(37a) Numerical and Analytical Coarse-Graining Strategies Based On Relative Entropy Minimization

A critical aspect of multiscale efforts is connecting coarse-grained models to atomic-level interactions in a quantitative manner. We propose that the relative entropy provides a fundamental and broad framework for such problems. This statistical-thermodynamic quantity measures, in some sense, the information lost upon coarse-graining; its minimization, therefore, provides an intuitive route for developing simplified and coarse-grained models. Of particular note, relative entropy minimization can be used not only to optimize intermolecular potentials of the kind targeted in other coarse-graining strategies like Boltzmann inversion and force-matching, but can be equally applied to problems in which simple analytical models are used to interpret molecular simulation results.

Here we describe general coarse-graining algorithms based on relative entropy minimization. We first develop robust numerical techniques for extracting coarse-grained intermolecular potentials from all-atom simulation results. These rely on conjugate gradient and generalized Newton minimization, and we discuss implementation issues such as parameter estimation, convergence detection, and error analysis. We demonstrate the approach with a case study involving a binary glass-former that is coarse-grained into a fluid whose interactions are governed by a soft-sphere potential. By doing so, we are able to use the optimized soft-sphere exponent to predict how dynamics in the original system scale with temperature and density.

We also show how the relative entropy facilitates the development and analysis of much simpler analytical and toy models of complex molecular systems. In particular, we also use the relative entropy to develop energy-landscape models of protein folding. Here, predicted structures for large proteins using computationally-intensive all-atom methods are projected onto analytical folding funnel landscapes. These models are able to quantify the degree of roughness in the landscape and, by minimizing the relative entropy with respect to the location of the funnel minimum, are actually able to filter out energy fluctuations to identify near-native structures.