Sensing Cell Shape at the Micron Scale with Reaction-Diffusion

Some dividing cells sense their shape by becoming polarized along their long axis. Polarity is often modeled by a reaction-diffusion system describing the proteins from the Rho GTPase family cycling between active membrane-bound forms and inactive cytosolic forms (a "wave-pinning" model). Does shape-sensing emerge from wave-pinning? We simulate wave-pinning on a curved surface and show that high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, a simple rule of minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. We find that as the surface becomes rough, the domains of the wave-pinning model are more robust in finding the peaks and troughs than predicted by the minimization rule. Why? The minimization rule models a sharp interface between the high and low activity regions whereas the wave-pinning model has a finite interface width. By changing the Rho GTPase diffusivity we can change the interface width and thus the robustness of the shape-sensing of the domains.