Free Access
Issue
Math. Model. Nat. Phenom.
Volume 4, Number 4, 2009
Morphogenesis
Page(s) 3 - 82
DOI https://doi.org/10.1051/mmnp/20094401
Published online 11 July 2009
  1. J. Adler. Chemotaxis in bacteria. Science, 153 (1966), 708–716. [CrossRef] [PubMed] [Google Scholar]
  2. G. Allen, R. Steene, M. Allen. A guide to angelfishes and butterflyfishes. Odyssey, 1998. [Google Scholar]
  3. K. Amonlirdviman, N. A. Khare, D. R. Tree, W. S. Chen, J. D. Axelrod, C. J. Tomlin. Mathematical modeling of planar cell polarity to understand domineering nonautonomy. Science, 307 (2005), No. 5708, 423–6. [Google Scholar]
  4. R. P. Araujo, D. L. S. McElwain. A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull. Math. Biol., 66 (2004), No. 5, 1039–1091. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  5. P. Arcuri, J. Murray. Pattern sensitivity to boundary and initial conditions in reaction-diffusion models. J. Math. Biol., 24 (1986), 141–165. [MathSciNet] [PubMed] [Google Scholar]
  6. R. Asai, E. Taguchi, Y. Kume, M. Saito, S. Kondo. Zebrafish leopard gene as a component of the putative reaction-diffusion system. Mech Dev, 89 (1999), 87–92. [CrossRef] [PubMed] [Google Scholar]
  7. M. Ashkenazi, H. G. Othmer. Spatial patterns in coupled biochemical oscillators. Jour. Math. Biol., 5 (1978), 305–350. [Google Scholar]
  8. S. Atkinson, C. Y. Chang, R. E. Sockett, M. Camara, P. Williams. Quorum sensing in yersinia enterocolitica controls swimming and swarming motility. J Bacteriol, 188 (2006), No. 4, 1451–61. [CrossRef] [PubMed] [Google Scholar]
  9. J. Bagnara, M. Hadley. Chromatophores and color change. Prentice-Hall, Eaglewood Cliffs, New Jersey. 1973. [Google Scholar]
  10. R. E. Baker, E. A. Gaffney, P. K. Maini. Partial differential equations for self-organization in cellular and developmental biology. Nonlinearity, 21 (2008), No. 11, 251–290. [CrossRef] [MathSciNet] [Google Scholar]
  11. R. E. Baker, P. K. Maini. A mechanism for morphogen-controlled domain growth. J Math Biol, 54 (2007), 597–622. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  12. J. Bard. A unity underlying the different zebra striping patterns. J. Zool., 183 (1977), 527–539. [Google Scholar]
  13. J. Bard. A model for generating aspects of zebra and other mammalian coat patterns. J. Theor. Biol., 93 (1981), 363–385. [CrossRef] [PubMed] [Google Scholar]
  14. J. Bard, V. French. Butterfly wing patterns: how good a determining mechanism is the simple diffusion of a single morphogen? J. Embryol. Exp. Morph., 84 (1984), 255–274. [Google Scholar]
  15. R. Barrio, C. Varea, J. Aragn, P. Maini. A two-dimensional numerical study of spatial pattern formation in interacting Turing systems. Bull. Math. Biol., 61 (1999), 483–505. [CrossRef] [PubMed] [Google Scholar]
  16. B. L. Bassler. How bacteria talk to each other: regulation of gene expression by quorum sensing. Curr Opin Microbiol, 2 (1999), No. 6, 582–587. [CrossRef] [PubMed] [Google Scholar]
  17. E. Ben-Jacob, I. Cohen, A. Czirok, T. Vicsek, D. L. Gutnick. Chemomodulation of cellular movement, collective formation of vortices by swarming bacteria, and colonial development. 238 (1997), No. 1, 181–. [Google Scholar]
  18. E. Ben-Jacob, I. Cohen, H. Levine. Cooperative self-organization of microorganisms. Advances in Physics, 49 (2000), No. 4, 395–554. [CrossRef] [Google Scholar]
  19. E. Ben-Jacob, I. Cohen, O. Shochet, I. Aranson, H. Levine. Complex bacterial patterns. 373 (1995), No. 6515, 566–557. [Google Scholar]
  20. E. Ben-Jacob, O. Schochet, A. Tenenbaum, I. Cohen, A. Czirok. Generic modelling of cooperative growth patterns in bacterial colonies. 368 (1994), No. 6466, 46–49. [Google Scholar]
  21. D. Ben-Zvi, B. Shilo, A. Fainsod, N. Barkai. Scaling of the BMP activation gradient in Xenopus embryos. Nature, 453 (2008), 1205–1211. [CrossRef] [PubMed] [Google Scholar]
  22. H. Berg. Random walks in biology 1983. [Google Scholar]
  23. H. C. Berg. Motile behavior of bacteria. Physics Today, 53 (2000), No. 1, 24–29. [CrossRef] [Google Scholar]
  24. H. C. Berg. http://webmac.rowland.org/labs/bacteria/movies/others/. (2008). [Google Scholar]
  25. M. D. Betterton, M. P. Brenner. Collapsing bacterial cylinders. Physical Review E, 64 (2001), 061904. [CrossRef] [Google Scholar]
  26. K. Bisset, C. Douglas. A continuous study of morphological phase in the swarm of proteus. J. Med. Microbiol., 9 (1976), 229–31. [CrossRef] [PubMed] [Google Scholar]
  27. V. Bolos, J. Grego-Bessa, J. L. de la Pompa. Notch signaling in development and cancer. Endocr Rev, 28 (2007), No. 3, 339–63. [CrossRef] [PubMed] [Google Scholar]
  28. J. T. Bonner. The development of Dictyostelium , Chapter Comparative Biology of Cellular Slime Molds. Academic Press 1982, 1–33. [Google Scholar]
  29. R. B. Bourret, K. A. Borkovich, M. I. Simon. Signal transduction pathways involving protein phosphorylation in prokaryotes. 60 (1991), 401–441. [Google Scholar]
  30. P. Brakefield, V. French. Eyespot development on butterfly wings: the epidermal response to damage. Dev. Biol., 168 (1995), 98–111. [CrossRef] [PubMed] [Google Scholar]
  31. M. P. Brenner, L. S. Levitov, E. O. Budrene. Physical mechanisms for chemotactic pattern formation by bacteria. 74 (1998), No. 4, 1677–1693. [Google Scholar]
  32. C. Brunetti, J. Selegue, A. Monteiro, V. French, P. Brakefield, S. Carroll. The generation and diversification of butterfly eyespot color patterns. Curr. Biol., 11 (2001), 1578–1585. [CrossRef] [PubMed] [Google Scholar]
  33. E. O. Budrene. Personal communication 2005. [Google Scholar]
  34. E. O. Budrene, H. C. Berg. Complex patterns formed by motile cells of Escherichia coli . Nature, 349 (1991), No. 6310, 630–633. [CrossRef] [PubMed] [Google Scholar]
