Tree leaf physics

"Explaining the size of tree leaves using physics"

by

Chris Lee

February 28th, 2013

Ars Technica

One of the most useless-but-cool things about physics are the post-hoc explanations it offers for the field of biology. I suspect that these physics-focused explanations really piss biologists off, and rightly so, since they tend to offer an explanation for observations, but have no predictive power to point us beyond what we already know.

Biology is a big field with lots of sub-fields, though. For the most part, biologists interested in mammals and birds have had to bear the burden of physicists' attention. But that's changing, as plant biologists have become the latest victims. Yes, physics now offers an explanation for why tall trees have such, ahem, tiny leaves.

Before we get to the trees and their pitifully small leaves, let's take a quick look at the sort of arguments and insights that physics offers biologists. For instance, it can help you determine why the smallest and largest land-dwelling mammals are about the size that they are—look no further than the balance of energy. Mammals are warm-blooded, and that costs some energy. And, to make matters worse, they lose energy through their skin simply through heat conduction. The smaller you are, the greater the proportion of your energy is spent maintaining your internal body temperature, simply because you have more surface area relative to body mass. At a certain size, you simply cannot eat fast enough to maintain an internal body temperature.

On the upper end of the scale, you have many more cells to supply with energy, so you spend more energy foraging for food. At some point, you end up spending more energy on foraging than the total available energy in the food being foraged. Hence, at the bottom of the scale shivers the shrew, scurrying around desperately eating to keep warm, while at the upper end, an elephant is constantly and slowly eating everything in sight.

It is possible to make similar arguments based on the strength of bone vs. total mass, blood pressure for height, and metabolic rate for average lifetime. Now, I know I made this look a bit of a joke in the introduction, because, truth be told, these sorts of arguments are fun. In all seriousness, though, this form of reasoning—figuring out limitations due to the way that physical processes scale with increases and decreases in particular parameters—is very handy. More fascinating is that they are at least broadly in agreement with observations.

Resistance makes growth futile

Now a pair of scientists have applied a scaling law arguments to the leaves of trees. The heart of the argument is about how fast sugar flows down the trunk of the tree. A tree has to be able to distribute energy from the leaves to the roots, and this is all governed by flow through a network of tubes, called phloem.

The speed of the flow is in turn governed by just a few factors: two different flow resistances, and the pressure difference between the top and the bottom of the tree. It turns out that the pressure difference is independent of height, but the flow resistances are a different matter.

Flow resistance in the trunk depends on the length and radius of the phloem tube. As the length increases, the flow resistance increases. But, as the radius increases, the resistance drops. For small trees, both the radius and the length increase with height, so the flow resistance may drop as the tree grows. Unfortunately, for taller trees, the radius tops out at 20 micrometers, leaving the flow resistance to continue increasing as the tree grows.

For leaves, the story is a little more complicated. The phloem has permeable membranes that are designed to accept increasing amounts of sugar from the leaf. The argument is that the increased permeability and phloem surface area result in greater concentration differences that drive a higher pressure differential within the leaf. Looking at the leaf as a black box, it simply appears that the internal resistance has dropped. The upshot is that the flow resistance of the leaf decreases as the leaf size increases.

Why does this matter? Both resistances contribute to slowing the flow but, if one is much larger than the other, then changing the smaller resistance has no impact. So, for a tall tree, increasing the leaf size doesn't result in faster flow, since the phloem's resistance dominates. For smaller trees, a larger leaf size does result in faster flow. And this changes the energy balance for the tree.

My, what expensive leaves you have

The speed of energy flow has a maximum when the leaf resistance is zero, but that maximum is approached very, very slowly. At some point, the cost of maintaining the larger leaf is greater than the marginal increase in energy flow. Once you choose a tree height and a cost factor, you end up with a maximum leaf size.

On the lower end of the scale, the tree requires a minimum flow of energy. This is governed by cell-to-cell diffusion of sugar-water. Once you plug this number in, for any given tree height, a minimum leaf size pops out.

As the tree grows taller, the minimum leaf size goes up, while the maximum leaf size goes down. At around 100m, the two cross, so that the minimum required leaf size is greater than the maximum useful leaf size. And, guess what, the tallest known trees top out at around 100m. More importantly though, the variation in leaf size decreases with increasing tree height, and this is exactly what the observational data shows.

In fact, the data fits remarkably well. For the largest leaf size, there is a cost-factor that they have to fit. But, considering that a single parameter fits the data for multiple species within the angiosperm family (flowering plants; the study didn't consider anything else), that is pretty good. The lower limit on leaf size is a lot rougher, but the data is also a lot noisier at the lower bound, so no simple curve fit that data well.

What I love about papers like this is how a few simple physical principles come together to provide a rough explanation for observed data. What makes it more interesting is that it sets criteria that can help us find exceptions, and exceptions make for interesting research.

[The article is by subscription/purchase only.  Physical Review Letters .]