Nonlinear traveling wave instability and arrested coarsening in a model for protein patterns on biomembranes

The formation of protein patterns inside biological cells is of crucial importance for the cell's spatial organization, growth and division. In many cases these dynamics can be described by coupled, mass-conserving reaction-diffusion equations. Motivated by the question how the generation of cell polarity is influenced by membrane heterogeneity, we study the dynamics emergent from the coupling of two well-known, mass-conserved systems.

System A is therein given by a simplified model for the emergence of cell polarity with a single pair of activated-inactivated proteins undergoing active phase separation [Mori et al., Biophys. J (2008)]. The inactive, fast-diffusing species in the bulk of the cell becomes activated when bound to the cell membrane and when further endowed with a positive feedback on the membrane-mediated activation, the system exhibits inherent polarizability. System B is described by the Cahn-Hilliard equation, i.e., two membrane species demix into a much smaller pattern of static droplets that slowly coarsen with time. By introducing a coupling between systems A and B due to a concentration-dependent interaction affinity, we observe the emergence of traveling waves in system A and likewise, a systematic slowing of the coarsening in system B - both mediated by the coupling strength between the systems. Our results elucidate the interaction between pattern forming systems of well-separated length scales and exemplify how complex behavior in biological systems arises from the coupling of simpler subsystems.