Modelling run-and-tumble chemotaxis in a shear flow

The biased random walk undergone by chemotactic bacteria such as Escherichia coli will be influenced at the microscopic level by flow in the ambient medium. In this paper, we model swimming bacteria being advected and rotated by a simple shear flow. Under certain scaling assumptions, we obtain an advection-diffusion equation for cell density, when the chemotactic response is small, which shows a coupling between the rotation and chemotaxis. We also present an alternative method for calculating the chemotactic flux in an unbounded region which is valid for more general chemotactic responses.