1
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Fall WS, Kolli HB, Mukherjee B, Chakrabarti B. Phase behavior of polymer dispersed liquid crystals, comparison between mean-field theory, and coarse-grained molecular dynamics simulations. SOFT MATTER 2024; 20:7735-7751. [PMID: 39308410 DOI: 10.1039/d4sm01005a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/03/2024]
Abstract
We report a simulation methodology to quantitatively predict the thermodynamic behaviour (phase diagrams) of polymer mixtures, that exhibit phases with broken orientational symmetry. Our system consists of a binary mixture of short oligomers (NA = 4) and long rod-like mesogens (NB = 8). Using coarse-grained molecular dynamics (CGMD) simulations we infer the topology of the temperature-dependent free energy landscape, from the probability distributions of the components for a range of compositions. The mixture exhibits nematic (N) and smectic phases (Sm-A) as a function of two temperature scales, Tc, that governs the demixing transition, and TNI the nematic-isotropic temperature. Thus in addition to the isotropic (I), a nematic (N) phases observed in simulations of similar systems earlier we report the formation of a new entropy-stabilized phase separated smectic-A (Sm-A) phase with alternating mesogen-rich and oligomer-rich layers. Using the mean-field free energy for polymer-dispersed liquid crystals (PDLCs), with suitably chosen parameter values, we construct a mean-field phase diagram that matches those obtained from CGMD simulations. Our results are applicable to mixtures of synthetic and biological macromolecules that undergo phase separation and are orientable, thereby giving rise to the liquid crystalline phases. Our proposed methodology has a distinct advantage over other computational techniques in its applicability to systems with complex molecular interactions and in capturing the coarsening dynamics of systems involving multiple order parameters.
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Affiliation(s)
- William S Fall
- Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, UK.
- Laboratoire de Physique des Solides - UMR 8502, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | | | - Biswaroop Mukherjee
- Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, UK.
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2
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Tamagnone S, Laio A, Gabrié M. Coarse-Grained Molecular Dynamics with Normalizing Flows. J Chem Theory Comput 2024. [PMID: 39223750 DOI: 10.1021/acs.jctc.4c00700] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 09/04/2024]
Abstract
We propose a sampling algorithm relying on a collective variable (CV) of midsize dimension modeled by a normalizing flow and using nonequilibrium dynamics to propose full configurational moves from the proposition of a refreshed value of the CV made by the flow. The algorithm takes the form of a Markov chain with nonlocal updates, allowing jumps through energy barriers across metastable states. The flow is trained throughout the algorithm to reproduce the free energy landscape of the CV. The output of the algorithm is a sample of thermalized configurations and the trained network that can be used to efficiently produce more configurations. We show the functioning of the algorithm first in a test case with a mixture of Gaussians. Then, we successfully tested it on a higher-dimensional system consisting of a polymer in solution with a compact state and an extended stable state separated by a high free energy barrier.
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Affiliation(s)
- Samuel Tamagnone
- International School for Advanced Studies (SISSA), Via Bonomea 265, Trieste 34136, Italy
| | - Alessandro Laio
- International School for Advanced Studies (SISSA), Via Bonomea 265, Trieste 34136, Italy
- The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste 34151, Italy
| | - Marylou Gabrié
- CMAP, CNRS, Institut Polytechnique de Paris, École Polytechnique, 91120 Palaiseau, France
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3
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Mazzaferro N, Sasmal S, Cossio P, Hocky GM. Good Rates From Bad Coordinates: The Exponential Average Time-dependent Rate Approach. J Chem Theory Comput 2024; 20:5901-5912. [PMID: 38954555 PMCID: PMC11270837 DOI: 10.1021/acs.jctc.4c00425] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2024] [Revised: 06/11/2024] [Accepted: 06/12/2024] [Indexed: 07/04/2024]
Abstract
Our ability to calculate rate constants of biochemical processes using molecular dynamics simulations is severely limited by the fact that the time scales for reactions, or changes in conformational state, scale exponentially with the relevant free-energy barrier heights. In this work, we improve upon a recently proposed rate estimator that allows us to predict transition times with molecular dynamics simulations biased to rapidly explore one or several collective variables (CVs). This approach relies on the idea that not all bias goes into promoting transitions, and along with the rate, it estimates a concomitant scale factor for the bias termed the "CV biasing efficiency" γ. First, we demonstrate mathematically that our new formulation allows us to derive the commonly used Infrequent Metadynamics (iMetaD) estimator when using a perfect CV, where γ = 1. After testing it on a model potential, we then study the unfolding behavior of a previously well characterized coarse-grained protein, which is sufficiently complex that we can choose many different CVs to bias, but which is sufficiently simple that we are able to compute the unbiased rate directly. For this system, we demonstrate that predictions from our new Exponential Average Time-Dependent Rate (EATR) estimator converge to the true rate constant more rapidly as a function of bias deposition time than does the previous iMetaD approach, even for bias deposition times that are short. We also show that the γ parameter can serve as a good metric for assessing the quality of the biasing coordinate. We demonstrate that these results hold when applying the methods to an atomistic protein folding example. Finally, we demonstrate that our approach works when combining multiple less-than-optimal bias coordinates, and adapt our method to the related "OPES flooding" approach. Overall, our time-dependent rate approach offers a powerful framework for predicting rate constants from biased simulations.
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Affiliation(s)
- Nicodemo Mazzaferro
- Department
of Chemistry, New York University, New York, New York 10003, United States
| | - Subarna Sasmal
- Department
of Chemistry, New York University, New York, New York 10003, United States
| | - Pilar Cossio
- Center
for Computational Mathematics, Flatiron
Institute, New York, New York 10010, United States
- Center
for Computational Biology, Flatiron Institute, New York, New York 10010, United States
| | - Glen M. Hocky
- Department
of Chemistry, New York University, New York, New York 10003, United States
- Simons
Center for Computational Physical Chemistry, New York University, New York, New York 10003, United States
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4
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Karvelis E, Swanson C, Tidor B. Substrate Turnover Dynamics Guide Ketol-Acid Reductoisomerase Redesign for Increased Specific Activity. ACS Catal 2024; 14:10491-10509. [PMID: 39050899 PMCID: PMC11264209 DOI: 10.1021/acscatal.4c01446] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2024] [Revised: 05/16/2024] [Accepted: 06/12/2024] [Indexed: 07/27/2024]
Abstract
The task of adapting enzymes for specific applications is often hampered by our incomplete ability to tune and tailor catalytic functions, particularly when seeking increased activity. Here, we develop and demonstrate a rational approach to address this challenge, applied to ketol-acid reductoisomerase (KARI), which has uses in industrial-scale isobutanol production. While traditional structure-based computational enzyme redesign strategies typically focus on the enzyme-bound ground state (GS) and transition state (TS), we postulated that additionally treating the underlying dynamics of complete turnover events that connect and pass through both states could further elucidate the structural properties affecting catalysis and help identify mutations that lead to increased catalytic activity. To examine the dynamics of substrate conversion with atomistic detail, we adapted and applied computational methods based on path sampling techniques to gather thousands of QM/MM simulations of attempted substrate turnover events by KARI: both productive (reactive) and unproductive (nonreactive) attempts. From these data, machine learning models were constructed and used to identify specific conformational features (interatomic distances, angles, and torsions) associated with successful, productive catalysis. Multistate protein redesign techniques were then used to select mutations that stabilized reactive-like structures over nonreactive-like ones while also meeting additional criteria consistent with enhanced specific activity. This procedure resulted in eight high-confidence enzyme mutants with a significant improvement in calculated specific activity relative to wild type (WT), with the fastest variant's increase in calculated k cat being (2 ± 1) × 104-fold. Collectively, these results suggest that introducing mutations designed to increase the population of reaction-promoting conformations of the enzyme-substrate complex before it reaches the barrier can provide an effective approach to engineering improved enzyme catalysts.
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Affiliation(s)
- Elijah Karvelis
- Department
of Biological Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, United States
- Computer
Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Chloe Swanson
- Department
of Biological Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, United States
- Computer
Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Bruce Tidor
- Department
of Biological Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, United States
- Computer
Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
- Department
of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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5
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Vervust W, Zhang DT, Ghysels A, Roet S, van Erp TS, Riccardi E. PyRETIS 3: Conquering rare and slow events without boundaries. J Comput Chem 2024; 45:1224-1234. [PMID: 38345082 DOI: 10.1002/jcc.27319] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2023] [Revised: 01/16/2024] [Accepted: 01/18/2024] [Indexed: 04/19/2024]
Abstract
We present and discuss the advancements made in PyRETIS 3, the third instalment of our Python library for an efficient and user-friendly rare event simulation, focused to execute molecular simulations with replica exchange transition interface sampling (RETIS) and its variations. Apart from a general rewiring of the internal code towards a more modular structure, several recently developed sampling strategies have been implemented. These include recently developed Monte Carlo moves to increase path decorrelation and convergence rate, and new ensemble definitions to handle the challenges of long-lived metastable states and transitions with unbounded reactant and product states. Additionally, the post-analysis software PyVisa is now embedded in the main code, allowing fast use of machine-learning algorithms for clustering and visualising collective variables in the simulation data.
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Affiliation(s)
- Wouter Vervust
- IBiTech-BioMMedA Group, Ghent University, Ghent, Belgium
| | - Daniel T Zhang
- Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway
| | - An Ghysels
- IBiTech-BioMMedA Group, Ghent University, Ghent, Belgium
| | - Sander Roet
- Department of Chemistry, Utrecht University, Utrecht, The Netherlands
| | - Titus S van Erp
- Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway
| | - Enrico Riccardi
- Department of Energy Resources, University of Stavanger, Stavanger, Norway
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6
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Louwerse MD, Sivak DA. Information thermodynamics of transition paths between multiple mesostates. Phys Rev E 2024; 109:064112. [PMID: 39020997 DOI: 10.1103/physreve.109.064112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2023] [Accepted: 04/23/2024] [Indexed: 07/20/2024]
Abstract
A central concern across the natural sciences is a quantitative understanding of the mechanism governing rare transitions between two metastable states. Recent research has uncovered a fundamental equality between the time-reversal asymmetry of the ensemble of such transition paths and the informativeness of system dynamics about the reactivity of a given trajectory, immediately leading to quantitative criteria for judging the importance of distinct system coordinates for the transition. Here we generalize this framework to multiple mesostates. We find that the main system-wide and coordinate-specific results generalize intuitively, while the combinatorial diversity of pairwise transitions raises new questions and points to new concepts. This work increases the previous framework's generality and applicability and forges connections to enhanced-sampling and coarse-grained dynamical approaches such as milestoning and Markov-state models.
