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In a challenging and provocative paper Gillooly et al. (2001) have proposed that the metabolism of all organisms can be described by a single equation,
Q = b0M3/4e−E/kT,
where Q = metabolic rate, M = body mass, E = the activation energy of metabolism (defined as the average activation energy for the rate-limiting enzyme catalysed biochemical reactions of metabolism), T = absolute temperature, k = Boltzmann's constant and b0 is a normalization constant independent of M and T. In deriving this equation Gillooly et al. (2001) start from the premise that metabolic rate scales with body mass as Q ∝ M3/4, based on the fractal-like design of exchange surfaces and distribution networks in plants and animals (West, Brown & Enquist 1997, 1999a,b). These arguments have stimulated some criticism (see for example Dodds, Rothman & Weitz 2001) but here I will concentrate on the derivation of the second part of the equation, namely the temperature dependence term.
Gillooly et al. (2001) called the temperature dependence term of this equation the Universal Temperature Dependence (UTD) of metabolism. Although there have been many statistical descriptions of the relationship between size, temperature and metabolism since the classic work of Hemmingsen (1950, 1960) and Kleiber (1950, 1961), the UTD differs from these in being explicitly derived from first principles, in the sense that the formulation of the temperature dependence term is derived from classical statistical thermodynamics. The UTD has subsequently been incorporated into explanations of developmental time in all organisms, and macroecological patterns including global-scale analyses of diversity and population density (Allen, Brown & Gillooly 2002; Belgrano et al. 2002; Gillooly et al. 2002).
Here I examine the assumptions underlying the formulation of the UTD, and test the relationship with a carefully assembled data set for teleost fish. In doing so I have distinguished between two philosophically different forms of the UTD, both of which are discussed but not explicitly distinguished by Gillooly et al. (2002). The first is where metabolic rate is determined mechanistically by temperature alone; this might be termed the hard UTD hypothesis. In the second form the UTD is simply a parameter-sparse statistical model describing the relationship between temperature and metabolic rate; this is the soft UTD hypothesis.
Temperature and metabolic rate
In deriving the UTD, Gillooly et al. (2002) start from the observation that temperature governs metabolism through its effects on rates of biochemical reactions. They then argue that this temperature dependence can be described by the equations of statistical thermodynamics, and specifically that reaction kinetics vary with temperature according to the Boltzmann factor e−E/kT. The metabolic rate of an organism involves the summation of many different biochemical reactions, so that Q = ΣRi,where Ri represents the rate of energy production by the individual reactions that constitute metabolism. Gillooly et al. (2001) argue that the rate of each of these reactions depends on three major variables:
Ri ∝ (concentration of reactants) (fluxes of reactants) (kinetic energy of the system)
The first two variables are constrained by the rates of supply of substrates and removal of products, and hence contain the majority of the body mass dependence of Ri. The third term contains the temperature dependence, which Gillooly et al. (2001) equate to a direct governance by the Boltzmann factor, e−E/kT, and from which they derive the UTD equation. It is important to recognize that although an increase in metabolic rate with temperature is observed universally in within-species (or within-individual) studies, the hard UTD hypothesis extrapolates this directly to the across-species relationship. The hard UTD hypothesis thus carries the implicit assumption that the within-species and across-species relationships between metabolic rate and temperature are identical, because the same physical mechanism underpins both. It thereby provides no opportunity or mechanism for laboratory acclimation, seasonal acclimatization or evolutionary adaptation, other than by a change in E, which thus becomes an empirical variable rather than a parameter of the UTD defined from first principles. Gillooly and coauthors do acknowledge variance about the UTD relationship associated with differences in ecology, but do not speculate on how this variability is achieved.
A key problem with this mechanistic derivation of the UTD is that while statistical thermodynamics provides a very successful description of the behaviour of a simple system where temperature is the only variable that changes, organismal metabolism is very different. Organismal metabolism involves a large number of physiological processes, each of which interacts with many others. Concerns over how best to apply concepts developed for simple systems to such complex entities go back to the earliest physiologists (for example Krogh 1916) and remain (Clarke 1983; Cossins & Bowler 1987; Hochachka & Somero 2002). The Stefan–Boltzmann distribution provides an excellent description of the distribution of kinetic energy in simple systems such as molecules in solution. Combined with the concept of the activation energy threshold introduced by Arrhenius (1889, 1915), it provides a very successful explanation for why a small rise in temperature, which increases mean kinetic energy by only a small amount, can cause a large change in reaction velocity. Critical here is the activation energy of the reaction, Ea, which provides a significant contribution to the Gibbs free energy of activation ΔG‡ = ΔH‡ −TΔS‡ and Ea ∼ ΔH‡ where ΔG‡ is the Gibbs free energy of activation, ΔH‡ and ΔS‡, respectively, the enthalpic and entropic contributions to this, and T is absolute temperature. The important point here is that the Arrhenius activation energy, Ea, is only one factor contributing to the overall Gibbs free energy of activation; entropy is also important. (For fuller discussions see Johnson, Eyring & Stover 1974; Clarke 1983; Cossins & Bowler 1987; Hochachka & Somero 2002.)
