Skip to main content

Advertisement

SpringerLink
Log in

Discrete Cilia Modelling with Singularity Distributions: Application to the Embryonic Node and the Airway Surface Liquid

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

We discuss in detail techniques for modelling flows due to finite and infinite arrays of beating cilia. An efficient technique, based on concepts from previous ‘singularity models’ is described, that is accurate in both near and far-fields. Cilia are modelled as curved slender ellipsoidal bodies by distributing Stokeslet and potential source dipole singularities along their centrelines, leading to an integral equation that can be solved using a simple and efficient discretisation. The computed velocity on the cilium surface is found to compare favourably with the boundary condition. We then present results for two topics of current interest in biology. 1) We present the first theoretical results showing the mechanism by which rotating embryonic nodal cilia produce a leftward flow by a ‘posterior tilt,’ and track particle motion in an array of three simulated nodal cilia. We find that, contrary to recent suggestions, there is no continuous layer of negative fluid transport close to the ciliated boundary. The mean leftward particle transport is found to be just over 1 μm/s, within experimentally measured ranges. We also discuss the accuracy of models that represent the action of cilia by steady rotlet arrays, in particular, confirming the importance of image systems in the boundary in establishing the far-field fluid transport. Future modelling may lead to understanding of the mechanisms by which morphogen gradients or mechanosensing cilia convert a directional flow to asymmetric gene expression. 2) We develop a more complex and detailed model of flow patterns in the periciliary layer of the airway surface liquid. Our results confirm that shear flow of the mucous layer drives a significant volume of periciliary liquid in the direction of mucus transport even during the recovery stroke of the cilia. Finally, we discuss the advantages and disadvantages of the singularity technique and outline future theoretical and experimental developments required to apply this technique to various other biological problems, particularly in the reproductive system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bartha, E., Liron, N., 1988a. Slender body interactions for low Reynolds numbers. Part 1. Body–wall interactions. SIAM J. Appl. Math. 48, 992–1008.

    Article  Google Scholar 

  • Bartha, E., Liron, N., 1988b. Slender body interactions for low Reynolds numbers. Part 2. Body–body interactions. SIAM J. Appl. Math. 48, 1262–1280.

    Article  Google Scholar 

  • Batchelor, G.K., 1970. Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.

    Article  MATH  MathSciNet  Google Scholar 

  • Blake, J.R., 1971. A note on the image system for a Stokeslet in a no slip boundary. Proc. Camb. Phil. Soc. 70, 303–310.

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Blake, J.R., 1973. Mucus flows. Math. Biosci. 17, 301–313.

    Article  Google Scholar 

  • Blake, J.R., Chwang, A.T., 1974. Fundamental singularities of viscous flow. Part 1. Image systems in the vicinity of a stationary no-slip boundary. J. Eng. Math. 8, 23–29.

    Article  MATH  Google Scholar 

  • Blake, J.R., Sleigh, M.A., 1974. Mechanics of ciliary locomotion. Biol. Rev. Camb. Philos. Soc. 49(1), 85–125.

    Google Scholar 

  • Boucher, R.C., 1994. Human airway ion transport. Part 1. Am. J. Respir. Crit. Care Med. 150, 271–281.

    Google Scholar 

  • Brokaw, C.J., 2005. Computer simulation of flagellar movement IX. Oscillation and symmetry breaking in a model for short flagella and nodal cilia. Cell Motil. Cytoskeleton 60, 35–47.

    Article  Google Scholar 

  • Brown, N.A., Wolpert, L., 1990. The development of handedness in left/right asymmetry. Development 109, 1–9.

    Google Scholar 

  • Buceta, J., Ibañes, M., Rasskin-Gutman, D., Okada, Y., Nobutaka, N., Izpisúa-Belmonte, J.C., 2005. Nodal cilia dynamics and the specification of the left/right axis in early vertebrate embryo development. Biophys. J. 89, 2199–2209.

