Preferential ligand binding to multi-state acceptor systems: the unexplored paradox of acceptor self-association that is ligand-mediated but detrimental to ligand binding

Consideration is given to the interactions of ligand with self-associating acceptor systems for which preferential ligand binding is an ambiguous term, in that the acceptor species with greater affinity for ligand possesses relatively fewer binding sites. A paradoxical situation wherein ligand-mediated self-association is seemingly detrimental to ligand binding is shown to be the predicted outcome for a transient range of ligand concentrations. This outcome reflects the existence of a critical point in the dependence of the extent of acceptor self-association upon ligand concentration that coincides with a cross-over point of ligand-binding curves for different, fixed total concentrations of acceptor. By classical differentiation methods the conditions for the existence of these critical points are established not only for two-state acceptor systems but also for three-state acceptor systems in which the ligand-binding form of monomer also undergoes reversible isomerization to an inactive state. Similar procedures are used to comment upon the forms of binding curves for the three-state acceptor systems, the Scatchard representations of which may exhibit as many as three critical points (two maxima and a minimum). This delineation of quantitative expressions for critical points and other distinctive features associated with the conflicting interplay of ligand-binding and self-association behaviour should provide a more definitive means of characterizing systems with one acceptor state the preferred binding form on affinity grounds but with the other the preferred state from the viewpoint of binding-site numbers.     

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.     

Preferential ligand binding to multi-state acceptor systems: comparisons of the calcium-binding and dimerization characteristics of prothrombin and fragment 1.

Consideration is given to the interactions of a ligand with self-associating acceptor systems for which preferential binding is an ambiguous term in that ligand-mediated self-association does not necessarily imply a greater binding constant for polymeric acceptor--even in instances where binding sites are preserved in the self-association process. This dilemma is shown to arise in situations involving the binding of ligand to monomeric and polymeric forms of an acceptor that also coexist in equilibrium with inactive isomeric states. For example, the ten-fold increase in the measured dimerization constant for prothrombin Fragment 1 in the presence of a saturating concentration of Ca2+ ion may well reflect the existence of a 12% greater binding constant for the interaction of metal ion with dimeric acceptor. However, that result, as well as the detailed form of the sigmoidal binding curve, are also reasonably described by another extreme model in which the monomeric and dimeric forms of the acceptor possess equal affinities for Ca2+ ion. Likewise, the fact that the same experimental dimerization constant applies to prothrombin and its Ca(2+)-saturated complex does not preclude the possibility that the active form of dimeric zymogen exhibits a 12% greater affinity for metal ion. Numerical simulations have established that characterization of the dimerization behaviour as a function of free ligand concentration should allow greater discrimination between such models of the interplay between calcium binding and self-association of prothrombin and Fragment 1. Finally, by illustrating the likelihood that the disparity in self-association behaviour of prothrombin and Fragment 1 merely reflects minor differences in the relative magnitudes of isomerization constants and/or binding constants for monomeric and dimeric states of the two acceptors, the present investigation serves to allay concern about the validity of employing the proteolytic fragment as a model of the intact zymogen.     

Interactions between micellar ligand systems and acceptors: Forms of binding curves

A previously formulated expression describing the competitive binding to an acceptor of two states of a ligand, monomeric and polymeric, coexisting in equilibrium is examined in terms of the different forms of Scatchard plots which may arise in cases of exclusive and of preferential binding of the ligand states. It is shown by differentiation of the binding equation written in Scatchard format, and by numerical examples, that exclusive binding of the monomeric form of ligand leads to Scatchard plots that are either sigmoidal or convex to the abscissa, whereas exclusive binding of the polymeric form results in plots concave to the abscissa and exhibiting a maximum. Particular attention is given to Scatchard plots which possess two critical points, a situation which is shown to be possible when the polymeric form of ligand binds preferentially (but not exclusively) to the acceptor. The two-state ligand concept is especially pertinent to solutes capable of globular micelle formation and several examples are cited of binding studies which have been conducted with such micellar systems. Of these, the chlorpromazine-brain tubulin system is given detailed consideration in order to illustrate the use of the present theory in describing the binding results which exhibit two critical points when plotted in Scatchard format.     

