Cooperative binding: a multiple personality

Simplifications and approximations in a single-gene circuit modeling

The absence of detailed knowledge about regulatory interactions makes the use of phenomenological assumptions mandatory in cell biology modeling. Furthermore, the challenges associated with the analysis of these models compel the implementation of mathematical approximations. However, the constraints these methods introduce to biological interpretation are sometimes neglected. Consequently, understanding these restrictions is a very important task for systems biology modeling. In this article, we examine the impact of such simplifications, taking the case of a single-gene autoinhibitory circuit; however, our conclusions are not limited solely to this instance. We demonstrate that models grounded in the same biological assumptions but described at varying levels of detail can lead to different outcomes, that is, different and contradictory phenotypes or behaviors. Indeed, incorporating specific molecular processes like translation and elongation into the model can introduce instabilities and oscillations not seen when these processes are assumed to be instantaneous. Furthermore, incorporating a detailed description of promoter dynamics, usually described by a phenomenological regulatory function, can lead to instability, depending on the cooperative binding mechanism that is acting. Consequently, although the use of a regulating function facilitates model analysis, it may mask relevant aspects of the system's behavior. In particular, we observe that the two cooperative binding mechanisms, both compatible with the same sigmoidal function, can lead to different phenotypes, such as transcriptional oscillations with different oscillation frequencies.

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.

Discriminating between negative cooperativity and ligand binding to independent sites using pre-equilibrium properties of binding curves

Negative cooperativity is a phenomenon in which the binding of a first ligand or substrate molecule decreases the rate of subsequent binding. This definition is not exclusive to ligand-receptor binding, it holds whenever two or more molecules undergo two successive binding events. Negative cooperativity turns the binding curve more graded and cannot be distinguished from two independent and different binding events based on equilibrium measurements only. The need of kinetic data for this purpose was already reported. Here, we study the binding response as a function of the amount of ligand, at different times, from very early times since ligand is added and until equilibrium is reached. Over those binding curves measured at different times, we compute the dynamic range: the fold change required in input to elicit a change from 10 to 90% of maximum output, finding that it evolves in time differently and controlled by different parameters in the two situations that are identical in equilibrium. Deciphering which is the microscopic model that leads to a given binding curve adds understanding on the molecular mechanisms at play, and thus, is a valuable tool. The methods developed in this article were tested both with simulated and experimental data, showing to be robust to noise and experimental constraints.

Homogeneous Molecular Systems are Positively Cooperative, but Charged Molecular Systems are Negatively Cooperative

Molecular systems bound together through noncovalent interactions are ubiquitous in nature, many of which are involved in essential life processes, yet little is known about the principles governing their structure, stability, and function. Cooperativity as one of the intrinsic properties in these systems plays a key role. In this work, on the basis of our recent quantification scheme of the cooperativity effect, we present a general pattern to identify which systems are positively cooperative and which are negatively cooperative. We show that cooperativity in homogeneous molecular systems is positive, but cooperativity in charged molecular systems is negative. We also employ analytical tools from energetics and information perspectives to appreciate the origin of the cooperativity effect. We find that positive cooperativity is dominated by the exchange-correlation interaction and steric effect, whereas negative cooperativity is governed by the electrostatic interaction. Our results should have strong implications for better understanding molecular recognition, protein folding, signal transduction, allosteric regulation, and other processes.

Decoupled molecules with binding polynomials of bidegree (n, 2)

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2(n!)^{2}$$\end{document}2(n!)2 decoupled molecules to a generic binding polynomial. For molecules with more binding sites for the second ligand, we provide computational results.

Quantification and origin of cooperativity: insights from density functional reactivity theory

Cooperativity is a widely used chemical concept whose existence is ubiquitous in chemical and biological systems but whose quantification is still controversial and origin much less appreciated. In this work, using the interaction energy of a molecular system, which is composed of multiple copies of a building block, we propose a quantitative measurement to evaluate the cooperativity effect. This quantification approach is then applied to six molecular systems, i.e., water cluster, argon cluster, protonated water cluster, zinc atom cluster, water cluster on top of a graphene sheet, and alpha helix of glycine amino acids, each with up to 20 copies of the building block. Cooperativity is seen in all these systems. Both positive and negative cooperativity effects are observed. Employing the two energy partition schemes in density functional theory and the information-theoretic quantities such as Shannon entropy, Fisher information, information gain, etc., we then examine the origin of the cooperativity effect for these systems. Strong linear correlations between the cooperativity measure and some of these theoretical quantities have been unveiled. With these correlations, we are able to quantitatively account for their origin of cooperativity. Our results show that the interactions governing the existence and validity of the cooperativity effect are complicated. An opposite mechanism in enthalpy-entropy compensation for positive and negative cooperativity has been unveiled. These results should provide new insights and understandings from a different viewpoint about the nature and origin of cooperativity to appreciate this vastly important chemical concept.

