  Synaptic transmission between neurons is governed by a cascade of stochastic
reaction-diffusion events that lead to calcium-induced vesicle release of
neurotransmitter. Since experimental measurements of such systems are
challenging due their nanometer and sub-millisecond scale, numerical
simulations remain the principal tool for studying calcium dependent synaptic
vesicle fusion, despite limitations of time-consuming calculations. In this
paper we develop an analytical solution to rapidly explore dynamical stochastic
reaction-diffusion problems, based on first-passage times. This is the first
analytical model that accounts simultaneously for relevant statistical features
of calcium ion diffusion, buffering, and its binding/unbinding reaction with a
vesicular sensor. In particular, unbinding kinetics are shown to have a major
impact on the calcium sensor's occupancy probability on a millisecond scale and
therefore cannot be neglected. Using Monte Carlo simulations we validated our
analytical solution for instantaneous calcium influx and that through
voltage-gated calcium channels. Overall we present a fast and rigorous
analytical tool to study simplified reaction-diffusion systems that allow a
systematic exploration of the biophysical parameters at a molecular scale,
while correctly accounting for the statistical nature of molecular interactions
within cells, that can also serve as a building block for more general cell
signaling simulators.
