ABSTRACT

The bilateral symmetry of flowers is a striking morphological achievement during floral evolution, providing high adaptation potential for pollinators. The symmetry can appear when floral organ primordia developmentally initiate. Primordia initiation at the ventral and dorsal sides of the floral bud is differentially regulated by several factors, including external organs of the flower and CYCLOIDEA (CYC) gene homologues, which are expressed asymmetrically on the dorso-ventral axis. It remains unclear how these factors control the diversity in the number and bilateral arrangement of floral organs. Here, we propose a mathematical model demonstrating that the relative strength of the dorsal-to-ventral inhibitions and the size of the floral stem cell region (meristem) determines the number and positions of the sepal and petal primordia. The simulations reproduced the diversity of monocots and eudicots, including snapdragon Antirrhinum majus and its cyc mutant, with respect to organ number, arrangement and initiation patterns, which were dependent on the inhibition strength. These theoretical results suggest that diversity in floral symmetry is primarily regulated by the dorso-ventral inhibitory field and meristem size during developmental evolution.

INTRODUCTION

Spatial positioning and number of organs (e.g. eyes, ears, nose and mouth in animals; carpels, stamens, petals and sepals in plants) represent one of the most fundamental differences among species. In flowering plants (angiosperms), the forms of flowers exhibit enormous diversity. Floral symmetry is an important example of this diversity, which affects the success of sexual reproduction via pollination, i.e. pollen transfer from male to female organs (Woźniak and Sicard, 2018). Because plants are immobile, these organisms entrust the transport of pollen to wind, water or, in the majority of flowering plants, to animals. A recent study suggested that approximately 87.5% of flowering plants are pollinated by animals, such as insects and birds (Ollerton et al., 2011); therefore, plant floral forms have evolved to attract and control pollinators. One mechanism to ensure the success of pollination is to fix the position of pollen attachment on the body of the pollinator. Some flowers have achieved this regulation of pollen attachment position by modifying floral organ number, positioning and form. In particular, zygomorphic (or bilateral) flowers, which have dorso-ventral (also called adaxial-abaxial or DV axis) asymmetry that corresponds to the DV axis of the pollinator (Fig. 1A,B, lateral flower; Endress, 1999), have developed in many plant species in various clades adapted to a variety of pollinator species, resulting in the diversification of floral morphologies (Sargent, 2004).

Fig. 1.

Structure and symmetry of flowers. (A) Schematic diagram of an inflorescence. Each lateral flower has a DV axis with respect to the main axis and a bract. (B) DV axis in an Antirrhinum majus lateral flower, corresponding to the DV axis of the pollinators. (C) Upper: typical arrangements of the outermost floral organs (sepals or outer tepals) with respect to the DV axis. Bottom: clade-specific number and arrangement of the outer organs (sepals and petals) with respect to the main axis (black circle) in a phylogenetic tree modified from APG IV (The Angiosperm Phylogeny Group, 2016). Monocots exhibit either dimery or trimery. Most trimerous species show the same arrangement (3B), although their initiating orders are not identical. For example, in Orchidaceae, the outer perianth initiates from the dorsal side (Pabón-Mora and González, 2008). The exceptionally opposite positioning of the trimerous perianth organs along the DV axis (3A) is found specifically in the order Dioscoreales and the family Smilacaceae (Liliales; Ronse De Craene, 2010), which exhibit sequential initiation for both inner and outer perianth organs. A dimerous arrangement with two lateral external tepals, one ventral tepal and one dorsal internal tepal, is found in several clades, including the genus Paepalanthus (Eriocaulaceae and Poales; de Lima Silva et al., 2016), but rarely in orchids [a few Japanese Dendrobium cultivars, abnormal flowers in Cattleya (Harshberger, 1907) and Cypripedium (Masters, 1887)]. The pentamerous and tetramerous flowers co-exist in many eudicot families, such as Plantaginaceae (e.g. 5A in A. majus and 4B in Veronica) and Fabaceae (e.g. 5A, 5B and 4A).

Fig. 1.

Structure and symmetry of flowers. (A) Schematic diagram of an inflorescence. Each lateral flower has a DV axis with respect to the main axis and a bract. (B) DV axis in an Antirrhinum majus lateral flower, corresponding to the DV axis of the pollinators. (C) Upper: typical arrangements of the outermost floral organs (sepals or outer tepals) with respect to the DV axis. Bottom: clade-specific number and arrangement of the outer organs (sepals and petals) with respect to the main axis (black circle) in a phylogenetic tree modified from APG IV (The Angiosperm Phylogeny Group, 2016). Monocots exhibit either dimery or trimery. Most trimerous species show the same arrangement (3B), although their initiating orders are not identical. For example, in Orchidaceae, the outer perianth initiates from the dorsal side (Pabón-Mora and González, 2008). The exceptionally opposite positioning of the trimerous perianth organs along the DV axis (3A) is found specifically in the order Dioscoreales and the family Smilacaceae (Liliales; Ronse De Craene, 2010), which exhibit sequential initiation for both inner and outer perianth organs. A dimerous arrangement with two lateral external tepals, one ventral tepal and one dorsal internal tepal, is found in several clades, including the genus Paepalanthus (Eriocaulaceae and Poales; de Lima Silva et al., 2016), but rarely in orchids [a few Japanese Dendrobium cultivars, abnormal flowers in Cattleya (Harshberger, 1907) and Cypripedium (Masters, 1887)]. The pentamerous and tetramerous flowers co-exist in many eudicot families, such as Plantaginaceae (e.g. 5A in A. majus and 4B in Veronica) and Fabaceae (e.g. 5A, 5B and 4A).

In the early stages of floral development, the first indication of perianth diversity appears in the number and arrangement of the floral organs, the sepals and the petals. The perianth of a flower typically consists of two circles or whorls of floral organs, and each whorl contains the same number of floral organs. Merosity describes the common organ number of perianth whorls and is usually clade specific (Fig. 1C; Ronse De Craene, 2010; Smyth, 2018). In eudicots, the largest clade of flowering plants, the common number is usually four or five, whereas in monocots, the sister clade to eudicots, it is three (Ronse De Craene and Brockington, 2013; Endress, 2010; Remizowa et al., 2010). Lateral flowers that bloom as the lateral branch of the main stem have two types of floral organ arrangements for each organ number with respect to the DV axis of the flower (Fig. 1C, upper panel). The arrangements along the DV axis are recognized by dividing the floral bud into three regions from the position closest to the main axis: dorsal, lateral and ventral regions (Fig. 1A,B). The model plant Arabidopsis thaliana exhibits tetramerous flowers with four sepals and four petals, and the sepal arrangement along the DV axis has two sepals in the lateral region and one each in the dorsal and ventral regions (type 4A; Fig. 1C; Smyth et al., 1990). The other type of tetramerous arrangement (type 4B) is also found in certain plants, including those in the genus Veronica (Plantaginaceae). Regarding pentamerous flowers, the majority of eudicot flowers have one dorsal, two lateral and two ventral sepals (type 5A), whereas flowers in several clades have reversed arrangements with two dorsal, two lateral and one ventral sepal (type 5B; e.g. the subfamily Papilionoideae or Fabaceae). The trimerous flowers in monocots typically have one inner and two outer tepals (perianth organs) in the dorsal region, and one outer and two inner tepals in the ventral region (type 3B; Rudall and Bateman, 2004). On the other hand, the reversed arrangement is a representative phenotype in several orders of monocots (type 3A; Ronse De Craene, 2010; Tobe et al., 2018). Additionally, dimerous flowers appear in several families in monocots and eudicots, and have two lateral sepals (outer tepals; type 2B). The developmental mechanisms that produce the clade-specific diversity of organ number and positioning along the DV axis have not been thoroughly elucidated.

