The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb

Bulletin of Mathematical Biology

Volumn 70, Issue 2, 2008, Pages 460-483

  Alber, Mark ^a   Glimm, Tilmann ^b   Hentschel, H G E ^c   Kazmierczak, Bogdan ^d   Zhang, Yong Tao ^a   Zhu, Jianfeng ^a   Newman, Stuart A ^e  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b WESTERN WASHINGTON UNIVERSITY   (United States)

^c EMORY UNIVERSITY   (United States)

^d INSTITUTE OF FUNDAMENTAL TECHNOLOGICAL RESEARCH   (Poland)

^e NEW YORK MEDICAL COLLEGE   (United States)

Author keywords

Chondrogenesis; Limb development; Mesenchymal condensation; Reaction diffusion model

Indexed keywords

ANIMAL; ARTICLE; BIOLOGICAL MODEL; CELL DIFFERENTIATION; CELL MOTION; CHRONOLOGY; COMPUTER SIMULATION; FEEDBACK SYSTEM; FINITE ELEMENT ANALYSIS; GENE EXPRESSION REGULATION; LIMB; MATHEMATICAL COMPUTING; MESENCHYMAL STEM CELL; METABOLISM; MORPHOGENESIS; PHYSIOLOGY; PRENATAL DEVELOPMENT; SIGNAL TRANSDUCTION; STATISTICS; SYSTEMS BIOLOGY; TISSUE DISTRIBUTION; TRANSPORT AT THE CELLULAR LEVEL; VERTEBRATE;

ANIMALS; BIOLOGICAL TRANSPORT; BODY PATTERNING; CELL DIFFERENTIATION; CELL MOVEMENT; CHRONOLOGY AS TOPIC; COMPUTER SIMULATION; EXTREMITIES; FEEDBACK, BIOCHEMICAL; FINITE ELEMENT ANALYSIS; GENE EXPRESSION REGULATION, DEVELOPMENTAL; MESENCHYMAL STEM CELLS; MODELS, BIOLOGICAL; NUMERICAL ANALYSIS, COMPUTER-ASSISTED; SIGNAL TRANSDUCTION; STOCHASTIC PROCESSES; SYSTEMS BIOLOGY; TISSUE DISTRIBUTION; VERTEBRATES;

VERTEBRATA;

EID: 38349112842     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-007-9264-3     Document Type: Article

References (71)

1. 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.
2. 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.
3. 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.
4. Brockes, J.P., Kumar, A., 2005. Appendage regeneration in adult vertebrates and implications for regenerative medicine. Science 310, 1919-1923.
5. 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.
6. 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.
7. 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.
8. Coates, M.I., Clack, J.A., 1990. Polydactyly in the earliest known tetrapod limbs. Nature 347, 66-69.
9. 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.
10. 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.
11. Cockburn, B., Shu, C.-W., 1998a. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199-224.
12. Cockburn, B., Shu, C.-W., 1998b. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440-2463.
13. 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.
14. 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.
15. 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.
16. 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.
17. Cross, G.W., 1978. Three types of matrix instability. J. Linear Algebra Appl. 20, 253-263.
18. 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.
19. 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.
20. Endo, T., Bryant, S.V., Gardiner, D.M., 2004. A stepwise model system for limb regeneration. Dev. Biol. 270, 135-145.
21. Entchev, E.V., Schwabedissen, A., Gonzalez-Gaitan, M., 2000. Gradient formation of the TGF-β homolog Dpp. Cell 103, 981-991.
22. 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.
23. Forgacs, G., Newman, S.A., 2005. Biological Physics of the Developing Embryo. Cambridge University Press, Cambridge.
24. 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.
25. 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.
26. 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.
27. 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.
28. Hartmann, D., Miura, T., 2006. Modelling in vitro lung branching morphogenesis during development. J. Theor. Biol. 242, 862-872.
29. 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.
30. Hinchliffe, J.R., 2002. Developmental basis of limb evolution. Int. J. Dev. Biol. 46, 835-845.
31. 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.
32. Johnson, C., 1987. Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge.
33. Lander, A.D., 2007. Morpheus unbound: reimagining the morphogen gradient. Cell 128, 245-256.
34. Lander, A.D., Nie, Q., Wan, F.Y., 2002. Do morphogen gradients arise by diffusion? Dev. Cell 2, 785-796.
35. 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.
36. 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.
37. 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.
38. Martin, G.R., 1998. The roles of FGFs in the early development of vertebrate limbs. Genes Dev. 12, 1571-1586.
39. Merkin, J.H., Sleeman, B.D., 2005. On the spread of morphogens. J. Math. Biol. 51, 1-17.
40. 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.
41. 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.
42. 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.
43. 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.
44. 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.
45. 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.
46. Murray, J.D., 1993. Mathematical Biology, 2nd edn. Springer, Berlin.
47. Myerscough, M.R., Maini, P.K., Painter, K.J., 1998. Pattern formation in a generalized chemotactic model. Bull. Math. Biol. 60, 1-26.
48. 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.
49. Newman, S.A., 1988. Lineage and pattern in the developing vertebrate limb. Trends Genet. 4, 329-332.
50. 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.
51. Newman, S.A., Bhat, R., Activator-inhibitor mechanisms of vertebrate limb pattern formation. Birth Defects Res C Embryo Today, in press.
52. Newman, S.A., Frisch, H., 1979. Dynamics of skeletal pattern formation in developing chick limb. Science 205, 662-668.
53. 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.
54. 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.
55. Pao, C.V., 1992. Nonlinear Parabolic and Elliptic Equations. Plenum, New York.
56. 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.
57. 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.
58. 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.
59. Salazar-Ciudad, I., Jernvall, J., Newman, S.A., 2003. Mechanisms of pattern formation in development and evolution. Development 130, 2027-2037.
60. Salazar-Ciudad, I., 2006. On the origins of morphological disparity and its diverse developmental bases. Bioessays 28, 1112-1122.
61. 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.
62. Satnoianu, R.A., Menzinger, M., Maini, P.K., 2000. Turing instabilities in general systems. J. Math. Biol. 41, 493-512.
63. 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.
64. 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.
65. Tickle, C., 2003. Patterning systems-from one end of the limb to the other. Dev. Cell 4, 449-458.
66. Turing, A.M., 1952. The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37-72.
67. Waddington, C.H., 1942. Canalization of development and the inheritance of acquired characters. Nature 150, 563-565.
68. Wagner, A., 2005. Robustness and Evolvability in Living Systems. Princeton University Press, Princeton.
69. 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.
70. Zhu, A.J., Scott, M.P., 2004. Incredible journey: how do developmental signals travel through tissue? Genes Dev. 18, 2985-2997.
71. 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.