(86i) Let's Not Use Histograms: Bayesian Inference of Potentials of Mean Force with Umbrella Sampling and Multistate Reweighting

Potentials of mean force (PMFs)—free energies along a selected set of collective variables—are ubiquitous in molecular simulation, and of significant value in understanding and engineering molecular behaviors. PMFs are most commonly estimated using variants of histogramming techniques, but such approaches obscures two important facets of the distribution. First, the empirical observations along the collective variable are defined by an ensemble of discrete observations and the coarsening of these observations into a histogram bins incurs unnecessary loss of information. Second, the potential of mean force is itself almost always a continuous function, and its representation by a histogram introduces inherent approximations due to the discretization. In this study, we relate the observed discrete observations to the underlying continuous probability distribution over the collective variables and derive histogram-free estimation techniques for the potential of mean force.

We reformulate PMF estimation to minimization of a Kullback-Leibler divergence between a continuous trial function and the discrete empirical distribution and show this is equivalent to likelihood maximization of a trial function and the sampled data. We then present a fully Bayesian treatment of this formalism to enable the incorporation of powerful Bayesian tools such as the inclusion of regularizing priors, uncertainty quantification, and model selection techniques. We demonstrate our new formalism in the analysis of umbrella sampling simulations for the χ torsion of a valine sidechain in the L99A mutant of T4 lysozyme with benzene bound in the cavity.