The multiscale physics of cilia and flagella

Abstract

Cilia and flagella are fundamental units of motion in cellular biology. These beating, hair-like organelles share a common basic structure but maintain widely varying functions in systems ranging from the isolated flagella of swimming algae to the dense ciliary carpets that pump fluid in the brains of mammals. Experiments and models have begun to elucidate the inner workings of single cilia as complex nonlinear oscillators, and the variety of hydrodynamical phenomena that result from beating dynamics. These results have shed light on complex locomotion strategies observed in single-celled microorganisms and collective phenomena observed in microbial suspensions. In animal systems, dense ciliary arrays exhibit a variety of emergent phenomena, including active filtration, noise robustness and metachronal waves. Surprising phenomena have been observed in neuronally controlled ciliary arrays, demonstrating the need for new physical models of cilia that include central control, defect dynamics and topology. We review the emergent physics of cilia across scales, starting from the microscale dynamics of single cilia, and then proceeding to microorganisms and animal systems.

Key points

  • The complex beating dynamics of cilia can be modelled as noisy, nonlinear oscillations driven by coupled chemical, mechanical and hydrodynamical forces.

  • Small numbers of coupled cilia can transiently synchronize and desynchronize in a manner analogous to that seen in classical studies of coupled oscillators.

  • The synchronization dynamics of cilia may play a role in facilitating locomotion and navigation by single-celled microorganisms.

  • Many animals have ‘carpets’ of densely packed cilia, which are used to pump mucous and other circulating fluids in the brain and lungs. The beating dynamics of these ciliary carpets exhibit physical phenomena that include travelling waves and topological defects.

  • Neuronally controlled cilia in certain animal systems exhibit a rich, and understudied, set of dynamical phenomena, making their study a promising research direction.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Ciliary motion across scales.
Fig. 2: Anatomy and structure of the axoneme.
Fig. 3: Ciliary geometries in animal systems.
Fig. 4: Ciliary hydrodynamics in animal systems.
Fig. 5: Ciliary parameters across animal systems.

References

  1. 1.

    Brennen, C. & Winet, H. Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339–398 (1977).

  2. 2.

    Satir, P., Mitchell, D. R. & Jékely, G. How did the cilium evolve? Curr. Top. Dev. Biol. 85, 63–82 (2008).

  3. 3.

    Marshall, W. F. & Nonaka, S. Cilia: tuning in to the cell’s antenna. Curr. Biol. 16, R604–R614 (2006).

  4. 4.

    Shah, A. S., Ben-Shahar, Y., Moninger, T. O., Kline, J. N. & Welsh, M. J. Motile cilia of human airway epithelia are chemosensory. Science 325, 1131–1134 (2009).

  5. 5.

    Goetz, J. G. et al. Endothelial cilia mediate low flow sensing during zebrafish vascular development. Cell Rep. 6, 799–808 (2014).

  6. 6.

    Clapham, D. E. TRP channels as cellular sensors. Nature 426, 517–524 (2003).

  7. 7.

    Sleigh, M. A. The Biology of Cilia and Flagella (Pergamon, 1962).

  8. 8.

    Wan, K. Y. Coordination of eukaryotic cilia and flagella. Essays Biochem. 62, 829–838 (2018).

  9. 9.

    Margulis, L., Chapman, M., Guerrero, R. & Hall, J. The last eukaryotic common ancestor (LECA): acquisition of cytoskeletal motility from aerotolerant spirochetes in the Proterozoic Eon. Proc. Natl Acad. Sci. USA 103, 13080–13085 (2006).

  10. 10.

    Mitchison, T. & Mitchison, H. Cell biology: How cilia beat. Nature 463, 308–309 (2010).

  11. 11.

    Satir, P. & Christensen, S. T. Overview of structure and function of mammalian cilia. Annu. Rev. Physiol. 69, 377–400 (2007).

  12. 12.

    Gray, J. The mechanism of ciliary movement.—VI. Photographic and stroboscopic analysis of ciliary movement. Proc. R. Soc. Lond. B 107, 313–332 (1930).

  13. 13.

    Machin, K. E. The control and synchronization of flagellar movement. Proc. R. Soc. Lond. B 158, 88–104 (1963).

  14. 14.

    Blake, J. R. & Sleigh, M. A. Mechanics of ciliary locomotion. Biol. Rev. 49, 85–125 (1974).

  15. 15.

    Blake, J. R. & Chwang, A. T. Fundamental singularities of viscous flow. J. Eng. Math. 8, 23–29 (1974).

  16. 16.

    Gray, J. & Hancock, G. J. The propulsion of sea-urchin spermatozoa. J. Exp. Biol. 32, 802–814 (1955).

  17. 17.

    Hand, W. G. & Haupt, W. Flagellar activity of the colony members of Volvox aureus Ehrbg. during light stimulation. J. Protozool. 18, 361–364 (1971).

  18. 18.

    Sleigh, M. A. The form of beat in cilia of Stentor and Opalina. J. Exp. Biol. 37, 1–10 (1960).

  19. 19.

    Schwartz, E. A., Leonard, M. L., Bizios, R. & Bowser, S. S. Analysis and modeling of the primary cilium bending response to fluid shear. Am. J. Physiol. Ren. Physiol. 272, F132–F138 (1997).

  20. 20.

    Wiggins, C. H. & Goldstein, R. E. Flexive and propulsive dynamics of elastica at low Reynolds number. Phys. Rev. Lett. 80, 3879 (1998).

  21. 21.

    Camalet, S. & Jülicher, F. Generic aspects of axonemal beating. N. J. Phys. 2, 24 (2000).

  22. 22.

    Xu, G. et al. Flexural rigidity and shear stiffness of flagella estimated from induced bends and counterbends. Biophys. J. 110, 2759–2768 (2016).

  23. 23.

    Bandyopadhyay, P. R. & Hansen, J. C. Breakup and then makeup: a predictive model of how cilia self-regulate hardness for posture control. Sci. Rep. 3, 1956 (2013).

  24. 24.

