Differential Binding Models for Isothermal Titration Calorimetry: Moving beyond the Wiseman Isotherm Isaac Herrera and Mitchell A. Winnik* Department of Chemistry, University of Toronto, 80 St. George Street, Toronto ON Canada M5S 3H6 * S Supporting Information ABSTRACT: We present a set of model-independent differ- ential 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·M T , where K is the binding constant and M T 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. 1. INTRODUCTION Isothermal titration calorimetry (ITC) is an analytical technique to measure the heat of reaction from a binding process between two or more species in solution. 1,2 Similar to other titrimetric methods, the experiment is carried out by injecting a small volume of titrant into an analyte solution while a physical property of the system “signals” the binding event. In ITC, the signal measured at each injection step is proportional to the heat of reaction, a thermodynamic property. Therefore, ITC allows for the direct determination of the enthalpy (ΔH), the free energy (ΔG), and the entropy (ΔS) of the binding process, in addition to the stoichiometry (n) and the binding constant (K), giving a complete thermodynamic and quantitative characterization of the system. There are several reviews and reports about the characterization of receptor- ligand interactions using ITC, most of which are aimed at the characterization of biologically relevant receptors such as proteins, 3 polysaccharides, 4 polynucleotides, 5 or amphiphilic 6 species with their respective ligand counterparts. In addition, the major developments in ITC are reviewed on a yearly basis, 7-9 which points to the growth of the field during the past decade. A crucial step in ITC is to analyze the binding data with a mathematical model that describes (i) the binding mechanism and (ii) the relation between the heat of reaction and the amount of titrant added. Commonly, the analysis of ITC experiments is carried out using the models embedded in the analysis software of the calorimeter. These built-in models can be applied to simple binding systems, but it is challenging to apply them to receptors with multiple binding sites. 10 There are some alternatives to analyze ITC experiments in the form of commercial software, 11 free software, 12-16 or mathematical models. 17-20 However, the binding models in these reports were constrained to represent receptors with a finite number of binding sites and specific binding interactions among the sites. For this purpose, we derive a general set of differential equations to analyze ITC experiments which can be applied to receptors with multiple binding sites and different binding interactions among the sites. The differential equations have the form df/dx, where the term f is a variable related to the binding species in solution and the term x is an independent variable related to the progress of the titration experiment. We refer to this type of differential equations to study ITC experiments as differential binding models (DBMs). The Wiseman isotherm (WI) 1 is the most well-known DBM, and it has been extensively applied to ligand-into-receptor Received: November 30, 2012 Revised: April 24, 2013 Published: July 10, 2013 Article pubs.acs.org/JPCB © 2013 American Chemical Society 8659 dx.doi.org/10.1021/jp311812a | J. Phys. Chem. B 2013, 117, 8659-8672