Fig 1.
Occurrence of plants with orixate phyllotaxis in the angiosperm phylogeny.
Plants with orixate phyllotaxis and their positions in the order-level phylogenetic tree of angiosperms based on Angiosperm Phylogeny Poster [33].
Fig 2.
Schematic views of the shoot apex with coordinates in DC models.
The shoot apex is considered as a plane in DC1 (A) and as a cone in DC2 (B).
Fig 3.
Color legend for the phyllotactic patterns generated in computer simulations.
The phyllotactic patterns generated in computer simulations were classified into an alternate pattern with a constant divergence angle or a two-cycle change in the divergence angle; a tetrastichous alternate pattern with a four-cycle change in the divergence angle; a whorled pattern; and other patterns. Whorled patterns were further classified into decussate (“opposite phyllotaxis” typified by true decussate), tricussate, and other whorled patterns. These patterns were distinguished using different colors. For regular alternate patterns with a constant divergence angle, the divergence angle was indicated by a color hue from cyan (0°) to red (180°). In the case of alternate patterns with a two-cycle divergence angle change, the color hue was assigned for the mean value of the successive divergence angles. In these two-cycle alternate patterns, small-to-large ratios of two successive plastochron times and two successive divergence angles were represented by lightness (full lightness for 0) and saturation (full saturation for 1), respectively. Tetrastichous alternate patterns with a four-cycle divergence angle change were similarly expressed by color brightness and saturation based on their ratios of plastochron times and divergence angles; however, instead of the divergence angles themselves, the absolute values of divergence angles were used to calculate the ratio of divergence angles. As the divergence angle of this type of alternate pattern changes in the sequence of p, q, −p, and −q (−180°<p,q≤180°), |q|/|p| gives the ratio of the absolute values of divergence angles if |p|>|q|. Typical examples of phyllotactic patterns are marked with circled numbers in the color legend and their schematic diagrams are shown at the bottom.
Fig 4.
Orixate phyllotaxis in the apical winter buds of Orixa japonica.
(A) Transverse section. O points to the summit of the SAM, and leaf primordia are designated as P1, P2, P3, etc., with P1 being the youngest visible primordium. Black lines represent orthostichies drawn by joining the gravity centers of leaf primorida and O. The four orthostichy lines can be roughly approximated by two orthogonal lines (pale gray broad lines). (B) Longitudinal section. I1 indicates the incipient primordium. (C) Scanning electron microscopic image. (D) Divergence angles measured using the transverse sections. Divergence angles close to 180° show opposite positioning of the successive primordia (blue), while angles near 90° or 270° show adjacent positioning (yellow). (E) The natural logs of plastochron ratios OP2/OP1 and OP3/OP2 are plotted based on whether the two primordia are located in an adjacent or opposite position. In (D) and (E), points linked by a line represent data from the same sample, and red points indicate data obtained from the section of (A).
Fig 5.
Phyllotactic patterns generated in computer simulations using DC1.
(A) Computer simulations using DC1 were performed under various settings of parameters G and η (101×101 conditions), and the patterns obtained are displayed according to the color legend shown in Fig 3. (B) The black and red dots indicate the absolute values of divergence angles of the tetrastichous alternate patterns generated in (A) and real orixate phyllotaxis observed for winter buds of O. japonica (data of P1~P2 and P2~P3 in Fig 4D), respectively. The blue dots show the averages determined from the real data of P1~P2 to P6~P7 (Fig 4D) for each winter bud of O. japonica. In this panel, alternate patterns with a four-cycle change in divergence angles in the sequence of p, q, −p, and −q (|p|>|q|) were plotted at the point (|p|,|q|). (C) An example of the tetrastichous alternate patterns, which was produced by computer simulation at G = 0.3 and η = 1.5. This pattern has a divergence angle change in the sequence of 165°, −91°, −165°, and 91° and, unlike orixate phyllotaxis, exhibits a distorted tetrastichy, rather than an orthogonal tetrastichy.
Fig 6.
Mathematical and computer simulation analysis of EDC1.
(A) Numerical solutions of parameters that fit the mathematical requirements for normal orixate phyllotaxis in EDC1. The two curves show the solutions obtained using various G values. The closed circles indicate the solutions obtained with G set at 0.1 intervals between 0.1 and 1.0. (B) Stable patterns generated in computer simulations using EDC1 under various parameter settings (201 settings for a, 101 settings for b, 3 settings for G, and thus 201×101×3 = 60,903 simulations in total). The patterns obtained are displayed according to the color legend shown in Fig 3. The white crosses (+) indicate the parameter conditions obtained as numerical solutions of , giving a minimum of I(θ) around θn−4i for normal orixate phyllotaxis, whereas white saltires (×) indicate the parameter conditions obtained as numerical solutions of
giving a maximum of I(θ). (C) Schematic diagrams of typical examples of the phyllotactic patterns generated in the computer simulations. The circled numbers relate the diagrams to the parameter conditions shown in (B).
