Indeed, in a mouse or blue whale, the respiratory rate is proportional to M-1/4, and the metabolic rate to M3/4. So too, the diameters of tree trunks and human aortas both scale as M3/8. All of this explains why elephants breathe more slowly and live longer than, say, chickens. But it doesn't answer why nature should use one ruler to draw all of life's plans.
Image: SANTE FE INSTITUTE
At last, the mystery might be solved. Researchers from New Mexico recently published a compelling explanation in Science. To construct their argument, James H. Brown, an ecologist at University of New Mexico, Albuquerque, Brian J. Enquist, a post-doc in his lab, and Geoffrey West, a particle physicist from Los Alamos National Laboratory, refined ideas they had put forth in a 1997 paper. In essence, they invoke fractal geometry to make an end-run around what had been one of the most baffling aspects of quarter-power scaling laws--namely, why they aren't cube-roots instead.
The German physiologist Max Rubner first made the case for cubes in 1883. Drawing on Euclidean geometry, he reasoned that if an animal is N times taller than another, then its surface area should be N2 greater and its mass N3 greater. Its total metabolic rate, then, which depends on the amount of heat it sheds, should vary according to its surface area, N2, which is proportional to M2/3. And the energy it burns per unit of mass--its specific metabolic rate--would be proportional to M2/3 divided by M, or M1/3. (A decade later, Rubner discovered a key fact about metabolism: the energy the body produces from food equals the energy a fire releases burning that food.)
Among very similar species, this cube-root law seemed to hold. In 1932, though, Max Kleiber proved it wrong by methodically plotting animal masses versus metabolic rates. He found correlations closer to quarter roots, measuring an exponent of 0.74 for total metabolic rate and -0.26 for the specific rate.
Image: UC DAVIS
For 50-odd years, the problem lay more or less where the Maxs' left it. Then in 1995, Brown, long interested in biological scaling, and Enquist, who uncovered quarter-power scaling laws in plants, went in search of someone with a heartier math background to help them out. Mike Simmons, a retired vice president of academic affairs at Santa Fe Institute, put them in touch with West.
For their first crack at the problem, the trio homed in on what all animals and plants share: some sort of vascular network for distributing nutrients. They assumed that these networks are invariably fractal and that the end branches are of a fixed size. An elephant's capillaries are the same width as a mouse's, and a redwood's match an orchid's. They further guessed that nature had tuned circulatory networks to use as little energy as possible. And from there, using the physics of capillaries and hydrodynamics, they created a model that predicted a number of quarter-power scalings, including a mammal's aorta size, capillary density and heart size, with remarkable precision.
Their work inspired others. Jayanth Banavar at Penn State and the University of Pittsburgh, along with two Italian scientists, Amos Maritan and Andrea Rinaldo, sought to simplify matters, maintaining that such a fundamental law should have a more fundamental basis. They stripped fractals from their argument, applied Rubner's line of thinking to circulatory systems and derived quarter-power laws as a feature of any efficient network. Their take, published in May in Nature, can also be modified to accurately predict cube-root scaling in river networks, but it not without gaps.
The new model from Brown, Enquist and West is also simpler than their last. But rather than abandon fractals like Banavar, they have eliminated the fluid dynamics from their earlier work. That's not to say that the new explanation is easy. It argues that an organism's internal area--the surfaces of its capillaries--is "maximally fractal" and so fills space so efficiently as to create, in effect, an added dimension. Instead of having two dimensions, this area scales as if it has three. So too, the internal volume--that of the circulatory network feeding into capillaries--gains an added fourth dimension. And it is this extra dimension that gives rise to universality of quarter roots in scaling. "Although living things occupy a three-dimensional space," writes the team, "their internal physiology and anatomy operate as if they were four-dimensional....Fractal geometry has literally given life an added dimension."
This argument too presents questions: If the internal length of circulatory systems scales at a quarter power, and so less rapidly than the outer length, which scales at a cube-root power, it is unclear how the two manage to keep in step. In other words, you might imagine that an elephants capillaries would fall short of feeding its wrinkly skin. But resovling this glitch may just be matter of finessing the model a little further--something Brown, Enquist and West have shown themselves to be quite good at.