    Chen, D. T., Heymann, M., Fraden, S., Nicastro, D. & Dogic, Z. ATP consumption of eukaryotic flagella measured at a single-cell level. Biophys. J. 109, 2562–2573 (2015).

  25. 25.

    Lindemann, C. B. Structural-functional relationships of the dynein, spokes, and central-pair projections predicted from an analysis of the forces acting within a flagellum. Biophys. J. 84, 4115–4126 (2003).

  26. 26.

    Jülicher, F., Ajdari, A. & Prost, J. Modeling molecular motors. Rev. Mod. Phys. 69, 1269 (1997).

  27. 27.

    Lindemann, C. B. A “geometric clutch” hypothesis to explain oscillations of the axoneme of cilia and flagella. J. Theor. Biol. 168, 175–189 (1994).

  28. 28.

    Brokaw, C. J. Molecular mechanism for oscillation in flagella and muscle. Proc. Natl Acad. Sci. USA 72, 3102–3106 (1975).

  29. 29.

    Elgeti, J., Winkler, R. G. & Gompper, G. Physics of microswimmers: single particle motion and collective behavior: a review. Rep. Prog. Phys. 78, 056601 (2015).

  30. 30.

    Vernon, G. G. & Woolley, D. M. Basal sliding and the mechanics of oscillation in a mammalian sperm flagellum. Biophys. J. 87, 3934–3944 (2004).

  31. 31.

    Riedel-Kruse, I. H., Hilfinger, A., Howard, J. & Jülicher, F. How molecular motors shape the flagellar beat. HFSP J. 1, 192–208 (2007).

  32. 32.

    Lin, J. & Nicastro, D. Asymmetric distribution and spatial switching of dynein activity generates ciliary motility. Science 360, eaar1968 (2018).

  33. 33.

    Brokaw, C. J. & Luck, D. J. L. Bending patterns of Chlamydomonas flagella: III. A radial spoke head deficient mutant and a central pair deficient mutant. Cell Motil. 5, 195–208 (1985).

  34. 34.

    Sartori, P., Geyer, V. F., Scholich, A., Jülicher, F. & Howard, J. Dynamic curvature regulation accounts for the symmetric and asymmetric beats of chlamydomonas flagella. eLife 5, e13258 (2016).

  35. 35.

    Hilfinger, A., Chattopadhyay, A. K. & Jülicher, F. Nonlinear dynamics of cilia and flagella. Phys. Rev. E 79, 051918 (2009).

  36. 36.

    Ishimoto, K. & Gaffney, E. A. An elastohydrodynamical simulation study of filament and spermatozoan swimming driven by internal couples. IMA J. Appl. Math. 83, 655–679 (2018).

  37. 37.

    Laskar, A. et al. Hydrodynamic instabilities provide a generic route to spontaneous biomimetic oscillations in chemomechanically active filaments. Sci. Rep. 3, 1964 (2013).

  38. 38.

    Lacey, S. E., He, S., Scheres, S. H. W. & Carter, A. P. Cryo-EM of dynein microtubule-binding domains shows how an axonemal dynein distorts the microtubule. eLife 8, e47145 (2019).

  39. 39.

    Ferreira, R. R., Vilfan, A., Jülicher, F., Supatto, W. & Vermot, J. Physical limits of flow sensing in the left-right organizer. eLife 6, e25078 (2017).

  40. 40.

    Chaaban, S. & Brouhard, G. J. A microtubule bestiary: structural diversity in tubulin polymers. Mol. Biol. Cell 28, 2924–2931 (2017).

  41. 41.

    Friedrich, B. Hydrodynamic synchronization of flagellar oscillators. Eur. Phys. J. Spec. Top. 225, 2353–2368 (2016).

  42. 42.

    Klindt, G. S., Ruloff, C., Wagner, C. & Friedrich, B. M. Load response of the flagellar beat. Phys. Rev. Lett. 117, 258101 (2016).

  43. 43.

    Okuno, M. & Hiramoto, Y. Mechanical stimulation of starfish sperm flagella. J. Exp. Biol. 65, 401–413 (1976).

  44. 44.

    Hill, D. B. et al. Force generation and dynamics of individual cilia under external loading. Biophys. J. 98, 57–66 (2010).

  45. 45.

    Machemer, H. Ciliary activity and the origin of metachrony in paramecium: effects of increased viscosity. J. Exp. Biol. 57, 239–259 (1972).

  46. 46.

    Gheber, L., Korngreen, A. & Priel, Z. Effect of viscosity on metachrony in mucus propelling cilia. Cell Motil. Cytoskelet. 39, 9–20 (1998).

  47. 47.

    Shingyoji, C., Higuchi, H., Yoshimura, M., Katayama, E. & Yanagida, T. Dynein arms are oscillating force generators. Nature 393, 711–714 (1998).

  48. 48.

    Jülicher, F. & Prost, J. Spontaneous oscillations of collective molecular motors. Phys. Rev. Lett. 78, 4510 (1997).

  49. 49.

    Eshel, D., Grossman, Y. & Priel, Z. Spectral characterization of ciliary beating: variations of frequency with time. Am. J. Physiol. Cell Physiol. 249, C160–C165 (1985).

  50. 50.

    Ma, R., Klindt, G. S., Riedel-Kruse, I. H., Jülicher, F. & Friedrich, B. M. Active phase and amplitude fluctuations of flagellar beating. Phys. Rev. Lett. 113, 048101 (2014).

  51. 51.

    Wan, K. Y. & Goldstein, R. E. Rhythmicity, recurrence, and recovery of flagellar beating. Phys. Rev. Lett. 113, 238103 (2014).

  52. 52.

    Han, J. & Peskin, C. S. Spontaneous oscillation and fluid–structure interaction of cilia. Proc. Natl Acad. Sci. USA 115, 4417–4422 (2018).

  53. 53.

    Gadêlha, H., Gaffney, E., Smith, D. & Kirkman-Brown, J. Nonlinear instability in flagellar dynamics: a novel modulation mechanism in sperm migration? J. R. Soc. Interface 7, 1689–1697 (2010).

