Scaling metabolic rate with body mass and inverse body temperature: a test of the Arrhenius fractal supply model


*Correspondence author. E-mail:


  • 1How body mass and body temperature influence metabolic rate has been of interest for decades. Today that interest can be seen in the form of debates over the proper scaling coefficients, and the mechanistic underpinnings of allometric models for metabolic rate in relation to body mass and body temperature. We tested explicit assumptions built into what we term the Arrhenius fractal supply (AFS) model of these relationships. This model, and its assumptions, is foundational to the controversial Metabolic Theory of Ecology.
  • 2In addition to predicting that the scaling exponent for body mass is 3/4, the AFS model originally predicted that metabolic responses to body temperature, measured as activation energies, should fall between 0·2 and 1·2 eV. More recently, the latter range was narrowed to 0·6 and 0·7 eV.
  • 3To test the AFS's predictions, we used multiple regression of ln(metabolic rate) as a function of ln(body mass) and 1/(body temperature) to fit the best scaling exponent for body mass to nine data sets of many diverse species.
  • 4For the majority of the data sets, in addition to not supporting a scaling exponent of 3/4, the analyses indicated that effects of body temperature sometimes fell outside the range of 0·6–0·7 eV, indicating that the predictions of the AFS model do not hold universally.
  • 5Effects of body temperature, however, did fall within the range of 0·2–1·2 eV. To aid interpretation of these results, we transformed activation energies into Q10s. At ecologically realistic temperatures, the values of Q10 that approximate activation energies of 0·2–1·2 eV ranged from c. 1·4 to 6·1 (where 6·1 is clearly unreasonably high). Hence, any model that predicts activation energies between 0·2 and 1·2 eV does not appear to be an informative scaling model at the organismal level.
  • 6The AFS model is foundational for the Metabolic Theory of Ecology. While we commend the attempt to incorporate scaling of metabolism into ecological theory, and the research it has inspired, we caution against using untested, and likely incorrect, assumptions as a foundation to a general theory of ecology. We recommend that scientists allow the data to determine the best model for incorporating energetics into ecological theory.


For more than a hundred years, scientists have debated how metabolic rate scales with body mass. Early hypotheses reasoned that metabolism of endotherms should scale with body mass raised to the 2/3 power (the surface law) because: (i) metabolic heat is generated within the volume of the animal and dissipated through its surface area, and (ii) objects (including animals), with common shapes, have surface areas scaling to the 2/3 power of volume (Rubner 1883; Sarrus and Rameaux 1839 cited by Brody 1945). However, later studies of small numbers of endothermic species found that basal metabolic rate scaled approximately with body mass raised to the 3/4 power (Kleiber 1932; Brody 1945). Three influential reviews reinforced the thinking that metabolic rate scaled with mass to the 3/4 power (Peters 1983; Calder 1984; Schmidt-Nielsen 1984; but cf. Heusner 1982, 1991). Recently, numerous papers rekindled the debate surrounding the scaling exponent of body mass with metabolism (Dodds, Rothman & Weitz 2001; White & Seymour 2003, 2004, 2005; Bokma 2004; Brown et al. 2004a,b; Kozlowski & Konarzewski 2004, 2005; Savage et al. 2004b; Farrell-Gray & Gotelli 2005; Glazier 2005; Reich et al. 2006; White, Phillips & Seymour 2006; White et al. 2007).

In addition to body mass, however, body temperature also potentially affects the scaling of metabolic rates. An increase in temperature, within limits, results in an acceleration of most processes (Schmidt-Nielsen 1997). Consequently, variation in body temperature may contribute to variation in metabolic rates. At least four analytical (or statistical) approaches can be used to model the relationship between body temperature and metabolic rate.

First, the relationship between body temperature and metabolism can be modelled in terms of a Q10 (e.g. White & Seymour 2003). Q10 predicts the effect of a 10 °C increase in temperature on metabolic rate by the equation:

image(eqn 1)

where B denotes metabolic rate and T denotes a temperature. The Q10 is used to adjust metabolism to a standardized temperature, and then the scaling exponent for body mass is determined for the adjusted metabolic rate by regressing log (Q10 adjusted metabolic rate) on log (body mass) (White & Seymour 2003). Q10 is well known to vary with temperature, but its value as a first order empirical generalization is robustly established (Withers 1992; Schmidt-Neilsen 1997; Gillooly et al. 2001).

