## SUMMARY

The importance of size as a determinant of metabolic rate (MR) was first
suggested by Sarrus and Rameaux over 160 years ago. Max Rubner's finding of a
proportionality between MR and body surface area in dogs (in 1883) was
consistent with Sarrus and Rameaux's formulation and suggested a
proportionality between MR and body mass (*M*_{b}) raised to
the power of 2/3. However, interspecific analyses compiled during the first
half of the 20th century concluded that mammalian basal MR (BMR, ml
O_{2} h^{-1}) was proportional to
*M*_{b}^{3/4}, a viewpoint that persisted for seven
decades, even leading to its common application to non-mammalian groups.
Beginning in 1997, the field was re-invigorated by three new theoretical
explanations for 3/4-power BMR scaling. However, the debate over which theory
accurately explains 3/4-power scaling may be premature, because some authors
maintain that there is insufficient evidence to adopt an exponent of 3/4 over
2/3. If progress toward understanding the non-isometric scaling of BMR is ever
to be made, it is first essential to know what the relationship actually is.
We re-examine previous investigations of BMR scaling by standardising units
and recalculating regression statistics. The proportion of large herbivores in
a data set is positively correlated both with the scaling exponent
(*b*, where BMR=*aM*_{b}^{b}) and the
coefficient of variation (CV: the standard deviation of ln-ln residuals) of
the relationship. Inclusion of large herbivores therefore both inflates
*b* and increases variation around the calculated trendline. This is
related to the long fast duration required to achieve the postabsorptive
conditions required for determination of BMR, and because peak post-feeding
resting MR (RMR_{pp}) scales with an exponent of 0.75±0.03 (95%
CI). Large herbivores are therefore less likely to be postabsorptive when MR
is measured, and are likely to have a relatively high MR if not
postabsorptive.

The 3/4 power scaling of RMR_{pp} is part of a wider trend where,
with the notable exception of cold-induced maximum MR
(*b*=0.65±0.05), *b* is positively correlated with the
elevation of the relationship (higher MR values scale more steeply). Thus
exercise-induced maximum MR (*b*=0.87±0.05) scales more steeply
than RMR_{pp}, field MR (*b*=0.73±0.04), thermoneutral
resting MR (RMR_{t}, *b*=0.712±0.013) and BMR. The
implication of this observation is that contamination of BMR data with
non-basal measurements is likely to increase the BMR scaling exponent even if
the contamination is randomly distributed with respect to
*M*_{b}. Artificially elevated scaling exponents can therefore
be accounted for by the inclusion of measurements that fail to satisfy the
requirements for basal metabolism, which are strictly defined (adult,
non-reproductive, postabsorptive animals resting in a thermoneutral
environment during the inactive circadian phase). Similarly, a positive
correlation between *M*_{b} and body temperature
(*T*_{b}) and between *T*_{b} and
mass-independent BMR contributes to elevation of *b*. While not
strictly a defined condition for the measurement of BMR, the normalisation of
BMR measurements to a common *T*_{b} (36.2°C) to achieve
standard metabolic rate (SMR) further reduces the CV of the relationship.
Clearly the value of the exponent depends on the conditions under which the
data are selected. The exponent for true BMR is 0.686 (±0.014),
*T*_{b} normalised SMR is 0.675 (±0.013) and
RMR_{t} is 0.712 (±0.013).

## Introduction

“*You can drop a mouse down a thousand-yard mine shaft; and, on
arriving at the bottom, it gets a slight shock and walks away, provided that
the ground is fairly soft. A rat is killed, a man is broken, a horse
splashes.*” `*On being the right size*', by J. B. S. Haldane
(1928).

