The initiation process

Following the earlier models [49], we represented the meristem as a circular disc with radius R₀ and the primordia as points (Fig 2A). A new primordium arises at the point along the edge of the meristem (R₀, θ), in polar coordinate with the origin at the meristem center, where θ gives the minimum value of the inhibition potential U_ini. As one of the simplest setups for sequential initiation [37], we followed the assumption of earlier models for spiral phyllotaxis [49], which state that new primordia arise sequentially with time intervals τ, as opposed to the simultaneous initiation studied previously for whorled phyllotaxis [19] (Fig 1). Although the structures outside of the flower, such as bracts and other flowers, as well as the position of the inflorescence axis, may affect the position of organ primordia, the pentamerous whorls appear despite their various arrangement [20]. Therefore, as the first step of modelling of floral organ arrangement, we assumed that whorl formation is independent of any positional information from structures outside of the flower. Thus, we calculated the inhibition potential only from floral organ primordia which are derived from a single floral meristem.

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Fig 2. Emergence of multiple whorls in model simulations.

A. Geometric assumptions of the model. B. The initiation process. A new primordium (i) is initiated at the edge of the floral meristem (FM; green circle) where the initiation potential U_ini takes the minimum value. i, i −1, and i −2 are the primordium indices that denote the initiation order. U_ini exponentially decreases with time (α) and the distance between primordia (λ_ini). C. The growth process. Each primordium (k) moves at the outside of the circular FM, depending on the growth potential U_{g, k}. Primordium k rarely moves against the gradient (grey thin arrow), but mostly follows the gradient (black thick arrow; see the Model section). D–F. Emergence of whorled-type pattern with increasing meristem radius R₀ and temporal decay rate α. Left panels: Spatial pattern after 15 primordia (red circles) initiated in an indexed order at the meristem edge (green circle; r = R₀). Middle panels: Radial distance (black) from the meristem center as a function of the primordium initiation index (left panel) averaged over 400 replicate Monte Carlo simulations. Error bars represent twice the S.D. Red circles are a set of representative samples. Right panels: Time evolution of the radial coordinates of each primordium averaged over 400 replicates. Error bars show 2 S.D. The arrowheads in D and F indicate the growth arrest of the fifth and sixth primordia, respectively. Colors denote the index of the primordia. Green line in the left, middle and right panels denotes the meristem edge. (R₀, α) = (20.0,0.0) in D, (5.0,0.0) in E and (20.0,2.0) in F. β = 1.0 × 10⁴, λ_ini = λ_g = 10.0, τ = 300, and σ_r = σ_θ = 0.05 in D–F.

https://doi.org/10.1371/journal.pcbi.1004145.g002

The potential functions for the initiation inhibition by preexisting primordia have been extensively analyzed in phyllotaxis models [16, 47, 49]. The potential decreases with increasing distance between an initiating primordium and the preexisting primordia account for the diffusion of inhibitors secreted by the preexisting primordia [27, 50], and the polar auxin transport in the epidermal layer, as proposed in previous models of phyllotaxis [15, 16, 31] and the flowers [43]. We employed an exponential function exp(−d_ij/λ_ini) as a function of θ, where d_ij denotes the distance between a new primordium i and a preexisting primordium j at (r_j, θ_j) as (1)

The function decreases spatially through the decay length λ_ini exponentially, induced by a mechanism proposed for the polar auxin transport, i.e., the up-the-gradient model [15, 16]. Up-the-gradient positive feedback amplifies local auxin concentration maxima and depletes auxin from the surrounding epidermis, causing spatially periodic concentration peaks to self-organize [15, 16] and thus determine the initiation position of the primordia [51]. The amplification and depletion work as short-range activation and long-range inhibition, respectively [52], which are common to Turing patterns of reaction-diffusion systems [18]. Since the interaction of local maxima in the reaction-diffusion systems follows the exponential potential [53, 54], the up-the-gradient model likely explains the exponential potential between the auxin maxima, while the rigorous derivation requires further research. The decay length λ_ini depends not only on the ratio of the auxin diffusion constant and the polar auxin-transport rate [15] but also on other biochemical parameters for polar transport and the underlying intracellular PIN1 cycling [55]. Another mechanism, referred to as the with-the-flux model [56, 57], has been proposed for the polar auxin transport. Although with-the-flux positive feedback can also produce spatial periodicity, the primordia position corresponds to auxin minima [57], which is inconsistent with observations [51].