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7
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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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8
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Bogetti A, Zwier MC, Chong LT. Revisiting Textbook Azide-Clock Reactions: A "Propeller-Crawling" Mechanism Explains Differences in Rates. J Am Chem Soc 2024; 146:12828-12835. [PMID: 38687173 PMCID: PMC11078601 DOI: 10.1021/jacs.4c03360] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2024] [Revised: 04/23/2024] [Accepted: 04/24/2024] [Indexed: 05/02/2024]
Abstract
An ongoing challenge to chemists is the analysis of pathways and kinetics for chemical reactions in solution, including transient structures between the reactants and products that are difficult to resolve using laboratory experiments. Here, we enabled direct molecular dynamics simulations of a textbook series of chemical reactions on the hundreds of ns to μs time scale using the weighted ensemble (WE) path sampling strategy with hybrid quantum mechanical/molecular mechanical (QM/MM) models. We focused on azide-clock reactions involving addition of an azide anion to each of three long-lived trityl cations in an acetonitrile-water solvent mixture. Results reveal a two-step mechanism: (1) diffusional collision of reactants to form an ion-pair intermediate; (2) "activation" or rearrangement of the intermediate to the product. Our simulations yield not only reaction rates that are within error of experiment but also rates for individual steps, indicating the activation step as rate-limiting for all three cations. Further, the trend in reaction rates is due to dynamical effects, i.e., differing extents of the azide anion "crawling" along the cation's phenyl-ring "propellers" during the activation step. Our study demonstrates the power of analyzing pathways and kinetics to gain insights on reaction mechanisms, underscoring the value of including WE and other related path sampling strategies in the modern toolbox for chemists.
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Affiliation(s)
- Anthony
T. Bogetti
- Department
of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
| | - Matthew C. Zwier
- Department
of Chemistry, Drake University, Des Moines, Iowa 50311, United States
| | - Lillian T. Chong
- Department
of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
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9
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Falkner S, Coretti A, Dellago C. Enhanced Sampling of Configuration and Path Space in a Generalized Ensemble by Shooting Point Exchange. PHYSICAL REVIEW LETTERS 2024; 132:128001. [PMID: 38579233 DOI: 10.1103/physrevlett.132.128001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2023] [Revised: 12/15/2023] [Accepted: 02/09/2024] [Indexed: 04/07/2024]
Abstract
The computer simulation of many molecular processes is complicated by long timescales caused by rare transitions between long-lived states. Here, we propose a new approach to simulate such rare events, which combines transition path sampling with enhanced exploration of configuration space. The method relies on exchange moves between configuration and trajectory space, carried out based on a generalized ensemble. This scheme substantially enhances the efficiency of the transition path sampling simulations, particularly for systems with multiple transition channels, and yields information on thermodynamics, kinetics and reaction coordinates of molecular processes without distorting their dynamics. The method is illustrated using the isomerization of proline in the KPTP tetrapeptide.
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10
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Jonas HJ, Schall P, Bolhuis PG. Activity affects the stability, deformation and breakage dynamics of colloidal architectures. SOFT MATTER 2024; 20:2162-2177. [PMID: 38351836 DOI: 10.1039/d3sm01255g] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/29/2024]
Abstract
Living network architectures, such as the cytoskeleton, are characterized by continuous energy injection, leading to rich but poorly understood non-equilibrium physics. There is a need for a well-controlled (experimental) model system that allows basic insight into such non-equilibrium processes. Activated self-assembled colloidal architectures can fulfill this role, as colloidal patchy particles can self-assemble into colloidal architectures such as chains, rings and networks, while self-propelled colloidal particles can simultaneously inject energy into the architecture, alter the dynamical behavior of the system, and cause the self-assembled structures to deform and break. To gain insight, we conduct a numerical investigation into the effect of introducing self-propelled colloids modeled as active Brownian particles, into self-assembling colloidal dispersions of dipatch and tripatch particles. For the interaction potential, we use a previously designed model that accurately can reproduce experimental colloidal self-assembly via the critical Casimir force [Jonas et al., J. Chem. Phys., 2021, 135, 034902]. Here, we focus primarily on the breakage dynamics of three archetypal substructures, namely, dimers, chains, and rings. We find a rich response behavior to the introduction of self-propelled particles, in which the activity can enhance as well as reduce the stability of the architecture, deform the intact structures and alter the mechanisms of fragmentation. We rationalize these findings in terms of the rate and mechanisms of breakage as a function of the direction and magnitude of the active force by separating the bond breakage process into two stages: escaping the potential well and separation of the particles. The results set the stage for investigating more complex architectures.
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Affiliation(s)
- H J Jonas
- van 't Hoff Institute for Molecular Sciences, University of Amsterdam, PO Box 94157, 1090 GD Amsterdam, The Netherlands.
| | - P Schall
- van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, PO Box 94485, 1090 GL Amsterdam, The Netherlands
| | - P G Bolhuis
- van 't Hoff Institute for Molecular Sciences, University of Amsterdam, PO Box 94157, 1090 GD Amsterdam, The Netherlands.
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11
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Walsh MR. Comparing brute force to transition path sampling for gas hydrate nucleation with a flat interface: comments on time reversal symmetry. Phys Chem Chem Phys 2024; 26:5762-5772. [PMID: 38214888 DOI: 10.1039/d3cp05059a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/13/2024]
Abstract
Fluid to solid nucleation is often investigated with the rare event method transition path sampling (TPS). I claim that the inherent irreversibility of solid nucleation, even at stationary conditions, calls into question TPS's applicability for determining solid nucleation mechanisms, especially for pre-critical behavior. Even when applied to a phenomenon which displays time reversal asymmetry like solid nucleation, TPS is a good means of exploring phase space and giving trends in post-critical structure, and its ability to facilitate nucleation rate and free energy calculations remains outstanding. Forward-only splitting and ratcheting methods such as forward flux sampling are more attractive for understanding nucleation mechanisms as they do not require time reversal symmetry, but at low driving forces may suffer from the same limitations as brute force: they may never make it to the first ratchet. Here I briefly summarize the TPS method and gas hydrate nucleation simulation literature, focusing on topics within both to facilitate a comparison of brute force hydrate nucleation to transition path sampling of hydrate nucleation. Perhaps anecdotally, the brute force technique results in more crystalline trajectories despite having higher driving forces than TPS. I maintain this difference is because of the inherent irreversibility of hydrate nucleation, meaning its pre-critical behavior cannot accurately be determined by the melting trajectories that comprise approximately half of the configurations in TPS's path ensemble.
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12
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Zhang DT, Baldauf L, Roet S, Lervik A, van Erp TS. Highly parallelizable path sampling with minimal rejections using asynchronous replica exchange and infinite swaps. Proc Natl Acad Sci U S A 2024; 121:e2318731121. [PMID: 38315841 PMCID: PMC10873605 DOI: 10.1073/pnas.2318731121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2023] [Accepted: 01/09/2024] [Indexed: 02/07/2024] Open
Abstract
Capturing rare yet pivotal events poses a significant challenge for molecular simulations. Path sampling provides a unique approach to tackle this issue without altering the potential energy landscape or dynamics, enabling recovery of both thermodynamic and kinetic information. However, despite its exponential acceleration compared to standard molecular dynamics, generating numerous trajectories can still require a long time. By harnessing our recent algorithmic innovations-particularly subtrajectory moves with high acceptance, coupled with asynchronous replica exchange featuring infinite swaps-we establish a highly parallelizable and rapidly converging path sampling protocol, compatible with diverse high-performance computing architectures. We demonstrate our approach on the liquid-vapor phase transition in superheated water, the unfolding of the chignolin protein, and water dissociation. The latter, performed at the ab initio level, achieves comparable statistical accuracy within days, in contrast to a previous study requiring over a year.
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Affiliation(s)
- Daniel T. Zhang
- Department of Chemistry, Norwegian University of Science and Technology, TrondheimN-7491, Norway
| | - Lukas Baldauf
- Department of Chemistry, Norwegian University of Science and Technology, TrondheimN-7491, Norway
| | - Sander Roet
- Department of Chemistry, Utrecht University, Utrecht3584 CH, Netherlands
| | - Anders Lervik
- Department of Chemistry, Norwegian University of Science and Technology, TrondheimN-7491, Norway
| | - Titus S. van Erp
- Department of Chemistry, Norwegian University of Science and Technology, TrondheimN-7491, Norway
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13
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Ono J, Matsumura Y, Mori T, Saito S. Conformational Dynamics in Proteins: Entangled Slow Fluctuations and Nonequilibrium Reaction Events. J Phys Chem B 2024; 128:20-32. [PMID: 38133567 DOI: 10.1021/acs.jpcb.3c05307] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2023]
Abstract
Proteins exhibit conformational fluctuations and changes over various time scales, ranging from rapid picosecond-scale local atomic motions to slower microsecond-scale global conformational transformations. In the presence of these intricate fluctuations, chemical reactions occur and functions emerge. These conformational fluctuations of proteins are not merely stochastic random motions but possess distinct spatiotemporal characteristics. Moreover, chemical reactions do not always proceed along a single reaction coordinate in a quasi-equilibrium manner. Therefore, it is essential to understand spatiotemporal conformational fluctuations of proteins and the conformational change processes associated with reactions. In this Perspective, we shed light on the complex dynamics of proteins and their role in enzyme catalysis by presenting recent results regarding dynamic couplings and disorder in the conformational dynamics of proteins and rare but rapid enzymatic reaction events obtained from molecular dynamics simulations.
