Chemotactic reaction enhancement in one dimension

Chemotaxis, which involves the directed movement of cells in response to a chemical gradient, plays a crucial role in a broad variety of biological processes. Examples include bacterial motion, the development of single-cell or multicellular organisms, and immune responses. Chemotaxis directs bacteria’s movement to find food (e.g., glucose) by swimming toward the highest concentration of food molecules. In multicellular organisms, chemotaxis is critical to early development (e.g., movement of sperm towards the egg during fertilization). Chemotaxis also helps mobilize phagocytic and immune cells at sites of infection, tissue injury, and thus facilitates immune reactions. In this paper, we study a PDE system that describes chemotactic processes in one dimension, which may correspond to a thin channel, the setting relevant in many applications: for example, spermatozoa progression to the ovum inside a Fallopian tube or immune response in a blood vessel. Our objective is to obtain qualitatively precise estimates on how chemotaxis improves reaction efficiency, when compared to purely diffusive situation. The techniques we use to achieve this goal include a variety of comparison principles and analysis of mass transport for a class of Fokker–Planck operators.

Keywords

chemotaxis modeling, Keller–Segel equation, reaction-diffusion systems, reaction enhancement

2010 Mathematics Subject Classification

35K57, 35Q92, 37N25, 92C17

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The authors acknowledge partial support of the NSF-DMS grants 1848790 and 2006372. AK has also been partially supported by Simons Foundation.

Received 12 April 2021

Received revised 15 November 2023

Accepted 15 November 2023

Published 15 July 2024