  35. E. O. Budrene, H. C. Berg. Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature, 376 (1995), No. 6535, 49–53. [Google Scholar]
  36. C. Caicedo-Carvajal, T. Shinbrot. In silico zebrafish pattern formation. Dev. Biol., 315 (2008), 397–403. [CrossRef] [PubMed] [Google Scholar]
  37. S. Carroll, J. Gates, D. Keys, S. Paddock, G. Panganiban, J. Selegue, J. Williams. Pattern formation and eyespot determination in butterfly wings. Science, 265 (1994), 109–114. [CrossRef] [PubMed] [Google Scholar]
  38. J. Casanova, G. Struhl. The torso receptor localizes as well as transduces the spatial signal specifying terminal body pattern in Drosophila. Nature, 362 (1993), 152–155. [CrossRef] [PubMed] [Google Scholar]
  39. V. Castets, E. Dulos, P. D. Kepper. Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. 64 (1990), No. 24, 2953–2956. [Google Scholar]
  40. C. M. Child. Patterns and problems of development. University of Chicago Press 1941. [Google Scholar]
  41. S. Childress, J. K. Percus. Nonlinear aspects of chemotaxis. 56 (1981) 217–237. [Google Scholar]
  42. J. H. Claxton. The determination of patterns with special reference to that of the central primary skin follicles in sheep. J. Theor. Biol., 7 (1964), 302–317. [CrossRef] [PubMed] [Google Scholar]
  43. J. H. Claxton. Developmental origin of even spacing between the microchaetes of Drosophila melanogaster. Aust J Biol Sci, 29 (1976) 131–135. [Google Scholar]
  44. P. Cluzel, M. Surette, S. Leibler. An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. Science, 287 (2000), 1652–1655. [CrossRef] [PubMed] [Google Scholar]
  45. E. Conway, D. Hoff, J. Smoller. Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math., (1977). [Google Scholar]
  46. M. Coppey, A. Berezhkovskii, Y. Kim, A. Boettiger, S. Shvartsman. Modeling the bicoid gradient: diffusion and reversible nuclear trapping of a stable protein. Dev. Biol., 312 (2007), 623–630. [CrossRef] [PubMed] [Google Scholar]
  47. M. Coppey, A. Boettiger, A. Berezhkovskii, S. Shvartsman. Nuclear trapping shapes the terminal gradient in the Drosophila embryo. Curr. Biol., 18 (2008), 915–919. [CrossRef] [PubMed] [Google Scholar]
  48. E. Crampin, W. Hackborn, P. Maini. Pattern formation in reaction-diffusion models with nonuniform domain growth. Bull. Math. Biol., 64 (2002), 747–769. [CrossRef] [PubMed] [Google Scholar]
  49. E. J. Crampin, E. A. Gaffney, P. K. Maini. Reaction and diffusion on growing domains: Scenarios for robust pattern formation. Bulletin of Mathematical Biology, 61 (1999), No. 6, 1093–1120. [CrossRef] [PubMed] [Google Scholar]
  50. O. Crauk, N. Dostatni. Bicoid determines sharp and precise target gene expression in the Drosophila embryo. Curr Biol, 15 (2005), No. 21, 1888–1898, comparative Study. [Google Scholar]
  51. F. H. Crick. Diffusion in embryogenesis. Nature, 225 (1970), 420–422. [CrossRef] [PubMed] [Google Scholar]
  52. M. C. Cross, P. C. Hohenberg. Pattern formation out of equilibrium. 65 (1993), No. 3, 851–1112. [Google Scholar]
  53. R. Daniels, J. Vanderleyden, J. Michiels. Quorum sensing and swarming migration in bacteria. FEMS Microbiol Rev., 28 (2004), 261–89. [CrossRef] [PubMed] [Google Scholar]
  54. P. de Kepper, V. Castets, E. Dulos, J. Boissonade. Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction. D 49 (1991) 161–169. [Google Scholar]
  55. R. Dilão, J. Sainhas. Modelling butterfly wing eyespot patterns. Proc. Biol. Sci., 271 (2004), 1565–1569. [CrossRef] [PubMed] [Google Scholar]
  56. R. Dillon, P. K. Maini, H. G. Othmer. Pattern formation in generalized Turing systems I. Steady-state patterns in systems with mixed boundary conditions. Journal of Mathematical Biology, 32 (1994), No. 4, 345–393. [CrossRef] [MathSciNet] [Google Scholar]
  57. R. Dillon, H. G. Othmer. A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud. J. Theor. Biol., 197 (1999), No. 3, 295–330. [CrossRef] [PubMed] [Google Scholar]
  58. W. Driever, C. Nüsslein-Volhard. A gradient of bicoid protein in Drosophila embryos. Cell, 54 (1988), No. 1, 83–93. [CrossRef] [PubMed] [Google Scholar]
  59. W. Driever, C. Nusslein-Volhard. The bicoid protein determines position in the Drosophila embryo in a concentration-dependent manner. Cell, 54 (1988), No. 1, 95–104. [CrossRef] [PubMed] [Google Scholar]
  60. S. Eglen. Development of regular cellular spacing in the retina: theoretical models. Mathematical Medicine and Biology, 23 (2006), No. 2, 79–99. [CrossRef] [Google Scholar]
  61. A. Eldar, R. Dorfman, D. Weiss, H. Ashe, B. Z. Shilo, N. Barkai. Robustness of the BMP morphogen gradient in drosophila embryonic patterning. Nature, 419 (2002), No. 6904, 304–308. [CrossRef] [PubMed] [Google Scholar]
  62. R. Erban. From individual to collective behavior in biological systems. Ph.D. thesis, University of Minnesota 2005. [Google Scholar]
  63. R. Erban, H. Othmer. From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology. 3 (2005), No. 2, 362–394. [Google Scholar]
  64. R. Erban, H. G. Othmer. From individual to collective behavior in bacterial chemotaxis. SIAM J. Appl. Math., 65 (2004), No. 2, 361–391. [CrossRef] [MathSciNet] [Google Scholar]
  65. R. Erban, H. G. Othmer. Taxis equations for amoeboid cells. J Math Biol, 54 (2007), 847–885. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  66. B. Ermentrout. Stripes or spots? nonlinear effects in bifurcation of reaction-diffusion equations on the square. Proc. Roy. Soc. Lond. A., 434 (1991), 413–417. [CrossRef] [Google Scholar]
  67. T. Evans, J. Marcus. A simulation study of the genetic regulatory hierarchy for butterfly eyespot focus determination. Evol. Dev., 8 (2006) 273–283. [Google Scholar]
