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Concepts of Phyllotaxis and the Genetics of Floral Symmetry

1.1. Types of floral symmetry

In the generally accepted classification system, four types of floral symmetry are defined (Endress 1999) (Figure 1.1):

- Actinomorphy (radial symmetry or polysymmetry): the flower has n axes of symmetry, where n > 2 is the number of organs in each whorl (e.g. six axes of symmetry; Figure 1.1(A)).

- Disymmetry: there are two perpendicular axes of symmetry (Figure 1.1(B)).

- Zygomorphy: (bilateral symmetry or monosymmetry): the flower is symmetric with respect to the dorsoventral axis and asymmetric with respect to the transverse axis (Figure 1.1(C)).

- Asymmetry: flowers have no axis of symmetry (Figure 1.1(D)).

Bilateral floral symmetry is predominant in certain species-rich families such as Lamiaceae and Fabaceae (legumes) in eudicots and Orchidaceae in monocots. Phylogenetic studies have demonstrated that zygomorphy has evolved many times from ancestors with radially symmetrical flowers and that reversals from bilateral to radial symmetry are frequent. A minimum of 130 shifts from radial to bilateral floral symmetry and 69 reversals to actinomorphy have been inferred within angiosperms. For example, in Lamiales, bilateral symmetry has evolved early in the diversification of the clade and has been followed by at least 10 reversals to radial symmetry (Reyes et al. 2016).

Figure 1.1. Different types of floral meristem symmetry before anthesis. (A) Actinomorphy: Antirrhinum majus (Plantaginaceae) cyc:dich mutant at an early developmental stage (from Luo et al. 1996) with six axes of symmetry. (B) Disymmetry: Lamprocapnos spectabilis (Papaveraceae) with two perpendicular axes of symmetry (from Damerval et al. 2013). (C) Zygomorphy: wild-type Antirrhinum majus (redrawn from Vincent and Coen 2004). Dashed lines represent the dorsoventral and transverse axes, which intersect at the center of the flower. (D) Flower meristems of Centranthus ruber redrawn from Roels and Smets (1996). The asymmetric flowers have a single stamen that is not on the dorsoventral axis. K = sepals from the German Kelch, not to confuse with S = stamen; P = petal; S = stamen

In flowers that have spur, the third dimension has to be taken into account, and symmetry is considered with respect to planes instead of axes (Figure 1.2(A)). In the case of Impatiens (Balsaminaceae; Figure 1.2(A)), the weight of the spur tilts the flower at 180° at anthesis (resupination).

Box 1.1. Drawings of developing and mature flowers by Jean-Baptiste Payer

Jean-Baptiste Payer was appointed professor of geology and mineralogy at the University of Rennes in 1840. In 1844, he obtained the chair of botany at the École Normale Supérieure in Paris. He received a doctorate in medicine from the Paris faculty in 1852 and was appointed that same year as professor of plant organography at the faculty of sciences. He published "On Comparative Plant Organogeny of the Flower" between 1854 and 1859.

Figure 1.2. (A) Impatiens glandulifera (Balsaminaceae) and (B) Dipsacus laciniatus (Caprifoliaceae) from Payer's "Traité d'organogénie végétale comparée de la fleur" (1857). General legend from Payer: fl = flower; ca = calyculus; s = sepal; p = petal; et = stamens (étamines); ep = spur (éperon); superscript: a = anterior; l = lateral; p = posterior. Dashed blue lines = axes of symmetry; straight lines = plane of symmetry

1.2. The bracteole theory

In 1875, the genius of botany A. W. Eichler studied at length the relationship between flowers and bracteoles (Vorblätter. Anschluss und Einsatz der Blüthe in Volume I of Blüthendiagramme (1875)) and described the different possible positions of the sepals. When only one bracteole is present - in particular, in monocots - the position of the first sepal/tepal (K1) is opposite the bracteole. When two bracteoles are present (a the older bracteole, and ß the younger bracteole), Eichler considered that the sepals were located on the same spiral as the bracteoles (Figure 1.4(A)).

