Supramolecular Chemistry: From Concepts to Applications 9783110595604

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Stefan Kubik Supramolecular Chemistry

Also of Interest Inorganic and Organometallic Polymers Narendra Pal Singh Chauhan, Narendra Singh Chundawat,  ISBN ----, e-ISBN ----, EPUB ----

Advanced Materials Theodorus van de Ven, Armand Soldera (Eds.),  ISBN ----, e-ISBN ----, EPUB ----

Reversible Deactivation Radical Polymerization Nikhil K. Singha and Jimmy Mays (Eds.),  ISBN ----, e-ISBN ----, EPUB ----

Silicon-based Polymers and Materials Jerzy J. Chruściel,  ISBN ----, e-ISBN ----, EPUB ----

Host-Guest Chemistry Brian D. Wagner,  ISBN ----, e-ISBN ----, EPUB ----

Stefan Kubik

Supramolecular Chemistry From Concepts to Applications

Author Prof. Dr. Stefan Kubik Technische Universität Kaiserslautern Fachbereich Chemie - Organische Chemie Erwin-Schrödinger-Str. 54 67663 Kaiserslautern Email: [email protected]

ISBN 978-3-11-059560-4 e-ISBN (PDF) 978-3-11-059561-1 e-ISBN (EPUB) 978-3-11-059357-0 Library of Congress Control Number: 2020943952 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 Walter de Gruyter GmbH, Berlin/Boston Cover image: Stefan Kubik Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

For my family Daniela, Miriam, and Jakob and my teachers Peter Klein, Günter Wulff, and Julius Rebek Jr.

Preface The year 2017 marked the 50th anniversary of supramolecular chemistry, whose origin is generally considered to date back to 1967, when Charles Pedersen’s first paper on crown ethers was published. One could argue that the field is, in reality, much older because supramolecular aspects were investigated even before 1967. Nonetheless, Pedersen’s publication undoubtedly served as a starting point for supramolecular chemistry to develop into the prominent research field it is today. With the almost explosive developments over the last five decades, supramolecular chemistry has played an important role in shaping the face of modern chemistry, partly because of its multidisciplinary character that allows bridging many scientific disciplines. A modern education in chemistry would therefore be incomplete without including some relevant aspects. I wrote this book with the intention to provide this basic knowledge. One of my objectives was to outline the differences between supramolecular chemistry and the chemistry taught in the early courses of chemistry studies. Supramolecular systems are, for example, typically held together by weak interactions, which causes entropy to have a profound influence on their thermodynamic stability. Moreover, the interactions are reversible and most systems are therefore highly dynamic, which requires a special view to understand their behavior. In addition to these fundamental aspects, I have devoted a large fraction of the book to the many fascinating topics that are associated with supramolecular chemistry today. Readers will thus obtain an overview of the field, but can also use the book as a reference since key concepts are easily identified by the questions distributed among the chapters. The first one is the following: How is this book organized?

The book starts with an introduction in which the term supramolecular chemistry is defined, and the historic development of the field and its current relevance explained. The next two chapters are then devoted to the fundamentals of molecular recognition processes. These chapters introduce the formalism of describing and characterizing the interactions of molecules and the different natures of these interactions. Many key concepts are explained in this context, and a good grasp of these aspects helps understanding the behavior of the systems described in later chapters. The fourth chapter focuses on the most important classes of receptors, their structures, syntheses, and binding properties. These receptors allow the complexation of a variety of different substrates and the chapter thus concludes with a classification indicating which receptor is used for which substrate type. The following chapters then deal with various specific topics of supramolecular chemistry, with the fifth chapter introducing strategies to assemble molecules and the sixth chapter concentrating on interlocked molecules. Chapters 7–10 address https://doi.org/10.1515/9783110595611-202

VIII

Preface

how to control molecular motion or develop supramolecular catalysts, carriers, or probes. These chapters also give first insight into applications of supramolecular systems. This aspect is examined in more detail in the final chapter to also outline potential directions into which the field might head in the future. Who could benefit from this book?

Although the book primarily addresses readers who are only starting to familiarize themselves with supramolecular chemistry and would like to obtain an accessible introduction, it may also be useful for those who already have a good understanding of the basic concepts and are interested in specific aspects. There is therefore no single way to read this book. Starting with the first three chapters is likely helpful to understand the concepts and terminology. All other chapters can be read independently, and additional information is hopefully located quickly with the available crossreferences. Are all aspects of supramolecular chemistry treated?

The short answer is no. The field is so large that I had to omit certain topics to retain the introductory character of the book. I decided to place the focus on supramolecular chemistry in solution, concentrating on discrete and structurally characterized complexes or assemblies, but to exclude or only briefly mention aspects relating to the solid state (clathrates, crystal engineering, molecular tectonics), materials (polymers, foldamers, gels, nanoparticles), or other larger assemblies (micelles, vesicles). I made this selection not because the latter aspects are less relevant but because I felt that they are outside the scope of an introductory textbook. I am aware that this view might not be shared by everyone, and I therefore apologize to all those who miss certain topics. I also apologize to all whose work or names I did not mention, although they made important contributions to the field. I had to make choices, which meant that I had to leave out many fine examples, even from friends and colleagues whose work I admire. I thank the people who supported me during the time of writing, most importantly my family. I thank Julia Bartl, Arne Lützen, and Konrad Tiefenbacher, who read parts of the book and made many helpful suggestions. I am also grateful to the team at DeGruyter for motivating me to embark on this venture and for their continuing support. I hope that this book will turn out to be a valued companion for many. Kaiserslautern, June 2020

Contents Preface

VII

Questions discussed

XV

1

Introduction and overview Bibliography 6

1

2 2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6

Analyzing complex formation 7 Thermodynamic and kinetic aspects 7 Analytical strategies and techniques 17 Strategies 17 NMR spectroscopy 23 UV–vis spectroscopy 28 Fluorescence spectroscopy 30 Potentiometry 30 Isothermal titration calorimetry 32 Bibliography 36

3 Understanding molecular recognition 37 3.1 Modes of binding 37 3.1.1 General considerations 37 3.1.2 Ion–ion interactions 40 3.1.3 Ion–dipole interactions 42 3.1.4 Dipole–dipole interactions 44 3.1.5 Hydrogen bonding 45 3.1.6 Halogen bonding 54 3.1.7 Cation–π interactions 56 3.1.8 Anion–π interactions 60 3.1.9 Aromatic–aromatic interactions 62 3.1.10 Dispersion interactions 64 3.2 Binding energies 66 3.2.1 General considerations 66 3.2.2 Trend analyses 67 3.2.3 Double-mutant cycles 69 3.2.4 Molecular balances 71 3.3 Solvent effects 72 3.4 Predicting binding strength in solution 80 3.5 Guidelines for receptor design 85 3.5.1 Complementarity, preorganization, and induced fit 3.5.2 Chelate effect and macrocyclic effect 87