    Google Scholar 

  • Burgers, J.M., 1938. On the motion of small particles of elongated form suspended in a viscous liquid. Kon. Ned. Akad. Wet. Verhand. (Eerste Sectie) 16, 113.

    Google Scholar 

  • Cartwright, J.H.E., Piro, O., Tuval, I., 2004. Fluid-dynamical basis of the embryonic development of left–right asymmetry in vertebrates. Proc. Natl. Acad. Sci. U.S.A. 101(19), 7234–7239.

    Article  Google Scholar 

  • Childress, S., 1981. Mechanics of Swimming and Flying. Cambridge University Press, New York.

    MATH  Google Scholar 

  • Chwang, A.T., Wu, T.Y., 1975. Hydrodynamics of the low-Reynolds number flows. Part 2. The singularity method for Stokes flows. J. Fluid Mech. 67, 787–815.

    Article  MATH  MathSciNet  Google Scholar 

  • Clarke, R.J., Jensen, O.E., Billingham, J., Williams, P.M., 2006. Three-dimensional flow due to a microcantilever oscillating near a wall: An unsteady slender-body analysis. Proc. R. Soc. A 462, 913–933.

    Article  MathSciNet  MATH  Google Scholar 

  • Cox, R.G., 1970. The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44, 791–810.

    Article  MATH  Google Scholar 

  • Eisenbach, M., Giojalas, L.C., 2006. Sperm guidance in mammals—an unpaved road to the egg. Nat. Rev. Mol. Cell Biol. 7, 276–285.

    Article  Google Scholar 

  • Fauci, L., Dillon, R., 2006. Biofluidmechanics of reproduction. Ann. Rev. Fluid Mech. 38(1), 371–394.

    Article  MathSciNet  Google Scholar 

  • Fulford, G.R., Blake, J.R., 1986a. Force distribution along a slender body straddling an interface. J. Aust. Math. Soc., Ser. B, Appl. Math. 27(3), 295–315.

    MATH  MathSciNet  Google Scholar 

  • Fulford, G.R., Blake, J.R., 1986b. Muco-ciliary transport in the lung. J. Theor. Biol. 121, 381–402.

    Article  Google Scholar 

  • Fulford, G.R., Katz, D.F., Powell, R.L., 1998. Swimming of spermatozoa in a linear viscoelastic fluid. Biorheology 35(4–5), 295–309.

    Article  Google Scholar 

  • Gueron, S., Levit-Gurevich, K., 2001. A three-dimensional model for ciliary motion based on the internal 9+2 structure. Proc. R. Soc. Lond. B 268, 599–607.

    Article  Google Scholar 

  • Gueron, S., Liron, N., 1992. Ciliary motion modeling, and dynamic multicilia interactions. Biophys. J. 63, 1045–1058.

    Article  Google Scholar 

  • Gueron, S., Liron, N., 1993. Simulations of three-dimensional ciliary beats and cilia interactions. Biophys. J. 65, 499–507.

    Google Scholar 

  • Hagiwara, H., Harada, S., Maeda, S., Aoki, T., Ohwada, N., Takata, K., 2002. Ultrastructural and immunohistochemical study of the basal apparatus of solitary cilia in the human oviduct epithelium. J. Anat. 200, 89.

    Article  Google Scholar 

  • Hancock, G.J., 1953. The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. B. 217, 96–121.

    Article  MATH  MathSciNet  Google Scholar 

  • Higdon, J.J.L., 1979. A hydrodynamic analysis of flagellar propulsion. J. Fluid Mech. 90, 685– 711.

    Article  MATH  MathSciNet  Google Scholar 

  • Johnson, R.E., 1980. An improved slender-body theory for Stokes flow. J. Fluid Mech. 99(2), 411–431.

    Article  MATH  MathSciNet  Google Scholar 

  • Kosaki, K., Casey, B., 1998. Genetics of human left–right axis malformations. Semin. Cell Dev. Biol. 9(1), 89–99.