  68. A. Filloux, I. Vallet. Biofilm: set-up and organization of a bacterial community. Med Sci (Paris), 19 (2003), No. 1, 77–83. [CrossRef] [EDP Sciences] [PubMed] [Google Scholar]
  69. P. R. Fisher, R. Merkl, G. Gerisch. Quantitative analysis of cell motility and chemotaxis in Dictyostelium discoideum by using an image processing system and a novel chemotaxis chamber providing stationary chemical gradients. J. Cell. Biol., 108 (1989), 973–984. [CrossRef] [PubMed] [Google Scholar]
  70. R. M. Ford, D. A. Lauffenburger. Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients. 53 (1991), No. 5, 721–749. [Google Scholar]
  71. R. M. Ford, D. A. Lauffenburger. Measurement of bacterial random motility and chemotaxis coefficients: II: Application of single-cell-based mathematical model. Biotechnol. Bioeng., 37 (1991), 661–672. [CrossRef] [PubMed] [Google Scholar]
  72. R. M. Ford, B. R. Phillips, J. A. Quinn, D. A. Lauffenburger. Measurement of bacterial random motility and chemotaxis coefficients: I. stopped–flow diffusion chamber assay. Biotechnol. Bioeng., 37 (1991) 647–660. [Google Scholar]
  73. C. Fowlkes, C. Hendriks, S. Kernen, G. Weber, O. Rbel, M. Huang, S. Chatoor, A. DePace, L. Simirenko, C. Henriquez, A. Beaton, R. Weiszmann, S. Celniker, B. Hamann, D. Knowles, M. Biggin, M. Eisen, J. Malik. A quantitative spatiotemporal atlas of gene expression in the Drosophila blastoderm. Cell, 133 (2008), 364–374. [CrossRef] [PubMed] [Google Scholar]
  74. H. Fricke. Juvenile-adult colour patterns and coexistence in the territorial coral reef fish Pomacanthus imperator. Marine Ecology, 1 (1980), 133–141. [CrossRef] [Google Scholar]
  75. A. Gierer, H. Meinhardt. A theory of biological pattern formation. 12 (1972), No. 1, 30–39. [Google Scholar]
  76. L. Glass. Stochastic generation of regular distributions. Science, 180 (1973), 1061–1063. [CrossRef] [PubMed] [Google Scholar]
  77. L. A. Goentoro, G. T. Reeves, C. P. Kowal, L. Martinelli, T. Schpbach, S. Y. Shvartsman. Quantifying the Gurken morphogen gradient in Drosophila oogenesis. Dev. Cell, 11 (2006), 263–272. [CrossRef] [PubMed] [Google Scholar]
  78. A. B. Goryachev, D. J. Toh, T. Lee. Systems analysis of a quorum sensing network: design constraints imposed by the functional requirements, network topology and kinetic constants. Biosystems, 83 (2006), No. 2–3, 178–87. [CrossRef] [PubMed] [Google Scholar]
  79. C. Graván, R. Lahoz-Beltra. Evolving morphogenetic fields in the zebra skin pattern based on turing's morphogen hypothesis. Int. J. Appl. Math. Comput. Sci., 14 (2004), 351–361. [MathSciNet] [Google Scholar]
  80. E. P. Greenberg. Bacterial communication: Tiny teamwork. Nature, 424 (2003), 134. [CrossRef] [PubMed] [Google Scholar]
  81. T. Gregor, W. Bialek, R. R. de Ruyter van Steveninck, D. W. Tank, E. F. Wieschaus. Diffusion and scaling during early embryonic pattern formation. Proc Natl Acad Sci U S A, 102 (2005), No. 51, 18403–7. [CrossRef] [PubMed] [Google Scholar]
  82. T. Gregor, D. W. Tank, E. F. Wieschaus, W. Bialek. Probing the limits to positional information. Cell, 130 (2007), No. 1, 153–164. [CrossRef] [PubMed] [Google Scholar]
  83. T. Gregor, E. F. Wieschaus, A. P. McGregor, W. Bialek, D. W. Tank. Stability and nuclear dynamics of the bicoid morphogen gradient. Cell, 130 (2007), No. 1, 141–152. [CrossRef] [PubMed] [Google Scholar]
  84. D. Grunwald, J. Eisen. Headwaters of the zebrafish – emergence of a new model vertebrate. Nat. Rev. Genet., 3 (2002), 717–724. [CrossRef] [PubMed] [Google Scholar]
  85. P. Haffter, J. Odenthal, M. C. Mullins, S. Lin, M. J. Farrell, L. Vogelsang, H. F., M. Brand, F. van Eeden, M. Furutani-Seiki, M. Granato, M. Hammerschmidt, C. P. Heisenberg, Y. J. Jiang, D. A. Kane, N. Kelsh, R. N. Hopkins, C. Nusslein-Volhard. Mutationa affecting pigmentation and shape of the adult zebrafish. Dev. Genes. Evol., 206 (1996), 260–276. [CrossRef] [PubMed] [Google Scholar]
  86. H. Berg, D. Brown. Chemotaxis in Escherichia Coli analyzed by three-dimensional tracking. Nature, 239 (1972), 502–507. [Google Scholar]
  87. D. Headon, K. J. Painter. Stippling the skin: Generation of anatomical periodicity by reaction-diffusion mechanisms. Submitted to MMNP, (2008). [Google Scholar]
  88. M. Herrero, J. Velázquez. Chemotactic collapse for the Keller-Segel model. J. Math. Biol., 35 (1996), 177–194. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  89. T. Hillen, H. G. Othmer. The diffusion limit of transport equation derived from velocity-jump processes. Siam J. Appl. Math., 61 (2000), No. 3, 751–775. [CrossRef] [MathSciNet] [Google Scholar]
  90. T. Hillen, K. Painter. A user's guide to pde models for chemotaxis. J. Math. Biol., 58 (2008), 183–217. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  91. M. Hirata, K.-i. Nakamura, T. Kanemaru, Y. Shibata, S. Kondo. Pigment cell organization in the hypodermis of zebrafish. Dev Dyn, 227 (2003) 497–503. [Google Scholar]
  92. D. Horstmann. From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I. Jahresbericht der DMV, 105 (2003), No. 3, 103–165. [Google Scholar]
  93. D. Horstmann. A constructive approach to traveling waves in chemotaxis. Journal of Nonlinear Science, 14 (2004), 1–25(25). [Google Scholar]
  94. B. Houchmandzadeh, E. Wieschaus, S. Leibler. Establishment of developmental precision and proportions in the early Drosophila embryo. Nature, 415 (2002), No. 6873, 798–802. [PubMed] [Google Scholar]
  95. L. Hufnagel, A. A. Teleman, H. Rouault, S. M. Cohen, B. I. Shraiman. On the mechanism of wing size determination in fly development. Proc Natl Acad Sci U S A, 104 (2007), No. 10, 3835–40, epub 2007 Feb 28. [Google Scholar]
