Shape optimization of metastable states

TL;DR

This paper introduces a shape-optimization approach for defining metastable states in molecular simulations, improving accuracy especially when entropic effects are significant.

Contribution

It develops a novel method based on shape-optimization of a timescale metric, with analytic expressions and algorithms for high-dimensional systems.

Findings

Significant improvement over conventional metastable state definitions.

Validated method on a biomolecular system.

Provides analytic expressions for shape variations of Dirichlet eigenvalues.

Abstract

The definition of metastable states is an ubiquitous task in the design and analysis of molecular simulations, and is a crucial input in a variety of acceleration methods for the sampling of long configurational trajectories. Although standard definitions based on local energy minimization procedures can sometimes be used, these definitions are typically suboptimal, or entirely inadequate when entropic effects are significant, or when the lowest energy barriers are quickly overcome by thermal fluctuations. In this work, we propose an approach to the definition of metastable states, based on the shape-optimization of a local separation of timescale metric directly linked to the efficiency of a family of accelerated molecular dynamics algorithms. To realize this approach, we derive analytic expressions for shape-variations of Dirichlet eigenvalues for a class of operators associated with reversible elliptic diffusions, and use them to construct a local ascent algorithm, explicitly treating the case of multiple eigenvalues. We propose two methods to make our method tractable in high-dimensional systems: one based on dynamical coarse-graining, the other on recently obtained low-temperature shape-sensitive spectral asymptotics. We validate our method on a benchmark biomolecular system, showcasing a significant improvement over conventional definitions of metastable states.