85

X

3.5.3 3.5.4

Contents

Multivalency and cooperativity Allosterism and cooperativity Bibliography 98

89 93

4 Hosting ions and molecules 101 4.1 Receptors 101 4.1.1 Crown ethers 101 4.1.2 Cryptands 114 4.1.3 Spherands 122 4.1.4 Cyclodextrins 125 4.1.5 Cyclophanes 137 4.1.6 Cyclotriveratrylenes 144 4.1.7 Calixarenes 153 4.1.8 Calixpyrroles 165 4.1.9 Resorcinarenes 169 4.1.10 Pillararenes 186 4.1.11 Cucurbiturils 193 4.1.12 Clefts and tweezers 204 4.1.13 Foldamers 215 4.2 Substrates 219 4.2.1 Inorganic and organic cations 220 4.2.2 Inorganic and organic anions 223 4.2.3 Zwitterions and ion pairs 228 4.2.4 Neutral organic molecules 231 Bibliography 235 5 5.1 5.2 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.4 5.4.1 5.4.2 5.4.3 5.5 5.5.1 5.5.2

Assembling molecules 245 Self-assembly and template effects 245 Self-assembly mediated by the hydrophobic effect 262 Self-assembly mediated by hydrogen bonds 266 Introduction 266 Rosettes 271 Capsules 278 Tubes 290 Self-assembly mediated by halogen bonds 295 Introduction 295 Helices 296 Capsules 297 Self-assembly mediated by coordination bonds 297 Introduction 297 Helices 302

Contents

5.5.3 5.5.4 5.5.5 5.6 5.6.1 5.6.2 5.6.3 5.7 5.7.1 5.7.2 5.7.3 5.7.4 5.8

Grids 308 Rings 309 Cages 314 Self-assembly mediated by covalent bonds Introduction 326 Rings 330 Cages 335 Dynamic combinatorial chemistry 341 Introduction 341 Casting 348 Molding 351 Self-assembly 356 Systems chemistry 358 Bibliography 360

326

6 6.1 6.2 6.2.1 6.2.2 6.3 6.3.1 6.3.2 6.3.3 6.4 6.4.1 6.4.2 6.5 6.5.1 6.5.2 6.5.3 6.6 6.6.1 6.7 6.7.1 6.7.2

Threading molecules 369 Molecular topology 369 Synthetic strategies 374 Molecular strategies 374 Supramolecular strategies 380 Syntheses using metal coordination 383 Catenanes 383 Knots 390 Rotaxanes 394 Syntheses using charge-transfer interactions 398 Catenanes 398 Rotaxanes 403 Syntheses using hydrogen bonds 404 Catenanes 404 Knots 414 Rotaxanes 416 Syntheses using halogen bonds 418 Rotaxanes 418 Syntheses using the hydrophobic effect 420 Knots 420 Rotaxanes 423 Bibliography 424

7 7.1 7.2

Controlling molecular motion 429 Introduction 429 Rotaxane-derived machines 439

XI

XII

7.3 7.4

Contents

Catenane-derived machines 451 Machines without mechanical bonds Bibliography 459

454

8 8.1 8.2 8.2.1 8.2.2 8.3 8.3.1 8.3.2 8.4

Mediating molecular transformations 463 Introduction 463 Stoichiometric transformations 471 Transformation by functional group participation Transformation by confinement 475 Catalytic transformations 481 Transformation by functional group participation Transformation by confinement 493 Self-replication 498 Bibliography 508

9 9.1 9.2 9.2.1 9.2.2 9.3 9.3.1 9.3.2 9.4

Transporting molecules 513 Introduction 513 Cation transport 518 Channels 518 Carriers 522 Anion transport 523 Channels 523 Carriers 525 Water transport 528 Bibliography 530

10 Detecting molecules 533 10.1 Introduction 533 10.2 Single analyte sensing 536 10.2.1 Direct optical sensing 536 10.2.2 Indirect optical sensing 542 10.2.3 Direct electrochemical sensing 10.3 Multiple analyte sensing 547 Bibliography 553

544

11 Applying supramolecular systems 555 11.1 Introduction 555 11.2 Applications in medicine 556 11.2.1 Drugs 556 11.2.2 Drug formulations 558 11.2.3 Imaging 559 11.3 Applications in separation processes

562

471

481

Contents

11.3.1 11.3.2 11.3.3 11.4 11.4.1 11.4.2 11.4.3 11.5 11.6 11.7 11.7.1 11.7.2 11.7.3

12 12.1 12.2

Index

Chromatography 562 Extraction 563 Precipitation 565 Applications in materials chemistry Self-assembled polymers 567 Polymer networks 569 Self-assembled gels 573 Applications in catalysis 574 Applications in molecular electronics Applications in consumer products Textiles 578 Food 578 Household 579 Bibliography 579

567

577 578

Appendices 583 Concentrations in a 1:2 binding equilibrium 583 Concentrations in an indicator displacement assay Bibliography 586 587

584

XIII

Questions discussed Preface VII How is this book organized? VII Who could benefit from this book? VIII Are all aspects of supramolecular chemistry treated? 1

2

3

4

5

6

Introduction and Overview 1 What is supramolecular chemistry? 1 How did supramolecular chemistry emerge and develop?

