Creating Phyllotaxis

The Stack-and-Drag Model

 

Frank M. J. van der Linden

[2007]

 

 

PhaseLab Foundation

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from the original publication, printed in B&W, in

MATHEMATICAL BIOSCIENCES 133:21-50 (1996)

by Elsevier, NY SSDI 0025-5564(95)00077-Q


Abstract

The genesis of phyllotaxis, the origin of the pattern of appendages on the surface just below the apical extreme of many plants, is an old unsolved puzzle. Whereas many models generate helices, the stack-and-drag model is the first to achieve this in an integral construction from seed to flower. Combination of the principle of gnomonic growth, where consecutive additions have comparable positions, with a "dragging" principle, where the developing zone follows the apical tip, provides a powerful tool in simulating a wide range of phyllotactic manifestations. The influence of three vital parameters for primordial size, compressibility, and canalization (or annular arrangement) helps in understanding the problem's nature.


1     Introduction

1.1    Observed patterns and some divergent models

Most plant bodies show a radial symmetry as a consequence of their axial development. Primordia (early appendages) are ordered in various phyllotactic patterns such as distichous (or opposite), decussate (or cross-opposite), in whorls, or in parastichies (or spirals and helices). Therefore it seems defendable to start a simulation for pattern generation from a centric growth principle. Investigations on pattern formation of epidermal cells in vegetative ontogeny [1-3], centric experiments [4], and centric surface models [5-8] show that use of (centric) surface models can be practical. However, studies on floral ontogeny [9, 10] and surgical experiments [11, 12] clearly contradict a purely centric theory by showing that centric phyllotaxis can be deduced from (or influenced after) peripheral pattern formation.

1.2    Differences from other models

This paper presents a phyllotactic model of which the spheres close-packing or stacking principle is traditional to some degree [13-15] but employs a dragging principle where primordium stacking below certain regions is prohibited. The use of a dragging aspect as a counterpart of gravity and adhesion is justifiable, for new material arises continuously at the top of the stem whereas the epidermal and lower layers have to follow. In the model, this dragging causes canalization of organ numbers or the tendency of organ numbers to remain constant in spite of genetic or environmental variation [16]. Canalization has not previously been incorporated in phyllotactic models for complete plant bodies. The arrangement of primordia in concentric circles on the shoot apex is also responsible for the annular differentiation of flower parts [17].

In contrast to existing theories that are based on observed geometrical consequences (like the Fibonacci angle of 137.5¡) [13-15, 18-30], the stack-and-drag model uses the gnomonic growth property [31], which makes the introduction of distances or angles to control correct unit positioning redundant. (For an understanding of the mechanism used in the construction of primordial units in the stack-and-drag model, a knowledge of the algorithms in the dislodgement model [7] is necessary. A structure grows gnomonically if it keeps its form during the process. An example is shown by the snail, which grows its shell by secreting ever larger amounts of chitin on the open side of the helical tube [32]. A more basic example is the planar stacking of growing circles from a center outward. Every circle is constructed against its predecessor in the same rotation direction around the initial circles. The result is a simple spiral of circles of increasing size. The dislodgement model [7, 33] obeys a slightly more complicated mechanism. Every new circle is initially constructed in a consequent position in relation to its predecessor. The structure's disk shape is maintained by adjusting configurations when a neighboring circle should intersect the new one. The stack-and-drag model resembles the dislodgement model, but has more biological justification because it shows (1) a stacking from base to top, (2) a predefined transition vegetative-generative stage, and (3) restricted settling of primordia in a zone.)

1.3    Goals and limitations of the theory

To demarcate the field of study, ramifications [34] are excluded. The model strives for

(1) Continuity in development from cotyledons to inflorescences and flowers (any initiating rule is consequently repeated from the very start, regardless of the motive behind it)

(2) A very limited set of easily recognizable parameters frequently mentioned in the literature

(3) Numerical and graphical output that can easily be compared with observations reported in the literature [19, 35, 36]

(4) Solutions of some open problems in phyllotaxis [35, 37-39]

(5) Consistency, according to recommendations for phyllotaxis theories [39].


