Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T20:25:03.426Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetry breaking cilia-driven flow in the zebrafish embryo

Published online by Cambridge University Press:  13 April 2012

Andrew A. Smith ,
Thomas D. Johnson ,
David J. Smith  and
John R. Blake
Andrew A. Smith
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Centre for Human Reproductive Science, Birmingham Women’s NHS Foundation Trust, Edgbaston, Birmingham B15 2TG, UK
Thomas D. Johnson
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Centre for Human Reproductive Science, Birmingham Women’s NHS Foundation Trust, Edgbaston, Birmingham B15 2TG, UK
David J. Smith*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Centre for Human Reproductive Science, Birmingham Women’s NHS Foundation Trust, Edgbaston, Birmingham B15 2TG, UK School of Engineering & Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, UK
John R. Blake
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Centre for Human Reproductive Science, Birmingham Women’s NHS Foundation Trust, Edgbaston, Birmingham B15 2TG, UK
*
†Email address for correspondence: D.J.Smith.2@bham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Fluid mechanics plays a vital role in early vertebrate embryo development, an example being the establishment of left–right asymmetry. Following the dorsal–ventral and anterior–posterior axes, the left–right axis is the last to be established; in several species it has been shown that an important process involved with this is the production of a left–right asymmetric flow driven by ‘whirling’ cilia. It has previously been established in experimental and mathematical models of the mouse ventral node that the combination of a consistent rotational direction and posterior tilt creates left–right asymmetric flow. The zebrafish organizing structure, Kupffer’s vesicle, has a more complex internal arrangement of cilia than the mouse ventral node; experimental studies show that the flow exhibits an anticlockwise rotational motion when viewing the embryo from the dorsal roof, looking in the ventral direction. Reports of the arrangement and configuration of cilia suggest two possible mechanisms for the generation of this flow from existing axis information: (a) posterior tilt combined with increased cilia density on the dorsal roof; and (b) dorsal tilt of ‘equatorial’ cilia. We develop a mathematical model of symmetry breaking cilia-driven flow in Kupffer’s vesicle using the regularized Stokeslet boundary element method. Computations of the flow produced by tilted whirling cilia in an enclosed domain suggest that a possible mechanism capable of producing the flow field with qualitative and quantitative features closest to those observed experimentally is a combination of posteriorly tilted roof and floor cilia, and dorsally tilted equatorial cilia.

