Apparently, the mysterious “3/4 law of metabolism” — proposed by Max Kleiber in 1932, printed in biology textbooks for decades, explained theoretically in Science in 1997 and described in a 2000 essay in Nature as “extended to all life forms” from bacteria to whales — is just plain wrong.
“Actually, it’s two-thirds,” says University of Vermont mathematician Peter Dodds. His paper in the January 29 edition of Physical Review Letters helps overturn almost eighty years of near-mystical belief in a 3/4 exponent used to describe the relationship between the size of animals and their resting metabolism.
To understand the debate between 2/3 and 3/4, assume a spherical cow. “That’s what a physicist would do,” Dodds says, laughing. Basic geometry shows that the surface area of this difficult-to-milk creature would increase as the square of its radius while the volume would increase as the cube of the radius. In other words, the exponent that describes the ratio of surface area to volume is 2/3.
Next, assume a spherical mouse. OK, now compare the resting metabolic rates of these sorry animals. Since the point of resting metabolism is to keep a warm-blooded animal warm (and alive!) with the lowest necessary energy use, both geometry and common sense suggest that the cow would have a lower rate of metabolism per cell than the mouse: the mouse, with more surface area relative to its volume, would lose heat faster than our cartoon cow…
“Kleiber’s original data is a mess, a complete mess,” says Dodds, “but it became something everyone believed in. The idea of quarter-powers begins to take on this spooky, magical quality. Nobody can explain it, but it’s a secret law of the universe. It’s quarterology!..”
In 1997, an elegant, though controversial, paper by Geoffrey West and colleagues was published in Science that claimed to derive 3/4 from first principles, drawing on ideas about fractals in networks and the growing length of tubes.
“The problem is their paper fell to pieces mathematically. It just didn’t work. Unfortunately, I showed that and published a paper with my adviser and a fellow student in 2001,” Dodds says…
A confluence of facts — greater understanding about how a network best minimizes volume, as evolution would favor in the costly production of blood supply; surface area geometry; and re-analysis of Kleiber’s and other data — seem to be pummeling the once-beguiling 3/4 law into dust.
“Especially for smaller guys,” Dodds says, “like birds, it’s just absolutely, stone-cold 2/3.”