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The Intersection of Theory and Application inElucidating Pattern Formation in Developmental Biology

Published online by Cambridge University Press:  11 July 2009

H. G. Othmer ,
K. Painter ,
D. Umulis  and
C. Xue
H. G. Othmer*
Affiliation:
School of Mathematics and Digital Technology Center, University of Minnesota, Minneapolis, MN 55455 USA
K. Painter
Affiliation:
Department of Mathematics, Department of Mathematics and Maxwell, Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK
D. Umulis
Affiliation:
Agricultural & Biological Engineering, Purdue University, West Lafayette, IN USA 47907 USA
C. Xue
Affiliation:
Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210 USA
*
othmer@math.umn.edu
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Abstract

We discuss theoretical and experimental approaches to three distinctdevelopmental systems that illustrate how theory can influence experimental workand vice-versa. The chosen systems – Drosophila melanogaster,bacterial pattern formation, and pigmentation patterns – illustrate thefundamental physical processes of signaling, growth and cell division, and cellmovement involved in pattern formation and development. These systems exemplifythe current state of theoretical and experimental understanding of how theseprocesses produce the observed patterns, and illustrate how theoretical andexperimental approaches can interact to lead to a better understanding ofdevelopment. As John Bonner said long ago
`We have arrived at the stage where models are useful to suggestexperiments, and the facts of the experiments in turn lead to new and improvedmodels that suggest new experiments. By this rocking back and forth between thereality of experimental facts and the dream world of hypotheses, we can moveslowly toward a satisfactory solution of the major problems of developmentalbiology.'

Keywords

robustnesschemotaxisbacterial patternsanimal coat markings

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Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 4: Morphogenesis , 2009 , pp. 3 - 82
Copyright
© EDP Sciences, 2009

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References

Adler, J.. Chemotaxis in bacteria. Science, 153 (1966), 708–716. CrossRef
G. Allen, R. Steene, M. Allen. A guide to angelfishes and butterflyfishes. Odyssey, 1998.
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.
Araujo, R. P., McElwain, D. L. S.. A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull. Math. Biol., 66 (2004), No. 5, 1039–1091. CrossRef
Arcuri, P., Murray, J.. Pattern sensitivity to boundary and initial conditions in reaction-diffusion models. J. Math. Biol., 24 (1986), 141–165. CrossRef
Asai, R., Taguchi, E., Kume, Y., Saito, M., Kondo, S.. Zebrafish leopard gene as a component of the putative reaction-diffusion system. Mech Dev, 89 (1999), 87–92. CrossRef
Ashkenazi, M., Othmer, H. G.. Spatial patterns in coupled biochemical oscillators. Jour. Math. Biol., 5 (1978), 305–350.
Atkinson, S., Chang, C. Y., Sockett, R. E., Camara, M., Williams, P.. Quorum sensing in yersinia enterocolitica controls swimming and swarming motility. J Bacteriol, 188 (2006), No. 4, 1451–61. CrossRef
J. Bagnara, M. Hadley. Chromatophores and color change. Prentice-Hall, Eaglewood Cliffs, New Jersey. 1973.
Baker, R. E., Gaffney, E. A., Maini, P. K.. Partial differential equations for self-organization in cellular and developmental biology. Nonlinearity, 21 (2008), No. 11, 251–290. CrossRef
Baker, R. E., Maini, P. K.. A mechanism for morphogen-controlled domain growth. J Math Biol, 54 (2007), 597–622. CrossRef
Bard, J.. A unity underlying the different zebra striping patterns. J. Zool., 183 (1977), 527–539.
Bard, J.. A model for generating aspects of zebra and other mammalian coat patterns. J. Theor. Biol., 93 (1981), 363–385. CrossRef
Bard, J., French, V.. Butterfly wing patterns: how good a determining mechanism is the simple diffusion of a single morphogen? J. Embryol. Exp. Morph., 84 (1984), 255–274.
Barrio, R., Varea, C., Aragn, J., Maini, P.. A two-dimensional numerical study of spatial pattern formation in interacting Turing systems. Bull. Math. Biol., 61 (1999), 483–505. CrossRef
Bassler, B. L.. How bacteria talk to each other: regulation of gene expression by quorum sensing. Curr Opin Microbiol, 2 (1999), No. 6, 582–587. CrossRef
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–.
Ben-Jacob, E., Cohen, I., Levine, H.. Cooperative self-organization of microorganisms. Advances in Physics, 49 (2000), No. 4, 395–554. CrossRef
E. Ben-Jacob, I. Cohen, O. Shochet, I. Aranson, H. Levine. Complex bacterial patterns. 373 (1995), No. 6515, 566–557.
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.
