The 3/4 Power Law

Biology professor James Brown explains how this fundamental law is helping scientists make predictions about organisms' behaviors, lifespans, and other characteristics.

by Russell Moore

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Look at a bottle of aspirin. The dosage information will likely suggest giving a fifty pound child about half as much aspirin as a hundred pound child, who in turn is advised to take about half as much as an adult man. That’s a linear dosage scale: twice the body size, twice the dose.

But that’s not the right way to determine dosage, says UNM Professor of Biology James Brown. Metabolic rate, not body mass, determines how much of a drug—or food, or any substance—an organism needs or can handle.

Aspirin isn’t a particularly dangerous drug. But what about antibiotics? Painkillers? Anesthetics? Over- or under-dosing can make a drug therapy program useless at best—and possibly even fatal.

Brown is investigating these and other implications of new research and theories. A recent discovery of why every animal’s metabolic rate equals its mass raised to the 3/4 power (M^3/4) is helping scientists make new predictions about organisms’ behaviors, lifespans, and other characteristics.

In 1997, Brown and two colleagues published a short paper that all but silenced critics of the 3/4 power law, which was first discovered in the 1930s. By finally showing why the 3/4 rule exists, the team was simultaneously able to explain deviation from the rule and enable scientists to compare varying forms of life.

Put simply, blood vessels have a finite amount of space to fill, so they do so in the most efficient way possible: as a fractal network of vessels branching out in diminishing size from the heart all the way to the capillaries.
(A fractal is a structure or pattern that is self-similar at different resolutions; for example, a deciduous tree’s branches are the same size and shape relative to the bigger limbs as the limbs are to the trunk. Break off a small piece from the end of a branch, and you’ll see the same structure.)

Complex math is involved, but the paper proposes that since heart rate, aorta size, and time of blood circulation all follow quarter-scaling patterns, metabolic rate must scale by 3/4.

The smallest level in a circulatory system, the capillary, is the same size in all animals. “We don’t use giant doorknobs or electrical outlets when we build skyscrapers,” Brown says. The ends of the network are always the same size.

Organisms aren’t the only things whose scale can be amplified or minimized in a predictable way. Professor of Biology Bruce Milne is examining how streams branch within a watershed—like blood vessels, in a fractal pattern. And like metabolic rate, the amount of runoff in a watershed is scaled exponentially. Furthermore, the point at which the amount of runoff shifts from a negative exponent to a positive one—also the point at which an area’s designation shifts from arid to humid—is believed to be where the most efficient plants grow: plants that maximize their conversion of water and solar radiation to plant mass.

Brown points out that his team’s explanation of why the 3/4 law exists also helps to show why not every animal displays a metabolic rate that follows the law exactly. Think of gravity, he says: the theory of gravity tells us that any two objects, regardless of size, will fall at exactly the same rate. But go outside with a hammer and a feather, and you’ll quickly see that they don’t hit the ground at the same time. External factors—air pressure, wind resistance—must be controlled or eliminated. For this experiment to uphold a law that we all accept, it must take place in a vacuum.

But life doesn’t exist in a vacuum, and we don’t know what external forces we must control in order to make the 3/4 law work, Brown says. He believes that if we could control environments the way we can control a vacuum, every single organism would display a metabolic rate exactly equal to M^3/4.

Brown explains metabolic rate is a “first-order predictor” of a huge range of behaviors. Lifespan, rates of evolution, even mating calls can all be determined once scientists know an animal’s size and temperature. These and other behaviors “are direct expressions of metabolic rate,” he says.

Since the 3/4 law appears to exist throughout biology, Brown says, it should be considered one of a few known fundamental unifying theories of life. He says the law is further evidence of a common ancestor, and additional support for Darwin’s theory of evolution.