Skip to main content

The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Bulletin of Mathematical Biology volume 73, pages 2527–2551 (2011)Cite this article

Abstract

Since its conception in 1952, the Turing paradigm for pattern formation has been the subject of numerous theoretical investigations. Experimentally, this mechanism was first demonstrated in chemical reactions over 20 years ago and, more recently, several examples of biological self-organisation have also been implicated as Turing systems. One criticism of the Turing model is its lack of robustness, not only with respect to fluctuations in the initial conditions, but also with respect to the inclusion of delays in critical feedback processes such as gene expression. In this work we investigate the possibilities for Turing patterns on growing domains where the morphogens additionally regulate domain growth, incorporating delays in the feedback between signalling and domain growth, as well as gene expression. We present results for the proto-typical Schnakenberg and Gierer–Meinhardt systems: exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important for pattern formation.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

43,34 €

Price includes VAT (Finland)
Tax calculation will be finalised during checkout.

References

  • Abdreeff, M., Goodrich, D. W., & Pardee, A. B. (2000). Cell proliferation, differentiation and apoptosis. In The Holland-frei cancer medicine (5th edn.). New York: Decker.

    Google Scholar 

  • Affolter, M., & Basler, K. (2007). The Decapentaplegic morphogen gradient: from pattern formation to growth regulation. Nat. Rev. Genet., 8, 663–674.

    Article  Google Scholar 

  • Al-Dewachi, H. S., Appleton, D. R., Watson, A. J., & Wright, N. A. (1977). Variation in the cell cycle time in the crypts of Lieberkiihn of the mouse. Virchows Arch., B Pathol., 31, 37–44.

    Article  Google Scholar 

  • Alberts, B., Johnson, A., Walter, P., Lewis, J., Raff, M., & Roberts, K. (2002). Molecular biology of the cell (5th edn.). New York: Garland Science.

    Google Scholar 

  • Baker, R. E., & Maini, P. K. (2007). A mechanism for morphogen-controlled domain growth. J. Math. Biol., 54, 597–622.

    MathSciNet  MATH  Article  Google Scholar 

  • Capdevila, J., & Belmonte, J. C. I. (2001). Patterning mechanisms controlling vertebrate limb development. Annu. Rev. Cell Dev. Biol., 17, 87–132.

    Article  Google Scholar 

  • Chiorino, G., Metz, J. A. J., Tomasoni, D., & Ubezio, P. (2001). Desynchronization rate in cell populations: mathematical modeling and experimental data. J. Theor. Biol., 208, 185–199.

    Article  Google Scholar 

  • Chisholm, J. C. (1998). Analysis of the fifth cell cycle of mouse development. J. Reprod. Fertil., 84, 29–36.

    Google Scholar 

  • Crampin, E. J., Gaffney, E. A., & Maini, P. K. (1999). Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol., 61, 1093–1120.

    Article  Google Scholar 

  • Crampin, E. J., Hackborn, W. W., & Maini, P. K. (2002). Pattern formation in reaction–diffusion models with nonuniform domain growth. Bull. Math. Biol., 64, 747–769.

    Article  Google Scholar 

  • Dillon, R., & Othmer, H. G. (1999). A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud. J. Theor. Biol., 197, 295–330.

    Article  Google Scholar 

  • Entchev, E. V., Schwabedissen, A., & Gonzalez-Gaitan, M. (2000). Gradient formation of the TGF-β homolog Dpp. Cell, 103, 981–991.

    Article  Google Scholar 

  • Fischer, J. A., Eun, S. H., & Doolan, B. T. (2006). Endocytosis, endosome trafficking, and the regulation of Drosophila development. Annu. Rev. Cell Dev. Biol., 22, 181–206.

    Article  Google Scholar 

  • Gaffney, E. A., & Monk, N. A. M. (2006). Gene expression time delays and Turing pattern formation systems. Bull. Math. Biol., 68, 99–130.

    MathSciNet  Article  Google Scholar 

  • Gierer, A., & Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik, 12, 30–39.

    Article  Google Scholar 

  • Gray, P., & Scott, S. K. (1983). Autocatalytic reactions in the isothermal, continuous stirred tank reactor. Chem. Eng. Sci., 38, 29–43.

    Article  Google Scholar 

  • Jones, C. M., Armes, N., & Smith, J. C. (1996). Signalling by TGF-β family members: short-range effects of Xnr-2 and BMP-4 contrast with the long-range effects of activin. Curr. Biol., 6, 1468–1475.

    Article  Google Scholar 

  • Kimmel, C. B., Ballard, W. W., Kimmel, S. R., Ullmann, B., & Schilling, T. F. (1995). Stages of embryonic development of the zebrafish. Dev. Dyn., 203, 253–310.

    Article  Google Scholar 

  • Kondo, S., & Asai, R. (1995). A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature, 376, 765–768.

    Article  Google Scholar 

  • Kondo, S., Iwashita, M., & Yamaguchi, M. (2009). How animals get their skin patterns: fish pigment pattern as a live Turing wave. Int. J. Dev. Biol., 53, 851–856.

    Article  Google Scholar 

  • Lewis, J. (2003). Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator. Curr. Biol., 13, 1398–1408.

    Article  Google Scholar 

  • Miura, T., & Shiota, K. (2000a). Extracellular matrix environment influences chondrogenic pattern formation in limb bud micromass culture: Experimental verification of theoretical models. Anat. Rec., 258, 100–107.