Ben-Zvi, D., Shilo, B., Fainsod, A., Barkai, N.. Scaling of the BMP activation gradient in Xenopus embryos. Nature, 453 (2008), 1205–1211. CrossRef
H. Berg. Random walks in biology 1983.
Berg, H. C.. Motile behavior of bacteria. Physics Today, 53 (2000), No. 1, 24–29. CrossRef
H. C. Berg. http://webmac.rowland.org/labs/bacteria/movies/others/ . (2008).
Betterton, M. D., Brenner, M. P.. Collapsing bacterial cylinders. Physical Review E, 64 (2001), 061904. CrossRef
Bisset, K., Douglas, C.. A continuous study of morphological phase in the swarm of proteus. J. Med. Microbiol., 9 (1976), 229–31. CrossRef
Bolos, V., Grego-Bessa, J., de la Pompa, J. L.. Notch signaling in development and cancer. Endocr Rev, 28 (2007), No. 3, 339–63. CrossRef
J. T. Bonner. The development of Dictyostelium , Chapter Comparative Biology of Cellular Slime Molds. Academic Press 1982, 1–33.
R. B. Bourret, K. A. Borkovich, M. I. Simon. Signal transduction pathways involving protein phosphorylation in prokaryotes. 60 (1991), 401–441.
Brakefield, P., French, V.. Eyespot development on butterfly wings: the epidermal response to damage. Dev. Biol., 168 (1995), 98–111. CrossRef
M. P. Brenner, L. S. Levitov, E. O. Budrene. Physical mechanisms for chemotactic pattern formation by bacteria. 74 (1998), No. 4, 1677–1693.
Brunetti, C., Selegue, J., Monteiro, A., French, V., Brakefield, P., Carroll, S.. The generation and diversification of butterfly eyespot color patterns. Curr. Biol., 11 (2001), 1578–1585. CrossRef
E. O. Budrene. Personal communication 2005.
Budrene, E. O., Berg, H. C.. Complex patterns formed by motile cells of Escherichia coli . Nature, 349 (1991), No. 6310, 630–633. CrossRef
E. O. Budrene, H. C. Berg. Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature, 376 (1995), No. 6535, 49–53.
Caicedo-Carvajal, C., Shinbrot, T.. In silico zebrafish pattern formation. Dev. Biol., 315 (2008), 397–403. CrossRef
Carroll, S., Gates, J., Keys, D., Paddock, S., Panganiban, G., Selegue, J., Williams, J.. Pattern formation and eyespot determination in butterfly wings. Science, 265 (1994), 109–114. CrossRef
Casanova, J., Struhl, G.. The torso receptor localizes as well as transduces the spatial signal specifying terminal body pattern in Drosophila. Nature, 362 (1993), 152–155. CrossRef
V. Castets, E. Dulos, P. D. Kepper. Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. 64 (1990), No. 24, 2953–2956.
C. M. Child. Patterns and problems of development. University of Chicago Press 1941.
S. Childress, J. K. Percus. Nonlinear aspects of chemotaxis. 56 (1981) 217–237.
Claxton, J. H.. 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
J. H. Claxton. Developmental origin of even spacing between the microchaetes of Drosophila melanogaster. Aust J Biol Sci, 29 (1976) 131–135.
Cluzel, P., Surette, M., Leibler, S.. An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells. Science, 287 (2000), 1652–1655. CrossRef
E. Conway, D. Hoff, J. Smoller. Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math., (1977).
Coppey, M., Berezhkovskii, A., Kim, Y., Boettiger, A., Shvartsman, S.. Modeling the bicoid gradient: diffusion and reversible nuclear trapping of a stable protein. Dev. Biol., 312 (2007), 623–630. CrossRef
Coppey, M., Boettiger, A., Berezhkovskii, A., Shvartsman, S.. Nuclear trapping shapes the terminal gradient in the Drosophila embryo. Curr. Biol., 18 (2008), 915–919. CrossRef
Crampin, E., Hackborn, W., Maini, P.. Pattern formation in reaction-diffusion models with nonuniform domain growth. Bull. Math. Biol., 64 (2002), 747–769. CrossRef
Crampin, E. J., Gaffney, E. A., Maini, P. K.. Reaction and diffusion on growing domains: Scenarios for robust pattern formation. Bulletin of Mathematical Biology, 61 (1999), No. 6, 1093–1120. CrossRef
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.
Crick, F. H.. Diffusion in embryogenesis. Nature, 225 (1970), 420–422. CrossRef
M. C. Cross, P. C. Hohenberg. Pattern formation out of equilibrium. 65 (1993), No. 3, 851–1112.
Daniels, R., Vanderleyden, J., Michiels, J.. Quorum sensing and swarming migration in bacteria. FEMS Microbiol Rev., 28 (2004), 261–89. CrossRef
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.
