Binding Polynomial Research Articles

Thermodynamic coupling between neighboring binding sites in homo-oligomeric ligand sensing proteins from mass resolved ligand-dependent population distributions.

Homo‐oligomeric ligand‐activated proteins are ubiquitous in biology. The functions of such molecules are commonly regulated by allosteric coupling between ligand‐binding sites. Understanding the basis for this regulation requires both quantifying the free energy ΔG transduced between sites, and the structural basis by which it is transduced. We consider allostery in three variants of the model ring‐shaped homo‐oligomeric trp RNA‐binding attenuation protein (TRAP). First, we developed a nearest‐neighbor statistical thermodynamic binding model comprising microscopic free energies for ligand binding to isolated sites ΔG 0, and for coupling between adjacent sites, ΔG α . Using the resulting partition function (PF) we explored the effects of these parameters on simulated population distributions for the 2 N possible liganded states. We then experimentally monitored ligand‐dependent population shifts using conventional spectroscopic and calorimetric methods and using native mass spectrometry (MS). By resolving species with differing numbers of bound ligands by their mass, native MS revealed striking differences in their ligand‐dependent population shifts. Fitting the populations to a binding polynomial derived from the PF yielded coupling free energy terms corresponding to orders of magnitude differences in cooperativity. Uniquely, this approach predicts which of the possible 2 N liganded states are populated at different ligand concentrations, providing necessary insights into regulation. The combination of statistical thermodynamic modeling with native MS may provide the thermodynamic foundation for a meaningful understanding of the structure–thermodynamic linkage that drives cooperativity.

The influence of excipients on the viscosity of monoclonal antibody solutions

The aggregation propensity of monoclonal antibodies can be modified by adding different cosolutes into the solution. A simple coarse-grained model in the combination with the thermodynamic perturbation theory was used to predict cluster distribution and viscosity of the solutions of IgG4 monoclonal anibody in the presence of L-Arginine Hydrochloride. The data were analysed using binding polynomial to describe the binding of cosolute (Arginine) to the antibody molecule. The results show that by binding to the antibody molecule the cosolute occupies some of the binding sites of the antibody, and in this way reduces the amount of binding sites available to other antibody molecules. The aggregation propensity of the antibody molecules is therefore reduced.

Cooperativity, absolute interaction, and algebraic optimization

We consider a measure of cooperativity based on the minimal interaction required to generate an observed titration behavior. We describe the corresponding algebraic optimization problem and show how it can be solved using the nonlinear algebra tool SCIP. Moreover, we compute the minimal interactions and minimal molecules for several binding polynomials that describe the oxygen binding of various hemoglobins under different conditions. We compare their minimal interaction with the maximal slope of the Hill plot, and discuss similarities and discrepancies with a view towards the shapes of the binding curves.

Binding Polynomial of a Graph

Binding Polynomial of a Graph

Influence of binding polynomials on feasibility of biochemical reactions

Influence of binding polynomials on feasibility of biochemical reactions

Tinkering with Binding Polynomials in Isothermal Titration Calorimetry.

Isothermal titration calorimetry (ITC) has become the preferred experimental technique for characterizing intermolecular interactions between biological molecules. Among the several advantages, the use of natural non-labeled molecules and the determination of the complete thermodynamic profile for the interaction in solution remain as the primary features that have promoted ITC to the forefront of experimental biophysics. The experimental design in ITC may range from studying a simple direct binary macromolecule-ligand interaction to studying the homotropic or heterotropic cooperative effect between ligands when interacting with a given macromolecule. The theory of the binding polynomial has proven to be an appropriate unifying framework for handling the complexities that can be encountered when studying macromolecule-ligand interactions, though it has been deemed troublesome. The goal of this chapter is to provide a quite simple and widely available set of training experiments aimed at mastering the formalism of the binding polynomial applied to isothermal titration calorimetry.

Decoupled molecules with binding polynomials of bidegree (n,\xa02)

We present a result on the number of decoupled molecules for systems binding two different types of ligands. In the case of n and 2 binding sites respectively, we show that there are 2(n!)^{2} decoupled molecules to a generic binding polynomial. For molecules with more binding sites for the second ligand, we provide computational results.