A model for ligand binding to hexacoordinate hemoglobins

Hexacoordinate hemoglobins are heme proteins capable of reversible intramolecular coordination of the ligand binding site by an amino acid side chain from within the heme pocket. Examples of these proteins are found in many living organisms ranging from prokaryotes to humans. The nonsymbiotic hemoglobins (nsHbs) are a class of hexacoordinate heme proteins present in all plants. The nsHb from rice (rHb1) has been used as a model system to develop methods for determining rate constants characterizing binding and dissociation of the His residue responsible for hexacoordination. Measurement of these reactions exploits laser flash photolysis to initiate the reaction from the unligated, pentacoordinate form of the heme protein. A model for ligand binding is presented that incorporates the reaction following rapid mixing with the reaction starting from the pentacoordinate hemoglobin (Hb). This model is based on results indicating that ligand binding to hexacoordinate Hbs is not a simple combination of competing first order (hexacoordination) and second order (exogenous ligand binding) reactions. Ligand binding following rapid mixing is a multiphasic reaction displaying time courses ranging from milliseconds to minutes. The new model incorporates a "closed", slow reacting form of the protein that is not at rapid equilibrium with the reactive conformation. It is also demonstrated that formation of the closed protein species is not dependent on hexacoordination.     

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.     

Cooperativity of ligand binding as a function of monomer-dimer equilibrium parameters and acceptor concentration

A general monomer-dimer equilibrium system involving ligand interactions ispresented. Cooperativity features of specific limited models are analyzed by selecting the appropriate family of equilibrium constants from this general scheme. Each system is then characterized in terms of Hill coefficient dependency on alterations in values of equilibrium constants and total acceptor concentration. This method permits comparison of predicted cooperativity trends between systems. Contrasting reports concerning cooperativity dependencies for certain defined equilibrium systems are compared and the discrepancies resolved. Characteristics of cooperativity binding patterns are shown to include symmetry about dimerization association constant values, both positive and negative cooperativity for a single set of parameters, and significant changes in cooperativity features with relatively small changes in equilibrium parameters.     

Autocrine ligand binding to cell receptors. Mathematical analysis of competition by solution "decoys".

Autocrine ligands have been demonstrated to regulate cell proliferation, cell adhesion, and cell migration in a number of different systems and are believed to be one of the underlying causes of malignant cell transformation. Binding of these ligands to their cellular receptors can be compromised by diffusive transport of ligand away from the secreting cell. Exogenous addition of antibodies or solution receptors capable of competing with cellular receptors for these autocrine ligands has been proposed as a means of inhibiting autocrine-stimulated cell behavioral responses. Such "decoys" complicate cellular binding by offering alternative binding targets, which may also be capable of aiding or abating transport of the ligand away from the cell surface. We present a mathematical model incorporating autocrine ligand production and the presence of competing cellular and solution receptors. We elucidate effects of key system parameters including ligand diffusion rate, binding rate constants, cell density, and secretion rate on the ability of solution receptors to inhibit cellular receptor binding. Both plated and suspension cell systems are considered. An approximate analytical expression relating the key parameters to the critical concentration of solution "decoys" required for inhibition is derived and compared to the numerical calculations. We find that in order to achieve essentially complete inhibition of surface receptor binding, the concentration of decoys may need to be as much as four to eight orders of magnitude greater than the equilibrium disociation constant for ligand binding to surface receptors.     

Negatively co-operative ligand binding

Simple systems are considered in which the binding of a ligand at a single site exhibits a doubly sigmoid curve when saturation is plotted against the logarithm of ligand concentration, i.e. where a fraction of the site exhibits one dissociation constant and the rest exhibits another. The condition for this to occur is that the ligand should also combine at another site on the binding molecule with comparable affinity and that the binding at one site should markedly lower affinity at the other. The protonation of simple compounds such as cysteine and 3-hydroxypyridine is taken as an example of this process and the equations derived are also applied to the binding of substrates to enzymes.     

A study of ligand binding to spleen myeloperoxidase

The ligand binding properties of spleen myeloperoxidase, a peroxidase formerly called "the spleen green hemeprotein", were studied as functions of temperature and pH, using chloride and cyanide as exogenous ligands. Ligand binding is influenced by a proton dissociable group with a pKa of 4. The protonated, uncharged form of cyanide binds to the unprotonated form of the enzyme, while chloride ion binds to the enzyme when this group is protonated. In both cyanide and chloride binding, the pH-dependent change in the apparent ligand affinity is due to a change in the apparent association rate with pH. The proton dissociable group on the enzyme involved in ligand binding has a delta H value of about 8 kcal . mol-1. The present results suggest that this ionizable group is the imidazole group of a histidine residue located near the ligand binding site.     

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.     