Dilão, R., Sainhas, J.. Modelling butterfly wing eyespot patterns. Proc. Biol. Sci., 271 (2004), 1565–1569. CrossRef
Dillon, R., Maini, P. K., Othmer, H. G.. 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
Dillon, R., Othmer, H. G.. A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud. J. Theor. Biol., 197 (1999), No. 3, 295–330. CrossRef
Driever, W., Nüsslein-Volhard, C.. A gradient of bicoid protein in Drosophila embryos. Cell, 54 (1988), No. 1, 83–93. CrossRef
Driever, W., Nusslein-Volhard, C.. The bicoid protein determines position in the Drosophila embryo in a concentration-dependent manner. Cell, 54 (1988), No. 1, 95–104. CrossRef
Eglen, S.. Development of regular cellular spacing in the retina: theoretical models. Mathematical Medicine and Biology, 23 (2006), No. 2, 79–99. CrossRef
Eldar, A., Dorfman, R., Weiss, D., Ashe, H., Shilo, B. Z., Barkai, N.. Robustness of the BMP morphogen gradient in drosophila embryonic patterning. Nature, 419 (2002), No. 6904, 304–308. CrossRef
R. Erban. From individual to collective behavior in biological systems. Ph.D. thesis, University of Minnesota 2005.
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.
Erban, R., Othmer, H. G.. From individual to collective behavior in bacterial chemotaxis. SIAM J. Appl. Math., 65 (2004), No. 2, 361–391. CrossRef
Erban, R., Othmer, H. G.. Taxis equations for amoeboid cells. J Math Biol, 54 (2007), 847–885. CrossRef
Ermentrout, B.. Stripes or spots? nonlinear effects in bifurcation of reaction-diffusion equations on the square. Proc. Roy. Soc. Lond. A., 434 (1991), 413–417. CrossRef
T. Evans, J. Marcus. A simulation study of the genetic regulatory hierarchy for butterfly eyespot focus determination. Evol. Dev., 8 (2006) 273–283.
Filloux, A., Vallet, I.. Biofilm: set-up and organization of a bacterial community. Med Sci (Paris), 19 (2003), No. 1, 77–83. CrossRef
Fisher, P. R., Merkl, R., Gerisch, G.. 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
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.
Ford, R. M., Lauffenburger, D. A.. Measurement of bacterial random motility and chemotaxis coefficients: II: Application of single-cell-based mathematical model. Biotechnol. Bioeng., 37 (1991), 661–672. CrossRef
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.
Fowlkes, C., Hendriks, C., Kernen, S., Weber, G., Rbel, O., Huang, M., Chatoor, S., DePace, A., Simirenko, L., Henriquez, C., Beaton, A., Weiszmann, R., Celniker, S., Hamann, B., Knowles, D., Biggin, M., Eisen, M., Malik, J.. A quantitative spatiotemporal atlas of gene expression in the Drosophila blastoderm. Cell, 133 (2008), 364–374. CrossRef
Fricke, H.. Juvenile-adult colour patterns and coexistence in the territorial coral reef fish Pomacanthus imperator. Marine Ecology, 1 (1980), 133–141. CrossRef
A. Gierer, H. Meinhardt. A theory of biological pattern formation. 12 (1972), No. 1, 30–39.
Glass, L.. Stochastic generation of regular distributions. Science, 180 (1973), 1061–1063. CrossRef
Goentoro, L. A., Reeves, G. T., Kowal, C. P., Martinelli, L., Schpbach, T., Shvartsman, S. Y.. Quantifying the Gurken morphogen gradient in Drosophila oogenesis. Dev. Cell, 11 (2006), 263–272. CrossRef
Goryachev, A. B., Toh, D. J., Lee, T.. 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
Graván, C., Lahoz-Beltra, R.. Evolving morphogenetic fields in the zebra skin pattern based on turing's morphogen hypothesis. Int. J. Appl. Math. Comput. Sci., 14 (2004), 351–361.
Greenberg, E. P.. Bacterial communication: Tiny teamwork. Nature, 424 (2003), 134. CrossRef
Gregor, T., Bialek, W., de Ruyter van, R. R. 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
Gregor, T., Tank, D. W., Wieschaus, E. F., Bialek, W.. Probing the limits to positional information. Cell, 130 (2007), No. 1, 153–164. CrossRef
Gregor, T., Wieschaus, E. F., McGregor, A. P., Bialek, W., Tank, D. W.. Stability and nuclear dynamics of the bicoid morphogen gradient. Cell, 130 (2007), No. 1, 141–152. CrossRef
Grunwald, D., Eisen, J.. Headwaters of the zebrafish – emergence of a new model vertebrate. Nat. Rev. Genet., 3 (2002), 717–724. CrossRef
Haffter, P., Odenthal, J., Mullins, M. C., Lin, S., Farrell, M. J., Vogelsang, L., 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
Berg, H., Brown, D.. Chemotaxis in Escherichia Coli analyzed by three-dimensional tracking. Nature, 239 (1972), 502–507. CrossRef
D. Headon, K. J. Painter. Stippling the skin: Generation of anatomical periodicity by reaction-diffusion mechanisms. Submitted to MMNP, (2008).
