Using the method of weighted residuals to compute potentials of mean force Eric C. Cyr * , Stephen D. Bond University of Illinois at Urbana-Champaign, Department of Computer Science, Urbana, IL 61801, USA Received 28 July 2006; received in revised form 19 December 2006; accepted 22 December 2006 Available online 10 January 2007 Abstract We propose a general framework for approximating the potential of mean force (PMF) along a reaction coordinate in conformational space. This framework, based on the method of weighted residuals, can be viewed as a generalization of thermodynamic integration and direct histogram methods. Using weighted residuals allows for higher-order approxima- tions to the PMF in the form of a global spectral method or a finite element method. In addition, the higher degree of continuity provided by spectral and higher-order elements makes weighted residual methods an attractive choice for use in tandem with biasing force methods. As an analysis tool, the weighted residuals framework provides a context for direct comparison of thermodynamic integration and histogram based methods. For validation of the new method, numerical experiments are performed on two systems: a simple double-well and alanine dipeptide in vacuum. Comparisons between the new weighted residual methods, thermodynamic integration, and WHAM are performed. When configuration space is perfectly sampled the high-order weighted residual methods are found to exhibit exponential convergence. For more realistic sampling, the weighted residual methods performed comparably to the other two. However, results suggest that spectral type methods are more robust with respect to parameter choices describing the solution space. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Potential of mean force; Method of weighted residuals; Free energy; Thermodynamic integration; Histogram methods 1. Introduction The potential of mean force (PMF) is one of the most important concepts in physical and biological chem- istry [1]. It describes the change in free energy along a ‘‘reaction coordinate’’ and determines the strength and likelihood of association in molecular systems [2]. Estimating the change in free energy between two molecular conformations is a challenging task due to the high dimensionality of phase space and complex structure of the energy landscape [3]. A variety of techniques have been developed to approximate the PMF including umbrella sampling [4], weighted histograms [5], free energy perturbation [6,7], thermodynamic integration [7,8], steered molecular 0021-9991/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2006.12.015 * Corresponding author. E-mail addresses: ericcyr@uiuc.edu (E.C. Cyr), sdbond@uiuc.edu (S.D. Bond). Journal of Computational Physics 225 (2007) 714–729 www.elsevier.com/locate/jcp