Minimizing memory as an objective for coarse-graining

Coarse-graining a molecular model is the process of integrating over degrees of freedom to obtain a reduced representation. This process typically involves two separate but related steps, selection of the coordinates comprising the reduced system and modeling their interactions. Both the coordinate selection and the modeling procedure present challenges. Here, we focus on the former. Typically, one seeks to integrate over the fast degrees of freedom and retain the slow degrees of freedom. Failure to separate timescales results in memory. With this motivation, we introduce a heuristic measure of memory and show that it can be used to compare competing coordinate selections for a given modeling procedure. We numerically explore the utility of this heuristic for three systems of increasing complexity. The first example is a four-particle linear model, which is exactly solvable. The second example is a sixteen-particle nonlinear model; this system has interactions that are characteristic of molecular force fields but is still sufficiently simple to permit exhaustive numerical treatment. The third example is an atomic-resolution representation of a protein, the class of models most often treated by relevant coarse-graining approaches; we specifically study an actin monomer. In all three cases, we find that the heuristic suggests coordinate selections that are physically intuitive and reflect molecular structure. The memory heuristic can thus serve as an objective codification of expert knowledge and a guide to sites within a model that requires further attention.