Measuring Entropy in Molecular Recognition by Proteins

Abstract

Molecular recognition by proteins is fundamental to the molecular basis of biology. Dissection of the thermodynamic landscape governing protein-ligand interactions has proven difficult because determination of various entropic contributions is quite challenging. Nuclear magnetic resonance relaxation measurements, theory, and simulations suggest that conformational entropy can be accessed through a dynamical proxy. Here, we review the relationship between measures of fast side-chain motion and the underlying conformational entropy. The dynamical proxy reveals that the contribution of conformational entropy can range from highly favorable to highly unfavorable and demonstrates the potential of this key thermodynamic variable to modulate protein-ligand interactions. The dynamical so-called entropy meter also refines the role of solvent entropy and directly determines the loss in rotational-translational entropy that occurs upon formation of high-affinity complexes. The ability to quantify the roles of entropy through an entropy meter based on measurable dynamical properties promises to highlight its role in protein function.

Keywords: NMR relaxation; conformational entropy; molecular recognition; protein dynamics; rotational–translational entropy; solvation entropy.

Figure 1

Correlation of the apparent contribution of conformational entropy to the binding of target domains by calmodulin with total binding entropy. The apparent contribution reported by a model-dependent local interpretation of changes in methyl side-chain dynamics in calmodulin in complex with six different calmodulin-binding domains (19). (a) Summed histogram of the distribution of methyl group $O_{\text{axis}}^{\text{2}}$values of calmodulin in various complexes. (b) Correlation of the fractional population of the three classes of methyl-group motion with the total binding entropy measured by isothermal titration calorimetry for the six calmodulin complexes: J-class (red circles), α-class (green diamonds), and ω-class (blue squares). (c) Application of the oscillator inventory approach to assess the role of conformational entropy of calmodulin in the binding of calmodulin-binding domains. The harmonic oscillator potential was used to convert changes in methyl-group motion to estimates of changes in local side-chain entropy. The simple sum of apparent entropies is plotted against the total binding entropy. A The p at the end of the terms in panel c signifies that the various calmodulin-binding domains are represented by peptides. dapted from Reference . Abbreviations: CaMK1p, calmodulin-binding domain of calcium/calmodulin-dependent protein kinase1; CaMKKαp, calmodulin-binding domain of calcium/calmodulin-dependent protein kinase kinase alpha; eNOSp, calmodulin-binding domain of endothelial nitric oxide synthase; nNOSp, calmodulin-binding domain of neuronal nitric oxide synthase; PDEp, calmodulin-binding domain of phosphodiesterase; smMLCKp; calmodulin-binding domain of smooth muscle myosin light chain kinase.

Figure 2

Correlation of the methyl rotamer entropy versus$O_{\text{axis}}^{\text{2}}$ from molecular dynamics (MD) simulations. Normalized entropy S_b/Nχ given for every side-chain methyl probe. The correlation was highly linear (R² of 0.77) with a slope of −0.88 ± 0.03 and an intercept of 0.78 ± 0.02. The different proteins are color coded as indicated in the inset key. Modified from Reference , copyright American Chemical Society. Abbreviations: ADBP, porcine procarboxypeptidase B; CaM, calmodulin; k_B, Boltzmann constant; HEWL, hen egg white lysozyme; nNOSp, calmodulin-binding domain of neuronal nitric oxide synthase; Nχ, total number of side-chain torsion angles; S_b, rotamer entropy; smMLCKp, calmodulin binding domain of the smooth muscle myosin light chain kinase.

Figure 3

The dynamic proxy of methyl groups is an excellent reporter of both methyl and total side-chain rotameric entropy. (Blue circles) The normalized methyl rotameric entropy for each protein is calculated as the summation of S_b for individual methyl-bearing amino acids divided by the number of associated rotamer angles (Nχ). (Red circles) The total rotameric entropy for each protein is calculated as the summation of S_b for all residues and is normalized by the respective total number of rotamer angles (Nχ). The average methyl $O_{\text{axis}}^{\text{2}}$parameter for all methyl-bearing residues, including Ala, is that obtained from MD simulations. A very high linear correlation is observed for both methyl side-chain rotamer entropy (slope = −1.16 ± 0.17, R² = 0.90) and total rotamer entropy (slope = −0.74 ± 0.10, R² = 0.91), indicating that methyl-group motion is a good predictor of total side-chain conformational entropy. Adapted from Reference .

Figure 4

Calibration of the dynamical proxy for protein conformational entropy. Fitting of Equation 6 to data provided by 28 protein–ligand associations. The difference in the measured total binding entropy and calculated solvent entropy is plotted against the change in the dynamical proxy upon binding of ligands. The dynamical proxy is the average Lipari-Szabo squared generalized order parameter of methyl-group symmetry axes (<$O_{\text{axis}}^{\text{2}}$>). The fitted slope (s_d) of −4.8 ± 0.5 J mol⁻¹ K⁻¹ allows for the conversion between measured changes in methyl-bearing side-chain motion and the associated conformational entropy. Other parameters of the entropy meter are summarized in Table 1. The CaM (calmodulin) and CAP (calmodulin-associated peptide) data subsets are shown in blue and green, respectively. Purple circles represent other binary complexes, as summarized elsewhere (7). The orange square represents the HBP(D24R)–histamine binary complex binding to serotonin, which has a dissociation constant in the millimolar range and was not used in the calibration. See text for details. Adapted from Reference .

Figure 5

Contribution of protein conformational entropy to the free energy of ligand binding to proteins. The broad range of contributions available to proteins for high-affinity binding of ligands is illustrated by the protein–ligand complexes used to calibrate the entropy meter (Figure 4). The 28 protein–ligand complexes are arranged in descending order of the contribution of conformational entropy (red bars) to the total free energy of binding (blue bars). Conformational entropy contributed by the response of amino acid side chains to the binding of a ligand can vary from highly unfavorable to negligible to highly favorable. In some cases, conformational entropy is essential for high-affinity binding. The structural origins of the variable utilization of conformational entropy in molecular recognition are unknown. In most cases, the change in solvent entropy remains a dominant contribution. Note that −TΔS_r–t, ΔS_ligand, and ΔS_solvent are not shown here. Adapted from Reference