Of Mice and Elephants: A Matter of Scale


January 12, 1999

Scientists, intent on categorizing everything around them, sometimes divide themselves into the lumpers and the splitters. The lumpers, many of whom flock to the unifying field of theoretical physics, search for hidden laws uniting the most seemingly diverse phenomena: Blur your vision a little and lightning bolts and static cling are really the same thing.

The splitters, often drawn to the biological sciences, are more taken with diversity, reveling in the 34,000 variations on the theme spider, or the 550 species of conifer trees.

But there are exceptions to the rule. When two biologists and a physicist, all three of the lumper persuasion, recently joined forces at the Santa Fe Institute, an interdisciplinary research center in northern New Mexico, the result was an advance in a problem that has bothered scientists for decades: the origin of biological scaling. How is one to explain the subtle ways in which various characteristics of living creatures -- their life spans, their pulse rates, how fast they burn energy -- change according to their body size?

As animals get bigger, from tiny shrew to huge blue whale, pulse rates slow down and life spans stretch out longer, conspiring so that the number of heartbeats during an average stay on Earth tends to be roughly the same, around a billion. A mouse just uses them up more quickly than an elephant.

Mysteriously, these and a large variety of other phenomena change with body size according to a precise mathematical principle called quarter-power scaling. A cat, 100 times more massive than a mouse, lives about 100 to the one-quarter power, or about three times, longer. (To calculate this number take the square root of 100, which is 10 and then take the square root of 10, which is 3.2.) Heartbeat scales to mass to the minus one-quarter power. The cat's heart thus beats a third as fast as a mouse's.

The Santa Fe Institute collaborators -- Geoffrey West, a physicist at Los Alamos National Laboratory, and two biologists at the University of New Mexico, Jim Brown and Brian Enquist -- have drawn on their different kinds of expertise to propose a model for what causes certain kinds of quarter-power scaling, which they have extended to the plant kingdom as well.

In their theory, scaling emerges from the geometrical and statistical properties of the internal networks animals and plants use to distribute nutrients. But almost as interesting as the details of this model, is the collaboration itself. It is rare enough for scientists of such different persuasions to come together, rarer still that the result is hailed as an important development.

"Scaling is interesting because, aside from natural selection, it is one of the few laws we really have in biology," said John Gittleman, an evolutionary biologist at the University of Virginia. "What is so elegant is that the work makes very clear predictions about causal mechanisms. That's what had been missing in the field."

Brown said: "None of us could have done it by himself. It is one of the most exciting things I've been involved in."

It might seem that because a cat is a hundred times more massive than a mouse, its metabolic rate, the intensity with which it burns energy, would be a hundred times greater -- what mathematicians call a linear relationship. After all, the cat has a hundred times more cells to feed.

But if this were so, the animal would quickly be consumed by a fit of spontaneous feline combustion, or at least a very bad fever. The reason: the surface area a creature uses to dissipate the heat of the metabolic fires does not grow as fast as its body mass. To see this, consider (like a good lumper) a mouse as an approximation of a small sphere. As the sphere grows larger, to cat size, the surface area increases along two dimensions but the volume increases along three dimensions. The size of the biological radiator cannot possibly keep up with the size of the metabolic engine.

If this was the only factor involved, metabolic rate would scale to body mass to the two-thirds power, more slowly than in a simple one-to-one relationship. The cat's metabolic rate would be not 100 times greater than the mouse's but 100 to the power of two-thirds, or about 21.5 times greater.

But biologists, beginning with Max Kleiber in the early 1930s, found that the situation was much more complex. For an amazing range of creatures, spanning in size from bacteria to blue whales, metabolic rate scales with body mass not to the two-thirds power but slightly faster -- to the three-quarter power.

Evolution seems to have found a way to overcome in part the limitations imposed by pure geometric scaling, the fact that surface area grows more slowly than size. For decades no one could plausibly say why.

Kleiber's law means that a cat's metabolic rate is not a hundred or 21.5 times greater than a mouse's, but about 31.6 -- 100 to the three-quarter power. This relationship seems to hold across the animal kingdom, from shrew to blue whale, and it has since been extended all the way down to single-celled organisms, and possibly within the cells themselves to the internal structures called mitochondria that turn nutrients into energy.

