Collective Dynamics in Self-propelled Bacteria Suspensions
Kuo-An Liu (劉國安), and Lin I (伊林)
Department of Physics and Center for Complex Systems, National Central University, Taiwan, Republic of China
What We Do
Method
The self-propelled bacteria can exhibit collective
behaviors like self-organized clustering and largescale coherent motions especially at high population
concentration. The dynamics of swimming bacteria
in a freely suspended thin liquid film was
experimentally investigated. We showed the
dynamics of the flow field and the statistical
properties. It was found that at high bacterial
concentration, the strong nonlinear bacteria
interactions lead to collective bacterial motions with
fluctuating vortices and the strong correlation
between the local velocity alignment and the velocity
magnitude.
Overnight cultures of wild type Escherichia coli RP437 in Luria Broth
were diluted 1/100 in T Broth, grown at 30 ℃ with 260 rpm shaking for
3 hrous, washed by centrifugation (2000 g for 10 min), and
resuspended to concentration 5 × 1010 ml-1 in motility medium (0.01 M
potassium phosphate, 0.067 M sodium chloride, 10-4 M EDTA, 0.01 M
glucose, and 0.002% Tween-20, pH 7.0).
A 1 ml drop of bacterial suspension was placed in a small square
formed by four Nylon monofilaments of diameter 20 mm in a chamber
with air saturated with water. Two of these fibers are movable. The
drop was stretched to area 5 × 5 mm2 by moving the fibers through
controlling the screw. The elasticity and surface tension created by the
surfactant Tween-20 were sufficient to maintain the film for few ten
minutes.
Vorticity (sec-1)
-6
-3
0
3
Speed (mm / sec)
6
100 mm
0
20
40
Velocity and vorticity
60
0.5 sec
wire
screw
film
Scale variation
Phase contrast images were recorded at 60
fps. The velocity field was extracted from
consecutive images by particle image
velocimetry (PIV) software MatPIV in Matlab.
Left and right columns show the vorticity and
speed of the flow respectively. The largest
vortex size can be up to 100 mm, which is
much larger than the scale at which the
energy is injected through the beating of
flagella. This large spatial coherent flow
results from collective motion of swimmers.
The flow fields show rich dynamics such as
vortex formation, propagation, fluctuation,
dissociation, combination, and dissipation.
Note that the high speed regions labeled by
red color in the right column usually exist in
the boundary of the vortex.
0 sec
chamber
The following two figures show the probability
distribution functions (PDFs) of the (a) speed and
(b) vorticity at different spatial coarse-grain scales.
(a) Different coarse-grain scales lead to different
distribution profiles. Without coarse grain, there
are two peaks at 15 and 40 mm/sec with a valley
at 25 mm/sec, indicating that the flow field either
flows slow or fast.
(b) The PDFs of vorticity are fitted by the
Gaussian distribution (solid curves). The data
have Gaussian core and non-Gaussian tail.
Interestingly, the tail without coarse-grain is longer
than that of Gaussian distribution (solid curve),
while PDFs with 5X5 and 7X7 coarse-grain have
shorter tails. The 3X3 PDF is more Gaussian-like.
The results show that the PDFs of velocity and
vorticity are not independent of the spatial scales.
(a)
(b)
140 sec
Definition of velocity field alignment
Definition of Correlation probability
correlation
probability
joint
probability
speed
〈〉 : average over 100 velocity
vectors in the square (60x 60
mm2) centered at
ordered
alignment
random
alignment
Joint probability (Arb. Units)
0.1
1
10
uncorrelated
probability
degree of alignment
C = 0 : uncorrelated
C > 0 : more likely to occur together
C < 0 : less likely to occur together
Correlation probability (Arb. Units)
-2
-1
0
1
2
Where is the high speed domain?
We introduced a local measurement Y(r) to represent the degree of alignment of
velocity vectors. If all 10 x 10 velocity vectors in a 60 X 60 mm2 square centered at r
have the same orientation, then Y(r) = 1. If the orientation is random, then Y(r) = 0. A
correlation probability was calculated by subtracting the uncorrelated one from the joint
probability. This method was used to study the relation between alignment Y(r) and
speed |v|. It was found that low speed regions are more likely to be associated with
poor alignment, while high speed regions prefer better alignment.
Discussion
degree of alignment y
The self-propelled system shows complicated dynamics. The PDFs of velocity
and vorticity are not universal at different spatial scales, indicating that the
classical self-similarity concept cannot simply describe this system.
Furthermore, our preliminary results about the correlation between field
alignments and velocity magnitudes may inspire future work on predicting the
formation of high speed domain from the spatiotemporal correlation.
References
Speed (mm / sec)
[1] Xiao-Lun Wu et al., Phys. Rev. Lett. 84, 3017 (2000).
[2] Andrey Sokolov et al., Phys. Rev. E 80, 031903 (2009).
[3] T. Ishikawa et al., Phys. Rev. Lett. 107, 028102 (2011).
[4] J. K. Sveen, An Introduction to MatPIV v.1.6.1, (Dept. of Mathematics, University of Oslo,
Oslo, 2004), http:// www.math.uio.no/~jks/matpiv.