Fascinating new scientific papers suggest how elementary geometry involving animals' physical dimensions is sufficient to shed light on some very basic biological phenomena. In particular, the papers attempt to determine the metabolic pace of all life and, in the process, help resolve a problem in evolutionary time measurement.
Hold on tight.
Let's start by considering why an animal can't be, say, five times its normal adult size. To understand that we can't simply multiply physical dimensions by a factor of five, imagine what would happen if a 6-foot, 160 pound man were scaled up to a height of 30 (6 x 5) feet. His weight, like his volume (in cubic feet), would increase not by a factor of five, but by a factor of 5^3 and thus would rise from 160 pounds to 20,000 pounds (160 x 5^3) -- 125 times as great as his original weight if he were proportioned similarly.
And what would hold up such a behemoth? The supporting cross-sectional area of his thighs, say 2 square feet originally, would increase not by a factor of five, but by a factor of 5^2 and would thus rise to 50 square feet (2 x 5^2) -- 25 times as great as the normal area if he were proportioned similarly. (The same would hold for his spine, knees and so on.)
But the pressure on his thighs -- his weight divided by the area of a cross-section of his thighs, i.e. 125 times his original weight divided by 25 times the original area -- would be five times as great. This would be a crushing pressure and the man would collapse. This is why heavy land animals like elephants and rhinos have such thick legs.
Metabolic Rates: Live Fast and Die
Mathematical considerations not too dissimilar to these also lie behind various scaling laws in biology relating animals' metabolic rates -- heart, breathing, twitching, etcetera -- to their surface areas and masses. Small animals' hearts, for example, beat faster than large animals' hearts and, more generally, they live faster and die younger than do large animals who measure out their energies at a more lumbering pace.
Since areas, including animals' surface areas, scale up with the 2nd power of their relative dimensions, and their masses or volumes scale up with the 3rd power, such considerations long ago led scientists to the belief that animals' metabolic rates were proportional to the surface areas of their skins or, equivalently, proportional to their masses to the 2/3rd power.
Much evidence suggests, however, that metabolic rates are proportional to animals' masses to the 3/4th power, not the 2/3rd power, and recent papers by ecologists Brian Enquist and James Brown and physicist Geoffrey West explain why this and other "quarter power scaling laws" make sense theoretically as well as empirically. (The short explanation is that the metabolic rate is affected not by how fast heat dissipates through the skin, but rather by how efficiently nutrients reach the body's cells and for this latter quantity a broader definition of "surface" area reflecting internal structure is needed.)
In the July 2004 issue of Ecology, Enquist, Brown and West go further and suggest that their quarter power formula (with some refinements involving not only mass but also temperature) describes the metabolic rates of all living organisms, plant as well as animal. As an associate, biologist James Gillooly has observed, "When you correct for size and temperature, the metabolic rates of a shark, a tomato plant and a tree are remarkably similar."
The authors even claim these metabolic considerations apply generally to biological phenomena on all scales ranging from the mutation rate for DNA to the speed with which ecosystems change. Again very simplistically put: small, hot phenomena proceed at a faster pace than do large, cold ones.
Resolving the Puzzle of Evolutionary Divergence
At the risk of losing readers who refuse to attend Imax theaters showing evolution-themed work, I note that it is this line of thought that also leads to a nice resolution of a problem in evolution. Why do various methods of calculating when two species branched apart often lead to quite different answers?
For example, by examining the DNA of rats and mice, seeing how many dissimilarities there are, and calculating how long it would take for this many mutations to come about, geneticists have placed the branching around 40 million years ago. But archaeologists looking at the fossils say that the divergence between rats and mice occurred much more recently, about 12 million years ago.
How can we reconcile these numbers? One answer suggested by Gillooly is based on the work above on metabolic rates. In the January issue of the Proceedings of the National Academy of Sciences, he and his colleagues stress that although small animals don't live as long as larger ones, their metabolic rates are such that their life spans, when measured by these rates rather than by physical times, are comparable to those of bigger animals.
In other words rats and mice, being small, live at a faster metabolic pace than do larger animals. Because of this and an associated quicker accumulation of mutation-inducing free radicals, their DNA mutates faster than that of larger animals, and hence they require less physical time to diverge as much as they have, not 40 million or so years but approximately 12 million years as the fossil record indicates. Similar reconciliations using animal-specific "metabolic clocks" rather than physical ones exist for other pairs of small animals.
Of course, these scaling laws are crude measuring instruments and admit of many exceptions, and not every biologist is convinced of their utility. Nevertheless, scaling laws do give us a rough handle on metabolic rates that is, on the whole, very suggestive. They also explain why there are no 30-foot-tall people walking around, except possibly in the imagination of those who believe that humans and dinosaurs co-existed.
-- Professor of mathematics at Temple University, John Allen Paulos is the author of best-selling books, including "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.