Universal Shape-Sensing Mechanisms of Protein Pattern Formation in Bulk-Boundary Coupled Systems

Protein pattern formation is crucial to many biological processes, including cell division, cell motility, and morphogenesis. A non-trivial shape can provide critical input to the pattern, thus guiding its orientation. However, the analysis of pattern-forming systems typically takes place in a highly symmetric geometry, which does not allow for the analysis of shape sensing, and oftentimes neglects important bulk gradients. In this study, we introduce geometric perturbation theory as a tool to systematically understand how protein patterns adapt to a given geometry and identify universal shape-sensing mechanisms. We investigate how various patterns exhibit shape-sensing in ellipsoids and deformed cylinders, which serve as models for elongated cells and membrane tubulations, respectively. Focusing on linear stability, we find that the analytical results agree with the numerical results over a large range of parameters. Finally, we extend our analysis by simulating the full pattern-forming system and identifying regimes in which the linear stability accurately predicts the orientation of the transient, nonlinear pattern. Our method offers the possibility of inferring transient patterns from linear stability alone in certain systems and generally studying shape-sensing mechanisms.