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The fractional diffusion limit of a kinetic model with biochemical pathway

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Abstract

Kinetic-transport equations that take into account the intracellular pathways are now considered as the correct description of bacterial chemotaxis by run and tumble. Recent mathematical studies have shown their interest and their relations to more standard models. Macroscopic equations of Keller–Segel type have been derived using parabolic scaling. Due to the randomness of receptor methylation or intracellular chemical reactions, noise occurs in the signaling pathways and affects the tumbling rate. Then comes the question to understand the role of an internal noise on the behavior of the full population. In this paper we consider a kinetic model for chemotaxis which includes biochemical pathway with noises. We show that under proper scaling and conditions on the tumbling frequency as well as the form of noise, fractional diffusion can arise in the macroscopic limits of the kinetic equation. This gives a new mathematical theory about how long jumps can be due to the internal noise of the bacteria.

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Authors and Affiliations

  1. Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, 75005, Paris, France

    Benoît Perthame

  2. Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC , V5A 1S6, Canada

    Weiran Sun

  3. Institute of Natural Sciences and Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Min Tang

Authors
  1. Benoît Perthame
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  2. Weiran Sun
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  3. Min Tang
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Corresponding author

Correspondence to Benoît Perthame.

Additional information

B. Perthame: This author has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant Agreement No. 740623). W. Sun: This author is partially supported by NSERC Discovery Grant No. R611626. M. Tang: This author is partially supported by NSFC 11301336 and 91330203.

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Perthame, B., Sun, W. & Tang, M. The fractional diffusion limit of a kinetic model with biochemical pathway. Z. Angew. Math. Phys. 69, 67 (2018). https://doi.org/10.1007/s00033-018-0964-3

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  • DOI: https://doi.org/10.1007/s00033-018-0964-3

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