Fractional Patlak-Keller-Segel Equations for Chemotactic Superdiffusion

The long range movement of certain organisms in the presence of a chemoattractant can be governed by long distance runs, according to an approximate Levy distribution. This article clarifies the form of biologically relevant model equations. We derive Patlak--Keller--Segel-like equations involving nonlocal, fractional Laplacians from a microscopic model for cell movement. Starting from a power-law distribution of run times, we derive a kinetic equation in which the collision term takes into account the long range behavior of the individuals. A fractional chemotactic equation is obtained in a biologically relevant regime. Apart from chemotaxis, our work has implications for biological diffusion in numerous processes.

[1]  L. Cesbron.  Anomalous Diffusion Limit of Kinetic Equations in Spatially Bounded Domains , 2016, Communications in Mathematical Physics.

[2]  Nicolas Vauchelet,et al.  Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway , 2015, Journal of Mathematical Biology.

[3]  W. Marsden.  I and J , 2012 .

[4]  J. Nieto,et al.  ABOUT THE KINETIC DESCRIPTION OF FRACTIONAL DIFFUSION EQUATIONS MODELING CHEMOTAXIS , 2016 .

[5]  J. Kennedy,et al.  Pheromone-Regulated Anemotaxis in Flying Moths , 1974, Science.

[6]  Yuhai Tu,et al.  How white noise generates power-law switching in bacterial flagellar motors. , 2005, Physical review letters.

[7]  J. Vilar,et al.  From molecular noise to behavioural variability in a single bacterium , 2004, Nature.

[8]  J. Burczak,et al.  SUPPRESSION OF BLOW UP BY A LOGISTIC SOURCE IN 2D KELLER-SEGEL SYSTEM WITH FRACTIONAL DISSIPATION , 2016, 1609.03935.

[9]  J. Dobnikar,et al.  E. coli superdiffusion and chemotaxis-search strategy, precision, and motility. , 2009, Biophysical journal.

[10]  H. Berg,et al.  Chemotaxis in Escherichia coli analysed by Three-dimensional Tracking , 1972, Nature.

[11]  J. Klafter,et al.  L\'evy walks , 2014, 1410.5100.

[12]  Wolfgang Alt,et al.  Singular perturbation of differential integral equations describing biased random walks. , 1981 .

[13]  Andrea J. Liu,et al.  Generalized Lévy walks and the role of chemokines in migration of effector CD8+ T cells , 2012, Nature.

[14]  M. Iijima,et al.  Signaling mechanisms for chemotaxis , 2011, Development, growth & differentiation.

[15]  M. Meerschaert,et al.  Fractional vector calculus for fractional advection–dispersion , 2006 .

[16]  Liang Li,et al.  Persistent Cell Motion in the Absence of External Signals: A Search Strategy for Eukaryotic Cells , 2008, PloS one.

[17]  Fractional diffusion equation for an n-dimensional correlated Lévy walk. , 2016, Physical review. E.

[18]  H. Berg.  Random Walks in Biology , 2018 .

[19]  J. Bonner,et al.  The social amoebae. , 2008, Scientific American.

[20]  J. Roquejoffre,et al.  The Influence of Fractional Diffusion in Fisher-KPP Equations , 2012, 1202.6072.

[21]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[22]  H. Othmer,et al.  Models of dispersal in biological systems , 1988, Journal of mathematical biology.

[23]  D. Dormann,et al.  Chemotactic cell movement during Dictyostelium development and gastrulation. , 2006, Current opinion in genetics & development.

[24]  Nicola Bellomo,et al.  Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues , 2015 .

[25]  H. Larralde,et al.  Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi) , 2003, Behavioral Ecology and Sociobiology.

[26]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[27]  P. Kareiva,et al.  Analyzing insect movement as a correlated random walk , 1983, Oecologia.

[28]  R. Macnab,et al.  The gradient-sensing mechanism in bacterial chemotaxis. , 1972, Proceedings of the National Academy of Sciences of the United States of America.

[29]  S. Ward.  Chemotaxis by the nematode Caenorhabditis elegans: identification of attractants and analysis of the response by use of mutants. , 1973, Proceedings of the National Academy of Sciences of the United States of America.

[30]  N. Wingreen,et al.  Accuracy of direct gradient sensing by single cells , 2008, Proceedings of the National Academy of Sciences.

[31]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[32]  M. Surette,et al.  Signal transduction in bacterial chemotaxis , 1992, The Journal of biological chemistry.

[33]  Injong Rhee,et al.  On the levy-walk nature of human mobility , 2011, TNET.

[34]  J. Timmer,et al.  Design principles of a bacterial signalling network , 2005, Nature.

[35]  S. Fedotov,et al.  Emergence of Lévy walks in systems of interacting individuals. , 2016, Physical review. E.

[36]  J. Bonner,et al.  Determination of the active portion of the folic acid molecule in cellular slime mold chemotaxis , 1975, Journal of bacteriology.

[37]  L. Segel,et al.  Model for chemotaxis. , 1971, Journal of theoretical biology.

[38]  Jeffrey Wyckoff,et al.  Cell migration in tumors. , 2005, Current opinion in cell biology.

[39]  D A Lauffenburger,et al.  Localized bacterial infection in a distributed model for tissue inflammation , 1983, Journal of mathematical biology.

[40]  P. V. van Haastert,et al.  Food Searching Strategy of Amoeboid Cells by Starvation Induced Run Length Extension , 2009, PloS one.

[41]  K. Painter,et al.  Travelling Waves in Hyperbolic Chemotaxis Equations , 2011, Bulletin of mathematical biology.

[42]  M. Pärtel,et al.  A synthesis of empirical plant dispersal kernels , 2017 .

[43]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations II: Chemotaxis Equations , 2002, SIAM J. Appl. Math..

[44]  Gil Ariel,et al.  Swarming bacteria migrate by Lévy Walk , 2015, Nature Communications.

[45]  Pablo A Iglesias,et al.  Chemoattractant signaling in dictyostelium discoideum. , 2004, Annual review of cell and developmental biology.

[46]  C. Patlak.  Random walk with persistence and external bias , 1953 .

[47]  W. Alt.  Biased random walk models for chemotaxis and related diffusion approximations , 1980, Journal of mathematical biology.

[48]  Lee A. Segel,et al.  A Theoretical Study of Receptor Mechanisms in Bacterial Chemotaxis , 1977 .

[49]  M. Levandowsky,et al.  Random movements of soil amebas , 1997 .