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Affiliation(s)
- Junichi Ono
- Waseda Research Institute for Science and Engineering (WISE), Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
| | - Yoshihiro Matsumura
- Institute for Chemical Reaction Design and Discovery (WPI-ICReDD), Hokkaido University, Sapporo, Hokkaido 001-0021, Japan
| | - Toshifumi Mori
- Institute for Materials Chemistry and Engineering, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
| | - Shinji Saito
- Institute for Molecular Science, Okazaki, Aichi 444-8585, Japan
- The Graduate University for Advanced Studies (SOKENDAI), Okazaki, Aichi 444-8585, Japan
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14
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Ji X, Wang R, Wang H, Liu W. On committor functions in milestoning. J Chem Phys 2023; 159:244115. [PMID: 38153148 DOI: 10.1063/5.0180513] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2023] [Accepted: 12/07/2023] [Indexed: 12/29/2023] Open
Abstract
As an optimal one-dimensional reaction coordinate, the committor function not only describes the probability of a trajectory initiated at a phase space point first reaching the product state before reaching the reactant state but also preserves the kinetics when utilized to run a reduced dynamics model. However, calculating the committor function in high-dimensional systems poses significant challenges. In this paper, within the framework of milestoning, exact expressions for committor functions at two levels of coarse graining are given, including committor functions of phase space point to point (CFPP) and milestone to milestone (CFMM). When combined with transition kernels obtained from trajectory analysis, these expressions can be utilized to accurately and efficiently compute the committor functions. Furthermore, based on the calculated committor functions, an adaptive algorithm is developed to gradually refine the transition state region. Finally, two model examples are employed to assess the accuracy of these different formulations of committor functions.
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Affiliation(s)
- Xiaojun Ji
- Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, Shandong 266237, People's Republic of China
- Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University, Qingdao, Shandong 266237, People's Republic of China
| | - Ru Wang
- Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, People's Republic of China
| | - Hao Wang
- Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, People's Republic of China
| | - Wenjian Liu
- Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, People's Republic of China
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15
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Lazzeri G, Jung H, Bolhuis PG, Covino R. Molecular Free Energies, Rates, and Mechanisms from Data-Efficient Path Sampling Simulations. J Chem Theory Comput 2023; 19:9060-9076. [PMID: 37988412 PMCID: PMC10753783 DOI: 10.1021/acs.jctc.3c00821] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2023] [Revised: 10/24/2023] [Accepted: 10/24/2023] [Indexed: 11/23/2023]
Abstract
Molecular dynamics is a powerful tool for studying the thermodynamics and kinetics of complex molecular events. However, these simulations can rarely sample the required time scales in practice. Transition path sampling overcomes this limitation by collecting unbiased trajectories and capturing the relevant events. Moreover, the integration of machine learning can boost the sampling while simultaneously learning a quantitative representation of the mechanism. Still, the resulting trajectories are by construction non-Boltzmann-distributed, preventing the calculation of free energies and rates. We developed an algorithm to approximate the equilibrium path ensemble from machine-learning-guided path sampling data. At the same time, our algorithm provides efficient sampling, mechanism, free energy, and rates of rare molecular events at a very moderate computational cost. We tested the method on the folding of the mini-protein chignolin. Our algorithm is straightforward and data-efficient, opening the door to applications in many challenging molecular systems.
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Affiliation(s)
- Gianmarco Lazzeri
- Frankfurt
Institute for Advanced Studies, Frankfurt am Main, 60438, Germany
- Goethe
University Frankfurt, Frankfurt
am Main, 60438, Germany
| | - Hendrik Jung
- Goethe
University Frankfurt, Frankfurt
am Main, 60438, Germany
- Department
of Theoretical Biophysics, Max Planck Institute
of Biophysics, Frankfurt
am Main, 60438, Germany
| | - Peter G. Bolhuis
- Van’t
Hoff Institute for Molecular Sciences, University
of Amsterdam, Amsterdam, 1090GD, The Netherlands
| | - Roberto Covino
- Frankfurt
Institute for Advanced Studies, Frankfurt am Main, 60438, Germany
- Goethe
University Frankfurt, Frankfurt
am Main, 60438, Germany
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16
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Liu Z. Accelerating Kinetics with Time-Reversal Path Sampling. Molecules 2023; 28:8147. [PMID: 38138635 PMCID: PMC10745403 DOI: 10.3390/molecules28248147] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2023] [Revised: 12/07/2023] [Accepted: 12/13/2023] [Indexed: 12/24/2023] Open
Abstract
In comparison to numerous enhanced sampling methods for equilibrium thermodynamics, accelerating simulations for kinetics and nonequilibrium statistics are relatively rare and less effective. Here, we derive a time-reversal path sampling (tRPS) method based on time reversibility to accelerate simulations for determining the transition rates between free-energy basins. It converts the difficult uphill path sampling into an easy downhill problem. This method is easy to implement, i.e., forward and backward shooting simulations with opposite initial velocities are conducted from random initial conformations within a transition-state region until they reach the basin minima, which are then assembled to give the distribution of transition paths efficiently. The effects of tRPS are demonstrated using a comparison with direct simulations of protein folding and unfolding, where tRPS is shown to give results consistent with direct simulations and increase the efficiency by up to five orders of magnitude. This approach is generally applicable to stochastic processes with microscopic reversibility, regardless of whether the variables are continuous or discrete.
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Affiliation(s)
- Zhirong Liu
- Beijing National Laboratory for Molecular Sciences (BNLMS), College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
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17
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Wang R, Wang H, Liu W, Elber R. Approximating First Hitting Point Distribution in Milestoning for Rare Event Kinetics. J Chem Theory Comput 2023; 19:6816-6826. [PMID: 37695680 DOI: 10.1021/acs.jctc.3c00315] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 09/13/2023]
Abstract
Milestoning is an efficient method for rare event kinetics calculation using short trajectory parallelization. Mean first passage time (MFPT) is the key kinetic output of Milestoning, whose accuracy crucially depends on the initial distribution of the short trajectory ensemble. The true initial distribution, i.e., the first hitting point distribution (FHPD), has no analytic expression in the general case. Here, we introduce two algorithms, local passage time weighted Milestoning (LPT-M) and Bayesian inference Milestoning (BI-M), to accurately and efficiently approximate FHPD for systems at equilibrium condition. Starting from sampling the Boltzmann distribution on milestones, we calculate the proper weighting factor for the short trajectory ensemble. The methods are tested on two model examples for illustration purpose. Both methods improve significantly over the widely used classical Milestoning (CM) method in terms of the accuracy of MFPT. In particular, BI-M covers the directional Milestoning method as a special case in deterministic Hamiltonian dynamics. LPT-M is especially advantageous in terms of computational costs and robustness with respect to the increasing number of intermediate milestones. Furthermore, a locally iterative correction algorithm for nonequilibrium stationary FHPD is developed for exact MFPT calculation, which can be combined with LPT-M/BI-M and is much cheaper than the exact Milestoning (ExM) method.
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Affiliation(s)
- Ru Wang
- Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, P. R. China
| | - Hao Wang
- Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, P. R. China
| | - Wenjian Liu
- Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, P. R. China
| | - Ron Elber
- Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78712, United States
- Department of Chemistry, University of Texas at Austin, Austin, Texas 78712, United States
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18
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Ray D, Parrinello M. Kinetics from Metadynamics: Principles, Applications, and Outlook. J Chem Theory Comput 2023; 19:5649-5670. [PMID: 37585703 DOI: 10.1021/acs.jctc.3c00660] [Citation(s) in RCA: 21] [Impact Index Per Article: 21.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/18/2023]
Abstract
Metadynamics is a popular enhanced sampling algorithm for computing the free energy landscape of rare events by using molecular dynamics simulation. Ten years ago, Tiwary and Parrinello introduced the infrequent metadynamics approach for calculating the kinetics of transitions across free energy barriers. Since then, metadynamics-based methods for obtaining rate constants have attracted significant attention in computational molecular science. Such methods have been applied to study a wide range of problems, including protein-ligand binding, protein folding, conformational transitions, chemical reactions, catalysis, and nucleation. Here, we review the principles of elucidating kinetics from metadynamics-like approaches, subsequent methodological developments in this area, and successful applications on chemical, biological, and material systems. We also highlight the challenges of reconstructing accurate kinetics from enhanced sampling simulations and the scope of future developments.
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Affiliation(s)
- Dhiman Ray
- Atomistic Simulations, Italian Institute of Technology, Via Enrico Melen 83, 16152 Genova, Italy
| | - Michele Parrinello
- Atomistic Simulations, Italian Institute of Technology, Via Enrico Melen 83, 16152 Genova, Italy
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19
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Van Speybroeck V, Bocus M, Cnudde P, Vanduyfhuys L. Operando Modeling of Zeolite-Catalyzed Reactions Using First-Principles Molecular Dynamics Simulations. ACS Catal 2023; 13:11455-11493. [PMID: 37671178 PMCID: PMC10476167 DOI: 10.1021/acscatal.3c01945] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2023] [Revised: 07/27/2023] [Indexed: 09/07/2023]
Abstract
Within this Perspective, we critically reflect on the role of first-principles molecular dynamics (MD) simulations in unraveling the catalytic function within zeolites under operating conditions. First-principles MD simulations refer to methods where the dynamics of the nuclei is followed in time by integrating the Newtonian equations of motion on a potential energy surface that is determined by solving the quantum-mechanical many-body problem for the electrons. Catalytic solids used in industrial applications show an intriguing high degree of complexity, with phenomena taking place at a broad range of length and time scales. Additionally, the state and function of a catalyst critically depend on the operating conditions, such as temperature, moisture, presence of water, etc. Herein we show by means of a series of exemplary cases how first-principles MD simulations are instrumental to unravel the catalyst complexity at the molecular scale. Examples show how the nature of reactive species at higher catalytic temperatures may drastically change compared to species at lower temperatures and how the nature of active sites may dynamically change upon exposure to water. To simulate rare events, first-principles MD simulations need to be used in combination with enhanced sampling techniques to efficiently sample low-probability regions of phase space. Using these techniques, it is shown how competitive pathways at operating conditions can be discovered and how broad transition state regions can be explored. Interestingly, such simulations can also be used to study hindered diffusion under operating conditions. The cases shown clearly illustrate how first-principles MD simulations reveal insights into the catalytic function at operating conditions, which could not be discovered using static or local approaches where only a few points are considered on the potential energy surface (PES). Despite these advantages, some major hurdles still exist to fully integrate first-principles MD methods in a standard computational catalytic workflow or to use the output of MD simulations as input for multiple length/time scale methods that aim to bridge to the reactor scale. First of all, methods are needed that allow us to evaluate the interatomic forces with quantum-mechanical accuracy, albeit at a much lower computational cost compared to currently used density functional theory (DFT) methods. The use of DFT limits the currently attainable length/time scales to hundreds of picoseconds and a few nanometers, which are much smaller than realistic catalyst particle dimensions and time scales encountered in the catalysis process. One solution could be to construct machine learning potentials (MLPs), where a numerical potential is derived from underlying quantum-mechanical data, which could be used in subsequent MD simulations. As such, much longer length and time scales could be reached; however, quite some research is still necessary to construct MLPs for the complex systems encountered in industrially used catalysts. Second, most currently used enhanced sampling techniques in catalysis make use of collective variables (CVs), which are mostly determined based on chemical intuition. To explore complex reactive networks with MD simulations, methods are needed that allow the automatic discovery of CVs or methods that do not rely on a priori definition of CVs. Recently, various data-driven methods have been proposed, which could be explored for complex catalytic systems. Lastly, first-principles MD methods are currently mostly used to investigate local reactive events. We hope that with the rise of data-driven methods and more efficient methods to describe the PES, first-principles MD methods will in the future also be able to describe longer length/time scale processes in catalysis. This might lead to a consistent dynamic description of all steps-diffusion, adsorption, and reaction-as they take place at the catalyst particle level.
