  We develop methods for investigating protein drift-diffusion dynamics in
heterogeneous cell membranes and the roles played by geometry, diffusion,
chemical kinetics, and phase separation. Our hybrid stochastic numerical
methods combine discrete particle descriptions with continuum-level models for
tracking the individual protein drift-diffusion dynamics when coupled to
continuum fields. We show how our approaches can be used to investigate
phenomena motivated by protein kinetics within dendritic spines. The spine
geometry is hypothesized to play an important biological role regulating
synaptic strength, protein kinetics, and self-assembly of clusters. We perform
simulation studies for model spine geometries varying the neck size to
investigate how phase-separation and protein organization is influenced by
different shapes. We also show how our methods can be used to study the roles
of geometry in reaction-diffusion systems including Turing instabilities. Our
methods provide general approaches for investigating protein kinetics and
drift-diffusion dynamics within curved membrane structures.
