  We study a minimal model of a crawling eukaryotic cell with a chemical
polarity controlled by a reaction-diffusion mechanism describing Rho GTPase
dynamics. The size, shape, and speed of the cell emerge from the combination of
the chemical polarity, which controls the locations where actin polymerization
occurs, and the physical properties of the cell, including its membrane
tension. We find in our model both highly persistent trajectories, in which the
cell crawls in a straight line, and turning trajectories, where the cell
transitions from crawling in a line to crawling in a circle. We discuss the
controlling variables for this turning instability, and argue that turning
arises from a coupling between the reaction-diffusion mechanism and the shape
of the cell. This emphasizes the surprising features that can arise from simple
links between cell mechanics and biochemistry. Our results suggest that similar
instabilities may be present in a broad class of biochemical descriptions of
cell polarity.

Biological Physics;Soft Condensed Matter;Cell Behavior