1
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Meraz VJ, Zou Z, Tiwary P. Simulating Crystallization in a Colloidal System Using State Predictive Information Bottleneck Based Enhanced Sampling. J Phys Chem B 2024; 128:8207-8214. [PMID: 39163635 DOI: 10.1021/acs.jpcb.4c02740] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/22/2024]
Abstract
We investigate crystal nucleation in supersaturated colloid suspensions using enhanced molecular dynamics simulations augmented with machine learning techniques. The simulations reveal that crystallization in the model colloidal system studied here, with particles interacting through a repulsive screened Coulomb Yukawa potential, proceeds from vapor to dense liquid droplet to crystalline phases across multiple high barriers. Employing a one-dimensional reaction coordinate derived from the State Predictive Information Bottleneck framework, our simulations capture back-and-forth phase transitions across multiple barriers effectively in biased metadynamics simulations. We obtain relative free energy differences between different phases and also quantify the roles of different molecular level features in driving the phase changes.
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Affiliation(s)
- Vanessa J Meraz
- Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, United States
| | - Ziyue Zou
- Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, United States
| | - Pratyush Tiwary
- Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, United States
- Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, United States
- University of Maryland Institute for Health Computing, Bethesda, Maryland 20852, United States
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2
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Zhang J, Zhang O, Bonati L, Hou T. Combining Transition Path Sampling with Data-Driven Collective Variables through a Reactivity-Biased Shooting Algorithm. J Chem Theory Comput 2024; 20:4523-4532. [PMID: 38801759 DOI: 10.1021/acs.jctc.4c00423] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/29/2024]
Abstract
Rare event sampling is a central problem in modern computational chemistry research. Among the existing methods, transition path sampling (TPS) can generate unbiased representations of reaction processes. However, its efficiency depends on the ability to generate reactive trial paths, which in turn depends on the quality of the shooting algorithm used. We propose a new algorithm based on the shooting success rate, i.e., reactivity, measured as a function of a reduced set of collective variables (CVs). These variables are extracted with a machine learning approach directly from TPS simulations, using a multitask objective function. Iteratively, this workflow significantly improves the shooting efficiency without any prior knowledge of the process. In addition, the optimized CVs can be used with biased enhanced sampling methodologies to accurately reconstruct the free energy profiles. We tested the method on three different systems: a two-dimensional toy model, conformational transitions of alanine dipeptide, and hydrolysis of acetyl chloride in bulk water. In the latter, we integrated our workflow with an active learning scheme to learn a reactive machine learning-based potential, which allowed us to study the mechanism and free energy profile with an ab initio-like accuracy.
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Affiliation(s)
- Jintu Zhang
- Innovation Institute for Artificial Intelligence in Medicine of Zhejiang University, College of Pharmaceutical Sciences, Zhejiang University, Hangzhou, Zhejiang 310058, China
- Atomistic Simulations, Italian Institute of Technology, Genova 16152, Italy
| | - Odin Zhang
- Innovation Institute for Artificial Intelligence in Medicine of Zhejiang University, College of Pharmaceutical Sciences, Zhejiang University, Hangzhou, Zhejiang 310058, China
| | - Luigi Bonati
- Atomistic Simulations, Italian Institute of Technology, Genova 16152, Italy
| | - TingJun Hou
- Innovation Institute for Artificial Intelligence in Medicine of Zhejiang University, College of Pharmaceutical Sciences, Zhejiang University, Hangzhou, Zhejiang 310058, China
- State Key Lab of CAD&CG, Zhejiang University, Hangzhou, Zhejiang 310058, China
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3
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Dalton BA, Kiefer H, Netz RR. The role of memory-dependent friction and solvent viscosity in isomerization kinetics in viscogenic media. Nat Commun 2024; 15:3761. [PMID: 38704367 PMCID: PMC11069540 DOI: 10.1038/s41467-024-48016-7] [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: 09/22/2023] [Accepted: 04/18/2024] [Indexed: 05/06/2024] Open
Abstract
Molecular isomerization kinetics in liquid solvent depends on a complex interplay between the solvent friction acting on the molecule, internal dissipation effects (also known as internal friction), the viscosity of the solvent, and the dihedral free energy profile. Due to the absence of accurate techniques to directly evaluate isomerization friction, it has not been possible to explore these relationships in full. By combining extensive molecular dynamics simulations with friction memory-kernel extraction techniques we consider a variety of small, isomerising molecules under a range of different viscogenic conditions and directly evaluate the viscosity dependence of the friction acting on a rotating dihedral. We reveal that the influence of different viscogenic media on isomerization kinetics can be dramatically different, even when measured at the same viscosity. This is due to the dynamic solute-solvent coupling, mediated by time-dependent friction memory kernels. We also show that deviations from the linear dependence of isomerization rates on solvent viscosity, which are often simply attributed to internal friction effects, are due to the simultaneous violation of two fundamental relationships: the Stokes-Einstein relation and the overdamped Kramers prediction for the barrier-crossing rate, both of which require explicit knowledge of friction.
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Affiliation(s)
| | - Henrik Kiefer
- Freie Universität Berlin, Fachbereich Physik, Berlin, Germany
| | - Roland R Netz
- Freie Universität Berlin, Fachbereich Physik, Berlin, Germany.
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4
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Müllender L, Rizzi A, Parrinello M, Carloni P, Mandelli D. Effective data-driven collective variables for free energy calculations from metadynamics of paths. PNAS NEXUS 2024; 3:pgae159. [PMID: 38665160 PMCID: PMC11044970 DOI: 10.1093/pnasnexus/pgae159] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/05/2023] [Accepted: 04/04/2024] [Indexed: 04/28/2024]
Abstract
A variety of enhanced sampling (ES) methods predict multidimensional free energy landscapes associated with biological and other molecular processes as a function of a few selected collective variables (CVs). The accuracy of these methods is crucially dependent on the ability of the chosen CVs to capture the relevant slow degrees of freedom of the system. For complex processes, finding such CVs is the real challenge. Machine learning (ML) CVs offer, in principle, a solution to handle this problem. However, these methods rely on the availability of high-quality datasets-ideally incorporating information about physical pathways and transition states-which are difficult to access, therefore greatly limiting their domain of application. Here, we demonstrate how these datasets can be generated by means of ES simulations in trajectory space via the metadynamics of paths algorithm. The approach is expected to provide a general and efficient way to generate efficient ML-based CVs for the fast prediction of free energy landscapes in ES simulations. We demonstrate our approach with two numerical examples, a 2D model potential and the isomerization of alanine dipeptide, using deep targeted discriminant analysis as our ML-based CV of choice.
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Affiliation(s)
- Lukas Müllender
- Department of Applied Physics, Science for Life Laboratory, KTH Royal Institute of Technology, SE-171 21 Solna, Sweden
- Computational Biomedicine, Institute of Advanced Simulations IAS-5/Institute for Neuroscience and Medicine INM-9, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany
- Department of Physics, RWTH Aachen University, 52062 Aachen, Germany
| | - Andrea Rizzi
- Computational Biomedicine, Institute of Advanced Simulations IAS-5/Institute for Neuroscience and Medicine INM-9, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany
- Atomistic Simulations, Italian Institute of Technology, 16163 Genova, Italy
| | - Michele Parrinello
- Atomistic Simulations, Italian Institute of Technology, 16163 Genova, Italy
| | - Paolo Carloni
- Computational Biomedicine, Institute of Advanced Simulations IAS-5/Institute for Neuroscience and Medicine INM-9, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany
- Department of Physics, RWTH Aachen University, 52062 Aachen, Germany
- Universitätsklinikum, RWTH Aachen University, 52062 Aachen, Germany
| | - Davide Mandelli
- Computational Biomedicine, Institute of Advanced Simulations IAS-5/Institute for Neuroscience and Medicine INM-9, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany
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5
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Sisk TR, Robustelli P. Folding-upon-binding pathways of an intrinsically disordered protein from a deep Markov state model. Proc Natl Acad Sci U S A 2024; 121:e2313360121. [PMID: 38294935 PMCID: PMC10861926 DOI: 10.1073/pnas.2313360121] [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: 08/10/2023] [Accepted: 11/22/2023] [Indexed: 02/02/2024] Open
Abstract
A central challenge in the study of intrinsically disordered proteins is the characterization of the mechanisms by which they bind their physiological interaction partners. Here, we utilize a deep learning-based Markov state modeling approach to characterize the folding-upon-binding pathways observed in a long timescale molecular dynamics simulation of a disordered region of the measles virus nucleoprotein NTAIL reversibly binding the X domain of the measles virus phosphoprotein complex. We find that folding-upon-binding predominantly occurs via two distinct encounter complexes that are differentiated by the binding orientation, helical content, and conformational heterogeneity of NTAIL. We observe that folding-upon-binding predominantly proceeds through a multi-step induced fit mechanism with several intermediates and do not find evidence for the existence of canonical conformational selection pathways. We observe four kinetically separated native-like bound states that interconvert on timescales of eighty to five hundred nanoseconds. These bound states share a core set of native intermolecular contacts and stable NTAIL helices and are differentiated by a sequential formation of native and non-native contacts and additional helical turns. Our analyses provide an atomic resolution structural description of intermediate states in a folding-upon-binding pathway and elucidate the nature of the kinetic barriers between metastable states in a dynamic and heterogenous, or "fuzzy", protein complex.
