1
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Blanc FEC, Houdusse A, Cecchini M. A weak coupling mechanism for the early steps of the recovery stroke of myosin VI: A free energy simulation and string method analysis. PLoS Comput Biol 2024; 20:e1012005. [PMID: 38662764 PMCID: PMC11086841 DOI: 10.1371/journal.pcbi.1012005] [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: 10/07/2023] [Revised: 05/10/2024] [Accepted: 03/15/2024] [Indexed: 05/12/2024] Open
Abstract
Myosin motors use the energy of ATP to produce force and directed movement on actin by a swing of the lever-arm. ATP is hydrolysed during the off-actin re-priming transition termed recovery stroke. To provide an understanding of chemo-mechanical transduction by myosin, it is critical to determine how the reverse swing of the lever-arm and ATP hydrolysis are coupled. Previous studies concluded that the recovery stroke of myosin II is initiated by closure of the Switch II loop in the nucleotide-binding site. Recently, we proposed that the recovery stroke of myosin VI starts with the spontaneous re-priming of the converter domain to a putative pre-transition state (PTS) intermediate that precedes Switch II closing and ATPase activation. Here, we investigate the transition from the pre-recovery, post-rigor (PR) state to PTS in myosin VI using geometric free energy simulations and the string method. First, our calculations rediscover the PTS state agnostically and show that it is accessible from PR via a low free energy transition path. Second, separate path calculations using the string method illuminate the mechanism of the PR to PTS transition with atomic resolution. In this mechanism, the initiating event is a large movement of the converter/lever-arm region that triggers rearrangements in the Relay-SH1 region and the formation of the kink in the Relay helix with no coupling to the active site. Analysis of the free-energy barriers along the path suggests that the converter-initiated mechanism is much faster than the one initiated by Switch II closure, which supports the biological relevance of PTS as a major on-pathway intermediate of the recovery stroke in myosin VI. Our analysis suggests that lever-arm re-priming and ATP hydrolysis are only weakly coupled, so that the myosin recovery stroke is initiated by thermal fluctuations and stabilised by nucleotide consumption via a ratchet-like mechanism.
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Affiliation(s)
- Florian E. C. Blanc
- Institut de Chimie de Strasbourg, UMR7177, CNRS, Université de Strasbourg, Strasbourg, France
- Structural Motility, Institut Curie, CNRS, UMR144, PSL Research University, Paris, France
| | - Anne Houdusse
- Structural Motility, Institut Curie, CNRS, UMR144, PSL Research University, Paris, France
| | - Marco Cecchini
- Institut de Chimie de Strasbourg, UMR7177, CNRS, Université de Strasbourg, Strasbourg, France
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2
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Wong CF. 15 Years of molecular simulation of drug-binding kinetics. Expert Opin Drug Discov 2023; 18:1333-1348. [PMID: 37789731 PMCID: PMC10926948 DOI: 10.1080/17460441.2023.2264770] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2023] [Accepted: 09/26/2023] [Indexed: 10/05/2023]
Abstract
INTRODUCTION Drug-binding kinetics has been increasingly recognized as an important factor to be considered in drug discovery. Long residence time could prolong the action of some drugs while produce toxicity on others. Early evaluation of the binding kinetics of drug candidates could reduce attrition rate late in the drug discovery process. Computational prediction of drug-binding kinetics is useful as compounds can be evaluated even before they are made. However, simulation of drug-binding kinetics is a challenging problem because of the long-time scale involved. Nevertheless, significant progress has been made. AREAS COVERED This review illustrates the rapid evolution of qualitative to quantitative molecular dynamics-based methods that have been developed over the last 15 years. EXPERT OPINION The development of new methods based on molecular dynamics simulations now enables computation of absolute association/dissociation rate constants. Cheaper methods capable of identifying candidates with fast or slow binding kinetics, or rank-ordering rate constants are also available. Together, these methods have generated useful insights into the molecular mechanisms of drug-binding kinetics, and the design of drug candidates with therapeutically favorable kinetics. Although predicting absolute rate constants is still expensive and challenging, rapid improvement is expected in the coming years with the continuing refinement of current technologies, development of new methodologies, and the utilization of machine learning.
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Affiliation(s)
- Chung F Wong
- Department of Chemistry and Biochemistry, University of Missouri-St. Louis, St. Louis, MO, USA
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3
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Evans L, Cameron MK, Tiwary P. Computing committors via Mahalanobis diffusion maps with enhanced sampling data. J Chem Phys 2022; 157:214107. [PMID: 36511548 DOI: 10.1063/5.0122990] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/03/2022] Open
Abstract
The study of phenomena such as protein folding and conformational changes in molecules is a central theme in chemical physics. Molecular dynamics (MD) simulation is the primary tool for the study of transition processes in biomolecules, but it is hampered by a huge timescale gap between the processes of interest and atomic vibrations that dictate the time step size. Therefore, it is imperative to combine MD simulations with other techniques in order to quantify the transition processes taking place on large timescales. In this work, the diffusion map with Mahalanobis kernel, a meshless approach for approximating the Backward Kolmogorov Operator (BKO) in collective variables, is upgraded to incorporate standard enhanced sampling techniques, such as metadynamics. The resulting algorithm, which we call the target measure Mahalanobis diffusion map (tm-mmap), is suitable for a moderate number of collective variables in which one can approximate the diffusion tensor and free energy. Imposing appropriate boundary conditions allows use of the approximated BKO to solve for the committor function and utilization of transition path theory to find the reactive current delineating the transition channels and the transition rate. The proposed algorithm, tm-mmap, is tested on the two-dimensional Moro-Cardin two-well system with position-dependent diffusion coefficient and on alanine dipeptide in two collective variables where the committor, the reactive current, and the transition rate are compared to those computed by the finite element method (FEM). Finally, tm-mmap is applied to alanine dipeptide in four collective variables where the use of finite elements is infeasible.