It is now clear from over two decades of comparative physiology that the thermodynamic activation parameters of homologous enzymes from related taxa living at different temperatures often differ (for the classic study of LDH (lactate dehydrogenase-A homologues) from barracuda, Sphyraena, which effectively defined the field see Graves & Somero 1982; Holland, McFall-Ngai & Somero 1997). We now know that in general key enzymes involved in ATP synthesis exhibit broadly similar activities (expressed as the catalytic rate constant, kcat, the rate at which substrate is converted to product per active site per unit time) in homologues isolated from species living at different temperatures, when measured at the ambient temperature for the organism in question. This is typically achieved by variation in the contributions of both ΔH‡ and ΔS‡ to ΔG‡ (although small changes in ΔG‡ may also be involved: Hochachka & Somero 2002). The net effect of these changes in catalytic properties is to achieve a relative independence of reaction rate from the enthalpic environment (that is, cellular temperature and hence kinetic energy). An important mechanism underpinning this result is that the rate-limiting step in enzyme-catalysed reactions is not the breaking and formation of covalent bonds (which happens relatively quickly because of the structural and ionic environment of the active site), but the binding and release of ligands. These are dependent on conformational changes in the enzyme tertiary structure, and are mediated largely through the breaking and formation of weak bonds. It is the relatively low free energies involved in these bonds that set the rate of reaction, and also the overall temperature sensitivity (for a succinct review of current knowledge see Hochachka & Somero 2002). These conformational changes are an important component of the entropic contribution to the overall Gibbs free energy of activation.
An alternative hypothesis: evolutionary trade-off
Current evidence thus indicates the existence of a range of evolutionary modifications that render reaction rate relatively independent of the temperature at which an organism is adapted to live (Hochachka & Somero 2002). So if the simple mechanistic link between resting metabolic rate and temperature proposed in the hard UTD hypothesis is incompatible with what we know about the physical chemistry of enzyme catalysis, why does resting metabolic rate exhibit a strong correlation with habitat temperature? The resting metabolic rate of an organism approximates to its basal metabolic rate, which is a measure of its cost of living; it is a summation of all those processes that preserve cellular and organismal integrity, and important components include the maintenance of proton and ion gradients, and macromolecular turnover (Rolfe & Brown 1997; Clarke & Fraser 2004). The resting metabolic rate of each species thus represents a quasi-independent evolutionary optimization to the energetic demands of maintaining organismal integrity at its environmental temperature, and of its ecology (lifestyle). This evolutionary adjustment occurs at two hierarchical levels within the organism. The first is in the relative size of organs and tissues of different inherent metabolic demand (muscle, brain, liver and so on: Daan, Masman & Groenewold 1990), and the second is adjustment of the metabolic demand of the individual tissues in response to an evolutionary trade-off between required function, energetic costs of maintenance and temperature (Clarke 1993). Neither the relative balance of the cellular processes contributing to resting metabolic rate nor how these costs vary with temperature can be predicted from first principles; all we can do is describe the evolutionary end result statistically. Nevertheless studies of particular taxa such as birds (Daan et al. 1990) indicate that the relative size of organs tends to be constrained within lineages, whereas all lineages respond similarly to the trade-offs involved in adapting to live at different temperatures (Clarke & Johnston 1999). The resting metabolic rate of an organism thus represents not a mechanistic response to temperature, but the energetic cost of evolutionary adaptation to a particular temperature and lifestyle. This might be termed the evolutionary trade-off hypothesis of temperature adaptation (Clarke 1993, 2003).
The evolutionary trade-off hypothesis thus regards the across-species relationship between resting metabolic rate and temperature as a statistical description of quasi-independent evolutionary optimizations to temperature and ecology (Clarke 1993, 2003). The data for different species are not fully independent because more closely related species tend to have more similar ecologies and lifestyles than do more distantly related organisms. This phylogenetic auto-correlation reduces the degrees of freedom and comparisons across taxa therefore need to use the suite of statistical techniques developed to handle these problems (reviewed by Harvey & Pagel 1991).
