We investigate pattern formation for a 2D PDE-ODE bulk-cell model, where
one or more bulk diffusing species are coupled to nonlinear intracellular
reactions that are confined within a disjoint collection of small
compartments. The bulk species are coupled to the spatially segregated
intracellular reactions through Robin conditions across the cell
boundaries. For this compartmental-reaction diffusion system, we show that
symmetry-breaking bifurcations leading to stable asymmetric steady-state
patterns, as regulated by a membrane binding rate ratio, occur even when
two bulk species have equal bulk diffusivities. This result is in distinct
contrast to the usual, and often biologically unrealistic, large
differential diffusivity ratio requirement for Turing pattern formation
from a spatially uniform state. Secondly, for the case of one-bulk
diffusing species in R^2, we derive a new memory-dependent ODE
integro-differential system that characterizes how intracellular
oscillations in the collection of cells are coupled through the PDE
bulk-diffusion field. By using a fast numerical approach relying on the
``sum-of-exponentials’‘ method to derive a time-marching scheme for this
nonlocal system, diffusion induced synchrony is examined for various
spatial arrangements of cells using the Kuramoto order parameter. This
theoretical modeling framework, relevant when spatially localized nonlinear
oscillators are coupled through a PDE diffusion field, is distinct from the
traditional Kuramoto paradigm for studying oscillator synchronization on
networks or graphs. (Joint work with Merlin Pelz, UBC and UMinnesota).