Bayesian ensemble refinement by replica simulations and reweighting

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 singlemolecule 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.

Combining Simulations and Solution Experiments as a Paradigm for RNA Force Field Refinement

“…It is important to notice that a similar equation could be obtained using theoretical approaches different from the one introduced in this paper. For instance, one could decide to maximize the posterior as a function of the residuals ε as it is done in Ref 29 , instead of computing their average value. More comments on this analogy can be found in Supporting Information (section 3).…”

“…Several Bayesian approaches have been discussed to model experimental errors in similar contexts (see e.g. 29,30,[61][62][63] ). Methods have been described to reweight a pre-computed ensemble of structures so as to match experimental averages 29,63,64 .…”

“…29,30,[61][62][63] ). Methods have been described to reweight a pre-computed ensemble of structures so as to match experimental averages 29,63,64 . We here apply the MaxEnt procedure on an extended system where fictitious variables are introduced that take into account the discrepancy between theory and experiment.…”

“…The value of σ thus tells us how much we trust in the original force field and is related to the value of θ introduced in Ref 29 . Non Gaussian prior distributions for ε can be used to better tolerate outliers.…”

“…For instance, White and Voth implemented the MaxEnt procedure in their experimentally directed simulation method (EDS) 27 . On top of this, recent works have underlined the importance of taking into account experimental errors when using these constraints [28][29][30] . This latter step is often done in a Bayesian framework, which is not designed to provide a single optimal model but to sample the posterior distribution thus generating ensembles of solutions compatible with the constraints.…”