VIII

4

Analyzing Complex Formation 7 How are 1:1 complexation equilibria formally described? 7 What happens when binding equilibria become more complex? 10 How does complex stability relate to binding enthalpy and entropy? 12 Is there a relationship between complex stability and the rate of complex formation? How can complex stoichiometry be determined? 17 How can complex stability be determined? 21 How can binding equilibria be analyzed? 23 Understanding Molecular Recognition 37 Which general parameters influence complex formation? 37 Which types of interactions cause molecules to stay together? How strong are intermolecular interactions? 66 How does the solvent influence complex stability? 72 How does water mediate molecular recognition? 76 Can binding strength be predicted? 80 Which strategies exist to achieve strong binding? 85

40

Hosting Ions and Molecules 101 Which strategies exist to favor macrocyclization reactions? 116 How does NMR spectroscopy help to characterize receptor-substrate complexes? How do water molecules in a receptor cavity contribute to complex formation? How does the counterion influence the binding of an ion to a receptor? 231 Assembling Molecules 245 What is the difference between preorganization and predisposition? How do templates work? 255 Can molecules be sorted? 259 How much space does a substrate usually occupy in a receptor cavity? Can a liquid be porous? 338 Does a template always amplify the best binder? 345 Are mixtures of molecules always messy? 358 Threading Molecules 369 What is the difference between topology and structure? 369 What is a mechanical bond? 373 What is a co-conformation? 386 When is a template passive and when is it active? 395

https://doi.org/10.1515/9783110595611-204

15

254

281

144 200

XVI

7

Questions discussed

Controlling Molecular Motion 429 What is a molecular machine? 429 What is the difference between a shuttle and a switch? Can a molecular motor power a car? 457

8

Mediating Molecular Transformations Do synthetic enzymes exist? 468 Is only DNA able to replicate? 498

9

Transporting Molecules 513 How do polar species cross cell membranes? How is membrane transport studied? 515

10 Detecting Molecules What is a sensor?

533 534

11 Applying Supramolecular Systems What is all of this good for? 555 Is this the end? 579 12 Appendices 583 How do I do the math?

463

583

555

513

437

1 Introduction and overview What is supramolecular chemistry?

A substantial part of the training in chemistry focuses on molecular chemistry. We learn in this context about the correlation between the structure of a molecule and its reactivity and about the synthetic methods available to form covalent bonds. The respective theoretical framework allows us to rather reliably predict how molecules react or how they are synthesized just by looking at their structural drawings. Let us take the reaction shown in Figure 1.1 as an example. We see that catechol and bis(2-chloroethyl) ether afford a macrocyclic product with six ether groups along the ring when treated with sodium hydroxide. Considering the intrinsic reactivity of phenols and alkyl halides under basic conditions, we can attribute the formation of each of the four new C–O bonds to the initial deprotonation of a catechol hydroxy group. Subsequently, the so-formed nucleophilic phenolate reacts with an electrophilic carbon atom in the reaction partner that releases a chloride ion as the leaving group. Each bond formation thus comprises a nucleophilic substitution and the formal generation of one equivalent of water and one equivalent of NaCl.

O OH 2

+

2 Cl

O

OH

NaOH

O

1-Butanol, reflux 44–48%

O

O

Cl O O

Figure 1.1: Reaction between catechol and bis(2-chloroethyl) ether under basic conditions that ultimately affords the depicted macrocyclic oligoether.

This mechanism adequately describes the actual formation of the C–O bonds, but does it also explain the formation of the macrocyclic product? Maybe if only two molecules of catechol and two molecules of bis(2-chloroethyl) ether would be present as Figure 1.1 suggests, but molecules are in reality rarely as lonely as in reaction schemes. Even reactions performed on a small scale involve the participation of an extremely large number of molecules when all reaction partners, reagents, and solvent molecules are considered. These molecules are moreover in constant motion and permanently bump into each other, rendering reaction mixtures very crowded and dynamic environments. As a consequence, the probability is high that a phenolate ion not only meets the correct partner in the above reaction but also the linear intermediates formed on the way to the product, potentially causing the composition of the solution to become very complex. Experimentally, however, the formation of https://doi.org/10.1515/9783110595611-001

2

1 Introduction and overview

the cyclic product proceeds surprisingly selectively and in good yields, even at a catechol concentration of 1.3 M, far from the high dilution conditions often used in cyclization reactions to favor ring formation over the unwanted chain elongation. The mechanism of ether formation, that is molecular chemistry, does not provide a straightforward explanation for this observation, indicating that effects that transcend the actual bond formation reaction might operate in this synthesis. As it turns out, these effects are related to the presence of the sodium ions that, although unimportant for C–O bond formation, are no innocent bystanders but play an active role in the reaction. At its onset, the sodium ions originating from the base preferentially interact with the solvent molecules. Once linear oligoethers start to appear in solution, however, they serve as additional and more potent binding partners. The respective interactions are mediated by the sequence of oxygen atoms along the chains of these intermediates, causing them to wrap around the cation. This situation is shown schematically in Figure 1.2 for the immediate precursor of the product. The resulting proximity of the phenolate group to the electrophilic carbon atom at the opposite end of the chain explains why the sodium ions facilitate the ring closure and improve the efficiency of the reaction.

O O

O Na

O

O O

Cl

Figure 1.2: Structure illustrating how a sodium ion preorganizes the linear precursor for the cyclization that affords the product in the reaction shown in Figure 1.1.

The principles underlying sodium ion complexation will be explained in later chapters, where many other systems in which similar interactions play a role will also be presented. In this introduction, the above example should just serve to illustrate that noncovalent interactions can exert characteristic effects on the outcome of a reaction. They are also responsible for the stereoselectivity of transformations mediated by certain catalysts, but their relevance extends far beyond reaction control. In biological systems, for example, such interactions induce protein folding, the substrate selectivity of enzymes, signal transduction, transport, the conservation and transmission of the genetic code, and many other fundamental biochemical processes. They are also important in various other areas of chemistry such as medicinal and materials chemistry, but to molecular chemistry, which is primarily concerned with creating covalent bonds, they are not central. The realm of intermolecular interactions, instead, lies at the heart of supramolecular chemistry, which is a field of chemistry whose name was coined by one of the pioneers, namely, Jean-Marie Lehn, who chose the Latin prefix supra to indicate that supramolecular chemistry transcends molecular chemistry. Lehn wrote in 1995, “Beyond molecular chemistry based

1 Introduction and overview

3

on the covalent bond there lies the field of supramolecular chemistry whose goal is to gain control over the intermolecular bond. It is concerned with the next step in increasing complexity beyond the molecule towards the supermolecule and organized molecular systems […]” [1]. This definition implies that supramolecular systems consist of structurally defined assemblies of interacting molecules whose formation is controlled by organizational principles encoded within the structures of the individual components. Figure 1.3 illustrates this idea.

Figure 1.3: Schematic representation of the formation of a supramolecular assembly from shape-complementary objects or molecules.

The shapes of some of the objects shown in this illustration allow them to arrange themselves such that the four curved segments surround the circle. The final arrangement is stabilized by attractive interactions between the individual components. These components thus recognize each other, ignoring the square-shaped constituents that cannot be incorporated into the product. An important prerequisite for this process to work is that errors occurring on the way to the final structure, which derive, for example, from incorrectly connected components, are constantly corrected. The interactions responsible for the assembly therefore need to be reversible, rendering the overall system dynamic and subject to thermodynamic control. Based on these considerations, we arrive at the following definition for supramolecular chemistry, which focuses on two basic principles, namely, recognition and reversibility, and covers most of the systems and processes discussed in this book. Supramolecular chemistry is a field in chemistry that deals with molecular recognition phenomena mostly under thermodynamic control.