The number and positioning of floral organs are mainly determined when the floral organ primordia initiate (Endress, 1999; Tucker, 1999; Spencer and Kim, 2018). The simplest case occurs when several primordia that make up a whorl (i.e. organs, such as petals, with the same identity) in a concentric circle initiate at once; however, this is not the case in many flowering plants, as early floral development is associated with non-synchronous initiation of the sepal primordia. The initiating order in the sepal whorl differs among species. The zygomorphic initiation patterns, such as unidirectional initiation along the DV axis and bidirectional initiation (Tucker, 2003), are unique to floral organ initiation in contrast to the spiral initiation sequence, which is also observed in phyllotaxis (the arrangement of leaves along the stem) as well as floral organs. Although the developmental mechanisms underlying the diversity of the initiation sequence, as well as the number and positioning of the floral organs, have been proposed for radially symmetric flowers (Kitazawa and Fujimoto, 2015); those that occur along the DV axis remain largely unknown for bilaterally symmetric flowers.

Floral symmetry depends on the position in the inflorescence (Fig. 1A). The lateral flowers are zygomorphic, whereas terminal flowers are actinomorphic (radially symmetric) in some peloria mutants of Lamiaceae (Rudall and Bateman, 2003). The merosity of lateral and terminal flowers can also be different with pentamery and tetramery, respectively, occurring in Adoxa (Adoxaceae, asterids; Roels and Smets, 1994) or the opposite case for Ruta (Rutaceae, rosids; Wei et al., 2012), where the organ initiation is zygomorphic in lateral flowers. Such differences between lateral and terminal flowers suggest that lateral floral bud polarity (Thoma and Chandler, 2015) affects zygomorphy. The relative position to the inflorescence meristem of the main axis (Fig. 1C) provides an asymmetric polarity field for floral organ initiation along the DV axis via signaling molecules and the polar transport of phytohormone auxin (Bowman et al., 2002; Wang and Jiao, 2018). This idea is similar to the inhibitory field theory that pre-existing organs regulate the initiation position of new organs, as this theory is widely accepted to explain phyllotaxis (Hofmeister, 1868; Snow and Snow, 1952, 1962; Adler et al., 1997; Dourdy and Couder, 1996a,b; Traas, 2013; Refahi et al., 2016; Kuhlemeier, 2017; Yonekura et al., 2019). Therefore, the inhibitory field may be key for understanding bilaterally symmetric floral organ arrangements.

Several genes that affect organ number and position along the DV axis have been identified. While wild-type flowers of the snapdragon Antirrhinum majus (Plantaginaceae; Fig. 1C) have one sepal at the dorsal side, two at the lateral side and two at the ventral side, a loss-of-function mutant of the CYCLOIDEA (CYC) gene (cyc mutant) exhibits two sepals via formation of an extra sepal at the dorsal side without reduction in the number of sepals on the other sides (type 6B; Fig. 1C; Luo et al., 1996). As the CYC gene is expressed on the dorsal side of the floral bud in wild-type flowers, this gene is thought to repress the initiation of dorsal sepal primordia (Luo et al., 1996). Although recent studies have suggested that the CYC/TB1 clade of class II TCP transcription factors, which include CYC, is involved in regulating shoot branching, floral transition, organ identity and growth (Dhaka et al., 2017), the precise mechanisms that affect organ number and arrangement still remains elusive. Generally, in the angiosperm species with zygomorphic flowers, the CYC gene is expressed on the dorsal side of the floral bud and is considered to be the central regulator of zygomorphy (Spencer and Kim, 2018). The generality of this CYC gene expression pattern, main axis positioning (Fig. 1A) and the suggested function on dorsal primordium initiation prompted us to investigate the contribution of the strength of the dorsal inhibitory field to the clade-specific diversity of organ number and positioning along the DV axis.

In addition, the genetic regulation of organ number and positioning at the ventral side has been identified. The double mutation of bop1 and bop2 in A. thaliana converted the arrangement from tetramerous (4A) to pentamerous (5A) via formation of extra sepals and bracts, which are specialized leaves surrounding a flower (Fig. 1A), at the abaxial (ventral) side (Hepworth et al., 2005; Khan et al., 2014; Norberg, 2005). The bract and sepal compete for ventral positioning via regulation of the genes LEAFY and PUCHI (Chandler and Werr, 2014), resulting in an alternate arrangement of sepals and a bract, as seen in many clades (e.g. type 4B flowers in Rutaceae, including Ruta, or rosids; type 4B and 5A flowers in Plantaginaceae, including Veronica and Antirrhinum, or asterids; Endress, 1999; Ronse De Craene, 2007, 2010). This suggests a ventral side inhibitory field by which the pre-existing bract inhibits sepal initiation. PERIANTHIA (PAN) may also be a related candidate, as this transcription factor interacts with BOP1 and BOP2 (Hepworth et al., 2005). The pan mutant of A. thaliana consistently yields arrangement type 5A with an extra ventral sepal (Fig. 1C; Running and Meyerowitz, 1996). PAN is expressed in the apical meristem, floral meristem and each whorl of the organ primordia during A. thaliana wild-type flower development (Maier et al., 2009). The floral meristem adaxial/abaxial (DV) polarity, which is controlled by the genes CYC, PAN and BOP, and external organs (bract and main axis), is indispensable for determination of the numbers and positions of floral organs (Thoma and Chandler, 2015) and is a candidate for the DV inhibitory field.

In this report, we present a design principle for diverse organ positioning and initiation sequence along the DV axis using numerical simulations of a mathematical model for phyllotaxis. Furthermore, we introduced the inhibitory field of organ initiation from the dorsal and/or ventral side of the floral meristem into the phyllotaxis model, which originally incorporated the inhibitory field from the pre-existing organ primordia (Douady and Couder, 1996,a,b) and was recently applied to floral development (Kitazawa and Fujimoto, 2015). The numerical simulation accounted for most of the observed floral organ positioning and numbers in a unified manner, and these positions and numbers depended not only on the inhibition strength of the dorsal and/or ventral sides but also on the meristem size. These results suggest that clade-dependent differences in DV inhibition lead to the diversification of organ number and positioning in angiosperms.

RESULTS

Organ number and arrangement depend on the strength of DV inhibition

First, we examined the simplest condition of the inhibitory field model without dorsal and ventral factors affecting the floral organ patterning, assuming the primordia as points and a floral meristem as a disc with radius R0 (Fig. 2; Materials and Methods). The angular position of the new primordium is determined by the inhibitory energy from the pre-existing primordia [U(θ) in Eqn 2; Fig. 2B], and the primordia move centrifugally according to tip growth (Eqn 1). One or more primordia arise at the meristem edge taking the local minimum energy below the threshold value (Fig. 2C). In silico development began with a quasi-simultaneous initiation of four primordia, where the second primordium appeared at the opposite side of the first primordium owing to the inhibition caused by the first primordia, and then simultaneous initiation of the two primordia between them followed (Fig. 3A; Fig. 2C), consistent with the previous model for radially symmetric flowers (Kitazawa and Fujimoto, 2015). We recognized these four primordia as a whorl, as the first four primordia appeared in successive steps of the numerical simulation and at nearly the same distance from the meristem center (Fig. 3A, right).

Fig. 2.

Extension of the inhibitory field model to floral development. (A) Model settings. The edge of the meristematic region is represented by a circle (green), and primordia are represented by points (red) in a 2D plane. (B) Inhibitory field of primordium initiation by the pre-existing primordia. The inhibition decays with an increase in the distance from the pre-existing primordium. (C) Snapshots of inhibitory field energy U in a 2D plane for cases of one (top), two (middle) and four (bottom) pre-existing primordia. (D) External inhibitory fields at the dorsal (blue) and ventral (orange) sides. Inhibition strength was controlled by changing the distance (rdorsal and rventral) between the floral center and inhibition sources.

Fig. 2.

