Combing different geometries for the modelling of realistic biolog-

With the technique presented in the previous section we are now able to correctly model diffusion on meshes which are set on either Cartesian, cylindrical, or spherical coordinates.

However, in order to use our approach for the simulation of chemical signalling in complex geometries, we have to combine different geometries. In the following, we will briefly show how this can be accomplished. In particular, we will present two simple examples in which set up more complex geometries. Namely we will construct a rod like bacterial cell and a dendrite with spines attached to it.

Figure 5.2: Left: Microscopic picture of a single E coli cell which is loaded with a fluorescent dye. Right: Idealised geometry used for modelling single cell organisms Bacterial cell

As a first example, we will construct a reaction medium that allows for the simulation of chemical signalling cascades in single bacterial cell such as Escherichia coli. Although bacterial cells only have a volume of approximately 1µm³, space nevertheless needs to be accounted for when modelling signalling cascades in these organisms. This is quite obvious if one for instance considers the Min proteins in E coli, which oscillate in order to find the middle of the cell for cell division [Fange and Elf, 2006]. But also other signalling cascades such as the cascades regulating bacterial chemotaxis depend on the shape and structure of the reaction medium [Lipkow et al., 2005]. Consequently a realistic representation of space for the modelling of chemical signalling cascades in bacterial cells is therefore highly desirable.

When taking a look at microscopic picture of E coli, such as the one on the left hand side of Figure 5.2, one immediately notices that the shape of a bacterial cell is well represented by a cigar or a rod. Hence, we can easily model such a shape by simply attaching two half spheres to a cylinder with radius r. Consequently we arrive at a reaction medium equal to the one visualised on the right hand side of Figure 5.2. The computational grid we obtain is similar to the one used by [Blom and Peletier, 2002].

However, unlike the grid presented there, we do not assume that the reaction medium is symmetric to the r-axis. Hence with our discretisation we are also able to account for variations of the molecules in theφ-direction. A visualisation of the grid is given in Figure 5.3, in which the left part shows a view that is obtained when cutting the bacterium in a sagital manner and in which the right part shows a section through the transverse plane of the bacterium. In particular the cylinder and the two half spheres are discretized in

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such a way, that each cylindrical grid cell at the interface between the cap and the main body of the cell neighbours only with one spherical grid cell. The hopping rates between the two grids are determined by the continuity of the particle currents (see part on Finite Volume Monte Carlo Method).

For testing whether diffusion is correctly modelled in our reaction medium, a Brownian

Figure 5.3: Computational grids used when modelling a single bacterial cell. The left part shows the longitudinal cut and the right part the cross section.

walker was allowed to freely diffused in the cell. As the quotient of the accumulated time which the particle spend in each gridcell t_i and the total simulation time T was equal to the quotient of the gridcell volume ωi and the total volume Ω, i.e. ti/T = ωi/Ω, we conclude that our reaction medium is well suited for modelling chemical signalling in bacterial cells.

Dendrite

After having presented a way for computationally representing the shape of single cell organisms such as Escherichia coli, we will now turn to our second example. Namely we will set up a reaction medium with which we can simulate chemical signalling in a dendrite that has spines attached to it. A possible application for our reaction medium is for instance the question of whether spines can really be regarded as isolated chemical reaction chambers or whether chemical crosstalk can occur among neighbouring spines [Ajay and Bhalla, 2006].

In the left part of Figure 5.4 we see a picture of a neuron with its dendritic tree. In the bottom part we further see a magnification of a piece of dendrite. As can be seen there, a dendrite is formed by two structures; namely by the parental dendrite and by the small protrusions, which are called spines. A spine is therefore nothing but a small membranous extrusions. Generally one further subdivides the spine into two parts. The first part is the bulbous head, which connects the dendrite with the axon of a presynaptic cell. The head also houses the signal transduction machinery of the postsynaptic side.

The second part of the spine is the so-called spine neck which simply connects the main dendrite with the head.

Figure 5.4: Left part: Two pictures showing a neuron and a magnified view of a piece of dendrite. Right part: A computational representation of a piece of dendrite to which spines are attached.

Generally spines come in a variety of different shapes and sizes. Basically one distin-guishes between three types: thin spines, mushroom spines, and stubby spines. In a first approximation, it is however reasonable to model both the main dendrite as well as the head and neck of the different spine types by cylinders of different sizes. The main den-drite normally has a diameter of3−4µm. Compared to this the spine necks are minute as they only have a diameter of 0.1−0.2µm. Typical diameters for the spine head lie around1µm and the overall height of a spine is approximately equal to2µm. Overall, if one models a dendrite as laid out above, one obtains a reaction medium as the one visu-alised in the right part of Figure 5.4. The discretisation of this shape is straightforward as it only contains cylinders. The only potential problem arises at the junction between the main dendrite because one would have to account for the curvature of the main dendrite in an exact representation. However, since the difference in diameter between the spine neck and the main dendrite are so large, the spine neck effectively sits on a flat surface.

Hence the interface area between these two structures is simply given by the footprint of the spine neck and the calculation of the distance between the bordering cells at the junction is also straightforward.

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