1
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Berezhkovskii AM, Bezrukov SM. Solute translocation probability, lifetime, and "rectification" in membrane channels with localized constriction. Phys Chem Chem Phys 2024; 26:15758-15764. [PMID: 38770832 DOI: 10.1039/d4cp00689e] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/22/2024]
Abstract
We study the translocation probability and lifetime of a solute molecule in a cylindrical membrane channel that contains a localized constriction at an arbitrary location. Using a one-dimensional continuous diffusion description of solute dynamics in the channel, we explore two models. The first one describes a molecule's interaction with the constriction in terms of a narrow rectangular barrier in the potential of mean force. The second novel model proposed here represents this interaction by introducing an infinitely thin permeable partition. It is shown that when the parameters of the two models are chosen to warrant the same translocation probability, both models predict the same mean lifetime of the molecule in the channel. While the translocation probability is independent of the constriction location, the mean lifetime is a function of the location. The benefit of the thin partition model is that it allows one to lump together the height and length of the potential barrier into a single parameter, which is the partition's permeability. It is shown that in the case of an asymmetric location of the localized constriction and strong repulsion between the solutes, the solute flux through the channel is a function of the direction in which it goes, analogous to the phenomenon known in ion channel electrophysiology as rectification.
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Affiliation(s)
- Alexander M Berezhkovskii
- Section on Molecular Transport, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, Maryland 20892, USA.
| | - Sergey M Bezrukov
- Section on Molecular Transport, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, Maryland 20892, USA.
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2
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Lorpaiboon C, Guo SC, Strahan J, Weare J, Dinner AR. Accurate estimates of dynamical statistics using memory. J Chem Phys 2024; 160:084108. [PMID: 38391020 PMCID: PMC10898919 DOI: 10.1063/5.0187145] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2023] [Accepted: 01/29/2024] [Indexed: 02/24/2024] Open
Abstract
Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasi-Markov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a two-dimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.
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Affiliation(s)
- Chatipat Lorpaiboon
- Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
| | - Spencer C. Guo
- Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
| | - John Strahan
- Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
| | - Jonathan Weare
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
| | - Aaron R. Dinner
- Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
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3
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Pravatto P, Fresch B, Moro GJ. Large barrier behavior of the rate constant from the diffusion equation. J Chem Phys 2023; 158:144110. [PMID: 37061487 DOI: 10.1063/5.0143522] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/17/2023] Open
Abstract
Many processes in chemistry, physics, and biology depend on thermally activated events in which the system changes its state by surmounting an activation barrier. Examples range from chemical reactions to protein folding and nucleation events. Parameterized forms of the mean field potential are often employed in the stochastic modeling of activated processes. In this contribution, we explore the alternative of employing parameterized forms of the equilibrium distribution by means of symmetric linear combination of two Gaussian functions. Such a procedure leads to flexible and convenient models for the landscape and the energy barrier whose features are controlled by the second moments of these Gaussian functions. The rate constants are examined through the solution of the corresponding diffusion problem, that is, the Fokker-Planck-Smoluchowski equation specified according to the parameterized equilibrium distribution. Numerical calculations clearly show that the asymptotic limit of large barriers does not agree with the results of the Kramers theory. The underlying reason is that the linear scaling of the potential, the procedure justifying the Kramers theory, cannot be applied when dealing with parameterized forms of the equilibrium distribution. A different kind of asymptotic analysis is then required and we introduce the appropriate theory when the equilibrium distribution is represented as a symmetric linear combination of two Gaussian functions: first in the one-dimensional case and afterward in the multidimensional diffusion model.
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Affiliation(s)
- Pierpaolo Pravatto
- Dipartimento di Scienze Chimiche, Università di Padova, Via Marzolo 1, 35131 Padova, Italy
| | - Barbara Fresch
- Dipartimento di Scienze Chimiche, Università di Padova, Via Marzolo 1, 35131 Padova, Italy
| | - Giorgio J Moro
- Dipartimento di Scienze Chimiche, Università di Padova, Via Marzolo 1, 35131 Padova, Italy
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4
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Wen Y, Li Z, Wang H, Zheng J, Tang J, Lai PY, Xu X, Tong P. Activity-assisted barrier crossing of self-propelled colloids over parallel microgrooves. Phys Rev E 2023; 107:L032601. [PMID: 37072954 DOI: 10.1103/physreve.107.l032601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2022] [Accepted: 03/01/2023] [Indexed: 04/20/2023]
Abstract
We report a systematic study of the dynamics of self-propelled particles (SPPs) over a one-dimensional periodic potential landscape U_{0}(x), which is fabricated on a microgroove-patterned polydimethylsiloxane (PDMS) substrate. From the measured nonequilibrium probability density function P(x;F_{0}) of the SPPs, we find that the escape dynamics of the slow rotating SPPs across the potential landscape can be described by an effective potential U_{eff}(x;F_{0}), once the self-propulsion force F_{0} is included into the potential under the fixed angle approximation. This work demonstrates that the parallel microgrooves provide a versatile platform for a quantitative understanding of the interplay among the self-propulsion force F_{0}, spatial confinement by U_{0}(x), and thermal noise, as well as its effects on activity-assisted escape dynamics and transport of the SPPs.
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Affiliation(s)
- Yan Wen
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
| | - Zhihao Li
- Department of Physics, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou, Guangdong 515063, China
| | - Haiqin Wang
- Department of Physics, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou, Guangdong 515063, China
- Technion-Israel Institute of Technology, Haifa 3200003, Israel
| | - Jing Zheng
- Department of Chemistry, The University of Hong Kong, Pokfulam, Hong Kong
| | - Jinyao Tang
- Department of Chemistry, The University of Hong Kong, Pokfulam, Hong Kong
| | - Pik-Yin Lai
- Department of Physics and Center for Complex Systems, National Central University, Taoyuan City 320, Taiwan
- Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
| | - Xinpeng Xu
- Department of Physics, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou, Guangdong 515063, China
- Technion-Israel Institute of Technology, Haifa 3200003, Israel
| | - Penger Tong
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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5
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Doi M, Xu X. Diffusion Limited Escape Rate of a Complex Molecule in Multi-dimensional Confinement. J Phys Chem B 2022; 126:6171-6179. [PMID: 35948046 DOI: 10.1021/acs.jpcb.2c03976] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The rate at which a Brownian particle confined in a closed space escapes from the space by passing through a narrow passage is called the escape rate. The escape rate is relevant to many diffusion limited processes in polymer and colloidal systems, such as colloidal aggregation, polymerization reaction, polymer translocation through a membrane, etc. Here, we propose a variational principle to calculate the escape rate of complex molecules doing Brownian motion in a multi-dimensional phase space. We propose a regional minimization method in which we divide the whole phase space into regions, conduct the minimization for each region, and combine the results to get the minimum in the entire space. As an example, we discuss (1) the escape rate of a point particle that escapes from a confinement passing through a long corridor and (2) the escape rate of a rod-like particle that escapes through a small hole made in the wall of the confinement.