Extension of the inhibitory field model to floral development. (A) Model settings. The edge of the meristematic region is represented by a circle (green), and primordia are represented by points (red) in a 2D plane. (B) Inhibitory field of primordium initiation by the pre-existing primordia. The inhibition decays with an increase in the distance from the pre-existing primordium. (C) Snapshots of inhibitory field energy U in a 2D plane for cases of one (top), two (middle) and four (bottom) pre-existing primordia. (D) External inhibitory fields at the dorsal (blue) and ventral (orange) sides. Inhibition strength was controlled by changing the distance (rdorsal and rventral) between the floral center and inhibition sources.

Fig. 3.

Representative arrangements obtained by numerical simulations. (A) A 2D arrangement (left) and radial coordinates as functions of primordium indices (right) without inhibition from the DV sides. (B-H) The 2D arrangements of primordia under different conditions: rdorsal=70 without ventral inhibition (Aventral=0.0) (B), rdorsal=50 and Aventral=0.0 (C), rdorsal=60 (D), rdorsal=40 (E), rdorsal=30 (F), rdorsal=60 (G) and rdorsal=40 and a=0.001 (H). The numbers in the red disks indicate the initiation order of primordia. a=0 (A-G) and rventral=90 (D) and =30 (E-H). (I) Dependency on dorsal and ventral inhibitions. The colors denote the number of primordia within the first whorl, whereas the black frames denote the arrangement along the DV axis (see Fig. 1C). The white characters in the phase diagram correspond to the parameters of Fig. 3B-G. As the model simulations do not include randomness, all results shown in A-I were reproducible once we set the model parameters to values as described above and in the Materials and Methods. R0=16.0 (A-I).

Fig. 3.

Representative arrangements obtained by numerical simulations. (A) A 2D arrangement (left) and radial coordinates as functions of primordium indices (right) without inhibition from the DV sides. (B-H) The 2D arrangements of primordia under different conditions: rdorsal=70 without ventral inhibition (Aventral=0.0) (B), rdorsal=50 and Aventral=0.0 (C), rdorsal=60 (D), rdorsal=40 (E), rdorsal=30 (F), rdorsal=60 (G) and rdorsal=40 and a=0.001 (H). The numbers in the red disks indicate the initiation order of primordia. a=0 (A-G) and rventral=90 (D) and =30 (E-H). (I) Dependency on dorsal and ventral inhibitions. The colors denote the number of primordia within the first whorl, whereas the black frames denote the arrangement along the DV axis (see Fig. 1C). The white characters in the phase diagram correspond to the parameters of Fig. 3B-G. As the model simulations do not include randomness, all results shown in A-I were reproducible once we set the model parameters to values as described above and in the Materials and Methods. R0=16.0 (A-I).

Next, we introduced external inhibition from the dorsal side (Fig. 2D; Eqn 3). The inhibition gradient of the Udorsal produced the local minima at the ventral side of the meristem edge, thereby generating the first primordium at the ventral side. The angular positions of subsequent primordia and the primordium number of the first whorl can differ, depending on the distance between the dorsal inhibition source and the floral apex (rdorsal). The tetramerous arrangement with one primordium at the dorsal side, two at the lateral side and one at the ventral side (type 4A; Fig. 1C) appeared at a certain strength of dorsal inhibition (rdorsal=70; Fig. 3B), which was consistent with sepal primordia initiation in A. thaliana (Smyth et al., 1990). Furthermore, the pentamerous whorl with two primordia on the dorsal side, two on the lateral side and one on the ventral side, is commonly found in many legume species (type 5B; Fig. 1C), and appeared with increased dorsal inhibition (rdorsal=50; Fig. 3C). Thus, two different arrangements typically found in different species were continuously induced by changing the strength of dorsal inhibition.

We next introduced ventral inhibition (Fig. 2D; Eqn 4), and weak inhibition reproduced the conversion from type 4A to the 5A pentamerous sepal arrangement via formation of an extra ventral organ (Fig. 3B,D), which was consistent with findings in the pan mutant (Running and Meyerowitz, 1996) and the bop1 bop2 double mutant (Hepworth et al., 2005) of A. thaliana. The initiation order was also consistent with the pan mutant, as the ventral and dorsal sepal primordia appeared first, followed by the two lateral sepal primordia.

Stronger ventral inhibition than dorsal inhibition reproduced another initiation order of the 5B arrangement (rdorsal=40, rventral=30; Fig. 3E) in other eudicot species, including wild-type A. majus and a few mimosoid legumes (type 5A; Fig. 1C). In these cases, the two lateral sepal primordia appeared first, followed by nearly simultaneous appearance of the remaining three (one dorsal and two ventral) sepal primordia (Luo et al., 1996; Ramírez-Domenech and Tucker, 1990). The number of organs within the first whorl was controlled by changing the dorsal inhibition, while fixing the other parameters. Increasing the dorsal inhibition to the same level as the ventral inhibition (rdorsal=30, rventral=30; Fig. 3F) added a ventral organ to the first whorl, resulting in the sepal arrangement of the cyc mutant of A. majus (type 6B; Fig. 1C). The resultant arrangement exhibited two primordia in the dorsal, lateral and ventral regions. In addition to these arrangements, the bidirectional initiation order of these six primordia in the first whorl was consistent with the cyc mutant of A. majus, wherein two lateral sepal primordia initiated first, followed by the appearance of the remaining four sepal primordia (Luo et al., 1996). A decrease in the dorsal inhibition (rdorsal=60, rventral=30) led to a decrease in the number of dorsal primordia from two to one, resulting in four primordia in total within the first whorl (Fig. 3G). This finding was consistent with that in Veronica (type 4B; Fig. 1C), which belongs to the same family as A. majus. The in silico development started with the initiation of two primordia in the dorsal region, followed by the initiation of two primordia in the ventral region. In these three arrangements, several primordia appeared at alternative positions to the first whorl, forming the second whorl upon the temporal decay of dorsal and ventral inhibitions (a non-zero value of a in Eqns 3 and 4; Fig. 3H). This positioning occurred because the inhibition from the first whorl on primordia initiation was stronger than that from the dorsal and ventral sides. These three arrangements and initiation sequences were observed in the family Plantaginaceae and were continuously altered by changing the degree of dorsal inhibition relative to the ventral inhibition (Fig. 3E-G,I).

The trimerous and dimerous arrangements observed in monocots (Fig. 1C) appeared in a small radius of the meristem (R0=8; Fig. 4A-D). The trimerous whorl with two dorsal and one ventral primordia, as observed in the external tepal whorl of most monocots (type 3B; Fig. 1C), appeared in two separate parameter regions. In these two regions, the initiation patterns were different, initiating from either the ventral (Fig. 4A) or the dorsal side (Fig. 4B). The former appeared with stronger dorsal inhibition than ventral inhibition (rdorsal=20 and rventral=50), whereas the latter appeared with a dorsal inhibition that was weaker than the ventral inhibition (rdorsal=70 and rventral=60), consistent with the initiation pattern of orchids (Pabón-Mora and González, 2008). A dimerous arrangement was obtained in the close parameters (type 2B; Figs 1C and 4C), as for several abnormal orchid flowers (Harshberger, 1907; Masters, 1887) and other order Poales (genus Paepalanthus; de Lima Silva et al., 2016). As an intermediate of the two separate parameter regions of type 3B, a reversed trimerous arrangement with one dorsal primordium appeared (type 3A; Figs 1C and 4D). This arrangement is found in Dioscoreales (monocots) tepals (Fig. 1C). Transition from type 3A to type 2B via loss of a dorsal primordium (Fig. 4C,D) was also consistent with a previous notion about the evolutionary transition of Paepalanthus to dimery (de Lima Silva et al., 2016; Fig. 1C). Therefore, the monocot diversity of organ number and arrangement appears even under conditions of constant meristem size.

Fig. 4.

