Effects of thermodynamic nonideality in ligand binding studies

Effects of thermodynamic nonideality are considered in relation to the quantitative characterization of the interaction between a small ligand. S, and a macromolecular acceptor. A, by two types of experimental procedure. The first involves determination of the concentration of ligand in dialysis equilibrium with the acceptor/ligand mixture, and the second, measurement of the concentration of unbound ligand in the reaction mixture by ultrafiltration or the rate of dialysis method. For each situation explicit expressions are formulated for the appropriate binding function with allowance for composition-dependent nonideality effects expressed in terms of molar volume, charge-charge interaction and covolume contributions. The magnitudes of these effects are explored with the aid of experimental studies on the binding of tryptophan and of methyl orange to bovine serum albumin. It is concluded for experiments conducted utilizing either equilibrium dialysis or frontal gel chromatography that, provided a correction is made for any Donnan redistribution of ligand, theoretically predicted acceptor-concentration dependence is likely to be negligible and that use of the conventional binding equation written for an ideal system is appropriate to the analysis of the results. Use of ultrafiltration or the rate of dialysis method requires examination of the assumption that the activity coefficient ratio y(A)y(s)/y(AS) for the reaction mixture approximates unity; but again reassurance is provided that nonideality manifested as a dependence of the binding function on acceptor concentration is unlikely to be significant.     

Studies of macromolecular heterogeneous associations involving cross-linking: a re-examination of the ovalbumin-lysozyme system

A system is considered in which a multivalent acceptor interacts with a bivalent ligand in solution to form an array of complexes via multiple binding and cross-linking reactions. With the use of reacted site probability functions expressions are derived in terms of a site binding constant which are of potential use in the interpretation of sedimentation equilibrium and binding results obtained with such systems. Their potential use is explored in relation to results obtained on the interacting ovalbumin-lysozyme system at pH 6.80, ionic strength 0.02. A comparison is made of this interpretation with that based on an interaction pattern involving only multiple binding of ligand in the absence of cross-linking effects. While both interpretations quantitatively describe certain results, it is shown, by invoking further experimental observations on apparent weight-average molecular weight and precipitation behavior, that the more favored interpretation is that involving the operation of a spectrum of forces leading to a large array of ovalbumin-lysozyme complexes, including those of the cross-linked type. It is stressed that the particular ovalbumin-lysozyme system is but one example of interaction between oppositely charged macromolecules and therefore that the derived equations may find wider application to such systems and those known to involve more specific cross-linking interactions.     

Ligand-affinity sedimentation equilibrium using an air-driven ultracentrifuge

An air-driven ultracentrifuge has been used to study the distribution of radioactive ligands at sedimentation equilibrium. In the presence of a suitable acceptor and under conditions where the ligand is essentially all bound the distribution of ligand can be analyzed to yield the molecular weight of the acceptor molecule. Suitable conditions can be chosen either experimentally by measuring the ratio of bound ligand compared to unbound ligand or theoretically for systems in which the ligand-binding affinity and number of acceptor binding sites is known. The method is applicable to the molecular characterization of binding proteins in crude mixtures and results are presented for the binding of various fatty acids to serum albumin samples.     

An examination of the forms of scatchard plots of binding results outside domains of sigmoidality

General expressions are formulated for the first and second derivatives of the Scatchard function, r/[S], with respect to the binding function, r, from an equation that describes the binding of a ligand to a two-state acceptor system (either isomerizing or polymerizing). The expressions are utilized to determine the sign of the second derivative for particular systems under conditions where the first derivative is negative for all r. The work therefore correlates with previous studies, which stressed conditions of existence of critical points in Scatchard plots, by examining more fully possible forms of binding curves outside such domains of sigmoidality. Particular attention is given to the condition, d(r/[S])/dr < 0 and d²(r/[S])/dr² > 0 for all r (which defines a Scatchard plot convex to the r-axis). In agreement with previous findings it is proven that the isomerizing acceptor model cannot give rise to this form of plot and is therefore distinguished from negatively co-operative allosteric models. On the other hand, the polymerizing acceptor model may yield such a Scatchard plot, a feature demonstrated by formulating explicit conditions for its manifestation when ligand binding is exclusive to the polymeric state, and illustrated numerically for a system in which ligand binds to both oligomeric states. Distinction between such systems and those exhibiting negative co-operativity is possible on the basis of the Scatchard plots, which exhibit dependence on acceptor concentration in the case of a polymerizing acceptor; indeed, an example is provided where variation of acceptor concentration for a system characterized by fixed interaction parameters effects a conversion from sigmoidal binding behaviour to that typified by a Scatchard plot convex to the r-axis.     

