Cell Size, Mechanical Tension, and GTPase Signaling in the Single Cell.

Cell polarization requires redistribution of specific proteins to the nascent front and back of a eukaryotic cell. Among these proteins are Rac and Rho, members of the small GTPase family that regulate the actin cytoskeleton. Rac promotes actin assembly and protrusion of the front edge, whereas Rho activates myosin-driven contraction at the back. Mathematical models of cell polarization at many levels of detail have appeared. One of the simplest based on "wave-pinning" consists of a pair of reaction-diffusion equations for a single GTPase. Mathematical analysis of wave-pinning so far is largely restricted to static domains in one spatial dimension. Here, we extend the analysis to cells that change in size, showing that both shrinking and growing cells can lose polarity. We further consider the feedback between mechanical tension, GTPase activation, and cell deformation in both static, growing, shrinking, and moving cells. Special cases (spatially uniform cell chemistry, the absence or presence of mechanical feedback) are analyzed, and the full model is explored by simulations in 1D. We find a variety of novel behaviors, including "dilution-induced" oscillations of Rac activity and cell size, as well as gain or loss of polarization and motility in the model cell.