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Turing Pattern Formation with Two Kinds of Cells and a Diffusive Chemical

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Abstract

We study a three-variable Turing system with two kinds of cells and a diffusive chemical, considering the formation and maintenance of fish skin patterns with multiple pigment cells. The two types of cells are produced from undifferentiated cells. They inhibit the supply rate of the other cell type, forming local clusters of the same cell type. In addition, the cells of one type can be maintained only in the presence of a diffusive chemical produced by the other cell type, resulting in the coexistence of two cell types in heterogeneous spatial patterns. We assume linear interaction among cells and the chemical, and cell supply rates constrained to be positive or zero. We derive the condition for diffusion-driven instability. In one-dimensional model, we examine how the wavelength of the periodic pattern depends on parameters. In the two-dimensional model, we study the condition for spot, stripe or reversed spot pattern to emerge (pattern selection). We discuss heuristic criteria for the pattern selection.

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References

  • Arcuri, P., Murray, J.D., 1986. Pattern sensitivity to boundary and initial conditions in reaction-diffusion models. J. Math. Biol. 24, 141–165.

    Article  MATH  MathSciNet  Google Scholar 

  • Asai, R., Taguchi, E., Kume, Y., Saito, M., Kondo, S., 1999. Zebrafish Leopard gene as a component of the putative reaction–diffusion system. Mech. Dev. 89, 87–92.

    Article  Google Scholar 

  • Barrio, R.A., Varea, C., Aragon, J.L., Maini, P.K., 1999. A two-dimensional numerical study of spatial pattern formation in interacting Turing systems. Bull. Math. Biol. 61, 483–505.

    Article  Google Scholar 

  • Cartwright, J.H.E., 2002. Labyrinthine Turing pattern formation in the cerebral cortex. J. Theor. Biol. 217, 97–103.

    Article  MathSciNet  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 

  • Dufiet, V., Boissonade, J., 1992. Numerical studies of Turing patterns selection in a two-dimensional system. Physica A 188, 158–171.

    Article  Google Scholar 

  • Ermentrout, B., 1991. Stripes or spots? Nonlinear effects in bifurcation of reaction–diffusion equations on the square. Proc. Roy. Soc. Lond. A 434, 413–417.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Hirata, M., Nakamura, K., Kanemaru, T., Shibata, Y., Kondo, S., 2003. Pigment cell organization in the hypodermis of zebrafish. Dev. Dyn. 227, 497–503.

    Article  Google Scholar 

  • Hirata, M., Nakamura, K., Kondo, S., 2005. Pigment cell distributions in different tissues of the zebrafish, with special reference to the striped pigment pattern. Dev. Dyn. 234, 293–300.

    Article  Google Scholar 

  • Hoshen, J., Kopelman, R., 1976. Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm. Phys. Rev. B 14, 3438–3445.

    Article  Google Scholar 

  • Jung, H.S., Francis-West, P.H., Widelitz, R.B., Jiang, T.X., Ting-Berreth, S., Tickle, C., Wolpert, L., Chuong, C.M., 1998. Local inhibitory action of BMPs and their relationships with activators in feather formation: implications for periodic patterning. Dev. Biol. 196, 11–23.

    Article  Google Scholar 

  • Koch, A.J., Meinhardt, H., 1994. Biological pattern formation: from basic mechanisms to complex structures. Rev. Mod. Phys. 66, 1481–1510.

    Article  Google Scholar 

  • Kondo, S., 2003. Chemical reaction generating biological patterns. In: Sekimura, T., Noji, S., Morita, R. (Eds.), Diversity in Pattern and Form of Biological System, pp. 67–74. Shokabo, Tokyo (in Japanese).

    Google Scholar 

  • Kondo, S., Asai, R., 1995. A reaction–diffusion wave on the marine angelfish Pomacanthus. Nature 376, 765–768.

    Article  Google Scholar 

  • Leppänen, T., Karttunen, M., Barrio, R.A., Kaski, K., 2004. Morphological transitions and bistability in Turing systems. Phys. Rev. E 70, 066202.

    Article  Google Scholar 

  • Lyons, M.J., Harrison, L.G., 1992. Stripe selection: an intrinsic property of some pattern-forming models with nonlinear dynamics. Dev. Dyn. 195, 201–215.

    Google Scholar 

  • Maderspacher, F., Nüsslein-Volhard, C., 2003. Formation of the adult pigment pattern in zebrafish requires leopard and obelix dependent cell interactions. Development 130, 3447–3457.

    Article  Google Scholar 

  • Meinhardt, H., 1982. Models of Biological Pattern Formation. Academic, London.