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Affiliation(s)
| | - Massimo Bocus
- Center for Molecular Modeling, Ghent University, Technologiepark 46, 9052 Zwijnaarde, Belgium
| | - Pieter Cnudde
- Center for Molecular Modeling, Ghent University, Technologiepark 46, 9052 Zwijnaarde, Belgium
| | - Louis Vanduyfhuys
- Center for Molecular Modeling, Ghent University, Technologiepark 46, 9052 Zwijnaarde, Belgium
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20
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Bolhuis PG, Brotzakis ZF, Keller BG. Optimizing molecular potential models by imposing kinetic constraints with path reweighting. J Chem Phys 2023; 159:074102. [PMID: 37581416 DOI: 10.1063/5.0151166] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2023] [Accepted: 06/19/2023] [Indexed: 08/16/2023] Open
Abstract
Empirical force fields employed in molecular dynamics simulations of complex systems are often optimized to reproduce experimentally determined structural and thermodynamic properties. In contrast, experimental knowledge about the interconversion rates between metastable states in such systems is hardly ever incorporated in a force field due to a lack of an efficient approach. Here, we introduce such a framework based on the relationship between dynamical observables, such as rate constants, and the underlying molecular model parameters using the statistical mechanics of trajectories. Given a prior ensemble of molecular dynamics trajectories produced with imperfect force field parameters, the approach allows for the optimal adaption of these parameters such that the imposed constraint of equally predicted and experimental rate constant is obeyed. To do so, the method combines the continuum path ensemble maximum caliber approach with path reweighting methods for stochastic dynamics. When multiple solutions are found, the method selects automatically the combination that corresponds to the smallest perturbation of the entire path ensemble, as required by the maximum entropy principle. To show the validity of the approach, we illustrate the method on simple test systems undergoing rare event dynamics. Next to simple 2D potentials, we explore particle models representing molecular isomerization reactions and protein-ligand unbinding. Besides optimal interaction parameters, the methodology gives physical insights into what parts of the model are most sensitive to the kinetics. We discuss the generality and broad implications of the methodology.
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Affiliation(s)
- Peter G Bolhuis
- van 't Hoff Institute for Molecular Sciences, University of Amsterdam, P.O. Box 94157, 1090 GD Amsterdam, The Netherlands
| | - Z Faidon Brotzakis
- Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom
| | - Bettina G Keller
- Department of Biology, Chemistry, Pharmacy, Freie Universität Berlin, Arnimallee 22, D-14195 Berlin, Germany
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21
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Faidon Brotzakis Z, Löhr T, Truong S, Hoff S, Bonomi M, Vendruscolo M. Determination of the Structure and Dynamics of the Fuzzy Coat of an Amyloid Fibril of IAPP Using Cryo-Electron Microscopy. Biochemistry 2023; 62:2407-2416. [PMID: 37477459 PMCID: PMC10433526 DOI: 10.1021/acs.biochem.3c00010] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2023] [Revised: 06/03/2023] [Indexed: 07/22/2023]
Abstract
In recent years, major advances in cryo-electron microscopy (cryo-EM) have enabled the routine determination of complex biomolecular structures at atomistic resolution. An open challenge for this approach, however, concerns large systems that exhibit continuous dynamics. To address this problem, we developed the metadynamic electron microscopy metainference (MEMMI) method, which incorporates metadynamics, an enhanced conformational sampling approach, into the metainference method of integrative structural biology. MEMMI enables the simultaneous determination of the structure and dynamics of large heterogeneous systems by combining cryo-EM density maps with prior information through molecular dynamics, while at the same time modeling the different sources of error. To illustrate the method, we apply it to elucidate the dynamics of an amyloid fibril of the islet amyloid polypeptide (IAPP). The resulting conformational ensemble provides an accurate description of the structural variability of the disordered region of the amyloid fibril, known as fuzzy coat. The conformational ensemble also reveals that in nearly half of the structural core of this amyloid fibril, the side chains exhibit liquid-like dynamics despite the presence of the highly ordered network backbone of hydrogen bonds characteristic of the cross-β structure of amyloid fibrils.
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Affiliation(s)
- Z. Faidon Brotzakis
- Centre
for Misfolding Diseases, Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, U.K.
| | - Thomas Löhr
- Centre
for Misfolding Diseases, Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, U.K.
| | - Steven Truong
- Centre
for Misfolding Diseases, Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, U.K.
| | - Samuel Hoff
- Department
of Structural Biology and Chemistry, Institut
Pasteur, Université Paris Cité CNRS UMR 3528, 75015 Paris, France
| | - Massimiliano Bonomi
- Department
of Structural Biology and Chemistry, Institut
Pasteur, Université Paris Cité CNRS UMR 3528, 75015 Paris, France
| | - Michele Vendruscolo
- Centre
for Misfolding Diseases, Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, U.K.
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22
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Vervust W, Zhang DT, van Erp TS, Ghysels A. Path sampling with memory reduction and replica exchange to reach long permeation timescales. Biophys J 2023; 122:2960-2972. [PMID: 36809877 PMCID: PMC10398259 DOI: 10.1016/j.bpj.2023.02.021] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Revised: 01/13/2023] [Accepted: 02/15/2023] [Indexed: 02/23/2023] Open
Abstract
Assessing kinetics in biological processes with molecular dynamics simulations remains a computational and conceptual challenge, given the large time and length scales involved. For kinetic transport of biochemical compounds or drug molecules, the permeability through the phospholipid membranes is a key kinetic property, but long timescales are hindering the accurate computation. Technological advances in high-performance computing therefore need to be accompanied by theoretical and methodological developments. In this contribution, the replica exchange transition interface sampling (RETIS) methodology is shown to give perspective toward observing longer permeation pathways. It is first reviewed how RETIS, a path-sampling methodology that gives in principle exact kinetics, can be used to compute membrane permeability. Next, recent and current developments in three RETIS aspects are discussed: several new Monte Carlo moves in the path-sampling algorithm, memory reduction by reducing pathlengths, and exploitation of parallel computing with CPU-imbalanced replicas. Finally, the memory reduction presenting a new replica exchange implementation, coined REPPTIS, is showcased with a permeant needing to pass a membrane with two permeation channels, either representing an entropic or energetic barrier. The REPPTIS results showed clearly that inclusion of some memory and enhancing ergodic sampling via replica exchange moves are both necessary to obtain correct permeability estimates. In an additional example, ibuprofen permeation through a dipalmitoylphosphatidylcholine membrane was modeled. REPPTIS succeeded in estimating the permeability of this amphiphilic drug molecule with metastable states along the permeation pathway. In conclusion, the presented methodological advances allow for deeper insight into membrane biophysics even if the pathways are slow, as RETIS and REPPTIS push the permeability calculations to longer timescales.
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Affiliation(s)
- Wouter Vervust
- IBiTech - Biommeda Research Group, Faculty of Engineering and Architecture, Ghent University, Gent, Belgium
| | - Daniel T Zhang
- Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway
| | - Titus S van Erp
- Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway
| | - An Ghysels
- IBiTech - Biommeda Research Group, Faculty of Engineering and Architecture, Ghent University, Gent, Belgium.
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23
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Van Speybroeck V. Challenges in modelling dynamic processes in realistic nanostructured materials at operating conditions. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220239. [PMID: 37211031 PMCID: PMC10200353 DOI: 10.1098/rsta.2022.0239] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2022] [Accepted: 01/23/2023] [Indexed: 05/23/2023]
Abstract
The question is addressed in how far current modelling strategies are capable of modelling dynamic phenomena in realistic nanostructured materials at operating conditions. Nanostructured materials used in applications are far from perfect; they possess a broad range of heterogeneities in space and time extending over several orders of magnitude. Spatial heterogeneities from the subnanometre to the micrometre scale in crystal particles with a finite size and specific morphology, impact the material's dynamics. Furthermore, the material's functional behaviour is largely determined by the operating conditions. Currently, there exists a huge length-time scale gap between attainable theoretical length-time scales and experimentally relevant scales. Within this perspective, three key challenges are highlighted within the molecular modelling chain to bridge this length-time scale gap. Methods are needed that enable (i) building structural models for realistic crystal particles having mesoscale dimensions with isolated defects, correlated nanoregions, mesoporosity, internal and external surfaces; (ii) the evaluation of interatomic forces with quantum mechanical accuracy albeit at much lower computational cost than the currently used density functional theory methods and (iii) derivation of the kinetics of phenomena taking place in a multi-length-time scale window to obtain an overall view of the dynamics of the process. This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.
