American Institute of Mathematical Sciences

January  2017, 14(1): 277-287. doi: 10.3934/mbe.2017018

A criterion of collective behavior of bacteria

 Institute of Computer Science, Pedagogical University, ul. Podchorazych 2, Krakow 30-084, Poland

* Corresponding author

Received  October 2015 Accepted  February 05, 2016 Published  October 2016

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.

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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References:
Double periodic cell $Q_{(0,0)}$ with segments
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 $0.15$
$\langle e_{44}\rangle$ 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 $\varrho = 0.35$)
Bacillus subtilis [18]
The values of $e_{44}$ for subsequent frames of the film
The averaged $e$-sums for various densities
 $\mathbf{\varrho}$ $\mathbf{\mbox{Re}\big[\langle e_{2}\rangle\big]}$ $\mathbf{\langle e_{22}\rangle}$ $\mathbf{\langle e_{33}\rangle}$ $\mathbf{\langle e_{44}\rangle}$ $0.05$ $3.12977$ $129.053$ $-3554.78$ $165787.0$ $0.1$ $3.14228$ $68.9110$ $-926.015$ $21743.5$ $\mathbf{0.15}$ $\mathbf{3.13271}$ $\mathbf{48.7003}$ $\mathbf{-424.611}$ $\mathbf{6725.43}$ $0.2$ $3.13447$ $38.8351$ $-251.143$ $3037.38$ $0.25$ $3.14641$ $33.0394$ $-167.170$ $1635.55$ $0.3$ $3.13646$ $28.9718$ $-121.079$ $1000.09$ $0.35$ $3.14165$ $26.3229$ $-93.1703$ $672.818$ $0.4$ $3.14652$ $24.2258$ $-73.9405$ $472.197$ $0.45$ $3.14838$ $22.7573$ $-61.2791$ $354.635$ $0.5$ $3.14157$ $21.4983$ $-51.8595$ $274.963$ $0.55$ $3.14517$ $20.5061$ $-44.5169$ $218.888$ $0.6$ $3.13946$ $19.7609$ $-39.4423$ $180.827$
 $\mathbf{\varrho}$ $\mathbf{\mbox{Re}\big[\langle e_{2}\rangle\big]}$ $\mathbf{\langle e_{22}\rangle}$ $\mathbf{\langle e_{33}\rangle}$ $\mathbf{\langle e_{44}\rangle}$ $0.05$ $3.12977$ $129.053$ $-3554.78$ $165787.0$ $0.1$ $3.14228$ $68.9110$ $-926.015$ $21743.5$ $\mathbf{0.15}$ $\mathbf{3.13271}$ $\mathbf{48.7003}$ $\mathbf{-424.611}$ $\mathbf{6725.43}$ $0.2$ $3.13447$ $38.8351$ $-251.143$ $3037.38$ $0.25$ $3.14641$ $33.0394$ $-167.170$ $1635.55$ $0.3$ $3.13646$ $28.9718$ $-121.079$ $1000.09$ $0.35$ $3.14165$ $26.3229$ $-93.1703$ $672.818$ $0.4$ $3.14652$ $24.2258$ $-73.9405$ $472.197$ $0.45$ $3.14838$ $22.7573$ $-61.2791$ $354.635$ $0.5$ $3.14157$ $21.4983$ $-51.8595$ $274.963$ $0.55$ $3.14517$ $20.5061$ $-44.5169$ $218.888$ $0.6$ $3.13946$ $19.7609$ $-39.4423$ $180.827$
