Fibonacci or quasi-symmetric phyllotaxis. Part I: why?

• Autorzy: Gole C. , Dumais J. , Douady S.
• Języki publikacji: EN

Abstrakty

EN

The study of phyllotaxis has focused on seeking explanations for the occurrence of consecutive Fibonacci numbers in the number of helices paving the stems of plants in the two opposite directions. Using the disk-accretion model, first introduced by Schwendener and justified by modern biological studies, we observe two distinct types of solutions: the classical Fibonacci-like ones, and also more irregular configurations exhibiting nearly equal number of helices in a quasi-square packing, the quasi-symmetric ones, which are a generalization of the whorled patterns. Defining new geometric tools allowing to work with irregular patterns and local transitions, we provide simple explanations for the emergence of these two states within the same elementary model. A companion paper will provide a wide array of plant data analyses that support our view.

• Wydawca: -
• Czasopismo: Acta Societatis Botanicorum Poloniae
• Rocznik: 2016
• Tom: 85
• Numer: 4
• Opis fizyczny: Article 3533 [34p.], fig.,ref.

Twórcy

autor

Gole C.

• Department of Mathematics, Smith College, Northampton, MA 01063, USA

autor

Dumais J.

• Faculty of Engineering and Sciences, Universidad Adolfo Ibanez, Vina del Mar, Chile

autor

Douady S.

• UMR 7057 Universite Paris Diderot - CNRS, Batiment Condorcet, CC 7057, 10 rue Alice Domon et Leonie Duquet, 75013 Paris, France