2     Approach and biological evidence

2.1    Cotyledons, stem, and flowering

The stack-and-drag model follows a tradition in the geometry of close-packing of spheres or circles in nonliving structures [40-42]. (Filling surfaces and bodies with circles and spheres has a long tradition that can be traced back to Plato, Archimedes, Kepler, and Kelvin.) The stacking is from two initial similar and touching spheres, like cotyledons, upwards. A phyllotactic unit is defined as a primordium or a sphere of influence including the "primordium"-forming cells. Primordial units may be shaped as spheres and arise one after the other with a nearly constant time interval (plastochron). The vegetative stem (Figure 1) is presented as an infinite cylinder (Figure 2).

The switch from vegetative to generative growth (flowering) [17] (Figure 3) is simulated by ending the cylinder stacking. Now, an apical hemisphere with radius 1 becomes the bearer of primordial spheres of constant radii [16] (Figure 4). For flower heads, the distance ratio becomes relatively small just before flowering, and units are nearly constant in size. In simulations of vegetative structures and flowers, units will soon fill up the hemisphere.

2.2    The gnomonic principle: biological justification and application in the model

In investigations and modeling in phyllotaxis, primordia are identified by numbering them in a logical way [37]. In general, it is an open question as to whether the numbers represent the actuaI succession of inception or not [16], but it is often seen that successively numbered, primordia are embedded in similar regions, in other words, subsequently arising primordia lean against subsequent environments. This empirical rule is equivalent to the rule that a plant grows gnomonically. Any primordium has generally the same difference in ordinal order or age from its lower contacts as the preceding primordium had from it. This holds for all primordia. Small positional and dimensional adjustments are of second order. In floral plant structures the gnomonical way of growing is not obvious. However, phyllotactic pattern generation takes place in early stages in plant development, while primordia are not differentiated and the apical dome keeps its form more or less [20], so there is gnomonical growth.

 

 

 

FIG.1. The origins of vegetative phyllotaxis. Left: Basically, dicotyledons show horizontal and vertical symmetry. Only dicotyledons, and only the shoot, are the subject of study. Right: Just below the extreme top, a cluster of cells with an active envelope is continuously pushed forward by the elongating and multiplying shoot cells.


 

 

FlG.2. The stack-and-drag model: initiation (left) and relation r(unit)/r(stem) in the vegetative stage.


 

 

 

FIG.3. Growth directions in vegetative and generative stages. Left: The mother cells are pushed upward, thereby dragging their mitotic active envelope, which is annular [43]. The very top is quiet and does not generate primordia. Right: While the apical dome expands, reproductive organs are generated. In flower heads, as shown, the active ring remains behind for some time before it rapidly closes on the top while primordia arise as little hemispheres on the surface.


 

 

FlG.4. The stack-and-drag model: from vegetative (left) to generative stage.


 

In the plant, the inception of a new primordium is a local event that depends on certain circumstances such as gravitation, surface tension, expansion, and chemical influences. More important here is the ascertainment that (1) circumstances are broadly stable in time and (2) heredity plays a modest part. (Phyllotaxis shows an important phenomenon. Different species may show the same phyllotaxis, and within a species or even an individual plant, different but stable phyllotactic patterns are possible. This indicates a geometric constraint to have dominance over differentiation-determining forces. An example of species overriding geometry is the ubiquity of Fibonacci numbers with typical anomalies like the Lucas numbers. In cases where phyllotaxis is unstable, the geometric constraint may be subsidiary to genetic or environmental variations.) In order to predict the site at which a new unit will arise, the stack-and-drag model makes use of the gnomonic growth principle. In other words, the place of a new unit in a configuration at a certain point in time tells much about the place in which the next unit will arise (Figure 5). By applying the gnomonic property, modeling is possible without the instant need of knowledge of underlying mechanisms. The stack-and-drag model can be said to include other models in the sense that it neither follows any of three well-known hypotheses, nor excludes them:

(1) New primordia should arise as far as possible from newly born primordia [14, 27, 44], which implies an inhibitor mechanism. Many phyllotactic patterns seem to obey this theory, but others, such as the monostichous spiral, contradict it [35].

