Chapter 5 Isothermal Titration Calorimetry: General Formalism Using Binding Polynomials

doi:10.1016/S0076-6879(08)04205-5

Methods in Enzymology

Volume 455, 2009, Pages 127-155

https://doi.org/10.1016/S0076-6879(08)04205-5 Get rights and content

Abstract

The theory of the binding polynomial constitutes a very powerful formalism by which many experimental biological systems involving ligand binding can be analyzed under a unified framework. The analysis of isothermal titration calorimetry (ITC) data for systems possessing more than one binding site has been cumbersome because it required the user to develop a binding model to fit the data. Furthermore, in many instances, different binding models give rise to identical binding isotherms, making it impossible to discriminate binding mechanisms using binding data alone. One of the main advantages of the binding polynomials is that experimental data can be analyzed by employing a general model‐free methodology that provides essential information about the system behavior (e.g., whether there exists binding cooperativity, whether the cooperativity is positive or negative, and the magnitude of the cooperative energy). Data analysis utilizing binding polynomials yields a set of binding association constants and enthalpy values that conserve their validity after the correct model has been determined. In fact, once the correct model is validated, the binding polynomial parameters can be immediately translated into the model specific constants. In this chapter, we describe the general binding polynomial formalism and provide specific theoretical and experimental examples of its application to isothermal titration calorimetry.

Introduction

The introduction of the binding polynomial theory several decades ago by Jeffries Wyman provided a general statistical thermodynamic framework for studying ligand binding to macromolecules (Wyman, 1948, Wyman, 1964, Wyman and Gill, 1990). Being equivalent to a partition function, the binding polynomial contains all the information about the system and allows derivation of all thermodynamic experimental observables (e.g., average number of ligand molecules bound, average excess enthalpy). Contrary to model‐dependent parameters, the parameters that define the binding polynomial have a general validity. Consequently, unless a binding model has been validated, the binding polynomial should be the preferred starting point for the analysis of complex binding situations. There are experimental situations that cannot be assigned to a particular model. For two or more binding sites, different binding models can give rise to mathematically equal binding equations. In those cases, the discrimination between models cannot be made on the basis of binding data alone and requires extrathermodynamic arguments.

The binding polynomial represents the basis for a general, model‐independent analysis of a binding experiment. The same methodology is applicable to a system with one or any arbitrary number of binding sites without the user having to decide on any particular binding mechanism. It is the preferred analysis protocol unless a specific binding model has been validated for the system under consideration.

Among the experimental techniques employed for studying ligand binding, isothermal titration calorimetry (ITC) exhibits several features that render it a unique experimental tool: (1) the signal measured (heat of reaction) is a universal probe, avoiding the use of nonnatural spectroscopic labels; (2) the interacting molecules are in solution; and (3) it allows determining simultaneously the association constant, the enthalpy, and the stoichiometry of binding in a single experiment. Accordingly, ITC data is ideally suited to be analyzed using the binding polynomial formalism. This chapter discusses the theoretical basis of the binding polynomials and their application to the analysis of ITC data.

Section snippets

The Binding Polynomial

The equilibrium of a ligand with a macromolecule with n ligand binding sites can be described in terms of two different sets of association constants, the overall association constants, β_i, or, alternatively the stepwise association constants, K_i: $\begin{array}{l} M + iL \to {ML}_{i} & β_{i} = \frac{[{ML}_{i}]}{[M] {[L]}^{i}} \\ {ML}_{i - 1} + L \to {ML}_{i} & K_{i} = \frac{[{ML}_{i}]}{[{ML}_{i - 1}] [L]} \end{array},$ The two sets of descriptors are equivalent, and they are related through the following relationships: $\begin{array}{l} β_{i} & = \prod_{j = 1}^{i} K_{j} \\ K_{i} & = \frac{β_{i}}{β_{i - 1}} \end{array} .$ Because the stepwise binding constants and the overall association constants

Microscopic Constants and Cooperativity

The overall and stepwise association constants are macroscopic association constants, and no mechanistic interpretation about the ligand binding can be inferred from them. Therefore, they are considered phenomenological or model‐free association constants. Besides macroscopic association constants, there is another type of association constants, the microscopic binding constants k_i, related intrinsically with the ligand binding to the different binding sites, and therefore reflecting the

Independent or Cooperative Binding?

In the analysis of systems with two or more binding sites, one of the most important questions is to assess whether the sites are independent of each other or whether cooperative interactions affect the ligand affinity of different sites. A qualitative analysis can be performed in a straightforward way once the overall association constants have been determined. For a macromolecule with n binding sites, a set of n‐1 parameters, ρ_i (i = 2, … , n), can be calculated from the macroscopic

Analysis of ITC Data Using Binding Polynomials

The binding polynomial ((5.5), (5.6)) provides the starting point in the analysis of ITC data. As in the standard analysis, the total concentration of ligand is written as the sum of the concentrations of free and bound ligand, and expressed in terms of the binding polynomial: ${[L]}_{T} = [L] + {[L]}_{B} = [L] + {[M]}_{T} n_{LB} = [L] + {[M]}_{T} \frac{\partial ln P}{\partial ln [L]} .$ Eq. (5.13) is the basis for the analysis of the binding experiment; knowing the total concentrations of ligand and macromolecule, the values of the macroscopic association

