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  • Open Access

Lagrangian transport by ciliary arrays

George T. Fortune* and Eric Lauga†

  • Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

  • *Contact author: gtf22@damtp.cam.ac.uk
  • †Contact author: e.lauga@damtp.cam.ac.uk
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Phys. Rev. Research 7, 043209 – Published 24 November, 2025

DOI: https://doi.org/10.1103/t41h-gsv9

Abstract

In many biological systems, from sinuses to airways, ciliary arrays are essential for transporting particles and bacteria. Challenging long-held assumptions, we show that the classical Eulerian view of cilia-induced flow overlooks a key feature of time-averaged transport. Using theoretical modeling and numerical simulations, we demonstrate that the drift generated by ciliary motion is not uniform, as previously predicted, but varies with distance from the ciliary surface in unexpected ways. This variation reveals drift stagnation points, offering a potential means to distinguish healthy from impaired ciliary beating and refines our understanding of fluid-driven transport.

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References (60)

  1. N. Klena and G. Pigino, Structural biology of cilia and intraflagellar transport, Annu. Rev. Cell Dev. Biol. 38, 100 (2022).
  2. C. Brennen and H. Winet, Fluid mechanics of propulsion by cilia and flagella, Annu. Rev. Fluid Mech. 9, 339 (1977).
  3. J. Blake, Fluid mechanics of cilliary propulsion, in Computational Modeling in Biological Fluid Mechanics, edited by L. J. Fauci and S. Gueron (IMA, New York, USA, 2001), Vol. 124, pp. 1–51.
  4. D. J. Smith, E. A. Gaffney, and J. R. Blake, Mathematical of cilia-driven transport of biological fluids, Proc. R. Soc. A. 465, 2417 (2009).
  5. S. Childress, Mechanics of Swimming and Flying (Cambridge University Press, Cambridge, 1981).
  6. E. Lauga, The Fluid Dynamics of Cell Motility (Cambridge University Press, Cambridge, 2020).
  7. G. T. Fortune, A. Worley, A. B. Sendova-Franks, N. R. Franks, K. C. Leptos, E. Lauga, and R. E. Goldstein, The fluid dynamics of collective vortex structures of plant-animal worms, J. Fluid Mech. 914, A20 (2021).
  8. J. R. Velez-Cordero and E. Lauga, Waving transport and propulsion in a generalized Newtonian fluid, J. Non-Newtonian Fluid Mech. 199, 37 (2013).
  9. D. J. Smith, E. A. Gaffney, and J. R. Black, Modelling mucociliary clearance, Respir. Physiol. Neurobiol. 163, 178 (2008).
  10. G. R. Fulford and J. R. Blake, Muco-ciliary transport in the lung, J. Theor. Biol. 121, 381 (1986).
  11. L. J. Fauci and R. Dillon, Biofluidmechanics of reproduction, Annu. Rev. Fluid Mech. 38, 371 (2006).
  12. T. J. Lardner and W. J. Shack, Cilia transport, Bull. Math. Biophys. 34, 325 (1972).
  13. C. O'Callaghan, K. Sikand, and M. A. Chilvers, Analysis of ependymal ciliary beat pattern and beat frequency using high speed imaging: Comparison with the photomultiplier and photodiode methods, Cilia 1, 8 (2012).
  14. I. Ibañez-Tallon, N. Heintz, and H. Omran, To beat or not to beat: Roles of cilia in development and disease, Hum. Mol. Genet. 12, 27R (2003).
  15. G. I. Taylor, Analysis of the swimming of microscopic organisms, Proc. R. Soc. A 209, 447 (1951).
  16. J. R. Blake, Infinite models for ciliary propulsion, J. Fluid Mech. 49, 209 (1971).
  17. J. E. Drummond, Propulsion by oscillating sheets and tubes in a viscous fluid, J. Fluid Mech. 25, 787 (1966).