(2) New primordia should arise at the first available space [45], which implies a tension field theory.

(3) A new primordium is positioned by mechanical pressure of its contacts [18, 46]. This does not account for the initiating location of a new primordium.


 

 

FIG.5. At time t+1 a new primordium will become visible, but where? In looking back at the structure at time t, we see that the new primordium there arose between or against two support units of which we know the ordinal numbers p and q (see Figure 8). Because, when turned, structure t+1 fits on structure t, the new primordium will arise in neighborhood t+1. Thus, at time t+1, we choose support units with an age of one plastochron younger (or with ordinal numbers p+1 and q+1) than the support units of the primordium that just arose. If positioning is impossible for any reason, we have to shift the new unit (see Section 3.1).


2.3    Separation of pattern formation and structural growth

 During formation of phyllotactic patterns, the plant is growing. In the model, pattern generation is separated from structural expansion. The geometric relationships between neighboring units and stem radius just at the inception of a new unit form the basis for the consequently arising patterns in which the stem width is measured by the distance ratio r(unit)/r(stem). Thus, instead of increasing the cylindrical radius while adding primordial units with increasing radii, the cylindrical radius may be frozen while the size of the units is decreased (see next paragraph). The maximum distance ratio is presumed to be 1, defining the initial cotyledons, and it diminishes during the stacking of new spheres. (Just below the apical tip lies a central cell region with relatively low mitotic activity: the "mother cells" [47] or the mŽristme d'attente [43, 48, 49]. This region is accompanied by another hypodermal one with high mitotic activity in which primordia are initiated (Buvat's anneau initial).  Its annular shape causes canalization. In flowers such as Ranunculus acris, stamen and carpel primordia are initiated in a spiral sequence [50].) The stem's absolute thickening has no topological significance. The settling height for new spheres depends on the configuration of the upper units.

The place of a new spherical unit is calculated as if it were the last (top) one to be constructed. At the very moment of placing the unit, the growth process is simulated topologically. Thus, the results of arising, developing, and positioning are copied while at the same time the possibility of stacking more units on the structure is maintained by its cylindrical shape. This is achieved by positioning units locally in the correct manner, with their size related not to each other but to their receptacle. Because in the model the stem is cylindrical with radius 1, it is easy to calculate correct unit sizes. When these are known for initiating primordia of any ordinal number, a construction or even a growth simulation for a certain point in time of development can be executed. (The stack-and-drag computer program is not only capable of producing images of the stacking iterations during calculation and the calculation results, but also shows the growth process from cotyledons to flowering adults.) To understand the relative sizing in the model, we should imagine the development on the apical dome as follows (Figure 6): The extreme top of the apical dome, being a point, is not the growth center; the perimeter of the cluster of mother cells determines a ring of development. In fact, there can never be a growth center without dimensions. One should speak rather of an expanded point.

 

 

FIG.6. Transition from vegetative to reproductive stage, interpreting Buvat's concept. Left: A new unit may arise in the initiation zone (shaded area) when there is no room left between the neighboring units just below it. A unit may not arise at a position significantly lower than a preceding unit. The initiation zone has a width that relates closely to the upper unit sizes. In vegetative structures, represented in the cylinder stage of the model, the initiation zone is relatively wide. Right: Determinate growth. The cylinder is closed by an apical hemisphere, being filled in with primordial units that meet at a point not necessarily lying exactly on the Z axis. Unit radii are related to the cylinder radius (see Figure 10).


2.4    The three S curves

The model makes use of three S curves [51]: the unit growth curve and two translating curves simulating axial and radial stem growth. (The vegetative apex promotes longitudinal growth, and the reproductive apex produces a meristematic envelope with a large surface area from which the parts of a flower or flowers develop [52]. The transition from vegetative to reproductive apex is gradual [52].) In the model, the moment of transition from indeterminate (vegetative) to determinate growth is preset. In contrast to the number of vegetative units, the exact number of units that fit on the apical hemisphere is not predictable.

