Microbial Dynamics in a Slit-Pore With Sticky Boundaries

The run and tumble behavior by which microbial motility is often characterized can be modeled using the concept of Levy motion. The Fokker-Planck equation for a Levy motion looks like the classical diffusion equation for Brownian motion except that the order of highest derivative is fractional. Fractional derivatives describe non-local processes and hence formulating and solving boundary value problems are more complicated than the classical cases. We examine Levy motion in a slit-pore (region between two infinite parallel plates) with sticky boundaries, i.e. boundaries that absorb particles for a random amount of time and then release them back into the system. The amount of sorbed time is allowed to vary over a wide range of values thus ensuring model's applicability to a variety of problems. The Lagrangian stochastic differential equation, driven by a Levy measure with a drift, is used to numerically compute, via a Monte Carlo simulation, the mean first passage time (MFPT) and other relevant statistical quantities associated with the breakthrough curves. Qualitative analysis of these quantities is performed with respect to non-dimensional parameters obtained by scaling the Eulerian Fokker-Planck equation. Further analysis leads to the development of a set of equations that can implicitly be used to compute the MFPT. Results from the theoretically derived equations match Monte Carlo simulations.