Size matters. Its effect is all pervasive, but it influences different
variables in different ways: given that the volume (*V*, and therefore
mass, *M*) of an object is proportional to the cube of some linear
dimension (*M*∝*L*^{3}), whilst its surface area
(*A*) is proportional to the square of a linear dimension
(A∝*L*^{2}), we can rearrange to find that
*L*∝*M*^{1/3}∝*A*^{1/2}, so
*A*∝*M*^{2/3}. If you double an object's length,
but keep its proportions the same, its surface increases fourfold, and its
volume eightfold. Surface area does not have an isometric relationship with
mass, and this is true of a great many physiological variables
(Huxley, 1932;
Gould, 1966;
Packard and Boardman, 1999).
Allometry (or scaling) is a technique used to describe this non-isometric
variation by regressing a variable of interest against body mass, thereby
describing the relationship. This relationship is often well described by a
linear regression of log(*y*) on log(*M*) such that
(1)

which can be rearranged to produce a power equation of the form: (2)

where *y* is the variable of interest, *a* is the allometric
coefficient, *M*_{b} is body mass and *b* is the
allometric exponent. Allometric procedures are used throughout the biological
sciences and a great many allometric relationships have been generated
(McMahon and Bonner, 1983;
Peters, 1983;
Calder, 1984;
Schmidt-Nielsen, 1984;
Brown and West, 2000).

One of the most frequently investigated allometric relationships is that
between *M*_{b} and basal metabolic rate (BMR). Given that heat
produced through metabolic processes must be lost through the body surface, it
can therefore be supposed that metabolic rate would also be proportional to
*M*_{b}^{2/3}, so that the rate of heat production
would be matched to the area of the surface over which it is dissipated. The
idea that the effect of body size on metabolism might reflect simple geometric
and physical processes was first suggested by Sarrus and Ramaeux in 1838
(cited by Brody, 1945) and
supported by Max Rubner
(1883). Rubner
(1883) found that the
metabolic rate of resting dogs was independent of mass when divided by surface
area. This finding, which came to be known as Rubner's surface law of
metabolism, stood largely unchallenged for almost 50 years, until publication
of Max Kleiber's influential monograph
(Kleiber, 1932). Kleiber
(1932,
1961) found that metabolic
rate was proportional to body mass raised to an exponent significantly greater
than 2/3, and a value of 3/4 was subsequently adopted. Famously, one of the
advantages of an exponent expressed as a simple fraction was the
simplification of slide rule calculations
(Schmidt-Nielsen, 1984).
Three-quarter power scaling of mammalian BMR is a central paradigm of
comparative physiology that has been accepted for over 70 years and remains in
widespread use. Kleiber's monograph
(Kleiber, 1932), for example,
continues to be cited in papers appearing in top journals
(Gillooly et al., 2001;
Niklas and Enquist, 2001;
Whitfield, 2001;
Darveau et al., 2002;
Marquet, 2002;
Weibel, 2002;
West et al., 2002b;
Cohen et al., 2003).

Since publication of Kleiber's monograph (Kleiber, 1938), a great deal of
effort has been invested in the investigation of both BMR scaling and the
adaptive significance of BMR variation. In the years following Kleiber (1938),
Benedict (1938) significantly
expanded the data set and Brody's famous mouse-to-elephant curve
(Brody, 1945) captured almost
all of the body mass variation available with terrestrial mammals: both
studies supported an exponent of 3/4. Hemmingsen
(1960) included unicellular
organisms and poikilothermic vertebrates, thereby expanding the range of body
masses to 18 orders of magnitude, and also supported an exponent of 3/4 (a
similar approach with a similar result was recently presented by
Gillooly et al., 2001). During
the last 20 years, investigation of BMR variation has gained prominence, and
many studies have taken the approach of inferring adaptive variation in BMR by
correlating it with traits of interest. Such studies have identified BMR
differences associated with, for example, phylogeny
(Hayssen and Lacy, 1985;
Elgar and Harvey, 1987), diet
(McNab, 1988), geography
(Lovegrove, 2000), aridity
(Lovegrove, 2000;
White, 2003), habitat
productivity (Lovegrove, 2000;
Mueller and Diamond, 2001),
climate (Lovegrove, 2003) and
relative organ masses (Konarzewski and
Diamond, 1995). In addition to identifying presumably adaptive
differences in BMR and presenting scaling relationships for specific groupings
of mammals, many of these studies also produced scaling relationships for all
data available to them (Table
1). Although calculation of a regression line conceals the
adaptive variation in BMR in a single average relationship between BMR and
*M*_{b}, the scaling of BMR independent of other factors has
continued to be of interest.