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Affiliation(s)
- Masao Doi
- Wenzhou Institute, University of Chinese Academy of Science,Wenzhou, Zhejiang 325000, China.,Oujiang Laboratory, Longwan District, Wenzhou, Zhejiang 325001, China
| | - Xianmin Xu
- LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
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6
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Berezhkovskii AM, Bezrukov SM. Intrinsic diffusion resistance of a membrane channel, mean first-passage times between its ends, and equilibrium unidirectional fluxes. J Chem Phys 2022; 156:071103. [DOI: 10.1063/5.0082482] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Alexander M. Berezhkovskii
- Section on Molecular Transport, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, Maryland 20892, USA
| | - Sergey M. Bezrukov
- Section on Molecular Transport, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, Maryland 20892, USA
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7
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Schleeh MM, Reiff J, García-Müller PL, Benito RM, Borondo F, Main J, Hernandez R. Mean first-passage times for solvated LiCN isomerization at intermediate to high temperatures. J Chem Phys 2022; 156:034103. [DOI: 10.1063/5.0065090] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Micha M. Schleeh
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Johannes Reiff
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Pablo L. García-Müller
- Departamento de Tecnología, Centro de Investigaciones Energéticas Medioambientales y Tecnológicas, Avda. Complutense 40, 28040 Madrid, Spain
| | - Rosa M. Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - Florentino Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
| | - Jörg Main
- Institut für Theoretische Physik I, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Rigoberto Hernandez
- Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA
- Departments of Chemical and Biomolecular Engineering and Materials Science and Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
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8
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Cherayil BJ. Particle dynamics in viscoelastic media: Effects of non-thermal white noise on barrier crossing rates. J Chem Phys 2021; 155:244903. [PMID: 34972363 DOI: 10.1063/5.0071206] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/07/2023] Open
Abstract
The growing interest in the dynamics of self-driven particle motion has brought increased attention to the effects of non-thermal noise on condensed phase diffusion. Thanks to data recently collected by Ferrer et al. on activated dynamics in the presence of memory [Phys. Rev. Lett. 126, 108001 (2021)], some of these effects can now be characterized quantitatively. In the present paper, the data collected by Ferrer et al. are used to calculate the extent to which non-thermal white noise alters the time taken by single micron-sized silica particles in a viscoelastic medium to cross the barrier separating the two wells of an optically created bistable potential. The calculation-based on a generalized version of Kramers's flux-over-population approach-indicates that the added noise causes the barrier crossing rate (compared to the noise-free case) to first increase as a function of the noise strength and then to plateau to a constant value. The precise degree of rate enhancement may depend on how the data from the experiments conducted by Ferrer et al. are used in the flux-over-population approach. As claimed by Ferrer et al., this approach predicts barrier crossing times for the original silica-fluid system that agree almost perfectly with their experimental counterparts. However, this near-perfect agreement between theory and experiment is only achieved if the theoretical crossing times are obtained from the most probable values of a crossing time distribution constructed from the distributions of various parameters in Kramers's rate expression. If the mean values of these parameters are used in the expression instead, as would be commonly done, the theoretical crossing times are found to be as much as 1.5 times higher than the experimental values. However, these times turn out to be consistent with an alternative model of viscoelastic barrier crossing based on a mean first passage time formalism, which also uses mean parameter values in its rate expression. The rate enhancements predicted for barrier crossing under non-thermal noise are based on these mean parameter values and are open to experimental verification.
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Affiliation(s)
- Binny J Cherayil
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, Karnataka, India
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9
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Chauhan K, Kumar S. Dynamics of a polymer chain translocating through varying cone-shaped channels. Phys Rev E 2021; 103:042501. [PMID: 34005922 DOI: 10.1103/physreve.103.042501] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2020] [Accepted: 03/04/2021] [Indexed: 11/07/2022]
Abstract
By employing the exact enumeration technique, we study consequences of different apex angles of a wedge-shaped channel on the mean first passage time and free-energy profile of a linear polymer chain translocating from the cis- to the trans-side through an interacting pore. We investigate effects of asymmetry arising in the free-energy profile due to the change in apex angles and its dependence on the first passage time. We report the combined effect of entropy (arising due to apex angles) and pore interaction on the nonmonotonic behavior of the translocation time. The effect of different solvent quality across the channel has also been explored. We show that the increase in monomer-monomer interaction leads to the formation of globules near the pore, which drives the process faster.
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Affiliation(s)
- Keerti Chauhan
- Department of Physics, Banaras Hindu University, Varanasi 221005, India
| | - Sanjay Kumar
- Department of Physics, Banaras Hindu University, Varanasi 221005, India
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10
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Das M, Kantz H. Role of thermal fluctuations in biological copying mechanisms. Phys Rev E 2021; 103:032110. [PMID: 33862776 DOI: 10.1103/physreve.103.032110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2020] [Accepted: 02/18/2021] [Indexed: 11/07/2022]
Abstract
During transcription, translation, or self-replication of DNA or RNA, information is transferred to the newly formed species from its predecessor. These processes can be interpreted as (generalized) biological copying mechanism as the new biological entities like DNA, RNA, or proteins are representing the information of their parent bodies uniquely. The accuracy of these copying processes is essential, since errors in the copied code can reduce the functionality of the next generation. Such errors might result from perturbations on these processes. Most important in this context is the temperature of the medium, i.e., thermal noise. Although a reasonable amount of experimental studies have been carried out on this important issue, theoretical understanding is truly sparse. In the present work, we illustrate a model study which is able to focus on the effect of the temperature on the process of biological copying mechanisms, as well as on mutation. We find for our paradigmatic models that, in a quite general scenario, the copying processes are most accurate at an intermediate temperature range; i.e., there exists an optimum temperature where mutation is most unlikely. This allows us to interpret the observations for some biological species with the aid of our model study.
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Affiliation(s)
- Moupriya Das
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| | - Holger Kantz
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
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11
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Moritz C, Sega M, Innerbichler M, Geissler PL, Dellago C. Weak scaling of the contact distance between two fluctuating interfaces with system size. Phys Rev E 2020; 102:062801. [PMID: 33465946 DOI: 10.1103/physreve.102.062801] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2020] [Accepted: 10/30/2020] [Indexed: 06/12/2023]
Abstract
A pair of flat parallel surfaces, each freely diffusing along the direction of their separation, will eventually come into contact. If the shapes of these surfaces also fluctuate, then contact will occur when their centers-of-mass remain separated by a nonzero distance ℓ. An example of such a situation is the motion of interfaces between two phases at conditions of thermodynamic coexistence, and in particular the annihilation of domain wall pairs under periodic boundary conditions. Here we present a general approach to calculate the probability distribution of the contact distance ℓ and determine how its most likely value ℓ^{*} depends on the surfaces' lateral size L. Using the Edward-Wilkinson equation as a model for interfaces, we demonstrate that ℓ^{*} scales weakly with system size, i.e., the dependence of ℓ^{*} on L for both (1+1)- and (2+1)-dimensional interfaces is such that lim_{L→∞}(ℓ^{*}/L)=0. In particular, for (2+1)-dimensional interfaces ℓ^{*} is an algebraic function of logL, a result that is confirmed by computer simulations of slab-shaped domains formed under periodic boundary conditions. This weak scaling implies that such domains remain topologically intact until ℓ becomes very small compared to the lateral size of the interface, contradicting expectations from equilibrium thermodynamics.
