# Scientists Explain a Fundamental Equation of Life

Since life first sprung from the primordial ooze about 3.6 billion years ago, it has evolved a multitude of fantastical forms. Towering redwood trees soar to heights of 300 feet, while hundreds of microscopic green algae can fit on the tip of your finger. Yet both are plants! A single blue whale weighs in at 190 tons, equaling the mass of about 21,000 red foxes. Yet both are animals! Life's diversity is truly incredible.

No less amazing are the features that we share in common. For example, all life -- as we know it -- is composed of cells, can reproduce, and can adapt or react to its surroundings. You may have noticed that those traits are more descriptive than concrete. That's because life is quite tricky to define. One of existence's most beautiful ironies is that we still have no unequivocal definition of life.

Scientists are ever on the hunt for commonalities, however. One of the most intriguing that they've discovered is a simple equation: *q*_{0} ~ *M*^{¾}. Termed Kleiber's Law (named for the Swiss biologist Max Kleiber), the equation states that an organism's basal metabolic rate -- the amount of energy it consumes at rest -- is roughly equal to its mass raised to the three-quarters power. That's quite impressive if you think about it, that such a simple equation can apply throughout the animal and plant kingdoms!

Even more awe-inspiring is one explanation for why this is so. In 1999, a trio of scientists led by Geoffrey West from Los Alamos National Laboratory postulated that the answer lies in fractals -- self-similar, repeating, geometric patterns. If that description seems in any way confusing or vague, don't worry. Like the concept of life, experts disagree on the precise definition of fractals. Mathematician Benoît Mandelbrot, who first devised the term, candidly characterized them only as "beautiful, damn hard, increasingly useful." But whether or not you know how to describe a fractal, you know one when you see it.

According to West and his colleagues, fractals are a facet of life.

"Unlike the genetic code, which has evolved only once in the history of life, fractal-like distribution networks that confer an additional effective fourth dimension have originated many times. Examples include extensive surface areas of leaves, gills, lungs, guts, kidneys, chloroplasts, and mitochondria, the whole-organism branching architectures of trees, sponges, hydrozoans, and crinoids, and the treelike networks of diverse respiratory and circulatory systems."

But why would life take on such a geometry? West explained that fractal-like shapes are terrific at maximizing surface area, which permits nutrients to be more efficiently transported within and throughout biological entities and structures.

"The vast majority of organisms exhibit scaling exponents very close to 3/4 for metabolic rate and to 1/4 for internal times and distances. These are the maximal and minimal values, respectively, for the effective surface area and linear dimensions for a volume-filling fractal-like network," West noted, before later coming to a spectacular conclusion. "Fractal geometry has literally given life an added dimension."

Understandably, not everyone was convinced by West's theoretical paper. Opponents of his claims countered that the role of fractal branching, particularly in the capillaries of the heart is not fundamental to the exponent 3/4. West later joined with some of his critics and admitted that fractals don't completely explain the 3/4 exponent, as it bears true even in organisms without apparent "fracticality" in their internal forms.

Last Tuesday, a international team of biologists and physicists politely set aside West's fractal idea and tendered a new explanation for Kleiber's Law: it's a mathematical expression of an evolutionary fact: that animals try to use energy as efficiently as possible.

"An organism is akin to an engine," they explained. "Part of the energy obtained from nourishment is used for organism function, growth, reproduction, while the rest is dissipated through its surface."

"Plant and animal geometries have evolved more or less in parallel," said University of Maryland botanist Todd Cooke, one of the authors. "The earliest plants and animals had simple and quite different bodies, but natural selection has acted on the two groups so the geometries of modern trees and animals are, remarkably, displaying equivalent energy efficiencies. They are both equally fit. And that is what Kleiber's Law is showing us."

Though compelling, Cooke and his colleagues' explanation of Kleiber's Law likely won't be the last.

**Sources: **

Jayanth R. Banavar, Todd J. Cooke, Andrea Rinaldo, and Amos Maritan Form, function, and evolution of living organisms *PNAS* 2014 ; published ahead of print February 18, 2014, doi:10.1073/pnas.1401336111

The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms. Geoffrey B. West, James H. Brown, and Brian J. Enquist *Science* 4 June 1999: 284 (5420), 1677-1679. [DOI:10.1126/science.284.5420.1677]

(Images: Wikimedia Commons, Max Kleiber)