On the other hand, the with-the-flux mechanism can explain auxin drain from the epidermal layer of the primordia to internal tissue [58]. Since the drain gets stronger as the primordia mature [58, 59], the auxin drain could cause decay of the potential depending on the primordia age. The auxin decrease in maturing organs can also be caused by controlling auxin biosynthesis [60, 61]. Therefore, we integrated another assumption, namely that the inhibition potential decreases exponentially with the primordia age at the decay rate α (Fig 2B). Temporally decaying inhibition was proposed previously to represent the degradation of some inhibitors [47, 48] and account for various types of phyllotaxis by simple extension of the inhibitory field model [33]. Taken together, the potential at the initiation of the i-th primordium is given by (2)

The growth process

Most phyllotaxis models have assumed, based on Hofmeister’s hypothesis, that the primordia move outward at a constant radial drift depending only on the distance from the floral center without angular displacement, which makes helical initiation result in spiral phyllotaxis [49]. Here, we assumed instead that all primordia repel each other, even after the initiation, except for movement into the meristematic zone (Fig 2C) following observation of the absence of auxin (DR5 expression) maxima at the center of the floral bud (e.g., [62]). Even at the peripheral zone away from the meristem, the growth is not limited. Hence there is no upper limit for the distance between primordia and the center. The repulsion exerted on the k-th primordium is represented by another exponentially decaying potential when there are i primordia (1 ≦ k ≦ i): (3) where the decay length, introduced as λ_g, can differ from λ_ini. The primordia descend along the gradient of potential U_g to find a location with weaker repulsion. The continuous repulsion can account for post-meristematic events such as the mechanical stress on epidermal cells caused by the enlargement of primordia [63, 64] or the gene expression that regulates the primordial boundary [42]. The present formulation (Eq 3) is similar to the contact pressure model, which has been proposed for re-correcting the divergence angle after initiation [25, 45, 46]. Another type of post-initiation angular rearrangement has been modeled as a function of the primordia age employed as i −j −1 in the present model (Eq 2) and the distance between primordia with some stochasticity [65]. Eq 3 accounts for not only the angular rearrangement but also the radial rearrangement with stochasticity in both directions as will be described in the next subsection.

Numerical experiments

We modeled the initiation process numerically by calculating the potential U_ini (Eq 2) for angular position θ incremented by 0.1 degree on the edge of the circular meristem. We introduced a new primordium at the position where the value of U_ini took the minimum, provided that the first primordium is initiated at θ = 0. We modeled the growth process by using a Monte Carlo method [66] to calculate the movement of primordia in the outside of the meristem depending on the potential U_{g, k} (Eq 3, Fig 2C). After the introduction of a new primordium, we randomly chose one primordium indexed by k from among the existing primordia and virtually moved its position (r_k, θ_k) to a new position $(r_k^{\prime },{\theta }_k^{\prime })$ in the outer meristem $(r_k,r_k^{\prime }\ge R_0)$. The new radius $r_k^{\prime }$ and the angle ${\theta }_k^{\prime }$ were chosen randomly following a two-dimensional Gaussian distribution whose mean and standard deviation were given by the previous position (r_k, θ_k) and by two independent parameters, (σ_r, σ_θ/r_k), respectively. Whether or not the k-th primordium moved to the new position was determined by the Metropolis algorithm [66]; the primordium moved if the growth potential (Eq 3) of the new position was lower than that of the previous position (i.e., $U_{g,k}(r_k^{\prime },{\theta }_k^{\prime })<U_{g,k}(r_k,{\theta }_k)$). Otherwise, it moved with the probability given by (4) where $\Delta U_g=U_{g,k}(r_k^{\prime },{\theta }_k^{\prime })-U_{g,k}(r_k,{\theta }_k)$ and β is a parameter for stochasticity. This stochasticity represents a random walk biased by the repulsion potential. A case P_MP = 0 represents that primordia movement always follows the potential (ΔU_g < 0). The first primordium stays at the meristem edge r = R₀ until the second one arises when P_MP = 0 because the growth potential is absent, while it can move randomly outside of the meristem when P_MP ≠ 0. To maintain the physical time interval of the initiation process at τ steps for each primordium, the number of iteration steps in the Monte Carlo simulation during each initiation interval was set to iτ, where i denotes the number of the primordia. We also studied the movement following U_g by numerical integration (fourth-order Runge-Kutta method) of ordinary differential equations to confirm the independence of the numerical methods (S1 Fig). All our programs were written in the C programming language and used the Mersenne Twister pseudo-random number generator (http://www.math.sci.hiroshima-u.ac.jp/m-mat/MT/emt.html) [67].

