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A criterion of collective behavior of bacteria

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  • 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.

    Mathematics Subject Classification: Primary: 92B15, 92B25; Secondary: 74Q15.

    Citation:

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  • Figure 1.  Double periodic cell $Q_{(0,0)}$ with segments

    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 $\varrho = 0.35$)

    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 $
    $0.3$$3.13646$$28.9718$$-121.079$$1000.09 $
    $ 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 $
    $0.5$$3.14157$$21.4983$$-51.8595$$274.963 $
    $0.55$$3.14517$$20.5061$$-44.5169$$218.888 $
    $0.6$$3.13946$$19.7609$$-39.4423$$180.827 $
     | Show Table
    DownLoad: CSV

    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 $$ 2351.56 $
    $ 2 $$ 2067 $$ 3.25984 $$ 36.6725 $$ -158.136 $$ 1920.47 $
    $ 3 $$ 2066 $$ 3.19667 $$ 34.8162 $$ -164.29 $$ 2071.58 $
    $ 4 $$ 2040 $$ 3.29149 $$ 35.4505 $$ -149.94 $$ 2060.21 $
    $ 5 $$ 2064 $$ 3.27662 $$ 33.9367 $$ -141.591 $$ 1627.76 $
    $ 6 $$ 2056 $$ 3.42917 $$ 37.4054 $$ -190.248 $$ 2867.12 $
    $ 7 $$ 2026 $$ 3.34495 $$ 35.6335 $$ -157.051 $$ 1811.85 $
    $ 8 $$ 2030 $$ 3.13718 $$ 34.0681 $$ -169.746 $$ 2077.70 $
    $ 9 $$ 2039 $$ 3.21947 $$ 34.6973 $$ -148.317 $$ 1675.23 $
    $ 10 $$ 2044 $$ 3.06423 $$ 37.2784 $$ -177.122 $$ 2865.54 $
    $ 11 $$ 2023 $$ 2.95417 $$ 32.9400 $$ -157.421 $$ 1695.34 $
    $ 12 $$ 2014 $$ 3.09097 $$ 36.1141 $$ -208.578 $$ 2967.78 $
    $ 13 $$ 2027 $$ 3.00734 $$ 36.0749 $$ -215.528 $$ 3292.64 $
    $ 14 $$ 2034 $$ 3.16291 $$ 35.3946 $$ -194.029 $$ 2697.51 $
    $ 15 $$ 2059 $$ 3.21142 $$ 35.7572 $$ -175.982 $$ 2647.37 $
    $ 16 $$ 2016 $$ 3.19012 $$ 36.9914 $$ -200.469 $$ 3200.68 $
    $ 17 $$ 2016 $$ 3.30939 $$ 35.3018 $$ -163.073 $$ 1911.99 $
    $ 18 $$ 2057 $$ 3.22744 $$ 38.7036 $$ -243.944 $$ 4057.40$
    $ 19 $$ 2055 $$ 3.18527 $$ 35.9201 $$ -144.187 $$ 1701.75 $
    $ 20 $$ 2071 $$ 3.31315 $$ 37.6613 $$ -152.177 $$ 2094.90 $
    $ 21 $$ 2066 $$ 3.2770 $$ 33.6304 $$ -131.371 $$ 1735.46 $
    $ 22 $$ 2073 $$ 3.3854 $$ 35.1252 $$ -129.436 $$ 1330.40 $
    $ 23 $$ 2040 $$ 3.24423 $$ 33.6249 $$ -126.809 $$ 1305.79 $
    $ 24 $$ 2080 $$ 3.30177 $$ 36.0663 $$ -159.988 $$ 1707.04 $
    $ 25 $$ 2077 $$ 3.19037 $$ 34.2243 $$ -168.806 $$ 1970.43 $
    $ 26 $$ 2065 $$ 3.39291 $$ 39.0489 $$ -186.748 $$ 2108.54 $
    $ 27 $$ 2062 $$ 3.17936 $$ 34.0767 $$ -138.028 $$ 1354.70 $
    $ 28 $$ 2024 $$ 3.11102 $$ 40.2420 $$ -202.873 $$ 3966.32 $
    $ 29 $$ 2068 $$ 3.12904 $$ 33.4322 $$ -155.213 $$ 1801.78 $
    $ 30 $$ 2059 $$ 3.28145 $$ 36.8591 $$ -176.772 $$ 2198.46 $
    $ 31 $$ 2042 $$ 3.24301 $$ 37.0932 $$ -208.055 $$ 2844.27 $
     | Show Table
    DownLoad: CSV

