Mathematical description of bacterial traveling pulses

TL;DR

This paper introduces a new macroscopic mathematical model for bacterial traveling pulses based on kinetic descriptions of individual bacterial movement, aligning well with experimental data and offering insights into collective bacterial behavior.

Contribution

The work develops a novel macroscopic model derived from kinetic descriptions of bacteria, accurately capturing experimental pulse features and enabling analysis and simulation.

Findings

Model reproduces pulse speed and tail size accurately

Captures asymmetry and transition in bacterial collective behavior

Coefficients derived from cellular scale influence macroscopic dynamics

Abstract

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on {\em E. coli} have shown precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at a macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This model can account for recent experimental observations with {\em E. coli}. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition we can capture quantitatively the main characteristics of the pulse such as the speed and the relative size of tails. This work opens several experimental and theoretical perspectives. Coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance the stiffness of the signal integration process turns out to have a strong effect on collective motion. Furthermore the bottom-up scaling allows to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion.