Bayesian Approach\nfor Computing Free Energy on Perturbation\nGraphs with Cycles

A common approach for computing free energy differences\namong multiple\nstates is to build a perturbation graph connecting the states and\ncompute free energy differences on all edges of the graph. Such perturbation\ngraphs are often designed to have cycles. Because free energy is a\nfunction of states, the free energy around any cycle is zero, which\nwe refer to as the cycle consistency condition. Since the cycle consistency\ncondition relates free energy differences on the edges of a cycle,\nit could be used to improve the accuracy of free energy estimates.\nHere, we propose a Bayesian method called the coupled Bayesian multistate\nBennett acceptance ratio (CBayesMBAR) that can properly couple the\ncalculations of free energy differences on the edges of cycles in\na principled way. We apply the CBayesMBAR to compute free energy differences\namong harmonic oscillators and relative protein–ligand binding\nfree energies. In both cases, the CBayesMBAR provides more accurate\nresults compared to methods that do not consider the cycle consistency\ncondition. Additionally, it outperforms the cycle closure correction\nmethod that also uses cycle consistency conditions.