ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

3.1.3 Cell proliferation
It is assumed that for very small aggregate sizes, cell growth is unlimited and exponential. Because
the mechanics and geometry of each cell is modeled and not the intracellular biology and
biochemistry, the cell cycle can be broken up into two important stages: increase of cell volume
and cytokinesis. For the cytokinesis we use a dumb-bell approximation like described in [4].
Ri
✲ ✲ ✲
Figure 1: Dumb-bell approach for modeling geometry of binary fission
We assume that the total cytoplasmic volume of the two spheres which represent the division
complex, remains constant. As the distance between the spheres, d, increases, the radii of the
individual spheres, r, decrease. From the assumption of constant volume and the initial volume Vi,
a relationship between r and d can be obtained:
with
CA =
⎛
⎝ 3Vi
8π +
r = d2
−
16CA
d
4 +CA, (5)
9Vi 2
64π2 + 3Vi d3 256π
⎠
64
+ d3
⎞
1/3
. (6)
The separation of the two dividing spheres is accomplished through a “division force”. Biologically,
this force results from the assembly of new cell cortex material. The division force is not a
constant force but a force that is adapted to the mechanical environment of the cell. The cytokinesis
can be mechanically inhibited if he division force exceeds a predefined threshold force Fd,max.
In the model, the division force is calculated as:
Fd = Kd (dr − d), (7)
in which Kd is a gain factor that is chosen in function of the discretization, and dr −d the difference
between the reference distance dr and the actual distance d. The reference distance is defined to
increase linearly in time during cytokinesis, until the spheres have no more overlap and make up
two independent cells. Aside from the “optimal” cytokinesis time, the model also sets a growth
time, i.e. the time in between two cytokinesis events. Using a fixed growth time, the cell cycles
would be exactly synchronized, giving rise to uncharacteristic forces and stress levels. In reality,
cell cycles within an aggregate are partially asynchronous. Therefore we introduce a normal
distribution on the growth time, with a preset standard deviation.
3.1.4 Cell migration
Due to the absence of an extracellular matrix in early expanding cell cultures, the only way cells
actively move is by crawling over the substrate or by crawling over other cells. Therefore, we
implemented cell migration as a tangential contact force acting between cells or between cells and
the substrate. Assuming that at small spatial scales no significant gradients in biochemical cues
should be present, the cells perform a random walk. The migration force is perpendicular to the
contact normal vector and in the direction of the velocity vector of the previous timestep, with a
small random rotation (estimated). Aside from active migration, the suspended cells also move as
a result of Brownian motion.