1
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Waigh TA, Korabel N. Heterogeneous anomalous transport in cellular and molecular biology. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2023; 86:126601. [PMID: 37863075 DOI: 10.1088/1361-6633/ad058f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2022] [Accepted: 10/20/2023] [Indexed: 10/22/2023]
Abstract
It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.
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Affiliation(s)
- Thomas Andrew Waigh
- Biological Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
| | - Nickolay Korabel
- Department of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom
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2
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Singh RK, Burov S. Universal to nonuniversal transition of the statistics of rare events during the spread of random walks. Phys Rev E 2023; 108:L052102. [PMID: 38115504 DOI: 10.1103/physreve.108.l052102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2023] [Accepted: 09/11/2023] [Indexed: 12/21/2023]
Abstract
Through numerous experiments that analyzed rare event statistics in heterogeneous media, it was discovered that in many cases the probability density function for particle position, P(X,t), exhibits a slower decay rate than the Gaussian function. Typically, the decay behavior is exponential, referred to as Laplace tails. However, many systems exhibit an even slower decay rate, such as power-law, log-normal, or stretched exponential. In this study, we utilize the continuous-time random walk method to investigate the rare events in particle hopping dynamics and find that the properties of the hop size distribution induce a critical transition between the Laplace universality of rare events and a more specific, slower decay of P(X,t). Specifically, when the hop size distribution decays slower than exponential, such as e^{-|x|^{β}} (β>1), the Laplace universality no longer applies, and the decay is specific, influenced by a few large events, rather than by the accumulation of many smaller events that give rise to Laplace tails.
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Affiliation(s)
- R K Singh
- Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel
| | - Stanislav Burov
- Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel
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3
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Höll M, Nissan A, Berkowitz B, Barkai E. Controls that expedite first-passage times in disordered systems. Phys Rev E 2023; 108:034124. [PMID: 37849182 DOI: 10.1103/physreve.108.034124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2022] [Accepted: 08/28/2023] [Indexed: 10/19/2023]
Abstract
First-passage time statistics in disordered systems exhibiting scale invariance are studied widely. In particular, long trapping times in energy or entropic traps are fat-tailed distributed, which slow the overall transport process. We study the statistical properties of the first-passage time of biased processes in different models, and we employ the big-jump principle that shows the dominance of the maximum trapping time on the first-passage time. We demonstrate that the removal of this maximum significantly expedites transport. As the disorder increases, the system enters a phase where the removal shows a dramatic effect. Our results show how we may speed up transport in strongly disordered systems exploiting scale invariance. In contrast to the disordered systems studied here, the removal principle has essentially no effect in homogeneous systems; this indicates that improving the conductance of a poorly conducting system is, theoretically, relatively easy as compared to a homogeneous system.
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Affiliation(s)
- Marc Höll
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
| | - Alon Nissan
- Institute of Environmental Engineering, ETH Zurich, Zurich, Switzerland
| | - Brian Berkowitz
- Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
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4
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Wen B, Li MG, Liu J, Bao JD. Ergodic Measure and Potential Control of Anomalous Diffusion. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1012. [PMID: 37509959 PMCID: PMC10377995 DOI: 10.3390/e25071012] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2023] [Revised: 06/15/2023] [Accepted: 06/27/2023] [Indexed: 07/30/2023]
Abstract
In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior.
