Continuous-Time Random Walks on Fractals

Blumen, A.; Klafter, J.; White, Bebo; Zumofen, G.

doi:10.1103/physrevlett.53.1301

Cited by 171 publications

(55 citation statements)

References 21 publications

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“…This is expected from diffusion on a fractal but not from a CTRW (8). In theory, the two processes can coexist, and thus we investigated the combination of a CTRW and a fractal (33,34). Meroz et al recently simulated a CTRW on a fractal structure (34).…”

Section: Resultsmentioning

confidence: 99%

Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking

Weigel

Simon

Tamkun

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

595

699

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Diffusion in the plasma membrane of living cells is often found to display anomalous dynamics. However, the mechanism underlying this diffusion pattern remains highly controversial. Here, we study the physical mechanism underlying Kv2.1 potassium channel anomalous dynamics using single-molecule tracking. Our analysis includes both time series of individual trajectories and ensemble averages. We show that an ergodic and a nonergodic process coexist in the plasma membrane. The ergodic process resembles a fractal structure with its origin in macromolecular crowding in the cell membrane. The nonergodic process is found to be regulated by transient binding to the actin cytoskeleton and can be accurately modeled by a continuous-time random walk. When the cell is treated with drugs that inhibit actin polymerization, the diffusion pattern of Kv2.1 channels recovers ergodicity. However, the fractal structure that induces anomalous diffusion remains unaltered. These results have direct implications on the regulation of membrane receptor trafficking and signaling.anomalous subdiffusion | Brownian motion | continuous time random walk | single-particle tracking T he plasma membrane is a highly complex system with a dynamic organization required to maintain many fundamental processes that include signal transduction, receptor recognition, endocytic transport, and cell-cell adhesion. The study of the diffusion pattern of transmembrane proteins and lipids grants biophysical information on membrane organization, structure, and dynamics. In particular, single-molecule tracking provides insight into the interaction of membrane proteins with their surroundings. Stochastic molecular transport is described by the probability distribution of displacements and how it evolves over time. This distribution is called a propagator, and in the case of Brownian motion it is Gaussian. Diffusion processes that deviate from Brownian motion are considered anomalous, and they involve propagators that may or may not be Gaussian. For example, diffusion in a fractal has a stretched Gaussian propagator (1).Experimental and theoretical work suggests there is a correlation between macromolecular crowding and anomalous diffusion (2-4). However, the mechanism behind hindered diffusion in living cells remains controversial. Several models have been proposed, including membrane compartmentalization, receptor and cytoskeleton binding, and membrane heterogeneity (lipid rafts). Particle trajectories are frequently characterized by their mean square displacement (MSD) (5). A Brownian particle in a 2D space yields an MSD hΔr 2 ðtÞi ¼ 4Dt. However, in many systems the MSD scales as a sublinear power law, indicating anomalous diffusion. In general hΔr 2 ðtÞi ∝ t γ , where γ is the anomaly exponent. Anomalous subdiffusion is manifested by a characteristic exponent γ < 1, and anomalous superdiffusion by γ > 1.Two biologically relevant processes are recognized to induce anomalous subdiffusion in the plasma membrane: (i) transient immobilization and (ii) geometrical i...

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Section: Resultsmentioning

confidence: 99%

Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking

Weigel

Simon

Tamkun

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

595

699

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“…First, subdiffusion could result in some systems from a combination of both the dynamic (CTRW) and static (diffusion on fractal) mechanisms. Interestingly, our approach can be adapted to study the example of CTRWs on a fractal that models such situations (54). Indeed, the same decomposition as in Eq.…”

Section: Discussionmentioning

confidence: 99%

“…20 (see also ref. 54). First-passage observables therefore permit us, in principle, to isolate and characterize each of the CTRW and fractal mechanisms even when they are both involved simultaneously.…”

Section: Discussionmentioning

confidence: 99%

Probing microscopic origins of confined subdiffusion by first-passage observables

