© 2018 Aaron A. King.

The allometric scaling of metabolic rate with body size

It has long been observed that an animal’s metabolic rate scales with its body size. That is, if \(W\) is the animal’s resting rate of energy consumption (basal metabolic rate) and \(m\) is its mass, then \[W \approx a\,m^b,\] where \(a\) and \(b\) are constants. A good deal of controversy surrounds the question of what \(b\) should be and what this means [1–8].

In particular, the so called “Kleiber’s Law” [9] states that \(b\) should be approximately 3/4, while other reasoning argues that \(b\) should be 2/3.

What do the data say?

Data on metabolic rate and body size in mammals

The following are data on the metabolic rates and sizes of an assemblage of 619 mammal species, ranging in size from shrews (c. \(2\,\mathrm{g}\)) to wildebeest and moose (c. \(3\times 10^{5}\,\mathrm{g}\)). These data are from White & Seymour [10].

Taking logs of both sides of the allometric equation gives \[\log{W} \approx \log{a}+b\,\log{m},\] i.e., a linear relationship between \(\log{W}\) and \(\log{m}\).

Accordingly, we plot basal metabolic rate vs. mass on a log-log scale:

ggplot(dat,aes(x=mass,y=bmr))+
  geom_point()+
  scale_x_log10()+
  scale_y_log10()+
  geom_smooth(method="lm",formula=y~x,se=TRUE)

And indeed, it does look like the data fall roughly along a line. What is the line? We can regress log BMR on log mass to obtain it.

fit1 <- lm(log(bmr)~log(mass),data=dat)

When we do so, we obtain the least-squares regression line \[\log{W}=1.5+0.672\,\log{m}.\] That is, a line with slope \(0.672\pm0.018\) is the best fitting line (using the least-squares criterion). Since this is closer to \(\frac{2}{3}\) than it is to \(\frac{3}{4}\), this might be taken as evidence in favor of a 2/3-power scaling and as evidence against Kleiber’s Law, which holds that \(b\approx\frac{3}{4}\).

A paradox

On the other hand, note that the allometric relationship above can be expressed equivalently as \[m\approx c\,W^{1/b},\] where \(b\) is as before and \(c\) is a different constant. This in turn implies \[\log{m}\approx \log{c}+\frac{1}{b}\,\log{W}\] and suggests we regress \(\log{m}\) on \(\log{W}\):

fit2 <- lm(log(mass)~log(bmr),data=dat)

The best-fitting line here is \[\log{m} = -1.51+1.34\,\log{W}.\] Hence, \(\frac{1}{b}\approx 1.34\pm0.018\). Since this close to \(\frac{4}{3}\), we are tempted to conclude that \(b\approx \frac{3}{4}\), in agreement with Kleiber’s Law.

How can this paradox be resolved?

ggplot(dat,aes(y=mass,x=bmr))+
  geom_point()+
  scale_x_log10()+
  scale_y_log10()+
  geom_smooth(method="lm",formula=y~x,se=TRUE)

References

1. Brown JH, Gillooly JF, Allen AP, Savage VM, West GB (2004) Toward a metabolic theory of ecology. Ecology 85: 1771–1789.

2. Dodds PS, Rothman DH, Weitz JS (2001) Re-examination of the 3/4-law of metabolism. Journal of Theoretical Biology 209: 9–27.

3. Glazier DS (2005) Beyond the 3/4-power law: Variation in the intra-and interspecific scaling of metabolic rate in animals. Biological Reviews 80: 611–662.

4. Kozłowski J, Konarzewski M (2004) Is West, Brown and Enquist’s model of allometric scaling mathematically correct and biologically relevant? Functional Ecology 18: 283–289.

5. West GB, Brown JH (2005) The origin of allometric scaling laws in biology from genomes to ecosystems: Towards a quantitative unifying theory of biological structure and organization. Journal of Experimental Biology 208: 1575–1592.

6. West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric scaling laws in biology. Science 276: 122–126.

7. West GB, Woodruff WH, Brown JH (2002) Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals. Proceedings of the National Academy of Sciences of the USA 99: 2473–2478.

8. White CR, Seymour RS (2005) Allometric scaling of mammalian metabolism. Journal of Experimental Biology 208: 1611–1619.

9. Kleiber M (1975) The fire of life : An introduction to animal energetics. Huntington, N.Y: R. E. Krieger Pub. Co.

10. White CR, Seymour RS (2003) Mammalian basal metabolic rate is proportional to body mass\(^{2/3}\). Proceedings of the National Academy of Sciences of the USA 100: 4046–4049. doi:10.1073/pnas.0436428100.