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.     

Statistical methods for estimating complexity from competition experiments between two populations

Often a screening or selection experiment targets a cell or tissue, which presents many possible molecular targets and identifies a correspondingly large number of ligands. We describe a statistical method to extract an estimate of the complexity or richness of the set of molecular targets from competition experiments between distinguishable ligands, including aptamers derived from combinatorial experiments (SELEX or phage display). In simulations, the non-parametric statistic provides a robust estimate of complexity from a 100 ×100 matrix of competition experiments, which is clearly feasible in high-throughput format. The statistic and method are potentially applicable to other ligand binding situations.     

Ligand binding analysis and screening by chemical denaturation shift

The identification of small molecule ligands is an important first step in drug development, especially drugs that target proteins with no intrinsic activity. Toward this goal, it is important to have access to technologies that are able to measure binding affinities for a large number of potential ligands in a fast and accurate way. Because ligand binding stabilizes the protein structure in a manner dependent on concentration and binding affinity, the magnitude of the protein stabilization effect elicited by binding can be used to identify and characterize ligands. For example, the shift in protein denaturation temperature (T_m shift) has become a popular approach to identify potential ligands. However, T_m shifts cannot be readily transformed into binding affinities, and the ligand rank order obtained at denaturation temperatures (?60 °C) does not necessarily coincide with the rank order at physiological temperature. An alternative approach is the use of chemical denaturation, which can be implemented at any temperature. Chemical denaturation shifts allow accurate determination of binding affinities with a surprisingly wide dynamic range (high micromolar to sub nanomolar) and in situations where binding changes the cooperativity of the unfolding transition. In this article, we develop the basic analytical equations and provide several experimental examples.     

Analysis of Competition Binding Assays: Assessment of the Range of Validity of a Commonly Invoked Assumption

Abstract

A common assumption invoked in the analysis of competition binding assays is that the fractional saturation of sites with the unlabeled ligand is given by 1 - (the concentration of bound labeled ligand in the presence of unlabeled ligand)/(the concentration of bound labeled ligand in the absence of unlabeled ligand). This assumption is critically evaluated in the context of several binding models: (a) binding of univalent ligands to multiple classes of equivalent and independent sites, with and without nonspecific binding; (b) cooperative binding of univalent ligands; and (c) binding of multivalent ligands to a single class of univalent acceptors. We show that the conventional assumption is only valid when the labeled ligand is mainly in the free form, occupies a small fraction of the total sites and binds univalently to all sites in an equivalent and independent manner, and when the unlabeled ligand forms l : l complexes with the acceptor sites. When these conditions are satisfied, the conventional assumption is valid even if the unlabeled ligand binds to nonequivalent sites or exhibits cooperativity. Finally, we apply the theory derived for case (a) above to the binding of fluoresceinated epidermal growth factor to A431 cells and demonstrate that the analysis of data obtained from both conventional and competition assays provides information which is difficult, if not impossible, to obtain from either assay alone.     

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.     

Control and stability of ligand receptor interaction in the presence of a competitive compound.

The law of mass action is almost universally applied to both endogenous ligands and drugs that interact with specific cellular receptors. The concentration, and hence the receptor binding, of a foreign drug molecule will depend on its pharmacokinetic properties, whereas an endogenous ligand is subject to intrinsic control since the concentration (z) remains within limits around an equilibrium level. We have previously examined this control for ligand-receptor interactions proceeding according to mass action in which the ligand is produced (rate F), eliminated exponentially (rate constant E) and controlled by a feedback function of receptor occupancy, phi (y), where y is the bound concentration. The current study examines the control of an endogenous ligand in the presence of a second compound (agonist or antagonist) that interacts with the same receptor. From a computer solution of the set of differential equations, illustrated in both time-plots and phase plane (y-z) plots, it is shown that if the second agent is a pure competitive antagonist the bound concentration of the endogenous agonist ligand may not decrease appreciably, even for large doses of the antagonist. Instead the level of binding depends on certain parameters of the control function. Further, the computer simulation shows how these parameters affect the time course of released ligand resulting from administration of an antagonist and the suppression of such release when the second compound is an agonist.     

Synthesis,binding and bioactivity of γ-methylene γ-lactam ecdysone receptor ligands: Advantages of QSAR models for flexible receptors

Nuclear hormone receptors, such as the ecdysone receptor, often display a large amount of induced fit to ligands. The size and shape of the binding pocket in the EcR subunit changes markedly on ligand binding, making modelling methods such as docking extremely challenging. It is, however, possible to generate excellent 3D QSAR models for a given type of ligand, suggesting that the receptor adopts a relatively restricted number of binding site configurations or ‘attractors’. We describe the synthesis, in vitro binding and selected in vivo toxicity data for γ-methylene γ-lactams, a new class of high-affinity ligands for ecdysone receptors from Bovicola ovis (Phthiraptera) and Lucilia cuprina (Diptera). The results of a 3D QSAR study of the binding of methylene lactams to recombinant ecdysone receptor protein suggest that this class of ligands is indeed recognised by a single conformation of the EcR binding pocket.     

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.     