  18. A. J. Reynolds, The swimming of minute organisms, J. Fluid Mech. 23, 241 (1965).
  19. E. O. Tuck, A note on a swimming problem, J. Fluid Mech. 31, 305 (1968).
  20. S. Childress, Inertial swimming as a singular perturbation, in Proceedings of the ASME 2008 Dynamic Systems and Control Conference (ASME, New York, USA, 2008), pp. 1413–1420.
  21. T. Y. T. Wu, Swimming of a waving plate, J. Fluid Mech. 10, 321 (1961).
  22. T. Y. T. Wu, Hydrodynamics of swimming propulsion. Part I. Swimming of a two-dimensional flexible plate at variable speeds in an inviscid fluid, J. Fluid Mech. 46, 337 (1971).
  23. G. I. Taylor, The action of waving cylindrical tails in propelling microscopic organisms, Proc. R. Soc. A 211, 225 (1952).
  24. T. K. Chaudhury, On swimming in a visco-elastic liquid, J. Fluid Mech. 95, 189 (1979).
  25. L. D. Sturges, Motion induced by a waving plate, J. Non-Newtonian Fluid Mech. 8, 357 (1981).
  26. G. R. Fulford, D. F. Katz, and R. L. Powell, Swimming of spermatozoa in a linear viscoelastic fluid, Biorheology 35, 295 (1998).
  27. E. Lauga, Propulsion in a viscoelastic fluid, Phys. Fluids 19, 083104 (2007)
  28. A. H. Shapiro, M. Y. Jaffrin, and S. L. Weinberg, Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech. 37, 799 (1969).
  29. M. Y. Jaffrin and A. H. Shapiro, Peristaltic pumping, Annu. Rev. Fluid Mech. 3, 13 (1971).
  30. C. Brennen, Oscillating boundary layer for ciliary propulsion, J. Fluid Mech. 65, 799 (1974).
  31. T. Ishikawa, Fluid dynamics of squirmers and ciliated microorganisms, Annu. Rev. Fluid Mech. 56, 119 (2024).
  32. H. Guo, H. Zhu, R. Liu, M. Bonnet, and S. Veerapaneni, Optimal ciliary locomotion of axisymmetric microswimmers, J. Fluid Mech. 927, A22 (2021).
  33. T. Omori, H. Ito, and T. Ishikawa, Swimming microorganisms acquire optimal efficiency with multiple cilia, Proc. Natl. Acad. Sci. USA 117, 30201 (2020).
  34. T. Nguyen, Y. Koga, T. Wakasugi, T. Kitamura, and H. Suzuki, Nasal polyps show decreased mucociliary transport despite vigorous ciliary beating, Braz. J. Otorhinolaryngol. 90, 101377 (2024).
  35. D. Palazzolo, J. M. Nelson, E. Ely, A. Crow, J. Distin, and S. Kunigelis, The effects of electronic cigarette (ECIG)-generated aerosol and conventional cigarette smoke on the mucociliary transport velocity (MTV) using the bullfrog (R. catesbiana) palate paradigm, Front. Physiol. 8, 1023 (2017).
  36. G. Schmidt, F. Borchers, S. Müller, A. Akbari, F. Edinger, M. Sander, C. Koch, and M. Henrich, Dobutamine, epinephrine, and milrinone accelerate particle transport velocity in murine tracheal epithelium via Ca2+ release from caffeine-sensitive internal stores, Cells, 14, 228 (2025).
  37. S. Kelly, P. Martinsen, and S. Tatkov, Rapid changes in mucociliary transport in the tracheal epithelium caused by unconditioned room air or nebulized hypertonic saline and mannitol are not determined by frequency of beating cilia, ICMx, 9, 8 (2021).
  38. The MathWorks, Inc., MATLAB and Image Processing Toolbox Release 2023b (The MathWorks Inc., Natick, MA, 2023).
  39. E. J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, 1991).
  40. B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev. 1, 701 (1920).
  41. A. A. Dorodnicyn, Asymptotic solution of Van der Pol's equation, Prikl. Mat. Mekh. 11, 313 (1947).
