1
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Skew brownian motion and complexity of the alps algorithm. J Appl Probab 2022. [DOI: 10.1017/jpr.2021.78] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
Abstract
Simulated tempering is a popular method of allowing Markov chain Monte Carlo algorithms to move between modes of a multimodal target density
$\pi$
. Tawn, Moores and Roberts (2021) introduces the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that, under appropriate assumptions, the ALPS algorithm mixes in time
$O(d [\log d]^2)$
or O(d), depending on which version is used.
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2
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Syed S, Bouchard‐Côté A, Deligiannidis G, Doucet A. Non‐reversible parallel tempering: A scalable highly parallel MCMC scheme. J R Stat Soc Series B Stat Methodol 2021. [DOI: 10.1111/rssb.12464] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Affiliation(s)
- Saifuddin Syed
- University of British Columbia Vancouver British Columbia Canada
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3
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Blaber S, Sivak DA. Skewed thermodynamic geometry and optimal free energy estimation. J Chem Phys 2020; 153:244119. [PMID: 33380076 DOI: 10.1063/5.0033405] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023] Open
Abstract
Free energy differences are a central quantity of interest in physics, chemistry, and biology. We develop design principles that improve the precision and accuracy of free energy estimators, which have potential applications to screening for targeted drug discovery. Specifically, by exploiting the connection between the work statistics of time-reversed protocol pairs, we develop near-equilibrium approximations for moments of the excess work and analyze the dominant contributions to the precision and accuracy of standard nonequilibrium free-energy estimators. Within linear response, minimum-dissipation protocols follow the geodesics of the Riemannian metric induced by the Stokes friction tensor. We find that the next-order contribution arises from the rank-3 supra-Stokes tensor that skews the geometric structure such that minimum-dissipation protocols follow the geodesics of a generalized cubic Finsler metric. Thus, near equilibrium, the supra-Stokes tensor determines the leading-order contribution to the bias of bidirectional free-energy estimators.
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Affiliation(s)
- Steven Blaber
- Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
| | - David A Sivak
- Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
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4
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Schälte Y, Hasenauer J. Efficient exact inference for dynamical systems with noisy measurements using sequential approximate Bayesian computation. Bioinformatics 2020; 36:i551-i559. [PMID: 32657404 PMCID: PMC7355286 DOI: 10.1093/bioinformatics/btaa397] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Abstract
MOTIVATION Approximate Bayesian computation (ABC) is an increasingly popular method for likelihood-free parameter inference in systems biology and other fields of research, as it allows analyzing complex stochastic models. However, the introduced approximation error is often not clear. It has been shown that ABC actually gives exact inference under the implicit assumption of a measurement noise model. Noise being common in biological systems, it is intriguing to exploit this insight. But this is difficult in practice, as ABC is in general highly computationally demanding. Thus, the question we want to answer here is how to efficiently account for measurement noise in ABC. RESULTS We illustrate exemplarily how ABC yields erroneous parameter estimates when neglecting measurement noise. Then, we discuss practical ways of correctly including the measurement noise in the analysis. We present an efficient adaptive sequential importance sampling-based algorithm applicable to various model types and noise models. We test and compare it on several models, including ordinary and stochastic differential equations, Markov jump processes and stochastically interacting agents, and noise models including normal, Laplace and Poisson noise. We conclude that the proposed algorithm could improve the accuracy of parameter estimates for a broad spectrum of applications. AVAILABILITY AND IMPLEMENTATION The developed algorithms are made publicly available as part of the open-source python toolbox pyABC (https://github.com/icb-dcm/pyabc). SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Yannik Schälte
- Helmholtz Zentrum München, Institute of Computational Biology, Neuherberg 85764, Germany
- Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technical University Munich, Garching 85748, Germany
| | - Jan Hasenauer
- Helmholtz Zentrum München, Institute of Computational Biology, Neuherberg 85764, Germany
- Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technical University Munich, Garching 85748, Germany
- Research Unit Biomathematics, University of Bonn, Bonn 53113, Germany
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5
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Guimarães MN, M. de Almeida M, Marques JMC, Prudente FV. A thermodynamic view on the microsolvation of ions by rare gas: application to Li+ with argon. Phys Chem Chem Phys 2020; 22:10882-10892. [DOI: 10.1039/d0cp01283a] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Parallel tempering Monte Carlo calculations on the Li+Arn microsolvation clusters have shown that the two peaks appearing in the heat capacity curve as a function of temperature correspond to the melting of the second and first solvation shells.
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Affiliation(s)
- M. N. Guimarães
- Instituto de Física
- Universidade Federal da Bahia
- 40170-115 Salvador
- Brazil
| | - M. M. de Almeida
- Instituto de Física
- Universidade Federal da Bahia
- 40170-115 Salvador
- Brazil
| | - J. M. C. Marques
- CQC
- Department of Chemistry
- University of Coimbra
- 3004-535 Coimbra
- Portugal
| | - F. V. Prudente
- Instituto de Física
- Universidade Federal da Bahia
- 40170-115 Salvador
- Brazil
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6
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Rozada I, Aramon M, Machta J, Katzgraber HG. Effects of setting temperatures in the parallel tempering Monte Carlo algorithm. Phys Rev E 2019; 100:043311. [PMID: 31770965 DOI: 10.1103/physreve.100.043311] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2019] [Indexed: 11/07/2022]
Abstract
Parallel tempering Monte Carlo has proven to be an efficient method in optimization and sampling applications. Having an optimized temperature set enhances the efficiency of the algorithm through more-frequent replica visits to the temperature limits. The approaches for finding an optimal temperature set can be divided into two main categories. The methods of the first category distribute the replicas such that the swapping ratio between neighboring replicas is constant and independent of the temperature values. The second-category techniques including the feedback-optimized method, on the other hand, aim for a temperature distribution that has higher density at simulation bottlenecks, resulting in temperature-dependent replica-exchange probabilities. In this paper, we compare the performance of various temperature setting methods on both sparse and fully connected spin-glass problems as well as fully connected Wishart problems that have planted solutions. These include two classes of problems that have either continuous or discontinuous phase transitions in the order parameter. Our results demonstrate that there is no performance advantage for the methods that promote nonuniform swapping probabilities on spin-glass problems where the order parameter has a smooth transition between phases at the critical temperature. However, on Wishart problems that have a first-order phase transition at low temperatures, the feedback-optimized method exhibits a time-to-solution speedup of at least a factor of two over the other approaches.