    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 $$ 6565.85 $
    $ 2 $$ 3.07985 $$ 50.6260 $$ -515.407 $$ 9617.15 $
    $ 3 $$ 3.36286 $$ 58.2470 $$ -629.653 $$ 11184.9 $
    $ 4 $$ 3.31838 $$ 47.8645 $$ -380.243 $$ 5763.63 $
    $ 5 $$ 3.01309 $$ 47.7780 $$ -435.587 $$ 6984.50 $
    $ 6 $$ 3.14305 $$ 47.8691 $$ -400.298 $$ 6207.25 $
    $ 7 $$ 3.20741 $$ 50.5550 $$ -433.739 $$ 6256.86 $
    $ 8 $$ 3.20946 $$ 45.6877 $$ -348.511 $$ 4868.42 $
    $ 9 $$ 3.08756 $$ 50.2205 $$ -485.495 $$ 8630.89 $
    $ 10 $$ 3.14825 $$ 51.9186 $$ -498.135 $$ 7884.83 $
    $ 11 $$ 3.15232 $$ 50.4770 $$ -407.538 $$ 5794.05 $
    $ 12 $$ 2.97260 $$ 48.3467 $$ -415.332 $$ 6423.79 $
    $ 13 $$ 3.18407 $$ 48.6382 $$ -406.544 $$ 6317.61 $
    $ 14 $$ 3.12623 $$ 43.5618 $$ -332.846 $$ 5012.32 $
    $ 15 $$ 2.96333 $$ 47.0048 $$ -403.513 $$ 6158.98 $
    $ 16 $$ 3.13992 $$ 49.2681 $$ -428.006 $$ 6764.48 $
    $ 17 $$ 3.16460 $$ 48.0914 $$ -402.791 $$ 6347.72 $
    $ 18 $$ 3.09493 $$ 53.3020 $$ -483.722 $$ 7700.97 $
    $ 19 $$ 3.12330 $$ 50.4108 $$ -415.444 $$ 6743.15 $
    $ 20 $$ 3.21182 $$ 49.3165 $$ -410.478 $$ 6876.66 $
    $ 21 $$ 3.21308 $$ 50.4445 $$ -476.521 $$ 8126.50 $
    $ 22 $$ 2.97221 $$ 48.6954 $$ -441.899 $$ 7384.68 $
    $ 23 $$ 3.23927 $$ 51.1514 $$ -466.984 $$ 6864.76 $
    $ 24 $$ 3.11142 $$ 43.8766 $$ -362.591 $$ 5776.80 $
    $ 25 $$ 2.84798 $$ 44.1550 $$ -383.563 $$ 5705.14 $
    $ 26 $$ 3.09189 $$ 44.8430 $$ -373.888 $$ 6020.28 $
    $ 27 $$ 3.11219 $$ 44.5645 $$ -331.345 $$ 4733.94 $
    $ 28 $$ 3.05673 $$ 50.1022 $$ -490.807 $$ 8516.17 $
    $ 29 $$ 3.09775 $$ 48.5431 $$ -416.398 $$ 6597.56 $
    $ 30 $$ 2.99318 $$ 47.1511 $$ -432.571 $$ 6636.21 $
    $ 31 $$ 3.01481 $$ 47.5799 $$ -400.869 $$ 6078.16 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV
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