Narrow escape time in biology

In biological systems, the main challenges in modeling transport processes can be summarized as being inside a heterogeneous medium, which is a fluctuating environment, and striving to reach to a chemically active receptor, which acts like an absorbing boundary while the other organelles of the interior of a living cell acting like active obstacles. These are very complex problems in general. The only viable approach known is to develop some stochastic models to take into account these aspects. A continuous random process, mostly Brownian motion, is commonly used to model the motion of chemicals in the intracellular transport. In certain cases, these chemicals display Brownian motion on the 2D surface of the cell. Therefore, the first passage time of such chemicals is the main determining mechanism for triggering critical biological processes.This requires studying a stochastic process on a two-dimensional surface which is topologically a sphere with small disks on them. These small disk like regions represent absorbing boundaries corresponding to the receptors. Some studies show that the calculated first passage times for such environments grow with the logarithm of the size of the disk like regions. In some cases, this time scale can be very long compared to the motion of the cell in its environment. For the dynamical model where the surface is fluctuating slowly as the particle executes Brownian motion on this surface, we can make use of a stochastic process with a variable background metric. Since the variations of the metric are slow we may use an adiabatic approximation. We analyze the variation of first passage times within this dynamical model.