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Affiliation(s)
- L Evans
- Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA
| | - M K Cameron
- Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA
| | - P Tiwary
- Department of Chemistry and Biochemistry and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
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4
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Louwerse MD, Sivak DA. Connections between efficient control and spontaneous transitions in an Ising model. Phys Rev E 2022; 106:064124. [PMID: 36671088 DOI: 10.1103/physreve.106.064124] [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: 09/14/2022] [Accepted: 12/02/2022] [Indexed: 06/17/2023]
Abstract
A system can be driven between metastable configurations by a time-dependent driving protocol, which uses external control parameters to change the potential energy of the system. Here we investigate the correspondence between driving protocols that are designed to minimize work and the spontaneous transition paths of the system in the absence of driving. We study the spin-inversion reaction in a 2D Ising model, quantifying the timing of each spin flip and heat flow to the system during both a minimum-work protocol and a spontaneous transition. The general order of spin flips during the transition mechanism is preserved between the processes, despite the coarseness of control parameters that are unable to reproduce more detailed features of the spontaneous mechanism. Additionally, external control parameters provide energy to each system component to compensate changes in internal energy, showing how control parameters are tuned during a minimum-work protocol to counteract underlying energetic features. This paper supports a correspondence between minimum-work protocols and spontaneous transition mechanisms.
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Affiliation(s)
- Miranda D Louwerse
- Department of Chemistry, Simon Fraser University, Burnaby, British Columbia, Canada V5A1S6
| | - David A Sivak
- Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A1S6
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5
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Ogden DS, Moradi M. Atomic-level characterization of the conformational transition pathways in SARS-CoV-1 and SARS-CoV-2 spike proteins. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2022:2022.11.29.518406. [PMID: 36482979 PMCID: PMC9727763 DOI: 10.1101/2022.11.29.518406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
Severe acute respiratory syndrome (SARS) coronaviruses 1 and 2 (SARS-CoV-1 and SARS-CoV-2) derive transmissibility from spike protein activation in the receptor binding domain (RBD) and binding to the host cell angiotensin converting enzyme 2 (ACE2). However, the mechanistic details that describe the large-scale conformational changes associated with spike protein activation or deactivation are still somewhat unknown. Here, we have employed an extensive set of nonequilibrium all-atom molecular dynamics (MD) simulations, utilizing a novel protocol, for the SARS-CoV-1 (CoV-1) and SARS-CoV-2 (CoV-2) prefusion spike proteins in order to characterize the conformational pathways associated with the active-to-inactive transition. Our results indicate that both CoV-1 and CoV-2 spike proteins undergo conformational transitions along pathways unique to each protein. We have identified a number of key residues that form various inter-domain saltbridges, suggesting a multi-stage conformational change along the pathways. We have also constructed the free energy profiles along the transition pathways for both CoV-1 and CoV-2 spike proteins. The CoV-2 spike protein must overcome larger free energy barriers to undergo conformational changes towards protein activation or deactivation, when compared to CoV-1.
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6
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Li W. Potential Energy Weighted Reactive Flux and Total Rate of Change of Potential Energy: Theory and Illustrative Applications. J Phys Chem A 2022; 126:7774-7786. [PMID: 36251005 DOI: 10.1021/acs.jpca.2c04886] [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
Reactive flux can be largely nonzero in a nonequilibrium ensemble of trajectories and provide insightful information for reactive transitions from the reactant state to the product state. Based on the reactive flux, a theoretical framework is proposed here for two quantities, the potential energy weighted reactive flux and the total rate of change of potential energy, which are useful for the identification of the mechanism from a nonequilibrium ensemble. From such quantities, two multidimensional free-energy analogues can be derived in the subspace of collective variables and they are equivalent in the regions where the reactive flux is divergence-free. These free-energy analogues are assumed to be closely related to the free energy in the subspace of collective variables, and they are reduced in the one-dimensional case to be the ensemble average of the potential energy weighted with reactive flux intensity, which was proposed recently [Li, W. J. Phys. Chem. A 2022, DOI: 10.1021/acs.jpca.2c04130] and could be decomposed into energy components at the per-coordinate level. In the subspace of collective variables, the decomposition of the multidimensional free-energy analogues at the per-coordinate level is theoretically possible and is numerically difficult to be calculated. Interestingly, the total rate of change of potential energy is able to identify the location of the transition state ensemble or the stochastic separatrix, in addition to the locations of the reactant and product states. The total rate of change of potential energy can be decomposed at the per-coordinate level, and its components can quantify the contribution of a coordinate to the reactive transition in the subspace of collective variables. We then illustrated the main insights and objects that can be provided by the approach in the applications to a two-dimensional system with various diffusion anisotropies and the alanine peptide in vacuum in various nonequilibrium ensembles of short trajectories, from which the results were found to be consistent.
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Affiliation(s)
- Wenjin Li
- Institute for Advanced Study, Shenzhen University, Shenzhen, 518060, China
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7
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Li W. Energy Decomposition along the Reaction Coordinate: Theory and Applications to Nonequilibrium Ensembles of Trajectories. J Phys Chem A 2022; 126:7763-7773. [PMID: 36214522 DOI: 10.1021/acs.jpca.2c04130] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
A theoretical framework is proposed for an energy decomposition scheme along the reaction coordinate, in which the ensemble average of the potential energy weighted with reactive flux intensity is decomposed into energy components at the per-coordinate level. The decomposed energy quantity is demonstrated to be closely related to the free energy along the reaction coordinate, and its connection to the emergent potential energy is provided. In the application to alanine dipeptide under vacuum, illustrative calculations were performed in three nonequilibrium ensembles of trajectories: (1) transition path ensemble sampled with transition path sampling; (2) ensemble of short trajectories initiated from configurations around the transition-state region; and (3) ensemble of short trajectories shooting from configurations in several transition paths. The energy components on each coordinate were found to be consistent among the three ensembles of trajectories, indicating a broad applicability of the approach in biomolecular studies. In addition, the free energies along an optimized reaction coordinate obtained with these nonequilibrium ensembles were largely overlapped with a reference free energy calculated from a long equilibrium trajectory.
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Affiliation(s)
- Wenjin Li
- Institute for Advanced Study, Shenzhen University, Shenzhen518060, China
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8
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Li W. Time-Lagged Flux in the Transition Path Ensemble: Flux Maximization and Relation to Transition Path Theory. J Phys Chem A 2022; 126:3797-3810. [PMID: 35670470 DOI: 10.1021/acs.jpca.2c02221] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The transition path ensemble is of special interest in reaction coordinate identification as it consists of reactive trajectories that start from the reactant state and end in the product one. As a theoretical framework for describing the transition path ensemble, the transition path theory has been introduced more than 10 years ago, and so far, its applications have only been illustrated in several low-dimensional systems. Given the transition path ensemble, expressions for calculating flux, current (a vector field), and principal curves are derived here in the space of collective variables from the transition path theory, and they are applicable to time series obtained from molecular dynamics simulations of high-dimensional systems, i.e., the position coordinates as a function of time in the transition path ensemble. The connection of the transition path theory is made to a density-weighted average flux, a quantity proposed in a previous work to appraise the relevance of a coordinate to the reaction coordinate [Li, W. J. Chem. Phys. 2022, 156, 054117]. Most importantly, as an extension of the existing quantities, time-lagged quantities such as flux and current are also proposed. The main insights and objects provided by these time-lagged quantities are illustrated in the application to the alanine peptide in vacuum.