Bibliografia

• 1. Zagórska-Marek B, Szpak M. Virtual phyllotaxis and real plant model cases. Funct Plant Biol. 2008;35:1025–1033. https://doi.org/10.1071/FP08076
• 2. Schwendener S. Mechanische Theorie der Blattstellungen. Leipzig: Engelmann; 1878.
• 3. van Iterson G. Mathematische und mikroskopisch-anatomische Studien über Blattstellungen nebst Betrachtungen über den Schalenbau der Miliolinen. Jena: Gustav Fischer Verlag; 1907. https://doi.org/10.5962/bhl.title.8287
• 4. Atela P. The geometric and dynamic essence of phyllotaxis. Math Model Nat Phenom. 2011;6:173–186. https://doi.org/10.1051/mmnp/20116207
• 5. Douady S. The selection of phyllotactic patterns. In: Jean RV, Barabé D, editors. Symmetry in plants. Singapore: World Scientific; 1998. p. 335–358. (Series in Mathematical Biology and Medicine; vol 4). https://doi.org/10.1142/9789814261074_0014
• 6. Atela P, Golé C. Rhombic tilings and primordia fronts of phyllotaxis [Preprint]. 2007 [cited 2016 Dec 30]. Available from: http://arxiv.org/abs/1701.01361
• 7. Mughal A, Weaire D. Phyllotaxis, disk packing and Fibonacci numbers [Preprint]. 2016 [cited 2016 Dec 31]. Available from: https://arxiv.org/abs/1608.05824
• 8. Turing A. The chemical basis of morphogenesis. Philos Trans R Soc Lond B. 1952;237(641):37–72. https://doi.org/10.1098/rstb.1952.0012
• 9. Veen AH. A computer model for phyllotaxis, a network of automata [Master thesis]. Philadelphia, PA: Computer and Information Sciences Graduate School of Arts and Sciences, University of Pennsylvania; 1973.
• 10. Williams R. Shoot apex and leaf growth. London: Cambridge University Press; 1974.
• 11. Meinhardt H, Koch A, Bernasconi G. Models of pattern formation applied to plant development. In: Jean RV, Barabé D, editors. Symmetry in plants. Singapore: World Scientific; 1998. p. 723–758. (Series in Mathematical Biology and Medicine; vol 4). https://doi.org/10.1142/9789814261074_0027
• 12. Pennybacker M, Shipman P, Newell A. Phyllotaxis: some progress, but a story far from over. Physica D: Nonlinear Phenomena. 2015;306:48–81. https://doi.org/10.1016/j.physd.2015.05.003
• 13. de Reuille PB, Bohn-Courseau I, Ljung K, Morin H, Carraro N, Godin C, et al. Computer simulations reveal properties of the cell-cell signaling network at the shoot apex in Arabidopsis. Proc Natl Acad Sci USA. 2006;103(5):1627–1632. https://doi.org/10.1073/pnas.0510130103
• 14. Smith R, Guyomarc’h S, Mandel T, Reinhardt D, Kuhlemeier C, Prusinkiewicz P. A plausible model of phyllotaxis. Proc Natl Acad Sci USA. 2006;103(5):1301–1306. https://doi.org/10.1073/pnas.0510457103
• 15. Jönsson H, Heisler MG, Shapiro B, Meyerowitz E, Mjolsness E. An auxin-driven polarized transport model for phyllotaxis. Proc Natl Acad Sci USA. 2006;103(5):1633–1638. https://doi.org/10.1073/pnas.0509839103
• 16. Reinhardt D, Mandel T, Kuhlemeier C. Auxin regulates the initiation and radial position of plant lateral organs. Plant Cell. 2000;12(4):507–518. https://doi.org/10.1105/tpc.12.4.507
• 17. Reinhardt D, Frenz M, Mandel T, Kuhlemeier C. Microsurgical and laser ablation analysis of interactions between the zones and layers of the tomato shoot apical meristem. Development. 2003;130(17):4073–4083. https://doi.org/10.1242/dev.00596
• 18. Hofmeister W. Allgemeine Morphologie der Gewächse. In: du Bary A, Irmisch TH, Sachs J, editors. Handbuch der Physiologischen Botanik. Leipzig: Engelman; 1868. p. 405–664.
• 19. Snow M, Snow R. Experiments on phyllotaxis. II. The effect of displacing a primordium. Philos Trans R Soc Lond B. 1932;222:353–400. https://doi.org/10.1098/rstb.1932.0019
• 20. Reinhardt D, Pesce E, Stieger P, Mandel T, Baltensperger K, Bennett M, et al. Regulation of phyllotaxis by polar auxin transport. Nature. 2003;426(6964):255–260. https://doi.org/10.1038/nature02081
• 21. Rueda-Contreras MD, Aragón JL. Alan turing’s chemical theory of phyllotaxis. Revista Mexicana de Física E. 2004;60(1):1–12.
• 22. Hotton S, Johnson V, Wilbarger J, Zwieniecki K, Atela P, Golé C, et al. The possible and the actual in phyllotaxis: bridging the gap between empirical observations and iterative models. J Plant Growth Regul. 2006;25:313–323. https://doi.org/10.1007/s00344-006-0067-9
