1
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Katznelson H, Rahav S. Nonuniform convergence in moment expansions of integral work relations. Phys Rev E 2022; 105:024123. [PMID: 35291177 DOI: 10.1103/physreve.105.024123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2021] [Accepted: 01/19/2022] [Indexed: 06/14/2023]
Abstract
Exponential averages that appear in integral fluctuation theorems can be recast as a sum over moments of thermodynamic observables. We use two examples to show that such moment series can exhibit nonuniform convergence in certain singular limits. The first example is a simple model of a process with measurement and feedback. In this example, the limit of interest is that of error-free measurements. The second system we study is an ideal gas particle inside an (infinitely) fast expanding piston. Both examples show qualitative similarities; the low-order moments are close to their limiting value, while high-order moments strongly deviate from their limit. As the limit is approached the transition between the two groups of moments is pushed toward higher and higher moments. Our findings highlight the importance of the ordering of limits in certain nonequilibrium-related calculations.
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Affiliation(s)
- Hila Katznelson
- Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 3200008, Israel
| | - Saar Rahav
- Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 3200008, Israel
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2
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Roussey NM, Dickson A. Enhanced Jarzynski free energy calculations using weighted ensemble. J Chem Phys 2020; 153:134116. [PMID: 33032408 PMCID: PMC7544513 DOI: 10.1063/5.0020600] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2020] [Accepted: 09/16/2020] [Indexed: 02/07/2023] Open
Abstract
The free energy of transitions between stable states is the key thermodynamic quantity that governs the relative probabilities of the forward and reverse reactions and the ratio of state probabilities at equilibrium. The binding free energy of a drug and its receptor is of particular interest, as it serves as an optimization function for drug design. Over the years, many computational methods have been developed to calculate binding free energies, and while many of these methods have a long history, issues such as convergence of free energy estimates and the projection of a binding process onto order parameters remain. Over 20 years ago, the Jarzynski equality was derived with the promise to calculate equilibrium free energies by measuring the work applied to short nonequilibrium trajectories. However, these calculations were found to be dominated by trajectories with low applied work that occur with extremely low probability. Here, we examine the combination of weighted ensemble algorithms with the Jarzynski equality. In this combined method, an ensemble of nonequilibrium trajectories are run in parallel, and cloning and merging operations are used to preferentially sample low-work trajectories that dominate the free energy calculations. Two additional methods are also examined: (i) a novel weighted ensemble resampler that samples trajectories directly according to their importance to the work of work and (ii) the diffusion Monte Carlo method using the applied work as the selection potential. We thoroughly examine both the accuracy and efficiency of unbinding free energy calculations for a series of model Lennard-Jones atom pairs with interaction strengths ranging from 2 kcal/mol to 20 kcal/mol. We find that weighted ensemble calculations can more efficiently determine accurate binding free energies, especially for deeper Lennard-Jones well depths.
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Affiliation(s)
- Nicole M. Roussey
- Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, Michigan 48823, USA
| | - Alex Dickson
- Author to whom correspondence should be addressed:
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3
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Mey ASJS, Allen BK, Macdonald HEB, Chodera JD, Hahn DF, Kuhn M, Michel J, Mobley DL, Naden LN, Prasad S, Rizzi A, Scheen J, Shirts MR, Tresadern G, Xu H. Best Practices for Alchemical Free Energy Calculations [Article v1.0]. LIVING JOURNAL OF COMPUTATIONAL MOLECULAR SCIENCE 2020; 2:18378. [PMID: 34458687 PMCID: PMC8388617 DOI: 10.33011/livecoms.2.1.18378] [Citation(s) in RCA: 114] [Impact Index Per Article: 28.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/13/2023]
Abstract
Alchemical free energy calculations are a useful tool for predicting free energy differences associated with the transfer of molecules from one environment to another. The hallmark of these methods is the use of "bridging" potential energy functions representing alchemical intermediate states that cannot exist as real chemical species. The data collected from these bridging alchemical thermodynamic states allows the efficient computation of transfer free energies (or differences in transfer free energies) with orders of magnitude less simulation time than simulating the transfer process directly. While these methods are highly flexible, care must be taken in avoiding common pitfalls to ensure that computed free energy differences can be robust and reproducible for the chosen force field, and that appropriate corrections are included to permit direct comparison with experimental data. In this paper, we review current best practices for several popular application domains of alchemical free energy calculations performed with equilibrium simulations, in particular relative and absolute small molecule binding free energy calculations to biomolecular targets.
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Affiliation(s)
- Antonia S. J. S. Mey
- EaStCHEM School of Chemistry, David Brewster Road, Joseph Black Building, The King’s Buildings, Edinburgh, EH9 3FJ, UK
| | | | - Hannah E. Bruce Macdonald
- Computational and Systems Biology Program, Sloan Kettering Institute, Memorial Sloan Kettering Cancer Center, New York NY, USA
| | - John D. Chodera
- Computational and Systems Biology Program, Sloan Kettering Institute, Memorial Sloan Kettering Cancer Center, New York NY, USA
| | - David F. Hahn
- Computational Chemistry, Janssen Research & Development, Turnhoutseweg 30, Beerse B-2340, Belgium
| | - Maximilian Kuhn
- EaStCHEM School of Chemistry, David Brewster Road, Joseph Black Building, The King’s Buildings, Edinburgh, EH9 3FJ, UK
- Cresset, Cambridgeshire, UK
| | - Julien Michel
- EaStCHEM School of Chemistry, David Brewster Road, Joseph Black Building, The King’s Buildings, Edinburgh, EH9 3FJ, UK
| | - David L. Mobley
- Departments of Pharmaceutical Sciences and Chemistry, University of California, Irvine, Irvine, USA
| | - Levi N. Naden
- Molecular Sciences Software Institute, Blacksburg VA, USA
| | | | - Andrea Rizzi
- Silicon Therapeutics, Boston, MA, USA
- Tri-Institutional Training Program in Computational Biology and Medicine, New York, NY, USA
| | - Jenke Scheen
- EaStCHEM School of Chemistry, David Brewster Road, Joseph Black Building, The King’s Buildings, Edinburgh, EH9 3FJ, UK
| | | | - Gary Tresadern
- Computational Chemistry, Janssen Research & Development, Turnhoutseweg 30, Beerse B-2340, Belgium
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4
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Copperman J, Aristoff D, Makarov DE, Simpson G, Zuckerman DM. Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture. J Chem Phys 2019; 151:174108. [PMID: 31703496 PMCID: PMC7043855 DOI: 10.1063/1.5120511] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2019] [Accepted: 10/14/2019] [Indexed: 01/04/2023] Open
Abstract
Probability currents are fundamental in characterizing the kinetics of nonequilibrium processes. Notably, the steady-state current Jss for a source-sink system can provide the exact mean-first-passage time (MFPT) for the transition from the source to sink. Because transient nonequilibrium behavior is quantified in some modern path sampling approaches, such as the "weighted ensemble" strategy, there is strong motivation to determine bounds on Jss-and hence on the MFPT-as the system evolves in time. Here, we show that Jss is bounded from above and below by the maximum and minimum, respectively, of the current as a function of the spatial coordinate at any time t for one-dimensional systems undergoing overdamped Langevin (i.e., Smoluchowski) dynamics and for higher-dimensional Smoluchowski systems satisfying certain assumptions when projected onto a single dimension. These bounds become tighter with time, making them of potential practical utility in a scheme for estimating Jss and the long time scale kinetics of complex systems. Conceptually, the bounds result from the fact that extrema of the transient currents relax toward the steady-state current.
