Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2016 | 85 | 4 |
Tytuł artykułu

The riddle of phyllotaxis: exquisite control of divergence angle

Treść / Zawartość
Warianty tytułu
Języki publikacji
Phyllotaxis studies published in German in the 1930s have reported intriguing regularity in the arrangement of incipient leaves on shoot apices of a wide variety of plant species. However, these studies have received little attention today, even though they provide a crucial evidence base for understanding this mathematical phenomena. Here I recapitulate the essential point by means of illustrative examples. It is emphasized that accurate control of apical divergence angle is at the heart of the numerical riddle of spiral phyllotaxis. The accurate patterning at the shoot apex has an unexpected evolutionary benefit of being optimally adaptive in the subsequent events of phyllotactic change to occur on an elongating shoot.
Słowa kluczowe
Opis fizyczny
Article 3527 [17p.], fig.,ref.
  • Graduate School of Integrated Science and Technology, Shizuoka University, Hamamatsu 432-8561, Japan
  • 1. van Iterson G. Mathematische und Mikroskopisch-Anatomische Studien über Blattstellungen. Jena: Gustav Fischer; 1907.
  • 2. Church AH. On the interpretation of phenomena of phyllotaxis. London: Humphrey Milford and Oxford University Press; 1920. (Botanical Memoirs).
  • 3. Hirmer M. Zur Lösung des Problems der Blattstellungen. Jena: Gustav Fischer; 1922.
  • 4. Snow M, Snow R. The interpretation of phyllotaxis. Biol Rev. 1934;9:132–137.
  • 5. Jean RV. Phyllotaxis: a systemic study in plant morphogenesis. Cambridge: Cambridge University Press; 1994.
  • 6. Hirmer M. Zur Kenntnis der Schraubenstellungen im Pflanzenreich. Planta. 1931;14:132–206.
  • 7. Schöffel K. Untersuchungen über den Blütenbau der Ranunculaceen. Planta. 1932;17:315–371.
  • 8. Bilhuber E. Beiträge zur Kenntnis der Organstellungen im Pflanzenreich. Botanisches Archiv. 1933;35:188–250.
  • 9. Breindl M. Zur Kenntnis der Baumechanik des Blütenkelches der Dicotylen. Botanisches Archiv. 1934;36:191–268.
  • 10. Barthelmess A. Über den Zusammenhang zwischen Blattstellung und Stelenbau unter besonderer Berücksichtigung der Koniferen. Botanisches Archiv. 1935;37:207–260.
  • 11. Hirmer M. Neue Untersuchungen auf dem Gebiet der Organstellungen. Ber Dtsch Bot Ges. 1934;52:26–72.
  • 12. Snow R. Problems of phyllotaxis and leaf determination. Endeavour. 1955;14:190–199.
  • 13. Snow R, Snow M. A theory of the regulation of phyllotaxis based on Lupinus albus. Philos Trans R Soc Lond B. 1962;244:483–513.
  • 14. Snow M, Snow R. Regulation of sizes of leaf primordia by older leaves. Proc R Soc Lond B Biol Sci. 1959;151:39–47.
  • 15. Adler I. A model of contact pressure in phyllotaxis. J Theor Biol. 1974;45:1–79.
  • 16. Jean RV. An interpretation of Fujita’s frequency diagrams in phyllotaxis. Bull Math Biol. 1986;48:77–86.
  • 17. Douady S, Couder Y. Phyllotaxis as a physical self-organized growth process. Phys Rev Lett. 1992;68:2098–2101.
  • 18. Newell AC, Shipman PD, Sun Z. Phyllotaxis: cooperation and competition between mechanical and biochemical processes. J Theor Biol. 2008;251:421–439.
  • 19. Fierz V. Aberrant phyllotactic patterns in cones of some conifers: a quantitative study. Acta Soc Bot Pol. 2015;84:261–265.
  • 20. Zagórska-Marek B. Phyllotactic patterns and transitions in Abies balsamea. Can J Bot. 1985;63:1844–1854.
  • 21. Zagórska-Marek B. Phyllotaxic diversity in Magnolia flowers. Acta Soc Bot Pol. 1994;63:117–137.