  54. 54.

    Bayly, P. V. & Dutcher, S. K. Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella. J. R. Soc. Interface 13, 20160523 (2016).

  55. 55.

    Hu, T. & Bayly, P. V. Finite element models of flagella with sliding radial spokes and interdoublet links exhibit propagating waves under steady dynein loading. Cytoskeleton 75, 185–200 (2018).

  56. 56.

    Ling, F., Guo, H. & Kanso, E. Instability-driven oscillations of elastic microfilaments. J. R. Soc. Interface 15, 20180594 (2018).

  57. 57.

    Bottier, M., Thomas, K. A., Dutcher, S. K. & Bayly, P. V. How does cilium length affect beating? Biophys. J. 116, 1292–1304 (2019).

  58. 58.

    Gray, J. Ciliary Movement. Cambridge Comparative Physiology (Cambridge Univ. Press, 1928).

  59. 59.

    Rothschild. Measurement of sperm activity before artificial insemination. Nature 163, 358–359 (1949).

  60. 60.

    Riedel, I. H., Kruse, K. & Howard, J. A self-organized vortex array of hydrodynamically entrained sperm cells. Science 309, 300–303 (2005).

  61. 61.

    Quaranta, G., Aubin-Tam, M.-E. & Tam, D. Hydrodynamics versus intracellular coupling in the synchronization of eukaryotic flagella. Phys. Rev. Lett. 115, 238101 (2015).

  62. 62.

    Wan, K. Y. & Goldstein, R. E. Coordinated beating of algal flagella is mediated by basal coupling. Proc. Natl Acad. Sci. USA 113, E2784–E2793 (2016).

  63. 63.

    Brumley, D. R., Wan, K. Y., Polin, M. & Goldstein, R. E. Flagellar synchronization through direct hydrodynamic interactions. eLife 3, e02750 (2014).

  64. 64.

    Gueron, S., Levit-Gurevich, K., Liron, N. & Blum, J. J. Cilia internal mechanism and metachronal coordination as the result of hydrodynamical coupling. Proc. Natl Acad. Sci. USA 94, 6001–6006 (1997).

  65. 65.

    Niedermayer, T., Eckhardt, B. & Lenz, P. Synchronization, phase locking, and metachronal wave formation in ciliary chains. Chaos 18, 037128 (2008).

  66. 66.

    Vilfan, A. & Jülicher, F. Hydrodynamic flow patterns and synchronization of beating cilia. Phys. Rev. Lett. 96, 058102 (2006).

  67. 67.

    Pikovsky, A., Rosenblum, M., Kurths, J. & Kurths, J. Synchronization: A Universal Concept in Nonlinear Sciences Vol. 12 (Cambridge Univ. Press, 2003).

  68. 68.

    Guo, H., Fauci, L., Shelley, M. & Kanso, E. Bistability in the synchronization of actuated microfilaments. J. Fluid Mech. 836, 304–323 (2018).

  69. 69.

    Kim, Y. W. & Netz, R. R. Pumping fluids with periodically beating grafted elastic filaments. Phys. Rev. Lett. 96, 158101 (2006).

  70. 70.

    Coy, R. & Gadêlha, H. The counterbend dynamics of cross-linked filament bundles and flagella. J. R. Soc. Interface 14, 20170065 (2017).

  71. 71.

    Lindemann, C. B., Macauley, L. J. & Lesich, K. A. The counterbend phenomenon in dynein-disabled rat sperm flagella and what it reveals about the interdoublet elasticity. Biophys. J. 89, 1165–1174 (2005).

  72. 72.

    Goldstein, R. E. Green algae as model organisms for biological fluid dynamics. Annu. Rev. Fluid Mech. 47, 343–375 (2015).

  73. 73.

    Goldstein, R. E., Polin, M. & Tuval, I. Noise and synchronization in pairs of beating eukaryotic flagella. Phys. Rev. Lett. 103, 168103 (2009).

  74. 74.

    Wan, K. Y., Leptos, K. C. & Goldstein, R. E. Lag, lock, sync, slip: the many ‘phases’ of coupled flagella. J. R. Soc. Interface 11, 20131160 (2014).

  75. 75.

    Geyer, V. F., Jülicher, F., Howard, J. & Friedrich, B. M. Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga. Proc. Natl Acad. Sci. USA 110, 18058–18063 (2013).

  76. 76.

    Elfring, G. J. & Lauga, E. Hydrodynamic phase locking of swimming microorganisms. Phys. Rev. Lett. 103, 088101 (2009).

  77. 77.

    Friedrich, B. M. & Jülicher, F. Flagellar synchronization independent of hydrodynamic interactions. Phys. Rev. Lett. 109, 138102 (2012).

  78. 78.

    Lauga, E. & Powers, T. R. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601 (2009).

  79. 79.

    Goldstein, R. E. Batchelor Prize Lecture Fluid dynamics at the scale of the cell. J. Fluid Mech. 807, 1–39 (2016).

  80. 80.

    Tam, D. & Hosoi, A. Optimal feeding and swimming gaits of biflagellated organisms. Proc. Natl Acad. Sci. USA 108, 1001–1006 (2011).

  81. 81.

    Wan, K. Y. et al. Reorganisation of complex ciliary flows around regenerating Stentor coeruleus. Preprint at bioRxiv https://doi.org/10.1101/681908 (2019).

  82. 82.

    Polin, M., Tuval, I., Drescher, K., Gollub, J. P. & Goldstein, R. E. Chlamydomonas swims with two gears in a eukaryotic version of run-and-tumble locomotion. Science 325, 487–490 (2009).

  83. 83.

    Rüffer, U. & Nultsch, W. Comparison of the beating of cis- and trans-flagella of Chlamydomonas cells held on micropipettes. Cell Motil. Cytoskelet. 7, 87–93 (1987).

  84. 84.

    Wan, K. Y. & Goldstein, R. E. Time irreversibility and criticality in the motility of a flagellate microorganism. Phys. Rev. Lett. 121, 058103 (2018).

  85. 85.