A measure to quantify the degree of cooperativity in overall titration curves

In the framework of the grand canonical ensemble, different definitions of cooperativity commonly used in the context of ligand binding are not equivalent. A unifying definition is the existence of non-real roots of the binding polynomial. Using this qualitative criterion, an open question is how to quantify the degree of cooperativity. In this work, we introduce a theoretical measure to quantify the degree of cooperativity of a titriation curve. Its definition is based on a minimal energy approach mapping a given binding polynomial to the minimal interaction energy which is required to generate it. We show that the degree of cooperativity can be calculated easily, if the molecule under consideration is assumed to consist of energetically identical binding sites. Moreover, the property of sub-multiplicativity allows us to determine upper bounds for the degree of cooperativity in asymmetric systems. The approach is consistent with the qualitative definition of cooperativity based on the existence of non-real roots of the binding polynomial, and thus helps to put the concept of cooperativity on a solid theoretical ground. It connects macro- and microstates, but takes here also into account that an infinite number of different molecules can cause the same macroscopic ligand binding behavior, which means that the underlying microsystem cannot be uniquely identified based on the titration curve only.

Mechanistic Models Fit to Variable Temperature Calorimetric Data Provide Insights into Cooperativity

Allostery pervades macromolecular function and drives cooperative binding of ligands to macromolecules. To decipher the mechanisms of cooperative ligand binding it is necessary to define at a microscopic level the structural and thermodynamic consequences of binding of each ligand to its allosterically coupled site(s). However, dynamic sampling of alternative conformations (microstates) in allosteric molecules complicates interpretation of both structural and thermodynamic data. Isothermal titration calorimetry has the potential to directly quantify the thermodynamics of allosteric interactions, but usually falls short of enabling mechanistic insight. This is because 1) its measurements reflect the sum of overlapping caloric processes involving binding-linked population shifts within and between microstates, and 2) data are generally fit with phenomenological binding polynomials that are underdetermined. Nevertheless, temperature-dependent binding data have the potential to resolve overlapping thermodynamic processes, while mechanistically constrained models enable hypothesis testing and identification of informative parameters. We globally fit temperature-dependent isothermal titration calorimetry data for binding of 11 tryptophan ligands to the homo-undecameric trp RNA-binding Attenuation Protein from Bacillus stearothermophilus using nearest-neighbor statistical thermodynamic models. This approach allowed us to distinguish alternative nearest-neighbor interaction models, and quantifies the thermodynamic contribution of neighboring ligands to individual binding sites. We also perform conventional Hill equation modeling and illustrate how comparatively limited it is in quantitative or mechanistic value. This work illustrates the potential of mechanistically constrained global fitting of binding data to yield the microscopic thermodynamic parameters essential for deciphering mechanisms of cooperativity in a wide range of ligand-regulated homo-oligomeric assemblies.

Conformational Dynamics and Binding Free Energies of Inhibitors of BACE-1: From the Perspective of Protonation Equilibria.

BACE-1 is the β-secretase responsible for the initial amyloidogenesis in Alzheimer’s disease, catalyzing hydrolytic cleavage of substrate in a pH-sensitive manner. The catalytic mechanism of BACE-1 requires water-mediated proton transfer from aspartyl dyad to the substrate, as well as structural flexibility in the flap region. Thus, the coupling of protonation and conformational equilibria is essential to a full in silico characterization of BACE-1. In this work, we perform constant pH replica exchange molecular dynamics simulations on both apo BACE-1 and five BACE-1-inhibitor complexes to examine the effect of pH on dynamics and inhibitor binding properties of BACE-1. In our simulations, we find that solution pH controls the conformational flexibility of apo BACE-1, whereas bound inhibitors largely limit the motions of the holo enzyme at all levels of pH. The microscopic pKa values of titratable residues in BACE-1 including its aspartyl dyad are computed and compared between apo and inhibitor-bound states. Changes in protonation between the apo and holo forms suggest a thermodynamic linkage between binding of inhibitors and protons localized at the dyad. Utilizing our recently developed computational protocol applying the binding polynomial formalism to the constant pH molecular dynamics (CpHMD) framework, we are able to obtain the pH-dependent binding free energy profiles for various BACE-1-inhibitor complexes. Our results highlight the importance of correctly addressing the binding-induced protonation changes in protein-ligand systems where binding accompanies a net proton transfer. This work comprises the first application of our CpHMD-based free energy computational method to protein-ligand complexes and illustrates the value of CpHMD as an all-purpose tool for obtaining pH-dependent dynamics and binding free energies of biological systems.

Cooperative binding: a multiple personality.

Cooperative binding has been described in many publications and has been related to or defined by several different properties of the binding behavior of the ligand to the target molecule. In addition to the commonly used Hill coefficient, other characteristics such as a sigmoidal shape of the overall titration curve in a linear plot, a change of ligand affinity of the other binding sites when a site of the target molecule becomes occupied, or complex roots of the binding polynomial have been used to define or to quantify cooperative binding. In this work, we analyze how the different properties are related in the most general model for binding curves based on the grand canonical partition function and present several examples which highlight differences between the cooperativity characterizing properties which are discussed. Our results mainly show that among the presented definitions there are not two which fully coincide. Moreover, this work poses the question whether it can make sense to distinguish between positive and negative cooperativity based on the macroscopic binding isotherm only. This article shall emphasize that scientists who investigate cooperative effects in biological systems could help avoiding misunderstandings by stating clearly which kind of cooperativity they discuss.

