Extent of enthalpy-entropy compensation in protein-ligand interactions

Abstract

The extent of enthalpy-entropy compensation in protein-ligand interactions has long been disputed because negatively correlated enthalpy (ΔH) and entropy (TΔS) changes can arise from constraints imposed by experimental and analytical procedures as well as through a physical compensation mechanism. To distinguish these possibilities, we have created quantitative models of the effects of experimental constraints on isothermal titration calorimetry (ITC) measurements. These constraints are found to obscure any compensation that may be present in common data representations and regression analyses (e.g., in ΔH vs. -TΔS plots). However, transforming the thermodynamic data into ΔΔ-plots of the differences between all pairs of ligands that bind each protein diminishes the influence of experimental constraints and representational bias. Statistical analysis of data from 32 diverse proteins shows a significant and widespread tendency to compensation. ΔΔH versus ΔΔG plots reveal a wide variation in the extent of compensation for different ligand modifications. While strong compensation (ΔΔH and -TΔΔS opposed and differing by < 20% in magnitude) is observed for 22% of modifications (twice that expected without compensation), 15% of modifications result in reinforcement (ΔΔH and -TΔΔS of the same sign). Because both enthalpy and entropy changes arise from changes to the distribution of energy states on binding, there is a general theoretical expectation of compensated behavior. However, prior theoretical studies have focussed on explaining a stronger tendency to compensation than actually found here. These results, showing strong but imperfect compensation, will act as a benchmark for future theoretical models of the thermodynamic consequences of ligand modification.

Figure 1

Thermodynamic parameters derived from ITC experiments for 171 protein–ligand interactions. The limitations of experimental procedures constrain measurements to the affinity window bounded by the dashed lines.

Figure 2

The effect of the affinity window and other experimental factors can be modeled using information from experimental measurements. (A) The distribution of all direct ITC measurements of ΔG reported in the PDBcal and SCORPIO databases is used to estimate the underlying probability of an experiment being successfully performed (bold line). This probability forms a basis for modeling the effects of the affinity window. (B) and (C) Experimental distributions of ΔH and −TΔS values found in the experimental dataset in Figure 1. Values for ΔH and −TΔS from the experimental dataset can be independently randomly sampled and then accepted with the probability defined in (A) to create a model distribution illustrating the effect of experimental constraints (D) on an otherwise random sample.

Figure 3

Illustrative models of the effect of experimental constraints and full/exact compensation on otherwise random distributions of the thermodynamic differences between ligands. (A), (B), and (C) Experimental distributions of ΔG for ligands and ΔΔH and −TΔΔS for pairs of ligands that bind the same protein. These can be sampled (see Methods section) to generate a model of unconstrained random changes in enthalpy and entropy (D and G), of the effect of experimental constraints of the affinity window plus correlated error (E and H), and of full compensation with measurement errors in (F and I). The ellipses surround 75% of the experimental datapoints ordered by the differences from the mean of their component coordinates.

Figure 4

The experimental relative thermodynamics of all pairs of ligands binding each protein have features consistent with the presence of enthalpy–entropy compensation. (A) ΔΔH versus −TΔΔS plot of all experimental differences for pairs of ligands binding to each protein (open circles) superimposed on a sample drawn from a constrained random model (points). The ellipses surrounds 75% of the experimental data ordered by difference from the mean (grey = experimental data, black = model). The experimental distribution is narrower than would be expected in the absence of compensation. (B) An alternative ΔΔH versus ΔΔG plot of the same data. The experimental data are closer in average behavior to that expected for an underlying compensation mechanism, but not equivalent to the fully compensated model (cf. Fig. 3I).

Figure 5

Samples of the experimental data are significantly different from an uncompensated constrained random model and show a strong tendency toward compensation. (A) The compensation coefficient θ relating ΔΔH and −TΔΔS of pairs of ligands is more strongly peaked for a sample drawn from the experimental data (histogram) around the values of −π/4 and 3π/4 radians, which correspond to exact enthalpy–entropy compensation, than the model (black line). (C) The experimental values of the compensation coefficient ϕ relating ΔΔH and ΔΔG are most likely to occur around the values of ±π/2 radians, which correspond to exact enthalpy–entropy compensation, unlike the model. The probability distribution functions of the Kuiper statistic for θ (B) and ϕ (D) generated from the constrained random model (histograms) with arrows indicating the values for four samples of the experimental data (grey arrows are the values of the samples in (A) and (C)). All experimental samples lie in the tails of the functions and are significantly different (e.g., the samples in (A) and (C) give P < 0.002 and P < 0.001, respectively).

Figure 6

The extent of compensation in protein–ligand interactions varies substantially within protein systems. (A) A schematic relating different regions of the ΔΔH versus ΔΔG plot to different degrees of compensation (10% compensation occurs where ΔΔH = 0.1 * TΔΔS and vice versa). Reinforcement occurs when ΔΔH and −TΔΔS have the same sign. (B) The experimental data from 674 ligand modifications in 32 protein systems (grey open circles) show an overall tendency toward higher degrees of compensation (68% are compensated to better than 10%, 22% to better than 80% − more than twofold greater than random). The experimental distribution is similar to a model created assuming an intrinsic correlation = −0.91 between enthalpy and entropy changes (black points). (C) Normalized frequency distributions of the compensation coefficient ϕ for all pairs of ligands for each of the 32 proteins illustrate the strong tendency to compensation compared to the expectation of a constrained random model and also the wide variation of degree of compensation observed experimentally. Each horizontal line represents a protein (or group of proteins in the bottom two cases), where black (100%) to white (0%) shading represents the proportion of ligands in each interval.