• 23. Golé C, Dumais J, Douady S. Fibonacci or quasi-symmetric. Part II: botanical observations? Acta Soc Bot Pol. 2016;85(4):3534. https://doi.org/10.5586/asbp.3534
• 24. Mitchison GJ. Phyllotaxis and the Fibonacci series. Science. 1977;196(4287):270–275. https://doi.org/10.1126/science.196.4287.270
• 25. Douady S, Couder Y. Phyllotaxis as a self organizing iterative process, Part II: the spontaneous formation of a periodicity and the coexistence of spiral and whorled patterns J Theor Biol. 1996;178:275–294. https://doi.org/10.1006/jtbi.1996.0025
• 26. A Weisse. Sketch of the mechanical hypothesis of leaf-position. In: K. Goebel’s Organography of plants. I. Oxford: Clarendon Press; 1900. p. 74–84.
• 27. Snow M, Snow R. Minimum areas and leaf determination. Proc R Soc Lond B Biol Sci. 1952;139(897):545–566. https://doi.org/10.1098/rspb.1952.0034
• 28. Couder Y. Initial transitions, order and disorder in phyllotactic patterns: the ontogeny of Helianthus annuus, a case study. Acta Soc Bot Pol. 1998;67:129–150. https://doi.org/10.5586/asbp.1998.016
• 29. Douady S, Couder Y. Phyllotaxis as a dynamical self organizing process Part I. J Theor Biol. 1996;178:255–274. https://doi.org/10.1006/jtbi.1996.0024
• 30. Zagórska-Marek B. Phyllotaxis triangular unit; phyllotactic transitions as the consequences of the apical wedge disclinations in a crystal-like pattern of the units. Acta Soc Bot Pol. 1987;56(2):229–255. https://doi.org/10.5586/asbp.1987.024
• 31. Bravais L, Bravais A. Essai sur la disposition des feuilles curvisériées. Annales des Sciences Naturelles. Botanique. 1837;7:42–110.
• 32. Bravais A. Mémoire sur les systèmes formés par des points distribués régulièrement sur un plan ou dans l’espace. Journal de l’École Polytechnique. 1850;19:1–128.
• 33. Douady S, Couder Y. Phyllotaxis as a physical self-organized growth process. Phys Rev Lett. 1992;68:2098–2101. https://doi.org/10.1103/PhysRevLett.68.2098
• 34. Atela P, Golé C, Hotton S. A dynamical system for plant pattern formation: a rigorous analysis. Journal of Nonlinear Science. 2002;12:641–676. https://doi.org/10.1007/s00332-002-0513-1
• 35. Adler I. A model of contact pressure in phyllotaxis. J Theor Biol. 1974;45:1–79. https://doi.org/10.1016/0022-5193(74)90043-5
• 36. Freeman E. Cylinder lattice applet. GeoGebra [Internet]. 2016 [cited 2016 Dec 31]. Available from: https://ggbm.at/NeHVks33
• 37. Mirabet V, Besnard F, Vernoux T, Boudaoud A. Noise and robustness in phyllotaxis. PLoS Comput Biol. 2012;8(2):e1002389. https://doi.org/10.1371/journal.pcbi.1002389
• 38. Douady S, Couder Y. Phyllotaxis as a dynamical self organizing process Part III: the simulation of the transient regimes of ontogeny. J Theor Biol. 1996;178:295–312. https://doi.org/10.1006/jtbi.1996.0026
• 39. Guédon Y, Refahi Y, Besnard F, Godin C, Vernoux, T. Pattern identification and characterization reveal permutations of organs as a key genetically controlled property of post-meristematic phyllotaxis, J Theor Biol. 2013;338:94–110. https://doi.org/10.1016/j.jtbi.2013.07.026
• 40. Bachmann K. Evolutionary genetics and the genetic control of morphogenesis in flowering plants. In: Evolutionary biology. Boston, MA: Springer; 1983. p. 157–208. https://doi.org/10.1007/978-1-4615-6971-8_5
• 41. Fredeen A, Horning J, Madill R. Spiral phyllotaxis of needle fascicles on branches and scales on cones in Pinus contorta var. latifolia: are they influenced by wood-grain spiral? Can J Bot. 2002;80(2):166–175. https://doi.org/10.1139/b02-002
• 42. Prusinkiewicz P. Graphical applications of L-systems. In: Green M, editor. Proceedings of Graphics Interface and Vision Interface. Vol. 86; 1986 May 26–30; Vancouver, Canada. Toronto, ON: Canadian Information Processing Society; 1986. p. 247–253. https://doi.org/10.20380/GI1986.44
• 43. Kunz M. Phyllotaxie, billards polygonaux et thorie des nombres [PhD thesis]. Lausanne: Université de Lausanne, Switzerland; 1997.
• 44. Fenichel N. Persistence and smoothness of invariant manifolds for flows. Indiana University Mathematics Journal. 1971;21(3):193–226. https://doi.org/10.1512/iumj.1972.21.21017
• 45. Levitov LS. Phyllotaxis of flux lattices in layered superconductors. Phys Rev Lett. 1991;66(2):224– 227. https://doi.org/10.1103/PhysRevLett.66.224

Identyfikatory

DOI

10.5586/asbp.3533