Ecology Letters, (2010) 13: 728–735
doi: 10.1111/j.1461-0248.2010.01461.x
LETTER
Why are metabolic scaling exponents so
controversial? Quantifying variance and testing
hypotheses
1,2
Nick J. B. Isaac * and Chris
Carbone1
1
Institute of Zoology, Zoological
Society of London, RegentÕs
Park, London NW1 4RY, UK
2
NERC Centre for Ecology and
Hydrology, Maclean Building,
Benson Lane, Crowmarsh
Gifford, Wallingford, Oxon
OX10 8BB, UK
*Correspondence: E-mail:
[email protected]
Abstract
The metabolic theory of ecology links physiology with ecology, and successfully predicts
many allometric scaling relationships. In recent years, proponents and critics of
metabolic theory have debated vigorously about the scaling of metabolic rate. We show
that the controversy arose, in part, because researchers examined the mean exponent
separately from the variance. We estimate both quantities simultaneously using linear
mixed-effects models and data from 1242 animal species. Metabolic rate scaling
converges on the predicted value of 3 ⁄ 4 but is highly heterogeneous: 50% of orders lie
outside the range 0.68–0.82. These findings are robust to several forms of statistical
uncertainty. We then test competing hypotheses about the variation. Metabolic theory is
currently unable to explain differences in scaling among orders, but the patterns are not
consistent with competing explanations either. We conclude that current theories are
inadequate to explain the full range of metabolic scaling patterns observed in nature.
Keywords
Allometry, body size, energetics, macroecology, metabolic rate, metabolic theory of
ecology, mixed models, taxonomy, variance, WBE model.
Ecology Letters (2010) 13: 728–735
INTRODUCTION
The metabolic theory of ecology (MTE) provides a powerful
framework for explaining properties of whole organisms in
terms of physiological processes (West et al. 1997; Gillooly
et al. 2001; Brown et al. 2004). Many life history parameters
are predicted to scale allometrically with body size by a
power law relationship in which the exponent is a multiple
of 1 ⁄ 4. MTE, however, remains highly controversial
(Martinez del Rio 2008). Proponents claim MTE has wide
support and can provide conceptual unification in ecology
(Brown et al. 2004). Detractors have criticized the assumptions and biological relevance of MTE, as well as the
supporting evidence and its claim to be mechanistic
(Kozlowski & Konarzewski 2004; OÕConnor et al. 2007).
The allometric scaling exponent for metabolic rate has
been a particular focus for this controversy (Dodds et al.
2001; White & Seymour 2003; Savage et al. 2004; FarrellGray & Gotelli 2005; Glazier 2005; Nagy 2005). This debate
has focused on two separate but related issues: whether the
metabolic scaling exponent takes a particular value (e.g. 3 ⁄ 4
or 2 ⁄ 3), and whether the variability among taxa is significant.
At the heart of metabolic theory is a model proposed by
2010 Blackwell Publishing Ltd/CNRS
West, Brown and Enquist (West et al. 1997, 2000,
henceforth WBE), which is based on the biophysical
properties of the vascular system and predicts 3 ⁄ 4 power
scaling in metabolic rate. Proponents of MTE have
presented evidence supporting this prediction (Gillooly
et al. 2001; Savage et al. 2004) and claim that deviations from
3 ⁄ 4 are either predictable exceptions (Allen & Gillooly
2007) or have limited generality (e.g. small numbers of
species, limited range in body size: Savage et al. 2004; FarrellGray & Gotelli 2005). Critics have presented numerous
counter-examples and concluded that the observed heterogeneity is sufficient to reject the notion that one ÔuniversalÕ
exponent applies to all organisms (Kozlowski et al. 2003;
Glazier 2005; White et al. 2006, 2007; Capellini et al. 2010). If
scaling is not universal then either WBE is incomplete
(Martinez del Rio 2008), or perhaps other theories of
metabolic scaling should be preferred.
In fact, WBE does predict that metabolic scaling
exponents should vary, albeit within a fairly narrow range.
