  We study the dynamics of membrane vesicle motor transport into dendritic
spines, which are bulbous intracellular compartments in neurons that play a key
role in transmitting signals between neurons. We consider the stochastic analog
of the vesicle transport model in [Park and Fai, The Dynamics of Vesicles
Driven Into Closed Constrictions by Molecular Motors. Bull. Math. Biol. 82, 141
(2020)]. The stochastic version, which may be considered as an agent-based
model, relies mostly on the action of individual myosin motors to produce
vesicle motion. To aid in our analysis, we coarse-grain this agent-based model
using a master equation combined with a partial differential equation
describing the probability of local motor positions. We confirm through
convergence studies that the coarse-graining captures the essential features of
bistability in velocity (observed in experiments) and waiting-time
distributions to switch between steady-state velocities. Interestingly, these
results allow us to reformulate the translocation problem in terms of the mean
first passage time for a run-and-tumble particle moving on a finite domain with
absorbing boundaries at the two ends. We conclude by presenting numerical and
analytical calculations of vesicle translocation.