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.

Uncovering the molecular and physiological processes of anticancer leads binding human serum albumin: A physical insight into drug efficacy

Human serum albumin (HSA) has its ability to bind drug molecules and influence their efficacies. Although anticancer leads NSC48693 and NSC290956 functioned at the same mechanism, the drug efficacies were obviously distinct. To gain insight into the distinct drug efficacy, the molecular and physiological processes of anticancer leads binding HSA have been investigated via a combined experimental and theoretical approach. The binding site, as characterized by fluorescence quenching and molecular modeling, is found to be located at site II in subdomain III A for NSC48693 with tight binding and at site FA1 in subdomain I B for NSC290956 with negatively cooperative binding, respectively. As indicated by the thermodynamic analysis, NSC48693 binds to HSA with an enthalpy driven mechanism, while NSC290956 binding with HSA is entropically driven. The further kinetic analysis indicates that the association rates appear to be similar to these two anticancer leads, however, the dissociation rate of NSC48693 is approximately 5-fold slower than that of NSC290956. For NSC48693, the pharmacodynamic efficacy is less than that of NSC290956, while its pharmacokinetic behavior is better than that of NSC290956. These parameters influence the pharmacodynamic efficacy and pharmacokinetic behavior, which will give further impacts on drug efficacy in vivo.

Cooperativity: a competition of definitions

Cooperativity as a concept is easy to grasp intuitively, but surprisingly hard to define. Two recent papers shed light on the issue and continue the debate on how best to define cooperative binding.

A Model for Carrier-Mediated Biological Signal Transduction Based on Equilibrium Ligand Binding Theory

Different variants of a mathematical model for carrier-mediated signal transduction are introduced with focus on the odor dose-electrophysiological response curve of insect olfaction. The latter offers a unique opportunity to observe experimentally the effect of an alteration in the carrier molecule composition on the signal molecule-dependent response curve. Our work highlights the role of involved carrier molecules, which have largely been ignored in mathematical models for response curves in the past. The resulting model explains how the involvement of more than one carrier molecule in signal molecule transport can cause dose-response curves as observed in experiments, without the need of more than one receptor per neuron. In particular, the model has the following features: (1) An extended sensitivity range of neuronal response is implemented by a system consisting of only one receptor but several carrier molecules with different affinities for the signal molecule. (2) Given that the sensitivity range is extended by the involvement of different carrier molecules, the model implies that a strong difference in the expression levels of the carrier molecules is absolutely essential for wide range responses. (3) Complex changes in dose-response curves which can be observed when the expression levels of carrier molecules are altered experimentally can be explained by interactions between different carrier molecules. The principles we demonstrate here for electrophysiological responses can also be applied to any other carrier-mediated biological signal transduction process. The presented concept provides a framework for modeling and statistical analysis of signal transduction processes if sufficient information on the underlying biology is available.

On Hill coefficients and subunit interaction energies

The study of cooperative ligand binding to multimeric proteins aims to explain complex cooperative binding phenomena using concepts derived from ideal binding isotherms. The purpose of such efforts is the dissection of the cooperative binding isotherm into its interacting components, a result with a clear mechanistic value. Historically, cooperative binding is usually quantified using the Hill coefficient, [Formula: see text], defined as the slope of the Hill plot at 50 % saturation. It was previously shown that the slope of the Hill plot throughout the titration is equal to the ratio of the binding variance in the system under study, to the binding variance of a reference non-interacting system. In the present contribution, this leads to a broader approach towards quantifying cooperativity, which empirically links cooperativity to the ensemble average of the subunit interaction energy. The resulting equations can be used to derive average differential subunit interaction energies directly from experimental binding isotherms. Combined with recent experimental advances in assessing binding distributions in multimeric proteins, these equations can also be used to calculate individual subunit interaction energies for specific n-ligated protein species.