Although this definition may not cover all aspects of supramolecular chemistry, since kinetically controlled processes also sometimes play a role, it is preferable to very broad definitions, which state, for example, that supramolecular chemistry comprises the development of functional molecules. While supramolecular systems are certainly functional, as we will see, it is questionable whether the reverse is always true. Acid–base indicators, for example, are functional because their optical properties depend on the pH of the solution, but this property and its use has no relation to supramolecular chemistry. In this book, a narrower view is therefore preferred.

4

1 Introduction and overview

How did supramolecular chemistry emerge and develop?

The advent of supramolecular chemistry is closely related to the reaction shown in Figure 1.1. This reaction occurred as an unwanted side reaction during the synthesis of bis[2-(2-hydroxyphenoxy)ethyl] ether, performed in 1962 by Charles J. Pedersen at DuPont in Wilmington, Delaware (Figure 1.4).

2 HO

+ Cl

O

O

O

O

NaOH

O

O

O

HO

OH

O

O

1-Butanol Cl

H+

O

O

O

O

Figure 1.4: Synthesis of bis[2-(2-hydroxyphenoxy)ethyl] ether performed by Charles J. Pedersen.

Pedersen was interested in this reaction because he wanted to use the VO+ complex of the product as a catalyst for olefin polymerization. He was aware that the available tetrahydropyranyl-protected starting material contained 10% of unprotected catechol, but he did not consider this impurity to be problematic because he expected it to give rise to easily separable oligomeric and polymeric by-products. Unexpectedly, the presence of catechol also afforded the macrocycle shown in Figure 1.1 in a yield of 1%, which could have easily gone unnoticed if the unusual properties of this product would not have sparked Pedersen’s curiosity. He observed, for example, that this compound is insoluble in methanol but readily dissolves after the addition of sodium hydroxide, which he correctly attributed to the binding of the sodium cation in the center of the ring. Pedersen subsequently investigated, in more detail, the syntheses and properties of such cyclic oligoethers, for which he coined the term “crown ethers,” and published an extensive account of this work in 1967 in the Journal of the American Chemical Society, which deals with almost 50 derivatives [2]. This paper is generally considered to mark the birth of supramolecular chemistry. Pedersen‘s work served as an inspiration for several other groups to invent novel low-molecular compounds, so-called receptors, that possess cavities available for the incorporation of suitable substrates. The next major step in this direction was made by the Lehn group with the development of cryptands, bi- or tricyclic analogs of crown ethers, whose affinities for alkali metal ions is typically significantly higher than those of crown ethers. The work on cryptands was only the first of numerous contributions from Lehn to the field of supramolecular chemistry. His work ranges from the development of various receptors and catalysts to polymetallic coordination compounds and organic materials. Accordingly, entire areas of supramolecular

1 Introduction and overview

5

chemistry and the associated concepts and terms, including the term supramolecular chemistry itself, can be traced back to him. The third pioneer of the field, Donald J. Cram, introduced the first chiral crown ethers and showed that these compounds are capable of enantioselective substrate recognition. Cram thus transferred a fundamental concept to synthetic systems that is a consequence of the homochirality of biomolecules in Nature. He then turned to the development of supramolecular catalysts that not only mimic the substrate affinity of enzymes but also their ability to transform the bound substrate. Cram furthermore developed various new classes of receptors, introduced many concepts of supramolecular chemistry such as that of preorganization, and he invented the term “host–guest chemistry” to describe the interaction of a receptor with its substrate. This term is still in use because it aptly illustrates the hosting of the substrate by a receptor (one could argue, however, that the receptor is a hostel rather than a host), but it refers to only part of the much broader field of supramolecular chemistry. In 1987, the Royal Swedish Academy of Sciences awarded the Nobel Prize in chemistry to Pedersen, Lehn, and Cram for their pioneering work, particularly for “their development and application of molecules with highly selective structurespecific interaction, i.e. molecules that can ‘recognize’ each other and choose with which other molecules they will form complexes” [3]. It was furthermore emphasized that the relevance of the molecules developed by Cram, Lehn, and Pedersen reaches far into life sciences, as they allow mimicking processes, which were previously the exclusive domain of biomolecules. It would be incorrect to say that there was no supramolecular chemistry prior to the work of the three Nobel Prize winners. Fundamental concepts underlying supramolecular chemistry were actually established well before Pedersen’s seminal paper, and groups were involved in what was later termed host-guest chemistry prior to 1967. Table 1.1 gives an overview of selected concepts and discoveries, which precede supramolecular chemistry and which are considered to have contributed to laying the groundwork of the field. In spite of this important early work, the design and development of synthetic supramolecular systems only really began after the publication of Pedersen’s paper. A further rapid increase in the worldwide research activities can be noted after the Nobel Prize was awarded in 1987 when many creative scientists joined the field. At the same time, the research became more and more diverse, touching organic, inorganic, physical, theoretical, materials, and analytical chemistry, often in combination with biochemistry. This interdisciplinary research not only led to a better understanding of thermodynamic and kinetic aspects of molecular recognition phenomena but also to the development of a wide variety of novel supramolecular systems. In many cases, the inspiration came from Nature, resulting in the transfer of biochemical concepts to synthetic systems such as catalysis, allosteric control of substrate binding, induced fit, cooperativity, multivalency, information storage, replication, transport, motion, and so on. The progressive improvement of

6

1 Introduction and overview

Table 1.1: Concepts and discoveries reported prior to 1967 that are relevant for supramolecular chemistry.  Alfred Werner describes the concepts of coordination chemistry.  Emil Fischer introduces the lock-and-key concept to rationalize the substrate selectivity of enzymes.  Hans Pringsheim reports on the complexation of organic compounds by cyclodextrins, natural macrocyclic receptors that were originally discovered by Antoine Villiers in .  Linus C. Pauling publishes his seminal paper in the Journal of the American Chemical Society entitled “The Nature of the Chemical Bond” in which he also alludes to the hydrogen bond.  James D. Watson and Francis H. C. Crick describe the structure of the DNA double helix.  Daniel E. Koshland Jr. introduces the “induced fit” concept, which refers to the conformational changes proteins undergo upon substrate binding.  Daryl H. Busch proposes a classification for template effects.

the understanding of these principles moreover facilitated moving further and further away from natural models and allowed making new discoveries. The corresponding extensive research activities caused supramolecular chemistry to develop into a prestigious and influential field of research. The current significance is also reflected in the fact that ca. 4,800 publications appeared in 2019 containing the term “supramolecular,” that is, more than 10 articles per day. The number of articles related in a broader sense to supramolecular chemistry is probably even higher because the term “supramolecular” often goes unmentioned today. Clearly, supramolecular chemistry has contributed substantially to shaping the face of modern chemistry and will continue to do so. It is therefore worth taking a closer look at the different facets of this fascinating field, but this is not possible without a firm understanding of the basics. The relevant concepts are explained in detail in the next chapters.