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Affiliation(s)
- Wenjin Li
- Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
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9
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Louwerse MD, Sivak DA. Information Thermodynamics of the Transition-Path Ensemble. PHYSICAL REVIEW LETTERS 2022; 128:170602. [PMID: 35570424 DOI: 10.1103/physrevlett.128.170602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2021] [Accepted: 03/10/2022] [Indexed: 06/15/2023]
Abstract
The reaction coordinate describing a transition between reactant and product is a fundamental concept in the theory of chemical reactions. Within transition-path theory, a quantitative definition of the reaction coordinate is found in the committor, which is the probability that a trajectory initiated from a given microstate first reaches the product before the reactant. Here we develop an information-theoretic origin for the committor and show how selecting transition paths from a long ergodic equilibrium trajectory induces entropy production which exactly equals the information that system dynamics provide about the reactivity of trajectories. This equality of entropy production and dynamical information generation also holds at the level of arbitrary individual coordinates, providing parallel measures of the coordinate's relevance to the reaction, each of which is maximized by the committor.
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Affiliation(s)
- Miranda D Louwerse
- Department of Chemistry, Simon Fraser University, Burnaby, British Columbia V5A1S6, Canada
| | - David A Sivak
- Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A1S6, Canada
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10
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Chen H, Ogden D, Pant S, Cai W, Tajkhorshid E, Moradi M, Roux B, Chipot C. A Companion Guide to the String Method with Swarms of Trajectories: Characterization, Performance, and Pitfalls. J Chem Theory Comput 2022; 18:1406-1422. [PMID: 35138832 PMCID: PMC8904302 DOI: 10.1021/acs.jctc.1c01049] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Abstract
The string method with swarms of trajectories (SMwST) is an algorithm that identifies a physically meaningful transition pathway─a one-dimensional curve, embedded within a high-dimensional space of selected collective variables. The SMwST algorithm leans on a series of short, unbiased molecular dynamics simulations spawned at different locations of the discretized path, from whence an average dynamic drift is determined to evolve the string toward an optimal pathway. However conceptually simple in both its theoretical formulation and practical implementation, the SMwST algorithm is computationally intensive and requires a careful choice of parameters for optimal cost-effectiveness in applications to challenging problems in chemistry and biology. In this contribution, the SMwST algorithm is presented in a self-contained manner, discussing with a critical eye its theoretical underpinnings, applicability, inherent limitations, and use in the context of path-following free-energy calculations and their possible extension to kinetics modeling. Through multiple simulations of a prototypical polypeptide, combining the search of the transition pathway and the computation of the potential of mean force along it, several practical aspects of the methodology are examined with the objective of optimizing the computational effort, yet without sacrificing accuracy. In light of the results reported here, we propose some general guidelines aimed at improving the efficiency and reliability of the computed pathways and free-energy profiles underlying the conformational transitions at hand.
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Affiliation(s)
- Haochuan Chen
- Research Center for Analytical Sciences, College of Chemistry, Tianjin Key Laboratory of Biosensing and Molecular Recognition, Nankai University, Tianjin 300071, China
- Theoretical and Computational Biophysics Group, NIH Center for Macromolecular Modeling and Bioinformatics, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States
- 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, 54506 Vandœuvre-lès-Nancy Cedex, France
| | - Dylan Ogden
- Department of Chemistry and Biochemistry, University of Arkansas, Fayetteville, Arkansas 72701, United States
| | - Shashank Pant
- Theoretical and Computational Biophysics Group, NIH Center for Macromolecular Modeling and Bioinformatics, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States
| | - Wensheng Cai
- Research Center for Analytical Sciences, College of Chemistry, Tianjin Key Laboratory of Biosensing and Molecular Recognition, Nankai University, Tianjin 300071, China
| | - Emad Tajkhorshid
- Theoretical and Computational Biophysics Group, NIH Center for Macromolecular Modeling and Bioinformatics, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States
- Department of Biochemistry and Center for Biophysics and Quantitative Biology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States
| | - Mahmoud Moradi
- Department of Chemistry and Biochemistry, University of Arkansas, Fayetteville, Arkansas 72701, United States
| | - Benoît Roux
- Department of Biochemistry and Molecular Biology, University of Chicago, Chicago, Illinois 60637, United States
| | - Christophe Chipot
- Theoretical and Computational Biophysics Group, NIH Center for Macromolecular Modeling and Bioinformatics, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States
- 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, 54506 Vandœuvre-lès-Nancy Cedex, France
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States
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11
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Guardiani C, Cecconi F, Chiodo L, Cottone G, Malgaretti P, Maragliano L, Barabash ML, Camisasca G, Ceccarelli M, Corry B, Roth R, Giacomello A, Roux B. Computational methods and theory for ion channel research. ADVANCES IN PHYSICS: X 2022; 7:2080587. [PMID: 35874965 PMCID: PMC9302924 DOI: 10.1080/23746149.2022.2080587] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2022] [Accepted: 05/15/2022] [Indexed: 06/15/2023] Open
Abstract
Ion channels are fundamental biological devices that act as gates in order to ensure selective ion transport across cellular membranes; their operation constitutes the molecular mechanism through which basic biological functions, such as nerve signal transmission and muscle contraction, are carried out. Here, we review recent results in the field of computational research on ion channels, covering theoretical advances, state-of-the-art simulation approaches, and frontline modeling techniques. We also report on few selected applications of continuum and atomistic methods to characterize the mechanisms of permeation, selectivity, and gating in biological and model channels.