2.5    Differentiation and the relationships between pattern, shape, and color

Closely connected with the expansion of a plant structure in radial and axial directions is the differentiation of primordial parts. (Many characteristics of flowers are related to the phyllotaxis of floral organs [53].) In general, differentiation in living structures can be explained by positioning signals [54]. Below, it will be shown how to get some essential shape and color differentiations as a by-product after translations. As a consequence, the stack-and-drag model relates pattern (phyllotaxis), shape (differentiation), and coloring (although it is not a pertinent variable) to each other.


3     Model description

The stack-and-drag model comprises two main processes: (1) creating patterns in which spheres of different sizes are positioned using the same algorithm throughout the entire construction and (2) shaping structures in which the centers of the spheres are translated and the spheres are resized to simulate axial and radial growth of stem and organs. In the first process a growth curve defines sphere radii in relation to receptacle radius (Figure 7). In the second two similar curves define translations in the distinguishable stem parts. [The S curve is defined by parameters b (minimum), h (half-life), and c (gradient) as Sn= (1-b) / (1+e c.(h-n)), in which n is an ordinal number, e = 2.71828... (the Euler number), b is the base or minimum relative sphere radius, h is half-life or the ordinal of the sphere of radius 0.5, and c is the growth constant or gradient. A unit defined as a primordium or sphere of influence including primordium-forming cells is represented as a sphere with its center on a cylindrical or hemispherical receptacle with radius 1. Unit radii can be expressed as Rn = 1- Sn.]

 


 

 

FIG.7. The relation (unit radius)/(stem radius). The unit shown is the last vegetative one, lying on the cylinder. Unit sizes are defined by the S curve [53].


 

 

FIG.8. Unit n will have contacts p and q when the previous unit (n -1) had units p -1 and q -1 as its contacts. Unit p lies counterclockwise around the positive Z axis against unit n. rp = r(n)+r(p); rq = r(n)+r(q). The center of unit n, M(n), is the intersection of a circle with center C with the cylinder. If a neighboring unit intersects unit n, it will replace unit p or q.

 

 

 

 

FIG.9. The highlighted unit (pp+j) will be counted, for unit n-j has unit pp as one of its support units. An increase in j indicates a decrease in sensitivity, for units in lower positions are counted.

 

3.1    Creating patterns

 Sizes of spheres as phyllotactic units are calculated; their centers are situated at distance 1 from the Z axis. The stacking direction is upwards from two initial touching spheres (Figure 2). These two spheres, representing cotyledons, have a unit radius of 1. Units are constructed using positioning algorithms of the dislodgement model (Figure 8). The main rule is that when unit n lies against the contacts with ordinal numbers p and q, unit n+1 will be positioned at the contacts p+1 and q+1. (In the dislodgement model, checks are needed to trace units as intersecting candidates [7, 32]. The stack-and-drag model uses some of these checks and decides which one of the support units has to be dislodged. If p=p(n) and q=q(n) are the ordinals of the support units of unit n, then (1) unit p may be replaced by one of the following units with ordinals p+n-q, q+q-p, p+q-n, p(q)+n-q, p(p)+n-p, and (2) unit q may be replaced by one of the units with ordinals q+n-p, p+p-q, q+p-n, q(p)+n-p, q(q)+n-q.) After the last unit on the cylindric stem, spheres close the surface in a transition from vertical to horizontal: the apical hemisphere with unit radius 1 (Figure 4). The parameters used are (i) tls, number of spheres on the cylinder, initiating spheres included; (ii) exp, expansion - spheres can be compressible or repulsive, depending on age; (iii) sens, sensitivity - the placement of spheres depends on past configurations (Figure 9); and (iv) canal, canalization - the ring in which a sphere is allowed to be constructed is chosen narrow to wide (Figure 10). (The expansion factor declines downward along the receptacle. One of the consequences is that misplacement of units caused by the static character of unit configurations in the simulation is neutralized [16]. High sensitivity means that only the locations of the upper spheres predict the settling of a new sphere. Low sensitivity means that lower spheres on the cylinder influence the settling of a new sphere. Strong canalization is caused by a narrow placement zone.)


