Analysis of a nonlocal and nonlinear Fokker-Planck model for cell crawling migration

Cell movement has essential functions in development, immunity, and cancer. Various cell migration patterns have been reported and a general rule has recently emerged, the so-called universal coupling between cell speed and cell persistence [P. Maiuri et al., Cell, 161 (2015), pp. 374-386]. This rule says that cell persistence, which quantifies the straightness of trajectories, is robustly coupled to migration speed. In [P. Maiuri et al., Cell, 161 (2015), pp. 374-386], the advection of polarity cues by a dynamic actin cytoskeleton undergoing ows at the cellular scale was proposed as a first explanation of this universal coupling. Here, following ideas proposed in that work, we present and study a simple model to describe motility initiation in crawling cells. It consists of a nonlinear and nonlocal Fokker-Planck equation with a coupling involving the trace value on the boundary. In the one-dimensional case we characterize the following behaviors: solutions are global if the mass is below the critical mass, and they can blow up in finite time above the critical mass. In addition, we prove a quantitative convergence result using relative entropy techniques.