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## Metabolism and Mathematics

Some very interesting mathematical biology:

Scientists have known for nearly two centuries that larger animals have relatively slower metabolisms than small ones. A mouse must eat about half its body weight every day not to starve; a human gets by on only 2%. The first theories to explain this trend, developed in the late nineteenth century by the German nutritionist Max Rubner and the French physiologist Charles Richet, were based on the ratio between an animal’s surface area, which changes with the square of its length, and its volume, which is proportional to its length cubed. So large animals have proportionately less surface area, lose heat more slowly, and, pound for pound, need less food. The square-versus-cube relationship makes the area of a solid proportional to the two-third power of its mass, so metabolic rate should also be proportional to mass2/3. For many years, most biologists thought that it was.

But in 1932, Max Kleiber, an animal physiologist working at the University of California’s agricultural station in Davis, re-examined the question, and found that, for mammals and birds, metabolic rate was mass0.73—closer to three quarters than two thirds. Kleiber looked at animals ranging in size from a rat to a steer. By the mid-1930s, other workers had put together a “mouse to elephant” curve that supported the three-quarter-power law, and by the 1960s, the plot had been extended for everything from microbes to whales, still seeming to show the same relationship. Quarter-power scaling also began to stretch beyond metabolic rate. Biological times, such as lifespan and heart rate, were found to be proportional to mass1/4, and fractions related to one-quarter show up in other scaling relationships: the diameter of the aorta and tree trunks is proportional to mass3/8, for example.

It was, however, much harder to find a theoretical reason for why metabolic rate should be proportional to mass3/4—and more generally, why quarter-power scaling laws should be so prevalent in biology. The impasse meant that by the mid-1980s interest in scaling had waned. But it sparked back into life in 1997, when two ecologists—James Brown of the University of New Mexico, Albuquerque, and his graduate student Brian Enquist, now at the University of Arizona, Tucson—and a physicist, Geoffrey West of the Santa Fe Institute, developed a new explanation of why metabolic rate should equal the three-quarter power of body mass.

West, Brown, and Enquist’s theory is based on the structure of biological distribution networks, such as blood vessels in vertebrates and xylem in plants. The trio assumed that metabolic rate equals the rate at which these networks deliver resources, and that evolution has minimized the time and energy needed to get materials from where they are taken up—the lungs or roots, for example—to the cells. They also assumed that, although organisms vary greatly in size, the terminal units in their distribution networks, such as blood capillaries or leaf stalks, do not.

Bigger plants and animals take longer to transport materials, and so use them more slowly. In West, Brown, and Enquist’s model, the maximally efficient network that serves every part of a body has a fractal structure, showing the same geometry at different scales. And the number of uniform terminal units in such a network—and so the rate at which resources are delivered to the cells—is proportional to the three-quarter power of body mass.

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