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Affiliation(s)
- Ignacio Rozada
- 1QB Information Technologies (1QBit), Vancouver, BC, V6C 2B5, Canada
| | - Maliheh Aramon
- 1QB Information Technologies (1QBit), Vancouver, BC, V6C 2B5, Canada
| | - Jonathan Machta
- Physics Department, University of Massachusetts, Amherst, Massachusetts 01003, USA.,Santa Fe Institute, Santa Fe, New Mexico 87501, USA
| | - Helmut G Katzgraber
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA.,Microsoft Quantum, Microsoft, Redmond, Washington 98052, USA.,Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
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7
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Radak BK, Suh D, Roux B. A generalized linear response framework for expanded ensemble and replica exchange simulations. J Chem Phys 2018; 149:072315. [PMID: 30134700 PMCID: PMC5984729 DOI: 10.1063/1.5027494] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2018] [Accepted: 05/15/2018] [Indexed: 11/15/2022] Open
Abstract
Expanded ensemble simulation is a powerful technique for enhancing sampling over a range of thermodynamic parameters. However, although the premise is relatively simple, running successful simulations in practice still presents something of an ad hoc challenge. Three main difficulties exist: (1) the selection of the thermodynamic states, (2) the selection of the sampling weights, and (3) efficient sampling of the expanded parameter space. Here we consider these problems in the context of a pairwise linear response approach to the work fluctuation theorem. The approach offers comprehensive tactics for addressing the three difficulties and can be used as either an alternative or a complement to replica exchange simulations. Importantly, the results are trivially implemented for multi-dimensional parameter spaces and they recover results from the literature aimed at the special cases of simulated/parallel tempering and replica exchange umbrella sampling. Illustrative examples are shown using the NAMD simulation engine.
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Affiliation(s)
- Brian K Radak
- Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2325, USA
| | - Donghyuk Suh
- Department of Chemistry, University of Chicago, Chicago, Illinois 60637-1454, USA
| | - Benoît Roux
- Department of Biochemistry and Molecular Biology, University of Chicago, Chicago, Illinois 60637-1454, USA
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8
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Bonati C, D'Elia M. Topological critical slowing down: Variations on a toy model. Phys Rev E 2018; 98:013308. [PMID: 30110775 DOI: 10.1103/physreve.98.013308] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2018] [Indexed: 06/08/2023]
Abstract
Numerical simulations of lattice quantum field theories whose continuum counterparts possess classical solutions with nontrivial topology face a severe critical slowing down as the continuum limit is approached. Standard Monte Carlo algorithms develop a loss of ergodicity, with the system remaining frozen in configurations with fixed topology. We analyze the problem in a simple toy model, consisting of the path integral formulation of a quantum mechanical particle constrained to move on a circumference. More specifically, we implement for this toy model various techniques which have been proposed to solve or alleviate the problem for more complex systems, like non-Abelian gauge theories, and compare them both in the regime of low temperature and in that of very high temperature. Among the various techniques, we propose an alternative algorithm which completely solves the freezing problem, but unfortunately is specifically tailored for this particular model and not easily exportable to more complex systems.
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Affiliation(s)
- Claudio Bonati
- Dipartimento di Fisica e Astronomia dell'Università di Firenze and INFN-Sezione di Firenze, Via Sansone 1, 50019, Sesto Fiorentino (FI), Italy
| | - Massimo D'Elia
- Dipartimento di Fisica dell'Università di Pisa and INFN-Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy
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9
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Kumar M, Kumar R, Weigel M, Banerjee V, Janke W, Puri S. Approximate ground states of the random-field Potts model from graph cuts. Phys Rev E 2018; 97:053307. [PMID: 29906939 DOI: 10.1103/physreve.97.053307] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2018] [Indexed: 11/07/2022]
Abstract
While the ground-state problem for the random-field Ising model is polynomial, and can be solved using a number of well-known algorithms for maximum flow or graph cut, the analog random-field Potts model corresponds to a multiterminal flow problem that is known to be NP-hard. Hence an efficient exact algorithm is very unlikely to exist. As we show here, it is nevertheless possible to use an embedding of binary degrees of freedom into the Potts spins in combination with graph-cut methods to solve the corresponding ground-state problem approximately in polynomial time. We benchmark this heuristic algorithm using a set of quasiexact ground states found for small systems from long parallel tempering runs. For a not-too-large number q of Potts states, the method based on graph cuts finds the same solutions in a fraction of the time. We employ the new technique to analyze the breakup length of the random-field Potts model in two dimensions.
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Affiliation(s)
- Manoj Kumar
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India.,Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
| | - Ravinder Kumar
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, England.,Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, 04009 Leipzig, Germany.,Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (𝕃4)
| | - Martin Weigel
- Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, England
| | - Varsha Banerjee
- Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
| | - Wolfhard Janke
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, 04009 Leipzig, Germany
| | - Sanjay Puri
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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10
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MacCallum JL, Muniyat MI, Gaalswyk K. Online Optimization of Total Acceptance in Hamiltonian Replica Exchange Simulations. J Phys Chem B 2018; 122:5448-5457. [PMID: 29584433 DOI: 10.1021/acs.jpcb.7b11778] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Abstract
Replica exchange is a widely used sampling strategy in molecular simulation. While a variety of methods exist to optimize parameters for temperature replica exchange, less is known about how to optimize parameters for more general Hamiltonian replica exchange simulations. We present an algorithm for the online optimization of total acceptance for both temperature and Hamiltonian replica exchange simulations using stochastic gradient descent. We optimize the total acceptance, a heuristic objective function capturing the efficiency of replica exchange. Our approach is general and has several desirable properties, including: (1) it makes few assumptions about the system of interest, (2) optimization occurs online without the requirement of presimulation, and (3) most importantly, it readily generalizes to systems where there are multiple control parameters (e.g., temperatures, force constants, etc.) that determine the Hamiltonian of each replica. We explore some general properties of the algorithm on a simple harmonic oscillator system, and demonstrate its effectiveness on a more complex data-guided protein folding simulation.