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Affiliation(s)
- Bao Wen
- Institutes of Science and Development, Chinese Academy of Sciences, Beijing 100190, China
- School of Public Policy and Management, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Ming-Gen Li
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Jian Liu
- Department of Physics, Beijing Technology and Business University, Beijing 100048, China
| | - Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
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5
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Sakai I, Akimoto T. Sample-to-sample fluctuations of transport coefficients in the totally asymmetric simple exclusion process with quenched disorder. Phys Rev E 2023; 107:054131. [PMID: 37328985 DOI: 10.1103/physreve.107.054131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2023] [Accepted: 02/17/2023] [Indexed: 06/18/2023]
Abstract
We consider the totally asymmetric simple exclusion processes on quenched random energy landscapes. We show that the current and the diffusion coefficient differ from those for homogeneous environments. Using the mean-field approximation, we analytically obtain the site density when the particle density is low or high. As a result, the current and the diffusion coefficient are described by the dilute limit of particles or holes, respectively. However, in the intermediate regime, due to the many-body effect, the current and the diffusion coefficient differ from those for single-particle dynamics. The current is almost constant and becomes the maximal value in the intermediate regime. Moreover, the diffusion coefficient decreases with the particle density in the intermediate regime. We obtain analytical expressions for the maximal current and the diffusion coefficient based on the renewal theory. The deepest energy depth plays a central role in determining the maximal current and the diffusion coefficient. As a result, the maximal current and the diffusion coefficient depend crucially on the disorder, i.e., non-self-averaging. Based on the extreme value theory, we find that sample-to-sample fluctuations of the maximal current and diffusion coefficient are characterized by the Weibull distribution. We show that the disorder averages of the maximal current and the diffusion coefficient converge to zero as the system size is increased and quantify the degree of the non-self-averaging effect for the maximal current and the diffusion coefficient.
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Affiliation(s)
- Issei Sakai
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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6
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Sakai I, Akimoto T. Non-self-averaging of current in a totally asymmetric simple exclusion process with quenched disorder. Phys Rev E 2023; 107:L052103. [PMID: 37329050 DOI: 10.1103/physreve.107.l052103] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2022] [Accepted: 05/01/2023] [Indexed: 06/18/2023]
Abstract
We investigate the current properties in the totally asymmetric simple exclusion process (TASEP) on a quenched random energy landscape. In low- and high-density regimes, the properties are characterized by single-particle dynamics. In the intermediate one, the current becomes constant and is maximized. Based on the renewal theory, we derive accurate results for the maximum current. The maximum current significantly depends on a disorder realization, i.e., non-self-averaging (SA). We demonstrate that the disorder average of the maximum current decreases with the system size, and the sample-to-sample fluctuations of the maximum current exceed those of current in the low- and high-density regimes. We find a significant difference between single-particle dynamics and the TASEP. In particular, the non-SA behavior of the maximum current is always observed, whereas the transition from non-SA to SA for current in single-particle dynamics exists.
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Affiliation(s)
- Issei Sakai
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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7
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Stojkoski V, Sandev T, Kocarev L, Pal A. Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process. Phys Rev E 2021; 104:014121. [PMID: 34412255 DOI: 10.1103/physreve.104.014121] [Citation(s) in RCA: 16] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 06/16/2021] [Indexed: 01/19/2023]
Abstract
We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for nonstationary and nonergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains nonergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the nonergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable state, and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand-alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.
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Affiliation(s)
- Viktor Stojkoski
- Faculty of Economics, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia.,Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
| | - Trifce Sandev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia.,Institute of Physics and Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany.,Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
| | - Ljupco Kocarev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia.,Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, P.O. Box 393, 1000 Skopje, Macedonia
| | - Arnab Pal
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel
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8
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Bao JD, Wang XR, Liu WM. Ergodic time scale and transitive dynamics in single-particle tracking. Phys Rev E 2021; 103:032136. [PMID: 33862786 DOI: 10.1103/physreve.103.032136] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2020] [Accepted: 02/26/2021] [Indexed: 11/07/2022]
Abstract
We investigate ergodic time scales in single-particle tracking by introducing a covariance measure Ω(Δ;t) for the time-averaged relative square displacement recorded in lag-time Δ at elapsed time t. The present model is established in the generalized Langevin equation with a power-law memory function. The ratio Ω(Δ;Δ)/Ω(Δ;t) is shown to obey a universal scaling law for long but finite times and is used to extract the effective ergodic time. We derive a finite-time-averaged Green-Kubo relation and find that, to control the deviations in measurement results from ensemble averages, the ratio Δ/t must be neither too small nor close to unity. Our paper connects the experimental self-averaging property of a tracer with the theoretic velocity autocorrelation function and sheds light on the transition to ergodicity.
