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

Christophe Golé, Jacques Dumais, Stéphane Douady

Abstract


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.

Keywords


phyllotaxis; Fibonacci; quasi-symmetry; disc-stacking model; irregular pattern

Full Text:

PDF

References


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

Schwendener S. Mechanische Theorie der Blattstellungen. Leipzig: Engelmann; 1878.

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

Atela P. The geometric and dynamic essence of phyllotaxis. Math Model Nat Phenom. 2011;6:173–186. https://doi.org/10.1051/mmnp/20116207

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

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

Mughal A, Weaire D. Phyllotaxis, disk packing and Fibonacci numbers [Preprint]. 2016 [cited 2016 Dec 31]. Available from: https://arxiv.org/abs/1608.05824

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

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.

Williams R. Shoot apex and leaf growth. London: Cambridge University Press; 1974.

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

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

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

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

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

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

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

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.

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

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

Rueda-Contreras MD, Aragón JL. Alan turing’s chemical theory of phyllotaxis. Revista Mexicana de Física E. 2004;60(1):1–12.

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

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

Mitchison GJ. Phyllotaxis and the Fibonacci series. Science. 1977;196(4287):270–275. https://doi.org/10.1126/science.196.4287.270

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

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.

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

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

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

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

Bravais L, Bravais A. Essai sur la disposition des feuilles curvisériées. Annales des Sciences Naturelles. Botanique. 1837;7:42–110.

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.

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

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

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

Freeman E. Cylinder lattice applet. GeoGebra [Internet]. 2016 [cited 2016 Dec 31]. Available from: https://ggbm.at/NeHVks33

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

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

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

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

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

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

Kunz M. Phyllotaxie, billards polygonaux et thorie des nombres [PhD thesis]. Lausanne: Université de Lausanne, Switzerland; 1997.

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

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