Dwell time of a Brownian molecule in a microdomain with traps and a small hole on the boundary

We calculate the mean time a Brownian particle spends in a domain with traps and the number of bonds it makes before escaping through a small hole in the boundary. This mean time, called the Dwell time, depends on the backward binding rate (with the trap, e.g., scaffolding molecules), the mean time to reach the trap (forward binding rate), and the size of the hole. We estimate the mean and variance of the number of bonds made prior to exit. In a biochemical context, a quantitative signal occurs when the mean number of bonds exceeds a certain threshold, which may initiate a cascade of chemical reactions that have physiological consequences. We apply the present results to obtain estimates of the mean time a Brownian receptor spends inside a synaptic domain, when it moves freely by lateral diffusion on the membrane of a neuron and interacts at a synapse with scaffolding molecules.