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24
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Blow KE, Tribello GA, Sosso GC, Quigley D. Interplay of multiple clusters and initial interface positioning for forward flux sampling simulations of crystal nucleation. J Chem Phys 2023; 158:2895225. [PMID: 37290068 DOI: 10.1063/5.0152343] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2023] [Accepted: 05/19/2023] [Indexed: 06/10/2023] Open
Abstract
Forward flux sampling (FFS) is a path sampling technique widely used in computer simulations of crystal nucleation from the melt. In such studies, the order parameter underpinning the progress of the FFS algorithm is often the size of the largest crystalline nucleus. In this work, we investigate the effects of two computational aspects of FFS simulations, using the prototypical Lennard-Jones liquid as our computational test bed. First, we quantify the impact of the positioning of the liquid basin and first interface in the space of the order parameter. In particular, we demonstrate that these choices are key to ensuring the consistency of the FFS results. Second, we focus on the frequently encountered scenario where the population of crystalline nuclei is such that there are multiple clusters of size comparable to the largest one. We demonstrate the contribution of clusters other than the largest cluster to the initial flux; however, we also show that they can be safely ignored for the purposes of converging a full FFS calculation. We also investigate the impact of different clusters merging, a process that appears to be facilitated by substantial spatial correlations-at least at the supercooling considered here. Importantly, all of our results have been obtained as a function of system size, thus contributing to the ongoing discussion on the impact of finite size effects on simulations of crystal nucleation. Overall, this work either provides or justifies several practical guidelines for performing FFS simulations that can also be applied to more complex and/or computationally expensive models.
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Affiliation(s)
- Katarina E Blow
- Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Gareth A Tribello
- Centre for Quantum Materials and Technologies, School of Mathematics and Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom
| | - Gabriele C Sosso
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - David Quigley
- Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
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25
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Goswami Y, Sastry S. Kinetic reconstruction of free energies as a function of multiple order parameters. J Chem Phys 2023; 158:144502. [PMID: 37061464 DOI: 10.1063/5.0144338] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/17/2023] Open
Abstract
A vast array of phenomena, ranging from chemical reactions to phase transformations, are analyzed in terms of a free energy surface defined with respect to a single or multiple order parameters. Enhanced sampling methods are typically used, especially in the presence of large free energy barriers, to estimate free energies using biasing protocols and sampling of transition paths. Kinetic reconstructions of free energy barriers of intermediate height have been performed, with respect to a single order parameter, employing the steady state properties of unconstrained simulation trajectories when barrier crossing is achievable with reasonable computational effort. Considering such cases, we describe a method to estimate free energy surfaces with respect to multiple order parameters from a steady state ensemble of trajectories. The approach applies to cases where the transition rates between pairs of order parameter values considered is not affected by the presence of an absorbing boundary, whereas the macroscopic fluxes and sampling probabilities are. We demonstrate the applicability of our prescription on different test cases of random walkers executing Brownian motion in order parameter space with an underlying (free) energy landscape and discuss strategies to improve numerical estimates of the fluxes and sampling. We next use this approach to reconstruct the free energy surface for supercooled liquid silicon with respect to the degree of crystallinity and density, from unconstrained molecular dynamics simulations, and obtain results quantitatively consistent with earlier results from umbrella sampling.
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Affiliation(s)
- Yagyik Goswami
- Theoretical Sciences Unit and School of Advanced Materials, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur Campus, Bengaluru 560064, India
| | - Srikanth Sastry
- Theoretical Sciences Unit and School of Advanced Materials, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur Campus, Bengaluru 560064, India
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26
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Reiner M, Bachmair B, Tiefenbacher MX, Mai S, González L, Marquetand P, Dellago C. Nonadiabatic Forward Flux Sampling for Excited-State Rare Events. J Chem Theory Comput 2023; 19:1657-1671. [PMID: 36856706 PMCID: PMC10061683 DOI: 10.1021/acs.jctc.2c01088] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2022] [Indexed: 03/02/2023]
Abstract
We present a rare event sampling scheme applicable to coupled electronic excited states. In particular, we extend the forward flux sampling (FFS) method for rare event sampling to a nonadiabatic version (NAFFS) that uses the trajectory surface hopping (TSH) method for nonadiabatic dynamics. NAFFS is applied to two dynamically relevant excited-state models that feature an avoided crossing and a conical intersection with tunable parameters. We investigate how nonadiabatic couplings, temperature, and reaction barriers affect transition rate constants in regimes that cannot be otherwise obtained with plain, traditional TSH. The comparison with reference brute-force TSH simulations for limiting cases of rareness shows that NAFFS can be several orders of magnitude cheaper than conventional TSH and thus represents a conceptually novel tool to extend excited-state dynamics to time scales that are able to capture rare nonadiabatic events.
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Affiliation(s)
- Madlen
Maria Reiner
- Research
Platform on Accelerating Photoreaction Discovery (ViRAPID), University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
- Vienna
Doctoral School in Physics, University of
Vienna, Boltzmanngasse
5, 1090 Vienna, Austria
| | - Brigitta Bachmair
- Research
Platform on Accelerating Photoreaction Discovery (ViRAPID), University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
- Vienna
Doctoral School in Chemistry, University
of Vienna, Währinger
Strasse 42, 1090 Vienna, Austria
| | - Maximilian Xaver Tiefenbacher
- Research
Platform on Accelerating Photoreaction Discovery (ViRAPID), University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
- Vienna
Doctoral School in Chemistry, University
of Vienna, Währinger
Strasse 42, 1090 Vienna, Austria
| | - Sebastian Mai
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
| | - Leticia González
- Research
Platform on Accelerating Photoreaction Discovery (ViRAPID), University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
| | - Philipp Marquetand
- Research
Platform on Accelerating Photoreaction Discovery (ViRAPID), University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
- Institute
of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
| | - Christoph Dellago
- Research
Platform on Accelerating Photoreaction Discovery (ViRAPID), University of Vienna, Währinger Strasse 17, 1090 Vienna, Austria
- Faculty
of Physics, University of Vienna, Kolingasse 14-16, 1090 Vienna, Austria
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27
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Zhang DT, Riccardi E, van Erp TS. Enhanced path sampling using subtrajectory Monte Carlo moves. J Chem Phys 2023; 158:024113. [PMID: 36641412 DOI: 10.1063/5.0127249] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
Abstract
Path sampling allows the study of rare events, such as chemical reactions, nucleation, and protein folding, via a Monte Carlo (MC) exploration in path space. Instead of configuration points, this method samples short molecular dynamics (MD) trajectories with specific start- and end-conditions. As in configuration MC, its efficiency highly depends on the types of MC moves. Since the last two decades, the central MC move for path sampling has been the so-called shooting move in which a perturbed phase point of the old path is propagated backward and forward in time to generate a new path. Recently, we proposed the subtrajectory moves, stone-skipping (SS) and web-throwing, that are demonstrably more efficient. However, the one-step crossing requirement makes them somewhat more difficult to implement in combination with external MD programs or when the order parameter determination is expensive. In this article, we present strategies to address the issue. The most generic solution is a new member of subtrajectory moves, wire fencing (WF), that is less thrifty than the SS but more versatile. This makes it easier to link path sampling codes with external MD packages and provides a practical solution for cases where the calculation of the order parameter is expensive or not a simple function of geometry. We demonstrate the WF move in a double-well Langevin model, a thin film breaking transition based on classical force fields, and a smaller ruthenium redox reaction at the ab initio level in which the order parameter explicitly depends on the electron density.
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Affiliation(s)
- Daniel T Zhang
- Norwegian University of Science and Technology, Department of Chemistry, NO-7491 Trondheim, Norway
| | - Enrico Riccardi
- Department of Informatics, UiO, Gaustadalléen 23B, 0373 Oslo, Norway
| | - Titus S van Erp
- Norwegian University of Science and Technology, Department of Chemistry, NO-7491 Trondheim, Norway
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28
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Aristoff D, Copperman J, Simpson G, Webber RJ, Zuckerman DM. Weighted ensemble: Recent mathematical developments. J Chem Phys 2023; 158:014108. [PMID: 36610976 PMCID: PMC9822651 DOI: 10.1063/5.0110873] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2022] [Accepted: 12/04/2022] [Indexed: 12/12/2022] Open
Abstract
Weighted ensemble (WE) is an enhanced sampling method based on periodically replicating and pruning trajectories generated in parallel. WE has grown increasingly popular for computational biochemistry problems due, in part, to improved hardware and accessible software implementations. Algorithmic and analytical improvements have played an important role, and progress has accelerated in recent years. Here, we discuss and elaborate on the WE method from a mathematical perspective, highlighting recent results that enhance the computational efficiency. The mathematical theory reveals a new strategy for optimizing trajectory management that approaches the best possible variance while generalizing to systems of arbitrary dimension.
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Affiliation(s)
- D. Aristoff
- Mathematics, Colorado State University, Fort Collins, CO 80521 USA
| | - J. Copperman
- Biomedical Engineering, Oregon Health and Science University, Portland, OR 97239 USA
| | - G. Simpson
- Mathematics, Drexel University, Philadelphia, Pennsylvania 19104 USA
| | - R. J. Webber
- Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California 91125 USA
| | - D. M. Zuckerman
- Biomedical Engineering, Oregon Health and Science University, Portland, OR 97239 USA
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29
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Roet S, Zhang DT, van Erp TS. Exchanging Replicas with Unequal Cost, Infinitely and Permanently. J Phys Chem A 2022; 126:8878-8886. [PMID: 36394633 PMCID: PMC9720720 DOI: 10.1021/acs.jpca.2c06004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2022] [Revised: 10/18/2022] [Indexed: 11/18/2022]
Abstract
We developed a replica exchange method that is effectively parallelizable even if the computational cost of the Monte Carlo moves in the parallel replicas are considerably different, for instance, because the replicas run on different types of processor units or because of the algorithmic complexity. To prove detailed-balance, we make a paradigm shift from the common conceptual viewpoint in which the set of parallel replicas represents a high-dimensional superstate, to an ensemble-based criterion in which the other ensembles represent an environment that might or might not participate in the Monte Carlo move. In addition, based on a recent algorithm for computing permanents, we effectively increase the exchange rate to infinite without the steep factorial scaling as a function of the number of replicas. We illustrate the effectiveness of this replica exchange methodology by combining it with a quantitative path sampling method, replica exchange transition interface sampling (RETIS), in which the costs for a Monte Carlo move can vary enormously as paths in a RETIS algorithm do not have the same length and the average path lengths tend to vary considerably for the different path ensembles that run in parallel. This combination, coined ∞RETIS, was tested on three model systems.
