# Mathematics of plant leaves

June 06, 2019

Leaves can be enjoyed for their shade, autumn colors, or taste, and the arrangement of leaves on a plant is a practical way to identify a species. However, the details of how plants control their leaf arrangement have remained a persistent mystery in botany. A Japanese plant species with a peculiar leaf pattern recently revealed unexpected insight into how almost all plants control their leaf arrangement.

"We developed the new model to explain one peculiar leaf arrangement pattern. But in fact, it more accurately reflects not only the nature of one specific plant, but the range of diversity of almost all leaf arrangement patterns observed in nature," said Associate Professor Munetaka Sugiyama from the University of Tokyo's Koishikawa Botanical Garden.

All in the angles

To identify the leaf arrangement of a plant species, botanists measure the angle between leaves, moving up the stem from oldest to youngest leaf.

Common patterns are symmetrical and have leaves arranged at regular intervals of 90 degrees (basil or mint), 180 degrees (stem grasses, like bamboo), or in Fibonacci golden angle spirals (like the needles on some spherical cacti, or the succulent spiral aloe).

The peculiar pattern that Sugiyama's research team studied is called "orixate" after the species Orixa japonica, a shrub native to Japan, China, and the Korean peninsula. O. Japonica is sometimes used as a hedge.

The angles between O. Japonica leaves are 180 degrees, 90 degrees, 180 degrees, 270 degrees, and then the next leaf resets the pattern to 180 degrees.

"Our research has the potential to truly understand beautiful patterns in nature," said Sugiyama.

The math of a plant

Sugiyama's research team began their investigation by doing exhaustive testing of the existing mathematical equation used to model leaf arrangement.

Leaf arrangement has been modeled mathematically since 1996 using an equation known as the DC2 (Douady and Couder 2). The equation can generate many, but not all, leaf arrangement patterns observed in nature by changing the value of different variables of plant physiology, such as the relationships between different plant organs or strength of chemical signals within the plant.

The DC2 has two shortcomings that researchers wanted to address:
1) No matter what values are put into the DC2 equation, certain uncommon leaf arrangement patterns are never calculated.

2) The Fibonacci spiral leaf arrangement pattern is by far the most common spiral pattern observed in nature, but is only modestly more common than other spiral patterns calculated by the DC2 equation.
A peculiar pattern

At least four unrelated plant species possess the unusual orixate leaf arrangement pattern. Researchers suspected that it must be possible to create the orixate pattern using the fundamental genetic and cellular machinery shared by all plants because the alternative possibility - that the same, very unusual leaf arrangement pattern evolved four or more separate times - seemed too unlikely.

One fundamental assumption used in the DC2 equation is that leaves emit a constant signal to inhibit the growth of other leaves nearby and that the signal gets weaker at longer distances. Researchers suspect that the signal is likely related to the plant hormone auxin, but the exact physiology remains unknown.

Rare patterns and common rules

"We changed this one fundamental assumption - inhibitory power is not constant, but in fact changes with age. We tested both increasing and decreasing inhibitory power with greater age and saw that the peculiar orixate pattern was calculated when older leaves had a stronger inhibitory effect," said Sugiyama.

This insight into the inhibitory signal power changing with age may be used to direct future studies of the genetics or physiology of plant development.

Researchers call this new version of the equation the EDC2 (Expanded Douady and Couder 2).

First author of the research paper, doctoral student Takaaki Yonekura, designed computer simulations to generate thousands of leaf arrangement patterns calculated by EDC2 and to count how often the same patterns were generated. Patterns that are more commonly observed in nature were more frequently calculated by the EDC2, further supporting the accuracy of the ideas used to create the formula.

"There are other very unusual leaf arrangement patterns that are still not explained by our new formula. We are now trying to design a new concept that can explain all known patterns of leaf arrangement, not just almost all patterns," said Sugiyama.

Do it yourself - ID the pattern

Experts recommend looking at a group of relatively new leaves when identifying a plant's leaf arrangement, or phyllotaxis, pattern. (In Greek, phyllon means leaf.) Older leaves may have turned (due to wind or sun exposure), which can make it difficult to identify their true angle of attachment to the stem.

Think of the stem as a circle and begin by carefully observing where on the circle the oldest and second-oldest leaves are attached. The angle between those two leaves is the first "angle of divergence." Continue identifying the angles of divergence between increasingly younger leaves on the stem. The pattern of angles of divergence is the leaf arrangement pattern.

