PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis


About Phyllotaxis:

I.       The Apical Meristem

II.      Classification of Apical Meristem growth

III.    Parastichous Spiral Phyllotaxis – Fibonacci – Golden Ratio and the Golden Angle

IV.    Mechanical dynamics of Apical Meristem growth

V.     Physiological Basis of Apical Mersitem growth

VI.    Genetic Basis of Apical Meristem growth

VII.   Evolutionary aspects of Phyllotaxis

VIII. References


About PhiTaxis:

GIRASOLE: Digital Fibonacci Sunflower

PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis

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About Phyllotaxis:


I.       The Apical Meristem

Meristems consists of undifferentiated cells (meristematic cells) which are found in plant segments where growth takes place; apical derives from the Latin apex (plural apices) meaning to be at the top – peak or summit of a structure.  Meristems consist of organogenic cells that are established during plant embryogenesis.  There are two apical meristems, the Shoot Apical Meristem (SAM) and the Root Apical Meristem (RAM), which produce respectively the aerial and subterranean parts of the plant body.  Cells of the apical meristem divide rapidly and are of an indeterminate nature being non-differentiated cells capable of carrying out continuous cellular divisions to remain meristematic or develop into specific organs or plant structures.  Meristematic cells are small and the protoplasm fills the cell completely: they contain large nuclei, small vacuoles and the protoplasm does not contain differentiated plastids (only present in a rudimentary form of proplastids).  Meristematic cells are packed closely together having no intercellular spaces; cell walls are very thin and consist only of a primary cell wall.

The Shoot Apical Meristem (SAM) consists of two outer cell layers, the epidermal L1 and subepidermal L2 layers, together called the 'tunica' (basic garment worn by both men and women in Ancient Rome), whereas the inner core L3 layer of the meristem is referred to as the ‘corpus’ (from the latin word for body) − also named central mother cells.  Cell divisions in the Shoot Apical Meristem are required to provide new cells for expansion and differentiation of tissues and initiation of new organs, providing the basic structure of the plant body.  The cells of the epidermal L1 layer undergo anticlinal divisions − new cell walls are oriented perpendicularly to the surface of the meristem.  The L1 gives rise to the epidermis of the entire shoot, including stomata and trichomes.  The cells of the subepidermal L2 layer undertake mostly anticlinal divisions, however, in the early process of organ initiation it has been observed that cells in the L2 layer also carry out periclinal divisions (new cell walls are oriented parallel to the meristem surface).  The L2 layer produces mesophyll, part of the ground tissues and gives rise to the sporogenic tissues present in flowers.  Therefore, the genotype of L2 cells determines the genotype of gametes, whereas the genotype of the L1 and L3 layers are not transmitted to the next generation.  Cells of the basal portion of the Shoot Apical Meristem (L3 layer) exhibit no preferred orientation and both anticlinal and periclinal cell divisions take place, increasing respectively organ circumference and girth.  The tissues derived from the L3 layer give rise to the central core tissues present in the stem and in lateral organs.


Tunica-Corpus model of the Shoot Apical Meristem: The epidermal (L1) and subepidermal (L2) layers form the outer layers called the tunica, the inner L3 layer is called the corpus or also central mother cells. Cells in the L1 and L2 layers undergo anticlinal divisions (perpendicular) keeping these layers distinct, whereas the L3 layer shows anticlinal and also periclinal (parallel) cell divisions.


Meristems are truly incredible self-organizing systems.  The Shoot Apical Meristem generates stem, leaves and lateral shoot meristems during the entire shoot ontogeny, vegetative leaves are generated in the vegetative developmental phase, while in the reproductive phase floral organs are formed.  Although the formation of meristems occurs during plant embryogenesis, numerous experimental studies have shown that Shoot Apical Meristems can spontaneously organize from undifferentiated tissues when given the proper set of hormonal signals.  This ability to form a SAM de novo postembryonically is the basis for plant tissue culture, in which plants are propagated vegetatively.  Remarkably, the newly formed Shoot Apical Meristems produce normal development patterns, demonstrating that pattern initiation is not restricted to embryogenesis.


II.      Classification of Apical Meristem growth

Phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon "leaf" and táxis "arrangement").  The basic phyllotactic patterns in the plant kingdom are either opposite, whorled or alternate.

Four major phyllotactic patterns depicted as plants and as top view line diagrams showing relative leaf positions. (i) and (ii) are opposite phyllotaxis; (i) distichous with a divergence angle of 180º (maize), (ii) decussate with pairs of leaves at 90º (Coleus sp). (iii) whorled with three or more leaves originating from the same node (Veronicastrum virginicum). (iv) alternate (spiral) with a divergence angle of 137.58º (sunflower). In line diagrams, lighter gray tones indicate progressively older leaves.


In opposite phyllotaxis leaf primordia grow one by one.  Opposite phyllotactic patterns may be further divided into distichous or decussated phyllotaxis.  In distichous phyllotaxis as in Maize, successive leaf primordia are placed each at 180 degrees from the previous one, in decussated phyllotaxis as in Coleus species, successive leaf pairs are perpendicular.

