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Helical trajectories of swimming cells with a flexible flagellar hook

Zonghao Zou, Wilson Lough, and Saverio Spagnolie*

  • Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706, USA

  • *spagnolie@math.wisc.edu
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Phys. Rev. Fluids 6, 103102 – Published 11 October, 2021

DOI: https://doi.org/10.1103/PhysRevFluids.6.103102

Abstract

The flexibility of the bacterial flagellar hook is believed to have substantial consequences for microorganism locomotion. Using a simplified model of a rigid flagellum and a flexible hook, we show that the paths of axisymmetric cell bodies driven by a single flagellum in Stokes flow are generically helical. Phase-averaged resistance and mobility tensors are produced to describe the flagellar hydrodynamics, and a helical rod model which retains a coupling between translation and rotation is identified as a distinguished asymptotic limit. A supercritical Hopf bifurcation in the flagellar orientation beyond a critical ratio of flagellar motor torque to hook bending stiffness, which is set by the spontaneous curvature of the flexible hook, the shape of the cell body, and the flagellum geometry, can have a dramatic effect on the cell's trajectory through the fluid. Although the equilibrium hook angle can result in a wide variance in the trajectory's helical pitch, we find a very consistent prediction for the trajectory's helical amplitude using parameters relevant to swimming P. aeruginosa cells.

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References (99)

