A recently proposed mathematical model of a “core” set of cellular and molecular interactions present in the developing vertebrate limb was shown to exhibit pattern-forming instabilities and limb skeleton-like patterns under certain restrictive conditions, suggesting that it may authentically represent the underlying embryonic process (Hentschel et al., Proc. R. Soc. B 271, 1713–1722, 2004). The model, an eight-equation system of partial differential equations, incorporates the behavior of mesenchymal cells as “reactors,” both participating in the generation of morphogen patterns and changing their state and position in response to them. The full system, which has smooth solutions that exist globally in time, is nonetheless highly complex and difficult to handle analytically or numerically. According to a recent classification of developmental mechanisms (Salazar-Ciudad et al., Development 130, 2027–2037, 2003), the limb model of Hentschel et al. is “morphodynamic,” since differentiation of new cell types occurs simultaneously with cell rearrangement. This contrasts with “morphostatic” mechanisms, in which cell identity becomes established independently of cell rearrangement. Under the hypothesis that development of some vertebrate limbs employs the core mechanism in a morphostatic fashion, we derive in an analytically rigorous fashion a pair of equations representing the spatiotemporal evolution of the morphogen fields under the assumption that cell differentiation relaxes faster than the evolution of the overall cell density (i.e., the morphostatic limit of the full system). This simple reaction–diffusion system is unique in having been derived analytically from a substantially more complex system involving multiple morphogens, extracellular matrix deposition, haptotaxis, and cell translocation. We identify regions in the parameter space of the reduced system where Turing-type pattern formation is possible, which we refer to as its “Turing space.” Obtained values of the parameters are used in numerical simulations of the reduced system, using a new Galerkin finite element method, in tissue domains with nonstandard geometry. The reduced system exhibits patterns of spots and stripes like those seen in developing limbs, indicating its potential utility in hybrid continuum-discrete stochastic modeling of limb development. Lastly, we discuss the possible role in limb evolution of selection for increasingly morphostatic developmental mechanisms.
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Alber, M., Hentschel, H.G.E., Kazmierczak, B., Newman, S.A., 2005a. Existence of solutions to a new model of biological pattern formation. J. Math. Anal. Appl. 308, 175–194.
Alber, M., Hentschel, H.G.E., Glimm, T., Kazmierczak, B., Newman, S.A., 2005b. Stability of n-dimensional patterns in a generalized Turing system: implications for biological pattern formation. Nonlinearity 18, 125–138.
Alberch, P., Gale, E.A., 1983. Size dependence during the development of the amphibian foot. Colchicine-induced digital loss and reduction. J. Embryol. Exp. Morphol. 76, 177–197.
Brockes, J.P., Kumar, A., 2005. Appendage regeneration in adult vertebrates and implications for regenerative medicine. Science 310, 1919–1923.
Chaturvedi, R., Huang, C., Kazmierczak, B., Schneider, T., Izaguirre, J.A., Newman, S.A., Glazier, J.A., Alber, M., 2005. On multiscale approaches to 3-dimensional modeling of morphogenesis. J. R. Soc. Interface 2, 237–253.
Cheng, Y., Shu, C.-W., 2007. A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput., posted on September 6, 2007, PII: S 0025-5718(07)02045-5, to appear in print.
Cickovski, T., Huang, C., Chaturvedi, R., Glimm, T., Hentschel, H.G.E., Alber, M., Glazier, J.A., Newman, S.A., Izaguirre, J.A., 2005. A framework for three-dimensional simulation of morphogenesis. IEEE/ACM Trans. Comput. Biol. Bioinf. 2, 273–288.
Coates, M.I., Clack, J.A., 1990. Polydactyly in the earliest known tetrapod limbs. Nature 347, 66–69.
Cockburn, B., Shu, C.-W., 1989. TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435.
Cockburn, B., Shu, C.-W., 1991. The Runge–Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. Math. Model. Numer. Anal. 25, 337–361.
Cockburn, B., Shu, C.-W., 1998a. The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224.
Cockburn, B., Shu, C.-W., 1998b. The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463.
Cockburn, B., Lin, S.-Y., Shu, C.-W., 1989. TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90–113.
Cockburn, B., Hou, S., Shu, C.-W., 1990. The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581.
Cooke, J., Summerbell, D., 1981. Control of growth related to pattern specification in chick wing-bud mesenchyme. J. Embryol. Exp. Morphol. 65(Suppl.), 169–185.
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.
Cross, G.W., 1978. Three types of matrix instability. J. Linear Algebra Appl. 20, 253–263.
