Multiple Equilibria in Proteins

By Jacinto Steinhardt, Jacqueline A. Reynolds and Nathan O. Kaplan

Multiple Equilibria in Proteins covers the multiple interactions between small ions and molecules and a protein molecule. The book also deals with the physicochemical mechanisms of this interaction and the information about protein structure and the forces stabilizing that structure. The text discusses the mathematical description of complex formation, the thermodynamic analysis of binding data, and various theoretical models which can be used to describe the phenomena of small molecule-macromolecule interactions. The measurement of complex formation; the binding of neutral molecules; and hydrogen-ion equilibria are also considered. The book further tackles metal-ion binding; the binding of organic ions by proteins; as well as protein-protein interaction. Chemists and biochemists will find the book useful.

• Language: English
• Publisher: Academic Press
• Release date: Jun 28, 2014
• ISBN: 9781483220307

Book preview

MULTIPLE EQUILIBRIA IN PROTEINS

Jacinto Steinhardt

DEPARTMENT OF CHEMISTRY, GEORGETOWN UNIVERSITY, WASHINGTON, D. C.

Jacqueline A. Reynolds

DEPARTMENTS OF BIOCHEMISTRY AND ANATOMY, DUKE UNIVERSITY MEDICAL CENTER, DURHAM, NORTH CAROLINA

Table of Contents

Cover image

Title page

Inside Front Cover

Copyright

Preface

Chapter I: Introduction

Publisher Summary

Chapter II: Thermodynamics and Model Systems

Publisher Summary

I Introduction

II Some Concepts in Thermodynamics

III Multiple Binding without Interaction between Sites

IV Binding with Interaction between Sites

V Method of Data Treatment

VI Thermodynamics of Multiple Equilibria

VII Binding-Induced Phase Transitions

VIII Linked Functions

Chapter III: The Measurement of Complex Formation

Publisher Summary

I Introduction

II Subtractive Methods

III Direct Measurements

IV Electrostatic Methods

V Other Methods

Chapter IV: Binding of Neutral Molecules

Publisher Summary

I Introduction

II Results Obtained with Particular Proteins

III The Hydration of Proteins

IV Interactions with Other Solvents

V Mechanism of Binding of Neutral Molecules

Chapter V: Hydrogen-Ion Equilibria

Publisher Summary

I Introduction

II Prototropic Groups in Proteins

III Some Experimental Details

IV An Experimental Titration Curve

V Anomalies Illustrated by Specific Proteins

VI Hydrogen-Ion Equilibria of Denatured Proteins

VII Unfolding as a Function of Protonation

VIII Summary

Chapter VI: Metal-Ion Binding

Publisher Summary

I Metal Ions and Their Complexes in Solution

II Experimental Methods and Data Treatment

III Proteins Containing Metal Ions Necessary for Biological Activity

IV Protein-Metal Complex Formation

Chapter VII: Binding of Organic Ions by Proteins

Publisher Summary

I Introduction

II Large Organic Ions–Ionic Detergents

III Dyes, Dye-like, and Other Cyclic Molecules Binding as Ions

IV Binding of Small Nonaromatic Ions to Proteins

Chapter VIII: Protein-Protein Interaction

Publisher Summary

I Introduction

II Quarternary Structure in Proteins

III Glutamic Dehydrogenase

IV Tobacco Mosaic Virus Protein

V Hemerythrin

Chapter IX: Summary and Conclusions

Publisher Summary

Author Index

Subject Index

Inside Front Cover

Molecular Biology

An International Series of Monographs and Textbooks

Edited by

BERNARD HORECKER,     Department of Molecular Biology, Albert Einstein College of Medicine, Yeshiva University, Bronx, New York

JULIUS MARMUR,     Department of Biochemistry, Albert Einstein College of Medicine, Yeshiva University, Bronx, New York

NATHAN O. KAPLAN,     Department of Chemistry, University of California At San Diego, La Jolla, California

HAROLD A. SCHERAGA,     Department of Chemistry, Cornell University, Ithaca, New York

HAROLD A. SCHERAGA. Protein Structure. 1961

STUART A. RICE AND MITSURU NAGASAWA. Polyelectrolyte Solutions: A Theoretical Introduction, with a contribution by Herbert Morawetz. 1961

SIDNEY UDENFRIEND. Fluorescence Assay in Biology and Medicine. Volume I—1962. Volume II—1969

J. HERBERT TAYLOR (Editor). Molecular Genetics. Part I—1963. Part II—1967

ARTHUR VEIS. The Macromolecular Chemistry of Gelatin. 1964

M. JOLY. A Physico-chemical Approach to the Denaturation of Proteins. 1965

SYDNEY J. LEACH (Editor). Physical Principles and Techniques of Protein Chemistry. Part A—1969. Part B in preparation

KENDRIC C. SMITH AND PHILIP C. HANAWALT. Molecular Photobiology: Inactivation and Recovery. 1969

RONALD BENTLEY. Molecular Asymmetry in Biology. Volume I.—1969. Volume II in preparation

JACINTO STEINHARDT AND JACQUELINE A. REYNOLDS. Multiple Equilibria in Proteins. 1969

DOUGLAS POLAND AND HAROLD A. SCHERAGA. Theory of Helix-Coil Transitions in Biopolymers. 1970

Copyright

COPYRIGHT © 1969, BY ACADEMIC PRESS, INC.