  96. J. Jaeger, J. Reinitz. On the dynamic nature of positional information. Bioessays, 28 (2006), No. 11, 1102–11. [CrossRef] [PubMed] [Google Scholar]
  97. J. Jaeger, S. Surkova, M. Blagov, H. Janssens, D. Kosman, K. Kozlov, Manu, E. Myasnikova, C. Vanario-Alonso, M. Samsonova, D. Sharp, J. Reinitz. Dynamic control of positional information in the early Drosophila embryo. Nature, 430 (2004), No. 430, 368–71. [CrossRef] [PubMed] [Google Scholar]
  98. W. Jäger, S. Luckhaus. On explosions of solutions to a system of partial differential equations modelling chemotaxis. 329 (1992), No. 2, 819–824. [Google Scholar]
  99. S. L. Johnson, D. Africa, C. Walker, J. A. Weston. Genetic control of adult pigment stripe development in zebrafish. Dev Biol, 167 (1995), 27–33. [CrossRef] [PubMed] [Google Scholar]
  100. H. S. Jung, P. H. Francis-West, R. B. Widelitz, T. X. Jiang, S. Ting-Berreth, C. Tickle, L. Wolpert, C. M. Chuong. Local inhibitory action of BMPs and their relationships with activators in feather formation: implications for periodic patterning. Dev Biol, 196 (1998), No. 1, 11–23. [Google Scholar]
  101. D. Kaiser. Coupling cell movement to multicellular development in myxobacteria. Nat. Rev. Microbiol., 1 (2003), No. 1, 45–54. [CrossRef] [PubMed] [Google Scholar]
  102. D. Kaiser. Myxococcus — from single-cell polarity to complex multicellular patterns. Annual Review of Genetics, 42 (2008), No. 1, 109–130. [CrossRef] [PubMed] [Google Scholar]
  103. E. F. Keller, G. M. Odell. Necessary and sufficient conditions for chemotactic bands. Mathematical Biosciences, 27 (1975), 309–317. [CrossRef] [MathSciNet] [Google Scholar]
  104. E. F. Keller, L. A. Segel. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol., 26 (1970), 399–415. [CrossRef] [PubMed] [Google Scholar]
  105. E. F. Keller, L. A. Segel. Model for chemotaxis. J. Theor. Biol., 30 (1971), 225–234. [CrossRef] [PubMed] [Google Scholar]
  106. E. F. Keller, L. A. Segel. Traveling bands of chemotactic bacteria: A theoretical analysis. J. Theor. Biol., 30 (1971), 235–248. [CrossRef] [PubMed] [Google Scholar]
  107. R. N. Kelsh, M. Brand, Y. J. Jiang, C. P. Heisenberg, S. Lin, P. Haffter, J. Odenthal, M. C. Mullins, F. J. van Eeden., M. Furutani-Seiki, M. Granato, M. Hammerschmidt, D. A. Kane, R. M. Warga, D. Beuchle, L. Vogelsang, C. Nusslein-Volhard. Zebrafish pigmentation mutations and the processes of neural crest development. Development, 123 (1996), 369–389. [PubMed] [Google Scholar]
  108. S.V. Keranen, C.C. Fowlkes, C.L. Luengo Hendriks, D. Sudar, D.W. Knowles, J. Malik, M.D. Biggin. Three-dimensional morphology and gene expression in the Drosophila blastoderm at cellular resolution II: dynamics. Genome Biol. 7 (1006), R124 [Google Scholar]
  109. M. Kerszberg, L. Wolpert. Mechanisms for positional signalling by morphogen transport: a theoretical study. J. Theor. Biol., 191 (1998), No. 1, 103–114. [CrossRef] [PubMed] [Google Scholar]
  110. A. Kicheva, P. Pantazis, T. Bollenbach, Y. Kalaidzidis, T. Bittig, F. Julicher, M. Gonzalez-Gaitan. Kinetics of morphogen gradient formation. Science, 315 (2007), No. 5811, 521–525. [CrossRef] [PubMed] [Google Scholar]
  111. P. Koch, D. Keys, T. Rocheleau, K. Aronstein, M. Blackburn, S. Carroll, R. ffrench Constant. Regulation of dopa decarboxylase expression during colour pattern formation in wild-type and melanic tiger swallowtail butterflies. Development, 125 (1998), 2303–2313. [PubMed] [Google Scholar]
  112. R. Kolter, E. P. Greenberg. Microbial sciences: The superficial life of microbes. Nature, 441 (2006), 300–302. [CrossRef] [PubMed] [Google Scholar]
  113. S. Kondo, R. Asai. A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature, 376 (1995), 675–768. [CrossRef] [Google Scholar]
  114. P. Kulesa, G. Cruywagen, S. Lubkin, P. Maini, J. Sneyd, M. Ferguson, J. Murray. On a model mechanism for the spatial patterning of teeth primordia in the alligator. J. Theor. Biol., 180 (1996), 287–296. [CrossRef] [Google Scholar]
  115. A. D. Lander. Morpheus unbound: reimagining the morphogen gradient. Cell, 128 (2007), No. 2, 245–256. [CrossRef] [PubMed] [Google Scholar]
  116. A. D. Lander, Q. Nie, F. Y. M. Wan. Do morphogen gradients arise by diffusion? Dev. Cell, 2 (2002), No. 6, 785–96. [Google Scholar]
  117. I. R. Lapidus, R. Schiller. A model for traveling bands of chemotactic bacteria. Biophys J., 22 (1978), No. 1, 1–13. [Google Scholar]
  118. D. Lauffenburger, C. R. Kennedy, R. Aris. Traveling bands of chemotactic bacteria in the context of population growth. Bulletin of Mathematical Biology, 46 (1984), No. 1, 19–40. [Google Scholar]
  119. N. Le Douarin, C. Kalcheim. The neural crest. CUP, Cambridge, 2nd edition 1999. [Google Scholar]
  120. I. Lengyel, I. R. Epstein. Modelling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system. Science, 251 (1991) 650–652. [Google Scholar]
  121. S. Liaw, C. Yang, R. Liu, J. Hong. Turing model for the patterns of lady beetles. Phys Rev E Stat Nonlin Soft Matter Phys, 64 (2001), 041909. [CrossRef] [PubMed] [Google Scholar]
  122. R. Liu, S. Liaw, P. Maini. Two-stage Turing model for generating pigment patterns on the leopard and the jaguar. Phys Rev E Stat Nonlin Soft Matter Phys, 74 (2006), 011914. [CrossRef] [PubMed] [Google Scholar]
  123. R. Lux, W. Shi. Chemotaxis-guided movements in bacteria. Crit. Rev. Oral. Biol. Med., 15 (2004), No. 4, 207–20. [CrossRef] [PubMed] [Google Scholar]
  124. M. Lyons, L. Harrison. Stripe selection: An intrinsic property of some pattern-forming models with nonlinear dynamics. Dev. Dyn., 195 (1992) 201–215. [Google Scholar]