    Article  Google Scholar 

  • Lauga, E., Brenner, M.P., Stone, H.A., 2005. Microfluidics: The no-slip boundary condition. In: Foss, J., Tropea, C., Yarin, A. (Eds.), Handbook of Experimental Fluid Dynamics. Springer, New York.

    Google Scholar 

  • Lighthill, M.J., 1976. Flagellar hydrodynamics. The Jon von Neumann lecture. SIAM Rev. 18(2), 161–230.

    Article  MATH  MathSciNet  Google Scholar 

  • Liron, N., 1978. Fluid transport by cilia between parallel plates. J. Fluid Mech. 86(4), 705–726.

    Article  MATH  MathSciNet  Google Scholar 

  • Liron, N., 1996. Stokes flow due to infinite arrays of Stokeslets in three dimensions. J. Eng. Math. 30(1–2), 267–297.

    Article  MATH  MathSciNet  Google Scholar 

  • Liron, N., 2001. The LGL (Lighthill–Gueron–Liron) Theorem–historical perspective and critique. Math. Methods Appl. Sci. 24, 1533–1540.

    Article  MATH  MathSciNet  Google Scholar 

  • Liron, N., Mochon, S., 1976a. The discrete-cilia approach to propulsion of ciliated micro-organisms. J. Fluid Mech. 75, 593–607.

    Article  MATH  MathSciNet  Google Scholar 

  • Liron, N., Mochon, S., 1976b. Stokes flow for a stokeslet between two parallel flat plates. J. Eng. Math. 10, 287–303.

    Article  MATH  Google Scholar 

  • Matsui, H., Davis, C.W., Tarran, R., Boucher, R.C., 2000. Osmotic water permeabilities of cultured, well-differentiated normal & cystic fibrosis airway epithelia. J. Clin. Invest. 105(10), 1419–1427.

    Google Scholar 

  • Matsui, H., Randell, S.H., Peretti, S.W., Davis, C.W., Boucher, R.C., 1998. Coordinated clearance of periciliary liquid and mucus from airway surfaces. J. Clin. Invest. 102(6), 1125–1131.

    Article  Google Scholar 

  • McGrath, J., Somlo, S., Makova, S., Tian, X., Brueckner, M., 2003. Two populations of node monocilia initiate left–right asymmetry in the mouse. Cell 114, 61–73.

    Article  Google Scholar 

  • Meyer, F.A., Silberberg, A., 1980. The rheology and molecular organization of epithelial mucus. Biorheology 17, 163–168.

    Google Scholar 

  • Nonaka, S., Shiratori, H., Saijoh, Y., Hamada, H., 2002. Determination of left–right patterning of the mouse embryo by artificial nodal flow. Nature 418, 96–98.

    Article  Google Scholar 

  • Nonaka, S., Tanaka, Y., Okada, Y., Takeda, S., Harada, A., Kanai, Y., Kido, M., Hirokawa, N., 1998. Randomization of left–right asymmetry due to loss of nodal cilia generating leftward flow of extraembryonic fluid in mice lacking KIF3B motor protein. Cell 95, 829–837.

    Article  Google Scholar 

  • Nonaka, S., Yoshiba, S., Watanabe, D., Ikeuchi, S., Goto, T., Marshall, W.F., Hamada, H., 2005. De novo formation of left–right asymmetry by posterior tilt of nodal cilia. PLoS Biol. 3(8), e268.

    Article  Google Scholar 

  • Okada, Y., Takeda, S., Tanaka, Y., Izpisúa Belmonte, J.-C., Hirokawa, N., 2005. Mechanism of nodal flow: A conserved symmetry breaking event in left–right axis determination. Cell 121, 633–644.

    Article  Google Scholar 

  • Orme, B.A.A., Otto, S.R., Blake, J.R., 2001. Enhanced efficiency of feeding and mixing due to chaotic flow patterns around choanoflagellates. Math. Med. Biol. 18(3), 293–325.