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
 $\mathbf{no.}$ $\mathbf N$ $\mathbf{\mbox{Re}[ e_2]}$ $\mathbf{ e_{22}}$ $\mathbf{ e_{33}}$ $\mathbf{ e_{44}}$ $1$ $2065$ $3.24113$ $35.3172$ $-166.312$ $2351.56$ $2$ $2067$ $3.25984$ $36.6725$ $-158.136$ $1920.47$ $3$ $2066$ $3.19667$ $34.8162$ $-164.29$ $2071.58$ $4$ $2040$ $3.29149$ $35.4505$ $-149.94$ $2060.21$ $5$ $2064$ $3.27662$ $33.9367$ $-141.591$ $1627.76$ $6$ $2056$ $3.42917$ $37.4054$ $-190.248$ $2867.12$ $7$ $2026$ $3.34495$ $35.6335$ $-157.051$ $1811.85$ $8$ $2030$ $3.13718$ $34.0681$ $-169.746$ $2077.70$ $9$ $2039$ $3.21947$ $34.6973$ $-148.317$ $1675.23$ $10$ $2044$ $3.06423$ $37.2784$ $-177.122$ $2865.54$ $11$ $2023$ $2.95417$ $32.9400$ $-157.421$ $1695.34$ $12$ $2014$ $3.09097$ $36.1141$ $-208.578$ $2967.78$ $13$ $2027$ $3.00734$ $36.0749$ $-215.528$ $3292.64$ $14$ $2034$ $3.16291$ $35.3946$ $-194.029$ $2697.51$ $15$ $2059$ $3.21142$ $35.7572$ $-175.982$ $2647.37$ $16$ $2016$ $3.19012$ $36.9914$ $-200.469$ $3200.68$ $17$ $2016$ $3.30939$ $35.3018$ $-163.073$ $1911.99$ $18$ $2057$ $3.22744$ $38.7036$ $-243.944$ $4057.40$ $19$ $2055$ $3.18527$ $35.9201$ $-144.187$ $1701.75$ $20$ $2071$ $3.31315$ $37.6613$ $-152.177$ $2094.90$ $21$ $2066$ $3.2770$ $33.6304$ $-131.371$ $1735.46$ $22$ $2073$ $3.3854$ $35.1252$ $-129.436$ $1330.40$ $23$ $2040$ $3.24423$ $33.6249$ $-126.809$ $1305.79$ $24$ $2080$ $3.30177$ $36.0663$ $-159.988$ $1707.04$ $25$ $2077$ $3.19037$ $34.2243$ $-168.806$ $1970.43$ $26$ $2065$ $3.39291$ $39.0489$ $-186.748$ $2108.54$ $27$ $2062$ $3.17936$ $34.0767$ $-138.028$ $1354.70$ $28$ $2024$ $3.11102$ $40.2420$ $-202.873$ $3966.32$ $29$ $2068$ $3.12904$ $33.4322$ $-155.213$ $1801.78$ $30$ $2059$ $3.28145$ $36.8591$ $-176.772$ $2198.46$ $31$ $2042$ $3.24301$ $37.0932$ $-208.055$ $2844.27$
 $\mathbf{no.}$ $\mathbf N$ $\mathbf{\mbox{Re}[ e_2]}$ $\mathbf{ e_{22}}$ $\mathbf{ e_{33}}$ $\mathbf{ e_{44}}$ $1$ $2065$ $3.24113$ $35.3172$ $-166.312$ $2351.56$ $2$ $2067$ $3.25984$ $36.6725$ $-158.136$ $1920.47$ $3$ $2066$ $3.19667$ $34.8162$ $-164.29$ $2071.58$ $4$ $2040$ $3.29149$ $35.4505$ $-149.94$ $2060.21$ $5$ $2064$ $3.27662$ $33.9367$ $-141.591$ $1627.76$ $6$ $2056$ $3.42917$ $37.4054$ $-190.248$ $2867.12$ $7$ $2026$ $3.34495$ $35.6335$ $-157.051$ $1811.85$ $8$ $2030$ $3.13718$ $34.0681$ $-169.746$ $2077.70$ $9$ $2039$ $3.21947$ $34.6973$ $-148.317$ $1675.23$ $10$ $2044$ $3.06423$ $37.2784$ $-177.122$ $2865.54$ $11$ $2023$ $2.95417$ $32.9400$ $-157.421$ $1695.34$ $12$ $2014$ $3.09097$ $36.1141$ $-208.578$ $2967.78$ $13$ $2027$ $3.00734$ $36.0749$ $-215.528$ $3292.64$ $14$ $2034$ $3.16291$ $35.3946$ $-194.029$ $2697.51$ $15$ $2059$ $3.21142$ $35.7572$ $-175.982$ $2647.37$ $16$ $2016$ $3.19012$ $36.9914$ $-200.469$ $3200.68$ $17$ $2016$ $3.30939$ $35.3018$ $-163.073$ $1911.99$ $18$ $2057$ $3.22744$ $38.7036$ $-243.944$ $4057.40$ $19$ $2055$ $3.18527$ $35.9201$ $-144.187$ $1701.75$ $20$ $2071$ $3.31315$ $37.6613$ $-152.177$ $2094.90$ $21$ $2066$ $3.2770$ $33.6304$ $-131.371$ $1735.46$ $22$ $2073$ $3.3854$ $35.1252$ $-129.436$ $1330.40$ $23$ $2040$ $3.24423$ $33.6249$ $-126.809$ $1305.79$ $24$ $2080$ $3.30177$ $36.0663$ $-159.988$ $1707.04$ $25$ $2077$ $3.19037$ $34.2243$ $-168.806$ $1970.43$ $26$ $2065$ $3.39291$ $39.0489$ $-186.748$ $2108.54$ $27$ $2062$ $3.17936$ $34.0767$ $-138.028$ $1354.70$ $28$ $2024$ $3.11102$ $40.2420$ $-202.873$ $3966.32$ $29$ $2068$ $3.12904$ $33.4322$ $-155.213$ $1801.78$ $30$ $2059$ $3.28145$ $36.8591$ $-176.772$ $2198.46$ $31$ $2042$ $3.24301$ $37.0932$ $-208.055$ $2844.27$
The $e$-sums calculated for 31 samples of DB sets. The parameters of distribution are $N = 2050$, $\varrho = 0.15$ and $\delta = \frac{l}{4}$
 $\mathbf{no.}$ $\mathbf{\mbox{Re}[ e_2]}$ $\mathbf{ e_{22}}$ $\mathbf{ e_{33}}$ $\mathbf{ e_{44}}$ $1$ $3.17987$ $46.6427$ $-393.453$ $6565.85$ $2$ $3.07985$ $50.6260$ $-515.407$ $9617.15$ $3$ $3.36286$ $58.2470$ $-629.653$ $11184.9$ $4$ $3.31838$ $47.8645$ $-380.243$ $5763.63$ $5$ $3.01309$ $47.7780$ $-435.587$ $6984.50$ $6$ $3.14305$ $47.8691$ $-400.298$ $6207.25$ $7$ $3.20741$ $50.5550$ $-433.739$ $6256.86$ $8$ $3.20946$ $45.6877$ $-348.511$ $4868.42$ $9$ $3.08756$ $50.2205$ $-485.495$ $8630.89$ $10$ $3.14825$ $51.9186$ $-498.135$ $7884.83$ $11$ $3.15232$ $50.4770$ $-407.538$ $5794.05$ $12$ $2.97260$ $48.3467$ $-415.332$ $6423.79$ $13$ $3.18407$ $48.6382$ $-406.544$ $6317.61$ $14$ $3.12623$ $43.5618$ $-332.846$ $5012.32$ $15$ $2.96333$ $47.0048$ $-403.513$ $6158.98$ $16$ $3.13992$ $49.2681$ $-428.006$ $6764.48$ $17$ $3.16460$ $48.0914$ $-402.791$ $6347.72$ $18$ $3.09493$ $53.3020$ $-483.722$ $7700.97$ $19$ $3.12330$ $50.4108$ $-415.444$ $6743.15$ $20$ $3.21182$ $49.3165$ $-410.478$ $6876.66$ $21$ $3.21308$ $50.4445$ $-476.521$ $8126.50$ $22$ $2.97221$ $48.6954$ $-441.899$ $7384.68$ $23$ $3.23927$ $51.1514$ $-466.984$ $6864.76$ $24$ $3.11142$ $43.8766$ $-362.591$ $5776.80$ $25$ $2.84798$ $44.1550$ $-383.563$ $5705.14$ $26$ $3.09189$ $44.8430$ $-373.888$ $6020.28$ $27$ $3.11219$ $44.5645$ $-331.345$ $4733.94$ $28$ $3.05673$ $50.1022$ $-490.807$ $8516.17$ $29$ $3.09775$ $48.5431$ $-416.398$ $6597.56$ $30$ $2.99318$ $47.1511$ $-432.571$ $6636.21$ $31$ $3.01481$ $47.5799$ $-400.869$ $6078.16$
 $\mathbf{no.}$ $\mathbf{\mbox{Re}[ e_2]}$ $\mathbf{ e_{22}}$ $\mathbf{ e_{33}}$ $\mathbf{ e_{44}}$ $1$ $3.17987$ $46.6427$ $-393.453$ $6565.85$ $2$ $3.07985$ $50.6260$ $-515.407$ $9617.15$ $3$ $3.36286$ $58.2470$ $-629.653$ $11184.9$ $4$ $3.31838$ $47.8645$ $-380.243$ $5763.63$ $5$ $3.01309$ $47.7780$ $-435.587$ $6984.50$ $6$ $3.14305$ $47.8691$ $-400.298$ $6207.25$ $7$ $3.20741$ $50.5550$ $-433.739$ $6256.86$ $8$ $3.20946$ $45.6877$ $-348.511$ $4868.42$ $9$ $3.08756$ $50.2205$ $-485.495$ $8630.89$ $10$ $3.14825$ $51.9186$ $-498.135$ $7884.83$ $11$ $3.15232$ $50.4770$ $-407.538$ $5794.05$ $12$ $2.97260$ $48.3467$ $-415.332$ $6423.79$ $13$ $3.18407$ $48.6382$ $-406.544$ $6317.61$ $14$ $3.12623$ $43.5618$ $-332.846$ $5012.32$ $15$ $2.96333$ $47.0048$ $-403.513$ $6158.98$ $16$ $3.13992$ $49.2681$ $-428.006$ $6764.48$ $17$ $3.16460$ $48.0914$ $-402.791$ $6347.72$ $18$ $3.09493$ $53.3020$ $-483.722$ $7700.97$ $19$ $3.12330$ $50.4108$ $-415.444$ $6743.15$ $20$ $3.21182$ $49.3165$ $-410.478$ $6876.66$ $21$ $3.21308$ $50.4445$ $-476.521$ $8126.50$ $22$ $2.97221$ $48.6954$ $-441.899$ $7384.68$ $23$ $3.23927$ $51.1514$ $-466.984$ $6864.76$ $24$ $3.11142$ $43.8766$ $-362.591$ $5776.80$ $25$ $2.84798$ $44.1550$ $-383.563$ $5705.14$ $26$ $3.09189$ $44.8430$ $-373.888$ $6020.28$ $27$ $3.11219$ $44.5645$ $-331.345$ $4733.94$ $28$ $3.05673$ $50.1022$ $-490.807$ $8516.17$ $29$ $3.09775$ $48.5431$ $-416.398$ $6597.56$ $30$ $2.99318$ $47.1511$ $-432.571$ $6636.21$ $31$ $3.01481$ $47.5799$ $-400.869$ $6078.16$
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 $0.108139$ $1.73937$ $27.9609$ $717.895$
 $\mathbf{\mbox{Re}\big[\langle e_{2}\rangle\big]}$ $\mathbf{\langle e_{22}\rangle}$ $\mathbf{\langle e_{33}\rangle}$ $\mathbf{\langle e_{44}\rangle}$ 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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