JFM classification

Biological Fluid Dynamics: Biomedical flows Low-Reynolds-number flows: Boundary integral methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 705: Special issue dedicated to Professor T. J. Pedley on his 70th birthday , 25 August 2012 , pp. 26 - 45
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
2. Afzelius, B. A. 1976 A human syndrome caused by immotile cilia. Science 193 (4250), 317–319.CrossRefGoogle ScholarPubMed
3. Ainley, J., Durkin, S., Embid, R., Boindala, P. & Cortez, R. 2008 The method of images for regularized Stokeslets. J. Comput. Phys. 227 (9), 4600–4616.Google Scholar
4. Berdon, W. E., McManus, C. & Afzelius, B. 2004 More on Kartageners syndrome and the contributions of Afzelius and A.K. Siewert. Pediatr. Radiol. 34 (7), 585–586.Google Scholar
5. Berdon, W. E. & Willi, U. 2004 Situs inversus, bronchiectasis, and sinusitis and its relation to immotile cilia: history of the diseases and their discoverers – Manes Kartagener and Bjorn Afzelius. Pediatr. Radiol. 34 (1), 38–42.CrossRefGoogle ScholarPubMed
6. Blake, J. R. 1971 A note on the image system for a stokeslet in a no-slip boundary. Math. Proc. Camb. Phil. Soc. 70, 303–310.Google Scholar
7. Blake, J. R. & Chwang, A. T. 1974 Fundamental singularities of viscous flow. J. Engng Maths 8 (1), 23–29.Google Scholar
8. 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. Cytoskel. 60 (1), 35–47.Google Scholar
9. Cartwright, J. H. E., Piro, N., Piro, O. & Tuval, I. 2007 Embryonic nodal flow and the dynamics of nodal vesicular parcels. J. R. Soc. Interface 4 (12), 49–55.CrossRefGoogle ScholarPubMed
10. Cartwright, J. H. E., Piro, N., Piro, O. & Tuval, I. 2008 Fluid dynamics of nodal flow and left–right patterning in development. Dev. Dyn. 237 (12), 3477–3490.CrossRefGoogle ScholarPubMed
11. 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. USA 101 (19), 7234–7239.Google Scholar
12. Cartwright, J. H. E., Piro, O. & Tuval, I. 2009 Fluid dynamics in developmental biology: moving fluids that shape ontogeny. HFSP J. 3 (2), 77–93.Google Scholar
13. Cortez, R. 2001 The method of regularized Stokeslets. SIAM J. Sci. Comput. 23 (4), 1204–1225.Google Scholar
14. Cortez, R., Fauci, L. & Medovikov, A. 2005 The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming. Phys. Fluids 17 (031504), 1–14.CrossRefGoogle Scholar
15. Drescher, K., Leptos, K. C., Tuval, I., Ishikawa, T., Pedley, T. J. & Goldstein, R. E. 2009 Dancing Volvox: hydrodynamic bound states of swimming algae. Phys. Rev. Lett. 102 (16), 168101.Google Scholar
16. Hashimoto, M., Shinohara, K., Wang, J., Ikeuchi, S., Yoshiba, S., Meno, C., Nonaka, S., Takada, S., Hatta, K., Wynshaw-Boris, A. & Hamada, H. 2010 Planar polarization of node cells determines the rotational axis of node cilia. Nat. Cell Biol. 12 (2), 170–176.Google Scholar
17. Hirokawa, N., Okada, Y. & Tanaka, Y. 2009 Fluid dynamic mechanism responsible for breaking the left–right symmetry of the human body: the nodal flow. Annu. Rev. Fluid Mech. 41, 53–72.Google Scholar
18. Ibañes, M. & Belmonte, J. C. I. 2009 Left-right axis determination. WIREs: Syst. Biol. Med. 1 (2), 210–219.Google ScholarPubMed
19. Ishikawa, T. & Pedley, T. J. 2007 Diffusion of swimming model micro-organisms in a semi-dilute suspension. J. Fluid Mech. 588, 437–462.Google Scholar
20. Kartagener, M. 1933 Zur Pathogenese der Bronchiektasien. Lung 84 (1), 73–85.Google Scholar
21. Kawakami, Y., Raya, Á, Raya, R. M., Rodríguez-Esteban, C. & Belmonte, J. C. I. 2005 Retinoic acid signalling links left–right asymmetric patterning and bilaterally symmetric somitogenesis in the zebrafish embryo. Nature 435 (7039), 165–171.Google Scholar
22. Kimmel, C. B., Ballard, W. W., Kimmel, S. R., Ullmann, B. & Schilling, T. F. 1995 Stages of embryonic development of the zebrafish. Am. J. Anat. 203 (3), 253–310.Google ScholarPubMed
23. Kramer-Zucker, A. G., Olale, F., Haycraft, C. J., Yoder, B. K., Schier, A. F. & Drummond, I. A. 2005 Cilia-driven fluid flow in the zebrafish pronephros, brain and Kupffer’s vesicle is required for normal organogenesis. Development 132 (8), 1907–1921.CrossRefGoogle ScholarPubMed
24. Kreiling, J. A., Prabhat, W. G. & Creton, R. 2007 Analysis of Kupffer’s vesicle in zebrafish embryos using a cave automated virtual environment. Dev. Dyn. 236 (7), 1963–1969.CrossRefGoogle ScholarPubMed
25. Nonaka, S., Shiratori, H., Saijoh, Y. & Hamada, H. 2002 Determination of left–right patterning of the mouse embryo by artificial nodal flow. Nature 418 (6893), 96–99.Google Scholar
26. 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 (6), 829–837.Google Scholar
27. 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), 1467–1472.CrossRefGoogle ScholarPubMed
28. Okabe, A., Boots, B. N., Sugihara, K. & Chiu, S. 1992 Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. J. Wiley.Google Scholar
29. Okabe, N., Xu, B. & Burdine, R. D. 2008 Fluid dynamics in zebrafish Kupffer’s vesicle. Dev. Dyn. 237 (12), 3602–3612.CrossRefGoogle ScholarPubMed
30. Okada, Y., Takeda, S., Tanaka, Y., Belmonte, J. C. I. & Hirokawa, N. 2005 Mechanism of nodal flow: a conserved symmetry breaking event in left–right axis determination. Cell 121 (4), 633–644.Google Scholar
31. Otto, S. R., Yannacopoulos, A. N. & Blake, J. R. 2001 Transport and mixing in Stokes flow: the effect of chaotic dynamics on the blinking stokeslet. J. Fluid Mech. 430, 1–26.CrossRefGoogle Scholar
32. Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24 (1), 313–358.CrossRefGoogle Scholar
33. Persson, P. O. & Strang, G. 2004 A simple mesh generator in MATLAB. SIAM Rev. 46 (2), 329–345.CrossRefGoogle Scholar
34. Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
35. Rawls, J. F., Mellgren, E. M. & Johnson, S. L. 2001 How the zebrafish gets its stripes. Dev. Biol. 240 (2), 301–314.CrossRefGoogle ScholarPubMed
36. Smith, D. J. 2009 A boundary element regularized Stokeslet method applied to cilia- and flagella-driven flow. Proc. R. Soc. Lond. A 465, 3605–3626.Google Scholar
37. Smith, D. J., Blake, J. R. & Gaffney, E. A. 2008 Fluid mechanics of nodal flow due to embryonic primary cilia. J. R. Soc. Interface 5 (22), 567–573.CrossRefGoogle ScholarPubMed
38. Smith, D. J., Gaffney, E. A. & Blake, J. R. 2007 Discrete cilia modelling with singularity distributions: application to the embryonic node and the airway surface liquid. Bull. Math. Biol. 69 (5), 1477–1510.CrossRefGoogle Scholar
39. Smith, D. J., Smith, A. A. & Blake, J. R. 2011 Mathematical embryology: the fluid mechanics of nodal cilia. J. Engng Maths 70, 255–279.Google Scholar
40. Sulik, K., Dehart, D. B., Inagaki, T., Carson, J. L., Vrablic, T., Gesteland, K. & Schoenwolf, G. C. 1994 Morphogenesis of the murine node and notochordal plate. Am. J. Anat. 201 (3), 260–278.Google Scholar
41. Supatto, W., Fraser, S. E. & Vermot, J. 2008 An all-optical approach for probing microscopic flows in living embryos. Biophys. J. 95 (4), 29–31.Google Scholar
42. Supatto, W. & Vermot, J. 2011 From cilia hydrodynamics to zebrafish embryonic development. Curr. Topics Dev. Biol. 95, 33.Google Scholar
43. Tanaka, Y., Okada, Y. & Hirokawa, N. 2005 FGF-induced vesicular release of Sonic hedgehog and retinoic acid in leftward nodal flow is critical for left–right determination. Nature 435 (7039), 172–177.CrossRefGoogle ScholarPubMed
44. Taylor, M. A., Wingate, B. A & Vincent, R. E. 2001 An algorithm for computing Fekete points in the triangle. SIAM J. Numer. Anal. 38 (5), 1707–1720.CrossRefGoogle Scholar