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Affiliation(s)
- Justin L MacCallum
- Department of Chemistry , University of Calgary , Calgary AB T2N 1N4 , Canada
| | - Mir Ishruna Muniyat
- Department of Chemistry , University of Calgary , Calgary AB T2N 1N4 , Canada
| | - Kari Gaalswyk
- Department of Chemistry , University of Calgary , Calgary AB T2N 1N4 , Canada
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11
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Bores C, Auerbach SM, Monson PA. Enhanced replica exchange reactive Monte Carlo simulations for constructing zeolite frameworks. MOLECULAR SIMULATION 2017. [DOI: 10.1080/08927022.2017.1399375] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Affiliation(s)
- Cecilia Bores
- Department of Chemical Engineering, University of Massachusetts Amherst, Amherst, MA, USA
| | - Scott M. Auerbach
- Department of Chemical Engineering, University of Massachusetts Amherst, Amherst, MA, USA
- Department of Chemistry, University of Massachusetts Amherst, Amherst, MA, USA
| | - Peter A. Monson
- Department of Chemical Engineering, University of Massachusetts Amherst, Amherst, MA, USA
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12
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Olson MA. On the Helix Propensity in Generalized Born Solvent Descriptions of Modeling the Dark Proteome. Front Mol Biosci 2017; 4:3. [PMID: 28197405 PMCID: PMC5281587 DOI: 10.3389/fmolb.2017.00003] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2016] [Accepted: 01/12/2017] [Indexed: 01/06/2023] Open
Abstract
Intrinsically disordered proteins that populate the so-called “Dark Proteome” offer challenging benchmarks of atomistic simulation methods to accurately model conformational transitions on a multidimensional energy landscape. This work explores the application of parallel tempering with implicit solvent models as a computational framework to capture the conformational ensemble of an intrinsically disordered peptide derived from the Ebola virus protein VP35. A recent X-ray crystallographic study reported a protein-peptide interface where the VP35 peptide underwent a folding transition from a disordered form to a helix-β-turn-helix topological fold upon molecular association with the Ebola protein NP. An assessment is provided of the accuracy of two generalized Born solvent models (GBMV2 and GBSW2) using the CHARMM force field and applied with temperature-based replica exchange dynamics to calculate the disorder propensity of the peptide and its probability density of states in a continuum solvent. A further comparison is presented of applying an explicit/implicit solvent hybrid replica exchange simulation of the peptide to determine the effect of modeling water interactions at the all-atom resolution.
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Affiliation(s)
- Mark A Olson
- Department of Cell Biology and Biochemistry, Molecular and Translational Sciences Division, United States Army Medical Research Institute for Infectious Diseases Fredrick, MD, USA
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13
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Olson MA, Legler PM, Goldman ER. Comparison of Replica Exchange Simulations of a Kinetically Trapped Protein Conformational State and its Native Form. J Phys Chem B 2016; 120:2234-40. [PMID: 26886055 DOI: 10.1021/acs.jpcb.6b00233] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Recently an X-ray crystallographic structure of a single-domain antibody was reported with the protein chain trapped in a rare homodimeric form. One of the conformers appears to exhibit a misfolded region, and thus presumably the configurational stability is less favorable. To investigate whether simulation methods can detect any difference between the conformers and buttress the notion that one conformation is trapped on a pathway that incurs lower activation energy to unfold, adaptive temperature-based replica exchange simulations were applied to each chain to model conformational transitions. Simulation results found that the observed crystallographic difference between the two chains in the complementarity determining region CDR2 induces a stark distinction in conformational populations on the energy landscape. An appraisal of the energetic difference between the CDR2 conformations at 300 K revealed a localized order-disorder free-energy transition of roughly equivalent to two peptide hydrogen bonds in solution. It was also found that interconversion between the conformers is slower than the rate to unfold and that near an unfolding transition temperature one conformer retained a greater fraction of native-like contacts and energy over a longer time span before fully populating the denatured state, thus verifying the coexistence of a metastable conformation in the crystallographic assembly.
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Affiliation(s)
- Mark A Olson
- Department of Cell Biology and Biochemistry, Molecular and Translational Sciences Division, USAMRIID , Frederick, Maryland 21702, United States
| | - Patricia M Legler
- Center for Bio/Molecular Science and Engineering, Naval Research Laboratory , 4555 Overlook Ave SW, Washington, D.C. 20375, United States
| | - Ellen R Goldman
- Center for Bio/Molecular Science and Engineering, Naval Research Laboratory , 4555 Overlook Ave SW, Washington, D.C. 20375, United States
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14
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Curotto E. Ion-Stockmayer clusters: Minima, classical thermodynamics, and variational ground state estimates of Li+(CH3NO2)n(n= 1–20). J Chem Phys 2015; 143:214301. [DOI: 10.1063/1.4936587] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- E. Curotto
- Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania 19038-3295, USA
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15
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16
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Hédin F, Plattner N, Doll JD, Meuwly M. Spatial Averaging: Sampling Enhancement for Exploring Configurational Space of Atomic Clusters and Biomolecules. J Chem Theory Comput 2014; 10:4284-96. [DOI: 10.1021/ct500529w] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
Affiliation(s)
- Florent Hédin
- Department
of Chemistry, University of Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland
| | - Nuria Plattner
- Department
of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, D-14195 Berlin, Germany
| | - J. D. Doll
- Department
of Chemistry, Brown University, Providence, Rhode Island 02912, United States
| | - Markus Meuwly
- Department
of Chemistry, University of Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland
- Department
of Chemistry, Brown University, Providence, Rhode Island 02912, United States
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17
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Olson MA, Lee MS. Evaluation of unrestrained replica-exchange simulations using dynamic walkers in temperature space for protein structure refinement. PLoS One 2014; 9:e96638. [PMID: 24848767 PMCID: PMC4029997 DOI: 10.1371/journal.pone.0096638] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/06/2014] [Accepted: 04/09/2014] [Indexed: 01/09/2023] Open
Abstract
A central problem of computational structural biology is the refinement of modeled protein structures taken from either comparative modeling or knowledge-based methods. Simulations are commonly used to achieve higher resolution of the structures at the all-atom level, yet methodologies that consistently yield accurate results remain elusive. In this work, we provide an assessment of an adaptive temperature-based replica exchange simulation method where the temperature clients dynamically walk in temperature space to enrich their population and exchanges near steep energetic barriers. This approach is compared to earlier work of applying the conventional method of static temperature clients to refine a dataset of conformational decoys. Our results show that, while an adaptive method has many theoretical advantages over a static distribution of client temperatures, only limited improvement was gained from this strategy in excursions of the downhill refinement regime leading to an increase in the fraction of native contacts. To illustrate the sampling differences between the two simulation methods, energy landscapes are presented along with their temperature client profiles.