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Affiliation(s)
- Thomas R. Sisk
- Department of Chemistry, Dartmouth College, Hanover, NH03755
| | - Paul Robustelli
- Department of Chemistry, Dartmouth College, Hanover, NH03755
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6
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Russell PPS, Maytin AK, Rickard MM, Russell MC, Pogorelov TV, Gruebele M. Metastable States in the Hinge-Bending Landscape of an Enzyme in an Atomistic Cytoplasm Simulation. J Phys Chem Lett 2024; 15:940-946. [PMID: 38252018 PMCID: PMC11180962 DOI: 10.1021/acs.jpclett.3c03134] [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] [Indexed: 01/23/2024]
Abstract
Many enzymes undergo major conformational changes to function in cells, particularly when they bind to more than one substrate. We quantify the large-amplitude hinge-bending landscape of human phosphoglycerate kinase (PGK) in a human cytoplasm. Approximately 70 μs of all-atom simulations, upon coarse graining, reveal three metastable states of PGK with different hinge angle distributions and additional substates. The "open" state was more populated than the "semi-open" or "closed" states. In addition to free energies and barriers within the landscape, we characterized the average transition state passage time of ≈0.3 μs and reversible substrate and product binding. Human PGK in a dilute solution simulation shows a transition directly from the open to closed states, in agreement with previous SAXS experiments, suggesting that the cell-like model environment promotes stability of the human PGK semi-open state. Yeast PGK also sampled three metastable states within the cytoplasm model, with the closed state favored in our simulation.
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Affiliation(s)
| | - Andrew K. Maytin
- Department of Physics, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Meredith M. Rickard
- Department of Chemistry, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Matthew C. Russell
- Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Taras V. Pogorelov
- Department of Chemistry, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
- Center for Biophysics and Computational Biology, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
- School of Chemical Sciences, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
- Beckman Institute for Advanced Science and Technology, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
- National Center for Supercomputing Applications, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Martin Gruebele
- Department of Chemistry, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
- Department of Physics, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
- Center for Biophysics and Computational Biology, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
- Beckman Institute for Advanced Science and Technology, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
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7
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Noel JK, Haglund E. Topological Reaction Coordinate Captures the Folding Transition State Ensemble in a Pierced Lasso Protein. J Phys Chem B 2024; 128:117-124. [PMID: 38118146 DOI: 10.1021/acs.jpcb.3c06678] [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: 12/22/2023]
Abstract
Proteins with a pierced lasso topology (PLT) have a covalent loop created by a disulfide bond, and the backbone circles back to thread the loop. This threaded topology has unique features compared to knotted topologies; notably, the topology is controlled by the chemical environment and the covalent loop remains intact even when denatured. In this work, we use the hormone leptin as our model PLT system and study its folding using molecular dynamics simulations that employ a structure-based (Go̅-like) model. We find that the reduced protein has a two-state folding mechanism with a transition state ensemble (TSE) that can be characterized by the reaction coordinate Q, the fraction of native contacts formed. In contrast, the oxidized protein, which must thread part of the polypeptide chain through a covalent loop, has a folding process that is poorly characterized by Q. Instead, we find that a topological coordinate that monitors the residue crossing the loop can identify the TSE of oxidized leptin. By precisely identifying the predicted TSE, one may now reliably calculate theoretical phi-values for the PLT protein, thereby enabling a comparison with experimental measurements. We find the loop-threading constraint leads to noncanonical phi-values that are uniformly small because this PLT protein has a flat energy landscape through the TSE.
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Affiliation(s)
- Jeffrey K Noel
- Structural Biology, Max Delbrück Center for Molecular Medicine, 13125 Berlin, Germany
| | - Ellinor Haglund
- Department of Chemistry, University of Hawaii, Manoa, Honolulu, Hawaii 96822, United States
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8
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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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9
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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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10
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Mouaffac L, Palacio-Rodriguez K, Pietrucci F. Optimal Reaction Coordinates and Kinetic Rates from the Projected Dynamics of Transition Paths. J Chem Theory Comput 2023; 19:5701-5711. [PMID: 37550088 DOI: 10.1021/acs.jctc.3c00158] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/09/2023]
Abstract
Finding optimal reaction coordinates and predicting accurate kinetic rates for activated processes are two of the foremost challenges of molecular simulations. We introduce an algorithm that tackles the two problems at once: starting from a limited number of reactive molecular dynamics trajectories (transition paths), we automatically generate with a Monte Carlo approach a sequence of different reaction coordinates that progressively reduce the kinetic rate of their projected effective dynamics. Based on a variational principle, the minimal rate accurately approximates the exact one, and it corresponds to the optimal reaction coordinate. After benchmarking the method on an analytic double-well system, we apply it to complex atomistic systems: the interaction of carbon nanoparticles of different sizes in water.
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Affiliation(s)
- Line Mouaffac
- Sorbonne Université, Muséum National d'Histoire Naturelle, UMR CNRS 7590, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, IMPMC, F-75005 Paris, France
| | - Karen Palacio-Rodriguez
- Sorbonne Université, Muséum National d'Histoire Naturelle, UMR CNRS 7590, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, IMPMC, F-75005 Paris, France
| | - Fabio Pietrucci
- Sorbonne Université, Muséum National d'Histoire Naturelle, UMR CNRS 7590, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, IMPMC, F-75005 Paris, France
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11
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Sisk T, Robustelli P. Folding-upon-binding pathways of an intrinsically disordered protein from a deep Markov state model. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.07.21.550103. [PMID: 37546728 PMCID: PMC10401938 DOI: 10.1101/2023.07.21.550103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/08/2023]
Abstract
A central challenge in the study of intrinsically disordered proteins is the characterization of the mechanisms by which they bind their physiological interaction partners. Here, we utilize a deep learning based Markov state modeling approach to characterize the folding-upon-binding pathways observed in a long-time scale molecular dynamics simulation of a disordered region of the measles virus nucleoprotein NTAIL reversibly binding the X domain of the measles virus phosphoprotein complex. We find that folding-upon-binding predominantly occurs via two distinct encounter complexes that are differentiated by the binding orientation, helical content, and conformational heterogeneity of NTAIL. We do not, however, find evidence for the existence of canonical conformational selection or induced fit binding pathways. We observe four kinetically separated native-like bound states that interconvert on time scales of eighty to five hundred nanoseconds. These bound states share a core set of native intermolecular contacts and stable NTAIL helices and are differentiated by a sequential formation of native and non-native contacts and additional helical turns. Our analyses provide an atomic resolution structural description of intermediate states in a folding-upon-binding pathway and elucidate the nature of the kinetic barriers between metastable states in a dynamic and heterogenous, or "fuzzy", protein complex.
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Affiliation(s)
- Thomas Sisk
- Dartmouth College, Department of Chemistry, Hanover, NH, 03755
| | - Paul Robustelli
- Dartmouth College, Department of Chemistry, Hanover, NH, 03755
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12
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Naleem N, Abreu CRA, Warmuz K, Tong M, Kirmizialtin S, Tuckerman ME. An exploration of machine learning models for the determination of reaction coordinates associated with conformational transitions. J Chem Phys 2023; 159:034102. [PMID: 37458344 DOI: 10.1063/5.0147597] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2023] [Accepted: 06/23/2023] [Indexed: 07/20/2023] Open
Abstract
Determining collective variables (CVs) for conformational transitions is crucial to understanding their dynamics and targeting them in enhanced sampling simulations. Often, CVs are proposed based on intuition or prior knowledge of a system. However, the problem of systematically determining a proper reaction coordinate (RC) for a specific process in terms of a set of putative CVs can be achieved using committor analysis (CA). Identifying essential degrees of freedom that govern such transitions using CA remains elusive because of the high dimensionality of the conformational space. Various schemes exist to leverage the power of machine learning (ML) to extract an RC from CA. Here, we extend these studies and compare the ability of 17 different ML schemes to identify accurate RCs associated with conformational transitions. We tested these methods on an alanine dipeptide in vacuum and on a sarcosine dipeptoid in an implicit solvent. Our comparison revealed that the light gradient boosting machine method outperforms other methods. In order to extract key features from the models, we employed Shapley Additive exPlanations analysis and compared its interpretation with the "feature importance" approach. For the alanine dipeptide, our methodology identifies ϕ and θ dihedrals as essential degrees of freedom in the C7ax to C7eq transition. For the sarcosine dipeptoid system, the dihedrals ψ and ω are the most important for the cisαD to transαD transition. We further argue that analysis of the full dynamical pathway, and not just endpoint states, is essential for identifying key degrees of freedom governing transitions.
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Affiliation(s)
- Nawavi Naleem
- Chemistry Program, Science Division, New York University, Abu Dhabi, UAE
| | - Charlles R A Abreu
- Chemical Engineering Department, Escola de Química, Universidade Federal do Rio de Janeiro, 21941-909 Rio de Janeiro, RJ, Brazil
| | - Krzysztof Warmuz
- Computer Science Program, Science Division, New York University, Abu Dhabi, UAE
| | - Muchen Tong
- Department of Chemistry, New York University (NYU), New York, New York 10003, USA
| | - Serdal Kirmizialtin
- Chemistry Program, Science Division, New York University, Abu Dhabi, UAE
- Department of Chemistry, New York University (NYU), New York, New York 10003, USA
- Center for Smart Engineering Materials, New York University, Abu Dhabi, UAE
| | - Mark E Tuckerman
- Department of Chemistry, New York University (NYU), New York, New York 10003, USA
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai, 3663 Zhongshan Rd. North, Shanghai 200062, China
- Simons Center for Computational Physical Chemistry at New York University, New York, New York 10003, USA
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13
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Abstract
Metal cofactors are critical centers for different biochemical processes of metalloproteins, and often, this metal coordination renders additional structural stability. In this study, we explore the additional stability conferred by the copper ion on azurin by analyzing both the apo and holo forms using temperature replica exchange molecular dynamics (REMD) data. We find a 14 K decrease in denaturation temperature for apo (406 K) azurin relative to that of holo (420 K), indicating a copper ion-induced additional thermal stability for holo azurin. The unfolding of apo azurin begins with the melting of α-helix and β-sheet V, similar to that of holo form. β-Sheets IV, VII, and VIII are comparatively more stable than other β-strands and melt at higher temperatures. Similar to holo azurin, the strong hydrophobic interactions among the apolar residues in the protein core is the key factor that renders high stability to apo protein as well. We construct free energy surfaces at different temperatures to capture the major conformations along the unfolding basins of the protein. Using contact maps from different basins we show the changes in the interaction between different residues along the unfolding pathway. Furthermore, we compare the Cα root-mean-square fluctuations (Cα-RMSF) and B-factor of all residues of apo and holo forms to understand the flexibility of different regions. The concerted displacement of α-helix and β-sheets V and VI from the protein core is another distinction we observe for apo compared to the holo form, where β-sheet VI was relatively stable.
