Plant leaves are arranged around the stem in a beautiful geometry that is called phyllotaxis. In the majority of plants, phyllotaxis exhibits a distichous, Fibonacci spiral, decussate, or tricussate pattern. To explain the regularity and limited variety of phyllotactic patterns, many theoretical models have been proposed, mostly based on the notion that a repulsive interaction between leaf primordia determines the position of primordium initiation. Among them, particularly notable are the two models of Douady and Couder (alternate-specific form, DC1; more generalized form, DC2), the key assumptions of which are that each leaf primordium emits a constant power that inhibits new primordium formation and that this inhibitory effect decreases with distance. It was previously demonstrated by computer simulations that any major type of phyllotaxis can occur as a self-organizing stable pattern in the framework of DC models. However, several phyllotactic types remain unaddressed. An interesting example is orixate phyllotaxis, which has a tetrastichous alternate pattern with periodic repetition of a sequence of different divergence angles: 180°, 90°, −180°, and −90°. Although the term orixate phyllotaxis was derived from Orixa japonica , this type is observed in several distant taxa, suggesting that it may reflect some aspects of a common mechanism of phyllotactic patterning. Here we examined DC models regarding the ability to produce orixate phyllotaxis and found that model expansion via the introduction of primordial age-dependent changes of the inhibitory power is absolutely necessary for the establishment of orixate phyllotaxis. The orixate patterns generated by the expanded version of DC2 (EDC2) were shown to share morphological details with real orixate phyllotaxis. Furthermore, the simulation results obtained using EDC2 fitted better the natural distribution of phyllotactic patterns than did those obtained using the previous models. Our findings imply that changing the inhibitory power is generally an important component of the phyllotactic patterning mechanism.

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The view is widely accepted that the inhibitory effect of existing leaf primordia on new primordium formation determines phyllotactic patterning. Previous studies have shown that mathematical models based on such inhibitory effect can generate most of phyllotactic patterns. However, a few types of phyllotaxis still remain unaddressed. A notable example is costoid phyllotaxis showing spiromonostichy, which is characterized by a steep spiral with a small divergence angle and is unique to Costaceae plants. Costoid phyllotaxis has been called a "genuine puzzle" because it seems to disagree with the inhibitory effect-based mechanism. In an attempt to produce a steep spiral pattern, we developed a new mathematical model assuming that each leaf primordium emits not only the inhibitory effect but also some inductive effect. Computer simulations with the new model successfully generated a steep spiral pattern when these two effects met a certain relationship. The obtained steep spiral matched the real costoid phyllotaxis observed with Costus megalobractea. We also found by the mathematical model analysis that the early phyllotactic transition in the seedlings of this plant can be explained by the SAM enlargement.

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