No Oscillations in Real Activator–Inhibitor Systems in Accomplishing Pattern Formation

Buikema, W. J., & Haselkorn, R. (2001). Expression of the Anabaena hetR gene from a copper-regulated promoter leads to heterocyst differentiation under repressing conditions. Proc. Natl. Acad. Sci. USA, 98, 2729–2734. Chen, Y., & Schier, A. F. (2002). Lefty proteins are long-range inhibitors of squint-mediated nodal signaling. Curr. Biol., 12, 2124–2128. Duboc, V., Rottinger, E., Besnardeau, L., & Lepage, T. (2004). Nodal and BMP2/4 signaling organizes the oral-aboral axis of the sea urchin embryo. Dev. Cell, 6, 397–410. Gierer, A., & Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik, 12, 30–39. Huang, X., Dong, Y., & Zhao, J. (2004). HetR homodimer is a DNA-binding protein required for heterocyst differentiation, and the DNA-binding activity is inhibited by PatS. Proc. Natl. Acad. Sci. USA, 101, 4848–4853. Irazoqui, J. E., Gladfelter, A. S., & Lew, D. J. (2003). Scaffold-mediated symmetry breaking by Cdc42p. Nat. Cell Biol., 5, 1062–1070. Mao, B., Wu, W., Li, Y., Hoppe, D., Stannek, P., Glinka, A., & Niehrs, C. (2001). LDL-receptor-related protein 6 is a receptor for Dickkopf proteins. Nature, 411, 321–325. Meinhardt, H. (1982). Models of biological pattern formation. London: Academic Press. Available at http://www.eb.tuebingen.mpg.de/meinhardt. Meinhardt, H. (1995). The algorithmic beauty of sea shells. Heidelberg: Springer. Meinhardt, H., & de Boer, P. A. J. (2001). Pattern formation in E.coli: a model for the pole-to-pole oscillations of min proteins and the localization of the division site. Proc. Natl. Acad. Sci. USA, 98, 14202–14207. Nakamura, Y., Tsiairis, C. D., Ozbek, S., & Holstein, T. W. (2011). Autoregulatory and repressive inputs localize hydra wnt3 to the head organizer. Proc. Natl. Acad. Sci. USA, 108, 9137–9142. Rentzsch, F., Fritzenwanker, J. H., Jens, H., Scholz, C. B., & Technau, U. (2008). FGF signalling controls formation of the apical sensory organ in the cnidarian Nematostella vectensis. Development, 135, 1761–1769. Risser, D. D., & Callahan, S. M. (2009). Genetic and cytological evidence that heterocyst patterning is regulated by inhibitor gradients that promote activator decay. Proc. Natl. Acad. Sci. USA, 106, 19884–19888. Schnakenberg, J. (1979). Simple chemical reaction system with limit cycle behavior. J. Theor. Biol., 31, 389–400. Seirin Lee, S., & Gaffney, E. A. (2010). Aberrant behaviours of reaction diffusion self-organisation models on growing domains in the presence of gene expression time delays. Bull. Math. Biol., 72, 2161–2179. Seirin Lee, S., Gaffney, E. A., & Monk, N. A. M. (2010). The influence of gene expression time delays on Gierer-Meinhardt pattern formation systems. Bull. Math. Biol., 72, 2139–2160. Seirin Lee, S., Gaffney, E. A., & Baker, R. E. (2011). The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays. Bull. Math. Biol., 73, 2527–2551. Yoon, H. S., & Golden, J. W. (1998). Heterocyst pattern-formation controlled by a diffusible peptide. Science, 282, 935–938.