We study the dynamics of a model of membrane vesicle transport into dendritic spines, which are bulbous intracellular compartments in neurons driven by molecular motors. We reduce the lubrication model proposed in [Fai et al  , Active elastohydrodynamics of vesicles in narrow, blind constrictions. Phys. Rev. Fluids, 2 (2017), 113601] to a fast-slow system, yielding an analytically and numerically tractable equation that is equivalent to the original model in the overdamped limit. The key parameters of the model are the ratio of motors that prefer to push toward the head of the dendritic spine to the ratio of motors that prefer to push in the opposite direction. We perform a numerical bifurcation analysis in these parameters and find that steady-state vesicle velocities appear and disappear through a number of saddle-node bifurcations. This  allows us to identify the region of parameter space in which multiple stable velocities exist. We show by direct calculations that for sufficiently close vesicle-to-spine diameter ratios, there can only be unidirectional motion . The critical vesicle-to-spine diameter ratio predicted by our analysis , at which there is a transition from unidirectional to bidirectional motion, is consistent with experimental observations of vesicle trajectories in the literature.
We study the dynamics of a model of membrane vesicle transport into dendritic spines, which are bulbous intracellular compartments in neurons driven by molecular motors. We reduce the lubrication model proposed in [Fai et al . , Active elastohydrodynamics of vesicles in narrow, blind constrictions. Phys. Rev. Fluids, 2 (2017), 113601] to a fast-slow system, yielding an analytically and numerically tractable equation equivalent to the original model in the overdamped limit. The model's key parameters are the ratio of motors that prefer to push toward the head of the dendritic spine to the ratio of motors that prefer to push in the opposite direction. We perform a numerical bifurcation analysis in these parameters and find that steady-state vesicle velocities appear and disappear through several saddle-node bifurcations. This process allows us to identify the region of parameter space in which multiple stable velocities exist. We show by direct calculations that there can only be unidirectional motion for sufficiently close vesicle-to-spine diameter ratios. Our analysis predicts the critical vesicle-to-spine diameter ratio , at which there is a transition from unidirectional to bidirectional motion, consistent with experimental observations of vesicle trajectories in the literature.
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