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Affiliation(s)
- Jeremy Copperman
- Department of Biomedical Engineering, Oregon Health and Science University, Portland, Oregon 97239, USA
| | - David Aristoff
- Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523, USA
| | - Dmitrii E Makarov
- Department of Chemistry and Oden Institute for Computational Engineering and Sciences, University of Texas, Austin, Texas 78712, USA
| | - Gideon Simpson
- Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104, USA
| | - Daniel M Zuckerman
- Department of Biomedical Engineering, Oregon Health and Science University, Portland, Oregon 97239, USA
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5
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Dinner AR, Mattingly JC, Tempkin JOB, Van Koten B, Weare J. Trajectory stratification of stochastic dynamics. SIAM REVIEW. SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS 2018; 60:909-938. [PMID: 34650314 PMCID: PMC8514164 DOI: 10.1137/16m1104329] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We present a general mathematical framework for trajectory stratification for simulating rare events. Trajectory stratification involves decomposing trajectories of the underlying process into fragments limited to restricted regions of state space (strata), computing averages over the distributions of the trajectory fragments within the strata with minimal communication between them, and combining those averages with appropriate weights to yield averages with respect to the original underlying process. Our framework reveals the full generality and flexibility of trajectory stratification, and it illuminates a common mathematical structure shared by existing algorithms for sampling rare events. We demonstrate the power of the framework by defining strata in terms of both points in time and path-dependent variables for efficiently estimating averages that were not previously tractable.
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Affiliation(s)
- Aaron R. Dinner
- James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
- Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, USA
| | - Jonathan C. Mattingly
- Departments of Mathematics and Statistical Science, Duke University, Durham, North Carolina 27708, USA
| | - Jeremy O. B. Tempkin
- James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
- Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, USA
| | - Brian Van Koten
- Department of Statistics, The University of Chicago, Chicago, Illinois 60637, USA
| | - Jonathan Weare
- James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
- Department of Statistics, The University of Chicago, Chicago, Illinois 60637, USA
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6
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Wang M, Mei Y, Ryde U. Predicting Relative Binding Affinity Using Nonequilibrium QM/MM Simulations. J Chem Theory Comput 2018; 14:6613-6622. [DOI: 10.1021/acs.jctc.8b00685] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
Affiliation(s)
- Meiting Wang
- State Key Laboratory of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
- Department of Theoretical Chemistry, Lund University, Chemical Centre, P.O. Box 124, SE-221 00 Lund, Sweden
| | - Ye Mei
- State Key Laboratory of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
- NYU−ECNU Center for Computational Chemistry at NYU Shanghai, Shanghai 200062, China
| | - Ulf Ryde
- Department of Theoretical Chemistry, Lund University, Chemical Centre, P.O. Box 124, SE-221 00 Lund, Sweden
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7
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Zhu L, Wang J. Calculating the free energy difference by applying the Jarzynski equality to a virtual integrable system. Phys Rev E 2018; 98:022117. [PMID: 30253520 DOI: 10.1103/physreve.98.022117] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2018] [Indexed: 11/07/2022]
Abstract
The Jarzynski equality (JE) provides a nonequilibrium method to measure and calculate the free energy difference (FED). Note that if two systems share the same Hamiltonian at two equilibrium states, respectively, they share the same FED between these two equilibrium states as well. Therefore the calculation of the FED of a system may be facilitated by considering instead another virtual system designed to this end. Taking advantage of this flexibility and the JE, we show that by introducing an integrable virtual system, the evolution problem involved in the JE can be solved. As a consequence, FED is expressed in the form of an equilibrium equality, in contrast with the nonequilibrium JE it is based on. Numerically, this result allows FED to be computed by sampling the canonical ensemble directly and the computational cost can be significantly reduced. The effectiveness and efficiency of this scheme are illustrated with numerical studies of several representative model systems.
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Affiliation(s)
- Liyun Zhu
- Department of Physics, Key Laboratory of Low Dimensional Condensed Matter Physics (Department of Education of Fujian Province), and Jiujiang Research Institute, Xiamen University, Xiamen 361005, Fujian, China
| | - Jiao Wang
- Department of Physics, Key Laboratory of Low Dimensional Condensed Matter Physics (Department of Education of Fujian Province), and Jiujiang Research Institute, Xiamen University, Xiamen 361005, Fujian, China
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8
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Li PF, Jia XY, Wang MT, Mei Y. Comparison of Accuracy and Convergence Rate between Equilibrium and Nonequilibrium Alchemical Transformations for Calculation of Relative Binding Free Energy. CHINESE J CHEM PHYS 2017. [DOI: 10.1063/1674-0068/30/cjcp1711204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
Affiliation(s)
- Peng-fei Li
- State Key Laboratory of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
| | - Xiang-yu Jia
- State Key Laboratory of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai, Shanghai 200062, China
| | - Mei-ting Wang
- State Key Laboratory of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
| | - Ye Mei
- State Key Laboratory of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai, Shanghai 200062, China
- Department of Chemistry and Biochemistry, University of Oklahoma, Norman OK 73019, USA
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9
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Wang M, Li P, Jia X, Liu W, Shao Y, Hu W, Zheng J, Brooks BR, Mei Y. Efficient Strategy for the Calculation of Solvation Free Energies in Water and Chloroform at the Quantum Mechanical/Molecular Mechanical Level. J Chem Inf Model 2017; 57:2476-2489. [DOI: 10.1021/acs.jcim.7b00001] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Meiting Wang
- State Key Laboratory
of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
| | - Pengfei Li
- State Key Laboratory
of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
| | - Xiangyu Jia
- State Key Laboratory
of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
| | - Wei Liu
- State Key Laboratory
of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
| | - Yihan Shao
- Q-Chem Inc., 6601 Owens Drive, Suite 105, Pleasanton, California 94588, United States
- Department of Chemistry and Biochemistry, University of Oklahoma, Norman Oklahoma 73019, United States
| | - Wenxin Hu
- The Computer
Center, School of Computer Science and Software Engineering, East China Normal University, Shanghai 200062, China
| | - Jun Zheng
- The Computer
Center, School of Computer Science and Software Engineering, East China Normal University, Shanghai 200062, China
| | - Bernard R. Brooks
- Laboratory of Computational Biology, National Institutes of Health, National Heart, Lung and Blood Institute, 5635 Fishers Lane, T-900 Suite, Rockville, Maryland 20852, United States
| | - Ye Mei
- State Key Laboratory
of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200062, China
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai, Shanghai 200062, China
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10
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Wan B, Yang C, Wang Y, Zhou X. Jarzynski matrix equality: Calculating the free-energy difference by nonequilibrium simulations with an arbitrary initial distribution. Phys Rev E 2016; 93:043312. [PMID: 27176433 DOI: 10.1103/physreve.93.043312] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2015] [Indexed: 06/05/2023]
Abstract
The Jarzynski equality (JE) method, which relates the work of a nonequilibrium process to the free-energy difference between its initial and final states, provides an efficient way to calculate free energies of thermodynamic systems in simulations or experiments. However, more extensive applications of the JE are hindered by the requirement that the initial state must be in equilibrium. In this work we extend the JE method to be the Jarzynski matrix equality (JME) method, which relates the work of trajectories connecting metastable conformational regions to their local free energies, and thus we can estimate the free energy from the nonequilibrium trajectories starting from an almost arbitrary initial distribution. We then apply the JME to toy models, Lennard-Jones fluids, and polymer chain models, demonstrating its efficiency in free-energy calculations with satisfactory accuracy. The JME extends the applicability of the nonequilibrium methods to complex systems whose initial equilibrium states are difficult to reach.