Dependency of number and arrangement on meristem size R0. (A-D) 2D arrangements of primordia at a=0.002, R0=8.0. rdorsal=20 and rventral=50 (A), rdorsal=70 and rventral=60 (B), rdorsal=20 and rventral=20 (C), and rdorsal=60 and rventral=70 (D). (E) Dependency on dorsal and ventral inhibitions at R0=8. The white characters in the phase diagram correspond to the parameters of Fig. 4A-D. (F) Dependency on dorsal inhibition and size of meristematic region R0. rdorsal=40. The colors indicate the arrangements along the DV axis (Fig. 1C) or the merosities (2 or >6) described in the diagram. As the simulations do not include randomness, all results presented in A-F are reproducible once we set the model parameters as described above, and in the Materials and Methods.

Fig. 4.

Dependency of number and arrangement on meristem size R0. (A-D) 2D arrangements of primordia at a=0.002, R0=8.0. rdorsal=20 and rventral=50 (A), rdorsal=70 and rventral=60 (B), rdorsal=20 and rventral=20 (C), and rdorsal=60 and rventral=70 (D). (E) Dependency on dorsal and ventral inhibitions at R0=8. The white characters in the phase diagram correspond to the parameters of Fig. 4A-D. (F) Dependency on dorsal inhibition and size of meristematic region R0. rdorsal=40. The colors indicate the arrangements along the DV axis (Fig. 1C) or the merosities (2 or >6) described in the diagram. As the simulations do not include randomness, all results presented in A-F are reproducible once we set the model parameters as described above, and in the Materials and Methods.

The current model reproduced both floral organ arrangements and initiation sequences in different clades of angiosperms, including the two largest clades eudicots and monocots as well as smaller clades within a family. A wide range of floral organ arrangements was reproduced by changing three parameters (rdorsal, rventral and R0), which were resistant to differences in energy functions (Eqns 2-4) and angle precision at the meristem edge (0.1 to 5°; Materials and Methods) in the inhibitory field model, while the functional form affected the parameter value of dorsal and ventral inhibition for each arrangement (Fig. S1B). These results suggest that the changes in DV inhibition as well as meristem size are potential primary regulators of the evolutionary changes of floral organ arrangement in angiosperms.

Comprehensive analysis of the model parameters

The dependence of floral organ arrangement on rdorsal, rventral and R0 further revealed changes in organ numbers within the first whorl (Figs 3I and 4E,F). An increase in R0 accompanied the monotonic increase in primordium number in the first whorl, as observed in previous phyllotaxis models without DV inhibition (Douady and Couder,1996b; Kitazawa and Fujimoto, 2015; van Mourik et al., 2012). For example, when the dorsal inhibition was slightly weaker than the ventral inhibition (rdorsal=40, rventral=30), the organ number in the first whorl was increased to four (type 4B at R0=12), five (type 5A at R0=16) and six (type 6B at R0≥20; dashed rectangle; Fig. 4F). This diversity of arrangements accounted for those in Plantaginaceae. In general, meristem size, and dorsal and ventral inhibitions normalized by the characteristic length of the inhibitory field from DV sources and pre-existing organs (R0/λ, rdorsal/λ, and rventral/λ) are the parameters that control organ number (Fig. S1A); this notion is consistent with the model for radially symmetric whorls (Douady and Couder,1996b) because the geometry of meristem and inhibitory field is scale invariant in the present model (see Materials and Methods). In addition, the parameter range of R0 for an arrangement type (e.g. type 5B) is wider in the presence of ventral inhibition (at rventral=60-70 in Fig. 4F) than in its absence (at Av=0 in Fig. 4F), suggesting that the robustness of positional arrangement to meristem size variation is promoted by the ventral inhibitory field. Therefore rdorsal/λ (Fig. 3I, Fig. S1A) and R0/λ (Fig. 4F) synergistically regulate organ arrangements.

The non-monotonic changes of organ numbers were found to be associated with rdorsal and rventral, when R0 was constant (Figs 3I and 4E). In addition, some of the arrangements occurred independently in several regions in the parameter space, yielding different orders of primordium initiation. For example, in one of the pentamerous arrangements (type 5B), unidirectional initiation from the dorsal side (rdorsal=10, rventral=80, R0=20), bidirectional initiation from the lateral side (rdorsal=30, rventral=40, R0=16 in Fig. 3I) and other bidirectional initiation events in the sequence of dorsal, ventral and lateral (rdorsal=40, rventral=80, R0=16 in Fig. 3I) were segregated depending on rdorsal, rventral and R0. As such different initiation orders that lead to the 5B arrangement were observed in legume flowers in different clades (Prenner, 2004), this model may predict the evolutionary path of Fabaceae; however, some issues, such as the observation that the change in these initiation orders did not occur with a continuous change of parameters in simulations, remain to be explored.

Design principle of organ arrangement and initiation order along the DV axis

In our model framework, the position of the first primordia was determined by the positional relationship between the meristematic region with a radius R0 and two inhibitory sources rdorsal and rventral. The inhibition energy along the DV axis reaches a local minimum at the middle of the two inhibitory sources (Voronoi edge; Fig. 5A, black line). When the Voronoi line crosses the edge of the meristem (Fig. 5A, green circle), the two primordia first appear at the intersection points (Fig. 5B-D). When the Voronoi line contacts or is outside the meristematic region, the global inhibition minimum on the edge of the meristem is located at the dorsal or ventral side and depends on rdorsal and rventral, as well as the meristem size R0 (Fig. 5E). Thus, the strength of the two inhibitory sources relative to the meristem size determines the number and position of the first primordia to appear.

Fig. 5.

Design principle of the arrangement of the first organs to initiate along the DV axis. (A) The energy minima (black solid lines) are located between the dorsal and ventral inhibitory fields, which are indicated by the blue and orange discs, respectively. When the line intersects the meristem edge (green circle), two primordia are formed at the first initiation at the intersection points (B-D). When the line contacts or does not intersect the meristem edge, one primordium is formed on either the dorsal or ventral side, depending on the closest inhibition source (E).

Fig. 5.

Design principle of the arrangement of the first organs to initiate along the DV axis. (A) The energy minima (black solid lines) are located between the dorsal and ventral inhibitory fields, which are indicated by the blue and orange discs, respectively. When the line intersects the meristem edge (green circle), two primordia are formed at the first initiation at the intersection points (B-D). When the line contacts or does not intersect the meristem edge, one primordium is formed on either the dorsal or ventral side, depending on the closest inhibition source (E).

The initiation of the next primordia to appear was determined by not only the two inhibitory sources but also by the one or two primordia that initiated first. When the intersection points occur at the middle of the meristem (Fig. 5A,B), the arrangement is symmetric to the DV axis. The number of the energy minima between the two intersection points can be either one or two, depending on whether the relative strength of the dorsal and ventral inhibition is stronger than the energy of the first primordia. When the DV inhibition is strong enough to affect the energy landscape of the inhibitory field, the two minima occur at both the dorsal and ventral sides, resulting in arrangement type 6B (Fig. 5B). When the intersection points do not occur at the middle of the meristem (Fig. 5A), the number of minima at the dorsal and ventral sides varies from zero to two. For example, when the intersection points are close to the dorsal side, the number of minima on the ventral side tends to be larger than that of the dorsal side (Fig. 5C,D). When the energy is low enough on the dorsal side, the arrangement becomes type 5A (Fig. 5C), but when the energy is not low enough, the arrangement becomes type 4B (Fig. 5D). For this mechanism, the inversion of the organ arrangements of the odd-numbered merosities, trimery (between types 3A and 3B in monocots; Fig. 1C) and pentamery (between types 5A and 5B in legumes), is reproduced by inverting the dorsal and ventral inhibition strengths (types 3A and 3B in Fig. 4B,D,E; types 5A and 5B in Fig. 3I).