    Article  MATH  Google Scholar 

  • Phan-Thien, N., Tran-Cong, T., Ramia, M., 1987. A boundary-element analysis of flagellar propulsion. J. Fluid Mech. 185, 533–549.

    Article  Google Scholar 

  • Salathe, M., O'Riordan, T.G., Wanner, A., 1997. Mucociliary clearance. In: Crystal, R.G., West, P.J., Barnes, P.J., Weibel, E.R. (Eds.), The Lung: Scientific Foundations. Lippincott–Raven, Philadelphia, pp. 2295–2308.

    Google Scholar 

  • Sanderson, M.J., Sleigh, M.A., 1981. Ciliary activity of cultured rabbit tracheal epithelium: Beat pattern and metachrony. J. Cell Sci. 47, 331–341.

    Google Scholar 

  • Smith, D.J., Gaffney, E.A., Blake, J.R., 2006. A viscoelastic traction layer model of muco-ciliary transport. Bull. Math. Biol. 69(1), 289–327.

    Google Scholar 

  • Smith, D.J., Gaffney, E.A., Blake, J.R., 2007. A model of tracer transport in airway surface liquid. Bull. Math. Biol. 69(3), 817–836.

    Google Scholar 

  • Staben, M.E., Zinchenko, A.Z., Davis, R.H., 2003. Motion of a particle between two plane parallel walls in low-Reynolds-number Poiseuille flow. Phys. Fluids 15(6), 1711–1733.

    Article  Google Scholar 

  • Tabin, C.J., Vogan, K.J., 2003. A two-cilia model for vertebrate left–right axis specification. Genes Dev. 17, 1–6.

    Article  Google Scholar 

  • Talbot, P., Geiske, C., Knoll, M., 1999. Oocyte pickup by the mammalian oviduct. Mol. Biol. Cell 10(1), 5–8.

    Google Scholar 

  • Tang, X.Z., Boozer, A.H., 1996. Finite time Lyapunov exponent and advection–diffusion equation. Physica D 95, 283–305.

    Article  MATH  MathSciNet  Google Scholar 

  • Tarran, R., Button, B., Boucher, R.C., 2006. Regulation of normal and cystic fibrosis airway surface liquid volume by phasic shear stress. Annu. Rev. Physiol. 68, 543–561.

    Article  Google Scholar 

  • Tillett, J.P.K., 1970. Axial and transverse stokes flow past slender axisymmetric bodies. J. Fluid Mech. 44, 401–417.

    Article  MATH  MathSciNet  Google Scholar 

  • Tornberg, A.-K., Shelley, M.J., 2004. Simulating the dynamics and interactions of flexible fibers in Stokes flows. J. Comput. Phys. 196, 8–40.

    Article  MATH  MathSciNet  Google Scholar 

  • Tuck, E.O., 1964. Some methods for flows past slender bodies. J. Fluid Mech. 18, 619.

    Article  MathSciNet  Google Scholar 

  • Williams, W.E., 1966. Boundary effects in Stokes flow. J. Fluid Mech. 24(2), 285–291.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. E. A. Gaffney

    Present address: Centre for Mathematical Biology, Mathematical Institute, 24–29 St. Giles, Oxford, OX1 3LB, UK

Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

    D. J. Smith, E. A. Gaffney & J. R. Blake

  2. Reproductive Biology and Genetics, School of Medicine, University of Birmingham, Birmingham, B15 2TT, UK

    D. J. Smith

  3. Assisted Conception Unit, Birmingham Women’s Hospital Trust, Birmingham, B15 2TG, UK

    D. J. Smith

Authors
  1. D. J. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. A. Gaffney
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. R. Blake
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. J. Smith.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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). https://doi.org/10.1007/s11538-006-9172-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-006-9172-y

Keywords

Search

Navigation