  125. F. Maderspacher, C. Nüsslein-Volhard. Formation of the adult pigment pattern in zebrafish requires leopard and obelix dependent cell interactions. Development, 130 (2003), 3447–3457. [CrossRef] [PubMed] [Google Scholar]
  126. A. Madzvamuse, P. Maini, A. Wathen, T. Sekimura. A predictive model for color pattern formation in the butterfly wing of papilio dardanus. Hiroshima Math. J., 32 (2002), 325–336. [MathSciNet] [Google Scholar]
  127. P. K. Maini, K. J. Painter, H. P. C. Nguyen. Spatial pattern formation in chemical and biological systems. J Chem Soc Faraday Trans, 93 (1997), No. 20, 3601–10. [CrossRef] [Google Scholar]
  128. N. V. Mantzaris, S. Webb, H. G. Othmer. Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol., 49 (2004), No. 2, 111–87. [Google Scholar]
  129. J. Marcus, T. Evans. A simulation study of mutations in the genetic regulatory hierarchy for butterfly eyespot focus determination. BioSystems, 93 (2008), 250–255. [CrossRef] [PubMed] [Google Scholar]
  130. M. McClure. Development and evolution of melanophore patterns in fishes of the genus Danio (Teleostei: Cyprinidae). J. Morphol., 241 (1999) 83–105. [Google Scholar]
  131. H. Meinhardt. Models of biological pattern formation. Academic Press, New York 1980. [Google Scholar]
  132. H. Meinhardt. Models for positional signalling with application to the dorsoventral patterning of insects and segregation into different cell types. Development, supplement (1989), 169–180. [Google Scholar]
  133. H. Meinhardt, P. Prusinkiewicz, D. Fowler. The algorithmic beauty of sea shells. Springer 2003. [Google Scholar]
  134. B. A. Mello, Y. Tu. Quantitative modeling of sensitivity in bacterial chemotaxis: the role of coupling among different chemoreceptor species. Proc. Nat. Acad. Sci. (USA), 100 (2003), 8223–8228. [CrossRef] [Google Scholar]
  135. D. Míguez, A. Muñuzuri. On the orientation of stripes in fish skin patterning. Biophys. Chem., 124 (2006), 161–167. [CrossRef] [PubMed] [Google Scholar]
  136. N. Milos, A. D. Dingle. Dynamics of pigment pattern formation in the zebrafish, Brachydanio rerio. I. Establishment and regulation of the lateral line melanophore stripe during the first eight days of development. J Exp Zool, 205 (1978), 205–216. [CrossRef] [Google Scholar]
  137. N. Mittal, E. O. Budrene, M. P. Brenner, A. Oudenaarden. Motility of Escherichia coli cells in clusters formed by chemotactic aggregation. Proc. Natl. Acad. Sci. (USA), 100 (2003), No. 23, 13259–63. [CrossRef] [Google Scholar]
  138. C. M. Mizutani, Q. Nie, F. Y. Wan, Y. T. Zhang, P. Vilmos, R. Sousa-Neves, E. Bier, J. L. Marsh, A. D. Lander. Formation of the BMP activity gradient in the Drosophila embryo. Dev. Cell, 8 (2005), No. 6, 915–24. [CrossRef] [PubMed] [Google Scholar]
  139. A. Monteiro, V. French, G. Smit, P. Brakefield, J. Metz. Butterfly eyespot patterns: evidence for specification by a morphogen diffusion gradient. Acta Biotheor., 49 (2001), 77–88. [CrossRef] [PubMed] [Google Scholar]
  140. J. R. Mooney, B. N. Nagorcka. Spatial patterns produced by a reaction-diffusion system in primary hair follicles. J. Theor. Biol., 115 (1985), 299–317. [CrossRef] [PubMed] [Google Scholar]
  141. J. Moreira, A. Deutsch. Pigment pattern formation in zebrafish during late larval stages: a model based on local interactions. Dev. Dyn., 232 (2005). [Google Scholar]
  142. C. Mou, B. Jackson, P. Schneider, P. A. Overbeek, D. J. Headon. Generation of the primary hair follicle pattern. Proc. Natl. Acad. Sci. U.S.A., 103 (2006), 9075–9080. [CrossRef] [PubMed] [Google Scholar]
  143. J. D. Murray. A pattern formation mechanism and its application to mammalian coat markings. volume 39 of Lecture Notes in Biomathematics, Springer, Berlin, Heidelberg, New York. 1979, (360–399). [Google Scholar]
  144. J. D. Murray. On pattern-formation mechanisms for leipdopteran wing patterns and mammalian coat markings. Phil. Trans. Roy. Soc. Lond. B, 295 (1981), 473–496. [CrossRef] [Google Scholar]
  145. J. D. Murray. A pre-pattern formation mechanism for animal coat markings. J. Theor. Biol., 88 (1981), 161–199. [CrossRef] [Google Scholar]
  146. J. D. Murray. Mathematical biology ii: Spatial models and biomedical applications. Springer, New York, 3rd edition 2003. [Google Scholar]
  147. J. D. Murray, D. Deeming, M. Ferguson. Size-dependent pigmentation-pattern formation in embryos of alligator-mississippiensis - time of initiation of pattern generation mechanism. Proc. Roy. Soc. Lond. B, 239 (1990), 279–293. [CrossRef] [Google Scholar]
  148. J. D. Murray, M. Myerscough. Pigmentation pattern formation on snakes. J. Theor. Biol., 149 (1991), 339–360. [CrossRef] [PubMed] [Google Scholar]
  149. J. D. Murray. Mathematical biology, volume 19 of Biomathematics 1989. [Google Scholar]
  150. B. N. Nagorcka, J. R. Mooney. The role of a reaction–diffusion system in the formation of hair fibres. J. Theor. Biol., 98 (1982) 575–607. [Google Scholar]
  151. B. N. Nagorcka, J. R. Mooney. The role of a reaction-diffusion system in the initiation of primary hair follicles. J. Theor. Biol., 114 (1985), 243–272. [CrossRef] [PubMed] [Google Scholar]
  152. T. Naitoh, A. Morioka, Y. Omura. Adaptation of a common freshwater goby, yoshinobori, rhinogobius brunneus temminck et schlegel to various backgrounds including those containing different sizes of black and white checkerboard squares. Zool. Sci., 2 (1985), 59. [Google Scholar]
  153. J. Nelson. Fishes of the world. John Wiley and Sons, New York, 3rd edition 1993. [Google Scholar]
  154. F. S. Neuman-Silberberg, T. Schupbach. The Drosophila dorsoventral patterning gene gurken produces a dorsally localized RNA and encodes a TGF alpha-like protein. Cell, 75 (1993), No. 1, 165–74. [PubMed] [Google Scholar]