West et al. (1997) predicted a steeper scaling exponent
among smaller-bodied organisms due to the higher proportion of the vascular system that is made up of cubic
branching vessels (e.g. capillaries), compared with the
Letter
area-preserving branching of larger vessels. Savage et al.
(2008) explored this in detail by relaxing the assumptions of
WBE: they predicted that an increase in body size should be
associated with either a curvilinear decrease in scaling
exponent and ⁄ or a decrease in the variance in scaling
exponents (but see Kozlowski & Konarzewski 2005).
A number of competing theories have been proposed to
explain why metabolic scaling should vary. Banavar et al.
(2002) proposed that variation derives from inefficiencies in
the balance between supply and demand for metabolites,
and from physiological compensating mechanisms. Two
theories make explicit predictions about the kinds of
organisms in which deviations should be found: the
metabolic level boundaries (MLB) and the cell metabolism
hypotheses. The MLB hypothesis states that the range of
possible metabolic states is defined by physiological
constraints (Glazier 2005, 2010). It predicts that taxa with
high metabolic rates exhibit relatively low scaling exponents,
which is related to an older suggestion that ectotherms
display steeper scaling than endotherms (Phillipson 1981).
The cell metabolism hypothesis (Kozlowski et al. 2003)
posits that whole organism metabolic rate is simply the sum
of the metabolic rates of constituent cells, and that the
metabolic rate of individual cells is determined by their size.
In this model, metabolic rate scaling is close to isometric
among organisms of the same mean cell size, but much
shallower (2 ⁄ 3 power) where body size differences are
entirely due to variation in cell size. This leads to the
prediction of a negative relationship between metabolic
scaling and genome size scaling, as genome size is an
important determinant of mean cell size (but see also
Starostova et al. 2009).
In this paper, we provide a systematic description of the
nature and magnitude of the variation in metabolic scaling
exponents among animals. We provide the first quantitative
estimate of the magnitude of variance in metabolic scaling
among taxa, and test which taxonomic level(s) exhibit most
variation. We use these findings to test two predictions from
WBE, namely that metabolic scaling is universal and
converges on 3 ⁄ 4 power. We then test a suite of hypotheses
proposed to explain the range of variation in scaling
exponents: a generalized version of WBE (Savage et al.
2008), the MLB hypothesis (Glazier 2005, 2010) and the cell
metabolism hypothesis (Kozlowski et al. 2003). Our primary
aim in describing this variation is to understand the
controversy surrounding metabolic scaling exponents.
MATERIALS AND METHODS
Data
We collated five datasets on resting metabolic rate (RMR)
and body mass for mammals (n = 626 species in 22 orders:
The metabolic scaling controversy 729
Savage et al. 2004), birds (n = 336, 21: Bennett & Harvey
1987) reptiles (n = 74, 3: Bennett & Dawson 1976), fish
(n = 69, 13: Clarke & Johnston 1999) and soil invertebrates
(n = 137, 5: Meehan 2006). Although field metabolic rate
data are available for some of these groups, we used RMR
for modelling because of the much larger availability of data
and because RMR has previously been used to support the
notion of a universal scaling exponent of 3 ⁄ 4 (Gillooly et al.
2001; Savage et al. 2004). All RMR data were converted to
species mean values in joules per second using standard
formulae (Gnaiger 1983). For reptiles, we used only data
where RMR was measured at 20 C. For fish, we used the
ÔbestÕ estimate of species mean RMR provided by Clarke &
Johnston (1999). For soil invertebrates, we standardized the
estimates of RMR to 20 C and for the mean mass of the
species (following Gillooly et al. 2001). We did not control
for body temperature in endotherms because many species
in our dataset lack body temperature data (Clarke & Rothery
2008). The range in endotherm body temperature is just a
few degrees and shows a strong phylogenetic pattern (Clarke
et al. 2010), such that variation within families is extremely
small (but see Appendix S1). All data were log-transformed
prior to analysis and log [mass] was additionally centred on
zero.
