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Locomotion of a single-flagellated bacterium

Published online by Cambridge University Press:  21 November 2018

Yunyoung Park ,
Yongsam Kim Open the ORCID record for Yongsam Kim [Opens in a new window]  and
Sookkyung Lim
Yunyoung Park
Affiliation:
Department of Mathematics, Chung-Ang University, Dongjakgu, Heukseokdong, Seoul 06974, Republic of Korea
Yongsam Kim*
Affiliation:
Department of Mathematics, Chung-Ang University, Dongjakgu, Heukseokdong, Seoul 06974, Republic of Korea
Sookkyung Lim
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, 4199 French Hall West, Cincinnati, OH 45221, USA
*
†Email address for correspondence: kimy@cau.ac.kr
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Abstract

Single-flagellated bacteria propel themselves by rotating a flagellar motor, translating rotation to the filament through a compliant hook and subsequently driving the rotation of the flagellum. The flagellar motor alternates the direction of rotation between counterclockwise and clockwise, and this leads to the forward and backward directed swimming. Such bacteria can change the course of swimming as the hook experiences its buckling caused by the change of bending rigidity. In this paper, we present a comprehensive model of a monotrichous bacterium as a free swimmer in a viscous fluid. We describe a cell body as a rigid body using the penalty method and a flagellum as an elastic rod using Kirchhoff rod theory. The hydrodynamic interaction of the bacterium is described by the regularized Stokes formulation. Our model of a single-flagellated micro-organism is able to mimic a swimming pattern that is well matched with the experimental observation. Furthermore, we find the critical thresholds of the rotational frequency of the motor and the bending modulus of the hook for the buckling instability, and investigate the dependence of the buckling angle and the reorientation of the swimming cell after buckling on the physical and geometrical parameters of the model.

JFM classification

Biological Fluid Dynamics: Micro-organism dynamics Biological Fluid Dynamics: Swimming/flying
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 859 , 25 January 2019 , pp. 586 - 612
Copyright
© 2018 Cambridge University Press 

