A Comprehensive Mathematical Model for Three-Body Binding Equilibria
Journal of the American Chemical Society (Impact Factor: 12.11). 04/2013; 135(16). DOI: 10.1021/ja311795d
Three-component systems are often more complex than their two-component counterparts. Although the reversible association of three components in solution is critical for a vast array of chemical and biological processes, no general physical picture of such systems has emerged. Here we have developed a general, comprehensive framework for understanding ternary complex equilibria, which relates directly to familiar concepts such as "EC50" and "IC50" from simpler (binary complex) equilibria. Importantly, application of our model to data from the published literature has enabled us to achieve new insights into complex systems ranging from coagulation to therapeutic dosing regimens for monoclonal antibodies. We also provide an Excel spreadsheet to assist readers in both conceptualizing and applying our models. Overall, our analysis has the potential to render complex three-component systems - which have previously been characterized as "analytically intractable" - readily comprehensible to theoreticians and experimentalists alike.
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ABSTRACT: 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.
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ABSTRACT: The cell interior is a complex and demanding environment. An incredible variety of molecules jockey to identify the correct position – the specific interactions that promote biology, which are often hidden among countless unproductive options. Ensuring that the business of the cell is successful requires sophisticated mechanisms to impose temporal and spatial specificity – both on transient interactions and their eventual outcomes. Two strategies employed to regulate macromolecular interactions in a cellular context are colocalization and compartmentalization. Macromolecular interactions can be promoted and specified by localizing the partners within the same subcellular compartment, or by holding them in proximity through covalent or non-covalent interactions with proteins, lipids, or DNA – themes that are familiar to any biologist. The net result of these strategies is an increase in effective molarity: the local concentration of a reactive molecule near its reaction partners. We will focus on this general mechanism, employed by nature and adapted in the lab, which allows delicate control in complex environments: the power of proximity to accelerate, guide, or otherwise influence the reactivity of signaling proteins and the information that they encode.
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ABSTRACT: In colorectal cancer cells, APC, a tumor suppressor protein, is commonly expressed in truncated form. Truncation of APC is believed to disrupt degradation of β-catenin[Formula: see text], which is regulated by a multiprotein complex called the destruction complex. The destruction complex comprises APC, Axin, β-catenin[Formula: see text], serine/threonine kinases, and other proteins. The kinases [Formula: see text] and [Formula: see text], which are recruited by Axin, mediate phosphorylation of β-catenin[Formula: see text], which initiates its ubiquitination and proteosomal degradation. The mechanism of regulation of β-catenin[Formula: see text] degradation by the destruction complex and the role of truncation of APC in colorectal cancer are not entirely understood. Through formulation and analysis of a rule-based computational model, we investigated the regulation of β-catenin[Formula: see text] phosphorylation and degradation by APC and the effect of APC truncation on function of the destruction complex. The model integrates available mechanistic knowledge about site-specific interactions and phosphorylation of destruction complex components and is consistent with an array of published data. We find that the phosphorylated truncated form of APC can outcompete Axin for binding to β-catenin[Formula: see text], provided that Axin is limiting, and thereby sequester β-catenin[Formula: see text] away from Axin and the Axin-recruited kinases [Formula: see text] and [Formula: see text]. Full-length APC also competes with Axin for binding to β-catenin[Formula: see text]; however, full-length APC is able, through its SAMP repeats, which bind Axin and which are missing in truncated oncogenic forms of APC, to bring β-catenin[Formula: see text] into indirect association with Axin and Axin-recruited kinases. Because our model indicates that the positive effects of truncated APC on β-catenin[Formula: see text] levels depend on phosphorylation of APC, at the first 20-amino acid repeat, and because phosphorylation of this site is mediated by [Formula: see text], we suggest that [Formula: see text] is a potential target for therapeutic intervention in colorectal cancer. Specific inhibition of [Formula: see text] is predicted to limit binding of β-catenin[Formula: see text] to truncated APC and thereby to reverse the effect of APC truncation.
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