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Affiliation(s)
- Biao Wan
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Cheng Yang
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Yanting Wang
- State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Xin Zhou
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
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11
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Giovannelli E, Cardini G, Chelli R. Elastic Barrier Dynamical Freezing in Free Energy Calculations: A Way To Speed Up Nonequilibrium Molecular Dynamics Simulations by Orders of Magnitude. J Chem Theory Comput 2016; 12:1029-39. [PMID: 26771534 DOI: 10.1021/acs.jctc.5b01117] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022]
Abstract
An important issue concerning computer simulations addressed to free energy estimates via nonequilibrium work theorems, such as the Jarzynski equality [Phys. Rev. Lett. 1997, 78, 2690], is the computational effort required to achieve results with acceptable accuracy. In this respect, the dynamical freezing approach [Phys. Rev. E 2009, 80, 041124] has been shown to improve the efficiency of this kind of simulations, by blocking the dynamics of particles located outside an established mobility region. In this report, we show that dynamical freezing produces a systematic spurious decrease of the particle density inside the mobility region. As a consequence, the requirements to apply nonequilibrium work theorems are only approximately met. Starting from these considerations, we have developed a simulation scheme, called "elastic barrier dynamical freezing", according to which a stiff potential-energy barrier is enforced at the boundaries of the mobility region, preventing the particles from leaving this region of space during the nonequilibrium trajectories. The method, tested on the calculation of the distance-dependent free energy of a dimer immersed into a Lennard-Jones fluid, provides an accuracy comparable to the conventional steered molecular dynamics, with a computational speedup exceeding a few orders of magnitude.
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Affiliation(s)
- Edoardo Giovannelli
- Dipartimento di Chimica, Università di Firenze , Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy
| | - Gianni Cardini
- Dipartimento di Chimica, Università di Firenze , Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy
| | - Riccardo Chelli
- Dipartimento di Chimica, Università di Firenze , Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy
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12
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Giovannelli E, Gellini C, Pietraperzia G, Cardini G, Chelli R. Nonequilibrium Candidate Monte Carlo Simulations with Configurational Freezing Schemes. J Chem Theory Comput 2015; 10:4273-83. [PMID: 26588124 DOI: 10.1021/ct500340b] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Nonequilibrium Candidate Monte Carlo simulation [Nilmeier et al., Proc. Natl. Acad. Sci. U.S.A. 2011, 108, E1009-E1018] is a tool devised to design Monte Carlo moves with high acceptance probabilities that connect uncorrelated configurations. Such moves are generated through nonequilibrium driven dynamics, producing candidate configurations accepted with a Monte Carlo-like criterion that preserves the equilibrium distribution. The probability of accepting a candidate configuration as the next sample in the Markov chain basically depends on the work performed on the system during the nonequilibrium trajectory and increases with decreasing such a work. It is thus strategically relevant to find ways of producing nonequilibrium moves with low work, namely moves where dissipation is as low as possible. This is the goal of our methodology, in which we combine Nonequilibrium Candidate Monte Carlo with Configurational Freezing schemes developed by Nicolini et al. (J. Chem. Theory Comput. 2011, 7, 582-593). The idea is to limit the configurational sampling to particles of a well-established region of the simulation sample, namely the region where dissipation occurs, while leaving fixed the other particles. This allows to make the system relaxation faster around the region perturbed by the finite-time switching move and hence to reduce the dissipated work, eventually enhancing the probability of accepting the generated move. Our combined approach enhances significantly configurational sampling, as shown by the case of a bistable dimer immersed in a dense fluid.
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Affiliation(s)
- Edoardo Giovannelli
- Dipartimento di Chimica, Università di Firenze , Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy
| | - Cristina Gellini
- Dipartimento di Chimica, Università di Firenze , Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy.,European Laboratory for Nonlinear Spectroscopy (LENS), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
| | - Giangaetano Pietraperzia
- Dipartimento di Chimica, Università di Firenze , Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy.,European Laboratory for Nonlinear Spectroscopy (LENS), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
| | - Gianni Cardini
- Dipartimento di Chimica, Università di Firenze , Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy.,European Laboratory for Nonlinear Spectroscopy (LENS), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
| | - Riccardo Chelli
- Dipartimento di Chimica, Università di Firenze , Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy.,European Laboratory for Nonlinear Spectroscopy (LENS), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
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13
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Acconcia TV, Bonança MVS. Degenerate optimal paths in thermally isolated systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042141. [PMID: 25974472 DOI: 10.1103/physreve.91.042141] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2015] [Indexed: 06/04/2023]
Abstract
We present an analysis of the work performed on a system of interest that is kept thermally isolated during the switching of a control parameter. We show that there exists, for a certain class of systems, a finite-time family of switching protocols for which the work is equal to the quasistatic value. These optimal paths are obtained within linear response for systems initially prepared in a canonical distribution. According to our approach, such protocols are composed of a linear part plus a function which is odd with respect to time reversal. For systems with one degree of freedom, we claim that these optimal paths may also lead to the conservation of the corresponding adiabatic invariant. This points to an interesting connection between work and the conservation of the volume enclosed by the energy shell. To illustrate our findings, we solve analytically the harmonic oscillator and present numerical results for certain anharmonic examples.