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Affiliation(s)
- Sander Roet
- Department of Chemistry, Norwegian University of Science and Technology (NTNU), N-7491Trondheim, Norway
| | - Daniel T. Zhang
- Department of Chemistry, Norwegian University of Science and Technology (NTNU), N-7491Trondheim, Norway
| | - Titus S. van Erp
- Department of Chemistry, Norwegian University of Science and Technology (NTNU), N-7491Trondheim, Norway
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30
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Baudel M, Guyader A, Lelièvre T. On the Hill relation and the mean reaction time for metastable processes. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.10.014] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
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31
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Abstract
The treatment of slow and rare transitions in the simulation of complex systems poses a great computational challenge. A powerful approach to tackle this challenge is the string method, which represents the transition path as a one-dimensional curve in a multidimensional space of collective variables. Commonly used strategies for pathway optimization include aligning the tangent of the string to the local mean force or to the mean drift determined from swarms of short trajectories. Here, a novel strategy is proposed, allowing the string to be optimized based on a variational principle involving the unidirectional reactive flux expressed in terms of the time-correlation function of the committor. The method is illustrated with model systems and then probed with the alanine dipeptide and a coarse-grained model of the barstar-barnase protein complex. Successive iterations variationally refine the string toward an optimal transition pathway following the gradient of the committor between two metastable states.
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Affiliation(s)
- Ziwei He
- Department of Chemistry, The University of Chicago, 5735 S. Ellis Avenue, Chicago60637, Illinois, United States
| | - Christophe Chipot
- Laboratoire International Associé Centre National de la Recherche Scientifique et University of Illinois at Urbana-Champaign, Unité Mixte de Recherche No. 7019, Université de Lorraine, B.P. 70239, Vandœuvre-lès-Nancy cedex54506, France
| | - Benoît Roux
- Department of Chemistry, The University of Chicago, 5735 S. Ellis Avenue, Chicago60637, Illinois, United States
- Department of Biochemistry and Molecular Biology, The University of Chicago, Chicago60637, IllinoisUnited States
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32
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Lei YK, Zhang Z, Han X, Yang YI, Zhang J, Gao YQ. Locating Transition Zone in Phase Space. J Chem Theory Comput 2022; 18:6124-6133. [PMID: 36135927 DOI: 10.1021/acs.jctc.2c00385] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Understanding the reaction mechanism is required for better control of chemical reactions and is usually achieved by locating transition states (TSs) along a proper one-dimensional coordinate called reaction coordinate (RC). The identification of RC can be very difficult for high-dimensional realistic systems. A number of methods have been proposed to tackle this problem. A machine learning method is developed here to incorporate the influence of velocity on the reaction process. The method is also free of the unbalanced label problem resulting from the rather low fraction of configurations near the TS and can be easily extended to large systems. It locates the transition zone in the phase space and defines the dividing surface with a high transmission coefficient. Moreover, considering that the reaction environment can not only change the reaction path but also activate the reactive mode through energy transfer, we devise two measures to quantify the influence of these two factors on the reaction process and find that solvents can assist the reaction by directly doing work along the reactive mode. Not surprisingly, there is a positive correlation between the efficiency of energy transfer into the reactive mode and the reaction rate.
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Affiliation(s)
- Yao-Kun Lei
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China.,Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China
| | - Zhen Zhang
- School of Physics and Technology, Tangshan Normal University, 063000 Tangshan, China
| | - Xu Han
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China.,Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China
| | - Yi Isaac Yang
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China
| | - Jun Zhang
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China
| | - Yi Qin Gao
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, 518055 Shenzhen, China.,Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China.,Biomedical Pioneering Innovation Center, Peking University, 100871 Beijing, China
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33
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Vani BP, Weare J, Dinner AR. Computing transition path theory quantities with trajectory stratification. J Chem Phys 2022; 157:034106. [PMID: 35868925 PMCID: PMC9296190 DOI: 10.1063/5.0087058] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2022] [Accepted: 06/23/2022] [Indexed: 11/15/2022] Open
Abstract
Transition path theory computes statistics from ensembles of reactive trajectories. A common strategy for sampling reactive trajectories is to control the branching and pruning of trajectories so as to enhance the sampling of low probability segments. However, it can be challenging to apply transition path theory to data from such methods because determining whether configurations and trajectory segments are part of reactive trajectories requires looking backward and forward in time. Here, we show how this issue can be overcome efficiently by introducing simple data structures. We illustrate the approach in the context of nonequilibrium umbrella sampling, but the strategy is general and can be used to obtain transition path theory statistics from other methods that sample segments of unbiased trajectories.
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Affiliation(s)
- Bodhi P. Vani
- Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
| | - Jonathan Weare
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
| | - Aaron R. Dinner
- Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
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34
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Lim CC, Lai SK. Metadynamics molecular dynamics and isothermal Brownian-type molecular dynamics simulations for the chiral cluster Au 18. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:325201. [PMID: 35580583 DOI: 10.1088/1361-648x/ac709f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2022] [Accepted: 05/17/2022] [Indexed: 06/15/2023]
Abstract
In an effort to gain insight into enantiomeric transitions, their transition mechanism, time span of transitions and distribution of time spans etc, we performed molecular dynamics (MD) simulations on chiral clusters Au10, Au15and Au18, and found that viable reaction coordinates can be deduced from simulation data for enlightening the enantiomeric dynamics for Au10and Au15, but not so for Au18. The failure in translating the Au18-L ⇌ Au18-R transitions by MD simulations has been chalked up to the thermal energykBTat 300 K being much lower than energy barriers separating the enantiomers of Au18. Two simulation strategies were taken to resolve this simulation impediment. The first one uses the well-tempered metadynamics MD (MMD) simulation, and the second one adeptly applies first a somewhat crude MMD simulation to locate a highly symmetrical isomer Au18Sand subsequently employed it as initial configuration in the MD simulation. In both strategies, we work in collective variable space of lower dimensionality. The well-tempered MMD simulation tactic was carried out aiming to offer a direct verification of Au18enantiomers, while the tactic to conduct MMD/MD simulations in two consecutive simulation steps was intended to provide an indirect evidence of the existence of enantiomers of Au18given that energy barriers separating them are much higher than ca.kBTat 300 K. This second tactic, in addition to confirming indirectly Au18-L and Au18-R starting from the symmetrical cluster Au18S, the simulation results shed light also on the mechanism akin to associative/nonassociative reaction transitions.
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Affiliation(s)
- C C Lim
- Complex Liquids Laboratory, Department of Physics, National Central University, Chungli 320, Taiwan
| | - S K Lai
- Complex Liquids Laboratory, Department of Physics, National Central University, Chungli 320, Taiwan
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35
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Ahmad K, Rizzi A, Capelli R, Mandelli D, Lyu W, Carloni P. Enhanced-Sampling Simulations for the Estimation of Ligand Binding Kinetics: Current Status and Perspective. Front Mol Biosci 2022; 9:899805. [PMID: 35755817 PMCID: PMC9216551 DOI: 10.3389/fmolb.2022.899805] [Citation(s) in RCA: 30] [Impact Index Per Article: 15.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2022] [Accepted: 05/09/2022] [Indexed: 12/12/2022] Open
Abstract
The dissociation rate (k off) associated with ligand unbinding events from proteins is a parameter of fundamental importance in drug design. Here we review recent major advancements in molecular simulation methodologies for the prediction of k off. Next, we discuss the impact of the potential energy function models on the accuracy of calculated k off values. Finally, we provide a perspective from high-performance computing and machine learning which might help improve such predictions.
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Affiliation(s)
- Katya Ahmad
- Computational Biomedicine (IAS-5/INM-9), Forschungszentrum Jülich, Jülich, Germany
| | - Andrea Rizzi
- Computational Biomedicine (IAS-5/INM-9), Forschungszentrum Jülich, Jülich, Germany
- Atomistic Simulations, Istituto Italiano di Tecnologia, Genova, Italy
| | - Riccardo Capelli
- Department of Applied Science and Technology (DISAT), Politecnico di Torino, Torino, Italy
| | - Davide Mandelli
- Computational Biomedicine (IAS-5/INM-9), Forschungszentrum Jülich, Jülich, Germany
| | - Wenping Lyu
- Warshel Institute for Computational Biology, School of Life and Health Sciences, The Chinese University of Hong Kong (Shenzhen), Shenzhen, China
- School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, China
| | - Paolo Carloni
- Computational Biomedicine (IAS-5/INM-9), Forschungszentrum Jülich, Jülich, Germany
- Molecular Neuroscience and Neuroimaging (INM-11), Forschungszentrum Jülich, Jülich, Germany
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36
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Baidakov VG. Stability of Metastable Phases and Kinetics of Nucleation in a Simple Single-Component System (Molecular Dynamics Simulation) (A Review). RUSS J GEN CHEM+ 2022. [DOI: 10.1134/s107036322204003x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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37
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Hall SW, Díaz Leines G, Sarupria S, Rogal J. Practical guide to replica exchange transition interface sampling and forward flux sampling. J Chem Phys 2022; 156:200901. [DOI: 10.1063/5.0080053] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Path sampling approaches have become invaluable tools to explore the mechanisms and dynamics of the so-called rare events that are characterized by transitions between metastable states separated by sizable free energy barriers. Their practical application, in particular to ever more complex molecular systems, is, however, not entirely trivial. Focusing on replica exchange transition interface sampling (RETIS) and forward flux sampling (FFS), we discuss a range of analysis tools that can be used to assess the quality and convergence of such simulations, which is crucial to obtain reliable results. The basic ideas of a step-wise evaluation are exemplified for the study of nucleation in several systems with different complexities, providing a general guide for the critical assessment of RETIS and FFS simulations.