  1. J. Lighthill, Flagellar hydrodynamics – JV Neumann lecture, 1975, SIAM Rev. 18, 161 (1976).
  2. C. Brennen and H. Winet, Fluid-mechanics of propulsion by cilia and flagella, Annu. Rev. Fluid Mech. 9, 339 (1977).
  3. R. M. Macnab, Bacterial flagella rotating in bundles: a study in helical geometry, Proc. Natl. Acad. Sci. USA 74, 221 (1977).
  4. F. A. Samatey, H. Matsunami, K. Imada, S. Nagashima, T. R. Shaikh, D. R. Thomas, J. Z. Chen, D. J. DeRosier, A. Kitao, and K. Namba, Structure of the bacterial flagellar hook and implication for the molecular universal joint mechanism, Nature (London) 431, 1062 (2004).
  5. S. M. Block, D. F. Blair, and H. C. Berg, Compliance of bacterial flagella measured with optical tweezers, Nature (London) 338, 514 (1989).
  6. S. M. Block, D. F. Blair, and H. C. Berg, Compliance of bacterial polyhooks measured with optical tweezers, Cytometry 12, 492 (1991).
  7. A. Sen, R. K. Nandy, and A. N. Ghosh, Elasticity of flagellar hooks, Microscopy 53, 305 (2004).
  8. T. C. Flynn and J. Ma, Theoretical analysis of twist/bend ratio and mechanical moduli of bacterial flagellar hook and filament, Biophys. J. 86, 3204 (2004).
  9. T. Pilizota, M. T. Brown, M. C. Leake, R. W. Branch, R. M. Berry, and J. P. Armitage, A molecular brake, not a clutch, stops the Rhodobacter sphaeroides flagellar motor, Proc. Natl. Acad. Sci. USA 106, 11582 (2009).
  10. K. Son, J. S. Guasto, and R. Stocker, Bacteria can exploit a flagellar buckling instability to change direction, Nat. Phys. 9, 494 (2013).
  11. M. T. Brown, B. C. Steel, C. Silvestrin, D. A. Wilkinson, N. J. Delalez, C. N. Lumb, B. Obara, J. P. Armitage, and R. M. Berry, Flagellar hook flexibility is essential for bundle formation in swimming Escherichia coli cells, J. Bacteriol. 194, 3495 (2012).
  12. I. Spöring, V. A. Martinez, C. Hotz, J. Schwarz-Linek, K. L. Grady, J. M. Nava-Sedeño, T. Vissers, H. M. Singer, M. Rohde, C. Bourquin et al., Hook length of the bacterial flagellum is optimized for maximal stability of the flagellar bundle, PLoS Biol. 16, e2006989 (2018).
  13. S. Kudo, N. Imai, M. Nishitoba, S. Sugiyama, and Y. Magariyama, Asymmetric swimming pattern of vibrio alginolyticus cells with single polar flagella, FEMS Microbiol. Lett. 242, 221 (2005).
  14. L. Xie, T. Altindal, S. Chattopadhyay, and X.-L. Wu, Bacterial flagellum as a propeller and as a rudder for efficient chemotaxis, Proc. Natl. Acad. Sci. USA 108, 2246 (2011).
  15. Y. Park, Y. Kim, and S. Lim, Locomotion of a single-flagellated bacterium, J. Fluid Mech. 859, 586 (2019).
  16. N. C. Darnton, L. Turner, S. Rojevsky, and H. C. Berg, On torque and tumbling in swimming Escherichia coli, J. Bacteriol. 189, 1756 (2007).
  17. M. Jabbarzadeh and H. C. Fu, Dynamic instability in the hook-flagellum system that triggers bacterial flicks, Phys. Rev. E 97, 012402 (2018).
  18. M. Jabbarzadeh and H. C. Fu, Large deformations of the hook affect free-swimming singly flagellated bacteria during flick motility, Phys. Rev. E 102, 033115 (2020).
  19. M. Palusa, J. De Graaf, A. Brown, and A. Morozov, Sedimentation of a rigid helix in viscous media, Phys. Rev. Fluids 3, 124301 (2018).
  20. H. O. Caldag and S. Yesilyurt, Trajectories of magnetically-actuated helical swimmers in cylindrical channels at low Reynolds numbers, J. Fluids Struct. 90, 164 (2019).
  21. K. Samsami, S. A. Mirbagheri, F. Meshkati, and H. C. Fu, Stability of soft magnetic helical microrobots, Fluids 5, 19 (2020).
  22. D. Das and E. Lauga, Transition to bound states for bacteria swimming near surfaces, Phys. Rev. E 100, 043117 (2019).
  23. K. Ishimoto, Bacterial spinning top, J. Fluid Mech. 880, 620 (2019).
  24. K. Ishimoto, Helicoidal particles and swimmers in a flow at low Reynolds number, J. Fluid Mech. 892, (2020).
