A design principle for floral organ number and arrangement in flowers with bilateral symmetry

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).

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.

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).

Next, we introduced external inhibition from the dorsal side (Fig. 2D; Eqn 3). The inhibition gradient of the U_dorsal 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 (r_dorsal). 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 (r_dorsal=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 (r_dorsal=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 (r_dorsal=40, r_ventral=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 (r_dorsal=30, r_ventral=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 (r_dorsal=60, r_ventral=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 (r_dorsal=20 and r_ventral=50), whereas the latter appeared with a dorsal inhibition that was weaker than the ventral inhibition (r_dorsal=70 and r_ventral=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.

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 (r_dorsal, r_ventral 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.

The dependence of floral organ arrangement on r_dorsal, r_ventral 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 (r_dorsal=40, r_ventral=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, r_dorsal/λ, and r_ventral/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 r_ventral=60-70 in Fig. 4F) than in its absence (at A_v=0 in Fig. 4F), suggesting that the robustness of positional arrangement to meristem size variation is promoted by the ventral inhibitory field. Therefore r_dorsal/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 r_dorsal and r_ventral, 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 (r_dorsal=10, r_ventral=80, R₀=20), bidirectional initiation from the lateral side (r_dorsal=30, r_ventral=40, R₀=16 in Fig. 3I) and other bidirectional initiation events in the sequence of dorsal, ventral and lateral (r_dorsal=40, r_ventral=80, R₀=16 in Fig. 3I) were segregated depending on r_dorsal, r_ventral 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.

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 r_dorsal and r_ventral. 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 r_dorsal and r_ventral, 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.

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).

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 r_dorsal=80, r_ventral=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.

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 (r_dorsal=50, r_ventral=50 to r_dorsal=50, r_ventral=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.

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.

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.

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 A_dorsal=A_ventral=1, and the threshold value of U to initiate a primordium was 0.5 unless specified in the texts or figure captions.

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.