3.2    Shaping structures

After forming the pattern on the receptacle, we displace units in the axial and radial directions. Translation depends on unit ordinal or age n, but also on d, the distance to the Z axis, and on z, the height from the XY plane. Stem growth causes shifting in the phyllotactic pattern. Translations follow the S curve (Figure 7), being a function of n but also of z or d, respectively.

Axial translations are expressed as

Axn = (1-Sn) . znLinkZ

in which zn is relative height (0 < zn £ 1) and LinkZ causes displacements to depend on n to zn (0 £ LinkZ £ 10). Parameter LinkZ leaves the dependence open, for there is no evidence to exclude either n or z. Note that axial translations and unit sizes use similar graphs when LinkZ = 0.

Radial translations are expressed as

Radn = Sn . dnLinkD

in which dn is relative distance (0 < dn £ 1) and LinkD causes displacements to depend on n to dn (0 £ LinkD £ 10). Notice that radial translations and unit sizes use horizontally mirrored graphs when LinkD = 0.

To simulate flowering according to the catastrophe theory [51], the three curves for unit growth, axial translations, and radial translations must work closely together. The distance ratio r(unit)/r(stem) is of decisive importance here. At the time of inflorescence, high values result in flowers and low values result in flower heads.

When spherical units are replaced by stem surface-filling polygons, which reveal vascular units [55-57], differentiation is simulated. (For defining leaf and internode as one unit, see also the leaf-skin model and the phytonic model in [39].) To understand this, we consider a given unit with both of its two support units and more lower units if necessary (Figure 11). The lower part of the top unit develops as an appendage, and the top parts of the lower units will contribute stem-filling parts. The length and top position of the appendage may be changed by specific parameters that are related to radial or axial translations. The use of these parameters leads to remarkable results when they are simply related to unit positions. In this way, differentiation and corresponding coloring in the distinguishable "plant" parts arise spontaneously, conditioned by the S curves. Adjacent polygons form a honeycomb structure, as in pineapple.


 

FIG.10. (a) Side view of unrolled cylinder. A new unit that fits or nearly fits between its predecessors (A, canalization zone) will be part of the existing ring. When it has been located significantly higher (B, free positioning zone) due to lack of space below, it will be the first unit in a new ring. Extreme high or low positions indicate an error (C, upper and lower error zone). (b) Bottom: Vertical section through the cylinder. The upper border of the canalization ring is settled by ca.r(n-1) or angle g, as well as by z(n-1). Angle d, where tan(d)=[r+r(n-1)]/r(cylinder), determines the upper border of the free positioning ring. A unit that has been placed within this ring retains its calculated position. Higher values for ca will narrow the ring. Top: Vertical section through the hemisphere. Angles a, where tan(a)=[z(n-1)-z(tls)]/d(n-1), and b, where tan(b)=[z-z(tls)]/d, define the ring's shifting. [z is the elevation of unit n; z(n-1) is the elevation of unit n-1; z(tls) is the elevation of the last vegetative unit; r(n) is the radius of unit n; ca is the canalization factor.]


 

 

FIG.11. Differentiations by the vascular unit [56]. Primordial shapes depend on the local configuration of units. (a) Defined by three primordial units of the model, the vascular unit PQN shows stem parts (yellow) and an appendage part (blue) [7]. (b) A vascular unit can, for example, be rhomboid. (c) The vascular unit of (b) is shown here in a configuration of primordial units in the stack-and-drag model. In this case, the central primordial unit Q is septagonal. (Appendage parts are blue.)


 

4     Results

My earlier dislodgement model [7] was sufficient for simulations with recognizable positioning of individual primordia on early flower heads [58].

Variable values for some vegetative and floral structures have been compared by using the stack-and-drag model (Table 1 and Figures 12-19) [59, 60]. The computer program ApexS [59], which generates patterns and structures, is able to produce very different phyllotactic patterns along the vegetative axis as well as in flowers and flower heads. Helices appear almost exclusively in Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, ...), and successive units make XY-projected angles of ~137.5¡, starting from right angles or the decussate arrangement at the base. Accessory series of the Fibonacci sequence may arise as sudden anomalies without particular parameter presettings. Canalization, which is induced by the mechanism as depicted in Figure 10, results in rings of units in fixed numbers like Fibonacci terms. The preference for fixed numbers in canalization follows from the preference for these numbers in parastichies. In contradistinction to stability for fixed numbers in parastichies, units often show canalization in deviations close to those numbers. (The computer program ApexS has a size of ~400 kBytes. It runs in the Windows environment of personal computers.)

