Conformational Sampling and Calculation of Molecular Free Energy Using Superposition Approximations

Abstract

The superposition approximations (SAs), first proposed in the distribution function theories of liquids, are a family of approximations to a multivariate probability distribution function (pdf) in terms of its lower order marginal pdfs. In this talk, we first present the relationship between various forms of SA, the measurement of correlation via mutual information, and approximations to the entropy of the full pdf via truncations of the Mutual Information Expansion.

Next, based on the SAs, a novel framework to construct computationally tractable approximations to the N-dimensional Boltzmann conformational distribution of molecule in terms of its low order marginal pdfs is presented. The marginal pdfs are obtained as normalized histograms of internal coordinates of a set of Boltzmann distributed conformations obtained by molecular dynamics (MD) simulation. We evaluate the accuracy of these approximate distributions constructed from marginal pdfs of order L <=3 for small molecules (<= 52 atoms) by using a novel conformational sampling algorithm to sample from them and comparing the samples with the original MD conformations used to populate the pdfs. We find that the triplet (L=3) level approximation has high conformational overlap with the physical Boltzmann distribution, and significantly better than that for the singlet (L=1) or doublet (L=2) level approximations. The results shed light on the relative importance of correlations of different orders.

The singlet (L=1) and doublet (L=2) level approximate distributions are then used to define reference systems with known free energies, and then to compute the physical free energy of molecules using the reference system approach. Free energies are computed for small peptides as test molecules, and it is found that the convergence of the free energy estimate using a doublet reference is dramatically faster than with the singlet reference, consistent with greater overlap of the doublet reference system with the physical system. Potential further developments and practical applications are discussed.