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Affiliation(s)
- Steven W. Hall
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA
| | - Grisell Díaz Leines
- Yusuf Hamied Department of Chemistry, University of Cambridge, Cambridgeshire CB2 1EW, United Kingdom
| | - Sapna Sarupria
- Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, USA
- Department of Chemical and Biomolecular Engineering, Clemson University, Clemson, South Carolina 29634, USA
| | - Jutta Rogal
- Department of Chemistry, New York University, New York, New York 10003, USA
- Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
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38
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Pérez de Alba Ortíz A, Vreede J, Ensing B. Sequence dependence of transient Hoogsteen base pairing in DNA. PLoS Comput Biol 2022; 18:e1010113. [PMID: 35617357 PMCID: PMC9177043 DOI: 10.1371/journal.pcbi.1010113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2021] [Revised: 06/08/2022] [Accepted: 04/19/2022] [Indexed: 11/19/2022] Open
Abstract
Hoogsteen (HG) base pairing is characterized by a 180° rotation of the purine base with respect to the Watson-Crick-Franklin (WCF) motif. Recently, it has been found that both conformations coexist in a dynamical equilibrium and that several biological functions require HG pairs. This relevance has motivated experimental and computational investigations of the base-pairing transition. However, a systematic simulation of sequence variations has remained out of reach. Here, we employ advanced path-based methods to perform unprecedented free-energy calculations. Our methodology enables us to study the different mechanisms of purine rotation, either remaining inside or after flipping outside of the double helix. We study seven different sequences, which are neighbor variations of a well-studied A⋅T pair in A6-DNA. We observe the known effect of A⋅T steps favoring HG stability, and find evidence of triple-hydrogen-bonded neighbors hindering the inside transition. More importantly, we identify a dominant factor: the direction of the A rotation, with the 6-ring pointing either towards the longer or shorter segment of the chain, respectively relating to a lower or higher barrier. This highlights the role of DNA's relative flexibility as a modulator of the WCF/HG dynamic equilibrium. Additionally, we provide a robust methodology for future HG proclivity studies.
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Affiliation(s)
- Alberto Pérez de Alba Ortíz
- Van ’t Hoff Institute for Molecular Sciences and Amsterdam Center for Multiscale Modeling, University of Amsterdam, Amsterdam, The Netherlands
- Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Utrecht, The Netherlands
| | - Jocelyne Vreede
- Van ’t Hoff Institute for Molecular Sciences and Amsterdam Center for Multiscale Modeling, University of Amsterdam, Amsterdam, The Netherlands
| | - Bernd Ensing
- Van ’t Hoff Institute for Molecular Sciences and Amsterdam Center for Multiscale Modeling, University of Amsterdam, Amsterdam, The Netherlands
- AI4Science Laboratory, University of Amsterdam, Amsterdam, The Netherlands
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39
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Abstract
The kinetics of a dynamical system dominated by two metastable states is examined from the perspective of the activated-dynamics reactive flux formalism, Markov state eigenvalue spectral decomposition, and committor-based transition path theory. Analysis shows that the different theoretical formulations are consistent, clarifying the significance of the inherent microscopic lag-times that are implicated, and that the most meaningful one-dimensional reaction coordinate in the region of the transition state is along the gradient of the committor in the multidimensional subspace of collective variables. It is shown that the familiar reactive flux activated dynamics formalism provides an effective route to calculate the transition rate in the case of a narrow sharp barrier but much less so in the case of a broad flat barrier. In this case, the standard reactive flux correlation function decays very slowly to the plateau value that corresponds to the transmission coefficient. Treating the committor function as a reaction coordinate does not alleviate all issues caused by the slow relaxation of the reactive flux correlation function. A more efficient activated dynamics simulation algorithm may be achieved from a modified reactive flux weighted by the committor. Simulation results on simple systems are used to illustrate the various conceptual points.
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Affiliation(s)
- Benoît Roux
- Department of Biochemistry and Molecular Biology, Department of Chemistry, The University of Chicago, 5735 S Ellis Ave., Chicago, Illinois 60637, USA
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40
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Lervik A, Svenum IH, Wang Z, Cabriolu R, Riccardi E, Andersson S, van Erp TS. The role of pressure and defects in the wurtzite to rock salt transition in cadmium selenide. Phys Chem Chem Phys 2022; 24:8378-8386. [PMID: 35332892 DOI: 10.1039/d1cp05051f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Using molecular dynamics and path sampling techniques we investigated the effect of pressure and defects in the wurtzite to rock salt transition in cadmium selenide (CdSe). In the pressure range 2-10 GPa, rate constants of transition are in the order of 10-23 to 105 s-1 for the transformation of a relatively small wurtzite crystal consisting of 1024 atoms with periodic boundary conditions. The transition paths predominantly evolve through an intermediate 5-coordinated structure, as reported before, though its typical lifetime within the transition paths is particularly long in the intermediate pressure range (4-6 GPa). The defects were created by removing Cd-Se pairs from an otherwise perfect crystal. The removals were either selected fully randomized or grouped in clusters (cavity creation). We find that the rate of transition due to the defects increases by several orders of magnitude even for a single pair removal. This is caused by a change in the transition mechanism that no longer proceeds via the intermediate 5-coordinated structure, when defects are present. Further, the cavity creation yields a lower rate than the fully randomized removal.
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Affiliation(s)
- Anders Lervik
- Department of Chemistry, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway.
| | - Ingeborg-Helene Svenum
- Department of Materials and Nanotechnology, SINTEF Industry, P.O. Box 4760 Torgarden, 7465 Trondheim, Norway
| | - Zhaohui Wang
- Department of Metal Production and Processing, SINTEF Industry, P.O. Box 4760 Torgarden, 7465 Trondheim, Norway
| | - Raffaela Cabriolu
- Department of Chemistry, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway. .,Department of Physics, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway
| | - Enrico Riccardi
- Department of Chemistry, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway. .,Department of Informatics, UiO, Gaustadalléen 23B, 0373 Oslo, Norway
| | - Stefan Andersson
- Department of Metal Production and Processing, SINTEF Industry, P.O. Box 4760 Torgarden, 7465 Trondheim, Norway
| | - Titus S van Erp
- Department of Chemistry, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway.
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41
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Paul TK, Taraphder S. Nonlinear Reaction Coordinate of an Enzyme Catalyzed Proton Transfer Reaction. J Phys Chem B 2022; 126:1413-1425. [PMID: 35138854 DOI: 10.1021/acs.jpcb.1c08760] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Abstract
We present an in-depth study on the theoretical calculation of an optimum reaction coordinate as a linear or nonlinear combination of important collective variables (CVs) sampled from an ensemble of reactive transition paths for an intramolecular proton transfer reaction catalyzed by the enzyme human carbonic anhydrase (HCA) II. The linear models are optimized by likelihood maximization for a given number of CVs. The nonlinear models are based on an artificial neural network with the same number of CVs and optimized by minimizing the root-mean-square error in comparison to a training set of committor estimators generated for the given transition. The nonlinear reaction coordinate thus obtained yields the free energy of activation and rate constant as 9.46 kcal mol-1 and 1.25 × 106 s-1, respectively. These estimates are found to be in quantitative agreement with the known experimental results. We have also used an extended autoencoder model to show that a similar analysis can be carried out using a single CV only. The resultant free energies and kinetics of the reaction slightly overestimate the experimental data. The implications of these results are discussed using a detailed microkinetic scheme of the proton transfer reaction catalyzed by HCA II.
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Affiliation(s)
- Tanmoy Kumar Paul
- Department of Chemistry, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
| | - Srabani Taraphder
- Department of Chemistry, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
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42
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Russo JD, Zhang S, Leung JMG, Bogetti AT, Thompson JP, DeGrave AJ, Torrillo PA, Pratt AJ, Wong KF, Xia J, Copperman J, Adelman JL, Zwier MC, LeBard DN, Zuckerman DM, Chong LT. WESTPA 2.0: High-Performance Upgrades for Weighted Ensemble Simulations and Analysis of Longer-Timescale Applications. J Chem Theory Comput 2022; 18:638-649. [PMID: 35043623 PMCID: PMC8825686 DOI: 10.1021/acs.jctc.1c01154] [Citation(s) in RCA: 15] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/13/2023]
Abstract
The weighted ensemble (WE) family of methods is one of several statistical mechanics-based path sampling strategies that can provide estimates of key observables (rate constants and pathways) using a fraction of the time required by direct simulation methods such as molecular dynamics or discrete-state stochastic algorithms. WE methods oversee numerous parallel trajectories using intermittent overhead operations at fixed time intervals, enabling facile interoperability with any dynamics engine. Here, we report on the major upgrades to the WESTPA software package, an open-source, high-performance framework that implements both basic and recently developed WE methods. These upgrades offer substantial improvements over traditional WE methods. The key features of the new WESTPA 2.0 software enhance the efficiency and ease of use: an adaptive binning scheme for more efficient surmounting of large free energy barriers, streamlined handling of large simulation data sets, exponentially improved analysis of kinetics, and developer-friendly tools for creating new WE methods, including a Python API and resampler module for implementing both binned and "binless" WE strategies.
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Affiliation(s)
- John D Russo
- Department of Biomedical Engineering, Oregon Health and Science University, Portland, Oregon 97239-3098, United States
| | - She Zhang
- OpenEye Scientific, Santa Fe, New Mexico 87508, United States
| | - Jeremy M G Leung
- Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
| | - Anthony T Bogetti
- Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
| | - Jeff P Thompson
- OpenEye Scientific, Santa Fe, New Mexico 87508, United States
| | - Alex J DeGrave
- Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
| | - Paul A Torrillo
- Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
| | - A J Pratt
- Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
| | - Kim F Wong
- Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
| | - Junchao Xia
- OpenEye Scientific, Santa Fe, New Mexico 87508, United States
| | - Jeremy Copperman
- Department of Biomedical Engineering, Oregon Health and Science University, Portland, Oregon 97239-3098, United States
| | - Joshua L Adelman
- Department of Biological Sciences, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
| | - Matthew C Zwier
- Department of Chemistry, Drake University, Des Moines, Iowa 50311-4505, United States
| | - David N LeBard
- OpenEye Scientific, Santa Fe, New Mexico 87508, United States
| | - Daniel M Zuckerman
- Department of Biomedical Engineering, Oregon Health and Science University, Portland, Oregon 97239-3098, United States
| | - Lillian T Chong
- Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
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43
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Abstract
Abstract
We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation of long time averages. We establish the consistency of weighted ensemble in this setting by proving an ergodic theorem for time averages. As part of the proof, we derive explicit variance formulas that could be useful for optimizing the method.
