Activation in Vesicle-Mediated Signaling Shaped by Batch Arrival Statistics

Vesicle-mediated secretion of ions or molecules is a central mechanism of cellular communication, for example in processes such as neurotransmission or hormone release. These events are inherently stochastic: vesicle fusions lead to bursts of variable sizes, releasing discrete packets of transmitters that are subsequently cleared or degraded. The dynamics break time-reversal symmetry due to the interplay of spontaneous bursts and continuous degradation. Using generating functions and a recursion relation, we derive an exact solution for the full time-dependent probability distribution of a general batch arrival-degradation model. This framework also enables a full analysis of first-passage times to a concentration threshold representing downstream activation. We show that activation kinetics are not determined by mean dynamics alone, but depend sensitively on the temporal statistics of arrival events, batch-size variability, and degradation. In particular, different arrival processes with identical mean rates can lead to qualitatively distinct first-passage behavior, reflecting the role of time-asymmetric fluctuations. We also discuss extensions incorporating vesicle depletion. Our results provide a transparent link between stochastic release dynamics and activation timing in vesicle-mediated signaling.