Herrero, M., Velázquez, J.. Chemotactic collapse for the Keller-Segel model. J. Math. Biol., 35 (1996), 177–194. CrossRef
Hillen, T., Othmer, H. G.. The diffusion limit of transport equation derived from velocity-jump processes. Siam J. Appl. Math., 61 (2000), No. 3, 751–775.
Hillen, T., Painter, K.. A user's guide to pde models for chemotaxis. J. Math. Biol., 58 (2008), 183–217. CrossRef
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.
Horstmann, D.. From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I. Jahresbericht der DMV, 105 (2003), No. 3, 103–165.
D. Horstmann. A constructive approach to traveling waves in chemotaxis. Journal of Nonlinear Science, 14 (2004), 1–25(25).
Houchmandzadeh, B., Wieschaus, E., Leibler, S.. Establishment of developmental precision and proportions in the early Drosophila embryo. Nature, 415 (2002), No. 6873, 798–802. CrossRef
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.
Jaeger, J., Reinitz, J.. On the dynamic nature of positional information. Bioessays, 28 (2006), No. 11, 1102–11. CrossRef
Jaeger, J., Surkova, S., Blagov, M., Janssens, H., Kosman, D., Kozlov, K., 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
W. Jäger, S. Luckhaus. On explosions of solutions to a system of partial differential equations modelling chemotaxis. 329 (1992), No. 2, 819–824.
Johnson, S. L., Africa, D., Walker, C., Weston, J. A.. Genetic control of adult pigment stripe development in zebrafish. Dev Biol, 167 (1995), 27–33. CrossRef
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.
Kaiser, D.. Coupling cell movement to multicellular development in myxobacteria. Nat. Rev. Microbiol., 1 (2003), No. 1, 45–54. CrossRef
Kaiser, D.. Myxococcus — from single-cell polarity to complex multicellular patterns. Annual Review of Genetics, 42 (2008), No. 1, 109–130. CrossRef
Keller, E. F., Odell, G. M.. Necessary and sufficient conditions for chemotactic bands. Mathematical Biosciences, 27 (1975), 309–317. CrossRef
Keller, E. F., Segel, L. A.. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol., 26 (1970), 399–415. CrossRef
Keller, E. F., Segel, L. A.. Model for chemotaxis. J. Theor. Biol., 30 (1971), 225–234. CrossRef
Keller, E. F., Segel, L. A.. Traveling bands of chemotactic bacteria: A theoretical analysis. J. Theor. Biol., 30 (1971), 235–248. CrossRef
Kelsh, R. N., Brand, M., Jiang, Y. J., Heisenberg, C. P., Lin, S., Haffter, P., Odenthal, J., Mullins, M. C., van Eeden, F. J.., 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
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
Kerszberg, M., Wolpert, L.. Mechanisms for positional signalling by morphogen transport: a theoretical study. J. Theor. Biol., 191 (1998), No. 1, 103–114. CrossRef
Kicheva, A., Pantazis, P., Bollenbach, T., Kalaidzidis, Y., Bittig, T., Julicher, F., Gonzalez-Gaitan, M.. Kinetics of morphogen gradient formation. Science, 315 (2007), No. 5811, 521–525. CrossRef
Koch, P., Keys, D., Rocheleau, T., Aronstein, K., Blackburn, M., Carroll, S., 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.
Kolter, R., Greenberg, E. P.. Microbial sciences: The superficial life of microbes. Nature, 441 (2006), 300–302. CrossRef
Kondo, S., Asai, R.. A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature, 376 (1995), 675–768. CrossRef
Kulesa, P., Cruywagen, G., Lubkin, S., Maini, P., Sneyd, J., Ferguson, M., Murray, J.. On a model mechanism for the spatial patterning of teeth primordia in the alligator. J. Theor. Biol., 180 (1996), 287–296. CrossRef
Lander, A. D.. Morpheus unbound: reimagining the morphogen gradient. Cell, 128 (2007), No. 2, 245–256. CrossRef
A. D. Lander, Q. Nie, F. Y. M. Wan. Do morphogen gradients arise by diffusion? Dev. Cell, 2 (2002), No. 6, 785–96.
I. R. Lapidus, R. Schiller. A model for traveling bands of chemotactic bacteria. Biophys J., 22 (1978), No. 1, 1–13.
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.
N. Le Douarin, C. Kalcheim. The neural crest. CUP, Cambridge, 2nd edition 1999.
I. Lengyel, I. R. Epstein. Modelling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system. Science, 251 (1991) 650–652.