DISCUSSION

Candidates for dorsal and ventral inhibition

The inhibitory field model reproduced organ arrangements of a wide range of angiosperms, supporting the notion that DV asymmetric inhibition of organ initiation is one of the key regulating factors of floral diversification. The primary candidates for involvement as DV asymmetric inhibition factors at the dorsal region are the CYCLOIDEA gene and its homologues. As the arrangement differences between the wild-type and cyc mutant of A. majus were dependent on only dorsal inhibition in our model, the strength of dorsal inhibition may directly account for differences in CYCLOIDEA expression. In light of the cyc-mutant phenotype with an extra dorsal sepal, an inhibitory effect on the dorsal side was suggested as a CYCLOIDEA function. The present model, however, suggests the opposite function, i.e. dorsal inhibition may be stronger in the cyc mutant because the stronger dorsal inhibition leads to the division of the energy of the dorsal side into two local minima, resulting in two primordia (Fig. 5A,B). Therefore, CYC gene homologues may similarly inhibit a dorsal inhibitory field, resulting in stronger dorsal inhibition in these mutants. In addition, our model accounted for the bidirectional initiation of Antirrhinum sepals with a lateral, dorsal and ventral sequence, and the alternate arrangement of two sepals and a bract at the ventral side, when both dorsal and ventral inhibition was incorporated (Fig. 3E). Therefore, two inhibition sources are likely to occur, supporting the idea that the bract is a ventral inhibitor. The family Plantaginaceae also includes tetramerous species, such as Veronica and Plantago (type 4B; Fig. 1C). The present model predicts that CYCLOIDEA expression is stronger on the dorsal side and that the dorsal inhibition is weaker (Fig. 3G,I) or the meristem size is smaller in these tetramerous flowers (Fig. 4F). Loss of a dorsal primordium that results in the 4B arrangement under these conditions in our model is consistent with the morphological observation-based hypothesis that the dorsal organs of the tetramerous Veronica and Plantago flowers are derived from the lateral organs of pentamerous flowers in Plantaginaceae (Endress, 1999).

Regarding the inhibitor candidates in the ventral regions, several mutants of PAN and BOP produced an extra sepal on the ventral side, thereby converting from the 4A arrangement to the 5A arrangement in A. thaliana (Running and Meyerowitz, 1996; Hepworth et al., 2005). The conversion as well as the initiation order (ventral, dorsal and then lateral organs) in these mutants were reproduced by increasing the ventral inhibition strength in the present model (Fig. 3B,D,I). Therefore, these genes may inhibit the ventral inhibitory field, resulting in a stronger inhibition in these mutants and playing an anti-symmetric role of CYC along the DV axis (Fig. 5A-C). In addition, the consistency with the alternate arrangement between two sepals and an extra bract at the ventral side of bop1 bop2 mutant verifies the idea that the bract is a ventral inhibitor. The present asymmetric regulation of organ initiation in the dorsal and ventral regions is indispensable for bilateral symmetry that occurs via determination of the numbers and positions of floral organs.

An increase or decrease in organ number correlated with an enlargement or reduction in floral meristem size has been observed in mutants of genes, such as CLAVATA and WUSCHEL, that function in the maintenance of the meristem population (Schoof et al., 2000) and related genes, such as ULTRAPETALA (Fletcher, 2001). The ultrapetala mutant Arabidopsis exhibits a 6A arrangement with increased meristem size (Fig. 1C, upper panel). While the sepal primordia form correctly on the dorsal and ventral sides, additional sepal primordia can also initiate with two primordia in the lateral positions, where only one forms in the wild type (type 4A; Fletcher, 2001). In our model, the transition from 4A to 6A occurred consistently by increasing the meristem size, resulting in addition of a primordium to each lateral side (e.g. R0=16 to R0=20 at rdorsal=80, rventral=70). Our model framework may explain not only the mutant phenotypes that result from the defects in DV polarity but also the meristem-size mutants that exhibit the diversity of floral numbers and organ arrangements along the DV axis. Accumulation of experimental evidence about additional mutants, as well as the external organs (bracts) that alter floral organ arrangement specifically at the dorsal, lateral or ventral side, will ultimately clarify the role of the DV inhibitory field in bilateral symmetry.

Consistency with the clade-specific diversity of DV patterning

Our model can be applied to the phylogenetic relationship within clades. The co-existence of merosity, especially that of pentamery and tetramery, is widely found in the eudicot clades (Smyth, 2018), not only in Plantaginaceae (Lamiales, Fig. 4A) but also in Dipsacales (e.g. Caprifoliaceae and Dipsacaceae), Fabales, Malpighiales (Matthews and Endress, 2013), Saxifragales, Ericales (e.g. Sapotaceae), Oxalidales (e.g. Caldcluvia paniculata (Fletcher, 2001; Matthews and Endress, 2002) and Brassicales (e.g. Tropaeolaceae, Caricaceae and Moringaceae). Different arrangements of the same number of sepals co-exist in some clades. For example, in legume species, type 5B arrangement exists in the subfamily Papilionoideae, and 5A occurs in other subfamilies: 4A occurs in Acacia [Mimosoideae (Prenner, 2011) in Fig. 1C]; tetramery and pentamery (Matthews and Endress, 2013) of both 5A and 5B (Zhang et al., 2010) co-exist in Malpighiaceae; 5A, 4B and 4A co-exist in Saxifragales (Ronse De Craene, 2010); and 4A, 5A and 6A occur in the family Sapotaceae (Kümpers et al., 2016). The causes of such changes are unclear, but the DV asymmetric expression of genes as well as bracts affect these arrangements. In Dipsacales, the pentamerous flowers are usually zygomorphic, whereas the tetramerous species include both radially symmetric (e.g. Symphorycalpos, Caprifoliaceae) and zygomorphic flowers (e.g. Knautia, Caprifoliaceae). The localized expression of CYCLOIDEA has been suggested to be responsible for zygomorphy in most of these mentioned clades, including Brassicales, Fabales, Dipsacales and Malpighiales (Hileman, 2014). Therefore, we expect that the clade-specific strength and expression domain of CYCLOIDEA plays a central role in the different floral organ numbers and arrangements among these clades, suggesting that the application of the present model is plausible. In the present model, changes in either dorsal or ventral inhibition accounted for the direct transition from 4A to 5A (rdorsal=50, rventral=50 to rdorsal=50, rventral=30; Fig. 3I). Such direct transitions among the 4A, 4B, 5A and 5B arrangement types occurred as the result of changes in either the dorsal or ventral inhibition or the meristem size (Figs 3I and 4F, Fig. S1). By combining our framework with observed organ arrangements and initiation orders, we can estimate the developmental parameters (i.e. dorsal or ventral inhibition and meristem size) that accompany evolutionary changes in floral development between clades.

Future implications for our model

Our model does not incorporate some known parameters, such as the geometrical shape, fusion and division of primordia, during floral development. The shape of the primordium is considered to be a regular disc in the present model, whereas the actual shape is more likely to be wider in the ventral dimension than in the dorsal dimension in many bilaterally symmetrical flowers (e.g. Luo et al., 1996). In some species, a decrease in organ number may occur due to the fusion of primordia (Endress, 1999; Rudall and Bateman, 2004; Woźniak and Sicard, 2018). For example, the evolutionary transition from pentamery (5B) to tetramery (4A) via the fusion of two dorsal sepals has been suggested in Dipsacaceae (Ronse De Craene, 2010). Similarly, some flowers, such as Pisum sativum (Fabaceae), exhibit co-initiation of two or more organs as a common primordium and subsequent division (Ferrandiz et al., 1999). Therefore, our model must be carefully compared not only with the organ arrangement of mature flowers but also with the early floral development. Furthermore, the timing of the termination of the floral meristem by formation of carpel(s) can affect the initiation order of floral organs, especially that of the inner organs, such as stamens. In some species, zygomorphy is established after an organ initiates and its fate is determined (Jabbour et al., 2009). The regulation of organ growth in later development, in part, by CYC homologues in the dorsal region, also contributes to zygomorphy (Chang et al., 2010; Hileman, 2014; Spencer and Kim, 2018). In addition, the bracts, which we considered here as a ventral inhibitor, also exist at the dorsal and/or lateral sides (Ronse De Craene, 2010). Properties such as organ size, shape, growth, timing of termination and zygomorphy establishment, and dorsal/lateral bracts are therefore factors that may be incorporated into our model. In future theoretical studies, these properties may contribute to the explanation of the observed predominance of patterns 3B and 5A among angiosperms, while these parameter ranges are as wide as those for patterns 3A and 5B, respectively, in the present model (Figs 3I and 4E).