A historical perspective of the biophysics of the thrombin–heparin system: an example of nonspecific binding and the consequent parking problem in action

Difficulties are encountered in the thermodynamic characterization of interactions between a protein ligand and a linear acceptor, such as a polynucleotide or a polysaccharide, because of the involvement of more than one unit of the polymer chain in each attachment of a protein molecule. Complications arise from the fact that random attachment of ligand to the polymer chain, each unit of which is a potential binding site, initially leads to suboptimal location of protein molecules along the polymer chain—a situation that has to be rectified before the attainment of thermodynamic equilibrium can be realized. Kinetic as well as thermodynamic consequences of such nonspecific binding, termed the parking problem, therefore need to be considered in any quantitative characterization of the interaction between a large ligand and a linear polymer acceptor chain. Results for the thrombin–heparin interaction have been used to illustrate a thermodynamic characterization of nonspecific binding that takes into account these consequences of the parking problem.     

The binding of an indefinitely associating ligand to acceptor: consideration of monovalent ligand species binding to a multivalent acceptor

Currently available binding theory is extended to incorporate the concept of indefinite self-association of the ligand. Binding equations are formulated in closed form for the case of the binding to a multivalent acceptor of a ligand capable of isodesmically indefinitely self-associating in a "head-to-tail" mode such that each ligand state bears one site capable of interacting with the acceptor. It is shown both mathematically and by way of numerical example that this system will give rise exclusively to binding curves convex to the r-axis in Scatchard format. Thus, the system provides another example of a binding mechanism capable of generating an apparent negatively co-operative binding response.     

Conditions for the existence of critical points in scatchard plots of binding results

An equation is presented in Scatchard format which describes the binding of a ligand to a two-state acceptor system (either isomerizing or polymerizing). It is used to formulate a general set of conditions for the existence of critical points in Scatchard plots and this set is explored in detail for particular cases. Explicit relations between the thermodynamic parameters governing the binding are thereby obtained which permit discussion of the boundaries of domains within which critical points may exist. Moreover, several items of evidence are presented which show that there is an exact correspondence between the appearance of critical points in Scatchard plots and points of inflexion in associated binding curves examples are presented where zero, one or more critical points and points of inflexion are observed. Finally, the effects on preferential binding of the variation of the environmental parameters, temperature and pressure (and for acceptor polymerization, the total acceptor concentration) are discussed in terms of the derived conditions of existence of critical points in Scatchard plots and their equivalent domains of sigmoidality of binding curves.     

Thermodynamics of metal ion binding. 1. Metal ion binding by wild-type carbonic anhydrase

Understanding the energetic consequences of molecular structure in aqueous solution is a prerequisite to the rational design of synthetic motifs with predictable properties. Such properties include ligand binding and the collapse of polymer chains into discrete three-dimensional structures. Despite advances in macromolecular structure determination, correlations of structure with high-resolution thermodynamic data remain limited. Here we compare thermodynamic parameters for the binding of Zn(II), Cu(II), and Co(II) to human carbonic anhydrase II. These calorimetrically determined values are interpreted in terms of high-resolution X-ray crystallographic data. While both zinc and cobalt are bound with a 1:1 stoichiometry, CAII binds two copper ions. Considering only the high-affinity site, there is a diminution in the enthalpy of binding through the series Co(II) --> Zn(II) --> Cu(II) that mirrors the enthalpy of hydration; this observation reinforces the notion that the thermodynamics of solute association with water is at least as important as the thermodynamics of solute-solute interaction and that these effects must be considered when interpreting association in aqueous solution. Additionally, DeltaC(p) data suggest that zinc binding to CAII proceeds with a greater contribution from desolvation than does binding of either copper or cobalt, suggesting Nature optimizes binding by optimizing desolvation.     

Effects of heterogeneity and cooperativity on the forms of binding curves for multivalent ligands

An exploratory investigation is made of the binding behavior that is likely to be encountered with multivalent ligands under circumstances where a single intrinsic binding constant does not suffice to describe all acceptor-ligand interactions. Numerical simulations of theoretical binding behavior have established that current criteria for recognizing heterogeneity and cooperativity of acceptor sites on the basis of the deviation of the binding curve from rectangular hyperbolic form for univalent ligands also apply to the interpretation of the corresponding binding curves for multivalent ligands. However, for systems in which the source of the departure from equivalence and independence of binding sites resides in the ligand, these criteria are reversed. On the basis of these observations a case is then made for attributing results of an experimental binding study of the interaction between pyruvate kinase and muscle myofibrils to positive cooperativity of enzyme sites rather than to heterogeneity or negative cooperativity of the myofibrillar sites.     