ALL RIGHTS RESERVED

NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by

ACADEMIC PRESS, INC. (LONDON) LTD.

Berkeley Square House, London W1X 6BA

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 79-86366

PRINTED IN THE UNITED STATES OF AMERICA

Preface

The authors of this book are keenly aware of the difficulties inherent in any effort to adequately treat a rapidly developing subject without erring on the side of lack of timeliness on the one hand, or on an uncritical emphasis on the most recent work on the other. We have often felt that we were on the verge of deducing general sets of ideas which would bring order and simplicity out of an appearance of multiple and almost arbitrary diversity. We hope that parts of this book will communicate such a feeling to the reader. Such a cliff-hanging feeling has twice led to a postponement of completion of the book, and therefore to a costly updating operation. Thus, we publish it now, still on the verge.

The authors are deeply indebted to Dr. Sherman Beychok of Columbia University who contributed importantly to the early stages of the formulation of the scope of this work, and whose assistance in the initial literature search was invaluable. Thanks are also due to Mrs. Ruth Frazier and Miss Susan Moss of Georgetown University for their patient assistance in the innumerable tasks that result in a final edited draft with verified citations. Without their help, and that of a number of others over a period of nearly five years, the manuscript would still be incomplete.

J.S.

J.A.R.

September, 1969

I

Introduction

Publisher Summary

This chapter provides an introduction to the book concerned with multiple interactions between small ions and molecules and a protein molecule. The book discusses the physicochemical mechanisms of this interaction and the information about protein structure and the forces stabilizing that structure, which can be obtained from binding studies. It is also concerned with binding isotherms that characterize the interactions referred to above, the dependence of these binding isotherms on the nature of the ligands involved and on environmental parameters, the nature of the binding sites, the forces involved in complex formation, and the extent to which ligands are bound to the propensity of the protein to undergo a binding-induced conformational change. This change is reflected in the isotherms themselves, because as a result of the conformational change some binding sites may be destroyed and new, often more numerous, sites may appear. There is an important difference between the multiple binding of hydrogen ions and of other substances, which is also discussed in the book.

This book is concerned with multiple interactions between small ions and molecules and a protein molecule. It deals with the physicochemical mechanisms of this interaction and the information about protein structure and the forces stabilizing that structure which can be obtained from binding studies. It is, therefore, also concerned with changes in protein structure brought about by small-ion or neutral-molecule binding as well as the converse, i.e., changes in binding induced by the alterations in protein structure. Preparation of this book has entailed an exhaustive literature search; the references selected for citation or discussion constitute a large fraction of the literature, but are not complete.

For the most part, we will be concerned with binding isotherms which characterize the interactions referred to above, the dependence of these binding isotherms on the nature of the ligands involved and on environmental parameters, the nature of the binding sites, and the forces involved in complex formation. Frequently we will relate the extent to which ligands are bound to the propensity of the protein to undergo a binding-induced conformational change. Such a change is reflected in the isotherms themselves, since as a result of the conformational change some binding sites may be destroyed and new, often more numerous, sites may appear.

We will usually exclude what may be called the characteristic reactions of enzymes, antibodies, and membrane transport proteins (Pardee, 1968) from the scope of this book. The reasons for doing so are numerous: (a) it would not be feasible to include all of them, (b) most of their characteristic binding reactions occur in molal ratios of 1 or 2 and are not examples of multiple equilibria, (c) the binding involved is often highly specific and implicates a specialized binding site peculiar to one protein and not a general property of proteins as a class.

Understanding the relation between complex formation and protein stability furnishes one of the strongest motives for studying multiple equilibria. If the initial binding of a ligand to a protein produces a tendency to bind more ligand (often referred to as cooperativity), we say that binding has destabilized the native conformation of the macromolecule. On the other hand, initial binding may result in stabilization of the native structure against conformational changes induced by other agents. The organization of the folded protein is therefore affected by ligand binding. This phenomenon has been clearly demonstrated in, for example, the hemoglobin molecule, by analyses of the oxygen binding isotherm and reaction rates (Wyman, 1948, 1965; Guidotti, 1967; Roughton et al., 1955; Antonini, 1965), by Perutz et al. (1965, 1968 a, b) by means of x-ray analysis in which clear differences are observed between oxy- and deoxyhemoglobin, and by measurements of the stabilizing effects of some covalently bound ligands of ferrihemoglobin (Steinhardt et al., 1963, Molday and Steinhardt, 1969).