Liaw, S., Yang, C., Liu, R., Hong, J.. Turing model for the patterns of lady beetles. Phys Rev E Stat Nonlin Soft Matter Phys, 64 (2001), 041909. CrossRef
Liu, R., Liaw, S., Maini, P.. 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
Lux, R., Shi, W.. Chemotaxis-guided movements in bacteria. Crit. Rev. Oral. Biol. Med., 15 (2004), No. 4, 207–20. CrossRef
M. Lyons, L. Harrison. Stripe selection: An intrinsic property of some pattern-forming models with nonlinear dynamics. Dev. Dyn., 195 (1992) 201–215.
Maderspacher, F., Nüsslein-Volhard, C.. Formation of the adult pigment pattern in zebrafish requires leopard and obelix dependent cell interactions. Development, 130 (2003), 3447–3457. CrossRef
Madzvamuse, A., Maini, P., Wathen, A., Sekimura, T.. A predictive model for color pattern formation in the butterfly wing of papilio dardanus. Hiroshima Math. J., 32 (2002), 325–336.
Maini, P. K., Painter, K. J., Nguyen, H. P. C.. Spatial pattern formation in chemical and biological systems. J Chem Soc Faraday Trans, 93 (1997), No. 20, 3601–10. CrossRef
N. V. Mantzaris, S. Webb, H. G. Othmer. Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol., 49 (2004), No. 2, 111–87.
Marcus, J., Evans, T.. A simulation study of mutations in the genetic regulatory hierarchy for butterfly eyespot focus determination. BioSystems, 93 (2008), 250–255. CrossRef
M. McClure. Development and evolution of melanophore patterns in fishes of the genus Danio (Teleostei: Cyprinidae). J. Morphol., 241 (1999) 83–105.
H. Meinhardt. Models of biological pattern formation. Academic Press, New York 1980.
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.
H. Meinhardt, P. Prusinkiewicz, D. Fowler. The algorithmic beauty of sea shells. Springer 2003.
Mello, B. A., Tu, Y.. 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
Míguez, D., Muñuzuri, A.. On the orientation of stripes in fish skin patterning. Biophys. Chem., 124 (2006), 161–167. CrossRef
Milos, N., Dingle, A. D.. 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
Mittal, N., Budrene, E. O., Brenner, M. P., Oudenaarden, A.. Motility of Escherichia coli cells in clusters formed by chemotactic aggregation. Proc. Natl. Acad. Sci. (USA), 100 (2003), No. 23, 13259–63. CrossRef
Mizutani, C. M., Nie, Q., Wan, F. Y., Zhang, Y. T., Vilmos, P., Sousa-Neves, R., Bier, E., Marsh, J. L., Lander, A. D.. Formation of the BMP activity gradient in the Drosophila embryo. Dev. Cell, 8 (2005), No. 6, 915–24. CrossRef
Monteiro, A., French, V., Smit, G., Brakefield, P., Metz, J.. Butterfly eyespot patterns: evidence for specification by a morphogen diffusion gradient. Acta Biotheor., 49 (2001), 77–88. CrossRef
Mooney, J. R., Nagorcka, B. N.. Spatial patterns produced by a reaction-diffusion system in primary hair follicles. J. Theor. Biol., 115 (1985), 299–317. CrossRef
J. Moreira, A. Deutsch. Pigment pattern formation in zebrafish during late larval stages: a model based on local interactions. Dev. Dyn., 232 (2005).
Mou, C., Jackson, B., Schneider, P., Overbeek, P. A., Headon, D. J.. Generation of the primary hair follicle pattern. Proc. Natl. Acad. Sci. U.S.A., 103 (2006), 9075–9080. CrossRef
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).
Murray, J. D.. On pattern-formation mechanisms for leipdopteran wing patterns and mammalian coat markings. Phil. Trans. Roy. Soc. Lond. B, 295 (1981), 473–496. CrossRef
Murray, J. D.. A pre-pattern formation mechanism for animal coat markings. J. Theor. Biol., 88 (1981), 161–199. CrossRef
J. D. Murray. Mathematical biology ii: Spatial models and biomedical applications. Springer, New York, 3rd edition 2003.
Murray, J. D., Deeming, D., Ferguson, M.. 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
Murray, J. D., Myerscough, M.. Pigmentation pattern formation on snakes. J. Theor. Biol., 149 (1991), 339–360. CrossRef
J. D. Murray. Mathematical biology, volume 19 of Biomathematics 1989.
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.
Nagorcka, B. N., Mooney, J. R.. The role of a reaction-diffusion system in the initiation of primary hair follicles. J. Theor. Biol., 114 (1985), 243–272. CrossRef
Naitoh, T., Morioka, A., Omura, Y.. 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.