Recent attention has focused on theoretical explanations for this quarter-power scaling based on the geometry of nutrient supply networks (West et al., 1997; Banavar et al., 1999, 2002b; Bejan, 2000), four-dimensional biology (West et al., 1999) and an allometric cascade that links cellular and whole animal metabolism (Darveau et al., 2002; Hochachka et al., 2003). Proponents of these theories remain unable to reach a consensus on which is correct, and have presented arguments disputing competing theories (Banavar et al., 2002a, 2003; West et al., 2002a, 2003; Darveau et al., 2003). However, the debate over which theory accurately explains quarter-power scaling may be premature, at least where mammalian BMR is concerned, because some authors maintain that there is insufficient evidence to adopt an exponent of 3/4 over 2/3 (e.g. Heusner, 1991; Dodds et al., 2001; White and Seymour, 2003). If progress toward understanding the non-isometric scaling of BMR is ever to be made, it is first essential to know what the relationship actually is.

## Regression model

Traditionally, allometric analyses are conducted with ordinary
least-squares (OLS) regression (e.g.
Huxley, 1932;
Gould, 1966;
Peters, 1983;
Calder, 1984;
Schmidt-Nielsen, 1984).
However, comparative data are likely not to meet two assumptions of this
model. Firstly, because of shared phylogenetic descent, species data are
likely not to represent statistically independent points. This results in
overestimation of degrees of freedom and an increased Type I error rate. A
large body of literature deals with both the documentation of this problem and
a discussion of how to properly account for it (e.g.
Felsenstein, 1985;
Harvey and Pagel, 1991;
Garland et al., 1992;
Rohlf, 2001). Although none of
the regressions presented here account for shared descent, BMR scaling
patterns observed for mammals and birds are not greatly altered by the
inclusion of such information (White and
Seymour, 2003; McKechnie and
Wolf, 2004). Secondly, OLS regression assumes that
*M*_{b} is independent of the variable of interest and measured
without error, which may not be the case. In such a situation, reduced major
axis (RMA) regression may be more appropriate for inferring functional
relationships (Sokal and Rohlf,
1995). Although the classic allometry studies use OLS regression
(Huxley, 1932;
Gould, 1966;
Peters, 1983;
Calder, 1984;
Schmidt-Nielsen, 1984), the
use of RMA regression is becoming more common (e.g.
Nunn and Barton, 2000;
Green, 2001;
Niklas, 2004). The RMA
exponent *b*_{RMA} can be calculated by dividing the OLS
exponent by the square root of the coefficient of determination
*r*^{2} (Sokal and Rohlf,
1995), so the difference between the regression models diminishes
as *r*^{2} increases. Where *r*^{2} is low,
however, the OLS exponent is likely to be an underestimate. Throughout this
review, OLS regression results are presented
(Table 1, Figs
1,
2,
3,
4,
5,
6,
7), and RMA regressions are
tabulated for the main findings (Table
2).

## Species composition and digestive state

A striking feature of the early analyses of mammalian BMR scaling (Kleiber, 1932; Benedict, 1938; Brody, 1945) is the consistent over-representation of large herbivorous lineages (Artiodactyla, Perissodactyla and Proboscidea). Whereas these groups (henceforth referred to as `large herbivores') represent only 5% of Mammalia (Nowak, 1999), they comprise 28-39% of the data in these seminal analyses (Table 1). Large herbivores are more appropriately represented in more recent studies, but considerable variation remains between analyses. A comparison of recent BMR scaling studies (since 1984) reveals that the inclusion of large herbivores both increases the coefficient of variation of the relationship (the standard deviation of ln-ln residuals; Garland, 1984) and increases the scaling exponent (Fig. 1). These observations can both be explained by the difficulty in obtaining the conditions necessary for the measurement of BMR, which are strictly defined (McNab, 1997). BMR measurements must be obtained in the inactive circadian phase for animals that are adult, non-reproductive, conscious, resting in a thermoneutral environment, and post-absorptive. Achieving this final condition is difficult in animals that use microbial fermentation to digest cellulose, and such animals are typically fasted for less than 72 h prior to measurement of oxygen consumption (e.g. Rogerson, 1968; Weiner, 1977; Renecker and Hudson, 1986). However, ruminants may require 7 days to arrive at a postabsorptive state (Blaxter, 1962), although such a state may in fact be unachievable (McNab, 1997).

