. Leuchtag and H. Duclohier, IEEE Trans. Dielect. Elect. Insul. 8(4):637-643, 2001.

CHAPTER 4

ELECTROPHYSIOLOGY OF THE AXON

Now that we have defined the problem of biological excitability, reviewed the informational structures of living organisms and perused the early history of animal electricity, it is time for us to get down to the specifics of electrophysiology, the study of the electrical behavior of cells, tissues and organs. We must follow this trail before we can descend to the molecular level. 1. EXCITABLE CELL PREPARATIONS A squid can squirt ink and swim forwards and backwards, by jet propulsion, expelling water backward or forward through its siphon. The anatomical studies of these marine cephalopods by L. W. Williams and J. Z. Young showed that they possess a pair of remarkable structures, which turned out to be axons almost a millimeter in diameter. 1 A single axon of the squid occupies a cross section that in a rabbit leg nerve would contain hundreds of nerve fibers. With this discovery of an axon so large that it has come to be referred to as a “giant” axon came the ability to study the properties of a single axon in isolation. This experimental preparation has given us great insight into nerve conduction. Although squid come in many sizes, even up to “sea monsters” several meters long, a typical squid used for neurophysiological experiments at the Marine Biological Laboratory in Woods Hole, Massachusetts is about 30 cm long. Connecting the squid's cerebral ganglion to its mantle is a pair of giant axons, about 600-800 m in diameter; see Figure 4.1.2

Figure 4.1. Squid, showing location of giant axons in the stellate nerves. Drawing by T. Inoué, from Guevara 2003.

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These axons can be impaled with “piggyback” double electrodes (one for current and one for voltage). They are also suitable for internal perfusion experiments, in which the axoplasm is rolled or flushed out and replaced with artificial solutions. This gives the experimenter the ability to control ion concentrations both internal and external to the membrane. When the axoplasm is extruded and analyzed, we learn that its ion concentrations are quite different from those of the fluid bathing the axon; see Table 4.1. The membrane potential difference Vm (intracellular relative to extracellular) is determined by the Nernst potentials VI of ions I, shown in the Table for the concentrations listed and temperature 20ºC.

Table 4.1. Intracellular and extracellular (seawater) ion concentrations and Nernst potentials of squid giant axon at 20ºC

Ion

K+ Na+ Ca2+ Cl-

In (mM) 400 50 0.0001 100

Out (mM)

VI (mV)

10 460 10 540

-93.1 56.0 145.3 -42.6

1.1. A squid giant axon experiment A squid is a thing of beauty, a slim foot-long cylindrical body swimming serenely backward and forward in the seawater tank. Its large eyes are on the side of its head; from the front, a thick bundle of tentacles emerges, always in motion. Its body is an ever-changing display of colored spots, chromatophores, which enlarge and contract, revealing the squid’s moods to its neighbors, swimming alongside it. We select a large male, catching it carefully because it has strong suckers on its tentacles and a sharp beak in its mouth. In its struggle to avoid the net it releases a cloud of ink. On a light table in the lab, the squid is quickly decapitated with scissors. The skinned mantle, illuminated from below, shows two faint lines diverging diagonally backward: the hindmost giant axons. Tied with fine threads into plastic dishes of seawater and dissected out, one axon is refrigerated for later use and the other carefully cleaned under the dissecting microscope to remove its connective tissue layer. Its diameter measured and the recorded, the axon is mounted in a plastic chamber filled with artificial seawater and placed in a Faraday cage—to shield it from extraneous electric fields—on a vibration-

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free table. The solutions have been prepared so as to avoid gradients of osmotic pressure. The temperature of the preparation is carefully controlled. Platinized platinum electrodes are inserted into the axon and the external solution. A stimulator provides the signal, and the voltage or current data being acquired are shown on an oscilloscope and captured on a computer disk. In an experiment on a carefully dissected squid giant axon, the resting potential is about --55 to --60 mV. Rapid depolarization of the membrane by about 10 mV stimulates an impulse. The action potential, measured from the resting potential, is about 100 mV, making the voltage inside the axon relative to the external solution roughly +45 mV at the peak of the action potential; see Figure 1.1 of Chapter 1. This region of positivity, called the overshoot, showed that Bernstein's potassium model was inadequate, demanding a revision of the accepted concept that the action potential is simply the elimination of the potassium potential. It made it necessary to consider the effects of sodium, as well as potassium, current through the membrane.3 1.2. Node of Ranvier As we saw in Chapter 3, the study of electrobiology began with the sciatic nerve of the frog. Vertebrate nerve is composed of many myelinated and unmyelinated fibers, enclosed in a sheath of connective tissue. In 1928, E. D. Adrian and D. W. Bronk recorded action potentials from individual nerve fibers by dissection of a rabbit nerve. Further experiments by Kaku (J. Kwak) in the laboratory of G. Kato developed the technique of recording the electrical activity of isolated nerve and muscle fibers. In motor nerve fibers of the toad, 10-15 m in diameter, the axon is covered by a myelin sheath interrupted about every 2 mm by a node of Ranvier. The fiber thins to about 1 m at the node, providing a narrow ring of contact between the axon and the external aqueous medium. By moving a stimulating electrode along a myelinated fiber, Ichiji Tasaki showed that the threshold for stimulating an action potential was a minimum at each node and rose steeply in the internodal myelinated regions. While electric current flows ohmically from node to node, the myelin sheath acts as an insulator, with extremely low dc conductance. Although myelinated axons are much smaller in diameter than squid axons, their geometry can be used to advantage in electrophysiological experiments. Since the myelin sheath is an effective insulator, current and voltage measurements can be made by isolating the nodes of Ranvier. A single axon dissected from a frog sciatic nerve is placed on a specially constructed chamber with three solution pools. The pools are insulated with petroleum jelly or air gaps, and electrodes are inserted into them. When an anesthetic solution, such as cocaine–Ringer’s, was introduced into the middle pool, the threshold of the middle node became unmeasurably high. Even though the node was inexcitable, however, action potentials traveled through the fiber. In some cases, action potentials were able to cross two inexcitable nodes, but never three. The interpretation given by Alan Hodgkin was that the action current at a node was strong enough to excite the unanesthetized nodes beyond the anesthetized zone. When the current drops below the threshold level for the nearest excitable node, conduction is blocked. The current pathway therefore must include the external fluid.

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By using glass pipette microelectrodes devised by Gilbert Ling and R. W. Gerard in 1949, W. L. Nastuk and Hodgkin in 1950 recorded action potentials from the interior of frog muscle fibers. This method is also used to record from nerve fibers, even in nerve trunks or the brain, but its application is limited by the injury to the fiber due to the penetration of the electrode. Studies of the node of Ranvier helped clarify the all-or-none law, according to which a small stimulus can produce a powerful response. When the stimulus is above threshold, the response is independent of stimulus strength—just as the power of a gunpowder explosion is independent of the size of the flame that touches it off. That the all-or-none law does not apply to the node when the duration of the stimulating current pulse is varied was shown by Ichiji Tasaki in 1956. With a new method of recording action potentials, Tasaki found that longer pulses produce smaller action potentials; see Figure 4.2.4 As the Figure shows, the shape of the nodal action potential differs from that of the squid axon. It is wider, with a shoulder at the end of the absolute refractory period, beyond which the voltage drops more steeply. Figure 4.2 also illustrates the bifurcation of the voltage traces into electrogenic and active responses, depending on slight differences in the stimuli. Studies of threshold stimulation by linearly rising voltage pulses in isolated myelinated fibers made it possible to investigate the process of accommodation. As du Bois–Reymond had noted, a slowly rising current fails to produce excitation even when its intensity rises well above the level that would excite the nerve when suddenly initiated or terminated. Action potentials are observed when the time rate of rise of the voltage exceeds a critical rate. 1.3. Molluscan neuron A great deal of progress has been achieved by the study of the nervous systems of invertebrates, chiefly annelids, arthropods and molluscs. Molluscan species in particular have the double advantage of a simple nervous system—as few as 10,000 neurons—and giant, easily impaled neurons. Neurons of snails and other molluscs showed that the inward current evoked by depolarization is not always carried only by Na+, but may, in addition, be carried by divalent ions. Action potentials with prominent Ca2+ components are found in effector processes such as secretion, contraction and bioluminescence. Susumu Hagiwara and his coworkers found a transient potassium current in a mollusc in 1961. This current, called the A current, IA, is distinctly different from the delayed potassium current of squid axon. Found in arthropods and vertebrates as well as molluscs, IA appears in encoder neurons, which transduce stimulus voltage into repetitive spikes.5 2. TECHNIQUES AND MEASUREMENTS The foundation of all research in biophysics is experiment. Theoretical models and hypotheses suggest experiments, but only the experimental results, properly interpreted,

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Figure 4.2. The action potential evoked in a node of Ranvier by a pulse of current decreases in amplitude (upper right of panels) as the duration of the pulses (lower right) decreases. The method is shown at the top. Excitable node N1 is located between inexcitable nodes N0 and N2, used as stimulating (S) and recording (V) electrodes. Note that the action potential has a shoulder, unlike that of a squid axon. From Tasaki, 1982.

can validate or reject them. Repeated experiments are necessary for reliability of results. 2.1. Space clamp The axon, with instabilities that complicated interpretation of the data, was tamed by techniques devised by George Marmont and Kenneth Cole. To analyze the behavior of

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the membrane, the traveling action potential had to be stopped. This was accomplished by the space clamp, an axial wire pushed into the axon to carry current, together with an external electrode to measure the voltage. Guard electrodes, maintained on both sides of the central measuring electrode at equal potential to it by electronic feedback, disposed of the troublesome time-varying currents at the electrode boundaries. The space clamp simplified the problem of measuring the action potential by holding the impulse fixed in space. 2.2. Current clamp The next step was to control the current through the membrane. This was accomplished by a feedback circuit known as a current clamp. Small current stimuli, inward or outward, produced linear, electrotonic, responses in the voltage, while outward currents displayed threshold behavior. A short-duration pulse of outward current above threshold led to a large positive voltage excursion with no external current passing through the membrane, essentially a stationary action potential; see Figure 1.2 of Chapter 1. But, while propagation had been removed from the measurement, the membrane voltage continued to refuse to stand still for a measurement. 6 2.3. Voltage clamp A technique that came to be of great importance in electrophysiology was devised to control the potential. Cole developed the voltage clamp, a feedback circuit that permitted the experimenter to vary the membrane voltage at will, while monitoring the current.7 It was hoped that this would yield an experimental preparation without the threshold behavior and instability associated with the current clamp. The voltage clamp was intended to answer the question, “What current will produce a given change of potential?” To accomplish this, the membrane voltage was connected to one input of an operational amplifier. A command voltage step was connected to the other input, and the difference between it and the actual membrane voltage, the so-called error voltage, was brought to zero by a negative feedback adjustment of the applied membrane current. The new techniques revolutionized neuro-physiology and allowed the excitable membrane to be analyzed on the basis of an equivalent electrical circuit. In spite of these improvements, spatial and temporal instabilities sometimes reassert themselves in the form of what has been called the “abominable notch” 8 and oscillation. Although improvements in amplifier bandwidth and electrode design have helped, the only remedy to the notch found to be reliable is to cut the external sodium concentration in half. 2.4. Internal perfusion The axoplasm can be removed by applying mechanical pressure with a roller, and replaced with an artificial solution. An alternative method of intracellular perfusion is

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the double cannulation technique. A narrow inlet cannula and a wider outlet cannula are introduced into a length of squid axon from opposite directions. The inlet cannula is inserted into the outlet cannula. The perfusion solution washes out the axoplasm and flows continuously during the experiment. If the pH, osmolarity and electrolyte composition of the internal and external solutions are properly chosen, the axon can maintain its ability to conduct action potentials for 10 hours or longer.9 The technique of internal perfusion makes possible the study of the axon under complete control of both intracellular and extracellular ion environment. It has been applied to muscle and other cells as well as squid and other giant axons. Substitution of anions has relatively small effects, while there is a high sensitivity to the substitution of cations. 3. RESPONSES TO VOLTAGE STEPS Measurements in which the response of the excitable membrane is observed to unfold in time (as opposed to frequency) are called time-domain measurements. The frequency-dependent impedance of the membrane can also help us distinguish between competing theories of channel function. This work will be discussed in Chapter 10. 3.1. The current–voltage curves The response of the potassium system to voltage variations is shown by a steady-state current–voltage curve. The squid axon exhibits an inward rectification; depolarizing voltages produce steadily increasing K+ currents, while voltage changes in the hyperpolarizing direction increase the current only slightly. Voltage-clamp current responses yield an I–V curve, also designated the I(V) curve to emphasize that V is the independent, and I the dependent, variable. This curve contains a region of “negative resistance,” in which the slope of the I–V curve becomes negative. As the voltage is increased from rest potential to about 60 mV above it, the inward current increases. This is due to the increasing permeability of the axonal membrane to sodium ions.

3.2. Step clamps and ramp clamps Although the voltage-clamp method allows the experimenter to impose any voltage function on the axon, a conventional technique soon developed: The voltage is initially maintained at a holding potential close to the resting potential. From there it is stepped discontinuously to one of a series of steady potentials, both hyperpolarizing and depolarizing. The step is maintained long enough to bring the axon to a new steady state, and then the axon is returned to holding. The constant voltage during a step eliminates voltage as a variable at each clamp level. The currents induced by the return step are called tail currents.

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Variations on the simple step clamp include brief prepulses, which change the subsequent membrane response, and other programmed steps.

Figure 4.3. Squid axon response to a slow (0.5 V/s) and fast (50 V/s) rising ramp clamp. (A) The succession of curves shows the effect of adding 50 mM tetraethyl ammonium to the internal perfusion solution of potassium fluoride. (B) The upward succession of curves shows the effect of replacing the external seawater solution with seawater plus 100 nM tetrodotoxin. From Fishman, 1970.

Another useful voltage function is the ramp clamp, which consists of a voltage rising or falling linearly with time. Harvey Fishman pointed out that continuous current–voltage characteristics of both the early and late currents could be obtained directly by means of a ramp clamp.10 Figure 4.3 shows that this method can be used to follow the time course of the slow and fast currents without pharmacological separation by the specific inhibitors of IK , tetraethyl ammonium (TEA), and of INa , tetrodotoxin (TTX). 3.3. Repetitive firing In many physiological activities, neurons fire at regularly repeated intervals, like a dripping faucet. Repetitive firing in excitable cells occurs in such rhythmic activities as the heartbeat, respiration and circadian rhythms, but is also associated with skeletal movement, peristalsis and sensory reception. These effects in pacemaker cells are endogenous to the cell and due to membrane feedback mechanisms. Beating pacemaker neurons fire single spikes at regular intervals, whereas bursting pacemaker neurons fire bursts of spikes in a regular pattern during the depolarized phase of their membrane potential. The ionic currents responsible for repetitive neuronal activity, first studied in molluscan neurons, have also been examined in crab axons and mammalian neurons such as the cerebellar Purkinje cells of guinea pigs.11

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Figure 4.4. Data from a bursting pacemaker neuron, R15, in the mollusc Aplysia californica. In voltage-clamp data (A) voltage steps from holding potential Vh are shown as light lines and current traces as heavy lines. Voltage calibration is 45 mV; current calibrations are 500, 200, 100 and 50 nA for graphs 1, 2, 3 and 4-5 respectively; time calibration is 2.5 s. The quasisteady-state I–V curve (B) shows a region of negative slope. The inset shows bursting pacemaker potential oscillations in the unclamped cell, with calibrations 50 mV, 26 s. From T. G. Smith, Jr., 1975, 1980. Reprinted by permission from MacMillan Publishers Ltd: Nature 253:450-452 copyright 1975.

The current–voltage relationship of pacemaker cells under voltage clamp shows a region of negative slope, in which depolarization leads to an increase in inward current. This current, which tends to depolarize the cell, is regenerative between about -50 mV and -35 mV; see Figure 4.4.12 A normally silent muscle cell or axon fires spontaneously if the external calcium concentration is lowered. The different behavior of pacemaker and nonpacemaker neurons in Aplysia has been shown to be related to differences in the accumulation of potassium ions at the external membrane surface.13 The ability of an excitable membrane to generate repeated pulses makes it comparable to an electrical oscillator. A depolarizing current that excites the membrane can produce repetitive firing until the process of accommodation occurs; in toad motor fibers the time constant of accommodation varies from 10 to 300 ms.14

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The repeat intervals are determined primarily by the intensity of the stimulus and the refractory period of the cell. It has been modeled as a relaxation oscillator, which is quite different from a resonant circuit. One simple model of a relaxation oscillator is a block sliding on a rough surface, being dragged by a soft spring, the other end of which is moving at constant speed. The phase of the rhythm may be reset by a brief electric pulse above threshold.15 In the neuronal pacemaker cycle of a molluscan neuron, the K+ flux increases with increasing Ca2+ concentration.16

Figure 4.5. The electric currents of a nerve impulse, traveling from left to right, form a pair of toroidal patterns riding on the axon. The schematic cross section shows sodium ions separating from the inner and outer membrane surfaces and permeating open sodium channels. The potassium currents are not shown. From C. Hammond, 1996.

