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Keller BG, Bolhuis PG. Dynamical Reweighting for Biased Rare Event Simulations. Annu Rev Phys Chem 2024; 75:137-162. [PMID: 38941527 DOI: 10.1146/annurev-physchem-083122-124538] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/30/2024]
Abstract
Dynamical reweighting techniques aim to recover the correct molecular dynamics from a simulation at a modified potential energy surface. They are important for unbiasing enhanced sampling simulations of molecular rare events. Here, we review the theoretical frameworks of dynamical reweighting for modified potentials. Based on an overview of kinetic models with increasing level of detail, we discuss techniques to reweight two-state dynamics, multistate dynamics, and path integrals. We explore the natural link to transition path sampling and how the effect of nonequilibrium forces can be reweighted. We end by providing an outlook on how dynamical reweighting integrates with techniques for optimizing collective variables and with modern potential energy surfaces.
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Affiliation(s)
- Bettina G Keller
- Department of Biology, Chemistry and Pharmacy, Freie Universität Berlin, Berlin, Germany;
| | - Peter G Bolhuis
- Van 't Hoff Institute for Molecular Sciences, University of Amsterdam, Amsterdam, The Netherlands
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2
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Donati L, Weber M. Assessing transition rates as functions of environmental variables. J Chem Phys 2022; 157:224103. [PMID: 36546809 DOI: 10.1063/5.0109555] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022] Open
Abstract
We present a method to estimate the transition rates of molecular systems under different environmental conditions that cause the formation or the breaking of bonds and require the sampling of the Grand Canonical Ensemble. For this purpose, we model the molecular system in terms of probable "scenarios," governed by different potential energy functions, which are separately sampled by classical MD simulations. Reweighting the canonical distribution of each scenario according to specific environmental variables, we estimate the grand canonical distribution, then use the Square Root Approximation method to discretize the Fokker-Planck operator into a rate matrix and the robust Perron Cluster Cluster Analysis method to coarse-grain the kinetic model. This permits efficiently estimating the transition rates of conformational states as functions of environmental variables, for example, the local pH at a cell membrane. In this work, we formalize the theoretical framework of the procedure, and we present a numerical experiment comparing the results with those provided by a constant-pH method based on non-equilibrium Molecular Dynamics Monte Carlo simulations. The method is relevant for the development of new drug design strategies that take into account how the cellular environment influences biochemical processes.
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Affiliation(s)
- Luca Donati
- Zuse Institute Berlin, Takustr. 7, D-14195 Berlin, Germany
| | - Marcus Weber
- Zuse Institute Berlin, Takustr. 7, D-14195 Berlin, Germany
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Rydzewski J, Chen M, Ghosh TK, Valsson O. Reweighted Manifold Learning of Collective Variables from Enhanced Sampling Simulations. J Chem Theory Comput 2022; 18:7179-7192. [PMID: 36367826 DOI: 10.1021/acs.jctc.2c00873] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022]
Abstract
Enhanced sampling methods are indispensable in computational chemistry and physics, where atomistic simulations cannot exhaustively sample the high-dimensional configuration space of dynamical systems due to the sampling problem. A class of such enhanced sampling methods works by identifying a few slow degrees of freedom, termed collective variables (CVs), and enhancing the sampling along these CVs. Selecting CVs to analyze and drive the sampling is not trivial and often relies on chemical intuition. Despite routinely circumventing this issue using manifold learning to estimate CVs directly from standard simulations, such methods cannot provide mappings to a low-dimensional manifold from enhanced sampling simulations, as the geometry and density of the learned manifold are biased. Here, we address this crucial issue and provide a general reweighting framework based on anisotropic diffusion maps for manifold learning that takes into account that the learning data set is sampled from a biased probability distribution. We consider manifold learning methods based on constructing a Markov chain describing transition probabilities between high-dimensional samples. We show that our framework reverts the biasing effect, yielding CVs that correctly describe the equilibrium density. This advancement enables the construction of low-dimensional CVs using manifold learning directly from the data generated by enhanced sampling simulations. We call our framework reweighted manifold learning. We show that it can be used in many manifold learning techniques on data from both standard and enhanced sampling simulations.
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Affiliation(s)
- Jakub Rydzewski
- Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Toruń, Poland
| | - Ming Chen
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States
| | - Tushar K Ghosh
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States
| | - Omar Valsson
- Department of Chemistry, University of North Texas, Denton, Texas 76201, United States
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Donati L, Weber M, Keller BG. Markov models from the square root approximation of the Fokker-Planck equation: calculating the grid-dependent flux. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:115902. [PMID: 33352543 DOI: 10.1088/1361-648x/abd5f7] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Molecular dynamics (MD) are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the MD as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker-Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive MD simulations. The square-root approximation of the Fokker-Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used MD simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to [Formula: see text] states can be obtained within seconds on a single high-performance compute server if regular grids are used.