A second approach uses multiple regression to regress log (metabolic rate) on log (body mass) and body temperature (not log temperature) as well as other predictors (e.g. activity; Robinson, Peters & Zimmermann 1983; Andrews & Pough 1985). This approach directly estimates the coefficients for body mass and body temperature instead of assuming a particular value for either coefficient a priori.

A third approach uses the Arrhenius equation to model the effects of body temperatures on metabolic rate. We refer to this approach as the Arrhenius fractal supply model (AFS), because the model relies on an Arrhenius approach to model body temperature effects and a fractal scaling approach to model effects of body mass (Gillooly et al. 2001). The AFS, an extension of a recent scaling model for body mass and metabolic rate, is based upon the explicit assumption that resources are transported through fractal-distribution networks of tubes and this represents the universal, rate-limiting step controlling metabolic rate in all organisms (West, Brown & Enquist 1997, 1999). This model assumes metabolic rate scales to the 3/4 power of body mass. In the AFS model, body temperature of the system influences metabolic rate according to the Boltzmann's factor (e−E/kT), such that:

image(eqn 2)

where M is body mass, e is the base of the natural logarithm, Ei is the average activation energy for the rate-limiting, enzyme-catalyzed biochemical reactions of metabolism, k is Boltzmann's constant (8·62 × 10−5 eV K−1), and T is body temperature in Kelvin. Some critics have questioned the validity of the model as a theory predicting metabolic rate (Clarke 2004, 2006; Clarke & Fraser 2004; Cottingham & Zens 2004; Cyr & Walker 2004; Koehl & Wolcott 2004; Marquet, Labra & Maurer 2004; but cf. Brown et al. 2004a,b; Gillooly et al. 2006), but here we focus on the predictions from this kind of model. For our purposes, the most important assumption of the AFS model is that eqn 2 fixes the scaling exponent for body mass at 3/4. Fixing the scaling exponent for mass at 3/4 potentially misleads analyses of the effects of body temperature, because if the true exponent is not 3/4 then the model is mis-specified and the regression coefficient for scaling temperature effects will be estimated incorrectly.

A fourth approach incorporates elements of the AFS model and of multiple regression to include inverse of body temperature as a predictor. This approach fits a multiple regression of ln(metabolic rate) on ln(body mass) and 1/(body temperature) simultaneously. Unlike the Q10 and AFS approaches, a multiple regression approach estimates scaling coefficients for body mass and body temperature from the best fit to the data. If the fitted scaling exponent for mass is not exactly 3/4, then this approach should produce different estimates for the effects of body temperature than the AFS model. An example of this approach is a recent test of the AFS model for five groups of soil and litter dwelling organisms (Meehan 2006).

We used multiple regression of ln(metabolic rate) on ln(body mass) and on inverse body temperature to reanalyze the data sets originally used to develop the AFS model. Our primary goal is to assess the range of activation energies derived from the temperature coefficients in the multiple regression in relation to activation energies derived from the AFS. The original AFS model ‘predict[s] that the value of Ei[activation energy] obtained from these plots will fall within the range of measured activation energies for metabolic reactions ... [that] vary between 0·2 and 1·2’ (Gillooly et al. 2001). In subsequent papers, the predicted range of activation energies was narrowed to between 0·6 and 0·7 eV (Brown et al. 2004b, p. 1774, fig. 1 caption, page 1775). We compare our results to both proposed ranges because both have been discussed in subsequent papers (e.g. Gillooly et al. 2006; Meehan 2006). Furthermore, examining both the narrow and wide ranges provides insight into how activation energy, particularly units of electron volts, relates to Q10. Our calculations about the electron volt equivalence to Q10 provide insight into the predictive power of the AFS model.

Materials and methods

We used several data sets to regress ln(metabolic rate) on ln(body mass) and inverse body temperature.

Omitting an error term for simplicity, the model we analyzed was:

image(eqn 3)

which log-transformed is:

image(eqn 4)

Because Boltzmann's constant is 8·62 × 10−5 eV K−1 and c = –Ei/1000k, the activation energy, Ei, can be calculated from the coefficient c in eqn 4.

