Diffusion in a Disk with a Circular Inclusion

Abstract

We consider diffusion in a disk, representing a cell with a circular interior compartment. Using bipolar coordinates, we perform exact calculations, not restricted by the size or location of the intracellular compartment. We find Green functions, hitting densities and mean times to move from the compartment to the cellular surface, and vice versa. For molecules with diffusivity D, mean times are proportional to R 2 /D, where R is the radius of the cell. We find explicit expressions for the dependence on a 2 (the fraction of the cell occupied by the intracellular compartment) and on the displacement of the compartment from the center of the cell. We consider distributions of initial conditions that are (i) uniform on the nuclear surface, (ii) uniform on the cellular surface, or (iii) given by the hitting density of particles diffusing from the nuclear to the cellular surface.

Diffusion in a disk with inclusion: Evaluating Green’s functions

“…Green’s function is the key to analytical solutions because it takes the shape of the domain and the boundary conditions into account. Quantities such as mean hitting times are obtained from it by standard integration, for any initial distribution [ 9 – 14 ]. It is also possible to model a surface with both absorbing and reflecting parts using Robin boundary conditions [ 15 – 17 ].…”

“…We consider the two cases where one circle is an absorbing boundary, the other is reflecting (reflecting inclusion inside a circular domain with absorbing boundary, and vice versa ). In [ 14 ], the circle of unit radius was referred to as the cellular surface and the inclusion as the cell’s nucleus. In two and three dimensions, Condamin et al .…”

“…We use bipolar coordinates, τ and σ ( Fig 1 ); the circle of unit radius has τ = τ 2 and the boundary of the inclusion has τ = τ 1 , where and We calculate Green’s functions on the rectangular domain in bipolar coordinates τ 2 ≤ τ ≤ τ 1 , 0 ≤ σ ≤ 2 π ; mean exit times are calculated by integrating over the coordinates ( τ , σ ) [ 14 ].…”

“…We may construct Green’s functions in nonconcentric domains from those in concentric domains, , which are also series expansions [ 27 ], as . However, the integrals needed to calculate mean exit times have only been performed using bipolar coordinates [ 14 ].…”

“…The resulting series solution can be summed to yield an explicit expression involving Jacobi Theta functions [ 29 ]. Heyda’s method has recently been applied to the problem where one circular boundary is absorbing and the other is reflecting [ 14 ]. Explicit exact solutions are useful, even when they are series, because they can be integrated to yield mean transport times, or expanded in small parameters to yield simple expressions, depending on the geometry and dynamics of the context of diffusion in confined geometries [ 8 , 30 – 35 ].…”