Coarse-graining bacterial diffusion in disordered media to surface states

Bacterial motility in spatially structured environments impacts a variety of natural and engineering processes. Constructing models to predict, control, and design bacterial motility for these processes remains challenging because bacteria and active swimmers have complex interactions with surfaces and because the precise environment geometry is unknown. Here, we present a method for deriving bacterial diffusion coefficients in disordered media in terms of cell and environmental parameters. The approach abstracts the dynamics in the full geometry to "surface states," which encode how cells interact with surfaces in the environment. Then, a long-time diffusion equation can be derived analytically from the state model. Applying this method to a run-and-tumble particle in a 2D Lorentz gas environment provides analytical predictions that show good agreement with particle simulations. Like past studies, we observe that the diffusivity depends nonmonotonically on the cell's run length. Using the analytical expressions, we derive the optimal run length, revealing an intuitive dependence on environmental length scales. Furthermore, we find that rescaling length and time by the average distance and time between trap events collapses all of the diffusivities onto a single curve, which we derive analytically. Thus, our approach extracts interpretable, macroscopic diffusive behavior from complex microscopic dynamics, and provides tools and intuitions for understanding bacterial diffusion in disordered media.