Determining Free-Energy Differences Through Variationally Derived Intermediates.

Martin Reinhardt,Helmut Grubmüller

doi:10.1021/acs.jctc.0c00106

Abstract

Free-energy calculations based on atomistic Hamiltonians and sampling are key to a first-principles understanding of biomolecular processes, material properties, and macromolecular chemistry. Here, we generalize the free-energy perturbation method and derive nonlinear Hamiltonian transformation sequences yielding free-energy estimates with minimal mean squared error with respect to the exact values. Our variational approach applies to finite sampling and holds for any finite number of intermediate states. We show that our sequences are also optimal for the Bennett acceptance ratio (BAR) method, thereby generalizing BAR to small sampling sizes and non-Gaussian error distributions.

Highlights

Free-energy calculations provide essential insights into numerous physical and biochemical systems
Our approach differs from previous ones in that it, first, optimizes the full mean squared error (MSE) with respect to the exact free-energy difference rather than the variance only
It directly optimizes the sequence of discrete states instead of a two step approach, where first a continuous thermodynamic integration (TI) path is optimized[17,26,27] and subsequently a discrete subset of states is chosen from this path

Summary

INTRODUCTION

Free-energy calculations provide essential insights into numerous physical and biochemical systems. Given the Hamiltonians H1(x) and HN(x) of two states 1 and N, where x ∈ 3M denotes the position of all M particles of the simulation system, the free-energy difference ΔG1,N between these states is given by the Zwanzig formula[12]. Alchemical transformations substantially reduce errors in the free-energy estimates[15,16] by introducing N − 2 intermediate states s and accumulating small free-energy differences between all adjacent states s and s + 1, N−1. Of the free-energy estimate ΔG(n) obtained through finite sampling with n sample points with respect to the exact freeenergy difference ΔG. As we will find, optimizing the sum of both variance and bias yields a conceptually improved result

THEORY

RESULTS AND DISCUSSION

ATOMISTIC TEST CASES

CONCLUSIONS

■ REFERENCES