Abstract
We study the dynamics of molecular motor-driven transport into dendritic spines, which are bulbous intracellular compartments in neurons that play a key role in transmitting signals between neurons. We further develop a stochastic model of vesicle transport in [Park, Singh, and Fai, SIAM J. Appl. Math. 82.3 (2022), pp. 793--820] by showing that second-order moments may be neglected. We exploit this property to significantly simplify the model and confirm through numerical simulations that the simplification retains key behaviors of the original agent-based myosin model of vesicle transport. We use the simplified model to explore the vesicle translocation time and probability through dendritic spines as a function of molecular motor parameters, which was previously not practically possible.
Relevance to Life Sciences: We find that thinner dendritic spine geometry can greatly reduce the probability of vesicle translocation to the post-synaptic density. The cell may alter molecular motor parameters to compensate, but only to a point. These findings are consistent with the biological literature, where brain disorders are often associated with an excess of long, thin dendritic spines.
Mathematical Content: We use a moment-generating function to deduce that second-order moments in motor attachment times may be neglected, and therefore the first-order moment is a sufficient approximation. Using only the mean attachment times and neglecting the variance yields a tractable master equation from which vesicle mean first passage times may be computed directly as a function of geometry and molecular motor parameters.