Fig 7.
Phyllotactic patterns generated in computer simulations using DC2.
(A) Computer simulations using DC2 were performed under various settings of parameters α and Γ (101 settings for α and 101 settings for Γ) with N = 1,1/3, or 1/5, and the resultant patterns are displayed for the cases of N = 1 and 1/3 according to the color legend shown in Fig 3. Simulations were started by placing a single primordium or two primordia at a central angle of 120° on the SAM periphery. (B) The regular alternate, two-cycle alternate, and tetrastichous four-cycle alternate patterns generated in computer simulations using DC2 in (A), including simulations with N = 1/5 as well as N = 1 and 1/3, were plotted using the ratio of absolute values of two successive divergence angles as the abscissa and the ratio of two successive plastochron times as the ordinate. The red dots indicate tetrastichous four-cycle patterns, while the black dots indicate regular alternate and two-cycle patterns. The blue dots show the data of real orixate phyllotaxis observed for winter buds of O. japonica (calculated from the data of P1~P2 and P2~P3 in Fig 4D). (C) Magnification of the lower-left corner of (B).
Fig 8.
Phyllotactic patterns generated in computer simulations using EDC2.
(A) Computer simulations using EDC2 were performed under various parameter settings (201 settings for −20≤A≤20, 101 settings for 0≤B≤1, and Γ = 1, 2, or 3) with fixed parameters α = 1 and N = 1/3, and the patterns obtained are displayed according to the color legend shown in Fig 3. Simulations were started by placing a single primordium on the SAM periphery. (B) Computer simulations using EDC2 were performed under various settings of parameters (101 settings for 0≤A≤20, 101 settings for 0≤B≤1, and Γ = 2, 2.5, or 3) with fixed parameters α = 1 and N = 1/3. The graph shows a scatter plot of alternate patterns with a constant divergence angle or a two-cycle change in the divergence angle (black), and tetrastichous alternate patterns with a four-cycle change in the divergence angle (red) generated in the computer simulations. In this graph, each pattern was plotted based on the ratio of absolute values of two successive divergence angles (abscissa) and the ratio of plastochron times (ordinate). The black dots surrounded by an orange circle represent semi-decussate-like patterns that occurred in the vicinities of orixate phyllotaxis in the parameter space, which are indicated by blue asterisks in S6A Fig. The blue dots indicate the data of real orixate phyllotaxis observed for winter buds of O. japonica (calculated from the data of P1~P2 and P2~P3 in Fig 4D).
Fig 9.
Effects of the inhibition range and increase in inhibitory power on phyllotactic patterns in EDC2.
Computer simulations were performed using EDC2 with α = 1, 2, or 4 under various settings of Γ and A (101×101 conditions), which reflect the maximum inhibition range of a primordium and the primordial age-dependent increase in the inhibitory power, respectively. The initial value of the inhibitory power was fixed to 0.047, i.e., A×B was fixed at 3. N was fixed at 1/3. The simulation was started by placing a single primordium on the SAM periphery. The patterns obtained are displayed according to the color legend shown in Fig 3.
Fig 10.
Characteristics of orixate patterns generated in computer simulations using EDC2.
(A) Contour map of the natural log of the inhibitory field strength I within the shoot apical region that generated orixate phyllotaxis in the computer simulation using EDC2. A value of 0 implies that the inhibitory field strength is equal to the threshold for primordium formation. (B) Relationship between plastochrons and divergence angles in orixate patterns generated in computer simulations using EDC2. For a pair of successive primordia, Lm and Lm+1, a standardized plastochron was calculated as tm+1−tm = ln(rm/rm+1). Orixate patterns were plotted based on their two standardized plastochrons: one for the pair of opposite primordia with a divergence angle of approximately 180°, and the other for the pair of adjacent primordia with a divergence angle of approximately ±90°.
Fig 11.
Schematic explanation of two conditions that enable orixate phyllotaxis formation.
(A) Gradual increase of the inhibitory power with a relatively small size of SAM. (B) Sudden decrease of the inhibitory power with a relatively large size of SAM. EDC1 can establish orixate phyllotaxis under either of these conditions while EDC2 can only under the former condition.