The presence of a prosthetic group is not essential to binding effects on stability. Stabilization and destabilization are found in the binding of other molecules and ions to proteins in which there are no prosthetic groups. Most of the evidence for such effects has been obtained with large organic ions, such as the anions of fatty acids and common detergents. This evidence will be presented and analyzed in Chapter VII. Strong indications have also been obtained that the binding of some uncharged molecules is also accompanied by conformation changes. Such evidence is of two kinds: (1) the simplest and most direct comes from uv difference spectra, optical rotation, and hydrodynamic properties; (2) the second kind of evidence rests on analysis of the binding itself, i.e., the way in which the molal binding ratio, v, varies with concentration of free ligand. While the results of all such analyses are not yet fully analyzed, the most pertinent ones, for at least one protein, serum albumin, indicate that after small numbers of sites (8-12) on the native protein are occupied by certain ions, a much larger number of sites appear, at least some of which were not initially present. Here again, the most clear-cut evidence based on electrophoresis measurements refers to ions rather than neutral molecules. Beyond certain low concentrations of free ligand, electrophoretic mobilities indicate that two classes of macro-ions are present, one binding only a few molecules, the other binding a very much larger number. This kind of what has been called cooperative (nonstatistical) binding can be explained only on the basis of a molecular transformation creating macromolecules containing a large number of binding sites. These latter binding sites may be characterized by association constants which may be smaller than, equal to, or larger than those responsible for the initial binding at the lowest ligand concentration. This conclusion is independent of whether or not microheterogeneity of the protein (Peterson and Foster, 1965) plays a part in the transformation. The situation just described is almost exactly analogous to the now well-known unmasking of prototropic groups, which occurs at both acid and alkaline extremes of the stability regions of numerous proteins. This phenomenon is described in detail for a number of proteins in Chapter V. The description of unmasking of prototropic groups in ferrihemoglobin, in which the effect is clear cut and large, is particularly instructive.

Except for the oxygen-hemoglobin equilibrium, almost all quantitative work on the multiple interactions of proteins prior to about 1940 consisted in the determination of the binding of hydrogen ions by proteins. That interaction remains the classical prototype for binding studies since it possesses to an outstanding degree every feature which is ever encountered in any protein binding process. That is:

(a) Very large numbers of binding sites reacting between pH 2 and 13 (about one for each five amino acid residues present); even more at greater extremes of acidity where amide groups and peptide groups protonate.

(b) Division of the binding sites into discrete sets, each member of which is characterized by the same intrinsic binding tendency (expressed as a dissociation or association constant).

(c) Interaction between sites, i.e., an effect of charges due to binding on one site on the apparent dissociation constants of the other sites.

(d) The existence of stability regions which may be characterized either in terms of the ligand concentrations which bracket them, or the amounts bound at these limits; outside these regions conformation changes occur which manifest themselves as changes in uv absorption, in shape-related parameters such as viscosity, sedimentation and diffusion velocity, and in binding properties. The changes are often but not always reversible.

(e) In numerous cases, some of the binding sites are inaccessible until the pH stability limits are exceeded. Only in the unfolded or disordered protein are the stoichiometric amounts of binding (as determined by amino acid composition) always observed.

Although the isotherms of ligands other than hydrogen ions have been much more recently studied, and are consequently less fully known or understood, most of these five features of hydrogen-ion binding are observed in every sufficiently studied case, although in any particular case, one or another of the features may be missing or barely observable. Interactions of charged sites are, of course, not observed when neutral molecules are bound, and they may be very small when the ligands are very long chain aliphatic ions. There are nevertheless very substantial differences between the binding of hydrogen ions and the binding of all other substances (with the possible exception of polyvalent metal ions) which will become quite clear in the chapters which follow. To begin with, the functional groups to which protons bind, or from which they are dissociated, are entirely known. They are the acidic and basic groups at the ends of the side chains of seven particular amino acids and are characteristic of the constituent polypeptide chains of proteins rather than of the way in which the chains are ordered (helices, pleated sheets, or folded into compact three-dimensional structures). It is the presence of abnormal (inaccessible) groups, rather than of normal (accessible) groups, which gives us insight into the native structure. With most other ligands of high affinity this is not the case. Binding sites are created by the arrangement of linearly well-separated elements on the folded chains or by the confluence of subunits. We deduce this to be the case because the high-affinity sites often disappear when the three-dimensional order is modified or destroyed, or because such binding has not been clearly demonstrated in smaller wholly covalent model compounds. Thus, the binding isotherms of ligands other than hydrogen ions are characteristic of elements of higher structure, and are charged with information about this structure if only we can read it.

