Complexity_ A Guided Tour - Melanie Mitchell [125]
This power law relation is now called Kleiber’s law. Such 3/4-power scaling has more recently been claimed to hold not only for mammals and birds, but also for the metabolic rates of many other living beings, such as fish, plants, and even single-celled organisms.
Kleiber’s law is based only on observation of metabolic rates and body masses; Kleiber offered no explanation for why his law was true. In fact, Kleiber’s law was baffling to biologists for over fifty years. The mass of living systems has a huge range: from bacteria, which weigh less than one one-trillionth of a gram, to whales, which can weigh over 100 million grams. Not only does the law defy simple geometric reasoning; it is also surprising that such a law seems to hold so well for organisms over such a vast variety of sizes, species types, and habitat types. What common aspect of nearly all organisms could give rise to this simple, elegant law?
Several other related scaling relationships had also long puzzled biologists. For example, the larger a mammal is, the longer its life span. The life span for a mouse is typically two years or so; for a pig it is more like ten years, and for an elephant it is over fifty years. There are some exceptions to this general rule, notably humans, but it holds for most mammalian species. It turns out that if you plot average life span versus body mass for many different species, the relationship is a power law with exponent 1/4. If you plot average heart rate versus body mass, you get a power law with exponent −1/4 (the larger an animal, the slower its heart rate). In fact, biologists have identified a large collection of such power law relationships, all having fractional exponents with a 4 in the denominator. For that reason, all such relationships have been called quarter-power scaling laws. Many people suspected that these quarter-power scaling laws were a signature of something very important and common in all these organisms. But no one knew what that important and common property was.
An Interdisciplinary Collaboration
By the mid-1990s, James Brown, an ecologist and professor at the University of New Mexico, had been thinking about the quarter-power scaling problem for many years. He had long realized that solving this problem—understanding the reason for these ubiquitous scaling laws—would be a key step in developing any general theory of biology. A biology graduate student named Brian Enquist, also deeply interested in scaling issues, came to work with Brown, and they attempted to solve the problem together.
Brown and Enquist suspected that the answer lay somewhere in the structure of the systems in organisms that transport nutrients to cells. Blood constantly circulates in blood vessels, which form a branching network that carries nutrient chemicals to all cells in the body. Similarly, the branching structures in the lungs, called bronchi, carry oxygen from the lungs to the blood vessels that feed it into the blood (figure 17.3). Brown and Enquist believed that it is the universality of such branching structures in animals that give rise to the quarter-power laws. In order to understand how such structures might give rise to quarter-power laws, they needed to figure out how to describe these structures mathematically and to show that the math leads directly to the observed scaling laws.
Most biologists, Brown and Enquist included, do not have the math background necessary to construct such a complex geometric and topological analysis. So Brown and Enquist went in search of a “math buddy”—a mathematician or theoretical physicist who could help them out with this problem but not simplify it so much that the biology would get lost in the process.
Left to right: Geoffrey West, Brian Enquist, and James Brown. (Photograph copyright © by Santa Fe Institute. Reprinted with permission.)
FIGURE 17.3. Illustration