The relationship with the helical (spiral) initiation order is another important issue for future analysis. For example, in the subfamily Papilionoideae (Fabaceae; type 5B in Fig. 1C), an evolutionary path from the ancestral helical initiation, which is found in other subfamilies of Fabaceae (Tucker, 2003), to simultaneous initiation via unidirectional initiation, which is typical in the subfamily Papilionoideae (Prenner, 2004), has been suggested. Evaluating this hypothesis using a theoretical approach and relating helical initiation to zygomorphic initiation will provide an effective insight into phylogenetic relationships between species through morphogenesis and allow estimation of the evolutionary history of floral ontogeny.

Conclusion

We describe an inhibitory field model along the DV axis, where CYC and the main axis are candidates for dorsal inhibitors, whereas PAN, BOP and a ventral bract are candidates for the ventral inhibitor. The model simulations on the number, positions and initiation patterns of the organ primordia verified not only cyc mutant of Antirrhinum as ventral inhibitor, but also pan and bop mutants of Arabidopsis and bract as ventral inhibitors. Model simulations further showed that the diversity of monocots and eudicots in the bilateral symmetry of the organ positions were dependent on the relative inhibition strength and meristem size, demonstrating a design principle for floral symmetry during developmental evolution.

MATERIALS AND METHODS

Model for perianth organ positioning

Numerous theoretical studies have targeted the organ patterning of plant aerial parts, especially phyllotaxis. More than a century ago, Hofmeister (1868), who described several rules of plant development, noted the periodic initiation of leaf primordia at the least crowded space around the apical meristem. This ‘Hofmeister's rule’ was later modified by Snow and Snow (1962), who performed surgical and chemical treatments to alter phyllotaxis and suggested that the initiating position of primordium is strongly affected by the neighboring primordia. These investigators insisted that the primordia appears when there is enough space to support initiation (Snow and Snow, 1952), as opposed to the periodic initiation suggested by Hofmeister. Douady and Couder(1996,a,b) developed two mathematical models that assumed an inhibitory energy imposed on primordium initiation by the existing primordia. The first model included periodic initiation following Hofmeister's observations, while the second model relied on Snow and Snow's modification. The substance responsible for initiation inhibition has been suggested to be related to the polar transport of phytohormone auxin (Fujita and Kawaguchi, 2018; Jönsson et al., 2006; de Reuille et al., 2006; Smith et al., 2006), the mechanical properties of epidermal tissues (e.g. buckling wavelength) (Green, 1996) or the direction of microtubule alignment (Heisler et al., 2010). Based on the second phyllotaxis model of Douady and Couder (1996b), we considered the organ primordia to be points and the apical meristem to be a disc with radius R0 (Fig. 2A). We denoted the position of the primordium j as (rj, θj) in polar coordinates. Owing to the growth of the apex, the primordia move centrifugally with a constant speed V. Thus,
formula
(1)
where Tj represents the time when primordium j appeared. A new primordium i arises at the edge of the apical meristem, namely ri=R0. The angular position of the new primordium is determined by the inhibitory energy from the other primordia, U(θ), given by
formula
(2)
where dij is the distance between an existing primordium j and the candidate position (R0, θ) for primordium i, and λ denotes the characteristic length of the decay of inhibitory energy (Fig. 2B). The position θ′ is employed as the position of a new primordium, when U(θ′) is below a threshold and has a local minimum of U(θ). Under these conditions, one or more primordia arise corresponding to the number of local minima that satisfy these criteria (Fig. 2C), whereas the apex continues to grow without generating any primordium when a sufficiently low minimum is not achieved. The major parameters of this model include the threshold value of U to initiate a primordium, decay length of inhibition λ and the size of meristematic region R0.
Following the suggested inhibitory effect on the initiation of primordia by gene expression along the DV axis (Luo et al., 1996), we employed external inhibitory sources to our model. The inhibition from the dorsal side of the floral meristem, Udorsal, was expressed in the same form as Eqn 2 for simplicity:
formula
(3)
where ddorsal denotes the distance between the candidate position of the initiating primordium and the external inhibitory factor at the dorsal side. Adorsal denotes the strength of inhibition relative to that of the older primordia (Eqn 2). We examined not only the condition when Udorsal was constant throughout development (a=0) but also the condition when Udorsal decreased with a constant a. The radial and angular positions of the dorsal inhibitory factor were given as (rdorsal, 0), and an increase in rdorsal decreased the inhibitory effect by this factor at the meristem edge (Fig. 2D). Inhibition by an external inhibitory factor at the ventral side at position (rventral, 180°) was formulated similarly:
formula
(4)
where ddorsal denotes the distance between the candidate position of the initiating primordium and the external inhibitory factor at the ventral side. Aventral denotes the strength of inhibition. When the x-y coordinate in the 2D meristem geometry (Fig. 2) is scaled by x→cx and y→cx (implying dijcdij, dventralcdventral, ddorsalcddorsal, R0cR0, λ→cλ, rdorsalcrdorsal and rdorsalcrdorsal), where c is a constant, the present model (Eqns 2, 3 and 4) has the scale invariant property, U(θ)→U(θ), Udorsal→Udorsal and Uventral→Uventral.

Numerical simulations

The potential energy of the initiating primordia was calculated for discrete angles with an interval of 0.1°. Without the dorsal and ventral inhibition sources, the potential is uniform over the edge of the meristem upon initiation of the first primordium/primordia. Therefore, we manually placed one primordium on the dorsal side. The whorl in Figs 3 and 4 and Fig. S1 was defined when the radial distance between successive organ primordia is V or less. In other cases, the position of the first primordium/primordia was specified according to the potential landscape. All programs were written in the C programming language. V=0.01, λ=10 and Adorsal=Aventral=1, and the threshold value of U to initiate a primordium was 0.5 unless specified in the texts or figure captions.

Acknowledgements

We thank Kimiko Yoshikawa for related simulations during the early stages of the work, and Akitoshi Iwamoto, Hirokazu Tsukaya and Hiroyuki Hirano for helpful discussions.

Footnotes

Author contributions

Conceptualization: A.N., M.S.K., K.F.; Methodology: A.N., M.S.K.; Validation: M.S.K.; Investigation: A.N., M.S.K.; Data curation: A.N., M.S.K.; Writing - original draft: A.N., M.S.K., K.F.; Writing - review & editing: M.S.K., K.F.; Supervision: K.F.; Project administration: K.F.; Funding acquisition: K.F.

Funding

This work was supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (17H06386, 16H01241, 16H06378 to K.F.).

