# Decoding the Mathematical Secrets of Plants’ Spiraling Leaf Patterns

Plant leaves are arranged in a beautiful geometric pattern around the stem, which is known as phyllotaxis. Phyllotaxis has common characteristics across plant species, which are commonly mathematically characterized and expressed in a small number of phyllotactic patterns.

One important premise in the study of phyllotaxis, or leaf patterns, is that leaves guard their personal space. Scientists have built models that can effectively reproduce many of nature’s typical patterns, based on the concept that existing leaves have an inhibitory influence on emerging ones, emitting a signal to prevent others from sprouting close. The enthralling Fibonacci sequence, for example, may be seen in anything from sunflower seed arrangements to nautilus shells to pine cones. The current agreement is that such patterns are caused by the motions of the growth hormone auxin1Auxins are a class of plant hormones (or plant-growth regulators) with some morphogen-like characteristics. Auxins play a cardinal role in coordination of many growth and behavioral processes in plant life cycles and are essential for plant body development. and the proteins that carry it throughout a plant.

Alternate phyllotaxis is a leaf arrangement with one leaf per node, whereas whorled phyllotaxis is a leaf arrangement with two or more leaves per node. Distichous phyllotaxis (bamboo) and Fibonacci spiral phyllotaxis – spiral with a divergence angle close to the golden angle of 137.5° (the succulent spiral aloe) are common alternate forms, while decussate phyllotaxis (basil or mint) and tricussate phyllotaxis (the succulent spiral aloe) are common whorled types (Nerium oleander, sometimes known as dogbane).

Certain leaf configurations, on the other hand, continue to confound standard plant growth models. This prompted researchers at the University of Tokyo to reconsider some phyllotactic models:

“In most plants, phyllotactic patterns have symmetry—spiral symmetry or radial symmetry,” says University of Tokyo plant physiologist Munetaka Sugiyama, senior author of the new study. “But in this special plant, Orixa japonica, the phyllotactic pattern is not symmetric, which is very interesting. More than 10 years ago, an idea came to me that some changes in the inhibitory power of each leaf primordium may explain this peculiar pattern.”

The divergence angles, or angles between consecutive leaves, are used by botanists to determine a plant’s phyllotaxis. The O. japonica shrub, which is endemic to Japan and other parts of East Asia, grows leaves in an alternating series of four angles: 180 degrees, 90 degrees, 180 degrees again, and 270 degrees. This pattern, named “orixate” phyllotaxis by the researchers, is not unique to this plant; other plants alternate their leaves in the same intricate sequence.

By adding the ages of the leaves as another variable to the Douady and Couder equations2The Douady and Couder equations equation make the fundamental assumption that leaves emit a constant signal to inhibit the growth of other leaves nearby and that the signal gets weaker at longer distances., the team created a novel model. Former models assumed that the inhibitory efficacy of leaves remained constant throughout time, but this was “not natural from a biological standpoint,” according to Sugiyama. Instead, Sugiyama’s team assumed that the potency of these “keep-away” signals could alter over time. The resulting models succeeded in recreating, through computerized growth, the intricate leaf arrangements of O. japonica.

All of the other frequent leaf patterns were likewise produced by the enlarged equations, which predicted the natural frequencies of these variations more accurately than earlier models. The new EDC2 model anticipated the Fibonacci spiral’s “super-dominance” over other arrangements, especially in spiral-patterned plants, but earlier models failed to explain why this particular shape appears everywhere in nature.

“Our model, EDC2, can generate orixate patterns in addition to all major types of phyllotaxis. This is clearly an advantage over the previous model,” Sugiyama says. “EDC2 also fits better to the natural occurrence of various patterns.”

The researchers aren’t sure what causes these growth patterns to be affected by leaf age, but Sugiyama speculates that it could be due to changes in the auxin transport system as a plant develops. Sugiyama and his colleagues are working to improve their model even more so that it can generate all known phyllotactic patterns. One “mystery” leaf pattern, a spiral with a small divergence angle, has yet to be predicted computationally, though Sugiyama believes they’re getting close.

Mathematical model studies of the comprehensive generation of major and minor phyllotactic patterns in plants with a predominant focus on orixate phyllotaxis, Takaaki Yonekura, Akitoshi Iwamoto, Hironori Fujita, Munetaka Sugiyama

Published: June 2019
DOI: https://doi.org/10.1371/journal.pcbi.1007044

• 1
Auxins are a class of plant hormones (or plant-growth regulators) with some morphogen-like characteristics. Auxins play a cardinal role in coordination of many growth and behavioral processes in plant life cycles and are essential for plant body development.
• 2
The Douady and Couder equations equation make the fundamental assumption that leaves emit a constant signal to inhibit the growth of other leaves nearby and that the signal gets weaker at longer distances.