A Typical Case: Macromolecule with Two Ligand‐Binding Sites

The binding polynomial for a macromolecule with two ligand binding sites is equal to (see Fig. 5.1): $P = 1 + β_{1} [L] + β_{2} {[L]}^{2} .$ The average number of ligand molecules bound per macromolecule, n_LB, and the average excess molar enthalpy, <ΔH>, are written in terms of the macroscopic association constants and binding enthalpies as follows: $\begin{array}{l} n_{LB} & = \frac{β_{1} [L] + 2 β_{2} {[L]}^{2}}{1 + β_{1} [L] + β_{2} {[L]}^{2}} = F_{1} + 2 F_{2} \\ 〈 Δ H 〉 & = \frac{β_{1} Δ H_{1} + β_{2} {[L]}^{2} Δ H_{2}}{1 + β_{1} [L] + β_{2} {[L]}^{2}} = F_{1} Δ H_{1} + F_{2} Δ H_{2} \end{array} .$ These expressions are completely general for any macromolecule with two ligand‐binding

Data Analysis

The equations described here have been implemented for an arbitrary number of binding sites in the analysis software distributed by manufacturers of isothermal titration calorimeters or can be employed as user‐defined fitting functions using commercially available software. Sometimes, stepwise association constants are estimated (e.g., MicroCal) rather than the overall association constants discussed here. In those cases, Eq. (5.2) should be used to calculate them. Thermodynamic binding

Data Interpretation

As discussed earlier, the precise binding mechanism for systems with two or more binding sites is difficult to derive and usually requires extrathermodynamic information. Statistical fitting of the data to a model does not validate the appropriateness of the model. Even for hemoglobin, the most widely studied binding system in history, there are still lingering questions about the exact oxygen‐binding mechanism (Holt and Ackers, 2005, Ackers and Holt, 2006).

Even a macromolecule with two binding

An Experimental Example

The binding of ferric ions to ovotransferrin is an example of a reaction that involves two sites. Fig. 5.5 shows a microcalorimetric titration of ovotransferrin with ferric ions, chelated with nitrilotriacetate, at 25 °C, 100 mM HEPES, pH 7.5. Nonlinear least squares analysis of the experimental data using the binding polynomials formalism yields: β₁ = 1.7·10⁶M⁻¹, ΔH₁ = −7.4 kcal/mol, β₂ = 1.1·10¹²M⁻² and ΔH₂ = −12.2 kcal/mol. The calculated value for ρ₂ is 1.5, suggesting positive

Experimental Situations from the Literature

Other examples of ITC studies of macromolecules with two ligand‐binding sites can be found in the literature. Several illustrative cases corresponding to different scenarios have been selected and are shown in Fig. 5.6. The published experimental titration data were digitally extracted and analyzed following the procedure outlined in this work.

In Fig. 5.6A, RSL is a fucose‐binding lectin from Ralstonia solanacearum, a gram‐negative β‐protobacterium causing lethal wilt in plants. RSL is a

Macromolecule with Three Ligand‐Binding Sites

As an additional example, three different possible models for a macromolecule with three ligand‐binding sites are illustrated in Fig. 5.7. In this case, a set of two parameters, ρ₂ and ρ₃ (see Eq. (5.11)), provides information about the behavior of the binding sites in the macromolecule.

The data analysis based on the binding polynomial formalism using overall association constants is fairly simple and straightforward. It requires solving a (n + 1)th‐order polynomial equation on the free ligand

Conclusions

The binding polynomial provides a general framework for describing ligand‐binding equilibria to biological macromolecules using tools from statistical thermodynamics. The methodology can be easily applied to nearly any kind of system. Binding experiments can be analyzed using a model‐free methodology, which allows determination of phenomenological overall macroscopic association constants, β_i, and binding enthalpies, ΔH_i. The particular values of a set of parameters ρ_i, provide information on

Acknowledgment

We acknowledge financial support from grant SAF2004‐07722 (Ministry of Education and Science) to A.V.‐C., and grants from the National Institutes of Health (GM56550 and GM57144) and the National Science Foundation (MCB0641252) to E.F. A.V.‐C. was supported by a Ramon y Cajal Research Contract from the Spanish Ministry of Science and Technology, and Fundación Aragón I+D (Diputación General de Aragón).

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Chapter 5 Isothermal Titration Calorimetry: General Formalism Using Binding Polynomials

Abstract

Introduction

Section snippets

The Binding Polynomial

Microscopic Constants and Cooperativity

Independent or Cooperative Binding?

Analysis of ITC Data Using Binding Polynomials

A Typical Case: Macromolecule with Two Ligand‐Binding Sites

Data Analysis

Data Interpretation

An Experimental Example

Experimental Situations from the Literature

Macromolecule with Three Ligand‐Binding Sites

Conclusions

Acknowledgment

J. Biol. Chem.

J. Biol. Chem.

J. Mol. Biol.

J. Biol. Chem.

J. Biol. Chem.

J. Mol. Biol.

Biophys. J.

Adv. Prot. Chem.

Adv. Prot. Chem.

Binding capacity: Cooperativity and buffering in biopolymers

Proc. Natl. Acad. Sci. USA