Model selection
We fitted a series of linear mixed-effect models in which the
slope (scaling exponent) and intercept (normalization
constant) varied among taxonomic groups (phylum, class,
order, family) as random effects (see Appendix S1). We
compared the fit of these models with ÔuniversalÕ models
that had a single slope but taxon-specific intercepts. This
approach allows us to estimate simultaneously the mean
scaling exponent and the variance among taxa (thus
reconciling the two parallel debates about metabolic scaling
exponents). The use of mixed models has two distinct
advantages over other approaches to allometric slopes
estimation (e.g. Warton et al. 2006). First, it allowed us to
explore variation in scaling exponents without prior
expectation of which taxonomic units to compare. Second,
taxon-specific scaling exponents are estimated with ÔshrinkageÕ (Pinheiro & Bates 2004), so the test for universal vs.
heterogeneous scaling is not unduly influenced by outliers
with small sample sizes (see Appendix S1 for further
details).
Quantifying the variance in scaling exponents implies an
over-arching null hypothesis that all taxa share the same
universal scaling exponent. However, we have no a priori
expectation of which taxonomic level (or levels) at which
variation will be found. Indeed, the hierarchical data
structure means that there is no unique universal model
(see below). We therefore adopted a mixture of null-hypoth 2010 Blackwell Publishing Ltd/CNRS
730 N. J. B. Isaac and C. Carbone
Letter
Table 1 Mixed models of the metabolism allometry
Rank
1
2
3
4
5
6
7
8
9*
11
12
13
27 Slope effects
P
C
Intercept effects
O
O
O
O
O
O
P
F
F
C
P
O
O
P
P
P
P
P
P
P
P
P
P
C
C
C
C
C
C
C
C
C
C
C
O
O
O
O
O
O
O
O
O
O
O
F
F
F
F
F
F
F
F
F
F
F
beta
LogLik
AIC
¶AIC
Weight
0.748
0.748
0.760
0.748
0.734
0.734
0.752
0.765
0.747
0.755
0.741
0.753
1.028
)521.56
)523.51
)521.47
)521.54
)542.51
)546.55
)552.42
)553.58
)562.83
)562.55
)587.79
)606.91
)2594.27
1061.11
1063.03
1064.95
1065.08
1107.01
1107.10
1122.84
1125.17
1139.65
1143.09
1191.58
1229.81
5194.55
0
1.92
3.84
3.97
45.9
46.0
61.7
64.1
78.5
82.0
130.5
168.7
4133
0.599
0.230
0.088
0.083
0
0
0
0
0
0
0
0
0
Rank denotes the rank fit among 27 candidate models (only 1 of 15 ÔuniversalÕ models is shown). Slope and Intercept columns indicate the
taxonomic levels at which each parameter was allowed to vary (P, phylum; C, class; O, order; F, family). Beta denotes the overall mean scaling
exponent of metabolic rate on body mass, LogLik is the log-restricted-likelihood, AIC is the Akaike Information Criterion, ¶AIC is the
difference in AIC compared with the best model, and Weight is the Akaike weight (an absolute measure of support that sums to 1 across all
candidate models).
*Best-fitting ÔuniversalÕ model.
Ordinary least squares regression.
esis significance testing and information theoretic model
comparison (Stephens et al. 2007). We employed a stepwise
model-fitting procedure (see Appendix S1 for full details)
that yielded a set of 27 candidate models, including the
ordinary least squares regression, 15 universal models and 11
alternate (heterogeneous) models. We compared these
models quantitatively using Akaike weights (Burnham &
Anderson 2002), which are based on the AIC scores for all
models in the candidate set. We report results from all
models with weights > 0.0005, and estimate the overall
mean scaling relationship using weighted model-averaging
(Burnham & Anderson 2002). Parameters were estimated by
restricted maximum likelihood (REML) using the lme4
package (Bates et al. 2008) version 0.999375-28 in R 2.8.1 (R
Development Core Team 2008). Further analyses were
based on the overall best-fitting model. We used a likelihood
ratio test (LRT) to compare the best alternate model with
the best universal model (these models are nested: Table 1)
and estimated confidence intervals for the mean scaling
exponent. We also tested whether our results are sensitive to
the statistical methods used, to small variations among
endotherms in body temperature, and to measurement error
(see Appendix S1 for details).