Because the initiation time interval is constant, one possible scenario for forming a whorled pattern should involve decreasing or arresting the radial displacement of primordia (Fig 1, forth row). Therefore, we focused on the change in radial position and velocity to find the whorled arrangement, while angular positions were not taken into account in the present manuscript.

Mutually repulsive growth promotes a whorled arrangement from sequential initiation at the proper meristem size

Numerical simulations showed that several whorls self-organized following the sequential initiation of primordia. Although several previous phyllotaxis models showed the transition between a spiral arrangement following sequential initiation and a whorled arrangement following simultaneous initiation [15, 16, 19], they were not able to reproduce the emergence of a whorled arrangement following sequential initiation, which is the situation observed in many eudicot flowers (Fig 1) [34, 37, 38, 40, 41]. In the present model, a tetramerous whorl appeared spontaneously that exhibited four primordia almost equidistant from the meristem center (Fig 2D, left and middle), by arresting radial movement of the fifth primordium at the meristem edge until the seventh primordium arose (arrowhead in Fig 2D, right). Likewise, subsequent primordia produced the same gap in radial distance for every four primordia (Fig 2D, middle and right), leading to several whorls comprising an identical number of primordia (Fig 2D). The radial positions of all primordia were highly reproducible despite stochasticity in the growth process (error bars in Fig 2D–2F, middle and right). Therefore, we identified the whorled arrangement by radial displacement arrest (arrowhead in Fig 2D, right).

The initiation order and angle of the first tetramerous whorl in the model reproduced those observed in A. thaliana sepals [68] (S2A Fig). The first primordium scarcely moved from the initiation point until the second primordium arose because growth repulsion was absent. The second primordium arose opposite the first, whereas the third and fourth primordia arose perpendicular to the preceding two. The angular position of the primordia did not change once the whorl was established because the primordia within a whorl blocked the angular displacement by the growth potential U_g (S3 Fig).

Introducing mutual repulsion among the primordia throughout the growth process caused the whorled arrangement to spontaneously emerge (Fig 2D). This was in contrast to the model of constant growth in which all primordia move away depending only on the distance from the floral apex [49]. A study of post-meristematic regulation by the organ-boundary gene CUP-SHAPED COTYLEDON2 (CUC2) showed that A. thaliana plants up-regulating CUC2 gene have an enlarged primordial margin and have whorled-like phyllotaxis following the normal helical initiation of primordia [42], suggesting that repulsive interactions among primordia after initiation are responsible for the formation of the floral whorls.

In the present model, the meristem size R₀ controls the transition from non-whorled (Fig 2E) to whorled arrangement (Fig 2D). Radial spacing of the primordia was regular when R₀ was small (Fig 2E, middle) because the older primordia pushed any new primordium across the meristem (Fig 2E, left), causing continuous movement at the same rate (Fig 2E, right). Above a threshold meristem size R₀, a tetramerous whorl appeared spontaneously. The primordium number within each whorl increased up to eight with increasing R₀, but the number tended to be more variable (S2B Fig). In the A. thaliana mutant wuschel, which has a decreased meristem size, the pattern of four sepals does not have square positions at the stage when the wild-type plant forms a tetramerous sepal whorl [69]. Conversely, the clavata mutant, which has an increased meristem size, has excessive floral organs with larger variation [69]. Our model consistently reproduced not only the transition from the non-whorled arrangement (Fig 2E) to the tetramerous whorled arrangement (Fig 2D) but also the variable increase in the primordia number within a whorl as the meristem size R₀ increased.

The pentamerous whorl stably appeared in the presence of temporal decay of initiation inhibition (α > 0 in Eq 2). The whorls comprising five primordia appeared in the same manner as the tetramerous whorls, namely, via the locking of the sixth primordium at the initiation site (Fig 2F, right; S2C Fig).

