Dwell time of a Brownian interacting molecule in a cellular microdomain

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.