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Hacked: The Mathematics of Leaf Vein Architecture

Researchers at the University of California - Los Angeles, have reverse engineered the mathematics behind the formation of leaf veins in growing and evolving plants – across all species. The rules have been distilled into a simple, reproducible form which can be used by third parties to reliably shape new species.

Leaf veins are not the typical target of plant modelling in most VR systems, although they will certainly be of great interest to many microscale simulations. However, when you consider that leaf vein formation governs leaf shape formation, it is no great leap from there to use the recently discovered maths as part of the process for procedurally modelling new species for use as the vegetation elsewhere in a VR.

Most likely these equations will also form part of the foundations for algorithms to create whole plants mathematically – and create new models that way.

Vein network in a tropical forest tree.
Credit: Michael Rawls, UCLA Life Sciences

The research, published May 15 in the journal Nature Communications, was intended for the study of global ecology and a method of allowing researchers to estimate original leaf sizes from just a fragment of a leaf. It is just a happy coincidence (as is so often the case) that this work has direct implications for virtual realities, and the creation of models to populate them. The intended initial hope for the work is that it will improve scientists' prediction and interpretation of climate in the deep past from leaf fossils.

Leaf veins are of tremendous importance in a plant's life, providing the nutrients and water that leaves need to conduct photosynthesis and supporting them in capturing sunlight. Leaf size is also of great importance for plants' adaptation to their environment, with smaller leaves being found in drier, sunnier places.

However, little has been known about what determines the architecture of leaf veins. Mathematical linkages between leaf vein systems and leaf size have the potential to explain important natural patterns. The new UCLA research focused on these linkages for plant species distributed around the globe.

"We found extremely strong, developmentally based scaling of leaf size and venation that has remained unnoticed until now," said Lawren Sack, a UCLA professor of ecology and evolutionary biology and lead author of the research.

How does the structure of leaf vein systems depend on leaf size? Sack and members of his laboratory observed striking patterns in several studies of just a few species. Leaf vein systems are made up of major veins (the first three branching "orders," which are large and visible to the naked eye) and minor veins, (the mesh embedded within the leaf, which makes up most of the vein length).

UCLA graduate student Christine Scoffoni, three UCLA undergraduate researchers and colleagues at other U.S. institutions measured hundreds of plant species worldwide using computer tools to focus on high-resolution images of leaves that were chemically treated and stained to allow sharp visualization of the veins.

The team discovered predictable relationships that hold across different leaves throughout the globe. Larger leaves had their major veins spaced further apart according to a clear mathematical equation, regardless of other variations in their structure (like cell size and surface hairiness) or physiological activities (like photosynthesis and respiration), Sack said.

"This scaling of leaf size and major veins has strong implications and can potentially explain many observed patterns, such as why leaves tend to be smaller in drier habitats, why flowering plants have evolved to dominate the world today, and how to best predict climates of the past," he said.

These leaf vein relationships can explain, at a global scale, the most famous biogeographical trend in plant form: the predomination of small leaves in drier and more exposed habitats. This global pattern was noted as far back as the ancient Greeks (by Theophrastus of Lesbos) and by explorers and scientists ever since. The classical explanation for why small leaves are more common in dry areas was that smaller leaves are coated by a thinner layer of still air and can therefore cool faster and prevent overheating. This would certainly be an advantage when leaves are in hot, dry environments, but it doesn't explain why smaller leaves are found in cool, dry places too, Sack noted.

How Smaller Leaves Lead to Stronger Plants

Last year, Scoffoni and Sack proposed that small leaves tend to have their major veins packed closely together, providing drought tolerance. That research, published in the journal Plant Physiology, pointed to an advantage for improving water transport during drought. To survive, leaves must open the stomatal pores on their surfaces to capture carbon dioxide, but this causes water to evaporate out of the leaves. The water must be replaced through the leaf veins, which pull up water through the stem and root from the soil. This drives a tension in the leaf vein "xylem pipes," and if the soil becomes too dry, air can be sucked into the pipes, causing blockage.

The team had found, using computer simulations and detailed experiments on a range of plant species, that because smaller leaves have major veins that are packed closer together — a higher major vein length per leaf area — they had more "superhighways" for water transport. The greater number of major veins in smaller leaves provides drought tolerance by routing water around blockages during drought.

This explanation is strongly supported by the team's new discovery of a striking global trend: higher major vein length per leaf area in smaller leaves.

The research also points to a new explanation for why leaf vein evolution allowed flowering plants to take over tens of millions of years ago from earlier evolved groups, such as cycads, conifers and ferns. Because, with few exceptions, only flowering plants have densely packed minor veins, and these allow a high photosynthetic rate providing water to keep the leaf cells hydrated and nutrients to fuel photosynthesis flowering plants can achieve much higher photosynthetic rates than earlier evolved groups, Sack said.

"While the major veins show close relationships with leaf size, becoming more spaced apart and larger in diameter in larger leaves, the minor veins are independent of leaf size and their numbers can be high in small leaves or large leaves," Sack said. "This uniquely gives flowering plants the ability to make large or small leaves with a wide range of photosynthetic rates. The ability of the flowering plants to achieve high minor-vein length per area across a wide range of leaf sizes allows them to adapt to a much wider range of habitats from shade to sun, from wet to dry, from warm to cold than any other plant group, helping them to become the dominant plants today."

The UCLA team explains that these patterns arise from the fact of a shared script or "program" for leaf expansion and the formation of leaf veins. The team reviewed the past 50 years of studies of isolated plant species and found striking commonalities across species in their leaf development. "Leaves develop in two stages," Sack said. "First, the tiny budding leaf expands slightly and slowly, and then it starts a distinct, rapid growth stage and expands to its final size."

Whilst this information is of course most pertinent to the plants of the physical world, incorporating it into plant designs for virtual environments will have the effect of greatly increasing believability and immersion, as the created plants, following these natural 'rules' will match aspects of the plants from wherever the user is from, increasing their tolerance for any strangeness in your world. In the long, long term, once we have a more complete picture, it may even help to enable plants following internal laws of physics and growth, even outside specialised research
VRs.

Why had these trends escaped notice until now?

"This is the time for plants," Sack said. "It's amazing what is waiting to be discovered in plant biology. It seems limitless right now. The previous century is known for exciting discoveries in physics and molecular biology, but this century belongs to plant biology. Especially given the centrality of plants for food and biosphere sustainability, more attention is being focused, and the more people look, the more fundamental discoveries will be made."

References

Hacking code of leaf vein architecture solves mysteries, allows predictions of past climate

Developmentally based scaling of leaf venation architecture explains global ecological patterns (Paper, Open Access)

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