Small-sample limit of the Bennett acceptance ratio method and the variationally derived intermediates.

Martin Reinhardt,Helmut Grubmüller

doi:10.1103/physreve.104.054133

Martin Reinhardt, Helmut Grubmüller

Open Access

https://doi.org/10.1103/physreve.104.054133

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Abstract

Free energy calculations based on atomistic Hamiltonians provide microscopic insight into the thermodynamic driving forces of biophysical or condensed matter systems. Many approaches use intermediate Hamiltonians interpolating between the two states for which the free energy difference is calculated. The Bennett acceptance ratio (BAR) and variationally derived intermediates (VI) methods are optimal estimator and intermediate states in that the mean-squared error of free energy calculations based on independent sampling is minimized. However, BAR and VI have been derived based on several approximations that do not hold for very few sample points. Analyzing one-dimensional test systems, we show that in such cases BAR and VI are suboptimal and that established uncertainty estimates are inaccurate. Whereas for VI to become optimal, less than seven samples per state suffice in all cases; for BAR the required number increases unboundedly with decreasing configuration space densities overlap of the end states. We show that for BAR, the required number of samples is related to the overlap through an inverse power law. Because this relation seems to hold universally and almost independent of other system properties, these findings can guide the proper choice of estimators for free energy calculations.

Highlights

Free energy differences provide detailed insights into the molecular driving forces of biophysical processes, and their accurate calculation is crucial for their successful application, e.g., in pharmaceutical ligand design or material science [1,2,3,4,5,6,7]
To calculate the free energy difference between, e.g., two potential drug molecules bound to a receptor, alchemical equilibrium techniques [8] based on simulations with atomistic Hamiltonians are among the most widely used methods
We have shown that for small sample sizes n the analytically calculated mean-squared error (MSE) of free energy estimates based on the Zwanzig formula become increasingly inaccurate due to approximations in its derivation

Summary

Introduction

Free energy differences provide detailed insights into the molecular driving forces of biophysical processes, and their accurate calculation is crucial for their successful application, e.g., in pharmaceutical ligand design or material science [1,2,3,4,5,6,7]. To calculate the free energy difference between, e.g., two potential drug molecules bound to a receptor, alchemical equilibrium techniques [8] based on simulations with atomistic Hamiltonians are among the most widely used methods. Two choices have to be made that critically affect the accuracy of free energy calculations: First, the choice of the estimator that is used to evaluate the free energy differences between the individual states. Whereas a number of estimators exist that have practical advantages in different situations [8,9,10], it has been shown that between two states the Bennett acceptance ratio (BAR) method [11] minimizes the variance, and the mean-squared error (MSE)

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