Explaining variation in scaling exponents
We then tested predictions about the variation in scaling
exponents among orders. In this section we describe these
tests in increasing order of complexity. We were able to test
2010 Blackwell Publishing Ltd/CNRS
most hypotheses with modifications to our best-fitting
model and using LRT.
We tested the MLB hypothesis (Glazier 2005, 2010) by
measuring the strength of correlation between order-specific
slopes and intercepts. Our best model contains a term
estimating the correlation between random effects for slope
and intercept, so we tested the significance of this term
using LRT by comparison with a model lacking this term
(i.e. the correlation term is constrained to zero). However,
the estimated correlation may be biased if the mass at which
intercepts are estimated is far from the range over which the
slopes are estimated (Glazier 2009, 2010). We therefore
performed the test on a modified dataset in which
log [mass] was centred on zero for each order (following
Enders & Tofighi 2007), such that intercepts (i.e. metabolic
level) are the fitted values of log [RMR] at the geometric
mean mass of each order. This procedure for estimating the
strength of correlation is more robust than comparing
slopes and intercepts extracted from a mixed-effects model
(Kliegl et al. 2009). We compared the scaling exponents of
endotherms and ectotherms (Phillipson 1981) by adding
fixed-effect terms for the mode of thermoregulation and its
interaction with log [mass] to our best model. For this
comparison, both models were fitted with maximum
likelihood (ML), rather than REML (following Pinheiro &
Bates 2004).
We fitted four additional models to test the prediction by
Savage et al. (2008) that metabolic scaling is a decreasing
function of mass. We first added a quadratic term (i.e.
RESULTS
We found strong evidence for heterogeneity in metabolic
scaling. The best universal (single exponent) model is a poor
description of the data when compared with models in
which the slope varies among taxonomic groups (Table 1).
Just 4 ⁄ 27 models have any statistical support (defined as an
Akaike weight > 0.0005), all of which contain a random
slopes term for orders. The best model overall has a
different slope for each order but not other taxonomic
levels. The evidence ratio (Burnham & Anderson 2002)
favouring this model over any universal model is > 1017
(LRT: v2 = 78.5, d.f. = 2, P < 0.0001).
While our models strongly reject the notion that a single
exponent can explain the metabolic scaling of all animals, our
results do support a mean value very close to 3 ⁄ 4 predicted
by West et al. (1997). The model-average slope is 0.749 and
the best-fitting model has a slope of 0.748 (Table 1). Support
for 3 ⁄ 4 power scaling is not dependent on allowing the
exponent to vary among taxa: most universal models have a
slope close to this value. The standard error around the
estimate from the best model is 0.017, with 95% confidence
intervals of 0.71–0.80. Order-specific exponents, however,
exhibit a much larger range of values (Fig. 1): the estimated
variance is 0.0111, indicating that 50% of orders are expected
to have scaling exponents outside the range 0.68–0.82 and
5% outside the range 0.54–0.95.
15
10
0
5
Number of orders
log [mass]2) to the fixed part of the model, and then to the
random part. We then added a cubic term to the fixed part
and finally to the random part. At each step, we used LRT to
compare the fit of each new model with the previous one,
using ML fits for pairs of models that differed in the fixedeffect part of the model, and using REML fits for models
that differed in the random-effect part (following Pinheiro
& Bates 2004).
For the other correlative tests, we estimated orderspecific scaling exponents (bj) from the best-fitting model
(see Appendix S1 for details). To test whether scaling is
more variable among small-bodied orders (Savage et al.
2008), we measured the correlation between mean(log
[mass]) and |bj )0.75|. We tested the cell metabolism
hypothesis by measuring the correlation between bj and
order-specific scaling exponents for genome size (following
Kozlowski et al. 2003), using data from the Animal Genome
Size Database (Gregory 2008). We selected a model of the
genome size allometry using the procedures described
above, with the additional constraint that all candidate
models contained a term for random slopes at the order
level. We then extracted order-specific scaling exponents for
genome size (see Appendix S1 for details) and estimated the
correlation with bj using standardized (reduced) major axis
(Warton et al. 2006).