References

Adler
,
I.
,
Barabe
,
D.
and
Jean
,
R. V.
(
1997
).
A history of the study of phyllotaxis
.
Ann. Bot.
80
,
231
-
244
.
Bowman
,
J. L.
,
Eshed
,
Y.
and
Baum
,
S. F.
(
2002
).
Establishment of polarity in angiosperm lateral organs
.
Trends Genet.
18
,
134
-
141
.
Chandler
,
J. W.
and
Werr
,
W.
(
2014
).
Arabidopsis floral phytomer development: auxin response relative to biphasic modes of organ initiation
.
J. Exp. Bot.
65
,
3097
-
3110
.
Chang
,
Y.-Y.
,
Kao
,
N.-H.
,
Li
,
J.-Y.
,
Hsu
,
W.-H.
,
Liang
,
Y.-L.
,
Wu
,
J.-W.
and
Yang
,
C.-H.
(
2010
).
Characterization of the possible roles for B class MADS box genes in regulation of perianth formation in orchid
.
Plant Physiol.
152
,
837
-
853
.
de Lima Silva
,
A.
,
Trovó
,
M.
and
Coan
,
A. I.
(
2016
).
Floral development and vascularization help to explain merism evolution in (Eriocaulaceae, Poales)
.
PeerJ
4
,
e2811
.
de Reuille
,
P. B.
,
Bohn-Courseau
,
I.
,
Ljung
,
K.
,
Morin
,
H.
,
Carraro
,
N.
,
Godin
,
C.
and
Traas
,
J.
(
2006
).
Computer simulations reveal properties of the cell–cell signaling network at the shoot apex in Arabidopsis
.
Proc. Natl. Acad. Sci. USA
103
,
1627
-
1632
.
Dhaka
,
N.
,
Bhardwaj
,
V.
,
Sharma
,
M. K.
and
Sharma
,
R.
(
2017
).
Evolving tale of TCPs: new paradigms and old lacunae
.
Front. Plant Sci.
8
,
479
.
Douady
,
S.
and
Couder
,
Y.
(
1996a
).
Phyllotaxis as a dynamical self organizing process part I: the spiral modes resulting from time-periodic iterations
.
J. Theor. Biol.
178
,
255
-
273
.
Douady
,
S.
and
Couder
,
Y.
(
1996b
).
Phyllotaxis as a dynamical self organizing process part II: the spontaneous formation of a periodicity and the coexistence of spiral and whorled patterns
.
J. Theor. Biol.
178
,
275
-
294
.
Endress
,
P. K.
(
1999
).
Symmetry in flowers: diversity and evolution
.
Int. J. Plant Sci.
160
,
S3
-
S23
.
Endress
,
P. K.
(
2010
).
Flower structure and trends of evolution in Eudicots and their major subclades
.
Ann. Mo. Bot. Gard.
97
,
541
-
583
.
Ferrandiz
,
C.
,
Navarro
,
C.
,
Gomez
,
M. D.
,
Canas
,
L. A.
and
Beltran
,
J. P.
(
1999
).
Flower development in pisum sativum: from the war of the whorls to the battle of the common primordia
.
Dev. Genet.
25
,
280
-
290
.
Fletcher
,
J. C.
(
2001
).
The ULTRAPETALA gene controls shoot and floral meristem size in Arabidopsis
.
Development
128
,
1323
-
1333
.
Fujita
,
H.
and
Kawaguchi
,
M.
(
2018
).
Spatial regularity control of phyllotaxis pattern generated by the mutual interaction between auxin and PIN1
.
PLoS Comput. Biol.
14
,
e1006065
.
Green
,
P.
(
1996
).
Phyllotactic patterns: a biophysical mechanism for their origin
.
Ann. Bot.
77
,
515
-
528
.
Harshberger
,
J. W.
(
1907
).
Teratologic notes
.
Plant World
10
,
186
-
189
. .
Heisler
,
M. G.
,
Hamant
,
O.
,
Krupinski
,
P.
,
Uyttewaal
,
M.
,
Ohno
,
C.
,
Jönsson
,
H.
,
Traas
,
J.
and
Meyerowitz
,
E. M.
(
2010
).
Alignment between PIN1 polarity and microtubule orientation in the shoot apical meristem reveals a tight coupling between morphogenesis and auxin transport
.
PLoS Biol.
8
,
e1000516
.
Hepworth
,
S. R.
,
Zhang
,
Y.
,
Mckim
,
S.
,
Li
,
X.
and
Haughn
,
G. W.
(
2005
).
BLADE-ON-PETIOLE-dependent signaling controls leaf and floral patterning in Arabidopsis
.
Plant Cell
17
,
1434
-
1448
.
Hileman
,
L. C.
(
2014
).
Bilateral flower symmetry — how, when and why?
Curr. Opin. Plant Biol.
17
,
146
-
152
.
Hofmeister
,
W.
(
1868
).
Allgemeine morphologie der gewächse
. In
Handbuch der Physiologischen Botanik
, Vol.
1
(ed.
W.
Hofmeister
), pp.
405
-
664
.
Leipzig
:
Engelmann
.
Jabbour
,
F.
,
Ronse De Craene
,
L. P.
,
Nadot
,
S.
and
Damerval
,
C.
(
2009
).
Establishment of zygomorphy on an ontogenic spiral and evolution of perianth in the tribe Delphinieae (Ranunculaceae)
.
Ann. Bot.
104
,
809
-
822
.
Jönsson
,
H.
,
Heisler
,
M. G.
,
Shapiro
,
B. E.
,
Meyerowitz
,
E. M.
and
Mjolsness
,
E.
(
2006
).
An auxin-driven polarized transport model for phyllotaxis
.
Proc. Natl. Acad. Sci. USA
103
,
1633
-
1638
.
Khan
,
M.
,
Xu
,
H.
and
Hepworth
,
S. R.
(
2014
).
BLADE-ON-PETIOLE genes: setting boundaries in development and defense
.
Plant Sci.
215-216
,
157
-
171
.
Kitazawa
,
M. S.
and
Fujimoto
,
K.
(
2015
).
A dynamical phyllotaxis model to determine floral organ number
.
PLoS Comput. Biol.
11
,
e1004145
.
Kuhlemeier
,
C.
(
2017
).
Phyllotaxis
.
Curr. Biol.
27
,
R882
-
R887
.
Kümpers
,
B. M. C.
,
Richardson
,
J. E.
,
Anderberg
,
A. A.
,
Wilkie
,
P.
and
Ronse De Craene
,
L. P.
(
2016
).
The significance of meristic changes in the flowers of Sapotaceae
.
Bot. J. Linn. Soc.
180
,
161
-
192
.
Luo
,
D.
,
Carpenter
,
R.
,
Vincent
,
C.
,
Copsey
,
L.
and
Coen
,
E.
(
1996
).
Origin of floral asymmetry in Antirrhinum
.
Nature
383
,
794
-
799
.
Maier
,
A. T.
,
Stehling-Sun
,
S.
,
Wollmann
,
H.
,
Demar
,
M.
,
Hong
,
R. L.
,
Haubeiss
,
S.
,
Weigel
,
D.
and
Lohmann
,
J. U.
(
2009
).
Dual roles of the bZIP transcription factor PERIANTHIA in the control of floral architecture and homeotic gene expression
.
Development
136
,
1613
-
1620
.
Masters
,
M. T.
(
1887
).
On the Floral Conformation of the Genus Cypripedium
.
J. Linn. Soc. Lond. Bot.
22
,
402
-
422
.
Matthews
,
M. L.
and
Endress
,
P. K.
(
2002
).
Comparative floral structure and systematics in Oxalidales (Oxalidaceae, Connaraceae, Brunelliaceae, Cephalotaceae, Cunoniaceae, Elaeocarpaceae, Tremandraceae)
.
Bot. J. Linn. Soc.
140
,
321
-
381
.
Matthews
,
M. L.
and
Endress
,
P. K.
(
2013
).
Comparative floral structure and systematics of the clade of Lophopyxidaceae and Putranjivaceae (Malpighiales)
.
Bot. J. Linn. Soc.
172
,
404
-
448
.
Norberg
,
M.
(
2005
).
The BLADE ON PETIOLE genes act redundantly to control the growth and development of lateral organs
.
Development
132
,
2203
-
2213
.
Ollerton
,
J.
,
Winfree
,
