The Power of Power Laws

Michael Trott, © Wolfram Research,Inc. The possibility of mathematical power laws governing the scaling of fundamental biological properties, such as metabolic rate, within a species group has been strongly suspected for almost a century. But since 1997, the laws have been confirmed by overwhelming experimental evidence and backed by convincing mathematical theory.

By | April 21, 2003

Michael Trott, © Wolfram Research,Inc.

The possibility of mathematical power laws governing the scaling of fundamental biological properties, such as metabolic rate, within a species group has been strongly suspected for almost a century. But since 1997, the laws have been confirmed by overwhelming experimental evidence and backed by convincing mathematical theory. Before, research biologists were puzzled by the fact that a wide range of ultimately related properties, such as aortal surface area in warm-blooded animals, and trunks or stems in plants, ranged in line with a fourth, rather than a third, power law. This latter law was established in 1932.

This long-awaited explanation of why biology scales in four rather than three dimensions emerged during the late 1990s, largely through the interdisciplinary work of physicist Geoffrey West, and ecologists James Brown and Brian Enquist, with their 1997 publication.1 Subsequent papers from this team showed that their theory is applicable to all life forms.2-5

The work has attracted attention--both positive and negative. "I think it's very important," says evolutionary biologist and popular science writer Richard Dawkins. "It's a powerful theory, explaining a wide range of biological scale rules with great economy." It tends, he says, "to be fully accepted by people who learn about it, and I would hope that it will become better known."

On the negative side, physicists such as Peter Dodds and others at the Massachusetts Institute of Technology claimed in 2001 that their analyses of data sets taken over a long period did not provide irrefutable evidence for 3/4 power scaling.6 Other physicists argue that the West team's work is based on assumptions that need not be made. Jayanth Banavar, a theoretical physicist at Pennsylvania State University, believes that Brown's assumption that a resource network must have a fractal branching structure is unnecessary. "The question we asked is, 'Which aspects of the problem are essential [for explaining the 3/4 power scaling] and which are really peripheral?'" says Banavar. "Fractality is not the underlying explanation."

Although the work is not yet well known among biologists, practical applications of these laws are emerging. Examples include determination of correct drug doses for humans based on trials on smaller animals, and acreage needs of animals of all sizes in the wild.

UNDERSTANDING POWER LAWS Here's a geometry refresher. A power law relates one variable to another raised to a constant power. The general form takes y = xa, where y and x are variables, and a is a constant exponent (hence power) such as 2. By contrast, an exponential relationship takes the form y = ax, where y varies, a is constant and usually the natural logarithm base e (about 2.7), with x the exponent as the variable on the right hand of the equation.

Courtesy of Phillip Hunter

In biology, the principal group of power laws relates a variety of fundamental properties to the mass, M, of the organism. Now M varies in line with the organism's volume and therefore as the cube of linear dimensions such as height. At first sight, it would seem that other geometrically based properties such as aorta diameter--and, in turn, functions such as metabolic rate (the rate at which each cell produces energy) that depend on these properties--should scale in line with the organism's volume as some third-power law. But instead, they scale in various forms of a fourth-power law, as if the organism's mass were a function of four rather than three linear dimensions.7

The oldest and best-known power relationship is Kleiber's law,7 devised by Swiss-American zoologist Max Kleiber in 1932. It states that the metabolic rate of individual cells scales as M-1/4 so that the metabolic capacity of a whole organism (calculated by multiplying the total number of cells by the average metabolic rate of each cell) scales as M3/4. If cellular metabolic rate were a function of the organism's volume, this would scale as M-1/3 and the total metabolic capacity as M2/3.

DO THE MATH The 1997 paper1 derived the quarter-power laws from the hydrodynamics and fractal geometry of the organism's branching network, in this case the blood circulation of mammals, that delivers resources to an organism's cells. Resources enter the blood stream via surfaces--the gut lining for nutrients and water, and the alveoli of the lungs for oxygen. The key point, and source of the fourth dimension, is the fact that the terminating units through which resources pass into the blood stream, and out into cells, are of fixed size independent of the mammal. Larger mammals just have more of them. Therefore, the area of exchange between each terminating unit remains the same. So, an organism's total area of exchange depends on the total number of these invariant terminating units, and these are packed into a volume, such as the lung or the gut's epithelial lining. The total area of resource exchange, comprising all the invariant terminating units, therefore scales in three dimensions.

But the average distance the blood travels to take resources from the source exchange point, such as the lung, to the destination cell, also increases with the organism's size, but in a single dimension. The total blood volume is then a product of the terminating unit's total area, which scales in three dimensions, and the average distance between source and destination, which scales in one. The result is a quantity that varies in four dimensions, but which also must range in line with the organism's mass if the ratio of blood volume to total tissue is to remain constant. This does not immediately yield the precise quarter-power scaling, but the relationships follow from consideration of the dynamics of branching, subject to the principle that natural selection evolves efficiently by transferring resources to the blood stream and then distributing it, Brown says.