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Affiliation(s)
- Albin Joy
- Department of Chemistry, Indian Institute of Technology Tirupati, Yerpedu 517619, Andhra Pradesh, India
| | - Rajib Biswas
- Department of Chemistry and Center for Atomic, Molecular and Optical Sciences & Technologies, Indian Institute of Technology Tirupati, Yerpedu 517619, Andhra Pradesh, India
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14
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Nagel D, Sartore S, Stock G. Selecting Features for Markov Modeling: A Case Study on HP35. J Chem Theory Comput 2023. [PMID: 37167425 DOI: 10.1021/acs.jctc.3c00240] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/13/2023]
Abstract
Markov state models represent a popular means to interpret molecular dynamics trajectories in terms of memoryless transitions between metastable conformational states. To provide a mechanistic understanding of the considered biomolecular process, these states should reflect structurally distinct conformations and ensure a time scale separation between fast intrastate and slow interstate dynamics. Adopting the folding of villin headpiece (HP35) as a well-established model problem, here we discuss the selection of suitable input coordinates or "features", such as backbone dihedral angles and interresidue distances. We show that dihedral angles account accurately for the structure of the native energy basin of HP35, while the unfolded region of the free energy landscape and the folding process are best described by tertiary contacts of the protein. To construct a contact-based model, we consider various ways to define and select contact distances and introduce a low-pass filtering of the feature trajectory as well as a correlation-based characterization of states. Relying on input data that faithfully account for the mechanistic origin of the studied process, the states of the resulting Markov model are clearly discriminated by the features, describe consistently the hierarchical structure of the free energy landscape, and─as a consequence─correctly reproduce the slow time scales of the process.
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Affiliation(s)
- Daniel Nagel
- Biomolecular Dynamics, Institute of Physics, University of Freiburg, 79104 Freiburg, Germany
| | - Sofia Sartore
- Biomolecular Dynamics, Institute of Physics, University of Freiburg, 79104 Freiburg, Germany
| | - Gerhard Stock
- Biomolecular Dynamics, Institute of Physics, University of Freiburg, 79104 Freiburg, Germany
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15
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Zanovello L, Löffler RJG, Caraglio M, Franosch T, Hanczyc MM, Faccioli P. Survival strategies of artificial active agents. Sci Rep 2023; 13:5616. [PMID: 37024516 PMCID: PMC10079664 DOI: 10.1038/s41598-023-32267-3] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2022] [Accepted: 03/24/2023] [Indexed: 04/08/2023] Open
Abstract
Artificial cells can be engineered to display dynamics sharing remarkable features in common with the survival behavior of living organisms. In particular, such active systems can respond to stimuli provided by the environment and undertake specific displacements to remain out of equilibrium, e.g. by moving towards regions with higher fuel concentration. In spite of the intense experimental activity aiming at investigating this fascinating behavior, a rigorous definition and characterization of such "survival strategies" from a statistical physics perspective is still missing. In this work, we take a first step in this direction by adapting and applying to active systems the theoretical framework of Transition Path Theory, which was originally introduced to investigate rare thermally activated transitions in passive systems. We perform experiments on camphor disks navigating Petri dishes and perform simulations in the paradigmatic active Brownian particle model to show how the notions of transition probability density and committor function provide the pivotal concepts to identify survival strategies, improve modeling, and obtain and validate experimentally testable predictions. The definition of survival in these artificial systems paves the way to move beyond simple observation and to formally characterize, design and predict complex life-like behaviors.
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Affiliation(s)
- Luigi Zanovello
- Physics Department, University of Trento, Via Sommarive 14, Povo, Trento, 38123, Italy
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, 6020, Innsbruck, Austria
| | - Richard J G Löffler
- Department of Cellular, Computational and Integrative Biology (CIBIO), University of Trento, Via Sommarive 9, Povo, Trento, 38123, Italy
| | - Michele Caraglio
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, 6020, Innsbruck, Austria
| | - Thomas Franosch
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, 6020, Innsbruck, Austria
| | - Martin M Hanczyc
- Department of Cellular, Computational and Integrative Biology (CIBIO), University of Trento, Via Sommarive 9, Povo, Trento, 38123, Italy.
- Department of Chemical and Biological Engineering, University of New Mexico, Albuquerque, NM, 87106, USA.
| | - Pietro Faccioli
- Physics Department, University of Trento, Via Sommarive 14, Povo, Trento, 38123, Italy.
- Trento Institute for Fundamental Physics and Applications (INFN-TIFPA), Via Sommarive 14, Povo, Trento, 38123, Italy.
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16
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Jung H, Covino R, Arjun A, Leitold C, Dellago C, Bolhuis PG, Hummer G. Machine-guided path sampling to discover mechanisms of molecular self-organization. NATURE COMPUTATIONAL SCIENCE 2023; 3:334-345. [PMID: 38177937 PMCID: PMC10766509 DOI: 10.1038/s43588-023-00428-z] [Citation(s) in RCA: 29] [Impact Index Per Article: 29.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2023] [Accepted: 03/10/2023] [Indexed: 01/06/2024]
Abstract
Molecular self-organization driven by concerted many-body interactions produces the ordered structures that define both inanimate and living matter. Here we present an autonomous path sampling algorithm that integrates deep learning and transition path theory to discover the mechanism of molecular self-organization phenomena. The algorithm uses the outcome of newly initiated trajectories to construct, validate and-if needed-update quantitative mechanistic models. Closing the learning cycle, the models guide the sampling to enhance the sampling of rare assembly events. Symbolic regression condenses the learned mechanism into a human-interpretable form in terms of relevant physical observables. Applied to ion association in solution, gas-hydrate crystal formation, polymer folding and membrane-protein assembly, we capture the many-body solvent motions governing the assembly process, identify the variables of classical nucleation theory, uncover the folding mechanism at different levels of resolution and reveal competing assembly pathways. The mechanistic descriptions are transferable across thermodynamic states and chemical space.
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Affiliation(s)
- Hendrik Jung
- Department of Theoretical Biophysics, Max Planck Institute of Biophysics, Frankfurt am Main, Germany
| | - Roberto Covino
- Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany
| | - A Arjun
- van 't Hoff Institute for Molecular Sciences, University of Amsterdam, Amsterdam, The Netherlands
| | | | | | - Peter G Bolhuis
- van 't Hoff Institute for Molecular Sciences, University of Amsterdam, Amsterdam, The Netherlands
| | - Gerhard Hummer
- Department of Theoretical Biophysics, Max Planck Institute of Biophysics, Frankfurt am Main, Germany.
- Institute of Biophysics, Goethe University Frankfurt, Frankfurt am Main, Germany.
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17
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Jaleel H, Mansha A, Usman M, Shah Y, Asim S, Yusaf A, Rehman U. DFT analysis on the reaction mechanism of Diels-Alder reaction between 2,4-hexane-1-ol and maleic anhydride. COMPUT THEOR CHEM 2023. [DOI: 10.1016/j.comptc.2023.114084] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/27/2023]
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18
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Durumeric AEP, Charron NE, Templeton C, Musil F, Bonneau K, Pasos-Trejo AS, Chen Y, Kelkar A, Noé F, Clementi C. Machine learned coarse-grained protein force-fields: Are we there yet? Curr Opin Struct Biol 2023; 79:102533. [PMID: 36731338 PMCID: PMC10023382 DOI: 10.1016/j.sbi.2023.102533] [Citation(s) in RCA: 25] [Impact Index Per Article: 25.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2022] [Revised: 12/14/2022] [Accepted: 12/18/2022] [Indexed: 02/04/2023]
Abstract
The successful recent application of machine learning methods to scientific problems includes the learning of flexible and accurate atomic-level force-fields for materials and biomolecules from quantum chemical data. In parallel, the machine learning of force-fields at coarser resolutions is rapidly gaining relevance as an efficient way to represent the higher-body interactions needed in coarse-grained force-fields to compensate for the omitted degrees of freedom. Coarse-grained models are important for the study of systems at time and length scales exceeding those of atomistic simulations. However, the development of transferable coarse-grained models via machine learning still presents significant challenges. Here, we discuss recent developments in this field and current efforts to address the remaining challenges.
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Affiliation(s)
- Aleksander E P Durumeric
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany
| | - Nicholas E Charron
- Department of Physics and Astronomy, Rice University, 6100 Main Street, Houston, 77005, Texas, USA; Department of Physics, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany; Center for Theoretical Biological Physics, Rice University, 6100 Main Street, Houston, 77005, Texas, USA
| | - Clark Templeton
- Department of Physics, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany. https://twitter.com/pbrun03
| | - Félix Musil
- Department of Physics, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany. https://twitter.com/FelixMusil
| | - Klara Bonneau
- Department of Physics, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany
| | - Aldo S Pasos-Trejo
- Department of Physics, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany. https://twitter.com/sayeg84
| | - Yaoyi Chen
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany. https://twitter.com/hello_yaoyi
| | - Atharva Kelkar
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany
| | - Frank Noé
- Microsoft Research AI4Science, Karl-Liebknecht Str. 32, Berlin, 10178, Berlin, Germany; Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany; Department of Physics, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany; Department of Chemistry, Rice University, 6100 Main Street, Houston, 77005, Texas, USA. https://twitter.com/FrankNoeBerlin
| | - Cecilia Clementi
- Department of Physics, Freie Universität Berlin, Arnimallee 12, 14195, Berlin, Germany; Center for Theoretical Biological Physics, Rice University, 6100 Main Street, Houston, 77005, Texas, USA; Department of Chemistry, Rice University, 6100 Main Street, Houston, 77005, Texas, USA; Department of Physics and Astronomy, Rice University, 6100 Main Street, Houston, 77005, Texas, USA.