    Kung, C. & Saimi, Y. The physiological basis of taxes in Paramecium. Annu. Rev. Physiol. 44, 519–534 (1982).

  86. 86.

    Mathijssen, A. J. T. M., Culver, J., Bhamla, M. S. & Prakash, M. Collective intercellular communication through ultra-fast hydrodynamic trigger waves. Nature 571, 560–564 (2019).

  87. 87.

    Bayless, B. A., Giddings, T. H. Jr, Winey, M. & Pearson, C. G. Bld10/Cep135 stabilizes basal bodies to resist cilia-generated forces. Mol. Biol. Cell 23, 4820–4832 (2012).

  88. 88.

    Coyle, S. M., Flaum, E., Li, H., Krishnamurthy, D. & Prakash, M. Coupled active systems encode an emergent hunting behavior in the unicellular predator Lacrymaria olor. Curr. Biol. 29, 3838–3850.e3 (2019).

  89. 89.

    Ainsworth, C. Cilia: tails of the unexpected. Nature 448, 638–641 (2007).

  90. 90.

    Grosberg, R. K. & Strathmann, R. R. The evolution of multicellularity: a minor major transition? Annu. Rev. Ecol. Evol. Syst. 38, 621–654 (2007).

  91. 91.

    Nielsen, C. Six major steps in animal evolution: are we derived sponge larvae? Evol. Dev. 10, 241–257 (2008).

  92. 92.

    Uchida, N. & Golestanian, R. Synchronization and collective dynamics in a carpet of microfluidic rotors. Phys. Rev. Lett. 104, 178103 (2010).

  93. 93.

    King, N. The unicellular ancestry of animal development. Dev. Cell 7, 313–325 (2004).

  94. 94.

    Nielsen, L. T. et al. Hydrodynamics of microbial filter feeding. Proc. Natl Acad. Sci. USA 114, 9373–9378 (2017).

  95. 95.

    Pettitt, M. E., Orme, B. A. A., Blake, J. R. & Leadbeater, B. S. C. The hydrodynamics of filter feeding in choanoflagellates. Eur. J. Protistol. 38, 313–332 (2002).

  96. 96.

    Higdon, J. J. L. The generation of feeding currents by flagellar motions. J. Fluid Mech. 94, 305–330 (1979).

  97. 97.

    Roper, M., Dayel, M. J., Pepper, R. E. & Koehl, M. Cooperatively generated stresslet flows supply fresh fluid to multicellular choanoflagellate colonies. Phys. Rev. Lett. 110, 228104 (2013).

  98. 98.

    Orme, B. A. A., Otto, S. R. & Blake, J. R. Chaos and mixing in micro-biological fluid dynamics: blinking stokeslets. Math. Methods Appl. Sci. 24, 1337–1349 (2001).

  99. 99.

    Kirkegaard, J. B., Marron, A. O. & Goldstein, R. E. Motility of colonial choanoflagellates and the statistics of aggregate random walkers. Phys. Rev. Lett. 116, 038102 (2016).

  100. 100.

    Kirkegaard, J. B., Bouillant, A., Marron, A. O., Leptos, K. C. & Goldstein, R. E. Aerotaxis in the closest relatives of animals. eLife 5, e18109 (2016).

  101. 101.

    Bidder, G. P. The relation of the form of a sponge to its currents. Q. J. Microsc. Sci. 67, 293–323 (1923).

  102. 102.

    Reiswig, H. M. Water transport, respiration and energetics of three tropical marine sponges. J. Exp. Mar. Biol. Ecol. 14, 231–249 (1974).

  103. 103.

    Mah, J. L., Christensen-Dalsgaard, K. K. & Leys, S. P. Choanoflagellate and choanocyte collar-flagellar systems and the assumption of homology. Evol. Dev. 16, 25–37 (2014).

  104. 104.

    Sogabe, S. et al. Pluripotency and the origin of animal multicellularity. Nature 570, 519–522 (2019).

  105. 105.

    LaBarbera, M. Principles of design of fluid transport systems in zoology. Science 249, 992–1000 (1990).

  106. 106.

    Asadzadeh, S. S., Larsen, P. S., Riisgård, H. U. & Walther, J. H. Hydrodynamics of the leucon sponge pump. J. R. Soc. Interface 16, 20180630 (2019).

  107. 107.

    Shapiro, O. H. et al. Vortical ciliary flows actively enhance mass transport in reef corals. Proc. Natl Acad. Sci. USA 111, 13391–13396 (2014).

  108. 108.

    Armon, S., Bull, M. S., Aranda-Diaz, A. & Prakash, M. Ultrafast epithelial contractions provide insights into contraction speed limits and tissue integrity. Proc. Natl Acad. Sci. USA 115, E10333–E10341 (2018).

  109. 109.

    Prakash, V., Bull, M. S. & Prakash, M. Motility induced fracture reveals a ductile to brittle crossover in the epithelial tissues of a simple animal. Preprint at bioRxiv https://doi.org/10.1101/676866 (2019).

  110. 110.

    Smith, C. L., Reese, T. S., Govezensky, T. & Barrio, R. A. Coherent directed movement toward food modeled in Trichoplax, a ciliated animal lacking a nervous system. Proc. Natl Acad. Sci. USA 116, 8901–8908 (2019).

  111. 111.

    Smith, C. L., Pivovarova, N. & Reese, T. S. Coordinated feeding behavior in Trichoplax, an animal without synapses. PLOS ONE 10, e0136098 (2015).

  112. 112.

    Varoqueaux, F. et al. High cell diversity and complex peptidergic signaling underlie placozoan behavior. Curr. Biol. 28, 3495–3501 (2018).

  113. 113.

    Emlet, R. B. Functional constraints on the evolution of larval forms of marine invertebrates: experimental and comparative evidence. Am. Zool. 31, 707–725 (1991).

  114. 114.

    Bick, C., Goodfellow, M., Laing, C. R. & Martens, E. A. Understanding the dynamics of biological and neural oscillator networks through mean-field reductions: a review. Preprint at arXiv https://arxiv.org/abs/1902.05307 (2019).

