Phyllotaxis or the properties of spiral lattices. III. An algebraic model of morphogenesis M. Kunz, F. Rothen To cite this version: M. Kunz, F. Rothen. Phyllotaxis or the properties of spiral lattices. III. An algebraic model of mor- phogenesis. Journal de Physique I, EDP Sciences, 1992, 2 (11), pp.2131-2172. 10.1051/jp1:1992273. jpa-00246692 HAL Id: jpa-00246692 https://hal.archives-ouvertes.fr/jpa-00246692 Submitted on 1 Jan 1992 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 213 1992, (1992) 2131-2172 Phys. France 2 I J. NOVEMBER PAGE Classification Physics Abstracts 61.00 87,10 spiral algebraic Phyllotaxis properties of lattices. III. An the or (*) morphogenesis model of M. Kunz and F. Rothen CH-l0l5Lausanne-Dorigny, Exp6rimentale, Physique Lausanne, Universit£ Institut de de Switzerland July 1992, 1991, accepted 24July (Received 1992) May 23 revised 22 phyllotactique d'une Rdsumd. Une les celles constituent dcailles structure que pomme comme pin prend apical qui naissance dtroit de croissance. local On de entoure centre sur un anneau un simple morphogenbse £tudie modble d'une £tablissant la de telle formation des relations un en algdbriques premiers chaque permutations correspondant h naturels, des nombres nombre entre n d'apparition l'ordre 6caille d'une d'une feuille. Ce modble conduit h la d6finition naturelle ou divergence divergence. prendre s'ordonnent d'une valeurs Les peut cette structure que en une hidrarchique parentd qui 6vidence des 6vidente l'arbre le r61e nombres nobles. Il de met y en a avec Farey long spirales alignds obtient classant de l'on les r6seaux cercles le de tangents que en 6quiangulaires. phyllotactical originates such of Abstract. the scales of fir-cone A the pattem arrangement as a ring along surrounding thin study simple growth. local of of model the We center a a a considering morphogenesis algebraic by botanical relations between of such structure permu- a given tations first corresponding the birth of the natural each numbers, number order of scale to a n divergence angIe possible leaf. model allows define in This The values of the natural to way. a or a according divergence places which be classified hierarchical the noble numbers to structure can a prominent position. study of Farey-tree in in the This construction is similar the obtained to a equiangular aligned along spirals. lattices of circles tangent Morphogenesis phyllotads. 1. and PHYSIOLOGISTS.-Phyllotaxis study spiral 1-1GEOMETERS the of helicoidal is AND or frequently objects spiral plants. instance the Its for pattems encounters among one are daisy florets sunflower, of the helicoidal distribution of in the scales of arrangement a or a or a pineapple. pine Fibonacci The has of the in these been pattems sequence emergence cone or a a (I). long Among speculation people of for time this those who interested in matter very were a (*) Panially supported by Swiss the (FNIRS). National Foundation (I) history phyuotaxis short of of A be found the [4]. in reference general For account may a phyllotaxis, [5-10]. references [I] introduction and to e-g- see PHYSIQUE JOURNAL N° DE I I I 2132 physicists, mathematicians, problem, crystallographers, notices naturalists, one numerous truly, who, Goethe biologists considered himself scientist. and such poets even as as a Facing biology, people essentially intrusion of arithmetic into this it see mere some as a following. They will Other of be called in the geometry. geometers consequence space geometrical phyllotactic plant scientists look for evidence of the in metabolism the of structure They growth plants. hope they during find hormones secreted there the of how pattems to are physiologists. morphogenesis. They deny during do But will called the role of be geometry not during emphasize inhibitors of they influence activators this and the want to stress process, or taking development chemical of individual contribution of in the the substance the part every plant. phyllotaxis point scientists different of Recently, third looked from still of category at a a physical physicists they consider which might because One call them systems view. pure point being far-off, self-organization starting [1-3]. similar botanical Our exhibit to very ones a longer this their work in discuss shall paper. we no opposition genetics attribute view, point between of the those who From to common a our morphogenesis, for that essential mechanisms determining in those who think and the role independent in the of selection of be discussed needs to nature, pattem not case are an phyllotaxis. splitting by [I I] 1935 Snow that the by observed indeed in Snow and in It parts two apex was having plant naturally phyllotaxis), often obtains Epilobium (a of hirsutium decussate two one phyllotaxis. requires clearly spiral approach experiment apices developing theoretical this A to independent genetic growth which information. of is of the model the a 1.2 APICAL emphasized geometrical Geometers have the RING AND MORPHOGENESIS.