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19
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Pompa-García I, Castilla R, Metzler R, Dagdug L. First-passage times in conical varying-width channels biased by a transverse gravitational force: Comparison of analytical and numerical results. Phys Rev E 2022; 106:064137. [PMID: 36671151 DOI: 10.1103/physreve.106.064137] [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: 09/02/2022] [Accepted: 12/20/2022] [Indexed: 06/17/2023]
Abstract
We study the crossing time statistic of diffusing point particles between the two ends of expanding and narrowing two-dimensional conical channels under a transverse external gravitational field. The theoretical expression for the mean first-passage time for such a system is derived under the assumption that the axial diffusion in a two-dimensional channel of smoothly varying geometry can be approximately described as a one-dimensional diffusion in an entropic potential with position-dependent effective diffusivity in terms of the modified Fick-Jacobs equation. We analyze the channel crossing dynamics in terms of the mean first-passage time, combining our analytical results with extensive two-dimensional Brownian dynamics simulations, allowing us to find the range of applicability of the one-dimensional approximation. We find that the effective particle diffusivity decreases with increasing amplitude of the external potential. Remarkably, the mean first-passage time for crossing the channel is shown to assume a minimum at finite values of the potential amplitude.
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Affiliation(s)
- Ivan Pompa-García
- Physics Department, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México, 09340, México
| | - Rodrigo Castilla
- Engineering Faculty, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México, 04510, México
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
| | - Leonardo Dagdug
- Physics Department, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México, 09340, México
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20
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Hassan A, Whitford PC. Identifying Strategies to Experimentally Probe Multidimensional Dynamics in the Ribosome. J Phys Chem B 2022; 126:8460-8471. [PMID: 36256879 DOI: 10.1021/acs.jpcb.2c05706] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023]
Abstract
The ribosome is a complex biomolecular machine that utilizes large-scale conformational rearrangements to synthesize proteins. For example, during the elongation cycle, the "head" domain of the ribosomal small subunit (SSU) is known to undergo transient rotation events that allow for movement of tRNA molecules (i.e., translocation). While the head may exhibit rigid-body-like properties, the precise relationship between experimentally accessible probes and multidimensional rotations has yet to be established. To address this gap, we perform molecular dynamics simulations of the translocation step of the elongation cycle in the ribosome, where the SSU head spontaneously undergoes rotation and tilt-like motions. With this data set (1250 simulated events), we used statistical and information-theory-based measures to identify possible single-molecule probes that can isolate SSU head rotation and head tilting. This analysis provides a molecular interpretation for previous single-molecule measurements, while establishing a framework for the design of next-generation experiments that may precisely probe the mechanistic and kinetic aspects of the ribosome.
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Affiliation(s)
- Asem Hassan
- Department of Physics, Northeastern University, Dana Research Center 111, 360 Huntington Avenue, Boston, Massachusetts02115, United States.,Center for Theoretical Biological Physics, Northeastern University, 360 Huntington Avenue, Boston, Massachusetts02115, United States
| | - Paul C Whitford
- Department of Physics, Northeastern University, Dana Research Center 111, 360 Huntington Avenue, Boston, Massachusetts02115, United States.,Center for Theoretical Biological Physics, Northeastern University, 360 Huntington Avenue, Boston, Massachusetts02115, United States
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21
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Ramos-Sánchez P, Harvey JN, Gámez JA. An automated method for graph-based chemical space exploration and transition state finding. J Comput Chem 2022; 44:27-42. [PMID: 36239971 DOI: 10.1002/jcc.27011] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2022] [Revised: 07/28/2022] [Accepted: 09/05/2022] [Indexed: 12/24/2022]
Abstract
Algorithms that automatically explore the chemical space have been limited to chemical systems with a low number of atoms due to expensive involved quantum calculations and the large amount of possible reaction pathways. The method described here presents a novel solution to the problem of chemical exploration by generating reaction networks with heuristics based on chemical theory. First, a second version of the reaction network is determined through molecular graph transformations acting upon functional groups of the reacting. Only transformations that break two chemical bonds and form two new ones are considered, leading to a significant performance enhancement compared to previously presented algorithm. Second, energy barriers for this reaction network are estimated through quantum chemical calculations by a growing string method, which can also identify non-octet species missed during the previous step and further define the reaction network. The proposed algorithm has been successfully applied to five different chemical reactions, in all cases identifying the most important reaction pathways.
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Affiliation(s)
- Pablo Ramos-Sánchez
- Digital R&D, Covestro Deutschland AG, Leverkusen, Germany.,Department of Chemistry, KU Leuven, Leuven, Belgium
| | | | - José A Gámez
- Digital R&D, Covestro Deutschland AG, Leverkusen, Germany
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22
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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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23
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Palacio-Rodriguez K, Pietrucci F. Free Energy Landscapes, Diffusion Coefficients, and Kinetic Rates from Transition Paths. J Chem Theory Comput 2022; 18:4639-4648. [PMID: 35899416 DOI: 10.1021/acs.jctc.2c00324] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We address the problem of constructing accurate mathematical models of the dynamics of complex systems projected on a collective variable. To this aim we introduce a conceptually simple yet effective algorithm for estimating the parameters of Langevin and Fokker-Planck equations from a set of short, possibly out-of-equilibrium molecular dynamics trajectories, obtained for instance from transition path sampling or as relaxation from high free-energy configurations. The approach maximizes the model likelihood based on any explicit expression of the short-time propagator, hence it can be applied to different evolution equations. We demonstrate the numerical efficiency and robustness of the algorithm on model systems, and we apply it to reconstruct the projected dynamics of pairs of C60 and C240 fullerene molecules in explicit water. Our methodology allows reconstructing the accurate thermodynamics and kinetics of activated processes, namely free energy landscapes, diffusion coefficients, and kinetic rates. Compared to existing enhanced sampling methods, we directly exploit short unbiased trajectories at a competitive computational cost.
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Affiliation(s)
- Karen Palacio-Rodriguez
- Muséum National d'Histoire Naturelle, UMR CNRS 7590, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, IMPMC, Sorbonne Université, F-75005 Paris, France
| | - Fabio Pietrucci
- Muséum National d'Histoire Naturelle, UMR CNRS 7590, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, IMPMC, Sorbonne Université, F-75005 Paris, France
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24
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Krivov SV. Additive eigenvectors as optimal reaction coordinates, conditioned trajectories, and time-reversible description of stochastic processes. J Chem Phys 2022; 157:014108. [DOI: 10.1063/5.0088061] [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
A fundamental way to analyze complex multidimensional stochastic dynamics is to describe it as diffusion on a free energy landscape—free energy as a function of reaction coordinates (RCs). For such a description to be quantitatively accurate, the RC should be chosen in an optimal way. The committor function is a primary example of an optimal RC for the description of equilibrium reaction dynamics between two states. Here, additive eigenvectors (addevs) are considered as optimal RCs to address the limitations of the committor. An addev master equation for a Markov chain is derived. A stationary solution of the equation describes a sub-ensemble of trajectories conditioned on having the same optimal RC for the forward and time-reversed dynamics in the sub-ensemble. A collection of such sub-ensembles of trajectories, called stochastic eigenmodes, can be used to describe/approximate the stochastic dynamics. A non-stationary solution describes the evolution of the probability distribution. However, in contrast to the standard master equation, it provides a time-reversible description of stochastic dynamics. It can be integrated forward and backward in time. The developed framework is illustrated on two model systems—unidirectional random walk and diffusion.
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Affiliation(s)
- Sergei V. Krivov
- University of Leeds, Astbury Center for Structural Molecular Biology, Faculty of Biological Sciences, University of Leeds, Leeds LS2 9JT, United Kingdom
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25
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Diez G, Nagel D, Stock G. Correlation-Based Feature Selection to Identify Functional Dynamics in Proteins. J Chem Theory Comput 2022; 18:5079-5088. [PMID: 35793551 DOI: 10.1021/acs.jctc.2c00337] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
To interpret molecular dynamics simulations of biomolecular systems, systematic dimensionality reduction methods are commonly employed. Among others, this includes principal component analysis (PCA) and time-lagged independent component analysis (TICA), which aim to maximize the variance and the time scale of the first components, respectively. A crucial first step of such an analysis is the identification of suitable and relevant input coordinates (the so-called features), such as backbone dihedral angles and interresidue distances. As typically only a small subset of those coordinates is involved in a specific biomolecular process, it is important to discard the remaining uncorrelated motions or weakly correlated noise coordinates. This is because they may exhibit large amplitudes or long time scales and therefore will be erroneously considered important by PCA and TICA, respectively. To discriminate collective motions underlying functional dynamics from uncorrelated motions, the correlation matrix of the input coordinates is block-diagonalized by a clustering method. This strategy avoids possible bias due to presumed functional observables and conformational states or variation principles that maximize variance or time scales. Considering several linear and nonlinear correlation measures and various clustering algorithms, it is shown that the combination of linear correlation and the Leiden community detection algorithm yields excellent results for all considered model systems. These include the functional motion of T4 lysozyme to demonstrate the successful identification of collective motion, as well as the folding of the villin headpiece to highlight the physical interpretation of the correlated motions in terms of a functional mechanism.