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References

Berg, H. C. 2003 The rotary motor of bacterial flagella. Annu. Rev. Biochem. 72 (1), 19–54.Google Scholar
Berg, H. C. & Anderson, R. A. 1973 Bacteria swim by rotating their flagellar filaments. Nature 245, 380–382.Google Scholar
Block, S. M., Blair, D. F. & Berg, H. C. 1989 Compliance of bacterial flagella measured with optical tweezers. Nature 338, 514–518.Google Scholar
Block, S. M., Blair, D. F. & Berg, H. C. 1991 Compliance of bacterial polyhooks measured with optical tweezers. Cytometry 12 (6), 492–496.Google Scholar
Chattopadhyay, S. & Wu, X. L. 2009 The effect of long-range hydrodynamic interaction on the swimming of a single bacterium. Biophys. J. 96 (5), 2023–2028.Google Scholar
Chwang, A. T. & Wu, T. Y. 1971 A note on the helical movement of micro-organisms. Proc. R. Soc. Lond. B 178 (1052), 327–346.Google Scholar
Cortez, R. 2001 The method of regularized Stokeslets. SIAM J. Sci. Comput. 23 (4), 1204–1225.Google Scholar
Flynn, T. S. & Ma, J. 2004 Theoretical analysis of twist/bend ratio and mechanical moduli of bacterial flagellar hook and filament. Biophys. J. 86 (5), 3204–3210.Google Scholar
Fujita, T. & Kawai, T. 2001 Optimum shape of a flagellated microorganism. JSME Intl J. 44 (4), 952–957.Google Scholar
Furuno, M., Atsumi, T., Yamada, T., Kojima, S., Nishioka, N., Kawagishi, I. & Homma, M. 1997 Characterization of polar-flagellar-length mutants in Vibrio alginolyticus . Microbiology 143 (5), 1615–1621.Google Scholar
Goto, T., Nakata, K., Baba, K., Nishimura, M. & Magariyama, Y. 2005 A fluid-dynamic interpretation of the asymmetric motion of singly flagellated bacteria swimming close to a boundary. Biophys. J. 89 (6), 3771–3779.Google Scholar
Hancock, G. J. 1953 The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. Lond. A 217 (1128), 96–121.Google Scholar
Higdon, J. J. L. 1979 The hydrodynamics of flagellar propulsion: helical waves. J. Fluid Mech. 94 (2), 331–351.Google Scholar
Homma, M., Oota, H., Kojima, S., Kawagishi, I. & Imae, Y. 1996 Chemotactic responses to an attractant and a repellent by the polar and lateral flagellar systems of Vibrio alginolyticus . Microbiology 142, 2777–2783.Google Scholar
Hsu, C. & Dillon, R. 2009 A 3D motile rod-shaped monotrichous bacterial model. Bull. Math. Biol. 71 (5), 1228–1263.Google Scholar
Ishikawa, T. 2009 Suspension biomechanics of swimming microbes. J. R. Soc. Interface 6 (39), 815–834.Google Scholar
Ishikawa, T., Sekiya, G., Imai, Y. & Yamaguchi, T. 2007 Hydrodynamic interactions between two swimming bacteria. Biol. J. 93 (6), 2217–2225.Google Scholar
Kawagishi, I., Imagawa, M., Imae, Y., McCarter, L. & Homma, M. 1996 The sodium-driven polar flagellar motor of marine Vibrio as the mechanosensor that regulates lateral flagellar expression. Mol. Microbiol. 20 (4), 693–699.Google Scholar
Kim, M. J., Bird, J. C., Parys, A. J. V., Breuer, K. S. & Powers, T. R. 2003 A macroscopic scale model of bacterial flagellar bundling. Proc. Natl Acad. Sci. USA 100 (26), 15481–15485.Google Scholar
Kim, Y. & Peskin, C. S. 2007 Penalty immersed boundary method for an elastic boundary with mass. Phys. Fluids 19 (5), 053103.Google Scholar
Kim, Y. & Peskin, C. S. 2016 A penalty immersed boundary method for a rigid body in fluids. Phys. Fluids 28 (3), 033603.Google Scholar
Ko, W., Lim, S., Lee, W., Kim, Y., Berg, H. C. & Peskin, C. S. 2017 Modeling polymorphic transformation of rotating bacterial flagella in a viscous fluid. Phys. Rev. E 95 (6), 063106.Google Scholar
Kudo, S., Imai, N., Nishitoba, M., Sugiyama, S. & Magariyama, Y. 2005 Asymmetric swimming pattern of Vibrio alginolyticus cells with single polar flagella. FEMS Microbiol. Lett. 242 (2), 221–225.Google Scholar
Lauga, E. 2016 Bacterial hydrodynamics. Annu. Rev. Fluid Mech. 48, 105–130.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.Google Scholar
Lee, W., Kim, Y., Olson, S. D. & Lim, S. 2014 Nonlinear dynamics of a rotating elastic rod in a viscous fluid. Phys. Rev. E 90 (3), 033012.Google Scholar
Lewis, C. L., Craig, C. C. & Senecal, A. G. 2014 Mass and density measurements of live and dead gram-negative and gram-positive bacterial populations. Appl. Environ. Microbiol. 80 (12), 3622–3631.Google Scholar