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Affiliation(s)
- Mark A. Olson
- Department of Cell Biology and Biochemistry, Molecular and Translational Sciences, USAMRIID, Fredrick, Maryland, United States of America
- Advanced Academic Programs, Zanvyl Krieger School of Arts and Sciences, Johns Hopkins University, Baltimore, Maryland, United States of America
| | - Michael S. Lee
- Computational Sciences Division, U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland, United States of America
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18
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Roberts GO, Rosenthal JS. Minimising MCMC variance via diffusion limits, with an application to simulated tempering. ANN APPL PROBAB 2014. [DOI: 10.1214/12-aap918] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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19
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20
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Jenkins R, Curotto E, Mella M. Replica exchange with Smart Monte Carlo and Hybrid Monte Carlo in manifolds. Chem Phys Lett 2013. [DOI: 10.1016/j.cplett.2013.10.053] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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21
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Sabri Dashti D, Roitberg AE. Optimization of Umbrella Sampling Replica Exchange Molecular Dynamics by Replica Positioning. J Chem Theory Comput 2013; 9:4692-9. [PMID: 26583388 DOI: 10.1021/ct400366h] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/03/2023]
Abstract
The positioning of sampling windows in an umbrella sampling simulation has an effect on the rate of convergence and computational efficiency. When such simulation is coupled with a Hamiltonian replica exchange setup, we show that such positioning can be optimized for maximal convergence of the results. We present a method for estimating the exchange acceptance ratio (EAR) between two arbitrary positions on a reaction coordinate in umbrella sampling replica exchange (USRE) molecular dynamics (MD). We designed a scoring function to optimize the position of the set of replicas (windows). By maximizing the scoring function, we make EAR the same for all neighbor replica pairs, increasing the efficiency of the method. We tested our algorithm by sampling a torsion for butane in implicit solvent and by studying a salt bridge in explicit solvent. We found that the optimized set of replicas recovers the correct free energy profile much faster than for equally spaced umbrellas.
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Affiliation(s)
- Danial Sabri Dashti
- Departments of Physics and ‡Chemistry and §Quantum Theory Project, University of Florida , Gainesville, Florida 32611-8435, United States
| | - Adrian E Roitberg
- Departments of Physics and ‡Chemistry and §Quantum Theory Project, University of Florida , Gainesville, Florida 32611-8435, United States
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22
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Malakis A, Papakonstantinou T. Comparative study of selected parallel tempering methods. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:013312. [PMID: 23944588 DOI: 10.1103/physreve.88.013312] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2013] [Revised: 05/21/2013] [Indexed: 06/02/2023]
Abstract
We review several parallel tempering schemes and examine their main ingredients for accuracy and efficiency. The present study covers two selection methods of temperatures and several choices for the exchange of replicas, including a recent novel all-pair exchange method. We compare the resulting schemes and measure specific heat errors and efficiency using the two-dimensional (2D) Ising model. Our tests suggest that an earlier proposal for using numbers of local moves related to the canonical correlation times is one of the key ingredients for increasing efficiency, and protocols using cluster algorithms are found to be very effective. Some of the protocols are also tested for efficiency and ground state production in 3D spin-glass models where we find that a simple nearest-neighbor approach using a local n-fold-way algorithm is the most effective. Finally, we present evidence that the asymptotic limits of the ground state energy for the isotropic case and for an anisotropic case of the 3D spin-glass model are very close and may even coincide.
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Affiliation(s)
- A Malakis
- Section of Solid State Physics, Department of Physics, University of Athens, Panepistimiopolis, GR 15784 Zografos, Athens, Greece
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23
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Doll JD, Plattner N, Freeman DL, Liu Y, Dupuis P. Rare-event sampling: occupation-based performance measures for parallel tempering and infinite swapping Monte Carlo methods. J Chem Phys 2013. [PMID: 23205986 DOI: 10.1063/1.4765060] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In the present paper we identify a rigorous property of a number of tempering-based Monte Carlo sampling methods, including parallel tempering as well as partial and infinite swapping. Based on this property we develop a variety of performance measures for such rare-event sampling methods that are broadly applicable, informative, and straightforward to implement. We illustrate the use of these performance measures with a series of applications involving the equilibrium properties of simple Lennard-Jones clusters, applications for which the performance levels of partial and infinite swapping approaches are found to be higher than those of conventional parallel tempering.
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Affiliation(s)
- J D Doll
- Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA
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24
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Roberts K, Sebsebie R, Curotto E. A rare event sampling method for diffusion Monte Carlo using smart darting. J Chem Phys 2012; 136:074104. [PMID: 22360233 DOI: 10.1063/1.3685453] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We identify a set of multidimensional potential energy surfaces sufficiently complex to cause both the classical parallel tempering and the guided or unguided diffusion Monte Carlo methods to converge too inefficiently for practical applications. The mathematical model is constructed as a linear combination of decoupled Double Wells [(DDW)(n)]. We show that the set (DDW)(n) provides a serious test for new methods aimed at addressing rare event sampling in stochastic simulations. Unlike the typical numerical tests used in these cases, the thermodynamics and the quantum dynamics for (DDW)(n) can be solved deterministically. We use the potential energy set (DDW)(n) to explore and identify methods that can enhance the diffusion Monte Carlo algorithm. We demonstrate that the smart darting method succeeds at reducing quasiergodicity for n ≫ 100 using just 1 × 10(6) moves in classical simulations (DDW)(n). Finally, we prove that smart darting, when incorporated into the regular or the guided diffusion Monte Carlo algorithm, drastically improves its convergence. The new method promises to significantly extend the range of systems computationally tractable by the diffusion Monte Carlo algorithm.
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Affiliation(s)
- K Roberts
- Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania 19038-3295, USA
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25
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Plattner N, Doll JD, Dupuis P, Wang H, Liu Y, Gubernatis JE. An infinite swapping approach to the rare-event sampling problem. J Chem Phys 2012; 135:134111. [PMID: 21992286 DOI: 10.1063/1.3643325] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We describe a new approach to the rare-event Monte Carlo sampling problem. This technique utilizes a symmetrization strategy to create probability distributions that are more highly connected and, thus, more easily sampled than their original, potentially sparse counterparts. After discussing the formal outline of the approach and devising techniques for its practical implementation, we illustrate the utility of the technique with a series of numerical applications to Lennard-Jones clusters of varying complexity and rare-event character.