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Affiliation(s)
- Georg Diez
- Biomolecular Dynamics, Institute of Physics, Albert-Ludwigs-Universität, 79104 Freiburg, Germany
| | - Daniel Nagel
- Biomolecular Dynamics, Institute of Physics, Albert-Ludwigs-Universität, 79104 Freiburg, Germany
| | - Gerhard Stock
- Biomolecular Dynamics, Institute of Physics, Albert-Ludwigs-Universität, 79104 Freiburg, Germany
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26
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Dang NL, Baranger AM, Beveridge DL. High Energy Channeling and Malleable Transition States: Molecular Dynamics Simulations and Free Energy Landscapes for the Thermal Unfolding of Protein U1A and 13 Mutants. Biomolecules 2022; 12:940. [PMID: 35883496 PMCID: PMC9312810 DOI: 10.3390/biom12070940] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2022] [Revised: 06/20/2022] [Accepted: 06/27/2022] [Indexed: 11/20/2022] Open
Abstract
The spliceosome protein U1A is a prototype case of the RNA recognition motif (RRM) ubiquitous in biological systems. The in vitro kinetics of the chemical denaturation of U1A indicate that the unfolding of U1A is a two-state process but takes place via high energy channeling and a malleable transition state, an interesting variation of typical two-state behavior. Molecular dynamics (MD) simulations have been applied extensively to the study of two-state unfolding and folding of proteins and provide an opportunity to obtain a theoretical account of the experimental results and a molecular model for the transition state ensemble. We describe herein all-atom MD studies including explicit solvent of up to 100 ns on the thermal unfolding (UF) of U1A and 13 mutants. Multiple MD UF trajectories are carried out to ensure accuracy and reproducibility. A vector representation of the MD unfolding process in RMSD space is obtained and used to calculate a free energy landscape for U1A unfolding. A corresponding MD simulation and free energy landscape for the protein CI2, well known to follow a simple two state folding/unfolding model, is provided as a control. The results indicate that the unfolding pathway on the MD calculated free energy landscape of U1A shows a markedly extended transition state compared with that of CI2. The MD results support the interpretation of the observed chevron plots for U1A in terms of a high energy, channel-like transition state. Analysis of the MDUF structures shows that the transition state ensemble involves microstates with most of the RRM secondary structure intact but expanded by ~14% with respect to the radius of gyration. Comparison with results on a prototype system indicates that the transition state involves an ensemble of molten globule structures and extends over the region of ~1-35 ns in the trajectories. Additional MDUF simulations were carried out for 13 U1A mutants, and the calculated φ-values show close accord with observed results and serve to validate our methodology.
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Affiliation(s)
| | | | - David L. Beveridge
- Department of Chemistry and Molecular Biophysics Program, Wesleyan University, Middletown, CT 06459, USA; (N.L.D.); (A.M.B.)
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27
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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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28
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Mehdi S, Wang D, Pant S, Tiwary P. Accelerating All-Atom Simulations and Gaining Mechanistic Understanding of Biophysical Systems through State Predictive Information Bottleneck. J Chem Theory Comput 2022; 18:3231-3238. [PMID: 35384668 PMCID: PMC9297332 DOI: 10.1021/acs.jctc.2c00058] [Citation(s) in RCA: 28] [Impact Index Per Article: 14.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/10/2023]
Abstract
An effective implementation of enhanced sampling algorithms for molecular dynamics simulations requires a priori knowledge of the approximate reaction coordinate describing the relevant mechanisms in the system. In this work, we focus on the recently developed artificial intelligence-based State Predictive Information Bottleneck (SPIB) approach and demonstrate how SPIB can learn such a reaction coordinate as a deep neural network even from undersampled trajectories. We exemplify its usefulness by achieving more than 40 times acceleration in simulating two model biophysical systems through well-tempered metadynamics performed by biasing along the SPIB-learned reaction coordinate. These include left- to right-handed chirality transitions in a synthetic helical peptide (Aib)9 and permeation of a small benzoic acid molecule through a synthetic, symmetric phospholipid bilayer. In addition to significantly accelerating the dynamics and achieving back and forth movement between different metastable states, the SPIB-based reaction coordinate gives mechanistic insights into the processes driving these two important problems.
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Affiliation(s)
- Shams Mehdi
- Biophysics Program and Institute for Physical Science and Technology, University of Maryland, College Park 20742, USA
| | - Dedi Wang
- Biophysics Program and Institute for Physical Science and Technology, University of Maryland, College Park 20742, USA
| | - Shashank Pant
- Center for Biophysics and Quantitative Biology, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
| | - Pratyush Tiwary
- Department of Chemistry and Biochemistry and Institute for Physical Science and Technology, University of Maryland, College Park 20742, USA
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29
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Kikutsuji T, Mori Y, Okazaki KI, Mori T, Kim K, Matubayasi N. Explaining reaction coordinates of alanine dipeptide isomerization obtained from deep neural networks using Explainable Artificial Intelligence (XAI). J Chem Phys 2022; 156:154108. [DOI: 10.1063/5.0087310] [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
A method for obtaining appropriate reaction coordinates is required to identify transition states distinguishing product and reactant in complex molecular systems. Recently, abundant research has been devoted to obtaining reaction coordinates using artificial neural networks from deep learning literature, where many collective variables are typically utilized in the input layer. However, it is difficult to explain the details of which collective variables contribute to the predicted reaction coordinates owing to the complexity of the nonlinear functions in deep neural networks. To overcome this limitation, we used Explainable Artificial Intelligence (XAI) methods of the Local Interpretable Model-agnostic Explanation (LIME) and the game theory-based framework known as Shapley Additive exPlanations (SHAP). We demonstrated that XAI enables us to obtain the degree of contribution of each collective variable to reaction coordinates that is determined by nonlinear regressions with deep learning for the committor of the alanine dipeptide isomerization in vacuum. In particular, both LIME and SHAP provide important features to the predicted reaction coordinates, which are characterized by appropriate dihedral angles consistent with those previously reported from the committor test analysis. The present study offers an AI-aided framework to explain the appropriate reaction coordinates, which acquires considerable significance when the number of degrees of freedom increases.
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Affiliation(s)
| | | | - Kei-ichi Okazaki
- Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Japan
| | - Toshifumi Mori
- Kyushu University Institute for Materials Chemistry and Engineering, Japan
| | - Kang Kim
- Graduate School of Engineering Science, Osaka University - Toyonaka Campus, Japan
| | - Nobuyuki Matubayasi
- Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Japan
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30
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Sun L, Vandermause J, Batzner S, Xie Y, Clark D, Chen W, Kozinsky B. Multitask Machine Learning of Collective Variables for Enhanced Sampling of Rare Events. J Chem Theory Comput 2022; 18:2341-2353. [PMID: 35274958 DOI: 10.1021/acs.jctc.1c00143] [Citation(s) in RCA: 17] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Computing accurate reaction rates is a central challenge in computational chemistry and biology because of the high cost of free energy estimation with unbiased molecular dynamics. In this work, a data-driven machine learning algorithm is devised to learn collective variables with a multitask neural network, where a common upstream part reduces the high dimensionality of atomic configurations to a low dimensional latent space and separate downstream parts map the latent space to predictions of basin class labels and potential energies. The resulting latent space is shown to be an effective low-dimensional representation, capturing the reaction progress and guiding effective umbrella sampling to obtain accurate free energy landscapes. This approach is successfully applied to model systems including a 5D Müller Brown model, a 5D three-well model, the alanine dipeptide in vacuum, and an Au(110) surface reconstruction unit reaction. It enables automated dimensionality reduction for energy controlled reactions in complex systems, offers a unified and data-efficient framework that can be trained with limited data, and outperforms single-task learning approaches, including autoencoders.
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Affiliation(s)
- Lixin Sun
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
| | - Jonathan Vandermause
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
| | - Simon Batzner
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
| | - Yu Xie
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
| | - David Clark
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
| | - Wei Chen
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
| | - Boris Kozinsky
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
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31
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Abstract
Proteins have dynamic structures that undergo chain motions on time scales spanning from picoseconds to seconds. Resolving the resultant conformational heterogeneity is essential for gaining accurate insight into fundamental mechanistic aspects of the protein folding reaction. The use of high-resolution structural probes, sensitive to population distributions, has begun to enable the resolution of site-specific conformational heterogeneity at different stages of the folding reaction. Different states populated during protein folding, including the unfolded state, collapsed intermediate states, and even the native state, are found to possess significant conformational heterogeneity. Heterogeneity in protein folding and unfolding reactions originates from the reduced cooperativity of various kinds of physicochemical interactions between various structural elements of a protein, and between a protein and solvent. Heterogeneity may arise because of functional or evolutionary constraints. Conformational substates within the unfolded state and the collapsed intermediates that exchange at rates slower than the subsequent folding steps give rise to heterogeneity on the protein folding pathways. Multiple folding pathways are likely to represent distinct sequences of structure formation. Insight into the nature of the energy barriers separating different conformational states populated during (un)folding can also be obtained by resolving heterogeneity.
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Affiliation(s)
- Sandhya Bhatia
- National Centre for Biological Sciences, Tata Institute of Fundamental Research, Bengaluru 560065, India.,Indian Institute of Science Education and Research, Pune 411008, India
| | - Jayant B Udgaonkar
- National Centre for Biological Sciences, Tata Institute of Fundamental Research, Bengaluru 560065, India.,Indian Institute of Science Education and Research, Pune 411008, India
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32
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Abstract
Constantly advancing computer simulations of biomolecules provide huge amounts of data that are difficult to interpret. In particular, obtaining insights into functional aspects of macromolecular dynamics, often related to cascades of transient events, calls for methodologies that depart from the well-grounded framework of equilibrium statistical physics. One of the approaches toward the analysis of complex temporal data which has found applications in the fields of neuroscience and econometrics is Granger causality analysis. It allows determining which components of multidimensional time series are most influential for the evolution of the entire system, thus providing insights into causal relations within the dynamic structure of interest. In this work, we apply Granger analysis to a long molecular dynamics trajectory depicting repetitive folding and unfolding of a mini β-hairpin protein, CLN025. We find objective, quantitative evidence indicating that rearrangements within the hairpin turn region are determinant for protein folding and unfolding. On the contrary, interactions between hairpin arms score low on the causality scale. Taken together, these findings clearly favor the concept of zipperlike folding, which is one of two postulated β-hairpin folding mechanisms. More importantly, the results demonstrate the possibility of a conclusive application of Granger causality analysis to a biomolecular system.