Synthetic phyllotactic shift by gibberellic acid [36] is simulated (see veg/gibb and Figure 12).

The Lucas sequence (1, 3, 4, 7, 11, 18,...) is generated under comp/lucas. It is not directly deducible from the parameter values that Lucas numbers will arise. Appearance of the numbers is caused by an early stacking accident [9] (Figure 16).

The early development of the composite flowering head of Microseris pygmaea [19] is simulated after parameter settings in the Table 1 column comp/micr (Figure 18).

The column cost/costus shows simulation values for Costus (Costaceae). Spiromonostichy in the species Costus scaber has been regarded as an unsolved problem for 120 years [35]. The apical dome is enclosed by sickle-shaped bract primordia, which show a lower and a simultaneously developing higher part. The higher part of a given bract overlaps the lower part of a previous bract. In the terms of our model, two spheres of influence define one primordium (Figure 19).
TABLE I: Comparison of parameters in the Stack-and-Drag Model for some structures

 

 

structure

 

veg

 

 

flow

 

 

 

 

comp

 

 

 

spike

cost

fruit

 

dec

gibb

canal

8532

52

333

1313

wild

lucas

sun1

sun2

micr

mais

cost

pinapp

tl

50

40

113

32

24

16

79

142

55

105

5939

67

311

202

261

tls veg

50

40

30

10

10

5

20

100

20

20

80

10

300

200

200

exp

0.70

1.27

0.50

1.10

1.10

0.80

1.05

1.00

1.00

1.00

0.95

0.77

1.10

1.40

1.05

sens

0.90

0.96

1.00

0.95

0.95

1.00

0.90

0.96

0.96

1.00

0.98

0.90

0.90

0.95

0.96

canal

0

0.30

1.00

0.70

0.85

0.58

0.65

0.70

0.50

0

0

0.20

0.90

0

0.09

base

1.00

0.63

0.15

0.24

0.28

0.33

0.16

0.19

0.17

0.13

0.015

0.17

0.42

0.50

0.16

hlife

0.52

0.50

0.30

0.60

0.50

0.40

0.50

0.90

0

0.50

0.44

0.80

0.05

0.15

0.05

const

0.10

-1.22

0.17

0.49

0.69

0.69

0.34

0.22

0.22

0.27

0.07

0.27

0.17

0.10

0.34

see fig

 

12

 

13

14a

14b

 

15

16

 

17

18

 

19

 

 

tl

calculated number of spheres on the cylinder and the hemisphere

tls veg

number of spheres to be constructed on the cylinder

exp (0.5-1.5)

units are compressible to repulsive

sens (0-1)

unit is settled with feedback with existing structure to strong sensitivity

canal (0-1)

units are stacked (leaning on neighbors) to dragged, or pulled (by Buvat's ring)

base (0-1)

growth function minimum

hlife (0-1...)

ordinal of unit with 0.5 x stem radius related to tls

const (0-1)

growth function steepness

Notes and remarks:

(1) Only pattern-defining parameters are shown.

(2) hlife may be in excess of 1. This is due to the definition of hlife as related to the vegetative number of units.

(3) cont may be negative in artificial conditions, where the S curve is reversed.

(4) Combinations of pattern-characterizing values are printed in italics.

(5) All patterns have been generated in the same single process. All structures have been translated, differentiated, and colored in a single progression. All figures are unedited output of the program ApexS [59] and have been drawn after a single command. Pattern calculation speed on a PC (Pentium 90) is approximately 150 units/s.