To examine the influence of body size on the fast duration required to achieve postabsorptive conditions, reference can be made to the mean retention time of particles in the digestive tract (MRT: the average time food requires to pass through the digestive tract), which has been suggested to correlate with required fast duration (Blaxter, 1989). In herbivorous mammals, MRT scales positively with an exponent of 0.17±0.05 (95% CI), regardless of whether the main site of fermentation is the foregut, hindgut or caecum (Fig. 2). A MRT of 12 h is therefore predicted for a 20 g herbivore, compared to nearly 3 days for a 500 kg one. However, the final appearance of faeces following feeding can take 4 times longer than the MRT (Blaxter, 1989), suggesting that MRT may approximate only the minimum time required to achieve a postabsorptive condition. Given that large herbivores are typically fasted for no more than 72 h, if at all, prior to determination of metabolic rate, it seems likely that such measurements fail to satisfy the conditions required for determination of BMR.

The possible confounding influence of ruminants was also recognized by
Kleiber (1932), who presented
his analysis both with and without these species. However, his fortuitous
selection of data (*r*^{2}=0.999,
Table 1) meant that removal of
any species altered the calculated regression little and rendered this
exclusion unproductive. Contrary to Kleiber's finding
(Kleiber, 1932), the exclusion
of large herbivores from recent data sets does decrease the scaling exponent
and also decreases variation around the regression line
(Fig. 1). To account for this
finding, it is first necessary to understand the scaling of metabolic rate
during digestion. Peak postfeeding resting metabolic rate (RMR_{pp})
is the highest metabolic rate measured in a resting animal following food
intake, and scales with an exponent of 0.75±0.03
(Fig. 3), which is higher than
all mammalian BMR scaling exponents produced since 1984
(Table 1). The peak factorial
increase in metabolism following feeding (=RMR_{pp}/BMR) is therefore
positively correlated with body mass in mammals, as has been documented
intraspecifically for Burmese pythons *Python molurus*
(Secor and Diamond, 1997) and
cane toads *Bufo marinus* (Secor
and Faulkner, 2002). Inclusion of non-postabsorptive measurements
is therefore likely to increase the scaling exponent for BMR, even if such
measurements are randomly distributed with respect to body mass. When
non-postabsorptive measurements for large herbivores are included, however,
the error is even greater, because larger species have greater factorial
increases in metabolic rate and, being near the upper limit of the mass range,
they have high leverage and exert a disproportionately large influence on
exponent calculation. To arrive at a precise estimate of the scaling exponent
relating BMR and *M*_{b}, it is therefore necessary to exclude
these large herbivores, together with other lineages for which basal
conditions are unlikely to be achieved (Macropodidae, Lagomorpha, and
Soricidae). Macropods and lagomorphs are excluded because they have microbial
fermentation in the stomach or hindgut
(Stevens and Hume, 1995),
while shrews (Soricidae) are excluded because resting and postabsorptive
conditions may be mutually exclusive
(Speakman et al., 1993). With
the exclusion of these suspected non-basal lineages, White and Seymour
(2003) found that BMR scales
with an exponent of 0.686±0.014
(Fig. 4). Relaxation of the
requirements for estimation of BMR to include species that may not be
postabsorptive (thermoneutral RMR or RMR_{t}:
Speakman et al., 2003) results
in an increase of the scaling exponent to 0.712±0.013 (recalculated
from Savage et al., 2004).

## Body temperature

Body mass and temperature are primary determinants of metabolic rate
(Gillooly et al., 2001).
Although there is no functional link in endotherms between metabolic rate and
*T*_{b} within the thermal neutral zone, where BMR is measured,
repeated attempts have been made to explain the BMR differences between birds
and mammals, and eutherians and marsupials, in terms of differences in
*T*_{b} (Hemmingsen,
1960; Dawson and Hulbert,
1970; White and Seymour, in press). In endotherms, the level of
BMR is determined by factors presently uncertain but apparently related to
cellular function (Hulbert and Else,
2000), while *T*_{b} is clearly regulated by the
central nervous system (Bligh,
1976). As defined, the measurement conditions for BMR do not
account for *T*_{b} differences between species, so descriptive
scaling of BMR should not be compensated for them. Compensation is necessary,
however, when comparing groups that differ in *T*_{b} or
searching for uniform explanations for scaling effects that do not depend on
*T*_{b}. *T*_{b} is significantly correlated
(*r*≈0.55) with the residual variation in mammalian BMR
(White and Seymour, 2003)
and, when normalised to a common *T*_{b} according to the
Q_{10} principle, the BMR values of birds and mammals do not differ in
allometric coefficient or exponent, nor do the BMR values of eutherians and
marsupials (White and Seymour, in press). An approach that accounts for
*T*_{b} differences has two further benefits: firstly, it
allows for investigation of the influence of body mass on BMR without the
confounding influence of *T*_{b}, which is also positively
correlated with body mass (Withers et
al., 2000; White and Seymour,
2003). Secondly, incorporation of *T*_{b} into
predictive multiple regression models allows for improved estimation of BMR
when both body mass and temperature are available.