3.4. The geometry of the nerve impulse The description of an axon or muscle fiber as a cylindrical tube is in many cases inadequate. In muscle fibers, the membrane is folded inward to form T tubules, in which the excitation wave couples to the contraction mechanism. The membrane of the ribbon-shaped giant axon of the mollusc Aplysia is also riddled with folds. By increasing the surface area, such infoldings slow the electrical response of the fiber to a stimulating pulse. Nevertheless, the picture of an axon as a circular cylinder is often a useful approximation. If we take a snapshot of the electric currents of the nerve impulse along a cylindrical axon, the sodium currents of the action potential can be described as a pair of toroids riding on the axon. At the leading edge, sodium ions are rushing back into

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the excited patch of the membrane externally and forward internally. Behind the region of open channels we see sodium currents flowing forward externally and backward internally; see Figure 4.5.17 Another pair of toroids, delayed and with directions reversed, would represent the potassium currents. 4. VARYING THE ION CONCENTRATIONS Under a given set of conditions, some species of ions permeate the membrane while others are excluded. The understanding of this property of selectivity is a central problem, along with that of gating, of excitability in membranes. The effects of varying ion concentrations on voltages can be explained quantitatively by an equation derived from electrodiffusion theory by the assumption of a uniform electric field across the membrane. This equation, the Goldman-Hodgkin-Katz equation, derived in Chapter 8, allows permeabilities to be calculated for Na+, K+ and Cl-. Let us review some of the experimental findings for the various current components of the action potential. 4.1. The early current The action potential in squid axon is completely abolished by the removal of external sodium ions. When Hodgkin and Katz changed the external solution from seawater to a dextrose solution free of Na+, the action potential flattened out in two minutes, returning slowly after the reapplication of seawater; see Figure 3.2 of Chapter 3.18 Note that the loss of Na+ also lengthens the time for the voltage to rise to a peak. In 1949, when Hodgkin and Katz made these measurements, the effect of varying the internal Na+ concentrations could not be determined, but they assumed that the action potential overshoot is determined solely by the Na+ concentration-ratio and given by the Nernst equation for sodium ions, VNa = (RT/zF) ln ([Na]o/[Na]i). Since the internal sodium-ion concentration was known to be approximately one-tenth that of seawater, the overshoot expected from the Nernst relation is about 58 mV, roughly consistent with the overshoot observed in seawater. Later experiments with internally perfused axons showed that, when part of the internal potassium was replaced with sodium, the action potential overshoot, while decreasing as [Na]i was increased, remained well above the values predicted by the Nernst relation.19 When the sodium ions are replaced by lithium, the currents are almost identical over time periods of less than one hour. Tasaki and collaborators also found that sodium ions could be replaced by a number of nitrogenous univalent cations in intracellularly perfused squid axons without losing excitability. These include hydrazinium ion, H2N-NH3+, hydroxylamine, guanidine and aminoguanidine. Experiments with these ions require increased external divalent ion concentrations to prevent membrane depolarization.20

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4.2. The delayed current Changes in the potassium concentration of the external solution affect the resting potential primarily. In a K +-free solution the resting potential is more negative than in seawater, while the peak of the action potential remains the same, thus increasing the height of the action potential from its base. If the K+ concentration is doubled, from 10 to 20 mM, the action potential shrinks both at the bottom and at the top. Axons become inexcitable by increases in external K+ long before they are completely depolarized. The different selectivities of the early and delayed currents pointed to two separate mechanisms. This was later confirmed with the discovery of different ion channels. The ammonium ion is unusual in that it permeates both the early and delayed channels. The substitution of Rb or Cs ions for K greatly prolongs the action potential. This suggests that the mobilities of these ions in the K channel is lower than that of K+ and therefore, according to a macromolecular interpretation, that they are more strongly bound by charged negative sites within the channel.21 4.3. Divalent ions A number of different types of calcium currents have been found in vertebrate as well as invertebrate cells, indicating the presence of different calcium channels. These channels are also permeable to other divalent ions, such as Ba2+ and Sr2+. The study of calcium channels is complicated by the low intracellular Ca2+ concentration, 10-7 M or less.22 An unusual property of calcium channels is their sensitivity to changes in the internal Ca2+ concentration. Increases in [Ca2+]in reduced the calcium conductance, decreasing the amplitude of subsequent calcium action potentials.23 A further complication is a potassium current activated by intracellular injection of calcium.24 Because of the high sensitivity of many intracellular enzymatic reactions to [Ca2+]in, a rise in it may result in cell death. Thus it is not surprising that cells have a high buffering capacity at physiological levels of [Ca2+]in, binding 99.95% of imposed Ca loads. Most of this buffering capacity appears to reside on the mitochondria and endoplasmic reticulum.25 4.4. Hydrogen ions Since metabolic processes in the cell produce acid, cells must have a mechanism for regulating pH. The regulation of the internal H+ concentration is similar to, but more complicated than, the sodium–potassium pump system.26 The early current in frog node is strongly dependent on hydrogen-ion concentration, but in a manner that is contrary to expectations. Since the membrane permeability calculated from reversal potentials for H+ is much greater than that for Na+, one might expect from the Independence Principle (see Chapter 9) an increase in external [H+] (lower pH) to increase the Na+ conductance, gNa. Actually, as Figure 4.6 shows, both gNa and gK are greatly reduced at low pH. In a model that approximately

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agrees with the data, Na channels are “blocked” by protonization of a single acid group with a pKa of 5.2 and K channels by a group with a pKa of 4.4.27

Figure 4.6. Titration of sodium (filled circles) and potassium (open circles) conductances of frog node of Ranvier as the pH of the external solution is varied. From Hille, 2001. Reproduced from the Journal of General Physiology 1968, 51:221-226 and 1973, 61:669-686. Copyright 1968, 1973 The Rockefeller University Press.

Contrary to the permeability calculated from the reversal potential and the Goldman–Hodgkin–Katz equation, the measured permeability of sodium channels to protons is minute, consistent with the slow dissociation of the acid groups with which the protons associate as they move through the channel. Possible roles for protons in the conduction mechanism have been proposed; see Chapters 14 and 20. 4.5. Varying the ionic environments To reduce the complexity of the axon system, the question arose, How simple can the electrolyte solutions be made without losing the axon’s ability to develop action potentials? The answer is that the salt of a single divalent cation outside and a single monovalent cation is sufficient. Calcium, strontium or barium are favorable external cations; the internal cation can be any alkali metal, tetramethylammonium, tetraethylammonium, choline or hydrazine, among others. The complete replacement of both intracellular potassium and extracellular sodium ions does not suppress axonal excitability. Magnesium cannot be used as the external cation without some calcium. The explanation of this is that the tendency of magnesium toward hydration causes the membrane colloid to swell. The difference in threshold concentration among ions of alkali metals is similarly attributed to their ability to loosen the compact structure of the membrane macromolecules (the voltage-sensitive ion channels). These cations form the following sequence according to their “depolarizing power”: K  Rb > Cs > NH4 > Na > Li

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The similarity in the stereochemical properties of K+ and Ca2+ may account for the great depolarizing power of K+. Both have a coordination number of 8, while Na+, with a coordination number of 6, would be less effective in displacing the Ca2+ in the membrane protein. The “bi-ionic” action potentials, with only two cation species, are characterized by long duration, abrupt termination and high resistance. A molecular interpretation is that, in the resting state, the membrane macromolecules are cross-linked by calcium ions; at a depolarization, some of these calcium bridges are broken and the membrane swells, reducing its resistance.28 5. MOLECULAR TOOLS Neurotoxins are useful tools for identifying and isolating the Na+ and other ion channels. During the course of evolution, certain organisms have developed these molecules as specialized weapons for defense. 5.1. The trouble with fugu One neurotoxin is tetrodotoxin (TTX), which is found in the liver and gonads of the pufferfish. The pufferfish is eaten as a delicacy in Japan, where it is called fugu. Although the fish are prepared in restaurants by specially trained cooks, fatal accidents occasionally happen. Even at micromolar concentrations, TTX effectively destroys the ability of sodium channels to conduct ions. Its action is reversible; when the TTX is washed off, the channels conduct normally again. A marine microorganism produces a similar toxin, saxitoxin, and a Central American tree frog produces yet another, chiriquitoxin. These toxins may have a common evolutionary origin—they may be synthesized in symbiotic bacteria within these hosts.29 TTX, STX and CTX inhibit the sodium current only when applied to the outside of the axon. Their structures are similar in that they all contain a guanidinium group, based on H2N+=C(NH2)2, a highly resonant, planar, positive ion. Because TTX and its congeners bind 1:1 to the voltage-sensitive sodium channel, they are used to measure the density of channels in various membranes. For example, the nodal regions of a myelinated rat axon have 700 Na channels per m2 and the (unmyelinated) squid axon has 330 Na channels per m2.30 These neurotoxins have been called “channel blockers” because in the approach to channels that considers them to be water-filled pores, the toxins are interpreted simply as plugs. In this approach the TTX molecule is thought of as binding to the channel and mechanically blocking it. Experiments show, however, that the action of these toxins depends critically on their molecular structure.31 The guanidinium group apparently has a special role, probably involving its positive charge. An alternative explanation is presented in Chapter 16. Figure 4.7 depicts the chemical structures of TTX and STX, as well as those of the local anesthetic procaine and the tetraethylammonium ion, a quaternary ammonium ion; the latter two ions inhibit ion conduction in potassium channels.32

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Figure 4.7. Molecular structures of tetrodotoxin, saxitoxin, procaine and tetraethyl ammonium ion. From Hille, 2001.

5.2. Lipid-soluble alkaloids There are other neurotoxins with various effects on the Na channel. The lipid-soluble alkaloids, aconitine and veratridine, found in species of the buttercup and lily families respectively, are sodium-channel poisons. Another, batrachotoxin, from Colombian arrow-poison frogs, eliminates the inactivation response from the membrane. Other classes of lipid-soluble activators of Na channels are grayanotoxins and pyrethroids such as allethrin; see Figure 4.8.33 Because these toxins facilitate the opening and delay the closing of Na channels, they are called Na-channel agonists.34 5.3. Quaternary ammonium ions No specific neurotoxin has been discovered for the potassium channel. However, certain ions fairly effectively impede the K+ channel when the potassium inside the axon is replaced with them. They include quaternary ammonium ions such as tetraethyl ammonium, TEA+, and tetramethyl ammonium, TMA+. These ions form bonds that are more stable than the bonds that the permeant K+ ions form with the pathway and compete for ion sites.

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Figure 4.8. The alkaloid toxins batrachotoxin, veratridine, aconitine and grayanotoxin that cause Na channels to remain open. From Conley and Brammar, 1999.

The TEA+ ion prolongs the falling phase of action potentials by interfering with the potassium current. When enough sites are filled with TEA+, the K+ permeation pathway no longer exists.35 Other agents that interfere with the potassium current are Cs+, Ba2+ , 4-aminopyridine and other organic cations with quaternary nitrogen atoms. The aqueous pore model describes the action of these ions as “blocking” or “ plugging” the channel.36 These are macroscopic, not molecular, concepts. These substances give us the means of suppressing one or both channels at will. When both ion conductances are suppressed, there remains a tiny displacement current, which because of its transient asymmetrical nature is assumed to be related to the dipolar shifts associated with the opening of the channel, and so is called the gating current; see Section 2.1 of Chapter 9. 5.4. Peptide toxins The venoms of scorpions and sea anemones contain mixtures of polypeptide toxins. A painful sting can lead to paralysis, cardiac arrhythmia and even death. When purified and sequenced, they are found to be single polypeptide chains held in compact structures by internal disulfide bonds. The scorpion peptides are 60-76 amino acids long, and the coelenterate peptides, 27-51. One class of these toxins, the -NaTx toxins, slow the inactivation of Na channels, leading to action potentials of much longer duration. Another class, the NaTx toxins, shift the voltage dependence of activation. By causing the channels to remain open at the normal resting potential, they produce long, repetitive trains of action potentials. A third class, members of the -KTx family, block potassium channels.37

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6. THERMAL PROPERTIES Modeling the action potential in terms of electrical circuits has its limitations. To gain more information about macromolecular changes occurring in axon membranes, electrical studies must be supplemented by experiments on nonelectrical signs of nervous excitation. These include thermal, optical and mechanical effects. Interest in the effect of temperature on bioelectric potentials goes back to Bernstein, who in 1902 assumed that ion permeability varied with temperature.38 Many investigators since Bernstein had studied the effect of temperature on electrical activity, but while they agreed that the resting potential has a low temperature coefficient, the data on spike amplitude was in conflict. Conventional studies of temperature dependence of axon parameters were limited to two temperatures. This suffices to calculate the quantity Q10, defined as the ratio by which a dependent variable increased when the temperature was raised by 10°C. However, in a number of studies the temperature is treated as a continuous variable. 6.1. The effect of temperature on electrical activity To resolve the question of temperature dependence, Hodgkin and Katz examined the effect of temperature on the giant axon of the squid.39 Changing the temperature of the seawater in which the axon was bathed, they found that their measurements were reversible as long as the temperature did not exceed 35°C, above which “the axons tended to fail progressively.” Propagation in squid axon is abolished at the heat-block temperature of about 38°C and may be restored when the fiber is cooled. The lowest temperature they achieved without risk of damaging the fiber was 1°C. In their experiments, the resting potential diminished only slightly while the spike amplitude dropped with increasing rapidity as the temperature was raised from 3 to almost 40°C. The positive phase of the action potential increases to a maximum at 2025°C and decreases again. The variation of action potential shapes shows a narrowing of the spikes with increasing temperature. The rate of fall of the action potential has a larger temperature coefficient than the rate of rise, although the Q10s of the conductances are similar. A. Krogh suggested that the transfer of sodium through cell membranes involves a transient reaction with membrane molecules, not a simple diffusion through pores.40 Thus, as Hodgkin and Katz have pointed out, the temperature coefficient for sodium ions may be larger than that for potassium and chloride. Spyropoulos found a cold-block temperature, the lower limit of excitability at which voltage spikes are abolished, at about 1°C in the squid Loligo vulgaris.41 However, cold block is species-dependent. In the squid Dorytheutis bleekeri, F. Kukita and S. Yamagishi found reversible cold block to occur below -20°C.42

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6.2. The effect of temperature on conduction speed The effect of temperature on the conduction velocity of axons in a nerve was studied in 1967 by R. A. Chapman. Figure 4.9 shows that conduction speed in Loligo vulgaris varies continuously, rising from cold block to a maximum at about 32 °C and decreasing sharply at the approach of heat block.43

Figure 4.9. Effect of temperature on conduction speed. The solid and dashed line is a model fit to the data. From R. A. Chapman, 1967. Reprinted by permission from MacMillan Publishers Ltd: Nature 213:1143-1144 copyright 1967.

6.3. Excitation threshold, temperature and accomodation One set of temperature measurements was of the threshold for excitation. Study of the threshold showed that the product of the stimulating current I and the duration of the pulse, t, is reasonably constant. This is the total charge Q0 = I t. For squid axon, Q0 was found by Rita Guttman to be 1.4 × 10-8 C/cm2, nearly independent of the temperature.44 Experiments with slowly rising deplorizations of squid axons in external media with lowered divalent ion concentrations show a relationship between accommodation and repetitive firing.45

6.4. Stability and thermal hysteresis Internal perfusion of giant axons made possible the further exploration of membranes.

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Figure 4.10. Cyclic temperature changes induce hysteresis in membrane potential. The top diagram shows the arrangement of inlet and outlet cannulas, ground (E) and recording (R) electrodes, and the thermocouple (TC). A cycle of temperature variation took about 3 minutes. Electrolyte compositions are given for loops A and B. From Y. Kobatake, 1975.

Under ionic conditions in which the cations of the external solution were exclusively divalent and the internal cations monovalent, stable action potentials were evoked. The gradual introduction of sodium ions into the external solution resulted in giant oscillations, culminating in jumps between the active and resting states.46 When the temperature of these axons was varied slowly, hysteresis loops were observed in the membrane potential; see Figure 4.10. These results suggest that a phase transition was taking place in the membrane system. The implications of this will be discussed in Chapters 14 to 21. 6.5. Temperature effects on current–voltage characteristics Voltage ramp clamp recordings of squid axon responses show the changes in current–voltage characteristics for slow and fast ramps as the temperature is varied from 5 to 25°C; see Figure 4.11. Slow-ramp speeds of 0.5, 1, 2 and 5 V/s and varying fastramp speeds, as labeled, were used. For comparison, step clamp data are also shown for the 10°C experiments.47

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Figure 4.11. Effect of temperature and ramp rate (marked) on axonal I(V) curves. The voltage was ramped (left) from resting potential RP to 150 mV above RP, and (right) from a hyperpolarization of -30 mV to 120 mV; see insets. From Fishman, 1970.

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6.6. Heat pulses modify ion currents A perturbed molecular system will adjust to its thermodynamic equilibrium in a process called relaxation. Studies of rate phenomena can often be described by a linear relaxation process characterized by one or more relaxation times. However, nonlinear and even irreversible processes can be investigated by relaxation methods, such as those induced by temperature jumps.48

Figure 4.12. Response of the Na+ current in frog node to a temperature jump induced by a laser-generated heat pulse. The perturbed current is superimposed on a control record. The temperature jump occurred 0.7 s after the depolarizing step of 25 mV. From Moore, 1975.

L. E. Moore and collaborators investigated ion-current changes in frog node induced by a laser-generated heat pulse applied to the nodal region.49 A sudden change in temperature defines a new state to which the ion conductances relax. The temperature jumps were estimated to be about 2-3 C°. The data were interpreted by linear relaxation theory applied to the Hodgkin–Huxley formalism, in which ion conductances depend on electric field, temperature, pressure and calcium concentration (see Chapter 9). Relaxation due to temperature changes may be expected to have the same time constants as relaxation due to voltage steps. Figure 4.12 shows the current relaxation induced by a temperature jump during the peak of the inward Na+ current.50 In steady-state measurements, the effect of heating was to increase the delayed currents. The inward transient currents increased at small depolarizations but decreased at large depolarizations. In the relaxation experiments, the sodium currents initially increased, then decreased, as Figure 4.12 shows. The responses suggest a process involving multiple relaxation times. Together with the results of other experiments, these findings are consistent with a membrane system dependent on voltage, temperature and Ca2+ concentration.

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6.7. Temperature and membrane capacitance Yoram Palti and William J. Adelman, Jr. used the ramp clamp as a method for determining the capacitance of the excitable membrane.51 From the ramp rate and the measured current they obtained data on the dependence of membrane capacitance with temperature in squid axon. Their results showed that membrane capacitance increases with temperature, rising steeply as the temperature is raised to near 40°C. An interpretation of this remarkable result in terms of a ferroelectric phase transition model is discussed in Chapter 16, Section 6.4. 6.8. Heat generation during an impulse Excitation in both invertebrate and vertebrate nerve exhibits a diphasic variation in temperature. Spyropoulos found that heat is produced during an action potential and absorbed after the membrane had returned to its resting value.52 Richard D. Keynes and collaborators showed that the excitation process of a nervous tissue involves an exothermic reaction.53 The observations that an action potential is accompanied by first a generation and then an absorption of heat54 suggest that membrane structures undergo a transition to a more highly ordered state during an action potential. Similar patterns occur in transitions of ferroelectric materials from a paraelectric to a ferroelectric state and the helix–random coil transitions of macromolecules. 7. OPTICAL PROPERTIES Since the conductance changes associated with an action potential must be associated with conformational changes in ion channels, it is useful to explore nonelectrical signs that may reflect these changes. Optical studies can provide independent information on these macromolecular transformations.55 Here we will discuss voltage-sensitive birefringence in excitable membranes and their sensitivity to ultraviolet light. Hervé Duclohier has reviewed techniques that combine fluorescence measurements with simultaneous electrical measurements in cell membranes and reconstituted systems.56 Light scattering spectroscopy experiments will be discussed in Chapter 11, Section 4.7. 7.1. Membrane birefringence The refractive index n of a material is the ratio of the speed of light in vacuum to that in the material. In birefringence or double refraction, the value of n depends on the direction of polarization of the light. Experiments by L. B. Cohen and collaborators demonstrated changes in turbidity and birefringence in excitable membranes during an action potential.57 The birefringence responses are diphasic: a decrease in light intensity is followed by an increase. The detection of optical signals from vitally stained nerve fibers made it possible to record optical signals simultaneously with electrical signals.58

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The layer of the axon directly under the axolemma consists of longitudinally oriented filamentous material, including neurofilaments and microtubules. Dye studies show that these filaments exhibit positive uniaxial birefringence, which undergoes rapid changes during the action potential. An interpretation of these results is that an initial depolymerization of filaments into monomers is followed by its reverse, polymerization.59 7.2. Ultraviolet effects Irradiation of frog node of Ranvier with ultraviolet light in the wavelength range of 240310 nm raises the stimulation threshold and diminishes the action current. The fractional change of the action potential decreases linearly with the duration of irradiation. Irradiation of the internode was ineffective. The effect was ascribed to a specific interference with a UV-sensitive excitation process.60 Voltage clamp experiments showed that this process follows the exponential kinetics of a first-order reaction. Ultraviolet radiation produces an irreversible shift of the steady-state sodium inactivation curve toward more negative voltages. The spectral sensitivity peaks at about 280 nm. The effect of UV irradiation does not alter the ionic selectivity or sensitivity to tetrodotoxin, and does not depend on temperature. The steady-state potassium current is left almost unchanged. The interference of UV with sodium channels is an all-or-none process in which a single photon disables one channel, while the remaining channels function normally. The effect of UV on the channel, referred to as a “selective blocking,” is subject to modification by external ion concentration changes. The UV sensitivity of the sodium system is increased by an increase of the external Ca2+ or H+ concentration, while hyperpolarizing the membrane reduces the UV sensitivity.61 UV irradiation cumulatively and irreversibly reduced the asymmetrical displacement (gating) currents of the sodium system without affecting its time constants. Increases in leakage and capacity currents became pronounced ~5 min after high doses of UV were applied.62 8. MECHANICAL PROPERTIES We have seen above that observations of light scattering have suggested a swelling of the axon during the passage of an action potential. Conversely, mechanical stimulation of an axon can lead to the production of action potentials. 8.1. Membrane swelling Tasaki and Iwasa recorded the mechanical response of the axon surface with a piezoelectric probe made of polyvinylidene fluoride (PVDF), a synthetic polymer. A narrow sheet of the piezofilm, maintained in tension by a nylon thread, contacted the axon by way of a few attached bristles (B). The pressure sensor was placed immediately above the internal wire that recorded the action potential; see Figure 4.13.63

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The transient pressure rise at a squid axon produced an electric signal in the piezofilm that was averaged over 500-2000 trials to improve the signal-to-noise ratio. The pressure increase of about 0.1 Pa (1 dyn/cm2) was immediately followed by a decrease, representing a shrinkage of the axon.