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Affiliation(s)
- Clemens Moritz
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
| | - Marcello Sega
- Forschungszentrum Jülich GmbH, Helmholtz Institute Erlangen-Nürnberg for Renewable Energy (IEK-11), Fürther Straße 248, 90429 Nürnberg, Germany
| | - Max Innerbichler
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
| | - Phillip L Geissler
- Department of Chemistry, University of California, Berkeley, California 94720, USA
| | - Christoph Dellago
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
- Erwin Schrödinger Institute for Mathematics and Physics, Boltzmanngasse 9, 1090, Vienna, Austria
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12
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Bel G, Zilman A, Kolomeisky AB. Different time scales in dynamic systems with multiple outcomes. J Chem Phys 2020; 153:054107. [PMID: 32770919 DOI: 10.1063/5.0018558] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Stochastic biochemical and transport processes have various final outcomes, and they can be viewed as dynamic systems with multiple exits. Many current theoretical studies, however, typically consider only a single time scale for each specific outcome, effectively corresponding to a single-exit process and assuming the independence of each exit process. However, the presence of other exits influences the statistical properties and dynamics measured at any specific exit. Here, we present theoretical arguments to explicitly show the existence of different time scales, such as mean exit times and inverse exit fluxes, for dynamic processes with multiple exits. This implies that the statistics of any specific exit dynamics cannot be considered without taking into account the presence of other exits. Several illustrative examples are described in detail using analytical calculations, mean-field estimates, and kinetic Monte Carlo computer simulations. The underlying microscopic mechanisms for the existence of different time scales are discussed. The results are relevant for understanding the mechanisms of various biological, chemical, and industrial processes, including transport through channels and pores.
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Affiliation(s)
- G Bel
- Department of Solar Energy and Environmental Physics, BIDR, and Department of Physics, Ben-Gurion University of the Negev, Sede Boqer Campus 8499000, Israel
| | - A Zilman
- Department of Physics and Institute for Biomaterials and Bioengineering, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
| | - A B Kolomeisky
- Department of Chemistry and Center for Theoretical Biological Physics, Department of Chemical and Biomolecular Engineering, Department of Physics and Astronomy, Rice University, Houston, Texas 77005-1892, USA
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13
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Sigg D, Voelz VA, Carnevale V. Microcanonical coarse-graining of the kinetic Ising model. J Chem Phys 2020; 152:084104. [PMID: 32113343 PMCID: PMC7042020 DOI: 10.1063/1.5139228] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2019] [Accepted: 02/06/2020] [Indexed: 11/15/2022] Open
Abstract
We propose a scheme for coarse-graining the dynamics of the 2-D kinetic Ising model onto the microcanonical ensemble. At subcritical temperatures, 2-D and higher-dimensional Ising lattices possess two basins of attraction separated by a free energy barrier. Projecting onto the microcanonical ensemble has the advantage that the dependence of the crossing rate constant on environmental conditions can be obtained from a single Monte Carlo trajectory. Using various numerical methods, we computed the forward rate constants of coarse-grained representations of the Ising model and compared them with the true value obtained from brute force simulation. While coarse-graining preserves detailed balance, the computed rate constants for barrier heights between 5 kT and 9 kT were consistently 50% larger than the true value. Markovianity testing revealed loss of dynamical memory, which we propose accounts for coarse-graining error. Committor analysis did not support the alternative hypothesis that microcanonical projection is incompatible with an optimal reaction coordinate. The correct crossing rate constant was obtained by spectrally decomposing the diffusion coefficient near the free energy barrier and selecting the slowest (reactive) component. The spectral method also yielded the correct rate constant in the 3-D Ising lattice, where coarse-graining error was 6% and memory effects were diminished. We conclude that microcanonical coarse-graining supplemented by spectral analysis of short-term barrier fluctuations provides a comprehensive kinetic description of barrier crossing in a non-inertial continuous-time jump process.
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Affiliation(s)
| | - Vincent A. Voelz
- Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122, USA
| | - Vincenzo Carnevale
- Institute for Computational Molecular Science, College of Science and Technology, Temple University, Philadelphia, Pennsylvania 19122, USA
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14
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Wei W, Elber R. ScMile: A Script to Investigate Kinetics with Short Time Molecular Dynamics Trajectories and the Milestoning Theory. J Chem Theory Comput 2020; 16:860-874. [PMID: 31922745 PMCID: PMC7031965 DOI: 10.1021/acs.jctc.9b01030] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
Abstract
Studies of complex and rare events in condensed phase systems continue to attract considerable attention. Milestoning is a useful theory and algorithm to investigate the long-time dynamics of activated molecular events. It is based on launching a large number of short trajectories and statistical analysis of the outcome. The implementation of the theory in a computer script is described that enables more efficient Milestoning calculation, reducing user time and errors, and automating a significant fraction of the algorithm. The script exploits a molecular dynamics engine, which at present is NAMD, to run the short trajectories. However, since the script is external to the engine, the script can be easily adapted to different molecular dynamics codes. The outcomes of the short trajectories are analyzed to obtain a kinetic and thermodynamic description of the entire process. While many examples of Milestoning were published in the past, we provide two simple examples (a conformational transition of alanine dipeptide in a vacuum and aqueous solution) to illustrate the use of the script.
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Affiliation(s)
- Wei Wei
- Oden Institute for Computational Engineering and Sciences , University of Texas at Austin , Austin , Texas 78712 , United States
| | - Ron Elber
- Oden Institute for Computational Engineering and Sciences , University of Texas at Austin , Austin , Texas 78712 , United States
- Department of Chemistry , The University of Texas at Austin , Austin , Texas 78712 , United States
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15
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Dahlke K, Zhao J, Sing CE, Banigan EJ. Force-Dependent Facilitated Dissociation Can Generate Protein-DNA Catch Bonds. Biophys J 2019; 117:1085-1100. [PMID: 31427067 DOI: 10.1016/j.bpj.2019.07.044] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2019] [Revised: 07/08/2019] [Accepted: 07/29/2019] [Indexed: 12/31/2022] Open
Abstract
Cellular structures are continually subjected to forces, which may serve as mechanical signals for cells through their effects on biomolecule interaction kinetics. Typically, molecular complexes interact via "slip bonds," so applied forces accelerate off rates by reducing transition energy barriers. However, biomolecules with multiple dissociation pathways may have considerably more complicated force dependencies. This is the case for DNA-binding proteins that undergo "facilitated dissociation," in which competitor biomolecules from solution enhance molecular dissociation in a concentration-dependent manner. Using simulations and theory, we develop a generic model that shows that proteins undergoing facilitated dissociation can form an alternative type of molecular bond, known as a "catch bond," for which applied forces suppress protein dissociation. This occurs because the binding by protein competitors responsible for the facilitated dissociation pathway can be inhibited by applied forces. Within the model, we explore how the force dependence of dissociation is regulated by intrinsic factors, including molecular sensitivity to force and binding geometry and the extrinsic factor of competitor protein concentration. We find that catch bonds generically emerge when the force dependence of the facilitated unbinding pathway is stronger than that of the spontaneous unbinding pathway. The sharpness of the transition between slip- and catch-bond kinetics depends on the degree to which the protein bends its DNA substrate. This force-dependent kinetics is broadly regulated by the concentration of competitor biomolecules in solution. Thus, the observed catch bond is mechanistically distinct from other known physiological catch bonds because it requires an extrinsic factor-competitor proteins-rather than a specific intrinsic molecular structure. We hypothesize that this mechanism for regulating force-dependent protein dissociation may be used by cells to modulate protein exchange, regulate transcription, and facilitate diffusive search processes.
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Affiliation(s)
- Katelyn Dahlke
- Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois
| | - Jing Zhao
- Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois
| | - Charles E Sing
- Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois.
| | - Edward J Banigan
- Institute for Medical Engineering and Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts.