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Affiliation(s)
- Thiago V Acconcia
- Instituto de Física 'Gleb Wataghin', Universidade Estadual de Campinas, 13083-859 Campinas, São Paulo, Brazil
| | - Marcus V S Bonança
- Instituto de Física 'Gleb Wataghin', Universidade Estadual de Campinas, 13083-859 Campinas, São Paulo, Brazil
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14
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Zerbetto M, Frezzato D. Towards bulk thermodynamics via non-equilibrium methods: gaseous methane as a case study. Phys Chem Chem Phys 2015; 17:1966-79. [DOI: 10.1039/c4cp03815k] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023]
Abstract
The equation of state of bulk materials is achieved via thermodynamic derivatives of the free energy yielded by nonequilibrium transformations and Jarzynski equality.
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Affiliation(s)
- Mirco Zerbetto
- Dipartimento di Scienze Chimiche
- Università degli Studi di Padova
- I-35131 Padova
- Italy
| | - Diego Frezzato
- Dipartimento di Scienze Chimiche
- Università degli Studi di Padova
- I-35131 Padova
- Italy
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15
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Le X, Gu Q, Xu J. Identifying MurI uncompetitive inhibitors by correlating decomposed binding energies with bioactivity. RSC Adv 2015. [DOI: 10.1039/c5ra03079j] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
Abstract
MurI uncompetitive inhibitors can be virtually identified by a new method that correlates decomposed binding free energies with the bioactivity.
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Affiliation(s)
- Xiu Le
- Research Center for Drug Discovery
- School of Pharmaceutical Sciences
- Sun Yat-Sen University
- Guangzhou 510006
- China
| | - Qiong Gu
- Research Center for Drug Discovery
- School of Pharmaceutical Sciences
- Sun Yat-Sen University
- Guangzhou 510006
- China
| | - Jun Xu
- Research Center for Drug Discovery
- School of Pharmaceutical Sciences
- Sun Yat-Sen University
- Guangzhou 510006
- China
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16
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Zerbetto M, Piserchia A, Frezzato D. Looking for some free energy? Call JEFREE (…). J Comput Chem 2014; 35:1865-81. [DOI: 10.1002/jcc.23701] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2013] [Revised: 04/09/2014] [Accepted: 07/13/2014] [Indexed: 11/09/2022]
Affiliation(s)
- Mirco Zerbetto
- Dipartimento di Scienze Chimiche; Università degli Studi di Padova; via Marzolo 1 I-35131 Padova Italy
| | - Andrea Piserchia
- Dipartimento di Scienze Chimiche; Università degli Studi di Padova; via Marzolo 1 I-35131 Padova Italy
| | - Diego Frezzato
- Dipartimento di Scienze Chimiche; Università degli Studi di Padova; via Marzolo 1 I-35131 Padova Italy
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17
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Bonança MVS, Deffner S. Optimal driving of isothermal processes close to equilibrium. J Chem Phys 2014; 140:244119. [DOI: 10.1063/1.4885277] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Marcus V. S. Bonança
- Instituto de Física “Gleb Wataghin,” Universidade Estadual de Campinas, 13083-859, Campinas, São Paulo, Brazil
- Department of Chemistry and Biochemistry and Institute of Physical Sciences and Technology, University of Maryland, College Park, Maryland 20742, USA
| | - Sebastian Deffner
- Department of Chemistry and Biochemistry and Institute of Physical Sciences and Technology, University of Maryland, College Park, Maryland 20742, USA
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18
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Hartmann AK. High-precision work distributions for extreme nonequilibrium processes in large systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052103. [PMID: 25353736 DOI: 10.1103/physreve.89.052103] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2013] [Indexed: 06/04/2023]
Abstract
The distributions of work for strongly nonequilibrium processes are studied using a very general form of a large-deviation approach, which allows one to study distributions down to extremely small probabilities of almost arbitrary quantities of interest for equilibrium, nonequilibrium stationary, and even nonstationary processes. The method is applied to quickly vary the external field in a wide range B = 3 ↔ 0 for a critical (T = 2.269) two-dimensional Ising system of size L × L = 128 × 128. To obtain free-energy differences from the work distributions, they must be studied in ranges where the probabilities are as small as 10^{-240}, which is not possible using direct simulation approaches. By comparison with the exact free energies, which are available for this model for the zero-field case, one sees that the present approach allows one to obtain the free energy with a very high relative precision of 10^{-4}. This works well also for a nonzero field, i.e., for a case where standard umbrella-sampling methods are not efficient to calculate free energies. Furthermore, for the present case it is verified that the resulting distributions of work for forward and backward processes fulfill Crooks theorem with high precision. Finally, the free energy for the Ising magnet as a function of the field strength is obtained.
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19
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Giovannelli E, Gellini C, Pietraperzia G, Cardini G, Chelli R. Combining path-breaking with bidirectional nonequilibrium simulations to improve efficiency in free energy calculations. J Chem Phys 2014; 140:064104. [DOI: 10.1063/1.4863999] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/30/2023] Open
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20
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Chelli R, Gellini C, Pietraperzia G, Giovannelli E, Cardini G. Path-breaking schemes for nonequilibrium free energy calculations. J Chem Phys 2014; 138:214109. [PMID: 23758360 DOI: 10.1063/1.4808037] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We propose a path-breaking route to the enhancement of unidirectional nonequilibrium simulations for the calculation of free energy differences via Jarzynski's equality [C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)]. One of the most important limitations of unidirectional nonequilibrium simulations is the amount of realizations necessary to reach suitable convergence of the work exponential average featuring the Jarzynski's relationship. In this respect, a significant improvement of the performances could be obtained by finding a way of stopping trajectories with negligible contribution to the work exponential average, before their normal end. This is achieved using path-breaking schemes which are essentially based on periodic checks of the work dissipated during the pulling trajectories. Such schemes can be based either on breaking trajectories whose dissipated work exceeds a given threshold or on breaking trajectories with a probability increasing with the dissipated work. In both cases, the computer time needed to carry out a series of nonequilibrium trajectories is reduced up to a factor ranging from 2 to more than 10, at least for the processes under consideration in the present study. The efficiency depends on several aspects, such as the type of process, the number of check-points along the pathway and the pulling rate as well. The method is illustrated through radically different processes, i.e., the helix-coil transition of deca-alanine and the pulling of the distance between two methane molecules in water solution.
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Affiliation(s)
- Riccardo Chelli
- Dipartimento di Chimica, Università di Firenze, Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy.