Bibliography [1] [2] [3]

Lehn JM. Supramolecular Chemistry – Concepts and Perspectives. Weinheim, VCH, 1995, p. 2. Pedersen CJ. Cyclic polyethers and their complexes with metal salts. J. Am. Chem. Soc. 1967, 89, 7017–36. The Nobel Prize in Chemistry 1987 (Accessed April 30, 2020, https://www.nobelprize.org/ nobel_prizes/chemistry/laureates/1987/press.html).

2 Analyzing complex formation CONSPECTUS: Before we come to the actual forces that hold supramolecular systems together, general principles are introduced in this chapter on how to mathematically describe and experimentally characterize complex formations. Thus, this chapter provides an understanding of the basic concepts and methods underlying the characterization of the supramolecular complexes presented later in the book.

2.1 Thermodynamic and kinetic aspects How are 1:1 complexation equilibria formally described?

Molecular recognition processes occur between structurally complementary molecules and lead to complexes that are characterized by their stability, and by the number and arrangement of the interacting subunits. These processes can involve any number of molecules but we restrict the discussion initially to only two, which we name receptor and substrate. This distinction is not necessary. Many supramolecular systems in fact comprise binding partners that cannot be easily classified into these terms, but the above distinction helps to understand the following considerations. Examples of receptors and substrates are the host–guest systems discussed in Chapter 4, in which the receptor is typically larger and has a defined cleft or cavity to host the substrate (Figure 2.1).

+

R

+

S

C

Figure 2.1: Schematic representation of the complexation of a substrate S by a structurally complementary receptor R and the respective reaction scheme.

The equilibrium arrow in Figure 2.1 indicates that the formation of complex C is reversible and that the receptor R, the substrate S, and the complex coexist in solution. The larger the extent to which receptor and substrate are bound in the complex and the smaller the amounts of uncomplexed receptor and substrate, the more efficient the interactions. This efficiency is described in quantitative terms by

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8

2 Analyzing complex formation

using Ka , the stability or association constant, which results from the law of mass action according to the following equation. Ka =

cC cR cS

(2:1)

Note that equation (2.1) specifies the amounts of receptor, substrate, and complex in concentrations (cR , cS , cC ) instead of dimensionless activities. This is more practical since activity constants are usually not available for the species involved in binding equilibria. The approximation of using concentrations is even justified to some extent because binding equilibria are often investigated in dilute solutions, but it causes the resulting stability constants to have dimensions. Stability constants associated with 1:1 equilibria, for example, have units of M−1 (L/mol) because the denominator in equation (2.1) contains a product of two concentrations. Supramolecular chemists prefer the use of stability constants to characterize binding equilibria, maybe because there is a direct correlation between magnitude and stability: the larger the Ka the more stable the respective complex. In biochemistry, binding efficiency is usually denoted in terms of dissociation constants Kd , which are the reciprocal values of stability constants (Kd = Ka− 1 ). Thus, Kd is expressed in units of M (mol/L) and becomes smaller with increasing complex stability. No matter which value one prefers, Ka and Kd belong to the key thermodynamic parameters to describe the stability of supramolecular complexes. They are characteristic for every receptor–substrate combination, but depend strongly on external influences such as temperature or solvent. Thus, a comparison of the performance of different receptors is only feasible if the stabilities of their complexes were quantified under comparable conditions. Equation (2.1) does not directly give access to the concentration of the complex because the concentrations of receptor and substrate in the equilibrium are unknown. These concentrations are, however, linked to the initial concentrations of receptor (c0R ) and substrate (c0S ) through the complex concentration (cC ) by the following mass balances. cR = c0R − cC

(2:2a)

cS = c0S − cC

(2:2b)

Combining equation (2.1) with (2.2a) and (2.2b) leads to (2.3), which is a quadratic equation in cC that can be solved in a straightforward manner. Ka =


cC =

c0R + c0S + Ka− 1 2

cC 


 − cC c0S − cC sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 0 2 cR + cS + Ka− 1 − − c0R c0S 4

c0R

(2:3)

(2:4)

9

2.1 Thermodynamic and kinetic aspects

Of the two possible solutions, only equation (2.4) with the minus sign in front of the square root correctly describes how cC depends on c0R , c0S , and Ka . The following arguments explain the reason: if Ka < 1 M−1, Ka− 1 becomes large and the terms in front and behind the minus sign approximately amount to Ka− 1 =2. As cC is almost zero under these conditions, the equation is only fulfilled if the two terms are subtracted. Equation (2.4) allows calculating how much of a 1:1 complex of known stability will form from a receptor and a substrate when starting from a mixture of both binding partners at given concentrations. To illustrate this relationship, Figure 2.2a shows how complex concentration changes when increasing amounts of a substrate are added to a receptor solution with c0R = 10−3 M for complexes with stability constants of 102, 103, 104, and 106 M−1.

(a)

1.0

0.08

cC (mM)

0.8

cC (mM)

(b)

0.10

0.6 0.4 0.2

0.06 0.04 0.02

0.0

0.00 0

1

2

3

4

5 c S 0 / c R0

0

1

2

3

4

5 c S 0 / c R0

Figure 2.2: Graphs showing how the complex concentration changes when increasing amounts of a substrate are added to a receptor solution of given concentration. The initial receptor concentration amounts to cR0 = 10−3 M (=1 mM) in diagram (a) and to cR0 = 10−4 M (=0.1 mM) in diagram (b). The curves represent complexes of different stability with the Ka amounting to 102 M−1 (orange), 103 M−1 (red), 104 M−1 (blue), and 106 M−1 (black). The dotted lines mark the 1:1 substrate/ receptor ratios in both graphs.