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Affiliation(s)
- Marcin Sobieraj
- Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland.,Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
| | - Piotr Setny
- Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
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33
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Wang W, Poe D, Yang Y, Hyatt T, Xing J. Epithelial-to-mesenchymal transition proceeds through directional destabilization of multidimensional attractor. eLife 2022; 11:74866. [PMID: 35188459 PMCID: PMC8920502 DOI: 10.7554/elife.74866] [Citation(s) in RCA: 15] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2021] [Accepted: 02/06/2022] [Indexed: 11/13/2022] Open
Abstract
How a cell changes from one stable phenotype to another one is a fundamental problem in developmental and cell biology. Mathematically, a stable phenotype corresponds to a stable attractor in a generally multi-dimensional state space, which needs to be destabilized so the cell relaxes to a new attractor. Two basic mechanisms for destabilizing a stable fixed point, pitchfork and saddle-node bifurcations, have been extensively studied theoretically; however, direct experimental investigation at the single-cell level remains scarce. Here, we performed live cell imaging studies and analyses in the framework of dynamical systems theories on epithelial-to-mesenchymal transition (EMT). While some mechanistic details remain controversial, EMT is a cell phenotypic transition (CPT) process central to development and pathology. Through time-lapse imaging we recorded single cell trajectories of human A549/Vim-RFP cells undergoing EMT induced by different concentrations of exogenous TGF-β in a multi-dimensional cell feature space. The trajectories clustered into two distinct groups, indicating that the transition dynamics proceeds through parallel paths. We then reconstructed the reaction coordinates and the corresponding quasi-potentials from the trajectories. The potentials revealed a plausible mechanism for the emergence of the two paths where the original stable epithelial attractor collides with two saddle points sequentially with increased TGF-β concentration, and relaxes to a new one. Functionally, the directional saddle-node bifurcation ensures a CPT proceeds towards a specific cell type, as a mechanistic realization of the canalization idea proposed by Waddington. Cells with the same genetic code can take on many different formss, or phenotypes, which have distinct roles and appearances. Sometimes cells switch from one phenotype to another as part of healthy growth or during disease. One such change is the epithelial-to-mesenchymal transition (EMT), which is involved in fetal development, wound healing and the spread of cancer cells. During EMT, closely connected epithelial cells detach from one another and change into mesenchymal cells that are able to migrate. Cells undergo a number of changes during this transition; however, the path they take to reach their new form is not entirely clear. For instance, do all cells follow the same route, or are there multiple ways that cells can shift from one state to the next? To address this question, Wang et al. studied individual lung cancer cells that had been treated with a protein that drives EMT. The cells were then imaged at regular intervals over the course of two to three days to see how they changed in response to different concentrations of protein. Using a mathematical analysis designed to study chemical reactions, Wang et al. showed that the cells transform into the mesenchymal phenotype through two main routes. This result suggests that attempts to prevent EMT, in cancer treatment for instance, would require blocking both paths taken by the cells. This information could be useful for biomedical researchers trying to regulate the EMT process. The quantitative approach of this study could also help physicists and mathematicians study other types of transition that occur in biology.
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Affiliation(s)
- Weikang Wang
- Department of Computational and Systems Biology, University of Pittsburgh, Pittsburgh, United States
| | - Dante Poe
- Department of Computational and Systems Biology, University of Pittsburgh, Pittsburgh, United States
| | - Yaxuan Yang
- Department of Computational and Systems Biology, University of Pittsburgh, Pittsburgh, United States
| | - Thomas Hyatt
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, United States
| | - Jianhua Xing
- Department of Computational and Systems Biology, University of Pittsburgh, Pittsburgh, United States
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34
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Li W. Optimizing reaction coordinate by flux maximization in the transition path ensemble. J Chem Phys 2022; 156:054117. [DOI: 10.1063/5.0079390] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Wenjin Li
- Institute for Advanced Study, Shenzhen University, Shenzhen, China
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35
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Rico-Pasto M, Alemany A, Ritort F. Force-Dependent Folding Kinetics of Single Molecules with Multiple Intermediates and Pathways. J Phys Chem Lett 2022; 13:1025-1032. [PMID: 35072478 PMCID: PMC9882750 DOI: 10.1021/acs.jpclett.1c03521] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/14/2023]
Abstract
Most single-molecule studies derive the kinetic rates of native, intermediate, and unfolded states from equilibrium hopping experiments. Here, we apply the Kramers kinetic diffusive model to derive the force-dependent kinetic rates of intermediate states from nonequilibrium pulling experiments. From the kinetic rates, we also extract the force-dependent kinetic barriers and the equilibrium folding energies. We apply our method to DNA hairpins with multiple folding pathways and intermediates. The experimental results agree with theoretical predictions. Furthermore, the proposed nonequilibrium single-molecule approach permits us to characterize kinetic and thermodynamic properties of native, unfolded, and intermediate states that cannot be derived from equilibrium hopping experiments.
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Affiliation(s)
- Marc Rico-Pasto
- Small
Biosystems Lab, Condensed Matter Physics Department, University of Barcelona, C/Martí i Franqués 1, Barcelona, 08028, Spain
| | - Anna Alemany
- Department
of Anatomy and Embryology, Leiden University
Medical Center, Leiden, 2333ZC, The Netherlands
| | - Felix Ritort
- Small
Biosystems Lab, Condensed Matter Physics Department, University of Barcelona, C/Martí i Franqués 1, Barcelona, 08028, Spain
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36
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Abstract
Unbiased molecular dynamics simulations of proteins can now capture spontaneous folding events. This provides a wealth of data reflecting information on folding mechanism, but raises the challenge of interpreting it in a meaningful way. Here, I describe how such simulations can be used to identify reactive states and reaction coordinates for describing folding, and how folding dynamics can be captured by projection onto those coordinates. Methods are described for quantifying the interactions important for defining the folding mechanism, and for comparison of simulations with experimental mechanistic probes, such as ϕ-values.
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Affiliation(s)
- Robert B Best
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD, USA.
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37
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Kameda T, Awazu A, Togashi Y. Molecular dynamics analysis of biomolecular systems including nucleic acids. Biophys Physicobiol 2022; 19:e190027. [DOI: 10.2142/biophysico.bppb-v19.0027] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2022] [Accepted: 08/18/2022] [Indexed: 12/01/2022] Open
Affiliation(s)
| | - Akinori Awazu
- Graduate School of Integrated Sciences for Life, Hiroshima University
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38
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Observing the base-by-base search for native structure along transition paths during the folding of single nucleic acid hairpins. Proc Natl Acad Sci U S A 2021; 118:2101006118. [PMID: 34853166 DOI: 10.1073/pnas.2101006118] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 10/02/2021] [Indexed: 12/25/2022] Open
Abstract
Biomolecular folding involves searching among myriad possibilities for the native conformation, but the elementary steps expected from theory for this search have never been detected directly. We probed the dynamics of folding at high resolution using optical tweezers, measuring individual trajectories as nucleic acid hairpins passed through the high-energy transition states that dominate kinetics and define folding mechanisms. We observed brief but ubiquitous pauses in the transition states, with a dwell time distribution that matched microscopic theories of folding quantitatively. The sequence dependence suggested that pauses were dominated by microbarriers from nonnative conformations during the search by each nucleotide residue for the native base-pairing conformation. Furthermore, the pauses were position dependent, revealing subtle local variations in energy-landscape roughness and allowing the diffusion coefficient describing the microscopic dynamics within the barrier to be found without reconstructing the shape of the energy landscape. These results show how high-resolution measurements can elucidate key microscopic events during folding to test fundamental theories of folding.
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39
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Marinelli F, Faraldo-Gómez JD. Force-Correction Analysis Method for Derivation of Multidimensional Free-Energy Landscapes from Adaptively Biased Replica Simulations. J Chem Theory Comput 2021; 17:6775-6788. [PMID: 34669402 DOI: 10.1021/acs.jctc.1c00586] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
A methodology is proposed for the calculation of multidimensional free-energy landscapes of molecular systems, based on analysis of multiple molecular dynamics trajectories wherein adaptive biases have been applied to enhance the sampling of different collective variables. In this approach, which we refer to as the Force-Correction Analysis Method (FCAM), local averages of the total and biasing forces are evaluated post hoc, and the latter are subtracted from the former to obtain unbiased estimates of the mean force across collective-variable space. Multidimensional free-energy surfaces and minimum free-energy pathways are then derived by integrating the mean-force landscape with a kinetic Monte Carlo algorithm. To evaluate the proposed method, a series of numerical tests and comparisons with existing approaches were carried out for small molecules, peptides, and proteins, based on all-atom trajectories generated with standard, concurrent, and replica-exchange metadynamics in collective-variable spaces ranging from one to six dimensional. The tests confirm the correctness of the FCAM formulation and demonstrate that calculated mean forces and free energies converge rapidly and accurately, outperforming other methods used to unbias this kind of simulation data.
