Stochastic simulation of di ffusion on unstructured meshes via first exit times.

Last modified: 2014-06-09

#### Abstract

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.

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.