Carolin Gietz

Prof. Dr. Leonid Berlyand (Pennsylvania State University): Sharp interface limit in a phase field model of cell motility

Wednesday, 04.05.2016 15:15 im Raum M5

Mathematik und Informatik

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.

Angelegt am Tuesday, 05.04.2016 11:40 von Carolin Gietz
Geändert am Thursday, 11.05.2017 18:31 von Frank Wübbeling
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