  The time spent by an interacting Brownian molecule inside a bounded
microdomain has many applications in cellular biology, because the number of
bounds is a quantitative signal, which can initiate a cascade of chemical
reactions and thus has physiological consequences. In the present article, we
propose to estimate the mean time spent by a Brownian molecule inside a
microdomain $\Omega$ which contains small holes on the boundary and agonist
molecules located inside. We found that the mean time depends on several
parameters such as the backward binding rate (with the agonist molecules), the
mean escape time from the microdomain and the mean time a molecule reaches the
binding sites (forward binding rate). In addition, we estimate the mean and the
variance of the number of bounds made by a molecule before it exits $\Omega$.
These estimates rely on a boundary layer analysis of a conditional mean first
passage time, solution of a singular partial differential equation. In
particular, we apply the present results to obtain an estimate of the mean time
spent (Dwell time) by a Brownian receptor inside a synaptic domain, when it
moves freely by lateral diffusion on the surface of a neuron and interacts
locally with scaffolding molecules.