 

veg/dec

Decussate leaf positioning with stem torsion caused by the phyllotactic pattern.

veg/gibb

See Figure 12.

vegl canal

Strong canalization leads to distinct whorls with many primordia when the unit constant is kept small. One of the rings has 16 (doubled Fibonacci number) units.

flow/8532

See Figure 13.

flow/52

See Figure 14a.

flow/333

See Figure 14b.

flow/1313

Flower with numeric canalization 13-13-8-5. Units in a canalized ring may show inequality in form and function.

flow/wild

See Figure 15.

comp/lucas

See Figure 16.

comp/sun1

Composite with a low number of florets.

comp/sun2

See Figure 17.

comp/micr

See Figure 18.

spike/mais

Grains are in 5 x 2 columns.

cost/costus

See Figure 19.

fruit/pinapp

A pineapple is regarded as a thickened stem. Leaves on the fruit are small, and fruit polygons are vascular units.

 

 

FIG.12. veg/gibb                    The phyllotactic shift decussate / spiral (Fibonacci) / distichous in Hedera helix L. after injection with gibberellic acid at the arising of unit 20 [36]. Right: Top view of the stacking, with spheres drawn small to show pattern change. Note that the numbering of primordial spheres does not preclude the possibility of primordia simultaneously arising in the actual plant (see Section 2.2.).


 

 

FIG.13. flow/8532                  Flower showing 8-5-3-2 canalization. Strong canalization with high unit constant results in a simple flower with distinguishable parts. Left: Appendages are expressed by polygons (vascular units with their upper tips in particular translated by the parameter leaf-length). From bottom left to center top: Spheres pattern with translated and resized units. Top right: Calculated stacking with 10 spherical units on the cylinder and 21 on the apical hemisphere (cylinder and hemisphere are not shown). Bottom center and right: Top view of the flower parts, which are divided by a horizontal plane between, and caused by, canalization rings. Note that (1) phyllotaxis is decussate (unit pairs 3+4, 5+6) near the cotyledons; (2) sepals, petals, and stamens/carpels show a three-way (vascular unit) division; and (3) in this case the flower is singularly minor symmetrical.


 

 

FIG.14. flow/52 and flow/333           Left: 5-2 flower after 10 vegetative units with flower diagram after translations of the calculated units. Right: 3-3-3 flower after only five stem units (cotyledons included) with a total of 16 units. Bottom: Calculated untranslated stacking, showing all units. Bottom right: Diagram showing the dissimilarity of the four whorls of flower parts, although there may be a six-petal symmetry.


 

 

FIG.15. flow/wild                   Wild phyllotaxis, with very regular flower. Left: The centers of the contacts (support units) of each unit are connected, showing (1) a stem pattern transition wild-decussate and (2) floral canalization. Center: Calculated stacking, translated spheres, leaves. Right, top to bottom: The vascular units for sepals, petals, stamens, and carpels, divided by horizontal canalization planes (see also Figure 13).


 

 

FIG.16. comp/lucas                Deviation of the Fibonacci phyllotaxis [9]. Center, bottom: The early deviation at n=6, which causes the Lucas phyllotaxis. Center, top: Follow-up from n=6. Corners: Connections of centers of units n-p, n-q, n-p(q), p-q.


 

 

FlG.17. comp/sun2                 Composite with a very high number of florets. Left: Possible vascular pattern: oblique; untranslated; and top view, translated. Right: Calculated spheres packing and top view of flower head with seeds. Center: Axial and radial translations, with every 13th unit highlighted.

 

 

FIG.18. comp/micr                 Topological/graphical results. Simulation, compared with observations in flower heads of Microseris pygmaea D. Don [19]. A special problem is the canalization jump after the first annular arrangement of 13 primordia. Observations (left) are compared with simulation results. (Top) Topological: Canalization jump out of the basical capitulum ring with 13 primordia. Left: Model according to observations [19]. Top right: Computed pattern, including (highlighted) 13-ring and canalization jump. (Bottom) Graphical: Characteristic patterns in three-dimensional distances between primordia. Left: Relations according to observations [19]. Right: Three-dimensional distances between units n and n-8. Note that (1) distances are dimensioned differently; in the model, they are related to the (untranslated) cylinder radius; (2) the simulation is done for an embryonic stage, so (radial) translations are small; and (3) to get optimal approximations, the simulation is begun by assuming rough values and fine-tuning is done by a trial-and-error procedure. Once graphs become similar, the characteristic values of the model parameters are known.

