A new mathematical model of phyllotaxis to solve the genuine puzzle spiromonostichy

Arrangement of plant leaves around the stem, termed phyllotaxis, exhibits beautiful and mysterious regularities and has been one of the most attractive subjects of biological pattern formation. After the long history of studies on phyllotaxis, it is now widely accepted that the inhibitory effect of existing leaf primordia on new primordium formation determines phyllotactic patterning. However, costoid phyllotaxis unique to Costaceae of Zingiberales, displaying spiromonostichy characterized by a steep spiral with a small divergence angle, seems to disagree with the inhibitory effect-based mechanism and has remained as a “genuine puzzle”. We developed a new mathematical model, hypothesizing that each leaf primordium exerts not only the inhibitory effect but also some inductive effect. Computer simulations with the new model successfully generated a spiromonostichous pattern when these two effects met a certain relationship. The obtained spiromonostichy matched the real costoid phyllotaxis observed with Costus megalobractea, particularly for the decrease of the divergence angle associated with the enlargement of the shoot apical meristem. The new model was also shown to be able to produce a one-sided distichous pattern that is seen in phyllotaxis of a few plants of Zingiberales and has never been addressed in the previous model studies. These results implicated inductive as well as inhibitory mechanisms in phyllotactic patterning, at least in Zingiberales. Supplementary Information The online version contains supplementary material available at 10.1007/s10265-023-01503-2.


Negative relationship between the SAM size and the divergence angle
From Eqn S1 and Eqn S2, we get the following equation: Eqn 18 of the main text is derived from Eqns S7.As considered above, the standardized plastochron  * can be determined independently of the meristem size  0 by solving Eqn 17 numerically.Therefore, Eqn 18 simply shows a negative relationship between the SAM size and the divergence angle, that is,  * becomes smaller when  0 becomes larger under the condition that all other parameters are fixed to constant values (Fig. 3c).

Comparison between the results of theoretical analysis and of computer simulations
In Fig. 3b, the solutions of  * and  * are plotted, if they exist, against   , and are compared with  and the divergence angle of the patterns generated by the new model simulations with varying   (Fig. 3b).This comparison demonstrates that the range of   where the solutions of  * and  * exist is almost correspondent to the   range where the new model produced the costoid or one-sided distichous pattern and that, in this range,  * and  * fit well to the  value and the divergence angle of the model-generated patterns, respectively.The good agreement validates our theoretical analysis with the simplified situation regarding the requirements for the generation of costoid or one-sided distichous pattern.Now the following explanation can be deduced.When   is enough small, the induction range always encompasses the inhibition range, and in such case, common types of phyllotaxis are generated.When   is moderately larger, the induction range is encompassed by the inhibition range at first and later it expands beyond the inhibition range somewhere on , and in such case, costoid phyllotaxis or one-sided distichous patterns are generated.When   is too large, the induction range is always encompassed by the inhibition range, and such case can never produce new primordia.
More accurately, however, there are some differences between the solutions obtained from theoretical analysis of the simplified situation and the simulation results.The  * values are slightly smaller than the  values (upper panels of Fig. 3b), which should reflect the small effect from the second youngest primordium and/or older primordia.Moreover, the   range generating costoid or one-sided distichous patterns in the simulation is a little different from the   range for the existence of the solutions of  * and  * (Fig. 3b).Such discrepancy is seen in the range of   where  * is smaller than   , the  value calculated from the simulation without considering the inductive effect (yellow zone in Fig. 3b), which is also attributable to the effect of the second youngest primordium and/or older primordium.

Fig. S2. Phyllotactic spiral directions in C. megalobractea seedlings
The upper panel shows a photograph of 37 days-after-sowing seedlings of C. megalobractea and the lower panel indicates their phyllotactic spiral directions.Cot, L, and R represent cotyledon, left-handed, and right-handed, respectively.Among these seedlings, 9 had a lefthanded spiral while 12 had a right-handed spiral.The bias in the spiral direction from the 1:1 ratio is not significant (p = 0.67, binomial test).

