Size is perhaps the most astonishing indicator of the variety of life on earth. The blue whale, the largest animal ever to exist, is 1021 times the size of the smallest microbe. Researchers since DaVinci have grappled with the biological implications of size in various creatures; even though it varies tremendously between species, size affects almost every aspect of life, from longevity to locomotion. In the past few decades, biologists have found an intriguing similarity underlies this incredible range of variation. An organism’s size and aspects of its physiology are tied together by simple scaling laws--laws that practically every living thing obeys. Characteristics like the metabolic rate of a mouse or an elephant, or their life span or their heart rate, or the cross- sectional area of a tree trunk, all vary in proportion to mass, following a simple mathematical formula.
A mouse that weighs 50 grams, for example, will have a metabolism that produces 5 kilocalories of heat per day. A 5000 kilogram elephant--a hundred thousand times the size of the mouse--produces 60,000 kilocalories per day. In 1932, biologist Max Kleiber of the University of California at Davis realized that when he compared the metabolic rates of mice, elephants--and just about every animal he knew--they all varied in proportion to their mass raised to the 3/4 power. What puzzled Kleiber, and many researchers since, is that this scaling law doesn’t reflect a simple geometric relationship. If, for example, metabolic rate were somehow dependent on surface area--perhaps to counteract heat loss through the skin--you would expect it to vary, like surface area, in proportion to mass raised to the 2/3 power. (The two and the three represent the number of dimensions in surface area and mass, respectively.) Other scaling laws have been discovered since Kleiber, and all of them use quarter powers, not thirds. Life span scales as mass to the 1/4 power, and heart rate as mass to the -1/4 power. For decades, no one has found a satisfactory explanation of these 1/4 power laws--until now, thanks to the collaborative efforts of physicist Geoffrey West of Los Alamos National Laboratory, and ecologists James Brown and Brian Enquist at the University of New Mexico. Scaling laws, the team says, stem from the branching transportation networks that all organisms use to deliver nutrients to all parts of the body.
Brown had been interested in scaling laws since his days as a graduate student, and was re-energized by the work of his own graduate student, Enquist, who realized that the metabolic rate of plants also followed a 3/4 power law. Earlier attempts to explain quarter power laws involved ideas like resistance to buckling forces, and were too specific too explain things like plant metabolism. They don’t account for the fact that you find these 1/4 powers almost universally, in nearly all organisms and over a wide range of biological structures and processes, says Brown. Enquist and Brown eventually came around to the idea that transportation of nutrients was the key to the problem. We needed to look at something that was not strictly geometric, and also biologically universal, Brown says. Getting essential resources around a body applies to every organism-- that’s a universal problem.
West, a theoretical physicist, was examining the role of scaling in physical interactions, and became interested in the problem of scaling in general. Whenever I gave talks to a general audience, I would try to find examples in other areas, says West. And then I came upon these wonderful scaling laws in biology. West started thinking about scaling laws of life span and longevity, and independently hit on the same idea. We’re all just a bunch of branches, West realized. We’ve all got to feed billions of cells. The three teamed up through the Santa Fe Institute, which specializes in interdisciplinary studies, about two years ago. With their combined knowledge of mathematics and living systems, they were able to hammer out a mathematical model of resource transport in living things, from which the power laws for various features could be derived, and other essential features could be predicted.
The model rests on three assumptions: first, in order for a system to supply resources to every corner of an organism’s body, it has to have a three-dimensional branching pattern. Also, they reasoned, the final branches of the network would be the same size in all organisms, because cells in most species are about the same size. And finally, the network should use a minimum amount of energy to distribute resources. The system that would best fit all these assumptions, they realized, was fractal. The repetitive patterns of fractal geometry have been used to model natural systems from rivers to snowflakes--so it seemed a likely candidate for living delivery systems, like an animal’s cardiovascular and respiratory systems, a plant’s vascular system, or an insect’s tracheae, the hollow tubes through which oxygen diffuses in an insect’s body. Once you get the idea that this may be a fractal, that becomes very attractive, says Brown, because fractals are known to produce other scaling exponents than the simple geometric ones, the multiples of one-third.
The key to deriving quarter powers came from the team’s discovery that in all distribution systems, area is preserved. DaVinci was the first to point out this phenomenon in trees. Suppose you have a tree that splits into two main branches. If you saw off each main branch, their combined cross-sectional area will add up to the area of the base. This phenomenon continues all the way up to the tiniest twigs in the treetops--if they were all bundled together, they would take up the same area as the tree trunk. Preservation of area in tree is easy to understand--trees are essentially bundles of tubes held tightly in the trunk, and allowed to splay outwards in the roots and the branches. But, as West says, Why the hell would the respiratory or cardiovascular system be the same way? Simply put, the physics of the air going through the lungs, and the blood flowing in our veins, forces these systems to preserve area just like a tree. And how does this translate into quarter powers? About three-and-a-half pages of dense mathematics provides the answer. In a way, it’s reassuring that it can’t be summed up in a sentence, says West. Otherwise I’m sure someone else would have figured this out 50 years ago.
Once the team worked out the mathematical and physical constraints of the system, they had a set of formulas that not only proved where the scaling laws originated, but also could predict other features of resource transport in living things, like the number of capillaries in a mammalian cardiovascular system, and the maximum height of a tree (pretty close to some of the tallest trees today, around 380 feet). It also predicts the amount of branching in a system. Even though a whale is 107 time heavier than a mouse, it only needs 70% more branches from its aorta to its capillaries than a mouse does. And, Brown says, if their model is sound, one-celled organisms should have a similar branching structure. We don’t know the whole internal workings of a cell, but we do know their interiors are chock full of structures like microtubules, says Brown. The model predicts that they should have essential fractal-like features.
Ever since the model was published, Brown has been getting correspondence from people in fields ranging from neurobiology to pharmacology to city planning. Nerve growth and development may follow a fractal pattern, and the model may provide a way to test this theory. It would also be a big help to people trying to predict the effects of certain chemicals entering a human body, when they can only test it on mice. Even artificial delivery systems, like water mains, phone lines, or highways, have a de facto fractal pattern, says Brown, and could benefit from an efficiency expert like Mother Nature. With a few minor modifications, this model can be applied to very wide range of systems, says Brown. The way nature has solved that problem is really so simple and elegant. Some of the mathematical details are pretty hairy, but the idea itself is so simple. This is one of the most fundamental themes about the way organisms work.