  Some dividing cells sense their shape by becoming polarized along their long
axis. Cell polarity is controlled in part by polarity proteins like Rho GTPases
cycling between active membrane-bound forms and inactive cytosolic forms,
modeled as a "wave-pinning" reaction-diffusion process. Does shape sensing
emerge from wave-pinning? We show that wave pinning senses the cell's long
axis. Simulating wave-pinning on a curved surface, we find 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. However, when we
introduce roughness to our surfaces, shape sensing can be disrupted, and
high-activity domains can become localized to locations other than the global
peaks and valleys of the surface. On rough surfaces, the domains of the
wave-pinning model are more robust in finding the peaks and troughs than the
minimization rule, though both can become trapped in steady states away from
the peaks and valleys. We can control the robustness of shape sensing by
altering the Rho GTPase diffusivity and the domain size. We also find that the
shape sensing properties of cell polarity models can explain how domains
localize to curved regions of deformed cells. Our results help to understand
the factors that allow cells to sense their shape - and the limits that
membrane roughness can place on this process.