J. Nelson. Fishes of the world. John Wiley and Sons, New York, 3rd edition 1993.
Neuman-Silberberg, F. S., Schupbach, T.. 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. CrossRef
Nijhout, H.. Wing pattern formation in lepidoptera: a model. J. Exp. Zool., 206 (1978), 119–136. CrossRef
Nijhout, H.. A comprehensive model for color pattern formation in butterflies. Proc. Roy. Soc. Lond. B, 239 (1990), 81–113. CrossRef
Nijhout, H., Maini, P. K., Madzvamuse, A., Wathen, A., Sekimura, T.. Pigmentation pattern formation in butterflies: experiments and models. C. R. Biol., 326 (2003), 717–727. CrossRef
O'Connor, M. B., Umulis, D. M., Othmer, H. G., Blair, S. S.. Shaping BMP morphogen gradients in the Drosophila embryo and pupal wing. Development, 133 (2006), 183–93. CrossRef
Odenthal, J., Rossnagel, K., Haffter, P., Kelsh, R. N., Vogelsang, E., Brand, M., van Eeden, F. J.., 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.
Otaki, J.. Physiologically induced color-pattern changes in butterfly wings: mechanistic and evolutionary implications. J. Insect Physiol., 54 (2008), 1099–1112. CrossRef
H. G. Othmer. Interactions of reaction and diffusion in open systems. Ph.D. thesis, University of Minnesota, Minneapolis 1969.
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).
H. G. Othmer. Synchronized and differentiated modes of cellular dynamics. In H. Haken, editor, Dynamics of Synergetic Systems, Springer-Verlag.
Othmer, H. G., Aldridge, J. A.. The effects of cell density and metabolite flux on cellular dynamics. J. Math. Biol., 5 (1978), 169–200. CrossRef
H. G. Othmer, S. R. Dunbar, W. Alt. Models of dispersal in biological systems. J. Math. Biol., 26 (1988), No. 3, 263–298.
Othmer, H. G., Hillen, T.. The diffusion limit of transport equations, Part II: chemotaxis equations. SIAM JAM, 62 (2002), 1222–1260.
Othmer, H. G., Pate, E. F.. Scale-invariance in reaction-diffusion models of spatial pattern formation. Proc Natl Acad Sci U S A, 77 (1980), No. 7, 4180–4184. CrossRef
Othmer, H. G., Schaap, P.. Oscillatory cAMP signaling in the development of Dictyostelium discoideum. Comments on Theoretical Biology, 5 (1998), 175–282.
Othmer, H. G., Scriven, L. E.. Instability and dynamic pattern in cellular networks. J. Theor. Biol., 32 (1971), 507–537. CrossRef
Othmer, H. G., Stevens, A.. Aggregation, blowup and collaps: The ABC's of generalized taxis. SIAM J. Appl. Math., 57 (1997), No. 4, 1044–1081.
Ouyang, Q., Swinney, H. L.. Transition from a uniform state to hexagonal and striped turing patterns. Nature, 352 (1991), 610–612. CrossRef
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).
Painter, K. J., Maini, P. K., Othmer, H. G.. Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc. Nat. Acad. Sci., 96 (1999), 5549–5554. CrossRef
Painter, K. J., Maini, P. K., Othmer, H. G.. Development and applications of a model for cellular response to multiple chemotactic cues. J. Math. Biol., 41 (2000), No. 4, 285–314. CrossRef
Parichy, D.. Homology and the evolution of novelty during Danio adult pigment pattern development. J. Exp. Zool. B Mol. Dev. Evol., 308 (2007), 578–590. CrossRef
Parichy, D. M., Ransom, D. G., Paw, B., Zon, L. I., Johnson, S. L.. 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
Parichy, D. M., Turner, J. M.. Temporal and cellular requirements for Fms signaling during zebrafish adult pigment pattern development. Development, 130 (2003), 817–833. CrossRef
Parichy, D. M., Turner, J. M.. Zebrafish puma mutant decouples pigment pattern and somatic metamorphosis. Dev Biol, 256 (2003), 242–257. CrossRef
Parichy, D. M., Turner, J. M., Parker, N. B.. Essential role for puma in development of postembryonic neural crest-derived cell lineages in zebrafish. Dev Biol, 256 (2003), 221–241. CrossRef
Pate, E., Othmer, H. G.. Differentiation, cell sorting and proportion regulation in the slug stage of Dictyostelium discoideum . J. Theor. Biol., 118 (1986), No. 3, 301–319. CrossRef
Patlak, C. S.. Random walk with persistence and external bias. Bull. of Math. Biophys., 15 (1953), 311–338. CrossRef
Pearson, J. E., Horsthemke, W.. Turing instabilities with nearly equal diffusion coefficients. Journal of Chemical Physics, 90 (1989), No. 3, 1588–1599. CrossRef
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.
