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:
[1]

N. I. Akhiezer, Elements of Theory of Elliptic Functions Nauka, 1970 (in Russian); Engl. transl. AMS, 1990.

[2]

R. CzaplaV. V. Mityushev and W. Nawalaniec, Effective conductivity of random two-dimensional composites with circular non-overlapping inclusions, Computational Materials Science, 63 (2012), 118-126.  doi: 10.1016/j.commatsci.2012.05.058.

[3]

R. CzaplaV. V. Mityushev and W. Nawalaniec, Simulation of representative volume elements for random 2D composites with circular non-overlapping inclusions, Theoretical and Applied Informatics, 24 (2012), 227-242. 

[4]

R. CzaplaV. V. Mityushev and N. Rylko, Conformal mapping of circular multiply connected domains onto segment domains, Electron. Trans. Numer. Anal., 39 (2012), 286-297. 

[5]

S. GluzmanD. A. Karpeev and L. V. Berlyand, Effective viscosity of puller-like microswimmers: A renormalization approach, J. R. Soc. Interface, 10 (2013), 1-10.  doi: 10.1098/rsif.2013.0720.

[6]

V. V. Mityushev, Representative cell in mechanics of composites and generalized Eisenstein--Rayleigh sums, Complex Variables, 51 (2006), 1033-1045.  doi: 10.1080/17476930600738576.

[7]

V. V. Mityushev and P. Adler, Longitudial permeability of a doubly periodic rectangular array of circular cylinders, I, ZAMM (Journal of Applied Mathematics and Mechanics), 82 (2002), 335-345.  doi: 10.1002/1521-4001(200205)82:5<335::AID-ZAMM335>3.0.CO;2-D.

[8]

V. V. Mityushev and W. Nawalaniec, Basic sums and their random dynamic changes in description of microstructure of 2D composites, Computational Materials Science, 97 (2015), 64-74.  doi: 10.1016/j.commatsci.2014.09.020.

[9]

V. V. Mityushev and N. Rylko, Optimal distribution of the non-overlapping conducting disks, Multiscale Model. Simul., 10 (2012), 180-190.  doi: 10.1137/110823225.

[10]

W. Nawalaniec, Algorithms for computing symbolic representations of basic e–sums and their application to composites Journal of Symbolic Computation 74 (2016), 328–345.

[11]

M. Potomkin, V. Gyrya, I. Aranson and L. Berlyand, Collision of microswimmers in viscous fluid Physical Review E 87 (2013), 053005. doi: 10.1103/PhysRevE.87.053005.

[12]

S. D. RyanL. BerlyandB. M. Haines and D. A. Karpeev, A kinetic model for semi-dilute bacterial suspensions, Multiscale Model. Simul., 11 (2013), 1176-1196.  doi: 10.1137/120900575.

[13]

S. D. Rayn, B. M. Haines, L. Berlyand, F. Ziebert and I. S. Aranson, Viscosity of bacterial suspensions: Hydrodynamic interactions and self-induced noise Rapid Communication to Phys. Rev. E 83 (2011), 050904(R). doi: 10.1103/PhysRevE.83.050904.

[14]

S. D. Ryan, A. Sokolov, L. Berlyand and I. S. Aranson, Correlation properties of collective motion in bacterial suspensions New Journal of Physics 15 (2013), 105021, 18pp. doi: 10.1088/1367-2630/15/10/105021.

[15]

N. Rylko, Representative volume element in 2D for disks and in 3D for balls, J. Mechanics of Materials and Structures, 9 (2014), 427-439.  doi: 10.2140/jomms.2014.9.427.

[16]

A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria Phys. Rev. Lett. 103 (2009), 148101. doi: 10.1103/PhysRevLett.103.148101.

[17]

A. Sokolov and I. S. Aranson, Physical properties of collective motion in suspensions of bacteria Phys. Rev. Lett. 109 (2012), 248109. doi: 10.1103/PhysRevLett.109.248109.

[18]

A. Sokolov, I. S. Aranson, J. O. Kessler and R. E. Goldstein, Concentration dependence of the collective dynamics of swimming bacteria Physical Review Letters 98 (2007), 158102. doi: 10.1103/PhysRevLett.98.158102.

[19]

A. Sokolov, R. E. Goldstein, F. I. Feldstein and I. S. Aranson, Enhanced mixing and spatial instability in concentrated bacteria suspensions, Phys. Rev. E, 80 (2009), 031903.

[20]

M. TournusL. V. BerlyandA. Kirshtein and I. Aranson, Flexibility of bacterial flagella in external shear results in complex swimming trajectories, Journal of the Royal Society Interface, 12 (2015), 1-11.  doi: 10.1098/rsif.2014.0904.