Our analyses focus on the data used to develop the AFS model (Gillooly et al. 2001). To allow for direct comparison, we analyzed all of the same data sets they analyzed (Table 1). However, the endotherm data set used by Gillooly et al. appeared to include 13 pairs of duplicate data points, so we ran regressions both with and without the duplicate points. In addition, the reptile, amphibian, fish and plant data sets appear to have multiple temperature readings from the same individuals. For direct comparison with the AFS model, we analyzed those data without correcting for the possible lack of independence that can result from replicate measurements on the same individuals.

Table 1.  Sample size, mass range (g) and temperature range (K) for each of the 11 data sets analyzed
 Data setSample sizeMass range (g)Temperature range (K)
Gillooly et al.unicellular 290·042 × 10−10–0·1278–313
multi-cellular invertebrates 203·6 × 10−4–500281·5–301
plants 670·125–202271·7–311
amphibians 640·3–387·2274–303
reptiles1051·7–53 000277–318
endotherms, no duplicates1312–9000274·2–313
White and Seymour mammals5072·4–325 000303–313·7
McKechnie and Wolf birds1095·5–100 000309–314·7
White and Seymour + McKechnie and Wolf endotherms6162·4–325 000303–314·7

The endothermic data set presented in Gillooly et al. (2001) was relatively small, so we also performed regressions on larger data sets for mammals and birds using the data from White & Seymour (2003) and McKechnie & Wolf (2004), respectively (Table 1). The McKechnie and Wolf data for birds did not contain data for body temperature, so we supplemented our analysis with body temperatures from original sources and from Prinzinger, Pressmar & Schleucher (1991). We also analyzed the pooled bird and mammal data sets (i.e. henceforth called the ‘comprehensive endotherm data set’). We did not include the endotherm data presented by Gillooly et al (2001) in the comprehensive endotherm data set to avoid including multiple values for individual species.

Multiple regressions of ln(metabolic rate) on ln(body mass) and 1000/body temperature were performed using sas (SAS 9·1 for Windows, Cary, NC). For the endotherm regression, we included a dummy variable to account for clade (mammals or birds). Before testing that model, we tested whether the slopes for ln(body mass) and 1000/T differed between birds and mammals. There was no significant (P > 0·05) heterogeneity of slopes, and there was no three-way interaction of clade, ln(body mass), and 1000/T. All interactions were then dropped from the model to test whether clade was significant (i.e. to test for different intercepts for birds and mammals). In the absence of a significant difference of intercept, a single model including ln(body mass) and 1000/T was used to estimate the ln(metabolic rate) of endotherms.

We also analyzed all data sets using the log-transformed AFS model of Gillooly et al. (2001), which assumes the scaling coefficient for body mass is 3/4:

image(eqn 5)


For all models of the endotherm data sets, clade (birds vs. mammals) was not statistically significant when body temperature was included in the model. Thus clade was not included in subsequent modelling.

In all analyses, the overall model, the scaling exponent for body mass, and the temperature coefficient were significantly different from zero (i.e. P < 0·05, Table 2). In the multiple regression model, the 95% confidence intervals for all of the data sets included the wide range of activation energies (0·2–1·2 eV), but only 6 of the 11 data sets included at least part of the narrower range of activation energies (0·6–0·7 eV, Fig. 1). Data for unicellular organisms, multicellular invertebrates, plants, amphibians and reptiles (Gillooly et al. 2001) fit the narrower range of activation energies as did the data for birds from McKechnie & Wolf (2004), but the amphibian data set overlaps the prediction only slightly. However, data for fishes, endotherms (Gillooly et al. 2001– with or without duplicate data points), White & Seymour's (2003) data on mammals and the comprehensive endotherm data set did not fit the narrower prediction range of activation energies (viz. 0·6–0·7 eV).

