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January 2017, 14(1): 277-287. doi: 10.3934/mbe.2017018
A criterion of collective behavior of bacteria
Roman Czapla and Vladimir V. Mityushev ,
Institute of Computer Science, Pedagogical University, ul. Podchorazych 2, Krakow 30-084, Poland
Received October 2015 Accepted February 05, 2016 Published October 2016
Figure(5) / Table(4)
It was established in the previous works that hydrodynamic interactions between the swimmers can lead to collective motion. Its implicit evidences were confirmed by reduction in the effective viscosity. We propose a new quantitative criterion to detect such a collective behavior. Our criterion is based on a new computationally effective RVE (representative volume element) theory based on the basic statistic moments ($e$-sums or generalized Eisenstein-Rayleigh sums). The criterion can be applied to various two-phase dispersed media (biological systems, composites etc). The locations of bacteria are modeled by short segments having a small width randomly embedded in medium without overlapping. We compute the $e$-sums of the simulated disordered sets and of the observed experimental locations of Bacillus subtilis. The obtained results show a difference between these two sets that demonstrates the collective motion of bacteria.
Keywords: Collective behavior of bacteria, disordered random sets, generalized Eisenstein-Rayleigh sums, RVE, non-overlapping segments on plane.
Mathematics Subject Classification: Primary: 92B15, 92B25; Secondary: 74Q15.
Citation: Roman Czapla, Vladimir V. Mityushev. A criterion of collective behavior of bacteria. Mathematical Biosciences & Engineering, 2017, 14 (1) : 277-287. doi: 10.3934/mbe.2017018
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Figure 1. Double periodic cell $Q_{(0,0)}$ with segments
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Figure 2. The real (circles) and imaginary (crosses) parts of the averaged directions for $N = 500$ and for the total number of distributions $M = 1500$ ($\varrho = 0.25$). All absolute values do not exceed $0.15$
Figure 3. $\langle e_{44}\rangle$ for $N = 500$ and for various densities a) $\varrho = 0.15$; b) $\varrho = 0.25$; c) $\varrho = 0.35$. Dashed lines show the deviation bounds $2\%$ (for $ \varrho = 0.15$), $1.5\%$ (for $\varrho = 0.25$) and $1\%$ (for $\varrho = 0.35$)
18]">Figure 4. Bacillus subtilis [18]
Figure 5. The values of $e_{44}$ for subsequent frames of the film
Table 1. The averaged $e$-sums for various densities
$\mathbf{\varrho}$ $\mathbf{\mbox{Re}\big[\langle e_{2}\rangle\big]}$ $\mathbf{\langle e_{22}\rangle}$ $\mathbf{\langle e_{33}\rangle}$ $\mathbf{\langle e_{44}\rangle}$
$0.05$ $3.12977$ $129.053$ $-3554.78$ $165787.0 $
$0.1$ $3.14228$ $68.9110$ $-926.015$ $21743.5 $
$\mathbf{0.15}$ $\mathbf{3.13271}$ $\mathbf{48.7003}$ $\mathbf{-424.611}$ $\mathbf{6725.43}$
$0.2$ $3.13447$ $38.8351$ $-251.143$ $3037.38 $
$0.25$ $3.14641$ $33.0394$ $-167.170$ $1635.55 $
$ 0.35$ $3.14165 $ $26.3229$ $-93.1703$ $672.818$
$0.4$ $3.14652$ $24.2258$ $-73.9405$ $472.197 $
$0.45$ $3.14838$ $22.7573$ $-61.2791$ $354.635 $
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Table 2. The $e$-sums for 31 film frames of Bacillus subtilis. The first column contains the number of the film frame, the second column contains the number of bacteria $N$ detected in the frame. The next columns show basic sums
$\mathbf{no.}$ $\mathbf N$ $\mathbf{\mbox{Re}[ e_2]}$ $\mathbf{ e_{22}}$ $\mathbf{ e_{33}}$ $\mathbf{ e_{44}}$
$ 1 $ $ 2065 $ $ 3.24113 $ $ 35.3172 $ $ -166.312 $ $ 2351.56 $
$ 3 $ $ 2066 $ $ 3.19667 $ $ 34.8162 $ $ -164.29 $ $ 2071.58 $
$ 10 $ $ 2044 $ $ 3.06423 $ $ 37.2784 $ $ -177.122 $ $ 2865.54 $
$ 18 $ $ 2057 $ $ 3.22744 $ $ 38.7036 $ $ -243.944 $ $ 4057.40$
$ 21 $ $ 2066 $ $ 3.2770 $ $ 33.6304 $ $ -131.371 $ $ 1735.46 $
Table 3. The $e$-sums calculated for 31 samples of DB sets. The parameters of distribution are $N = 2050$, $\varrho = 0.15$ and $\delta = \frac{l}{4}$
$\mathbf{no.}$ $\mathbf{\mbox{Re}[ e_2]}$ $\mathbf{ e_{22}}$ $\mathbf{ e_{33}}$ $\mathbf{ e_{44}}$
$ 1 $ $ 3.17987 $ $ 46.6427 $ $ -393.453 $ $ 6565.85 $
$ 3 $ $ 3.36286 $ $ 58.2470 $ $ -629.653 $ $ 11184.9 $
$ 10 $ $ 3.14825 $ $ 51.9186 $ $ -498.135 $ $ 7884.83 $
Table 4. Comparison of the averaged $e$-sums for the observed bacteria locations with the $e$-sums computed for the DB sets $(\varrho = 0.15)$ from Table 2 and Table 3
$\mathbf{\mbox{Re}\big[\langle e_{2}\rangle\big]}$ $\mathbf{\langle e_{22}\rangle}$ $\mathbf{\langle e_{33}\rangle}$ $\mathbf{\langle e_{44}\rangle}$
averaged $e$-sums for theoretical distributions $ 3.11721 $ $ 48.6107 $ $ -425.941 $ $ 6791.75 $
standard deviation of the $e$-sums for theoretical distributions $ 0.107542 $ $ 3.02546 $ $ 60.3803 $ $ 1366.42 $
averaged $e$-sums for distributions of bacteria $ 3.22092$ $ 35.7922 $ $ -169.75 $ $ 2255.47 $
standard deviation of the $e$-sums for distributions of bacteria $ 0.108139 $ $ 1.73937 $ $ 27.9609 $ $ 717.895 $
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Roman Czapla Vladimir V. Mityushev