Long before meeting Brown and Enquist, West was interested in how scaling manifests itself in the world of subatomic particles. The strong nuclear force, which binds quarks into neutrons, protons and other particles, is weaker, paradoxically, when the quarks are closer together, but stronger as they are pulled farther apart -- the opposite of what happens with gravity or electromagnetism.

Scaling also shows up in Heisenberg's Uncertainty Principle: the more finely you measure the position of a particle, viewing it on a smaller and smaller scale, the more uncertain its momentum becomes.

"Everything around us is scale dependent," West said. "It's woven into the fabric of the universe."

The lesson he took away from this was that you cannot just naively scale things up. He liked to illustrate the idea with Superman. In two panels labeled "A Scientific Explanation of Clark Kent's Amazing Strength," from Superman's first comic book appearance in 1938, the artists invoked a scaling law: "The lowly ant can support weights hundreds of times its own. The grasshopper leaps what to man would be the space of several city blocks." The implication was that on the planet Krypton, Superman's home, strength scaled to body mass in a simple linear manner: If an ant could carry a twig, a Superman or Superwoman could carry a giant ponderosa pine.

But in the real universe, the scaling is actually much slower. Body mass increases along three dimensions, but the strength of legs and arms, which is proportional to their cross-sectional area, increases along just two dimensions. If a man is a million times more massive than an ant, he will be only 1,000,000 to the two-thirds power stronger: about 10,000 times, allowing him to lift objects weighing up to a hundred pounds, not thousands.

Things behave differently at different scales, but there are orderly ways -- scaling laws -- that connect one realm to another. "I found this enormously exciting," West said. "That's what got me thinking about scaling in biology."

At some point he ran across Kleiber's law. "It is truly amazing because life is easily the most complex of complex systems," West said. "But in spite of this, it has this absurdly simple scaling law. Something universal is going on."

Enquist became hooked on scaling as a student at Colorado College in Colorado Springs in 1988. When he was looking for a graduate school to study ecology, he chose the University of New Mexico in Albuquerque partly because a professor there, Brown, specialized in how scaling occurred in ecosystems.

There are obviously very few large species, like elephants and whales, and a countless number of small species. But who would have expected, as Enquist learned in one of Brown's classes, that if one drew a graph with the size of animals on one axis and the number of species on the other axis, the slope of the resulting line would reveal another quarter-power scaling law? Population density, the average number of offspring, the time until reproduction -- all are dependent on body size scaled to quarter-powers.

"As an ecologist you are used to dealing with complexity -- you're essentially embedded in it," Enquist said. "But all these quarter-power scaling laws hinted that something very general and simple was going on."

The examples Brown had given all involved mammals. "Has anyone found similar laws with plants?" Enquist asked. Brown said, "I have no idea. Why don't you find out?"

After sifting through piles of data compiled over the years in agricultural and forestry reports, Enquist found that the same kinds of quarter-power scaling happened in the plant world. He even uncovered an equivalent to Kleiber's law.

It was surprising enough that these laws held among all kinds of animals. That they seemed to apply to plants as well was astonishing. What was the common mechanism involved? "I asked Jim whether or not we could figure it out," Enquist recalled. "He kind of rubbed his head and said, 'Do you know how long this is going to take?"'

They assumed that Kleiber's law, and maybe the other scaling relationships, arose because of the mathematical nature of the networks both animals and trees used to transport nutrients to all their cells and carry away the wastes. A silhouette of the human circulatory system and of the roots and branches of a tree look remarkably similar.

But they knew that precisely modeling the systems would require some very difficult mathematics and physics. And they wanted to talk to someone who was used to trafficking in the idea of general laws.

"Physicists tend to look for universals and invariants whereas biologists often get preoccupied with all the variations in nature," Brown said. He knew that the Santa Fe Institute had been established to encourage broad-ranging collaborations. He asked Mike Simmons, then an institute administrator, whether he knew of a physicist interested in tackling biological scaling laws.