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Affiliation(s)
- C. Guardiani
- Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Rome, Italy
| | - F. Cecconi
- CNR - Istituto dei Sistemi Complessi, Rome, Italy and Istituto Nazionale di Fisica Nucleare, INFN, Roma1 section. 00185, Roma, Italy
| | - L. Chiodo
- Department of Engineering, Campus Bio-Medico University, Rome, Italy
| | - G. Cottone
- Department of Physics and Chemistry-Emilio Segrè, University of Palermo, Palermo, Italy
| | - P. Malgaretti
- Helmholtz Institute Erlangen-Nürnberg for Renewable Energy (IEK-11), Forschungszentrum Jülich, Erlangen, Germany
| | - L. Maragliano
- Department of Life and Environmental Sciences, Polytechnic University of Marche, Ancona, Italy, and Center for Synaptic Neuroscience and Technology, Istituto Italiano di Tecnologia, Genova, Italy
| | - M. L. Barabash
- Department of Materials Science and Nanoengineering, Rice University, Houston, TX 77005, USA
| | - G. Camisasca
- Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Rome, Italy
- Dipartimento di Fisica, Università Roma Tre, Rome, Italy
| | - M. Ceccarelli
- Department of Physics and CNR-IOM, University of Cagliari, Monserrato 09042-IT, Italy
| | - B. Corry
- Research School of Biology, The Australian National University, Canberra, ACT 2600, Australia
| | - R. Roth
- Institut Für Theoretische Physik, Eberhard Karls Universität Tübingen, Tübingen, Germany
| | - A. Giacomello
- Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Rome, Italy
| | - B. Roux
- Department of Biochemistry & Molecular Biology, University of Chicago, Chicago IL, USA
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Abstract
![]()
The kinetics of
a dynamical system comprising two metastable states
is formulated in terms of a finite-time propagator in phase space
(position and velocity) adapted to the underdamped Langevin equation.
Dimensionality reduction to a subspace of collective variables yields
familiar expressions for the propagator, committor, and steady-state
flux. A quadratic expression for the steady-state flux between the
two metastable states can serve as a robust variational principle
to determine an optimal approximate committor expressed in terms of
a set of collective variables. The theoretical formulation is exploited
to clarify the foundation of the string method with swarms-of-trajectories,
which relies on the mean drift of short trajectories to determine
the optimal transition pathway. It is argued that the conditions for
Markovity within a subspace of collective variables may not be satisfied
with an arbitrary short time-step and that proper kinetic behaviors
appear only when considering the effective propagator for longer lag
times. The effective propagator with finite lag time is amenable to
an eigenvalue-eigenvector spectral analysis, as elaborated previously
in the context of position-based Markov models. The time-correlation
functions calculated by swarms-of-trajectories along the string pathway
constitutes a natural extension of these developments. The present
formulation provides a powerful theoretical framework to characterize
the optimal pathway between two metastable states of a system.
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Affiliation(s)
- Benoît Roux
- Department of Biochemistry and Molecular Biology, The University of Chicago, Chicago, Illinois 60637, United States.,Department of Chemistry, The University of Chicago, 5735 S. Ellis Avenue, Chicago, Illinois 60637, United States
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13
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Molecular Dynamics-Based Thermodynamic and Kinetic Characterization of Membrane Protein Conformational Transitions. Methods Mol Biol 2021; 2302:289-309. [PMID: 33877634 DOI: 10.1007/978-1-0716-1394-8_16] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Molecular dynamics (MD) simulations are routinely used to study structural dynamics of membrane proteins. However, conventional MD is often unable to sample functionally important conformational transitions of membrane proteins such as those involved in active membrane transport or channel activation process. Here we describe a combination of multiple MD based techniques that allows for a rigorous characterization of energetics and kinetics of large-scale conformational changes in membrane proteins. The methodology is based on biased, nonequilibrium, collective-variable based simulations including nonequilibrium pulling, string method with swarms of trajectories, bias-exchange umbrella sampling, and rate estimation techniques.
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14
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Lev B, Allen TW. Simulating ion channel activation mechanisms using swarms of trajectories. J Comput Chem 2020; 41:387-401. [PMID: 31743478 DOI: 10.1002/jcc.26102] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Revised: 10/16/2019] [Accepted: 10/17/2019] [Indexed: 12/14/2022]
Abstract
Atomic-level studies of protein activity represent a significant challenge as a result of the complexity of conformational changes occurring on wide-ranging timescales, often greatly exceeding that of even the longest simulations. A prime example is the elucidation of protein allosteric mechanisms, where localized perturbations transmit throughout a large macromolecule to generate a response signal. For example, the conversion of chemical to electrical signals during synaptic neurotransmission in the brain is achieved by specialized membrane proteins called pentameric ligand-gated ion channels. Here, the binding of a neurotransmitter results in a global conformational change to open an ion-conducting pore across the nerve cell membrane. X-ray crystallography has produced static structures of the open and closed states of the proton-gated GLIC pentameric ligand-gated ion channel protein, allowing for atomistic simulations that can uncover changes related to activation. We discuss a range of enhanced sampling approaches that could be used to explore activation mechanisms. In particular, we describe recent application of an atomistic string method, based on Roux's "swarms of trajectories" approach, to elucidate the sequence and interdependence of conformational changes during activation. We illustrate how this can be combined with transition analysis and Brownian dynamics to extract thermodynamic and kinetic information, leading to understanding of what controls ion channel function. © 2019 Wiley Periodicals, Inc.
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Affiliation(s)
- Bogdan Lev
- School of Science, RMIT University, Melbourne, Victoria, 3000, Australia
| | - Toby W Allen
- School of Science, RMIT University, Melbourne, Victoria, 3000, Australia
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15
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Suh D, Jo S, Jiang W, Chipot C, Roux B. String Method for Protein-Protein Binding Free-Energy Calculations. J Chem Theory Comput 2019; 15:5829-5844. [PMID: 31593627 DOI: 10.1021/acs.jctc.9b00499] [Citation(s) in RCA: 29] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023]
Abstract
A powerful computational strategy to determine the equilibrium association constant of two macromolecules with explicit-solvent molecular dynamics (MD) simulations is the "geometric route", which considers the reversible physical separation of the bound complex in solution. Nonetheless, multiple challenges remain to render this type of methodology reliable and computationally efficient in practice. In particular, in one, formulation of the geometric route relies on the potential of mean force (PMF) for physically separating the two binding partners restrained along a straight axis, which must be selected prior to the calculation. However, practical applications indicate that the calculation of the separation PMF along the predefined rectilinear pathway may be suboptimal and slowly convergent. Recognizing that a rectilinear straight separation pathway is generally not representative of how the protein complex physically separates in solution, we put forth a novel theoretical framework for binding free-energy calculations, leaning on the optimal curvilinear minimum free-energy path (MFEP) determined from the string method. The proposed formalism is validated by comparing the results obtained using both rectilinear and curvilinear pathways for a prototypical host-guest complex formed by cucurbit[7]uril (CB[7]) binding benzene, and for the barnase-barstar protein complex. On the basis of multi-microsecond MD calculations, we find that the calculations following the traditional rectilinear pathway and the string-based curvilinear pathway agree quantitatively, but convergence is faster with the latter.