Common leaf arrangement patterns are distichous (regular 180 degrees, bamboo), Fibonacci spiral (regular 137.5 degrees, the succulent Graptopetalum paraguayense), decussate (regular 90 degrees, the herb basil), and tricussate (regular 60 degrees, Nerium oleander sometimes known as dogbane).
-end-
Journal Article

Takaaki Yonekura, Akitoshi Iwamoto, Hironori Fujita, Munetaka Sugiyama. 2019. Mathematical model studies of the comprehensive generation of major and minor phyllotactic patterns in plants with a predominant focus on orixate phyllotaxis. PLOS Computational Biology. DOI: 10.1371/journal.pcbi.1007044 https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1007044

Related Links

Sugiyama lab website: https://www.bg.s.u-tokyo.ac.jp/common/research/sugi-lab/index.html

Koishikawa Botanical Garden: https://www.bg.s.u-tokyo.ac.jp/koishikawa/eng/

Graduate School of Science: https://www.s.u-tokyo.ac.jp/en/

Research contact

Associate Professor Munetaka Sugiyama
Koishikawa Botanical Gardens, Graduate School of Science, The University of Tokyo
Tel: +81-(0)3-3814-0368
Email: sugiyama@ns.bg.s.u-tokyo.ac.jp

Press Contacts

Ms. Caitlin Devor
Division for Strategic Public Relations, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8654, JAPAN
Tel: +81-3-5841-0876
Email: press-releases.adm@gs.mail.u-tokyo.ac.jp

Office of Communication, Graduate School of Science, The University of Tokyo
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 JAPAN
Tel: +81-3-5841-8856
E-mail: kouhou.s@gs.mail.u-tokyo.ac.jp

About the University of Tokyo

The University of Tokyo is Japan's leading university and one of the world's top research universities. The vast research output of some 6,000 researchers is published in the world's top journals across the arts and sciences. Our vibrant student body of around 15,000 undergraduate and 15,000 graduate students includes over 2,000 international students. Find out more at http://www.u-tokyo.ac.jp/en/ or follow us on Twitter at @UTokyo_News_en.

University of Tokyo

Related Plant Species Articles:

Study: One-third of plant and animal species could be gone in 50 years
University of Arizona researchers studied recent extinctions from climate change to estimate the loss of plant and animal species by 2070.
Scientists challenge notion of binary sexuality with naming of new plant species
A collaborative team of scientists from the US and Australia has named a new plant species from the remote Outback.
Plant lineage points to different evolutionary playbook for temperate species
An ancient, cosmopolitan lineage of plants is shaking up scientists' understanding of how quickly species evolve in temperate ecosystems and why.
Native plant species may be at greater risk from climate change than non-natives
A study led by researchers at Indiana University's Environmental Resilience Institute has revealed that warming temperatures affect native and non-native flowering plants differently, which could change the look of local landscapes over time.
'Specialized' microbes within plant species promote diversity
A Yale-led research team conducted an experiment that suggests microbes can specialize within plant species, which can promote plant species diversity and increased seed dispersal.
New machine learning method predicts additions to global list of threatened plant species
A new method uses machine learning and open-access data to predict whether species are eligible for at-risk status on the IUCN Red List.
Bioactive novel compounds from endangered tropical plant species
A Japan-based research team led by Kanazawa University has isolated 17 secondary metabolites, including three novel compounds from the valuable endangered tropical plant species Alangium longiflorum.
Global study finds taller plant species taking over as mountains and the Arctic warm
A study by more than 100 global researchers, including Simon Fraser University biologist David Hik, is linking the effects of climate change to new and taller plant species in the Arctic and alpine tundra.
New plant species discovered in museum is probably extinct
A single non-photosynthetic plant specimen preserved in a Japanese natural history museum has been identified as a new species.
Plant virus alters competition between aphid species
In the world of plant-feeding insects, who shows up first to the party determines the overall success of the gathering; yet viruses can disrupt these intricate relationships, according to researchers at Penn State.

## Trending Science News

Current Coronavirus (COVID-19) News

## Top Science Podcasts

We have hand picked the top science podcasts of 2020.
Now Playing: TED Radio Hour

Uncharted
There's so much we've yet to explorefrom outer space to the deep ocean to our own brains. This hour, Manoush goes on a journey through those uncharted places, led by TED Science Curator David Biello.
Now Playing: Science for the People

#556 The Power of Friendship
It's 2020 and times are tough. Maybe some of us are learning about social distancing the hard way. Maybe we just are all a little anxious. No matter what, we could probably use a friend. But what is a friend, exactly? And why do we need them so much? This week host Bethany Brookshire speaks with Lydia Denworth, author of the new book "Friendship: The Evolution, Biology, and Extraordinary Power of Life's Fundamental Bond". This episode is hosted by Bethany Brookshire, science writer from Science News.
Now Playing: Radiolab

Dispatch 2: Every Day is Ignaz Semmelweis Day
It began with a tweet: "EVERY DAY IS IGNAZ SEMMELWEIS DAY." Carl Zimmer  tweet author, acclaimed science writer and friend of the show  tells the story of a mysterious, deadly illness that struck 19th century Vienna, and the ill-fated hero who uncovered its cure ... and gave us our best weapon (so far) against the current global pandemic. This episode was reported and produced with help from Bethel Habte and Latif Nasser. Support Radiolab today at Radiolab.org/donate.