MULTICELLULAR TO UNICELLULAR The theory of the quarter-power law has turned out to apply even more widely than its pioneers expected. According to Brown, it was not clear whether quarter-power scaling applied to such a wide range of phenomena--metabolic rate, aortal diameter--that the underlying theory had to be generalized. Consequently, the work led to a flurry of experiments and measurements showing that these quarter-power scaling laws were more ubiquitous than almost anyone expected.4

Kleiber's law scales from the largest to the smallest mammal, that is, from blue whales to shrews, which covers about six orders of magnitude of mass, with the same applicable to birds. But, says West, it is now known through their work and others that Kleiber's law scales right down to unicellular organisms, increasing the span of mass-magnitude order to 20, and it also applies to cold-blooded creatures, plants and trees with exactly the same power law exponent of 3/4.

It goes further, as West explains. "A year or two ago, we looked down intracellularly within mammalian cells at the mitochondria [the energy-producing organelles within cells], and then went inside the mitochondria. This comprises a bunch of membranes, and in those is a whole sequence of complex molecules called respiratory complexes" (where ATP, the currency of energy, is manufactured aerobically combining oxygen and glucose). Brown continues, "We then went further still because the respiratory complex consists of groups of molecules, some of which are enzymes that control the reaction."

Even within the respiratory complex, West et al. found the metabolic rate to scale as a 3/4 power of mass.2,3 This extends the range of Kleiber's law to 27 orders of mass magnitude, from large trees or blue whales down to individual metabolic components at the molecular level. In effect, Kleiber's law now embraces almost all biological processes relying on distribution of metabolic resources, such as unicellular organisms and even the mitochondria within them.

Tammy Irvine, Rear View Illustration
 SCALES AND THEN SOME: Kleiber's law scales from the largest to the smallest mammal, namely the blue whale to the shrew, which covers about six orders of magnitude of mass.

This development created a new theoretical problem, because some of the systems now embraced by the power laws did not have anything resembling a fractal branching network, which is a branching set of connections that looks the same at different scales. This led Brown, West, and Enquist to generalize their theory to any hierarchical form of distribution in which resources are transported from a central location to all points of the system--that is, the network fills the system's space, whether this is a mammalian body or a single cell.2

The West/Brown/Enquist theory still leans heavily on fractal geometry, even though it is now applied in a virtual sense to networks that may not have any physical branches at all, such as a cell's vesicles. As pointed out earlier, some physicists, including Banavar, are unhappy that the theory must rely on the assumption that resource distribution is based on a branching hierarchy, whether virtual or not, in which supply lines divide progressively into smaller and smaller tributaries. Some argue that the laws should apply for any network that transmits resources from sources to destinations, subject to a few fundamental constraints, such as tissue density and cell size being the same for all organisms (for example, mammals) within a group.

But for most biologists the point is that the case for power laws now has been established beyond all reasonable doubt, says Brown. The challenge is to apply them in their research.

POWER LAWS AND SAVANNAS According to Brown, the quarter-power laws apply equally on the scale of ecosystems, which can be regarded as resource distribution networks serving the fauna within them. Brown is working to exploit the power laws to develop a complete "Metabolic Theory of Ecology," which could be used for predictive modeling of ecosystems based on three key variables: size, temperature, and availability of critical resources. "Size of the body determines the rate at which you can supply resources to an animal," says Brown. "The second thing is the temperature of the bath, which means environmental temperature, except for birds and mammals. These two variables, size and temperature, account for most, but not all, of the variations in a wide range of phenomena, from growth rates and life-span, to rates of molecular evolution, to rates of speciation and extinction, and the rates at which ecological interactions come in," Brown adds. The effects of temperature are determined by Stefan Boltzmann's law, equating chemical reaction time to e-E/KT, where e, again, is the natural logarithm base, E is the potential reaction energy dependent upon the nature of the reaction ingredients, T the absolute temperature of the environment, and K a constant.

The temperature law is well known, but Brown has been studying the predominant cause of the remaining ecological variation--resource availability. As oxygen and carbon dioxide are ubiquitous, the primary resources that can run short are water (over land), phosphorus for nucleic acids, and nitrogen for protein. A shortage of any of these reduces an ecosystem's productivity, but the question was by how much and whether any precise scaling rule analogous to the quarter-power law could be derived. The early evidence, says Brown, suggests that ecosystem productivity scales with water according to a one-to-one linear relationship. "That still needs to be worked on, but it appears there is a linear scaling with water at least," says Brown. That means that where water is an issue, halving its availability will halve the biomass of the ecosystem.