Figure 4.13. Mechanical response of a squid giant axon during passage of an action potential. (Left) Schematic diagram of the experimental setup, showing the stimulating (S) and recording (R) electrodes. (Right) The top trace shows the rise and subsequent fall of the axonal pressure. The bottom trace shows the action potential (peak about 110 mV above resting potential). From Tasaki and Iwasa, 1983.

Membrane swelling was also observed by optical means, in which light reflected from gold particles on the axon surface was measured. The surface displacement was about 0.5 nm.64 8.2. Mechanoreception Cells such as arterial baroreceptors and the vestibular hair cells of the ear are sensitive to external pressure and vibration. These mechanoreceptor cells contain stressactivated (or stress-inactivated) channels, in which the probability of channel opening is a function of the applied mechanical stress. The resulting receptor potential is translated by the cell into action potentials firing at variable frequencies. The adaptation found in the molecular mechanisms of some mechanoreceptors maximizes their sensitivity over a broad stimulus domain.65

NOTES AND REFERENCES 1. L.W. Williams, The Anatomy of the Common Squid, Loligo Pealii (Leseur), Leiden, 1909; John Z. Young, Cold Spring Harbor Symposia on Quantitative Biology 4: 1-6, 1936. 2. Michael R. Guevara, in Nonlinear Dynamics in Physiology and Medicine, edited by Anne Beuter, Leon Glass, Michael C. Mackey and Michèle S. Titcombe, 2003, 87-121. With kind permission of Springer Science and Business Media. 3. Kenneth S. Cole, Membranes, Ions, and Impulses, University of California, Berkeley, 1972, 145-148. 4. Reprinted from Ichiji Tasaki, Physiology and Electrochemistry of Nerve Fibers, Academic Press, New York, 1982, 37-64, by permission from Elsevier. 5. John A. Connor, in Molluscan Nerve Cells: From Biophysics to Behavior, edited by John Koester and John H. Byrne, Cold Spring Harbor Laboratory, 1980, 125-133. 6. G. Marmont, J. Cell Comp. Physiol. 34: 351-382, 1949. 7. K.S. Cole, Arch. Sci. Physiol. 3:253-258, 1949.

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8. Cole, ref. 3, 325ff. 9. Irwin Singer and Ichiji Tasaki, in Biological Membranes: Physical Fact and Function, vol. 1, Academic Press, London, 1968, 347-410. 10. H. M. Fishman, Biophys. J. 10:799-817, 1970. 11. Thomas G. Smith, Jr., in Koester and Byrne, Molluscan Nerve Cells: From Biophysics to Behavior, Cold Spring Harbor Laboratory, 1980, 135-143; Rodolfo Llinás, in Koester and Byrne, 145-155. 12. Thomas G. Smith, Jr., Jeffery L. Barker and Harold Gainer, Nature 253: 450 -452, 1975; Smith, op. cit., 136. 13. Douglas Junge, Nerve and Muscle Excitation, Sinauer Associates, Inc., Sunderland, MA, 1981, 115132. 14. J. J. B. Jack, D. Noble and R. W. Tsien, Electric Current Flow in Excitable Cells, Clarendon Press, Oxford, 1983, 305-378. 15. Tasaki, 114-129. 16. Anthony L. F. Gorman, Anton Hermann and Martin V. Thomas , in Koester and Byrne, Molluscan Nerve Cells: From Biophysics to Behavior, Cold Spring Harbor Laboratory, 1980, 169-180. 17. Reprinted from Constance Hammond, Cellular and Molecular Neurobiology, Academic Press, San Diego, 1996, 150, with permission from Elsevier. 18. A. L. Hodgkin and B. Katz, J. Physiol. 108:37-77, 1949; Constance Hammond., Cellular and Molecular Biology, Academic Press, San Diego, 1996, p. 119. 19. Tasaki, 140-143 and 216-218. 20. Tasaki, 211f. 21. Tasaki, 273. 22. S. Hagiwara and G. Yellen, in Koester and Byrne, Molluscan Nerve Cells: From Biophysics to Behavior, Cold Spring Harbor Laboratory, 1980, 33-40. 23. D. Tillotson, in Koester and Byrne, Molluscan Nerve Cells: From Biophysics to Behavior, Cold Spring Harbor Laboratory, 1980, 41-48. 24. H. Dieter Lux, in Koester and Byrne, Molluscan Nerve Cells: From Biophysics to Behavior, Cold Spring Harbor Laboratory, 1980, 105-114. 25. Floyd J. Brinley, in Koester and Byrne, Molluscan Nerve Cells: From Biophysics to Behavior, Cold Spring Harbor Laboratory, 1980, 73-80. 26. Roger C. Thomas, in Koester and Byrne, Molluscan Nerve Cells: From Biophysics to Behavior, Cold Spring Harbor Laboratory, 1980, 65-72. 27. Bertil Hille, Ion Channels of Excitable Membranes, Sinauer, Sunderland, Mass., 2001, 476f. 28. Tasaki, 232-255. 29. Hille, 63. 30. Hille, 396. 31. C. Y. Kao and S. E. Walker, J. Physiol. 323, 619, 1982. 32. Hille, 63. 33. Reprinted from Edward C. Conley and William J. Brammar, The Ion Channel FactsBook: Voltage Gated Channels, Academic, San Diego, 1999, 822, with permission from Elsevier. 34. Hille, 641 ff. 35. C. M. Armstrong, J. Gen. Physiol. 50:491-503, 1966; 54:553-575, 1969; 58:413-437, 1971. 36. Hille, 65, 503-537. 37. Hille, 635-645. 38. J. Bernstein, Pflüg. Arch. ges. Physiol. 92:521, 1902. 39. A. L. Hodgkin and B. Katz, J. Physiol. 109:240-249, 1949. 40. A. Krogh, Proc. Roy. Soc. B 133:140, 1946. 41. C. S. Spyropoulos, J. gen. Physiol. 48:49-53, 1965. 42. F. Kukita and S. Yamagishi, Biophys. J. 35:243, 1981. 43. R. A. Chapman, Nature 213:1143-1144, 1967. 44. Rita Guttman, in Biophysics and Physiology of Excitable Membranes, edited by W. J. Adelman Jr., Van Nostrand Reinhold, New York, 1971, 320-336. 45. Eric Jacobsson and Rita Guttman, in The Biophysical Approach to Excitable Systems, edited by William J. Adelman, Jr. and David E. Goldman, Plenum, New York, 1981, 197-211. 46. Y. Kobatake, in Membranes, Dissipative Structures, and Evolution, edited by G. Nicolis and R. Lefever, John Wiley, New York, 1975, 319-340. 47. Harvey M. Fishman, Biophys. J. 10:799-817, 1970.

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48. M. Eigen and L. de Maeyer, in Investigation of Rates and Mechanisms of Reactions, Part II, edited by S. L. Friess, E. S. Lewis and A. Weissberger, Interscience, New York, 1963, 895-1054. 49. L. E. Moore, J. P. Holt, Jr. and B. D. Lindley, Biophys. J. 12:157-174, 1972. 50. Reprinted from L. E. Moore, Biochim. Biophys Acta 375:115-123, 1975, with permission from Elsevier. 51. Y. Palti and W. J. Adelman, Jr., J. Membr. Biol. 1:431-458, 1969. 52. C. S. Spyropoulos, J. Gen. Physiol. 48: 49-53, 1965. 53. J. V. Howarth, R. D. Keynes and J. M. Ritchie, J. Physiol. 194:745-793, 1968. 54. I. Singer and I. Tasaki, in Biological Membranes: Physical Fact and Function, volume 1, edited by Dennis Chapman, Academic, London, 1968, 347-410. 55. L. B. Cohen and D. Landowne, in Biophysics and Physiology of Excitable Membranes, edited by W. J. Adelman Jr., Van Nostrand Reinhold, New York, 1971, 247-263. 56. Hervé Duclohier, J. Fluorescence 10:127-134, 2000. 57. L. B. Cohen, R. D. Keynes and B. Hille, Nature (London) 218:438-441, 1968. 58. Tasaki, 305. 59. Tasaki,308-310. 60. Hans-Christoph Lüttgau, Pflügers Archiv 262:244-255, 1956. 61. J. M. Fox, Pflügers Archiv 351:287-301; 303-314, 1974. 62. J. M. Fox, B. Neumcke, W. Nonner and R. Stämpfli, Pflügers Archiv 364:143-145, 1976. 63. I. Tasaki and K. Iwasa, in Structure and Function in Excitable Cells, edited by D.C. Chang, I. Tasaki, W. J. Adelman, Jr. and H. R. Leuchtag, Plenum, N. Y., 1983, 307-319. With kind permission of Springer Science and Business Media. 64. K. Iwasa and I. Tasaki, Biochem. Biophys. Res. Comm. 95:1328-1331, 1980. 65. Owen P. Hamill and Don W. McBride, Jr., NIPS 9:53-59, 1994.

CHAPTER 5

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The electric behavior of excitable membranes reflects the properties of the ion channels embedded in their lipid matrix and the composition of the aqueous media. To further our goal of explaining the behavior of the voltage-sensitive ion channels, we will review the physics of condensed materials with an eye to possible clues to the understanding of these channels. If we are to have a fair start in understanding their behavior in basic physical terms, we should be sure we understand how the behavior of simpler forms of matter is explained by condensed matter physics. We will clarify our ideas about the quantum basis of matter and review the way material structures undergo cooperative changes, phase transitions, in their largescale conformation. From there we will explore the way these macroscopic phenomena arise from the atomic structure of matter. These elementary principles should help us apply the principles of physics to the configurational transitions that ion channels undergo. 1. THE LANGUAGE OF PHYSICS Newton's classical mechanics has now been supplanted by quantum mechanics, which has become the accepted formulation for the analysis of microscopic systems and even some large systems, such as superconductors, superfluids and quantum interference devices. As B. S. Chandrasekhar1 put it, Just as English is the language of Shakespeare, quantum mechanics is the language of physics. Quantum mechanics applies to the microscopic aspects of ion channels along with all other forms of matter. 1.1. The Schrödinger equation We saw in Chapter 1 that the application of Planck's relation, E = ћ 7, to the electron's motion saved atomic physics from the deep contradictions of the classical theory. The electronic orbitals were quantized, and energy could be emitted from (or absorbed by) an atom only in whole quanta. Quantum mechanics developed from the realization that particles of matter had wave properties, analogous to light. Erwin Schrödinger wrote a wave equation for the wavefunction 5 of a nonrelativistic particle in a potential U(r), 89

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where the Hamiltonian operator, representing the total energy of the system, is given by (1.2)

The Laplacian operator /2 reduces to d2/dx2 in one dimension. The time-independent Schrödinger equation for a stationary system with energy E is (1.3) The first step in solving any dynamical problem in quantum mechanics is usually to write down the appropriate Hamiltonian for the system. The first term on the right of Equation 1.2 corresponds to the kinetic energy of the system, the second term the potential energy. Since Schrödinger's equation is a differential equation, its solution involves integrations. Integration operations in quantum mechanics are often represented in the form of diagrams known as Feynman diagrams. In any bounded system, Schrödinger's equation has solutions for only discrete values of E, the eigenvalues En. To each eigenvalue corresponds a function that solves Schrödinger's equation, called an eigenfunction. Different eigenfunctions may have the same eigenvalue, often as a result of the symmetry of the system; these functions are called degenerate. The number of degenerate eigenfunctions generally increases as the energy of the system rises. 1.2. The Uncertainty Principle Werner Heisenberg arrived at an alternative formulation of quantum mechanics by asking, What happens to the electron between two discrete states such as Bohr orbits? He showed that any attempt to follow the electron by, say shining a light beam on it, would severely perturb the position of the electron, perhaps even jar it completely out of the atom. Since we can never follow the electron's path, we should be willing to admit that the concept of a path between states is without foundation. Heisenberg proceeded to bracket the limits of our ignorance in his famous Uncertainty Principle: The position x and momentum p of a particle can not be measured precisely, but only to within uncertainties x and p, which must be large enough so that their product is no smaller than a quantity of the order of Planck's constant. A similar relationship exists between the uncertainties in energy and time.

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The interpretation of the wavefunction 5 is still controversial, but it is agreed that the square of the absolute value of this complex number, |5|2, must correspond to the probability of finding the electron in a given quantum state, as Max Born found. 1.3. Spin and the hydrogen atom Spectral and magnetic measurements showed that another number, beyond x and p, was needed to describe the electron's motion. This is the electron's spin angular momentum. Measured in units of ħ = h/2%, spin had the unexpected property of being expressed in half integers, such as 0, ±½, ±1, etc. The exclusion principle of Wolfgang Pauli stated that only one electron could exist in a state described by a set of quantum numbers. In this way quantum mechanics provided a way of explaining the hydrogen atom, and eventually all the atoms of the periodic table. The hydrogen atom is described by four quantum numbers, n, l, m and s, called respectively the principal, azimuthal, magnetic and spin quantum numbers. The Pauli exclusion principle states that no two electrons can exist in the same quantum state, that is, with the same values of n, l, m and s. The lowest energy state consistent with the exclusion principle, its most stable state, called the ground state, is used to characterize not only hydrogen but all atoms. With the conventional notation of s, p, d and f to designate states with l = 0, 1, 2 and 3, respectively, an atom's electronic configuration can be described compactly. A 1s orbital is a spherically symmetric distribution with n = 1 and l = 0. A 2s orbital, with n = 2, has the same symmetry but has a larger radius and more energy. As an example, we can write the ground state of a sodium atom as 1s22s22p63s1, meaning that the n = 1 "shell" contains two electrons (of opposite spin), the n = 2 shell has two electrons with l = 0 and six electrons (two in each of the three space directions) with l = 1, and the n = 3 shell has one electron. Note that the removal of the n = 3 electron by ionization leaves the very stable neon configuration, 1s22s22p6. The ionization of two elements and the electrostatic attraction of their ions accounts for their ionic bonding. Consider, for example, the sodium chloride molecule. Since the electron configuration of sodium in the ground state is 1s22s22p63s1, it has a single 3s electron outside a closed subshell, which is weakly bound to the atom. It takes only 5.1 eV to remove this electron,2 leaving the atom a positive ion, Na+. The chlorine atom in its ground state has the configuration 1s22s22p63s23p5; the neutral atom lacks one electron of filling the 3p subshell. The addition of one electron lowers the atom's energy by 3.8 eV. Thus the addition of only 1.3 eV suffices to transfer an electron from an Na to a Cl atom, and this energy is easily supplied by the electrostatic attraction, since the Coulomb potential energy equals -1.8 eV at a separation of 4.0 Å.3 By 1927 the concepts of quantum mechanics were applied to molecules. The atomic orbitals overlap to form a new orbital called a molecular orbital, which like an atomic orbital contains two electrons of opposite spin. The description of diatomic molecules includes a principal quantum number n and a quantum number , which gives the component of angular momentum along the axis joining the two nuclei. In analogy to the atomic terminology, states with  = 0, 1, and 2 respectively are designated ), % and .

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A covalent bond in which the overlap is concentrated along the internuclear axis, as in the hydrogen molecule H2, is called a sigma ()) bond. The combination of different s and p orbitals forms hybrid orbitals. The combination of one 2s orbital and two 2p orbitals, called sp2 hybrid orbitals, form three lobes lying in a plane and an unhybridized 2p orbital perpendicular to the plane. These orbitals may form double bonds: a sigma bond lying in the plane of the two nuclei, and a bond, called a pi (%) bond, formed by the overlapping p orbitals. The concept of quantum mechanical resonance is of great importance in determining the structure of multiatomic molecules. One of the most important applications of quantum mechanics is the study of condensed (i.e., non-gaseous) matter. In later chapters we will see that a number of topics of condensed-matter physics are of special significance to the study of ion channels. 1.4. Identical particles—why matter exists One of the surprising ways in which quantum mechanics differs from classical mechanics is in the way particles interact when they bounce off each other in scattering experiments. Suppose a stream of  particles, which are helium nuclei, are fired at a target of oxygen nuclei. It is convenient to think of this collision as happening in a moving frame of reference, so that the particles move toward each other while the center of mass remains fixed. After the collision they will be moving in opposite directions again, but at some angle  from the original directions, which we may take to be 0° for the  and 180° for the oxygen nucleus. Suppose detectors mark the arrival of each particle without distinguishing whether it is an  or an oxygen nucleus. Then the probability of a particle arriving at a particular one of the detectors is simply the sum of two probabilities: the probability that the particle is scattered through an angle  (and is an ) plus the probability that the particle is scattered through an angle 180° -  (and is an oxygen nucleus). The same thing happens when the target is hydrogen, carbon, or anything else—except helium. When helium nuclei bounce off helium nuclei, the results are different, by as much as a factor of two when  is 90°. That is because with identical particles there is no way in principle to tell which of the particles entered the detector. So quantum mechanics tells us that we must add the wavefunction amplitudes 5, not the probabilities 52. When we do that, we obtain results that agree with experiment. However, the particles can interfere with either the same phase or with opposite phase. This means that there are two kinds of particles, called Bose and Fermi particles. With Bose particles, such as photons and  particles, the amplitude of the particle that is exchanged simply adds to the amplitude of the particle that goes directly into the detector. However, with Fermi particles, such as electrons and sodium ions, the amplitude of the particle that is exchanged subtracts from the amplitude of the particle that goes directly into the detector. Thus bosons and fermions, as these particles are called, act in completely different ways. Fermions obey the Pauli exclusion principle, as we have already seen for electrons. Without fermions, matter could not exist at all. Without the exclusion principle, matter would simply collapse: electrons would be drawn into the nucleus,

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making the existence of atomic matter impossible. Two fermions are never found in the same state—you might say they are antisocial. The bosons, on the other hand, are very social. Any number of Bose particles can exist in the same state. The result is a phenomenon called Bose condensation, in which bosons congregate in large numbers in the same positional or motional state. The interaction of large numbers of particles can lead to correlated motions, called cooperative (or collective) phenomena. Such behavior is seen in superconductors and in helium at low temperatures, as well as in phase transitions. The two isotopes of helium, of mass 3 and mass 4, present an example of the difference on collective behavior of fermions and bosons. Helium at low temperatures is nature’s simplest, most orderly liquid. Liquid helium-4 has a phase transition at 2.19 K, below which it is superfluid, and the motion of the Bose particles is quantized. Helium-3, a fermion, becomes superfluid at a much lower temperature, a few thousandths of a degree, because a pair of fermions behaves like a single boson. 1.5. Tunneling If a particle is held in place by walls that rise so high that the particle's energy is insufficient to scale them, it will be trapped, at least within the scheme of classical mechanics. However, quantum mechanics provides an exception to this rule: the probability of passage depends not on the height but on the thickness of the wall. We recall that the particle can be described as a wave, of wavelength h/mv. If the wall, or more precisely the potential barrier, is thin in comparison with the wavelength of the particle, the particle can diffract through it, so that there is some probability of its appearing on the other side. This phenomenon, called tunneling, has been observed to occur, particularly in low-mass particles such as electrons and protons. Quantum tunneling is considered a limiting factor in the development of information technology. Transistors have been developed with key dimensions as small as 50 nm, but the shrinking of nanostructures is expected to reach the physical limit of quantum tunneling at “gate lengths” of 10-20 nm.4 When we compare this to the 5-nm thickness of a biological membrane, we see that quantum tunneling effects in ion channels cannot be ruled out. 1.6. Quantum mechanics and classical mechanics Quantum mechanics has a totally different look from classical mechanics, as Richard Feynman pointed out:5 Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like ... anything that you have ever seen. ... We know how large objects will act, but things on a small scale just do not act that way.

