The interface scattering law for run-and-tumble kinetic models

Abstract

For microorganisms with run-and-tumble motility, kinetic models provide a well-established description of transport in smoothly varying environments. Sharp signal changes, however, can induce asymmetric interface responses. A motivating example is the response to dark-light boundaries: cells moving into the light largely maintain their direction, whereas those entering the dark are effectively scattered back. Through an interface-layer analysis of pathway-based kinetic equations, we formally derive effective interface conditions for the limiting run-and-tumble model. These velocity-nonlocal conditions capture non-adiabatic scattering near signal discontinuities, linking intracellular sensory dynamics to transport across sharp environmental boundaries. For a two-velocity model, they reduce to an explicit transmission matrix. We assess the interface law numerically using biologically motivated parameters for E. coli chemotaxis.