(37c) Coarse-Graining Proteins Using Relative Entropy Theory

The development of accurate coarse-grained molecular models is central to many efforts in soft-condensed matter, materials, and biophysics that seek to understand large length and time scale properties.  We recently proposed a broad new statistical-mechanical framework for such problems based the relative entropy, a phase-space functional measuring the “information” lost upon coarse graining and hence the (inverse) fitness of a given coarse-grained model [1].  We suggest that minimization of the relative entropy provides a universal variational principle for coarse-graining. 

Here, we show how this approach provides a quantitative and systematic approach to the development of coarse-grained models of proteins.  We first describe the development of stable, efficient, and robust numerical coarse-graining algorithms based upon relative entropy minimization [2].  In particular, these approaches are able to locate optimal models in very high dimensional parameter spaces, here for one-, two-, and three-bead (per amino acid) models of peptides.  We show that the relative entropy framework offers exquisite control over coarse-graining errors, and is able to generate coarse peptide models whose structural correlations and folding behavior are in excellent agreement with more expensive all-atom simulations.  We also demonstrate how these models are well-suited to large-scale simulations of peptide self-assembly.  Second, we show that the relative entropy approach provides an a priori prediction of coarse-graining errors (i.e., errors in the properties predicted by coarse-grained models), and gives a systematic strategy for designing coarse-grained potentials [3].  In particular, we use this theory to develop simple analytical energy-landscape models of protein folding.  Here, an ensemble of predicted structures for a large protein is projected onto a coarse-grained funnel landscape.  The relative entropy quantifies which structure, when considered at the bottom of the funnel, best describes the energy-structure relationships, in turn predicting which structure is nearest to the true native structure.  We present results on several model protein systems and from a recent community-wide blind structure prediction competition.

[1] M. S. Shell, “The relative entropy is fundamental to multiscale and inverse thermodynamic problems,” J. Chem. Phys. 129, 144108 (2008).

[2] A. Chaimovich and M.S. Shell, “Anomalous waterlike behavior in spherically-symmetric water models  optimized with the relative entropy,” Phys. Chem. Chem. Phys 11, 1901 (2009).

[3] A. Chaimovich and M. S. Shell, “Coarse-graining errors and optimization using a relative entropy framework,” J. Chem. Phys. 134, 094112 (2011).