Developmental preference for particular organ number within a whorl

In order to study the organ number within each whorl extensively, known as the merosity [70], we counted the number of primordia existing prior to the arrest of primordium displacement, which corresponds to the merosity of the first whorl (arrowheads in Fig 2D and 2F, right). We defined arrest of primordium displacement as occurring when the ratio of the initial radial velocity of a new primordium immediately after initiation to that of the previous primordium was lower than 0.2. The definition does not affect the following results as long as the ratio is between 0.1 and 0.6. We found that the key parameter for merosity is the relative value of R₀ normalized by the average radial velocity $V={\sigma }_r/\sqrt{2\pi }$ (see S1 Text) and the initiation time interval τ (Fig 3). The arrest of radial displacement did not occur below a threshold of R₀/Vτ (the left region colored red in Fig 3A), whereas the whorled arrangement appeared above the threshold value of R₀/Vτ. As R₀/Vτ increased further, tetramery, pentamery, hexamery, heptamery, and octamery appeared, successively (Fig 3A).

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Fig 3. Merosity of the first whorl.

A, B. The number of primordia before the first arrest (arrowheads in Fig 2C and 2E) is depicted by colors in the legend. The red region indicates a non-whorled pattern. For simplicity, we set P_MP = 0 (Eq 4) so that primordia could not move against the potential gradient U_{g, k}. λ_ini = λ_g = 10.0, σ_r = σ_θ = 0.05. α = 0.0 (A) and α = 2.0 (B). The four panels between A and B are representative examples of each merosity where the arrowhead indicates the third primordium. C. Phase diagram of the first-whorl merosity according to α and R₀/Vτ at $V\tau =(-0.5R_0+50)/\sqrt{2\pi }$ (white line in A). The color code is the same as that in A and B. The region of dimerous arrangement (green) increases as α increases, because the previous primordium becomes the most dominant inhibitor so that the new primordium initiates just opposite to the previous one and its growth is arrested by the second previous one.

https://doi.org/10.1371/journal.pcbi.1004145.g003

The present model showed dominance of special merosity, i.e., tetramery and octamery in the absence of temporal decay of inhibition (α = 0 in Eq 2; Fig 3A); pentamery in the presence of temporal decay (α > 0; Fig 3B and 3C), in contrast to previous phyllotaxis models for whorled arrangement in which the parameter region leading to each level of merosity decreased monotonically with increasing merosity [19]. The major difference between α = 0 and α > 0 was that θ₃, the angular position of the third primordium, took an average value of 90 degrees when α = 0 (arrowhead in Fig 3A bottom magenta panel) and decreased significantly as α increased (arrowhead in Fig 3A bottom cyan panel). In a pentamerous flower Silene coeli-rosa, the third primordium is located closer to the first primordium than the second one [34]. This is consistent with the third primordium position at α > 0, indicating the necessity of α, as we will discuss in the next section. The parameter region R₀/Vτ for pentamery expanded with increasing α, whereas the border between the whorled and non-whorled arrangements was weakly dependent on α (Fig 3C). The tetramery, pentamery, and octamery arrangements were more robust to R₀/Vτ and α than the hexamery and heptamery arrangements. Dominance of the particular number also appears in the ray-florets within a head inflorescence of Asteraceae [71], in which radial positions show the whorled-type arrangement [72]. Meanwhile, the leaf number in a single vegetative pseudo-whorl transits between two to six by hormonal control without any preference [73].

Moreover, the transition between the different merosities occurred directly, without the transient appearance of the non-whorled arrangement. This is in contrast to an earlier model [19] in which the transition between different merosity always involved transient spiral phyllotaxis. The fact that the merosity can change while keeping its whorled nature in flowers (e.g., the flowers of Trientalis europaea[74]) supports our results. To our knowledge, ours is the first model showing direct transitions between whorled patterns with different merosities as well as preferences for tetramery and pentamery, the most common merosities in eudicot flowers.

