Statistically Optimal Continuous Free Energy Surfaces from Biased Simulations and Multistate Reweighting

Abstract

Free energies as a function of a selected set of collective variables are commonly computed in molecular simulation and of significant value in understanding and engineering molecular behavior. These free energy surfaces are most commonly estimated using variants of histogramming techniques, but such approaches obscure two important facets of these functions. 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 bin incurs unnecessary loss of information. Second, the free energy surface 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 from biased simulations to the inferred underlying continuous probability distribution over the collective variables and derive histogram-free techniques for estimating this free energy surface. We reformulate free energy surface estimation as minimization of a Kullback–Leibler divergence between a continuous trial function and the discrete empirical distribution and show that this is equivalent to likelihood maximization of a trial function given a set of sampled data. We then present a fully Bayesian treatment of this formalism, which enables the incorporation of powerful Bayesian tools such as the inclusion of regularizing priors, uncertainty quantification, and model selection techniques. We demonstrate this new formalism in the analysis of umbrella sampling simulations for the χ torsion of a valine side chain in the L99A mutant of T4 lysozyme with benzene bound in the cavity.

Network analysis reveals how lipids and other cofactors influence membrane protein allostery

“…Each density is represented by a Gaussian mixture model (GMM) [29]; a linear combination of Gaussian basis functions. Because Gaussians are continuous, the density estimation is more stable where the data is sparse and does not introduce unnecessary approximations compared to discrete methods such as histogramming [30,31]. Given the number of Gaussians, GMM parameters (amplitudes, means and covariances) are estimated with expectation-maximization [29].…”

Accelerated Computation of Free Energy Profile at Ab Initio Quantum Mechanical/Molecular Mechanics Accuracy via a Semiempirical Reference Potential. 3. Gaussian Smoothing on Density-of-States

“…The numerical difficulty arises in the nonparametric summing-up step and may have an impact on the second step. We noticed some recent progress in Bayesian interference for free energy calculations. − The output free energy profiles are guaranteed to be smooth since continuous basis functions are used. How this Gaussian smoothing process will affect the free energy profile from these parametric fitting methods will be investigated in our future studies.…”

“…We noticed some recent progress in Bayesian interference for free energy calculations. [55][56][57][58][59] The…”

Accelerated Computation of Free Energy Profile at Ab Initio QM/MM Accuracy via a Semi-Empirical Reference-Potential. III. Gaussian Smoothing on Density-of-States

“…We noticed some recent progress in Bayesian interference for free energy calculations. [55][56][57][58][59] The output free energy profiles are guaranteed to be smooth since continuous basis functions are used. How this Gaussian smoothing process will affect the free energy profile from these parametric fitting methods will be investigated in our future studies.…”