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16
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Kappler J, Hinrichsen VB, Netz RR. Non-Markovian barrier crossing with two-time-scale memory is dominated by the faster memory component. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2019; 42:119. [PMID: 31494784 DOI: 10.1140/epje/i2019-11886-7] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2019] [Accepted: 08/06/2019] [Indexed: 06/10/2023]
Abstract
We investigate non-Markovian barrier-crossing kinetics of a massive particle in one dimension in the presence of a memory function that is the sum of two exponentials with different memory times, [Formula: see text] and [Formula: see text] . Our Langevin simulations for the special case where both exponentials contribute equally to the total friction show that the barrier-crossing time becomes independent of the longer memory time if at least one of the two memory times is larger than the intrinsic diffusion time. When we associate memory effects with coupled degrees of freedom that are orthogonal to a one-dimensional reaction coordinate, this counterintuitive result shows that the faster orthogonal degrees of freedom dominate barrier-crossing kinetics in the non-Markovian limit and that the slower orthogonal degrees become negligible, quite contrary to the standard time-scale separation assumption and with important consequences for the proper setup of coarse-graining procedures in the non-Markovian case. By asymptotic matching and symmetry arguments, we construct a crossover formula for the barrier crossing time that is valid for general multi-exponential memory kernels. This formula can be used to estimate barrier-crossing times for general memory functions for high friction, i.e. in the overdamped regime, as well as for low friction, i.e. in the inertial regime. Typical examples where our results are important include protein folding in the high-friction limit and chemical reactions such as proton-transfer reactions in the low-friction limit.
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Affiliation(s)
- Julian Kappler
- Freie Universität Berlin, Fachbereich Physik, Berlin, Germany
| | | | - Roland R Netz
- Freie Universität Berlin, Fachbereich Physik, Berlin, Germany.
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17
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Chen J, Yeh K, Chang W. Insights into the Kramers’ flux‐over‐population rate for chemical reactions in liquid phases through the matrix transport equation. J CHIN CHEM SOC-TAIP 2019. [DOI: 10.1002/jccs.201800228] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
Affiliation(s)
- Jenn‐Shing Chen
- Department of Applied ChemistryNational Chiao‐Tung University Hsin‐Chu Taiwan
| | - Kwei‐Tin Yeh
- Department of Applied ChemistryNational Chiao‐Tung University Hsin‐Chu Taiwan
| | - Wen‐Yih Chang
- Department of Applied ChemistryNational Chiao‐Tung University Hsin‐Chu Taiwan
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18
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Sahoo AK, Bagchi B, Maiti PK. Unfolding Dynamics of Ubiquitin from Constant Force MD Simulation: Entropy–Enthalpy Interplay Shapes the Free-Energy Landscape. J Phys Chem B 2019; 123:1228-1236. [DOI: 10.1021/acs.jpcb.8b09318] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022]
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19
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Hori N, Denesyuk NA, Thirumalai D. Frictional Effects on RNA Folding: Speed Limit and Kramers Turnover. J Phys Chem B 2018; 122:11279-11288. [PMID: 30179471 DOI: 10.1021/acs.jpcb.8b07129] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/19/2023]
Abstract
We investigated frictional effects on the folding rates of a human telomerase hairpin (hTR HP) and H-type pseudoknot from the Beet Western Yellow Virus (BWYV PK) using simulations of the Three Interaction Site (TIS) model for RNA. The heat capacity from TIS model simulations, calculated using temperature replica exchange simulations, reproduces nearly quantitatively the available experimental data for the hTR HP. The corresponding results for BWYV PK serve as predictions. We calculated the folding rates ( kF) from more than 100 folding trajectories for each value of the solvent viscosity (η) at a fixed salt concentration of 200 mM. By using the theoretical estimate (∝ √N where N is the number of nucleotides) for folding free energy barrier, kF data for both the RNAs are quantitatively fit using one-dimensional Kramers's theory with two parameters specifying the curvatures in the unfolded basin and the barrier top. In the high-friction regime (η ≳ 10-5 Pa·s), for both HP and PK, kF values decrease as 1/η, whereas in the low friction regime, kF values increase as η increases, leading to a maximum folding rate at a moderate viscosity (∼10-6 Pa·s), which is the Kramers turnover. From the fits, we find that the speed limit to RNA folding at water viscosity is between 1 and 4 μs, which is in accord with our previous theoretical prediction as well as results from several single molecule experiments. Both the RNA constructs fold by parallel pathways. Surprisingly, we find that the flux through the pathways could be altered by changing solvent viscosity, a prediction that is more easily testable in RNA than in proteins.
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Affiliation(s)
- Naoto Hori
- Department of Chemistry , University of Texas , Austin , Texas 78712 , United States
| | - Natalia A Denesyuk
- Biophysics program, Institute for Physical Science and Technology , University of Maryland , College Park , Maryland 20742-2431 , United States
| | - D Thirumalai
- Department of Chemistry , University of Texas , Austin , Texas 78712 , United States
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20
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Kappler J, Daldrop JO, Brünig FN, Boehle MD, Netz RR. Memory-induced acceleration and slowdown of barrier crossing. J Chem Phys 2018; 148:014903. [PMID: 29306292 DOI: 10.1063/1.4998239] [Citation(s) in RCA: 31] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
Abstract
We study the mean first-passage time τMFP for the barrier crossing of a single massive particle with non-Markovian memory by Langevin simulations in one dimension. In the Markovian limit of short memory time τΓ, the expected Kramers turnover between the overdamped (high-friction) and the inertial (low-friction) limits is recovered. Compared to the Markovian case, we find barrier crossing to be accelerated for intermediate memory time, while for long memory time, barrier crossing is slowed down and τMFP increases with τΓ as a power law τMFP∼τΓ2. Both effects are derived from an asymptotic propagator analysis: while barrier crossing acceleration at intermediate memory can be understood as an effective particle mass reduction, slowing down for long memory is caused by the slow kinetics of energy diffusion. A simple and globally accurate heuristic formula for τMFP in terms of all relevant time scales of the system is presented and used to establish a scaling diagram featuring the Markovian overdamped and the Markovian inertial regimes, as well as the non-Markovian intermediate memory time regime where barrier crossing is accelerated and the non-Markovian long memory time regime where barrier crossing is slowed down.
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Affiliation(s)
- Julian Kappler
- Freie Universität Berlin, Fachbereich Physik, 14195 Berlin, Germany
| | - Jan O Daldrop
- Freie Universität Berlin, Fachbereich Physik, 14195 Berlin, Germany
| | - Florian N Brünig
- Freie Universität Berlin, Fachbereich Physik, 14195 Berlin, Germany
| | - Moritz D Boehle
- Freie Universität Berlin, Fachbereich Physik, 14195 Berlin, Germany
| | - Roland R Netz
- Freie Universität Berlin, Fachbereich Physik, 14195 Berlin, Germany
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21
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Michaels TCT, Liu LX, Curk S, Bolhuis PG, Šarić A, Knowles TPJ. Reaction rate theory for supramolecular kinetics: application to protein aggregation. Mol Phys 2018. [DOI: 10.1080/00268976.2018.1474280] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
Affiliation(s)
- Thomas C. T. Michaels
- Department of Chemistry, University of Cambridge, Cambridge, UK
- Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
| | - Lucie X. Liu
- Department of Chemistry, University of Cambridge, Cambridge, UK
| | - Samo Curk
- Department of Physics and Astronomy, Institute for the Physics of Living Systems, University College London, London, UK
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia
| | - Peter G. Bolhuis
- van't Hoff Institute for Molecular Sciences, University of Amsterdam, Amsterdam, Netherlands
| | - Anđela Šarić
- Department of Physics and Astronomy, Institute for the Physics of Living Systems, University College London, London, UK
| | - Tuomas P. J. Knowles
- Department of Chemistry, University of Cambridge, Cambridge, UK
- Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge, UK
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22
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Wierenga H, Ten Wolde PR, Becker NB. Quantifying fluctuations in reversible enzymatic cycles and clocks. Phys Rev E 2018; 97:042404. [PMID: 29758603 DOI: 10.1103/physreve.97.042404] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2018] [Indexed: 11/07/2022]
Abstract
Biochemical reactions are fundamentally noisy at a molecular scale. This limits the precision of reaction networks, but it also allows fluctuation measurements that may reveal the structure and dynamics of the underlying biochemical network. Here, we study nonequilibrium reaction cycles, such as the mechanochemical cycle of molecular motors, the phosphorylation cycle of circadian clock proteins, or the transition state cycle of enzymes. Fluctuations in such cycles may be measured using either of two classical definitions of the randomness parameter, which we show to be equivalent in general microscopically reversible cycles. We define a stochastic period for reversible cycles and present analytical solutions for its moments. Furthermore, we associate the two forms of the randomness parameter with the thermodynamic uncertainty relation, which sets limits on the timing precision of the cycle in terms of thermodynamic quantities. Our results should prove useful also for the study of temporal fluctuations in more general networks.