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Affiliation(s)
- Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Xiang-Rong Wang
- Department of Physics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China
| | - Wu-Ming Liu
- Institute of Physics, Chinese Academy of Science, Beijing 100190, China
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9
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Akimoto T, Barkai E, Radons G. Infinite invariant density in a semi-Markov process with continuous state variables. Phys Rev E 2020; 101:052112. [PMID: 32575214 DOI: 10.1103/physreve.101.052112] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2019] [Accepted: 04/09/2020] [Indexed: 06/11/2023]
Abstract
We report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant density, in a semi-Markov process where the state is determined by the interevent time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized Lévy walk model and the energy of a particle in the trap model. In our model, the interevent-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long interevent times imply small state values. We find two scaling laws describing the density for the state value, which accumulates in the vicinity of zero in the long-time limit. These laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density. Moreover, we provide two distributional limit theorems for time-averaged observables in these nonstationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.
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Affiliation(s)
- Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Eli Barkai
- Department of Physics, Bar-Ilan University, Ramat-Gan
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
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10
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Akimoto T, Saito K. Trace of anomalous diffusion in a biased quenched trap model. Phys Rev E 2020; 101:042133. [PMID: 32422836 DOI: 10.1103/physreve.101.042133] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2020] [Accepted: 04/07/2020] [Indexed: 06/11/2023]
Abstract
Diffusion in a quenched heterogeneous environment in the presence of bias is considered analytically. The first-passage-time statistics can be applied to obtain the drift and the diffusion coefficient in periodic quenched environments. We show several transition points at which sample-to-sample fluctuations of the drifts or the diffusion coefficients remain large even when the system size becomes large, i.e., non-self-averaging. Moreover, we find that the disorder average of the diffusion coefficient diverges or becomes 0 when the corresponding annealed model generates superdiffusion or subdiffusion, respectively. This result implies that anomalous diffusion in an annealed model is traced by anomaly of the diffusion coefficients in the corresponding quenched model.
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Affiliation(s)
- Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Keiji Saito
- Department of Physics, Keio University, Yokohama 223-8522, Japan
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11
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Pronin KA. Fluctuations and self-averaging in random trapping transport: The diffusion coefficient. Phys Rev E 2020; 101:022132. [PMID: 32168707 DOI: 10.1103/physreve.101.022132] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2018] [Accepted: 02/07/2020] [Indexed: 11/07/2022]
Abstract
On the basis of a self-consistent cluster effective-medium approximation for random trapping transport, we study the problem of self-averaging of the diffusion coefficient in a nonstationary formulation. In the long-time domain, we investigate different cases that correspond to the increasing degree of disorder. In the regular and subregular cases the diffusion coefficient is found to be a self-averaging quantity-its relative fluctuations (relative standard deviation) decay in time in a power-law fashion. In the subdispersive case the diffusion coefficient is self-averaging in three dimensions (3D) and weakly self-averaging in two dimensions (2D) and one dimension (1D), when its relative fluctuations decay anomalously slowly logarithmically. In the dispersive case, the diffusion coefficient is self-averaging in 3D, weakly self-averaging in 2D, and non-self-averaging in 1D. When non-self-averaging, its fluctuations remain of the same order as, or larger than, its average value. In the irreversible case, the diffusion coefficient is non-self-averaging in any dimension. In general, with the decreasing dimension and/or increasing disorder, the self-averaging worsens and eventually disappears. In the cases of weak self-averaging and, especially, non-self-averaging, the reliable reproducible experimental measurements are highly problematic. In all the cases under consideration, asymptotics with prefactors are obtained beyond the scaling laws. Transition between all cases is analyzed as the disorder increases.
