Negative co-operativity in the EGF receptor

Scatchard analyses of the binding of EGF (epidermal growth factor) to its receptor (EGFR) yield concave up Scatchard plots, indicative of some type of heterogenity in ligand-binding affinity. This was typically interpreted as being due to the presence of two independent binding sites: one of high affinity representing ≤10% of the receptor population, and one of low affinity making up the bulk of the receptors. However, the concept of two independent binding sites is difficult to reconcile with the X-ray structures of the dimerized EGFR that show symmetrical binding of the two ligands. A new approach to the analysis of 125I-EGF-binding data combined with the structure of the singly-occupied Drosophila EGFR have now shown that this heterogeneity is due to the presence of negative co-operativity in the EGFR. Concerns that negative co-operativity precludes ligand-induced dimerization of the EGFR confuse the concepts of linkage and co-operativity. Linkage refers to the effect of ligand on the assembly of dimers, whereas co-operativity refers to the effect of ligand binding to one subunit on ligand binding to the other subunit within a preassembled dimer. Binding of EGF to its receptor is positively linked with dimer assembly, but shows negative co-operativity within the dimer.     

Absence of positive co-operativity in the binding of 2,3,7,8-tetrachlorodibenzo-p-dioxin to its cytosolic receptor protein.

The role of positive co-operativity in stabilizing the binding of 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) to the rat hepatic cytosolic TCDD receptor protein (Ah receptor) was investigated. The binding mechanism of TCDD was determined by kinetic means through equilibrium and saturation binding studies, and Scatchard and Hill plot analysis. In all studies, the slope of the Hill plot was close to 1.0, indicating the absence of positive co-operativity. Interpretation of the Scatchard plot was however complicated by the fact that both linear and nonlinear plots were experimentally obtained. The nonlinearity was shown to be an experimental artifact and a consequence not of co-operativity, but of high levels of nonspecific binding. The high level of nonspecific binding could be attributed to: (1) lipophilicity of the TCDD ligand, and (2) inefficient competition of receptor-bound [3H]TCDD. When nonspecific binding was minimized, the Scatchard slope was linear and in agreement with the Hill coefficient, thus indicating the lack of positive co-operativity in the binding of TCDD to the Ah receptor.     

A simplified analysis of Scatchard plots for systems with two interacting binding sites

A simple graphical method for calculating stoichiometric and site binding constants for systems with two initially equivalent interacting sites is derived from a modified Scatchard equation. The binding constants can be calculated from Scatchard plots (r/[A] as a function of r) using the values of r/[A] (r is the molar ratio of bound ligands to total protein and [A] is the equilibrium concentration of free ligand) when r = 0 and r = 1 (half-saturation). The applicability of the method to the adsorption of bilirubin by peptide pendants immobilized on a polyacrylamide support is demonstrated.     

Use of binding site neighbor-effect parameters to evaluate the interactions between adjacent ligands on a linear lattice. Effects on ligand-lattice association.

A method using binding site "neighbor-effect" parameters (NEPs) is introduced to evaluate the effects of interaction between adjacent ligands on their binding to an infinite linear lattice. Binding site overlap is also taken into account. This enables the conditional probability approach of McGhee & von Hippel to be extended to more complex situations. The general equation for the isotherm is v/LF = SFKF, where v is the ratio of bound ligands to lattice residues, LF is the free ligand concentration, SF is the fraction of binding sites that are free, and KF is the average association constant of a free site. Solutions are derived for three cases: symmetric ligands, and asymmetric ligands on isotropic or anisotropic lattices. For symmetric ligands there is one NEP, E, which is the ratio of the average binding affinity of a free site if the status of the lattice residue neighboring one end of the site is unspecified (left to chance) to the affinity when this residue is free (holding the other neighbor constant). Thus KF is KE2, where K is the affinity of an isolated site. If a site is n residues long, SF is f ffn-1, where f = 1 - nv is the fraction of residues that are free and ff is the conditional probability that a free residue is bordered on a given side by another free residue. The expression for ff is 1/(1 + x/E), where x is v/f, E is (1 - x + [(1 - x)2 + 4x omega]1/2)/2, and omega is the co-operativity parameter. The binding of asymmetric ligands to an isotropic lattice is described by two NEPs; the last case involves four NEPs and a bound ligand orientation parameter. For each case, the expected length distribution of clusters of bound ligands can be calculated as a function of v. When Scatchard plots with the same intercepts and initial slope are compared, it is found that ligand asymmetry lowers the isotherm (relative to the corresponding symmetric ligand isotherm), whereas lattice anisotrophy raises it.     

