Identifying low variance pathways for free energy calculations of molecular transformations in solution phase

Improving the efficiency of free energy calculations is important for many biological and materials design applications, such as protein-ligand binding affinities in drug design, partitioning between immiscible liquids, and determining molecular association in soft materials. We show that for any pair potential, moderately accurate estimation of the radial distribution function for a solute molecule is sufficient to accurately estimate the statistical variance of a sampling along a free energy pathway. This allows inexpensive analytical identification of low statistical error free energy pathways. We employ a variety of methods to estimate the radial distribution function (RDF) and find that the computationally cheap two-body "dilute gas" limit performs as well or better than 3D-RISM theory and other approximations for identifying low variance free energy pathways. With a RDF estimate in hand, we can search for pairwise interaction potentials that produce low variance. We give an example of a search minimizing statistical variance of solvation free energy over the entire parameter space of a generalized "soft core" potential. The free energy pathway arising from this optimization procedure has lower curvature in the variance and reduces the total variance by at least 50% compared to the traditional soft core solvation pathway. We also demonstrate that this optimized pathway allows free energies to be estimated with fewer intermediate states due to its low curvature. This free energy variance optimization technique is generalizable to solvation in any homogeneous fluid and for any type of pairwise potential and can be performed in minutes to hours, depending on the method used to estimate g(r).