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Affiliation(s)
- K A Pronin
- Institute of Biochemical Physics, Russian Academy of Sciences, Moscow, Kosygin Street 4, 119 334, Russia
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12
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Luo L, Yi M. Quenched trap model on the extreme landscape: The rise of subdiffusion and non-Gaussian diffusion. Phys Rev E 2019; 100:042136. [PMID: 31770896 DOI: 10.1103/physreve.100.042136] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2019] [Indexed: 11/07/2022]
Abstract
Non-Gaussian diffusion has been intensively studied in recent years, which reflects the dynamic heterogeneity in the disordered media. The recent study on the non-Gaussian diffusion in a static disordered landscape suggests novel phenomena due to the quenched disorder. In this paper, we further investigate the random walk on this landscape under various effective temperatures μ, which continuously modulate the dynamic heterogeneity. We show in the long-time limit, the trap dynamics on the landscape is equivalent to the quenched trap model in which subdiffusion appears for μ<1. The non-Gaussian distribution of displacement has been analytically estimated for short t of which the stretched exponential tail is expected for μ≠1. Due to the localization in the ensemble of trajectory segments, an additional peak arises in P(x,t) around x=0 even for μ>1. Evolving in different timescales, the peak and the tail of P(x,t) are well split for a wide range of t. This theoretical paper reveals the connections among the subdiffusion, non-Gaussian diffusion, and the dynamic heterogeneity in the static disordered medium. It also offers an insight on how the cell would benefit from the quasistatic disordered structures.
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Affiliation(s)
- Liang Luo
- Department of Physics, Huazhong Agricultural University, Wuhan 430070, China.,Institute of Applied Physics, Huazhong Agricultural University, Wuhan 430070, China
| | - Ming Yi
- School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
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13
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Akimoto T, Saito K. Exact results for first-passage-time statistics in biased quenched trap models. Phys Rev E 2019; 99:052127. [PMID: 31212478 DOI: 10.1103/physreve.99.052127] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2019] [Indexed: 11/07/2022]
Abstract
We provide exact results for the mean and variance of first-passage times (FPTs) of making a directed revolution in the presence of a bias in heterogeneous quenched environments where the disorder is expressed by random traps on a ring with period L. FPT statistics are crucially affected by the disorder realization. In the large-L limit, we obtain exact formulas for the FPT statistics, which are described by the sample mean and variance for waiting times of periodically arranged traps. Furthermore, we find that these formulas are still useful for nonperiodic heterogeneous environments; i.e., the results are valid for almost all disorder realizations. Our findings are fundamentally important for the application of FPT to estimate diffusivity of a heterogeneous environment under a bias.
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Affiliation(s)
- Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Keiji Saito
- Department of Physics, Keio University, Yokohama 223-8522, Japan
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14
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Hou R, Cherstvy AG, Metzler R, Akimoto T. Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing. Phys Chem Chem Phys 2018; 20:20827-20848. [PMID: 30066003 DOI: 10.1039/c8cp01863d] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023]
Abstract
We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.
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Affiliation(s)
- Ru Hou
- School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China.
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15
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Akimoto T, Cherstvy AG, Metzler R. Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks. Phys Rev E 2018; 98:022105. [PMID: 30253516 DOI: 10.1103/physreve.98.022105] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2018] [Indexed: 06/08/2023]
Abstract
Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case, the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance; i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed.
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Affiliation(s)
- Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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16
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Comolli A, Dentz M. Impact of diffusive motion on anomalous dispersion in structured disordered media: From correlated Lévy flights to continuous time random walks. Phys Rev E 2018; 97:052146. [PMID: 29906927 DOI: 10.1103/physreve.97.052146] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2018] [Indexed: 11/07/2022]
Abstract
We elucidate the impact of diffusive motion on the nature of anomalous dispersion in layered and fibrous disordered media. We consider two types of disorder characterized by quenched random velocities and quenched random retardation properties. Purely advective particle motion is ballistic in both disorder models. This changes dramatically in the presence of transverse diffusion, which leads to dimension-dependent disorder sampling. For d≤3 dimensions, heavy-tailed velocity distributions render large-scale particle motion a correlated Lévy flight, while transport in the quenched random retardation model behaves as a biased continuous time random walk with correlated time increments.
