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The study of the diffusive motion of ions or molecules in confined biological microdomains requires the derivation of the explicit dependence of quantities, such as the decay rate of the population or the forward chemical reaction rate constant on the geometry of the domain. Here, we obtain this explicit dependence for a model of a Brownian particle (ion, molecule, or protein) confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. We call the calculation of the mean escape time the narrow escape problem. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. Here, we present asymptotic formulas for the mean escape time in several cases, including regular domains in two and three dimensions and in some singular domains in two dimensions. The mean escape time comes up in many applications, because it represents the mean time it takes for a molecule to hit a target binding site. We present several applications in cellular biology: calcium decay in dendritic spines, a Markov model of multicomponent chemical reactions in microdomains, dynamics of receptor diffusion on the surface of neurons, and vesicle trafficking inside a cell. molecular trafficking ͉ mean first passage time ͉ random motion ͉ cellular biology ͉ small hole T he function of biological microdomains, and specifically neurobiological microstructures, such as dendritic spines, is largely unknown, and much effort has been spent in the last 20 years to unravel the molecular pathways responsible for the maintenance or modulation of cellular functions and, ultimately, to extract fundamental principles (1, 2). The cytoplasm of eukaryotic cells is a complex environment where dynamic organelles, cytoskeletal network, and soluble macromolecules are organized in heterogenous structures and local microdomains (3). These submicrometer domains may contain only a small number of molecules, of the order between just a few and up to hundreds. This is the case in microdomains like endosomes (4), synapses (5), and the sensory compartments of cells, such as the outer segment of photoreceptors, but at such low molecular number, the addition of external chemical binding dye molecules, necessary for experimental purposes, may alter the signaling pathway and thus modify the physiological phenomenology. This circumstance calls for physical and mathematical modeling to separate the interfering effects, and ultimately the physical-mathematical model is expected to be a fundamental tool for both the quantitative and qualitative study of chemical reactions in microdomains.Because of the small number of molecules involved in chemical reactions occurring in extremely confined domains (such as endoplasmic reticulum, caveolae, and mitochondria, to obtain quantitative information about chemical processes, modeling and simulations seem to be inevitable to reconstruct the microdomain's environment and to obtain precise quantitative information about the mole...

Stochastic trajectories are described that underly classical diffusion between known concentrations. The description of those experimental boundary conditions requires a phase space using the full Langevin equation, with displacement and velocity as state variables, even if friction entirely dominates the dynamics of diffusion, because the incoming and outgoing trajectories have to be told apart. The conditional flux, probabilities, mean first-passage times, and contents (of the reaction region) of the four types of trajectories—the trans trajectories LR and RL and the cis trajectories LL and RR—are expressed in terms of solutions of the Fokker–Planck equation in phase space and are explicitly calculated in the Smoluchowski limit of high friction. With these results, diffusion in a region between fixed concentrations can be described exactly as a chemical reaction for any potential function in the region, made of any combination of high or low barriers or wells.

A Brownian particle with diffusion coefficient D is confined to a bounded domain of volume V in R 3 by a reflecting boundary, except for a small absorbing window. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. We construct an asymptotic approximation for the case of an elliptical window of large semi axis a ≪ V 1/3

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