  115. 115.

    Panaggio, M. J. & Abrams, D. M. Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28, R67 (2015).

  116. 116.

    Jékely, G. Origin and early evolution of neural circuits for the control of ciliary locomotion. Proc. R. Soc. B 278, 914–922 (2010).

  117. 117.

    Bezares-Calderon, L. A. et al. Neural circuitry of a polycystin-mediated hydrodynamic startle response for predator avoidance. eLife 7, e36262 (2018).

  118. 118.

    Verasztó, C. et al. Ciliomotor circuitry underlying whole-body coordination of ciliary activity in the Platynereis larva. eLife 6, e26000 (2017).

  119. 119.

    Lenz, P. & Ryskin, A. Collective effects in ciliar arrays. Phys. Biol. 3, 285 (2006).

  120. 120.

    Leoni, M. & Liverpool, T. B. Hydrodynamic synchronization of nonlinear oscillators at low Reynolds number. Phys. Rev. E 85, 040901 (2012).

  121. 121.

    Guirao, B. & Joanny, J.-F. Spontaneous creation of macroscopic flow and metachronal waves in an array of cilia. Biophys. J. 92, 1900–1917 (2007).

  122. 122.

    Knight-Jones, E. W. Relations between metachronism and the direction of ciliary beat in metazoa. J. Cell Sci. 3, 503–521 (1954).

  123. 123.

    Sleigh, M. A., Blake, J. R. & Liron, N. The propulsion of mucus by cilia. Am. Rev. Respir. Dis. 137, 726–741 (1988).

  124. 124.

    Elgeti, J. & Gompper, G. Emergence of metachronal waves in cilia arrays. Proc. Natl Acad. Sci. USA 110, 4470–4475 (2013).

  125. 125.

    Babataheri, A., Roper, M., Fermigier, M. & Du Roure, O. Tethered fleximags as artificial cilia. J. Fluid Mech. 678, 5–13 (2011).

  126. 126.

    Shields, A. R. et al. Biomimetic cilia arrays generate simultaneous pumping and mixing regimes. Proc. Natl Acad. Sci. USA 107, 15670–15675 (2010).

  127. 127.

    Hanasoge, S., Hesketh, P. J. & Alexeev, A. Microfluidic pumping using artificial magnetic cilia. Microsyst. Nanoeng. 4, 11 (2018).

  128. 128.

    Gheber, L. & Priel, Z. Ciliary activity under normal conditions and under viscous load. Biorheology 27, 547–557 (1990).

  129. 129.

    Guo, H. & Kanso, E. Evaluating efficiency and robustness in cilia design. Phys. Rev. E 93, 033119 (2016).

  130. 130.

    Smith, D. J., Gaffney, E. A. & Blake, J. R. Modelling mucociliary clearance. Respir. Physiol. Neurobiol. 163, 178–188 (2008).

  131. 131.

    Osterman, N. & Vilfan, A. Finding the ciliary beating pattern with optimal efficiency. Proc. Natl Acad. Sci. USA 108, 15727–15732 (2011).

  132. 132.

    Guo, H., Nawroth, J., Ding, Y. & Kanso, E. Cilia beating patterns are not hydrodynamically optimal. Phys. Fluids 26, 091901 (2014).

  133. 133.

    Spagnolie, S. E. & Lauga, E. The optimal elastic flagellum. Phys. Fluids 22, 031901 (2010).

  134. 134.

    Gueron, S. & Levit-Gurevich, K. Energetic considerations of ciliary beating and the advantage of metachronal coordination. Proc. Natl Acad. Sci. USA 96, 12240–12245 (1999).

  135. 135.

    Chateau, S., Favier, J., D’ortona, U. & Poncet, S. Transport efficiency of metachronal waves in 3D cilium arrays immersed in a two-phase flow. J. Fluid Mech. 824, 931–961 (2017).

  136. 136.

    Datt, C., Natale, G., Hatzikiriakos, S. G. & Elfring, G. J. An active particle in a complex fluid. J. Fluid Mech. 823, 675–688 (2017).

  137. 137.

    Brokaw, C. J. & Simonick, T. F. Mechanochemical coupling in flagella. V. Effects of viscosity on movement and ATP-dephosphorylation of Triton-demembranated sea-urchin spermatozoa. J. Cell Sci. 23, 227–241 (1977).

  138. 138.

    Mettot, C. & Lauga, E. Energetics of synchronized states in three-dimensional beating flagella. Phys. Rev. E 84, 061905 (2011).

  139. 139.

    Ding, Y., Nawroth, J. C., McFall-Ngai, M. J. & Kanso, E. Mixing and transport by ciliary carpets: a numerical study. J. Fluid Mech. 743, 124–140 (2014).

  140. 140.

    Blake, J. A model for the micro-structure in ciliated organisms. J. Fluid Mech. 55, 1–23 (1972).

  141. 141.

    Smith, D. J., Gaffney, E. A. & Blake, J. R. Discrete cilia modelling with singularity distributions: application to the embryonic node and the airway surface liquid. Bull. Math. Biol. 69, 1477–1510 (2007).

  142. 142.

    Quek, R., Lim, K. M. & Chiam, K. H. Three-Dimensional Simulations of Ciliary Flow 197–218 (Springer, 2014).

  143. 143.

    Supatto, W., Fraser, S. E. & Vermot, J. An all-optical approach for probing microscopic flows in living embryos. Biophys. J. 95, L29–L31 (2008).

  144. 144.

    Ramirez-San Juan, G. R. et al. Multi-scale spatial heterogeneity enhances particle clearance in airway ciliary arrays. Preprint at bioRxiv https://doi.org/10.1101/665125 (2019).

  145. 145.

    Schneiter, M., Ricka, J. & Frenz, M. Self-organization of self-clearing beating patterns in an array of locally interacting ciliated cells formulated as an adaptive boolean network. Theory Biosci. https://doi.org/10.1007/s12064-019-00299-x (2019).

  146. 146.