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Affiliation(s)
- Donghyuk Suh
- Department of Chemistry , University of Chicago , Chicago , Illinois 60637-1454 , United States
| | - Sunhwan Jo
- Computational Science Division , Argonne National Laboratory , Argonne , Illinois 60439-8643 , United States
| | - Wei Jiang
- Computational Science Division , Argonne National Laboratory , Argonne , Illinois 60439-8643 , United States
| | - Chris Chipot
- Laboratoire International Associé Centre National de la Recherche Scientifique et University of Illinois at Urbana-Champaign , Unité Mixte de Recherche n°7019, Université de Lorraine , B.P. 70239, 54506 Vandoeuvre-lès-Nancy cedex , France.,Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology , University of Illinois at Urbana-Champaign , Urbana , Illinois 61801-2325 , United States.,Department of Physics , University of Illinois at Urbana-Champaign , Urbana , Illinois 61801-2325 , United States
| | - Benoît Roux
- Department of Chemistry , University of Chicago , Chicago , Illinois 60637-1454 , United States.,Department of Biochemistry and Molecular Biology , University of Chicago , Chicago , Illinois 60637-1454 , United States.,Center for Nanoscale Materials , Argonne National Laboratory , Argonne , Illinois 60439-8643 , United States
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16
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Flood E, Boiteux C, Lev B, Vorobyov I, Allen TW. Atomistic Simulations of Membrane Ion Channel Conduction, Gating, and Modulation. Chem Rev 2019; 119:7737-7832. [DOI: 10.1021/acs.chemrev.8b00630] [Citation(s) in RCA: 65] [Impact Index Per Article: 13.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
Affiliation(s)
- Emelie Flood
- School of Science, RMIT University, Melbourne, Victoria 3000, Australia
| | - Céline Boiteux
- School of Science, RMIT University, Melbourne, Victoria 3000, Australia
| | - Bogdan Lev
- School of Science, RMIT University, Melbourne, Victoria 3000, Australia
| | - Igor Vorobyov
- Department of Physiology & Membrane Biology/Department of Pharmacology, University of California, Davis, 95616, United States
| | - Toby W. Allen
- School of Science, RMIT University, Melbourne, Victoria 3000, Australia
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17
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Zinovjev K, Tuñón I. Adaptive Finite Temperature String Method in Collective Variables. J Phys Chem A 2017; 121:9764-9772. [DOI: 10.1021/acs.jpca.7b10842] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Kirill Zinovjev
- Departament de Química
Física, Universitat de València, 46100 Burjassot, Spain
| | - Iñaki Tuñón
- Departament de Química
Física, Universitat de València, 46100 Burjassot, Spain
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18
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Skeel RD, Zhao R, Post CB. A minimization principle for transition paths of maximum flux for collective variables. Theor Chem Acc 2017; 136. [PMID: 29225509 DOI: 10.1007/s00214-016-2041-3] [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: 10/20/2022]
Abstract
Considered is the construction of transition paths of conformational changes for proteins and other macromolecules, using methods that do not require the generation of dynamics trajectories. Special attention is given to the use of a reduced set of collective variables for describing such paths. A favored way to define transition paths is to seek channels through the transition state having cross sections with a high reactive flux (density of last hitting points of reactive trajectories). Given here is a formula for reactive flux that is independent of the parameterization of "collective variable space." This formula is needed for the principal curve of the reactive flux (as in the revised finite temperature string method) and for the maximum flux transition (MaxFlux) path. Additionally, a resistance functional is derived for narrow tubes, which when minimized yields a MaxFlux path. A strategy for minimization is outlined in the spirit of the string method. Finally, alternative approaches based on determining trajectories of high probability are considered, and it is observed that they yield paths that depend on the parameterization of collective variable space, except in the case of zero temperature, where such a path coincides with a MaxFlux path.
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Affiliation(s)
- Robert D Skeel
- Department of Computer Science, Purdue University, West Lafayette, IN, 47906 USA
| | - Ruijun Zhao
- Department of Mathematics and Statistics, Minnesota State University, Mankato, MN, 56001 USA
| | - Carol Beth Post
- Department of Medicinal Chemistry and Molecular Pharmacology, Purdue University, West Lafayette, IN, 47906 USA
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19
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Tiwary P, Berne BJ. Predicting reaction coordinates in energy landscapes with diffusion anisotropy. J Chem Phys 2017; 147:152701. [PMID: 29055314 PMCID: PMC5446309 DOI: 10.1063/1.4983727] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2017] [Accepted: 05/03/2017] [Indexed: 01/21/2023] Open
Abstract
We consider a range of model potentials with metastable states undergoing molecular dynamics coupled to a thermal bath in the high friction regime and consider how the optimal reaction coordinate depends on the diffusion anisotropy. For this we use our recently proposed method "spectral gap optimization of order parameters (SGOOP)" [P. Tiwary and B. J. Berne, Proc. Natl. Acad. Sci. U. S. A. 113, 2839 (2016)]. We show how available information about dynamical observables in addition to static information can be incorporated into SGOOP, which can then be used to accurately determine the "best" reaction coordinate for arbitrary anisotropies. We compare our results with transmission coefficient calculations and published benchmarks wherever applicable or available, respectively.
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Affiliation(s)
- Pratyush Tiwary
- Department of Chemistry, Columbia University, New York, New York 10027, USA
| | - B J Berne
- Department of Chemistry, Columbia University, New York, New York 10027, USA
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20
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Zinovjev K, Tuñón I. Reaction coordinates and transition states in enzymatic catalysis. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2017. [DOI: 10.1002/wcms.1329] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/16/2023]
Affiliation(s)
- Kirill Zinovjev
- Departament de Química FísicaUniversitat de València Valencia Spain
| | - Iñaki Tuñón
- Departament de Química FísicaUniversitat de València Valencia Spain
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21
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Fakharzadeh A, Moradi M. Effective Riemannian Diffusion Model for Conformational Dynamics of Biomolecular Systems. J Phys Chem Lett 2016; 7:4980-4987. [PMID: 27973909 DOI: 10.1021/acs.jpclett.6b02208] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We present a Riemannian formalism for effective diffusion of biomolecules in collective variable spaces that provides a robust framework for conformational free energy calculation methods. Unlike their Euclidean counterparts, the Riemannian potential of mean force (PMF) and minimum free energy path (MFEP) are invariant under coordinate transformations. The presented formalism can be readily employed to modify the collective variable based enhanced sampling techniques, such as umbrella sampling (US) commonly used in biomolecular simulations, to take into account the role of intrinsic geometry of collective variable space. Although our model is mathematically equivalent to a Euclidean diffusion with a position-dependent diffusion tensor, the Riemannian formulation provides a more convenient framework for free energy calculation methods and path-finding algorithms aimed at characterizing the effective conformational dynamics of biomolecules. A simple three-dimensional toy model and a pentapeptide (met-enkephalin) simulated in an explicit solvent environment are used to illustrate the workings of the formalism and its implementation.