Reconstructing the Silene coeli-rosa pentamerous whorl arrangement

To further validate our model of the pentamerous whorl arrangement, we quantitatively compared its results with the radial distances and divergence angles in eudicot flowers. Here we focus on a Scanning Electron Microscope (SEM) image of the floral meristem of S. coeli-rosa, Caryophyllaceae (Fig 4A–4C) [34], because S. coeli-rosa exhibits not only five sepals and five petals in alternate positions, which is the most common arrangement in eudicots, but also the helical initiation of these primordia, which we targeted in the present model. In addition, to our knowledge, this report by Lyndon is the only publication showing a developmental sequence for both the divergence angle Δθ_{k, k+1} = θ_k+1 −θ_k (0 ≤ Δθ_{k, k+1} < 360) and the ratio of the radial position, r_k/r_k+1, referred to as the plastochron ratio [75], in eudicot floral organs. Reconstructing such developmental sequences of both radial and angular positions is an unprecedented theoretical challenge, while those which describe the angular position alone for the ontogeny of spiral phyllotaxis (180 degree, 90 degree and finally convergence to 137 degree [16, 33]; the ‘M-shaped’ motif, i.e., 137, 275, 225, 275 and 137 degrees [76, 77]) have been reproduced numerically.

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Fig 4. Reconstructing pentamerous floral development.

A. Flower of Silene coeli-rosa (Caryophyllaceae). B. Reproduction of the S. coeli-rosa floral meristem traced from an SEM image by Lyndon [34]; the colors were modified. Numbers indicate the initiation order. K (sepals), C (petals), St (stamens), AB (axillary bud). C. Average position of the S. coeli-rosa floral primordia reconstructed from the divergence angle and plastochron ratio (E) measured by Lyndon (Table 1 in [34]). The number of measured apices is 9 for sepals, 5 for petals, 7 for stamens, and 2 for carpels. The positions of sepals and petals are depicted in large squares, and those of stamens and carpels are depicted in small squares. D. Spatial pattern of the model simulation. The first ten primordia are shown by large circles, and the subsequent ten primordia are shown by small circles. τ = 600, R₀ = 30.0, α = 3.0, σ_r = 0.05, σ_θ = 5.0, λ_ini = λ_g = 20.0, P_MP = 0. E. Divergence angle (top panel) and plastochron ratio (middle) between two succeeding primordia, and the distance from the center of the apex (bottom panel) in S. coeli-rosa (blue squares) and in the model simulation (red circles). The order of petal initiation was estimated from that of the adjacent stamens (St6-St10 in B) following the experimental report [34]. The measurements agree with the model until the ninth primordium (open arrowhead). Error bars for the divergence angle and plastochron ratio of S. coeli-rosa denote the standard errors. Because the absolute values of the S. coeli-rosa primordia radii were not published, the distance from the center is normalized by the radius of the first sepal. The values of the parameters are the same as those in D. The green line (D and E bottom panel) indicates the meristem boundary in the simulation.

https://doi.org/10.1371/journal.pcbi.1004145.g004

By substituting the initial divergence angle between the first and second sepals of S. coeli-rosa into Δθ_1,2 = 156 but not any plastochron data into the simulation (θ₁ = 0 and θ₂ = 156 degree), we numerically calculated the positions of the subsequent organs (Fig 4D). The observed divergence angle Δθ_2,3 = 132 degree indicates α > 0, because Δθ_2,3 = Δθ_1,3 = (360−156)/2 = 102 degree at α = 0, in the present model setting r₁ ≅ r₂. Even when r₁ > r₂, the divergence angle was calculated as Δθ_2,3 = 113 degree (r₁ = R₀+2Vτ, r₂ = R₀+Vτ, R₀ = 1, Vτ = 0.14, and λ_ini = 0.05 estimated from the S. coeli-rosa SEM image [34]; see S4 Fig for detail), which is still less than the observed value. As α became larger, the inhibition from the second primordium became stronger than that from the first one, making Δθ_2,3 consistent with the observed value in S. coeli-rosa (Fig 4E, top).