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Affiliation(s)
| | | | - Nils B Becker
- DKFZ, Bioquant, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany
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23
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Magazzù L, Hänggi P, Spagnolo B, Valenti D. Quantum resonant activation. Phys Rev E 2017; 95:042104. [PMID: 28505843 DOI: 10.1103/physreve.95.042104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2017] [Indexed: 06/07/2023]
Abstract
Quantum resonant activation is investigated for the archetype setup of an externally driven two-state (spin-boson) system subjected to strong dissipation by means of both analytical and extensive numerical calculations. The phenomenon of resonant activation emerges in the presence of either randomly fluctuating or deterministic periodically varying driving fields. Addressing the incoherent regime, a characteristic minimum emerges in the mean first passage time to reach an absorbing neighboring state whenever the intrinsic time scale of the modulation matches the characteristic time scale of the system dynamics. For the case of deterministic periodic driving, the first passage time probability density function (pdf) displays a complex, multipeaked behavior, which depends crucially on the details of initial phase, frequency, and strength of the driving. As an interesting feature we find that the mean first passage time enters the resonant activation regime at a critical frequency ν^{*} which depends very weakly on the strength of the driving. Moreover, we provide the relation between the first passage time pdf and the statistics of residence times.
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Affiliation(s)
- Luca Magazzù
- Institute of Physics, University of Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany
| | - Peter Hänggi
- Institute of Physics, University of Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany
- Nanosystems Initiative Munich, Schellingstraße 4, D-80799 München, Germany
- Department of Physics, National University of Singapore, Singapore 117546
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
| | - Bernardo Spagnolo
- Dipartimento di Fisica e Chimica, Group of Interdisciplinary Theoretical Physics, Università di Palermo and CNISM, Unità di Palermo, Viale delle Scienze, Edificio 18, I-90128 Palermo, Italy
- Radiophysics Department, Lobachevsky State University of Nizhny Novgorod, Russia
- Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Italy
| | - Davide Valenti
- Dipartimento di Fisica e Chimica, Group of Interdisciplinary Theoretical Physics, Università di Palermo and CNISM, Unità di Palermo, Viale delle Scienze, Edificio 18, I-90128 Palermo, Italy
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24
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Pierse CA, Dudko OK. Distinguishing Signatures of Multipathway Conformational Transitions. PHYSICAL REVIEW LETTERS 2017; 118:088101. [PMID: 28282172 PMCID: PMC6863504 DOI: 10.1103/physrevlett.118.088101] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2016] [Indexed: 05/16/2023]
Abstract
The folding and binding of biomolecules into functional conformations are thought to be commonly mediated by multiple pathways rather than a unique route. Yet even in experiments where one can "see" individual conformational transitions, their stochastic nature generally precludes one from determining whether the transitions occurred through one or multiple pathways. We establish model-free, observable signatures in the response of macromolecules to force that unambiguously identify multiple pathways-even when the pathways themselves cannot be resolved. The unified analytical description reveals that, through multiple pathways, the response of molecules to external forces can be shaped in diverse ways, resulting in a rich design space for a tailored biological function already at the single-molecule level.
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Affiliation(s)
- Christopher A Pierse
- Department of Physics, University of California at San Diego, La Jolla, California 92093, USA
| | - Olga K Dudko
- Department of Physics, University of California at San Diego, La Jolla, California 92093, USA
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25
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Sharma A, Wittmann R, Brader JM. Escape rate of active particles in the effective equilibrium approach. Phys Rev E 2017; 95:012115. [PMID: 28208481 DOI: 10.1103/physreve.95.012115] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2016] [Indexed: 06/06/2023]
Abstract
The escape rate of a Brownian particle over a potential barrier is accurately described by the Kramers theory. A quantitative theory explicitly taking the activity of Brownian particles into account has been lacking due to the inherently out-of-equilibrium nature of these particles. Using an effective equilibrium approach [Farage et al., Phys. Rev. E 91, 042310 (2015)PLEEE81539-375510.1103/PhysRevE.91.042310] we study the escape rate of active particles over a potential barrier and compare our analytical results with data from direct numerical simulation of the colored noise Langevin equation. The effective equilibrium approach generates an effective potential that, when used as input to Kramers rate theory, provides results in excellent agreement with the simulation data.
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Affiliation(s)
- A Sharma
- Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland
| | - R Wittmann
- Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland
| | - J M Brader
- Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland
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26
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Suárez E, Adelman JL, Zuckerman DM. Accurate Estimation of Protein Folding and Unfolding Times: Beyond Markov State Models. J Chem Theory Comput 2016; 12:3473-81. [PMID: 27340835 PMCID: PMC5022777 DOI: 10.1021/acs.jctc.6b00339] [Citation(s) in RCA: 41] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022]
Abstract
Because standard molecular dynamics (MD) simulations are unable to access time scales of interest in complex biomolecular systems, it is common to "stitch together" information from multiple shorter trajectories using approximate Markov state model (MSM) analysis. However, MSMs may require significant tuning and can yield biased results. Here, by analyzing some of the longest protein MD data sets available (>100 μs per protein), we show that estimators constructed based on exact non-Markovian (NM) principles can yield significantly improved mean first-passage times (MFPTs) for protein folding and unfolding. In some cases, MSM bias of more than an order of magnitude can be corrected when identical trajectory data are reanalyzed by non-Markovian approaches. The NM analysis includes "history" information, higher order time correlations compared to MSMs, that is available in every MD trajectory. The NM strategy is insensitive to fine details of the states used and works well when a fine time-discretization (i.e., small "lag time") is used.
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Affiliation(s)
- Ernesto Suárez
- Department of Computational and Systems Biology, University of Pittsburgh , Pittsburgh, Pennsylvania 15260, United States
| | - Joshua L Adelman
- Department of Biological Sciences, University of Pittsburgh , Pittsburgh, Pennsylvania 15260, United States
| | - Daniel M Zuckerman
- Department of Computational and Systems Biology, University of Pittsburgh , Pittsburgh, Pennsylvania 15260, United States
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27
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28
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Dynamic heterogeneity controls diffusion and viscosity near biological interfaces. Nat Commun 2015; 5:3034. [PMID: 24398864 PMCID: PMC3971065 DOI: 10.1038/ncomms4034] [Citation(s) in RCA: 43] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2013] [Accepted: 11/29/2013] [Indexed: 11/14/2022] Open
Abstract
At a nanometer scale, the behavior of biological fluids is largely governed by interfacial physical chemistry. This may manifest as slowed or anomalous diffusion. Here we describe how measures developed for studying glassy systems allow quantitative measurement of interfacial effects on water dynamics, showing that correlated motions of particles near a surface result in a viscosity greater than anticipated from individual particle motions. This effect arises as a fundamental consequence of spatial heterogeneity on nanometer length scales and applies to any fluid near any surface. Increased interfacial viscosity also causes the classic finding that large solutes such as proteins diffuse much more slowly than predicted in bulk water. This has previously been treated via an empirical correction to the solute size: the hydrodynamic radius. Using measurements of quantities from theories of glass dynamics, we can now calculate diffusion constants from molecular details alone, eliminating the empirical correction factor.
