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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.vander
bilt.edu/biolib/hc/infant_nutrition/images/rubner.jpg&imgrefurl=
http://www.mc.vanderbilt.edu/biolib/hc/infant_nutrition/hcphoto
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%26client%3Dsafari%26rls%3Den%26sa%3DN
• http://images.google.com/imgres?imgurl=http://animalscience.u
cdavis.edu/memorial/MaxKleiber.gif&imgrefurl=http://animalsc
ience.ucdavis.edu/memorial/kleiber.htm&h=297&w=216&sz=6
0&hl=en&start=1&um=1&tbnid=A5aTbB2IRYLJ8M:&tbnh=11
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