Publisher's Note: Binding polynomial in molecular self-assembly [Phys. Rev. E89, 062138 (2014)

Received 19 February 2015DOI:https://doi.org/10.1103/PhysRevE.91.029906©2015 American Physical Society

Using Sedimentation Velocity to Investigate the Nucleotide-Linked Assembly of E. coli ClpA

The AAA+ (ATPases Associated with various cellular Activities) group of proteins is a large superfamily whose members are present in single-celled and multi-celled organisms. This family is subdivided into two classes; Class I ATPases contain two ATPase domains per monomer while Class II ATPases contain one ATPase domain per monomer. Both classes utilize ATP binding and hydrolysis to undergo conformational changes that allow them to perform mechanical work. Further, the biologically active form for most members of the AAA+ family is a hexamer. Quantitatively examining the biologically-relevant activities for AAA+ proteins requires a precise determination of the concentration of hexamers in solution. The difficulty in determining the hexamer concentration is that the oligomeric state of interest often resides in a dynamic equilibrium. This leads to a distribution of oligomeric states in solution. Therefore, insight into nucleotide-linked assembly is required to be able to understand how any of the AAA+ motors function. ClpA is a Class I AAA+ protein. Apparent self-association equilibrium constants were obtained using sedimentation velocity and were subsequently analyzed using binding polynomials that incorporate either zero cooperativity or infinite cooperativity into the nucleotide binding process to yield apparent nucleotide binding constants. Here we show that the population of ClpA dimers, tetramers, and hexamers is dependent upon nucleotide concentration. Further, apparent nucleotide binding constants were found for each oligomeric state. The above-stated information was used to generate a prediction of ClpA hexamer population that takes into account nucleotide-linked assembly. We anticipate the results of this study will set the stage for being able to predict the concentration of hexamers at any given nucleotide concentration, therefore allowing us to probe the kinetic and energetic properties of ClpA as well as other members of the AAA+ superfamily.

A unified framework based on the binding polynomial for characterizing biological systems by isothermal titration calorimetry

Isothermal titration calorimetry (ITC) has become the gold-standard technique for studying binding processes due to its high precision and sensitivity, as well as its capability for the simultaneous determination of the association equilibrium constant, the binding enthalpy and the binding stoichiometry. The current widespread use of ITC for biological systems has been facilitated by technical advances and the availability of commercial calorimeters. However, the complexity of data analysis for non-standard models is one of the most significant drawbacks in ITC. Many models for studying macromolecular interactions can be found in the literature, but it looks like each biological system requires specific modeling and data analysis approaches. The aim of this article is to solve this lack of unity and provide a unified methodological framework for studying binding interactions by ITC that can be applied to any experimental system. The apparent complexity of this methodology, based on the binding polynomial, is overcome by its easy generalization to complex systems.

Protocols utilizing constant pH molecular dynamics to compute pH-dependent binding free energies.

In protein–ligand binding, the electrostatic environments of the two binding partners may vary significantly in bound and unbound states, which may lead to protonation changes upon binding. In cases where ligand binding results in a net uptake or release of protons, the free energy of binding is pH-dependent. Nevertheless, conventional free energy calculations and molecular docking protocols typically do not rigorously account for changes in protonation that may occur upon ligand binding. To address these shortcomings, we present a simple methodology based on Wyman’s binding polynomial formalism to account for the pH dependence of binding free energies and demonstrate its use on cucurbit[7]uril (CB[7]) host–guest systems. Using constant pH molecular dynamics and a reference binding free energy that is taken either from experiment or from thermodynamic integration computations, the pH-dependent binding free energy is determined. This computational protocol accurately captures the large pKa shifts observed experimentally upon CB[7]:guest association and reproduces experimental binding free energies at different levels of pH. We show that incorrect assignment of fixed protonation states in free energy computations can give errors of >2 kcal/mol in these host–guest systems. Use of the methods presented here avoids such errors, thus suggesting their utility in computing proton-linked binding free energies for protein–ligand complexes.

Binding polynomial in molecular self-assembly.

In the present work the concept of a binding polynomial is revisited for the most widely used case of self-assembly of identical molecular units and results in the re-construction of a link to the grand partition function of such a system. It is found that if the self-assembly process is not pronounced (i.e., the product of the equilibrium constant and the monomer concentration is close to zero), the binding polynomial has the meaning of a molecular partition function that is given by the summation over energy levels of any molecule in the system. In other cases the concept of a binding polynomial may be misleading.

