Abstract
Understanding the mechanisms governing and regulating self-organisation in the developing embryo is a key challenge that has puzzled and fascinated scientists for decades. Since its conception in 1952 the Turing model has been a paradigm for pattern formation, motivating numerous theoretical and experimental studies, though its verification at the molecular level in biological systems has remained elusive. In this work, we consider the influence of receptor-mediated dynamics within the framework of Turing models, showing how non-diffusing species impact the conditions for the emergence of self-organisation. We illustrate our results within the framework of hair follicle pre-patterning, showing how receptor interaction structures can be constrained by the requirement for patterning, without the need for detailed knowledge of the network dynamics. Finally, in the light of our results, we discuss the ability of such systems to pattern outside the classical limits of the Turing model, and the inherent dangers involved in model reduction.
Similar content being viewed by others
References
Abreu, J. G., Ketpura, N. I., Reversade, B., & De Robertis, E. M. (2002). Connective-tissue growth factor (CTGF) modulates cell signalling by BMP and TGF-β. Nat. Cell Biol., 4(8), 599–604.
Alber, M., Tilmann, T. G., Hentschel, H. G. E., Kazmierczak, B., Zhang, Y., Zhu, J., & Newman, S. A. (2008). The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb. Bull. Math. Biol., 70(2), 460–483.
Alberts, B., Johnson, A., Walter, P., Lewis, J., Raff, M., & Roberts, K. (2002). Molecular biology of the cell (5th ed.). New York: Garland Science.
Baker, R. E., Gaffney, E. A., & Maini, P. K. (2008). Partial differential equations for self-organization in cellular and developmental biology. Nonlinearity, 21, R251–R290.
Bard, J., & Lauder, I. (1974). How well does Turing’s theory of morphogenesis work? J. Theor. Biol., 45, 501–531.
Bunow, B., Kernevez, J. P., Joly, G., & Thomas, D. (1980). Pattern formation by reaction-diffusion instabilities: applications to morphogenesis in Drosophila. J. Theor. Biol., 84, 629–649.
Carroll, S. B. (2005). Evolution at two levels: on genes and form. PLoS Biol., 3(7), e245.
Chen, Y., & Schier, A. F. (2002). Lefty proteins are long-range inhibitors of squint-mediated nodal signaling. Curr. Biol., 12, 2124–2128.
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.
Davidson, D. (1983). The mechanism of feather pattern development in the chick.1. the time of determination of feather position. J. Embryol. Exp. Morphol., 74, 245–259.
Ermentrout, B., & Lewis, M. (1997). Pattern formation in systems with one spatially distributed species. Bull. Math. Biol., 59, 533–549.
Gaffney, E. A., & Monk, N. A. M. (2006). Gene expression time delays and Turing pattern formation system. Bull. Math. Biol., 68, 99–130.
Gierer, A., & Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik, 12, 30–39.
Gilbert, S. F. (2006). Developmental biology (8th ed.). Sunderland: Sinauer.
Gregor, T., Bialek, W., de Ruyter van Steveninck, R., Tank, D., & Wieschaus, E. (2005). Kinetics of morphogen gradient formation. Proc. Natl. Acad. Sci. USA, 102, 18403–18407.
Harris, M. P., Williamson, S., Fallon, J. F., Meinhardt, H., & Prum, R. O. (2005). Molecular evidence for an activator–inhibitor mechanism in development of embryonic feather branching. Proc. Natl. Acad. Sci. USA, 102(33), 11734–11739.
Hentschel, H., Glimm, T., Glazier, J., & Newman, S. (2004). Dynamical mechanisms for skeletal pattern formation in the vertebrate limb. Proc. R. Soc. Lond. B, Biol. Sci., 271(1549), 1713–1722.
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(1), 11–23.
Kicheva, A., Pantazis, P., Bollenbach, T., Kalaidzidis, Y., Bittig, T., Jülicher, F., & González-Gaitán, M. (2007). Kinetics of morphogen gradient formation. Science, 315, 521–525.
Leuzinger, S., Hirth, F., Gerlich, D., Acampora, D., Simeone, A., Gehring, W., Finkelstein, R., Furukubo-Tokunaga, K., & Reichert, H. (1998). Equivalence of the fly orthodenticle gene and the human OTX genes in embryonic brain development of Drosophila. Development, 125(9), 1703–1710.
Levy, V., Lindon, C., Harfe, B. D., & Morgan, B. A. (2005). Distinct stem cell populations regenerate the follicle and interfollicular epidermis. Dev. Cell, 9(6), 855–861.
Lin, C.-M., Jiang, T. X., Baker, R. E., Maini, P. K., Widelitz, R. B., & Chuong, C.-M. (2009). Spots and stripes: pleomorphic patterning of stem cells via p-ERK-dependent cell chemotaxis shown by feather morphogenesis and mathematical simulation. Dev. Biol., 334(2), 369–382.
Lodish, H., Berk, A., Kaiser, C. A., Krieger, M., Scott, M. P., Bretscher, A., Ploegh, H., & Matsudaira, P. (2007). Molecular cell biology (6th ed.). New York: Freeman.