  155. H. Nijhout. Wing pattern formation in lepidoptera: a model. J. Exp. Zool., 206 (1978), 119–136. [CrossRef] [Google Scholar]
  156. H. Nijhout. A comprehensive model for color pattern formation in butterflies. Proc. Roy. Soc. Lond. B, 239 (1990), 81–113. [CrossRef] [Google Scholar]
  157. H. Nijhout, P. K. Maini, A. Madzvamuse, A. Wathen, T. Sekimura. Pigmentation pattern formation in butterflies: experiments and models. C. R. Biol., 326 (2003), 717–727. [CrossRef] [PubMed] [Google Scholar]
  158. M. B. O'Connor, D. M. Umulis, H. G. Othmer, S. S. Blair. Shaping BMP morphogen gradients in the Drosophila embryo and pupal wing. Development, 133 (2006), 183–93. [CrossRef] [PubMed] [Google Scholar]
  159. J. Odenthal, K. Rossnagel, P. Haffter, R. N. Kelsh, E. Vogelsang, M. Brand, F. J. van Eeden., M. Furutani-Seiki, M. Granato, M. Hammerschmidt, C. P. Heisenberg, Y. J. Jiang, D. A. Kane, M. C. Mullins, C. Nusslein-Volhard. Mutations affecting xanthophore pigmentation in the zebrafish, Danio rerio. Development, 123 (1996), 391–398. [PubMed] [Google Scholar]
  160. J. Otaki. Physiologically induced color-pattern changes in butterfly wings: mechanistic and evolutionary implications. J. Insect Physiol., 54 (2008), 1099–1112. [CrossRef] [PubMed] [Google Scholar]
  161. H. G. Othmer. Interactions of reaction and diffusion in open systems. Ph.D. thesis, University of Minnesota, Minneapolis 1969. [Google Scholar]
  162. H. G. Othmer. Current problems in pattern formation. In Some mathematical questions in biology, volume VIII, Amer. Math. Soc., Providence, R.I. 1977, (57–85). [Google Scholar]
  163. H. G. Othmer. Synchronized and differentiated modes of cellular dynamics. In H. Haken, editor, Dynamics of Synergetic Systems, Springer-Verlag. [Google Scholar]
  164. H. G. Othmer, J. A. Aldridge. The effects of cell density and metabolite flux on cellular dynamics. J. Math. Biol., 5 (1978), 169–200. [PubMed] [Google Scholar]
  165. H. G. Othmer, S. R. Dunbar, W. Alt. Models of dispersal in biological systems. J. Math. Biol., 26 (1988), No. 3, 263–298. [Google Scholar]
  166. H. G. Othmer, T. Hillen. The diffusion limit of transport equations, Part II: chemotaxis equations. SIAM JAM, 62 (2002), 1222–1260. [Google Scholar]
  167. H. G. Othmer, E. F. Pate. Scale-invariance in reaction-diffusion models of spatial pattern formation. Proc Natl Acad Sci U S A, 77 (1980), No. 7, 4180–4184. [CrossRef] [PubMed] [Google Scholar]
  168. H. G. Othmer, P. Schaap. Oscillatory cAMP signaling in the development of Dictyostelium discoideum. Comments on Theoretical Biology, 5 (1998), 175–282. [Google Scholar]
  169. H. G. Othmer, L. E. Scriven. Instability and dynamic pattern in cellular networks. J. Theor. Biol., 32 (1971), 507–537. [CrossRef] [PubMed] [Google Scholar]
  170. H. G. Othmer, A. Stevens. Aggregation, blowup and collaps: The ABC's of generalized taxis. SIAM J. Appl. Math., 57 (1997), No. 4, 1044–1081. [CrossRef] [MathSciNet] [Google Scholar]
  171. Q. Ouyang, H. L. Swinney. Transition from a uniform state to hexagonal and striped turing patterns. Nature, 352 (1991), 610–612. [CrossRef] [Google Scholar]
  172. K. Painter. Mathematical models for biological pattern formation, chapter Modelling of pigment patterns in fish. Number 121 in IMA Volumes in Mathematics and its Applications, Springer-Verlag, Berlin 2000, (59–82). [Google Scholar]
  173. K. J. Painter, P. K. Maini, H. G. Othmer. Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc. Nat. Acad. Sci., 96 (1999), 5549–5554. [CrossRef] [Google Scholar]
  174. K. J. Painter, P. K. Maini, H. G. Othmer. Development and applications of a model for cellular response to multiple chemotactic cues. J. Math. Biol., 41 (2000), No. 4, 285–314. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  175. D. Parichy. Homology and the evolution of novelty during Danio adult pigment pattern development. J. Exp. Zool. B Mol. Dev. Evol., 308 (2007), 578–590. [CrossRef] [PubMed] [Google Scholar]
  176. D. M. Parichy, D. G. Ransom, B. Paw, L. I. Zon, S. L. Johnson. 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 (2000), 3031–3044. [PubMed] [Google Scholar]
  177. D. M. Parichy, J. M. Turner. Temporal and cellular requirements for Fms signaling during zebrafish adult pigment pattern development. Development, 130 (2003), 817–833. [CrossRef] [PubMed] [Google Scholar]
  178. D. M. Parichy, J. M. Turner. Zebrafish puma mutant decouples pigment pattern and somatic metamorphosis. Dev Biol, 256 (2003), 242–257. [CrossRef] [PubMed] [Google Scholar]
  179. D. M. Parichy, J. M. Turner, N. B. Parker. Essential role for puma in development of postembryonic neural crest-derived cell lineages in zebrafish. Dev Biol, 256 (2003), 221–241. [CrossRef] [PubMed] [Google Scholar]
  180. E. Pate, H. G. Othmer. Differentiation, cell sorting and proportion regulation in the slug stage of Dictyostelium discoideum. J. Theor. Biol., 118 (1986), No. 3, 301–319. [CrossRef] [PubMed] [Google Scholar]
  181. C. S. Patlak. Random walk with persistence and external bias. Bull. of Math. Biophys., 15 (1953), 311–338. [CrossRef] [Google Scholar]
  182. J. E. Pearson, W. Horsthemke. Turing instabilities with nearly equal diffusion coefficients. Journal of Chemical Physics, 90 (1989), No. 3, 1588–1599. [CrossRef] [MathSciNet] [Google Scholar]
  183. F. Peri. The role of EGF and TGF-b signaling in specifying the polarity of the Drosophila egg and embryo. Doctoral, University of Cologne 2001. [Google Scholar]
