The fractional diffusion limit of a kinetic model with biochemical pathway

TL;DR

This paper investigates how internal biochemical noise in bacterial chemotaxis models leads to fractional diffusion in the macroscopic limit, revealing the impact of internal randomness on population movement.

Contribution

It introduces a kinetic model incorporating biochemical pathway noise and demonstrates that fractional diffusion emerges under specific scaling and noise conditions.

Findings

Fractional diffusion appears in the macroscopic limit due to internal noise.

Proper scaling and noise conditions are crucial for fractional diffusion emergence.

The study provides a new mathematical framework linking internal noise to long-range bacterial movement.

Abstract

Kinetic-transport equations that take into account the intra-cellular 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 intra-cellular 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.