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21
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Computing Equilibrium Free Energies Using Non-Equilibrium Molecular Dynamics. ENTROPY 2013. [DOI: 10.3390/e16010041] [Citation(s) in RCA: 50] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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22
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Nicolini P, Frezzato D, Gellini C, Bizzarri M, Chelli R. Toward quantitative estimates of binding affinities for protein-ligand systems involving large inhibitor compounds: a steered molecular dynamics simulation route. J Comput Chem 2013; 34:1561-76. [PMID: 23620471 DOI: 10.1002/jcc.23286] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2012] [Revised: 03/01/2013] [Accepted: 03/05/2013] [Indexed: 11/09/2022]
Abstract
Understanding binding mechanisms between enzymes and potential inhibitors and quantifying protein-ligand affinities in terms of binding free energy is of primary importance in drug design studies. In this respect, several approaches based on molecular dynamics simulations, often combined with docking techniques, have been exploited to investigate the physicochemical properties of complexes of pharmaceutical interest. Even if the geometric properties of a modeled protein-ligand complex can be well predicted by computational methods, it is still challenging to rank with chemical accuracy a series of ligand analogues in a consistent way. In this article, we face this issue calculating relative binding free energies of a focal adhesion kinase, an important target for the development of anticancer drugs, with pyrrolopyrimidine-based ligands having different inhibitory power. To this aim, we employ steered molecular dynamics simulations combined with nonequilibrium work theorems for free energy calculations. This technique proves very powerful when a series of ligand analogues is considered, allowing one to tackle estimation of protein-ligand relative binding free energies in a reasonable time. In our cases, the calculated binding affinities are comparable with those recovered from experiments by exploiting the Michaelis-Menten mechanism with a competitive inhibitor.
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Affiliation(s)
- Paolo Nicolini
- Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
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23
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Perišić O. Pulling-spring modulation as a method for improving the potential-of-mean-force reconstruction in single-molecule manipulation experiments. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:013303. [PMID: 23410456 DOI: 10.1103/physreve.87.013303] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2012] [Revised: 10/22/2012] [Indexed: 06/01/2023]
Abstract
The free-energy calculation is usually limited to close to equilibrium perturbation regimes because faster perturbations introduce a bias into the estimate. The Jarzynski equality offers a solution to this problem by directly connecting the free-energy difference and the external work, regardless how far from equilibrium that work may be. However, a limited sampling coupled to the fast perturbation introduces a slowly converging bias into the Jarzynski free-energy estimate also. In this paper we present two perturbation protocols devised with the intention to overcome the convergence issues of the Jarzynski-based potential of mean force estimation in the single-molecule, constant velocity manipulation experiments. The protocols are designed to improve the convergence issues by increasing the variation of the external work through the modulation of the spring used to pull a molecule. Of the two methods, the one which continuously changes the amplitude of the spring stiffness offers an excellent reconstruction and requires less than one tenth of the samples required by the normal, constant spring pulling to produce the same quality of the reconstruction.
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24
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Seifert U. Stochastic thermodynamics, fluctuation theorems and molecular machines. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2012; 75:126001. [PMID: 23168354 DOI: 10.1088/0034-4885/75/12/126001] [Citation(s) in RCA: 1198] [Impact Index Per Article: 99.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/18/2023]
Abstract
Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.
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Affiliation(s)
- Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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25
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Ozer G, Quirk S, Hernandez R. Thermodynamics of Decaalanine Stretching in Water Obtained by Adaptive Steered Molecular Dynamics Simulations. J Chem Theory Comput 2012; 8:4837-44. [DOI: 10.1021/ct300709u] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Gungor Ozer
- Center for Computational and
Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
United States
| | - Stephen Quirk
- Kimberly-Clark Corporation, Atlanta, Georgia 30076-2199, United States
| | - Rigoberto Hernandez
- Center for Computational and
Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
United States
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26
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Ozer G, Quirk S, Hernandez R. Adaptive steered molecular dynamics: validation of the selection criterion and benchmarking energetics in vacuum. J Chem Phys 2012; 136:215104. [PMID: 22697572 DOI: 10.1063/1.4725183] [Citation(s) in RCA: 62] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The potential of mean force (PMF) for stretching decaalanine in vacuum was determined earlier by Park and Schulten [J. Chem. Phys. 120, 5946 (2004)] in a landmark article demonstrating the efficacy of combining steered molecular dynamics and Jarzynski's nonequilibrium relation. In this study, the recently developed adaptive steered molecular dynamics (ASMD) algorithm [G. Ozer, E. Valeev, S. Quirk, and R. Hernandez, J. Chem. Theory Comput. 6, 3026 (2010)] is used to reproduce the PMF of the unraveling of decaalanine in vacuum by averaging over fewer nonequilibrium trajectories. The efficiency and accuracy of the method are demonstrated through the agreement with the earlier work by Park and Schulten, a series of convergence checks compared to alternate SMD pulling strategies, and an analytical proof. The nonequilibrium trajectories obtained through ASMD have also been used to analyze the intrapeptide hydrogen bonds along the stretching coordinate. As the decaalanine helix is stretched, the initially stabilized i → i + 4 contacts (α-helix) is replaced by i → i + 3 contacts (3(10)-helix). No significant formation of i → i + 5 hydrogen bonds (π-helix) is observed.
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Affiliation(s)
- Gungor Ozer
- Center for Computational and Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, USA
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27
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Kapri R. Hysteresis and nonequilibrium work theorem for DNA unzipping. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:041906. [PMID: 23214614 DOI: 10.1103/physreve.86.041906] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2012] [Revised: 06/09/2012] [Indexed: 06/01/2023]
Abstract
We study by using Monte Carlo simulations the hysteresis in unzipping and rezipping of a double stranded DNA (dsDNA) by pulling its strands in opposite directions in the fixed force ensemble. The force is increased at a constant rate from an initial value g(0) to some maximum value g(m) that lies above the phase boundary and then decreased back again to g(0). We observed hysteresis during a complete cycle of unzipping and rezipping. We obtained probability distributions of work performed over a cycle of unzipping and rezipping for various pulling rates. The mean of the distribution is found to be close (the difference being within 10%, except for very fast pulling) to the area of the hysteresis loop. We extract the equilibrium force versus separation isotherm by using the work theorem on repeated nonequilibrium force measurements. Our method is capable of reproducing the equilibrium and the nonequilibrium force-separation isotherms for the spontaneous rezipping of dsDNA.
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Affiliation(s)
- Rajeev Kapri
- Indian Institute of Science Education and Research Mohali, Knowledge City, Sector 81, SAS Nagar - 140 306, Punjab India.