To standardise the metabolic rates of mammals to a common
*T*_{b}, it is necessary to determine the relationship between
MR and *T*_{b}, find the mean *T*_{b}, and then
apply the appropriate Q_{10} to the temperature difference
(*T*_{b} - normalised BMR is henceforth referred to as standard
metabolic rate: SMR). White and Seymour
(2003) determined an
appropriate Q_{10} for this procedure by calculating the
*r*^{2} value for the linear regression relating log(SMR) and
log(mass), normalised to a *T*_{b} of 36.2°C using
Q_{10} values ranging from 2.0 to 4.0. A value of 3.0 was finally
selected because it minimised *r*^{2}. Within the range
2.0-4.0, Q_{10} has a quantifiable effect on the scaling exponent
estimated for the relationship between SMR and body mass, but none of the
exponents differs significantly from 2/3 (White and Seymour, in press).
Alternatively, Q_{10} can be estimated from the relationship between
the natural log of mass-independent BMR and *T*_{b} (=
BMR×*M*_{b}^{-b}, where
SMR=a*M*_{b}^{b}. However, estimation of Q_{10}
in this manner requires the scaling exponent for SMR (not BMR), which cannot
be obtained without Q_{10}. The relationship between BMR and
*T*_{b} is described by the equation:
(3)

which can be rearranged to form: (4)

The quantity *e*^{10c} then represents the factorial change
in BMR associated with a 10°C change in body temperature, or
Q_{10}. Determination of Q_{10} in this manner is advantageous
because, by making use of linear regression, it is possible to produce
confidence limits for *c*, and therefore also for Q_{10}. A
drawback of this method, however, is that it requires an estimate of the SMR
scaling exponent. Where such an estimate is not available it can be derived
using a multiple regression approach (described below) or by iteratively
solving for both *b* and *c*. The latter approach entails first
estimating *b* by normalising data to a common *T*_{b}
using a reasonable Q_{10} (e.g. 2.5). This value of *b* is then
used in the approach described above to estimate a new Q_{10}, which
is in turn used to produce a new estimate of *b*, and so on. By
accounting for the effect of *T*_{b}, White and Seymour
(2003) found that mammalian
SMR scales with an exponent of 0.675±0.013, which is not significantly
different from 2/3 (Fig. 5).
This exponent was subsequently used to estimate the Q_{10} relating
*T*_{b} and BMR, which was equal to 2.8 (White and Seymour, in
press).

Separate determination of b and Q_{10} is useful in situations
where it is of particular interest to identify one or the other. This allows
for derivation of relationships between BMR and either *T*_{b}
or *M*_{b}, an approach that provides intuitively simple
results. However, one of the great strengths of scaling is in the predictive
power that it provides. For species that are difficult to obtain or measure,
BMR can be estimated only from *M*_{b}, because
*M*_{b} alone accounts for 94% of the variation in BMR
(White and Seymour, 2003).
Such predictions can be further improved by incorporating
*T*_{b} in a multiple regression approach. By regressing
ln(BMR) against *T*_{b} and ln(*M*_{b}), it is
possible to derive both *b* and Q_{10} from a relationship of
the form:
(5)

which can be rearranged to form: (6)

Interestingly, the Q_{10} for euthermic mammals (2.8; 95% CI,
2.4-3.2) is not significantly different to that found for hibernating
(Q_{10}=2.2; 95% CI, 1.7-2.8) or torpid ones (Q_{10}=2.4; 95%
CI, 1.6-3.5), although the exact values for hibernating and torpid animals are
quite uncertain. Once normalised to a euthermic *T*_{b}
(36.2°C), the metabolic rate of torpid mammals is equivalent to that of
euthermic mammals, whilst small hibernating mammals show an additional
suppression of metabolic rate. As such, the SMR scaling exponent for
hibernating mammals (*b*=0.87; 95% CI, 0.79-0.94) is significantly
different to that of euthermic (*b*=0.675; 95% CI, 0.663-0.688) and
torpid ones (*b*=0.67; 95% CI, 0.56-0.77;
Fig. 5).

