c - IARIA Journals

International Journal on Advances in Systems and Measurements, vol 5 no 3 & 4, year 2012, http://www.iariajournals.org/systems_and_measurements/
Figure 2. Space-time plot of bioluminescence measured along the contact
line of the cylindrical vessel [12], [16].
contact line. The images were captured at different times of
the population evolution.
In general, the dynamic processes in unstirred cultures are
rather complicated and need to be modeled in three dimensional
space [2], [11], [12]. Since luminous cells concentrate
near the contact line, the three-dimensional processes were
simulated in one dimension (quasi-one dimensional rings in
Figure 1). Figure 2 shows the corresponding space-time plot
of quasi-one-dimensional bioluminescence intensity.
By varying the model parameters the simulation results
were analyzed with a special emphasis to achieving a
spatiotemporal pattern similar to the experimentally obtained
pattern shown in Figure 2. Figure 3 shows the results of the
informal pattern fitting, where Figures 3a and 3b present the
simulated space-time plots of the dimensionless cell density
u and the chemoattractant concentration v, respectively. The
corresponding valuesū and ¯v averaged on the circumference
of the vessel are depicted in Figure 3c,
1
ū(t) = u(x,t)dx,
0
1
¯v(t) = v(x,t)dx.
0
(20)
Regular oscillations as well as chaotic fluctuations similar
to the experimental ones were computationally simulated.
Accepting the constant form of the chemotactic sensitivity
(χ(u,v) = χ0) and the simple gradient, the dynamics of the
bacterial population was simulated at the following values
of the model parameters [16]:
a) x
b)
x
2012, © Copyright by authors, Published under agreement with IARIA - www.iaria.org
t
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 0.5 1
ū, v̄
2.5
0.0
t
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 0.5 1
1.0
0.9
0.8
0.7
0.6
0.5
ū
0.4
0.3
0.2
0.1
v̄
0.0
0 0.1 0.2 0.3 0.4 0.5 0.6
c)
Figure 3. Simulated space-time plots of the dimensionless cell density u
(a) as well as the chemoattractant concentration v (b) and the dynamics of
the corresponding averaged values ū and ¯v (c). Values of the parameters
are as defined in (21).
D = 0.1, χ0 = 6.2, ρ = 0, r = 1,
φ = 0.73, s = 625, m = 0.
t
1.0
0.0
(21)
A spatially-varying random perturbation ε(x) of the dimensionless
cell density u of 10% was applied for the initial
distribution of bacteria near the three phase contact line
when simulating the spatiotemporal patterns.
Due to a relatively great number of model parameters,
there is no guarantee that the values (21) mostly approach the
pattern shown in Figure 2. Similar patterns were achieved at
different values of the model parameters. The linearization
and the stability analysis of homogenous solutions of the
Keller-Segel model showed similar effects [32], [33]. An
increase in one parameter can be often compensated by
decreasing or increasing another one. Because of this, it is
important to investigate the influence of the model parameters
on the pattern formation and to develop a mathematical
model containing a minimal number of parameters and
ensuring a qualitative analysis of bacterial pattern formation
in a liquid medium [6], [10], [17], [21].
V. RESULTS AND DISCUSSION
By varying the input parameters the output results were
analyzed with a special emphasis on the influence of the
158