Rather than dealing with measurable real quantities, quantum mechanics deals with complex wavefunctions that are not measurable quantities. By taking the absolute square of a wavefunction, 5 2, we obtain the probability of finding the system in a

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given state. While classical mechanics yields dynamical quantities, quantum mechanics yields only probabilities. Another important difference between classical and quantum mechanics is that in quantum mechanics the object of study cannot be isolated from the rest of the world. In particular, when we wish to determine some property of an object, that is, make a measurement, we interact with it. Thus both we and the object are changed in the process. In the quantum world there are no events, only probabilities. Events only come into being in the process of measurement, the interaction of a quantum system with a classical detector. The laws of classical mechanics can be obtained from quantum mechanics by taking the limit in which Planck's constant goes to zero. (Of course, the value of h does not change, but quantum effects in most cases become negligible for large systems, for which the action terms are much greater than h.) This concept is called the Correspondence Principle. It states that the classical theory accounts for phenomena in the limit at which quantum discontinuities may be considered negligibly small. Accordingly, a formal analogy must exist between quantum mechanics and classical mechanics. In an abstract formulation of quantum mechanics, the wave functions are viewed as vectors in a mathematical function space. These vectors are acted upon by operators such as the Hamiltonian. In distinction to classical mechanics, particles (or quasiparticles) are not necessarily conserved in quantum mechanics. Operators called creation and annihilation operators can change the number of particles in a system. An example of this process is the formation of an electron–positron pair coupled with the annihilation of a photon. 1.7. Quantum mechanics and ion channels If we could solve Schrödinger's equation for an ion channel, we should be able to derive all its properties from that solution. But there is a difficulty. Even if we knew the exact structure of a voltage-sensitive ion channel—and at this time we don't—we cannot solve Schrödinger's equation for a molecule with hundreds of amino acid residues, tens of thousands of atoms, millions of electrons. Moreover, even if we had such a solution, we would be so inundated with data that we couldn't make sense of it. Although the only problems physicists have been able to solve exactly have been restricted to two-particle problems such as the hydrogen atom, they have developed powerful approximation methods. These usually take as their starting point the exact solution known for a simpler system and treating the additional complexity as a small perturbation of it. Among the systems that have been successfully studied in this way are crystals, which can be greatly simplified because of their symmetries. At a much lower level of order than crystals are glasses. Glasses possess properties of both liquids and solids in that they appear to be rigid but are disordered and capable of flowing, although extremely slowly. Paradoxically, these amorphous solids exhibit many of the same properties as crystalline solids. The fact that many of the concepts developed for crystals apply quite well to glasses shows that this behavior is robust, and not dependent on the structural details.

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Biological molecules are highly complex objects, products of billions of years of adaptation to changing environments. They are neither highly symmetrical like crystals nor totally disordered like glasses. Yet the fact that the same principles apply to both systems gives us reason to hope that these principles may be applicable to ion channels. 2. CONDENSED MATTER The traditional phases of matter, solid, liquid and gas, are all present in living organisms. Solids maintain their shapes under stress; they may be crystalline, their atoms forming regularly repeated patterns in space. Crystals in the human body give rigidity and strength to the teeth and bones in which they lodge. Liquids in the body include blood, lymph, cerebrospinal fluid, saliva, urine and stomach acid. Gases are found in the ears, lungs and abdomen, as well as the swim bladders of fish and the bones of birds. However, the bodies of living organisms clearly contain many tissues that do not fall into the three traditional categories of solid, liquid and gas, so that a classification scheme that contained only the three classical states of matter would not go far toward describing the complexity of the body. Fortunately we need not look far: Physics recognizes that there are intermediate phases between the long-range order of crystals and the disorder that characterizes liquids. These phases, between liquids and crystalline solids, are called mesophases or liquid crystals. A molecular liquid crystal, when heated, may lose longrange order in one or two dimensions but retain it in the other two or one. Solids, liquid crystals and liquids belong to the category of condensed matter. 2.1. Liquids and solids Liquids are disordered phases that flow freely when not confined but tend to maintain a constant volume. Liquids frequently act as solvents, and when their solutes are ions, they are good electrical conductors. Further heating destroys even the short-range order that holds the molecules of a liquid together, leading to the gas phase. In a liquid, molecules attract each other strongly enough to cohere, but not enough to snap into a tight crystalline order. Thus, in a liquid as in a gas, all directions are equivalent; these phases are highly symmetrical. A crystal, on the other hand, has specific symmetry properties involving a set of translation vectors. In a molecular crystal the centers of mass of the molecules are located on a three-dimensional lattice. This means that there are three vectors such that the molecule will be exactly reproduced by moving integer steps along any of these three directions. If a is a vector from a specific point of a molecule (in its mean configuration) to the corresponding point of a neighboring molecule, we can (disregarding the boundaries of the crystal) move by a finite number of a-translations without changing any property of the crystal. Because the crystal has three dimensions, there are two other independent vectors, b and c, not necessarily orthogonal to each other or to a, with the same translation invariance, so that we write

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where f is any quantitative property of the crystal, such as the quantum wavefunction or electric field, r a displacement from an arbitrary origin, and l, m and n arbitrary integers. The study of crystalline solids is one of the most productive applications of quantum mechanics. The symmetry of a crystal allows us to think of it as a single unit cell repeated over and over in three directions. This periodicity of the crystal—and all of its physical properties—allows us to study it with the help of Fourier analysis.6 Since this symmetry property is a subset of the much broader symmetry of a liquid or a gas, we can say that a solid has lower symmetry than a liquid or gas. In the formation of a crystal from a liquid by freezing, the liquid's symmetry is said to be broken. Broken symmetries are interesting because they result in the formation of new directions that were selected from the continuum of possible directions. A biological example of a broken symmetry is the development of the animal and vegetal poles in a spherical egg by the process of fertilization. Phase transitions from higher to lower symmetry always involve broken symmetries. A symmetry of the crystalline type is said to exhibit long-range order. Heating a crystal to the melting point destroys this order. It converts the material to a liquid, which displays only short-range order. Metallic solids are good electrical conductors because their valence electrons are not localized but are free to travel through the metal. Nonmetals are generally insulators, and transition elements such as germanium and silicon form semiconductors. Solids may also be ionic conductors, as we will see in Chapter 6. The structure of real crystals is not always a perfect lattice, but may be interrupted by defects. The simplest type is located at one atom, and is called a point defect. The atom may be missing, or an additional atom inserted, or the atom may be replaced by one of a different type (impurity). A defect that involves a line of atoms is called a line defect. 2.2. Polymorphism A solid that is a lattice of atoms of one type is an atomic crystal; a lattice of molecules is a molecular crystal. More than one structure is possible in either case. The term polymorphism, diversity of nature, is used in crystallography as the possibility of at least two different arrangements of the molecules of a compound in the solid state. The different forms are characterized by different crystal habits, melting points and other properties. For a material with two forms, one will be stable at low temperature and the other at high; the forms are separated by a specific transition temperature.7 An example of polymorphism in an atomic crystal is carbon, which, under different conditions, can crystallize as graphite or diamond. Many examples exist of polymorphism in molecular crystals, including proteins. Polymorphism is particularly prevalent in macromolecular crystals, including proteins such as the forms of human hemoglobin. Conformational transitions in proteins will be discussed in Chapter 12.

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2.3. Quasicrystals Quasicrystals are solids with a forbidden symmetry, such as the icosahedral form, which cannot exhibit translational symmetry. Such substances have been found; e.g., alloys of different metals. The growth of quasicrystals cannot be based on the local pattern of atoms, but must be nonlocal. The mechanism taking place must be the evolution of a linear superposition of alternative arrangements of many atoms, followed, when the difference between energy levels reaches one quantum, by the singling out of one arrangement that becomes actualized in the quasicrystal. Roger Penrose has suggested that brain plasticity is based on the growth and contraction of dendritic spines, causing activation and deactivation of synapses. He further speculates that this growth (or contraction) is governed by quantum-mechanical processes such as those involved in quasicrystal growth.8 2.4. Phonons A simple solid of neutral atoms such as argon can be described as a regular lattice of equilibrium positions about which the atoms vibrate in thermal motion. These vibrations are quantized, so that the crystal is characterized by an excitation spectrum. The analysis of this system is greatly simplified by the symmetry relation (2.1). Let us visualize the system as a lattice of masses connected by springs to their nearest neighbors. Fourier analysis yields the types of wave motions that the solid can sustain. From this analysis, physical properties of the solid, such as its specific heat, can be calculated. The coherent motion of large numbers of particles is called a collective excitation. A sound wave propagating in a background of thermally vibrating molecules is an example of a collective excitation. The unceasing motion of the lattice may be described as a system of waves traveling through the crystal, with different amplitudes, directions and frequencies. These waves are characterized by the relation of the velocity of monochromatic waves to their wavelengths, called the dispersion relation. This relation often displays two branches, describing two modes of motion. This distinction may be illustrated by a linear diatomic chain, in which pairs of ions of unequal mass constitute a primitive unit cell. In the acoustic mode, the ions within a primitive cell move together, while in the optical mode, they move 180° out of phase. The optical branch of the dispersion relation is higher in frequency than the acoustic branch. In the acoustic mode the ions within a primitive cell move as a unit, as in a sound wave; in the optical mode the ion vibrations can couple with an electromagnetic wave. If we send energy into the crystal, say in the form of x rays, this energy can only be absorbed by the electrons in discrete quanta. These quantized excitations are called phonons; they may be acoustic or optical. The absorption and emission of a phonon can be visualized as a particle collision, with the significant exception that the law of conservation of momentum does not hold, since the crystal as a whole can absorb momentum.

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2.5. Liquid crystals Liquid crystals are phases more ordered than liquids and less ordered than solids. They are formed by rod-shaped or disk-shaped molecules. Common phases of uniform matter include nematic, cholesteric and smectic; the cholesteric form may be considered a modified form of the nematic. Transitions from one phase to another, such as melting, freezing, evaporation and condensation, are usually accompanied by the release or absorption of energy. Such energy changes are also observed in transitions involving the mesophases. Liquid crystals undergo transitions at definite temperatures, such as: isotropic liquid Û nematic Û smectic Û crystalline solid These delicate phases of matter are of importance to the biophysics of membranes and ion channels.

Figure 5.1. A lipid bilayer enters a liquid crystalline phase at transition temperature Tc. From Adam et al., 1977.

Lipid bilayers exhibit transition temperatures at which phase transitions occur. For example, lecithin with a saturated C18 fatty acid chain undergoes a transition at Tc = 41°C. Below that temperature the bilayer is fairly rigid, with its carbohydrate chains extended in parallel. Above 41°C, the order of the chains is disturbed and the bilayer is fluid; see Figure 5.1.9 A compound that exists in different phases due to its interaction with a solvent is lyotropic. If its phase depends on both temperature and solvent concentration, as in excitable membranes, it is called amphotropic. Lyotropic mesogens are molecular species capable of forming a lyotropic mesophase. In addition to their high molecular mass and nonspherical shape, properties that they share with thermotropic mesogens, lyophilic mesogens are amphiphilic: The molecule is separated into a hydrophilic part (soluble in polar solvents such as water) and a hydrophobic part (soluble in nonpolar solvents but poorly soluble in water).10 Lipids, with polar heads and nonpolar tails, are examples of lyophilic mesogens. Molecules with both hydrophilic and hydrophobic parts, called amphiphilic, act as surface active agents or surfactants. The molecules may be anionic (with negative headgroup), cationic (positive) or zwitterionic (both charges). The shapes of the molecules determine the configurations of the aggregates they form, including micelles and bilayers, as shown in Figure 5.2.11 Bulky headgroups favor the formation

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of micelles, spherical aggregates with strong surface curvature. The formation of different shapes from the shape of the monomer is modeled by the concept of a packing parameter, introduced by J. Israelachvili.12 3. REVIEW OF THERMODYNAMICS Thermodynamics uses macroscopic concepts to discuss the large-scale behavior of matter. Its statistical foundations are discussed in statistical mechanics, reviewed later in this chapter and in Chapter 15. 3.1. Laws of thermodynamics The sensation of warmer is the basis of our physical concept of temperature. To make it quantitative, we construct a thermometer consisting of a mass of gas in a cylinder fitted with a piston. The position of the piston tells us the volume V of the gas, and the force on it divided by its area is the gas pressure P. If two bodies, insulated from their environment, are brought into contact, heat flows from the warmer to the colder. After they have been in contact for a long time, they are in thermal equilibrium and we say they are at the same empirical temperature t. If body A is in thermal equilibrium with body B, tA = tB. If, for three bodies A, B and C, it is true that tA = tB and tB = tC, then the zeroth law of thermodynamics tells us that tA = tC. This property allows us to construct a scale of temperature with an arbitrary zero point. Monatomic gases tend to behave similarly at high temperatures, giving us the notion of an ideal gas, which obeys the law (3.1) where N is the number of molecules of gas and k is Boltzmann’s constant. The temperature at which the volume of an ideal gas tends to zero at constant pressure is defined as the absolute zero of temperature, and T is called the absolute temperature; it is measured in kelvins. An equation such as (3.1), which gives the temperature as a function of the state variables of the system, is called a thermal equation of state. Induction from a large number of careful experiments shows that energy is conserved. Heat Q, work W and internal energy U are three forms of energy, and for a closed system the first law of thermodynamics can be expressed as (3.2) where dW is the differential of work done by the system.

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Figure 5.2. The shapes of lyotropic liquid crystal molecules determine the configurations of the aggregates they form. The concept of a packing parameter of J. Israelachvili explains the origin of aggregate shape in the space requirements of hydrophobic and hydrophilic moieties. SDS, sodium dodecyl sulfate; CTAB, hexadecyltrimethylammonium bromide. From K. Hiltrop, 2001, after Israelachvili, 1985.

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An equation that gives the internal energy U in terms of the state variables of the system is called a caloric equation of state. A simple thermodynamic system of one chemically pure component has two degrees of freedom, say P and V, with dW = PdV. It is characterized by a thermal and a caloric equation of state. Such simple systems also differ in the nature of their phase transitions: A mass of hydrogen may exist in three kinds of phase, solid, liquid and gas, while for helium there exists an additional, superfluid, phase. A molecular substance may also exist in one or more liquid crystalline phases. The internal energy function U has the useful property that, for a process carried out in a sequence of infinitesimally small steps for which the system can be assumed to be in equilibrium, reversing the process will bring it back to its original value. Such processes are said to be quasi-static. Thus, in a reversible cyclic process, the internal energy makes a closed loop. This is not true in general for the functions Q or W; unlike dU, dQ and dW are not perfect differentials. However, mathematical analysis suggests that an integrating factor exists that will turn these functions into reversible functions. For Q, that factor was shown by Sadi Carnot's experiments on heat engines and subsequent experiments to be 1/T. Thus we can write the first law in the form (3.3) We define (3.4)

where S is the entropy. Unlike dQ or dW, dS is a perfect differential. From the way entropy has been defined, we know that quasi-static processes leave the entropy unchanged. For all other processes in a closed system, experiment shows that the entropy always increases. That is the second law of thermodynamics.13 As V or T becomes large, the system behaves like a perfect gas; classical mechanics works well here. This is not true near absolute zero. According to the second law and the definition of entropy, Equation 3.4, the T = 0 isotherm is singular. In the neighborhood of absolute zero, where the energy of the system is low, the fact that energy is quantized leads to significant effects. It is observed that, when the temperature is lowered to zero with other variables held constant, the value approached by entropy is independent of all other variables and may be equated to zero. The isentropic (constant-entropy) and isothermal characteristics approach coincidence as the temperature tends toward zero; this is the third law of thermodynamics. It is also expressed in the statement that T = 0 cannot be attained in any finite number of steps.

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From the statistical viewpoint that entropy measures the disorder of a system, the third law indicates that matter becomes more highly organized as the temperature is lowered. The shift toward higher order may occur continuously or by way of a discrete phase transition. 3.2. Characteristic functions Thermodynamic states of the system correspond to definite values of certain characteristic functions, which depend only on the initial and final states of the system and not on the details of the process connecting them. Such characteristic functions include V, P, S, U and T. Other characteristic functions may be defined, of which the most useful to us here is the Gibbs function G, (3.5) Note that the products PV and TS pair one extensive variable (V or S), which scales with the size of the system, with one intensive variable (P or T), which does not scale with size. Since the first and second laws may be combined to give (3.6) we can write (3.7) and, by differentiating 3.5 and substituting 3.7, (3.8) For reversible processes, in which the equals sign holds, Equations 3.7 and 3.8 imply (3.9) (3.10)

For each characteristic function there is a thermal and a caloric equation. By taking second derivatives, we can obtain the conditions of compatibility known as the Maxwell relations. For example, from Equation 3.10 we obtain

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(3.11)

When T and P are the independent variables it is convenient to choose G as the characteristic function. The extension of thermodynamics to open systems, linear irreversible thermodynamics, is discussed in Chapter 15, Section 6. 4. PHASE TRANSITIONS For homogeneous materials, we divide the total energy, total mass, total magnetic or electric moment by the volume V to obtain the energy density, mass density, magnetization or electric polarization. These quantities, mechanical variables, are mostly continuous as external fields, such as pressure, temperature, magnetic field or electric field, are varied. There are regions in which a mechanical variable is not uniquely determined but has a choice between options. We are familiar with the fact that, at T = 373 K and P = 1 atm, the density ' of density of H2O may be high (water phase) or low (steam phase). This happens on a subset of points of the PT plane, a line that extends from the triple point, where ice, water and steam are in equilibrium, to the critical point, Tc = 647 K and Pc = 218 atm, at which water and steam have the same density.14 A second example is the ferromagnetic state of the metals iron, cobalt and nickel. The magnetization vector M is not fixed at low temperature when the applied magnetic field H is zero; it may point in different directions. This free choice of directions ceases when the temperature T exceeds Tc, the critical temperature known as the ferromagnetic Curie temperature. A third example is the ferroelectric state of dielectric crystals, such as Rochelle salt and triglycine sulfate, or certain liquid crystals, such as some chiral organic polymers; see Chapter 17. In this case the polarization vector P is not fixed when the applied electric field E is zero; it may point in different directions. Polarization vanishes at zero field when the temperature T exceeds Tc, the critical temperature known as the ferroelectric Curie temperature. Thermodynamic effects due to electric fields will be considered in the next chapter. Phenomena that occur near a critical point are called critical phenomena; these are discussed in Chapter 15. 4.1. Phase transitions in thermodynamics The Gibbs function G is a smooth function over most of the PT plane, but, as we just saw, there may be discontinuities along certain lines in its first derivatives, S and V, given by Equation 3.10. These are called first order phase transitions. If S and V are continuous as well as G, but the second derivatives are discontinuous, the transition is said to be of second order.

