Synchronized Oscillatory Dynamics for a 1-D Model of Membrane Kinetics Coupled by Linear Bulk Diffusion

Abstract

Spatial-temporal dynamics associated with a class of coupled membrane-bulk PDE-ODE models in one spatial dimension is analyzed using a combination of linear stability theory, numerical bifurcation software, and full time-dependent simulations. In our simplified one-dimensional setting, the mathematical model consists of two dynamically active membranes, separated spatially by a distance $2L$, that are coupled together through a linear bulk diffusion field, with a fixed bulk decay rate. The coupling of the bulk and active membranes arises through both nonlinear flux boundary conditions for the bulk diffusion field and from feedback terms, depending on the local bulk concentration, to the dynamics on each membrane. For this class of models, it is shown both analytically and numerically that bulk diffusion can trigger a synchronous oscillatory instability in the temporal dynamics associated with the two active membranes. For the case of a single active component on each membrane, and in the limit $L\to \infty$, rigorous spectral results for the linearization around a steady-state solution, characterizing the possibility of Hopf bifurcations and temporal oscillations in the membranes, are obtained. For finite $L$, a weakly nonlinear theory, accounting for eigenvalue-dependent boundary conditions appearing in the linearization, is developed to predict the local branching behavior near the Hopf bifurcation point. The analytical theory, together with numerical bifurcation results and full numerical simulations of the PDE-ODE system, is undertaken for various coupled membrane-bulk systems, including two specific biologically relevant applications. Our results show the existence of a wide parameter range where stable synchronous oscillatory dynamics in the two membranes can occur.