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Affiliation(s)
- Nuria Plattner
- Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA
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26
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Lee MS, Olson MA. Comparison of two adaptive temperature-based replica exchange methods applied to a sharp phase transition of protein unfolding-folding. J Chem Phys 2011; 134:244111. [PMID: 21721616 DOI: 10.1063/1.3603964] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Temperature-based replica exchange (T-ReX) enhances sampling of molecular dynamics simulations by autonomously heating and cooling simulation clients via a Metropolis exchange criterion. A pathological case for T-ReX can occur when a change in state (e.g., folding to unfolding of a protein) has a large energetic difference over a short temperature interval leading to insufficient exchanges amongst replica clients near the transition temperature. One solution is to allow the temperature set to dynamically adapt in the temperature space, thereby enriching the population of clients near the transition temperature. In this work, we evaluated two approaches for adapting the temperature set: a method that equalizes exchange rates over all neighbor temperature pairs and a method that attempts to induce clients to visit all temperatures (dubbed "current maximization") by positioning many clients at or near the transition temperature. As a test case, we simulated the 57-residue SH3 domain of alpha-spectrin. Exchange rate equalization yielded the same unfolding-folding transition temperature as fixed-temperature ReX with much smoother convergence of this value. Surprisingly, the current maximization method yielded a significantly lower transition temperature, in close agreement with experimental observation, likely due to more extensive sampling of the transition state.
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Affiliation(s)
- Michael S Lee
- U S Army Research Laboratory, Computational Sciences and Engineering Branch, Aberdeen Proving Ground, Maryland 21005, USA.
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27
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Fiore CE. Comparing different protocols of temperature selection in the parallel tempering method. J Chem Phys 2011; 135:114107. [DOI: 10.1063/1.3636382] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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28
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Araque JC, Panagiotopoulos AZ, Robert MA. Lattice model of oligonucleotide hybridization in solution. I. Model and thermodynamics. J Chem Phys 2011; 134:165103. [PMID: 21528982 DOI: 10.1063/1.3568145] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023] Open
Abstract
A coarse-grained lattice model of DNA oligonucleotides is proposed to investigate the general mechanisms by which single-stranded oligonucleotides hybridize to their complementary strands in solution. The model, based on a high-coordination cubic lattice, is simple enough to allow the direct simulation of DNA solutions, yet capturing how the fundamental thermodynamic processes are microscopically encoded in the nucleobase sequences. Physically relevant interactions are considered explicitly, such as interchain excluded volume, anisotropic base-pairing and base-stacking, and single-stranded bending rigidity. The model is studied in detail by a specially adapted Monte Carlo simulation method, based on parallel tempering and biased trials, which is designed to overcome the entropic and enthalpic barriers associated with the sampling of hybridization events of multiple single-stranded chains in solution. This methodology addresses both the configurational complexity of bringing together two complementary strands in a favorable orientation (entropic barrier) and the energetic penalty of breaking apart multiple associated bases in a double-stranded state (enthalpic barrier). For strands with sequences restricted to nonstaggering association and homogeneous pairing and stacking energies, base-pairing is found to dominate the hybridization over the translational and conformational entropy. For strands with sequence-dependent pairing corresponding to that of DNA, the complex dependence of the model's thermal stability on concentration, sequence, and degree of complementarity is shown to be qualitatively and quantitatively consistent both with experiment and with the predictions of statistical mechanical models.
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Affiliation(s)
- Juan C Araque
- Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
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29
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Prakash MK, Barducci A, Parrinello M. Replica Temperatures for Uniform Exchange and Efficient Roundtrip Times in Explicit Solvent Parallel Tempering Simulations. J Chem Theory Comput 2011; 7:2025-7. [DOI: 10.1021/ct200208h] [Citation(s) in RCA: 57] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Meher K. Prakash
- Department of Chemistry and Applied Biosciences, ETH Zurich USI Campus Via Giuseppe Buffi 13, CH 6900 Lugano, Switzerland
| | - Alessandro Barducci
- Department of Chemistry and Applied Biosciences, ETH Zurich USI Campus Via Giuseppe Buffi 13, CH 6900 Lugano, Switzerland
| | - Michele Parrinello
- Department of Chemistry and Applied Biosciences, ETH Zurich USI Campus Via Giuseppe Buffi 13, CH 6900 Lugano, Switzerland
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30
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Fiore CE, da Luz MGE. Comparing parallel- and simulated-tempering-enhanced sampling algorithms at phase-transition regimes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:031104. [PMID: 21230022 DOI: 10.1103/physreve.82.031104] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2010] [Revised: 07/05/2010] [Indexed: 05/30/2023]
Abstract
Two important enhanced sampling algorithms, simulated (ST) and parallel (PT) tempering, are commonly used when ergodic simulations may be hard to achieve, e.g., due to a phase space separated by large free-energy barriers. This is so for systems around first-order phase transitions, a case still not fully explored with such approaches in the literature. In this contribution we make a comparative study between the PT and ST for the Ising (a lattice gas in the fluid language) and the Blume-Emery-Griffiths (a lattice gas with vacancies) models at phase-transition regimes. We show that although the two methods are equivalent in the limit of sufficiently long simulations, the PT is more advantageous than the ST with respect to all the analysis performed: convergence toward the stationarity; frequency of tunneling between phases at the coexistence; and decay of time-displaced correlation functions of thermodynamic quantities. Qualitative arguments for why one may expect better results from the PT than the ST near phase-transitions conditions are also presented.
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Affiliation(s)
- Carlos E Fiore
- Departamento de Física, Universidade Federal do Paraná, CP 19044, 81531-980 Curitiba, PR, Brazil.
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31
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Noon DP, Orkoulas G. Simulation of phase transitions via spatial updating and tempering. MOLECULAR SIMULATION 2010. [DOI: 10.1080/08927021003671574] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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32
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Spinney RE, Bowler DR, Gillan MJ. Spatially local parallel tempering for thermal-equilibrium sampling. J Chem Phys 2010; 132:104106. [PMID: 20232946 DOI: 10.1063/1.3334616] [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
Parallel tempering (PT) is a set of techniques for accelerating thermal-equilibrium sampling in systems where the exploration of configuration space is hindered by energy barriers. With standard PT algorithms, the computational effort scales unfavorably with system size, so that it is difficult to apply them to large systems. We propose local PT algorithms, for which the computational effort is proportional to the number of degrees of freedom. We demonstrate the effectiveness of the new algorithms on two one-dimensional model systems, showing that results for selected observables are correctly reproduced, and that practical linear scaling is achieved. We show also that the algorithms are readily applied to systems in higher dimensions. We note the prospects for studying large extended systems, including surfaces and interfaces.
