Unexpectedly Intriguing!
February 16, 2010

Spherical Cow Let us tell you a story from the world of science that has barely received much attention from the U.S. media, but which seriously undermines what has come to be commonly accepted as incontrovertible fact by scientists and laypeople alike. Decades of data and observations were held up as proof of the fact, and the fact, having been peer-reviewed and disseminated through countless journals and incorporated into basic science textbooks, was such that the science was believed to be settled and the time for debate about its validity over.

And then, just earlier this year, the fact simply stopped being true. The scientific analysis supporting it could no longer be supported by the available data and evidence. Instead, it turned out that the widely believed scientific fact had been based upon some not so great data and some pretty sloppy analysis. But it produced a compelling relationship that seemed to describe things that were being observed in the real world that could also be used to make predictions.

The only problem though was that nobody could quite figure out the fundamental reasons behind it. Worse, as more research was done, real world observations weren't agreeing with what the mathematical model behind the scientists' compelling relationship would predict.

Believe it or not, we're not talking about the theory of Anthropogenic Global Warming (AGW), where the evidence of an increasing amount of carbon dioxide (CO2) in the atmosphere is believed to be a prime driver of climate change on a global scale, which then impacts virtually every living creature on the planet Earth. While this seemingly simple relationship between the change in the levels of atmospheric CO2 and average global temperatures and the scandals now coming to light related to it certainly has parallels to the story we're telling today, we're instead talking about a simple scientific fact that has enjoyed even stronger support from the scientific community over a longer period of time, while also applying to virtually every living creature on the planet Earth.

We're talking about the relationship between an animal's body mass and its resting metabolism. Physorg's Joshua E. Brown explains (we've added the links for reference):

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.

That does sound quite a bit like the recent Climategate scandal. As you'll see next, the parallels regarding the handling, or mishandling of data are indeed very close. The following discussion will also hopefully explain the image we posted in the upper right hand corner of our post today!

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!"

Again, we're amazed by the parallels with the Climategate scandal. Next, we'll see otherwise rational scientists fall into line over the apparent relationship. Again, we've added the links for reference:

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."

Instead of explaining it, in the 1960s a Scottish conference on energy metabolism simply voted, 29-0, to enshrine 3/4 as the official exponent. Then, 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 advisor and a fellow student in 2001," Dodds says. They also reanalyzed data from Kleiber and six other scientists and concluded that there is little empirical evidence for rejecting 2/3 in favor of 3/4. "But we didn't have a better theory," Dodds says, "or some way to clean it all up."

Until now. Dodds's new paper explores the geometry of branching networks -- like blood supply -- to show how a material, like blood, can be most efficiently delivered. "If you're going to build organisms with a central source, like a heart, that places physical constraints that evolution has to run up against," he says. "These constraints won't let the ratio be too far away from 2/3."

"My new paper follows the argument that was put forth in 1997 -- that, somehow, networks give rise to the 3/4 law. They were right that supply networks are key to understanding the metabolic limitations of animals. Except my paper shows that networks give rise to the 2/3 law, actually," Dodds says, "If you do the math properly."

And here's the deviation from the Climategate scandal. Where animal body mass and metabolism are concerned, science worked actively to correct itself in the light of new data and stronger analysis.

But then, the financial incentive for those involved in promoting the flawed "3/4 law of metabolism" aren't anywhere near the level of those promoting the AGW climate change theory....

Previously on Political Calculations

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