Compute-Intensive Simulations for Cellular Models - Parallel Computing for Bioinformatics and Computational Biology - page 110

Finally we conclude with some applications and discussion of the ultimate aims of
modeling colonies of cells.

4.7.1 Modelling the Forces Between Cells

In [60] colonies of cells are modeled by considering each cell to be a circle in two
dimensions (see also [61, 62, 64]). In its most basic form each cell is characterized
by its position, radius, and a list of neighbors with which it is bonded. Each cell can
then exert a force on the neighbors to which it is bonded.
For a collection of n cells, the forces acting on the i th cell are modeled using
Newton's second law as follows:

n

m i d 2 u i
d t 2

c i d u i

+

d t =

F ij , (4.27)

j = 1

where u i is the physical displacement of the i th cell, m i its mass, c i a damping constant,
and F ij the force between the i th and j th cells. Note that F ij is only nonzero for cells
that are bonded.
The i th cell has a radius r i , which can change as the cell grows, and the mass is
proportional to the area of the cell. The damping constant is also proportional to the
mass.
The force F ij between the i th and j th cells is assumed to be related to the separation

as

F ij = r ij (r ij − (r i + r j )) ,

where r ij is the separation of the centers of cells i and j . Hence, when the surfaces
of a pair of cells are just touching there is no force between them, while overlapping
cells experience a repulsive force, and separated cells an attractive force. Note that
the pairs of bonded cells that are most distant experience the greatest attractive force.
However, as the forces only apply to bonded pairs this is not unreasonable. Another
approach to intercellular forces is taken in [64] where the force peaks when the cells
are in close contact, and tails off quickly with greater distance.
The next step is to consider the components of the forces along the x and y axes.
Using elementary geometry, the components of Eq. (4.27 ) along the axes are readily
obtained, giving a total of 2 n differential equations. To solve these the standard Euler
approximations

d u i , x
d t

u i , x (t) − u i , x (t − δt)

= ,

δt

d 2 u i , x
d t 2

u i , x (t + δt) − 2 u i , x (t) + u i , x (t − δt)

=

(δt) 2

may be used, where δt is a small time interval and u i , x the component of u i along the
x -axis, and similarly for the y -components. This gives a system of 2 n linear equations
in 2 n unknowns.