In molecular biology it is of interest, besides reactions, to
also simulate
spatial phenomena in the cell, that is di ffusion. Biochemical systems in
cell biology often only contain very low copy numbers of certain species.
As a result the reaction-di ffusion equation, a macroscopic or deterministic
partial di fferential equation describing the concentrations, is
inaccurate and
does not reproduce experimental data. Hence, a stochastic and individual-
based description is needed [1]. The di ffusion of the molecules is then
given
by Brownian dynamics and the reactions between them occur with certain
probabilities.
For stochastic simulation of the di ffusion, the cell is partitioned
into compartments or voxels in a mesoscopic model. The number of
molecules in a voxel is recorded and the molecules can jump between
neighbouring voxels to model di ffusion. In order to accurately
represent the geometry of the cell including outer and inner curved
boundaries it is helpful to use unstructured grids, meaning
triangulations, for the voxels. The probabilities to jump between the
voxels are given in [2] by a discretization of the Laplacian with the
finite element method (FEM) on the mesh. Solutions of the di ffusion
equation with FEM can encounter problems on unstructured grids of poor
quality. The maximum principle may not be satised by the FEM solution
and the derived jump coe cients may be negative, which no longer allows
for an interpretation as jump propensities. Ignoring the negative
coe cients and setting them to zero leads to an incorrect di ffusion speed.
Presenting the speed of di ffusion correctly from the inner part of the
cell to the outer boundary can be of importance for example in
signalling, where a signal is released in the nucleus and propagates to
the cell membrane. We therefore derive the jump coe cients from the
theory of first exit times. In order to demonstrate the method of
global first exit times (GFET) we will show results on skewed
unstructured meshes in 2D and compare with
analytical results. Then the method is applied to di ffusion on truly
irregular grids in 3D.
1. A. Mahmutovic, D. Fange, O. G. Berg, and J. Elf, Lost in presumption:
stochastic reactions in spatial models, Nature Methods 9, 1163-1166, 2012.
2. S. Engblom, L. Ferm, A. Hellander, and P. Lotstedt. Simulation of
stochastic reaction-di ffusion processes on unstructured meshes. SIAM J.
Sci.
Comput., 31(3):1774-1797, 2009.