Eichler tried to explain sepal arrangement "mechanically" using Hofmeister's rule (1868), that is, a new primordium is initiated in the largest space available between pre-existing primordia. This rule predicts an alternation of bracteoles and sepals. However, the many exceptions to this rule forced Eichler to abandon the path opened by Hofmeister.

Figure 1.3. (A) Initiation of the bract (= B), floral meristem (FM), bracteoles (a and ß the older bracteole a being larger) and the first sepal (K1) in Aquilegia vulgaris (redrawn from the SEM of Tucker and Hodges 2005). (B) Illustration of a flower of Helleborus niger (Thomé 1885). Bracteole a is below ß on the pedicel

Our model is based on the equations of Douady and Couder (1996a) that simulate the effect of an inhibition potential (Box 1.2) generated by pre-existing primordia on the location of the subsequent primordium. If we hypothesize that bracteoles determine the arrangement of sepals, this dynamical model of phyllotaxis, based on an inhibition field, makes it possible to account for the numerous arrangements (trimery, tetramery and pentamery) found in monocots and eudicots. This hypothesis suggests that determining sepal arrangement is the first biological function of bracteoles, as intuited by Eichler in 1875. Our model of phyllotaxis explains Eichler's different cases in a "purely mechanical" way, thus responding to the challenge launched by Eichler 150 years ago (Walch and Blaise 2022b).

Box 1.2. What is a potential?

Consider a meteoroid located 400 km away from Earth: the force of universal attraction pulls the meteoroid towards the center of our planet. In the absence of this meteoroid, something remains: it is the attraction field created by the Earth. The gravitational potential is a measure of this attraction field, and it takes a second object, like a meteoroid, for the potential force to materialize into an actual force.

Like universal attraction, the inhibition generated by a primordium decreases as the distance from that primordium increases (Figure 1.4(C)).

1.3. Phyllotaxis

1.3.1. Phyllotaxis models

Our phyllotaxis model and its parameterization have been described in detail in Walch (2022) and Walch and Blaise (2023). Flower organs (sepals, petals, stamens and carpels) originate from meristems as small primordia and are initiated in a crown at the periphery of the central zone (CZ), or "front", of the meristem at a certain distance from its center (Figure 1.4(A); Godin et al. 2020).

Figure 1.4. (A) Arrangement and initiation sequence of the bracteoles and sepals in Ranunculus repens (from Meicenheimer 1979). K1 to K5 = sepals numbered from the oldest to the youngest. Dashed circle = periphery of the central zone of the meristem or "front" where organs are initiated. R0 = radius of the front circle. Main trigonometric angles are shown. (B) Model of the calyx of Ranunculus repens. B = bract; a ß bracteoles. Colors represent the "effective distance" (R; see Box 2.1) of the inhibition sources from the center of the flower, and the intensity of the inhibition increasing, the closer the organ producing the inhibition is to the flower center. (C) The inhibition profile along the periphery of the CZ (or "front": dashed circle in Figure 1.4(A)) at the initiation of K1. Local maxima are produced by existing primordia (here, the bract and the two bracteoles). K1 is located at the potential minimum (? = 316°)

Table 1.1 Modeling parameters for quincuncial arrangement of Figure 1.4. Parameters are polar coordinates of organs

For example, to model the perianth of Ranunculus repens (Ranunculaceae), whose initiation is described by Meicenheimer (1979) (Figure 1.4(A)), we considered three sources of inhibition: the bract and the two bracteoles, a and ß, initiated in this chronological order; therefore, ß is closer to the CZ than a (and a is closer to the CZ than the bract). The two bracteoles are located in the continuity of the vegetative spiral, and therefore have an angular distance of 138° (Walch 2022). According to the model, the inhibition potential at the front is higher the closer primordia are to the front (corresponding to younger primordia, here bracteole ß. The subsequent primordium (primordium K1)...