For the subsequent sepals and petals, the model faithfully reproduced the period-five oscillation of the divergence angle and the plastochron ratio until the ninth primordium (Fig 4E), notably in the deviation of the divergence angle from regular pentagon (144 degree) and the increase of plastochron ratio at the boundary between the sepal and petal whorls. Moreover, a similar increase in the plastochron ratio occurred weakly between the second and third primordia in the first whorl (closed arrowhead in Fig 4E), indicating a hierarchically whorled arrangement (i.e., whorls within a whorl). Such weak separation of the two outer primordia from the three inner ones within a whorl is consistent with the quincuncial pattern of sepal aestivation that reflects spiral initiation in many of eudicots with pentamerous flowers (e.g., Fig 2D–E in [21]). Even with an identical set of parameters, the order of initiation in the first pentamerous whorl can vary depending on the stochasticity in the growth process. The variations of the initiation order in simulations may be caused by the absence of the outer structure, because the axillary bud seems to act as a positional information for the first primordia in S. coeli-rosa floral development (Fig 4B). The positioning of the five primordia in the first whorl was reproducible in 70% of the numerical replicates, within less than 20 degrees of that in S. coeli-rosa or that of the angles in a regular pentagon. Mismatches in the inner structure (from the tenth primordium, i.e., the last primordium in petal whorl) might be due to an increase in the rate of successive primordia initiation later in development [35], which we did not assume in our model. The agreements between our model and actual S. coeli-rosa development of sepals and petals in both the angular and the radial positions suggests that the S. coeli-rosa pentamerous whorls are caused by decreasing inhibition from older primordia.

Mechanism for the tetramerous whorl emergence

A possible mechanism to arrest the radial displacement of a new primordium, a key process for whorl formation (arrowheads in Fig 2D and 2F), involves an inward-directed gradient of the growth potential U_{g, k} (Eq 3) of a new primordium so that its radial movement is prevented. To confirm this for tetramerous whorl formation (Fig 3A), we analytically derived the parameter region such that the radial gradient of the growth potential at the angle of the fifth primordium U_g,5 (Eq 3), which is determined by the positions of the preceding four primordia, is inward-directed. For ease in the analytical calculation, we set α = 0 and P_MP = 0. The first four primordia positions were intuitively estimated (see S2 Text) as (5) which agreed with the numerical results with an error of less than several percent regardless of the parameter spaces. Hereafter we demonstrate a case Vτ = 6.0. The position of the fifth primordium derived from the positions of four existing primordia (Eq 5) becomes θ₅ = 90 when R₀ ≤ 2, whereas θ₅ ∼ 135 when R₀ > 2 (S5 Fig). Next, we calculated the potential for the fifth primordium in radial direction by substituting Eq 5 and the position of the fifth primordium θ₅ into Eq 3. The function becomes (6) The potential exhibits a unimodal (2 < R₀ < 10; Fig 5A) or bi-modal (R₀ < 2, R₀ > 10; Fig 5B and 5C) shape. At R₀ < 10, the potential gradient at the initiation position of the fifth primordium ∂U_g,5(r, θ₅)/∂r∣_{r = R0} is outward-directed (Fig 5A), providing almost constant growth resulting a non-whorled arrangement in the simulations (Fig 3A, red region). At R₀ > 10, we defined the radial position of the local maximum closest to the fifth primordium as r_max (open arrowhead in Fig 5B and 5C; red squares in the upper half of Fig 5D) and the local minimum as r_min (blue circles in Fig 5D; 0 < r_min < r_max). The potential gradient ∂U_g,5(r, θ₅)/∂r∣_{r = R0} has a negative value when R₀ < r_min or r_max < R₀ (Fig 5C), causing the fifth primordium to constantly move outward. On the other hand, the potential gradient is positive, i.e., directed inward (Fig 5B), when r_min < R₀ < r_max (between the two solid arrowheads in Fig 5D), causing the arrest of radial movement of the fifth primordium. The values of r_min and r_max, analytically calculated as function of R₀ and τ (solid black line in Fig 5E), were faithfully consistent with the parameter boundaries between the non-whorled pattern and the tetramerous-whorled pattern and between the tetramerous-whorled and pentamerous-whorled patterns, respectively, in the numerical simulations (Fig 5E). The assumption r₁ = r₂ (Eq 5) according to our numerical results (Fig 2D), which is a similar setup to co-initiation of two primordia, is not a necessary condition for consistency (S6 Fig). Thus the inward-directed gradient of the growth potential (Eq 3), which works as a barrier to arrest the outward displacement of the fifth primordium, causes the formation of tetramerous whorl.

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Fig 5. The potential landscape captures tetramerous whorl formation.

