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Ghasemi Nezhadhaghighi M, Chechkin A, Metzler R. Numerical approach to unbiased and driven generalized elastic model. J Chem Phys 2014; 140:024106. [PMID: 24437864 DOI: 10.1063/1.4858425] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
From scaling arguments and numerical simulations, we investigate the properties of the generalized elastic model (GEM) that is used to describe various physical systems such as polymers, membranes, single-file systems, or rough interfaces. We compare analytical and numerical results for the subdiffusion exponent β characterizing the growth of the mean squared displacement 〈(δh)(2)〉 of the field h described by the GEM dynamic equation. We study the scaling properties of the qth order moments 〈∣δh∣(q)〉 with time, finding that the interface fluctuations show no intermittent behavior. We also investigate the ergodic properties of the process h in terms of the ergodicity breaking parameter and the distribution of the time averaged mean squared displacement. Finally, we study numerically the driven GEM with a constant, localized perturbation and extract the characteristics of the average drift for a tagged probe.
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Affiliation(s)
| | - A Chechkin
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - R Metzler
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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Metzler R, Jeon JH, Cherstvy AG, Barkai E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys 2014; 16:24128-64. [DOI: 10.1039/c4cp03465a] [Citation(s) in RCA: 1046] [Impact Index Per Article: 104.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.
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Affiliation(s)
- Ralf Metzler
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
- Physics Department
- Tampere University of Technology
| | - Jae-Hyung Jeon
- Physics Department
- Tampere University of Technology
- Tampere, Finland
- Korean Institute for Advanced Study (KIAS)
- Seoul, Republic of Korea
| | - Andrey G. Cherstvy
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
| | - Eli Barkai
- Physics Department and Institute of Nanotechnology and Advanced Materials
- Bar-Ilan University
- Ramat Gan, Israel
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53
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Kursawe J, Schulz J, Metzler R. Transient aging in fractional Brownian and Langevin-equation motion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062124. [PMID: 24483403 DOI: 10.1103/physreve.88.062124] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2013] [Indexed: 06/03/2023]
Abstract
Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin-equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin-equation motion under external confinement are transiently nonergodic-time and ensemble averages behave differently-from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient aging, that is, physical observables such as the time-averaged mean-squared displacement depend on the time lag between the initiation of the system at time t=0 and the start of the measurement at the aging time t(a). In particular, it turns out that for fractional Langevin-equation motion the aging dependence on t(a) is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time-averaged mean-squared displacement and present a numerical analysis of this transient aging phenomenon.
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Affiliation(s)
- Jochen Kursawe
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG
| | - Johannes Schulz
- Physics Department, Technical University of Munich, 85747 Garching, Germany
| | - Ralf Metzler
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG and Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany and Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
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Javanainen M, Hammaren H, Monticelli L, Jeon JH, Miettinen MS, Martinez-Seara H, Metzler R, Vattulainen I. Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes. Faraday Discuss 2013; 161:397-417; discussion 419-59. [PMID: 23805752 DOI: 10.1039/c2fd20085f] [Citation(s) in RCA: 133] [Impact Index Per Article: 12.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Lateral diffusion plays a crucial role in numerous processes that take place in cell membranes, yet it is quite poorly understood in native membranes characterized by, e.g., domain formation and large concentration of proteins. In this article, we use atomistic and coarse-grained simulations to consider how packing of membranes and crowding with proteins affect the lateral dynamics of lipids and membrane proteins. We find that both packing and protein crowding have a profound effect on lateral diffusion, slowing it down. Anomalous diffusion is observed to be an inherent property in both protein-free and protein-rich membranes, and the time scales of anomalous diffusion and the exponent associated with anomalous diffusion are found to strongly depend on packing and crowding. Crowding with proteins also has a striking effect on the decay rate of dynamical correlations associated with lateral single-particle motion, as the transition from anomalous to normal diffusion is found to take place at macroscopic time scales: while in protein-poor conditions normal diffusion is typically observed in hundreds of nanoseconds, in protein-rich conditions the onset of normal diffusion is tens of microseconds, and in the most crowded systems as large as milliseconds. The computational challenge which results from these time scales is not easy to deal with, not even in coarse-grained simulations. We also briefly discuss the physical limits of protein motion. Our results suggest that protein concentration is anything but constant in the plane of cell membranes. Instead, it is strongly dependent on proteins' preference for aggregation.
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Affiliation(s)
- Matti Javanainen
- Department of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finland
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56
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Olah MJ, Stefanovic D. Superdiffusive transport by multivalent molecular walkers moving under load. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062713. [PMID: 23848721 DOI: 10.1103/physreve.87.062713] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2012] [Revised: 04/10/2013] [Indexed: 06/02/2023]
Abstract
We introduce a model for translational molecular motors to demonstrate that a multivalent catalytic walker with flexible, uncoordinated legs can transform the free energy of surface-bound substrate sites into mechanical work and undergo biased, superdiffusive motion, even in opposition to an external load force. The walker in the model lacks any inherent orientation of body or track, and its legs have no chemomechanical coupling other than the passive constraint imposed by their connection to a common body. Yet, under appropriate kinetic conditions, the walker's motion is biased in the direction of unvisited sites, which allows the walker to move nearly ballistically away from the origin as long as a local supply of unmodified substrate sites is available. The multivalent random walker model is mathematically formulated as a continuous-time Markov process and is studied numerically. We use Monte Carlo simulations to generate ensemble estimates of the mean squared displacement and mean work done for this nonergodic system. Our results show that a residence time bias between visited and unvisited sites leads to superdiffusive motion over significant times and distances. This mechanism can be used to adapt any enzyme-substrate system with appropriate kinetics for use as a functional chemical implementation of a molecular motor, without the need for structural anisotropy or conformationally mediated chemomechanical coordination.
