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

3.3.1 Centers of mass
Table 1: Base parameter set for the individual-based model
Parameter Symbol Units Value
timestep ∆t s 0.5 ·10 −3
simulation time Tend s 65 ·10 3
conjugate gradient precision emax m/s 10 ·10 −12
division size R0 m 4.5 ·10 −6
division size sdev δdiv m 0.1 ·10 −6
division force threshold Fd,max N 0.5 ·10 −9
division time Td s 3.6 ·10 3
growth time Tg s 7.2 ·10 3
attr constant FLO + -FLO + σ++ J/m 2 8 ·10 −6
attr constant FLO + - f lo − σ+− J/m 2 4.5 ·10 −6
attr constant f lo − - f lo − σ−− J/m 2 1 ·10 −6
youngs modulus Ec Pa 1600
poisson ratio νc 0.4
perpendicular friction coef. γ ⊥ kg/s 0.06 ·10 −6
parallel friction coef. γ kg/s 0.2 ·10 −6
viscosity µ Pa·s 0.7 ·10 −3
temperature T K 310
The easiest is to compare the centers of mass for the different constituents. We define
ε =
Nf lo− xi
∑
i=0
mi
NFLO +
xi
− ∑
i=0
mi
/ 3
Voltotal · 3
4π
which is normalized with the theoretical minimum radius of a cell-mass with the same volume for
better comparison (see figure 3 (e)). For an ideally sorted and spherical flock, ε approaches zero.
A problem of this method is, that ε becomes zero for completely randomly mixed flocks as well.
3.3.2 Radial distribution function
Another generic description of particle distribution in such a system that is often used, is the
radial distribution function (also called pair-correlation or two-point correlation function). We
encountered two problems with this approach: The statistics are even at the end of the simulation
not very good, and the normalization is questionable, since the algorithm we employed uses the
smallest rectangular box aligned with the global coordinate system, into which the whole colony
fits. This leads to results comparable to figure 2, but both problems make it impossible to compare
more than the first peaks of different realizations of the system. Therefore, we prefer to directly
compare the average coordination numbers of the constituents (for which the first peak of the
radial distribution function is indicative):
3.3.3 Coordination number
The average coordination number can be calculated by counting how many other cells a given cell
has contact with - here we define contact as being closer than 102 % of the added radii of the cells.
The precise number does not influence the outcome (see figure 3 (a) - (d)). We distinguish contacts
between a FLO1 expressing cell (FLO + ) and a non-expressing cell ( f lo − ), etc.
(5)