  This letter is concerned with asymptotic analysis of a PDE model for motility
of a eukaryotic cell on a substrate. This model was introduced in [1], where it
was shown numerically that it successfully reproduces experimentally observed
phenomena of cell-motility such as a discontinuous onset of motion and shape
oscillations. The model consists of a parabolic PDE for a scalar phase-field
function coupled with a vectorial parabolic PDE for the actin filament network
(cytoskeleton). We formally derive the sharp interface limit (SIL), which
describes the motion of the cell membrane and show that it is a volume
preserving curvature driven motion with an additional nonlinear term due to
adhesion to the substrate and protrusion by the cytoskeleton. In a 1D model
problem we rigorously justify the SIL, and, using numerical simulations,
observe some surprising features such as discontinuity of interface velocities
and hysteresis. We show that nontrivial traveling wave solutions appear when
the key physical parameter exceeds a certain critical value and the potential
in the equation for phase field function possesses certain asymmetry.
