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König G, Glaser N, Schroeder B, Kubincová A, Hünenberger PH, Riniker S. An Alternative to Conventional λ-Intermediate States in Alchemical Free Energy Calculations: λ-Enveloping Distribution Sampling. J Chem Inf Model 2020; 60:5407-5423. [PMID: 32794763 DOI: 10.1021/acs.jcim.0c00520] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023]
Abstract
Alchemical free energy calculations typically rely on intermediate states to bridge between the relevant phase spaces of the two end states. These intermediate states are usually created by mixing the energies or parameters of the end states according to a coupling parameter λ. The choice of the procedure has a strong impact on the efficiency of the calculation, as it affects both the encountered energy barriers and the phase space overlap between the states. The present work builds on the connection between the minimum variance pathway (MVP) and enveloping distribution sampling (EDS). It is shown that both methods can be regarded as special cases of a common scheme referred to as λ-EDS, which can also reproduce the behavior of conventional λ-intermediate states. A particularly attractive feature of λ-EDS is its ability to emulate the use of soft core potentials (SCP) while avoiding the associated computational overhead when applying efficient free energy estimators such as the multistate Bennett's acceptance ratio (MBAR). The method is illustrated for both relative and absolute free energy calculations considering five benchmark systems. The first two systems (charge inversion and cavity creation in a dipolar solvent) demonstrate the use of λ-EDS as an alternative coupling scheme in the context of thermodynamic integration (TI). The three other systems (change of bond length, change of dihedral angles, and cavity creation in water) investigate the efficiency and optimal choice of parameters in the context of free energy perturbation (FEP) and Bennett's acceptance ratio (BAR). It is shown that λ-EDS allows larger steps along the alchemical pathway than conventional intermediate states.
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Affiliation(s)
- Gerhard König
- Laboratory of Physical Chemistry, Department of Chemistry and Applied Biosciences, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Nina Glaser
- Laboratory of Physical Chemistry, Department of Chemistry and Applied Biosciences, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Benjamin Schroeder
- Laboratory of Physical Chemistry, Department of Chemistry and Applied Biosciences, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Alžbeta Kubincová
- Laboratory of Physical Chemistry, Department of Chemistry and Applied Biosciences, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Philippe H Hünenberger
- Laboratory of Physical Chemistry, Department of Chemistry and Applied Biosciences, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Sereina Riniker
- Laboratory of Physical Chemistry, Department of Chemistry and Applied Biosciences, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
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Li Y, Nam K. Repulsive Soft-Core Potentials for Efficient Alchemical Free Energy Calculations. J Chem Theory Comput 2020; 16:4776-4789. [PMID: 32559374 DOI: 10.1021/acs.jctc.0c00163] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023]
Abstract
In alchemical free energy (FE) simulations, annihilation and creation of atoms are generally achieved with the soft-core potential that shifts the interparticle separations. While this soft-core potential eliminates the numerical instability occurring near the two end states of the transformation, it makes the hybrid Hamiltonian vary nonlinearly with respect to the parameter λ, which interpolates between the Hamiltonians representing the two end states. This complicates FE estimation by Bennett acceptance ratio (BAR), free energy perturbation (FEP), and thermodynamic integration (TI) methods, thus reducing their calculation efficiency. In this work, we develop a new type of repulsive soft-core potential, called Gaussian soft-core (GSC) potential, with two parameters controlling its maximum and width. The main advantage of this potential is the linearity of the hybrid Hamiltonian with respect to λ, thus permitting the direct application of BAR, FEP, TI, and other variant FE methods. The accuracy and efficiency of the GSC potential are demonstrated by comparing the free energies of annihilation determined for 13 small molecules and an alchemical mutation of a protein side chain. In addition, in combination with a TI integrand (∂H/∂λ) estimation strategy, we show that GSC can considerably reduce the number of λ simulations compared to the commonly used separation-shifted soft-core potential.
