Statistically optimal analysis of samples from multiple equilibrium states

TL;DR

This paper introduces MBAR, a statistically optimal estimator for analyzing samples from multiple equilibrium states, improving accuracy and efficiency over previous methods in free energy calculations.

Contribution

The paper presents the MBAR estimator, a novel method that unifies and improves upon existing techniques for analyzing multi-state equilibrium data without histogram discretization.

Findings

MBAR provides unbiased estimates with minimal variance in large samples.

It reduces computational complexity compared to histogram-based methods.

Demonstrated by precise free energy estimation in DNA hairpin experiments.

Abstract

We present a new estimator for computing free energy differences and thermodynamic expectations as well as their uncertainties from samples obtained from multiple equilibrium states via either simulation or experiment. The estimator, which we term the multistate Bennett acceptance ratio (MBAR) estimator because it reduces to the Bennett acceptance ratio when only two states are considered, has significant advantages over multiple histogram reweighting methods for combining data from multiple states. It does not require the sampled energy range to be discretized to produce histograms, eliminating bias due to energy binning and significantly reducing the time complexity of computing a solution to the estimating equations in many cases. Additionally, an estimate of the statistical uncertainty is provided for all estimated quantities. In the large sample limit, MBAR is unbiased and has the lowest variance of any known estimator for making use of equilibrium data collected from multiple states. We illustrate this method by producing a highly precise estimate of the potential of mean force for a DNA hairpin system, combining data from multiple optical tweezer measurements under constant force bias.