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Affiliation(s)
- R E Spinney
- London Centre for Nanotechnology, UCL, Gordon Street, London WC1H 0AH, United Kingdom.
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33
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Lee A, Yau C, Giles MB, Doucet A, Holmes CC. On the Utility of Graphics Cards to Perform Massively Parallel Simulation of Advanced Monte Carlo Methods. J Comput Graph Stat 2010; 19:769-789. [DOI: 10.1198/jcgs.2010.10039] [Citation(s) in RCA: 175] [Impact Index Per Article: 12.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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34
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Shenfeld DK, Xu H, Eastwood MP, Dror RO, Shaw DE. Minimizing thermodynamic length to select intermediate states for free-energy calculations and replica-exchange simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:046705. [PMID: 19905480 DOI: 10.1103/physreve.80.046705] [Citation(s) in RCA: 57] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2009] [Revised: 08/27/2009] [Indexed: 05/28/2023]
Abstract
In computational thermodynamics, a sequence of intermediate states is commonly introduced to connect two equilibrium states. We consider two cases where the choice of intermediate states is particularly important: minimizing statistical error in free-energy difference calculations and maximizing average acceptance probabilities in replica-exchange simulations. We derive bounds for these quantities in terms of the thermodynamic distance between the intermediates, and show that in both cases the intermediates should be chosen as equidistant points along a geodesic connecting the end states.
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35
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Kim J, Keyes T, Straub JE. Replica exchange statistical temperature Monte Carlo. J Chem Phys 2009; 130:124112. [PMID: 19334813 DOI: 10.1063/1.3095422] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Abstract
The replica exchange statistical temperature Monte Carlo algorithm (RESTMC) is presented, extending the single-replica STMC algorithm [J. Kim, J. E. Straub, and T. Keyes, Phys. Rev. Lett. 97, 050601 (2006)] to alleviate the slow convergence of the conventional temperature replica exchange method (t-REM) with increasing system size. In contrast to the Gibbs-Boltzmann sampling at a specific temperature characteristic of the standard t-REM, RESTMC samples a range of temperatures in each replica and achieves a flat energy sampling employing the generalized sampling weight, which is automatically determined via the dynamic modification of the replica-dependent statistical temperature. Faster weight determination, through the dynamic update of the statistical temperature, and the flat energy sampling, maximizing energy overlaps between neighboring replicas, lead to a considerable acceleration in the convergence of simulations even while employing significantly fewer replicas. The performance of RESTMC is demonstrated and quantitatively compared with that of the conventional t-REM under varying simulation conditions for Lennard-Jones 19, 31, and 55 atomic clusters, exhibiting single- and double-funneled energy landscapes.
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Affiliation(s)
- Jaegil Kim
- Department of Chemistry, Boston University, Boston, Massachusetts 02215, USA.
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36
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Kamberaj H, van der Vaart A. An optimized replica exchange molecular dynamics method. J Chem Phys 2009; 130:074906. [PMID: 19239315 DOI: 10.1063/1.3077857] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We introduce a new way to perform swaps between replicas in replica exchange molecular dynamics simulations. The method is based on a generalized canonical probability distribution function and flattens the potential of mean force along the temperature coordinate, such that a random walk in temperature space is achieved. Application to a Go model of protein A showed that the method is more efficient than conventional replica exchange. The method results in a constant probability distribution of the replicas over the thermostats, yields a minimum round-trip time between extremum temperatures, and leads to faster ergodic convergence.
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Affiliation(s)
- Hiqmet Kamberaj
- Department of Chemistry and Biochemistry, Center for Biological Physics, Arizona State University, Tempe, Arizona 85287, USA
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37
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Kim J, Straub JE. Optimal replica exchange method combined with Tsallis weight sampling. J Chem Phys 2009; 130:144114. [PMID: 19368436 PMCID: PMC2736612 DOI: 10.1063/1.3108523] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2009] [Accepted: 03/08/2009] [Indexed: 02/03/2023] Open
Abstract
A unified framework integrating the generalized ensemble sampling associated with the Tsallis weight [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] and the replica exchange method (REM) has been proposed to accelerate the convergence of the conventional temperature REM (t-REM). Using the effective temperature formulation of the Tsallis weight sampling, it is shown that the average acceptance probability for configurational swaps between neighboring replicas in the combination of Tsallis weight sampling and REM (Tsallis-REM) is directly proportional to an overlap integral of the energy distributions of neighboring replicas as in the t-REM. Based on this observation, we suggest a robust method to select optimal Tsallis parameters in the conventional parametrization scheme and present new parametrization schemes for the Tsallis-REM, which significantly improves the acceptance of configurational swaps by systematically modulating energy overlaps between neighboring replicas. The distinguished feature of our method is that all relevant parameters in the Tsallis-REM are automatically determined from the equilibrium phase simulation using the t-REM. The overall performance of our method is explicitly demonstrated for various simulation conditions for the Lennard-Jones 31 atom clusters, exhibiting a double-funneled energy landscape.
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Affiliation(s)
- Jaegil Kim
- Department of Chemistry, Boston University, Boston, Massachusetts 02215, USA.
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38
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Woodard DB, Schmidler SC, Huber M. Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions. ANN APPL PROBAB 2009. [DOI: 10.1214/08-aap555] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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39
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Denschlag R, Lingenheil M, Tavan P. Optimal temperature ladders in replica exchange simulations. Chem Phys Lett 2009. [DOI: 10.1016/j.cplett.2009.03.053] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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40
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Fiore CE. First-order phase transitions: a study through the parallel tempering method. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:041109. [PMID: 18999381 DOI: 10.1103/physreve.78.041109] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2008] [Revised: 07/11/2008] [Indexed: 05/27/2023]
Abstract
We study the applicability of the parallel tempering (PT) method in the investigation of first-order phase transitions. In this method, replicas of the same system are simulated simultaneously at different temperatures, and the configurations of two randomly chosen replicas can occasionally be interchanged. We apply the PT method for the Blume-Emery-Griffiths model, which displays strong first-order transitions at low temperatures. A precise estimate of coexistence lines is obtained, revealing that the PT method may be a successful tool for the characterization of discontinuous transitions.
