A criterion of collective behavior of bacteria

Roman Czapla; Vladimir V. Mityushev; Roman Czapla; Vladimir V. Mityushev

doi:10.3934/mbe.2017018

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 277-287. doi: 10.3934/mbe.2017018

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A criterion of collective behavior of bacteria

Roman Czapla ,
Vladimir V. Mityushev ^,

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

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.
- Collective behavior of bacteria,
- disordered random sets,
- generalized Eisenstein-Rayleigh sums,
- RVE,
- non-overlapping segments on plane
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

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Abstract

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.

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