  25. J. Zhang, M. Chinappi, and L. Biferale, Base flow decomposition for complex moving objects in linear hydrodynamics: Application to helix-shaped flagellated microswimmers, Phys. Rev. E 103, 023109 (2021).
  26. S. Jung, K. Mareck, L. Fauci, and M. J. Shelley, Rotational dynamics of a superhelix towed in a Stokes fluid, Phys. Fluids 19, 103105 (2007).
  27. O. S. Pak, S. E. Spagnolie, and E. Lauga, Hydrodynamics of the double-wave structure of insect spermatozoa flagella, J. R. Soc. Interface 9, 1908 (2012).
  28. J. T. Pham, A. Morozov, A. J. Crosby, A. Lindner, and O. du Roure, Deformation and shape of flexible, microscale helices in viscous flow, Phys. Rev. E 92, 011004(R) (2015).
  29. M. K. Jawed, N. K. Khouri, F. Da, E. Grinspun, and P. M. Reis, Propulsion and Instability of a Flexible Helical Rod Rotating in a Viscous Fluid, Phys. Rev. Lett. 115, 168101 (2015).
  30. B. Chakrabarti, Y. Liu, J. LaGrone, R. Cortez, L. Fauci, O. du Roure, D. Saintillan, and A. Lindner, Flexible filaments buckle into helicoidal shapes in strong compressional flows, Nat. Phys. 16, 689 (2020).
  31. H. S. Jennings, On the significance of the spiral swimming of organisms, Am. Naturalist 35, 369 (1901).
  32. H. C. Crenshaw, Orientation by helical motion I. Kinematics of the helical motion of organisms with up to six degrees of freedom, Bull. Math. Biol. 55, 197 (1993).
  33. B. Liu, M. Gulino, M. Morse, J. X. Tang, T. R. Powers, and K. S. Breuer, Helical motion of the cell body enhances Caulobacter crescentus motility, Proc. Natl. Acad. Sci. USA 111, 11252 (2014).
  34. Y. Hyon, T. R. Powers, R. Stocker, and H. C. Fu, The wiggling trajectories of bacteria, J. Fluid Mech. 705, 58 (2012).
  35. H. Shum and E. A. Gaffney, The effects of flagellar hook compliance on motility of monotrichous bacteria: A modeling study, Phys. Fluids 24, 061901 (2012).
  36. M. A. Constantino, M. Jabbarzadeh, H. C. Fu, Z. Shen, J. G. Fox, F. Haesebrouck, S. K. Linden, and R. Bansil, Bipolar lophotrichous Helicobacter suis combine extended and wrapped flagella bundles to exhibit multiple modes of motility, Sci. Rep. 8, 14415 (2018).
  37. M. Rossi, G. Cicconofri, A. Beran, G. Noselli, and A. DeSimone, Kinematics of flagellar swimming in euglena gracilis: Helical trajectories and flagellar shapes, Proc. Natl. Acad. Sci. USA 114, 13085 (2017).
  38. N. Giuliani, M. Rossi, G. Noselli, and A. DeSimone, How euglena gracilis swims: Flow field reconstruction and analysis, Phys. Rev. E 103, 023102 (2021).
  39. A. Thawani and M. S. Tirumkudulu, Trajectory of a model bacterium, J. Fluid Mech. 835, 252 (2018).
  40. O. A. Croze, V. A. Martinez, T. Jakuszeit, D. Dell'Arciprete, W. C. K. Poon, and M. A. Bees, Helical and oscillatory microswimmer motility statistics from differential dynamic microscopy, New J. Phys. 21, 063012 (2019).
  41. D. Cortese and K. Y. Wan, Control of Helical Navigation by Three-Dimensional Flagellar Beating, Phys. Rev. Lett. 126, 088003 (2021).
  42. R. Vogel and H. Stark, Motor-driven bacterial flagella and buckling instabilities, Eur. Phys. J. E 35, 15 (2012).
  43. F. T. M. Nguyen and M. D. Graham, Buckling instabilities and complex trajectories in a simple model of uniflagellar bacteria, Biophys. J. 112, 1010 (2017).
  44. F. T. M. Nguyen and M. D. Graham, Impacts of multiflagellarity on stability and speed of bacterial locomotion, Phys. Rev. E 98, 042419 (2018).
  45. J. B. Keller and S. I. Rubinow, Swimming of flagellated microorganisms, Biophys. J. 16, 151 (1976).
  46. S. Childress, Mechanics of Swimming and Flying (Cambridge University Press, Cambridge, UK, 1981).
  47. J. D. Meiss, Differential Dynamical Systems (SIAM, Philadelphia, 2007).
  48. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Prentice Hall, Englewood Cliffs, NJ, 1965).
  49. E. M. Purcell, Life at low Reynolds-number, Am. J. Phys. 45, 3 (1977).
  50. J. J. L. Higdon, The hydrodynamics of flagellar propulsion: helical waves, J. Fluid Mech. 94, 331 (1979).