A–C. Color-coded potential landscape (upper panel; legend) and the section (bottom panel) at the angle where U_ini takes the global minimum so that the fifth primordium arises (white dashed line in upper panel; Eq 6). The green line shows the meristem edge with a diameter of R₀. The direction of the potential at the position where the fifth primordium arises, denoted by the red circle, is inward in B but outward in A and C (bottom panel). R₀ = 5.0 (A), 15.0 (B), 45.0 (C). D. Radial positions that take the local minima (r_min, blue circles) and maxima (r_max, red squares) of potential U_g,5 (Eq 6). Between R₀ = r_min and r_max, indicated by the two arrowheads, the potential at the meristem edge decreases inward as in B. Black diamonds correspond to the initiating position of the fifth primordium of A–C. Vτ = 6.0, σ_θ = 0.0, λ_ini = λ_g = 10.0, and P_MP = 0 in A–D. E. Superposition of the analytical result onto the numerical results (Fig 3A). Solid lines show the crossovers r_min = R₀ and r_max = R₀, respectively (arrowheads in D). λ_ini = λ_g = 10.0 and P_MP = 0. σ_r = σ_θ = 0.05 for numerical result, σ_θ = 0.0 for analytical result.

https://doi.org/10.1371/journal.pcbi.1004145.g005

Mechanism for the pentamerous whorl emergence

The inward radial gradient of the potential U_{g, k} (Eq 3) also accounted for the emergence of pentamerous whorls at α > 0. Unlike the case of α = 0, the angular position of the third primordium θ₃ at the global minimum of U_ini decreases from 90 degrees as α increases (Fig 6A). For example, the recursive calculations for the minimum of U_ini gave the angular positions of the two subsequent primordia, θ₃ ≅ 62 and θ₄ ≅ 267, respectively, at α = 2.0 (Vτ = 6.0, R₀ = 20.0, and P_MP = 0). Those angular positions were consistent with the numerical results (e.g., Fig 2F and S2B Fig). The gradient of the growth potential ∂U_g,5(r, θ₅)/∂r at the edge of the meristem for the fifth primordium that arises at θ₅ ≅ 129 is negative (Fig 6B). Therefore, the fifth primordium moves outward at constant velocity so that the tetramerous whorl is unlikely to emerge. The inward-directed potential at the position of the new primordium first appears when the sixth primordium arises around 343 degrees, which was derived by the recursive calculation (Fig 6C). The first primordium (the rightmost potential peak in Fig 6C) prevents the outward movement of the sixth primordium (red circle in Fig 6C). Arrest of radial displacement of the sixth primordium is maintained until the seventh primordium arises to allow the radial gap between these primordia to appear (i.e., a pentamerous whorl emerged). After the appearance, the growth potential gradients of the sixth and the seventh primordia become outward-directed, providing their constant growth with keeping the radial gap to the first whorl. Likewise, the other merosities can be explained by similar recursive calculations of the angular position from the initiation potential (Eq 2) and the radial gradient of the growth potential (Eq 3).

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Fig 6. The potential landscape captures pentamerous whorl formation.

A. The angular position of the third primordium as a function of α. B–C. Color-coded growth-potential landscape (top) and the section (bottom) at the angle where the fifth (B) and the sixth (C) primordia arise (white dashed line in the top panel). α = 2.0, Vτ = 6.0, R₀ = 20.0, σ_θ = 0.0, λ_ini = λ_g = 10.0, and P_MP = 0 in A–C.

https://doi.org/10.1371/journal.pcbi.1004145.g006

Relevance of the whorled arrangement to the phyllotactic parameter G

Based on these analytical results (Figs 5 and 6) and the dimensionless parameter G = τV/R₀, which represents the natural logarithm of the average plastochron ratio [49, 75], we quantitatively compared the present model against previous phyllotaxis models assuming simultaneous initiation based on the initiation potential [19]. The tetramerous and pentamerous whorls appeared in, at most, 1.3-fold and 1.2-fold ranges of G, respectively, in the earlier study (Fig 4D in [19]); however, they appeared in much wider ranges in our model (i.e., 3-fold to 5-fold and 1.2-fold to 5-fold ranges of G, respectively; Fig 3C). Here, another key parameter is the temporal decay rate of the initiation inhibition α that shorten the transient process approaching to the golden angle (Fig 6A) than those of spiral phyllotaxis [32, 33]. λ_ini, representing the gradient of the initiation potential (Eq 2), little affected the border between the whorled and non-whorled arrangements at α = 0 (Fig 7A and 7B); λ_ini affected the border only when α ≠ 0 (Fig 7C and 7D). The independency of λ_ini at α = 0 is consistent with the result shown by the previous model, which did not incorporate temporal decay of the potential and indicated that the phyllotactic pattern depends little on the functional type of initiation potential [49]. On the other hand, the gradient of the growth potential (Eq 3) regulated by λ_g caused a drastic transition between the whorled and non-whorled arrangements (Fig 7E and 7F). Unlike G, λ_ini, and α (Fig 6A), λ_g hardly affects the angular position, as demonstrated in the previous sections, but it controls how far the growth potential works as a barrier to determine the merosities of the whorls (Fig 7E and 7F). Thus, λ_g, α, and G differentially regulate phyllotaxis of the floral organs, suggesting the involvement of distinct molecular or physiological underpinnings.

