![]() 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. 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. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. We call the calculation of the mean escape time the narrow escape problem. 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. 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. ![]() In this model the receptors are first inserted into the extrasynaptic plasma membrane and then random walk in and out of corrals through narrow openings on their way to their final destination. Following their synthesis in the endoplasmic reticulum, receptors are trafficked to their postsynaptic sites on dendrites and axons. The synaptic receptors are trafficked in and out of synapses by a diffusion process. At a molecular level, the synaptic weight is determined by the number and properties of protein channels (receptors) on the synapse. In particular, it is believed that the memory state in the brain is stored primarily in the pattern of synaptic weight values, which are controlled by neuronal activity. This process underlies synaptic plasticity, which relates to learning and memory. This approach provides a framework for the theoretical study of receptor trafficking on membranes. We use the results to estimate the confinement time as a function of the parameters and also the time it takes for a diffusing receptor to be anchored at its final destination on the postsynaptic membrane, after it is inserted in the membrane. Using this approach, it is possible to describe the Brownian motion of a random particle in an environment containing domains with small openings by a coarse grained diffusion process. We compute the mean confinement time of the Brownian particle in the asymptotic limit of a narrow opening and calculate the probability to exit through a given small opening, when the boundary contains more than one. We model the motion of a receptor on the membrane surface of a synapse as free Brownian motion in a planar domain with intermittent trappings in and escapes out of corrals with narrow openings. Our preconditioned system uses only 13.6 million “compressed” degrees of freedom and a few dozen GMRES iterations. For that case, adaptive discretization of each patch would lead to a dense linear system with about 360 million degrees of freedom. We demonstrate the method with several numerical examples, and are able to achieve highly accurate solutions with 100 000 patches in one hour on a 60-core workstation. Our method is insensitive to the patch size, and the total cost scales with the number N of patches as O(NlogN), after a precomputation whose cost depends only on the patch size and not on the number or arrangement of patches. We develop a hierarchical, fast multipole method-like algorithm to accelerate each matrix-vector product. This system is solved iteratively using GMRES. A block-diagonal preconditioner together with a multiple scattering formalism leads to a well-conditioned system of second-kind integral equations and a very efficient approach to discretization. They are numerically challenging for two main reasons: (1) the solutions are non-smooth at Dirichlet-Neumann interfaces, and (2) they involve adaptive mesh refinement and the solution of large, ill-conditioned linear systems when the number of small patches is large.īy using the Neumann Green's functions for the sphere, we recast each boundary value problem as a system of first-kind integral equations on the collection of patches. Mathematically, these give rise to mixed Dirichlet/Neumann boundary value problems of the Poisson equation. The narrow escape problem is the dual problem: it models the behavior of a Brownian particle confined to the interior of a sphere whose surface is reflective, except for a collection of small patches through which it can escape. The narrow capture problem models the equilibrium behavior of a Brownian particle in the exterior of a sphere whose surface is reflective, except for a collection of small absorbing patches. We present an efficient method to solve the narrow capture and narrow escape problems for the sphere.
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