  51. N. Phan-Thien, T. Tran-Cong, and M. Ramia, A boundary-element analysis of flagellar propulsion, J. Fluid Mech. 184, 533 (1987).
  52. S. Chattopadhyay and X.-L. Wu, The effect of long-range hydrodynamic interaction on the swimming of a single bacterium, Biophys. J. 96, 2023 (2009).
  53. R. R. Trivedi, R. Maeda, N. L. Abbott, S. E. Spagnolie, and D. B. Weibel, Bacterial transport of colloids in liquid crystalline environments, Soft Matter 11, 8404 (2015).
  54. S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications (Dover, Mineola, NY, 1991).
  55. J. Gray and G. J. Hancock, The propulsion of sea-urchin spermatozoa, J. Exp. Biol. 32, 802 (1955).
  56. R. E. Johnson and C. J. Brokaw, Flagellar hydrodynamics—Comparison between resistive-force theory and slender-body theory, Biophys. J. 25, 113 (1979).
  57. K. Ishimoto and E. Lauga, The N-flagella problem: elastohydrodynamic motility transition of multi-flagellated bacteria, Proc. R. Soc. A 475, 20180690 (2019).
  58. B. Rodenborn, C.-H. Chen, H. L. Swinney, B. Liu, and H. P. Zhang, Propulsion of microorganisms by a helical flagellum, Proc. Natl. Acad. Sci. USA 110, E338 (2013).
  59. L. Li and S. E. Spagnolie, Swimming and pumping of rigid helical bodies in viscous fluids, Phys. Fluids 26, 041901 (2014).
  60. M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, and E. Grinspun, Discrete elastic rods, in ACM SIGGRAPH (ACM Press, New York, 2008), pp. 1–12.
  61. O. M. O'Reilly, Modeling Nonlinear Problems in the Mechanics of Strings and Rods (Springer, Berlin, 2017).
  62. D. G. Morgan, R. M. Macnab, N. R. Francis, and D. J. DeRosier, Domain organization of the subunit of the Salmonella typhimurium flagellar hook, J. Mol. Biol. 229, 79 (1993).
  63. T. Hirano, S. Yamaguchi, K. Oosawa, and S.-I. Aizawa, Roles of FliK and FlhB in determination of flagellar hook length in Salmonella typhimurium, J. Bacteriol. 176, 5439 (1994).
  64. K. Kobayashi, T. Saitoh, D. S. H. Shah, K. Ohnishi, I. G. Goodfellow, R. E. Sockett, and S.-I. Aizawa, Purification and characterization of the flagellar basal body of Rhodobacter sphaeroides, J. Bacteriol. 185, 5295 (2003).
  65. M. Koike, H. Terashima, S. Kojima, and M. Homma, Isolation of basal bodies with C-ring components from the Na+-driven flagellar motor of Vibrio alginolyticus, J. Bacteriol. 192, 375 (2010).
  66. M. Jabbarzadeh and H. C. Fu, A numerical method for inextensible elastic filaments in viscous fluids, J. Comput. Phys. 418, 109643 (2020).
  67. L. Li, H. Manikantan, D. Saintillan, and S. E. Spagnolie, The sedimentation of flexible filaments, J. Fluid Mech. 735, 705 (2013).
  68. Y. Man and E. Kanso, Morphological transitions of axially-driven microfilaments, Soft Matter 15, 5163 (2019).
  69. O. Du Roure, A. Lindner, E. N. Nazockdast, and M. J. Shelley, Dynamics of flexible fibers in viscous flows and fluids, Annu. Rev. Fluid Mech. 51, 539 (2019).
  70. M. Fujii, S. Shibata, and S.-I. Aizawa, Polar, peritrichous, and lateral flagella belong to three distinguishable flagellar families, J. Mol. Biol. 379, 273 (2008).
  71. S. M. Vater, S. Weiße, S. Maleschlijski, C. Lotz, F. Koschitzki, T. Schwartz, U. Obst, and A. Rosenhahn, Swimming behavior of Pseudomonas aeruginosa studied by holographic 3d tracking, PLoS ONE 9, e87765 (2014).
  72. J. H. Long Jr. and K. S. Nipper, The importance of body stiffness in undulatory propulsion, Am. Zool. 36, 678 (1996).
  73. S. Alben, P. G. Madden, and G. V. Lauder, The mechanics of active fin-shape control in ray-finned fishes, J. R. Soc. Interface 4, 243 (2007).
  74. S. E. Spagnolie, L. Moret, M. J. Shelley, and J. Zhang, Surprising behaviors in flapping locomotion with passive pitching, Phys. Fluids 22, 041903 (2010).
  75. S. J. Ebbens and J. R. Howse, In pursuit of propulsion at the nanoscale, Soft Matter 6, 726 (2010).