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28
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Chelli R. Local Sampling in Steered Monte Carlo Simulations Decreases Dissipation and Enhances Free Energy Estimates via Nonequilibrium Work Theorems. J Chem Theory Comput 2012; 8:4040-52. [DOI: 10.1021/ct300348w] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/24/2023]
Affiliation(s)
- Riccardo Chelli
- Dipartimento di Chimica,
Università di Firenze,
Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy
- European Laboratory for Nonlinear Spectroscopy (LENS),
Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
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29
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Ballard AJ, Jarzynski C. Replica exchange with nonequilibrium switches: Enhancing equilibrium sampling by increasing replica overlap. J Chem Phys 2012; 136:194101. [DOI: 10.1063/1.4712028] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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30
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Davie SJ, Reid JC, Searles DJ. The free energy of expansion and contraction: Treatment of arbitrary systems using the Jarzynski equality. J Chem Phys 2012; 136:174111. [DOI: 10.1063/1.4707348] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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31
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Paliwal H, Shirts MR. A Benchmark Test Set for Alchemical Free Energy Transformations and Its Use to Quantify Error in Common Free Energy Methods. J Chem Theory Comput 2011; 7:4115-34. [PMID: 26598357 DOI: 10.1021/ct2003995] [Citation(s) in RCA: 115] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
There is a significant need for improved tools to validate thermophysical quantities computed via molecular simulation. In this paper we present the initial version of a benchmark set of testing methods for calculating free energies of molecular transformation in solution. This set is based on molecular changes common to many molecular design problems, such as insertion and deletion of atomic sites and changing atomic partial charges. We use this benchmark set to compare the statistical efficiency, reliability, and quality of uncertainty estimates for a number of published free energy methods, including thermodynamic integration, free energy perturbation, the Bennett acceptance ratio (BAR) and its multistate equivalent MBAR. We identify MBAR as the consistently best performing method, though other methods are frequently comparable in reliability and accuracy in many cases. We demonstrate that assumptions of Gaussian distributed errors in free energies are usually valid for most methods studied. We demonstrate that bootstrap error estimation is a robust and useful technique for estimating statistical variance for all free energy methods studied. This benchmark set is provided in a number of different file formats with the hope of becoming a useful and general tool for method comparisons.
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Affiliation(s)
- Himanshu Paliwal
- Department of Chemical Engineering, University of Virginia , Charlottesville, Virginia 22904-4741, United States
| | - Michael R Shirts
- Department of Chemical Engineering, University of Virginia , Charlottesville, Virginia 22904-4741, United States
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32
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Perišić O, Lu H. Efficient free-energy-profile reconstruction using adaptive stochastic perturbation protocols. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:056705. [PMID: 22181545 DOI: 10.1103/physreve.84.056705] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2009] [Revised: 08/07/2011] [Indexed: 05/31/2023]
Abstract
The Jarzynski-relation-based free-energy calculation is limited by the very slow convergence of the estimate when dissipation is high. We present two novel perturbation protocols able to significantly improve the quality of the potential of mean force (PMF) calculation by reducing the estimate's bias without increasing the number of samples. The protocols are directly applicable with both numerical simulations and real-life experiments. The improvement is achieved through the intentional but controlled widening of the distribution of the external work used to perturb a given system. Our protocols can achieve the same accuracy in PMF estimation as the normal constant velocity pulling with less than 10% of the required samples.
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Affiliation(s)
- Ognjen Perišić
- Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA
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33
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Rogers DM, Beck TL, Rempe SB. An Information Theory Approach to Nonlinear, Nonequilibrium Thermodynamics. JOURNAL OF STATISTICAL PHYSICS 2011; 145:385-409. [PMID: 22966210 PMCID: PMC3436205 DOI: 10.1007/s10955-011-0358-9] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Using the problem of ion channel thermodynamics as an example, we illustrate the idea of building up complex thermodynamic models by successively adding physical information. We present a new formulation of information algebra that generalizes methods of both information theory and statistical mechanics. From this foundation we derive a theory for ion channel kinetics, identifying a nonequilibrium 'process' free energy functional in addition to the well-known integrated work functionals. The Gibbs-Maxwell relation for the free energy functional is a Green-Kubo relation, applicable arbitrarily far from equilibrium, that captures the effect of non-local and time-dependent behavior from transient thermal and mechanical driving forces. Comparing the physical significance of the Lagrange multipliers to the canonical ensemble suggests definitions of nonequilibrium ensembles at constant capacitance or inductance in addition to constant resistance. Our result is that statistical mechanical descriptions derived from a few primitive algebraic operations on information can be used to create experimentally-relevant and computable models. By construction, these models may use information from more detailed atomistic simulations. Two surprising consequences to be explored in further work are that (in)distinguishability factors are automatically predicted from the problem formulation and that a direct analogue of the second law for thermodynamic entropy production is found by considering information loss in stochastic processes. The information loss identifies a novel contribution from the instantaneous information entropy that ensures non-negative loss.
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Affiliation(s)
- David M. Rogers
- Center for Biological and Materials Sciences, MS 0895, Sandia National Laboratories, Albuquerque, NM 87185, USA
| | - Thomas L. Beck
- Department of Chemistry, University of Cincinnati, Cincinnati, OH 45221-0172, USA
| | - Susan B. Rempe
- Center for Biological and Materials Sciences, MS 0895, Sandia National Laboratories, Albuquerque, NM 87185, USA
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34
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St-Pierre JF, Karttunen M, Mousseau N, Róg T, Bunker A. Use of Umbrella Sampling to Calculate the Entrance/Exit Pathway for Z-Pro-Prolinal Inhibitor in Prolyl Oligopeptidase. J Chem Theory Comput 2011; 7:1583-94. [PMID: 26596426 DOI: 10.1021/ct1007058] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023]
Abstract
Prolyl oligopeptidase (POP), a member of the prolyl endopeptidase family, is known to play a role in several neurological disorders. Its primary function is to cleave a wide range of small oligopeptides, including neuroactive peptides. We have used force biased molecular dynamics simulation to study the binding mechanism of POP. We examined three possible binding pathways using Steered Molecular Dynamics (SMD) and Umbrella Sampling (US) on a crystal structure of porcine POP with bound Z-pro-prolinal (ZPP). Using SMD, an exit pathway between the first and seventh blade of the β-propeller domain of POP was found to be a nonviable route. US on binding pathways through the β-propeller tunnel and the TYR190-GLN208 flexible loop at the interface between both POP domains allowed us to isolate the flexible loop pathway as the most probable. Further analysis of that pathway suggests a long-range covariation of the interdomain H-bond network, which indicates the possibility of large-scale domain reorientation observed in bacterial homologues and hypothesized to also occur in human POP.