  42. K. Ma, N. Pujara, and J.-L. Thiffeault, Reaching for the surface: Spheroidal microswimmers in surface gravity waves, Phys. Rev. Fluids 7, 014310 (2022).
  43. E. A. Gaffney, M. P. Dalwadi, C. Moreau, K. Ishimoto, and B. J. Walker, Canonical orbits for rapidly deforming planar microswimmers in shear flow, Phys. Rev. Fluids 7, L022101 (2022).
  44. B. J. Walker, K. Ishimoto, E. A. Gaffney, C. Moreau, and M. P. Dalwadi, Effects of rapid yawing on simple swimmer models and planar Jeffery's orbits, Phys. Rev. Fluids 7, 023101 (2022).
  45. B. J. Walker, K. Ishimoto, and E. A. Gaffney, Systematic of minimal models of microswimming, Phys. Rev. Fluids 8, 034102 (2023).
  46. M. P. Dalwadi, C. Moreau, E. A. Gaffney, K. Ishimoto, and B. J. Walker, Generalised Jeffery's equations for rapidly spinning particles. Part 1: Spheroids, J. Fluid Mech. 979, A1 (2024).
  47. E. A. Gaffney, K. Ishimoto, and B. J. Walker, Motility and rotation of multi-timescale microswimmers in linear background flows, J. Fluid Mech. 1022, A13 (2025).
  48. D. R. Brumley, K. Y. Wan, M. Polin, and R. E. Goldstein, Flagellar synchronization through direct hydrodynamic interactions, eLife 3, e02750 (2014).
  49. T. Niedermayer, B. Eckhardt, and P. Lenz, Synchronization, phase locking, and metachronal wave formation in ciliary chains, Chaos 18, 037128 (2008).
  50. N. Uchida and R. Golestanian, Generic conditions for hydrodynamic synchronization, Phys. Rev. Lett. 106, 058104 (2011).
  51. N. Uchida and R. Golestanian, Hydrodynamic synchronization between objects with cyclic rigid trajectories, Eur. Phys. J. E 35, 135 (2012).
  52. F. Boselli, J. Jullien, E. Lauga, and R. Goldstein, Fluid mechanics of mosaic ciliated tissues, Phys. Rev. Lett. 127, 198102 (2021).
  53. W. Wang, I. Tanasijevic, J. Zhang, E. Lauga, and I. Cohen, Electronically actuated artificial hinged cilia for efficient bidirectional pumping, Lab Chip 24, 4549 (2024).
  54. M. Chioccioli, L. Feriani, Q. Nguyen, J. Kotar, S. D. Dell, V. Mennella, I. Amirav, and P. Cicuta, A how-to guide to: Quantitative high-speed video profiling to discriminate between variants of primary ciliary dyskinesia, bioRxiv, https://doi.org/10.1101/614966.
  55. C. A. Schneider, W. S. Rasband, and K. W. Eliceiri, NIH Image to ImageJ: 25 years of image analysis, Nat. Methods 9, 671 (2012).
  56. J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid, J.-Y. Tinevez, D. J. White, V. Hartenstein, K. Eliceiri, P. Tomancak, and A. Cardona, Fiji: an open-source platform for biological-image analysis, Nat. Methods 9, 676 (2012).
  57. M. Mewe, D. Tielker, R. Schönberg, M. Schachner, K.-E. Jaeger, and U. Schumacher, Pseudomonas aeruginosa lectins I and II and their interaction with human airway cilia, J. Laryngol. Otol. 119, 595 (2005).
  58. J. Galli, L. Calò, F. Ardito, M. Imperiali, E. Bassotti, G. C. Passali, G. La Torre, G. Paludetti, and G. Fadda, Damage to ciliated epithelium in chronic rhinosinusitis: What is the role of bacterial biofilms? Ann. Otol. Rhinol. Laryngol 117, 902 (2008).
  59. G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proc. R. Soc. A. 102, 161 (1922).
  60. T. D. Montenegro-Johnson and E. Lauga, Optimal swimming of a sheet, Phys. Rev. E 89, 060701(R) (2014).

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Phys. Rev. Research 7, 043209– Published 24 November, 2025
  • Received 19 December 2024
  • Accepted 19 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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