The effect of ligand heterogeneity on the Scatchard plot. Particular relevance to lipoprotein binding analysis

Computer simulations of equilibrium binding studies of a mixture of two labeled ligands binding competitively to a single class of identical and independent sites (receptors) were performed to investigate how ligand heterogeneity affects the observed data in such studies. The simulated data are presented in Scatchard plots. Ligand heterogeneity was generally found to be indistinguishable from the case of a homogeneous ligand when usual experimental conditions applied (that is, Scatchard plots of the data were straight lines). Some factors that increased the probability of recognizing heterogeneity in the system were identified, however. These are 1) a large difference between the dissociation constants of the two ligands, 2) a high concentration of receptors relative to the dissociation constant of the higher-affinity ligand, 3) a high concentration of the lower-affinity ligand relative to that of the higher-affinity ligand, 4) a high specific activity of the lower-affinity ligand relative to that of the higher-affinity ligand, and 5) lack of experimental error. When ligand heterogeneity (under certain conditions) did cause curvilinearity in the Scatchard plot, the curve formed was always concave-downwards. Thus, ligand heterogeneity may occasionally mimic positive cooperativity, but never mimics negative cooperativity or multiple classes of binding sites. Implications of these findings for equilibrium binding studies involving lipoproteins (which are generally isolated as heterogeneous mixtures of particles) are discussed in detail. These findings are also relevant to equilibrium binding studies using ligands which are mixtures of stereoisomers or which contain chemical or radiochemical impurities.     

Simulation of Association Curves and ‘Scatchard’ Plots of Binding Reactions Where Ligand and Receptor are Degraded or Internalized

Abstract

Frequently during the course of binding to a receptor, ligand is degraded. In some preparations receptor is degraded. And with isolated cell preparations, ligand and/or receptor may be internalized. Here we present a mathematical model in which the binding and other reactions are combined. The resulting set of differential equations is solved numerically to simulate association curves and the resulting values of bound and free ligand are used to construct Scatchard plots. Where non-ideal conditions exist, the Scatchard plots are generally curvilinear. Dependence of this curvilinearity - on time of measurement of free and bound ligand, on degradation and internalization of ligand, and on degradation and internalization of receptor - is shown. Equilibrium constants derived from the Scatchard plots are generally incorrect but the derived receptor concentration is often correct. The simulations suggest experimental possibilities for distinction among the several side reactions in ligand-receptor binding systems.     

Simulation of Association Curves and ‘Scatchard’ Plots of Binding Reactions where Ligand and Receptor are Degraded or Internalized

Abstract

Frequently during the course of binding to a receptor, ligand is degraded. In some preparations receptor is degraded. And with isolated cell preparations, ligand and/or receptor are internalized. Here we present a mathematical model for the combined binding and other reactions which gives useful information about the behaviour of such systems. The set of differential equations is solved numerically to simulate association curves and the resulting values of bound and free ligand are used to construct Scatchard plots. Where non-ideal conditions exist, the Scatchard plots are generally curvilinear. Dependence of this curvilinearity on time of measurement of free and bound ligand, on degradation and internalization of ligand and on degradation and internalization of receptor is shown. Equilibrium constants derived from the Scatchard plots are generally incorrect but the derived receptor concentration is often correct. The simulations lead to possibilities for distinction among the several side reactions in ligand-receptor binding systems.     

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.     

Interaction of protein with a self-associating ligand. Deviation from a hyperbolic binding curve and the appearance of apparent co-operativity in the Scatchard plot

We derived a general formula to analyze a binding system in which a ligand self-associates, in terms of experimentally determinable quantities, i.e. r, the average number of bound ligands per protein molecule, and Lft, the total free ligand concentration, which are expressed as a ligand monomer unit. The limiting behaviors of the Scatchard plot (r/Lft vs r plot), that is, the intercepts on the r-axis and the r/Lft-axis, and the limiting slopes, are generally given. Three models that may be encountered are considered in detail. Numerical examples are also presented to illustrate how the self-association of a ligand affects the binding curves. The ligand self-association alone can cause deviation of the profile of the binding curve (r vs Lft plot) from a hyperbola, resulting in a nonlinear Scatchard plot. Therefore, analysis of the binding data without consideration of ligand self-association may lead to erroneous conclusions as to the numbers and classes of binding sites, co-operativity among the sites and binding parameter values.     