Perkins, T. J., Jaeger, J., Reinitz, J., Glass, L.. Reverse engineering the gap gene network of Drosophila melanogaster. PLoS Comput Biol, 2 (2006), No. 5, 0417–28. CrossRef
Polezhaev, A. A., Pashkov, R. A., Lobanov, A. I., Petrov, I. B.. Spatial patterns formed by chemotactic bacteria escherichia coli. Int J Dev Biol, 50 (2006), No. 2-3, 309–14. CrossRef
Prum, R., Torres, R.. Structural colouration of mammalian skin: convergent evolution of coherently scattering dermal collagen arrays. J. Exp. Biol., 207 (2004), 2157–2172. CrossRef
Prum, R., Williamson, S.. Reaction-diffusion models of within-feather pigmentation patterning. Proc. Biol. Sci., 269 (2002), 781–792. CrossRef
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.
Rauprich, O., Matsushita, M., Weijer, C. J., Siegert, F., Esipov, S. E., Shapiro, J. A.. Periodic phenomena in proteus mirabilis swarm colony development. J Bacteriol, 178 (1996), No. 22, 6525–38. CrossRef
Reeves, G., Kalifa, R., Klein, D., Lemmon, M., Shvartsman, S.. Computational analysis of EGFR inhibition by Argos. Dev. Biol., 284 (2005), 523–535. CrossRef
Reeves, G. T., Muratov, C. B., Schupbach, T., Shvartsman, S. Y.. Quantitative models of developmental pattern formation. Dev Cell, 11 (2006), No. 3, 289–300. CrossRef
Roose, T., Chapman, S. J., Maini, P. K.. Mathematical models of avascular tumor growth. SIAM Review, 49 (2007), No. 2, 179–208. CrossRef
Ruxton, G.. The possible fitness benefits of striped coat coloration for zebra. Mammal. Rev., 32 (2002), 237–244. CrossRef
Saxton, M. J.. A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model. Biophysical Journal, 92 (2007), No. 4, 1178. CrossRef
Scher, H., Lax, M.. Stochastic transport in a disordered solid. I. theory. Phys. Rev. B, 7 (1973), No. 10, 4491–502. CrossRef
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.
Segall, J. E., Block, S. M., Berg, H. C.. Temporal comparisons in bacterial chemotaxis. Proc. Nat. Acad. Sci. USA, 83 (1986), 8987–8991. CrossRef
Segel, L. A.. A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM Journal on Applied Mathematics, 32 (1977), 653–665. CrossRef
Sekimura, T., Madzvamuse, A., Wathen, A., Maini, P. K.. A model for colour pattern formation in the butterfly wing of Papilio dardanus. Proc. Biol. Sci., 267 (2000), 851–859. CrossRef
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.
Shapiro, J. A.. Thinking about bacterial populations as multicellular organisms. Ann Rev Microbiol, 52 (1998), 81–104. CrossRef
Shimizu, T. S., Aksenov, S. V., Bray, D.. A spatially extended stochastic model of the bacterial chemotaxis signalling pathway. J. Mol. Biol., 329 (2003), 291–309. CrossRef
Shimmi, O., Umulis, D., Othmer, H. G., O'Connor, M. B.. 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
Shoji, H., Iwasa, Y., Mochizuki, A., Kondo, S.. Directionality of stripes formed by anisotropic reaction-diffusion models. J. Theor. Biol., 214 (2002), 549–561. CrossRef
Shoji, H., Mochizuki, A., Iwasa, Y., Hirata, M., Watanabe, T., Hioki, S., Kondo, S.. Origin of directionality in the fish stripe pattern. Dev. Dyn., 226 (2003), 627–633. CrossRef
Shvartsman, S. Y., Muratov, C. B., Lauffenburger, D. A.. Modeling and computational analysis of EGF receptor-mediated cell communication in Drosophila oogenesis. Development, 129 (2002), 2577–2589.
Sick, S., Reinker, S., Timmer, J., Schlake, T.. WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism. Science, 314 (2006), No. 5804, 1447–50. CrossRef
Silver, D., Hou, L., Pavan, W.. The genetic regulation of pigment cell development. Adv. Exp. Med. Biol., 589 (2006), 155–169. CrossRef
Singh, R., Paul, D., Jain, R. K.. Biofilms: implications in bioremediation. J. Math. Biol., 14 (2006), No. 9, 389–97.
Spiro, P. A., Parkinson, J. S., Othmer, H. G.. A model of excitation and adaptation in bacterial chemotaxis. PNAS., 94 (1997), No. 14, 7263–7268. CrossRef
Spirov, A., Fahmy, K., Schneider, M., Frei, E., Noll, M., Baumgartner, S.. Formation of the bicoid morphogen gradient: an mRNA gradient dictates the protein gradient. Development., 136 (2009), 605-614. CrossRef
Steinberg, M.. Differential adhesion in morphogenesis: a modern view. Curr. Opin. Genet. Dev., 17 (2007), 281–286. CrossRef
Süffert, F.. Die ausbildung des imaginalen flügelschnittes in der schmetterlingspuppe. Z. Morph. Ökol. Tiere, 14 (1929), 338–359. CrossRef
Sugimoto, M.. 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.
