Correlation as a determinant of configurational entropy in supramolecular and protein systems.

For biomolecules in solution, changes in configurational entropy are thought to contribute substantially to the free energies of processes like binding and conformational change. In principle, the configurational entropy can be strongly affected by pairwise and higher-order correlations among conformational degrees of freedom. However, the literature offers mixed perspectives regarding the contributions that changes in correlations make to changes in configurational entropy for such processes. Here we take advantage of powerful techniques for simulation and entropy analysis to carry out rigorous in silico studies of correlation in binding and conformational changes. In particular, we apply information-theoretic expansions of the configurational entropy to well-sampled molecular dynamics simulations of a model host-guest system and the protein bovine pancreatic trypsin inhibitor. The results bear on the interpretation of NMR data, as they indicate that changes in correlation are important determinants of entropy changes for biologically relevant processes and that changes in correlation may either balance or reinforce changes in first-order entropy. The results also highlight the importance of main-chain torsions as contributors to changes in protein configurational entropy. As simulation techniques grow in power, the mathematical techniques used here will offer new opportunities to answer challenging questions about complex molecular systems.

Sampling conformations in high dimensions using low-dimensional distribution functions.

We present an approximation to a molecule's N-dimensional conformational probability density function (pdf) in terms of marginal pdfs of highest order l, where l is much less than N. The approximation is constructed as a product of conditional pdfs derived by recursive application of the generalized Kirkwood superposition approximation. Furthermore, an algorithm is presented to sample conformations from the approximate full-dimensional pdf based upon all input marginal pdfs. The sampling algorithm is tested for three small molecule systems by using the algorithm to sample conformations at levels l=1, 2, or 3 and comparing the distributions of sampled conformations with those from the molecular dynamics (MD) simulations. The distributions of conformations sampled at third (l=3) order resemble the MD distributions rather well and significantly better than those sampled at second (l=2) or first (l=1) order. In addition to highlighting the importance of correlations among internal degrees of freedom, these results suggest that low-order correlations suffice to describe most of the conformational fluctuations of molecules in a thermal environment.

Configurational entropy in protein-peptide binding: computational study of Tsg101 ubiquitin E2 variant domain with an HIV-derived PTAP nonapeptide.

Configurational entropy is thought to influence biomolecular processes, but there are still many open questions about this quantity, including its magnitude, its relationship to molecular structure, and the importance of correlation. The mutual information expansion (MIE) provides a novel and systematic approach to extracting configurational entropy changes due to correlated motions from molecular simulations. We present the first application of the MIE method to protein-ligand binding using multiple molecular dynamics simulations to study the association of the ubiquitin E2 variant domain of the protein Tsg101 and an HIV-derived nonapeptide. This investigation utilizes the second-order MIE approximation, which accounts for correlations between all pairs of degrees of freedom. The computed change in configurational entropy is large and has a major contribution from changes in pairwise correlation. The results also reveal intricate structure-entropy relationships. Thus, the present analysis suggests that in order for a model of binding to be accurate, it must include a careful accounting of configurational entropy changes.

Efficient calculation of configurational entropy from molecular simulations by combining the mutual-information expansion and nearest-neighbor methods.

Changes in the configurational entropies of molecules make important contributions to the free energies of reaction for processes such as protein-folding, noncovalent association, and conformational change. However, obtaining entropy from molecular simulations represents a long-standing computational challenge. Here, two recently introduced approaches, the nearest-neighbor (NN) method and the mutual-information expansion (MIE), are combined to furnish an efficient and accurate method of extracting the configurational entropy from a molecular simulation to a given order of correlations among the internal degrees of freedom. The resulting method takes advantage of the strengths of each approach. The NN method is entirely nonparametric (i.e., it makes no assumptions about the underlying probability distribution), its estimates are asymptotically unbiased and consistent, and it makes optimum use of a limited number of available data samples. The MIE, a systematic expansion of entropy in mutual information terms of increasing order, provides a well-characterized approximation for lowering the dimensionality of the numerical problem of calculating the entropy of a high-dimensional system. The combination of these two methods enables obtaining well-converged estimations of the configurational entropy that capture many-body correlations of higher order than is possible with the simple histogramming that was used in the MIE method originally. The combined method is tested here on two simple systems: an idealized system represented by an analytical distribution of six circular variables, where the full joint entropy and all the MIE terms are exactly known, and the R,S stereoisomer of tartaric acid, a molecule with seven internal-rotation degrees of freedom for which the full entropy of internal rotation has been already estimated by the NN method. For these two systems, all the expansion terms of the full MIE of the entropy are estimated by the NN method and, for comparison, the MIE approximations up to third order are also estimated by simple histogramming. The results indicate that the truncation of the MIE at the two-body level can be an accurate, computationally nondemanding approximation to the configurational entropy of anharmonic internal degrees of freedom. If needed, higher-order correlations can be estimated reliably by the NN method without excessive demands on the molecular-simulation sample size and computing time.

Extraction of configurational entropy from molecular simulations via an expansion approximation.

A method is presented for extracting the configurational entropy of solute molecules from molecular dynamics simulations, in which the entropy is computed as an expansion of multidimensional mutual information terms, which account for correlated motions among the various internal degrees of freedom of the molecule. The mutual information expansion is demonstrated to be equivalent to estimating the full-dimensional configurational probability density function (PDF) using the generalized Kirkwood superposition approximation (GKSA). While the mutual information expansion is derived to the full dimensionality of the molecule, the current application uses a truncated form of the expansion in which all fourth- and higher-order mutual information terms are neglected. Truncation of the mutual information expansion at the nth order is shown to be equivalent to approximating the full-dimensional PDF using joint PDFs with dimensionality of n or smaller by successive application of the GKSA. The expansion method is used to compute the absolute (classical) configurational entropy in a basis of bond-angle-torsion internal coordinates for several small molecules as well as the change in entropy upon binding for a small host-guest system. Convergence properties of the computed entropy values as a function of simulation time are investigated and comparisons are made with entropy values from the second generation Mining Minima software. These comparisons demonstrate a deviation in -TS of no more than about 2 kcal/mol for all cases in which convergence has been obtained.