## Introduction

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 *Q*_{10} (e.g. White & Seymour 2003). *Q*_{10} predicts the effect of a 10 °C increase in temperature on metabolic rate by the equation:

where *B* denotes metabolic rate and *T* denotes a temperature. The *Q*_{10} 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 (*Q*_{10} adjusted metabolic rate) on log (body mass) (White & Seymour 2003). *Q*_{10} 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:

where *M* is body mass, *e* is the base of the natural logarithm, *E _{i}* 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 *Q*_{10} 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 *E _{i}*[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

*Q*

_{10}. Our calculations about the electron volt equivalence to

*Q*

_{10}provide insight into the predictive power of the AFS model.