Parameter-space topology of models for cell polarity

Abstract

Reaction–diffusion systems have been widely successful in the theoretical description of biological patterning phenomena, giving rise to numerous models based on differing mechanisms, mathematical implementations and parameter choices. However, even for models with common design features, the diversity of mathematical realizations may hinder the identification of common behavior. Here, we analyze three different reaction–diffusion models for cell polarity that feature conservation of mass, rapid cytoplasmic diffusion and bistability via a cusp bifurcation of uniform states. In all three models, the nonuniform polar states are front solutions, and growth of domains ceases through stalling of a propagating front. For these three models we find a characteristic parameter space topology, comprising a region of linear instability that loops around the cusp point and that is enclosed by a ‘comet-shaped’ region of nonuniform domain states. We propose a minimal model based on the cusp bifurcation normal form that includes essential characteristics of all cell polarity models considered. For this minimal model, we provide a complete analytical description of the parameter space topology, and find that the instability loop appears as a generic property of the cusp bifurcation. This topological analysis provides a unifying understanding of earlier mathematically distinct models and is suitable to classify future models.

Keywords

• cell polarity
• reaction-diffusion equations
• cusp bifurcation