Bayesian ensemble refinement by replica simulations and reweighting

TL;DR

This paper reviews Bayesian ensemble refinement methods for characterizing dynamic biomolecular structures, introducing new approaches that improve sampling efficiency and convergence by combining replica simulations, reweighting, and adaptive algorithms.

Contribution

It unifies various Bayesian ensemble refinement techniques, derives optimal distributions, and proposes methods to enhance sampling and convergence in biomolecular simulations.

Findings

Optimal Bayesian ensemble distribution derived for discrete configurations

Restraint strength should scale linearly with number of replicas for convergence

Combined replica and reweighting methods accelerate convergence to the optimal ensemble

Abstract

We describe different Bayesian ensemble refinement methods, examine their interrelation, and discuss their practical application. With ensemble refinement, the properties of dynamic and partially disordered (bio)molecular structures can be characterized by integrating a wide range of experimental data, including measurements of ensemble-averaged observables. We start from a Bayesian formulation in which the posterior is a functional that ranks different configuration space distributions. By maximizing this posterior, we derive an optimal Bayesian ensemble distribution. For discrete configurations, this optimal distribution is identical to that obtained by the maximum entropy "ensemble refinement of SAXS" (EROS) formulation. Bayesian replica ensemble refinement enhances the sampling of relevant configurations by imposing restraints on averages of observables in coupled replica molecular dynamics simulations. We show that the strength of the restraint should scale linearly with the number of replicas to ensure convergence to the optimal Bayesian result in the limit of infinitely many replicas. In the "Bayesian inference of ensembles" (BioEn) method, we combine the replica and EROS approaches to accelerate the convergence. An adaptive algorithm can be used to sample directly from the optimal ensemble, without replicas. We discuss the incorporation of single-molecule measurements and dynamic observables such as relaxation parameters. The theoretical analysis of different Bayesian ensemble refinement approaches provides a basis for practical applications and a starting point for further investigations.