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Systems are, in practice, often found that are not in equilibrium but are in socalled metastable states. These states are near, but not on, the equilibrium curves. For example, a vapor can be compressed to a pressure beyond its vapor pressure if nuclei for initiating condensation are lacking. Such a metastable vapor, called supersaturated, is unstable against disturbances such as the passage of an airplane, which may leave a “vapor trail.” Similarly, a liquid may be heated to a temperature above the boiling point, becoming superheated. Metastable states play essential roles in phase transitions of real systems. 4.2. Transitions of first order We can now derive equations for a first-order transition. For changes at constant pressure and temperature, dP = dT = 0, Equation 3.8 shows that the Gibbs function must be nonincreasing, (4.1) where the equals sign applies to reversible changes. In an irreversible change, G will tend to move toward the minimum value consistent with the constraints on the system. We assume that the necessary mechanisms for lowering the Gibbs function are available, so that the process will not stop at a metastable point, as in supercooling. For a system consisting of two phases characterized by specific Gibbs functions g1 and g2, we can write

for the Gibbs function G and the mass M of the total system. The only constraint on the system is the conservation of mass. If g1 < g2, the value of G will be lowered to its minimum Gmin if phase 2 converts to phase 1. Then

The system will be stable with arbitrary amounts m1 and m2 only it the specific Gibbs functions are equal, so the condition for phase stability is (4.2) From the differential form of equation 4.2, dg1 = dg2, we can write (4.3)

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Using the notation ( ) = ( )1 - ( )2, we obtain from Equation 4.3

With equations 3.10, this becomes the Clausius–Clapeyron relation (4.4) where L = T S is the latent heat of the transition per molecule and V indicates a packing difference between the two phases. The Clausius–Clapeyron equation determines the rate at which the equilibrium pressure varies with the equilibrium temperature in the P-T diagram. For example, if phase 1 is liquid and phase 2 solid, S and T will typically be positive, so that dP/dT (and its reciprocal, dT/dP) is positive. One exception is water, since the specific volume of the liquid is less than that of the solid at melting, and ice floats. Then the Clausius–Clapeyron equation implies that the freezing point of water decreases with increasing pressure, a fact that makes ice-skating possible. The specific volumes of the liquid and gas phases may be made equal by heating at constant P or increasing the pressure at constant T. At a critical point, labeled (Pc, Tc), the two phases become indistinguishable, and since V = 0, dP/dT cannot be determined from the Clausius–Clapeyron equation, Equation 4.4. 4.3. Chemical potentials and phase diagrams In an equilibrium between two phases, such as a vapor–liquid equilibrium, a substance is not necessarily homogeneous, even if it is chemically pure. The liquid droplets and the vapor phase are homogeneous parts of a condensing system, characterized by separate Gibbs potentials G1 and G2. Each phase is characterized by the number of its molecules, N1 and N2, as well as the common variables P and T. The total Gibbs potential is the sum of the Gibbs potentials for each phase, G = G1 + G2, and it will be a minimum for equilibrium. The total number of molecules, N = N1 + N2, remains constant as the number in each phase, N1 and N2 vary. The phase equilibrium is specified by

The Gibbs potential per molecule,  = (0G/0N)P, T, is called the chemical potential.15 With this definition, the equilibrium between two phases can be written (4.5)

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This equation defines a relation between P and T for every given value of . The relation is a curve dividing the PT plane into areas occupied by the different phases. This plot is called a phase diagram, which can be thought of as a section through a three-dimensional PT graph, with the phases separated by intersecting surfaces. The intersection of the surfaces of 1 and 2 above the PT plane define the line along which the phases are in equilibrium.

Figure 5.3. A phase diagram, showing the triple point B, at which the solid, liquid and gas states are in equilibrium, and the critical point, C, at which the liquid and vapor phases become indistinguishable. From Finkelstein, 1969.

In an isotropic substance with three phases, solid, liquid and vapor, the surfaces may intersect pairwise to give three equilibrium curves: solid–vapor, solid–liquid and liquid–vapor. At a point at which all three surfaces intersect, the triple point, all three phases are in equilibrium. A phase diagram can help us see the way the chemical potentials of two coexisting phases behave in the vicinity of equilibrium; see Figure 5.3. 4.4 Transitions of second order In a second-order phase transition, not only G but also its first derivatives are continuous. For such a transition, we know from (3.10) that (4.6) (4.7) Expressing equation 4.6 in differential form, we have

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from which we obtain (4.8)

where  is the expansion coefficient and K the compressibility, (4.9) (4.10) Similarly, taking the differential form of Equation 4.7, we have (4.11)

With 4.11, the Maxwell relation 3.11, Equation 4.9 and the definition of CP, the specific heat at constant pressure,

we have (4.12)

Equations 4.8 and 4.12 are called the Ehrenfest equations. Unfortunately, they only apply when the discontinuities are finite, which is often not the case. However, modified forms are useful even when the specific heat becomes infinite at the transition.16

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4.5. Qualitative aspects of phase transitions Matter becomes more highly organized as its temperature is lowered. This tendency summarizes the statistical viewpoint that entropy measures the "disorder" of a physical system. The molecules of a gas are not as highly correlated as those of a liquid, and those of a liquid are less ordered than those of a solid. As we saw in Section 2 above, liquid crystals undergo transitions between these, the degree of order increasing with transitions from isotropic liquid to nematic, then to smectic and finally to crystalline solid. As independent variables such as temperature and pressure are changed, the degree of organization usually changes continuously. However, if we visualize horizontal or vertical lines cutting across the PT plane of Figure 5.3, we see that discontinuous changes appear in the state of aggregation for a first-order transition. The change of organization that occurs during a transition is associated with nonanalytic behavior of the thermodynamic functions. These functions measure the difference in the symmetry of the separate phases. Solution of this difficult problem requires, first, relating the quantum mechanical energy spectrum with its degeneracies to the symmetry of the corresponding phase and, second, relating the thermodynamic functions to this spectrum by the principles of statistical mechanics. 5. FROM STATISTICS TO THERMODYNAMICS A number of experimental facts, particularly about phase transitions, cannot be explained by the laws of thermodynamics alone. They require a consideration of the actual dynamical behavior of the system in equilibrium. Whereas thermodynamics describes a closed system in thermodynamic equilibrium by a small number of variables such as P and V, the microscopic point of view tells us that the number of degrees of freedom of a system of macroscopic size is of the order of Avogadro's number. For a macromolecule such as a sodium channel of 250 kDa, the number of degrees of freedom is of the order of 105. Clearly, the thermodynamic description is an enormous abbreviation of the microscopic system. The connection between the microscopic and macroscopic points of view, studied by Ludwig Boltzmann, Josiah Gibbs and others, is called statistical mechanics. 5.1. Phase space Mechanics describes a system of particles by their generalized coordinates and momenta, whose equation of motion is given in terms of the energy operator, the Hamiltonian. Each of the N particles can be described as moving in a space of six dimensions, three for the coordinates and three for the momenta. More abstractly, the entire system can be described as a single point in a phase space of 6N dimensions. Within its constraints, the system can occupy many such points; the set of such points is called the Gibbs ensemble. The laws of mechanics, classical or quantum, tell us the way in which the Gibbs ensemble moves in phase space. A number of integrals of the motion of

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individual points may be obtained, such as energy, electric charge and angular momentum. When energy is conserved, each point must move on a hypersurface called the ergodic surface. Except for special symmetries or constraints, there is no reason to believe that the system will prefer one region of the ergodic surface over another. Such unrestricted motions, which come arbitrarily close to every point of the ergodic surface, are termed quasi-ergodic. We can postulate that, in the subspace of the Gibbs ensemble in which energy and other integrals of the motion are constant, the density of points is constant. In other words, all parts of the phase space allowed by the given energy and external constraints represent equally probable states of the system. Two important expressions for ensembles that depend only on the energy integral are the canonical and the grand canonical ensembles. We can divide the phase space into cells for comparison to experiment. A mathematical analysis permits us to equate the statistical mechanical terms to their corresponding thermodynamic terms, with the following results: For a closed system in equilibrium, the density &i of the ith cell is given by the canonical ensemble,17

(5.1) where (5.2) The quantity Q, called the partition function, may in principle be calculated from the exact microscopic Hamiltonian by Equation 5.2. The link between statistical mechanics and thermodynamics is clearly seen by the free energy A, the work done in a reversible isothermal process, (5.3) which is related to the partition function by (5.4) When A is known, the thermal and caloric equations of state of the system can be obtained from the formulas (5.5) (5.6)

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5.2. The canonical distribution In the present analysis the physical system has been represented by a single point in phase space. It is nevertheless possible to obtain information about the distributions of individual molecules from this approach. From the canonical distribution we can obtain the spread in energy of a gas by integration. The probability that the system as a whole has energy E is (5.7) To find the distribution of one particular molecule, say molecule 1, we must integrate over the position and momentum coordinates of the other molecules. Under reasonable assumptions, this integration gives for the probability that the first molecule has momentum between p1 and p1 + dp1 (5.8) For a canonical Gibbs ensemble, the molecules thus are distributed in momentum space by the Maxwell–Boltzmann distribution. Correlations between positions, momenta or other variables of molecules can be obtained by appropriate integrations of the probability function. 5.3. Open systems For an open system, one that can exchange particles, mass, energy and momentum with its surroundings, the grand canonical distribution is the appropriate formalism. The probability that the system is characterized by the energy E and the number of particles N is (5.9)

This equation is comparable to the canonical distribution of Equation 5.1. Open system differ from closed systems in internal equilibrium in their time behavior. While in the equilibrium system every flux is compensated by an opposite flux and every gain cancelled by a corresponding loss, this is not true of the open system. Here the symmetry of the system is broken by fluxes that pass through the system, moving it to a state far from equilibrium. We recall that biological systems are open systems. Another open system is the laser, in which an active material such as ruby held between two parallel mirrors is illuminated by light coming from an ordinary lamp. When the power of the lamp is greater than a certain threshold, the light in the cavity forms a coherent wave. When one of the mirrors is semi-transparent, a beam of

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coherent light is emitted.18 A simpler example is a flute or recorder, in which a stream of air hitting a sharp edge resonates in an open tube to produce a sound of discrete frequencies determined by the geometry of the tube. 5.4. Thermodynamics of quantum systems Josiah Gibbs, the American physicist who developed the principles of statistical mechanics, applied them to classical mechanics and hence was unsuccessful in describing real physical systems. However, the deep correspondence between quantum and classical mechanics (Section 1.6 of this Chapter) allows us to apply the principles of statistical thermodynamics to quantum systems with little change to the formalism. The quantum partition function may be obtained from the matrix formulation of quantum mechanics; here, following Finkelstein's book, we will simply postulate it.19 The Hamiltonian operator for a system of structureless particles may be written

(5.10)

where the momentum operator pi = -iW h /i, and the potential energy Uij is a function of the magnitude of the displacement vector between the ith and jth particles. The timeindependent Schrödinger equation is given by Equation 1.3, H5 = E5. If a system is fully described by a set of quantum numbers a1, a2, ..., the energy spectrum of the solution to Schrödinger's equation can be written as a function of the quantum numbers E(a1, a2, ...). We now define the partition function as (5.11) where the sum extends over the complete spectrum. According to the Correspondence Principle, we postulate that the connection with thermodynamics is given by Equations 5.4-5.6, which give the thermal and caloric equations of state. Remarkably, these equations, together with the knowledge of the atomic structure of the hydrogen atom, and taking account of the Coulomb field between electrons and protons, make it possible in principle to determine the equations of state of all the phases and phase transitions of hydrogen. The partition function Q may be written in a more abbreviated form than Equation 5.12 if we symbolize the set of states with quantum numbers a1, a2, ... as .

(5.12) The set of states often has symmetry, so that different states possess the same

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energy. The number of states gn with energy level En is called the degeneracy of that level. The degeneracy generally increases with energy. In terms of the distinct energy levels n, the partition function may be written

(5.13) 5.5. Phase transitions in statistical mechanics In statistical mechanics, all thermodynamic properties are supposed to be deduced from the atomic structure by the application of Schrödinger’s equation and the partition function. This general formalism should yield the phase transitions and the equations of state of the separate phases. The detailed structure of the condensed phases should fall out of this formalism. The degeneracy of the energy states determines the dimensionality of the system’s representation. The degeneracy is high in the gas phase, but decreases as the energy is decreased. We can illustrate this problem by considering a mass of hydrogen at a pressure such that only the solid and vapor forms exist. The qualitative features of the spectrum are shown in Figure 5.4.20

Figure 5.4. Spectrum of a system that may exist in either a gas or solid phase. From Finkelstein, 1969.

The spectrum shows that at low temperature the hydrogen is frozen into a solid, in which only rotational states of the macroscopic crystal appear. At somewhat higher

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temperatures, the solid exhibits the universal Debye spectrum, with its specific heat proportional to the cube of the temperature. With rising temperature, the detailed crystal structure with its symmetries (labeled ) is observed. Above the sublimation temperature Ts, hydrogen is a gas. At very high temperature, T4, the degeneracy is so high that the spectrum is well approximated by the states of an ideal gas. 5.6. Structural transitions in ion channels We have seen that quantum mechanics is the correct formulation for the analysis of microscopic systems such as ion channels. A condensed matter system can exist in different phases, characterized by different physical properties. Which of its possible phases a region of matter occupies depends on its temperature, pressure and the magnitude and direction of any fields applied to it. We will discuss the application of these concepts to fluctuations in Chapter 11 and to structural transitions such as the open–close transitions of voltage-sensitive ion channels in the chapter on critical phenomena, Chapter 15. NOTES AND REFERENCES 1. B. S. Chandrasekhar, Why Things Are the Way They Are, Cambridge University, 1998, 5. 2. The electron volt (eV) is the change of potential energy of an electronic charge moved against the potential difference of one volt. 3. R. T. Weidner and R. L. Sells, Elementary Modern Physics, Allyn and Bacon, Boston, 1960, 426f. 4. T. N. Theis and P. M. Horn, Physics Today 56 (7):44-49, July 2003. 5. Richard P. Feynman, Robert B. Leighton, Matthew Sands, The Feynman Lectures on Physics, vol. III, Addison-Wesley, Reading, Mass., 1965, 1-1. 6. Fourier analysis is discussed in Chapter 10. 7. Joel Bernstein, Polymorphism in Molecular Crystals, Clarendon, Oxford, 2002. 8. Roger Penrose, The Emperor's New Mind, Oxford, 1989, 437f. 9. G. Adam, P. Läuger and G. Stark, Physikalische Chemie und Biophysik, Springer, Berlin, 1977, 274f. With kind permission of Springer Science and Business Media. 10. Alexander G. Petrov, The Lyotropic State of Matter: Molecular Physics and Living Matter Physics, Gordon and Breach, Amsterdam, 1999. 11. K. Hiltrop, in Heinz-Siegfried Kitzerow and Christian Bahr, in Chirality in Liquid Crystals, edited by Heinz-Siegfried Kitzerow and Christian Bahr, Springer, New York, 2001, 447-480. With kind permission of Springer Science and Business Media. 12. Reprinted from J. Israelachvili, Intermolecular and Surface Forces, Academic, London, 1985, with permission from Elsevier. 13. This law has been expressed in several ways: A transformation whose only final result is to transform into work heat extracted from a source which is at the same temperature throughout is impossible (Kelvin). A transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature is impossible (Clausius). 14. S. K. Ma, Modern Theory of Critical Phenomena, W. A. Benjamin, Inc., 1976, 2ff. 15. Minoru Fujimoto, The Physics of Phase Transitions, Springer, New York, 1997, 5. 16. Robert J. Finkelstein, Thermodynamics and Statistical Physics: A Short Introduction, W. H. Freeman, San Francisco, 1969, 49-52. 17. See, e.g., Finkelstein, 101-129. 18. See, e.g., Giorgio Careri, Order and Disorder in Matter, Benjamin/Cummings, Menlo Park, 1984, 98103. 19. Finkelstein, 122. 20. Finkelstein, 188.

CHAPTER 7

IONS DRIFT AND DIFFUSE

Coming to the heart of membrane science, we now focus on electrodiffusion, a theoretical model that has played a central role in the development of our understanding of ion currents through membranes. 1. THE ELECTRODIFFUSION MODEL Simplicity is a powerful idea, as important in science as in music, art and lifestyle. As the Shaker tune has it, 'Tis a gift to be simple. Physicists in particular treasure the simple: “Try simplest cases first” is a common admonition. So it comes as no surprise that the first theory on the behavior of ions in membranes was a simple one. This chapter is devoted to the mathematical development of this model, the electrodiffusion model. An equivalent model in solid-state physics is called drift–diffusion. Electrodiffusion deals with problems of ion flow from a macroscopic perspective, that is, without worrying about the details of the environments of the individual ions; it treats the membrane as a continuous medium with two bounding surfaces. While electrodiffusion is important as one of the principal starting points of excitablemembrane theory, it is by no means an endpoint. It may be thought that modeling a membrane as a featureless wall cannot be a valid approach, since we know that the membrane has a richly complex structure, as we saw in Chapter 2. To be sure, if we could isolate each component and study it separately, it would seem preferable to do so. Unfortunately, this can not be done for the protein components. Their conduction properties can only be studied when they are embedded in a lipid bilayer, and not in isolation. The technique of patch clamping has taken us part of the way toward the goal of channel isolation, but it must be remembered that even in a micrometer-size patch, the channels only occupy a small fraction of the membrane area. So we really can't study the function of protein channels separately. And even if we could, the channel properties would still have to be recombined with the lipid properties to obtain a holistic model of a membrane’s behavior between two aqueous phases. It is perfectly valid to apply a continuum approach to the study of membranes, as long as we are aware of the limitations of the method. After all, we use the gas laws, which treat gases as continua, even though we are aware that gases consist of individual

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molecules colliding with each other and the walls of the container. The gas laws are accurate for macroscopic measurements, and continuum models of membrane conduction, if we get them right (and that's a big if!), should be accurate for membrane measurements with large (not micropatch) electrodes. We just have to realize that the results will give us an average in which the individual channel behavior will be diluted and smoothed out. This dilution and smoothing is seen in macroelectrode experiments, which give records in which single-channel currents can not be distinguished. 1.1. The postulates of the model An ion's motion, as postulated by Nernst and Planck, is ruled by two tendencies. One is the tendency of a particle to diffuse, bouncing drunkenly from a region of high concentration to one of low concentration. The other tendency of the ion is to drift, like a tumbleweed on a windy prairie, migrating in the direction that the force of the electric field exerts on its electric charge. The ion's movement, according to the electrodiffusion model, is made up of these two components, diffusion and migration (or drift). Electrodiffusion is one of the key concepts of membrane science; it plays a central role in the science of voltage-sensitive ion channels because of its mathematical simplicity and consistency. As we already saw in Chapter 3, it led to the development of the membrane hypothesis in 1890-1902. Also, as we will see in the next Chapter, it was the starting point for a paper by Hodgkin and Katz that led to the important 1952 model of Hodgkin and Huxley, which provided the first mathematical description of an action potential. And, as we will see in Chapter 14, electrodiffusion provides a clue that connects ion-channel science to a branch of condensed-state physics, ferroelectricity. Like all models, electrodiffusion is a simplification of the real situation. We will be making some rather unrealistic assumptions, but that is the price we have to pay for the insights we can obtain from a mathematical model. Diffusion and electrodiffusion are macroscopic models, models that deal with matter as a continuum. The results of analyses of macroscopic models are validly applied to measurements at a scale that includes a huge number of molecules, so that the measured variables can be treated as continuous functions. As mentioned above, one example of a continuum model is the gas laws, which treat pressure as a continuous variable. It took a long time after these macroscopic laws were discovered by Boyle and Charles before the kinetic theory of Boltzmann, Maxwell, Clausius and others explained the nature of pressure as an emergent property of numerous discrete molecular collisions. Application of the electrodiffusion theory is sometimes criticized because of its disregard of the molecular structure of the membrane. However, since the structure of the molecules that constitute the membrane is still largely unknown, it is just this feature that makes it useful. The application of the electrodiffusion model to a membrane does not mean that one is asserting that the actual membrane is a featureless wall; rather, it means that one is taking a macroscopic view of the membrane.