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Affiliation(s)
- Ashkan Fakharzadeh
- Department of Physics, North Carolina State University , Raleigh, North Carolina 27695, United States
| | - Mahmoud Moradi
- Department of Chemistry and Biochemistry, University of Arkansas , Fayetteville, Arkansas 72701, United States
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22
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Affiliation(s)
- Baron Peters
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106;
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23
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Zinovjev K, Tuñón I. Transition state ensemble optimization for reactions of arbitrary complexity. J Chem Phys 2015; 143:134111. [DOI: 10.1063/1.4931596] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023] Open
Affiliation(s)
- Kirill Zinovjev
- Departament de Química Física, Universitat de València, 46100 Burjassot, Spain
| | - Iñaki Tuñón
- Departament de Química Física, Universitat de València, 46100 Burjassot, Spain
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24
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Atomic-level characterization of transport cycle thermodynamics in the glycerol-3-phosphate:phosphate antiporter. Nat Commun 2015; 6:8393. [PMID: 26417850 PMCID: PMC4598623 DOI: 10.1038/ncomms9393] [Citation(s) in RCA: 68] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/06/2015] [Accepted: 08/18/2015] [Indexed: 01/09/2023] Open
Abstract
Membrane transporters actively translocate their substrate by undergoing large-scale structural transitions between inward- (IF) and outward-facing (OF) states ('alternating-access' mechanism). Despite extensive structural studies, atomic-level mechanistic details of such structural transitions, and as importantly, their coupling to chemical events supplying the energy, remain amongst the most elusive aspects of the function of these proteins. Here we present a quantitative, atomic-level description of the functional thermodynamic cycle for the glycerol-3-phosphate:phosphate antiporter GlpT by using a novel approach in reconstructing the free energy landscape governing the IF↔OF transition along a cyclic transition pathway involving both apo and substrate-bound states. Our results provide a fully atomic description of the complete transport process, offering a structural model for the alternating-access mechanism and substantiating the close coupling between global structural transitions and local chemical events.
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25
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Fujisaki H, Moritsugu K, Matsunaga Y, Morishita T, Maragliano L. Extended Phase-Space Methods for Enhanced Sampling in Molecular Simulations: A Review. Front Bioeng Biotechnol 2015; 3:125. [PMID: 26389113 PMCID: PMC4558547 DOI: 10.3389/fbioe.2015.00125] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2015] [Accepted: 08/10/2015] [Indexed: 01/12/2023] Open
Abstract
Molecular Dynamics simulations are a powerful approach to study biomolecular conformational changes or protein-ligand, protein-protein, and protein-DNA/RNA interactions. Straightforward applications, however, are often hampered by incomplete sampling, since in a typical simulated trajectory the system will spend most of its time trapped by high energy barriers in restricted regions of the configuration space. Over the years, several techniques have been designed to overcome this problem and enhance space sampling. Here, we review a class of methods that rely on the idea of extending the set of dynamical variables of the system by adding extra ones associated to functions describing the process under study. In particular, we illustrate the Temperature Accelerated Molecular Dynamics (TAMD), Logarithmic Mean Force Dynamics (LogMFD), and Multiscale Enhanced Sampling (MSES) algorithms. We also discuss combinations with techniques for searching reaction paths. We show the advantages presented by this approach and how it allows to quickly sample important regions of the free-energy landscape via automatic exploration.
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Affiliation(s)
| | - Kei Moritsugu
- Graduate School of Medical Life Science, Yokohama City University, Yokohama, Japan
| | | | - Tetsuya Morishita
- National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan
| | - Luca Maragliano
- Center for Synaptic Neuroscience, Istituto Italiano di Tecnologia, Genova, Italy
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26
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Affiliation(s)
- Baron Peters
- Department
of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
- Department
of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States
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27
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Ma W, Schulten K. Mechanism of substrate translocation by a ring-shaped ATPase motor at millisecond resolution. J Am Chem Soc 2015; 137:3031-40. [PMID: 25646698 PMCID: PMC4393844 DOI: 10.1021/ja512605w] [Citation(s) in RCA: 43] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022]
Abstract
Ring-shaped, hexameric ATPase motors fulfill key functions in cellular processes, such as genome replication, transcription, or protein degradation, by translocating a long substrate through their central pore powered by ATP hydrolysis. Despite intense research efforts, the atomic-level mechanism transmitting chemical energy from hydrolysis into mechanical force that translocates the substrate is still unclear. Here we employ all-atom molecular dynamics simulations combined with advanced path sampling techniques and milestoning analysis to characterize how mRNA substrate is translocated by an exemplary homohexameric motor, the transcription termination factor Rho. We find that the release of hydrolysis product (ADP + Pi) triggers the force-generating process of Rho through a 0.1 millisecond-long conformational transition, the time scale seen also in experiment. The calculated free energy profiles and kinetics show that Rho unidirectionally translocates the single-stranded RNA substrate via a population shift of the conformational states of Rho; upon hydrolysis product release, the most favorable conformation shifts from the pretranslocation state to the post-translocation state. Via two previously unidentified intermediate states, the RNA chain is seen to be pulled by six K326 side chains, whose motions are induced by highly coordinated relative translation and rotation of Rho's six subunits. The present study not only reveals in new detail the mechanism employed by ring-shaped ATPase motors, for example the use of loosely bound and tightly bound hydrolysis reactant and product states to coordinate motor action, but also provides an effective approach to identify allosteric sites of multimeric enzymes in general.
