  In the framework of a phase field model of a single cell crawling on a
substrate, we investigate how the properties of the cell membrane affect the
shape and motility of the cell. Since the membrane influences the cell dynamics
on multiple levels and provides a nontrivial feedback, we consider the
following fundamental interactions: (i) the reduction of the actin
polymerization rate by membrane tension; (ii) area conservation of the cell's
two-dimensional cross-section vs. conservation of its circumference (i.e.
membrane inextensibility); and (iii) the contribution from the membrane's
bending energy to the shape and integrity of the cell. As in experiments, we
investigate two pertinent observables -- the cell's velocity and its aspect
ratio. We find that the most important effect is the feedback of membrane
tension on the actin polymerization. Bending rigidity has only minor effects,
visible mostly in dynamic reshaping events, as exemplified by collisions of the
cell with an obstacle.