    Faubel, R., Westendorf, C., Bodenschatz, E. & Eichele, G. Cilia-based flow network in the brain ventricles. Science 353, 176–178 (2016).

  147. 147.

    Veening, J. G. & Barendregt, H. P. The regulation of brain states by neuroactive substances distributed via the cerebrospinal fluid; a review. Cerebrospinal Fluid Res. 7, 1 (2010).

  148. 148.

    Pellicciotta, N. et al. Synchronization of mammalian motile cilia in the brain with hydrodynamic forces. Preprint at bioRxiv https://doi.org/10.1101/668459 (2019).

  149. 149.

    Devenport, D. The cell biology of planar cell polarity. J. Cell Biol. 207, 171–179 (2014).

  150. 150.

    Vladar, E. K., Lee, Y. L., Stearns, T. & Axelrod, J. D. in Methods in Cell Biology Vol. 127 Ch. 3 (eds Basto, R. & Marshall, W. F.) 37–54 (Elsevier, 2015).

  151. 151.

    Hilfinger, A. & Jülicher, F. The chirality of ciliary beats. Phys. Biol. 5, 016003 (2008).

  152. 152.

    Kim, M. J. & Breuer, K. S. Microfluidic pump powered by self-organizing bacteria. Small 4, 111–118 (2008).

  153. 153.

    Mathijssen, A. J., Guzmán-Lastra, F., Kaiser, A. & Löwen, H. Nutrient transport driven by microbial active carpets. Phys. Rev. Lett. 121, 248101 (2018).

  154. 154.

    Golestanian, R., Yeomans, J. M. & Uchida, N. Hydrodynamic synchronization at low Reynolds number. Soft Matter 7, 3074–3082 (2011).

  155. 155.

    Uchida, N., Golestanian, R. & Bennett, R. R. Synchronization and collective dynamics of flagella and cilia as hydrodynamically coupled oscillators. J. Phys. Soc. Jpn. 86, 101007 (2017).

  156. 156.

    Brumley, D. R. et al. Long-range interactions, wobbles, and phase defects in chains of model cilia. Phys. Rev. Fluids 1, 081201 (2016).

  157. 157.

    Solon, A. & Tailleur, J. Revisiting the flocking transition using active spins. Phys. Rev. Lett. 111, 078101 (2013).

  158. 158.

    Gilpin, W., Prakash, V. N. & Prakash, M. Vortex arrays and ciliary tangles underlie the feeding–swimming trade-off in starfish larvae. Nat. Phys. 13, 380–386 (2017).

  159. 159.

    Bourland, W. A., Wendell, L., Hampikian, G. & Vdancỳ, P. Morphology and phylogeny of Bryophryoides ocellatus ng, n. sp.(Ciliophora, Colpodea) from in situ soil percolates of Idaho, USA. Eur. J. Protistol. 50, 47–67 (2014).

  160. 160.

    Feriani, L. et al. Assessing the collective dynamics of motile cilia in cultures of human airway cells by multiscale DDM. Biophys. J. 113, 109–119 (2017).

  161. 161.

    Brumley, D. R., Polin, M., Pedley, T. J. & Goldstein, R. E. Metachronal waves in the flagellar beating of Volvox and their hydrodynamic origin. J. R. Soc. Interface 12, 20141358 (2015).

  162. 162.

    Nielsen, C. Structure and function of metazoan ciliary bands and their phylogenetic significance. Acta Zool. 68, 205–262 (1987).

  163. 163.

    Strathmann, R. R. The evolution and loss of feeding larval stages of marine invertebrates. Evolution 32, 894–906 (1978).

  164. 164.

    Agassiz, A. North American Starfishes Vol. 5 (Welch, Bigelow, and Company, Univ. Press, 1877).

  165. 165.

    Riisgård, H. U. & Larsen, P. S. Minireview: Ciliary filter feeding and bio-fluid mechanics—present understanding and unsolved problems. Limnol. Oceanogr. 46, 882–891 (2001).

  166. 166.

    Jorgensen, C. B. Fluid mechanical aspects of suspension feeding. Mar. Ecol. Prog. Ser. 11, 89–103 (1983).

  167. 167.

    Rubenstein, D. I. & Koehl, M. A. R. The mechanisms of filter feeding: some theoretical considerations. Am. Naturalist 111, 981–994 (1977).

  168. 168.

    Mathijssen, A. J., Jeanneret, R. & Polin, M. Universal entrainment mechanism controls contact times with motile cells. Phys. Rev. Fluids 3, 033103 (2018).

  169. 169.

    Ding, Y. & Kanso, E. Selective particle capture by asynchronously beating cilia. Phys. Fluids 27, 121902 (2015).

  170. 170.

    Gilpin, W., Prakash, V. N. & Prakash, M. Dynamic vortex arrays created by starfish larvae. Phys. Rev. Fluids 2, 090501 (2017).

  171. 171.

    Gilpin, W., Prakash, V. N. & Prakash, M. Rapid behavioral transitions produce chaotic mixing by a planktonic microswimmer. Preprint at arXiv https://arxiv.org/abs/1804.08773 (2018).

  172. 172.

    Gilpin, W., Prakash, V. N. & Prakash, M. Flowtrace: simple visualization of coherent structures in biological fluid flows. J. Exp. Biol. 220, 3411–3418 (2017).

  173. 173.

    von Dassow, G., Emlet, R. & Grünbaum, D. Boundary effects on currents around ciliated larvae. Nat. Phys. 13, 520–521 (2017).

  174. 174.

    Gilpin, W., Prakash, V. N. & Prakash, M. Reply to ‘Boundary effects on currents around ciliated larvae’. Nat. Phys. 13, 521–522 (2017).

  175. 175.

    Krishnamurthy, D. et al. Scale-free vertical tracking microscopy: Towards bridging scales in biological oceanography. Preprint at bioRxiv https://doi.org/10.1101/610246 (2019).

  176. 176.

    Bruot, N. & Cicuta, P. Realizing the physics of motile cilia synchronization with driven colloids. Annu. Rev. Condens. Matter Phys. 7, 323–348 (2016).

