Tzeng and Kalodimos (Nature 488, 236-240; 2012) showed that variants of the Escherichia coli catabolite activator protein (CAP) bind DNA with markedly different affinities despite essentially identical binding interfaces, and that the difference is carried by conformational entropy: across the variant series, the conformational entropy change derived from NMR methyl order parameters, −TΔSconf-T\Delta S_{\mathrm{conf}}, scales near-linearly with the calorimetric total entropy change, −TΔS-T\Delta S - an empirical relation that has not been derived from microscopic theory. The microscopic basis of this linearity, the atomic-resolution determinants of the observed enthalpy-entropy compensation, and the assumed invariance of non-conformational (solvation and translational/rotational) entropy contributions remain unresolved. The problem is to supply a microscopic, quantitative account of the entropic decomposition of ligand binding that explains the CAP phenomenology. Later work has validated and broadened the NMR dynamics-to-entropy calibration, demonstrated entropy-entropy compensation involving solvent in other systems, and contested the pure-conformational interpretation of the CAP entropies, but no derivation of the linear scaling or settled partition of the CAP entropy changes exists, so the core remains open.

Background

Allostery is the regulation of a protein's activity at one site by the binding of a molecule at a distant site; classically it was explained by structural (induced-fit or conformational-selection) changes, but it can also be communicated purely through changes in protein dynamics. The catabolite activator protein (CAP, also called CRP) is an Escherichia coli transcription factor and a paradigm for such dynamically driven allostery: binding of the signaling ligand cyclic AMP (cAMP) at a regulatory domain controls DNA binding at a distant domain, and Popovych, Sun, Ebright and Kalodimos (Nat. Struct. Mol. Biol. 13, 831-838; 2006) showed that the negative cooperativity between the two cAMP sites of the CAP dimer occurs with essentially no structural change, implying a predominantly entropic mechanism. Tzeng and Kalodimos (Nature 462, 368-372; 2009) then showed that cAMP activation of CAP proceeds through shifts in a slow (microsecond-millisecond) conformational equilibrium among excited states.

Conformational entropy denotes the entropy stored in the ensemble of a protein's internal conformations; it changes when binding alters the amplitude of fast internal motions on the picosecond-nanosecond timescale. Such amplitudes are measured by nuclear magnetic resonance (NMR) relaxation experiments, which yield generalized order parameters (S2S^2, ranging from 0 for a fully mobile bond vector to 1 for a rigid one), typically read out at methyl-bearing side chains. Frederick, Marlow, Valentine and Wand (Nature 448, 325-329; 2007) calibrated an empirical 'entropy meter': a roughly linear map from changes in methyl order parameters to changes in conformational entropy, derived from molecular dynamics simulations, which lets NMR dynamics report the conformational-entropy contribution to binding.

Tzeng and Kalodimos (Nature 488, 236-240; 2012) combined this NMR proxy with isothermal titration calorimetry (ITC, which measures the binding enthalpy ΔH\Delta H and, with the affinity, the Gibbs energy ΔG\Delta G and total entropy change −TΔS-T\Delta S) for a series of CAP variants binding the same DNA duplex. Although the CAP-DNA interfaces in the complexes were essentially identical and the binding free energies spanned only a few kJ/mol, the enthalpies ranged from about −92-92 to +44+44 kJ/mol with compensating entropy factors from about −54-54 to +79+79 kJ/mol - a dramatic instance of enthalpy-entropy compensation, the widespread tendency of ΔH\Delta H and −TΔS-T\Delta S changes to offset each other. Across these variants the NMR-derived −TΔSconf-T\Delta S_{\mathrm{conf}} correlated near-linearly with the calorimetric −TΔS-T\Delta S, which the authors used to argue that conformational entropy dominates the differences; the microscopic basis of this linearity has not been derived, and a closely related interpretive caveat is documented for a calmodulin-peptide system by Frederick et al. (2007): such a linear correlation implies either that the conformational-entropy change is a major contribution to the binding entropy or that the various sources of entropy change in concert. The linear interpretation presupposes that the remaining contributions - solvation entropy (release and ordering of water and ions) and the translational/rotational entropy lost on complex formation - are effectively invariant across variants; the 2012 analysis supported this by inferring the solvation contribution from the difference between ITC measurements in H2O\mathrm{H_2O} and D2O\mathrm{D_2O}, which was found to be similar across the variants, so that −TΔS-T\Delta S denotes the calorimetric total entropy change throughout. Together the fast-dynamics (conformational-entropy) channel and the slow-dynamics (excited-state population) channel form a two-timescale mechanism by which CAP activity is regulated without change in the ground-state structure.

The unresolved issue is the microscopic foundation of this phenomenology: why the dynamics-derived conformational entropy should track the total binding entropy linearly, what sets the magnitude of compensation in each complex, and how the total entropy is actually partitioned among conformational, solvation, and translational/rotational terms.

Problem Statement

Provide a microscopic, quantitative account of the entropic component of protein-ligand binding thermodynamics that explains the CAP phenomenology of Tzeng and Kalodimos (2012). The following open questions arise from the source data and their contested interpretation: (1) explain, from statistical-mechanical or atomic-resolution theory, why the conformational entropy change derived from NMR methyl order parameters (−TΔSconf-T\Delta S_{\mathrm{conf}}) scales linearly with the calorimetric total entropy change (−TΔS-T\Delta S) across the CAP variants, whose DNA-binding interfaces are essentially identical - an empirical correlation whose microscopic basis has not been derived; (2) identify the structural and dynamical determinants that fix, complex by complex, the magnitude of enthalpy-entropy compensation observed in the series; and (3) establish how the total entropy change is partitioned among conformational, solvation, and translational/rotational contributions - in particular whether the non-conformational terms are effectively invariant across the variants, as the linear interpretation assumes, or vary in concert with conformational entropy as the solvation-based alternative proposes. These questions are nested rather than independent: a partition of the total entropy change that fixes whether the non-conformational terms are invariant across the variants (3) would itself account for the linearity (1), whereas the per-complex determinants of the compensation magnitude (2) stand on their own.

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