Distichous      Decussated


In whorled phyllotaxis, at least three leaf primordia grow at the same node on the stem.  Primordia in a node are evenly spread around the stem, midway between those in the previous node.

Rosette of Leaves


More than 80% of the 250,000 higher plant species have an alternate phyllotaxis, as in the case of potato, araucaria, yucca or sunflowers.  Alternate phyllotaxis can be further divided into spiral or multijugate phyllotaxis.

In spiral phyllotaxis, leaf primordia grow one per node, each at a constant divergence angle of 137.5 degrees from the previous.

        Spiral Phyllotaxis

In multijugate phyllotaxis, two or more leaf primordia grow at the same node. Leaf primordia of a node are spread evenly around the stem, each group of leaf primordia of a node is at a constant divergence angle of 137.5 degrees from the leaf primordia group of the previous node.

 Multijugate Phyllotaxis

Often, multijugate patterns look very similar to spiral patterns and the only way to detect them is to count the number of spirals visible in the pattern (called parastichies).  If the parastichy numbers have no common divisor other than 1, the pattern is a spiral phyllotaxis. If the parastichy numbers do have a common divisor k, then the pattern is multijugate (more precisely k-jugate) and there are k elements at each node.


Aonium has parastichy numbers (2, 3), since 1 is the only common divisor of 2 and 3, this is a spiral pattern. Since spiral phyllotaxis can be viewed as 1-jugate, the notation 1(2, 3) is also used for this pattern.


Gymnocalycium has parastichy numbers (10, 16), which have the common divisor k=2. Hence this is a multijugate pattern (more precisely 2-jugate). The notation 2(5, 8) is also used to classify this pattern.

Below are shown the scanning electron micrographs of shoot apical meristems of a gymnosperm and an angiosperm:

 Norway Spruce (8, 13)        Artichoke (34, 55)

Other examples of Spiral Phyllotaxis include:

 Aloe       Cabbage

 Pine     Sunflower


III.    Parastichous Spiral Phyllotaxis – Fibonacci – Golden Ratio and the Golden Angle

The white Lily has 3 petals, Ranunculus has 5 petals, Marigold has 13 petals, Aster has 21 petals and Daisies and Sunflowers have 34, 55 or 89 petals depending on variety.  Equally, depending on variety, Daisies and Sunflowers have floral primordia arranged in 34 clock-wise spirals and 55 counter-clock spirals - denominated parastichous spiral phyllotaxis (34, 55), some daisy and sunflower varieties have a (55, 89) phyllotaxis while others have a (89, 144) phyllotaxis. 

All of the above mentioned numbers of petals and seed spirals belong to the Fibonacci Series (named after Leonardo of Pisa, son of Bonacci, c. 1170 – c. 1250):

Fib(1) = 1

Fib(2) = 1

Fib(n) = Fib(n-2) + Fib(n-1)

→ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, .....

Fib(n-1) divided by Fib(n-2) converges to the golden ratio, denoted by the lowercase Greek letter phi (φ) = 1.618033988…, at about Fib(24)/Fib(23) or 46368/28657.  The irrational number phi (φ) possesses the unique characteristic such that φ is equal to one plus the reciprocal of φ, which yields the algebraic solution:

φ = 1 + 1/φ             →
φ 2φ – 1 = 0         →
φ = (1 + √5)/2 = 1.618033988…

The formula φ = 1 + 1/φ can be expanded recursively to obtain a continued fraction for the golden ratio:

The convergents of the continued fraction (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...) are ratios of successive Fibonacci numbers.  The golden ratio has the simplest expression − yet slowest convergence − as a continued fraction expansion of any irrational number.  Multiplying the formula by φ yields the equation φ2 = 1 + φ which results in the continued square root form of the golden ratio:

Mathematicians since Euclid 300 B.C. have studied the properties of the golden ratio, including its appearance in the dimensions of line segments, the golden triangle, the regular pentagon, the pentagram and the golden rectangle:  

Line segments:



Golden triangle:


Regular Pentagon:




Golden Rectangle:



To the left above is shown the relationship between the golden rectangle and the Fibonacci spiral resulting from the tangents of the successive numbers of the series, to the right, Greek temples and other constructions were based on rectangles adjusted to the golden ratio, because it allowed successive subdivisions into squares and rectangles with the exact same proportions as the original.

In plants with spiral phyllotaxis, successive leaf or floral primordia are separated by the golden angle that divides the circle of the meristem according to the golden ratio:

360º – 360º/φ = 360º/φ2 = 137.5077641 degrees = Golden Angle.