West liked to joke that if Galileo had been a biologist, he would have written volumes cataloging how objects of different shapes fall from the leaning tower of Pisa at slightly different velocities. He would not have seen through the distracting details to the underlying truth: if you ignore air resistance, all objects fall at the same rate regardless of their weight.

But at their first meeting in Santa Fe, he was impressed that Brown and Enquist were interested in big, all-embracing theories. And they were impressed that West seemed like a biologist at heart. He wanted to know how life worked.

It took them a while to learn each other's languages, but before long they were meeting every week at the Santa Fe Institute. West would show the biologists how to translate the qualitative ideas of biology into precise equations. And Brown and Enquist would make sure West was true to the biology. Sometimes he would show up with a neat model, a physicist's dream. No, Brown and Enquist would tell him, real organisms do not work that way.

"When collaborating across that wide a gulf of disciplines, you're never going to learn everything the collaborator knows," Brown said. "You have to develop an implicit trust in the quality of their science. On the other hand, you learn enough to be sure there are not miscommunications."

They started by assuming that the nutrient supply networks in both animals and plants worked according to three basic principles: the networks branched to reach every part of the organism and the ends of the branches (the capillaries and their botanical equivalent) were all about the same size. After all, whatever the species, the sizes of cells being fed were all roughly equivalent. Finally they assumed that evolution would have tuned the systems to work in the most efficient possible manner.

What emerged closely approximated a so-called fractal network, in which each tiny part is a replica of the whole. Magnify the network of blood vessels in a hand and the image resembles one of an entire circulatory system. And to be as efficient as possible, the network also had to be "area-preserving."

If a branch split into three daughter branches, their cross-sectional areas had to add up to that of the parent branch. This would insure that blood or sap would continue to move at the same speed throughout the organism.

The scientists were delighted to see that the model gave rise to three-quarter-power scaling between metabolic rate and body mass. But the system worked only for plants. "We worked through the model and made clear predictions about mammals," Brown said, "every single one of which was wrong."

In making the model as simple as possible, the scientists had hoped they could ignore the fact that blood is pumped by the heart in pulses and treat mammals as though they were trees. After studying hydrodynamics, the nature of liquid flow, they realized they needed a way to slow the pulsing blood as the vessels got tinier and tinier.

These finer parts of the network would not be area-preserving but area-increasing: the cross sections of the daughter branches would add up to a sum greater than the parent branch, spreading the blood over a larger area.

After adding these and other complications, they found that the model also predicted three-quarter-power scaling in mammals. Other quarter-power scaling laws also emerged naturally from the equations. Evolution, it seemed, has overcome the natural limitations of simple geometric scaling by developing these very efficient fractal-like webs.

Sometimes it all seemed too good to be true. One Friday night, West was at home playing with the equations when he realized to his chagrin that the model predicted that all mammals must have about the same blood pressure. That could not be right, he thought. After a restless weekend, he called Brown, who told him that indeed this was so.

The model was revealed, about two years after the collaboration began, on April 4, 1997, in an article in Science. A follow-up last fall in Nature extended the ideas further into the plant world.

More recently the three collaborators have been puzzling over the fact that a version of Kleiber's law also seems to apply to single cells and even to the energy-burning mitochondria inside cells. They assume this is because the mitochondria inside the cytoplasm and even the respiratory components inside the mitochondria are arranged in fractal-like networks.

For all the excitement the model has caused, there are still skeptics. A paper published last year in American Naturalist by two scientists in Poland, Dr. Jan Kozlowski and Dr. January Weiner, suggests the possibility that quarter-power scaling across species could be nothing more than a statistical illusion. And biologists persist in confronting the collaborators with single species in which quarter-power scaling laws do not seem to hold.

West is not too bothered by these seeming exceptions. The history of physics is replete with cases where an elegant model came up against some recalcitrant data, and the model eventually won. He is now working with other collaborators to see whether river systems, which look remarkably like circulatory systems, and even the hierarchical structure of corporations obey the same kind of scaling laws.

The overarching lesson, West says, is that as organisms grow in size they become more efficient. "That is why nature has evolved large animals," he said. "It's a much better way of utilizing energy. This might also explain the drive for corporations to merge. Small may be beautiful but it is more efficient to be big."