The growth patterns generated by bacterial colonies form a kind of structure
that is neither regular nor random, but which is a self-similar structure
called a fractal. The dimension of a fractal, i.e. the fractal dimension,
can be defined/measured by different methods. The most popular one is called
the box-counting method.
(1)Box counting
The box dimension of a pattern d is defined by[5]
To calculate d, we divide the pattern into grid of non-overlapping
blocks of size e´e
and let the computer scan the whole pattern, counting the number of blocks
covering the pattern (or occupied by walkers).We
define the number as Ne.
Then we subdivide the grid into e/2´e/2
blocks and repeat the previous procedure until e=ap,
the pixel size. We then use the least squares method to determine the slope
of the logarithm of the counts
Ne
versus the block size e:
(2) Mass distribution
For fractals generated from a growth process, as in our case, the fractal
dimension can also be defined by: where
M
is the "mass" of the patterns with linear size R. Equivalently, we can
write it as: To
get evenly spaced data points of d, we let Rn=2n
and Mn=M(Rn)-M(Rn-1). It is easy to see
that where
the upper limit of Rn is taken to be the gyration radius of
the pattern Rg, which is defined as: where
the summation is over all walkers (including stationary ones). One problem
with this method, as we shall see in the results part, is caused by the
inoculation at the center of the lattice.We
will have thousands or tens of thousands of walkers packed in the center
region, and the first a few data with Rn=20, 21,
22 will be biased towards the high limit (i.e. 2) and must be
disposed of.Since we only have
5 or 6 points to fit the fractal dimension, we will have too few points
for fitting by throwing away the first a few.
One might argue that instead of having R=2n, we can use evenly
spaced
dr and use to
avoid the bias without counting the mass distribution in the center.However,
to make the above relation valid, dr
should be as small as possible.This
causes severe oscillations in the data dM(R) in comparison to M(R),
which is the cumulative function of dM(R).
We will compare the results from both the box counting method and the
mass distribution method and see the advantages and disadvantages of both.
where
Ne
is the minimal number of identical small objects (of linear size e
each) needed to cover the pattern.
.
In our simulation the active walkers perform off-lattice movement,
and the pixel size is not necessarily the lattice constant a0.Since
the area of each bacterium is about 1mm2
and each walker represents 5´103
bacteria, the size of each walker is ~0.1mm, smaller than a0~0.25mm.However,
in the final stage of the simulation, there are about 105walkers
in a limited area (R~50a0), and it is not a bad approximation
to view the walkers as if they are on the lattice in the final stage.Our
program also provides options to locate the final walkers on a fine grid
of lattice spacing ap with a0 = ap·sub
(sub is an integer >1).
