XXIV ICTAM, 21-26 August 2016, Montreal, Canada
HIEARCHICAL STRUCTURE OF SPATIALLY LOCALIZED BIOCONVECTION OF
PHOTOSENSITIVE MICROORGANISM
Makoto Iima ∗ 1 , Takayuki Yamaguchi2 , Takuma Ogawa1 , Nobuhiko Suematsu3 , Hiraku Nishimori1 , Akinori
Awazu1 , and Shunsuke Izumi1
1
Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Japan
2
Graduate School of BioMedical & Health Science, Hiroshima University, Hiroshima, Japan
3
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Tokyo, Japan
Summary It is known that the suspension of Euglena gracilis exhibits macroscopic spatially localized patterns when it is illuminated from
below with strong light. Such structure is hierarchical ranging from µm scale of microorganism to cm scale of the convection cells. We
discuss the localization mechanism in terms of the photosensitive behavior of Euglena gracilis in both scales. Individual’s response to
light gradient is analyzed experimentally, and the result is related to the flux of the number density in macroscopic scale to construct a
hydrodynamic model. The model has spatially localized steady solutions, and the bifurcation structure has a bistable region, which often
observed in spatially localized structures in dissipative systems.
INTRODUCTION
Euglena gracilis is a photosensitive microorganism with flagella; the body is approximately 10 µm wide and 50-100
µm long(Fig.1(a)). They can swim by the flagellum, and the statistical property of the motion depends on the surrounding
light environment. In particular, they swim away from the light source if the light intensity is stronger than a critical value
(negative phototaxis), while forward to the light source if the light intensity is less than the critical value (positive phototaxis).
When a suspension of Euglena gracilis is illuminated from the bottom with strong light, they form ordered convection
patterns called bioconvection (Fig.1(b)). Unlike other bioconvection patterns known so far, the bioconvection pattern of
Euglena gracilis is peculiar because it is spatially localized[1]. Shoji et al.[2] succeeded in obtaining the elementary structures
of the localized bioconvection. The aim of this paper is to investigate the hierarchical structure of the localized bioconvection
pattern and its dynamical property to understand the localization mechanism.
EXPERIMENTS
A key mechanism of the localization is the photomovement of Euglena gracilis due to the light intensity gradient. We
will show experimental results of the characteristics in both microscopic and macroscopic scales. In microscopic motion, we
track the orbit of individuals in the light gradient environment to construct a statistical model. A simple Markov model is
constructed to explain the response to the light gradient, and the equilibrium state of the model agrees with the observation.
In macroscopic motion, number density flux of the suspension of the microorganism has been measured for light intensity
gradient .
For experiments of the macroscopic convection pattern, we prepared an annular container to suppress the complex patterns
in the radial direction and to exclude the wall effect in the azimuthal direction. So far, two types of localized convection
patterns were observed[2]. One pattern consists of a single region of high microorganism density sandwiched with two
counter-rotating convection rolls (‘bioconvection unit’). Another pattern is a spatially localized traveling wave, a wave of high
density region in confined region. These two typical patterns are similar to those observed in thermal convection of binary fluid
mixtures (e.g., mixtures of water and alcohol): so-called “convecton” and “localized traveling wave”, respectively[3], although
physical mechanisms of the binary fluid convection is the (positive) buoyancy and the Soret effect, which are different from
the bioconvection. For the bioconvection unit, several bound states are observed. In particular, two (or three) bioconvection
units exists for a long time before disappearing one by one. Such interaction of bioconvection units will be discussed.
NUMERICAL MODEL
Based on the experimental results, we construct a hydrodynamic model in which the number density flux depends on the
light intensity gradient. The model reproduces a localized bioconvection pattern similar to the bioconvection unit. Linear
stability analysis indicates that the effect of the light gradient on the number density flux greatly shifts the critical value of the
onset of the convection. The bifurcation analysis using a branch-tracking technique reveals that a bistability region is observed
when the aspect ratio Γ is large (Γ = 8), while such region is not obtained when Γ is small (Γ = 2).
∗ Corresponding
author. Email: [email protected]
CONCLUDING REMARKS
The experimental results of the localized convection patterns suggests that the localized patterns are similar to those in the
binary fluid convection. The bioconvection unit, a minimal localized structure, and their dynamics are similar to those in the
reaction-diffusion systems. Such similarities are not directly owing to the similarities of the physical mechanism. However,
the mathematical similarities are suggested to explain the similarities. Bistability is the first evidence of the conjecture and the
further research is needed. Interaction between the localized bioconvections is an interesting topic and a theoretical approach
based on the dynamical system theory may be useful[4, 5].
(a)
(b)
(c)
Figure 1: (a) Euglena gracilis. (b) Localized convection pattern of suspension of Euglena gracilis illuminated below. (c)
Steady solution representing bioconvection unit.
References
[1] Suematsu N.J., Awazu A., Izumi S., Noda S., Nakata S. and Nishimori H.: Localized Bioconvection of Euglena Caused by Phototaxis in the Lateral
Direction. J. Phys. Soc. Jpn. 80: 064003, 2011.
[2] Shoji E., Nishimori H., Awazu A., Izumi S. and Iima M.: Localized Bioconvection Patterns and Their Initial State Dependency in Euglena gracilis
Suspensions in an Annular Container. J. Phys. Soc. Jpn 83:043001, 2014.
[3] Watanabe T., Iima M. and Nishiura Y.: Spontaneous formation of traveling localized structures and their asymptotic behaviour in binary fluid convection.
J. Fluid Mech 712: 219–243, 2012.
[4] Yamaguchi T. and Iima M.: Numerical analysis of transient orbits by the pullback method for covariant Lyapunov vector. Theor. Appl. Mech. Jpn
63:91–96, 2015.
[5] Iima M., Yamaguchi T., Watanabe T., Kawaharada A., Tasaka Y. and Shoji E.: Towards understanding global flow structure. Proc. Int. Conf. on
Mathematical Fluid Dynamics, Present and Future, 2016 (in press)