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1.2. A mathematical membrane Although electrodiffusion has been used in various ways, we will begin with the simplest case, that of a uniform membrane bounded by planar surfaces. Electrodiffusion is based on certain simple model assumptions. The real membrane is of course nonuniform and its surface may well be quite uneven; so, since we know these assumptions are not true, why bother with electrodiffusion at all? The answer is that electrodiffusion is a simple powerful tool to examine a number of features that a real membrane must possess. It gives us a mathematical representation of an idealized membrane with the microscopic details blurred out by an implicit averaging. These assumptions need not be true, but the inexorable logic of mathematical reasoning endows the model with a consistent structure. That means that, if the conclusions of our model turn out to be contradicted by experimental fact, we will know that the source of the error must be one or more of the assumptions. That can lead us to the next step in our reasoning. Our model consists of two aqueous phases, separated by a membrane of uniform material. In three dimensions, the membrane will be in the xy plane, and the z axis will be along the outward membrane normal. For simplicity we consider the membrane to be bounded by parallel planar surfaces. The membrane thickness is L. On one side, labeled I, is the internal, cytoplasmic solution; on the other side, II, is an aqueous solution corresponding to the external fluid bathing the axon. We will follow the convention that the external potential is zero and the internal potential is the membrane voltage Vm. We assume that the aqueous solutions are electrically neutral, at least in the bulk.1 Since the membrane is thin and not necessarily a good electrical conductor, it need not be electrically neutral. 1.3. Boundary conditions Within the membrane, the number density of ions, N, will vary continuously from its value at the left boundary, N(0) = NI, to its value at the right boundary, N(L) = NII; these boundary values are collectively called the boundary conditions. We must not think, however, that the ion concentration just inside the membrane at the boundary will be equal to that just outside. After all, the solubility of an ion or molecule will be different in different media. The ratio of solubility of the membrane to that of water is called its partition coefficient and given the symbol .2 The partition coefficient of a membrane is not directly measurable. We can write

(1.1) where Ni and No are the concentrations of the ion in the aqueous solutions inside and outside the cell, respectively. It should be noted that we are here assuming that the aqueous media inside and out are “well stirred,” so that their boundary values are equal to their bulk values. This

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is more likely to be the case when the axon is internally perfused than in vivo. Ion accumulation and depletion may play significant roles in electrical behavior, as we saw in Chapter 6. 2. ONE ION SPECIES, STEADY STATE We shall begin with several additional simplifications. Of the various ions in the system of a real membrane, we shall focus on only one ionic species that can permeate through the membrane. We shall also assume that the system is in a steady state, so that the flow of ions is not changing in time. Note that a steady state does not imply equilibrium, since at equilibrium the flow of ions would cease. Time dependence and multi-ion electrodiffusion will be treated in Chapter 8. 2.1. The Nernst–Planck equation Let us temporarily disregard the charge of the ion and consider the diffusion of a neutral molecule, such as a molecule of sugar. Our particle is repeatedly colliding with other particles of its own kind, which in general are not uniformly distributed. The majority of the collisions are with particles arriving from the direction of highest concentration. These tend to impel the particle in the other direction, so the average movement is from higher to lower concentration, down the concentration gradient. In one dimension, we can say that the ion flux due to diffusion is proportional to the negative derivative of concentration with distance. (2.1) This equation is Fick's first law. The derivative is a one-dimensional component of the concentration gradient; the scalar flux is a component of a vector. N is the number density of the ions.3 Flux is the number of ions crossing a unit area in unit time, so that its mks units are m-2s-1. The proportionality constant D is called the diffusion coefficient. Since the units of N are m-3, the units of D are m2s-1, square meters per second. An empirical relation, valid when the boundary concentrations are not too dissimilar, is (2.2) where P is the permeability of the given ion in the membrane. Now let us restore the ion's charge and compute the migration component. The force F on an ion of charge q due to the electric field E at its location is qE.4 The field is the negative gradient of the electric potential V, (2.3)

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Averaging over many collisions, we say that the ion is dragged by the force F through a viscous medium, so that its velocity v is proportional to the force, v = uF. The proportionality constant u is called the mechanical mobility5; its units are mN-1s-1. Therefore, the average drift velocity of an ion is uqE. The flux is the ion concentration times the average velocity, so that (2.4) Adding the two contributions, Equations 2.1 and 2.4, we obtain the net flux due to both diffusion and migration.

Since each ion carries charge q, the current density J is q times the net flux. (2.5) The units of current density are amperes per square meter, Am-2. In the stationary state, J is uniform across the membrane, as we will show in Section 8.2, so we can treat J as a constant here. The charge q of the ion is the product of its (signed) valency Z times proton charge e, (2.6) The constants D and u both describe the motion of the ion, so it is not surprising that they are related. As Einstein showed, the diffusion coefficient equals the mobility times the temperature, expressed in energy units, (2.7) Using the boundary conditions, Equation 1.1, with Equations 2.2 and 2.7, we can show that the empirical permeability P is (2.8) Using Einstein's relation to eliminate D from (2.5) and rearranging, we obtain (2.9)

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This is the Nernst–Planck equation, introduced in Chapter 2. Using the values of constants, we can calculate the value of kT/e; see the Box on Chemical Notation on page 146. The Nernst–Planck equation describes the ion current as made up of two contributions, the first depending on the gradient of the ion concentration and the second depending on the product of field and concentration. The first term, the concentration gradient, will be negative if NI is greater than NII, as is usually the case for the potassium ion. Because of the negative sign on the right-hand term, the diffusion contribution to the current will therefore be positive, representing an outward flux. Because N is always positive, the direction of the migration current will be determined by E and the sign of the ionic charge. We note that E = -dV/dz and the voltage across the membrane is the potential inside relative to that outside, (2.10) because the outside potential is by convention set equal to zero. Thus, for the membrane at resting potential difference, when the potential inside is negative relative to the outside, the membrane voltage will be negative and the field will be inward. Because the two negative signs cancel, the migration contribution to the ion current for a positive ion will be inward at resting potential. 2.2. Electrical equilibrium We see that the diffusion and migration components of the current density may be in opposite directions, as for example for the potassium ion at resting potential. When they cancel out exactly, the membrane system is in electrical equilibrium. No net current flows, J = 0, so that Equation 2.9 becomes (2.11)

Rearranging, we find that (2.12)

where we have used the fact that the derivative of the natural logarithm is the reciprocal of the dependent variable times its derivative. Using (2.4), we obtain

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We integrate both sides to obtain (2.13)

Using Vm = VI - VII, we rearrange this equation to find

(2.14) This is the equation for the Nernst potential; notice that the units of concentration cancel. It is often convenient to write it in terms of the common logarithm, with Equation 2.6,

(2.15)

Taking the exponential of both sides of (2.14) yields (2.16) This equation can also be derived directly from the partition function; see Section 5 of Chapter 5. Here is an example of the use of the Nernst equation; the answers are given in Table 4.1 of Chapter 4: Calculate Vm for (a) Ni = 400 mM; No = 10 mM, Z = 1; (b) Ni = 50 mM; No = 460 mM, Z = 1; (c) Ni = 0.0001 mM; No = 10 mM, Z = 2; (d) Ni = 100 mM; No = 540 mM, Z = -1. Assume a temperature of 20ºC. These values are typical for potassium, sodium, calcium and chloride respectively in squid axon membranes. How are the boundary conditions applied? Let us suppose, for concreteness, that the internal solution (i) is 400 mM KCl and the external solution (o) is 10 mM KCl. To keep things simple, let us suppose that the membrane is permeable to K+ ions but not to Cl- ions. If we ignore electrical forces for now, it is clear that the potassium ions will tend to diffuse outward. Since the KCl is completely dissociated, there are 10 times 10-3 times Avogadro's number potassium ions in a liter, 103 cubic centimeters, of external solution. Therefore the number density of potassium ions in o is No = 10×10-3×6.02×1023/(103 cm3) = 6.02×1018 cm-3. In MKS (meter-kilogram-second) units we can write No = 6.02×1024 m-3 Ni = 40×6.02×1024 m-3 = 1.20×1026 m-3

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If we know that the value of the ionic current density and concentration at the left and right boundaries, we could use the Nernst–Planck and Gauss equations to find the concentration a short distance to the right of the left boundary, say one hundredth of the membrane thickness, and then use these values to go another short distance, and so on. Such numerical methods are often useful.6 Starting from one end and working toward the other, called a shooting method, has the disadvantage that it may be difficult to end up at the correct boundary value on the other side. When both boundary conditions are specified the problem is a two-point boundary value problem. We will not use numerical analysis here, but seek a general solution. This solution will have arbitrary constants, and we can set these to values for which the boundary conditions are satisfied. 3. THE CONSTANT FIELD APPROXIMATION The Nernst–Planck equation 2.9 and Gauss's law together form a system of differential equations. We solve the problem exactly below, but for now we can simplify the math by making a reasonable assumption based on an examination of the physical situation. We are dealing with a very thin membrane, and while it is true that the charges within the membrane will change the electric field, this contribution to the field will often be small compared to the external field across the membrane. Thus the field E can be taken as uniform along the space variable z. As a function of z, therefore, the field will be assumed to be constant.

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3.1. Linearizing the equations Within this constant-field approximation, Gauss's law is replaced by the simple equation (3.1) With this substitution, the Nernst–Planck equation (2.9) becomes (3.2)

where the E has been absorbed into the constant term in parentheses. This differential equation is linear, unlike the situation in the more general equation 2.9, which is made nonlinear by the term in EN. Because of the linearity of this equation, its solution is straightforward. Let (3.3)

Then, since J is constant,

so that Equation 3.2 becomes (3.4)

This equation can be solved for y in the same way we solved for the Nernst equation in Equation 2.12.

Integrating, we have

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where c is a constant. Using (3.3) to eliminate y, we obtain

We can evaluate c by applying the left-hand boundary condition, N = NI at z = 0:

Substituting and rearranging, we find that

(3.5)

where the dependence of N on z is explicitly indicated. Taking the exponential of both sides and simplifying,

Rearranging, we obtain

(3.6)

Equation 3.6 shows that under the constant field assumption the ion concentration has an exponentially rising or falling profile across the membrane, and that the sign of the prefactor of the variable term depends on whether J is less than or greater than q2uENI. If they are equal, N is independent of z. 3.2. The current–voltage relationship To calculate the important current-voltage relationship, we apply the right-hand boundary condition, N(L) = NII, to Equation 3.6,

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and solve for J to obtain

Since the membrane voltage Vm = EL, we can write after some rearrangement

(3.7)

where we have used Equations 1.1. According to Equation 3.7, the J–V curve approaches linear asymptotes as the membrane voltage attains large positive and negative values. For Vm  +, J/Vm  q2uNI/L and, as Vm  -, J/Vm  q2uNII/L. The ratio of these quantities, known as the rectification ratio, thus is equal to NI/NII. By examining the behavior of Equation 3.7 as we approach zero membrane voltage, we can compare the results to the flux of a neutral molecule through a membrane. Noting that, for small values of x, ex 1 + x, we see that Equations 2.2 and 2.8 can be recovered. 3.3. Comparison with axonal membrane data Now that we have an approximate solution to the stationary electrodiffusion problem for a single ion, we can compare it with data. Figure 7.1 shows steady state and peak inward current–voltage characteristics measured at 5, 10 and 20°C.7 Stationary theory of course does not apply to the peak inward case, but even the steady state data show conflict with the electrodiffusion results. This is seen primarily in the rectification ratio, which is much greater in the axon than in the mathematical model. The results of the comparison are disappointing: The squid axon is a much better rectifier than the classical electrodiffusion model predicts. The electrodiffusion model in this approximation does not do justice to the potassium system.

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Figure 7.1. Current-voltage characteristics for squid axon membrane under voltage clamp at 5, 10 and 20 °C. Steady state currents are labeled Iss and early inward peak currents, Ip. From Cole, 1972.

When the inside and outside concentrations are equal, electrodiffusion theory predicts a straight line for the current–voltage curve: In theory, this system is ohmic. It turns out, however, that, while the I–V characteristic is linear over a limited region near the origin, the nonlinearity of the membrane asserts itself in bumps elsewhere along the voltage axis.8 When we consider the sodium system, the failure of the model is even more pronounced, as it fails to explain the negative resistance of the sodium current. In a 1965 review assessing electrodiffusion, Cole wrote that “such simple models and such elementary analyses ... are not adequate.”9 By that time, many electrophysiologists had already abandoned electrodiffusion in favor of the Hodgkin–Huxley model; see Chapter 9. But have we really learned all we can from the electrodiffusion model? What features of the solution have we lost in the linearization inherent in the constant field approximation? Let us

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proceed to the exact solutions, which have been known since 1936.10 4. AN EXACT SOLUTION The simple assumption that the electric field is a constant across the membrane cannot be valid. The ions are, after all, charged, and so they must contribute to the field. The relationship that expresses this fact is Gauss's law, which we have already seen in Chapter 6. 4.1. One-ion steady-state electrodiffusion In three dimensions, in rationalized mks (SI) units, Gauss's law is (4.1)

where D is the electric displacement vector and qN is the charge density. The relation between D and the electric field E can be written, as in Equation 1.4 of Chapter 6, (4.2) where the relative permittivity  is assumed, in classical electrodiffusion, to be a constant. Inserting (4.2) into (4.1), we have (4.3) In one dimension, this becomes (4.4)

This becomes the companion equation to the Nernst–Planck equation (2.9). There are two dependent variables, N and E, and two equations relating them, so the problem is, as mathematicians say, well posed. These equations can be solved simply for the special case J = 0, but we will proceed to the general case of a finite current first.11 4.2. Finite current Equations 2.9 and 4.4 form a complete system of differential equations, in the sense that they contain enough information to specify a solution, given the boundary conditions. We could solve them just as they stand, but they look a little cluttered. What if the constants could all be made to disappear? Of course, we can't just set them equal to

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one, but we can get the same effect by absorbing the constants into the variables. Let N0 be an arbitrary unit of concentration. Then we can define a distance unit

(4.5)

a quantity closely related to the Debye length. We will replace E, N and J by their lower-case equivalents, and convert z to s, by using the following equations for our transformation of variables: (4.6) (4.7) (4.8) (4.9) (4.10) Substituting (4.6) to (4.10) into Equations (2.9), (4.4) and (2.3), we arrive at the dimensionless equations (4.11) (4.12) (4.13)

We can see that they are like the earlier equations but with the constants set equal to one (or two), and it was done quite legitimately! When we are finished solving them, we can simply use the scaling transformation (4.6)-(4.10) in reverse to put the constants back in. Now let's solve the equations. Substituting (4.12) into (4.11) we eliminate n to obtain (4.14)

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a second-order differential equation. We integrate it to obtain the first-order equation (4.15) where g = eI2 - nI is a constant. Since the value of s0 in Equation 4.6 has not yet been assigned, we now set it so that (4.16)

With this assignment, the constant g in (4.15) vanishes, and we have (4.17) This is a form of Riccati's equation. To simplify it, we substitute (4.18) where we have used (4.13). Differentiating the first of these equations twice, we obtain (4.19) (4.20)

Since from Equation 4.17 the quantity in parentheses equals js, we arrive at (4.21)

We assume that j g 0; the zero-current case will be solved below. To convert Equation 4.21 to a standard form, we absorb the constant j by the substitution (4.22) Then, by the chain rule, (4.23)

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so that (4.21) becomes (4.24) This is the Airy equation. Since the Airy equation is an equation of second order, it has two linearly independent solutions; these are called the Airy functions and written Ai()) and Bi()). They and their first derivatives Ai'()) and Bi'()) are tabulated, and their properties

Figure 7.2. The Airy function Ai(x). From Wolfram, 2002.

listed, in mathematical tables and handbooks.12 Computer routines for them are available. Higher derivatives can be calculated from these by using (4.24); e.g., Ai''()) = ) Ai()). The Airy functions are oscillatory for negative arguments. When the argument ) is positive, Ai()) decays and Bi()) grows rapidly. Figure 7.2 shows the function Ai(x).13 The general solution to Equation 4.24 is (4.25) where a and b are arbitrary constants to be determined by the boundary conditions. From (4.18) we obtain for the electric potential (4.26) Factoring out the a and substituting (4.22) results in (4.27) where R = b/a. The constant a is determined by the reference potential.

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For the dimensionless field e we obtain from (4.19)

(4.28)

where the as have canceled. This result could be obtained more directly by differentiating the potential (4.27). For the ion concentration we use, from (4.12) and (4.17), (4.29) a parabolic relation, which with (4.25) results in

(4.30)

Figure 7.3. Exact solutions of the electrodiffusion equations: plots of ion concentration n, electric field e and voltage v, as functions of distance s, for the positive dimensionless current density j = 1. The electric field at s = 0, Ie = 2, -2, as labeled. Left: g = -2; Middle: g = 0; Right: g = 2. Adapted from Leuchtag and Swihart, 1977.

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As often happens with nonlinear differential equations, these solutions are not well behaved. From Equations 4.27, 4.28 and 4.30 we see that singularities occur at points at which (4.31) These singular points separate the functions into branches or domains, as in the Maxwell -Wagner effect; see Figure 6.3(c), page 128. Figure 7.3 shows a set of these functions for j = 1 and g = -2, 0 and 2.14 The graphs display the logarithmic singularity of the potential, the positive and negative infinities of the field and the positive infinity of the concentration. Whether these singularities have any physical interpretation is not clear. When we use the boundary conditions to apply these equations to a membrane at a stationary state, we accept only solutions that do not contain singularities. Negative values of n, which are unphysical but appear in the solutions, must also be rejected. 4.3. Reclaiming the dimensions These equations contain the dimensionless quantities jb, js and jas. When we apply the transformation equations (4.5-4.10) back to the dimensionful solutions, we arrive at

(4.32)

where 1 and 2 are units of length. The field equation (4.28) becomes

(4.33)

and the dimensionful equations for N and V can be obtained similarly from 4.32, 4.8, 4.30, 4.10 and 4.27. 4.4. Electrical equilibrium Since the case of zero current density was excluded from these solutions, we must deal with it separately. We can call it the condition of electrical equilibrium.

