Parameter-space topology of models for cell polarity

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.