 

 

FIG.19. cost/costus                Small divergence angles; scarcely translated structure showing the small torsion as a phyllotactic, not physiologic, quality in Costus scaber [35]. Left: Because of its embracing sickle shape, a primordium is defined here by two units. Right: Schematic presentation in top view, with connections between centers of subsequent units. Note that of the two near lower units, one is always not a direct contact.


5     Discussion

5.1    The model

The stack-and-drag model has been built on three basic size/shape assumptions (Figure 20a-c):

(1) There are three plant parts: the initiating cotyledons (represented by two spheres), the vegetative stem (represented by a cylinder), and the generative apex (represented by a hemisphere).

(2) Plant appendages are regarded in relation to internodal stem parts. A primordial unit is always combined with two supporting lower units. Adjoining parts of these three spheres form the vascular unit [57].

(3) Three interacting S curves determine the growth of primordia and plant body: one for unit growth, one for axial expansion, and one for radial expansion.

FIG.20. Basics in the stack-and-drag model. (a) The three parts of a plant reflected by the model. (b) Two spheres P and Q determine the location of a third sphere, N. Translations determine shape and color of the vascular unit PQN. (c) The unit growth curve (left) is important for pattern development, and the other curves define the receptacle's growth. (d) Three pattern-defining parameters simulate different levels of interaction between units.


5.2    The parameters

Three parameters can be isolated to produce a wide range of recognizable phyllotactic patterns: b determines the minimum relative size of a primordial unit (Figure 7), exp gives the compressibility of units, and canal is a measure of the sharpness of the annular dragging zone in which units may arise (Figure 20d). Three additional parameters are of less importance: h and c together with b determine the path of the S curve for the size of early primordial units, and tls is the number of vegetative units that indirectly defines the unit size just before inflorescence and the unit growth rate.

5.3    The drag component

The status of the drag parameter canal is a moot point. As depicted above, in the small physical scale of phyllotactic pattern formation we realized the necessity for a counterpart of gravity, the influence of which diminishes with size. We interpreted Buvat's embryonal ring as a zone of pressure from the hypodermal shoot apical meristem on the epidermal layers and combined this with the assumption that outer layers have to follow after central elongation. The biological nature of the drag parameter has to be specified.

5.4    Branching

 Bud formation is possible by permitting a structure to branch outside its cylindrical casing. Fractal theories are not satisfactory for obtaining branching patterns because they lack sufficient biological evidence. One approach might be to integrate the L-systems [62]. On the other hand, the fractal property should be a qualification like the angle of 137.5¡ is.

5.5    Phyllotaxis and differentiation

The stack-and-drag model tries to reduce the question of the origin of phyllotaxis to the question of primordial differentiation. What we know about differentiation in a developing living structure is that, given the potential of any embryonic cell to specialize in any adult organ function, morphological changes in cell clusters are strongly directed by chemical diffusion and physical tension from their environment. The geometrical orientation of cell clusters in their environment (phyllotaxis) is crucial to the way these influences act upon them [63]. Phyllotaxis is governed by annular differentiation of the apex, and after that, the differentiation of primordia is governed by phyllotaxis [17].

We may consider physical and chemical forces to indirectly shape the shoot apex including the primordia arising on it. Shape influences differentiations and therefore new initiating points for primordia. If the shape is known and the influence of forces is translated into universal geometrical parameters, a continuation of the process of pattern development can be deduced.

Although the true character of differentiation is still unknown, we do have knowledge of the character of the arising phyllotactic patterns. By reducing the number of parameters to only those that are recognizable geometrical parameters, the problems of differentiation and phyllotaxis have been disconnected. The a-b-c model for specification of organ identity in flower development [17, 63] seems to link up closely with this conclusion and may be useful in the future development of determinate differentiation in the stack-and-drag model.

 

I thank Konrad Bachmann for his push toward the present model and his proofreading, Johannes Battjes for the feedback from his observations and interpreting, Roger Jean for the exchange of some opposing thoughts, and the independent reviewers for their critical comments on the manuscript and consequent improvements.


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