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Affiliation(s)
- Luca Donati
- Department of Biology, Chemistry, Pharmacy, Freie Universität Berlin, Takustraße 3, D-14195 Berlin, Germany
| | - Marcus Weber
- Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany
| | - Bettina G Keller
- Department of Biology, Chemistry, Pharmacy, Freie Universität Berlin, Takustraße 3, D-14195 Berlin, Germany
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Sikorski A, Weber M, Schütte C. The Augmented Jump Chain. ADVANCED THEORY AND SIMULATIONS 2021. [DOI: 10.1002/adts.202000274] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Affiliation(s)
- Alexander Sikorski
- Zuse Institute Berlin Mathematics for Life and Materials Sciences Takustr. 7 D‐14195 Berlin Germany
| | - Marcus Weber
- Zuse Institute Berlin Mathematics for Life and Materials Sciences Takustr. 7 D‐14195 Berlin Germany
| | - Christof Schütte
- Zuse Institute Berlin Mathematics for Life and Materials Sciences Takustr. 7 D‐14195 Berlin Germany
- Freie Universität Berlin Department of Mathematics and Computer Science Biocomputing Group Arnimallee 6 D‐14195 Berlin Germany
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Kieninger S, Keller BG. Path probability ratios for Langevin dynamics-Exact and approximate. J Chem Phys 2021; 154:094102. [PMID: 33685138 DOI: 10.1063/5.0038408] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Abstract
Path reweighting is a principally exact method to estimate dynamic properties from biased simulations-provided that the path probability ratio matches the stochastic integrator used in the simulation. Previously reported path probability ratios match the Euler-Maruyama scheme for overdamped Langevin dynamics. Since molecular dynamics simulations use Langevin dynamics rather than overdamped Langevin dynamics, this severely impedes the application of path reweighting methods. Here, we derive the path probability ratio ML for Langevin dynamics propagated by a variant of the Langevin Leapfrog integrator. This new path probability ratio allows for exact reweighting of Langevin dynamics propagated by this integrator. We also show that a previously derived approximate path probability ratio Mapprox differs from the exact ML only by O(ξ4Δt4) and thus yields highly accurate dynamic reweighting results. (Δt is the integration time step, and ξ is the collision rate.) The results are tested, and the efficiency of path reweighting is explored using butane as an example.
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Affiliation(s)
- S Kieninger
- Department of Biology, Chemistry, Pharmacy, Freie Universität Berlin, Arnimallee 22, D-14195 Berlin, Germany
| | - B G Keller
- Department of Biology, Chemistry, Pharmacy, Freie Universität Berlin, Arnimallee 22, D-14195 Berlin, Germany
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Ray S, Sunkara V, Schütte C, Weber M. How to calculate pH-dependent binding rates for receptor–ligand systems based on thermodynamic simulations with different binding motifs. MOLECULAR SIMULATION 2020. [DOI: 10.1080/08927022.2020.1839660] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- Sourav Ray
- Computational Drug Design, Zuse Institute Berlin, Berlin, Germany
| | - Vikram Sunkara
- Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany
- Computational Physiology and Anatomy, Zuse Institute Berlin, Berlin, Germany
| | - Christof Schütte
- Computational Drug Design, Zuse Institute Berlin, Berlin, Germany
- Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany
- Computational Physiology and Anatomy, Zuse Institute Berlin, Berlin, Germany
| | - Marcus Weber
- Computational Drug Design, Zuse Institute Berlin, Berlin, Germany
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Kieninger S, Donati L, Keller BG. Dynamical reweighting methods for Markov models. Curr Opin Struct Biol 2020; 61:124-131. [PMID: 31958761 DOI: 10.1016/j.sbi.2019.12.018] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2019] [Revised: 12/13/2019] [Accepted: 12/26/2019] [Indexed: 12/21/2022]
Abstract
Conformational dynamics is essential to biomolecular processes. Markov State Models (MSMs) are widely used to elucidate dynamic properties of molecular systems from unbiased Molecular Dynamics (MD). However, the implementation of reweighting schemes for MSMs to analyze biased simulations is still at an early stage of development. Several dynamical reweighing approaches have been proposed, which can be classified as approaches based on (i) Kramers rate theory, (ii) rescaling of the probability density flux, (iii) reweighting by formulating a likelihood function, (iv) path reweighting. We present the state-of-the-art and discuss the methodological differences of these methods, their limitations and recent applications.
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Affiliation(s)
- Stefanie Kieninger
- Freie Universität Berlin, Department of Biology, Chemistry, Pharmacy, Arnimallee 22, D-14194 Berlin, Germany
| | - Luca Donati
- Freie Universität Berlin, Department of Biology, Chemistry, Pharmacy, Arnimallee 22, D-14194 Berlin, Germany
| | - Bettina G Keller
- Freie Universität Berlin, Department of Biology, Chemistry, Pharmacy, Arnimallee 22, D-14194 Berlin, Germany.
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