Exact model of aerotactic band: From Fokker-Planck equation to band structure and fluid flow.

Abstract

A variety of bacterial species are able to spontaneously assemble into an aerotactic band, a local accumulation at a fixed distance from the air-water interface. Although the phenomenon is long known, its modeling so far is limited to mesoscopic, one-dimensional or numerical descriptions. We investigate band properties at the microscopic scale using exact solutions to the Fokker-Planck equation. First, we show that the interplay between oxygen consumption and tumbling modulation is governed by a third-order nonlinear differential equation relating the oxygen concentration to the aerotactic response. For two model aerotactic behaviors, we present analytical solutions and discuss the resulting band structure. Second, we investigate how an aerotactic band of magnetotactic bacteria in a magnetic field induces a fluid flow, as observed in experiments [Marmol et al., Phys. Rev. Lett. (2025), doi: 10.1103/qrgn-m91t]. In the low-field limit, we determine the bacterial distribution and the active stress tensor. Using the Green function of the hydrodynamic problem, we obtain a prediction for the fluid flow that is both simple and consistent with observations. Altogether, our results provide a model system of aerotactic band and solid ground for analyzing aerotaxis-driven self-organization.