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Affiliation(s)
- Fabrizio Marinelli
- Theoretical Molecular Biophysics Laboratory, National Heart, Lung and Blood Institute, National Institutes of Health, Bethesda, Maryland 20814, United States
| | - José D Faraldo-Gómez
- Theoretical Molecular Biophysics Laboratory, National Heart, Lung and Blood Institute, National Institutes of Health, Bethesda, Maryland 20814, United States
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40
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Tsai ST, Smith Z, Tiwary P. SGOOP-d: Estimating Kinetic Distances and Reaction Coordinate Dimensionality for Rare Event Systems from Biased/Unbiased Simulations. J Chem Theory Comput 2021; 17:6757-6765. [PMID: 34662516 DOI: 10.1021/acs.jctc.1c00431] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/29/2022]
Abstract
Understanding kinetics including reaction pathways and associated transition rates is an important yet difficult problem in numerous chemical and biological systems, especially in situations with multiple competing pathways. When these high-dimensional systems are projected on low-dimensional coordinates, which are often needed for enhanced sampling or for interpretation of simulations and experiments, one can end up losing the kinetic connectivity of the underlying high-dimensional landscape. Thus, in the low-dimensional projection, metastable states might appear closer or further than they actually are. To deal with this issue, in this work, we develop a formalism that learns a multidimensional yet minimally complex reaction coordinate (RC) for generic high-dimensional systems. When projected along this RC, all possible kinetically relevant pathways can be demarcated and the true high-dimensional connectivity is maintained. One of the defining attributes of our method lies in that it can work on long unbiased simulations as well as biased simulations often needed for rare event systems. We demonstrate the utility of the method by studying a range of model systems including conformational transitions in a small peptide Ace-Ala3-Nme, where we show how two-dimensional and three-dimensional RCs found by our previously published spectral gap optimization method "SGOOP" [Tiwary, P. and Berne, B. J. Proc. Natl. Acad. Sci. 2016, 113, 2839] can capture the kinetics for 23 and all 28 out of the 28 dominant state-to-state transitions, respectively.
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Affiliation(s)
- Sun-Ting Tsai
- Department of Physics and Institute for Physical Science and Technology, University of Maryland, College Park 20742, Maryland, United States
| | - Zachary Smith
- Biophysics Program and Institute for Physical Science and Technology, University of Maryland, College Park 20742, Maryland, United States
| | - Pratyush Tiwary
- Department of Chemistry and Biochemistry and Institute for Physical Science and Technology, University of Maryland, College Park 20742, Maryland, United States
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41
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Sharpe DJ, Wales DJ. Nearly reducible finite Markov chains: Theory and algorithms. J Chem Phys 2021; 155:140901. [PMID: 34654307 DOI: 10.1063/5.0060978] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
Abstract
Finite Markov chains, memoryless random walks on complex networks, appear commonly as models for stochastic dynamics in condensed matter physics, biophysics, ecology, epidemiology, economics, and elsewhere. Here, we review exact numerical methods for the analysis of arbitrary discrete- and continuous-time Markovian networks. We focus on numerically stable methods that are required to treat nearly reducible Markov chains, which exhibit a separation of characteristic timescales and are therefore ill-conditioned. In this metastable regime, dense linear algebra methods are afflicted by propagation of error in the finite precision arithmetic, and the kinetic Monte Carlo algorithm to simulate paths is unfeasibly inefficient. Furthermore, iterative eigendecomposition methods fail to converge without the use of nontrivial and system-specific preconditioning techniques. An alternative approach is provided by state reduction procedures, which do not require additional a priori knowledge of the Markov chain. Macroscopic dynamical quantities, such as moments of the first passage time distribution for a transition to an absorbing state, and microscopic properties, such as the stationary, committor, and visitation probabilities for nodes, can be computed robustly using state reduction algorithms. The related kinetic path sampling algorithm allows for efficient sampling of trajectories on a nearly reducible Markov chain. Thus, all of the information required to determine the kinetically relevant transition mechanisms, and to identify the states that have a dominant effect on the global dynamics, can be computed reliably even for computationally challenging models. Rare events are a ubiquitous feature of realistic dynamical systems, and so the methods described herein are valuable in many practical applications.
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Affiliation(s)
- Daniel J Sharpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - David J Wales
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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42
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Meshkin H, Zhu F. Toward Convergence in Free Energy Calculations for Protein Conformational Changes: A Case Study on the Thin Gate of Mhp1 Transporter. J Chem Theory Comput 2021; 17:6583-6596. [PMID: 34523931 DOI: 10.1021/acs.jctc.1c00585] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Abstract
It has been challenging to obtain reliable free energies for protein conformational changes from all-atom molecular dynamics simulations, despite the availability of many enhanced sampling techniques. To alleviate the difficulties associated with the enormous complexity of the conformational space, here we propose a few practical strategies for such calculations, including (1) a stringent method to examine convergence by comparing independent simulations starting from different initial coordinates, (2) adoption of multistep schemes in which the complete conformational change consists of multiple transition steps, each sampled using a distinct reaction coordinate, and (3) application of boundary restraints to simplify the conformational space. We demonstrate these strategies on the conformational changes between the outward-facing and outward-occluded states of the Mhp1 membrane transporter, obtaining the equilibrium thermodynamics of the relevant metastable states, the kinetic rates between these states, and the reactive trajectories that reveal the atomic details of spontaneous transitions. Our approaches thus promise convergent and reliable calculations to examine intuition-based hypotheses and to eventually elucidate the underlying molecular mechanisms of reversible conformational changes in complex protein systems.
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Affiliation(s)
- Hamed Meshkin
- Department of Physics, Indiana University Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202, United States
| | - Fangqiang Zhu
- Department of Physics, Indiana University Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202, United States
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43
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Sharpe DJ, Wales DJ. Numerical analysis of first-passage processes in finite Markov chains exhibiting metastability. Phys Rev E 2021; 104:015301. [PMID: 34412280 DOI: 10.1103/physreve.104.015301] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Accepted: 05/29/2021] [Indexed: 12/19/2022]
Abstract
We describe state-reduction algorithms for the analysis of first-passage processes in discrete- and continuous-time finite Markov chains. We present a formulation of the graph transformation algorithm that allows for the evaluation of exact mean first-passage times, stationary probabilities, and committor probabilities for all nonabsorbing nodes of a Markov chain in a single computation. Calculation of the committor probabilities within the state-reduction formalism is readily generalizable to the first hitting problem for any number of alternative target states. We then show that a state-reduction algorithm can be formulated to compute the expected number of times that each node is visited along a first-passage path. Hence, all properties required to analyze the first-passage path ensemble (FPPE) at both a microscopic and macroscopic level of detail, including the mean and variance of the first-passage time distribution, can be computed using state-reduction methods. In particular, we derive expressions for the probability that a node is visited along a direct transition path, which proceeds without returning to the initial state, considering both the nonequilibrium and equilibrium (steady-state) FPPEs. The reactive visitation probability provides a rigorous metric to quantify the dynamical importance of a node for the productive transition between two endpoint states and thus allows the local states that facilitate the dominant transition mechanisms to be readily identified. The state-reduction procedures remain numerically stable even for Markov chains exhibiting metastability, which can be severely ill-conditioned. The rare event regime is frequently encountered in realistic models of dynamical processes, and our methodology therefore provides valuable tools for the analysis of Markov chains in practical applications. We illustrate our approach with numerical results for a kinetic network representing a structural transition in an atomic cluster.
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Affiliation(s)
- Daniel J Sharpe
- Department of Chemistry, University of Cambridge, Lensfield Road, and Cambridge CB2 1EW, United Kingdom
| | - David J Wales
- Department of Chemistry, University of Cambridge, Lensfield Road, and Cambridge CB2 1EW, United Kingdom
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44
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Abstract
Protein-folding kinetics is often described as Markovian (i.e., memoryless) diffusion in a one-dimensional free energy landscape, governed by an instantaneous friction coefficient that is fitted to reproduce experimental or simulated folding times. For the α-helix forming polypeptide alanine9 and a specific reaction coordinate that consists of the summed native hydrogen-bond lengths, we demonstrate that the friction extracted from molecular dynamics simulations exhibits significant memory with a decay time that is in the nanosecond range and thus, of the same order as the folding and unfolding times. Our non-Markovian modeling not only reproduces the molecular dynamics simulations accurately but also demonstrates that memory friction effects lead to anomalous and drastically accelerated protein kinetics. We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide alanine9 for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote–Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects.
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45
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Xi K, Hu Z, Wu Q, Wei M, Qian R, Zhu L. Assessing the Performance of Traveling-salesman based Automated Path Searching (TAPS) on Complex Biomolecular Systems. J Chem Theory Comput 2021; 17:5301-5311. [PMID: 34270241 DOI: 10.1021/acs.jctc.1c00182] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
Though crucial for understanding the function of large biomolecular systems, locating the minimum free energy paths (MFEPs) between their key conformational states is far from trivial due to their high-dimensional nature. Most existing path-searching methods require a static collective variable space as input, encoding intuition or prior knowledge of the transition mechanism. Such information is, however, hardly available a priori and expensive to validate. To alleviate this issue, we have previously introduced a Traveling-salesman based Automated Path Searching method (TAPS) and demonstrated its efficiency on simple peptide systems. Having implemented a parallel version of this method, here we assess the performance of TAPS on three realistic systems (tens to hundreds of residues) in explicit solvents. We show that TAPS successfully located the MFEP for the ground/excited state transition of the T4 lysozyme L99A variant, consistent with previous findings. TAPS also helped identifying the important role of the two polar contacts in directing the loop-in/loop-out transition of the mitogen-activated protein kinase kinase (MEK1), which explained previous mutant experiments. Remarkably, at a minimal cost of 126 ns sampling, TAPS revealed that the Ltn40/Ltn10 transition of lymphotactin needs no complete unfolding/refolding of its β-sheets and that five polar contacts are sufficient to stabilize the various partially unfolded intermediates along the MFEP. These results present TAPS as a general and promising tool for studying the functional dynamics of complex biomolecular systems.
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Affiliation(s)
- Kun Xi
- Warshel Institute for Computational Biology, School of Life and Health Sciences, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong 518172, P. R. China.,School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
| | - Zhenquan Hu
- Warshel Institute for Computational Biology, School of Life and Health Sciences, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong 518172, P. R. China.,School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
| | - Qiang Wu
- School of Science and Engineering, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong 518172, P. R. China
| | - Meihan Wei
- Warshel Institute for Computational Biology, School of Life and Health Sciences, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong 518172, P. R. China
| | - Runtong Qian
- Warshel Institute for Computational Biology, School of Life and Health Sciences, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong 518172, P. R. China
| | - Lizhe Zhu
- Warshel Institute for Computational Biology, School of Life and Health Sciences, The Chinese University of Hong Kong (Shenzhen), Shenzhen, Guangdong 518172, P. R. China
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46
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Fernández Del Río B, Rey A. Behavior of Proteins under Pressure from Experimental Pressure-Dependent Structures. J Phys Chem B 2021; 125:6179-6191. [PMID: 34100621 PMCID: PMC8478274 DOI: 10.1021/acs.jpcb.1c03313] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Structure-based models are coarse-grained representations of the interactions responsible for the protein folding process. In their simplest form, they use only the native contact map of a given protein to predict the main features of its folding process by computer simulation. Given their limitations, these models are frequently complemented with sequence-dependent contributions or additional information. Specifically, to analyze the effect of pressure on the folding/unfolding transition, special forms of these interaction potentials are employed, which may a priori determine the outcome of the simulations. In this work, we have tried to keep the original simplicity of structure-based models. Therefore, we have used folded structures that have been experimentally determined at different pressures to define native contact maps and thus interactions dependent on pressure. Despite the apparently tiny structural differences induced by pressure, our simulation results provide different thermodynamic and kinetic behaviors, which roughly correspond to experimental observations (when there is a possible comparison) of two proteins used as benchmarks, hen egg-white lysozyme and dihydrofolate reductase. Therefore, this work shows the feasibility of using experimental native structures at different pressures to analyze the global effects of this physical property on the protein folding process.
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Affiliation(s)
- Beatriz Fernández Del Río
- Departamento de Química Física, Facultad de Ciencias Químicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain
| | - Antonio Rey
- Departamento de Química Física, Facultad de Ciencias Químicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain
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47
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Freitas FC, Fuchs G, de Oliveira RJ, Whitford PC. The dynamics of subunit rotation in a eukaryotic ribosome. BIOPHYSICA 2021; 1:204-221. [PMID: 37484008 PMCID: PMC10361705 DOI: 10.3390/biophysica1020016] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 07/25/2023]
Abstract
Protein synthesis by the ribosome is coordinated by an intricate series of large-scale conformational rearrangements. Structural studies can provide information about long-lived states, however biological kinetics are controlled by the intervening free-energy barriers. While there has been progress describing the energy landscapes of bacterial ribosomes, very little is known about the energetics of large-scale rearrangements in eukaryotic systems. To address this topic, we constructed an all-atom model with simplified energetics and performed simulations of subunit rotation in the yeast ribosome. In these simulations, the small subunit (SSU; ~1MDa) undergoes spontaneous and reversible rotations (~8°). By enabling the simulation of this rearrangement under equilibrium conditions, these calculations provide initial insights into the molecular factors that control dynamics in eukaryotic ribosomes. Through this, we are able to identify specific inter-subunit interactions that have a pronounced influence on the rate-limiting free-energy barrier. We also show that, as a result of changes in molecular flexibility, the thermodynamic balance between the rotated and unrotated states is temperature-dependent. This effect may be interpreted in terms of differential molecular flexibility within the rotated and unrotated states. Together, these calculations provide a foundation, upon which the field may begin to dissect the energetics of these complex molecular machines.
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Affiliation(s)
- Frederico Campos Freitas
- Laboratório de Biofísica Teórica, Departamento de Física, Instituto de Ciências Exatas, Naturais e Educação, Universidade Federal do Triângulo Mineiro, Uberaba, MG, Brazil
| | - Gabriele Fuchs
- Department of Biological Sciences, The RNA Institute, University at Albany 1400 Washington Ave, Albany, NY,12222
| | - Ronaldo Junio de Oliveira
- Laboratório de Biofísica Teórica, Departamento de Física, Instituto de Ciências Exatas, Naturais e Educação, Universidade Federal do Triângulo Mineiro, Uberaba, MG, Brazil
| | - Paul Charles Whitford
- Department of Physics, Northeastern University, 360 Huntington Ave, Boston, MA 02115
- Center for Theoretical Biological Physics, Northeastern University, 360 Huntington Ave, Boston, MA 02115
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Fraser A, Prokhorov NS, Jiao F, Pettitt BM, Scheuring S, Leiman PG. Quantitative description of a contractile macromolecular machine. SCIENCE ADVANCES 2021; 7:7/24/eabf9601. [PMID: 34117062 PMCID: PMC8195476 DOI: 10.1126/sciadv.abf9601] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2020] [Accepted: 04/23/2021] [Indexed: 05/14/2023]
Abstract
Contractile injection systems (CISs) [type VI secretion system (T6SS), phage tails, and tailocins] use a contractile sheath-rigid tube machinery to breach cell walls and lipid membranes. The structures of the pre- and postcontraction states of several CISs are known, but the mechanism of contraction remains poorly understood. Combining structural information of the end states of the 12-megadalton R-type pyocin sheath-tube complex with thermodynamic and force spectroscopy analyses and an original modeling procedure, we describe the mechanism of pyocin contraction. We show that this nanomachine has an activation energy of 160 kilocalories/mole (kcal/mol), and it releases 2160 kcal/mol of heat and develops a force greater than 500 piconewtons. Our combined approach provides a quantitative and experimental description of the membrane penetration process by a CIS.
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Affiliation(s)
- Alec Fraser
- Department of Biochemistry and Molecular Biology, Sealy Center for Structural Biology and Molecular Biophysics (SCSB), The University of Texas Medical Branch at Galveston, Galveston, TX 77555, USA
| | - Nikolai S Prokhorov
- Department of Biochemistry and Molecular Biology, Sealy Center for Structural Biology and Molecular Biophysics (SCSB), The University of Texas Medical Branch at Galveston, Galveston, TX 77555, USA
| | - Fang Jiao
- Department of Anesthesiology, Weill Cornell Medicine, 1300 York Avenue, New York, NY 10065, USA
- Department of Physiology and Biophysics, Weill Cornell Medicine, 1300 York Avenue, New York, NY 10065, USA
| | - B Montgomery Pettitt
- Department of Biochemistry and Molecular Biology, Sealy Center for Structural Biology and Molecular Biophysics (SCSB), The University of Texas Medical Branch at Galveston, Galveston, TX 77555, USA
| | - Simon Scheuring
- Department of Anesthesiology, Weill Cornell Medicine, 1300 York Avenue, New York, NY 10065, USA.
- Department of Physiology and Biophysics, Weill Cornell Medicine, 1300 York Avenue, New York, NY 10065, USA
| | - Petr G Leiman
- Department of Biochemistry and Molecular Biology, Sealy Center for Structural Biology and Molecular Biophysics (SCSB), The University of Texas Medical Branch at Galveston, Galveston, TX 77555, USA.
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Manuchehrfar F, Li H, Tian W, Ma A, Liang J. Exact Topology of the Dynamic Probability Surface of an Activated Process by Persistent Homology. J Phys Chem B 2021; 125:4667-4680. [PMID: 33938737 DOI: 10.1021/acs.jpcb.1c00904] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
To gain insight into the reaction mechanism of activated processes, we introduce an exact approach for quantifying the topology of high-dimensional probability surfaces of the underlying dynamic processes. Instead of Morse indexes, we study the homology groups of a sequence of superlevel sets of the probability surface over high-dimensional configuration spaces using persistent homology. For alanine-dipeptide isomerization, a prototype of activated processes, we identify locations of probability peaks and connecting ridges, along with measures of their global prominence. Instead of a saddle point, the transition state ensemble (TSE) of conformations is at the most prominent probability peak after reactants/products, when proper reaction coordinates are included. Intuition-based models, even those exhibiting a double-well, fail to capture the dynamics of the activated process. Peak occurrence, prominence, and locations can be distorted upon subspace projection. While principal component analysis accounts for conformational variance, it inflates the complexity of the surface topology and destroys the dynamic properties of the topological features. In contrast, TSE emerges naturally as the most prominent peak beyond the reactant/product basins, when projected to a subspace of minimum dimension containing the reaction coordinates. Our approach is general and can be applied to investigate the topology of high-dimensional probability surfaces of other activated processes.
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Affiliation(s)
- Farid Manuchehrfar
- Center for Bioinformatics and Quantiative Biology and Department of Bioengneering, University of Illinois at Chicago, Chicago, Illinois 60607, United States
| | - Huiyu Li
- Center for Bioinformatics and Quantiative Biology and Department of Bioengneering, University of Illinois at Chicago, Chicago, Illinois 60607, United States
| | - Wei Tian
- Center for Bioinformatics and Quantiative Biology and Department of Bioengneering, University of Illinois at Chicago, Chicago, Illinois 60607, United States
| | - Ao Ma
- Center for Bioinformatics and Quantiative Biology and Department of Bioengneering, University of Illinois at Chicago, Chicago, Illinois 60607, United States
| | - Jie Liang
- Center for Bioinformatics and Quantiative Biology and Department of Bioengneering, University of Illinois at Chicago, Chicago, Illinois 60607, United States
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Abstract
The ability to make sense of the massive amounts of high-dimensional data generated from molecular dynamics simulations is heavily dependent on the knowledge of a low-dimensional manifold (parameterized by a reaction coordinate or RC) that typically distinguishes between relevant metastable states, and which captures the relevant slow dynamics of interest. Methods based on machine learning and artificial intelligence have been proposed over the years to deal with learning such low-dimensional manifolds, but they are often criticized for a disconnect from more traditional and physically interpretable approaches. To deal with such concerns, in this work we propose a deep learning based state predictive information bottleneck approach to learn the RC from high-dimensional molecular simulation trajectories. We demonstrate analytically and numerically how the RC learnt in this approach is connected to the committor in chemical physics and can be used to accurately identify transition states. A crucial hyperparameter in this approach is the time delay or how far into the future the algorithm should make predictions about. Through careful comparisons for benchmark systems, we demonstrate that this hyperparameter choice gives useful control over how coarse-grained we want the metastable state classification of the system to be. We thus believe that this work represents a step forward in systematic application of deep learning based ideas to molecular simulations.
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Affiliation(s)
- Dedi Wang
- Biophysics Program and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
| | - Pratyush 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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