Chapter 5 Isothermal Titration Calorimetry: General Formalism Using Binding Polynomials
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, βi, or, alternatively the stepwise association constants, Ki:The two sets of descriptors are equivalent, and they are related through the following relationships: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 ki, 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, ρi (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: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):The average number of ligand molecules bound per macromolecule, nLB, and the average excess molar enthalpy, <ΔH>, are written in terms of the macroscopic association constants and binding enthalpies as follows: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, βi, and binding enthalpies, ΔHi. The particular values of a set of parameters ρi, 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).
References (21)
- et al.
Asymmetric cooperativity in a symmetric tetramer: Human hemoglobin
J. Biol. Chem.
(2006) The hemoglobin system, VI: The oxygen dissociation curve of hemoglobin
J. Biol. Chem.
(1925)- et al.
Thermodynamic characterization of binding Oxytricha nova single strand telomere DNA with the alpha protein n‐terminal domain
J. Mol. Biol.
(2006) - et al.
Thermodynamics of cyclic nucleotide binding to the cAMP receptor protein and its T127L mutant
J. Biol. Chem.
(1995) - et al.
The Fucose‐binding lectinfrom Ralstonia solanacearum. A new type of β‐propeller architecture formed by oligomerization and interacting with fucoside, fucosyllactose, and plant xyloglucan
J. Biol. Chem.
(2005) - et al.
Optimization of the palindromic order of the TtgR operator enhances binding cooperativity
J. Mol. Biol.
(2007) - et al.
Exact analysis of heterotropic interactions in proteins: Characterization of cooperative ligand binding by isothermal titration calorimetry
Biophys. J.
(2006) Heme proteins
Adv. Prot. Chem.
(1948)Linked functions and reciprocal effects in hemoglobin: A second look
Adv. Prot. Chem.
(1964)- et al.
Binding capacity: Cooperativity and buffering in biopolymers
Proc. Natl. Acad. Sci. USA
(1988)
Cited by (139)
Molecular insight into how the position of an abasic site modifies DNA duplex stability and dynamics
2024, Biophysical JournalPITB: A high affinity transthyretin aggregation inhibitor with optimal pharmacokinetic properties
2023, European Journal of Medicinal ChemistryThermodynamics and kinetics of DNA and RNA dinucleotide hybridization to gaps and overhangs
2023, Biophysical JournalUnexpected thermodynamic signature for the interaction of hydroxymethylated DNA with MeCP2
2023, International Journal of Biological Macromolecules