The metabolic scaling controversy 731
20
Letter
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Scaling exponent
Figure 1 Histogram of metabolic scaling exponents among 54
orders. Values were extracted from the best-fitting model. The blue
lines indicate the mean (vertical bar) and 95% confidence intervals
for the overall mean scaling exponent. The red line indicates the
95% prediction intervals for order-specific exponents.
The conclusion that scaling varies among orders is robust
to several forms of uncertainty. Our findings are qualitatively
unchanged (1) under more conservative assumptions about
the statistical penalties for complex models (Table S1), (2)
when analyses are restricted to ectotherms at 20 C or the
subset of endotherms that could be standardized to 39 C
(Table S2) and (3) measurement error in the estimates of
metabolic rate (Fig. S1, see Appendix S1 for details).
We then tested several explanations for why orders vary
in metabolic scaling, starting with predictions from the
generalized version of WBE. We found no correlation
between body size and variability in scaling exponents
among orders (r = 0.02, P = 0.89). We also found no
evidence for curvilinearity in metabolic scaling for the full
dataset (P > 0.2 for all tests), but we did find a significant
quadratic term in some classes. Both mammals (v2 = 6.44,
d.f. = 1, P = 0.011) and the Malacostraca (v2 = 8.47,
d.f. = 1, P = 0.004) show curvilinearity in the opposite
direction to that predicted by WBE: metabolic scaling is
steeper at larger body size. Only insects (v2 = 4.61, d.f. = 1,
P = 0.032) show nonlinear scaling in the direction predicted
by WBE; a simple power law is sufficient to describe scaling
in other classes. We found no evidence that orders with
steeper scaling have lower metabolic rates (the MLB
hypothesis: q = 0.16, v2 = 0.48, d.f. = 1, P = 0.49,
Fig. 2), nor when each of the eight classes are analysed
separately (P > 0.33 in all cases). We also found no
significant differences in the scaling of endotherms vs.
ectotherms (b = 0.06, v2 = 4.28, d.f. = 2, P = 0.12). Con 2010 Blackwell Publishing Ltd/CNRS
732 N. J. B. Isaac and C. Carbone
Letter
Figure 2 Variation in metabolic
scaling
among orders. Intercepts (left panel) are
expressed as fitted values of metabolic rate
(log scale) at the geometric mean mass of
each order; slopes (right panel) are expressed
as departures from the overall mean scaling
exponent of 0.748. Horizontal bars indicate
the 95% prediction intervals. Values were
extracted from a model in which log (mass)
is centred on zero for each order.
trary to predictions, our test of the cell metabolism
hypothesis revealed a weak positive relationship between
metabolic scaling and genome size scaling (b = 0.0094, 95%
CI: 0.0058, 0.013).
DISCUSSION
This study represents the most comprehensive quantitative
analysis of scaling patterns in metabolism to date. Overall,
we found that metabolic scaling converges almost exactly on
a value of 3 ⁄ 4 predicted by West et al. (1997) with
confidence intervals excluding the Figure of 2 ⁄ 3 expected
from surface-area to volume considerations (Fig. 1). The
results, however, clearly indicate that metabolic scaling is not
universal but dependent on the taxonomic level at which it
is measured. Although other studies have reached similar
conclusions (Glazier 2005; Kozlowski & Konarzewski 2005;
White et al. 2006, 2007; Capellini et al. 2010), our results
provide the first estimate of the variance among taxa and the
taxonomic level at which that variance is manifested.
What do our findings mean for metabolic theory and
WBE? On the one hand, we have found strong support for
3 ⁄ 4 scaling of metabolism on average, and there are
relatively few orders for whom the 95% prediction intervals
2010 Blackwell Publishing Ltd/CNRS
exclude 3 ⁄ 4 (Fig. 2). On the contrary, we showed that
scaling is neither universal nor explicable by relaxing the
assumptions of the underlying pseudo-fractal resource
distribution model (West et al. 1997; Savage et al. 2008).