There is another important difference between the multiple binding of hydrogen ions and of other substances. When hydrogen ion dissociates from carboxyl, imidazole, phenolic, sulfhydryl, and amino groups, partially covalent bonds of measurable strength (low in the case of carboxyl, moderate with imidazole and sulfhydryl, fairly high in the others) are broken, and work is done. This work is largely of enthalpic origin (it is larger–in aqueous solutions–in the case of the e-amino groups of lysine, even though no work of charge separation is involved, than in the case of carboxyl groups where charge separation is required). When other ions or molecules are bound, whether or not coulombic interactions are involved, enthalpy effects are nearly always very small, and are commonly positive. These substances therefore cannot be bound in the same way as hydrogen ion, by large changes in molecular orbitals. The work required to separate a dodecylsulfate ion from native serum albumin is larger than the work required to dissociate a hydrogen ion from the phenolic group in one of its constituent tyrosines, but it arises from changes in entropy rather than from changes in enthalpy (Kauzmann, 1959).¹

There is another difference between the interaction of proteins with hydrogen ions and their interactions with other ions: hydrogen ions exist in solution partially covalently bound to water as H3O +, H5O2+, etc. Thus the association of a proton to a protein site is accompanied by its dissociation from hydronium ion. All of the thermodynamic and kinetic parameters associated with this reaction contain partial quantities related to both the association and dissociation partial reactions. The main effect to which we call attention here is that the total energy of activation of the reaction is bound to be small. Thus, the reaction will proceed to equilibrium exceedingly rapidly in either direction. A consequence of this feature is that protons can redistribute themselves (under the influence of an outside force such as an approaching charge) very rapidly over vacant sites, giving rise to physicochemical behavior equivalent to that of charges on a conducting hollow body.² Such a resemblance is the basis of the Linderström-Lang model for describing protein acid-base titration curves (Chapter II). Although other ions are hydrated in aqueous solution, there is no obvious analog to the symmetrical acid-base interchange of protons that prevails with hydrogen ion, especially when the ions contain large hydrocarbon tails, as do the detergents. Binding of such ions to proteins therefore may not be characterized by similar low activation energies, essentially diffusion-limited rates, and rapid charge fluctuations. The relatively small number of binding sites in the native protein would also give a much cruder approach to the conducting hollow body than would the much larger number of hydrogen-ion binding sites.

which is analogous to (1 – α) represents the ratio of bound ions to the total number of molecules. Unlike α, its value may greatly exceed unity. Unlike α also, it represents the average is at all large. Thus, for example, Edsall (in Cohn and Edsall, 1943) has shown that the number of instantaneously isoelectric protein ions in an isoelectric protein solution may be very small. The class is smaller, the steeper the titration curve at the pH of interest. It is apparent, therefore, that a uniform population (with respect to the extent of binding) is to be found only when the concentration of free ligand is far removed from the intrinsic dissociation constant of any sets of binding sites on either high or low side of it. Just how far it must be depends on the presence or absence of intersite interaction. In the absence of such interactions, a factor of 100 gives a high degree of charge uniformity. The foundation of the quantitative aspect of these relations is presented in Chapters II and V.

Multiple equilibria, when accompanied by conformation changes which make initially inaccessible sites in the native protein available to ligand, furnish an opportunity to study analogs to the conformational adaptability (Koshland and Neet, 1968) or allostery (Monod et al., 1965) which has been invoked in the study of enzymes and other proteins which undergo subunit dissociation. Monod postulated an essential role of molecular symmetry in cooperative interactions with substrate. It will be shown in Chapter VII, following ideas developed in Foster’s laboratory and by Steinhardt and his collaborators, that even in the absence of subunit structure, cooperativity is manifested whenever (1) inaccessible groups are made available by ligand-induced unfolding and (2) the new binding sites are much more numerous or have a higher affinity for ligand (substrate) than the sites being filled at the ligand concentration where unfolding occurs. For example, if acid unfolds a protein at pH 4, a cooperative type isotherm will be found near pH 4 provided the initially hidden groups have pK values appreciably above 4, and therefore combine practically quantitatively at pH 4.³ This concept is experimentally demonstrable only in the case of multiple equilibria which do not involve dissociation into subunits (as in serum albumin) and has not been invoked in the case of the enzymes and respiratory proteins to which symmetry arguments have been applied.

It will be clear from the foregoing that unmasking may occur without overt manifestations if unfolding makes accessible only combining groups which do not react until higher ligand concentrations are reached than are required to unfold. The absence of cooperative effects from a binding isotherm can never be taken as an indication that all normally reactive groups are accessible to solvent and dissolved ligand in the native protein. Only comparison of the isotherms of native and denatured proteins after allowance for possible differences in the interactions between sites, in the two states–usually requiring fast reaction techniques–can establish that there are no hidden sites in the native protein.