Generation of continuous ligand–macromolecule binding isotherms. Use of exponential concentration gradients

Continuous exponential concentration gradients, accurate, precise, and reproducible to better than 1% over at least a 10³–10⁴ concentration range, can be easily produced. In fact, it appears that these gradients may be substantially better than this. Such gradients have been used to generate a continuous ligand binding isotherm for the system Mg⁺⁺ + Eriochrome Black T over a 10⁶ ligand concentration range. This continuous isotherm is shown to be sufficiently precise to provide a first derivative function Γ(y), which is precise to 0.4%. The moments of Γ(y) were calculated and found to be extremely accurate. For example, the second moment about the mean was calculated to be 3.32 and 3.30 for two different experiments compared to a theoretical value of 3.29. The general application of these techniques to generate continuous ligand binding isotherms is discussed.     

Reconciliation of classical and reacted‐site probability approaches to allowance for ligand multivalence in binding studies

The objective of this investigation is to engender greater confidence in the validity of binding equations derived for multivalent ligands on the basis of reacted‐site probability theory. To that end, a demonstration of the theoretical interconnection between expressions derived by the classical stepwise equilibria and reacted‐site probability approaches for univalent ligands is followed by the use of the traditional stepwise procedure to derive binding equations for bivalent and trivalent ligands. As well as demonstrating the unwieldy nature of the classical binding equation for multivalent ligand systems, that exercise has allowed numerical simulation to be used to illustrate the equivalence of binding curves generated by the two approaches. The advantages of employing a redefined binding function for multivalent ligands are also confirmed by subjecting the simulated results to a published analytical procedure that has long been overlooked. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Cooperation effects in binding of large ligands to DNA. II. Contact interactions between adsorbed ligands

Cooperative effects arising upon binding of biologically active ligands to DNA are considered. Equations are derived which enable one to describe the binding of two different ligands to DNA. We also consider the case when ligand can form two type of DNA complexes. The cooperative binding of the ligand in the vicinity of saturation level of binding can be described with a good accuracy by equation derived for the non-cooperative adsorption of the same ligand with some effective binding constant Keff. It is shown that cooperative effects arising upon binding of proteins and other ligands to DNA can be divided into two groups depending on the symmetry of interactions between the bound ligand molecules. In particular, if such interactions favor the formation of dimeric ligand species on the DNA, Keff approximately a1/2, where a is the ligand-ligand interaction constant. If cooperative interactions favor the formation of aggregates of unrestricted size, then Keff approximately aL+Y, where L is the size of the binding site for the ligand on DNA.     

Folding energetics of ligand binding proteins. I. Theoretical model

Heat capacity curves as obtained from differential scanning calorimetry are an outstanding source for molecular information on protein folding and ligand-binding energetics. However, deconvolution of C(p) data of proteins in the presence of ligands can be compromised by indeterminacies concerning the correct choice of the statistical thermodynamic ensemble. By convent, the assumption of constant free ligand concentration has been used to derive formulae for the enthalpy. Unless the ligand occurs at large excess, this assumption is incorrect. Still the relevant ensemble is the grand canonical ensemble. We derive formulae for both constraints, constancy of total or free ligand concentration and illustrate the equations by application to the typical equilibrium Nx <=> N + x <=> D + x. It is demonstrated that as long as the thermodynamic properties of the ligand can be completely corrected for by performing a reference measurement, the grand canonical approach provides the proper and mathematically significantly simpler choice. We demonstrate on the two cases of sequential or independent ligand-binding the fact, that similar binding mechanisms result in different and distinguishable heat capacity equations. Finally, we propose adequate strategies for DSC experiments as well as for obtaining first estimates of the characteristic thermodynamic parameters, which can be used as starting values in a global fit of DSC data.     

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.     

Circular Permutation Provides an Evolutionary Link between Two Families of Calcium-dependent Carbohydrate Binding Modules

The microbial deconstruction of the plant cell wall is a critical biological process, which also provides important substrates for environmentally sustainable industries. Enzymes that hydrolyze the plant cell wall generally contain non-catalytic carbohydrate binding modules (CBMs) that contribute to plant cell wall degradation. Here we report the biochemical properties and crystal structure of a family of CBMs (CBM60) that are located in xylanases. Uniquely, the proteins display broad ligand specificity, targeting xylans, galactans, and cellulose. Some of the CBM60s display enhanced affinity for their ligands through avidity effects mediated by protein dimerization. The crystal structure of vCBM60, displays a β-sandwich with the ligand binding site comprising a broad cleft formed by the loops connecting the two β-sheets. Ligand recognition at site 1 is, exclusively, through hydrophobic interactions, whereas binding at site 2 is conferred by polar interactions between a protein-bound calcium and the O2 and O3 of the sugar. The observation, that ligand recognition at site 2 requires only a β-linked sugar that contains equatorial hydroxyls at C2 and C3, explains the broad ligand specificity displayed by vCBM60. The ligand-binding apparatus of vCBM60 displays remarkable structural conservation with a family 36 CBM (CBM36); however, the residues that contribute to carbohydrate recognition are derived from different regions of the two proteins. Three-dimensional structure-based sequence alignments reveal that CBM36 and CBM60 are related by circular permutation. The biological and evolutionary significance of the mechanism of ligand recognition displayed by family 60 CBMs is discussed.     