[21]

A. Weil, Elliptic Functions According to Eisenstein and Kronecker Springer-Verlag, 1976.

show all references

References:
[1]

N. I. Akhiezer, Elements of Theory of Elliptic Functions Nauka, 1970 (in Russian); Engl. transl. AMS, 1990.

[2]

R. CzaplaV. V. Mityushev and W. Nawalaniec, Effective conductivity of random two-dimensional composites with circular non-overlapping inclusions, Computational Materials Science, 63 (2012), 118-126.  doi: 10.1016/j.commatsci.2012.05.058.

[3]

R. CzaplaV. V. Mityushev and W. Nawalaniec, Simulation of representative volume elements for random 2D composites with circular non-overlapping inclusions, Theoretical and Applied Informatics, 24 (2012), 227-242. 

[4]

R. CzaplaV. V. Mityushev and N. Rylko, Conformal mapping of circular multiply connected domains onto segment domains, Electron. Trans. Numer. Anal., 39 (2012), 286-297. 

[5]

S. GluzmanD. A. Karpeev and L. V. Berlyand, Effective viscosity of puller-like microswimmers: A renormalization approach, J. R. Soc. Interface, 10 (2013), 1-10.  doi: 10.1098/rsif.2013.0720.

[6]

V. V. Mityushev, Representative cell in mechanics of composites and generalized Eisenstein--Rayleigh sums, Complex Variables, 51 (2006), 1033-1045.  doi: 10.1080/17476930600738576.

[7]

V. V. Mityushev and P. Adler, Longitudial permeability of a doubly periodic rectangular array of circular cylinders, I, ZAMM (Journal of Applied Mathematics and Mechanics), 82 (2002), 335-345.  doi: 10.1002/1521-4001(200205)82:5<335::AID-ZAMM335>3.0.CO;2-D.

[8]

V. V. Mityushev and W. Nawalaniec, Basic sums and their random dynamic changes in description of microstructure of 2D composites, Computational Materials Science, 97 (2015), 64-74.  doi: 10.1016/j.commatsci.2014.09.020.

[9]

V. V. Mityushev and N. Rylko, Optimal distribution of the non-overlapping conducting disks, Multiscale Model. Simul., 10 (2012), 180-190.  doi: 10.1137/110823225.

[10]

W. Nawalaniec, Algorithms for computing symbolic representations of basic e–sums and their application to composites Journal of Symbolic Computation 74 (2016), 328–345.

[11]

M. Potomkin, V. Gyrya, I. Aranson and L. Berlyand, Collision of microswimmers in viscous fluid Physical Review E 87 (2013), 053005. doi: 10.1103/PhysRevE.87.053005.

[12]

S. D. RyanL. BerlyandB. M. Haines and D. A. Karpeev, A kinetic model for semi-dilute bacterial suspensions, Multiscale Model. Simul., 11 (2013), 1176-1196.  doi: 10.1137/120900575.

[13]

S. D. Rayn, B. M. Haines, L. Berlyand, F. Ziebert and I. S. Aranson, Viscosity of bacterial suspensions: Hydrodynamic interactions and self-induced noise Rapid Communication to Phys. Rev. E 83 (2011), 050904(R). doi: 10.1103/PhysRevE.83.050904.

[14]

S. D. Ryan, A. Sokolov, L. Berlyand and I. S. Aranson, Correlation properties of collective motion in bacterial suspensions New Journal of Physics 15 (2013), 105021, 18pp. doi: 10.1088/1367-2630/15/10/105021.

[15]

N. Rylko, Representative volume element in 2D for disks and in 3D for balls, J. Mechanics of Materials and Structures, 9 (2014), 427-439.  doi: 10.2140/jomms.2014.9.427.

[16]

A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria Phys. Rev. Lett. 103 (2009), 148101. doi: 10.1103/PhysRevLett.103.148101.

[17]

A. Sokolov and I. S. Aranson, Physical properties of collective motion in suspensions of bacteria Phys. Rev. Lett. 109 (2012), 248109. doi: 10.1103/PhysRevLett.109.248109.

[18]

A. Sokolov, I. S. Aranson, J. O. Kessler and R. E. Goldstein, Concentration dependence of the collective dynamics of swimming bacteria Physical Review Letters 98 (2007), 158102. doi: 10.1103/PhysRevLett.98.158102.

[19]

A. Sokolov, R. E. Goldstein, F. I. Feldstein and I. S. Aranson, Enhanced mixing and spatial instability in concentrated bacteria suspensions, Phys. Rev. E, 80 (2009), 031903.

[20]

M. TournusL. V. BerlyandA. Kirshtein and I. Aranson, Flexibility of bacterial flagella in external shear results in complex swimming trajectories, Journal of the Royal Society Interface, 12 (2015), 1-11.  doi: 10.1098/rsif.2014.0904.

[21]

A. Weil, Elliptic Functions According to Eisenstein and Kronecker Springer-Verlag, 1976.

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 $
$\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 $
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 $$ 2844.27 $
$\mathbf{no.}$$\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 $
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 $$ 6078.16 $
$\mathbf{no.}$$\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 $
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 $$ 717.895 $
$\mathbf{\mbox{Re}\big[\langle e_{2}\rangle\big]}$$\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]

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