Gibbs Sampler-Based λ-Dynamics and Rao–Blackwell Estimator for Alchemical Free Energy Calculation

Abstract

λ-dynamics is a generalized ensemble method for alchemical free energy calculations. In traditional λ-dynamics, the alchemical switch variable λ is treated as a continuous variable ranging from 0 to 1 and an empirical estimator is utilized to approximate the free energy. In the present paper, we describe an alternative formulation of λ-dynamics that utilizes the Gibbs sampler framework which we call Gibbs Sampler λ-dynamics (GSLD). GSLD, like traditional λ-dynamics, can be readily extended to calculate free energy differences between multiple ligands in one simulation. We also introduce a new free energy estimator, the Rao-Blackwell estimator (RBE) for use in conjunction with GSLD. Compared with the current empirical estimator, the advantage of RBE is that RBE is an unbiased estimator and its variance is usually smaller than the current empirical estimator. We also show that the multistate Bennett acceptance ratio (MBAR) equation or the unbinned weighted histogram analysis method (UWHAM) equation can be derived using the RBE. We illustrate the use and performance of this new free energy computational framework by application to a simple harmonic system as well as relevant calculations of small molecule relative free energies of solvation and binding to a protein receptor. Our findings demonstrate consistent and improved performance compared with conventional alchemical free energy methods.

Approaching protein design with multisite λ dynamics: Accurate and scalable mutational folding free energies in T4 lysozyme

“…Alchemical free energy methods can make rigorous estimates of thermodynamic stability using MD simulations. Free energy methods include free energy perturbation (FEP), thermodynamic integration (TI), multisite λ dynamics (MS λ D), enveloping distribution sampling, and many others . These methods have most commonly been applied to compute solvation free energies of ligands, drug binding free energies, and pKa values of titratable groups in proteins .…”

Estudo da enzima beta-lactamase e sua relação com a resistência aos antibióticos

“…O cálculo de ligação de proteína-ligantes possui sua importância na descoberta de medicamentos, principalmente, em estágios de geração e otimização de compostos com atividade terapêutica. Vários métodos de cálculo da energia livre foram desenvolvidos nas últimas décadas, como perturbação de energia, integração termodinâmica, amostragens de distribuição envolvente e dinâmica (Ding, Vilseck, Hayes & Brooks, 2017 Estudos realizados no Sudão em 2020 (Dirar, Bilal, Ibrahim & Hamid, 2020) corroboram estes resultados, pois mostram que enzimas AmpC foi identificada em uma porcentagem de 49,3% de bactérias, desta forma as bactérias apresentaram experimentalmente Research,Society and Development,v. 9,n.…”

Free energy change estimation: The Divide and Conquer MBAR method

“…The Multistate Bennett Acceptance Ratio (MBAR) 1 method, which could be considered as the binless version of the Weighted Histogram Analysis Method (WHAM), 2–5 is widely applied to predict the free energy changes of various biological and chemical processes based on the data sampled by independent or coupled parallel simulations 6–9 . MBAR is also found to be related to the unbiased Rao–Blackwell estimator 10–13 . Tan proposed one update scheme by Rao‐Blackwellization in the Self Adjusted Mixture Sampling (SAMS) method and the offline free energies could be obtained by solving the MBAR equations 11 .…”

“…Tan proposed one update scheme by Rao‐Blackwellization in the Self Adjusted Mixture Sampling (SAMS) method and the offline free energies could be obtained by solving the MBAR equations 11 . Ding et al introduced the Rao–Blackwell estimator for use in conjunction with Gibbs sampler‐based λ ‐dynamics (GSLD) and gives a derivation of the MBAR equations using Rao–Blackwell estimator directly 12 . In a word, MBAR is provably the lowest variance unbiased estimator of both free energies and ensemble averages to date 14…”