  177. 177.

    Amemiya, S. et al. Development of ciliary bands in larvae of the living isocrinid sea lily Metacrinus rotundus. Acta Zool. 96, 36–43 (2015).

  178. 178.

    Nasouri, B. & Elfring, G. J. Hydrodynamic interactions of cilia on a spherical body. Phys. Rev. E 93, 033111 (2016).

  179. 179.

    Ghorbani, A. & Najafi, A. Symplectic and antiplectic waves in an array of beating cilia attached to a closed body. Phys. Rev. E 95, 052412 (2017).

  180. 180.

    Panaggio, M. J. & Abrams, D. M. Chimera states on a flat torus. Phys. Rev. Lett. 110, 094102 (2013).

  181. 181.

    Kuramoto, Y. & Battogtokh, D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5, 380–385 (2002).

  182. 182.

    Abrams, D. M. & Strogatz, S. H. Chimera states for coupled oscillators. Phys. Rev. Lett. 93, 174102 (2004).

  183. 183.

    Wollin, C. & Stark, H. Metachronal waves in a chain of rowers with hydrodynamic interactions. Eur. Phys. J. E 34, 42 (2011).

  184. 184.

    Nawroth, J. C. et al. Motile cilia create fluid-mechanical microhabitats for the active recruitment of the host microbiome. Proc. Natl Acad. Sci. USA 114, 9510–9516 (2017).

  185. 185.

    Childress, S. & Dudley, R. Transition from ciliary to flapping mode in a swimming mollusc: flapping flight as a bifurcation in Reω. J. Fluid Mech. 498, 257–288 (2004).

  186. 186.

    Reiten, I. et al. Motile-cilia-mediated flow improves sensitivity and temporal resolution of olfactory computations. Curr. Biol. 27, 166–174 (2017).

  187. 187.

    Pfeifer, R., Lungarella, M. & Iida, F. Self-organization, embodiment, and biologically inspired robotics. Science 318, 1088–1093 (2007).

  188. 188.

    Mohren, T. L., Daniel, T. L., Brunton, S. L. & Brunton, B. W. Neural-inspired sensors enable sparse, efficient classification of spatiotemporal data. Proc. Natl Acad. Sci. USA 115, 10564–10569 (2018).

  189. 189.

    Hauser, H., Ijspeert, A. J., Füchslin, R. M., Pfeifer, R. & Maass, W. Towards a theoretical foundation for morphological computation with compliant bodies. Biol. Cybern. 105, 355–370 (2011).

  190. 190.

    Mayne, R. & den Toonder, J. Atlas of Cilia Bioengineering and Biocomputing (River Publishers, 2018).

  191. 191.

    Guérin, T., Prost, J. & Joanny, J.-F. Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models. Eur. Phys. J. E 34, 60 (2011).

  192. 192.

    Guérin, T., Prost, J. & Joanny, J.-F. Bidirectional motion of motor assemblies and the weak-noise escape problem. Phys. Rev. E 84, 041901 (2011).

  193. 193.

    Van Kampen, N. G. Stochastic Processes in Physics and Chemistry Vol. 1 (Elsevier, 1992).

  194. 194.

    Uchida, N. & Golestanian, R. Generic conditions for hydrodynamic synchronization. Phys. Rev. Lett. 106, 058104 (2011).

  195. 195.

    Mirzakhanloo, M. & Alam, M.-R. Flow characteristics of chlamydomonas result in purely hydrodynamic scattering. Phys. Rev. E 98, 012603 (2018).

  196. 196.

    Wei, D., Dehnavi, P. G., Aubin-Tam, M.-E. & Tam, D. Is the zero Reynolds number approximation valid for ciliary flows? Phys. Rev. Lett. 122, 124502 (2019).

  197. 197.

    Hong, H. & Strogatz, S. H. Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators. Phys. Rev. Lett. 106, 054102 (2011).

  198. 198.

    Zakharova, A., Kapeller, M. & Schöll, E. Chimera death: symmetry breaking in dynamical networks. Phys. Rev. Lett. 112, 154101 (2014).

  199. 199.

    Gilpin, W. Self-organized avalanches in globally-coupled phase oscillators. Preprint at arXiv https://arxiv.org/abs/1906.05514 (2019).

  200. 200.

    Kawamura, Y. Chimera Ising walls in forced nonlocally coupled oscillators. Phys. Rev. E 75, 056204 (2007).

  201. 201.

    Ottino-Löffler, B. & Strogatz, S. H. Volcano transition in a solvable model of frustrated oscillators. Phys. Rev. Lett. 120, 264102 (2018).

  202. 202.

    Abrams, D. M., Mirollo, R., Strogatz, S. H. & Wiley, D. A. Solvable model for chimera states of coupled oscillators. Phys. Rev. Lett. 101, 084103 (2008).

  203. 203.

    Totz, J. F., Rode, J., Tinsley, M. R., Showalter, K. & Engel, H. Spiral wave chimera states in large populations of coupled chemical oscillators. Nat. Phys. 14, 282–285 (2018).

  204. 204.

    Brumley, D. R., Polin, M., Pedley, T. J. & Goldstein, R. E. Hydrodynamic synchronization and metachronal waves on the surface of the colonial alga Volvox carteri. Phys. Rev. Lett. 109, 268102 (2012).

  205. 205.

    Kotar, J. et al. Optimal hydrodynamic synchronization of colloidal rotors. Phys. Rev. Lett. 111, 228103 (2013).

  206. 206.

    Dreyfus, R. et al. Microscopic artificial swimmers. Nature 437, 862–865 (2005).

  207. 207.

    Darnton, N., Turner, L., Breuer, K. & Berg, H. C. Moving fluid with bacterial carpets. Biophys. J. 86, 1863–1870 (2004).

  208. 208.

    Wadhwa, N., Phillips, R. & Berg, H. C. Torque-dependent remodeling of the bacterial flagellar motor. Proc. Natl Acad. Sci. USA 116, 11764–11769 (2019).

  209. 209.