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29
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Abstract
A new theory and an exact computer algorithm for calculating kinetics and thermodynamic properties of a particle system are described. The algorithm avoids trapping in metastable states, which are typical challenges for Molecular Dynamics (MD) simulations on rough energy landscapes. It is based on the division of the full space into Voronoi cells. Prior knowledge or coarse sampling of space points provides the centers of the Voronoi cells. Short time trajectories are computed between the boundaries of the cells that we call milestones and are used to determine fluxes at the milestones. The flux function, an essential component of the new theory, provides a complete description of the statistical mechanics of the system at the resolution of the milestones. We illustrate the accuracy and efficiency of the exact Milestoning approach by comparing numerical results obtained on a model system using exact Milestoning with the results of long trajectories and with a solution of the corresponding Fokker-Planck equation. The theory uses an equation that resembles the approximate Milestoning method that was introduced in 2004 [A. K. Faradjian and R. Elber, J. Chem. Phys. 120(23), 10880-10889 (2004)]. However, the current formulation is exact and is still significantly more efficient than straightforward MD simulations on the system studied.
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Affiliation(s)
- Juan M Bello-Rivas
- Department of Chemistry, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78712, USA
| | - Ron Elber
- Department of Chemistry, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78712, USA
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30
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Bello-Rivas JM, Elber R. Simulations of thermodynamics and kinetics on rough energy landscapes with milestoning. J Comput Chem 2015; 37:602-13. [PMID: 26265358 DOI: 10.1002/jcc.24039] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2015] [Revised: 06/26/2015] [Accepted: 07/14/2015] [Indexed: 01/22/2023]
Abstract
We investigated by computational means the kinetics and stationary behavior of stochastic dynamics on an ensemble of rough two-dimensional energy landscapes. There are no obvious separations of temporal scales in these systems, which constitute a simple model for the behavior of glasses and some biomaterials. Even though there are significant computational challenges present in these systems due to the large number of metastable states, the Milestoning method is able to compute their kinetic and thermodynamic properties exactly. We observe two clearly distinguished regimes in the overall kinetics: one in which diffusive behavior dominates and another that follows an Arrhenius law (despite the absence of a dominant barrier). We compare our results with those obtained with an exactly-solvable one-dimensional model, and with the results from the rough one-dimensional energy model introduced by Zwanzig. © 2015 Wiley Periodicals, Inc.
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Affiliation(s)
- Juan M Bello-Rivas
- Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas, 78712
| | - Ron Elber
- Department of Chemistry, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas, 78712
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31
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Gernert R, Emary C, Klapp SHL. Waiting time distribution for continuous stochastic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062115. [PMID: 25615052 DOI: 10.1103/physreve.90.062115] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2014] [Indexed: 06/04/2023]
Abstract
The waiting time distribution (WTD) is a common tool for analyzing discrete stochastic processes in classical and quantum systems. However, there are many physical examples where the dynamics is continuous and only approximately discrete, or where it is favourable to discuss the dynamics on a discretized and a continuous level in parallel. An example is the hindered motion of particles through potential landscapes with barriers. In the present paper we propose a consistent generalization of the WTD from the discrete case to situations where the particles perform continuous barrier crossing characterized by a finite duration. To this end, we introduce a recipe to calculate the WTD from the Fokker-Planck (Smoluchowski) equation. In contrast to the closely related first passage time distribution (FPTD), which is frequently used to describe continuous processes, the WTD contains information about the direction of motion. As an application, we consider the paradigmatic example of an overdamped particle diffusing through a washboard potential. To verify the approach and to elucidate its numerical implications, we compare the WTD defined via the Smoluchowski equation with data from direct simulation of the underlying Langevin equation and find full consistency provided that the jumps in the Langevin approach are defined properly. Moreover, for sufficiently large energy barriers, the WTD defined via the Smoluchowski equation becomes consistent with that resulting from the analytical solution of a (two-state) master equation model for the short-time dynamics developed previously by us [Phys. Rev. E 86, 061135 (2012)]. Thus, our approach "interpolates" between these two types of stochastic motion. We illustrate our approach for both symmetric systems and systems under constant force.
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Affiliation(s)
- Robert Gernert
- Institut für Theoretische Physik, Sekr. EW 7-1, Technische Universität Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany
| | - Clive Emary
- Department of Physics and Mathematics, University of Hull, Kingston-upon-Hull, HU6 7RX, United Kingdom
| | - Sabine H L Klapp
- Institut für Theoretische Physik, Sekr. EW 7-1, Technische Universität Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany
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32
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Choudhary OP, Paz A, Adelman JL, Colletier JP, Abramson J, Grabe M. Structure-guided simulations illuminate the mechanism of ATP transport through VDAC1. Nat Struct Mol Biol 2014; 21:626-32. [PMID: 24908397 PMCID: PMC4157756 DOI: 10.1038/nsmb.2841] [Citation(s) in RCA: 80] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2013] [Accepted: 05/14/2014] [Indexed: 01/02/2023]
Abstract
The voltage-dependent anion channel (VDAC) mediates the flow of metabolites and ions across the outer mitochondrial membrane of all eukaryotic cells. The open channel passes millions of ATP molecules per second, whereas the closed state exhibits no detectable ATP flux. High-resolution structures of VDAC1 revealed a 19-stranded β-barrel with an α-helix partially occupying the central pore. To understand ATP permeation through VDAC, we solved the crystal structure of mouse VDAC1 (mVDAC1) in the presence of ATP, revealing a low-affinity binding site. Guided by these coordinates, we initiated hundreds of molecular dynamics simulations to construct a Markov state model of ATP permeation. These simulations indicate that ATP flows through VDAC through multiple pathways, in agreement with our structural data and experimentally determined physiological rates.