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44
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Thin film breakage in oil–in–water emulsions, a multidisciplinary study. Colloids Surf A Physicochem Eng Asp 2022. [DOI: 10.1016/j.colsurfa.2021.127808] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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45
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Ray D, Stone SE, Andricioaei I. Markovian Weighted Ensemble Milestoning (M-WEM): Long-Time Kinetics from Short Trajectories. J Chem Theory Comput 2021; 18:79-95. [PMID: 34910499 DOI: 10.1021/acs.jctc.1c00803] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
We introduce a rare-event sampling scheme, named Markovian Weighted Ensemble Milestoning (M-WEM), which inlays a weighted ensemble framework within a Markovian milestoning theory to efficiently calculate thermodynamic and kinetic properties of long-time-scale biomolecular processes from short atomistic molecular dynamics simulations. M-WEM is tested on the Müller-Brown potential model, the conformational switching in alanine dipeptide, and the millisecond time-scale protein-ligand unbinding in a trypsin-benzamidine complex. Not only can M-WEM predict the kinetics of these processes with quantitative accuracy but it also allows for a scheme to reconstruct a multidimensional free-energy landscape along additional degrees of freedom, which are not part of the milestoning progress coordinate. For the ligand-receptor system, the experimental residence time, association and dissociation kinetics, and binding free energy could be reproduced using M-WEM within a simulation time of a few hundreds of nanoseconds, which is a fraction of the computational cost of other currently available methods, and close to 4 orders of magnitude less than the experimental residence time. Due to the high accuracy and low computational cost, the M-WEM approach can find potential applications in kinetics and free-energy-based computational drug design.
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Affiliation(s)
- Dhiman Ray
- Department of Chemistry, University of California Irvine, Irvine, California 92697, United States
| | - Sharon Emily Stone
- Department of Chemistry, University of California Irvine, Irvine, California 92697, United States
| | - Ioan Andricioaei
- Department of Chemistry, University of California Irvine, Irvine, California 92697, United States.,Department of Physics and Astronomy, University of California Irvine, Irvine, California 92697, United States
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46
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Vlachas PR, Zavadlav J, Praprotnik M, Koumoutsakos P. Accelerated Simulations of Molecular Systems through Learning of Effective Dynamics. J Chem Theory Comput 2021; 18:538-549. [PMID: 34890204 DOI: 10.1021/acs.jctc.1c00809] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
Simulations are vital for understanding and predicting the evolution of complex molecular systems. However, despite advances in algorithms and special purpose hardware, accessing the time scales necessary to capture the structural evolution of biomolecules remains a daunting task. In this work, we present a novel framework to advance simulation time scales by up to 3 orders of magnitude by learning the effective dynamics (LED) of molecular systems. LED augments the equation-free methodology by employing a probabilistic mapping between coarse and fine scales using mixture density network (MDN) autoencoders and evolves the non-Markovian latent dynamics using long short-term memory MDNs. We demonstrate the effectiveness of LED in the Müller-Brown potential, the Trp cage protein, and the alanine dipeptide. LED identifies explainable reduced-order representations, i.e., collective variables, and can generate, at any instant, all-atom molecular trajectories consistent with the collective variables. We believe that the proposed framework provides a dramatic increase to simulation capabilities and opens new horizons for the effective modeling of complex molecular systems.
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Affiliation(s)
- Pantelis R Vlachas
- Computational Science and Engineering Laboratory, ETH Zurich, CH-8092, Switzerland
| | - Julija Zavadlav
- Professorship of Multiscale Modeling of Fluid Materials, TUM School of Engineering and Design, Technical University of Munich, 85748 Garching bei München, Germany.,Munich Data Science Institute, Technical University of Munich, 85748 Munich, Germany
| | - Matej Praprotnik
- Laboratory for Molecular Modeling, National Institute of Chemistry, SI-1001 Ljubljana, Slovenia.,Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
| | - Petros Koumoutsakos
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
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Zuckerman DM, Russo JD. A gentle introduction to the non-equilibrium physics of trajectories: Theory, algorithms, and biomolecular applications. AMERICAN JOURNAL OF PHYSICS 2021; 89:1048-1061. [PMID: 35530173 PMCID: PMC9075726 DOI: 10.1119/10.0005603] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2021] [Accepted: 06/25/2021] [Indexed: 06/14/2023]
Abstract
Despite the importance of non-equilibrium statistical mechanics in modern physics and related fields, the topic is often omitted from undergraduate and core-graduate curricula. Key aspects of non-equilibrium physics, however, can be understood with a minimum of formalism based on a rigorous trajectory picture. The fundamental object is the ensemble of trajectories, a set of independent time-evolving systems, which easily can be visualized or simulated (e.g., for protein folding) and which can be analyzed rigorously in analogy to an ensemble of static system configurations. The trajectory picture provides a straightforward basis for understanding first-passage times, "mechanisms" in complex systems, and fundamental constraints on the apparent reversibility of complex processes. Trajectories make concrete the physics underlying the diffusion and Fokker-Planck partial differential equations. Last but not least, trajectory ensembles underpin some of the most important algorithms that have provided significant advances in biomolecular studies of protein conformational and binding processes.
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Chen M. Collective variable-based enhanced sampling and machine learning. THE EUROPEAN PHYSICAL JOURNAL. B 2021; 94:211. [PMID: 34697536 PMCID: PMC8527828 DOI: 10.1140/epjb/s10051-021-00220-w] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2021] [Accepted: 10/03/2021] [Indexed: 05/14/2023]
Abstract
ABSTRACT Collective variable-based enhanced sampling methods have been widely used to study thermodynamic properties of complex systems. Efficiency and accuracy of these enhanced sampling methods are affected by two factors: constructing appropriate collective variables for enhanced sampling and generating accurate free energy surfaces. Recently, many machine learning techniques have been developed to improve the quality of collective variables and the accuracy of free energy surfaces. Although machine learning has achieved great successes in improving enhanced sampling methods, there are still many challenges and open questions. In this perspective, we shall review recent developments on integrating machine learning techniques and collective variable-based enhanced sampling approaches. We also discuss challenges and future research directions including generating kinetic information, exploring high-dimensional free energy surfaces, and efficiently sampling all-atom configurations. GRAPHIC ABSTRACT
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Affiliation(s)
- Ming Chen
- Department of Chemistry, Purdue University, West Lafayette, IN 47907 USA
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Roet S, Daub CD, Riccardi E. Chemistrees: Data-Driven Identification of Reaction Pathways via Machine Learning. J Chem Theory Comput 2021; 17:6193-6202. [PMID: 34555907 PMCID: PMC8515787 DOI: 10.1021/acs.jctc.1c00458] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/03/2022]
Abstract
![]()
We propose to analyze
molecular dynamics (MD) output via a supervised machine
learning (ML) algorithm, the decision tree.
The approach aims to identify the predominant geometric features which
correlate with trajectories that transition between two arbitrarily
defined states. The data-driven algorithm aims to identify these features
without the bias of human “chemical intuition”. We demonstrate
the method by analyzing the proton exchange reactions in formic acid
solvated in small water clusters. The simulations were performed with ab initio MD combined with a method to efficiently sample
the rare event, path sampling. Our ML analysis identified relevant
geometric variables involved in the proton transfer reaction and how
they may change as the number of solvating water molecules changes.
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Affiliation(s)
- Sander Roet
- Department of Chemistry, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway
| | - Christopher D Daub
- Department of Chemistry, University of Helsinki, P.O. Box 55, FI-00014 Helsinki, Finland
| | - Enrico Riccardi
- Department of Informatics, UiO, Gaustadalléen 23B, 0373 Oslo, Norway
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Liwo A, Czaplewski C, Sieradzan AK, Lipska AG, Samsonov SA, Murarka RK. Theory and Practice of Coarse-Grained Molecular Dynamics of Biologically Important Systems. Biomolecules 2021; 11:1347. [PMID: 34572559 PMCID: PMC8465211 DOI: 10.3390/biom11091347] [Citation(s) in RCA: 25] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2021] [Revised: 09/03/2021] [Accepted: 09/09/2021] [Indexed: 12/16/2022] Open
Abstract
Molecular dynamics with coarse-grained models is nowadays extensively used to simulate biomolecular systems at large time and size scales, compared to those accessible to all-atom molecular dynamics. In this review article, we describe the physical basis of coarse-grained molecular dynamics, the coarse-grained force fields, the equations of motion and the respective numerical integration algorithms, and selected practical applications of coarse-grained molecular dynamics. We demonstrate that the motion of coarse-grained sites is governed by the potential of mean force and the friction and stochastic forces, resulting from integrating out the secondary degrees of freedom. Consequently, Langevin dynamics is a natural means of describing the motion of a system at the coarse-grained level and the potential of mean force is the physical basis of the coarse-grained force fields. Moreover, the choice of coarse-grained variables and the fact that coarse-grained sites often do not have spherical symmetry implies a non-diagonal inertia tensor. We describe selected coarse-grained models used in molecular dynamics simulations, including the most popular MARTINI model developed by Marrink's group and the UNICORN model of biological macromolecules developed in our laboratory. We conclude by discussing examples of the application of coarse-grained molecular dynamics to study biologically important processes.
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Affiliation(s)
- Adam Liwo
- Faculty of Chemistry, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland; (C.C.); (A.K.S.); (A.G.L.); (S.A.S.)
| | - Cezary Czaplewski
- Faculty of Chemistry, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland; (C.C.); (A.K.S.); (A.G.L.); (S.A.S.)
| | - Adam K. Sieradzan
- Faculty of Chemistry, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland; (C.C.); (A.K.S.); (A.G.L.); (S.A.S.)
| | - Agnieszka G. Lipska
- Faculty of Chemistry, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland; (C.C.); (A.K.S.); (A.G.L.); (S.A.S.)
| | - Sergey A. Samsonov
- Faculty of Chemistry, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland; (C.C.); (A.K.S.); (A.G.L.); (S.A.S.)
| | - Rajesh K. Murarka
- Department of Chemistry, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhopal 462066, MP, India;
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