By GEORGE JOHNSON
January 12, 1999
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
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
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
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
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
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
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
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
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."