###### February 18, 2014

# Kleiber’s Law Helps Explain The Shape Of Evolution

**April Flowers for redOrbit.com - Your Universe Online**

During its lifetime, the heart of a mouse beats about the same number of times as an elephant. However, the mouse only lives about a year, while the elephant might live to the age of 70. Scientists have also observed that small plants and animals mature faster than large ones, and that nature has created radically different forms for life—from the loose-limbed beauty of a flowering tree to the fearful symmetry of a tiger.

Understanding these mysteries has escaped scientists since ancient times. An international team of researchers from the University of Maryland and the University of Padua in Italy has proposed an answer based on a famous mathematical formula that had been accepted as true for generations, but never fully understood.The findings of this study, published in the *Proceedings of the National Academy of Sciences*, offers a rethinking of the formula—known as Kleiber's Law—as a mathematical expression of an evolutionary fact. This suggests that plants' and animals' widely different forms evolved in parallel, as ideal ways to solve the problem of how to use energy efficiently.

Kleiber's Law—metabolism equals mass to the three-quarter power—is taught to high school and college biology students. This formula is one of the few widely held tenets in biology. Named after the Swiss biologist Max Kleiber who formulated it in the 1930s, the law shows that as living things get larger, their metabolism and their life spans increase at predictable rates. Scientists have used the formula to predict everything from an animal's energy intake to the number of young they will bear, and to calculate the correct human dosage of a medicine tested on mice, among other things.

Although generations of scientists have hunted for a simple, convincing explanation for why the formula holds true, one hasn't been found. The new study proposes that the shapes of both plants and animals evolved in response to mathematical and physical principles. The team worked through the logic underlying Kleiber's mathematical formula, and applied it separately to the geometry of plants and animals. This allowed them to explain decades’ worth of real-world observations.

"Plant and animal geometries have evolved more or less in parallel," said UMD botanist Todd Cooke. "The earliest plants and animals had simple and quite different bodies, but natural selection has acted on the two groups so the geometries of modern trees and animals are, remarkably, displaying equivalent energy efficiencies. They are both equally fit. And that is what Kleiber's Law is showing us."

To understand, picture two organisms—a tree and a tiger. Looking at them from an evolutionary aspect, the tree has an easier job. It has to convert sunlight to energy and move it within a body that more or less stays put. The tree has evolved into a branching shape with many surfaces—its leaves—to make this process as efficient as possible.

"The tree's surface area and the volume of space it occupies are nearly the same," said physicist Jayanth Banavar, dean of the UMD College of Computer, Mathematical, and Natural Sciences. "The tree's nutrients flow at a constant speed, regardless of its size."

The team used these variables to calculate the relationship between the mass of different tree species and their metabolisms, finding that the relationship conformed to Kleiber's Law.

An animal needs fuel to nourish its mass, and burning that fuel generates heat. The excess heat has to be released somehow. An obvious answer would be surface cooling, but this doesn't work for the tiger. The tiger's surface area is proportionally smaller than its mass, meaning that the surface is not up to the task. The hide would become so hot that the tiger's coat would burst into flames.

As an animal increases in size, its metabolism must increase at a slower rate than their volume. Otherwise, they would be unable to shed the excess heat. The animal's metabolism would increase as its size increased—at the rate of its mass to the two-thirds power—if surface area was the only thing that mattered. Kleiber's Law, however, says that the actual rate is mass to the three-quarters power. This has been established by many sets of observations.

The scientists, realizing that there has to be a missing factor, have pored over the data, trying to find it. Different theories have been put forward. One proposes that the missing part of the equation has to do with the space occupied by internal organs, while another focuses on the fractal, or branching, form that is common to tree limbs and animals' blood vessels, but added in new assumptions about the volume of fluids contained in those fractal networks.

The current study authors argue that a critical variable has been overlooked, specifically the speed at which nutrients are carried throughout the animals' bodies and heat is carried away. They calculated the rate at which animals' hearts pump blood, finding that the velocity of blood flow was equal to the animals' mass to the one-twelfth power.

"The information was there all along, but its significance had been overlooked," said hydrologist Andrea Rinaldo, of Italy's University of Padua and Switzerland's Ecole Polytechnique Federale. "Animals need to adjust the flow of nutrients and heat as their mass changes to maintain the greatest possible energy efficiency. That is why animals need a pump – a heart – and trees do not."

The research team plugged that data into Kleiber's Law and found that they had attained a complete explanation for the equation.

"An elegant answer sometimes is the right one, and there's an elegance to this in the sense that it uses very simple geometric arguments," said physicist Amos Maritan, of the University of Padua. "It doesn't call for any specialized structures. It has very few preconditions. You have these two lineages, plants and animals, that are very different and they arrive at the same conclusion. That is what's called convergent evolution, and the stunning result is that it's being driven by the underlying physics and the underlying math."