We therefore have two quite different hypotheses for the relationship between resting metabolic rate and temperature. The hard UTD hypothesis proposes that the relationship is dictated directly by the temperature kinetics of enzyme reactions, with ecology contributing to variance about the UTD line (Gillooly et al. 2001). The evolutionary trade-off hypothesis regards the relationship as a statistical description of evolutionary optimizations to temperature and ecology (Clarke 1993, 2003). These two hypotheses lead to subtly different predictions, which allow for a test with empirical data. The hard UTD hypothesis assumes that precisely the same physical processes underpin enzyme and organismal thermal behaviour: it follows logically that the within- and across-species relationships between resting metabolic rate and temperature should be the same. Gillooly et al. (2001) tested this by using a range of observed enzyme activation energies to predict ‘the activation energy of organismal metabolism’ (that is the slope of the regression line fitted to an Arrhenius model, which is the logarithm of reaction rate as a function of inverse temperature). They concluded that empirical data fitted the prediction sufficiently well for the UTD hypothesis to be supported. The evolutionary trade-off hypothesis implies that the acute effect of temperature on an individual's resting metabolic rate (which is a direct thermodynamic response to temperature, albeit in a complex system) and the across-species relationships are fundamentally different, but cannot make any quantitative predictions concerning the latter.
We are able to test the prediction of the hard UTD hypothesis using a data set for resting metabolic rate in teleost fish (Clarke & Johnston 1999). Data from 69 taxa exhibited a range of mass exponent values (Fig. 1a), with a mean value of 0·793 and 95% confidence limits of 0·77–0·82. These empirical data thus exclude (but only just) the theoretical value of 0·75 derived by West, Brown & Enquist (1999a,b). The value of b varies significantly between families (ancova: F = 3·50, P < 0·001) and orders (F = 3·48, P = 0·01), although the outcome of the analysis is influenced strongly by single low values for two particular families (Clarke & Johnston 1999). Data for resting metabolic rate were found to be well described by an Arrhenius model (Fig. 1b); the slope of the fitted regression line was, however, significantly less than the within-species relationship. The overall across-species relationship was equivalent to a Q10 of 1·83, calculated over the range 0–30 °C. For 14 studies where a within-species Q10 was reported, the mean value was 2·36 with 95% confidence intervals of 1·89–2·81 (Fig. 1c). We therefore conclude that the data for teleost fish do not support the hard UTD hypothesis, either in the value of mass exponent or of temperature sensitivity, although the differences from the hard UTD prediction are in both cases small. Whatever factors are dictating the relationship between resting metabolic rate and temperature, it is not driven simply by the Boltzmann factor.
The predictions of the hard UTD hypothesis could also be tested using the Arrhenius formulation. To do so, however, would involve conversion of the observational data (oxygen consumption) to rates of energy utilization, and this requires knowledge of metabolic substrates and experimental conditions. Because these were mostly unknown, comparison was limited to the Q10 data reported in the original studies.
The soft UTD hypothesis
An alternative approach is to view the UTD as simply a useful statistical description of the across-species relationship between resting metabolic rate and temperature (the full equation was described by Gillooly et al. 2001 as the zeroth-order model that describes the effect of size and temperature on metabolic rate as primary); this is the soft UTD hypothesis. The validity of an Arrhenius model for mass-corrected resting metabolic rate data in across-species studies is clear (Fig. 1b), and as such the soft UTD hypothesis is useful in that metabolic rate is defined by only two variables (M and T). The full equation does, however, include two further terms which have to be determined empirically, namely the scaling factor b0 and the Arrhenius parameter for resting metabolic rate, called by Gillooly et al. (2001) the ‘activation energy of metabolism’. An Arrehenius plot is, however, only one of several statistical models that linearize the relationship between resting metabolic rate and temperature. The data for teleost fish, for example, are also linearized by log/linear and log/log models (Clarke & Johnston 1999). Once it is recognized that neither the UTD nor the Arrhenius model has any primacy in terms of a theoretical underpinning, then the choice between statistical models is purely subjective when, as here, they provide equally parsimonious descriptions of the data. Gillooly et al. (2001) reiterate the long-recognized point that the Arrhenius parameter and Q10 are not linearly related (as commented by van’t Hoff 1896); there is, however, no theoretical justification for preferring one over the other.
Ecologists need broad-scale descriptions of nature to handle its complexity. Without valid generalizations we can never hope to improve our understanding, or to incorporate biology into ecosystem models. As such the UTD is a valuable statistical generalization, although it is mathematically identical to previous descriptions of the relationship between reaction rate and temperature (Johnson et al. 1974). It is, however, not a fundamental mechanistic model for explaining metabolism in purely physical terms; evolutionary optimization at all levels has broken the direct link between the thermal behaviour of simple systems and that of complex entities. This also removes the theoretical underpinning for a purely mechanistic scaling of physiology to assemblage or macroecological scales (Allen et al. 2002; Belgrano et al. 2002), or for extending arguments based on resting metabolism in postlarval organisms to the very different process of development (Gillooly et al. 2002). It does not, however, invalidate the use of this or related equations as broad statistical descriptions of the world in order to probe large-scale patterns. A statistical description of the world is not a theory in itself, as has been suggested by Peters (1983). At present we lack a clear understanding of the relationship between temperature and metabolism at the organismal scale (Clarke 2003). We can explain biochemical behaviour on the small scale, and generate statistical regularities at the large scale; interpolating physiology at the intermediate scale remains a significant challenge.