  The small GTPases Rac and Rho are known to regulate eukaryotic cell shape,
promoting front protrusion (Rac) or rear retraction (Rho) of the cell edge.
Such cell deformation changes the contact and adhesion of cell to the
extracellular matrix (ECM), while ECM signaling through integrin receptors also
affects GTPase activity. We develop and investigate a model for this three-way
feedback loop in 1D and 2D spatial domains, as well as in a fully deforming 2D
cell shape. The model consists of reaction-diffusion equations solved
numerically with open-source software, Morpheus, and with custom-built cellular
Potts model simulations. We find a variety of patterns and cell behaviors,
including persistent polarity, flipped front-back cell polarity oscillations,
and random protrusion-retraction. We show that the observed spatial patterns
depend on the cell shape, and vice versa. The cell stiffness and biophysical
properties also affect patterning, overall cell migration phenotypes.