In all cases, the complex concentration progressively increases with increasing substrate concentration. The black curve, representing the most stable complex, exhibits an almost linear rise until a 1:1 substrate/receptor ratio is reached, showing that every substrate molecule added to the solution is consumed in the complex. Once saturation is reached, further substrate molecules do not cause any change because no more free receptor molecules are available. The shape of this curve is consistent with equation (2.4) when Ka becomes large and Ka− 1 is therefore negligible. Assuming that c0S = xc0R with x denoting the substrate/receptor ratio and Ka− 1 ~ 0, the rearrangement of equation (2.4) yields cC = 0.5½1 + x − absð1 − xÞc0R . The complex concentration cC thus increases linearly between 0 ≤ x ≤ 1 and remains equal to c0R if x > 1.

10

2 Analyzing complex formation

The curves describing the less stable complexes in Figure 2.2a become progressively shallower as Ka decreases. As a consequence, substantial amounts of uncomplexed receptor and substrate are still present in solution even if an excess of the substrate is added. For comparison, 97% of the receptor is complexed at a 1:1 substrate/receptor ratio in the case of the complex with a Ka of 106 M−1, whereas the corresponding fractions amount to 73%, 38%, and 8% for the complexes with stability constants of 104, 103, and 102 M−1, respectively. Figure 2.2b shows that the decrease of the receptor concentration also causes the curves to become shallower. As a consequence, higher amounts of the substrate are required to reach saturation with respect to the situation in more concentrated solutions. We conclude from these considerations that in order to saturate the receptor with the substrate one can either: – add an excess of the substrate, whereby the exact amount is determined by complex stability, – or increase the concentrations of receptor and substrate.

What happens when binding equilibria become more complex?

For complexes that do not have a 1:1 stoichiometry, the law of mass action in equation (2.1) is no more valid. However, the general strategy to mathematically describe such equilibria is not very different from that explained earlier. Let us assume that the receptor is able to bind a second substrate molecule, leading to the formation of a 1:2 receptor–substrate complex. In this case, complex formation is a stepwise process, which involves the initial formation of the 1:1 complex that is subsequently converted into the 1:2 complex. The overall reaction thus comprises two equilibria of which each is associated with an individual equilibrium constant. The following reaction schemes illustrate the two steps. The corresponding laws of mass action are specified in the equations (2.5a,b). R +

S Ð C11 Ka11 =

C11

+

c11 C cR cS

S Ð C12 Ka12 =

c12 C c11 C cS

(2:5a,b)

In addition to these laws of mass action, we again need mass balances that establish a relationship between the initial concentrations of the substrate c0S and the receptor 12 c0R , and the concentrations of the 1:1 complex c11 C , the 1:2 complex cC , the substrate cS , and the receptor cR in the equilibrium. These mass balances are specified in equations (2.6a) and (2.6b). Note the factor 2 in equation (2.6b), which reflects the fact that the 1:2 complex contains two substrate molecules. 12 cR = c0R − c11 C − cC

(2:6a)

11

2.1 Thermodynamic and kinetic aspects

12 cS = c0S − c11 C − 2 cC

(2:6b)

Combining equations (2.5) and (2.6a) and (2.6b) ultimately affords a cubic equation 12 that can be solved in c11 C and cC . This solution is derived in Appendix 12.1. Here, we only qualitatively assess how the concentrations of the different complex species vary depending on the receptor–substrate ratio and the stepwise binding constants by using the two examples shown in Figure 2.3.

(a)

(b)

1.0

0.8

cC (1:1/1:2) (mM)

cC (1:1/1:2) (mM)

1.0

0.6 0.4 0.2

0.8 0.6 0.4 0.2

0.0

0.0 0

1

2

3

4

5 c S0 / c R 0

0

1

2

3

4

5 c S0 / c R 0

Figure 2.3: Graphs showing the concentrations of the 1:1 complex (orange) and the 1:2 receptor–substrate complex (red) when different amounts of a substrate are added to a 10−3 M (=1 mM) receptor solution. The black curves denote the sum of the concentrations of the 1:1 and the 1:2 complexes. In diagram (a) Ka11 = 100 M−1 and Ka12 = 10,000 M−1 and in diagram (b) Ka11 = 10,000 M−1 and Ka12 = 100 M−1.

The graphs in Figure 2.3a are associated with a stepwise equilibrium characterized by a Ka11 that is smaller than Ka12 . The 1:1 complex is therefore only formed to a small extent, but this amount is converted almost completely into the 1:2 complex. As a consequence, the overall equilibrium is dominated by the 1:2 complex and the curve describing its formation has a clear sigmoidal shape. Figure 2.3b shows graphs describing a two-step equilibrium in which Ka11 > Ka12 . The 1:1 complex is in this case almost the only species in solution until the substrate is present in excess. Only at higher substrate concentrations, significant amounts of the 1:2 complex are formed at the expense of the 1:1 complex. All curves in Figures 2.2 and 2.3 differ characteristically in shape. They therefore provide crucial information about the strength and stoichiometry of the underlying interactions. Accordingly, binding equilibria can be characterized by following how the concentrations of one or more species involved in complex formation change when the receptor–substrate ratio is varied. The fitting of the obtained binding isotherms to a suitable mathematical model then allows assessing whether the assumed stoichiometry is correct, ultimately also affording the stability constants of the investigated system. An obvious mismatch between the experimental and the theoretical

12

2 Analyzing complex formation

isotherms immediately indicates that assumptions made when selecting the binding model, for example the expected complex stoichiometry, were incorrect. Sigmoidal binding isotherms, for example, should not be evaluated on the basis of 1:1 equilibria. Before we look at the methods available for following binding equilibria in more detail in Section 2.2, other pertinent thermodynamic parameters associated with the stability of a receptor–substrate complex must be introduced. How does complex stability relate to binding enthalpy and entropy?