Ligand competition curves as a diagnostic tool for delineating the nature of site-site interactions: theory

A few molecular models have been developed in recent years to explain the mechanism of cooperative ligand binding. The concerted model of Monod, Wyman and Changeux and the sequential model of Koshland, Némethy and Filmer were formulated to account for positively cooperative binding. The pre-existent asymmetry model and the sequential model can account for negatively cooperative ligand binding. In most cases, however, it is virtually impossible to deduce the molecular mechanism of ligand binding solely from the shape of the binding isotherm. In the present study we suggest a new strategy for delineating the molecular mechanism responsible for cooperative ligand binding from binding isotherms. In this approach one examines the effect of one ligand on the cooperativity observed in the binding of another ligand, where the two ligands compete for the same set of binding sites. It is demonstrated that the cooperativity of ligand binding can be modulated when a competitive ligand is present in the protein-ligand binding mixture. A general mathematical formulation of this modulation is presented in thermodynamic terms, using model-independent parameters. The relation between the Hill coefficient at 50% ligand saturation with respect to ligand X in the absence, h(x), and in the presence of a competing ligand Z, h(x,z), is expressed in terms of the thermodynamic parameters characterizing the binding of the two ligands. Then the relationship between h(x) and h(x,z), in terms of the molecular parameters of the different allosteric models, is explored. This analysis reveals that the different allosteric models predict different relationships between h(x,z) and h(x). These differences are especially focused when Z binds non-cooperatively. Thus, it becomes possible, on the basis of ligand binding experiments alone, to decide which of the allosteric models best fits a set of experimental data.     

Effects of ligand multivalency in binding studies: a general counterpart of the Scatchard analysis

A general counterpart of the Scatchard analysis has been developed which takes into account the valence of the ligand. Its use is first demonstrated by application to binding data obtained by exclusion chromatography of mixtures of Dextran T2000 and concanavalin A (a bivalent ligand) on a column of porous glass beads (Glyceryl-CPG 170) equilibrated at 5 degrees C with phosphate-chloride buffer (pH 5.5), I 0.5. A recycling partition equilibrium study with Sephadex G-100 as gel phase then provides a quantitative evaluation of the interaction between haemoglobin and a monoclonal mouse antihaemoglobin antibody preparation in 0.1 M phosphate (pH 7.0) in order to emphasize the ability of the present analysis to consider collectively binding results obtained with a range of acceptor concentrations. Finally, the use of the generalized Scatchard analysis to assess acceptor site homogeneity is illustrated by reappraisal of results for the binding of glyceraldehyde-3-phosphate dehydrogenase to erythrocyte membranes.     

Binding equations for interacting systems comprising multivalent acceptor and bivalent ligand: application to antigen-antibody systems

Consideration is given to the reversible interaction of a bivalent ligand, B, with a multivalent acceptor, A (possessing f reactive sites) which leads to the formation of a series of complexes, A_iB_j, comprising networks of alternating acceptor and ligand molecules. A binding equation is derived on the basis of a site association constant, k, defined in terms of reacted site probability functions. This equation, which relates the binding function, r (the moles of ligand bound per mole of acceptor) to the concentration of unbound ligand, m_b, is used to show that plots of r vs. 2km_B constructed with fixed but different values of $\text{k}\text{m}{}_{\mathrm{A}}$ intersect at the point ($\text{m}{}_{\mathrm{B}}\text{=}\text{1}\text{2k}\text{, r =}\text{f}\text{2}$) where the extent of reaction and the concentrations of those complexes for which $\text{j}\text{i}\text{=}\text{f}\text{2}$ attain maximal values. Corresponding Scatchard plots are shown by numerical example to be non-linear, their second derivative being positive for all r. It follows that such deviations from linearity cannot be taken alone as evidence for site heterogeneity in cross-linking systems. The binding equation obtained directly is shown to be identical with that obtained with f = 2 by summation procedures involving the general expression for concentrations of complexes, m_AiBj, formulated in terms of appropriate statistical factors. In this way, previous findings on precipitation and gel formation in cross-linking systems are correlated with the present development of binding theory.     

Effects of ligand multivalency in binding studies: a general counterpart of the Scatchard analysis

A general counterpart of the Scatchard analysis has been developed which takes into account the valence of the ligand. Its use is first demonstrated by application to binding data obtained by exclusion chromatography of mixtures of Dextran T2000 and concanavalin A (a bivalent ligand) on a column of porous glass beads (Glyceryl-CPG 170) equilibrated at 5°C with phosphate-chloride buffer (pH 5.5), I 0.5. A recycling partition equilibrium study with Sephadex G-100 as gel phase then provides a quantitative evaluation of the interaction between haemoglobin and a monoclonal mouse antihaemoglobin antibody preparation in 0.1 M phosphate (pH 7.0) in order to emphasize the ability of the present analysis to consider collectively binding results obtained with a range of acceptor concentrations. Finally, the use of the generalized Scatchard analysis to assess acceptor site homogeneity is illustrated by reappraisal of results for the binding of glyceraldehyde-3-phosphate dehydrogenase to erythrocyte membranes.     

