A free boundary problem for cell motion

The ability of a large variety of eukaryotic cells to actively move along different substrates plays a vital role in many biological processes. A key player in these processes is the cytoskeleton. In [7] we introduced a minimal hyperbolic-parabolic model for the reorganization of the actin cytoskeleton of a generic cell resting on a flat substrate and turning into a polarized state upon some external cue. In this paper we derive moving boundary conditions for the same cytoskeleton model and by this allow for the description of actual motion. For the free boundary problem we prove short time well-posedness for a wide class of initial conditions and analyze the emergence of Dirac measures in the densities of actin filament tips. These have a direct biophysical interpretation as sharp polymerization fronts which are experimentally observed in [19], for example. Further, numerical results will illustrate both, the motion of an initially symmetric resting cell and the emergence of sharp fronts of actin filaments from initially smooth distributions.