The thermodynamic driving force of complex formation is described in quantitative terms with the following equation. ΔG0 = − RTlnKa

(2:7)

This equation establishes a correlation between the association constant Ka of a complex and the Gibbs free energy of its formation ΔG0 . The other parameters are the gas constant R and the temperature T. Complexation processes are thus exergonic, that is, associated with a negative Gibbs free energy if Ka > 1. These reactions occur spontaneously, with the extent to which the complex forms depending on the difference in the Gibbs free energy of the system prior to complex formation and in the equilibrium. The more negative the ΔG0 , the more stable the complex. Note that ΔG0 describes the thermodynamic driving force of complex formation if all binding partners are in their standard states, which for dissolved species means that their concentrations amount to 1 M. Relating the thermodynamics of complex formation to this standard state has the advantage that different systems can be compared, thus rendering ΔG0 a useful alternative to Ka to quantify complex stability. In contrast to Ka , however, ΔG0 refers to only a single and in most cases relatively unrealistic situation. Experimentally, complex formation is rarely investigated in 1 M solutions and even if both binding partners are (or can be) mixed at this concentration, their interaction causes the system to immediately leave the standard state because the progressive formation of the complex causes the concentrations of the free binding partners to decrease. As a consequence, the absolute value of ΔG (note the absence of the index 0 , which indicates that this value refers to states of the system other than the standard state) progressively decreases until the equilibrium is reached where ΔG equals zero. The effect of the external conditions on the ΔG of complex formation can qualitatively be derived by comparing the graphs in Figure 2.2a and b. The blue curve in Figure 2.2a indicates, for example, that complex formation is 73% complete at equimolar concentrations of the binding partners, whereas the extent of complex formation goes down to 38% if the initial concentrations of receptor and substrate are reduced from 10−3 to 10−4 M (Figure 2.2b). The thermodynamic driving force of complex formation, that is, the exergonicity of the reaction, is thus much lower in the

2.1 Thermodynamic and kinetic aspects

13

dilute solution although the Ka of the complex is unchanged. There even exist ratios of receptor, substrate, and complex, were the reverse reaction, namely, complex dissociation becomes favorable. We therefore have to carefully distinguish between the ΔG0 that refers to the standard state and allows a comparison of different systems and the ΔG that refers to the thermodynamics of the system under the experimental conditions, which not only has a different absolute value than ΔG0 but can even have a different sign. In this context, it should be emphasized that the term system refers to more than just the receptor and the substrate. The thermodynamics of complex formation are strongly affected by all molecules present in the mixture, most importantly the solvent molecules whose concentration substantially exceeds that of the binding partners. These solvent molecules characteristically mediate the strength of receptor–substrate interactions and therefore have a profound effect on the thermodynamics of the reaction. If, for example, the Gibbs free energy required to desolvate the binding partners cannot be compensated by the Gibbs free energy gained during the complex formation, the corresponding complex will not form in the respective solvent although it might be very stable in an environment in which solvation is not so strong. It is therefore useful to separate the overall Gibbs free energy of binding ΔG0 into the intrinsic free energy of complexation in the gas phase ΔG0intr and the free energies of solvation of the receptor ΔG0solv ðRÞ, the substrate ΔG0solv ðSÞ, and the complex ΔG0solv ðCÞ. The respective treatment yields equation (2.8), which shows that the binding process only becomes exergonic if the intrinsic binding strength overcompensates the difference between the free energies of solvation of the complex and of its components (ΔG0solv ðCÞ − ΔG0solv ðRÞ − ΔG0solv ðSÞ). ΔG0 = ΔG0intr + ΔG0solv ðCÞ − ΔG0solv ðRÞ − ΔG0solv ðSÞ

(2:8)

We will come back to the influence of the solvent on the thermodynamics of complex formation in Section 3.3. Correlating binding strength with the Gibbs free energy ΔG0 allows further breaking down the energetics of complex formation into binding enthalpy and entropy. The corresponding underlying formalism is based on the following Gibbs–Helmholtz equation. ΔG0 = ΔH 0 − TΔS0

(2:9)

The change in enthalpy is defined as the heat change of the system during complex formation at a constant pressure. Again, ΔH 0 refers to the standard state in which receptor and substrate are present at 1 M concentrations. Heat changes during a molecular recognition process are to a first approximation related to the actual interactions between the molecules involved in the complexation process, with an attractive receptor–substrate interaction producing a favorable exothermic ΔH 0 contribution to ΔG0 . The direct receptor–substrate interactions are, however, not

14

2 Analyzing complex formation

the only factors influencing ΔH 0 . Further contributions come from solvent effects that could add adverse effects to ΔH 0 if the free binding partners are more strongly solvated than their complex. Moreover, unfavorable enthalpic contributions also result from strained receptor or substrate conformations in the complex. The overall binding enthalpy associated with complex formation therefore depends on a balance of a variety of factors and can end up to be exothermic (ΔH 0 < 0) or endothermic (ΔH 0 > 0). In the latter case, complex formation must be associated with a sufficiently large positive entropy to become overall exergonic. Entropy refers to the order of a system, with a positive ΔS0 denoting the increase of disorder, which promotes complex formation. There are a number of factors that contribute to the entropy. A fundamental one is that any binding process involving two or more molecules coming together to form a complex is necessarily entropically unfavorable because the individual components lose degrees of freedom, the most important ones being translational and rotational mobility. The question therefore arises whether it is at all possible for a complexation process to be entropically favored. The answer once again lies in the solvent contributions, namely, the reorganization of solvent molecules, which are released from the solvation shells of receptor and substrate when they interact. As a consequence, complexation processes, which intrinsically lead to ordered assemblies, increase the disorder of the overall systems by allowing solvent molecules to gain freedom. Global disorder can therefore cause local order, which is a very important concept in supramolecular chemistry in general and selfassembly in particular as we will see in Chapter 5. More detailed aspects of solvent effects are treated in Section 3.3. Note that entropy is temperature dependent, which has consequences when complex formation is investigated at different temperatures. Another important aspect is that the entropy of a solution increases upon dilution. This effect explains why complexation equilibria shift toward the dissociated species if the concentration is reduced as shown in Figure 2.2. It should also be noted that entropy more strongly influences the formation of supramolecular complexes than the formation of covalent bonds. The reason is that the interactions that stabilize supramolecular complexes are much weaker than covalent bonds and the corresponding ΔH 0 therefore smaller. The substantial contributions of both enthalpy and entropy to the formation of a complex is the cause for a peculiar phenomenon that can have very frustrating consequences when trying to optimize the performance of a receptor by increasing the binding strength, that is, by making the binding enthalpy more negative. The characterization of the newly designed and sometimes laboriously synthesized receptor not seldom reveals that it is actually not much better than the previous receptor because the improvement of ΔH 0 at the same time causes the binding entropy to become less favorable. The marginal change of ΔG0 is therefore caused

2.1 Thermodynamic and kinetic aspects

15

by the opposing directions into which ΔH 0 and ΔS0 develop, an effect called enthalpy–entropy compensation. This effect is qualitatively explained as follows: if binding becomes stronger, the complex also becomes “tighter.” As a consequence, conformational degrees of freedom of the receptor and the substrate are lost, which affects ΔS0 unfavorably. Conversely, weak binding is entropically beneficial since the binding partners retain flexibility in the complex. Whether enthalpy–entropy compensation is a real thermodynamic phenomenon is a controversial topic. Compensating effects are indeed often observed for supramolecular systems, but it is unclear, given the many factors that contribute to ΔH 0 and ΔS0 , why a gain in one parameter should almost exactly be canceled out by a loss in the other. Is there a relationship between complex stability and the rate of complex formation?