- properties phyllotactic highly symmetric of the thought these pattem structures to emerge : are during growth particular point without call for adopted (2). mechanism view This of in any was by [lo] Koch and of found; where references be two papers many one us can as a morphogenesis problem of phyllotactic only the side of issue in structures consequence, was a [lo]. However, references references [12-14] in idea which take Marzec of and up an [15], Kappraff morphogenesis considered is point the physiologists of from the of view the : moulding primordia growth apical of (as the and the called) shoots the the young new on are ring modelled. is fact, producing differentiation As of cellular place shoot new-bom does take matter not at a a ring-shaped tip (apex), surrounding within of the the stalk but region [7]. the the stem or apex a apical ring This remains linked the during growth. For the observer the to apex apex, on new an (Fig. radially I). shoots to appear go away interval Let that time growth successive and the of the between shoots two rate suppose us phyllotaxis problem settled down stable value. We follows then the of have to state may a as : apical ring completely explain why successive (the the shoots in ordered the to appear way a on becoming angle approximately after between successive shoots transient short constant, two a period) why, generically, angular golden ~137.5°...). and this angle distance is the angular divided explaining by divergence. is This 2 the In called distance aim at we ar « » giving growth apical ring, dynamical is, deterministic model of define the the that to on a a physical interpretation generically which chemical does has reasonable and system some or « » reproduce phyllotactic observed. the pattem phyllotaxis is of mechanism that be We here that the modelled in result the assume a can plant. As introduction, in discussed basic in the in every same manner was assume we particular independent genetics. of it is that exist, special Or, interest if mechanism does is (2) such it hidden and of geometers. to no a 2133 MORPHOGENESIS MODEL OF ALGEBRAIC PHYLLOTAXIS N° II AN Apex ~~~~~~~~ ° Q Q growth ~ QO O Q Q growing surrounding of ring-shaped region the bom inside Fig. shoots I. The stem. apex a are a new ring radially where from the shoots moving the newbom with the For observer to go away appear apex, an appeared. they first ofmorphogenesis. dynamical chemical the model 1.2. A circle In order I system to on as a apical [12-15] illustrate growth, existence the of of references authors advocate the the process interpreted acting growth inhibitory of hydra of factor substance in the a as an as case development [16]. following. Such called substance will be inhibitor in the The inhibitor a » « specify Turing morphogen by [17] is of word coined chemical substance type to a a one », « playing growth morphogenetic during the role process. some hydra, high neighbourhood inhibitor concentration is In the of the in the the of case a sprouting long value, and this concentration exceeds head threshold wherever as, as some no develop. morphogen is head elapsing, by is diffusion able As time this fades to away new or description through has also Thomley by [16]. chemical used been reaction This model the to a growth phyllotactic [18]. of pattem a profile apical ring, already placed inhibitor of Consider the leaves the the result ; on a new leaf its minimum. Because of this concentration of inhibitor will the sprout, at appear new profile locally (through reaction) increases. fades diffusion chemical Then the whole away or until the leaf next appears. Kapraff [15] [14] Sadik Koch, Bemasconi Guerreiro, and have studied Marzec and and a dynamical Douady [3] proposed which illustrates and Couderc idea this have system a physical experiment dynamical analogous exhibits behaviour. shall this that We refer to an dynamic general of construction in the model. scheme our algorithm The growth permutation model and the describe 1.2.2 A. We the to want process defining Thomley's simple algorithm apical ring, following by deterministic the a on [18]. scheme analytical study dynamics [12-15] proposed of references The of in the the is systems simplify why extremely difficult this is model further.