  76. B. J. Nelson, I. K. Kaliakatsos, and J. J. Abbott, Microrobots for minimally invasive medicine, Annu. Rev. Biomed. Eng. 12, 55 (2010).
  77. S. Sengupta, M. E. Ibele, and A. Sen, Fantastic voyage: Designing self-powered nanorobots, Angew. Chem. Int. Ed. 51, 8434 (2012).
  78. J. Li, B. Esteban-Fernández de Ávila, W. Gao, L. Zhang, and J. Wang, Micro/nanorobots for biomedicine: Delivery, surgery, sensing, and detoxification, Sci. Robot. 2, eaam6431 (2017).
  79. L. Ricotti, B. Trimmer, A. W. Feinberg, R. Raman, K. K. Parker, R. Bashir, M. Sitti, S. Martel, P. Dario, and A. Menciassi, Biohybrid actuators for robotics: A review of devices actuated by living cells, Sci. Robot. 2, eaaq0495 (2017).
  80. C. Hu, S. Pane, and B. J. Nelson, Soft micro- and nanorobotics Annu. Rev. Cont. Robotics Auton. Syst. 1, 53 (2018).
  81. A. C. H. Tsang, E. Demir, Y. Ding, and O. S. Pak, Roads to smart artificial microswimmers, Adv. Intell. Syst. Comput. 2, 1900137 (2020).
  82. R. W. Carlsen and M. Sitti, Bio-hybrid cell-based actuators for microsystems, Small 10, 3831 (2014).
  83. L. Zhu and H. A. Stone, Harnessing elasticity to generate self-oscillation via an electrohydrodynamic instability, J. Fluid Mech. 888, A31 (2020).
  84. M. J. Kim and K. S. Breuer, Enhanced diffusion due to motile bacteria, Phys. Fluids 16, L78 (2004).
  85. K. C. Leptos, J. S. Guasto, J. P. Gollub, A. I. Pesci, and R. E. Goldstein, Dynamics of Enhanced Tracer Diffusion in Suspensions of Swimming Eukaryotic Microorganisms, Phys. Rev. Lett. 103, 198103 (2009).
  86. Z. Lin, J.-L. Thiffeault, and S. Childress, Stirring by squirmers, J. Fluid Mech. 669, 167 (2011).
  87. J.-L. Thiffeault, Distribution of particle displacements due to swimming microorganisms, Phys. Rev. E 92, 023023 (2015).
  88. H. C. Fu, T. R. Powers, and H. C. Wolgemuth, Theory of Swimming Filaments in Viscoelastic Media, Phys. Rev. Lett. 99, 258101 (2007).
  89. H. C. Fu, C. W. Wolgemuth, and T. R. Powers, Swimming speeds of filaments in nonlinearly viscoelastic fluids, Phys. Fluids 21, 033102 (2009).
  90. L. Li and S. E. Spagnolie, Swimming and pumping by helical waves in viscous and viscoelastic fluids, Phys. Fluids 27, 021902 (2015).
  91. B. Liu, T. R. Powers, and K. S. Breuer, Force-free swimming of a model helical flagellum in viscoelastic fluids, Proc. Natl. Acad. Sci. USA 108, 19516 (2011).
  92. S. E. Spagnolie, B. Liu, and T. R. Powers, Locomotion of Helical Bodies in Viscoelastic Fluids: Enhanced Swimming at Large Helical Amplitudes, Phys. Rev. Lett. 111, 068101 (2013).
  93. V. A. Martinez, J. Schwarz-Linek, M. Reufer, L. G. Wilson, A. N. Morozov, and W. C. K. Poon, Flagellated bacterial motility in polymer solutions, Proc. Natl. Acad. Sci. USA 111, 17771 (2014).
  94. A. Zöttl and J. M. Yeomans, Enhanced bacterial swimming speeds in macromolecular polymer solutions, Nat. Phys. 15, 554 (2019).
  95. J. P. Binagia, A. Phoa, K. D. Housiadas, and E. S. G. Shaqfeh, Swimming with swirl in a viscoelastic fluid, J. Fluid Mech. 900, A4 (2020).
  96. S. Gómez, F. A. Godínez, E. Lauga, and R. Zenit, Helical propulsion in shear-thinning fluids, J. Fluid Mech. 812, R3 (2017).
  97. E. Demir, N. Lordi, Y. Ding, and O. S. Pak, Nonlocal shear-thinning effects substantially enhance helical propulsion, Phys. Rev. Fluids 5, 111301(R) (2020).
  98. Z. Qu and K. S. Breuer, Effects of shear thinning viscosity and viscoelastic stresses on flagellated bacteria motility, Phys. Rev. Fluids 5, 073103 (2020).
  99. M. B. Short, C. A. Solari, S. Ganguly, T. R. Powers, J. O. Kessler, and R. E. Goldstein, Flows driven by flagella of multicellular organisms enhance long-range molecular transport, Proc. Natl. Acad. Sci. USA 103, 8315 (2006).

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Phys. Rev. Fluids 6, 103102– Published 11 October, 2021
  • Received 16 June 2021
  • Accepted 23 September 2021
Article part of CHORUS
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