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Affiliation(s)
- Jean-François St-Pierre
- Département de Physique and Regroupement Québécois sur les Matériaux de Pointe, Université de Montréal , C.P. 6128, succursale centre-ville, Montréal (Québec), Canada H3C 3J7
| | - Mikko Karttunen
- Department of Applied Mathematics, The University of Western Ontario , 1151 Richmond Street North, London (Ontario), Canada N6A 5B7
| | - Normand Mousseau
- Département de Physique and Regroupement Québécois sur les Matériaux de Pointe, Université de Montréal , C.P. 6128, succursale centre-ville, Montréal (Québec), Canada H3C 3J7
| | - Tomasz Róg
- Department of Physics, Tampere University of Technology , P.O. Box 692, FI-33101 Tampere, Finland
| | - Alex Bunker
- Centre for Drug Research, Faculty of Pharmacy, University of Helsinki , P.O. Box 56, FI-00014, University of Helsinki, Finland.,Department of Chemistry, Aalto University , PO Box 6100, FI-02015, Aalto, Finland
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35
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Nicolini P, Frezzato D, Chelli R. Exploiting Configurational Freezing in Nonequilibrium Monte Carlo Simulations. J Chem Theory Comput 2011; 7:582-93. [DOI: 10.1021/ct100568n] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Paolo Nicolini
- Dipartimento di Chimica, Università di Firenze, Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy
| | - Diego Frezzato
- Dipartimento di Scienze Chimiche, Università di Padova, Via Marzolo 1, I-35131 Padova, Italy
| | - Riccardo Chelli
- Dipartimento di Chimica, Università di Firenze, Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy
- European Laboratory for Nonlinear Spectroscopy (LENS), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
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36
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37
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Minh DDL, Vaikuntanathan S. Density-dependent analysis of nonequilibrium paths improves free energy estimates II. A Feynman–Kac formalism. J Chem Phys 2011; 134:034117. [DOI: 10.1063/1.3541152] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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38
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Hahn AM, Then H. Measuring the convergence of Monte Carlo free-energy calculations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041117. [PMID: 20481687 DOI: 10.1103/physreve.81.041117] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2009] [Revised: 02/24/2010] [Indexed: 05/29/2023]
Abstract
The nonequilibrium work fluctuation theorem provides the way for calculations of (equilibrium) free-energy based on work measurements of nonequilibrium, finite-time processes, and their reversed counterparts by applying Bennett's acceptance ratio method. A nice property of this method is that each free-energy estimate readily yields an estimate of the asymptotic mean square error. Assuming convergence, it is easy to specify the uncertainty of the results. However, sample sizes have often to be balanced with respect to experimental or computational limitations and the question arises whether available samples of work values are sufficiently large in order to ensure convergence. Here, we propose a convergence measure for the two-sided free-energy estimator and characterize some of its properties, explain how it works, and test its statistical behavior. In total, we derive a convergence criterion for Bennett's acceptance ratio method.
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Affiliation(s)
- Aljoscha M Hahn
- Institut für Physik, Carl von Ossietzky Universität, 26111 Oldenburg, Germany
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39
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Geiger P, Dellago C. Optimum protocol for fast-switching free-energy calculations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:021127. [PMID: 20365550 DOI: 10.1103/physreve.81.021127] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2009] [Indexed: 05/29/2023]
Abstract
Free-energy differences computed from fast-switching simulations or measurements according to the Jarzynski equation are independent of the particular protocol specifying how the control parameter is changed in time. In contrast, the average work carried out on the system as well the accuracy of the resulting free energy strongly depend on the protocol. Recently, Schmiedl and Seifert [Phys. Rev. Lett. 98, 108301 (2007)] found that protocols that minimize the average work for a given duration of the switching process have discrete steps at the beginning and the end. Here we determine numerically the protocols that minimize the statistical error in the free energy estimate and find that such minimum error protocols have similar discrete jumps. Our analysis shows that the reduction in computational effort achieved by the use of steplike protocols can be considerable. Such large savings of computing time, however, typically occur for parameter ranges in which an application of the Jarzynski equation is impractical due to large statistical errors arising from the exponential work average.
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Affiliation(s)
- Philipp Geiger
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
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40
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Minh DDL, Chodera JD. Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages. J Chem Phys 2009; 131:134110. [PMID: 19814546 PMCID: PMC2771048 DOI: 10.1063/1.3242285] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2009] [Accepted: 09/13/2009] [Indexed: 11/14/2022] Open
Abstract
Existing optimal estimators of nonequilibrium path-ensemble averages are shown to fall within the framework of extended bridge sampling. Using this framework, we derive a general minimal-variance estimator that can combine nonequilibrium trajectory data sampled from multiple path-ensembles to estimate arbitrary functions of nonequilibrium expectations. The framework is also applied to obtain asymptotic variance estimates, which are a useful measure of statistical uncertainty. In particular, we develop asymptotic variance estimates pertaining to Jarzynski's equality for free energies and the Hummer-Szabo expressions for the potential of mean force, calculated from uni- or bidirectional path samples. These estimators are demonstrated on a model single-molecule pulling experiment. In these simulations, the asymptotic variance expression is found to accurately characterize the confidence intervals around estimators when the bias is small. Hence, the confidence intervals are inaccurately described for unidirectional estimates with large bias, but for this model it largely reflects the true error in a bidirectional estimator derived by Minh and Adib.
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Affiliation(s)
- David D L Minh
- Laboratory of Chemical Physics, NIDDK, National Institutes of Health, Bethesda, Maryland 20892, USA.
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41
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Nicolini P, Chelli R. Improving fast-switching free energy estimates by dynamical freezing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:041124. [PMID: 19905290 DOI: 10.1103/physreve.80.041124] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2009] [Revised: 09/01/2009] [Indexed: 05/28/2023]
Abstract
An important limitation of nonequilibrium pulling experiments/simulations in recovering free energy differences is the poor convergence of path-ensemble averages. Therefore, a large number of fast-switching trajectories needs to achieve free energy estimates with acceptable accuracy. We propose a method to improve free energy estimates by drastically lowering the computational cost of steered molecular dynamics simulations employed to realize such trajectories. This is accomplished by generating trajectories where the particles not directly involved in the driven process are dynamically frozen. Such a freezing is dynamical rather than thermal because it is reached by a synchronous scaling of atomic masses and velocities keeping the kinetic energy of each particle unchanged. The forces between dynamically frozen particles can then be calculated rarely. Thus, it is possible to generate realizations of a process whose computational cost is not correlated with the size of the whole system, but only with that of the reaction site. The method is illustrated on a simple model system and its general applicability is discussed.
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Affiliation(s)
- Paolo Nicolini
- Dipartimento di Chimica, Università di Firenze, Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy
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Hahn AM, Then H. Characteristic of Bennett's acceptance ratio method. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:031111. [PMID: 19905066 DOI: 10.1103/physreve.80.031111] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2009] [Indexed: 05/28/2023]
Abstract
A powerful and well-established tool for free-energy estimation is Bennett's acceptance ratio method. Central properties of this estimator, which employs samples of work values of a forward and its time-reversed process, are known: for given sets of measured work values, it results in the best estimate of the free-energy difference in the large sample limit. Here we state and prove a further characteristic of the acceptance ratio method: the convexity of its mean-square error. As a two-sided estimator, it depends on the ratio of the numbers of forward and reverse work values used. Convexity of its mean-square error immediately implies that there exists a unique optimal ratio for which the error becomes minimal. Further, it yields insight into the relation of the acceptance ratio method and estimators based on the Jarzynski equation. As an application, we study the performance of a dynamic strategy of sampling forward and reverse work values.