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When J = 0, Equation 4.4 can be used to eliminate N from Equation 2.11, resulting in the second-order differential equation

(4.34) This can be integrated to the first-order equation (4.35) where ±A2 is a constant of integration. With (4.4), this becomes (4.36) Note that the E–N relationship is parabolic. The vertex of the parabola may be above, on or below the E-axis, depending on whether the constant, ±A2, is positive, zero or negative. Evaluating (4.36) at the left boundary, we see that (4.37) Thus the upper sign applies when the diffusion term is dominant in the current density, Equation 2.5, while the lower sign applies when the drift effect is dominant. Equation 4.35 can be rearranged to read (4.38) For the case in which diffusion and drift effects are balanced, A2 = 0, this is integrated to (4.39) giving for this case the solution

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(4.40)

Note that this solution has a singularity at z = 2kT/qEI. In the osmodominant case, i.e., with the upper sign, Equation 4.38 has the multibranched solution

(4.41)

For the electrodominant case, with the lower sign in Equation 4.36, a physically meaningful solution can only exist when E2 > A2 since N must be positive. This solution is

(4.42)

Equations 4.41 and 4.42 describe functions composed of branches separated

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by singularities. They are thus qualitatively different from those obtained by the constant field assumption. Singularities appear because of the limitations of the model as a descriptor of the physical system. However, even for a valid model, singularities are difficult to observe experimentally, due to instrumental resolution.15 4.5. Applying the boundary conditions Applying the boundary conditions (1.1) to the exact solutions of the one-ion steadystate classical electrodiffusion equations, we obtain membrane profiles of voltage, electric field and ion concentration for a given current. The resulting I–V curve is continuous if the current distribution in the membrane remains continuous; however, for certain sets of the thickness parameter l and the boundary concentrations n1 and n2, the electrodiffusion equations have no continuous solution. From the physical point of view, the membrane is too thick or the ion concentrations too low to carry the required current. It is not possible to write out an explicit current–voltage relation, since the current enters the solution (4.39) in an involved way. Instead, current–voltage curves are obtained by applying the boundary conditions for each value of the current separately and iterating.16 Figure 7.4 shows data of Cole and Moore fitted by this method.17 For positive currents, the fit is good up to about 50 mV. At higher outward currents, saturation occurs. This is qualitatively explainable by ion accumulation in the space between the axon and the Schwann-cell layer. Below the potassium reversal potential of about -50 mV, the fit becomes very poor. Although Cole and Moore give no data points for negative currents, rectification is known to be very high, so that the current density would remain close to the negative voltage axis. Thus we must conclude that single-ion electrodiffusion is unable to fit the rectification data. This is further discussed in reference 9, a review of electrodiffusion by Cole. However, it is necessary to point out that this model is not appropriately applied to squid axon in seawater, since it neglects the other ions present. Calcium ions in particular can interfere with monovalent ions, as we saw in Chapter 6, Figure 6.10. This would be a factor in inward currents but not outward, because external [Ca2+] is substantial while internal [Ca2+] is negligible. This consideration alone is qualitatively sufficient to explain why outward currents are well fitted by the model and inward currents are not. In the profiles, the field e increases from left to right. The concentration profiles drop down sharply at the left boundary for large negative (inward) currents, then shift toward the outer membrane surface as the currents become outward and increase. The consistency between the profiles is demonstrated by the steepening of the field profiles as the concentration of positive ions within the membrane increases: More force is required to push an ion against the repulsion of the ions already present in the membrane.

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Figure 7.4. Exact single-ion electrodiffusion applied to 1960 squid axon data of Cole and Moore. (Top) Current density versus membrane potential difference. The dimensionless fitting parameters and temperature are given on the graph. Note that the model does not fit the expected rectification. (Bottom) Membrane profile of electric field (left) and ion concentration (right) for dimensionless current densities varying from 1.5 to 9.0.

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4.6. Equal potassium concentrations An attempt to fit a series of current–voltage data obtained by Eduardo Rojas and Gerald Ehrenstein18 was again only partially successful; see Figure 7.5. Rojas and Ehrenstein obtained data from a squid axon intracellularly perfused with 600 mM potassium fluoride in seawater (open circles), then immersed in 600 mM KCl (squares), and lastly again in seawater (filled circles). The steady-state current–voltage curve for equal internal and external potassium-ion concentrations was nearly linear. The data points were fitted with exact electrodiffusion using parameters shown in the Figure.

Figure 7.5. Computed fit of current–voltage data obtained from squid axon by Rojas and Ehrenstein (1965) by exact solutions for single-ion electrodiffusion. The observed rectification in seawater is much greater than that calculated by single-ion electrodiffusion. From H.R. Leuchtag, unpublished calculations.

The calculated fits did not properly account for the rectification property in artificial seawater; although no data points are given for the seawater runs below -60

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mV, the currents for these membrane potentials would be expected to be much smaller than those predicted by single-ion electrodiffusion because of the rectification problem mentioned above. As we saw, the failure to fit the seawater data for inward currents should not be surprising, since seawater contains the divalent ion Ca2+, which may affect monovalent ion conductivity. It is significant that the rectification dilemma does not occur in the axon immersed in 600 mM KCl, a solution without divalent cations. Separate calculations with the constant field approximation gave fits almost identical to those given be the exact electrodiffusion computations. Although of course the field profiles are different, the current–voltage curve for the exact solution is very similar to the one obtained from the constant-field approximation. Like the approximate solution, the exact solution of the classical electrodiffusion equations does not resolve the problem of the rectification ratio. We will discuss this failure in Section 6 of Chapter 14.

NOTES AND REFERENCES 1. As we saw in Chapter 6, aqueous solutions are not neutral all the way to the membrane or other boundary. There is a double layer, of counterions and coions, within a few tenths of a nanometer of the surface. 2. A. L. Hodgkin and B. Katz, J. Physiol. (Lond.) 108:37-77, 1949. 3. We have here assumed that no interaction exists between the ion and the membrane; such interactions are often accounted for by replacing N by N, where  is the activity coefficient. 4. In some works, the symbol E is used to stand for voltage. 5. The mechanical mobility is the reciprocal of the coefficient of friction. 6. See, e.g., R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, AddisonWesley, Reading, MA, 1963, vol. I, Chapter 9. 7. Kenneth S. Cole, Membranes, Ions and Impulses, University of California, Berkeley, 1972, 450. By permission of University of California Press. 8. G. Ehrenstein and H. Lecar, Ann. Rev. Biophys. Bioeng. 1:347, 1972. 9. K. S. Cole, Physiol. Rev. 45:340-379, 1965. 10. F. Borgnis, Z. Physik 100:117, 478, 1936. 11. H. R. Leuchtag and J. C. Swihart, Biophys. J. 17:24-46, 1977. 12. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University, 1962; H. A. Antosiewitz, in Applied Math. Series 55: Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, U. S. National Bureau of Standards, Washington, 1964. 13. Stephen Wolfram, A New Kind of Science, Wolfram Media, Inc., Champaign, Ill., 2002, 145. 14. H. R. Leuchtag and J. C. Swihart, Biophys. J. 17:27-46, 1977. 15. Nigel Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, Reading, MA, 1992, 133. 16. N. Sinharay and B. Meltzer, Solid-State Electron. 7:125, 1964. 17. K. S. Cole and J. W. Moore, J. Gen. Physiol. 43:971-980, 1960; details of the data fitting are in H. Richard Leuchtag, Indiana University Ph. D. thesis, University Microfilms International, Ann Arbor, Mich., 1974, 142-162. 18. E. Rojas and G. Ehrenstein, J. Cell. Comp. Physiol. 66 (Suppl. 2):71-77, 1965.

CHAPTER 8

MULTI-ION AND TRANSIENT ELECTRODIFFUSION

In Chapter 7, we reviewed the theory of electrodiffusion as applied to a membrane traversed by a steady current of a single species of ions. Let us now consider two complications: multi-ion electrodiffusion and time-dependent electrodiffusion. We will also note that the system of electrodiffusion equations obeys a set of scaling rules. We close this chapter with a critique of the classical electrodiffusion model as applied to membranes containing voltage-sensitive ion channels. 1. MULTIPLE SPECIES OF PERMEANT IONS, STEADY STATE We now extend the discussion of the classical electrodiffusion model to the multi-ion case, which is important because membranes are permeated by multiple species of ions, and because of the role multiple ions play in ion-channel theory. We see from Equations 2.9 and 4.4 of Chapter 7, to be designated 7.2.9 and 7.4.4, that we have characterized ions by two properties, charge q (or valency Z) and mobility u. Let us begin by considering two ion species of the same charge; of course they will generally have different mobilities, say u1 and u2. We can say that they belong to the same charge class, and that they have the same valency. For example, Na+, K+, Li+ and NH4+ belong to the charge class with valency +1. Ions with valency -1, e.g. Cland Br- form a different charge class, and those with valency +2, e.g. Ca2+ and Mg2+, a different one yet. The electrodiffusion problem of a system of ions in the same charge class, albeit of different mobilities, is much simpler than the more general problem in which valency also differs. 1.1. Ions of the same charge We can write the generalizations of Equations 7.2.9 and 7.4.4 for two ion species of the same charge class by the equations

163

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Adding Equations 7.1.1 and 7.1.2, with N = N1 + N2, we obtain

(1.4)

Thus we end up with essentially the same Equations as 7.2.9 and 7.4.4, provided we define J = J1 + J2 and u' such that (1.5)

We can conclude that for two (and, by extension, any number of) ion species of the same charge, the forms of the solutions will be the same as those we already worked out for a single charge, although the problem is complicated by the fact that the boundary conditions for each ion species will be different. 1.2. Ions of different charges The situation is much more complex if both the valency and the mobility are different. We have derived the Nernst–Planck equation for this case from thermodynamics, Equation 4.9 of Chapter 6, (1.6)

A detailed mathematical analysis shows that, instead of the first-order Equation 7.4.17, we obtain a second-order equation when two charge classes are present, a third-order with three charges, and so on.1 The multi-ion problem is simpler when we consider the situation at electrical equilibrium. This problem was solved in the constant field approximation by David

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Goldman,2 with later modifications by Alan Hodgkin and Bernhard Katz.3 1.3. The Goldman–Hodgkin–Katz equation For the multi-ion case, the current equation for a uniform field, Equation 7.3.7, can be written

(1.7)

Let us consider the case in which the membrane is permeated by ions of only two charges, q1 = e and q2 = -e. There will be two ions of the first class, Na+ and K+, labeled 11 and 12 respectively, and one of the second class, Cl-, ion 21. Then

(1.8)

where we have used the definition of the partition function  in Equation 7.1.1 and that of permeability in 7.2.8. Similarly, for the potassium ion,

(1.9)

For the chloride ion, we find

(1.10)

For the net current density J = JNa + JK + JCl , we obtain from Equations 1.8-

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1.10, using the definition of permeability P given in Equation 7.2.8,

For the equilibrium case, J = 0, this equation is solved for Vm to obtain the Goldman–Hodgkin–Katz equation, (1.11) We recall from the Box on Chemical Notation (page 146) that the factor kT/e can be replaced with RT/F. Figure 8.1 shows a current–voltage characteristic calculated by Douglas Junge4 based on data obtained by Douglas Eaton and colleagues for an Aplysia giant neuron at 6°C.5 The ion concentrations given by Junge for the solutions, in mM, are [K]i = 280, [Na]o = 485, [Na]i = 61, [Cl]o = 485, [Na]i = 51, and the permeability ratios are PNa/PK = 0.12 and PCl/PK = 1.44. In these calculations, the external 10 mM Ca2+ concentration was neglected. In Equation 1.11, the absolute values of the permeabilities are not required; it is sufficient to specify their relative proportions. For a resting axonal membrane, the proportions PK : PNa : PCl = 1 : 0.04 : 0.45 were established by fitting data to the equation. At the peak of the action potential, the proportions become 1 : 20 : 0.45, showing a great increase in sodium ion permeability while the other permeabilities are unchanged. Note that the application of Equation 1.11 to the active membrane is contrary to the assumption of a steady state. While it is true that the first derivative of the current is zero at the maximum point, the second derivative is negative. Therefore the membrane is not in a steady state and the application of Equation 1.11 to the active membrane is invalid. Nevertheless, it has become commonplace, and we shall use it in the following example: For an axon (Table 4.1) with an internal solution of 400 mM K+, 50 mM Na+ and 100 mM Cl-, and an external solution of 10 mM K+, 460 mM Na+ and 540 mM Cl-, and with the permeabilities given in the text, the resting and action potentials may be calculated from the Goldman equation.6 2. TIME-DEPENDENT ELECTRODIFFUSION We now extend our discussion of simple electrodiffusion models to include time variation, particularly transient responses to currents. Time-dependent electrodiffusion in excitable membrane systems has been studied by a number of authors.7 For the timedependent case, we have to specify initial conditions as well as boundary conditions.

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Figure 8.1. Current–voltage curves predicted by the constant-field approximation for four external potassium ion concentrations. The ordinate variable E is the membrane voltage. Based on data by Eaton et al., 1975, for the Aplysia giant neuron. From Junge, 1981.

We will limit our discussion to the case of a single permeant species of ions. The Nernst–Planck equation now becomes, in partial derivative form, (2.1) Ions entering or leaving the region of interest change the number of ions enclosed within; this fact is summarized by the equation of continuity, (2.2) We can verify whether the ionic current density is uniform in the steady state: When 0N/0t = 0, 0Jion/0z must also vanish. Gauss's law, Equation 7.4.4, describes the effect of the ionic charge on the electric field.

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2.1. Scaling of variables While linear homogeneous differential equations, such as Equation 3.4, have the property that any solution multiplied by a constant, or any sum of solutions, is also a solution, this is not the case for nonlinear equations. However, these also exhibit certain symmetries. The system of electrodiffusion equations, Equations 6.1-6.3, obeys a set of rules according to which new solutions can be generated from existing ones. Given a real nonzero number , a new solution can be generated by the scaling rules:

(2.3)

We can verify the validity of these rules by substituting them into the equations: the s cancel. Note that the voltage V is invariant under transformation 2.3; so are combinations such as Jz3 and Nt. Such groupings have appeared as the arguments of functions in our exact solutions, such as Equation 4.27. The transformation keeps t and N positive even for negative values of . The scaling rules can be used, for example, to represent the solution for an electrodiffusion membrane of a certain thickness with that for a thicker or thinner one. 2.2. The Burgers equation If we substitute Gauss's law, Equation 7.4.4, into the equation of continuity, 2.2, we obtain

(2.4)

which can be integrated over z to give (2.5) Since we are dealing with partial derivatives, instead of a constant of integration we have an arbitrary function of time, J(t). Relation 2.5, due to Maxwell, states that the ionic current plus the displacement current8 equals the total current J(t), which must be equal to the external current applied to the electrodes. Therefore this formulation is directly applicable to a current-clamp experiment.

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When we substitute Equations 2.1 and 2.2 into Equation 2.5 and simplify, we obtain (2.6)

As usual in classical electrodiffusion, we assume mobility u and relative permittivity  to be constant. Equation 2.6, in dimensionless form, is a well known partial differential equation. Its homogeneous form, with J(t) = 0, called Burgers's equation,9 has been used to model turbulence, shock waves and other nonlinear dissipative phenomena. The full equation, with a nonzero forcing function J(t), called the forced or driven Burgers equation, has been applied to wind-driven water waves.10 2.3. A simple case Because the membrane boundary conditions and initial condition specify values of N, essentially the first derivative of E, rather than E itself, the problem is complicated. The problem in which the variable is specified at the two boundaries is called a Dirichlet problem, and the problem in which the first derivative of the variable is specified is a Neumann problem. To examine some of the features of Equation 2.6, we will explore a low-temperature approximation with no forcing function. When we look at Equation 2.6, we see that the second-order, diffusive term is the only one containing the temperature T. As T  0, that term must become negligible compared to the other terms, so it is useful to explore the equation without the second-order term. To further simplify matters, we will consider the case in which a finite membrane current is switched to zero at time t = 0, so that J(t) = 0, leaving (2.7) This equation has a simple general solution (2.8) where F is an arbitrary analytical function, as can be verified by differentiation of Equation 2.8 with respect to z and t. Remarkably, this equation represents a wave of electric field E in which the wave velocity v is directly proportional to E; if the quantity in parentheses is written as z - vt, we see that v = quE. The wave advances forward across the membrane when qE is positive and backward in the opposite case. The wave is dispersive, in the sense that it moves more rapidly the larger the field becomes. Because different parts of the waveform therefore travel at different speeds, the waveform tends to sharpen as the wave travels. The property of dispersion is seen in the breaking of ocean waves on a beach.

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The characteristics of Equation 2.8 illustrate the properties of the homogeneous Burgers equation. If we restore the diffusion term, its effect is to soften the sharp bends in the ion profile. We will see equations of this type again in the next chapter and in Chapter 18, which discusses solitons in liquid crystals. 3. INADEQUACY OF THE CLASSICAL MODEL We began Chapter 7 with the Shaker quotation, “'Tis a gift to be simple.” Yet the simple premise that the ion flux is the sum of diffusion and migration fluxes has, through the magic of mathematical analysis, turned into web of considerable intricacy, which we have here only begun to explore. The tree of electrodiffusion has borne many fruits: such important results as the Nernst potential, the Goldman and Goldman–Hodgkin–Katz equations, the strange singularities of the exact solutions, the unexpected similarity of time-dependent electrodiffusion to shock waves, and other insights into the way ions might move through membranes. But something is missing. With a few bold assumptions we have been able to obtain a deep insight into the problem of an abstract electrodiffusion membrane. But the price we have had to pay is to lose touch with the real membrane. We found that out when we saw that the predictions of the model do not agree with the experimental data. As pointed out by Cole in 1965, classical electrodiffusion has been unable to explain many of the properties of real excitable membranes, such as the negative differential resistance of the sodium channel and the unusual rectification characteristics of the potassium channel.11 We have seen that some of these difficulties arise from the inappropriate application of single-ion theory to a multi-ion experiment. In particular, the data fits attempted have not accounted for the presence of calcium ions in the solutions. Nevertheless, all the difficulties may not be resolvable simply by accounting for the interaction of the calcium ions with the other ions in the membrane. While classical electrodiffusion fails to account for the behavior of excitable membranes, we know that it contains special assumptions only justified by simplicity of the analysis. These include the assumption of constancy of the dielectric permittivity  and the ionic mobility u. Because these assumptions greatly narrow the scope of classical electrodiffusion, we cannot peremptorily reject the electrodiffusion approach. Rather, we must call upon our intuitive skills and background knowledge to alter the assumptions so as to steer our model closer to reality. Where have we gone wrong? Which assumptions are faulty? We will leave these questions for the reader to ponder as we move on, but we will take them up again in Section 6 of Chapter 14. NOTES AND REFERENCES The two-charge case was analyzed by L. Bass, Trans. Farad. Soc. 60:1656-1663, 1964 and L. J. Bruner, Biophys. J. 5:867-886, 1965. H. R. Leuchtag, J. Math. Physics 22:1317-1321, 1981 generalized the analysis to an arbitrary number of charge classes. 2. D. E. Goldman, J. Gen. Physiol. 27:37-60, 1943. 3. A. L. Hodgkin and B. Katz, J. Physiol. (London) 108:37-77, 1949. 4. Douglas Junge, Nerve and Muscle Excitation, Second Edition, Sinauer Associates, Sunderland, 1981, 38. 1.

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5. D. C. Eaton, J. M. Russell and A. M. Brown, J. Membrane Biol. 21:353-374, 1975. With kind permission of Springer Science and Business Media. 6. Resting potential = -58.7 mV; action potential = 43.6 mV. 7. See Kenneth S. Cole, Physiol. Rev. 45:340-379, 1965, Jarl V. Hägglund, J. Membrane Biol. 10:153170, 1972, and references cited therein. 8. The vector quantity D = 0  E is called the electric displacement. 9. This equation belongs to the quasilinear parabolic class of partial differential equations. It appeared in a 1915 paper by Harry Bateman (Monthly Weather Review 43:163-170) and a 1950 paper by J. M. Burgers (Proc. Kron. Ned. Acad. 53:247-261), and is extensively discussed in Burgers's book, The Nonlinear Diffusion Equation. Asymptotic Solutions and Statistical Problems, D. Reidel, Dordrecht, 1974. Solutions to the Burgers equation are listed in E. R. Benton and G. W. Platzman, Appl. Math. 30:195-212, 1972. 10. H. R. Leuchtag and H. M. Fishman, in Structure and Function in Excitable Cells, edited by D. C. Chang, I. Tasaki, W. A. Adelman Jr. and H. R. Leuchtag, 415-434, Plenum, New York, 1983. 11. K. S. Cole, Physiol. Rev. 45:340-379, 1965.