Quantitative considerations of the consequences of an interplay between ligand binding and reversible adsorption of a macromolecular solute

Explicit expressions are derived which determine the equilibrium composition of mixtures comprising a multivalent, insoluble matrix, a multivalent, macromolecular solute (acceptor) and a univalent ligand. With three-reactant mixtures of this type a range of combinations of interactions is possible wherein the ligand interacts with either the acceptor or the matrix, in either event perturbing the acceptor-matrix equilibria. Theory encompassing this range of possibilities is written in terms of a single site-binding constant for each type of interaction to account, in general terms, for both multiple binding and crosslinking effects. These explicit thermodynamic relationships are discussed, with the use of reported findings on several biological systems, in two frameworks. First, it is established that the theory is applicable to the quantitative interpretation of affinity chromatography experiments designed to elucidate the thermodynamic interaction parameters governing the various types of interacting system. Second, it is emphasized that the relationships are also relevant to metabolite-induced changes in the subcellular distribution of macromolecular species.     

Zeeman-Stark modeling of the RF EMF interaction with ligand binding

The influence of radiofrequency electromagnetic exposure on ligand binding to hydrophobic receptor proteins is a plausible early event of the interaction mechanism. A comprehensive quantum Zeeman-Stark model has been developed which takes into account the energy losses of the ligand ion due to its collisions inside the receptor crevice, the attracting nonlinear endogenous force due to the potential energy of the ion in the binding site, the out of equilibrium state of the ligand-receptor system due to the basal cell metabolism, and the thermal noise. The biophysical "output" is the change of the ligand binding probability that, in some instances, may be affected by a suitable low intensity exogenous electromagnetic "input" exposure, e.g., if the depth of the potential energy well of a putative receptor protein matches the energy of the radiofrequency photon. These results point toward both the possibility of the electromagnetic control of biochemical processes and the need for a new database of safety standards.     

Analysis of ligand binding curves in terms of species fractions

The ligand binding curve for a macromolecular system presents the average number or ligand molecules bound per macromolecule as a function of the chemical potential or the logarithm of the ligand concentration. We show that various observable properties of this curve, for example its asymptotes and derivatives, are expressible in terms of linear combinations of the mole fractions α_i of macromolecules binding i molecules of ligand. Whenever enough such properties of the binding curve are known, the linear equations in α_i can be solved to give the mole fractions of each of the various macromolecular species. An application of these results is that a Hill plot for hemoglobin-ligand equilibrium where the asymptotes approach unit slope can be made to yield the four Adair constants by a simple algebraic method. A second use is that a knowledge of the first and second derivatives of the binding curve at points along the curve can yield the species fractions as functions of the degree of saturation without direct knowledge of the ligand binding constants. These methods are illustrated by some numerical examples.     

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.     

Graphical analysis of binding data reflecting competition between two ligands for the same acceptor sites

A theoretical expression is derived for the analysis of results from competitive binding studies in which two multivalent ligands compete for acceptor sites, and a linear transform is suggested for simple graphical representation and assessment of experimental results. The protocol is illustrated by application to competitive binding data, obtained by ultrafiltration, on the interactions of bovine serum albumin with two structurally similar organic anions, methyl orange and methyl red. In a second experimental study the present approach is then used to establish that lactate dehydrogenase and aldolase compete for the same myofibrillar sites of bovine cardiac muscle. Finally, numerically simulated behavior of systems with additional binding sites for either ligand is used to emphasize that the criterion for classical (complete) competition is agreement between an experimentally determined equilibrium constant for ligand binding and the apparent value deduced from competitive binding studies. Nevertheless, the present analysis of competitive binding data should still offer considerable scope for screening quantitatively the cross-reactivities of drug and antigen analogs for their respective specific protein-acceptor sites.     

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.     

Crystallographic studies of ligand binding by Zn-alpha2-glycoprotein

Zn-alpha2-glycoprotein (ZAG) is a 41 kDa soluble protein that is present in most bodily fluids. The previously reported 2.8 A crystal structure of ZAG isolated from human serum demonstrated the structural similarity between ZAG and class I major histocompatibility complex (MHC) molecules and revealed a non-peptidic ligand in the ZAG counterpart of the MHC peptide-binding groove. Here we present crystallographic studies to explore further the nature of the non-peptidic ligand in the ZAG groove. Comparison of the structures of several forms of recombinant ZAG, including a 1.95 A structure derived from ZAG expressed in insect cells, suggests that the non-peptidic ligand in the current structures and in the structure of serum ZAG is a polyethylene glycol (PEG), which is present in the crystallization conditions used. Further support for PEG binding in the ZAG groove is provided by the finding that PEG displaces a fluorophore-tagged fatty acid from the ZAG binding site. From these results we hypothesize that our purified forms of ZAG do not contain a bound endogenous ligand, but that the ZAG groove is capable of binding hydrophobic molecules, which may relate to its function.