Sensing the shape of a cell: reaction-diffusion and energy minimization

How can a cell sense its own shape? How can cells pattern their proteins over a rough complex surface? Some dividing cells sense their shape, becoming polarized along their long axis. We study polarity arising from Rho GTPase proteins cycling between active membrane-bound forms and inactive cytosolic forms – a wave-pinning reaction-diffusion process. We show that wave pinning senses the cell's long axis. Simulating wave-pinning on a curved surface, high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, simply minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, on rough surfaces, shape sensing is disrupted, and high-activity domains localize to locations other than the global peaks and valleys of the surface. Simultaneously, the perimeter minimization rule fails. We study how domains evolve on rough surfaces, falling into local minima in a corrugated energy landscape. The effectiveness of the shape sensing can be controlled by altering the protein's diffusion coefficient. Our results help understand the factors that allow cells to sense their own shape – and the limits that membrane roughness can place on this process.