M. Sugimoto. Morphological color changes in fish: regulation of pigment cell density and morphology. Microsc. Res. Tech., 58 (2002) 496–503.
Tosney, K.. Long-distance cue from emerging dermis stimulates neural crest melanoblast migration. Dev. Dyn., 229 (2004), 99–108. CrossRef
Trainor, P.. Specification of neural crest cell formation and migration in mouse embryos. Semin. Cell Dev. Biol., 16 (2005), 683–693. CrossRef
Tsimring, L., Levine, H., Aranson, I., Ben-Jacob, E., Cohen, I., Shochet, O., Reynolds, W. N.. Aggregation patterns in stressed bacteria. Phys. Rev. Letts, 75 (1995), No. 9, 1859–1862. CrossRef
Turing, A. M.. The chemical basis of morphogenesis. Phil. Trans. R. Soc. London, 237 (1952), 37–72. CrossRef
Tyson, R., Lubkin, S., Murray, J.. Model and analysis of chemotactic bacterial patterns in a liquid medium. J Math Biol, 38 (1999), 359–375. CrossRef
Tyson, R., Lubkin, S. R., Murray, J. D.. A minimal mechanism for bacterial pattern formation. Proc. R. Soc. Lond. B, 266 (1999), 299–304. CrossRef
Tyson, R., Stern, L., LeVeque, R.. Fractional step methods applied to a chemotaxis model. J. Math. Biol., 41 (2000), 455–75. CrossRef
D. Umulis, M. O'Connor, H. Othmer. Robustness of embryonic spatial patterning in Drosophila melanogaster. Curr. Top. Dev. Biol., 81 (2008) 65–111.
Umulis, D. M., Serpe, M., O'Connor, M. B., Othmer, H. G.. 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
Varea, C., Aragon, J. L., Barrio, R. A.. Confined turing patterns in growing systems. Phys. Rev. E., 56 (1997), 1250–1253. CrossRef
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.
von Dassow, G., Meir, E., Munro, E. M., Odell, G. M.. The segment polarity network is a robust developmental module. Nature, 406 (2000), No. 6792, 188–92. CrossRef
Wadhams, G. H., Armitage, J. P.. Making sense of it all: bacterial chemotaxis. Nat. Rev. Mol. Cell Biol., 5 (2004), No. 12, 1024–37. CrossRef
M. Walters, V. Sperandio. Quorum sensing in Escherichia coli and Salmonella. Int J Med Microbiol., 296 (2006), No. 2-3, 125–31.
Wang, X., Harris, R., Bayston, L., Ashe, H.. Type IV collagens regulate BMP signalling in Drosophila. Nature, 455 (2008), 72–77. CrossRef
Wang, Y. C., Ferguson, E. L.. Spatial bistability of Dpp-receptor interactions during Drosophila dorsal-ventral patterning. Nature, 434 (2005), No. 7030, 229–34. CrossRef
M. P. Weir, C. W. Lo. Gap-junctional communication. compartments in the Drosophila wing imaginal disc. Developmental Biology, 102 (1984) 130–146.
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.
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.
Wolpert, L.. Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol., 25 (1969), No. 1, 1–67. CrossRef
Wolpert, L.. Positional information and pattern formation. Curr. Topics in Dev. Biol., 6 (1971), 183–224. CrossRef
Woodward, D. E., Tyson, R., Myerscough, M. R., Murray, J. D., Budrene, E. O., Berg, H. C.. Spatio-temporal patterns generated by salmonella typhimurium . Biophys. J., 68 (1995), No. 5, 2181–2189. CrossRef
C. Xue. Mathematical models of taxis-driven bacterial pattern formation. Ph.D. thesis, University of Minnesota 2008.
C. Xue, H. G. Othmer. Radial and spiral streams formation in bacterium Proteus mirabilis colonies 2009, preprint.
C. Xue, H. G. Othmer. Multiscale models of taxis-driven patterning in bacterial populations. SIAM J. Appl. Math., to appear (2009).
Yakoby, N., Bristow, C. A., Gouzman, I., Rossi, M. P., Gogotsi, Y., Schpbach, T., Shvartsman, S. Y.. Systems-level questions in Drosophila oogenesis. Syst Biol (Stevenage), 152 (2005), 276–284. CrossRef
Yamaguchi, M., Yoshimoto, E., Kondo, S.. 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
Young, D.. A local activator-inhibitor model of vertebrate skin patterns. Math. Biosci., 72 (1984), 51–58. CrossRef
Zhang, Y., Lander, A., Nie, Q.. Computational analysis of BMP gradients in dorsal-ventral patterning of the zebrafish embryo. J. Theor. Biol., 248 (2007), 579–589. CrossRef