FEMLAB - KTH

Since the thickness of the cytoplasm and nucleus are of the order of a 1000
times that of the membranes, the graphical representation of the membranes in the
femlab Draw mode would be very thin. Therefore the triangular elements in these
thin domains would be very tiny, and thus the model would be too computationally
expensive.
One way to solve this problem would be to approximate the membranes as
simple boundaries, by a technique known as thin film approximation [4]. However,
the example in the FEMLAB manual is straightforward and involves no partition
coefficient, whereas in the present case, there is a partition coefficient because the
membrane should act as a reservoir for A and U. Of course, this is possible, but
rather inconvenient. Therefore, I chose a different approach.
The extracellular solution, as will be shown below, is 27.748 times thicker than
the nucleus whose scaled radius is 1: If we denote the scaled radius of the entire
cell by r, and that of the medium surrounding it by R, then equating the ratio of
their volumes to 168, we have
R3 = 168 (5.1)
3
r
If we then substitute r by 4 in (5.1), we find R to be 22, and therefore the scaled
thickness of the external solution alone is 22 − 4 = 18.
Therefore our computational domain consists of:
• A rectangle which represents the extracellular solution
• A thin outer arc which represents the cell membrane
• A thicker inner arc which represents the cytoplasm. The thousand of tiny
membranes in the cytoplasm are not represented in the domain because they
are handled by homogenisation.
• A thin inner arc which represents the nuclear membrane
• Finally, a thicker central circle which represents the nucleus
All the arcs are concentric with the central circle and only 22.5 o is taken, from
the centre of the central circle, as seen in Figure 5.1.
In the FEMLAB draw mode, the scaled domain in Figure 5.1 is drawn. The
representation of a domain in FEMLAB is called a geometry [8, p.157].
5.3 Subdomain properties, equations and constants
From the multiphysics menu, a physics is chosen depending on which subdomain,
and appropriate parameters are entered in the subdomain settings dialog box
[8, p.157] to specify the diffusion-reaction equation in that subdomain. The coefficient
form of the PDE mode is used.
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