Thus, we can say that WBE is incomplete at best. Most
predictions of MTE do not assume a constant metabolic
scaling exponent, but those which do should therefore be
reassessed. But perhaps our concept of universal scaling
exponents is a straw man. We have used universal scaling to
mean Ôa single exponent for allÕ, but perhaps the term is
analogous to another cornerstone of metabolic theory, life
history invariants (Charnov 1997). Invariants are not fixed,
but rather are uncorrelated with body size and the subset of
biologically interesting invariants exhibit unimodal central
tendency with a limited range of variation (Savage et al.
2006). Thus, Ô3 ⁄ 4 scaling of metabolism is universalÕ should
perhaps be read as shorthand for Ô3 ⁄ 4 scaling of metabolism
is the central tendencyÕ. This raises the question of how
much heterogeneity is permitted before universality (or
invariance) no longer holds. Semantic issues aside, our
findings present a challenge to WBE: it is not sufficient to
predict accurately the mean scaling exponent because other
explanations for 3 ⁄ 4 power scaling (as the central tendency)
are at least as parsimonious (Banavar et al. 2002; Darveau et
Letter
al. 2002; Ginzburg & Damuth 2008). Facing this challenge
has already begun: recent work has shown that relaxing the
assumptions of WBE leads to a range of predictions about
allometric relationships (Price et al. 2007; Savage et al. 2008).
Price et al. (2009) showed that such flexible approaches
outperform universal models in predicting plant morphological traits. We look forward to new derivations of WBE
that explore multiple physiological constraints and make a
wider range of testable predictions about scaling exponents
(Martinez del Rio 2008).
We found no evidence to support two competing
hypotheses about variation in metabolic scaling. We found
no correlation between scaling exponent and metabolic level
(the MLB hypothesis, Glazier 2005, 2010), in contrast with
the negative correlations reported by Glazier and colleagues
for diverse taxa (Glazier 2008, 2009, 2010; Killen et al. 2010).
One explanation is that mass-specific metabolic rate
(GlazierÕs preferred measure of metabolic level) is a
decreasing function of mass itself (because scaling is
generally shallower than isometric). We have observed that
scaling is steeper in larger malacostracans and mammals (see
also Kozlowski & Konarzewski 2005; Clarke et al. 2010):
thus, large-bodied taxa in these groups have both steeper
scaling and lower mass-specific metabolic rates, leading to a
negative correlation between mass-specific metabolic rate
and scaling. However, this pattern is not widespread
(indeed, insects show the opposite pattern), so additional
explanations are required to explain the discrepancy in
results. Much of the evidence for the MLB hypothesis
comes from ontogenetic (i.e. intraspecific) scaling patterns
(Glazier 2009, 2010; Killen et al. 2010), which we have not
examined, and it is possible that different physiological
processes operate at this level (Glazier 2005). Further
exploration of the MLB hypothesis (both assumptions and
predictions) is therefore keenly anticipated. However, we
recommend that future tests of the MLB hypothesis define
metabolic level, as here, using data centred within groups
(Enders & Tofighi 2007), to avoid the problems associated
with mass-specific metabolic rate. We also found no support
for the cell metabolism hypothesis (Kozlowski et al. 2003), at
least when genome size is applied as a proxy for cell size. In
fact, we found a positive correlation between order-specific
scaling of RMR and genome size, contrasting with the
significant negative correlation reported by Kozlowski et al.
(2003) among orders of birds and mammals (our results
were unchanged using this subset). There are two broad
reasons for this discrepancy. One is that heterogeneity
among orders in genome size scaling is extremely small
compared with the heterogeneity within (by 3 orders of
magnitude, results not shown). The other is the particular
set of orders used for comparison by Kozlowski et al.: the
correlation in birds was driven by two outliers, both of
which are based on extremely small samples (Charadrifor-
The metabolic scaling controversy 733
mes nC = 4 species; Gruiformes nMR = 4): removing either
causes the relationship to become nonsignficant. In mammals, the correlation disappears if data from the Xenarthra
are included (nMR = 15, nC = 10).
It is not clear why the variation in scaling should be found
among orders and no other taxonomic level. Orders are
somewhat arbitrarily-defined and differ greatly in age (Avise
& Johns 1999), so we caution against over interpreting this
result. However, we can be reasonably confident that
differences among orders reflect divergent responses to
natural selection. An alternative explanation would be that
imbalances in the supply and demand of metabolites can
explain variation in scaling exponents (Banavar et al. 2002).