A general framework improving teaching ligand binding to a macromolecule

The interaction of a ligand with a macromolecule has been modeled following different theories. The tenants of the induced fit model consider that upon ligand binding, the protein–ligand complex undergoes a conformational change. In contrast, the allosteric model assumes that only one among different coexisting conformers of a given protein is suitable to bind the ligand optimally. In the present paper, we propose a general framework to model the binding of ligands to a macromolecule. Such framework built on the binding polynomial allows opening new ways to teach in a unified manner ligand binding, enzymology and receptor binding in pharmacology. Moreover, we have developed simple software that allows building the binding polynomial from the schematic description of the biological system under study. Taking calmodulin as a canonical example, we show here that the proposed tool allows the easy retrieval of previously experimental and computational reports. This article is part of a Special Issue entitled: Calcium Signaling in Health and Disease. Guest Editors: Geert Bultynck, Jacques Haiech, Claus W. Heizmann, Joachim Krebs, and Marc Moreau.

Atteinte cardiaque au cours de dystrophinopathies et de sarcoglycanopathies : étude de série marocaine

The interaction of a ligand with a macromolecule has been modeled following different theories. The tenants of the induced fit model consider that upon ligand binding, the protein–ligand complex undergoes a conformational change. In contrast, the allosteric model assumes that only one among different coexisting conformers of a given protein is suitable to bind the ligand optimally. In the present paper, we propose a general framework to model the binding of ligands to a macromolecule. Such framework built on the binding polynomial allows opening new ways to teach in a unified manner ligand binding, enzymology and receptor binding in pharmacology. Moreover, we have developed simple software that allows building the binding polynomial from the schematic description of the biological system under study. Taking calmodulin as a canonical example, we show here that the proposed tool allows the easy retrieval of previously experimental and computational reports. This article is part of a Special Issue entitled: Calcium Signaling in Health and Disease. Guest Editors: Geert Bultynck, Jacques Haiech, Claus W. Heizmann, Joachim Krebs, and Marc Moreau.

An Explicit Formulation Approach for the Analysis of Calcium Binding to EF-Hand Proteins Using Isothermal Titration Calorimetry

We present an improved and extended version of a recently proposed mathematical approach for modeling isotherms of ligand-to-macromolecule binding from isothermal titration calorimetry. Our approach uses ordinary differential equations, solved implicitly and numerically as initial value problems, to provide a quantitative description of the fraction bound of each competing member of a complex mixture of macromolecules from the basis of general binding polynomials. This approach greatly simplifies the formulation of complex binding models. In addition to our generalized, model-free approach, we have introduced a mathematical treatment for the case where ligand is present before the onset of the titration, essential for data analysis when complete removal of the binding partner may disrupt the structural and functional characteristics of the macromolecule. Demonstration programs playable on a freely available software platform are provided. Our method is experimentally validated with classic calcium (Ca2+) ion-selective potentiometry and isotherms of Ca2+ binding to a mixture of chelators with and without residual ligand present in the reaction vessel. Finally, we simulate and compare experimental data fits for the binding isotherms of Ca2+ binding to its canonical binding site (EF-hand domain) of polycystin 2, a Ca2+-dependent channel with relevance to polycystic kidney disease.

Differential Binding Models for Isothermal Titration Calorimetry: Moving beyond the Wiseman Isotherm

We present a set of model-independent differential equations to analyze isothermal titration calorimetry (ITC) experiments. In contrast with previous approaches that begin with specific assumptions about the number of binding sites and the interactions among them (e.g., sequential, independent, cooperative), our derivation makes more general assumptions, such that a receptor with multiple sites for one type of ligand species (homotropic binding) can be studied with the same analytical expression. Our approach is based on the binding polynomial formalism, and the resulting analytical expressions can be extended to account for any number of binding sites and any type of binding interaction among them. We refer to the set of model-independent differential equations to study ITC experiments as a differential binding model (DBM). To demonstrate the flexibility of our DBM, we present the analytical expressions to study receptors with one or two binding sites. The DBM for a receptor with one site is equivalent to the Wiseman isotherm but with a more intuitive representation that depends on the binding polynomial and the dimensionless parameter c = K·MT, where K is the binding constant and MT the total receptor concentration. In addition, we show how to constrain the general DBM for a receptor with two sites to represent sequential, independent, or cooperative binding interactions between the sites. We use the sequential binding model to study the binding interaction between Gd(III) and citrate anions. In addition, we simulate calorimetry titrations of receptors with positive, negative, and noncooperative interactions between the two binding sites. Finally, we derive a DBM for titrations of receptors with n-independent binding sites.