Table 2.  Activation energy (Ei ± SE), scaling coefficient (b ± SE, where appropriate) and all 11 data sets using the multiple regression model inline image and the Arrhenius-allometric model inline image Model, Ei, and b P-values were all highly significant (≤ 0·0005)
Model Multiple regressionArrhenius-allometric (i.e. b = 0·75) Ei ± SE
Data set (n)b ± SEEi ± SE
Gillooly et al.unicellular (29)0·717 ± 0·0200·766 ± 0·1130·757 ± 0·117
multi-cellular invertebrates (20)0·741 ± 0·0340·768 ± 0·1700·789 ± 0·146
plants (67)0·692 ± 0·0290·644 ± 0·0460·656 ± 0·047
fishes (113)0·753 ± 0·0300·433 ± 0·0390·433 ± 0·039
amphibians (64)0·688 ± 0·0400·494 ± 0·0560·497 ± 0·056
reptiles (105)0·708 ± 0·0190·748 ± 0·0420·757 ± 0·043
endotherms (142)0·687 ± 0·0250·776 ± 0·0240·784 ± 0·025
endotherms, no duplicates (131)0·682 ± 0·0250·767 ± 0·0270·776 ± 0·026
White and Seymour mammals (507)0·673 ± 0·0060·940 ± 0·0600·856 ± 0·068
McKechnie and Wolf birds (109)0·672 ± 0·0150·752 ± 0·2091·005 ± 0·212
White and Seymour + McKechnie and Wolf endotherms (616)0·676 ± 0·0060·889 ± 0·0470·899 ± 0·054
Figure 1.

95% confidence intervals for the mass scaling coefficient, b, and activation energy (panels A and B, respectively) for 11 data sets determined from the multiple regression model inline image in which b was calculated as part of the regression. Panel C shows the 95% confidence intervals for activation energy for all 11 data sets determined from the AFS model inline image in which b is fixed at a b = 3/4. Black horizontal bars in panel A indicate data sets with confidence intervals that include b = 3/4. Shaded vertical areas on panels B and C indicate the narrow range of predicted activation energies (0·6–0·7 eV, Brown et al. 2004b). Black horizontal bars on panels B and C indicate data sets with confidence intervals that include at least part of the narrow range of activations energies. All of the data sets for both models have confidence intervals for activation energies that fall within 0·2–1·2 eV (Gillooly et al. 2001).

The scaling of metabolic rate with mass has been amply examined elsewhere. Consequently, we report mass scaling exponents in brief only to provide a more thorough description of our results. The 95% confidence interval for the mass-scaling exponent did not include 3/4 for Gillooly et al.'s endotherm data (duplicate data points included or excluded), their plant data, their reptile data, White and Seymour's mammal data, McKechnie and Wolf's bird data, and the comprehensive endotherm data set. Only unicellular organisms, multicellular invertebrates, fishes, and amphibians, had confidence intervals for the scaling exponent for body mass that included 3/4 (Fig. 1). The 95% confidence interval included 2/3 for all data sets except the unicellular organisms and fish (from Gillooly et al. 2001). The confidence interval included both 2/3 and 3/4 for unicellular organisms, multicellular invertebrates and amphibians (Fig. 1).

When the data were analyzed using the AFS model with the scaling exponent for body mass fixed at 3/4, all of data sets fit the larger range of activation energies (0·2–1·2 eV). Similarly analyses of unicellular organisms, multicellular invertebrates, plants and reptile data sets from Gillooly et al. (2001), and bird data from McKechnie & Wolf (2004) fit the narrower range of activation energies (Fig. 1). Conversely, the data sets for fishes, amphibians, endotherms (with and without duplicates) from Gillooly et al. (2001), mammal data from White & Seymour (2003), as well as the comprehensive endotherm data did not fit the narrow range of activation energies.


Our analyses indicate that using a multiple regression to study the scaling of metabolic rate with body mass and body temperature may be superior to assuming that past analyses have correctly determined scaling constants. Multiple regression avoids making unnecessary assumptions about mechanisms and coefficients. For example, it is unnecessary to assume that the scaling coefficient for body mass is 3/4 or that a single Q10 applies over the range of body temperatures under study. Additionally, multiple regression helps avoid the omitted variable problem (Maddala 1988 pp. 121–123) wherein a regression that omits a variable that is required of the model (i.e. the model is mis-specified) can affect coefficients for the other predictors. In the present context, if inverse temperature is a significant predictor of ln(metabolic rate) and it is omitted from the model, then the estimated scaling exponent for body mass will be biased. This suggests that a new direction in estimating metabolic scaling models may be an initial search for all the universal, biologically important predictor variables that belong in the model.

