Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations

Frédéric Paquin-Lefebvre
University of British Columbia, Vancouver

In recent years, there has been an interest in the theory of coupled bulk-surface semilinear partial differential equations and their applications in cell biology. One of the reasons behind this is the natural compartmentalization of cytosolic (bulk) and membrane-bound species that such a class of models allows. Here we consider the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of spatial dimensionality and diffusion on the formation of spatio-temporal patterns. In a simplified geometry consisting of a 1-D spatial bulk domain, the model reduces to a coupled PDE-ODE system, with passive diffusion inside the cell coupled to two systems of nonlinear ODEs for binding kinetics at each end of the cell. On a 2-D circular bulk domain, species can also also move along the membrane and thus surface diffusion is added to the nonlinear ODEs. In both cases, the coupling is modeled with nonlinear Robin-type boundary conditions. Finally, our analysis of the 1-D case reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation-type oscillations existing in the limit of long membrane-residence time, whereas in 2-D we observe the formation of stationary Turing patterns, rotating waves and standing waves. This is joint work with Bin Xu, Kelsey DiPietro, Alan Lindsay and Alexandra Jilkine, from the University of Notre Dame.