Condamin

Tejedor

Voituriez

et al. 2008

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

193

165

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Subdiffusive motion of tracer particles in complex crowded environments, such as biological cells, has been shown to be widespread. This deviation from Brownian motion is usually characterized by a sublinear time dependence of the mean square displacement (MSD). However, subdiffusive behavior can stem from different microscopic scenarios that cannot be identified solely by the MSD data. In this article we present a theoretical framework that permits the analytical calculation of first-passage observables (mean first-passage times, splitting probabilities, and occupation times distributions) in disordered media in any dimensions. This analysis is applied to two representative microscopic models of subdiffusion: continuous-time random walks with heavy tailed waiting times and diffusion on fractals. Our results show that first-passage observables provide tools to unambiguously discriminate between the two possible microscopic scenarios of subdiffusion. Moreover, we suggest experiments based on first-passage observables that could help in determining the origin of subdiffusion in complex media, such as living cells, and discuss the implications of anomalous transport to reaction kinetics in cells.anomalous diffusion ͉ cellular transport ͉ reaction kinetics ͉ random motion I n the past few years, subdiffusion has been observed in an increasing number of systems (1, 2), ranging from physics (3, 4) or geophysics (5) to biology (6, 7). In particular, living cells provide striking examples for systems where subdiffusion has been repeatedly observed experimentally, either in the cytoplasm (6-9), the nucleus (10, 11), or the plasmic membrane (12-14). However, the microscopic origin of subdiffusion in cells is still debated, even if believed to be caused by crowding effects in a wide sense, as indicated by in vitro experiments (15-18).The subdiffusive behavior significantly deviates from the usual Gaussian solution of the simple diffusion equation, and is usually characterized by a mean square displacement (MSD) (1) that scales as ͗⌬r 2 ͘ ϳ t ␤ with ␤ Ͻ 1. Such a scaling law can be obtained from a few models based on different underlying microscopic mechanisms. Here, we focus on two possibilities (a third classical model of subdiffusion is given by the fractional Brownian motion that concerns processes with long-range correlations): (i) The first class of models that we consider stems from continuous time random walks (CTRWs) (1, 19) and their continuous limit described by fractional diffusion equations (1, 20). The anomalous behavior in these models originates from a heavy-tailed distribution of waiting times (21): at each step the walker lands on a trap, where it can be trapped for extended periods of time. When dealing with a tracer particle, traps can be out-of-equilibrium chemical binding configurations (22, 23), and the waiting times are then the dissociation times; traps can also be realized by the free cages around the tracer in a hard sphere-like crowded environment, and the waiting times are the life times of the ...

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“…Namely, both components with long life and with high back correlated motion contribute to the slowing down of the decay of F s (k,t). The combination of the temporal and spatial mechanism is thoretically treated by Blumen et al 6 These two mechanisms are distinguishable in a microscopic point of view. Results of further analysis of these mechanisms will be shown in a separate paper.…”

Section: B Fractal Dimension Analysismentioning

confidence: 99%

Fracton excitation and Lévy flight dynamics in alkali silicate glasses

1997

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We have examined the relaxation behavior of alkali metal ions in lithium metasilicate glasses by means of molecular dynamics simulation. We have observed a change of slope of the mean squared displacement at ϳ300 ps. In shorter time regions, localized motion of lithium ions within neighboring sites is observed, which is caused by the small fracton dimension ͑fracton excitation͒. On the other hand, an accelerated motion of particles due to cooperative jumps is found, which characterizes the diffusion and conduction mechanisms of the alkali metal ions in longer time regions. The dynamics of accelerated motion is discussed in relation to Lévy flight dynamics. ͓S0163-1829͑97͒03510-8͔

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Continuous-Time Random Walks on Fractals

Cited by 171 publications

References 21 publications

Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking

Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking

Probing microscopic origins of confined subdiffusion by first-passage observables

Fracton excitation and Lévy flight dynamics in alkali silicate glasses

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