IV.    Mechanical dynamics of Apical Meristem growth

Because of the way plants grow due to consecutive cell divisions, with anticlinal (perpendicular) and periclinal (parallel) cell divisions increasing respectively organ circumference and girth, leaf or floral primordia migration at the apex becomes faster as the radius increases, with velocity proportional to the radius.  In Sunflowers the n-th seed is located from the apex centre at a distance in relation to the square root of n.  Velocity of growth, determined by cellular divisions and cell elongation − both triggered by the phytohormone auxin, together with the direction of cellular divisions, allow for organized organ growth by placing consecutive primordia at a golden angle which results in leaves or seeds to be packaged with a maximum efficiency.  The result is evolutionary convergence towards the golden angle which allows for more primordia in less space.  Leonardo da Vinci was the first to suggest that the adaptive advantage of the Fibonacci pattern in the plant kingdom is to maximize number of primordia in space as to optimize their access to moisture, rainfall and sunlight [1].  Goethe − the German poet, philosopher and natural Scientist − postulated in 1830 the existence of a general ‘spiral tendency in plant vegetation’ [2].  Previously, Goethe had published in 1790 “Versuch die Metamorphose der Pflanzen zu erklären”, correctly deducing the serial homologous nature of leaf organs in plants, from cotyledons − to photosynthetic leaves − to sepals and petals of a flower, suggesting that the constituent parts of a flower are all structurally modified leaves which are functionally specialized for reproduction or protection.

Insight into the Phyllotactic spiral mechanism was gained when Wilhelm Hofmeister proposed a model in 1868: a primordium forms at the least crowded part of the apical meristem so that the golden angle between successive primordia is the blind result of physical forces pushing primordia to guarantee that none of them ever follow the same radial line from the center to the edge [3].

Compare a scanning electron micrograph of the shoot apical meristem of a Norway spruce (Picea abies) with the computer generated arrangement of primordia based on the mathematical model which implements the rules of Hofmeister.  Primordia are numbered according to their age - the higher the number, the older the primordium.  Notice the remarkable similarity between the plant and the mathematical model.

In 1878 Simon Schwendener published his book “Mechanische Theorie der Blattstellungen” [4], he proposed that different leaf arrangements result from the contact pressure that each growing leaf primordium exerts on its neighbours.  To develop his argument he used force diagrams, ruler and compass constructions, together with a mechanical apparatus that served essentially as an analog computer which simulated the conditions tested.  In 1904 Arthur Harry Church established the Shoot Apical Meristem as the origin of pattern and described the relation of phyllotaxis to mechanical laws [5], he proposed the idea that parastichies are lines of force, undulating cellular masses with different wavelengths produce the patterns of parastichies.  He put forward a mathematical treatment of these ideas naming his theory equipotential phyllotaxis, describing the mathematical aspects and properties of parastichies as lines of force in analogy to the hydrodynamic lines of force in vortex movements or of similar lines of force present in electromagnetic fields.  Church’s metaphysical view considered that life energy follows lines of force comparable to those of electric fields; however, Church failed to recognize that patterns can be generated merely as a consequence of regular apex growth.

It was Mary and Robert Snow who first recognized that models of phyllotaxis up to 1931 had fallen into two categories: those who proposed pattern to be determined by an unknown property derived from the stem and those who had advocated leaf arrangement to be dictated by contact with existing primordia.  The Snows tested the validity of the models – assuming that if those in the second group were correct, the manipulation of primordia as they arose would consequently influence the position of subsequent primordia.  Whereas previous researchers had only observed and calculated, Snow and Snow experimentally used a cataract knife and a dissecting microscope to perform microsurgery on Lupinus albus shoot apical meristems [6].  The couple observed that isolating a primordium or its site caused phyllotactic spirals to be reversed with primordia forming at opposite sides to those expected had a normal phyllotaxy been retained.  This led the Snows to propose a concept of "first available space", in line with the ideas previously proposed by Hofmeister in 1868.  Existing primordia determined where a new primordium formed – and this was in the widest gap, furthest away from the growing apex [6, 7, 8].  The next researcher to carry out microsurgery was Claude Wilson Wardlaw in 1949 [9].  Wardlaw manipulated the large meristems and widely spaced primordia of the fern Dryopteris using comparable microsurgery techniques and arrived at similar results to those of the Snows.  He did not reject the hypothesis of the Snows regarding the "first available space"; however, he drew inhibitory fields across the meristem interpreting "space" according to the hypothesis laid out by Schoute in 1913 who had proposed that the placement of primordia was determined by a chemical inhibitor produced by preceding primordia [10].  Wardlaw argued that space itself did not "execute" anything in particular and was not a "causal factor" in morphogenesis.  A new primordium emerged at the position of the weakest inhibition with the most recently formed primordium as the inhibitor source [9].  After Wardlaw, the assumption that phyllotactic patterns could be explained by inhibitory fields remained an enduring view.


The Snows and Wardlaw.  (a) and (b) Diagrams by Snow and Snow of transverse sections of the shoot meristem of Lupinus albus [6]. (a) The straight line across a normal bud marks the position of the incision made to isolate the area of tissue (I1 region), about to initiate the next primordium. (b) Phyllotactic alteration caused. Note the different positions of I4 and I5 after incision. P1, P2, P3, etc. = existing primordia, in order of increasing age. I1, I2, etc. = incipient primordia, invisible at the time of operation, in order of appearance.  (c) and (d) Representations of Wardlaw of the shoot apex of Dryopteris [9]. (c) Position of leaf primordia and incipient primordia. (d) Possible "inhibitory fields" associated with these "growth centres". ac = apical cell; b = bud rudiments; m =  lower limits of apical meristem. The dashed line indicates the approximate base of the apical cone. I1, I2, etc., as above. Numbers = existing primordia (increasing age).