Daeschler, E.B., Shubin, N.H., Jenkins, F.A. Jr., 2006. A Devonian tetrapod-like fish and the evolution of the tetrapod body plan. Nature 440, 757–763.
De Joussineau, C., Soule, J., Martin, M., Anguille, C., Montcourrier, P., Alexandre, D., 2003. Delta-promoted filopodia mediate long-range lateral inhibition in Drosophila. Nature 426, 555–559.
Endo, T., Bryant, S.V., Gardiner, D.M., 2004. A stepwise model system for limb regeneration. Dev. Biol. 270, 135–145.
Entchev, E.V., Schwabedissen, A., Gonzalez-Gaitan, M., 2000. Gradient formation of the TGF-β homolog Dpp. Cell 103, 981–991.
Filion, R.J., Popel, A.S., 2004. A reaction–diffusion model of basic fibroblast growth factor interactions with cell surface receptors. Ann. Biomed. Eng. 32, 645–663.
Forgacs, G., Newman, S.A., 2005. Biological Physics of the Developing Embryo. Cambridge University Press, Cambridge.
Franssen, R.A., Marks, S., Wake, D., Shubin, N., 2005. Limb chondrogenesis of the seepage salamander, Desmognathus aeneus (Amphibia: Plethodontidae). J. Morphol. 265, 87–101.
Frenz, D.A., Jaikaria, N.S., Newman, S.A., 1989. The mechanism of precartilage mesenchymal condensation: a major role for interaction of the cell surface with the amino-terminal heparin-binding domain of fibronectin. Dev. Biol. 136, 97–103.
Fujimaki, R., Toyama, Y., Hozumi, N., Tezuka, K., 2006. Involvement of Notch signaling in initiation of prechondrogenic condensation and nodule formation in limb bud micromass cultures. J. Bone Miner. Metab. 24, 191–198.
Gehris, A.L., Stringa, E., Spina, J., Desmond, M.E., Tuan, R.S., Bennett, V.D., 1997. The region encoded by the alternatively spliced exon IIIA in mesenchymal fibronectin appears essential for chondrogenesis at the level of cellular condensation. Dev. Biol. 190, 191–205.
Hartmann, D., Miura, T., 2006. Modelling in vitro lung branching morphogenesis during development. J. Theor. Biol. 242, 862–872.
Hentschel, H.G.E., Glimm, T., Glazier, J.A., Newman, S.A., 2004. Dynamical mechanisms for skeletal pattern formation in the vertebrate limb. Proc. R. Soc. B 271, 1713–1722.
Hinchliffe, J.R., 2002. Developmental basis of limb evolution. Int. J. Dev. Biol. 46, 835–845.
Izaguirre, J.A., Chaturvedi, R., Huang, C., Cickovski, T., Coffland, J., Thomas, G., Forgacs, G., Alber, M., Hentschel, G., Newman, S.A., Glazier, J.A., 2004. CompuCell, a multi-model framework for simulation of morphogenesis. Bioinformatics 20, 1129–1137.
Johnson, C., 1987. Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge.
Lander, A.D., 2007. Morpheus unbound: reimagining the morphogen gradient. Cell 128, 245–256.
Lander, A.D., Nie, Q., Wan, F.Y., 2002. Do morphogen gradients arise by diffusion? Dev. Cell 2, 785–796.
Leonard, C.M., Fuld, H.M., Frenz, D.A., Downie, S.A., Massagu, J., Newman, S.A., 1991. Role of transforming growth factor-β in chondrogenic pattern formation in the embryonic limb: stimulation of mesenchymal condensation and fibronectin gene expression by exogenous TGF-β and evidence for endogenous TGF-β-like activity. Dev. Biol. 145, 99–109.
Litingtung, Y., Dahn, R.D., Li, Y., Fallon, J.F., Chiang, C., 2002. Shh and Gli3 are dispensable for limb skeleton formation but regulate digit number and identity. Nature 418, 979–983.
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.
Martin, G.R., 1998. The roles of FGFs in the early development of vertebrate limbs. Genes Dev. 12, 1571–1586.
Merkin, J.H., Sleeman, B.D., 2005. On the spread of morphogens. J. Math. Biol. 51, 1–17.
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.
Miura, T., Shiota, K., 2000a. TGF-β2 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., 2000b. 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., 2002. Depletion of FGF acts as a lateral inhibitory factor in lung branching morphogenesis in vitro. Mech. Dev. 116, 29–38.
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.
Moftah, M.Z., Downie, S.A., Bronstein, N.B., Mezentseva, N., Pu, J., Maher, P.A., Newman, S.A., 2002. Ectodermal FGFs induce perinodular inhibition of limb chondrogenesis in vitro and in vivo via FGF receptor 2. Dev. Biol. 249, 270–282.
