An Analytic Approach to the Modeling of Microbial Locomotion in Porous Media

Much research has been done to study swimming cell motility in bulk fluids. However, many swimming cells’ environments are rife with porous, microstructured materials and boundaries. The goal of the overarching project (which this study is a part of) is to establish a quantitative framework that models microbial locomotion in porous media. In this study, we focus on an analytic approach that complements the overarching project’s lab experiments and computer models. In the low particle density case, we consider the microscopic model of a random walk with one-step memory on the dual graph defined by a porous media’s fluid pockets and tunnels. We study the long-term macroscopic particle distribution and present an analytic solution to our generalization of the overarching project’s computer-generated numerical solutions. We present properties of our analytic solution that can be tested in an experimental setting to determine the microscopic model’s accuracy and validity. Our approach uses Markov analysis to prove that the stationary distribution of a large class of n-dimensional porous media can be split into n+ 1 probability classes, so that the ratio of interior particles to boundary particles is proportional to the number