R.
and
Tarrant
,
S.
(
2011
).
How many flowering plants are pollinated by animals?
Oikos
120
,
321
-
326
.
Pabón-Mora
,
N.
and
González
,
F.
(
2008
).
Floral ontogeny of Telipogon spp. (Orchidaceae) and insights on the Perianth symmetry in the family
.
Int. J. Plant Sci.
169
,
1159
-
1173
.
Prenner
,
G.
(
2004
).
New aspects in floral development of Papilionoideae: initiated but suppressed bracteoles and variable initiation of sepals
.
Ann. Bot.
93
,
537
-
545
.
Prenner
,
G.
(
2011
).
Floral ontogeny of Acacia celastrifolia: an enigmatic mimosoid legume with pronounced polyandry and multiple carpels
. In
Flowers on the Tree of Life
(ed.
L.
Wanntorp
and
L. P.
Ronse De Craene
), pp.
256
-
278
.
Cambridge University Press
.
Ramírez-Domenech
,
J. I.
and
Tucker
,
S. C.
(
1990
).
Comparative ontogeny of the perianth in mimosoid legumes
.
Am. J. Bot.
77
,
624
-
635
.
Refahi
,
Y.
,
Brunoud
,
G.
,
Farcot
,
E.
,
Jean-Marie
,
A.
,
Pulkkinen
,
M.
,
Vernoux
,
T.
and
Godin
,
C.
(
2016
).
A stochastic multicellular model identifies biological watermarks from disorders in self-organized patterns of phyllotaxis
.
eLife
5
,
e14093
.
Remizowa
,
M. V.
,
Sokoloff
,
D. D.
and
Rudall
,
P. J.
(
2010
).
Evolutionary history of the monocot flower
.
Ann. Mo. Bot. Gard.
97
,
617
-
645
.
Roels
,
P.
and
Smets
,
E.
(
1994
).
A comparative floral ontogenetical study between Adoxa moschatellina and Sambucus ebulus
.
Belg. J. Bot.
127
,
157
-
170
. .
Ronse De Craene
,
L. P.
, (
2007
).
Are petals sterile stamens or bracts? the origin and evolution of petals in the core Eudicots
.
Ann. Bot.
100
,
621
-
630
.
Ronse De Craene
,
L. P.
(
2010
).
Floral Diagrams: An Aid to Understanding Flower Morphology and Evolution
.
Cambridge University Press
.
Ronse De Craene
,
L. P.
and
Brockington
,
S. F.
(
2013
).
Origin and evolution of petals in angiosperms
.
Plant Ecol. Evol.
146
,
5
-
25
.
Rudall
,
P. J.
and
Bateman
,
R. M.
(
2003
).
Evolutionary change in flowers and inflorescences: evidence from naturally occurring terata
.
Trends Plant Sci.
8
,
76
-
82
.
Rudall
,
P. J.
and
Bateman
,
R. M.
(
2004
).
Evolution of zygomorphy in monocot flowers: iterative patterns and developmental constraints
.
New Phytol.
162
,
25
-
44
.
Running
,
M. P.
and
Meyerowitz
,
E. M.
(
1996
).
Mutations in the PERIANTHIA gene of Arabidopsis specifically alter floral organ number and initiation pattern
.
Development
122
,
1261
-
1269
.
Sargent
,
R. D.
(
2004
).
Floral symmetry affects speciation rates in angiosperms
.
Proc. Biol. Sci.
271
,
603
-
608
.
Schoof
,
H.
,
Lenhard
,
M.
,
Haecker
,
A.
,
Mayer
,
K. F. X.
,
Jürgens
,
G.
and
Laux
,
T.
(
2000
).
The stem cell population of Arabidopsis shoot meristems is maintained by a regulatory loop between the CLAVATA and WUSCHEL Genes
.
Cell
100
,
635
-
644
.
Smith
,
R. S.
,
Guyomarc'h
,
S.
,
Mandel
,
T.
,
Reinhardt
,
D.
,
Kuhlemeier
,
C.
and
Prusinkiewicz
,
P.
(
2006
).
A plausible model of phyllotaxis
.
Proc. Natl. Acad. Sci. USA
103
,
1301
-
1306
.
Smyth
,
D. R.
(
2018
).
Evolution and genetic control of the floral ground plan
.
New Phytol.
220
,
70
-
86
.
Smyth
,
D. R.
,
Bowman
,
J. L.
and
Meyerowitz
,
E. M.
(
1990
).
Early flower development in Arabidopsis
.
Plant Cell
2
,
755
-
767
.
Snow
,
M.
and
Snow
,
G. R. S.
(
1952
).
Minimum areas and leaf determination
.
Proc. R. Soc. Lond. B Biol. Sci.
139
,
545
-
566
.
Snow
,
M.
and
Snow
,
G. R. S.
(
1962
).
A theory of the regulation of phyllotaxis based on Lupinus albus
.
Philos. Trans. R. Soc. Lond. B Biol. Sci.
244
,
483
-
513
.
Spencer
,
V.
and
Kim
,
M.
(
2018
).
Re“CYC”ling molecular regulators in the evolution and development of flower symmetry
.
Semin. Cell Dev. Biol.
79
,
16
-
26
.
The Angiosperm Phylogeny Group
(
2016
).
An update of the angiosperm phylogeny Group classification for the orders and families of flowering plants: APG IV
.
Bot. J. Linn. Soc.
181
,
1
-
20
.
Thoma
,
R.
and
Chandler
,
J. W.
(
2015
).
Polarity in the early floral meristem of Arabidopsis
.
Plant Signal. Behav.
10
,
e992733
.
Tobe
,
H.
,
Huang
,
Y.-L.
,
Kadokawa
,
T.
and
Tamura
,
M. N.
(
2018
).
Floral structure and development in Nartheciaceae (Dioscoreales), with special reference to ovary position and septal nectaries
.
J. Plant Res.
131
,
411
-
428
.
Traas
,
J.
(
2013
).
Phyllotaxis
.
Development
140
,
249
-
253
.
Tucker
,
S. C.
(
1999
).
Evolutionary lability of symmetry in early floral development
.
Int. J. Plant Sci.
160
,
S25
-
S39
.
Tucker
,
S. C.
(
2003
).
Floral development in legumes
.
Plant Physiol.
131
,
911
-
926
.
Van Mourik
,
S.
,
Kaufmann
,
K.
,
Van Dijk
,
A. D. J.
,
Angenent
,
G. C.
,
Merks
,
R. M. H.
and
Molenaar
,
J.
(
2012
).
Simulation of organ patterning on the floral meristem using a polar auxin transport model
.
PLoS ONE
7
,
e28762
.
Wang
,
Y.
and
Jiao
,
Y.
(
2018
).
Auxin and above-ground meristems
.
J. Exp. Bot.
69
,
147
-
154
.
Wei
,
L.
,
Wang
,
Y. Z.
and
Li
,
Z. Y.
(
2012
).
Floral ontogeny of Ruteae (Rutaceae) and its systematic implications
.
Plant Biol.
14
,
190
-
197
.
Woźniak
,
N. J.
and
Sicard
,
A.
(
2018
).
Evolvability of flower geometry: convergence in pollinator-driven morphological evolution of flowers
.
Semin. Cell Dev. Biol.
79
,
3
-
15
.
Yonekura
,
T.
,
Iwamoto
,
A.
,
Fujita
,
H.
and
Sugiyama
,
M.
(
2019
).
Mathematical model studies of the comprehensive generation of major and minor phyllotactic patterns in plants with a predominant focus on orixate phyllotaxis
.
PLoS Comput. Biol.
15
,
e1007044
.
Zhang
,
W.
,
Kramer
,
E. M.
and
Davis
,
C. C.
(
2010
).
Floral symmetry genes and the origin and maintenance of zygomorphy in a plant-pollinator mutualism
.
Proc. Natl. Acad. Sci. USA
107
,
6388
-
6393
.

Competing interests

The authors declare no competing or financial interests.

Supplementary information