Supplementary material from "The unified rule of phyllotaxis explaining both spiral and non-spiral arrangements"

Leaf-like appendages of different plant groups are arranged in common phyllotaxis patterns categorized into two types: spiral and non-spiral arrangements. The adaptive reason for this morphological convergence is unknown. In the non-spiral arrangement, divergence angle between successive leaves is a simple fraction of 360°, e.g. distichy, decussate and whorled phyllotaxis. In the spiral arrangement, the divergence angle of nascent leaves at the shoot apex is fixed at the golden angle 137.5°, whereas those of the developed leaves varies within a sequence of Fibonacci fractions, such as 1/3, 2/5, 3/8, 5/13, etc. The optimality of the golden angle has been shown recently by assuming that the pattern of developed leaves varies depending on the divergence angle of nascent leaves. Here we propose a unified rule of phyllotaxis to explain both types of arrangement: the developed leaves form vertical rows along the stem. In the non-spiral arrangement, nascent to developed leaves always follow this rule, so that the number of leaf rows is kept constant irrespective of stem growth. In the spiral arrangement, developed leaves attain this rule by adjusting the divergence angle from the golden angle. The spiral arrangement is adaptive in that the number of leaf rows varies during ontogeny depending on shoot thickness.