CHAPTER 9

MODELS OF MEMBRANE EXCITABILITY

In this chapter we will examine models of the onset and propagation of a traveling action potential, including the powerful phenomenological model, the Hodgkin and Huxley (HH) model.1 This milestone work, based on quantitative interpretation of experimental data, has determined the direction of progress in electrophysiology for decades, led to numerous experimental discoveries and spawned many related models. We will review its assumptions, the way its equations are interpreted and the predictions this model has produced. While a successful phenomenological model is a great step forward, it is not the solution of the problem. The HH and related models, based as they are on macroscopic measurements, cannot be expected to explain what is happening at the molecular level. Still, it was not long before speculative models of the workings of the ion channel molecules began to appear. A scientific problem is solved only when the behavior of the system can be derived from a priori principles, and that still remains to be done for the processes underlying nerve and muscle excitability. 1. THE MODEL OF HODGKIN AND HUXLEY In Chapter 4 we noted the development of electronic instrumentation for the squid axon, particularly the voltage clamp, which gave the experimenter control over the membrane voltage. Hodgkin visited Cole at the University of Chicago to acquaint himself with the new device, and returned to Cambridge to improve the voltage clamp and pursue a highly focused series of measurements with his colleagues, Bernhard Katz and Andrew Huxley. It had become clear to them that the electrodiffusion theory (at least as then formulated) was inadequate to explain the behavior of nerve membranes (see Chapter 8). Hodgkin and Katz showed that the sodium ion was responsible for the excitation and overshoot of the action potential in squid axon.2 They became convinced that the action potential, which not only neutralized the resting potential but actually became positive, was expressing an increased membrane permeability to sodium ions. This was a new concept, because up to that time Bernstein's idea that only potassium ions played a role in nerve conduction had prevailed.3

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1.1. Ion-current separation and ion conductances By replacing Na+ with an impermeant ion in the external solution, Hodgkin, Huxley and Katz were able to separate the current measured with the voltage clamp into its ionic components. This approach later became supplanted by the use of toxins and other channel blockers. With the picture on classical electrodiffusion in confusion due to its failure to fit axonal data, the Cambridge group tried an eclectic approach. Borrowing from electrodiffusion, chemical kinetics and circuit theory, they developed a new formalism, which they could adjust to the demands of the data they were generating. Hodgkin and Huxley developed techniques for breaking down the problem into smaller steps; at each step, they examined their voltage-clamp data to see what mathematical forms were needed to provide a fit. At the culmination of their model-building, they arrived at a partial differential equation that predicted a traveling wave. The profile of their computed wave provided a good fit to the measured action potential, a remarkable achievement that provided a sense of closure to the problem. 1.2. The current equation Hodgkin and Huxley observed that the electric current crossing a membrane could be described as a sum of ionic components, of which the currents carried by sodium, INa, and potassium, IK, were the most important. With the other components lumped together as IL, called the leakage current, the net ionic current was (1.1) In a series of careful experiments, they found that these currents depended on voltage approximately linearly for an instantaneous response over a limited range, according to the relationship (1.2) where the subscript i refers to Na, K or L. Because gi is a current divided by a finite voltage increment, it is called a chord conductance, as opposed to a slope conductance dI/dV. Slope and chord conductances are in general different and need not even be of the same sign. Since the ith ion current is zero and the current changes sign when V = Vi, the constant Vi is called the reversal potential. This is roughly equal to the Nernst potential of the ion at equilibrium, Equation 7.2.14. Equation 1.2 states that the current through the channels permeable to ion i in a given patch of membrane is the product of the ion chord conductance gi and the driving force V - Vi. Although the equation has an ohmic form, it is not linear: The conductances are not constant, but were found to be functions of V as well as calcium ion concentration and even time. The dielectric properties of the bilayer were modeled by a membrane capacitance c, so that the net ion current was described by the equation

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175 (1.3)

Current, capacitance and conductances are usually referred to unit area of the axonal membrane. Hodgkin and Huxley took gL to be a constant; it was later found to be voltage-dependent. Equation 1.3 can be represented by a schematic circuit of parallel capacitive and conductive branches, in which the sodium and potassium conductances are represented as variable resistors in series with their ionic reversal voltages; see Figure 9.1, from Hille’s adaptation4 of the Hodgkin and Huxley diagram.5

Figure 9.1. Equivalent circuit of the Hodgkin–Huxley current equation. The symbol E stands for the reversal voltage of the respective branch. The variable resistances are controlled by membrane voltage. From Hille, 2001, after Hodgkin and Huxley, 1952d.

1.3. The independence principle Hodgkin and Huxley treated the axonal currents carried by different ions as traversing separate entities acting in parallel. This concept, based on data from isotopic ion measurements analyzed by an electrodiffusion analysis by Hans Ussing,6 became known as the independence principle. Ussing’s calculations predict that the ratio of simultaneous influx and efflux of an ion at one potential is equal to the ratio of the external to internal electrochemical activities of the ion.

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1.4. Linear kinetic functions In their squid axon experiments, Hodgkin and Huxley demonstrated that the instantaneous current-voltage relationship is linear, and separated the fast (Na) and slow (K) chord conductances as nonlinear functions of first-order kinetic functions of membrane voltage. In these experiments, the state of the ion-conducting pathways was altered by prepulses of different size and sign. For the fast current, the rise and spontaneous fall of the conductance was modeled by separate linear kinetic functions for activation (m) and inactivation (h). The delayed current, carried by potassium ions did not appear to inactivate substantially and so was modeled by the single activation function n. The value of this function is given by the linear differential equation (1.4) The solution to this first-order linear equation can be obtained directly. We rewrite it in the form (1.5)

where (1.6) (1.7)

Equation 1.5 is solved to obtain (1.8) where n0 is the value of n at zero time, when the membrane potential is displaced from an equilibrium state. 1.5. Activation and inactivation The early current declines from its peak during prolonged depolarizations. A conditioning pulse of depolarization prior to the depolarizing clamp shows that this inactivation is delayed by several milliseconds (like the rise of the potassium current). The sodium current exhibits both activation m and inactivation h, given by the equations

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177 (1.9) (1.10)

The functions m and h have solutions of the same form as Equation 1.8. The actual functions differ, however, because of the different values of the parameters. As these linear first-order functions were not adequate to fit the voltage-clamp data, Hodgkin and Huxley generated nonlinear function from their powers and products. For the K+ current, data fitting gave the relationship

(1.11) between the potassium conductance and its activation function; gK is the maximum potassium conductance. The fast current was fitted by the equation (1.12) The coefficients  and  in Equations 1.4, 1.6 and 1.7 are functions of membrane voltage, calcium-ion concentration and temperature; gNa is the maximum sodium conductance. The parameters m0, -m, m, h0, -h and h are defined analogously to Equations 1.6 to 1.8. The voltage dependence of the parameters was modeled by modifying the Goldman–Hodgkin–Katz equation to obtain

(1.13 )

where V equals membrane voltage minus resting voltage. The values of the HH parameters may be adjusted for different experimental preparations. Note that -m is an order of magnitude smaller than either -h or -n.

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These functions are given a probability interpretation similar to that for n, given in Section 2.2 below. Three simultaneous events of probability m are assumed to be necessary to open the sodium channel, while a single event of probability (1 - h) suffices to block it. 1.6. The partial differential equation of Hodgkin and Huxley When Equations 1.11 and 1.12 are substituted into 1.3, we obtain (1.14) From our discussion of cable theory in Chapter 6, particularly Equation 3.5 there, the current density of a nerve fiber in a large volume of external conducting fluid can be written as (1.15) A voltage impulse V traveling at constant speed  must be describable by a function V(x - t). Therefore 0V/0t = -  0V/0x and 02V/0t2 = 2 02V/0x2, so that we obtain

(1.16)

This, Hodgkin and Huxley’s differential equation, determines the time course of the model action potential. Note that the voltage dependence in this equation is both explicit and implicit; n = n(t, V, [Ca2+]), and similarly for m and h. 1.7. Closing the circle The final paper of the four Hodgkin and Huxley published in 1952 contains a calculation of the wave profile obtained by the data fit, closing the circle.7 The calculation gave a remarkably good fit to a measured action potential, providing a worthy conclusion to their work. Surprisingly, Huxley’s later calculations revealed that in addition to the propagated action potential the HH equations also predict an unstable slow wave.8 Figure 9.2, from a book by Alwyn Scott, shows both.9

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Figure 9.2. A propagating action potential and an unstable threshold impulse generated by the HH partial differential equation. From Scott, 2003.

2. EXTENSIONS AND INTERPRETATIONS OF THE HH MODEL By providing a quantitative framework for analyzing excitability in nerve and muscle cells, the Hodgkin and Huxley model revolutionized membrane biophysics. While it does not answer all questions about ion currents across membranes, it became the basis for a host of new measurements and theoretical developments. 2.1. The gating current The HH equivalent circuit tells us that there are specialized structures within the membrane that respond to electric fields and selectively conduct specific ions: the voltage-sensitive ion channels. We noted in Chapter 1 that voltage plays two roles, one in impelling a change in the conformation of the channel molecules, and the other in helping to drive the ions across the membrane. This suggests that the conformational change may induce a tiny current, called the gating current, even when no ions are available to cross the membrane. (The word gating in this traditional term is used here in a metaphorical sense and should not be interpreted as implying the presence of material “gates” in the channel; see the discussion in Chapter 14.) As the underlying mechanism of the open–close transition, Hodgkin and Huxley proposed that the voltage dependence of the conductance change is the result of movements of a charged or

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dipolar component of the membrane.12 Such movements, apparently necessary for the transition to occur, can be expected to induce displacement currents (see Chapter 8). The prediction stimulated a search for a membrane current not due to the crossing of ions. Of course, these tiny currents normally would be masked by the much larger currents of ion transfer. Hodgkin and Huxley predicted that when the permeant ions are removed from the system, the charges or dipoles that control the gating process would be detectible, making possible an examination of the underlying processes. This experiment was done by three groups, who sought to detect these gating currents by eliminating permeant ions from the membrane or by blocking conducting systems.10 Any shift in the distribution of charges or dipole moments could elicit a gating current. The measurement of the gating current, which required rather sophisticated protocols, showed that there were indeed minute currents, equivalent to positive charges moving outward. By applying equations based on certain assumptions, they determined that the gating current was equivalent to a small number of elementary charges flowing outward. The number of these gating charges was determined to be about six. The interpretation of these findings is clearly critical to the understanding of channel function. One current view of the way channels respond to electrical stimulation is that charge movements drive conformational changes in the channel.11 This driving was often interpreted as an electrostatic force or torque that caused a hypothetical cluster within the channel, called the gate, to move in such as way as to relieve an obstruction of the supposed aqueous pore. This hypothetical mechanism will be discussed in Chapter 14. 2.2. Probability interpretation of the conductance functions Equation 1.11 has been interpreted to mean that four charged particles are required to move to a certain region of the membrane under the influence of the electric field to allow a path for potassium to form. If n is the probability for one n particle to be in place, the probability for all four to be is n4. This has been modeled by the following sequential reaction scheme:

 CLOSED

0

:

4

2 1

: 3

3 2

:

2

4 3

:

4

OPEN



An analogous probability interpretation of Equation 1.12 requires three m particles to be present and the h particle to be absent for the sodium channel to be open. 2.3. The Cole–Moore shift In 1960, Kenneth S. Cole and John W. Moore performed a series of experiments with results that conflicted with the Hodgkin–Huxley equations. The squid axon was initially

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voltage clamped to a hyperpolarizing potential, ranging from -52 to -212 mV, before being raised to its depolarizing potential of +60 mV. To inhibit the activation of sodium channels, Cole and Moore set the depolarizing potential at a value equal to the equilibrium potential of the sodium system, V = VNa, so that no sodium current would flow. The residual current after the capacitance transients was the sigmoidal potassium current. As Figure 9.3 shows, the negative preconditioning voltage pulses produces a series of records shifted in time, parallel to themselves. The greater the hyperpolarizing prepulse, the longer the delay period.12

Figure 9.3. Transient current density across squid giant axon membrane after a voltage clamp to a potential near the sodium reversal potential from seven different holding potentials, as indicated. In the data of Cole and Moore (1960) , the current rise is delayed by an interval that increases with the magnitude of the hyperpolarization. From Cole, 1972.

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While Cole and Moore found their results in agreement with the HH assumption that the potassium current is a function of a single variable of state, n, they found the HH fourth power inadequate to fit the observed delays. Instead, they found that a 25th power function fitted the data well, and proposed to replace Equation 1.11 with the equation (2.1) The sequential scheme of Section 2.2 turned out to be inadequate to model the parallel rises of the potassium current together with the measured inductive delays.13 Alternative theoretical models proposed to overcome these difficulties will be discussed in Section 5 of Chapter 14. 2.4. Mathematical extensions of the Hodgkin and Huxley equations Among the many influences of the Hodgkin and Huxley model were the many mathematical developments spawned by its equations. The model made it possible to simulate the response of an axon on a digital computer. The effects of temperature were dealt with by scaling the capacitance and time, while leaving the s and s (or -s) unchanged.14 At a Celsius temperature T, C becomes 1C and the unit of time becomes 1/1 where, for Q10 = 3, 1 = 3(T - 6.3)/10. The HH equations belong to the class of reaction–diffusion systems,15 as does an early model of impulse conduction by Franklin Offner and collaborators.16 Simplified versions of the HH equations were studied by Richard FitzHugh17 and JinIchi Nagumo and collaborators18 and by V. S. Markin and Y. A. Chizmadzhev.19 Catherine Morris and Harold Lecar adapted the HH model to membrane switching by calcium rather than sodium ions in their study of voltage oscillations in barnacle muscle.20 The Burgers equation, which we studied in the section on time-dependent electrodiffusion of Chapter 8, is a related nonlinear diffusion equation. 2.5. The propagated action potential is a soliton Solitary waves described by nonlinear partial differential equations are studied in mathematical physics under the name of solitons. Rigorous solitons are characterized as stable propagated waves with finite energy that are continuous, bounded and localized in space, with certain special properties: They collide elastically, keeping their identity after a collision, and a soliton may decompose into two or more solitons. Nonlinear waves in the real world do not usually possess these special properties, and rigorous solitons are considered a special case of solitons.21 Thus we can say that the propagated action potential is a soliton. It is not a rigorous soliton, since two colliding action potentials, far from passing through each other, will destroy one another because of the property of inactivation. But it is a soliton, quantitatively described by the Hodgkin and Huxley partial differential equation. We will further explore the concept

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of solitons in Chapter 18, where we explore solitons in liquid crystals. 2.6. Action potential as a vortex pair The action potential consists of a pair of vortical rings (“smoke-rings”), oppositely directed; see Figure 9.4.22 Vortices are of interest in certain areas of physics, such as Benard cells and superfluidity, where microscopic quantized vortices have been postulated; see Chapter 15.

Figure 9.4. The nerve impulse as a vortex pair. a) An impulse travels toward the depolarized end on the left of a squid axon. The membrane potential shows a positive overshoot. b) The current (only shown for the upper membrane surface) flows in toroidal closed circuits, with a double toroid at the action potential and a single vortex, the injury current, at the cut end of the axon. From Cole, 1965.

2.7. Catastrophe theory version of the Hodgkin and Huxley model An interesting mathematical development that grew out of the Hodgkin and Huxley model is an analysis based on a general theory of nonlinear differential equations, catastrophe theory, developed by French mathematician René Thom.23 Thom's ideas were applied to the nerve impulse, as well as the heartbeat, by E. C. Zeeman, whose analysis provides a mathematical insight into the Hodgkin and Huxley equations.24 Thom recognized that certain essential characteristics of differential equations could be seen by a geometrical approach. He categorized equations according to their topology into groups with a fairly small set of characteristic shapes. One of these is the cusp, the sharp change of direction of a point on the rim of a rolling wheel. The key features of a differential equation can be seen from a consideration of its display as a sheet in three-dimensional space, suspended over a table. The surface of the table is

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called the control plane. The folds of the sheet will determine the motion of a point moving along it. Zeeman applied catastrophe theory to the nerve impulse as described by a simplified version of the Hodgkin and Huxley formalism. He models the current flow along the axon and the flow of ions through the membrane by two separate equations, rather than combining them into a single differential equation, as Hodgkin and Huxley do. This is because the propagation wave and the repolarization wave are physically different: the former can be stopped by cutting the axon but the latter continues, determined by local events. In the analogy of Chapter 1, a domino would continue to fall after tipping, even if the line of dominoes were disrupted by a gap. Starting with the Hodgkin–Huxley equations, Zeeman obtained simpler equations that will not be reproduced here. These equations may be plotted on the control plane to look like Figure 9.5.25 The state of the axon is represented by a point in three-dimensional space (a, b, x). Variables a and b determine the lower, control, plane, where a is a linear function of the potassium conductance gK and b is a linear function of the voltage above resting potential, V. The vertical coordinate x represents the negative sodium conductance, -gNa . The system point moves on the upper sheet, which is a single-valued attractor outside the cusp region of the control plane, and triple-valued within that region, with two attractors separated by a repellor. The membrane is in equilibrium at the resting potential, V = 0. In a threshold depolarization, the system jumps off the upper attractor to land on the lower attractor. This fast action is followed by a smooth return to equilibrium as the potassium conductance rises and then falls slowly. The application of a step by a voltage clamp raises V to Vc , displacing the state from equilibrium, point E, to position F in the clamp plane. The fast equation carries the state to point G on the slow manifold and the slow equation moves it to H, where the slow vector is perpendicular to the clamp plane. The return follows the dotted flow lines slowly to equilibrium. These concepts give us a new, pictorial way to look at the action potential. Their application qualitatively fits the data. The application of catastrophe theory to the HH description of the action potential suggests that the HH equations have captured certain important aspects of the problem, and that the actual form of the HH equations is not very critical to their ability to provide this description. The fitting of the action potential by the HH equations does not prove that they are correct in every detail. Catastrophe theory has also been applied to a molecular model of excitability, as we will see in Chapter 18. While the correctness of Thom’s mathematical analysis is undisputed, the application of catastrophe theory to the action potential and other biological and social science problems has been severely criticized.26 The model of the nerve impulse is faulted for disagreeing with the HH data, denying “universally accepted concepts” and leading to the wrong propagation speed for the action potential. Whether the application of catastrophe theory to the nerve impulse is a “blind alley” or a challenging approach is left for the reader to decide.

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Figure 9.5. Catastrophe theory model of the nerve impulse. From Zeeman, 1977.

2.8. Beyond the squid axon After the successful description of squid-axon data by the HH formalism, the question naturally arose, Will it work for other preparations? On the basis of Ichiji Tasaki’s observation that the node of a myelinated axon already is fitted with two useful connections, the insulated internodes, innovative methods were developed for voltage clamping the node.27 A Swedish investigator, Bernhard Frankenhaeuser, and his collaborators extended the method to the frog node of Ranvier. Frankenhaeuser, and with Dodge, used negative feedback to prevent longitudinal current flow into a node so as to measure the resting and action potentials.28 In 1963, he used this method to develop a quantitative description of potassium currents in the toad Xenopus by equations closely related to those of Hodgkin and Huxley. As in those equations, the

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potassium permeability PK was assumed to be a power function of a first-order equation, but the data gave an exponent of 2 for the potassium activation n. (2.2) PK is a constant. They found that the potassium current also displays where  inactivation, although much slower than that of the sodium current. Because of the slowness of the inactivation k, it can be absorbed into the constant prefactor for short pulses (