Mathematical Study of Rho GTPases in Motile Cells

Rho family GTPases play an essential role in cell motility, transducing extracellular signal to the actin cytoskeleton. This thesis presents the formulation and analysis of several models concerning three members of the Rho family: Cdc42, Rac, and Rho. Experimental observations of the spatio-temporal dynamics of these proteins give evidence of spatial polarization and mutual exclusion between Cdc42/Rac and Rho, which suggests that the Rho protein system is bistable. Biochemical evidence shows that there is crosstalk between the Rho proteins. However, current biological literature includes multiple (and often contradictory) hypotheses about these mutual interactions. We idealize the indirect pathways between the Rho proteins as direct effects (positive or negative feedback from one protein to another), and investigate the various proposed interactions to determine which of them can give rise to a bistable ”toggle switch”, alternating between a high level of Cdc42/Rac, low level of Rho steady state and low level of Cdc42/Rac, high level of Rho steady state. We discuss the biological interpretation of our findings. Next, we investigate a spatial variant of a model with suitable interactions to determine if it can give rise to spatial segregation of Rho proteins. We find that in a suitable parameter regime, a small initial gradient of one of the proteins will become amplified and give rise to long-lasting spatial segregation of Cdc42/Rac and Rho. On a longer timescale, the system eventually reaches a spatially uniform high Cdc42/Rac and low Rho steady state. Finally, a multi-compartment model for the activation of a single Rho family GTPase that includes a greater level of biological detail is developed. Parameter values for this model are obtained from biochemical literature. This compartment model can be used as a subunit in more realistic models of cell polarization and motility when further details of the biochemistry and interactions of the Rho proteins emerge from experimental data.