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Affiliation(s)
- Alessandro Comolli
- Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain.,Department of Civil and Environmental Engineering, Technical University of Catalonia (UPC), 08034 Barcelona, Spain.,Associated Unit: Hydrogeology Group (UPC-CSIC), 08034 Barcelona, Spain
| | - Marco Dentz
- Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain.,Associated Unit: Hydrogeology Group (UPC-CSIC), 08034 Barcelona, Spain
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17
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Akimoto T, Barkai E, Saito K. Non-self-averaging behaviors and ergodicity in quenched trap models with finite system sizes. Phys Rev E 2018; 97:052143. [PMID: 29906876 DOI: 10.1103/physreve.97.052143] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2018] [Indexed: 06/08/2023]
Abstract
Tracking tracer particles in heterogeneous environments plays an important role in unraveling material properties. These heterogeneous structures are often static and depend on the sample realizations. Sample-to-sample fluctuations of such disorder realizations sometimes become considerably large. When we investigate the sample-to-sample fluctuations, fundamental averaging procedures are a thermal average for a single disorder realization and the disorder average for different disorder realizations. Here we report on non-self-averaging phenomena in quenched trap models with finite system sizes, where we consider the periodic and the reflecting boundary conditions. Sample-to-sample fluctuations of diffusivity greatly exceed trajectory-to-trajectory fluctuations of diffusivity in the corresponding annealed model. For a single disorder realization, the time-averaged mean square displacement and position-dependent observables converge to constants because of the existence of the equilibrium distribution. This is a manifestation of ergodicity. As a result, the time-averaged quantities depend neither on the initial condition nor on the thermal histories but depend crucially on the disorder realization.
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Affiliation(s)
- Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Eli Barkai
- Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel
| | - Keiji Saito
- Department of Physics, Keio University, Yokohama 223-8522, Japan
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18
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Abstract
Non-Gaussian diffusion is commonly considered as a result of fluctuating diffusivity, which is correlated in time or in space or both. In this work, we investigate the non-Gaussian diffusion in static disordered media via a quenched trap model, where the diffusivity is spatially correlated. Several unique effects due to quenched disorder are reported. We analytically estimate the diffusion coefficient D_{dis} and its fluctuation over samples of finite size. We show a mechanism of population splitting in the non-Gaussian diffusion. It results in a sharp peak in the distribution of displacement P(x,t) around x=0, that has frequently been observed in experiments. We examine the fidelity of the coarse-grained diffusion map, which is reconstructed from particle trajectories. Finally, we propose a procedure to estimate the correlation length in static disordered environments, where the information stored in the sample-to-sample fluctuation has been utilized.
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Affiliation(s)
- Liang Luo
- Department of Physics, Huazhong Agricultural University, Wuhan 430070, China
- Institute of Applied Physics, Huazhong Agricultural University, Wuhan 430070, China
| | - Ming Yi
- Department of Physics, Huazhong Agricultural University, Wuhan 430070, China
- Institute of Applied Physics, Huazhong Agricultural University, Wuhan 430070, China
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19
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Meyer P, Barkai E, Kantz H. Scale-invariant Green-Kubo relation for time-averaged diffusivity. Phys Rev E 2017; 96:062122. [PMID: 29347404 DOI: 10.1103/physreve.96.062122] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2017] [Indexed: 06/07/2023]
Abstract
In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time- and ensemble-averaged mean-squared displacement are remarkably different. The ensemble-averaged diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale-invariant nonstationary velocity correlation function with the transport coefficient. Here we obtain the relation between time-averaged diffusivity, usually recorded in single-particle tracking experiments, and the underlying scale-invariant velocity correlation function. The time-averaged mean-squared displacement is given by 〈δ^{2}[over ¯]〉∼2D_{ν}t^{β}Δ^{ν-β}, where t is the total measurement time and Δ is the lag time. Here ν is the anomalous diffusion exponent obtained from ensemble-averaged measurements 〈x^{2}〉∼t^{ν}, while β≥-1 marks the growth or decline of the kinetic energy 〈v^{2}〉∼t^{β}. Thus, we establish a connection between exponents that can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant D_{ν}. We demonstrate our results with nonstationary scale-invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. If the averaged kinetic energy is finite, β=0, the time scaling of 〈δ^{2}[over ¯]〉 and 〈x^{2}〉 are identical; however, the time-averaged transport coefficient D_{ν} is not identical to the corresponding ensemble-averaged diffusion constant.
