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According to a new mathematical analysis, the mysterious “3/4 law of metabolism” — proposed by Max Kleiber in 1932 and later described as “extended to all life forms” from bacteria to whales — is wrong.

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 80 years of near-mystical belief in a 3/4 exponent used to describe the relationship between the size of animals and their resting metabolism.

Two-thirds or three-quarters?

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.

And what about in real animals? In 1883, a German physiologist named Max Rubner measured the heat output of some dogs ranging from a few pounds to nearly seventy. He plotted these numbers to show that the dog’s metabolic rates were proportional to their mass with an exponent of 2/3 — just like the geometry of an imaginary spherical beast would suggest.



But, in 1932, Swiss agricultural chemist Max Kleiber presented a paper with a now-famous graph. It plotted, on a logarithmic scale, the body weight of 13 mammals, ranging from rats to cows, against their resting metabolism. Strangely, the line traced through the data points did not conform to Rubner’s observation nor common sense. Instead, it hewed to a line with a somewhat steeper slope of about .73. To make it easier for slide rule use, he rounded the exponent to a neat .75. Kleiber’s 3/4-power law was born.

“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!”

Over the next decades, hundreds of animals’ resting metabolisms were measured or estimated, from microbes to whales. The results in various groups of animals ranged from slopes of less than 2/3 to greater than 1. But as Vaclav Smil wrote in a sweeping “millennium essay” for Nature they were “close enough to the 0.75 line,” and concluded that “the 3/4 slope is representative for all” animals.

“Some data seems to fit this 3/4 line — if you’re looking for it!” says Dodds. “It was pre-supposed to be true — and became a universal overarching law that somebody needs to explain.”

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