Thermodynamics and kinetics of co-operative protein-nucleic acid binding: I. General aspects of analysis of data

The process under consideration is the binding of a ligand to a linear polymer of equivalent subunits such that each bound molecule of ligand occupies n subunits. Interactions between bound ligand molecules are also considered. Some useful points regarding the evaluation of raw data without recourse to any specific binding mechanism are discussed first. For a treatment in terms of appropriate thermodynamic parameters a simple model is examined in greater detail. It assumes that interactions are limited to those between ligands bound to nearest-neighbour positions on the polymer.Exact expressions for some basic binding properties of this model at equilibrium are developed. The relations can be considerably simplified in the case of pronounced positive co-operativity which is frequently encountered in practice. Appropriate plots of the data to test the model and to evaluate its parameters are proposed.A simple but consistent kinetic scheme is also introduced. It allows calculation of relaxation times as they can be measured by means of special techniques.     

Large-ligand adsorption to membranes: I. Linear ligands as a limiting case

It is shown that the usual evaluation of binding data in terms of linear Scatchard plots or using simple mass-action equations is not applicable to the binding of ligands covering more than one receptor on a membrane. As a first step in a general treatment of large-ligand adsorption to membranes, a simple formula is derived which applies to linear chain molecules binding to a membrane without cooperative interactions. The formula may be considered as an extension to surface problems of the well-known one-dimensional treatment of Mc Ghee and Von Hippel (Mc Ghee, J.D. and Von Hippel, P.H. (1974) J. Mol. Biol. 86, 469–489). Being applied to non-linear ligand molecules, our treatment yields a lower limit estimate of the stoichiometric number.     

Cooperative effects during the binding of large ligands with DNA. Non-contact interaction between adsorbed ligands

Equations are derived to describe the cooperative binding of large ligands to DNA. A mathematical approach is developed which enables one to give a simple probabilistic interpretation of binding equations and to solve them in the general case when long-range interactions are allowed between bound ligands. These interactions can be mediated by conformation changes induced in the DNA in the course of binding process and transformed over some distances beyond the DNA region immediately covered by a bound ligand molecule (allosteric effect of DNA). Interactions between ligand molecules can be formally described in terms of model potential characterizing pairwise interactions between bound ligands. A procedure is developed which allows one to determined the form of such potential from experimentally measured binding isotherms. It is based on a comparison of experimental binding isotherms with the appropriate curves calculated for the case of non-interacting ligands.     

Co-operativity and enzymatic activity in polymer-activated enzymes. A one-dimensional piggy-back binding model and its application to the DNA-dependent ATPase of DNA gyrase

The binding of a ligand to a one-dimensional lattice in the presence of a second ("rider") ligand, which binds only to the first ligand (piggy-back binding), is studied. A model derived from this study is used to analyze the effects of co-operativity on the reaction rates of enzymes activated by polymeric cofactors that provide multiple binding sites for the enzyme. It is found that in the presence of strong co-operativity, the steady-state reaction rates of polymer-activated enzymes can be very different from the Michaelis-Menten paradigm. By adjusting the co-operativity parameters and the binding constants of the ligands, the model can generate apparent auto-catalytic enhancement by substrates at low substrate concentrations and apparent substrate inhibition at high substrate concentrations. The model is shown to be able to explain the differences in the rates of ATP hydrolysis by DNA gyrase in the presence of long versus short DNA molecules and in the presence of long DNA molecules at different gyrase to DNA ratios.     

Thermodynamic model of cooperativity in a dimeric protein: unique and independent parameters formulation.

