A simple model for conserved intracellular dynamics exhibits multiscale pattern formation, traveling protein domains and arrested coarsening of lipid domains

We model the spatiotemporal dynamics of cellular protein concentrations relevant to cell polarity near membranes composed of different lipids. Therefore, we consider a three-variable continuum model for membrane-bound protein, cytosolic protein, and the local composition of a binary lipid membrane. The model contains two globally conserved quantities: the total protein content and the average fractions of the two lipid species. It combines a conserved reaction-diffusion model for the protein dynamics, undergoing an active phase separation, with a Cahn-Hilliard equation for lipid demixing. Linear stability analysis of the homogeneous steady state and direct numerical simulations show that the lipid dynamics undergoes classical phase separation, whereas the protein dynamics exhibits oscillatory phase separation for intermediate total protein contents, associated with a long-wavelength instability and traveling domains. In parameter regions where both instabilities are present, we find multiscale patterns with larger-scale traveling and rotating protein domains coexisting with smaller-scale stationary lipid domains. In this regime, traveling protein domains coexist with arrested coarsening of stationary lipid domains above a critical coupling. We further show that the main instabilities and phase diagram are well captured by an extension of a recently proposed conserved FitzHugh-Nagumo model for non-reciprocal pattern formation. The extended model consists of two non-reciprocally coupled Cahn-Hilliard equations with different interface tensions, reflecting the distinct physical properties of lipids and proteins. This also explains the observed asymmetry between static lipid patterns and traveling protein patterns.