## Scaling of non-basal metabolic rate

Investigation of the influence of body mass on metabolic rate is certainly
not unique to the basal condition. Sufficient data have accumulated for
examination of the scaling of metabolic rate under a range of conditions, some
of which are discussed elsewhere in this issue or received recent attention
(Nagy et al., 1999;
Savage et al., 2004;
Weibel et al., 2004). In
addition to RMP_{pp}, RMR_{t} and SMR, these include maximum
metabolic rate (MMR) induced either by exercise (MMR_{e};
Seeherman et al., 1981) or
cold exposure (MMR_{c}, measured in an atmosphere comprising 21%
oxygen in helium; Rosenmann and Morrison,
1974; Thomas et al.,
1998), and field metabolic rate (FMR; measured by the doubly
labeled water technique; Speakman,
1997). Discussion of the specifics of each of these relationships
is well beyond the scope of this review; however, examination of the scaling
of non-basal metabolic rate can provide insight into the effect that the
inclusion of non-basal data may have on the scaling of basal metabolic rate.
Perhaps not surprisingly, MMR_{c} scales with an exponent not
significantly different from 2/3 (Fig.
6B), and therefore scales with an exponent similar to surface area
(Reynolds, 1997). This might
be expected given that MMR_{c} is measured under conditions that
maximise heat loss. Unfortunately, measurements of both body surface area and
MMR_{c} are available for too few species to determine if there is any
relationship between the residual variation in these traits. For the remaining
MR values, *b* is significantly positively correlated with the
elevation of the relationship (Fig.
7), so higher MRs scale more steeply: MMR_{e}
(*b*=0.87±0.05; Fig.
6A) scales with an exponent greater than RMR_{pp}
(*b*=0.75±0.03; Fig.
3), FMR (*b*=0.73±0.04;
Nagy et al., 1999),
RMR_{t} (*b*=0.712±0.013;
Savage et al., 2004), BMR
(*b*=0.686±0.014; Fig.
4) and SMR (*b*=0.675±0.013;
Fig. 5). This pattern is
observed both for OLS and RMA regression exponents
(Table 2). As is the case for
the inclusion of non-postabsorptive measurements, the inclusion of non-basal
measurements is therefore likely to increase the scaling exponent even if
these measurements are randomly distributed with respect to body mass. Whereas
it might be expected that non-basal measurements would increase only the
elevation of the scaling relationship and leave the exponent unaffected, past
failures to strictly adhere to the requirements for the measurement of BMR are
therefore likely to have contributed to the generation of inflated scaling
exponents.

## Conclusion

Basal metabolic rate is, and is likely to remain, a benchmark measurement in comparative physiology. It represents the minimum energy cost of steady-state existence and is a useful index of energy expenditure: after the removal of body mass effects, BMR is significantly correlated with the residuals for a variety of physiological and ecological variables including maximum metabolic rate, field metabolic rate, resting heart rate, lifespan, litter size and population density (White and Seymour, in press). Part of the attraction of BMR is its relative ease of measurement and the fact that it for allows direct comparison of different species by placing them in the same physiological state. However, to achieve this aim it is necessary to adhere rigorously to the conditions specified for the measurement of BMR. Doing so results in determination of a scaling exponent for true BMR of 0.686±0.014. Similarly, McKechnie and Wolf (2004) have recently shown that when BMR data for birds are rigorously selected, BMR scales with an exponent of 0.669.

We thank Hans Hoppeler, Ewald Weibel, and all of the participants in the
`*Size as a determinant of biological processes*' conference for an
enjoyable meeting, and a series of entertaining and informative talks and
discussions. Two referees provided comments that improved an earlier version
of this manuscript.

- © The Company of Biologists Limited 2005