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Affiliation(s)
- Yaozong Li
- Department of Chemistry, Umeå University, SE-901 87 Umeå, Sweden.,Department of Biochemistry, University of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland
| | - Kwangho Nam
- Department of Chemistry, Umeå University, SE-901 87 Umeå, Sweden.,Department of Chemistry and Biochemistry, University of Texas at Arlington, Arlington, Texas 76019-0065, United States
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Ruiter AD, Oostenbrink C. Extended Thermodynamic Integration: Efficient Prediction of Lambda Derivatives at Nonsimulated Points. J Chem Theory Comput 2016; 12:4476-86. [PMID: 27494138 DOI: 10.1021/acs.jctc.6b00458] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Thermodynamic integration (TI) is one of the most commonly used free-energy calculation methods. The derivative of the Hamiltonian with respect to lambda, ⟨∂H/∂λ⟩, is determined at multiple λ-points. Because a numerical integration step is necessary, high curvature regions require simulations at densely spaced λ-points. Here, the principle of extended TI is introduced, where ⟨∂H/∂λ⟩ values are predicted at nonsimulated λ-points. On the basis of three model systems, it is shown that extended TI requires significantly fewer λ-points than regular TI to obtain similar accuracy.
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Affiliation(s)
- Anita de Ruiter
- Institute for Molecular Modeling and Simulation, University of Natural Resources and Life Sciences (BOKU) Vienna, 1180 Wien, Austria
| | - Chris Oostenbrink
- Institute for Molecular Modeling and Simulation, University of Natural Resources and Life Sciences (BOKU) Vienna, 1180 Wien, Austria
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Olsson MA, Söderhjelm P, Ryde U. Converging ligand-binding free energies obtained with free-energy perturbations at the quantum mechanical level. J Comput Chem 2016; 37:1589-600. [PMID: 27117350 PMCID: PMC5074236 DOI: 10.1002/jcc.24375] [Citation(s) in RCA: 40] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2016] [Revised: 03/08/2016] [Accepted: 03/09/2016] [Indexed: 12/15/2022]
Abstract
In this article, the convergence of quantum mechanical (QM) free-energy simulations based on molecular dynamics simulations at the molecular mechanics (MM) level has been investigated. We have estimated relative free energies for the binding of nine cyclic carboxylate ligands to the octa-acid deep-cavity host, including the host, the ligand, and all water molecules within 4.5 Å of the ligand in the QM calculations (158-224 atoms). We use single-step exponential averaging (ssEA) and the non-Boltzmann Bennett acceptance ratio (NBB) methods to estimate QM/MM free energy with the semi-empirical PM6-DH2X method, both based on interaction energies. We show that ssEA with cumulant expansion gives a better convergence and uses half as many QM calculations as NBB, although the two methods give consistent results. With 720,000 QM calculations per transformation, QM/MM free-energy estimates with a precision of 1 kJ/mol can be obtained for all eight relative energies with ssEA, showing that this approach can be used to calculate converged QM/MM binding free energies for realistic systems and large QM partitions. © 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.
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Affiliation(s)
- Martin A Olsson
- Department of Theoretical Chemistry, Lund University, Chemical Centre, P. O. Box 124, Lund, SE-221 00, Sweden
| | - Pär Söderhjelm
- Department of Biophysical Chemistry, Lund University, Chemical Centre, P. O. Box 124, Lund, SE-221 00, Sweden
| | - Ulf Ryde
- Department of Theoretical Chemistry, Lund University, Chemical Centre, P. O. Box 124, Lund, SE-221 00, Sweden
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Min D, Zheng L, Harris W, Chen M, Lv C, Yang W. Practically Efficient QM/MM Alchemical Free Energy Simulations: The Orthogonal Space Random Walk Strategy. J Chem Theory Comput 2015; 6:2253-66. [PMID: 26613484 DOI: 10.1021/ct100033s] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The difference between free energy changes occurring at two chemical states can be rigorously estimated via alchemical free energy (AFE) simulations. Traditionally, most AFE simulations are carried out under the classical energy potential treatment; then, accuracy and applicability of AFE simulations are limited. In the present work, we integrate a recent second-order generalized ensemble strategy, the orthogonal space random walk (OSRW) method, into the combined quantum mechanical/molecular mechanical (QM/MM) potential based AFE simulation scheme. Thereby, within a commonly affordable simulation length, accurate QM/MM alchemical free energy simulations can be achieved. As revealed by the model study on the equilibrium of a tautomerization process of hydrated 3-hydroxypyrazole and by the model calculations of the redox potentials of two flavin derivatives, lumichrome (LC) and riboflavin (RF) in aqueous solution, the present OSRW-based scheme could be a viable path toward the realization of practically efficient QM/MM AFE simulations.
