Scaling Exponent: 2/3 or 3/4?

B=aM^b

Scaling Exponents

• Should it be 2/3 or 3/4? (theoretical controversy)

• Is it 2/3 or 3/4? (empirical controversy)• How universal is this metabolic scaling

anyway? What is the limiting factor in metabolism?

• Can we really distinguish between 2/3 and 3/4?

The original debate

• Max Rubner (1883) vs Max Kleiber (1932)

Review of Assumptions

• 2/3 scaling based on geometry/dimensional analysis:– Organisms are essentially the same shape– The rate limiting factor is the ability to

dissipate heat through the outer surface -or- that the relevant metabolic surface area also ~2/3 the volume.

• 3/4 power law scaling based on fractal networks:– Organisms distribute necessary molecules for

metabolism through fractal networks– Organisms optimize for space filling networks,

small fluid volume– Capillary/network endpoint size and capacity are

independent of the size of the overall organism– Rate determining step is the ability to get the

nutrients to the cells through this network

Is it 3/4…

• Savage, 2004– Found exponent of 3/4 over birds, mammals, and plants

• Farrel-Gray and Gotelli, 2005.– Did analysis of analyses, and found that exponent was

not correlated with sample size, mid point of mass, or range of mass of species studied, and that the exponent was most likely to be 3/4.

Savage - Example Data

2/3…White and Seymour, 2002. They argue that the 3/4

power law is wrong. To really look at metabolic rate, we need to compare similar things.

BMR, in a post-absorptive, resting, conscious state.

619 species

Adjusted for temperature dependence of metabolism using BMRc=BMR*10^(Tc-Tb)log(Q10/10)

Excluded certain species

The most restrictive data set gave them b=.68, which was significantly different from 3/4 but not 2/3

Neither…• Dodds et al: re-evaluated a lot of data from

commonly cited studies, including Kleiber (13 species), Brody, 1945 (67 mammals), Bennet and Harvey, 1987 (398 birds), and Heusner, 1991 (391 species)

• Determined that there is no universal exponent• Could not reject their “null hypothesis” of a 2/3

exponent, particularly for birds and animals with M<10-20 kg

• Considered a break off point of M=10-20 kg where scaling factor changes

Or … one?

• Reich et al (including Jose-Luis Machado) found that night time respiration in plants ~ mass within a set environment, and that over different environments, nitrogen supply was more predictive of respiration than mass was

• Enquist, Brown, Gillooly, West, et al responded, arguing that in fact for small plants R is predicted to scale like M (ie b=1) and that there is a transition point above which they scale like b=3/4

Data (Reich et al)

Data, Brown et al. Small plants show different metabolic scaling than big plants.

Reconciliation?

• Some studies (eg Heusner, 1991) suggest that 2/3 is the valid exponent for comparing within species, while 3/4 is good for comparing between species

• Different concerns may be important for organisms that metabolize differently

Can we even tell, with this data?

Hui and Jackson, 2007. A mathematical analysis of the impact of sample size, measurement error and line fit technique on the determined exponent

Used data from Savage et al, 2004.

Sample Size• Using the “population” as all of Savage’s data, with

the b=.711, they extracted samples of various sizes (500 of each sample size) and calculated the probability of accepting 2/3 or 3/4 (both outside Savage’s 95% confidence interval) and of rejecting the “true” value of .711.

• In order to have P(reject true value) < .05, they had to sample more than 61% of the population of 626 species

• For sample sizes > 30% of the population, P(accept false value)<.05

• Most studies suggesting “universal” relations aren’t this comprehensive, especially for big animals!

Measurement Error• created data sets using measured mass distribution

from Savage and ideal relationship (.711, 2/3 and 3/4), plus normally distributed random errors (for 2/3 and 3/4, this was set at (BMR)=0.2BMR, (M)=0.4M); 500 data points per set

• 14% probability of rejecting “true” 2/3 value• 15% probability of rejecting “true” 3/4 value• 8% probability of accepting “false” 2/3 value when

data was for 3/4 relationship• 500 data points is bigger than most studies I’ve seen

So, there probably is a scaling dependence, but…

• It’s almost certainly more complicated than a universal 2/3 or 3/4

• We may not be able to tell yet what it is, where it changes, or what it means

• Remember the issues with log-log plots. Stay tuned for the power series presentation.

Sources• White, Craig R. and Roger Seymour, “Mammalian basal metabolic rate

is proportional to body mass^2/3,” www.pnas.org, 2002.

• Reich, et al. “Universal scaling of respiratory metabolism, size, and nitrogen in plants,” Nature, 2006.

• Farrel-Gray and Gotelli, “Allometric exponents support a 3/4-power scaling law.” Ecology, 2005.

• Dodds et al, “Re-examination of the “3/4 law” of metabolism.” Journal of Theoretical Biology, 2001.

• Brown, et al. “Does the exception prove the rule?” Nature, 2007.

• Hui and Jackson. “Uncertainty in allometric exponent estimation: a case study in scaling metabolic rate with body mass.” Journal of Theoretical Biology, 2007.

• Savage, et al. “The predominance of quarter power scaling in biology.” Ecology. 2004.

Image Sources• http://images.google.com/imgres?imgurl=http://

www.mc.vanderbilt.edu/biolib/hc/infant_nutrition/images/rubner.jpg&imgrefurl=http://www.mc.vanderbilt.edu/biolib/hc/infant_nutrition/hcphototest.html&h=422&w=282&sz=23&hl=en&start=1&um=1&tbnid=7KT3BHAlDXpn_M:&tbnh=126&tbnw=84&prev=/images%3Fq%3Dimage%2Bmax%2Brubner%26um%3D1%26hl%3Den%26client%3Dsafari%26rls%3Den%26sa%3DN

• http://images.google.com/imgres?imgurl=http://animalscience.ucdavis.edu/memorial/MaxKleiber.gif&imgrefurl=http://animalscience.ucdavis.edu/memorial/kleiber.htm&h=297&w=216&sz=60&hl=en&start=1&um=1&tbnid=A5aTbB2IRYLJ8M:&tbnh=116&tbnw=84&prev=/images%3Fq%3Dimage%2Bmax%2Bkleiber%26um%3D1%26hl%3Den%26client%3Dsafari%26rls%3Den%26sa%3DG