Cot
Cot

Fig. S8. Computer simulations with the new model over a wide range of combinations of seven parameters focusing on the inhibitory effect (3)
Computer simulations were performed using the new model under various settings of six parameters: 51 settings for   (0 ≤   ≤ 20), 51 settings for   (−1 ≤   ≤ 1), 9 settings for   and   (   ,   = 1.5, 2 , 2, 2 , 2.5, 2 , 2, 2.5 , 2.5, 2.5 , 3, 2.5 , 2.5, 3 , 3, 3 , or 3, 3.5 ), 4 settings for   (  = 0, 1, 5, or 10), and 3 settings for   (  = 0, −0.5, or 0.5).,   , and   were fixed to 1/3, 2, and 3, respectively.The patterns obtained are displayed in the     space according to the color legend shown in 1 3,   = 4,   = 2,   = 3.5,   = 3,   = 20,   = 0.64,   = 10, and   = 0. l-q, One-sided distichous phyllotaxis generated by the new model under  = Τ 1 3,   = 3,   = 3,   = 3.5,   = 3,   = 20,   = 0.52,   = 10, and   = 0.The contour maps 0.1 standardized time unit before (a, f, l), immediately before (b, g, m), and immediately after a new primordium arises (c, h, n) are shown.The blue, brown, and pink areas represent the region where the inductive field strength exceeds a given threshold, the region where the inhibitory field strength exceeds a given threshold, and the region where both the inductive and inhibitory field strengths exceed given thresholds, respectively.The inductive (blue) and inhibitory (red) field strengths on the circle  0.1 standardized time unit before (d, i, o) and immediately before (e, j, p) a new primordium arises are shown with angles as abscissa.Dashed line indicates the threshold level (field strength = 1), and arrows point the positions which a new primordium can arise, that is, the inductive field strength exceeds a given threshold while the inhibitory field strength falls below a given threshold.In j and p, there are two positions where the inductive and inhibitory field strengths almost equivalent and also close to the thresholds.Which of these positions is chosen for the site of new primordium formation is determined by the effect of the second youngest primordium P2. k and q show only the inductive (blue) and inhibitory (red) field strengths derived from P2.The vertical dotted lines indicate the substructions between the inductive and inhibitory effects of P2 at the positions where the inhibitory field strength is equivalent to its given threshold.In k, as the angle-dependent change of the P2 effect is larger in inhibition than in induction on the circle  due to   >   , the position far from P2 receives a smaller inhibitory effect from P2 than the position close to P2 and is therefore chosen as the site of new primordium formation, resulting the generation of costoid phyllotaxis.In q, as the angle-dependent change of the P2 effect is larger in induction than in inhibition on the circle  due to   ≤   , the position close to P2 receives a larger inductive effect from P2 than the position far from P2 and is therefore chosen as the site of new primordium formation, resulting the generation of one-sided phyllotaxis phyllotaxis.When   ≤   (a), the effect of P2 is more "inductive" and P0 arises at the side close to P2, leading to one-sided distichy.When   >   (b), the effect of P2 is more "inhibitory" and P0 arises at the side far from P2, leading to costoid phyllotaxis.
−  ( * −  ) = 1.(S1)On the other hand, ( * ) is calculated as the modified Euclidian distance between ( 0 indicate the relationships to be satisfied by  * and  * .Numerical determination of the standardized plastochronBy eliminating ( * ) , Eqn 17 of the main text is derived from Eqn S1.The standardized plastochron  * is obtained as the solution of Eqn 17.It is important to note that the  * is determined independently of the meristem size  0 and the shape  and is dependent on the parameters characterizing the inhibitory and inductive effects.Eqn 17 cannot be solved analytically for  * .Before numerical solution, let us consider how many solutions Eqn 17 has from the extremum points of (

Fig
Fig. S1.Color legend for the phyllotactic patterns generated in computer simulations.Phyllotactic patterns generated in computer simulations were classified into four categories: alternate patterns with a constant divergence angle or a two-cycle change in the divergence angle, tetrastichous alternate patterns with a four-cycle change in the divergence angle, whorled patterns, and other patterns.Whorled patterns were further classified into decussate, tricussate, and other whorled patterns.Different patterns are displayed by different colors in the parameter space of the model used for computer simulations.For regular alternate patterns with a constant divergence angle, the divergence angle is indicated by a color hue from cyan (0°) to red (180°).In the case of alternate patterns with a two-cycle divergence angle change with a constant absolute value of the divergence angle, the color hue is assigned for the absolute value of the divergence angle.In the case of other alternate patterns with a two-cycle divergence angle change, the color hue is assigned for the mean absolute value of the successive divergence angles.In these two-cycle alternate patterns, small-to-large ratios of two successive plastochrons and two successive divergence angles are represented by lightness (full lightness for 0) and saturation (full saturation for 1), respectively.Tetrastichous alternate patterns with a four-cycle divergence angle change are similarly expressed by color lightness 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 are used to calculate the ratio of divergence angles.As the divergence angle of this type of alternate pattern changes in the sequence of , , −, and − (−180°< ,  ≤ 180°), ||/|| gives the ratio of the absolute values of divergence angles if  >  .When no new primordia were formed in simulation, it is indicated by the saltire mark (×).Typical examples of phyllotactic patterns are marked with circled numbers in the color legend and their schematic diagrams are shown at the bottom; 1. distichous, 2. Fibonacci spiral, 3. Lucas spiral, 4. semi-decussate, 5. decussate, 6. orixate, 7. costoid, and 8. one-sided distichous.The color assignment is modified from Yonekura et al., (2019).
Fig. S10.Difference of the induction and inhibition ranges between conditions that produce one-sided distichous and costoid phyllotaxesSchematic views of the difference of the induction and inhibition ranges between parameter conditions that produce one-sided distichous and costoid phyllotaxes at the time of the initiation of a new primordium (P0).Blue and red lines show the induction and inhibition ranges, respectively.Dashed lines indicate the induction/inhibition ranges of P1 without considering the effects of the older primordium, whereas solid lines indicate the induction/inhibition range considering the effect of P2.When   ≤   (a), the effect of P2 is more "inductive" and P0 arises at the side close to P2, leading to one-sided distichy.When   >   (b), the effect of P2 is more "inhibitory" and P0 arises at the side far from P2, leading to costoid phyllotaxis.