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Affiliation(s)
- Carlos E Fiore
- Departamento de Física, Universidade Federal do Paraná, Curitiba, Paraná, Brazil
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41
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Nadler W, Hansmann UHE. Optimized explicit-solvent replica exchange molecular dynamics from scratch. J Phys Chem B 2008; 112:10386-7. [PMID: 18671362 DOI: 10.1021/jp805085y] [Citation(s) in RCA: 53] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Replica exchange molecular dynamics (REMD) simulations have become an important tool to study proteins and other biological molecules in silico. However, such investigations require considerable, and often prohibitive, numerical effort when the molecules are simulated in explicit solvents. In this communication we show that in this case the cost can be minimized by choosing the number of replicas as N(opt) approximately 1+0.594 radical C ln(Tmax/Tmin), where C is the specific heat, and the temperatures distributed according to Ti(opt) approximately T min(Tmax/Tmin)(i-1)/(N-1).
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Affiliation(s)
- Walter Nadler
- John-von-Neumann Institute for Computing, Forschungszentrum Julich, D-52425 Julich, Germany.
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42
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Zheng W, Andrec M, Gallicchio E, Levy RM. Simple continuous and discrete models for simulating replica exchange simulations of protein folding. J Phys Chem B 2008; 112:6083-93. [PMID: 18251533 PMCID: PMC2978075 DOI: 10.1021/jp076377+] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The efficiency of temperature replica exchange (RE) simulations hinge on their ability to enhance conformational sampling at physiological temperatures by taking advantage of more rapid conformational interconversions at higher temperatures. While temperature RE is a parallel simulation technique that is relatively straightforward to implement, kinetics in the RE ensemble is complicated, and there is much to learn about how best to employ RE simulations in computational biophysics. Protein folding rates often slow down above a certain temperature due to entropic bottlenecks. This "anti-Arrhenius" behavior represents a challenge for RE. However, it is far from straightforward to systematically explore the impact of this on RE by brute force molecular simulations, since RE simulations of protein folding are very difficult to converge. To understand some of the basic mechanisms that determine the efficiency of RE, it is useful to study simplified low dimensionality systems that share some of the key characteristics of molecular systems. Results are presented concerning the efficiency of temperature RE on a continuous two-dimensional potential that contains an entropic bottleneck. Optimal efficiency was obtained when the temperatures of the replicas did not exceed the temperature at which the harmonic mean of the folding and unfolding rates is maximized. This confirms a result we previously obtained using a discrete network model of RE. Comparison of the efficiencies obtained using the continuous and discrete models makes it possible to identify non-Markovian effects, which slow down equilibration of the RE ensemble on the more complex continuous potential. In particular, the rate of temperature diffusion and also the efficiency of RE is limited by the time scale of conformational rearrangements within free energy basins.
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Affiliation(s)
- Weihua Zheng
- Department of Physics and Astronomy Rutgers, the State University of New Jersey, 136 Frelinghuysen Road, Piscataway NJ 08854, USA
| | - Michael Andrec
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology Rutgers, the State University of New Jersey, 610 Taylor Road, Piscataway NJ 08854, USA
| | - Emilio Gallicchio
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology Rutgers, the State University of New Jersey, 610 Taylor Road, Piscataway NJ 08854, USA
| | - Ronald M. Levy
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology Rutgers, the State University of New Jersey, 610 Taylor Road, Piscataway NJ 08854, USA
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43
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Sabo D, Meuwly M, Freeman DL, Doll JD. A constant entropy increase model for the selection of parallel tempering ensembles. J Chem Phys 2008; 128:174109. [DOI: 10.1063/1.2907846] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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44
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Hritz J, Oostenbrink C. Optimization of replica exchange molecular dynamics by fast mimicking. J Chem Phys 2008; 127:204104. [PMID: 18052416 DOI: 10.1063/1.2790427] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present an approach to mimic replica exchange molecular dynamics simulations (REMD) on a microsecond time scale within a few minutes rather than the years, which would be required for real REMD. The speed of mimicked REMD makes it a useful tool for "testing" the efficiency of different settings for REMD and then to select those settings, that give the highest efficiency. We present an optimization approach with the example of Hamiltonian REMD using soft-core interactions on two model systems, GTP and 8-Br-GTP. The optimization process using REMD mimicking is very fast. Optimization of Hamiltonian-REMD settings of GTP in explicit water took us less than one week. In our study we focus not only on finding the optimal distances between neighboring replicas, but also on finding the proper placement of the highest level of softness. In addition we suggest different REMD simulation settings at this softness level. We allow several replicas to be simulated at the same Hamiltonian simultaneously and reduce the frequency of switching attempts between them. This approach allows for more efficient conversions from one stable conformation to the other.
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Affiliation(s)
- Jozef Hritz
- Leiden Amsterdam Center for Drug Research (LACDR), Division of Molecular Toxicology, Vrije Universiteit, Amsterdam, The Netherlands
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45
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Nadler W, Hansmann UHE. Dynamics and optimal number of replicas in parallel tempering simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:065701. [PMID: 18233891 DOI: 10.1103/physreve.76.065701] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2007] [Indexed: 05/25/2023]
Abstract
We study the dynamics of parallel tempering simulations, also known as the replica exchange technique, which has become the method of choice for simulation of proteins and other complex systems. Recent results for the optimal choice of the control parameter discretization allow a treatment independent of the system in question. By analyzing mean first passage times across the control parameter space, we find an expression for the optimal number of replicas in simulations covering a given temperature range. Our results suggest a particular protocol to optimize the number of replicas in actual simulations.
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Affiliation(s)
- Walter Nadler
- Department of Physics, Michigan Technological University, Houghton, Michigan, USA.
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46
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Zheng W, Andrec M, Gallicchio E, Levy RM. Simulating replica exchange simulations of protein folding with a kinetic network model. Proc Natl Acad Sci U S A 2007; 104:15340-5. [PMID: 17878309 PMCID: PMC2000486 DOI: 10.1073/pnas.0704418104] [Citation(s) in RCA: 85] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2007] [Indexed: 11/18/2022] Open
Abstract
Replica exchange (RE) is a generalized ensemble simulation method for accelerating the exploration of free-energy landscapes, which define many challenging problems in computational biophysics, including protein folding and binding. Although temperature RE (T-RE) is a parallel simulation technique whose implementation is relatively straightforward, kinetics and the approach to equilibrium in the T-RE ensemble are very complicated; there is much to learn about how to best employ T-RE to protein folding and binding problems. We have constructed a kinetic network model for RE studies of protein folding and used this reduced model to carry out "simulations of simulations" to analyze how the underlying temperature dependence of the conformational kinetics and the basic parameters of RE (e.g., the number of replicas, the RE rate, and the temperature spacing) all interact to affect the number of folding transitions observed. When protein folding follows anti-Arrhenius kinetics, we observe a speed limit for the number of folding transitions observed at the low temperature of interest, which depends on the maximum of the harmonic mean of the folding and unfolding transition rates at high temperature. The results shown here for the network RE model suggest ways to improve atomic-level RE simulations such as the use of "training" simulations to explore some aspects of the temperature dependence for folding of the atomic-level models before performing RE studies.