    Google Scholar 

  • Meinhardt, H., 1983. Digits, segments, somites—the superposition of sequential and periodic structures. In: Jäger, W., Murray, J.D. (Eds.), Lecture Notes in Biomathematics, vol. 55, pp. 228–245. Springer, Berlin.

    Google Scholar 

  • Meinhardt, H., 1995. The Algorithmic Beauty of Sea Shells. Springer, Berlin.

    Google Scholar 

  • Miura, T., Maini, P.K., 2004. Speed of pattern appearance in reaction–diffusion models: implications in the pattern formation of limb bud mesenchyme cells. Bull. Math. Biol. 66, 627–649.

    Article  MathSciNet  Google Scholar 

  • Miura, T., Shiota, K., Morriss-Kay, G., Maini, P.K., 2006. Mixed-mode pattern in Doublefoot mutant mouse limb—turing reaction–diffusion model on a growing domain during limb development. J. Theor. Biol. 240, 562–573.

    Article  MathSciNet  Google Scholar 

  • Moreira, J., Deutsch, A., 2005. Pigment pattern formation in zebrafish during late larval stages: a model based on local interactions. Dev. Dyn. 232, 33–42.

    Article  Google Scholar 

  • Murray, J.D., 1981. A pre-pattern formation mechanism for animal coat markings. J. Theor. Biol. 88, 161–199.

    Article  Google Scholar 

  • Murray, J.D., 1989. Mathematical Biology. Springer, New York.

    MATH  Google Scholar 

  • Okuno, T., Kume, H., Haga, T., Yoshizawa, T., 1971. Multivariate Analysis. JUSE Press, Tokyo (in Japanese).

    Google Scholar 

  • Painter, K.J., Maini, P.K., Othmer, H.G., 1999. Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc. Natl. Acad. Sci. USA 96, 5549–5554.

    Article  Google Scholar 

  • Parichy, D.M., 2006. Evolution of danio pigment pattern development. Heredity 97, 200–210.

    Article  Google Scholar 

  • Parichy, D.M., Turner, J.M., 2003. Temporal and cellular requirements for Fms signaling during zebrafish adult pigment pattern development. Development 130, 817–833.

    Article  Google Scholar 

  • Parichy, D.M., Ransom, D.G., Paw, B., Zon, L.I., Johnson, S.L., 2000. An orthologue of the kit-related gene fms is required for development of neural crest-derived xanthophores and a subpopulation of adult melanocytes in the zebrafish, Danio rerio. Development 127, 3031–3044.

    Google Scholar 

  • Quigley, A.K., Turner, J.M., Nuckles, R.J., Manuel, J.L., Budi, E.H., MacDonald, E.L., Parichy, D.M., 2004. Pigment pattern evolution by differential deployment of neural crest and post-embryonic melanophore lineages in Danio fishes. Development 131, 6053–6069.

    Article  Google Scholar 

  • Shoji, H., Iwasa, Y., 2005. Labyrinthine versus straight-striped patterns generated by two-dimensional turing systems. J. Theor. Biol. 237, 104–116.

    Article  MathSciNet  Google Scholar 

  • Shoji, H., Iwasa, Y., Mochizuki, A., Kondo, S., 2002. Directionality of stripes formed by anisotropic reaction-diffusion models. J. Theor. Biol. 214, 549–561.

    Article  MathSciNet  Google Scholar 

  • Shoji, H., Mochizuki, A., Iwasa, Y., Hirata, M., Watanabe, T., Hioki, S., Kondo, S., 2003a. Origin of directionality in the fish stripe pattern. Dev. Dyn. 226, 627–633.

    Article  Google Scholar 

  • Shoji, H., Iwasa, Y., Kondo, S., 2003b. Stripes, spots, or, reversed spots in two-dimensional Turing systems. J. Theor. Biol. 224, 339–350.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Yamaguchi, M., Yoshimoto, E., Kondo, S., 2007. Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism. Proc. Nati. Acad. Sci. USA 104(12), 4790–4793.

    Article  Google Scholar 

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  1. Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka, 812-8581, Japan

    Koichiro Uriu & Yoh Iwasa

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  1. Koichiro Uriu
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  2. Yoh Iwasa
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Correspondence to Koichiro Uriu.

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Uriu, K., Iwasa, Y. Turing Pattern Formation with Two Kinds of Cells and a Diffusive Chemical. Bull. Math. Biol. 69, 2515–2536 (2007). https://doi.org/10.1007/s11538-007-9230-0

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  • DOI: https://doi.org/10.1007/s11538-007-9230-0

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