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Affiliation(s)
- Mark J Olah
- Department of Computer Science, University of New Mexico, MSC01 1130, 1 University of New Mexico, Albuquerque, New Mexico 87131-0001, USA.
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57
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Godec A, Metzler R. Finite-time effects and ultraweak ergodicity breaking in superdiffusive dynamics. PHYSICAL REVIEW LETTERS 2013; 110:020603. [PMID: 23383882 DOI: 10.1103/physrevlett.110.020603] [Citation(s) in RCA: 50] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2012] [Indexed: 06/01/2023]
Abstract
We study the ergodic properties of superdiffusive, spatiotemporally coupled Lévy walk processes. For trajectories of finite duration, we reveal a distinct scatter of the scaling exponents of the time averaged mean squared displacement δx2 around the ensemble value 3-α (1<α<2) ranging from ballistic motion to subdiffusion, in strong contrast to the behavior of subdiffusive processes. In addition we find a significant dependence of the average of δx2 over an ensemble of trajectories as a function of the finite measurement time. This so-called finite-time amplitude depression and the scatter of the scaling exponent is vital in the quantitative evaluation of superdiffusive processes. Comparing the long time average of the second moment with the ensemble mean squared displacement, these only differ by a constant factor, an ultraweak ergodicity breaking.
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Affiliation(s)
- Aljaž Godec
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany.
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Cherstvy AG, Metzler R. Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes. Phys Chem Chem Phys 2013; 15:20220-35. [DOI: 10.1039/c3cp53056f] [Citation(s) in RCA: 97] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/19/2023]
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Fritsch CC, Langowski J. Kinetic lattice Monte Carlo simulation of viscoelastic subdiffusion. J Chem Phys 2012; 137:064114. [PMID: 22897262 DOI: 10.1063/1.4742909] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Abstract
We propose a kinetic Monte Carlo method for the simulation of subdiffusive random walks on a cartesian lattice. The random walkers are subject to viscoelastic forces which we compute from their individual trajectories via the fractional Langevin equation. At every step the walkers move by one lattice unit, which makes them differ essentially from continuous time random walks, where the subdiffusive behavior is induced by random waiting. To enable computationally inexpensive simulations with n-step memories, we use an approximation of the memory and the memory kernel functions with a complexity O(log n). Eventual discretization and approximation artifacts are compensated with numerical adjustments of the memory kernel functions. We verify with a number of analyses that this new method provides binary fractional random walks that are fully consistent with the theory of fractional brownian motion.
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Affiliation(s)
- Christian C Fritsch
- BIOMS Center for Modeling and Simulation in the Biosciences, D-69120 Heidelberg, Germany
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Jeon JH, Monne HMS, Javanainen M, Metzler R. Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. PHYSICAL REVIEW LETTERS 2012; 109:188103. [PMID: 23215336 DOI: 10.1103/physrevlett.109.188103] [Citation(s) in RCA: 154] [Impact Index Per Article: 12.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2012] [Indexed: 05/08/2023]
Abstract
Combining extensive molecular dynamics simulations of lipid bilayer systems of varying chemical compositions with single-trajectory analyses, we systematically elucidate the stochastic nature of the lipid motion. We observe subdiffusion over more than 4 orders of magnitude in time, clearly stretching into the submicrosecond domain. The lipid motion depends on the lipid chemistry, the lipid phase, and especially the presence of cholesterol. We demonstrate that fractional Langevin equation motion universally describes the lipid motion in all phases, including the gel phase, and in the presence of cholesterol. The results underline the relevance of anomalous diffusion in lipid bilayers and the strong effects of the membrane composition.
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Affiliation(s)
- Jae-Hyung Jeon
- Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
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61
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Lucena D, Tkachenko DV, Nelissen K, Misko VR, Ferreira WP, Farias GA, Peeters FM. Transition from single-file to two-dimensional diffusion of interacting particles in a quasi-one-dimensional channel. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:031147. [PMID: 22587078 DOI: 10.1103/physreve.85.031147] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2011] [Revised: 11/01/2011] [Indexed: 05/31/2023]
Abstract
Diffusive properties of a monodisperse system of interacting particles confined to a quasi-one-dimensional channel are studied using molecular dynamics simulations. We calculate numerically the mean-squared displacement (MSD) and investigate the influence of the width of the channel (or the strength of the confinement potential) on diffusion in finite-size channels of different shapes (i.e., straight and circular). The transition from single-file diffusion to the two-dimensional diffusion regime is investigated. This transition [regarding the calculation of the scaling exponent (α) of the MSD (Δx(2)(t) ∝ t(α)] as a function of the width of the channel is shown to change depending on the channel's confinement profile. In particular, the transition can be either smooth (i.e., for a parabolic confinement potential) or rather sharp (i.e., for a hard-wall potential), as distinct from infinite channels where this transition is abrupt. This result can be explained by qualitatively different distributions of the particle density for the different confinement potentials.
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Affiliation(s)
- D Lucena
- Departamento de Física, Universidade Federal do Ceará, Fortaleza, Ceará, Brazil
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