The hypotheses put forward by Hofmeister (1868) for the formation of primordia were verified in 1992-1996 by Stéphane Douady and Yves Couder in a compelling physical experiment which involved magnetized drops of ferrofluid dripped onto a magnetized dish filled with silicone oil, drops repelled each other and the physical experiment produced phyllotactic patterns with Fibonacci numbers [11, 12].  The dynamics of the phyllotactic patterns was found to be controlled by only one adimensional parameter G = V * T/R; with V equal to Velocity (controlled by the magnetic field gradient), T corresponding to Time (controlled by drip frequency) and R to Radius from the centre.  Parameter G was found to be equivalent to Richards’ plastochrone ratio [13], which characterizes apex growth by a = rn-1/rn as the proportion of radial distances of two successive primordia from the apex centre.  Richards’ index is still currently used for the description and quantitative analysis of phyllotaxis.  The results of Douady and Couder clearly demonstrated that all dynamical models in which a new primordium forms in a position determined by either repulsion or inhibition from the previous elements will give rise to the organization of Fibonacci order and spiral Phyllotaxis with a convergence towards the golden angle for decreasing G values [11, 12].


Three photographs seen from above of typical phyllotactic patterns formed by ferrofuid drops for different values of the control parameter G [11, 12](a) For G ≈ 1 each new drop is repelled only by the previous one and a distichous mode is obtained, φ = 180º. (b) For G ≈ 0.7 the successive drops move away from each other with a divergence angle φ = 150º (between drop three and four).  Drops define an anti-clockwise spiral shown as a dashed line with parastichy numbers (1, 2).  (c) For smaller G values (G ≈ 0.1) higher order Fibonacci modes are obtained.  Here φ = 139º and parastichy numbers are (5, 8).


On the other hand, in line with the ideas initially suggested by Schwendener (1878) and Church (1904), Paul B. Green proposed during the period of 1992-1996 that biophysical forces such as tension, compression and shear in the L1 layer regulate organ formation and meristem pattern.  According to the hypothesis of Green, no differential regulation of cell identity is required and there is no necessity for specific chemical signals in the meristem to establish phyllotaxis [14, 15].  Biophysical regulation was thought to be based on the tensile and compressive forces within the meristem: forces result from the geometry and growth of the apex operating on the entire meristem including the L1 layer.  Green believed that growth is regulated at a global meristem level and patterns emerge as a consequence of local changes in biophysical parameters that result from the growth and development of the apex as a whole, similar to the observable fact which gives “wrinkles in wet skin”.  Computational modeling of these biophysical forces recreated natural phyllotactic patterns [14, 15]. 


Green: undulations [14, 15].  (a) Wild-type floral meristem of Antirrhinum. Stamens arising in whorl 3. (b) Floral meristem of a deficiens mutant of Antirrhinum, which makes sepals in the first two whorls of the flower, followed by carpels. (c) and (d) Mechanical simulations of undulations in a fixed circular region. It is proposed that in the meristem, undulations depend on the ratio of corpus (flexible) to tunica (rigid). A shift from out-of-plane undulations, giving stamens, to in-plane undulations will give rise to carpels, mimicking the effects of the deficiens mutant. s, sepals; p, petals; st, stamens; c, carpels.


Hernandez and Green proceeded to apply these principles of biophysical forces in vivo by growing sunflower heads in constrained environments, resulting not only in predictable alterations in phyllotactic patterns but also in altered organ identities [16].  Green interpreted these results as an "abnormal buckling" phenomena in which physical processes have an influence on gene expression levels [16] and a “tissue-level physical process appears to be upstream of organ-specific gene expression” [17].


Experimental Setup and Resulting Abnormalities of Constrained Sunflower Capitula [16].  (A) The device used to constrain young capitula seen after 9 days of treatment showing the head's oval shape. (B) Anomalies in a mature capitulum with flower corollas removed. Three bracts subtend two ovaries (upper arrow). Two bracts nest together (lower arrows).  (C) Comparable to (B), but in another region of the capitulum. Between the arrows is an anomalous row of bracts.  (D) An unusual "disc trumpet" not previously seen on other discs. It has the dimensions of a ray floret with the radial symmetry of a normal disc flower.  (E) Central region with bract primordia. Note that all lamina are at right angles to the capitulum center (black dot). This presumably reflects a causal relation between the orientation of the generative front line and the blade line. Bars in (A) and (B) = 5 mm; bar in (D) = 3 mm; bar in (E) = 250 μm.