When unfolding complicates the interpretation of binding data it is important to distinguish whether unfolding occurs simply because the algebraic conditions given in Chapter VII are fulfilled (a consequence of many sites remaining inaccessible as long as the protein remains folded), or whether some correlate of the binding process itself, such as the accumulation of a large net electrostatic charge, overcomes the cohesive forces that hold the molecule together. With anion binding, the destruction may be established by counteracting the accumulating net charge of the molecule due to anions by changing the amounts of hydrogen ion bound. Very few such studies have been made (Reynolds and Steinhardt, 1969), and they are incomplete. This recourse is not available in the case of proton binding, and the destruction of structure may be operationally vague.

In the foregoing pages an attempt has been made to delineate the scope of the subject of this book, to indicate the unique features of the reactions it embraces, to mention some of its difficulties, possibilities, and the gaps in our knowledge; and above all, to indicate why the subject is important to both molecular biology and to physical chemistry. The arrangement of the book follows as simple a sequence as possible. Chapter II sets forth the basic physicochemical concepts and relationships which are essential to the development of the subject. It is not an exhaustive treatment, and the serious student is advised to supplement it by making use of articles by Klotz (1953), Edsall and Wyman (1958), Tanford (1961), Beychok and Steinhardt (1964), and Joly (1965) for more complete accounts of portions of the material, as well as of the specific references given there. Chapter III presents an enumeration and critical discussion of most of the methods which have been used to determine extents of binding and complete binding isotherms. To avoid at the outset some of the difficulties and uncertainties inherent in electrostatic interactions between sites, Chapter IV gathers together our present knowledge and hypotheses of the multiple interactions of proteins with uncharged molecules. The principal arrangement is by individual proteins with appropriate subsections for classes of ligands. Large parts of this chapter are necessarily concerned with the serum albumins and hemoglobins. Rather arbitrarily, most of the quantitative work on zwitterions is included here because the net charge is the same as that of a neutral molecule. Many sulfonamides exist in solution largely in the undissociated form; it has been a convenience to arbitrarily collect all the data on sulfonamides in a single chapter although some are ionic at neutral pH.

Chapter V is devoted to the most familiar multiple equilibria, those represented by hydrogen-ion titration curves. Since a number of excellent reviews of this subject are still reasonably up to date (Tanford, 1962; Steinhardt and Beychok, 1964) the subject is developed in a summary fashion. An effort has been made to include all of the more recent data, at least in tabular form. Detailed examples are given of unusual isotherms complicated by the intrusion of binding-induced conformation changes and unmasking of hidden groups.

Chapter VI deals with the binding of metallic ions of valence higher than one. This group of substances merits a separate chapter because the binding of metal ions is more closely interrelated to the binding isotherms of hydrogen ions than is the case with any other ions. Equilibria involving these ions are often important in the catalytic function of enzymes, sometimes because they are determinants of enzyme conformation in the active state. Metallic ions are also important, because, with few exceptions, they constitute the only large class of cations the binding of which to proteins has been subjected to adequate study.

Chapter VII considers the binding of all other ions by proteins. Its principal organization is by three classes of ligand: (a) long-chain hydrocarbons with ionic terminal groups (detergents); (b) dyes, and other predominantly aromatic ions; and (c) small ions, including monoatomic ions. This organization has been adopted in an effort to separate and distinguish the several principal types of binding forces which are involved in the binding of all except the smallest ions: hydrophobic interactions; coulombic interactions in which entropic effects on polarized water molecules are importantly involved; and hydrogen bonding or charge delocalizations. An effort is made to develop and apply the algebra of a general model of all multiple equilibria in which binding-induced conformation changes occur. Considerable attention is given to the physicochemical criteria of such unfolding.

In Chapter VIII a limited number of examples of biologically important multiple equilibria are given, as well as a very brief treatment of protein-protein interactions, including dissociation into subunits. Chapter IX is a summary which attempts to draw together the principal characteristics of multiple equilibria in proteins, the nature of binding sites, the interrelations between complex formation and induced changes in structure, and suggestions for further work.

It may help the reader in going through the book if he keeps before him what we consider to be the central questions relating to multiple equilibria in proteins. These questions are:

(A) What are the differences, if any, between the processes that are responsible for multiple equilibria (especially when whole families of substances are involved) and those responsible for the highly specific binding which is characteristic of enzyme-substrate, membrane-protein substrate, and hapten-antibody reactions?

(B) What characteristics must a protein have to be a strong binder of many equivalents of a large variety of ligands or of groups of particular types of ligands (obviously lysozyme lacks whatever the characteristics are that are responsible for the binding of hydrocarbons)?

(C) Within homologous groups of chemical substrates, e.g., organic anions, what are the characteristics (parameters) which may make them ligands for proteins in general or for particular proteins?

