Determining Free-Energy Differences Through Variationally Derived Intermediates

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.

[1]  J. Kirkwood.  Statistical Mechanics of Fluid Mixtures , 1935 .

[2]  R. Zwanzig.  High‐Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases , 1954 .

[3]  Charles H. Bennett,et al.  Efficient estimation of free energy differences from Monte Carlo data , 1976 .

[4]  M. Parrinello,et al.  Polymorphic transitions in single crystals: A new molecular dynamics method , 1981 .

[5]  Wang,et al.  Replica Monte Carlo simulation of spin glasses. , 1986, Physical review letters.

[6]  T. Straatsma,et al.  Separation‐shifted scaling, a new scaling method for Lennard‐Jones interactions in thermodynamic integration , 1994 .

[7]  A. Mark,et al.  Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations , 1994 .

[8]  Grubmüller,et al.  Predicting slow structural transitions in macromolecular systems: Conformational flooding. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  C. Jarzynski.  Nonequilibrium Equality for Free Energy Differences , 1996, cond-mat/9610209.

[10]  Xiao-Li Meng,et al.  Simulating Normalizing Constants: From Importance Sampling to Bridge Sampling to Path Sampling , 1998 .

[11]  John W. Wilkins,et al.  Simple bias potential for boosting molecular dynamics with the hyperdynamics scheme , 1998 .

[12]  Sidney Yip,et al.  Optimized free-energy evaluation using a single reversible-scaling simulation , 1999 .

[13]  J. A. White,et al.  Lennard-Jones as a model for argon and test of extended renormalization group calculations , 1999 .

[14]  David A. Kofke,et al.  Accuracy of free-energy perturbation calculations in molecular simulation. II. Heuristics , 2001 .

[15]  David A. Kofke,et al.  Accuracy of free-energy perturbation calculations in molecular simulation. I. Modeling , 2001 .

[16]  A. Laio,et al.  Escaping free-energy minima , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Thomas B Woolf,et al.  Theory of a systematic computational error in free energy differences. , 2002, Physical review letters.

[18]  Michael R. Shirts,et al.  Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods. , 2003, Physical review letters.

[19]  F. Ritort,et al.  Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Arnaud Blondel,et al.  Ensemble variance in free energy calculations by thermodynamic integration: Theory, optimal “Alchemical” path, and practical solutions , 2004, J. Comput. Chem..

[21]  Thomas B Woolf,et al.  Using overlap and funnel sampling to obtain accurate free energies from nonequilibrium work measurements. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Thomas B. Woolf,et al.  Improving the efficiency and reliability of free energy perturbation calculations using overlap sampling methods , 2004, J. Comput. Chem..

[23]  B. Berne,et al.  Replica exchange with solute tempering: a method for sampling biological systems in explicit water. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Di Wu,et al.  Phase-space overlap measures. II. Design and implementation of staging methods for free-energy calculations. , 2005, The Journal of chemical physics.

[25]  Michael R. Shirts,et al.  Comparison of efficiency and bias of free energies computed by exponential averaging, the Bennett acceptance ratio, and thermodynamic integration. , 2005, The Journal of chemical physics.

[26]  David L Mobley,et al.  Nonlinear scaling schemes for Lennard-Jones interactions in free energy calculations. , 2007, The Journal of chemical physics.

[27]  Wilfred F van Gunsteren,et al.  Enveloping distribution sampling: a method to calculate free energy differences from a single simulation. , 2007, The Journal of chemical physics.

[28]  Christoph Dellago,et al.  Optimum bias for fast-switching free energy calculations , 2007, Comput. Phys. Commun..

[29]  Wilfred F van Gunsteren,et al.  Multiple free energies from a single simulation: extending enveloping distribution sampling to nonoverlapping phase-space distributions. , 2008, The Journal of chemical physics.

[30]  C. Jarzynski,et al.  Escorted free energy simulations: improving convergence by reducing dissipation. , 2008, Physical review letters.

[31]  Michael R. Shirts,et al.  Statistically optimal analysis of samples from multiple equilibrium states. , 2008, The Journal of chemical physics.

[32]  Michael P Eastwood,et al.  Minimizing thermodynamic length to select intermediate states for free-energy calculations and replica-exchange simulations. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Michael R. Shirts,et al.  Optimal pairwise and non-pairwise alchemical pathways for free energy calculations of molecular transformation in solution phase. , 2012, The Journal of chemical physics.

[34]  Michael Habeck,et al.  Bayesian estimation of free energies from equilibrium simulations. , 2012, Physical review letters.

[35]  Michele Parrinello,et al.  Variational approach to enhanced sampling and free energy calculations. , 2014, Physical review letters.

[36]  Thomas Fox,et al.  Accuracy Assessment and Automation of Free Energy Calculations for Drug Design , 2014, J. Chem. Inf. Model..

[37]  Daisy Y. Kyu,et al.  Calculating Partition Coefficients of Small Molecules in Octanol/Water and Cyclohexane/Water. , 2016, Journal of chemical theory and computation.

[38]  Z. Tan.  Optimally Adjusted Mixture Sampling and Locally Weighted Histogram Analysis , 2017 .

[39]  Bryce K. Allen,et al.  Relative Binding Free Energy Calculations in Drug Discovery: Recent Advances and Practical Considerations , 2017, J. Chem. Inf. Model..

[40]  Matteo Aldeghi,et al.  Accurate Estimation of Ligand Binding Affinity Changes upon Protein Mutation , 2018, ACS central science.

[41]  M. Marinica,et al.  Unsupervised Calculation of Free Energy Barriers in Large Crystalline Systems. , 2018, Physical review letters.

[42]  Elizabeth Yuriev,et al.  Free Energy Methods in Drug Design: Prospects of "Alchemical Perturbation" in Medicinal Chemistry. , 2017, Journal of medicinal chemistry.