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Affiliation(s)
- Philipp Meyer
- Max Planck Institute for the Physics of Complex Systems Noethnitzer Strasse 38 D 01187 Dresden, Germany
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel
| | - Holger Kantz
- Max Planck Institute for the Physics of Complex Systems Noethnitzer Strasse 38 D 01187 Dresden, Germany
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20
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Russian A, Dentz M, Gouze P. Self-averaging and weak ergodicity breaking of diffusion in heterogeneous media. Phys Rev E 2017; 96:022156. [PMID: 28950545 DOI: 10.1103/physreve.96.022156] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2017] [Indexed: 06/07/2023]
Abstract
Diffusion in natural and engineered media is quantified in terms of stochastic models for the heterogeneity-induced fluctuations of particle motion. However, fundamental properties such as ergodicity and self-averaging and their dependence on the disorder distribution are often not known. Here, we investigate these questions for diffusion in quenched disordered media characterized by spatially varying retardation properties, which account for particle retention due to physical or chemical interactions with the medium. We link self-averaging and ergodicity to the disorder sampling efficiency R_{n}, which quantifies the number of disorder realizations a noise ensemble may sample in a single disorder realization. Diffusion for disorder scenarios characterized by a finite mean transition time is ergodic and self-averaging for any dimension. The strength of the sample to sample fluctuations decreases with increasing spatial dimension. For an infinite mean transition time, particle motion is weakly ergodicity breaking in any dimension because single particles cannot sample the heterogeneity spectrum in finite time. However, even though the noise ensemble is not representative of the single-particle time statistics, subdiffusive motion in q≥2 dimensions is self-averaging, which means that the noise ensemble in a single realization samples a representative part of the heterogeneity spectrum.
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Affiliation(s)
- Anna Russian
- Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano, Milano, Italy
| | - Marco Dentz
- Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain
| | - Philippe Gouze
- Géosciences, Université de Montpellier, CNRS, Montpellier, France
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21
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Yamamoto E, Akimoto T, Kalli AC, Yasuoka K, Sansom MSP. Dynamic interactions between a membrane binding protein and lipids induce fluctuating diffusivity. SCIENCE ADVANCES 2017; 3:e1601871. [PMID: 28116358 PMCID: PMC5249258 DOI: 10.1126/sciadv.1601871] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2016] [Accepted: 12/07/2016] [Indexed: 05/08/2023]
Abstract
Pleckstrin homology (PH) domains are membrane-binding lipid recognition proteins that interact with phosphatidylinositol phosphate (PIP) molecules in eukaryotic cell membranes. Diffusion of PH domains plays a critical role in biological reactions on membrane surfaces. Although diffusivity can be estimated by long-time measurements, it lacks information on the short-time diffusive nature. We reveal two diffusive properties of a PH domain bound to the surface of a PIP-containing membrane using molecular dynamics simulations. One is fractional Brownian motion, attributed to the motion of the lipids with which the PH domain interacts. The other is temporally fluctuating diffusivity; that is, the short-time diffusivity of the bound protein changes substantially with time. Moreover, the diffusivity for short-time measurements is intrinsically different from that for long-time measurements. This fluctuating diffusivity results from dynamic changes in interactions between the PH domain and PIP molecules. Our results provide evidence that the complexity of protein-lipid interactions plays a crucial role in the diffusion of proteins on biological membrane surfaces. Changes in the diffusivity of PH domains and related membrane-bound proteins may in turn contribute to the formation/dissolution of protein complexes in membranes.
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Affiliation(s)
- Eiji Yamamoto
- Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Takuma Akimoto
- Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Antreas C. Kalli
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
- Leeds Institute of Cancer and Pathology, School of Medicine, St. James’s University Hospital, University of Leeds, Leeds LS9 7TF, UK
| | - Kenji Yasuoka
- Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Mark S. P. Sansom
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
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