Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution

Fedotov, Sergei; Han, Daniel; Zubarev, A. Yu.; Johnston, Mark; Allan, Victoria

doi:10.1098/rsta.2020.0317

Cited by 7 publications

(4 citation statements)

References 40 publications

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“…This idea for standard diffusion has been considered by theories 129 of "diffusing diffusivity" [50][51][52] and such heterogeneity was 130 demonstrated to be advantageous for biochemical processes 131 triggered by first arrival [29]. Moreover, heterogeneity can 132 be modeled in many ways such as a nonconstant diffusion 133 coefficient [53][54][55] or a nonconstant anomalous exponent 134 [41,48,[56][57][58]. Dichotomously alternating force fields in the 135 fractional Fokker-Planck equation have also been used to 136 model temporal heterogeneity [59].…”

Section: Fractional Master Equation With Random Transition Probabilitiesmentioning

confidence: 99%

“…The aim of this paper is to explore the effects of popula-64 tion heterogeneity, characterized by a distribution in transition 65 probability, on the fractional master equation. Below, we 66 demonstrate how heterogeneity changes the fundamental 67 characteristic of the fractional master equation, used in 68 modeling many biological processes that exhibit anoma-69 lous trapping [41,42]. The effective underlying random walk 70 exhibits self-reinforcement due to the ensemble-averaged 71 conditional transition rates increasing as previous steps ac-72 cumulate.…”

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confidence: 95%

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Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects

Fedotov

Han

2023

Phys. Rev. E

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We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble selfreinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.

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Section: Fractional Master Equation With Random Transition Probabilitiesmentioning

confidence: 99%

mentioning

confidence: 95%

Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects

Fedotov

Han

2023

Phys. Rev. E

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“…The authors demonstrate that the appearance of the clusters can significantly decrease the thermal effect. Application of the theory of nonlinear transport to biological systems is continued by Fedotov et al [30]. The authors formulate the space-dependent variable-order fractional master equation to model clustering of particles, organelles, inside living cells.…”

Section: The General Content Of the Issuementioning

confidence: 99%

Transport phenomena in complex systems (part 1)

Alexandrov

Zubarev

2021

Phil. Trans. R. Soc. A.

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The issue, in two parts, is devoted to theoretical, computational and experimental studies of transport phenomena in various complex systems (in porous and composite media; systems with physical and chemical reactions and phase and structural transformations; in biological tissues and materials). Various types of these phenomena (heat and mass transfer; hydrodynamic and rheological effects; electromagnetic field propagation) are considered. Anomalous, relaxation and nonlinear transport, as well as transport induced by the impact of external fields and noise, is the focus of this issue. Modern methods of computational modelling, statistical physics and hydrodynamics, nonlinear dynamics and experimental methods are presented and discussed. Special attention is paid to transport phenomena in biological systems (such as haemodynamics in stenosed and thrombosed blood vessels magneto-induced heat generation and propagation in biological tissues, and anomalous transport in living cells) and to the development of a scientific background for progressive methods in cancer, heart attack and insult therapy (magnetic hyperthermia for cancer therapy, magnetically induced circulation flow in thrombosed blood vessels and non-contact determination of the local rate of blood flow in coronary arteries). The present issue includes works on the phenomenological study of transport processes, the derivation of a macroscopic governing equation on the basis of the analysis of a complicated internal reaction and the microscopic determination of macroscopic characteristics of the studied systems. This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

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“…Notice finally that similar approach to interacting particle systems leading to the new variable order fractional kinetic equations was developed in [11] and [12]. Quite different variable order kinetic equations are analysed in [3].…”

Section: Introductionmentioning

confidence: 99%

CTRW approximations for fractional equations with variable order

Kolokoltsov¹

2022

Preprint

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The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different however. The diffusion coefficient can be made responsible for the size of jumps or for the intensity of jumps. The "rough" diffusion limit does not feel the difference. The situation changes, if we model jump-type approximations via CTRW with nonexponential waiting times. If we make the diffusion coefficient responsible for the size of jumps and take waiting times from the domain of attraction of an α-stable law with a constant intensity α, then the standard scaling would lead in the limit of small jumps and large intensities to the most standard fractional diffusion equation. However, if we choose the CTRW approximations with fixed jump sizes and use the diffusion coefficient to distinguish intensities at different points, then we obtain in the limit the equations with variable position-dependent fractional derivatives. In this paper we build rigorously these approximations and prove their convergence to the corresponding fractional equations for the cases of multidimensional diffusions and more general Feller processes.

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Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution

Cited by 7 publications

References 40 publications

Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects

Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects

Transport phenomena in complex systems (part 1)

CTRW approximations for fractional equations with variable order

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