Phase field models are very efficient in computational studies of moving deformable in- terfaces. We will present a phase field system that models the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters. This system consists of two PDEs: the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. Two key features of this system are (i) the gradients coupling and (ii) volume preservation constraints.

We pass to the sharp interface limit (SIL), which reduces the system to a single scalar equation and show that the motion of the cell boundary is the mean curvature motion modified by a novel nonlinear term. Numerical and analytical studies of the SIL equation reveal the existence of two distinct regimes of the physical parameters. The subcritical regime was studied numerically and analytically by my Ph. D. student M. Mizuhara. Our main focus is the supercritical regime. Here we established surprising features of the motion of the interface such as discontinuities of velocities, hysteresis in the 1D model, instability of the circular shape, rise of asymmetry in the 2D model, and existence of non-trivial traveling waves.

Because of features (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. Instead, our derivation of SIL is based on a special asymptotic ansatz.

This is joint work with V. Rybalko and M. Potomkin.