Our analyses suggest that the range of activation energies for metabolic rate may be narrower than 0·2–1·2 eV (Gillooly et al. 2001), but is certainly wider than 0·6 and 0·7 eV (Brown et al. 2004b). The unfamiliarity of the unit of electron volt has made it challenging to put this result, as well as those of previous biological papers examining the AFS, in context. So an obvious question is, how accurate are these predictions? One way to frame this question is to translate eV into units commonly familiar to physiologists. Early work by van't Hoff (1896) and more recent work by others (Withers 1992, p. 124; Schmidt-Nielsen 1997, p. 589; Gillooly et al. 2001, 2006) have shown mathematically that when the activation energy is held constant the Q10 must vary depending on the range of temperatures over which it is calculated.

Despite this limitation, the Q10 is a helpful and widely used way to conceptualize the magnitude of temperature effects. We calculated the Q10 for a given activation energy at temperatures of 0, 10, 20 and 30 °C (i.e. 273, 283, 293 and 303 K), following the formula for relating Q10 to activation energy in Gillooly et al. (2001) (Dixon & Webb 1964): inline image where T and T0 are temperature in Kelvins. Activation energies of 0·2 eV and 1·2 eV are very different (Table 3). At 0 °C, with a 10 °C temperature increase, an activation energy of 0·2 eV corresponds to a Q10 of 1·35 and an activation energy of 1·2 eV corresponds to a Q10 of 6·06 (extraordinarily high for whole body metabolism). At 30 °C with a 10 °C temperature increase, an activation energy of 0·2 eV corresponds to a Q10 of 1·28 and an activation energy of 1·2 eV corresponds to a Q10 of 4·34. Hence the broad range of predicted activation energies by Gillooly et al. (2001) indicates huge differences in thermal sensitivity, including values that seem incorrectly high. By comparison with the more familiar Q10, it is apparent that the prediction by Gillooly et al. (2001) concerning the magnitude of activation energies is not overly prescriptive (i.e. it is not a narrow range). Essentially, the AFS prediction is a weak prediction because few metabolic processes would fail to conform to it. How much narrower the expected range of activation energies is remains unclear. Brown et al.'s suggestion of a range of 0·6–0·7 eV is quite narrow and does not seem to apply universally (Table 3). Perhaps some intermediate range, smaller than that suggested in Gillooly et al. (2001), but broader than hypothesized in Brown et al. (2004b), may best capture the range of temperature coefficients exhibited by most metabolic scaling relationships of whole organisms. An activation energy range of 0·46–0·96 eV (or more roundly 0·5–1·0 eV) might be appropriate because it corresponds to the typical range of Q10 for whole body metabolism (i.e. Q10 c. 2–3 over the range of 0–40 °C).

Table 3.  The relationship between Q10 and activation energy (Ei) for four temperature ranges. Base temperature is the smaller number in a 10 °C temperature (T) change for which Q10 was calculated using the formula inline image. The range in Ei shown (0·2–1·2) is the wide range of activation energies predicted for the Arrhenius-allometric model by Gillooly et al. (2001)
EiBase temperature (°C)

The AFS model has been influential in rekindling an interest in the scaling of metabolic rate and has recently been expanded into The Metabolic Theory of Ecology (Brown et al. 2004b). The theory has been used to explain, among others, molecular clocks (Gillooly et al. 2005), development times (Gillooly et al. 2002), population density (Allen, Brown & Gillooly 2002), and population growth (Savage et al. 2004a). However, this theory explicitly predicts a range of activation energies (Brown et al. 2004b) not supported by the data. Thus, this laudable attempt to use scaling metabolism in ecology should avoid untested, and likely incorrect assumptions.

Our analyses show that when the scaling exponent for body mass is appreciably different from 3/4, the effect on estimating activation energies can be substantive. We think that a better approach to the study of scaling relations in ecology is to allow the data to define the scaling exponent(s). If biologists allow the data to determine the best fit of a scaling model, then we should be better able to test whether the Metabolic Theory of Ecology is a major breakthrough or a model in need of revision.


We thank J. Gillooly, A. McKechnie, R.A. Prinzinger for sharing data. J. Gillooly kindly reviewed a draft of this manuscript. The comments from anonymous reviewers and A. Clarke were also invaluable in clarifying early versions of this manuscript. CRT was partially supported by the Nevada Biodiversity Initiative. JPH was partially supported by award IOS 03-44994 from the National Science Foundation.