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Affiliation(s)
- Aljoscha M Hahn
- Institut für Physik, Carl von Ossietzky Universität, 26111 Oldenburg, Germany
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Engel A. Asymptotics of work distributions in nonequilibrium systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:021120. [PMID: 19792090 DOI: 10.1103/physreve.80.021120] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2008] [Revised: 06/24/2009] [Indexed: 05/28/2023]
Abstract
The asymptotic behavior of the work distribution in driven nonequilibrium systems is determined using the method of optimal fluctuations. For systems described by Langevin dynamics the corresponding Euler-Lagrange equation together with the appropriate boundary conditions and an equation for the leading pre-exponential factor are derived. The method is applied to three representative examples and the results are used to improve the accuracy of free-energy estimates based on the application of the Jarzynski equation.
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Affiliation(s)
- A Engel
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany
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Minh DDL. Density-dependent analysis of nonequilibrium paths improves free energy estimates. J Chem Phys 2009; 130:204102. [PMID: 19485432 PMCID: PMC2832053 DOI: 10.1063/1.3139189] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2009] [Accepted: 04/29/2009] [Indexed: 11/15/2022] Open
Abstract
When a system is driven out of equilibrium by a time-dependent protocol that modifies the Hamiltonian, it follows a nonequilibrium path. Samples of these paths can be used in nonequilibrium work theorems to estimate equilibrium quantities such as free energy differences. Here, we consider analyzing paths generated with one protocol using another one. It is posited that analysis protocols which minimize the lag, the difference between the nonequilibrium and the instantaneous equilibrium densities, will reduce the dissipation of reprocessed trajectories and lead to better free energy estimates. Indeed, when minimal lag analysis protocols based on exactly soluble propagators or relative entropies are applied to several test cases, substantial gains in the accuracy and precision of estimated free energy differences are observed.
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Affiliation(s)
- David D L Minh
- Laboratory of Chemical Physics, NIDDK, National Institutes of Health, Bethesda, Maryland 20892-0520, USA.
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Harris NC, Kiang CH. Velocity convergence of free energy surfaces from single-molecule measurements using Jarzynski's equality. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:041912. [PMID: 19518261 DOI: 10.1103/physreve.79.041912] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2008] [Revised: 03/09/2009] [Indexed: 05/27/2023]
Abstract
We studied the velocity dependence of mechanical unfolding of single protein molecules with the atomic force microscope. We showed that with enough realizations, the free energy surfaces reconstructed from Jarzynski's equality converge with respect to pulling velocity, in good agreement with theory. Using the I27 domain of titin as an example, we estimated the required number of realizations for a given pulling velocity, and suggested the optimal range of velocities for single-molecule experiments. The results demonstrate that Jarzynski's equality is a powerful and practical tool for reconstructing free energy landscapes.
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Affiliation(s)
- Nolan C Harris
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
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Chelli R, Procacci P. A potential of mean force estimator based on nonequilibrium work exponential averages. Phys Chem Chem Phys 2009; 11:1152-8. [DOI: 10.1039/b810914c] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Gomez-Marin A, Schmiedl T, Seifert U. Optimal protocols for minimal work processes in underdamped stochastic thermodynamics. J Chem Phys 2008; 129:024114. [PMID: 18624523 DOI: 10.1063/1.2948948] [Citation(s) in RCA: 85] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
For systems in an externally controllable time-dependent potential, the optimal protocol minimizes the mean work spent in a finite-time transition between two given equilibrium states. For overdamped dynamics which ignores inertia effects, the optimal protocol has been found to involve jumps of the control parameter at the beginning and end of the process. Including the inertia term, we show that this feature not only persists but that even delta-peak-like changes of the control parameter at both boundaries make the process optimal. These results are obtained by analyzing two simple paradigmatic cases: First, a Brownian particle dragged by a harmonic optical trap through a viscous fluid and, second, a Brownian particle subject to an optical trap with time-dependent stiffness. These insights could be used to improve free energy calculations via either thermodynamic integration or "fast growth" methods using Jarzynski's equality.
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Affiliation(s)
- Alex Gomez-Marin
- Facultat de Fisica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
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Protein-protein interaction investigated by steered molecular dynamics: the TCR-pMHC complex. Biophys J 2008; 95:3575-90. [PMID: 18621828 DOI: 10.1529/biophysj.108.131383] [Citation(s) in RCA: 89] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
We present a novel steered molecular dynamics scheme to induce the dissociation of large protein-protein complexes. We apply this scheme to study the interaction of a T cell receptor (TCR) with a major histocompatibility complex (MHC) presenting a peptide (p). Two TCR-pMHC complexes are considered, which only differ by the mutation of a single amino acid on the peptide; one is a strong agonist that produces T cell activation in vivo, while the other is an antagonist. We investigate the interaction mechanism from a large number of unbinding trajectories by analyzing van der Waals and electrostatic interactions and by computing energy changes in proteins and solvent. In addition, dissociation potentials of mean force are calculated with the Jarzynski identity, using an averaging method developed for our steering scheme. We analyze the convergence of the Jarzynski exponential average, which is hampered by the large amount of dissipative work involved and the complexity of the system. The resulting dissociation free energies largely underestimate experimental values, but the simulations are able to clearly differentiate between wild-type and mutated TCR-pMHC and give insights into the dissociation mechanism.
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Vaikuntanathan S, Jarzynski C. Escorted free energy simulations: improving convergence by reducing dissipation. PHYSICAL REVIEW LETTERS 2008; 100:190601. [PMID: 18518431 DOI: 10.1103/physrevlett.100.190601] [Citation(s) in RCA: 57] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2008] [Indexed: 05/25/2023]
Abstract
Nonequilibrium, "fast switching" estimates of equilibrium free energy differences DeltaF are often plagued by poor convergence due to dissipation. We propose a method to improve these estimates by generating trajectories with reduced dissipation. Introducing an artificial flow field that couples the system coordinates to the external parameter driving the simulation, we derive an identity for DeltaF in terms of the resulting trajectories. When the flow field effectively escorts the system along a near-equilibrium path, the free energy estimate converges efficiently and accurately. We illustrate our method on a model system and discuss the general applicability of our approach.
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Minh DDL, Adib AB. Optimized free energies from bidirectional single-molecule force spectroscopy. PHYSICAL REVIEW LETTERS 2008; 100:180602. [PMID: 18518359 PMCID: PMC2504746 DOI: 10.1103/physrevlett.100.180602] [Citation(s) in RCA: 59] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2008] [Indexed: 05/08/2023]
Abstract
An optimized method for estimating path-ensemble averages using data from processes driven in opposite directions is presented. Based on this estimator, bidirectional expressions for reconstructing free energies and potentials of mean force from single-molecule force spectroscopy-valid for biasing potentials of arbitrary stiffness-are developed. Numerical simulations on a model potential indicate that these methods perform better than unidirectional strategies.
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Affiliation(s)
- David D L Minh
- Laboratory of Chemical Physics, NIDDK, National Institutes of Health, Bethesda, Maryland 20892, USA.
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