Next time you go outside, take a minute to look at your local leaf arrangements. You’ll probably notice a few different patterns. In basil plants, each leaf is about 90 degrees--a quarter-turn--from the last, a template called “decussate.”

Bamboo leaves are directly opposite each other, or “distichous,” while the spiral aloe plant forms a swirl that follows the Fibonacci sequence.

And then there’s Orixa japonica. The shrub, which is common in Japan, has glossy green leaves that are arranged asymmetrically, in a kind of spinning stagger-step.

If you begin with the oldest leaf and move up the twig, the next will be 180 degrees away. The third leaf is 90 degrees from the second, the fourth 180 degrees from the third, and the fifth 270 degrees from the fourth. After that, the sequence starts again.

A few other unrelated plants, including the red-flowered torch lily of South Africa and a popular flowering tree called the crepe myrtle, also display this leaf layout, which is called “orixate” after its main showcase.

It’s a “peculiar pattern” previously unexplained by science, said Munetaka Sugiyama, a plant physiologist at the University of Tokyo. In a study published Thursday in PLOS Computational Biology, Sugiyama and his colleagues present the first mathematical model that successfully accounts for this unusual arrangement.

Sugiyama, who often walks past Orixa japonica shrubs at his university’s botanical gardens, has long been intrigued by leaf arrangements, or phyllotaxis. But it was “just my hobby,” he said, until he found a kindred spirit in Takaaki Yonekura, now a graduate student. About five years ago, he joined Sugiyama’s lab, and the two began studying orixate patterns.

“I was so excited at the topic,” Yonekura said.

The researchers started with an existing phyllotaxis equation called the Douady and Couder 2 model, or DC2. Developed in 1996, the DC2 model is based on the assumption that each leaf exerts a chemical “inhibitory power” on the area surrounding it--a sort of force field that prevents other leaves from growing. The force peters off with distance until it disappears, allowing new leaves to form.

If you plug information about a particular species--like basil or the spiral aloe--into the DC2 model, it will almost always spit out the pattern that the plant actually displays in nature. But it doesn’t work for Orixa japonica.

Why not? Sugiyama had long thought that the answer might lie in “some changes in the inhibitory power of the developing leaves,” he said.

So the researchers decided to add another variable: leaf age. They tweaked the model so that older leaves possess a larger “force field” than younger ones. This time, when they put in Orixa japonica’s stats, the right shape came out.

It also worked for all the patterns DC2 already had covered. The researchers call their new model Extended DC2, or EDC.

The study “gives you a real feeling of the space of possibility” for the study of natural patterns, said Stéphane Douady, the co-creator of the DC2 model, who was not involved in the new study but reviewed it before publication. (Douady was himself inspired to study phyllotaxis by an encounter with Romanesco broccoli.)

Sugiyama hopes their discovery will “contribute to understanding the beauty of nature.” But he and Yonekura have already moved on to the plant world’s next strange and unexplained pattern: “spiromonostichy,” which is found in perennial Costus plants, making them look like tight spiral staircases.

“For the researchers of phyllotaxis, this pattern is so mysterious,” Sugiyama said. “We are now trying to modify our model.”

And so science continues to leaf out.

(June 6, 2019)