Lim, S., Ferent, A., Wang, X. S. & Peskin, C. S. 2008 Dynamics of a closed rod with twist and bend in fluid. SIAM J. Sci. Comput. 31 (1), 273–302.Google Scholar
Lim, S. & Peskin, C. S. 2012 Fluid-mechanical interaction of flexible bacterial flagella by the immersed boundary method. Phys. Rev. E 85, 036307.Google Scholar
Magariyama, Y., Masuda, S., Takano, Y., Ohtani, T. & Kudo, S. 2001 Difference between forward and backward swimming speeds of the single polar-flagellated bacterium, Vibrio alginolyticus . FEMS Microbiol. Lett. 205 (2), 343–347.Google Scholar
McCarter, L. L. 2001 Polar flagellar motility of the Vibrionaceae . Microbiol. Mol. Biol. Rev. 65 (3), 445–462.Google Scholar
Olson, S., Lim, S. & Cortez, R. 2013 Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation. J. Comput. Phys. 238, 169–187.Google Scholar
Park, Y., Kim, Y., Ko, W. & Lim, S. 2017 Instabilities of a rotating helical rod in a viscous fluid. Phys. Rev. E 95 (2), 022410.Google Scholar
Phan-Thien, N., Tran-Cong, T. & Ramia, M. 1987 A boundary-element analysis of flagellar propulsion. J. Fluid Mech. 184, 533–549.Google Scholar
Purcell, E. M. 1997 The efficiency of propulsion by rotating flagellum. Proc. Natl Acad. Sci. USA 94 (21), 11307–11311.Google Scholar
Ramia, M., Tullock, K. L. & Phan-Thien, N. 1993 The role of hydrodynamic interaction in the locomotion of microorganisms. Biophys. J. 65 (2), 755–778.Google Scholar
Rodenborn, B., Chen, C., Swinney, H. L., Liu, B. & Zhang, H. P. 2013 Propulsion of microorganisms by helical flagellum. Proc. Natl Acad. Sci. USA 110 (5), 338–347.Google Scholar
Sen, A., Nandy, R. K. & Ghosh, A. N. 2004 Elasticity of flagellar hooks. J. Electron Microsc. 53 (3), 305–309.Google Scholar
Shum, H. & Gaffney, E. A. 2012 The effects of flagellar hook compliance on motility of monotrichous bacteria: a modeling study. Phys. Fluids 24 (6), 061901.Google Scholar
Shum, H., Gaffney, E. A. & Smith, D. J. 2010 Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry. Proc. R. Soc. Lond. A 466, 1725–1748.Google Scholar
Son, K., Guasto, J. S. & Stocher, R. 2013 Bacteria can exploit a flagellar buckling instability to change direction. Nat. Phys. 9, 494–498.Google Scholar
Stocker, R. 2011 Reverse and flick: hybrid locomotion in bacteria. Proc. Natl Acad. Sci. USA 108 (7), 2635–2636.Google Scholar
Takano, Y., Yoshida, K., Kudo, S., Nishitoba, M. & Magariyama, Y. 2003 Analysis of small deformation of helical flagellum of swimming Vibrio alginolyticus . JSME. Intl J. 46 (4), 1241–1247.Google Scholar
Taylor, G. I. 1952 The action of waving cylindrical tails in propelling microscopic organisms. Proc. R. Soc. Lond. A 211 (1105), 225–239.Google Scholar
Thawani, A. & Tirumkudulu, M. S. 2017 Trajectory of a model bacterium. J. Fluid Mech. 835, 252–270.Google Scholar
Timoshenko, S. 1961 Theory of Elastic Stability, 2nd edn. McGraw-Hill.Google Scholar
Weng, Y., Delgado, F. F., Son, S., Burg, T. P., Wasserman, S. C. & Manalis, S. R. 2011 Mass sensors with mechanical traps for weighing single cells in different fluids. Lab on a Chip 11, 4174–4180.Google Scholar
Xie, L., Altindal, T., Chattopadhyay, S. & Wu, X. L. 2010 Bacterial flagellum as a propeller and as a rudder for efficient chemotaxis. Proc. R. Soc. Lond. A 108 (6), 2246–2251.Google Scholar

Park et al. supplementary movie 1

The backward and forward swimming motions of a monotrichous bacterium without a hook. The motor first rotates CW till t=50 ms and then switches to CCW rotation. When the motor rotates CW (CCW), the cell body counterrotates and the bacterium swims backward (forward).

Download Park et al. supplementary movie 1(Video)
Video 9.4 MB

Park et al. supplementary movie 2

The run-reverse-flick movement of a bacterium with a flexible hook. The rotation of motor changes CW to CCW at t= 20 ms, and the hook is more flexible (relaxed) from t=20 ms till t=50 ms. During this time period, there occurs a buckling instability of the hook and the flicking of the cell body. At t=50 ms, the hook becomes less flexible (loaded) again, and the helical filament begins to be aligned with the cell body.

Download Park et al. supplementary movie 2(Video)
Video 9.5 MB