Molecular interaction analysis in ligand design using mass transport,kinetic and thermodynamic methods

Ligand design in biotechnology is underpinned by the control of molecular affinity. Hence, measuring binding interactions is a key component in designing ligands for such uses as therapeutics, diagnostics, biomaterials and separation science. Mass transport, kinetic and thermodynamic methods have been used for macromolecular interaction analysis but also have potential applicability as direct methods for measuring small molecular interactions. They can enhance the ligand design process by providing the ability to choose ligands based on both their kinetic and thermodynamic binding properties.     

On the need to consider kinetic as well as thermodynamic consequences of the parking problem in quantitative studies of nonspecific binding between proteins and linear polymer chains

Attention is drawn to a need for caution in the thermodynamic characterization of nonspecific binding of a large ligand to a linear acceptor such as a polynucleotide or a polysaccharide-because of the potential for misidentification of a transient (pseudoequilibrium) state as true equilibrium. The time course of equilibrium attainment during the binding of a large ligand to nonspecific three-residue sequences of a linear acceptor lattice has been simulated, either by numerical integration of the system of ordinary differential equations or by a Monte Carlo procedure, to identify the circumstances under which the kinetics of elimination of suboptimal ligand attachment (called the parking problem) create such difficulties. These simulations have demonstrated that the potential for the existence of a transient plateau in the time course of equilibrium attainment increases greatly (i) with increasing extent of acceptor saturation (i.e., with increasing ligand concentration), (ii) with increasing magnitude of the binding constant, and (iii) with increasing length of the acceptor lattice. Because the capacity of the polymer lattice for ligand is most readily determined under conditions conducive to essentially stoichiometric interaction, the parameter so obtained is thus likely to reflect the transient (irreversible) rather than equilibrium binding capacity. A procedure is described for evaluating the equilibrium capacity from that irreversible parameter; and illustrated by application to published results [M. Nesheim, M.N. Blackburn, C.M. Lawler, K.G. Mann, J. Biol. Chem. 261 (1986) 3214-3221] for the stoichiometric titration of heparin with thrombin.     

The binding of a ligand to an acceptor undergoing indefinite self-association

Explicit expressions are derived which describe the binding of a univalent ligand to equivalent and independent sites on each state of an acceptor undergoing indefinite self-association that is governed by an isodesmic equilibrium constant KI. From considerations of systems in which the same site-binding constant kA applies to all acceptor-ligand interactions, the general forms of binding curves and Scatchard plots are deduced for situations in which binding sites are either created or lost at each monomer-monomer interface. Greater generality is then introduced into the model by allowing ligand interactions with polymeric acceptor states to be governed by a site-binding constant kp that differs in magnitude from that for monomeric acceptor kA. Finally, experimental results with the glutamate dehydrogenase-GTP and lysozyme-saccharide systems are used to illustrate ways in which the present quantitative expressions may be applied to the characterization of inteactions between a ligand and an indefinitely self-associating acceptor.     

Selective immobilization of multivalent ligands for surface plasmon resonance and fluorescence microscopy

Cell surface multivalent ligands, such as proteoglycans and mucins, are often tethered by a single attachment point. In vitro, however, it is difficult to immobilize multivalent ligands at single sites due to their heterogeneity. Moreover, multivalent ligands often lack a single group with reactivity orthogonal to other functionality in the ligand. Biophysical analyses of multivalent ligand-receptor interactions would benefit from the availability of strategies for uniform immobilization of multivalent ligands. To this end, we report the design and synthesis of a multivalent ligand that has a single terminal orthogonal functional group and we demonstrate that this material can be selectively immobilized onto a surface suitable for surface plasmon resonance (SPR) experiments. The polymeric ligand we generated displays multiple copies of 3,6-disulfogalactose, and it can bind to the cell adhesion molecules P- and L-selectin. Using SPR measurements, we found that surfaces displaying our multivalent ligands bind specifically to P- and L-selectin. The affinities of P- and L-selectin for surfaces displaying the multivalent ligand are five- to sixfold better than the affinities for a surface modified with the corresponding monovalent ligand. In addition to binding soluble proteins, surfaces bearing immobilized polymers bound to cells displaying L-selectin. Cell binding was confirmed by visualizing adherent cells by fluorescence microscopy. Together, our results indicate that synthetic surfaces can be created by selective immobilization of multivalent ligands and that these surfaces are capable of binding soluble and cell-surface-associated receptors with high affinity.