Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria

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Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria

Daiki Nishiguchi, Ken H. Nagai, Hugues Chaté, and Masaki Sano

Phys. Rev. E 95, 020601(R) – Published 7 February 2017

Abstract

We study the collective dynamics of elongated swimmers in a very thin fluid layer by devising long filamentous nontumbling bacteria. The strong confinement induces weak nematic alignment upon collision, which, for large enough density of cells, gives rise to global nematic order. This homogeneous but fluctuating phase, observed on the largest experimentally accessible scale of millimeters, exhibits the properties predicted by standard models for flocking, such as the Vicsek-style model of polar particles with nematic alignment: true long-range nematic order and nontrivial giant number fluctuations.

Received 31 May 2016
Revised 26 September 2016

DOI:https://doi.org/10.1103/PhysRevE.95.020601

Physics Subject Headings (PhySH)

Collective behavior

Bacteria Living matter & active matter

Fluorescence spectroscopy Imaging & optical processing

Statistical Physics & ThermodynamicsPhysics of Living SystemsPolymers & Soft MatterGeneral PhysicsInterdisciplinary PhysicsNonlinear Dynamics

Authors & Affiliations

Daiki Nishiguchi^1,*, Ken H. Nagai², Hugues Chaté^3,4, and Masaki Sano¹

¹Department of Physics, The University of Tokyo, Hongo 7-3-1, Tokyo 113-0033, Japan
²School of Materials Science, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, Japan
³Service de Physique de l'Etat Condensé, CEA, CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
⁴Beijing Computational Science Research Center, Beijing 100094, China

^*nishiguchi@daisy.phys.s.u-tokyo.ac.jp

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Issue

Vol. 95, Iss. 2 — February 2017

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Images

Figure 1
(a) Aligning collision events between two bacteria. Top: acute angle collision leading to alignment. Bottom: obtuse angle collision leading to antialignment. Dashed line: mean outgoing angle. The mean incoming angle is not shown as it is only slightly different from the outgoing angle. (b) Binary collision statistics: difference between the incoming angle and the outgoing angle $Δ θ$ vs the incoming angle $θ_{in}$ . Green circles: 2214 individual collision events. Red circles: mean $Δ θ$ obtained via binning $θ_{in}$ . Error bars: standard deviation. Collision events with $| Δ θ | > 40^{\circ}$ are not shown for visibility. See the Supplemental Material for details [26].
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Figure 2
Typical snapshots. (a) Zoom of the disordered phase at low density in a $2 − μ m$ thin experiment. (b) Zoom of the disordered phase at high density in a $10 − μ m$ thick experiment. (c) Zoom of the nematically ordered phase at high density in a thin experiment with superimposed manually tracked 10-s trajectories of a few cells. (d) Full field of view in the same experiment as in (c). See the Supplemental Material movies [26].
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Figure 3
(a) Log-log plot of nematic order parameter $〈 C 〉$ vs area $S$ of the ROI. Red circles: the globally nematically ordered state at high density in a very thin sample shown in Figs. 2 and 2. Blue squares: the disordered state at low density. Cyan solid line: slope of exponent $- 0.5$ as a guide to the eye [42]. The nematic order $〈 C 〉$ stays at high values over the whole field of view in the ordered state (the red data). (b) The same data as in (a) for the ordered state in a magnified range (log-log scale). The curvature in this log-log plot indicates slower decay than a power law. The last three points were excluded from the fit because they were not reliable due to longer correlation times and inhomogeneities at such large scales. Inset: the same data from which the estimated asymptotic value of $C_{\infty} = 0.505$ has been subtracted (log-log scale). Magenta solid lines: fit $〈 C 〉 = C_{\infty} + k S^{β}$ with $C_{\infty} = 0.505, β = - 0.66$ , and $k = 4.6$ . Error bars in (a) and (b): standard error. (c) Scaling of number fluctuations $Δ N / \sqrt{〈 N 〉}$ vs $〈 N 〉$ on the log-log scale. Blue squares: normal fluctuations in the disordered low-density phase. Red circles: anomalous “giant” fluctuations recorded in the high-density nematically ordered state of Figs. 2 and 2. Cyan dashed line: normal fluctuations $Δ N \propto {〈 N 〉}^{0.5}$ as a guide to the eye. Magenta dashed line: fitted curve $Δ N \propto {〈 N 〉}^{0.63}$ for the ordered state.
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Figure 4
(a) Color map of the correlation function $Corr (R)$ of the director fluctuations ${δ n}_{⊥} = n - n_{0}$ . The global mean director $n_{0}$ is aligned in the $x$ direction. (b) Log-log plot of the correlation function $Corr (R)$ in the longitudinal direction (along $n_{0})$ . The red solid line is a slope with the exponent $- 0.3$ just to guide the eye.
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