A criterion of collective behavior of bacteria

  • Received: 01 October 2015 Accepted: 05 February 2016 Published: 01 January 2017
  • MSC : Primary: 92B15, 92B25; Secondary: 74Q15

  • 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[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 277-287. doi: 10.3934/mbe.2017018

    Related Papers:

  • 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.


    加载中
    [1] [ N. I. Akhiezer, Elements of Theory of Elliptic Functions Nauka, 1970 (in Russian); Engl. transl. AMS, 1990.
    [2] [ R. Czapla,V. V. Mityushev,W. Nawalaniec, Effective conductivity of random two-dimensional composites with circular non-overlapping inclusions, Computational Materials Science, 63 (2012): 118-126.
    [3] [ R. Czapla,V. V. Mityushev,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. Czapla,V. V. Mityushev,N. Rylko, Conformal mapping of circular multiply connected domains onto segment domains, Electron. Trans. Numer. Anal., 39 (2012): 286-297.
    [5] [ S. Gluzman,D. A. Karpeev,L. V. Berlyand, Effective viscosity of puller-like microswimmers: A renormalization approach, J. R. Soc. Interface, 10 (2013): 1-10.
    [6] [ V. V. Mityushev, Representative cell in mechanics of composites and generalized Eisenstein--Rayleigh sums, Complex Variables, 51 (2006): 1033-1045.
    [7] [ V. V. Mityushev,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.
    [8] [ V. V. Mityushev,W. Nawalaniec, Basic sums and their random dynamic changes in description of microstructure of 2D composites, Computational Materials Science, 97 (2015): 64-74.
    [9] [ V. V. Mityushev,N. Rylko, Optimal distribution of the non-overlapping conducting disks, Multiscale Model. Simul., 10 (2012): 180-190.
    [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.
    [12] [ S. D. Ryan,L. Berlyand,B. M. Haines,D. A. Karpeev, A kinetic model for semi-dilute bacterial suspensions, Multiscale Model. Simul., 11 (2013): 1176-1196.
    [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).
    [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.
    [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.
    [16] [ A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria Phys. Rev. Lett. 103 (2009), 148101.
    [17] [ A. Sokolov and I. S. Aranson, Physical properties of collective motion in suspensions of bacteria Phys. Rev. Lett. 109 (2012), 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.
    [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. Tournus,L. V. Berlyand,A. Kirshtein,I. Aranson, Flexibility of bacterial flagella in external shear results in complex swimming trajectories, Journal of the Royal Society Interface, 12 (2015): 1-11.
    [21] [ A. Weil, Elliptic Functions According to Eisenstein and Kronecker Springer-Verlag, 1976.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2631) PDF downloads(520) Cited by(14)

Article outline

Figures and Tables

Figures(5)  /  Tables(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog