
-
Previous Article
On a mathematical model of bone marrow metastatic niche
- MBE Home
- This Issue
-
Next Article
Sufficient optimality conditions for a class of epidemic problems with control on the boundary
A criterion of collective behavior of bacteria
Institute of Computer Science, Pedagogical University, ul. Podchorazych 2, Krakow 30-084, Poland |
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.
References:
[1] |
N. I. Akhiezer,
Elements of Theory of Elliptic Functions Nauka, 1970 (in Russian); Engl. transl. AMS, 1990. |
[2] |
R. Czapla, V. 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. Czapla, V. 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. Czapla, V. 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. Gluzman, D. 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. Ryan, L. Berlyand, B. 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. Tournus, L. V. Berlyand, A. 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. Czapla, V. 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. Czapla, V. 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. Czapla, V. 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. Gluzman, D. 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. Ryan, L. Berlyand, B. 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. Tournus, L. V. Berlyand, A. 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. |




| |||||
| |||||
| ||||
| ||||
averaged | ||||
standard deviation of the | ||||
averaged | ||||
standard deviation of the |
averaged | ||||
standard deviation of the | ||||
averaged | ||||
standard deviation of the |
[1] |
Alexander Blokh, Lex Oversteegen, Vladlen Timorin. Non-degenerate locally connected models for plane continua and Julia sets. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5781-5795. doi: 10.3934/dcds.2017251 |
[2] |
Michael Blank. Emergence of collective behavior in dynamical networks. Discrete and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 |
[8] |
Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233 |
[9] |
Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523 |
[10] |
Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations and Control Theory, 2022, 11 (1) : 259-282. doi: 10.3934/eect.2021002 |
[11] |
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 |
[12] |
Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020134 |
[13] |
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 |
[14] |
Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 |
[15] |
Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161 |
[16] |
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 |
[17] |
Rich Stankewitz, Toshiyuki Sugawa, Hiroki Sumi. Hereditarily non uniformly perfect sets. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2391-2402. doi: 10.3934/dcdss.2019150 |
[18] |
Piotr Oprocha, Pawel Wilczynski. Distributional chaos via isolating segments. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 347-356. doi: 10.3934/dcdsb.2007.8.347 |
[19] |
Elena Beretta, Markus Grasmair, Monika Muszkieta, Otmar Scherzer. A variational algorithm for the detection of line segments. Inverse Problems and Imaging, 2014, 8 (2) : 389-408. doi: 10.3934/ipi.2014.8.389 |
[20] |
Giuseppe Da Prato, Arnaud Debussche. Asymptotic behavior of stochastic PDEs with random coefficients. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1553-1570. doi: 10.3934/dcds.2010.27.1553 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]