DNA bis-intercalation: Theoretical calculation of binding curves

A statistical mechanical calculation of the binding properties of DNA bis-intercalators is presented, based on the sequence-generating function method of Lifson. The effects of binding by intercalation of one or both chromophores of a bifunctional intercalating agent are examined. The secular equation for a general model that includes the effects of neighbor (nearest and non-nearest) exclusion and/or cooperativity in the binding of both singly and doubly intercalated ligands is derived. Numerical results for binding curves are presented for a more restricted model in which each type of bound ligand rigorously excludes its nearest neighbor and the total number of sites covered by a doubly intercalated ligand is variable. At low values of free ligand concentration bis-intercalation dominates the binding process, while at high value of free ligand concentration, intercalation of only one chromophore per ligand becomes significant due to the unavailability of contiguous free sites required for bis-intercalation. Also, depending on the binding parameters, the free energy of the system can be lowered by a loss of doubly intercalated ligands in favor of singly intercalated ones. Corresponding to this transition in binding mode, the average number of sites occupied by a bound ligand decreases from that characteristic of bis-intercalation to that characteristic of mono-intercalation as free ligand concentration increases. An analysis of Scatchard plots describing bis-intercalation is presented.     

Congeneric but still distinct: how closely related trypsin ligands exhibit different thermodynamic and structural properties

A congeneric series of benzamidine-type ligands with a central proline moiety and a terminal cycloalkyl group—linked by a secondary amine, ether, or methylene bridge—was synthesized as trypsin inhibitors. This series of inhibitors was investigated by isothermal titration calorimetry, crystal structure analysis in two crystal forms, and molecular dynamics simulations. Even though all of these congeneric ligands exhibited essentially the same affinity for trypsin, their binding profiles at the structural, dynamic, and thermodynamic levels are very distinct. The ligands display a pronounced enthalpy/entropy compensation that results in a nearly unchanged free energy of binding, even though individual enthalpy and entropy terms change significantly across the series. Crystal structures revealed that the secondary amine-linked analogs scatter over two distinct conformational families of binding modes that occupy either the inside or of the outside the protein's S3/S4 specificity pocket. In contrast, the ether-linked and methylene-linked ligands preferentially occupy the hydrophobic specificity pocket. This also explains why the latter ligands could only be crystallized in the conformationally restricting closed crystal form whereas the derivative with the highest residual mobility in the series escaped our attempts to crystallize it in the closed form; instead, a well-resolved structure could only be achieved in the open form with the ligand in disordered orientation. These distinct binding modes are supported by molecular dynamics simulations and correlate with the shifting enthalpic/entropic signatures of ligand binding. The examples demonstrate that, at the molecular level, binding modes and thermodynamic binding signatures can be very different even for closely related ligands. However, deviating binding profiles provide the opportunity to optimally address a given target.     

Nuclear magnetic resonance studies of flexible opiate conformations at monoclonal antibody binding sites. Quantitative interproton distances obtained from comparing theoretical and experimental transferred nuclear Overhauser enhancement: correlation with antibody sequence

A previous publication described the use of qualitative intramolecular 1H-transferred nuclear Overhauser effect measurements to determine the conformations of flexible ligands at monoclonal anti-opiate antibody binding sites. This paper concentrates on the quantitative interpretation of experiments of this type using the ligand nalorphine (N-allyl morphine) and a single anti-opiate monoclonal antibody. I compare the experimental unidimensional driven nuclear Overhauser effect buildup curves to theoretical curves derived with a knowledge of the fixed interproton distances in the ligand. The discussion covers the potential accuracies of derived distances and concentrates on two problem areas associated with determining structures from this type of experiment. The most serious one is the case where, because of particular multiproton spatial distributions, spin diffusion is so rapid that it cannot be determined experimentally and where numerical fits of theoretical calculations are misleading. The results show that, while intraligand spin diffusion complicates the interpretation for some proton pairs, with many others accuracies within about 0.3 A for interproton distances from 2 to 4 A are attainable. The results confirm the earlier report that the conformation of nalorphine in this antibody binding site differs from the major one present in solution or in the crystal. An important aspect of the work is that theoretical prediction of nuclear Overhauser effect time-dependence is an important practical tool for recognizing cases where interpretation of experiments will be difficult. Initial data on protein-to-ligand transferred nuclear Overhauser effect are presented, which show that at least one aromatic amino acid residue is closely involved in the binding of the ligand. The companion paper presents the primary sequences of the variable regions of the antibodies being used in our studies. In this paper, these and associated immunochemical studies are correlated with the nuclear magnetic resonance results. The combination of data presented in the two papers provides a basis for future work on protein-ligand interproton distances in the range 1 to 5 A using both transferred nuclear Overhauser effect (for rapidly exchanging ligands) and isotope-edited, indirectly detected nuclear Overhauser effect (for tightly bound ligands).