    Article  Google Scholar 

  • Miura, T., & Shiota, K. (2000b). TGFβ2 acts as an activator molecule in a reaction–diffusion model and is involved in cell sorting phenomenon in mouse limb micromass culture. Dev. Dyn., 217, 241–249.

    Article  Google Scholar 

  • Mou, C., Jackson, B., Schneider, P., Overbeek, P. A., & Headon, D. J. (2006). Generation of the primary hair follicle pattern. Proc. Natl. Acad. Sci. USA, 103, 9075–9080.

    Article  Google Scholar 

  • Murray, J. D. (1993). Mathematical biology (2nd edn.). Berlin: Springer.

    MATH  Book  Google Scholar 

  • Nakamasu, A., Takahashi, G., Kanbe, A., & Kondo, S. (2009). Interactions between zebrafish pigment cells responsible for the generation of Turing patterns. Proc. Natl. Acad. Sci. USA, 106, 8429–8434.

    Article  Google Scholar 

  • Neville, A., Matthews, P., & Byrne, H. M. (2006). Interactions between pattern formation and domain growth. Bull. Math. Biol., 68, 1975–2003.

    MathSciNet  Article  Google Scholar 

  • Pardee, A. B. (1974). A restriction point for control of normal animal cell proliferation. Proc. Natl. Acad. Sci. USA, 71, 1286–1290.

    Article  Google Scholar 

  • Piddini, E., & Vincent, J. (2003). Modulation of developmental signals by endocytosis: different means and many ends. Curr. Opin. Cell Biol., 15, 474–481.

    Article  Google Scholar 

  • Ramis, J. M., Collart, C., & Smith, J. C. (2007). Xnrs and Activin regulate distinct genes during Xenopus development: Activin regulates cell division. PLoS ONE, 2, e213.

    Article  Google Scholar 

  • Rogulja, D., & Irvine, K. D. (2005). Regulation of cell proliferation by a morphogen gradient. Cell, 123, 449–461.

    Article  Google Scholar 

  • Roy, C. L., & Wrana, J. L. (2005). Clathrin- and nonclathrin-mediated endocytic regulation of cell signaling. Nat. Rev. Mol. Cell Biol., 6, 112–126.

    Article  Google Scholar 

  • Sakuma, R., Ohnishi, Y., Meno, C., Fujii, H., Juan, H., Takeuchi, J., Ogura, T., Li, E., Miyazono, K., & Hamada, H. (2002). Inhibition of Nodal signalling by Lefty mediated through interaction with common receptors and efficient diffusion. Genes Cells, 7, 401–412.

    Article  Google Scholar 

  • Schnakenberg, J. (1979). Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol., 81, 389–400.

    MathSciNet  Article  Google Scholar 

  • Seirin-Lee, S., & Gaffney, E. A. (2010). Aberrant behaviours of reaction diffusion self-organisation models on growing domains in the presence of gene expression time delays. Bull. Math. Biol., 72, 2161–2179.

    MathSciNet  MATH  Article  Google Scholar 

  • Seirin-Lee, S., Gaffney, E. A., & Monk, N. A. M. (2010). The influence of gene expression time delays on Gierer-Meinhardt pattern formation systems. Bull. Math. Biol., 72, 2139–2160.

    MathSciNet  Article  Google Scholar 

  • Shraiman, B. I. (2005). Mechanical feedback as a possible regulator of tissue growth. Proc. Natl. Acad. Sci. USA, 102, 3318–3323.

    Article  Google Scholar 

  • Sick, S., Reinker, S., Timmer, J., & Schlake, T. (2006). WNT and DKK determine hair follicle spacing through a reaction–diffusion mechanism. Science, 314(5804), 1447–1450.

    Article  Google Scholar 

  • Solnica-Krezel, L. (2003). Vertebrate development: Taming the nodal waves. Curr. Biol., 13, R7–R9.

    Article  Google Scholar 

  • Sorkin, A., & von Zastrow, M. (2002). Signal transduction and endocytosis: close encounters of many kinds. Nat. Rev. Mol. Cell Biol., 3, 600–614.

    Article  Google Scholar 

  • Tabata, T., & Takei, Y. (2004). Morphogens, their identification and regulation. Development, 131, 703–712.

    Article  Google Scholar 

  • Tennyson, C. N., Klamut, H. J., & Worton, R. G. (1995). The human dystrophin gene requires 16 hr to be transcribed and is cotranscriptionally spliced. Nat. Genet., 9, 184–190.

    Article  Google Scholar 

  • Turing, A. M. (1952). The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, 237, 37–72.

    Article  Google Scholar 

  • Wolfe, S. L. (1995). Introduction to cell and molecular biology. Belmont: Wadsworth.

    Google Scholar 

  • Zetterberg, A., Larsson, O., & Wiman, K. G. (1995). What is the restriction point? Curr. Opin. Cell Biol., 7, 835–842.

    Article  Google Scholar 

Download references

Author information

Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 153-8914, Japan

    S. Seirin Lee

  2. Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    E. A. Gaffney & R. E. Baker

Authors
  1. S. Seirin Lee
    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. R. E. Baker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Seirin Lee.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Seirin Lee, S., Gaffney, E.A. & Baker, R.E. The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays. Bull Math Biol 73, 2527–2551 (2011). https://doi.org/10.1007/s11538-011-9634-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-011-9634-8

Keywords

  • Turing pattern formation
  • Growing domains
  • Time delays