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Fig 7. Effects of λ_ini, λ_g, and α on merosities.

Superposition of the analytical result (the solid lines are identical to Fig 5E: λ_ini = λ_g = 10.0, α = 0.0, P_MP = 0) and the numerical result (σ_r = σ_θ = 0.05; the following parameters are different from the solid line: A. λ_ini = 5.0, B. λ_ini = 20.0, C. α = 2.0 D. α = 2.0, λ_ini = 20.0, E. λ_g = 5.0, F. λ_g = 20.0). The colors follow Fig 3. λ_g and α affect the boundary lines between each whorl as well as that between non-whorls and tetramerous whorls (C, E and F), whereas λ_ini hardly affects at α = 0 (A and B). At α ≠ 0, α and λ_ini synergistically affect the phase boundaries (C and D).

https://doi.org/10.1371/journal.pcbi.1004145.g007

Predictions

We have seen that both the mutual repulsion of growth regulated by λ_g and the temporal decay of initiation inhibition controlled by α are responsible for the formation of tetramerous and pentamerous whorls following sequential initiation. These mechanisms can be experimentally verified by tuning λ_g and α. Here, we discuss several candidates for the molecular and physiological underpinnings.

Future problems

Future studies should also clarify the limits and applicability of the common developmental principle elucidated here by exploring more complex development in a wide variety of flowers. Because our model assumes sequential initiation of the primordia, it does not cover the floral development of all eudicots; sepal primordia arise simultaneously in some eudicot clades (Fig 1; e.g., mimosoid legume [94]). Likewise, in later development, several primordia arise at once in the stamen and carpel whorls (e.g., Ranunculaceae [35]). The transitions between simultaneous and sequential development have two additional intriguing implications for evolutionary developmental biology. First, the initiation types may affect the stochastic variation of floral organ numbers, possibly caused by the absence or presence of pseudo-whorls (Fig 1) and the noisy expression domain of homeotic genes [95]. Second, such transitions occur even in animal body segmentation [3, 4], possibly caused by evolution of both gene regulatory network topologies and embryonic growth [7, 9–11]. The limitations of the model can be reduced by introducing initiation whenever and wherever the potential (Eq 2) is below a threshold, allowing simultaneous as well as sequential initiation [19]. The threshold model exhibiting both types of initiation does not by itself result in the dominance of particular merosities [19]. Incorporating two mechanisms, mutual growth repulsion and temporally decreasing inhibition at the point of initiation, into the threshold model could explain the dominance of particular merosities following both the sequential and the simultaneous initiation of floral organ primordia (Fig 1). Another problem is the absence of trimerous whorls in the present model (Fig 3). The transition between the trimery and tetramery or pentamery, and vice versa, occurred multiple times during the evolution of angiosperms. Therefore, trimerous flowers are scattered across the basal angiosperms, monocots, and a few families of eudicots [96, 97]. Elucidating the developmental mechanisms underlying the transitions between the different merosities, as well as those between sequential and simultaneous initiation, will be an important avenue for future studies.

Conclusion

One problem in determining floral organ number is how to generate whorls comprised of a specific number of organs. By introducing a growth assumption (i.e., continuous repulsion among primordia throughout development, which was originally proposed as the contact pressure model [25, 45, 46] and is supported by experimental observations [42]) into a dynamical model of phyllotaxis [49], we showed that the whorled arrangement arises spontaneously from sequential initiation. Moreover, when we allowed the inhibition to decay over time [33, 47, 48], pentamerous whorls became the dominant pattern. The merosity tended to be four or five in much larger parameter spaces than those in which it tended to be six or seven. The emergence of tetramerous and pentamerous whorls could be verified experimentally by tuning the two parameters α and λ_g.