If this is correct, then ecosystem productivity varies, with mass according to the quarter-power law, with temperature according to Boltzmann's Law, and with resource availability according to a simple linear relationship. But for warm-blooded animals, which have constant body temperature, the only relevant scaling is the quarter-power law of size, assuming there is no resource shortage.

This finding is now being applied in a multinational study of the South African tropical savanna, with Han Olff, professor of conservation ecology at the University of Groningen, Netherlands, a lead participant. "We're looking at the full range of mammal sizes from large herbivores, such as rhinos, down to small hares, to see if power laws allow these species to coexist," says Olff. At first sight, there sometimes appear to be insufficient resources for the existing species within a particular habitat. "We think that large species and small species view the world at a different scale, so that the bigger you get, the more detail you omit. This opens up opportunities for smaller species to have the leftovers of the larger ones," says Olff. This finding may seem obvious, but the power law makes it possible to quantify the relationship between species and habitat sizes. "We have found that bigger species have a disproportionate need for area, greater than you would expect from their size alone," says Olff. This has obvious implications for conservationists attempting to maximize the protection for species given constraints over available land area.

 PERCEPTION v. REALITY: Spanish Catchfly (top) was once considered rare because of its narrow distribution. Deptford Pink, despite its wider distribution, is actually more rare. Power laws make such predictions possible.

POWER LAWS AND CARNATIONS Another notable ecological application of power laws is in determining how vulnerable a rare species is to extinction. In the United Kingdom, a team led by William Kunin, senior lecturer at University of Leeds, is studying how apparent rarity of plant varieties scales across different resolutions. Put simply, a plant can be numerically rare, but common in the sense that it is found in many different locations, or it can be abundant in just one small spot. The more abundant plant might then be more vulnerable to extinction through a single event, such as building on its only habitat. This is an extreme example, but Kunin found something similar happening in the United Kingdom with two varieties of carnation, the Deptford Pink and the Spanish Catchfly. Only the latter was listed as rare, on the grounds that it was found within only about six adjacent 10 kilometer-square grids within one region, East Anglia. The Deptford Pink was deemed more common because it had been spotted in about 16 cells scattered across the whole country. But when Kunin's team studied the plants in detail, they found that the Spanish Catchfly was more abundant, by a factor of 200, because of its greater density within the 10 km grids. So, by replotting the occurrence of the two plants at a scale of 100 meter-square grids rather than 10 km grids, the relative rarities would be reversed. It all depends on how closely one looks.

The relevance here is that plant distribution ranges with grid size according to a rule closely resembling a quarter-power law, with a slight deviation resulting from other factors. The big prize is that the power law makes it possible to predict with some degree of accuracy how a plant variety is distributed in detail at a fine level of resolution by considering sparse data at a coarser level, which is easier to obtain. "If that follows a power law, or indeed any other coherent relationship, then looking at these relatively coarse scale things, you should be able to make some sort of prediction about what's happening at finer scales," Kunin explains. The same principle should apply to animals as well as plants, although complicated by seasonal and random migration.

POWER LAWS AND DRUGS Power laws also have potentially highly valuable applications at the level of single organisms, for example, in the pharmaceutical industry. For years, researchers have used rats to test new drugs, with dosage levels then scaled up to humans on the basis of a linear mass relationship. This leads to overdosage if the decline in cellular metabolic rate with size is not taken into account. The situation is complicated further by the fact that some drugs, such as local anesthetics, act in a small area, in which case the effective organism size is the immediate region of the injection. According to West, drug companies are now showing interest in this potential application of power laws.

Meanwhile, the underlying mathematical debate relating to the 3/4 power scaling law remains to be settled, with the quest being to circumvent fractal geometry and develop a general scaling theory that holds for any resource network that delivers materials within some structure. But such a theory must also embrace the wide variety of biological phenomena that obey the 3/4 law.

Philip Hunter ( is a freelance writer in London.

1. G.B. West et al., "A general model for the origin of allometric scaling laws in biology," Science, 276:122-6,1997.

2. G.B. West et al., "The fourth dimension of life, fractal geometry, and allometric scaling of organisms," Science, 284:1677-9, 1999.

3. G.B. West et al., "A general model for ontogenetic growth," Nature, 413:628-31, 2001.

4. G.B. West et al., "Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals," Proc Natl Acad Sci, 99:2473-8, 2002.

5. G.B. West et al., "Why does metabolic rate scale with body size?" Nature, 421:713, Feb. 13, 2003.

6. P.S. Dodds et al., "Reexamination of the 3/4-law of metabolism," Theoretical Biol, 209:9-27, 2001.

7. M. Kleiber, "Body size and metabolism," Hilgardia, 6:315-53, 1932.

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