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The polarization and motility of eukaryotic cells depends on assembly andcontraction of the actin cytoskeleton and its regulation by proteins calledGTPases. The activity of GTPases causes assembly of filamentous actin (byGTPases Cdc42, Rac), resulting in protrusion of the cell edge. Mathematicalmodels for GTPase dynamics address the spontaneous formation of patterns andnonuniform spatial distributions of such proteins in the cell. Here we revisitthe wave-pinning model for GTPase-induced cell polarization, together with anumber of extensions proposed in the literature. These include introduction ofsources and sinks of active and inactive GTPase (by the group of A. Champneys),and negative feedback from F-actin to GTPase activity. We discuss theseextensions singly and in combination, in 1D, and 2D static domains. We thenshow how the patterns that form (spots, waves, and spirals) interact with cellboundaries to create a variety of interesting and dynamic cell shapes andmotion.

Spots, strips, and spiral waves in models for static and motile cells