A model of the cooperative interaction of ligand binding to a dimeric protein is presented based upon the unique and independent parameters (UIP) thermodynamic formulation (Gutheil and McKenna, Biophys. Chem. 45 (1992) 171-179). The analysis is developed from an initial model which includes coupled conformational and ligand binding equilibria. This completely general model is then restricted to focus on conformationally mediated cooperative interactions between the ligands and the expressions for the apparent ligand binding constant and the apparent ligand-ligand interaction constant are derived. The conditions under which there is no cooperative interaction between the ligands are found as roots to a polynomial equation. Consideration of the distribution of species among the various conformational states in this general model leads to a set of inequalities which can be represented as a two dimensional plot of boundaries. By superimposing a contour plot of the value of the apparent ligand-ligand interaction constant over the plot of boundaries a complete graphical representation of this system is achieved similar to a phase diagram. It is found that the parameter space homologous to Koshland-Nemethy-Filmer type of model is most consistent with both positive and negative cooperativity in this model. The maximal amount of positive and negative cooperativity are found to be simple functions of Kc, the equilibrium constant associated with the change of a subunit and ligand from the unligated to ligated conformation. It is shown that under certain limiting conditions the apparent allosteric interaction between ligands is equal to the conformational interaction between subunits. The methods presented are generally applicable to the theoretical analysis of thermodynamic interactions in complex systems.     

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.     

On the analysis of linear binding effects associated with curved Scatchard plots.

First the question is examined as to which binding data, especially if given as Scatchard plots, can be described in terms of a basic model mechanism. This referes to a linear lattice of equivalent binding sites (as for example located on a linear biopolymer) which can exert cooperative interaction between nearest neighbors. It is shown that the effect of overlapping of potential site (so-called "multiple-contact"binding), as may occur with larger ligands, will largely be compensated by a higher degree of cooperativity. Therefore, in practice such properties can scarcely be separated by means of oridnary binding experiments. A pronounced inflection point in the Scatchard plot turns out to be clearly indicating a more complex mechanism involving at least two rather antagonistic cooperative interactions which may, however, occur even between equivalent binding sites. Finally some consequences of different classes of bindings sites are considered. In particular a simple approach is introduced by which the binding to mutually exclusive classes of sites may be described. Such a model is of interest for multiple-mode binding of ionic ligands to oppositely charged polymers.     

The reversible cross-linking of receptors by ligands: theory for the prediction of binding responses

The general concept of receptor aggregation through the action of a cross-linking ligand is considered, three models being examined in detail. In two, ligand self-interaction leads to the formation of receptor cross-links whereas, in the third, receptor cross-links are formed through a single ligand bridge. Binding equations in closed form are formulated for each case and are shown to predict both concave and convex Scatchard plots in binding studies conducted with such systems. The significant point which emerges is that while each of the systems gives rise to very different types of binding responses, the form of the binding response is always strongly dependent on the total concentration of receptor in the system.     

Co-operativity and the methods of plotting binding and steady-state kinetic data.

The features that distinguish positive from negative co-operativity in double-reciprocal Eadie-Hofstee-Scatchard and Hanes plots, often incorrectly stated to be the sign of curvature or second derivatives, are explained. It is shown how to determine the 'Hill exponent' and interaciton free energies from curves in these plots, and in the simple plot of ligand binding or velocity against free ligand or substrate concentration. New types of plots, where the kind of co-operative behaviour is more obvious than in the traditional ones, are proposed.     

An extended Scatchard analysis for competitively bound ligands and its application to cyclic adenosine-3',5'-monophosphate and cyclic 5'-amido-5'-deoxyadenosine-3', 5'-monophosphate binding to beef skeletal muscle protein.

A method for determining the dissociation constants of ligands and ligand analogs is described. It is based on competition binding studies in the presence of an isotope-labeled, or otherwise measurable, ligand and suitable analog concentrations.The steps used are determination of (1) the maximal amount of radioactive ligand that can be bound, (2) the slopes and intercepts from Scatchard plots at different analog concentrations and (3) the values for the dissociation constants of radioactive ligand and ligand analog from replots of the reciprocals of the slopes and intercepts obtained from the Scatchard plots. Application of the method to a cyclic AMP-binding protein from beef muscle is demonstrated, yielding dissociation constants of 2.10-9 M for cyclic (3H) AMP and cyclic AMP, and 3.10-5 M for cyclic 5'-amido-5'-deoxyadenosine-3', 5'-monophosphate.