V.      Physiological Basis of Apical Meristem growth

Johannes Cornelis Schoute published in 1913 his paper “Beitrage zur Blattstellungslehre” proposing the theory that placement of a primordium is determined by a chemical inhibitor secreted by previous primordia that prevents a new primordium from emerging too close [10].  This theory was dominant during the first half of the twentieth century.  The link between chemical gradients resulting from genetic instructions and the physical laws that govern morphogenesis were first expressed by Alan Turing, who also worked as a cryptanalyst during WWII and cracked the secret ciphers of the German Enigma machine using a mathematical method called bombe, although he is better known as the father figure in the birth of computers and artificial intelligence (he invented the Turing machine, which is the simplest form of a computer he also helped in forming algorithms and concepts which had a main role in the creation of the modern computer).  During his later years his central interest was in understanding Fibonacci phyllotaxis: the existence of Fibonacci numbers in plant structures.  Turing suggested “that a system of chemical substances called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis”.  In 1952 Turing used mathematical analysis and what is perhaps the first example of computer simulation in developmental biology, he showed that a pattern could emerge from a group of identical cells, all operating with identical rules.  He used reaction - diffusion equations which are central to the field of pattern formation and correctly predicted that the diffusion of two different chemical signals, one activating growth and diffusing slowly together with one deactivating growth and diffusing rapidly, would set up different patterns of development – he proved mathematically that certain chemical reactions can lead to non-homogeneous concentrations of chemicals which result in plant morphogenesis and pattern creation [18].  In 1996 Green, Steele and Rennich demonstrated that physical stress is also capable of producing phyllotactic patterns similar to those caused by the morphogens predicted by Turing [17].

The importance of chemical gradients and transcriptional controls in the establishment of pattern was beautifully worked out in Drosophila melanogaster in the 1980s (see for an introduction to developmental patterning in fruit flies); now it is know that similar processes occur in plants.  In the early 1950s when Alan Turing established the mathematical basis of reaction - diffusion equations, Auxin had already been characterized and isolated in 1937 as the first Phytohormone by Frits Went and Kenneth V. Thimann [19].  As early as 1880 Charles Darwin had published in his book “The Power of Movement in Plants” a description of plant phototropism and gravitropism responses [20], these are now recognized to be controlled by Auxin gradients, in addition, Auxin has also been implicated in many other aspects of plant development such as hydrotropism, root development, vascular differentiation, leaf venation, apical dominance, embryo axis formation and somatic embryogenesis.  Nonetheless, the end of the 20th century had to be reached until Auxin was finally shown to also induce leaf and flower formation [21] at the Shoot Apical Meristem (SAM).  

The role of Auxin in organ initiation and phyllotaxis was studied in tomato using a specific inhibitor of polar Auxin transport and also in the Arabidopsis Auxin transport mutant pin-formed1.  Didier Reinhardt et al. (2000) showed that the Auxin transport inhibitor, NPA, specifically inhibited leaf initiation in tomato meristems [21].  However, meristem perpetuation and stem growth were not affected, since the apices continued to grow, forming NPA pin shoots which lacked leaf primordia.  Thus, Auxin transport is specifically required for leaf initiation, but not for general meristem growth.  The separation of organ formation from general meristem growth made it possible to analyze these processes independently.  To establish the role of Auxin in leaf formation, small droplets of lanolin containing indole-3-acetic acid (IAA) were administered to the flank of the tomato NPA pin shoots, at a distance from the summit that corresponded to the distance in natural leaf formation.  The treatment induced leaf formation in the tomato NPA pin shoots after only one day [21].


Inhibition of tomato leaf primordium initiation by the Auxin Transport Inhibitor NPA [21].  (A) Culture of tomato apices on NPA-containing medium results in a naked pin (arrowhead), outgrowth of axillary meristems (A), and altered development of preexisting primordia (P1 and P2, with P1 being the youngest primordium at the beginning of the experiment). (B) Close-up of an NPA pin visualized by low-vacuum scanning electron microscopy.


Induction of tomato leaves on NPA Pins by Auxin (IAA) [21].  (A) NPA pin 4 days after treatment with control paste (white arrowhead) at the flank of the meristem (black arrowhead). (B) to (D) Induction of a leaf primordium on NPA pin treated with red lanolin paste containing 10 mM IAA (white arrowheads are IAA treatments and black arrowheads point to the meristem). The same pin was photographed after 1 day (B), 2 days (C), and 4 days (D). White Arrows point to leaf primordia (P). (E) Leaf induced by IAA on NPA pin.


Arabidopsis mutations in the pin-formed1 (pin-1) gene lead to a defect in organ formation in the inflorescence, resulting in pin-shaped stalks resembling the above tomato apices cultured on Auxin transport inhibitors [22].  This mutation has been traced to a putative Auxin efflux carrier [23], indicating that, as with leaf formation in tomato, flower initiation in the Arabidopsis inflorescence requires a polar Auxin transport system.  Reinhardt et al. (2000) administered droplets containing IAA to the flank of pin-1 meristems inducing flower primordia [21].  In both tomato and Arabidopsis shoot pins, organs were only induced by IAA application at the flank of the meristems as in the case of the normal control meristems, on the contrary, organs were never induced by IAA on pin-1 meristems at the summit or below the flank of the meristems [21].


