A free boundary problem for cell motion. (English) Zbl 1363.92004

Summary: The ability 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 [the first author et al., “Initiation of cytoskeletal asymmetry for cell polarization and movement”, J. Theor. Biol. 249, 278–288 (2007)], 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 that 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 biophysical interpretation as sharp polymerization fronts which are experimentally observed in [A. Ponti et al., “Two distinct actin networks drive the protrusion of migrating cells”, Science 305, 1782–1786 (2004)], for example. Further, simulations illustrate the motion of an initially symmetric resting cell and the emergence of sharp actin fronts from initially smooth distributions.


92C17 Cell movement (chemotaxis, etc.)
35K20 Initial-boundary value problems for second-order parabolic equations
35L45 Initial value problems for first-order hyperbolic systems