Wave-pinned patterns for cell polarity — a catastrophe theory explanation

A class of four-component reaction-diffusion systems are studied in one spatial dimension, with one of four specific reaction kinetics. Models of this type seek to capture the interaction between active and inactive forms of two G-proteins, known as ROPs in plants, thought to underly cellular polarity formation. The systems conserve total concentration of each ROP, which enables reduction to simple canonical forms when one seeks conditions for homogeneous equilibria or heteroclinic connections between them. Transitions between different multiplicities of such states are classified using a novel application of catastrophe theory. For the time-dependent problem, the heteroclinic connections represent so-called wave-pinned states that separate regions of the domain with different ROP concentrations. It is shown numerically how the form of wave-pinning reached can be predicted as a function of the domain size and initial total ROP concentrations. This leads to state diagrams of different polarity forms as a function of total concentrations and system parameters.