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Affiliation(s)
- Donghong Min
- Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306
| | - Lianqing Zheng
- Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306
| | - William Harris
- Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306
| | - Mengen Chen
- Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306
| | - Chao Lv
- Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306
| | - Wei Yang
- Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306
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Mikulskis P, Genheden S, Ryde U. A large-scale test of free-energy simulation estimates of protein-ligand binding affinities. J Chem Inf Model 2014; 54:2794-806. [PMID: 25264937 DOI: 10.1021/ci5004027] [Citation(s) in RCA: 52] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
We have performed a large-scale test of alchemical perturbation calculations with the Bennett acceptance-ratio (BAR) approach to estimate relative affinities for the binding of 107 ligands to 10 different proteins. Employing 20-Å truncated spherical systems and only one intermediate state in the perturbations, we obtain an error of less than 4 kJ/mol for 54% of the studied relative affinities and a precision of 0.5 kJ/mol on average. However, only four of the proteins gave acceptable errors, correlations, and rankings. The results could be improved by using nine intermediate states in the simulations or including the entire protein in the simulations using periodic boundary conditions. However, 27 of the calculated affinities still gave errors of more than 4 kJ/mol, and for three of the proteins the results were not satisfactory. This shows that the performance of BAR calculations depends on the target protein and that several transformations gave poor results owing to limitations in the molecular-mechanics force field or the restricted sampling possible within a reasonable simulation time. Still, the BAR results are better than docking calculations for most of the proteins.
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Affiliation(s)
- Paulius Mikulskis
- Department of Theoretical Chemistry, Lund University, Chemical Centre , P.O. Box 124, SE-221 00 Lund, Sweden
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Abstract
Free energy calculations are extremely useful for investigating small-molecule biophysical properties such as protein-ligand binding affinities and partition coefficients. However, these calculations are also notoriously difficult to implement correctly. In this chapter, we review standard methods for computing free energy via simulation, discussing current best practices and examining potential pitfalls for computational researchers performing them for the first time. We include a variety of examples and tips for how to set up and conduct these calculations, including applications to relative binding affinities and small-molecule solvation free energies.
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Affiliation(s)
- Michael R Shirts
- Department of Chemical Engineering, University of Virginia, Charlottesville, VA, USA.
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Abstract
Free energy calculations are increasingly of interest for computing biophysical properties of novel small molecules of interest in drug design, such as protein-ligand binding affinities and small molecule partition coefficients. However, these calculations are also notoriously difficult to implement correctly. In this article, we review standard methods for computing free energy differences via simulation, discuss current best practices, and examine potential pitfalls for computational researchers without extensive experience in such calculations. We include a variety of examples and tips for how to set up and conduct these calculations, including applications to relative binding affinities and absolute binding free energies.
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Affiliation(s)
- Michael R Shirts
- Department of Chemical Engineering, University of Virginia, Charlottesville, VA, USA.
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Brooks B, Brooks C, MacKerell A, Nilsson L, Petrella R, Roux B, Won Y, Archontis G, Bartels C, Boresch S, Caflisch A, Caves L, Cui Q, Dinner A, Feig M, Fischer S, Gao J, Hodoscek M, Im W, Kuczera K, Lazaridis T, Ma J, Ovchinnikov V, Paci E, Pastor R, Post C, Pu J, Schaefer M, Tidor B, Venable RM, Woodcock HL, Wu X, Yang W, York D, Karplus M. CHARMM: the biomolecular simulation program. J Comput Chem 2009; 30:1545-614. [PMID: 19444816 PMCID: PMC2810661 DOI: 10.1002/jcc.21287] [Citation(s) in RCA: 6061] [Impact Index Per Article: 404.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
CHARMM (Chemistry at HARvard Molecular Mechanics) is a highly versatile and widely used molecular simulation program. It has been developed over the last three decades with a primary focus on molecules of biological interest, including proteins, peptides, lipids, nucleic acids, carbohydrates, and small molecule ligands, as they occur in solution, crystals, and membrane environments. For the study of such systems, the program provides a large suite of computational tools that include numerous conformational and path sampling methods, free energy estimators, molecular minimization, dynamics, and analysis techniques, and model-building capabilities. The CHARMM program is applicable to problems involving a much broader class of many-particle systems. Calculations with CHARMM can be performed using a number of different energy functions and models, from mixed quantum mechanical-molecular mechanical force fields, to all-atom classical potential energy functions with explicit solvent and various boundary conditions, to implicit solvent and membrane models. The program has been ported to numerous platforms in both serial and parallel architectures. This article provides an overview of the program as it exists today with an emphasis on developments since the publication of the original CHARMM article in 1983.