  184. T. J. Perkins, J. Jaeger, J. Reinitz, L. Glass. Reverse engineering the gap gene network of Drosophila melanogaster. PLoS Comput Biol, 2 (2006), No. 5, 0417–28. [Google Scholar]
  185. A. A. Polezhaev, R. A. Pashkov, A. I. Lobanov, I. B. Petrov. Spatial patterns formed by chemotactic bacteria escherichia coli. Int J Dev Biol, 50 (2006), No. 2-3, 309–14. [CrossRef] [PubMed] [Google Scholar]
  186. R. Prum, R. Torres. Structural colouration of mammalian skin: convergent evolution of coherently scattering dermal collagen arrays. J. Exp. Biol., 207 (2004), 2157–2172. [CrossRef] [PubMed] [Google Scholar]
  187. R. Prum, S. Williamson. Reaction-diffusion models of within-feather pigmentation patterning. Proc. Biol. Sci., 269 (2002), 781–792. [CrossRef] [PubMed] [Google Scholar]
  188. C. V. Rao, J. R. Kirby, A. P. Arkin. Design and diversity in bacterial chemotaxis: A comparative study in Escherichia coli and Bacillus subtilis. PLoS Biol, 2 (2004), No. 2, E49. [Google Scholar]
  189. O. Rauprich, M. Matsushita, C. J. Weijer, F. Siegert, S. E. Esipov, J. A. Shapiro. Periodic phenomena in proteus mirabilis swarm colony development. J Bacteriol, 178 (1996), No. 22, 6525–38. [PubMed] [Google Scholar]
  190. G. Reeves, R. Kalifa, D. Klein, M. Lemmon, S. Shvartsman. Computational analysis of EGFR inhibition by Argos. Dev. Biol., 284 (2005), 523–535. [CrossRef] [PubMed] [Google Scholar]
  191. G. T. Reeves, C. B. Muratov, T. Schupbach, S. Y. Shvartsman. Quantitative models of developmental pattern formation. Dev Cell, 11 (2006), No. 3, 289–300. [CrossRef] [PubMed] [Google Scholar]
  192. T. Roose, S. J. Chapman, P. K. Maini. Mathematical models of avascular tumor growth. SIAM Review, 49 (2007), No. 2, 179–208. [CrossRef] [MathSciNet] [Google Scholar]
  193. G. Ruxton. The possible fitness benefits of striped coat coloration for zebra. Mammal. Rev., 32 (2002), 237–244. [CrossRef] [Google Scholar]
  194. M. J. Saxton. A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model. Biophysical Journal, 92 (2007), No. 4, 1178. [CrossRef] [PubMed] [Google Scholar]
  195. H. Scher, M. Lax. Stochastic transport in a disordered solid. I. theory. Phys. Rev. B, 7 (1973), No. 10, 4491–502. [CrossRef] [MathSciNet] [Google Scholar]
  196. B. Schwanwitsch. On the groundplan of the wing pattern in nymphalids and certain other families of rhopalocara. Proc. Zool. Sci. Lond., 34 (1924) 509–528. [Google Scholar]
  197. J. E. Segall, S. M. Block, H. C. Berg. Temporal comparisons in bacterial chemotaxis. Proc. Nat. Acad. Sci. USA, 83 (1986), 8987–8991. [CrossRef] [Google Scholar]
  198. L. A. Segel. A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM Journal on Applied Mathematics, 32 (1977), 653–665. [CrossRef] [Google Scholar]
  199. T. Sekimura, A. Madzvamuse, A. Wathen, P. K. Maini. A model for colour pattern formation in the butterfly wing of Papilio dardanus. Proc. Biol. Sci., 267 (2000), 851–859. [CrossRef] [PubMed] [Google Scholar]
  200. M. Serpe, D. Umulis, A. Ralston, J. Chen, D. Olson, A. Avanesov, H. Othmer, M. O'Connor, S. Blair. The BMP-binding protein Crossveinless 2 is a short-range, concentration-dependent, biphasic modulator of BMP signaling in Drosophila. Dev. Cell, 14 (2008) 940–953. [Google Scholar]
  201. J. A. Shapiro. Thinking about bacterial populations as multicellular organisms. Ann Rev Microbiol, 52 (1998), 81–104. [CrossRef] [Google Scholar]
  202. T. S. Shimizu, S. V. Aksenov, D. Bray. A spatially extended stochastic model of the bacterial chemotaxis signalling pathway. J. Mol. Biol., 329 (2003), 291–309. [CrossRef] [PubMed] [Google Scholar]
  203. O. Shimmi, D. Umulis, H. G. Othmer, M. B. O'Connor. Facilitated transport of a Dpp/Scw heterodimer by Sog/Tsg leads to robust patterning of the Drosophila blastoderm embryo. Cell, 120 (2005), No. 6, 873–86. [CrossRef] [PubMed] [Google Scholar]
  204. H. Shoji, Y. Iwasa, A. Mochizuki, S. Kondo. Directionality of stripes formed by anisotropic reaction-diffusion models. J. Theor. Biol., 214 (2002), 549–561. [CrossRef] [PubMed] [Google Scholar]
  205. H. Shoji, A. Mochizuki, Y. Iwasa, M. Hirata, T. Watanabe, S. Hioki, S. Kondo. Origin of directionality in the fish stripe pattern. Dev. Dyn., 226 (2003), 627–633. [CrossRef] [PubMed] [Google Scholar]
  206. S. Y. Shvartsman, C. B. Muratov, D. A. Lauffenburger. Modeling and computational analysis of EGF receptor-mediated cell communication in Drosophila oogenesis. Development, 129 (2002), 2577–2589. [PubMed] [Google Scholar]
  207. S. Sick, S. Reinker, J. Timmer, T. Schlake. WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism. Science, 314 (2006), No. 5804, 1447–50. [CrossRef] [PubMed] [Google Scholar]
  208. D. Silver, L. Hou, W. Pavan. The genetic regulation of pigment cell development. Adv. Exp. Med. Biol., 589 (2006), 155–169. [CrossRef] [PubMed] [Google Scholar]
  209. R. Singh, D. Paul, R. K. Jain. Biofilms: implications in bioremediation. J. Math. Biol., 14 (2006), No. 9, 389–97. [Google Scholar]
  210. P. A. Spiro, J. S. Parkinson, H. G. Othmer. A model of excitation and adaptation in bacterial chemotaxis. PNAS., 94 (1997), No. 14, 7263–7268. [CrossRef] [Google Scholar]
  211. A. Spirov, K. Fahmy, M. Schneider, E. Frei, M. Noll, S. Baumgartner. Formation of the bicoid morphogen gradient: an mRNA gradient dictates the protein gradient. Development., 136 (2009), 605-614. [CrossRef] [PubMed] [Google Scholar]
  212. M. Steinberg. Differential adhesion in morphogenesis: a modern view. Curr. Opin. Genet. Dev., 17 (2007), 281–286. [CrossRef] [PubMed] [Google Scholar]
  213. F. Süffert. Die ausbildung des imaginalen flügelschnittes in der schmetterlingspuppe. Z. Morph. Ökol. Tiere, 14 (1929), 338–359. [CrossRef] [Google Scholar]