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28
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Sanchez-Martinez M, Field M, Crehuet R. Enzymatic minimum free energy path calculations using swarms of trajectories. J Phys Chem B 2014; 119:1103-13. [PMID: 25286154 DOI: 10.1021/jp506593t] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
Abstract
The development of approaches for simulating rare events in complex molecular systems is a central concern in chemical physics. In recent work, Roux and co-workers proposed a novel, swarms of trajectories (SoT) method for determining the transition paths of such events. It consists of the dynamical refinement on the system's free energy surface of a putative transition path that is parametrized in terms of a set of collective variables (CVs) that are identified as being important for the transition. In this work, we have implemented the SoT method and used it to investigate the catalytic mechanisms of two enzymatic reactions using hybrid QM/MM potentials. Our aim has been to test the performance of SoT for enzyme systems and to devise robust simulation protocols that can be employed in future studies of this type. We identify the conditions under which converged results can be obtained using inertial and Brownian dynamical evolutions of the CVs, show that the inclusion of several CVs can give significant additional insight into the mechanisms of the reactions, and show that the use of minimum energy paths as starting guesses can greatly accelerate path refinement.
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Affiliation(s)
- Melchor Sanchez-Martinez
- Institute of Advanced Chemistry of Catalonia (IQAC), CSIC , Jordi Girona 18-26, 08034, Barcelona, Spain
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29
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Zinovjev K, Tuñón I. Exploring chemical reactivity of complex systems with path-based coordinates: Role of the distance metric. J Comput Chem 2014; 35:1672-81. [DOI: 10.1002/jcc.23673] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2014] [Revised: 05/16/2014] [Accepted: 06/18/2014] [Indexed: 01/06/2023]
Affiliation(s)
- Kirill Zinovjev
- Departament de Química Física; Universitat de València; 46100 Burjassot Spain
| | - Iñaki Tuñón
- Departament de Química Física; Universitat de València; 46100 Burjassot Spain
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30
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Li Y, Li X, Ma W, Dong Z. Conformational Transition Pathways of Epidermal Growth Factor Receptor Kinase Domain from Multiple Molecular Dynamics Simulations and Bayesian Clustering. J Chem Theory Comput 2014; 10:3503-3511. [PMID: 25136273 PMCID: PMC4132868 DOI: 10.1021/ct500162b] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2014] [Indexed: 01/15/2023]
Abstract
![]()
The
epidermal growth factor receptor (EGFR) is aberrantly activated
in various cancer cells and an important target for cancer treatment.
Deep understanding of EGFR conformational changes between the active
and inactive states is of pharmaceutical interest. Here we present
a strategy combining multiply targeted molecular dynamics simulations,
unbiased molecular dynamics simulations, and Bayesian clustering to
investigate transition pathways during the activation/inactivation
process of EGFR kinase domain. Two distinct pathways between the active
and inactive forms are designed, explored, and compared. Based on
Bayesian clustering and rough two-dimensional free energy surfaces,
the energy-favorable pathway is recognized, though DFG-flip happens
in both pathways. In addition, another pathway with different intermediate
states appears in our simulations. Comparison of distinct pathways
also indicates that disruption of the Lys745-Glu762 interaction is
critically important in DFG-flip while movement of the A-loop significantly
facilitates the conformational change. Our simulations yield new insights
into EGFR conformational transitions. Moreover, our results verify
that this approach is valid and efficient in sampling of protein conformational
changes and comparison of distinct pathways.
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Affiliation(s)
- Yan Li
- The Hormel Institute, University of Minnesota , Austin, Minnesota 55912, United States
| | - Xiang Li
- Department of Physiology and Pathophysiology, School of Basic Medical Sciences, Zhengzhou University , 450001 Zhengzhou, Henan, China
| | - Weiya Ma
- The Hormel Institute, University of Minnesota , Austin, Minnesota 55912, United States
| | - Zigang Dong
- The Hormel Institute, University of Minnesota , Austin, Minnesota 55912, United States
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31
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Haas KR, Yang H, Chu JW. Fisher information metric for the Langevin equation and least informative models of continuous stochastic dynamics. J Chem Phys 2014; 139:121931. [PMID: 24089743 DOI: 10.1063/1.4820491] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The evaluation of the Fisher information matrix for the probability density of trajectories generated by the over-damped Langevin dynamics at equilibrium is presented. The framework we developed is general and applicable to any arbitrary potential of mean force where the parameter set is now the full space dependent function. Leveraging an innovative Hermitian form of the corresponding Fokker-Planck equation allows for an eigenbasis decomposition of the time propagation probability density. This formulation motivates the use of the square root of the equilibrium probability density as the basis for evaluating the Fisher information of trajectories with the essential advantage that the Fisher information matrix in the specified parameter space is constant. This outcome greatly eases the calculation of information content in the parameter space via a line integral. In the continuum limit, a simple analytical form can be derived to explicitly reveal the physical origin of the information content in equilibrium trajectories. This methodology also allows deduction of least informative dynamics models from known or available observables that are either dynamical or static in nature. The minimum information optimization of dynamics is performed for a set of different constraints to illustrate the generality of the proposed methodology.
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Affiliation(s)
- Kevin R Haas
- Department of Chemical and Biomolecular Engineering, University of California-Berkeley, Berkeley, California 94720, USA
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32
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Abstract
Proteins are fascinating supramolecular structures, which are able to recognize ligands transforming binding information into chemical signals. They can transfer information across the cell, can catalyse complex chemical reactions, and are able to transform energy into work with much more efficiency than any human engine. The unique abilities of proteins are tightly coupled with their dynamic properties, which are coded in a complex way in the sequence and carefully refined by evolution. Despite its importance, our experimental knowledge of protein dynamics is still rather limited, and mostly derived from theoretical calculations. I will review here, in a systematic way, the current state-of-the-art theoretical approaches to the study of protein dynamics, emphasizing the most recent advances, examples of use and the expected lines of development in the near future.
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Affiliation(s)
- Modesto Orozco
- Institute for Research in Biomedicine (IRB Barcelona), Baldiri i Reixac 8, Barcelona 08028, Spain.
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33
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34
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Maragliano L, Roux B, Vanden-Eijnden E. Comparison between Mean Forces and Swarms-of-Trajectories String Methods. J Chem Theory Comput 2014; 10:524-33. [PMID: 26580029 PMCID: PMC6980172 DOI: 10.1021/ct400606c] [Citation(s) in RCA: 32] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The original formulation of the string method in collective variable space is compared with a recent variant called string method with swarms-of-trajectories. The assumptions made in the original method are revisited and the significance of the minimum free energy path (MFEP) is discussed in the context of reactive events. These assumptions are compared to those made in the string method with swarms-of-trajectories, and shown to be equivalent in a certain regime: in particular an expression for the path identified by the swarms-of-trajectories method is given and shown to be closely related to the MFEP. Finally, the algorithmic aspects of both methods are compared.
