Prof. Dr. Leonid Berlyand (Pennsylvania State University): Sharp interface limit in a phase field model of cell motility
Mittwoch, 04.05.2016 15:15 im Raum M5
We study the motion of a eukaryotic cell on a substrate and investigate the dependence
of this motion on key physical parameters such as strength of protrusion by actin filaments
and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn
equation for the scalar phase field function coupled with a vectorial parabolic equation for
the orientation of the actin filament network.
The two key properties of this system are (i) presence of gradients in the coupling terms
and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive
the equation of the motion of the cell boundary, which is mean curvature motion modifed
by a novel nonlinear term. We establish the existence of two distinct regimes of the physical
parameters. In the subcritical regime, the well-posedness of the problem is proved by my
Ph. D. student M. Mizuhara. Our main focus is the supercritical regime where we established
surprising features of the motion of the interface such as discontinuities of velocities, and
hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in
the 2D model. We also proved existence of traveling waves.
Because of properties (i)-(ii), classical comparison principle techniques do not apply to
this system. Furthermore, the system can not be written in a form of gradient
ow, which is why T-convergence techniques also can not be used. This is joint work with V. Rybalko
and M. Potomkin.