Influence of boundary geometry on active patterns

Mechanochemical patterns arising in the actomyosin cortex drive many cellular processes. Here we consider a hydrodynamic model for the actomyosin cortex of cells and study the sensitivity of the emergent patterns to both physical parameters and the geometry of the confining domain. We first establish a general framework for the Galerkin analysis of such patterns far from the linear stability regime on an arbitrary two-dimensional domain. In the case of a circular disk, our analytical results predict transitions from isotropic to anisotropic patterns upon changing the strength of the active stress and the turnover rate. We confirm the existence of this secondary bifurcation of the homogeneous state by an explicit numerical analysis of our model. Extending our numerical analysis to harmonic deformations of the circular disk, we show that the emergent patterns are also sensitive to the curvature of the domain. In particular, the actomyosin patterns resulting from our study closely resemble those seen in cells confined to micropatterned substrates. Our study demonstrates the role of geometry in controlling patterns within the context of a simple model for the actomyosin cortex.