Equation of the Month

Equation of the Month

A blog run by the

Theoretical Population Ecology and Evolution Group,

Biology Dept.,

Lund University

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

Wednesday, June 27, 2012

The Kleiber Law


(from Hemmingsen, 1960)

What it means
Larger animals have relatively slower metabolisms than small ones. A mouse must eat about a third of its body mass every day not to starve whereas a human can survive on only 2%. The relationship follows a power law: basal metabolic rate (R) is proportional to the ¾ power of an animal's mass (M). This relationship, the Kleiber Law (Kleiber 1947), can be drawn as a straight line on a log-log plot (see Fig.). Mysteriously, this simple relationship holds, from simple organisms to most complex ones, from microbes to giant blue whales across 18 orders of magnitude in body mass.

Where does it come from
Max Kleiber, an ecologist from Switzerland, discovered the law in the early 1930s. After its initial publication, other workers added additional species to his original figure. They extended it to a ‘mouse-elephant-curve’ and subsequently even further, to whales and microbes, confirming the Kleiber law’s surprising validity. Before Kleiber published his law, first explanations for why metabolic rate would change with body mass were already around. These were based on an organism’s body surface to volume ratio. Large animals have proportionately less surface area per unit volume, they hence lose body heat more slowly, and, so it was argued, need proportionately less food and have a relatively slower metabolism. But, following this argumentation, metabolic rate should scale with mass to the power of 2/3, not 3/4. A causal explanation for the ¾-law has been lacking until scientist in the 1990s, using mathematical models, proposed that the geometry and particularly the fractal structure of an animal’s circulatory system could be the reason for the ¾ exponent (West et al. 1997, West et al. 1999). One problem with these models is that the derivations build on considerations of blood flow, but the Kleiber law also holds for organisms without a blood circulatory system, like bacteria or corals.

Applicability and importance
Whether metabolic rate always scales with body mass to the power of 3/4 is still debated - some researchers think that no single exponent fits all the data, and some believe it should be 2/3 instead. The ¾-law though, favored by the majority of biologists and fitting the data best, seems to be one of the few examples of a generally applicable ‘law’ in biology. Biologists are not used to finding general rules of this kind within their domain. I first learnt about it in a course on animal physiology during my graduate studies, and I remember clearly how much its simplicity and generality fascinated me. In the past few years, researchers have come up with a new theory for ecology along these lines, which names metabolism as its basic principle (Brown et al. 2004). The ‘metabolic theory of ecology’ posits that the way animals use energy should be considered a unifying principle of ecology. It states that metabolism provides the fundamental constraints by which ecological processes are governed. Supporters of the theory suggest that processes at all levels of organization, from single organism’s life-history strategies to population dynamics and ecosystem processes could possibly be explained in terms of constraints imposed by metabolic rate.

Barbara Fischer

Further reading
Kleiber M. (1947) Body size and metabolic rate. Physiological Reviews 27 (4): 511–541.

West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric scaling laws in biology. Science 276: 122–6

West, G.B., Brown, J.H., & Enquist, B.J. (1999). The fourth dimension of life: Fractal geometry and allometric scaling of organisms. Science 284 (5420): 1677–9.

Brown, J. H., Gillooly, J. F., Allen, A. P., Savage, V. M., & G. B. West (2004) Toward a metabolic theory of ecology. Ecology 85: 1771–1789