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Non-homogeneous Random Walks, Subdiffusive Migration of Cells and Anomalous Chemotaxis

Published online by Cambridge University Press:  24 April 2013

S. Fedotov ,
A. O. Ivanov  and
A. Y. Zubarev
S. Fedotov*
Affiliation:
School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK
A. O. Ivanov
Affiliation:
Department of Mathematical Physics, Ural Federal University, Ekaterinburg, 620083, Russia
A. Y. Zubarev
Affiliation:
Department of Mathematical Physics, Ural Federal University, Ekaterinburg, 620083, Russia
*
⋆ Corresponding author. E-mail: sergei.fedotov@manchester.ac.uk
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Abstract

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We derive the fractional Fokker-Planck equation for the density of cells and apply this equation to the anomalous chemotaxis problem. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.

Keywords

anomalous random walkscell migrationaggregation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 28 - 43
Copyright
© EDP Sciences, 2013

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References

Abad, E., Yuste, S. B., Lindenberg, K.. Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous-time random walks. Phys. Rev. E 81 (2010), 031115. CrossRefGoogle ScholarPubMed
Anomalous transport: foundations and applications. Eds. R. Klages, G. Radons, I. M. Sokolov (Wiley-VCH, 2008).
Campos, D., Fedotov, S., Méndez, V.. Anomalous reaction-transport processes: The dynamics beyond the law of mass action. Phys. Rev. E 77 (2008), 061130. CrossRefGoogle ScholarPubMed
Baker, R. E., Yates, Ch. A., Erban, R.. From microscopic to macroscopic descriptions of cell migration on growing domains. Bull. Math. Biology 72 (2010), 719–762. CrossRefGoogle ScholarPubMed
Chechkin, A. V., Gorenflo, R., Sokolov, I. M.. Fractional diffusion in inhomogeneous media. J. Phys. A: Math. Gen 38 (2005), L679. CrossRefGoogle Scholar
D. R. Cox, H. D. Miller. The Theory of Stochastic Processes (Methuen, London, 1965).
Dieterich, P., Klages, R., Preuss, R., Schwab, A.. Anomalous dynamics of cell migration. PNAS J 105 (2008), 459-463. CrossRefGoogle ScholarPubMed
Erban, R., Othmer, H.. From individual to collective behaviour in bacterial chemotaxis. SIAM J. Appl. Math. 65 (2004), No. 2, 361–391. CrossRefGoogle Scholar
Fedotov, S., Iomin, A.. Migration and proliferation dichotomy in tumor-cell invasion. Phys. Rev. Lett. 98 (2007), 118101. CrossRefGoogle Scholar
Fedotov, S., Iomin, A.. Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion. Phys. Rev. E 77 (2008), 031911. CrossRefGoogle ScholarPubMed
Fedotov, S.. Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts. Phys. Rev. E 81 (2010), 011117. CrossRefGoogle ScholarPubMed
Fedotov, S.. Subdiffusion, chemotaxis, and anomalous aggregation. Phys. Rev. E 83 (2011), 021110. CrossRefGoogle Scholar
Fedotov, S., Iomin, A., Ryashko, L.. Non-Markovian models for migration-proliferation dichotomy of cancer cells: Anomalous switching and spreading rate. Phys. Rev. E 84 (2011), 061131. CrossRefGoogle ScholarPubMed
Fedotov, S., Falconer, S.. Subdiffusive master equation with space-dependent anomalous exponent and structural instability Phys. Rev. E 85 (2012), 031132. Google Scholar
W. Feller. An introduction to probability theory and its applications. Volume 2 (Wiley, NY, 1971).
