# 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

RM3/4

(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

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

1. In spite of the importance of the law and the subsequent MTE I'd like to stress that it is even more important to step back and look at what MTE is based on:

The data used are measurements of basal and standard metabolic rate (BMR and SMR), meaning animals at rest. Are animals resting all the time? It is a fact that 'normal metabolic rate' have a different scaling, as e.g. the maximal metabolic rate scales with an exponent significantly larger than 3/4 (e.g. Killen et al. 2007). This effect is due to all kinds of activity.

Currently we're missing quantitative theories of metabolism that can accurately assess both magnitude and changes in slope with activity levels. As an example the fishes spend about 80% of their energy on activity which is not dealt with in MTE. We should thus work on expanding the metabolic theory and incorporating it into trophic food web models such that we make quantitative predictions of energy exchanges in natural systems.

DEB models (Kooijman 2000) aim at solving this problem. However, these models are mostly based on fit to growth curves, which do not allow for proper estimation of energy intake and expenditures as they fit to the difference between the two. We need to focus on descriptions of both uptake and expenditures to provide the desired quantitative description of energy exchanges.

Lastly we should not forget that even though the mass-specific rate is strikingly similar across body size there is still 2 orders of magnitude in difference across taxa (Makarieva, 2008)... a difference that indeed is significant if we ever hope to quantitatively descripe energy exchanges in natual systems.

MTE has provided a lot of knowledge on interspecific scaling of metabolism. It's now time to focus on intra-specific relationships.

Just a few thoughts,
Martin Hartvig

References

Killen, S. S.; Costa, I.; Brown, J. A. & Gamperl, A. K. Little left in the tank: metabolic scaling in marine teleosts and its implications for aerobic scope. Proc. R. Soc. B, 2007, 274, 431-438

Kooijman, S. A. L. M. Dynamic Energy and Mass Budgets in Biological Systems. Cambridge University Press, 2000

Makarieva, A.; Gorshkov, V.; Li, B.; Chown, S.; Reich, P. & Gavrilov, V. Mean mass-specific metabolic rates are strikingly similar across life's major domains: evidence for life's metabolic optimum. Proc. Natl. Acad. Sci. USA, 2008, 105, 16994

2. The way that the allometric scaling exponents are derived has also come under fire. A straight line is fitted to the data even if this is not the best relationship from a statistical or information theory standpoint. The log-log plots used to calculate the metabolism-mass slope do not well describe the arithmetic relationship, although the latter is what we use to discuss allometry. The allure of a finding some sort of constant throughout all biology has made us somewhat selective about what data we choose to include/exclude in meta-analyses, and we often forget about mathematical and statistical rigor.

An articles for the interested:

Packard, G.C., Boardman, T.J., 2008. Model selection and logarithmic transformation in allometric analysis. Physiological and Biochemical Zoology 81, 496-507.

3. Kleiber's law, or any other scaling law, is an attempt to mathematically describe an observed pattern. Similar models can do this good or less well. Those models do not, however, necessarily tell us anything about from where this pattern emerges. Where does "3/4" or "2/3" really come from?
My understanding is that MTE's ambition is to do precisely that - to figure out where scaling comes from in the first place. Wether it has succeed is beyond me to judge.

Kleiber found the pattern, others have tried to refine the description of it and now we should find out where it comes from.

Nice to see so many comments on this post, by the way!

4. Great to see that my guest entry got so many interesting comments!
As Per said above, the Kleiber law describes an observed pattern – a remarkable one, at least in my view – which says absolutely nothing about the underlying processes that produce it. Of course metabolic rate is far from being constant for any species. But without taking such a big-picture view that ignores all the variation over time for an individual and all the between-individual variation for any species, it might have never been noticed! This does of course not mean that all this variation is not important – it depends on the questions you are asking.

West et al. tried to come up with a causal explanation for the pattern. Given that the Kleiber law ignores all intraspecific variation in metabolic rate, it’s not a big surprise that their suggested explanation and the MTE that has emerged from it do that as well. So I agree, as almost always in science, there is clearly room for future work here! Maybe someone will at some point even come up with an alternative causal explanation for the Kleiber law that doesn’t require the fractal structure of the circulatory system as an ingredient.

5. In the late 1960s an academic grandchild of Kleiber's invited him to offer a seminar on this topic at Kansas State University, in the school of Veterinary Medicine. As a graduate student in a program for Bio-Environmental Engineering, it was my privilege to attend. It offered not only an opportunity to meet Kleiber, but to buy his text on the topic. At the moment I'm trying to develop a nutritional requirement for a pair of dogs. Likely I'll use my own size and caloric needs to calculate the constants needed to get those canine needs. Might there be a better approach? Comments?