Modeling and asymptotic analysis of the concentration difference in a nanoregion between an influx and outflux diffusion across narrow windows

Abstract

When a flux of Brownian particles is injected in a narrow window located on the surface of a bounded domain, these particles diffuse and can eventually escape through a cluster of narrow windows. At steady-state, we compute asymptotically the distribution of concentration between the different windows. The solution is obtained by solving Laplace's equation using Green's function techniques and second order asymptotic analysis, and depends on the influx amplitude, the diffusion properties as well as the geometrical organization of all the windows, such as their distances and the mean curvature. We explore the range of validity of the present asymptotic expansions using numerical simulations of the mixed boundary value problem. Finally, we introduce a length scale to estimate how deep inside a domain a local diffusion current can spread. We discuss some applications in biophysics.

Figures (11)

• Figure 1: Schematic representation of the influx-outfluxA: Influx/outflux diffusion on an arbitrary domain. B: Spherical geometry with Neumann boundary conditions. C: Spherical geometry with mixed Neumann-Dirichlet boundary conditions. D: Illustration of the a free tetrahedral meshing of a unit ball with two windows generated by COMSOL Multiphysics Version 5.2a comsol used to solve numerically the distribution of the concentration.
• Figure 2: Concentration drop in a unit ball with two windows with Neumann boundary conditions (formula \ref{['eq:sphere2']}).A: An influx and an outflux currents are applied respectively on two patches centered around the North $(x_1)$ and South $(x_2)$ Poles. B: Concentration difference as a function of the radius $\varepsilon$. C: Relative error as a function of the radius $\varepsilon$. D: By increasing the colatitude of a patch centered in $x_2$ on the trivial azymuth, we study the effect of varying the distance $l = \|x_1-x_2\|$. E: Concentration difference as a function of the distance $l$ for $\varepsilon = 0.02,\,0.05,\, 0.1$. F: Relative error as a function of the distance $l$.
• Figure 3: Concentration drop in a unit ball with a single absorbing window (formula \ref{['eq:sphere2abs']}).A: The absorbing window is centered at the South Pole $(x_2)$. B: Concentration difference as a function of the radius $\varepsilon$. C: Relative error as a function of the radius $\varepsilon$. D: By increasing the colatitude of a patch centered in $x_2$ on the trivial azymuth, we study the effect of varying the distance $l = \|x_1-x_2\|$. E: Concentration difference as a function of the distance $l$ for $\varepsilon = 0.01,\,0.05,\, 0.1$. F: Relative error as a function of the distance $l$.
• Figure 4: Several absorbing holes (formula \ref{['eq:sphereN']}).A: There are 6 equidistant holes on the sphere, located at the North and South Poles with the last four on the equator. B: Normalized concentration drop as a function of the radius $\varepsilon$ for $N=2,\,4,\,6$ holes. C: Relative error as a function of $\varepsilon$. D: We consider $9$ holes located on a concentric ring of radius $l$ centered at the North Pole (where the usual influx patch is located). E: Concentration drop as a function of the distance $l$. F: Relative error as a function of the distance $l$.
• Figure 5: Concentration drop due to a linear configuration of absorbing windows (formula \ref{['eq:sphereN']}).A: There are $5$ absorbing windows located on the trivial azymuth, with a distance of $2.5\varepsilon$ between each neighbor. B: Concentration drop as a function of the distance from the North pole to the center of the nearest window on the line. C: Relative error as a function of the distance $l$. D: Here $l$ represents the distance between each neighboring window. The center of the second window remains fixed on the Equator. E: Concentration drop as a function of $l$. F: Relative error as a function of $l$.