An extension of generalized Taylor dispersion in unbounded homogeneous shear flows to run-and-tumble chemotactic bacteria

Abstract

In the absence of flow, the biased random walk of bacteria such as Escherichia coli is modeled by straight runs punctuated by random changes in direction, tumbles. We model chemotaxis by allowing the tumble rate of a run to depend on the component of swimming velocity in the direction of the chemoattractant gradient. In the well-studied situation of weak bias in tumble rate, bacteria disperse over a diffusive time scale, and the evolution of density satisfies the classic Keller–Segel advection-diffusion equation. In this paper, we model swimming bacteria being advected and rotated by an unbounded homogeneous shear flow. The flow field alters the trajectories of individuals and thus affects the macroscopic dispersion of a population. We adapt the formal framework of generalized Taylor dispersion theory to make it applicable for run-and-tumble bacteria with an arbitrary bias in tumble rate. This enables us to obtain a macroscopic description of the dispersion of bacteria. For the particular case of simple shear flow, we calculate explicitly the effect of flow on the diffusion tensor and mean swimming velocity.