In this model, steeper scaling (b > 0.75) derives from
inefficient network design, but it does not seem reasonable
to suggest that entire orders are metabolically inefficient as
each by definition is an adaptive radiation. By contrast,
shallow scaling is said to derive from metabolic demand
exceeding supply, which is thought to be most likely during
ontogenetic development or among closely related species
(Banavar et al. 2002).
We conclude that the controversy surrounding metabolic
scaling exponents has arisen, in part, because researchers
have tried to estimate the mean and variance in scaling
exponents separately. One part of the literature has compared
theoretical predictions (2 ⁄ 3 or 3 ⁄ 4) with the (mean) scaling
exponent across a large span of body masses (Dodds et al.
2001; White & Seymour 2003; Savage et al. 2004), without
admitting the possibility that scaling might not be universal.
Other studies have estimated the scaling exponent at finer
taxonomic scales in order to demonstrate that certain groups
are exceptions, or that scaling is heterogeneous (Glazier 2005;
Kozlowski & Konarzewski 2005; Nagy 2005; White et al.
2006, 2007). Our study provides a synthesis of these parallel
debates: the exponent in animals is 3 ⁄ 4 on average, but the
variation among taxa is extensive and real (i.e. departures
from 3 ⁄ 4 have strong statistical support and cannot be
explained as exceptions). Estimating both mean and variance
in scaling reveals that extremely steep (b > 1) or shallow
(b < 0.5) scaling should not be unexpected, nor inconsistent
with the overall ÔruleÕ of 3 ⁄ 4 power.
Our attempts to understand the variation in metabolic
scaling have been greatly enhanced by testing multiple
predictions using a single dataset, and we encourage this
pluralistic approach to be adopted more widely (see also
Price et al. 2009). Unfortunately, the search for a mechanistic
understanding of metabolic scaling is hampered by the fact
that candidate theories do not make predictions that are
mutually exclusive. For example, convergence on 3 ⁄ 4 power
scaling could emerge from simple considerations about
space-time dimensionality (Ginzburg & Damuth 2008), or
as a mid-point between boundary conditions (Glazier 2005,
2010). Curve-fitting, however sophisticated, is unlikely to
2010 Blackwell Publishing Ltd/CNRS
734 N. J. B. Isaac and C. Carbone
provide resolution (McGill 2003). The lack of overwhelming
support for any one theory indicates that considerable
theoretical and empirical work remains to be carried out to
explain the diversity of scaling patterns observed in nature,
and the physiological and ecological factors that drive them.
ACKNOWLEDGEMENTS
We are grateful to Peter Bennett, Andrew Clarke, Peter
Cotgreave, John Damuth, T. Ryan Gregory and Tim
Meehan for making their data available, and to Douglas
Bates, Mark Brewer, Stephen Freeman, Reinhold Kliegl and
Andrew Robinson for statistical advice. We also thank
David Atkinson, Tim Blackburn, Jim Brown, Isabella
Capellini, Andrew Clarke, Brian Enquist, Kate Jones, David
Storch and five anonymous referees for comments on
previous versions of this manuscript. NI is supported by a
NERC Fellowship (NE ⁄ D009448 ⁄ 2).
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SUPPORTING INFORMATION
Additional Supporting Information may be found in the
online version of this article:
Figure S1 Simulation results.
Table S1 Summary of the best model of metabolic scaling
under four different information criteria and penalty terms.
Table S2 Summary of the best model of metabolic scaling
for ectotherms and endotherms, both with and without
temperature-correction.
Appendix S1 Methodological details and further analyses.
As a service to our authors and readers, this journal provides
supporting information supplied by the authors. Such
materials are peer-reviewed and may be re-organized for
online delivery, but are not copy-edited or typeset. Technical
support issues arising from supporting information (other
than missing files) should be addressed to the authors.
Editor, David Storch
Manuscript received 8 January 2010
First decision made 3 February 2010
Manuscript accepted 19 February 2010
2010 Blackwell Publishing Ltd/CNRS