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.
Miura, T., & Shiota, K. (2000b). TGFβ2 acts as an activator molecule in reaction-diffusion model and is involved in cell sorting phenomenon in mouse limb micromass culture. Dev. Dyn., 217, 241–249.
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.
Moreo, P., Gaffney, E. A., Garcia-Aznar, J. M., & Doblare, M. (2010). On the modelling of biological patterns with mechanochemical models: insights from analysis and computation. Bull. Math. Biol., 72(2), 400–431.
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(24), 9075–9080.
Mou, C., Thomason, H. A., Willan, P. M., Clowes, C., Harris, W. E., Drew, C. F., Dixon, J., Dixon, M. J., & Headon, D. J. (2008). Enhanced ectodysplasin-a receptor (EDAR) signaling alters multiple fiber characteristics to produce the east Asian hair form. Human Mutat., 29(12), 1405–1411.
Mou, C., Pitel, F., Gourichon, D., Vignoles, F., Tzika, A., Tato, P., Yu, L., Burt, D. W., Bed’hom, B., Tixier-Boichard, M., Painter, K. J., & Headon, D. J. (2011). Cryptic patterning of avian skin confers a developmental facility for loss of neck feathering. PLoS Biol., 9(3), e1001028.
Murray, J. D. (2003). Mathematical biology II: spatial models and biochemical applications (Vol. II, 3rd ed.). Berlin: Springer.
Plikus, M., Wang, W. P., Liu, J., Wang, X., Jiang, T. X., & Chuong, C. M. (2008). Morpho-regulation of ectodermal organs: integument pathology and phenotypic variations in K14-Noggin engineered mice through modulation of bone morphogenic protein pathway. Am. J. Pathol., 164(3), 1099–1114.
Pummila, M., Fliniaux, I., Jaatinen, R., James, M. J., Laurikkala, J., Schneider, P., Thesleff, I., & Mikkola, M. L. (2007). Ectodysplasin has a dual role in ectodermal organogenesis: inhibition of BMP activity and induction of Shh expression. Development, 134(1), 117–125.
Qian, H., & Murray, J. D. (2001). A simple method of parameter space determination for diffusion-driven instability with three species. Appl. Math. Lett., 14, 405–411.
Rauch, E. M., & Millonas, M. M. (2004). The role of trans-membrane signal transduction in Turing-type cellular pattern formation. J. Theor. Biol., 226, 401–407.
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.
Satnoianu, R. A., Menzinger, M., & Maini, P. K. (2000). Turing instabilities in general systems. J. Math. Biol., 41, 493–512.
Schmidt-Ullrich, R., & Paus, R. (2005). Molecular principles of hair follicle induction and morphogenesis. BioEssays, 27(3), 247–261.
Segel, L. A., & Jackson, J. L. (1972). Dissipative structure—explanation and an ecological example. J. Theor. Biol., 37, 545–559.
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.
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, 2319–2360.
Seirin-Lee, S., Gaffney, E. A., & Baker, R. E. (2011). The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays. Bull. Math. Biol. doi:10.1007/s11538-011-9634-8.
Sengel, P. (1990). Pattern formation in skin development. Int. J. Dev. Biol., 34(1), 33–50.
Sharov, A. A., Sharova, T. Y., Mardaryev, A. N., Tommasi di Vignano, A., Atoyan, R., Weiner, L., Yang, S., Brissette, J. L., Dotto, G. P., & Botchkarev, V. A. (2006). Bone morphogenetic protein signaling regulates the size of hair follicles and modulates the expression of cell cycle-associated genes. Proc. Natl. Acad. Sci. USA, 103(48), 18166–18171.
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.
Solnica-Krezel, L. (2003). Vertebrate development: taming the nodal waves. Curr. Biol., 13, R7–R9.
Turing, A. (1952). The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, Biol. Sci., 237, 37–72.
White, K. A. J., & Gilligan, C. A. (1998). Spatial heterogeneity in three species, plant-parasite-hyperparasite, systems. Philos. Trans. R. Soc. B, 353, 543–557.
Wolpert, L. (1994). Positional information and pattern formation in development. Dev. Genet., 15(6), 485–490.
Wolpert, L. (2002). Principles of development (2nd ed.). London: Oxford University Press.
Zhang, Y., Tomann, P., Andl, T., Gallant, N. M., Huelsken, J., Jerchow, B., Birchmeier, W., Paus, R., Piccolo, S., Mikkola, M. L., Morrisey, E. E., Overbeek, P. A., Scheidereit, C., Millar, S. E., & Schmidt-Ullrich, R. (2009). Reciprocal requirements for EDA/EDAR/NF-kappaB and Wnt/beta-catenin signaling pathways in hair follicle induction. Dev. Cell, 17(1), 49–61.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Klika, V., Baker, R.E., Headon, D. et al. The Influence of Receptor-Mediated Interactions on Reaction-Diffusion Mechanisms of Cellular Self-organisation. Bull Math Biol 74, 935–957 (2012). https://doi.org/10.1007/s11538-011-9699-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-011-9699-4