## Introduction

The interactions between proteins and their ligands can be characterized by the free energy, enthalpy, and entropy changes associated with the binding reaction. The Gibb's free energy (Δ*G*) for a reaction carried out at a temperature *T* is related to the change in enthalpy (Δ*H*) and the change in entropy (Δ*S*) of that reaction by

It is often observed that the range of Δ*G* values for groups of related reactions is much smaller than the ranges of their associated changes in Δ*H* and *T*Δ*S*. This has led to the idea that the differences in the enthalpic and entropic contributions are negatively correlated or “compensated” as a result of some shared features of the physical reaction mechanism.1 However, it is also known that negative correlations of enthalpy and entropy changes arise from experiment design, random measurement errors, or from the methods of analysis of measurements.1–3 Claims for enthalpy–entropy compensation in protein–ligand interactions often fail to examine other possible source of correlation and in cases where statistical analyses have been performed, the observability of compensation has remained in doubt.4–6 In addition to being an issue of fundamental interest, it remains important to establish the presence (and extent if present) of compensation in experimental studies of protein–ligand binding because such data would impinge on the assessment of models of molecular interaction and may affect how thermodynamic information is used in rational drug design.7 Consequently, here we revaluate the evidence for enthalpy–entropy compensation in protein–ligand interactions using the large quantity of isothermal titration calorimetry (ITC) data that has been produced in recent years.

Much of the historical difficulty in investigating, and consequent controversy with respect to the existence/extent of, compensation in protein reactions has its origin in the dominant early role played by van't Hoff (and Arrhenius) analyses. Until relatively recently, the simplest way to find the enthalpy and entropy changes of a reaction was via a plot of the logarithm of the equilibrium constant *K* (= e^{−ΔG/RT}) against the reciprocal of the temperature. Rearranging (1) gives the van't Hoff equation describing such a plot.

The slope of the line is −Δ*H* and the intercept is Δ*S* (divided by the gas constant). However, this approach introduces relatively large errors in Δ*H* compared to the magnitude of ΔG. Because errors in the slope are correlated with errors in the intercept, errors alone can produce highly correlated changes in Δ*H* and Δ*S* for a series of reactions.2, 3 Statistical tests have been proposed to discriminate cases of compensation from these artefactual correlations.4, 8, 9 Using such tests, it was found that many reported instances of high correlation between Δ*H* and ΔS for a variety of chemical reactions are indistinguishable from experimental artefacts,8 including several examples of the interactions of individual proteins with series of ligands.4–6

ITC measures the Δ*H* of a binding reaction directly through the heat output or input associated with a titrated reaction at constant temperature and Δ*G* is found from a nonlinear regression analysis of the titration curve.10 Unlike a van't Hoff analysis, these measurements are essentially independent and usually precise (e.g., mean reported errors for Δ*H* and ΔG are 1.5 and 0.5 kJ mol^{−1}, respectively, in the SCORPIO database11 of ITC data, and 1.7 and 0.4 kJ mol^{−1} in a recent systematic analysis of replicated experiments on many protein–ligand systems12). Consequently, enthalpy–entropy correlation arising from measurement errors, which in the case of ITC results from the use of Eq. (1) to determine *T*Δ*S*, is much smaller than that for a van't Hoff analysis. Indeed, the precision of ITC measurements is such that it has again become common to assume that statistical testing is unnecessary and that a high-degree correlation in a Δ*H* versus *T*Δ*S* plot alone is sufficient evidence for compensation.13–15

Unfortunately, there are several sources of potential correlation in ITC data, which must be eliminated or accounted for in any analysis. In addition to the small correlation due to measurement errors, Cooper *et al*.16 have pointed out that the range of Δ*G* values that are accurately measurable using the most common direct ITC method is limited by the necessity to obtain an analyzable sigmoidal titration curve within the constraints of protein solubility and instrument sensitivity. This “affinity window” is narrower for direct ITC measurements than that for many other methods for monitoring binding and thus poses a particular problem for rigorous analysis of compensation. In addition, correlation can arise from “extra-experimental” factors, that is, biases in the nature of system that are selected for study.4 For example, interactions with cognate ligands are constrained in their affinity because they are usually required to be reversible and have a significant bound population at biological concentrations.4, 16 Also, studies of protein with synthetic ligands often involve a series of similar changes being made to the ligand. These may each result in similar changes to Δ*H* and *T*Δ*S* and introduce confounding correlations into the data.4, 17 As a consequence of these issues, careful data selection and statistical analysis of the effects of errors and experimental factors are required for analysis of enthalpy–entropy relationships in ITC data.

Here, we combine ITC data from many proteins to investigate whether compensation is an observable feature of protein–ligand interactions. In selecting data from a wide range of systems, we minimize the potential for extra-experimental chemical biases affecting our conclusions. To enable statistical testing, we create models of the correlation expected to arise as a result of errors and the ITC affinity window; putting previous qualitative arguments16 about these factors on a quantitative footing. We show that these experimental sources of correlation are so large in traditional Δ*H* versus *T*Δ*S* plots as to render them of no use in identifying compensation, reinforcing earlier analyses.6, 18 However, we find that a new approach based on analysis of the distributions of the relative thermodynamic values (ΔΔ*H*, *T*ΔΔ*S*, and ΔΔG) of all pairs of ligands that bind to each protein does allow the effects of enthalpy–entropy compensation to be distinguished from other sources of correlation. Our analysis shows that there is significant, widespread and strong tendency to enthalpy–entropy compensation in protein–ligand interactions. However, it is also clear that there is a range of degrees of compensation observed within protein–ligand systems. We reconsider previously suggested theoretical models for compensation and conclude that the prior theoretical emphasis on explaining perfect compensation is unwarranted by/inconsistent with the experimental data and that improved theories are needed to explain the varied extent of compensation actually observed.