Murray, J.D., 1993. Mathematical Biology, 2nd edn. Springer, Berlin.
Myerscough, M.R., Maini, P.K., Painter, K.J., 1998. Pattern formation in a generalized chemotactic model. Bull. Math. Biol. 60, 1–26.
Nelson, C.M., Vanduijn, M.M., Inman, J.L., Fletcher, D.A., Bissell, M.J., 2006. Tissue geometry determines sites of mammary branching morphogenesis in organotypic cultures. Science 314, 298–300.
Newman, S.A., 1988. Lineage and pattern in the developing vertebrate limb. Trends Genet. 4, 329–332.
Newman, S.A., 2003. From physics to development: the evolution of morphogenetic mechanisms. In: G.B. Müller, S.A. Newman (Eds.), Origination of Organismal Form: Beyond the Gene in Developmental and Evolutionary Biology. MIT Press, Cambridge, pp. 221–239.
Newman, S.A., Bhat, R., Activator-inhibitor mechanisms of vertebrate limb pattern formation. Birth Defects Res C Embryo Today, in press.
Newman, S.A., Frisch, H., 1979. Dynamics of skeletal pattern formation in developing chick limb. Science 205, 662–668.
Newman, S.A., Müller, G.B., 2005. Origination and innovation in the vertebrate limb skeleton: an epigenetic perspective. J. Exp. Zoolog. B Mol. Dev. Evol. 304, 593–609.
Nijhout, H.F., 2003. Gradients, diffusion and genes in pattern formation. In: G.B. Müller, S.A. Newman (Eds.), Origination of Organismal Form: Beyond the Gene in Developmental and Evolutionary Biology. MIT Press, Cambridge, pp. 165–181.
Pao, C.V., 1992. Nonlinear Parabolic and Elliptic Equations. Plenum, New York.
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.
Salazar-Ciudad, I., Jernvall, J., 2005. Graduality and innovation in the evolution of complex phenotypes: insights from development. J. Exp. Zool. B (Mol. Dev. Evol.) 304B, 619–631.
Salazar-Ciudad, I., Newman, S.A., Solé, R., 2001. Phenotypic and dynamical transitions in model genetic networks. I. Emergence of patterns and genotype-phenotype relationships. Evol. Dev. 3, 84–94.
Salazar-Ciudad, I., Jernvall, J., Newman, S.A., 2003. Mechanisms of pattern formation in development and evolution. Development 130, 2027–2037.
Salazar-Ciudad, I., 2006. On the origins of morphological disparity and its diverse developmental bases. Bioessays 28, 1112–1122.
Satnoianu, R.A., van den Driessche, P., 2005. Some remarks on matrix stability with application to Turing instability. J. Linear Algebra Appl. 398, 69–74.
Satnoianu, R.A., Menzinger, M., Maini, P.K., 2000. Turing instabilities in general systems. J. Math. Biol. 41, 493–512.
Shubin, N.H., Daeschler, E.B., Jenkins, F.A. Jr., 2006. The pectoral fin of Tiktaalik roseae and the origin of the tetrapod limb. Nature 440, 764–771.
Stark, R.J., Searls, R.L., 1973. A description of chick wing bud development and a model of limb morphogenesis. Dev. Biol. 33, 138–153.
Tickle, C., 2003. Patterning systems-from one end of the limb to the other. Dev. Cell 4, 449–458.
Turing, A.M., 1952. The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72.
Waddington, C.H., 1942. Canalization of development and the inheritance of acquired characters. Nature 150, 563–565.
Wagner, A., 2005. Robustness and Evolvability in Living Systems. Princeton University Press, Princeton.
Williams, P.H., Hagemann, A., Gonzalez-Gaitan, M., Smith, J.C., 2004. Visualizing long-range movement of the morphogen Xnr2 in the Xenopus embryo. Curr. Biol. 14, 1916–1923.
Zhu, A.J., Scott, M.P., 2004. Incredible journey: how do developmental signals travel through tissue? Genes Dev. 18, 2985–2997.
Zykov, V., Engel, H., 2004. Dynamics of spiral waves under global feedback in excitable domains of different shapes. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 70, 016201.
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Alber, M., Glimm, T., Hentschel, H.G.E. et al. The Morphostatic Limit for a Model of Skeletal Pattern Formation in the Vertebrate Limb. Bull. Math. Biol. 70, 460–483 (2008). https://doi.org/10.1007/s11538-007-9264-3
- Limb development
- Mesenchymal condensation
- Reaction–diffusion model