# 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
##### References:

show all references

##### 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 $0.108139$ $1.73937$ $27.9609$ $717.895$
 [1] Alexander Blokh, Lex Oversteegen, Vladlen Timorin. Non-degenerate locally connected models for plane continua and Julia sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5781-5795. doi: 10.3934/dcds.2017251 [2] Michael Blank. Emergence of collective behavior in dynamical networks. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 313-329. doi: 10.3934/dcdsb.2013.18.313 [3] Guang-hui Cai. Strong laws for weighted sums of i.i.d. random variables. Electronic Research Announcements, 2006, 12: 29-36. [4] Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature. Mathematical Biosciences & Engineering, 2017, 14 (2) : 437-453. doi: 10.3934/mbe.2017027 [5] Laurent Boudin, Francesco Salvarani. The quasi-invariant limit for a kinetic model of sociological collective behavior. Kinetic & Related Models, 2009, 2 (3) : 433-449. doi: 10.3934/krm.2009.2.433 [6] Alessandro Ferriero, Nicola Fusco. A note on the convex hull of sets of finite perimeter in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 103-108. doi: 10.3934/dcdsb.2009.11.103 [7] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 [8] Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523 [9] Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020233 [10] A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100 [11] Víctor Jiménez López, Gabriel Soler López. A topological characterization of ω-limit sets for continuous flows on the projective plane. Conference Publications, 2001, 2001 (Special) : 254-258. doi: 10.3934/proc.2001.2001.254 [12] Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 [13] Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161 [14] Rich Stankewitz, Toshiyuki Sugawa, Hiroki Sumi. Hereditarily non uniformly perfect sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2391-2402. doi: 10.3934/dcdss.2019150 [15] Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687 [16] Jing Wang, Feng Xie. On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2767-2784. doi: 10.3934/dcdsb.2016072 [17] Karsten Matthies, George Stone. Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3299-3355. doi: 10.3934/dcds.2018143 [18] Giuseppe Da Prato, Arnaud Debussche. Asymptotic behavior of stochastic PDEs with random coefficients. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1553-1570. doi: 10.3934/dcds.2010.27.1553 [19] Piotr Oprocha, Pawel Wilczynski. Distributional chaos via isolating segments. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 347-356. doi: 10.3934/dcdsb.2007.8.347 [20] Elena Beretta, Markus Grasmair, Monika Muszkieta, Otmar Scherzer. A variational algorithm for the detection of line segments. Inverse Problems & Imaging, 2014, 8 (2) : 389-408. doi: 10.3934/ipi.2014.8.389

2018 Impact Factor: 1.313