Leaf and Inflorescence Phenotype of the Arabidopsis pin1-1 Mutant and Induction of Flowers by IAA.  (A) Wild-type Arabidopsis seedling with two leaf primordia.  (B) pin1-1 seedling with one fused leaf primordium.  (C) pin1-1 seedling with a cup-shaped leaf primordium.  (D) Inflorescence apex of a pin1-1 mutant plant devoid of flowers. The arrowhead indicates the meristem.  (E) to (G) Induction of flowers by treatment with 1 mM IAA (red paste) at the flank. Apices were analyzed 38 hr (arrow points to local bulge) (E), 4 days (F), 7 days (G) after treatment. Arrowheads in (E) to (G) denote the meristem.


The results discussed above show that the initiation of leaves and flowers share a common mechanism involving auxin, therefore they may not represent fundamentally different processes.  As a trigger of organ initiation, Auxin plays a role in determining organ position and therefore phyllotaxis [21]. 



VI.    Genetic Basis of Apical Meristem growth

Plant organ growth is a complex quantitative trait with a genetic control that involves a large number of genes and regulatory elements which are up and down regulated in concert due to the interplay between phytohormones and various environmental signals (24, 25), in addition, auxin − as discussed above − together with genes and regulatory elements specify leaf or floral primordia formation, but the genetic regulatory network (GRN) of apex growth does not specify the gap between consecutive primordia; that is taken care by the physical dynamics of plant growth.  In other words, plant morphogenesis and development are an association between genetics and physical laws.  In 1994 Brian C. Goodwin expressed that genes are important, but only as part of a process constrained by physical laws, the environment and the universal tendencies of complex adaptive systems (14).

Physical aspects of Apical Mersitem growth have been briefly reviewed above, with regard to the environment: as plant development occurs exclusively postembryonically it is an environmentally plastic process, meaning that information about the environment is integrated with genetic programs to affect the pattern of growth.  Environmental plasticity ensures that the plant body forms in a way that is appropriate for its environment, even when the environment is continually changing.  Unlike the shoot and root systems, leaf and floral structures are usually determinate organs that grow to a predetermined finite size and often only function for a finite period.  For instance, deciduous trees produce and discard leaves in annual cycles, in a regular and reproducible fashion as the genetic pathways that underlie leaf formation and maturation.


With regard to the genetic control of meristem size, the process is rigorously controlled and is determined by the rate of cell production and the rate at which cells leave the meristem as leaf primordia.  In the Arabidopsis clavata1 mutant, cell proliferation in the meristem is unrestrained, the meristem grows abnormally large and phyllotactic patterns are disrupted.  CLAVATA1 is a component of the feedback loop that restricts the size of the meristem.  Two mutants in monocot species, abphyl in maize (Zea mays) and decussate in rice (Oryza sativa), have a change in phyllotaxy so that leaves are produced in pairs rather than alternately.  ABPHYL and DECUSSATE affect cytokinin signaling.  Cytokinin regulates cell proliferation in the SAM and both these mutants have an increased SAM size.  Therefore, the altered phyllotactic patterns are probably a secondary consequence of changes in meristem size that affect the communication between primordia.  (REFERENCES).

Another step in experimental phyllotaxis was reached when molecular biologists identified genes that promote a change in phyllotaxis from spiral to whorled in Anthirrhinum (Carpenter et al., 1995).


VII.   Evolutionary aspects of Phyllotaxis

The prevalence of regular phyllotactic patterns in nature suggests that there must be a selective advantage of regular versus random arrangement of organs.   In the case of flowers, it is conceivable that regular architecture is important for the attraction of pollinators or for an optimal packaging of seeds as in sunflower heads, in the case of leaves, possibly to avoid shading and have an optimal access to light and moisture.  Nonetheless, the reoccurrence of several distinct phyllotactic patterns in nature may indicate that regularity itself, rather than the specific phyllotactic arrangement, represents a selective advantage over random leaf arrangement.

As has been seen above, the process of leaf initiation is intimately linked with the phyllotactic positioning and several mutants affected in the process of leaf formation also exhibit defects in leaf positioning.  Therefore, it may be the process of leaf formation at the meristem, rather than the function of the final arrangement of the mature foliage, that requires regularity for optimal function.  Besides the formation of leaves and flowers, the meristem carries out at least two additional important functions, namely self-maintenance and the formation of the internodes.  The stem cells required for self-maintenance reside in the central zone of the meristem, where they continuously produce new cell material for organogenesis.  Founder cells are selected at specific sites in the peripheral zone.  Thus, the meristem has to provide exactly the right number of cells at the right position and at the right time to replenish the cells engaged in organogenesis.  Here, regular phyllotaxis may represent a selective advantage over random leaf position, allowing the meristem to optimally allocate founder cells, thus avoiding depletion or over-proliferation of organogenic cells on one side of the apex.  Since the meristem is responsible for the formation of the stem, an imbalance of cell number could potentially affect shoot architecture as a whole.  For example, if cells were depleted on one side, there would be fewer cells for the formation of the internode.  The resulting internode would therefore be curved.  In this context, it is interesting to note that mutants with irregular phyllotaxis frequently exhibit irregular and sometimes curved internodes ([22], 20).  Thus, regular phyllotaxis may be important for optimal development at the shoot tip as well as for the final architecture of the entire shoot (23).