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Affiliation(s)
- Om P Choudhary
- 1] Joint Carnegie Mellon University-University of Pittsburgh PhD Program in Computational Biology, Pittsburgh, Pennsylvania, USA. [2]
| | - Aviv Paz
- 1] Department of Physiology, David Geffen School of Medicine, University of California, Los Angeles, Los Angeles, California, USA. [2]
| | - Joshua L Adelman
- 1] Department of Biological Sciences, University of Pittsburgh, Pittsburgh, Pennsylvania, USA. [2]
| | - Jacques-Philippe Colletier
- 1] Université Grenoble Alpes, Institut de Biologie Structurale, Grenoble, France. [2] Centre National de la Recherche Scientifique, Institut de Biologie Structurale, Grenoble, France. [3] Commissariat à l'Énergie Atomique et aux Énergies Alternatives, Direction des Sciences du Vivant, Institut de Biologie Structurale, Grenoble, France. [4]
| | - Jeff Abramson
- 1] Department of Physiology, David Geffen School of Medicine, University of California, Los Angeles, Los Angeles, California, USA. [2] Institute for Stem Cell Biology and Regenerative Medicine, National Centre for Biological Sciences-Tata Institute of Fundamental Research, Bangalore, India
| | - Michael Grabe
- 1] Department of Biological Sciences, University of Pittsburgh, Pittsburgh, Pennsylvania, USA. [2] Cardiovascular Research Institute, Department of Pharmaceutical Chemistry, University of California, San Francisco, San Francisco, California, USA
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33
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Mazo JJ, Fajardo OY, Zueco D. Thermal activation at moderate-to-high and high damping: Finite barrier effects and force spectroscopy. J Chem Phys 2013; 138:104105. [DOI: 10.1063/1.4793983] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
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34
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Jiménez-Aquino JI, Romero-Bastida M. Non-Markovian stationary probability density for a harmonic oscillator in an electromagnetic field. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:061115. [PMID: 23367901 DOI: 10.1103/physreve.86.061115] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2012] [Indexed: 06/01/2023]
Abstract
We calculate the exact solution of the Fokker-Planck equation for the stationary-state probability density of a harmonic oscillator embedded in an electromagnetic field. The magnetic field is assumed to be a constant and the electric field an external stochastic force with the properties of a Gaussian and exponentially correlated noise (Ornstein-Uhlenbeck process). In this work, we first study the problem in the absence of the magnetic field, then we obtain the complete solution and corroborate that the latter reduces to the former when the magnetic field is suppressed.
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Affiliation(s)
- J I Jiménez-Aquino
- Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa 09340, Distrito Federal, Mexico.
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35
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Affiliation(s)
- Riccardo Scalco
- Department of Biochemistry, University of Zurich, Winterthurerstrasse
190, CH-8057 Zurich, Switzerland
| | - Amedeo Caflisch
- Department of Biochemistry, University of Zurich, Winterthurerstrasse
190, CH-8057 Zurich, Switzerland
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36
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Maurer P, Thomas V, Iftimie R. A computational study of ultrafast acid dissociation and acid–base neutralization reactions. II. The relationship between the coordination state of solvent molecules and concerted versus sequential acid dissociation. J Chem Phys 2011; 134:094505. [DOI: 10.1063/1.3554654] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
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37
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Zhu P, You X, Pratt LR, Papadopoulos KD. Generalizations of the Fuoss approximation for ion pairing. J Chem Phys 2011; 134:054502. [DOI: 10.1063/1.3544680] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Wang XX, Bao JD. Master equation approach to time-dependent escape rate over a periodically oscillating barrier. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:011127. [PMID: 21405681 DOI: 10.1103/physreve.83.011127] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2010] [Revised: 12/08/2010] [Indexed: 05/30/2023]
Abstract
We propose a master equation approach to investigate the transition function and the escape dynamics of a general damping particle in a metastable potential. The transition function in the master equation is obtained analytically from the Langevin dynamics. We apply it to the oscillating barrier problem, in which the potential is structured by a harmonic potential smoothly linking with an inverse harmonic one, and thus both the barrier height and the curvature of potential change periodically with time. We use a Monte Carlo method to simulate the master equation and then calculate a time-dependent escape rate. This can decrease the coarse grain and save more computing time in comparison with the Langevin simulation. Our result has shown that there is a resonant activation phenomenon for the escape rate in the underdamped case.
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Affiliation(s)
- Xin-Xin Wang
- Department of Physics, Beijing Normal University, Beijing, People's Republic of China
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39
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Getfert S, Reimann P. Thermally activated escape far from equilibrium: A unified path-integral approach. Chem Phys 2010. [DOI: 10.1016/j.chemphys.2010.04.028] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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40
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Mondal D, Ray DS. Diffusion over an entropic barrier: non-Arrhenius behavior. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:032103. [PMID: 21230121 DOI: 10.1103/physreve.82.032103] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2010] [Indexed: 05/30/2023]
Abstract
We have examined the temperature dependence of noise-induced escape rate of brownian particles between two distinct regions in a two-dimensional enclosure through a narrow bottleneck. Varying cross section of the confinement results in an effective entropic barrier in reduced dimension. The strong non-Arrhenius dependence of the escape rate is observed, which can be explained in terms of a crossover between entropy-dominated regime and energy-dominated regime in presence of an interplay of gravitational bias and thermal motion.
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Affiliation(s)
- Debasish Mondal
- Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
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41
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Kim C, Talkner P, Lee EK, Hänggi P. Rate description of Fokker–Planck processes with time-periodic parameters. Chem Phys 2010. [DOI: 10.1016/j.chemphys.2009.10.027] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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43
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Abstract
We present a new general method to trace back a lot of valuable information from direct simulations and experiments of activated processes. In particular, it allows the reconstruction of the free-energy landscape for an arbitrary reaction coordinate directly from the out-of-equilibrium dynamics of the process. We demonstrate the power of this concept by its application to a molecular dynamics simulation of nucleation of a Lennard-Jones vapor. The same method can be also applied to Brownian dynamics and stochastic simulations.
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Affiliation(s)
- Jan Wedekind
- Departament de Física Fonamental, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
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44
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Deng ML, Zhu WQ. Fermi resonance and its effect on the mean transition time and rate. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:061114. [PMID: 18643224 DOI: 10.1103/physreve.77.061114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2007] [Indexed: 05/26/2023]
Abstract
Fermi resonance and its effect on the mean transition time and rate are studied. The necessary frequency ratio 1:2 for Fermi resonance to occur is explained by applying the deterministic averaging method to the two-dimensional conservative Pippard system, and a more frequent fluctuation of energy process due to Fermi resonance is shown by using the samples obtained from digital simulation of the stochastic Pippard system. In the case of weak coupling, the mean transition time of the reacting oscillator energy is evaluated for both nonresonance and Fermi resonance by using the standard (Stratonovich) stochastic averaging method. The theoretical results for the mean transition time in the case of Fermi resonance and nonresonance is then extended to the stochastic system with bistable potential, and the effects of frequency ratio and coupling coefficient on the mean reaction rate are analyzed. In the case of strong coupling, it is pointed out that the exciting oscillator and reacting oscillator move together like one oscillator and no Fermi resonance can occur. In this case, the mean transition rate of the system total energy is studied by using the stochastic averaging method for quasi-non-integrable Hamiltonian systems. All the theoretical results are confirmed through comparison with those from digital simulation, and the effect of Fermi resonance on the transition time and rate is discussed.
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Affiliation(s)
- M L Deng
- Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, China
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45
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Dybiec B, Gudowska-Nowak E, Hänggi P. Escape driven by alpha-stable white noises. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:021109. [PMID: 17358315 DOI: 10.1103/physreve.75.021109] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2006] [Indexed: 05/14/2023]
Abstract
We explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the Brownian particle is driven by white Lévy noise in a dynamical regime where inertial effects can safely be neglected. The behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise-induced discontinuity which is caused by the generalized Brownian paths that jump beyond the barrier location without actually hitting it. This fact implies that the boundary conditions for the mean first passage time (MFPT) are no longer determined by the well-known local boundary conditions that characterize the case with normal diffusion. By numerically implementing properly the set up boundary conditions, we investigate the survival probability and the average escape time as a function of the corresponding Lévy white noise parameters. Depending on the value of the skewness beta of the Lévy noise, the escape can either become enhanced or suppressed: a negative asymmetry parameter beta typically yields a decrease for the escape rate while the rate itself depicts a non-monotonic behavior as a function of the stability index alpha that characterizes the jump length distribution of Lévy noise, exhibiting a marked discontinuity at alpha=1. We find that the typical factor of 2 that characterizes for normal diffusion the ratio between the MFPT for well-bottom-to-well-bottom and well-bottom-to-barrier-top no longer holds true. For sufficiently high barriers the survival probabilities assume an exponential behavior versus time. Distinct non-exponential deviations occur, however, for low barrier heights.