  • 22. Goodwin B. How the leopard changed its spots: the evolution of complexity. London: Weidenfeld & Nicolson; 1994.
  • 23. Arthur W. Evolution: a developmental approach. Hoboken, NJ: Wiley-Blackwell; 2011.
  • 24. Bravais L, Bravais A. Essai sur la disposition des feuilles curvisériées. Annales des Sciences Naturelles. Botanique. 1837;7:42–110.
  • 25. Reinhardt D, Pesce ER, Stieger P, Mandel T, Baltensperger K, Bennett M, et al. Regulation of phyllotaxis by polar auxin transport. Nature. 2003;426:255–260.
  • 26. Besnard F, Refahi Y, Morin V, Marteaux B, Brunoud G, Chambrier P, et al. Cytokinin signalling inhibitory fields provide robustness to phyllotaxis. Nature. 2014;505:417–421.
  • 27. Ludwig F. Die Anzahl der Strahlenblüten bei Chrysanthemum leucanthemum und anderen Kompositen. Deutsche Botanische Monatsschrift. 1887;5:52–58.
  • 28. Weisse A. Die Zahl der Randblüten an Compositenköpfchen in ihrer Beziehung zur Blattstellung und Ernährung. Jahrbücher für Wissenschaftliche Botanik. 1894;30:453–483.
  • 29. Fujita T. Über die Reihe 2, 5, 7, 12… in der schraubigen Blattstellung und die mathematische Betrachtung verschiedener Zahlenreihensysteme. Botanical Magazine Tokyo. 1937;51:298–307.
  • 30. Fujita T. Statistische Untersuchungern über den Divergenzwinkel bei den schraubigen Organstellungen. Botanical Magazine Tokyo. 1939;53:194–199.
  • 31. Davies PA. Leaf position in Ailanthus altissima in relation to the Fibonacci series. Am J Bot. 1939;26:67–74.
  • 32. Kumazawa M, Kumazawa M. Periodic variations of the divergence angle, internode length and leaf shape, revealed by correlogram analysis. Phytomorphology. 1971;21:376–389.
  • 33. Ryan GW, Rouse JL, Bursill LA. Quantitative analysis of sunflower seed packing. J Theor Biol. 1991;147:303–328.
  • 34. Bachmann K, Battjes J, Vischer NOE. Capitulum phyllotaxis and numerical canalization in Microseris pygmaea (Asteraceae: Lactuceae). Am J Bot. 1993;80:419–428.
  • 35. Okabe T. Systematic variations in divergence angle. J Theor Biol. 2012;313:20–41.
  • 36. Braun A. Vergleichende Untersuchung über die Ordnung der Schuppen an den Tannenzapfen. Nova acta Academiae Caesareae Leopoldino-Carolinae. 1831;15:195–402.
  • 37. Okabe T. Extraordinary accuracy in floret position of Helianthus annuus. Acta Soc Bot Pol. 2015;84:79–85.
  • 38. Komai T, Yamazi I. Order found in the arrangement of organs and zooids in some medusae. Annot Zool Jpn. 1945;23:1–7.
  • 39. Okabe T, Yoshimura J. Optimal hash arrangement of tentacles in jellyfish. Sci Rep. 2016;6:27347.
  • 40. Berrill NJ. Growth, development, and pattern. San Francisco, CA: W. H. Freeman Co.; 1961.
  • 41. Knuth DE. The art of computer programming. Volume 3: sorting and searching. 2nd ed. Redwood City, CA: Addison Wesley Longman Publishing Co., Inc.; 1998.
  • 42. Marzec C, Kappraff J. Properties of maximal spacing on a circle related to phyllotaxis and to the golden mean. J Theor Biol. 1983;103:201–226.
  • 43. Puławska Z. Correlations in the development of the leaves and leaf traces in the shoot of Actinidia arguta Planch. Acta Soc Bot Pol. 1965;34:697–712.
  • 44. Endress PK. Floral structure and relationships of Hortonia (Monimiaceae). Plant Syst Evol. 1980;133:199–221.