Phyllotaxis in the terminal ear1 (te1) mutant of maize (ref. 20).  (A) Top view of a te1 plant.  Instead of distichous phyllotaxis te1 exhibits irregular phyllotaxis.  (B) Lateral view of a te1 shoot axis with irregular internodes (leaves removed).


VII.   References

[1] I. Adler, D. Barabe, R. V. Jean (1997).  A History of the Study of Phyllotaxis.  Annals of Botany 80: 231-244.

[2] J.W. Goethe (1830).  Über die Spiraltendenz der Vegetation, in Schriften zur Botanik und Wissenschafttslehre, dtv Gesamtausgabe 1963, München, Germany.

[3] W. Hofmeister (1868).  Allgemeine Morphologie der Gewachse.  Handbuch der Physiologischen Botanik 1: 405-664. Leipzig, Engelmann.

[4] S. Schwendener (1878).  Mechanische Theorie der Blattstellungen. Leipzig: Engelmann.

[5] A.H. Church (1904).  On the relation of Phyllotaxis to Mechanical Laws.  Williams & Norgate, London.

[6] M. Snow and R. Snow (1931). Experiments on phyllotaxis. I. The effect of isolating a primordium. Phil. Trans. R. Soc. London B221: 1–43.

[7] M. Snow and R. Snow (1933). Experiments on Phyllotaxis. II. The Effect of Displacing a Primordium. Phil. Trans. R. Soc. London B222: 353-400.

[8] M. Snow and R. Snow (1935). Experiments on Phyllotaxis. III. Diagonal Splits through Decussate Apices. Phil. Trans. R. Soc. London B225: 63-94.

[9] C.W. Wardlaw (1949). Experiments on organogenesis in ferns. Growth Supplement 13: 93–131.

[10] J.C. Schoute (1913).  Beitrage zur Blattstellunglehre. I. Die Theorie. Recueilde Travaux Botániques Néerlandais 10: 153-339.

[11] S. Douady and Y. Couder (1992).  Phyllotaxis as a physical self-organized growth process. Physical Review Letters 68: 2098-2101.

[12] S. Douady and Y. Couder (1996).  Phyllotaxis as a Dynamical Self Organizing Process (Part I, II, III).  J. theor. Biol. 139: 178-312.

[13] F.J. Richards (1951). Phyllotaxis: its quantitative expression and relation to growth in the apex. Phil. Trans. R. Soc. London B114: 498-453.

[14] P.B. Green (1992). Pattern formation in shoots: A likely role for minimal energy configurations of the tunica. Int. J. Plant Sci. 153: 59-75.

[15] P.B. Green (1996). Expression of form and pattern in plants – a role for biophysical fields. Semin. Cell Dev. Biol. 7: 903-911.

[16] L.F Hernandez and P.B. Green (1993).  Transductions for the expression of structural pattern: analysis in sunflower. Plant Cell 5: 1725–1738.

[17] P.B. Green, C.S. Steele, S.C. Rennich (1996).  Phyllotactic patterns: a biophysical mechanism for their origin. Annals of Botany 77: 515-527.

[18] A.M. Turing (1952).  The chemical basis of morphogenesis. Phil. Trans. R. Soc. London B139: 545-566.

[19] F.W. Went and K.V. Thimann (1937).  Phytohormones.  The Macmillan Company, New York.

[20] C.R. Darwin (1880).  The power of movement in plants. London: John Murray.

[21] D. Reinhardt, T. Mandel, C. Kuhlemeier (2000).  Auxin Regulates the Initiation and Radial Position of Plant Lateral Organs. The Plant Cell 12: 507-518.

[22] K. Okada, J. Ueda, M.K. Komaki, C.J. Bell, Y.Shimura (1991).  Requirement of the auxin polar transport system in early stages of Arubidopsis floral bud formation. Plant Cell 3: 677-684.

[23] L. Gälweiler, C Guan, A. Muller, E. Wisman, K. Mendgen, A. Yephremov, K. Palme (1998).  Regulation of polar auxin transport by AtPIN1 in Arabidopsis vascular tissue. Science 282: 2226-2230.



14) B. Goodwin (1994).  How the Leopard Changed its Spots: The Evolution of Complexity.  Charles Scribner’s Sons, New York, 1994.

20) B. Veit, S.P. Briggs, R.J. Schmidt, M.F. Yanofsky, S. Hake (1998).  Regulation of leaf initiation by Terminal Ear1 gene of maize. Nature 393: 166-168.

Not Used Yet - 22) D. Reinhardt, E.R. Pesce, P. Stieger, T. Mandel, K. Baltensperger, M. Bennett, J. Traas, J. Friml, C. Kuhlemeier (2003).  Regulation of phyllotaxis by polar auxin transport. Nature 426: 255-260.

23) D. Reinhardt (2005).  Regulation of Phyllotaxis.  Int. J. Dev. Biol. 49: 539-546.