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Affiliation(s)
- Weihua Zheng
- Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854; and
| | - Michael Andrec
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology, Rutgers, The State University of New Jersey, 610 Taylor Road, Piscataway, NJ 08854
| | - Emilio Gallicchio
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology, Rutgers, The State University of New Jersey, 610 Taylor Road, Piscataway, NJ 08854
| | - Ronald M. Levy
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology, Rutgers, The State University of New Jersey, 610 Taylor Road, Piscataway, NJ 08854
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47
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Sharapov VA, Mandelshtam VA. Solid−Solid Structural Transformations in Lennard-Jones Clusters: Accurate Simulations versus the Harmonic Superposition Approximation. J Phys Chem A 2007; 111:10284-91. [PMID: 17685597 DOI: 10.1021/jp072929c] [Citation(s) in RCA: 49] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We consider systems undergoing very-low-temperature solid-solid transitions associated with minima of similar energy but different symmetry, and separated by a high potential barrier. In such cases the well-known "broken-ergodicity" problem is often difficult to overcome, even using the most advanced Monte Carlo (MC) techniques, including the replica exchange method (REM). The methodology that we develop in this paper is suitable for the above specified cases and is numerically accurate and efficient. It is based on a new MC move implemented within the REM framework, in which trial points are generated analytically using an auxiliary harmonic superposition system that mimics well the true system at low temperatures. Due to the new move, the low-temperature random walks are able to frequently switch the relevant potential energy funnels leading to an efficient sampling. Numerically accurate results are obtained for a number of Lennard-Jones clusters, including those that have so far been treated only by the harmonic superposition approximation (HSA). The latter is believed to provide good estimates for low-temperature equilibrium properties but is manifestly uncontrollable and is difficult to validate. The present results provide a good test for the HSA and demonstrate its reliability, particularly for estimation of the solid-solid transition temperatures in most cases considered.
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Affiliation(s)
- Vladimir A Sharapov
- Chemistry Department, University of California at Irvine, Irvine, California 92697, USA
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48
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Predescu C. Entropic effects in large-scale Monte Carlo simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:016704. [PMID: 17677591 DOI: 10.1103/physreve.76.016704] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2007] [Indexed: 05/16/2023]
Abstract
The efficiency of Monte Carlo samplers is dictated not only by energetic effects, such as large barriers, but also by entropic effects that are due to the sheer volume that is sampled. The latter effects appear in the form of an entropic mismatch or divergence between the direct and reverse trial moves. We provide lower and upper bounds for the average acceptance probability in terms of the Rényi divergence of order 1/2 . We show that the asymptotic finitude of the entropic divergence is the necessary and sufficient condition for nonvanishing acceptance probabilities in the limit of large dimension. Furthermore, we demonstrate that the upper bound is reasonably tight by showing that the exponent is asymptotically exact for systems made up of a large number of independent and identically distributed subsystems. For the last statement, we provide an alternative proof that relies on the reformulation of the acceptance probability as a large deviation problem. The reformulation also leads to a class of low-variance estimators for strongly asymmetric distributions. We show that the entropy divergence causes a decay in the average displacements with the number of dimensions n that are simultaneously updated. For systems that have a well-defined thermodynamic limit, the decay is demonstrated to be n(-1/2) for random-walk Monte Carlo and n(-1/6) for smart Monte Carlo (SMC). Numerical simulations of the Lennard-Jones 38 (LJ(38)) cluster show that SMC is virtually as efficient as the Markov chain implementation of the Gibbs sampler, which is normally utilized for Lennard-Jones clusters. An application of the entropic inequalities to the parallel tempering method demonstrates that the number of replicas increases as the square root of the heat capacity of the system.
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Affiliation(s)
- Cristian Predescu
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA.
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Sharapov VA, Meluzzi D, Mandelshtam VA. Low-temperature structural transitions: circumventing the broken-ergodicity problem. PHYSICAL REVIEW LETTERS 2007; 98:105701. [PMID: 17358547 DOI: 10.1103/physrevlett.98.105701] [Citation(s) in RCA: 59] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2006] [Indexed: 05/14/2023]
Abstract
We consider systems undergoing very-low-temperature solid-solid transitions, exhibiting the well-known "broken-ergodicity" problem that is often so severe that even the replica exchange method converges too slowly. We propose an improvement of the latter, which consists of coupling the lower-temperature random walks to analytically generated random walks corresponding to an auxiliary harmonic superposition system. Numerically accurate results are obtained for several Lennard-Jones clusters, which have so far been treated only by the harmonic superposition approximation.
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Affiliation(s)
- Vladimir A Sharapov
- Chemistry Department, University of California at Irvine, Irvine, California 92697, USA
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Nadler W, Hansmann UHE. Generalized ensemble and tempering simulations: a unified view. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:026109. [PMID: 17358396 DOI: 10.1103/physreve.75.026109] [Citation(s) in RCA: 55] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2006] [Revised: 12/15/2006] [Indexed: 05/14/2023]
Abstract
From the underlying master equations we derive one-dimensional stochastic processes that describe generalized ensemble simulations as well as tempering (simulated and parallel) simulations. The representations obtained are either in the form of a one-dimensional Fokker-Planck equation or a hopping process on a one-dimensional chain. In particular, we discuss the conditions under which these representations are valid approximate Markovian descriptions of the random walk in order parameter or control parameter space. They allow a unified discussion of the stationary distribution on, as well as of the stationary flow across, each space. We demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space and discuss the consequences of our results for optimizing simulations. Finally, we point out the limitations of these representations under conditions of broken ergodicity.
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Affiliation(s)
- Walter Nadler
- Department of Physics, Michigan Technological University, 1400 Townsend Drive, Houghton, Michigan 49931-1295, USA.
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