  • 45. Kubitzki K. Origin and significance of trimerous flowers. Taxon. 1987;36:21–28.
  • 46. Endress PK. Patterns of floral construction in ontogeny and phylogeny. Biol J Linn Soc Lond. 1990;39:153–175.
  • 47. Battjes J, Vischer NOE, Bachmann K. Meristem geometry and heritable variation in numbers of florets and involucral bracts in Microseris pygmaea (Asteraceae, Lactuceae). Acta Botanica Neerlandica. 1993;42:255–268.
  • 48. Vlot EC, Bachmann K. Genetics of the number of achenes and involucral bracts in interstrain hybrid D37 of Microseris douglasii (Asteraceae, Lactuceae). Biol Zent Bl. 1993;112:7–17.
  • 49. Barabé D. Phyllotaxis: open and closed systems. Journal of Biological Systems. 1995;3:917–927.
  • 50. Richards FJ. The geometry of phyllotaxis and its origin. Symp Soc Exp Biol. 1948;2:217–245.
  • 51. Schwendener S. Mechanische Theorie der Blattstellungen. Leipzig: Engelmann; 1878.
  • 52. Plantefol L. La théorie des hélices foliaires multiples: fondements d’une théorie phyllotaxique nouvelle. Paris: Masson; 1948. (Annales des Sciences Naturelles. Botanique et Biologie Végétale; vol 7–8).
  • 53. Richards FJ. Über einige Fragen betreffend die Messung von Blattstellungen. Planta. 1955;45:198–207.
  • 54. Richards FJ. Spatial and temporal correlations involved in leaf pattern production at the apex. In: Milthorpe FL, editor. The growth of leaves. London: Butterworths; 1956. p. 66–76.
  • 55. Richards FJ, Schwabe WW. Phyllotaxis: a problem of growth and form. In: Steward FC, editor. Plant physiology: a treatise, volume VA. New York, NY: Academic Press; 1969. p. 77–116.
  • 56. Snow M, Snow R. Experiments on phyllotaxis. I. The effects of isolating a primordium. Philos Trans R Soc Lond B. 1932;221:1–43.
  • 57. Okabe T. Physical phenomenology of phyllotaxis. J Theor Biol. 2011;280:63–75.
  • 58. Okabe T. Biophysical optimality of the golden angle in phyllotaxis. Sci Rep. 2015;5:15358.
  • 59. D’Arcy Thompson W. On growth and form. Cambridge: Cambridge University Press; 1942.
  • 60. Sachs J, Ward HM. Lectures on the physiology of plants. Oxford: Clarendon Press; 1887.
  • 61. Sachs J, Balfour IB, Garnsey HEF. History of botany (1530–1860). Oxford: Clarendon Press; 1906.
  • 62. Sendtner O. Kritische Vergleichung der Lehren über die Blattstellung von Schimper und den Gebrüdern Bravais. Flora. 1847;13:201–214,217–226,233–242.
  • 63. Wikimedia Commons. Corkscrew beggarticks, flower head close up [Internet]. 2003 [cited 2016 Dec 15]. Available from:
  • 64. Wikimedia Commons. Galinsoga parviflora [Internet]. 2013 [cited 2016 Dec 15]. Available from:
  • 65. Wikimedia Commons. A poster of twelve different species of flowers of the Asteraceae family, belonging to the three most representative subfamilies: Asteroideae, Cichorioideae and Carduoideae [Internet]. 2008 [cited 2016 Dec 15]. Available from:
  • 66. Wikimedia Commons. Rudbeckia laciniata [Internet]. 2006 [cited 2016 Dec 15]. Available from:
  • 67. Wikimedia Commons. Yellow sunflower [Internet]. 2010 [cited 2016 Dec 15]. Available from:
  • 68. Botanical Society of America. Sequoia sempervirens, shoot apex, x.s. [Internet]. 2016 [cited 2016 Dec 15]. Available from:
  • 69. Wolfram Demonstrations Project. Phyllotaxis of plant shoots [Internet]. 2014 [cited 2016 Dec 15]. Available from:
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.