Animal lifespans and space-filling curves

Science News has a review article on the 3/4 law of animal lifespans and metabolism.

In 1883, German physiologist Max Rubner proposed that an animal's metabolic rate is proportional to its mass raised to the 2/3 power. This idea was rooted in simple geometry. If one animal is, say, twice as big as another animal in each linear dimension, then its total volume, or mass, is 23 times as large, but its skin surface is only 22 times as large. Since an animal must dissipate metabolic heat through its skin, Rubner reasoned that its metabolic rate should be proportional to its skin surface, which works out to mass to the 2/3 power.

In 1932, however, animal scientist Max Kleiber of the University of California, Davis looked at a broad range of data and concluded that the correct exponent is 3/4, not 2/3. In subsequent decades, biologists have found that the 3/4-power law appears to hold sway from microbes to whales, creatures of sizes ranging over a mind-boggling 21 orders of magnitude. …

Rubner was on the right track in comparing surface area with volume, but that an animal's metabolic rate is determined not by how efficiently it dissipates heat through its skin but by how efficiently it delivers fuel to its cells.

Rubner should have considered an animal's "effective surface area," which consists of all the inner surfaces across which energy and nutrients pass from blood vessels to cells, says West. These surfaces fill the animal's entire body, like linens stuffed into a laundry machine.

The idea, West says, is that a space-filling surface scales as if it were a volume, not an area. If you double each of the dimensions of your laundry machine, he observes, then the amount of linens you can fit into it scales up by 23, not 22. Thus, an animal's effective surface area scales as if it were a three-dimensional, not a two-dimensional, structure.

This creates a challenge for the network of blood vessels that must supply all these surfaces. In general, a network has one more dimension than the surfaces it supplies, since the network's tubes add one linear dimension. But an animal's circulatory system isn't four dimensional, so its supply can't keep up with the effective surfaces' demands. Consequently, the animal has to compensate by scaling back its metabolism according to a 3/4 exponent.

Though the original 1997 model applied only to mammals and birds, researchers have refined it to encompass plants, crustaceans, fish, and other organisms. The key to analyzing many of these organisms was to add a new parameter: temperature.

Mammals and birds maintain body temperatures between about 36°C and 40°C, regardless of their environment. By contrast, creatures such as fish, which align their body temperatures with those of their environments, are often considerably colder. Temperature has a direct effect on metabolism—the hotter a cell, the faster its chemical reactions run.

In 2001, after James Gillooly, a specialist in body temperature, joined Brown at the University of New Mexico, the researchers and their collaborators presented their master equation, which incorporates the effects of size and temperature. An organism's metabolism, they proposed, is proportional to its mass to the 3/4 power times a function in which body temperature appears in the exponent. The team found that its equation accurately predicted the metabolic rates of more than 250 species of microbes, plants, and animals. These species inhabit many different habitats, including marine, freshwater, temperate, and tropical ecosystems. …

A single equation predicts so much, the researchers contend, because metabolism sets the pace for myriad biological processes. An animal with a high metabolic rate processes energy quickly, so it can pump its heart quickly, grow quickly, and reach maturity quickly.

Unfortunately, that animal also ages and dies quickly, since the biochemical reactions involved in metabolism produce harmful by-products called free radicals, which gradually degrade cells.

"Metabolic rate is, in our view, the fundamental biological rate," Gillooly says. There is a universal biological clock, he says, "but it ticks in units of energy, not units of time." …

The team's master equation may resolve a longstanding controversy in evolutionary biology: Why do the fossil record and genetic data often give different estimates of when certain species diverged? …

The problem is that there is no universal clock that determines the rate of genetic mutations in all organisms, Gillooly and his colleagues say. They propose in the Jan. 4 Proceedings of the National Academy of Sciences that, instead, the mutation clock—like so many other life processes—ticks in proportion to metabolic rate rather than to time.

The DNA of small, hot organisms should mutate faster than that of large, cold organisms, the researchers argue. An organism with a revved-up metabolism generates more mutation-causing free radicals, they observe, and it also produces offspring faster, so a mutation becomes lodged in the population more quickly.

When the researchers use their master equation to correct for the effects of size and temperature, the genetic estimates of divergence times—including those of rats and mice—line up well with the fossil record.