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Affiliation(s)
- Luca Maragliano
- Department of Biochemistry and Molecular Biology, University of Chicago, Chicago, Illinois 60637, United States
| | - Benoît Roux
- Department of Biochemistry and Molecular Biology, University of Chicago, Chicago, Illinois 60637, United States
- Biosciences Division, Argonne National Laboratory, Argonne, Illinois 60439, United States
| | - Eric Vanden-Eijnden
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, United States
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35
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Peters B, Bolhuis PG, Mullen RG, Shea JE. Reaction coordinates, one-dimensional Smoluchowski equations, and a test for dynamical self-consistency. J Chem Phys 2013; 138:054106. [PMID: 23406097 DOI: 10.1063/1.4775807] [Citation(s) in RCA: 65] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
Abstract
We propose a method for identifying accurate reaction coordinates among a set of trial coordinates. The method applies to special cases where motion along the reaction coordinate follows a one-dimensional Smoluchowski equation. In these cases the reaction coordinate can predict its own short-time dynamical evolution, i.e., the dynamics projected from multiple dimensions onto the reaction coordinate depend only on the reaction coordinate itself. To test whether this property holds, we project an ensemble of short trajectory swarms onto trial coordinates and compare projections of individual swarms to projections of the ensemble of swarms. The comparison, quantified by the Kullback-Leibler divergence, is numerically performed for each isosurface of each trial coordinate. The ensemble of short dynamical trajectories is generated only once by sampling along an initial order parameter. The initial order parameter should separate the reactants and products with a free energy barrier, and distributions on isosurfaces of the initial parameter should be unimodal. The method is illustrated for three model free energy landscapes with anisotropic diffusion. Where exact coordinates can be obtained from Kramers-Langer-Berezhkovskii-Szabo theory, results from the new method agree with the exact results. We also examine characteristics of systems where the proposed method fails. We show how dynamical self-consistency is related (through the Chapman-Kolmogorov equation) to the earlier isocommittor criterion, which is based on longer paths.
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Affiliation(s)
- Baron Peters
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA
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36
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Jimenez-Cruz CA, Garcia AE. Reconstructing the Most Probable Folding Transition Path from Replica Exchange Molecular Dynamics Simulations. J Chem Theory Comput 2013; 9:3750-5. [DOI: 10.1021/ct400170x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Camilo Andres Jimenez-Cruz
- Department of Physics,
Applied Physics and Astronomy,
and Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, New York 12180,
United States
| | - Angel E. Garcia
- Department of Physics,
Applied Physics and Astronomy,
and Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, New York 12180,
United States
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37
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Berezhkovskii AM, Szabo A. Diffusion along the splitting/commitment probability reaction coordinate. J Phys Chem B 2013; 117:13115-9. [PMID: 23777371 DOI: 10.1021/jp403043a] [Citation(s) in RCA: 46] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
The splitting or commitment probabilities of states in the region of configuration space that separates reactants and products play an important role in the theory of chemical reactions. Assuming that the splitting probability changes more slowly than any other coordinate, we project multidimensional diffusive dynamics onto it. The resulting one-dimensional diffusion equation is not exact because the assumed separation of time scales does not hold in general. Nevertheless, this equation has the remarkable property that it always predicts the exact value of the number of transitions between reactants and products per unit time at equilibrium and hence the exact reaction rate. In the special case of two deep basins separated by a harmonic saddle, this equation is equivalent to the one that describes diffusion along a coordinate perpendicular to the transition state, defined as the surface starting from which reactants and products are reached with equal probability.
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Affiliation(s)
- Alexander M Berezhkovskii
- Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology, National Institutes of Health , Bethesda, Maryland 20892, United States
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38
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Adelman JL, Grabe M. Simulating rare events using a weighted ensemble-based string method. J Chem Phys 2013; 138:044105. [PMID: 23387566 PMCID: PMC3568092 DOI: 10.1063/1.4773892] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2012] [Accepted: 12/17/2012] [Indexed: 11/14/2022] Open
Abstract
We introduce an extension to the weighted ensemble (WE) path sampling method to restrict sampling to a one-dimensional path through a high dimensional phase space. Our method, which is based on the finite-temperature string method, permits efficient sampling of both equilibrium and non-equilibrium systems. Sampling obtained from the WE method guides the adaptive refinement of a Voronoi tessellation of order parameter space, whose generating points, upon convergence, coincide with the principle reaction pathway. We demonstrate the application of this method to several simple, two-dimensional models of driven Brownian motion and to the conformational change of the nitrogen regulatory protein C receiver domain using an elastic network model. The simplicity of the two-dimensional models allows us to directly compare the efficiency of the WE method to conventional brute force simulations and other path sampling algorithms, while the example of protein conformational change demonstrates how the method can be used to efficiently study transitions in the space of many collective variables.
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Affiliation(s)
- Joshua L Adelman
- Department of Biological Sciences, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA.
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39
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Faccioli P, Pederiva F. Microscopically computing free-energy profiles and transition path time of rare macromolecular transitions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:061916. [PMID: 23367984 DOI: 10.1103/physreve.86.061916] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2012] [Indexed: 06/01/2023]
Abstract
We introduce a rigorous method to microscopically compute the observables which characterize the thermodynamics and kinetics of rare macromolecular transitions for which it is possible to identify a priori a slow reaction coordinate. In order to sample the ensemble of statistically significant reaction pathways, we define a biased molecular dynamics (MD) in which barrier-crossing transitions are accelerated without introducing any unphysical external force. In contrast to other biased MD methods, in the present approach the systematic errors which are generated in order to accelerate the transition can be analytically calculated and therefore can be corrected for. This allows for a computationally efficient reconstruction of the free-energy profile as a function of the reaction coordinate and for the calculation of the corresponding diffusion coefficient. The transition path time can then be readily evaluated within the dominant reaction pathways approach. We illustrate and test this method by characterizing a thermally activated transition on a two-dimensional energy surface and the folding of a small protein fragment within a coarse-grained model.
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Affiliation(s)
- P Faccioli
- Physics Department, University of Trento, Via Sommarive 14, Povo, I-38129 Trento, Italy
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