  214. M. Sugimoto. Morphological colour changes in the medaka, oryzias latipes, after prolonged background adaptation — i. changes in the population and morphology of the melanophores. Comp. Biochem. Physiol., 104A (1993), 513. [Google Scholar]
  215. M. Sugimoto. Morphological color changes in fish: regulation of pigment cell density and morphology. Microsc. Res. Tech., 58 (2002) 496–503. [Google Scholar]
  216. K. Tosney. Long-distance cue from emerging dermis stimulates neural crest melanoblast migration. Dev. Dyn., 229 (2004), 99–108. [CrossRef] [PubMed] [Google Scholar]
  217. P. Trainor. Specification of neural crest cell formation and migration in mouse embryos. Semin. Cell Dev. Biol., 16 (2005), 683–693. [CrossRef] [PubMed] [Google Scholar]
  218. L. Tsimring, H. Levine, I. Aranson, E. Ben-Jacob, I. Cohen, O. Shochet, W. N. Reynolds. Aggregation patterns in stressed bacteria. Phys. Rev. Letts, 75 (1995), No. 9, 1859–1862. [CrossRef] [PubMed] [Google Scholar]
  219. A. M. Turing. The chemical basis of morphogenesis. Phil. Trans. R. Soc. London, 237 (1952), 37–72. [CrossRef] [Google Scholar]
  220. R. Tyson, S. Lubkin, J. Murray. Model and analysis of chemotactic bacterial patterns in a liquid medium. J Math Biol, 38 (1999), 359–375. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  221. R. Tyson, S. R. Lubkin, J. D. Murray. A minimal mechanism for bacterial pattern formation. Proc. R. Soc. Lond. B, 266 (1999), 299–304. [CrossRef] [Google Scholar]
  222. R. Tyson, L. Stern, R. LeVeque. Fractional step methods applied to a chemotaxis model. J. Math. Biol., 41 (2000), 455–75. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  223. D. Umulis, M. O'Connor, H. Othmer. Robustness of embryonic spatial patterning in Drosophila melanogaster. Curr. Top. Dev. Biol., 81 (2008) 65–111. [Google Scholar]
  224. D. M. Umulis, M. Serpe, M. B. O'Connor, H. G. Othmer. Robust, bistable patterning of the dorsal surface of the Drosophila embryo. Proc Natl Acad Sci U S A, 103 (2006), No. 31, 11613–8. [CrossRef] [PubMed] [Google Scholar]
  225. C. Varea, J. L. Aragon, R. A. Barrio. Confined turing patterns in growing systems. Phys. Rev. E., 56 (1997), 1250–1253. [CrossRef] [Google Scholar]
  226. B. J. Varnum-Finney, E. Voss, D. R. Soll. Frequency and orientation of pseudopod formation of Dictyostelium discoideum amebae chemotaxing in a spatial gradient: Further evidence for a temporal mechanism. 8 (1987), No. 1, 18–26. [Google Scholar]
  227. G. von Dassow, E. Meir, E. M. Munro, G. M. Odell. The segment polarity network is a robust developmental module. Nature, 406 (2000), No. 6792, 188–92. [CrossRef] [PubMed] [Google Scholar]
  228. G. H. Wadhams, J. P. Armitage. Making sense of it all: bacterial chemotaxis. Nat. Rev. Mol. Cell Biol., 5 (2004), No. 12, 1024–37. [CrossRef] [PubMed] [Google Scholar]
  229. M. Walters, V. Sperandio. Quorum sensing in Escherichia coli and Salmonella. Int J Med Microbiol., 296 (2006), No. 2-3, 125–31. [Google Scholar]
  230. X. Wang, R. Harris, L. Bayston, H. Ashe. Type IV collagens regulate BMP signalling in Drosophila. Nature, 455 (2008), 72–77. [CrossRef] [PubMed] [Google Scholar]
  231. Y. C. Wang, E. L. Ferguson. Spatial bistability of Dpp-receptor interactions during Drosophila dorsal-ventral patterning. Nature, 434 (2005), No. 7030, 229–34. [CrossRef] [PubMed] [Google Scholar]
  232. M. P. Weir, C. W. Lo. Gap-junctional communication. compartments in the Drosophila wing imaginal disc. Developmental Biology, 102 (1984) 130–146. [Google Scholar]
  233. L. Werdelin, L. Olsson. How the leopard got its spots: a phylogenetic view of the evolution of felid coat patterns. Biol. J. Linn. Soc., 62 (1997) 383–400. [Google Scholar]
  234. N. A. Whitehead, A. M. Barnard, H. Slater, N. J. Simpson, G. P. Salmond. Quorum-sensing in gram-negative bacteria. FEMS Microbiol Rev., 25 (2001), No. 4, 365–404. [Google Scholar]
  235. L. Wolpert. Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol., 25 (1969), No. 1, 1–67. [CrossRef] [PubMed] [Google Scholar]
  236. L. Wolpert. Positional information and pattern formation. Curr. Topics in Dev. Biol., 6 (1971), 183–224. [CrossRef] [Google Scholar]
  237. D. E. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. O. Budrene, H. C. Berg. Spatio-temporal patterns generated by salmonella typhimurium . Biophys. J., 68 (1995), No. 5, 2181–2189. [CrossRef] [PubMed] [Google Scholar]
  238. C. Xue. Mathematical models of taxis-driven bacterial pattern formation. Ph.D. thesis, University of Minnesota 2008. [Google Scholar]
  239. C. Xue, H. G. Othmer. Radial and spiral streams formation in bacterium Proteus mirabilis colonies 2009, preprint. [Google Scholar]
  240. C. Xue, H. G. Othmer. Multiscale models of taxis-driven patterning in bacterial populations. SIAM J. Appl. Math., to appear (2009). [Google Scholar]
  241. N. Yakoby, C. A. Bristow, I. Gouzman, M. P. Rossi, Y. Gogotsi, T. Schpbach, S. Y. Shvartsman. Systems-level questions in Drosophila oogenesis. Syst Biol (Stevenage), 152 (2005), 276–284. [PubMed] [Google Scholar]
  242. M. Yamaguchi, E. Yoshimoto, S. Kondo. Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism. Proc. Natl. Acad. Sci. U.S.A., 104 (2007), 4790–4793. [CrossRef] [PubMed] [Google Scholar]
  243. D. Young. A local activator-inhibitor model of vertebrate skin patterns. Math. Biosci., 72 (1984), 51–58. [CrossRef] [MathSciNet] [Google Scholar]
  244. Y. Zhang, A. Lander, Q. Nie. Computational analysis of BMP gradients in dorsal-ventral patterning of the zebrafish embryo. J. Theor. Biol., 248 (2007), 579–589. [CrossRef] [PubMed] [Google Scholar]

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