Life on the scales: simple mathematical relationships underpin much of biology and ecology

Science News,  Feb 12, 2005  by Erica Klarreich

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A mouse lives just a few years, while an elephant can make it to age 70. In a sense, however, both animals fit in the same amount of life experience. In its brief life, a mouse squeezes in, on average, as many heartbeats and breaths as an elephant does. Compared with those of an elephant, many aspects of a mouse's life--such as the rate at which its cells burn energy, the speed at which its muscles twitch, its gestation time, and the age at which it reaches maturity--are sped up by the same factor as its life span is. It's as if in designing a mouse, someone had simply pressed the fast-forward button on an elephant's life. This pattern relating life's speed to its length also holds for a sparrow, a gazelle, and a person--virtually any of the birds and mammals, in fact. Small animals live fast and die young, while big animals plod through much longer lives.

"It appears as if we've been gifted with just so much life," says Brian Enquist, an ecologist at the University of Arizona in Tucson. "You can spend it all at once or slowly dribble it out over a long time."

Scientists have long known that most biological rates appear to bear a simple mathematical relationship to an animal's size: They are proportional to the animal's mass raised to a power that is a multiple of 1/4. These relationships are known as quarter power scaling laws. For instance, an animal's metabolic rate appears to be proportional to mass to the 3/4 power, and its heart rate is proportional to mass to the--1/4 power.

The reasons behind these laws were a mystery until 8 years ago, when Enquist, together with ecologist James Brown of the University of New Mexico in Albuquerque and physicist Geoffrey West of Los Alamos (N.M.) National Laboratory proposed a model to explain quarter-power scaling in mammals (SN: 10/16/99, p. 249). They and their collaborators have since extended the model to encompass plants, birds, fish and other creatures. In 2001, Brown, West, and several of their colleagues distilled their model to a single formula, which they call the master equation, that predicts a species' metabolic rate in terms of its body size and temperature.

"They have identified the basic rate at which life proceeds," says Michael Kaspari, an ecologist at the University of Oklahoma in Norman.

In the July 2004 Ecology, Brown, West, and their colleagues proposed that their equation can shed light not just on individual animals' life processes but on every biological scale, from subcellular molecules to global ecosystems. In recent months, the investigators have applied their equation to a host of phenomena, from the mutation rate in cellular DNA to Earth's carbon cycle.

Carlos Martinez del Rio, an ecologist at the University of Wyoming in Laramie, hails the team's work as a major step forward. "I think they have provided us with a unified theory for ecology," he says.

THE BIOLOGICAL CLOCK 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.

For most of the past 70 years, ecologists had no explanation for the 3/4 exponent. "One colleague told me in the early '90s that he took 3/4-scaling as 'given by God,'" Brown recalls.

The beginnings of an explanation came in 1997, when Brown, West, and Enquist described metabolic sealing in mammals and birds in terms of the geometry of their circulatory systems. It turns out, West says, that 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.