(D) Why does the binding of a small number of equivalents of certain ligands, such as the longest-chain detergents, result in large conformation changes with most proteins? The total concentration of some such substances which is required to produce extensive unfolding of bovine serum albumin (BSA)in 0.1% solution is considerably less than 10−3 M; thus these substances are far more potent initiators of unfolding than the common reagents, urea and guanidine hydrochloride, used as denaturants.

(E) How are the properties enumerated above utilized physiologically, as for example, in such a versatile transport protein as serum albumin?

We will return to these questions in Chapter IX.

REFERENCES

Antonini, E.A. Science. 1965; 158:1417.

Cohn, E.J., Edsall, J.T.Proteins, Amino Acids, and Peptides as Ions and Dipolar Ions,. New York: Reinhold, 1943. [Chapter 20].

Edsall, J.T., Wyman, J. Biophys. Chem. 1958; 1 [Chapter 11].

Guidotti, G. J. Biol. Chem. 1967; 242:3794.

Joly, M.Physical-Chemical Approach to Denaturation of Proteins.. New York: Academic Press, 1965.

Kauzmann, W. Advan. Protein Chem. 1959; 14:1.

Klotz, I.M.Neurath, H., Bailey, K., eds. The Proteins, 1B. New York: Academic Press, 1953. [Chapter 8].

Koshland, D.E., Jr., Neet, K.E. Biochem. Am. R. 1968; 37:359.

Lovrien, R., Jr. Polymer Preprints. 1968; 9:219.

Michaelis, L. J. Biol. Chem. 1932; 96:703.

Michaelis, L. Cold Spring Harbor Symp. Quant. Biol. 1933; 1:224.

Michaelis, L. Cold Spring Harbor Symp. Quant. Biol. 1939; 7:33.

Michaelis, L.Green D.E., ed. Biochemical Research. New York: Wiley (Interscience), 1946.

Molday, R., and Steinhardt, J. (1969). B.B.A. (in press).

Monod, J., Wyman, J., Changeux, J.P. J. Mol. Biol. 1965; 12:88.

Pardee, A. Science. 1968; 162:632.

Perutz, M.F. J. Mol. Biol. 1965; 13:646.

Perutz, M.F., Muirhead, H., Cox, J.M., Goaman, L.C.G., Mathews, F.S., McGandy, E.L., Webb, L.E. Nature. 1968; 219:29.

Perutz, M.F., Muirhead, H., Cox, J.M., Goaman, L.C.G. Nature. 1968; 219:131.

Peterson, H.A., Foster, J.F. J. Biol. Chem. 1965; 240:2503, 3858.

Reynolds, J., Steinhardt, J. Biochem. 1969; [(in press)].

Roughton, F.J.W., Otis, A.B., Lyster, R.H.J. Proc. Roy. Soc. London. 1955; B144:29.

Steinhardt, J., Beychok, S.Neurath, H., eds. The Proteins, II. New York: Academic Press, 1964. [Chapter 8].

Steinhardt, J., Ona, R., Beychok, S., Ho, C. Biochemistry. 1963; 2:256.

Tanford, C. Physical Chemistry of Macromolecules,. New York: Wiley, 1961; 337–344.

Tanford, C. Advan. Protein. Chem. 1962; 17:69.

Wyman, J., Jr. Advan. Protein Chem. 1948; 4:407.

Wyman, J., Jr. J. Mol. Biol. 1965; 11:631.

---

¹It will be shown in Chapter VII that exception must be made for one of the binding sites in some proteins in which the free energy of binding anions is largely enthalpic in origin (Lovrien, 1968).

²Conduction occurs in the innermost water layer rather than on the surface of the ion.

³Parallels may be found in the two-step oxidation of certain hydroquinones, where the removal of the first hydrogen leaves a free radical which is more easily oxidized than the original substance, and is therefore observed only under special circumstances (Michaelis, 1932, 1933, 1939, 1946).

II

Thermodynamics and Model Systems

Publisher Summary

This chapter presents the mathematical description of complex formation, the thermodynamic analysis of binding data, and various theoretical models that can be used to describe the phenomena of small molecule–macromolecule interactions. The experimental quantities of interest in the study of multiple equilibria are the molal binding ratio, defined as the average number of moles of ligand bound per mole of protein, and the ligand concentration in equilibrium with the ligand complex. These two experimentally determined quantities permit one to calculate the association constants of ligand for the site or sites on the protein as well as the free energy of the complex formation. When binding is measured as a function of temperature, other thermodynamic quantities such as enthalpy and entropy of binding can also be determined. Chemical equilibria are determined by the changes in free energy that occur when reactants combine or move from one phase to another. The binding of small organic and inorganic molecules to biopolymers a common method of handling the binding data makes use of linear equations. Both of these equations suffer from the limitation of a relatively uncertain extrapolation and the error inherent in taking the slope of a line drawn through experimental points, which are themselves subject to a finite uncertainty.