As we have seen, thermodynamics predicts that a negative ΔG0 causes complex formation to proceed spontaneously. This does not necessarily imply that complexation is also fast, although many binding equilibria in supramolecular chemistry are indeed associated with small activation barriers and are hence often (almost) diffusion controlled. There are, however, exceptions of which one is illustrated in Figure 2.4.

(a)

(b)

Free binding partners

Constrictive binding

G0

G0

G‡ G‡

G0

Complex Reaction coordinate

Free binding partners

G0 Complex Reaction coordinate

Figure 2.4: Energy profiles associated with complexation equilibria of which one has a small activation barrier (small ΔG‡) and leads to a stable complex (large ΔG0 ) (a) and the other has a large activation barrier (large ΔG‡) but leads to thermodynamically not very stable complex (small ΔG0 ) (b).

Figure 2.4a shows the energy profile of a complexation reaction where binding is associated with a substantial exergonic stabilization of the complex and a small activation barrier ΔG‡ that is easily overcome. In such a case, the equilibrium lies far

16

2 Analyzing complex formation

on the side of the complex. However, the system is dynamic, meaning that the complexes constantly form and dissociate. The situation shown in the energy profile in Figure 2.4b is different. In this case, the complex is thermodynamically not significantly favored over the free binding partners. However, complex formation and dissociation have to overcome substantial activation barriers. Such a situation arises, for example, if the receptor and the substrate have to adopt strained conformations to allow substrate exchange as in the hemicarcerands described in Section 4.1.9. Such complexes are therefore inert although their thermodynamic stability is low. To describe the behavior of these systems in quantitative terms, Cram introduced the term constrictive binding, which relates the Gibbs free energy of the transition state to the Gibbs free energy of the binding partners (Figure 2.4b) [1]. In other words, constrictive binding is the free energy that must be invested to reach the transition state from the uncomplexed state, rendering it also a measure for the rate of complex formation. The reaction profile in Figure 2.4a exhibits a small activation barrier and we therefore expect complex formation to be fast. But what does fast exactly mean in this context? To obtain information in this respect, it is useful to correlate the stability constant Ka of the complex with the rate constants associated with its formation and dissociation. This treatment is based on the rate equations (2.10a,b) in which kon represents the rate constant of complex formation and koff that of dissociation. R + S

kon

Ð koff

dcC = kon cR cS dt

C

−

dcC = koff cC dt

(2:10a,b)

Once the reaction reaches the thermodynamic equilibrium, the rates of complex formation and dissociation are the same (steady state), allowing us to write the following expression. kon cR cS = koff cC

(2:11)

The rearrangement of (2.11) yields equation (2.12), which shows that the stability constant Ka is given by a ratio of rate constants. kon cC = = Ka koff cR cS

(2:12)

Stable complexes thus form more rapidly than they dissociate. Note that this statement also applies to the steady state of an equilibrium, where the rates of complex formation (kon cR cS ) and dissociation (koff cC ) end up being the same because the rate constants are multiplied with concentrations. In the case of very stable complexes, for example, a large kon is multiplied with two very small concentrations, while a smaller koff is multiplied with a large cC .

2.2 Analytical strategies and techniques

17

The correlation in equation (2.12) allows us to estimate the lifetime of typical supramolecular complexes in solution. Assuming that complex formation is diffusion controlled (kon ~ 109 M−1 s−1) and that the complex has a Ka of 106 M−1, which represents a rather stable complex according to Figure 2.2a, we end up with a koff of 103 s−1. Thus, the complex has a lifetime in the order of milliseconds, which is rather short on the human timescale. The important lesson is that even highly stable complexes rapidly form and dissociate in solution and that the static picture suggested by reaction schemes such as the one in Figure 2.1 is misleading.

2.2 Analytical strategies and techniques 2.2.1 Strategies The only value required to derive the stability constant from the law of mass action in equation (2.1) is the complex concentration cC (in the case of higher complexes, the concentrations of all complexes present) because the concentrations of the receptor cR and the substrate cS are accessible from the mass balances (equations (2.2)). Unfortunately, most analytical techniques used for binding studies do not yield absolute concentrations but rather provide information to what extent the equilibrium shifts relative to the initial state if the concentrations of one or both binding partners are changed. Therefore, binding studies generally involve titrations during which the concentrations of the binding partners are varied while measuring a physical property that correlates with cC . These titrations afford binding isotherms, such as those shown in Figures 2.2 and 2.3, and fitting these isotherms to the mathematical model underlying complex formation then yields Ka . How can complex stoichiometry be determined?

The crucial aspect in this context is choosing the correct model because meaningful stability constants are only obtained if all aspects of complex formation, particularly complex stoichiometry, have been correctly taken into account in the mathematical treatment. It is therefore helpful to know the composition of the complex prior to recording binding isotherms, although information in this respect can sometimes be derived from the isotherms themselves. The sharp bend of the isotherms associated with the most stable complex in Figure 2.2a clearly indicates that this complex has a 1:1 stoichiometry, for example. Conversely, the black isotherm in Figure 2.2b reaches saturation approximately at a 1:2 receptor–substrate ratio, suggesting that two substrate molecules are bound to the receptor. Once the complexes become less stable, these correlations are, however, not reliable. Moreover, sigmoidal shapes of binding isotherms are important indications that higher complexes

18

2 Analyzing complex formation

are present, but they provide little information about the actual stoichiometry. Therefore, independent methods are usually needed to determine the composition of the complex. A classical approach relies on Job’s method of continuous variations, which Paul Job introduced in 1928 to determine the stability and stoichiometry of coordination complexes [2]. The underlying strategy involves the preparation of a series of solutions containing the receptor and the substrate at the same overall concentration but in different ratios, followed by measuring the amount of complex (or a property that correlates with complex concentration) in each solution. The idea underlying this method is illustrated for a 1:1 and a 1:2 complex in Figure 2.5. 0.50 c C (RS) c S0

(a)

c 0R 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction X

0.7

0.8

0.9

1.0

0.67 c C (RS2) c S0

(b)

c 0R 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction X

0.7

0.8

0.9

1.0

Fi