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Affiliation(s)
- B Dybiec
- M. Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagellonian University, ul. Reymonta 4, 30-059 Kraków, Poland.
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46
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Managing free-energy barriers in nuclear pore transport. J Biol Phys 2006; 32:465-72. [PMID: 19669451 DOI: 10.1007/s10867-006-9029-5] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2006] [Accepted: 11/12/2006] [Indexed: 10/23/2022] Open
Abstract
The Nuclear Pore Complexes (NPC) facilitate highly selective gateways for transport of macromolecules across the Nuclear Envelope (NE). Based on the current accumulated knowledge of the architecture of NPC we have established a minimal physical model of the pore and the transport mechanism. The barrier properties of the NPC model are analyzed by the recently established Wang-Landau Monte Carlo computer simulation technique and the transport properties are extracted by employing Kramers' theory of reaction rates. We show that our physical model can account for a range of characteristics observed for nuclear pore transport.
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Shalloway D, Faradjian AK. Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation. J Chem Phys 2006; 124:054112. [PMID: 16468856 DOI: 10.1063/1.2161211] [Citation(s) in RCA: 41] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023] Open
Abstract
The generalized master equation or the equivalent continuous time random walk equations can be used to compute the macroscopic first passage time distribution (FPTD) of a complex stochastic system from short-term microscopic simulation data. The computation of the mean first passage time and additional low-order FPTD moments can be simplified by directly relating the FPTD moment generating function to the moments of the local FPTD matrix. This relationship can be physically interpreted in terms of steady-state relaxation, an extension of steady-state flow. Moreover, it is amenable to a statistical error analysis that can be used to significantly increase computational efficiency. The efficiency improvement can be extended to the FPTD itself by modelling it using a gamma distribution or rational function approximation to its Laplace transform.
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Affiliation(s)
- David Shalloway
- Biophysics Program, Department of Molecular Biology and Genetics, Cornell University, Ithaca, New York 14853, USA.
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Mierzejewski M, Dajka J, Łuczka J, Talkner P, Hänggi P. Dynamical bimodality in equilibrium monostable systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:041102. [PMID: 17155017 DOI: 10.1103/physreve.74.041102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/04/2006] [Indexed: 05/12/2023]
Abstract
General features of the stochastic dynamics of classical systems approaching a thermodynamic equilibrium Gibbs state are studied via the numerical analysis of time-dependent solutions of the Fokker-Planck equation for an overdamped particle in various monostable potentials. A large class of initial states can dynamically bifurcate during its time evolution into bimodal transient states, which in turn wear off when approaching the long-time regime. Suitable quantifiers characterizing this transient dynamical bimodality, such as its lifetime, the positions of maxima, and the time-dependent well depth of the probability distribution, are analyzed. Some potential applications are pointed out that make use of this interesting principle which is based on an appropriately chosen initial preparation procedure.
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Affiliation(s)
- M Mierzejewski
- Institute of Physics, University of Silesia, 40-007 Katowice, Poland
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Pollak E, Talkner P. Reaction rate theory: what it was, where is it today, and where is it going? CHAOS (WOODBURY, N.Y.) 2005; 15:26116. [PMID: 16035918 DOI: 10.1063/1.1858782] [Citation(s) in RCA: 157] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius [Z. Phys. Chem. 4, 226 (1889)], we follow especially the formulation of transition state theory by Wigner [Z. Phys. Chem. Abt. B 19, 203 (1932)] and Eyring [J. Chem. Phys. 3, 107 (1935)]. Transition state theory (TST) made it possible to obtain quick estimates for reaction rates for a broad variety of processes even during the days when sophisticated computers were not available. Arrhenius' suggestion that a transition state exists which is intermediate between reactants and products was central to the development of rate theory. Although Wigner gave an abstract definition of the transition state as a surface of minimal unidirectional flux, it took almost half of a century until the transition state was precisely defined by Pechukas [Dynamics of Molecular Collisions B, edited by W. H. Miller (Plenum, New York, 1976)], but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the question as to the definition of the activation energy for which Arrhenius became famous. In 1978, Chandler [J. Chem. Phys. 68, 2959 (1978)] finally showed that especially when considering condensed phases, the activation energy is a free energy, it is the barrier height in the potential of mean force felt by the reacting system. Parallel to the development of rate theory in the chemistry community, Kramers published in 1940 [Physica (Amsterdam) 7, 284 (1940)] a seminal paper on the relation between Einstein's theory of Brownian motion [Einstein, Ann. Phys. 17, 549 (1905)] and rate theory. Kramers' paper provided a solution for the effect of friction on reaction rates but left us also with some challenges. He could not derive a uniform expression for the rate, valid for all values of the friction coefficient, known as the Kramers turnover problem. He also did not establish the connection between his approach and the TST developed by the chemistry community. For many years, Kramers' theory was considered as providing a dynamic correction to the thermodynamic TST. Both of these questions were resolved in the 1980s when Pollak [J. Chem. Phys. 85, 865 (1986)] showed that Kramers' expression in the moderate to strong friction regime could be derived from TST, provided that the bath, which is the source of the friction, is handled at the same level as the system which is observed. This then led to the Mel'nikov-Pollak-Grabert-Hanggi [Mel'nikov and Meshkov, J. Chem. Phys. 85, 1018 (1986); Pollak, Grabert, and Hanggi, ibid. 91, 4073 (1989)] solution of the turnover problem posed by Kramers. Although classical rate theory reached a high level of maturity, its quantum analog leaves the theorist with serious challenges to this very day. As noted by Wigner [Trans. Faraday Soc. 34, 29 (1938)], TST is an inherently classical theory. A definite quantum TST has not been formulated to date although some very useful approximate quantum rate theories have been invented. The successes and challenges facing quantum rate theory are outlined. An open problem which is being investigated intensively is rate theory away from equilibrium. TST is no longer valid and cannot even serve as a conceptual guide for understanding the critical factors which determine rates away from equilibrium. The nonequilibrium quantum theory is even less well developed than the classical, and suffers from the fact that even today, we do not know how to solve the real time quantum dynamics for systems with "many" degrees of freedom.
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Affiliation(s)
- Eli Pollak
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel
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50
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Bier M. Critical exponents near a bifurcation point in noise-induced escape over a fluctuating barrier. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:011108. [PMID: 15697581 DOI: 10.1103/physreve.71.011108] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/23/2004] [Indexed: 05/24/2023]
Abstract
If a bistable system is brought closer to the bifurcation point at A= A(c) , the rate of noise-activated escape to the other stable equilibrium increases as ln k proportional, variant- mid R:A- A(c) mid R:(3/2) . Recently, Dykman, Golding, and Ryvkine derived that upon the addition of an oscillating force, the exponent equals 2 in an intermediate-frequency regime [Phys. Rev. Lett. 92, 080602 (2004)]; Phys. Rev. E 69, 061102 (2004)]]. In this paper we study the critical exponents near a bifurcation point for an analytically solvable case; that of a piecewise linear well with an imposed dichotomous fluctuation. It is shown how an exponent 2 emerges when a parameter is changed to move a system from nonadiabaticity to adiabaticity. It is, furthermore, shown that the power law breaks down for low barriers. Finally, it is explained why fundamental differences occur between smooth and piecewise linear setups.
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Affiliation(s)
- Martin Bier
- Department of Physics, East Carolina University, Greenville, North Carolina 27858, USA
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