I

Introduction

This chapter deals with the mathematical description of complex formation, the thermodynamic analysis of binding data, and various theoretical models which can be used to describe the phenomena of small molecule-macromolecule interactions. The discussion includes binding with and without interaction between multiple identical sites. In addition, the problem of binding-induced conformational changes is treated.

The experimental quantities of interest in the study of multiple equilibria are the molal binding ratio, defined as the average number of moles of ligand bound per mole of protein, and the ligand concentration in equilibrium with the ligand complex. These two experimentally determined quantities permit one to calculate the association constants of ligand for the site or sites on the protein as well as the free energy of the complex formation. When binding is measured as a function of temperature, other thermodynamic quantities such as enthalpy and entropy of binding can also be determined.

Recent general treatments of parts of these subjects, and more extended treatments of some of them are to be found in Edsall and Wyman (1958), Tanford (1961), and Steinhardt and Beychok (1964).

II Some Concepts in Thermodynamics

Chemical equilibria are determined by the changes in free energy (ΔF) which occur when reactants combine or move from one phase to another.

From the second law of thermodynamics it is known that

(2-1)

where dH is the change in enthalpy in the system and dS is the change in entropy in the system. The free energy of any system is a function of temperature, T, pressure, P, and quantities of chemical components, ni,

(2-2)

and

(2-3)

where (part;F/∂ni)T,P is the increase in free energy per mole of added component, ni, when the amount of ni added is sufficiently small that the composition of the solution remains essentially unchanged. At constant T and P, Eq. (2-3) becomes

(2-4)

where (∂F/∂ni)T,P ≡ µi is the partial molal free energy of the ith component, called by Gibbs the chemical potential.

At equilibrium dFT,P = 0, and Eq. (2-4) becomes

(2-5)

The composition-dependent free energy of a system at a particular constant T and P is

(2-6)

Differentiation of Eq. (2-6) gives

(2-7)

Combining Eqs. (2-5) and (2-7)

(2-8)

which is the well-known Gibbs-Duhem equation for the equilibrium state.

Thus, when a multicomponent system at equilibrium undergoes an infinitesimal displacement, the sum of the products of concentration of each component and its change in chemical potential is zero.

An important relationship between chemical potentials is suggested by the ideal gas law, PV = -nRT. It can be shown that for a mixture of two ideal gases

(2-9)

where p1 and p2 are the partial pressures of each gas and are proportional to the molar concentrations. Each gas has been referred to a standard reference state of chemical potential µ0. That is,

(2-10)

In an ideal solution the chemical potential of each component may be defined in a manner similar to Eq. (2-10)

(2-11)

where mi is the molar concentration of component iis a standard chemical potential chosen arbitrarily.

For a real solution (nonideal) we can define a quantity, ai, which we call the activity of a specific component and which is a function of the molar concentration, mi

(2-12)

where γi is customarily chosen equal to unit activity for real solutions, and ai is made to approach mi at infinite dilution of real solutions. Thus Eq. (2-11) becomes for real solutions

(2-13)

and the difference in chemical potential for any component in two states (e.g., dissolved and crystalline) in a solution at equilibrium may be written

(2-14)

Application of Eq. (2-14) to any chemical system in equilibrium as defined by the law of mass action leads to the relation

(2-15)

where ΔF° is the standard free energy change in the reaction and K, the mass action equilibrium constant.

III Multiple Binding without Interaction between Sites

The mathematical description of simple binding may be approached from a very general standpoint. For further discussion the reader is referred to a number of comprehensive reviews (Klotz, 1953; Edsall and Wyman, 1958; Tanford, 1962; and Weber, 1965). In addition to these references the early works of Michaelis (1925) and Langmuir (1918) as well as a more recent derivation by Hill (1960) are instructive accounts of various approaches to the problem treated in this section.

Given a number of uniform particles (S) in solution each with n independent, indistinguishable, and identical sites, the following equations define the equilibrium between a ligand and the sites on the particles:

(2-16)

or alternatively, following the stepwise addition of ligand

(2-17)

where at equilibrium

(2-18)

It is important to note that the above equations and the discussion to follow are strictly applicable only when the association is thermodynamically reversible. Furthermore, since we are discussing real solutions, the quantities [SLi], [SLi-1], and [L] properly refer to activities as discussed in Section II of this chapter and not to concentrations. In general the activity coefficients are not readily ascertainable, but at constant ionic strength, they are combined with the apparent association constants k0 in K[in Eq. (2-23) and (2-26)].