24) L. Bögre, Z. Magyar, E. López-Juez (2008).  New clues to organ size control in plants.  Genome Biology 9: 226.  (

25) K. Johnson and M. Lenhard (2011).  Genetic control of plant organ growth.  New Phytologist 191: 319–333.


About PhiTaxis:


GIRASOLE: Digital Fibonacci Sunflower

GIRASOLE: Digital Fibonacci Sunflower – is a simulation of spiral apex growth with a (13, 21) phyllotaxis, resulting in successive inter-connected floral primordia numbered clock-wise 1,14,27,40,53... with a successive increase of 13, while counter-clock-wise inter-connected floral primordia are numbered 1,22,43,64,85... with a successive increase of 21.  To place initially Fib(7) = 13 primordia with a radius of 1 mm, a minimum orbit radius of Fib(6) = 8 mm is necessary.  To maintain organized growth, a primordium radius increase of 1 mm and an orbit radius increase of the square root of n are necessary, where n is the last primordium formed on the new orbit.  This means that as 13 new floral primordia are formed with a radius of 1 mm, the previous 13 have moved a distance proportional to the Sqrt of 26 mm away from the origin turning by the golden angle while their radius has increased to 2 mm.

The Turbo Pascal algorithm of GIRASOLE: Digital Fibonacci Sunflower is:


function fib (f : integer) : integer;                                               {Recursive Fibonacci Function}


if (f=1) or (f=2) then

fib := 1


fib := fib(f-2) + fib(f-1)



fi := (1+sqrt(5))/2;                                                                          {Golden ratio}

alfa := (2*Pi)-((2*Pi)/fi);                                                                {Golden angle in radians}

beta := alfa;

orx := (GetMaxX div 2);

ory := (GetMaxY div 2);

OrbitRadius := fib(6);                                                                   {8: Initial Orbit Radius}

PrimordiumRadius := fib(2);                                                       {1: Initial Primordium Radius}


for o := 1 to fib(8) do                                                                     {21: Number of Orbits}


for p := 1 to fib(7) do                                                           {13: Number of Primordia per Orbit}


if beta > (2*Pi) then

beta := beta-(2*Pi);

geKat := round(sin(beta)*OrbitRadius);

anKat := round(cos(beta)*OrbitRadius);

newx := orx+anKat; newy := ory-geKat;

circle(newx, newy, PrimordiumRadius);          {Draw Primordium}

beta := beta+alfa;                                                  {Turn by Golden Angle}


OrbitRadius := OrbitRadius+sqrt((o+1)*fib(7));              {Inrease Orbit Radius by Sqrt n}

PrimordiumRadius := PrimordiumRadius+fib(2);         {1: Increment of Primordium Radius}



PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis

PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis − is controlled by six parameters.  Parameter 1 codes for the total number of primordia per orbit, parameter 2 the initial primordium radius, parameter 3 controls the way in which the primordium radius increases and has two variants: either by the Fibonacci series or by a constant number.  Parameter 4 codes for the total number of orbits with primordia, parameter 5 specifies the initial orbit radius and parameter 6 controls the way in which the orbit radius increases and has three variants: either by the Fibonacci series, by a constant number or by the square root of n, where n is the last primordium formed on the next orbit.  Random spiral Phyllotaxis can be simulated with PhiTaxis to observe possible patterns seen and that also could have evolved in nature.


Content Credits, Licensing & Attribution of

PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis

Content Credits:

Text and Images regarding the Phyllotaxis study material of PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis have been recompiled and edited by Dr. Richard Mario Fratini from a wide range of different Public Domain Internet Sources, a list of all Resources and Authors would prove to be endlessly long, yet detailed references have been provided for those interested in pursuing further knowledge concerning Phyllotaxis.  The Author is grateful and indebted to all the providers of information around the web.  A special mention is owed to Dr. Pau Atela and Dr. Chris Golé of the Department of Mathematics of Smith College in MA, USA, for providing the Orbital pictures and photographs of Aonium, Gymnocalycium, Artichoke, Norway Spruce and the Hofmeister Computer replication (

Graphics Algorithm of PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis written by Dr. Richard Mario Fratini for Microsoft DOS system using the 16 bits compiler of Turbo Pascal 7.0 by Borland and upgraded to 32 bits using the Free Pascal compiler.  Adaptation and implementation of the Graphics Algorithm of PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis to the 64 bits Microsoft WINDOWS XP, Vista and WINDOWS 7 systems by Dr. Richard Mario Fratini using the Java NetBeans IDE 6.5 and NetBeans Platform of Sun Microsystems.


Text and Images of PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis are in the public domain and thus free of any copyright restrictions. However, as is the norm in scientific publishing and as a matter of courtesy, any user should credit the content provider for any public or private use of Text or Images whenever possible.  Please cite as: R. Fratini. (2012).  PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis.

PhiTaxis: Fibonacci digital simulation of spiral Phyllotaxis is free software and distributed under the BSD license, in the hope of being a useful tool to understand Apical Meristem Structure and Spiral Phyllotaxis, with an ultimate aim to stimulate the conception of new plant ideotypes for the advancement of a creative and innovative Plant Breeding.


Dr. Richard Mario Fratini.

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