    Sanchez, T., Welch, D., Nicastro, D. & Dogic, Z. Cilia-like beating of active microtubule bundles. Science 333, 456–459 (2011).

  210. 210.

    Sanchez, T., Chen, D. T. N., DeCamp, S. J., Heymann, M. & Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434 (2012).

  211. 211.

    DiPetrillo, C. G. & Smith, E. F. Pcdp1 is a central apparatus protein that binds Ca2+-calmodulin and regulates ciliary motility. J. Cell Biol. 189, 601–612 (2010).

  212. 212.

    Lin, J., Heuser, T., Song, K., Fu, X. & Nicastro, D. One of the nine doublet microtubules of eukaryotic flagella exhibits unique and partially conserved structures. PLOS ONE 7, e46494 (2012).

  213. 213.

    Shoemark, A. & Hogg, C. Electron tomography of respiratory cilia. Thorax 68, 190–191 (2013).

  214. 214.

    Odate, T., Takeda, S., Narita, K. & Kawahara, T. 9 + 0 and 9 + 2 cilia are randomly dispersed in the mouse node. Microscopy 65, 119–126 (2016).

  215. 215.

    Wilkerson, C. G., King, S. M., Koutoulis, A., Pazour, G. J. & Witman, G. B. The 78,000 M(r) intermediate chain of Chlamydomonas outer arm dynein is a WD-repeat protein required for arm assembly. J. Cell Biol. 129, 169–178 (1995).

  216. 216.

    Austin-Tse, C. et al. Zebrafish ciliopathy screen plus human mutational analysis identifies C21orf59 and CCDC65 defects as causing primary ciliary dyskinesia. Am. J. Hum. Genet. 93, 672–686 (2013).

  217. 217.

    Stokes, M. Larval locomotion of the lancelet. J. Exp. Biol. 200, 1661–1680 (1997).

  218. 218.

    Bone, Q., Carre, C. & Chang, P. Tunicate feeding filters. J. Mar. Biol. Assoc. U. K. 83, 907–919 (2003).

  219. 219.

    Sutherland, K. R., Madin, L. P. & Stocker, R. Filtration of submicrometer particles by pelagic tunicates. Proc. Natl Acad. Sci. USA 107, 15129–15134 (2010).

  220. 220.

    Petersen, J. K., Mayer, S. & Knudsen, M. Beat frequency of cilia in the branchial basket of the ascidian Ciona intestinalis in relation to temperature and algal cell concentration. Mar. Biol. 133, 185–192 (1999).

  221. 221.

    Riisgård, H. U., Nielsen, C. & Larsen, P. S. Downstream collecting in ciliary suspension feeders: the catch-up principle. Mar. Ecol. Prog. Ser. 207, 33–51 (2000).

  222. 222.

    Okabe, N., Xu, B. & Burdine, R. D. Fluid dynamics in zebrafish Kupffer’s vesicle. Dev. Dyn. 237, 3602–3612 (2008).

Download references

Acknowledgements

The authors thank G. Ramirez-San Juan, A. Mathijssen and the other members of the Prakash Lab for their helpful comments on the manuscript. W.G. was supported by the NSF-Simons Center for Mathematical and Statistical Analysis of Biology at Harvard University, NSF grant no. DMS-1764269, the Harvard FAS Quantitative Biology Initiative, the U. S. Department of Defense National Defense Science and Engineering Graduate (NDSEG) Fellowship Program and the National Geographic Society ‘Young Explorers’ program. M.S.B. thanks the NSF Graduate Research Fellowship Program and the Stanford Bio-X fellowship. M.P. thanks the NSF Careers Program, NSF ‘Center for Cellular Construction’ program (DBI-1548297), NIH Directors New Innovator Award, HHMI-Gates Faculty Fellows program, the W. M. Keck Foundation, the Gordon and Betty Moore Foundation and the Chan Zuckerberg Biohub Investigators program for supporting this work.

Author information

All authors contributed to the preparation of the manuscript.

Correspondence to William Gilpin or Manu Prakash.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Glossary

Microtubule

A tube-shaped protein assembly found in the cytoplasm of many cells. Microtubules allow cells to maintain their shape and internal arrangements, and they can aggregate to form specialized structures, including cilia.

Hopf bifurcations

A Hopf bifurcation is a phenomenon occurring in many nonlinear dynamical systems, in which a periodic orbit spontaneously appears or disappears as a control parameter is varied.

Volvox

Commonly known as ‘globe algae’, these single-celled green algae form spherical colonies containing up to 50,000 cells.

Counterbend

A phenomenon in deforming elastic beams — and a deviation from classical Euler–Bernoulli beam theory — in which an applied curvature in one location induces a compensatory curvature elsewhere along the beam.

Run-and-tumble

A navigation strategy employed by bacteria and other microorganisms, in which an organism follows nutrient gradients by intermittently switching between directional swimming and random reorientation.

Reynolds numbers

Dimensionless quantity expressing the ratio of inertial to viscous forces in a fluid dynamics problem. Navigation and locomotion strategies are qualitatively different in the low-Reynolds-number (overdamped) regime and high-Reynolds-number (turbulent) regime.

Ependymal

Referring to a thin membrane of cells lining the ventricles of the brain and the central canal of the spinal cord. These cells play a central role in supporting neuronal function.

Coarsening

A phenomenon in the dynamics of spatially varying scalar fields, in which small-wavelength features are gradually consolidated into larger-wavelength patterns.

Solitons

Travelling, bounded wave packets occurring in nonlinear media that propagate at fixed velocity.

Lagrangian chaos

Chaotic motion of tracer particles in a fluid, which readily occurs in high-Reynolds-number flows (such as turbulence). Under certain conditions, it can also occur in the low-Reynolds-number flows produced by ciliated microorganisms.

Morphospace

An abstract coordinate system describing all possible forms or shapes of an organism, parameterized by a small number of independent variables.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gilpin, W., Bull, M.S. & Prakash, M. The multiscale physics of cilia and flagella. Nat Rev Phys 2, 74–88 (2020). https://doi.org/10.1038/s42254-019-0129-0

Download citation