Fenchel, T., Blackburn, N.. Motile chemosensory behaviour of phagotrophic protists: mechanisms for and efficiency in congregating at food patches. Protist 160 (1999), 325–336. CrossRefGoogle Scholar
Henry, B. I., Langlands, T. A. M.. Fractional chemotaxis diffusion equations. Phys. Rev. E 81 (2010), 051102. Google Scholar
Henry, B. I., Langlands, T. A. M., Wearne, S. L.. Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations. Phys. Rev. E 74 (2006) , 031116. CrossRefGoogle ScholarPubMed
Hillen, Th., Othmer, H. G.. The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61 (2000), No. 3, 751–775. Google Scholar
Iomin, A.. A toy model of fractal glioma development under RF electric field treatment. Eur. Phys. J. E 35 (2012), 42. CrossRefGoogle Scholar
Johnston, S. T., Simpson, M. J., Baker, R. E.. Mean-field descriptions of collective migration with strong adhesion. Phys. Rev. E 85 (2012), 051922. CrossRefGoogle Scholar
van Kampen, N. G.. Composite stochastic processes. Physica A 96 (1979) 435-453. CrossRefGoogle Scholar
Khain, E., Katakowski, M., Hopkins, S., Szalad, A., Zheng, X., Jiang, F., Chopp, M.. Collective behavior of brain tumor cells: The role of hypoxia. Phys. Rev. E 83 (2011), 031920. CrossRefGoogle ScholarPubMed
M. M. Meerschaert, A. Sikorskii. Stochastic models for fractional calculus (De Gruyter, Berlin, 2012).
Metzler, R., Barkai, E., Klafter, J.. Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82 (1999), 3563-3567. CrossRefGoogle Scholar
Metzler, R., Klafter, J.. The Random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Reports, 339 (2000) 1-77. CrossRefGoogle Scholar
Mierke, C. T., Frey, B., Fellner, M., Herrmann, M., Fabry, B., Integrin α5β1 facilitates cancer cell invasion through enhanced contractile forces. J. Cell Science, 124 (2011), 369-383. CrossRefGoogle ScholarPubMed
V. Méndez, S. Fedotov, W. Horsthemke, Reaction-transport systems: mesoscopic foundations, fronts, and spatial instabilities. (Springer, Berlin 2010).
Méndez, V., Campos, D., Pagonabarraga, I., Fedotov, S.. Density-dependent dispersal and population aggregation patterns. J. Theor. Biology, 309 (2012), 113-120. CrossRefGoogle Scholar
Nec, Y., Nepomnyashchy, A. A.. Turing instability in sub-diffusive reaction–diffusion systems. J. Phys. A: Math. Theor. 40 (2007), 14687. CrossRefGoogle Scholar
Othmer, H. G., Dunbar, S. R., Alt, W.. Models of dispersal in biological systems. J. Math. Biol. 26 (1988), No. 3, 263–298. CrossRefGoogle ScholarPubMed
Othmer, H. G., Stevens, A.. Aggregation, blow-up and collapse. The ABC’s of generalized taxis, SIAM J. Appl. Math. 57 (1997), 1044–1081. Google Scholar
Orsingher, E., Polito, F.. On a fractional linear birth–death process. Bernoulli 17 (2011), No. 1, 114-137. CrossRefGoogle Scholar
Ridley, A. J., Schwartz, M. A., Burridge, K., Firtel, R. A., Ginsberg, M. H., Borisy, G., Parsons, J. T., Horwitz, A. R.. Cell migration: integrating signals from front to back. Science 302 (2003), 1704-1709. CrossRefGoogle ScholarPubMed
Sagues, F., Shkilev, V. P., Sokolov, I. M., Reaction-subdiffusion equations for the A ⇆ B reaction. Phys. Rev. E 77 (2008), 032102. CrossRefGoogle Scholar
Shkilev, V. P.. Propagation of a subdiffusion reaction front and the “aging” of particles. J. Exp. Theor. Physics, 112 (2011), 711-716. CrossRefGoogle Scholar
Volpert, V. A., Nec, Y., Nepomnyashchy, A. A.. Fronts in anomalous diffusion–reaction systems. Phil. Trans. R. Soc. A 371 (2013), 20120179.CrossRefGoogle ScholarPubMed