The number of possible combinations of n binding sites taken exactly i at a time (or alternatively, the number of equally probable forms of the complex [SLi]) is

(2-19)

Thus, for example, there are n equally probable forms of SL1 and only one form of SLn. Since all the sites are by definition equal, they all must have, if present alone, the same intrinsic association constant for ligand which we will call k0. Taking into account the number of equally probable forms of each complex, we may define the ki‘s in terms of an invariant equilibrium constant k0 as follows:

(2-20)

The average number of ligand molecules bound may be defined as

(2-21)

And, by systematic application of Eq. (2-18),

(2-22)

Substituting the relationship between ki and k0 defined by Eq. (2-20) in Eq. (2-22) we obtain

(2-23)

Note that the denominator of Eq. (2-23) is the general expression of the binomial theorem [Eq. (2-24)] and the numerator is the first derivative thereof [Eq. (2-25)]:

(2-24)

(2-25)

Substituting Eqs. (2-24) and (2-25) in (2-23), we obtain (after substituting K for the product of the combined activity coefficients and k0)

(2-26)

/n = θ or degree of saturation. Equation (2-26) can be rearranged to give

(2-27)

(2-28)

(2-29)

The three linear equations given above provide a convenient means for determining K when the molal binding ratio is known as a function of free ligand concentration. Note that in applications of Eq. (2-29) an assumption must be made with respect to the value of n. If n is known then Eq. (2-29) should give a linear plot of log [(n )/v] vs. log [L] with a slope of − 1. Slopes other than − 1 or failure to obtain a straight line suggest that the simple model described here is incorrect. The interpretation is often made that such deviations indicate heterogeneity of binding sites, but it will be shown that other causes such as interaction between sites, and initiation of conformation changes may produce these effects.

If there is more than one set of sites on the surface of the macromolecule such that set 1 has n1 sites and an intrinsic association constant K1, set 2 has n2 sites and an intrinsic association constant K2, etc., Eq. (2-26) may be generalized in the following manner:

(2-30)

This treatment assumes complete independence of sets of sites, in particular the absence of any conformational change of the surface as a result of binding, such that a previously masked set is exposed or an exposed set destroyed. If the values of Ki are separated by more than about 10⁴, then at any particular value of [L] only one term of in each such region is determined by one particular Ki.

If there is only one set of sites but two ligands competing for this set, each with a different association constant, Eq. (2-26) becomes

(2-31)

thus taking account of the fact that part of the n sites will be filled by each ligand. Algebraic combination of the above equations leads to

(2-32)

A special case of sites which are similar but not identical, with a statistical distribution of K’s around a mean, was made use of by Karush and Sonnenberg (1949) in an attempt to fit some anion binding data obtained with bovine serum albumin. These authors proposed a Gaussian distribution function to describe the site energy distribution, and Sips (1948, 1950) later showed that a distribution function close to Gaussian leads explicitly to the following equation:

(2-33)

where K is the intrinsic association constant, and a is an index of dispersion of K around the mean. /n = K[L]a which does not reach a saturation limit at high values of [L]. There is no direct evidence that either of these two models represents physical reality.

IV Binding with Interaction between Sites

Sets of otherwise identical sites will not behave in accordance with Eq. (2-26) if the state of each site (combined or uncombined) affects the binding properties of the other sites. Interaction between binding sites in the same molecule can result from one or more of the following factors:

(1) Steric interference between bound molecules may occur.

(2) When the ligand is charged, electrostatic interaction may arise from the increasing charge on the protein as more ions are bound.

(3) Conformational changes of the binding body may be induced by interaction with the ligand.

In the present discussion steric interference is not dealt with at length. It can be treated in the manner of electrostatic interaction by increasing the exponent in the free energy law [Eq. (2-36)] to a large number.

Conformational changes induced by binding have been known for many years. Early examples are the expansion of linear polyelectrolytes with increasing charge (Arnold and Overbeek, 1950; Harris and Rice, 1952) and the unfolding of proteins as a function of hydrogen ions bound (Steinhardt and Zaiser, 1955). This subject is discussed in some detail in Section VII of this chapter. This section is concerned primarily with the problem of electrostatic interaction.

It is intuitively apparent that if the ligand is a charged species, the first bound ion will have a repulsive effect on the approach of the second ligand molecule to the same neighborhood, or if the surface is a conducting one, to any point on the surface. This repulsive effect must be taken into account when determining the association constant of the charged ligand for the specific sites on the surface. In the case of small ions early efforts to take this effect into account were made by Bjerrum (1923), who estimated the electrostatic contribution to the second ionization constant of dibasic acids. Gane and Ingold (1931) and Kirkwood and Westheimer (1938) later refined these calculations.

The general case of identical and indistinguishable sites is considered here first.¹ The standard free energy of the reaction defined by Eq. (2-16) is defined by

(2-34)

where we now write K’ for K. The additional contribution to the free energy made by electrostatic interaction when the ligand is charged is just the difference in the charging energy between [SLi-1] and [SLi]–in other words, the electrical work done in increasing the charge on the surface by an amount equivalent to the binding of one charged ion. This may be expressed formally by defining a