Modeling active cellular transport as a directed search process with stochastic resetting and delays

TLDR: A probabilistic renewal method is used to explicitly calculate the splitting probabilities and conditional mean first passage times (MFPTs) for capture by a finite array of contiguous targets and shows that both models have the same splitting probabilities, and that increasing the resetting rate r increases (reduces) the splitting probability for proximal (distal) targets.

Abstract: We show how certain active transport processes in living cells can be modeled in terms of a directed search process with stochastic resetting and delays. Two particular examples are the motor-driven intracellular transport of vesicles to synaptic targets in the axons and dendrites of neurons, and the cytonemebased transport of morphogen to target cells during embryonic development. In both cases, the restart of the search process following reset has a finite duration with two components: a finite return time and a refractory period. We use a probabilistic renewal method to explicitly calculate the splitting probabilities and conditional mean first passage times (MFPTs) for capture by a finite array of contiguous targets. We consider two different search scenarios: bounded search on the interval [0, L], where L is the length of the array, with a refractory boundary at x = 0 and a reflecting boundary at x = L (model A), and partially bounded search on the half-line (model B). In the latter case there is a non-zero probability of failure to find a target in the absence of resetting. We show that both models have the same splitting probabilities, and that increasing the resetting rate r increases (reduces) the splitting probability for proximal (distal) targets. On the other hand the MFPTs for model A are monotonically increasing functions of r, whereas the MFPTs of model B are non-monotonic with a minimum at an optimal resetting rate. We also formulate multiple rounds of search-and-capture events as a G/M/∞ queue and use this to calculate the steady-state accumulation of resources in the targets. Directed search process with stochastic resetting and delays 2