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Affiliation(s)
- B.R. Brooks
- Laboratory of Computational Biology, National Heart, Lung, and
Blood Institute, National Institutes of Health, Bethesda, MD 20892
| | - C.L. Brooks
- Departments of Chemistry & Biophysics, University of
Michigan, Ann Arbor, MI 48109
| | - A.D. MacKerell
- Department of Pharmaceutical Sciences, School of Pharmacy,
University of Maryland, Baltimore, MD, 21201
| | - L. Nilsson
- Karolinska Institutet, Department of Biosciences and Nutrition,
SE-141 57, Huddinge, Sweden
| | - R.J. Petrella
- Department of Chemistry and Chemical Biology, Harvard University,
Cambridge, MA 02138
- Department of Medicine, Harvard Medical School, Boston, MA
02115
| | - B. Roux
- Department of Biochemistry and Molecular Biology, University of
Chicago, Gordon Center for Integrative Science, Chicago, IL 60637
| | - Y. Won
- Department of Chemistry, Hanyang University, Seoul
133–792 Korea
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | - M. Karplus
- Department of Chemistry and Chemical Biology, Harvard University,
Cambridge, MA 02138
- Laboratoire de Chimie Biophysique, ISIS, Université de
Strasbourg, 67000 Strasbourg France
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Random walk in orthogonal space to achieve efficient free-energy simulation of complex systems. Proc Natl Acad Sci U S A 2008; 105:20227-32. [PMID: 19075242 DOI: 10.1073/pnas.0810631106] [Citation(s) in RCA: 249] [Impact Index Per Article: 15.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022] Open
Abstract
In the past few decades, many ingenious efforts have been made in the development of free-energy simulation methods. Because complex systems often undergo nontrivial structural transition during state switching, achieving efficient free-energy calculation can be challenging. As identified earlier, the "Hamiltonian" lagging, which reveals the fact that necessary structural relaxation falls behind the order parameter move, has been a primary problem for generally low free-energy simulation efficiency. Here, we propose an algorithm by achieving a random walk in both the order parameter space and its generalized force space; thereby, the order parameter move and the required conformational relaxation can be efficiently synchronized. As demonstrated in both the alchemical transition and the conformational transition, a leapfrog improvement in free-energy simulation efficiency can be obtained; for instance, (i) it allows us to solve a notoriously challenging problem: accurately predicting the pK(a) value of a buried titratable residue, Asp-66, in the interior of the V66E staphylococcal nuclease mutant, and (ii) it allows us to gain superior efficiency over the metadynamics algorithm.
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Steinbrecher T, Mobley DL, Case DA. Nonlinear scaling schemes for Lennard-Jones interactions in free energy calculations. J Chem Phys 2008; 127:214108. [PMID: 18067350 DOI: 10.1063/1.2799191] [Citation(s) in RCA: 236] [Impact Index Per Article: 14.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Alchemical free energy calculations provide a means for the accurate determination of free energies from atomistic simulations and are increasingly used as a tool for computational studies of protein-ligand interactions. Much attention has been placed on efficient ways to deal with the "endpoint singularity" effect that can cause simulation instabilities when changing the number of atoms. In this study we compare the performance of linear and several nonlinear transformation methods, among them separation shifted "soft core" scaling, for a popular test system, the hydration free energy of an amino acid side chain. All the nonlinear methods yield similar results if extensive sampling is performed, but soft core scaling provides smooth lambda curves that are best suited for commonly used numerical integration schemes. Additionally, results from a more flexible solute, hexane, will also be discussed.
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Affiliation(s)
- Thomas Steinbrecher
- Department of Molecular Biology, The Scripps Research Institute, La Jolla, San Diego, California 92037, USA.
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