Temperature and the metabolic theory of ecology

The fundamental equation of the Metabolic Theory of Ecology links the metabolic rate of an organism, Q, to its mass, M, and temperature, T, as: Q = b0Mb e−E/kT. This equation comprises two components, a power relationship describing the mass dependency (scaling) and a Boltzmann temperature correction. The value of the scaling exponent b is taken to be ∼0·75, derived from the physics of distribution networks in animals (West, Brown and Enquist 1997) and plants (West, Brown & Enquist 1999), and b0 is a normalization constant which is fitted empirically (Brown et al. 2004a). Gillooly et al. (2001) called the temperature component of the equation the Universal Temperature Dependence of Metabolism (UTD), and its formulation follows from the application of statistical thermodynamics to whole-organism metabolism. Here k is Boltzmann's constant and E the ‘mean activation energy of metabolism’, its value being estimated empirically from measurements of enzyme kinetics in vitro (Gillooly et al. 2001). Recently Clarke (2004) and Clarke & Fraser (2004) have argued that although the UTD is one of several useful statistical descriptions of the relationship between temperature and whole-organism resting metabolism rate, it cannot represent a direct mechanistic dependence of metabolic rate on temperature. This is because of the complex nature of metabolism, and the feedbacks involved in its control. Gillooly et al. (2006) have responded robustly, and here we reply to their concerns. Gillooly et al. (2006) make a number of points in their critique, the key ones being: The Boltzmann temperature correction is mechanistic and not statistical (phenomenological), and hence is superior to any other description of the relationship between temperature and metabolic rate. Boltzmann kinetics override any effects of the complexity of cellular physiology. The rate of ATP generation by isolated mitochondria in vitro exhibits a temperature sensitivity closely similar to that of whole organisms. The MTE equation can accommodate acclimation or evolutionary adjustment through variation in the normalization constant b0, and this means that interspecific (between-species) scaling may differ from intraspecific (within-species) scaling. I suspect that our two views of the relationship between temperature and metabolic rate are closer than the above would imply. It would therefore be sensible to outline where we agree, to distinguish those areas where we do not. Firstly, it is universally recognized that an acute change of temperature produces a corresponding change in metabolic rate, and it is generally observed that this acute effect is ameliorated by compensatory processes. Secondly, when comparison is made across taxa that have adapted over evolutionary time to live with different body temperatures, there is a strong positive and monotonic relationship between resting metabolic rate and temperature. This relationship is typically close to exponential in shape, and can be linearized by a variety of statistical models. Where we disagree is in considering the relationship between resting metabolic rate and temperature observed across taxa as simple and mechanistic whereby an increase in temperature is the only factor involved in determining resting metabolic rate (as expressed formally in the MTE equation, and what Clarke (2004) termed the hard UTD hypothesis). This strict interpretation of the MTE equation would allow for the effect of temperature on metabolic rate to be ameliorated through adjustments to the mean activation energy, E; Gillooly et al. (2006), however, argue that the value of E will remain relatively constant in the range 0·6–0·7 eV. This difference of viewpoint does not undermine any exploration of the consequences of metabolic scaling for ecology or diversity (e.g. Allen, Brown & Gillooly 2002; Savage et al. 2004a; Gillooly et al. 2005). Indeed there is a long history of applying allometric relationships to ecology (Peters 1983), the usefulness of which depends essentially on the validity of the ancillary assumptions. However, if the underpinning scaling relationship is statistical rather than mechanistic, we must accept a limitation in our understanding of the circumstances under which this relationship fails to match reality. Here I reply to the key issues raised by Gillooly et al. (2006) in their critique. But first I make some general points to set the context for my response. West & Brown (2004) describe the MTE equation as the zeroth order relationship linking metabolic rate, body mass and temperature. In other words, after correcting for body mass and temperature, all organisms from bacteria to whales, and from unicellular algae to trees, have the same resting metabolic rate. They do not, of course, as has been known ever since physiologists compared the metabolic rates of ectotherms and endotherms (Fig. 1a). Gillooly et al. (2001) recognize this quite explicitly, but the MTE equation cannot predict these differences; it can only describe them statistically through an empirical fit of the normalization constant b0 to data. An important point here, to which I return later, is the taxonomic level at which the normalization constant is fitted. West et al. (1997, 1999) present examples broadly at the level of taxonomic kingdom or class, and the comparison of resting metabolic rates in Fig. 1(a) involves the fitting of the normalization constant at the level of class (reptiles and mammals). Scaling of resting metabolic rate in vertebrates. MR = metabolic rate. Data plotted in original units. (a) Reptiles and mammals, scaling of resting metabolic rate with body mass (data from White & Seymour 2003). Both variables log-transformed, with no correction for temperature. All data sets contain only one data point per species, assigned to a median body mass and temperature. Individual regression lines fitted with a General Linear Model (GLM). The GLM is equivalent to fitting the normalization constant at the taxonomic level of class, and indicates statistically significant heterogeneity in slopes (P < 0·001) and in elevation (P < 0·001) for a fitted common slope. (b) Teleost fish, scaling of resting metabolic rate with temperature. Here the GLM is equivalent to fitting the normalization constant at the taxonomic level of order (data from Clarke & Johnston 1999). Data are resting metabolic rate calculated for a fish of standard wet mass 50 g, using a scaling coefficient b = 0·789. The data are plotted in Arrhenius form. The GLM indicates no heterogeneity in slopes (P > 0·05; that is all orders exhibit similar temperature sensitivity) but highly significant heterogeneity in elevation (P < 0·001). As a second example, data on the resting metabolic rate of teleost fish reveal a strong scaling with body mass (Clarke & Johnston 1999). The overall mean scaling relationship has a value of b = 0·79, the 95% confidence intervals of which exclude (albeit only just) the theoretical value of 0·75 predicted by West et al. (1997). When the fish resting metabolism data are broken down by order it is found that the slope of the relationship is always statistically indistinguishable from the mean value calculated for all fish (General Linear Model, GLM: P > 0·05) whereas the elevation characteristics of each order differs significantly (GLM, P < 0·05). This GLM analysis is thus fitting the normalization constant at the level of taxonomic order, describing significant heterogeneity within a taxonomic class (Fig. 1b). The important point here is whether an ecologist or physiologist is interested in the underlying relationship or the variance. The MTE provides a robust description of an important central tendency in organismal physiology, and such descriptions are of enormous theoretical and practical importance. They allow us to model fundamental aspects of organisms using relatively few variables and parameters and hence to make broad ecological generalizations. The MTE cannot explain why a given species does a particular thing, but this is not its aim. What it does do is provide a powerful description of an important central tendency in biology, against which we can compare the variable real world (West & Brown 2004). As Harte (2004) has argued cogently, the MTE provides a valuable null theory, but one where much of the interesting biology lies in the deviations from the model. The MTE has thus provided a valuable pointer towards interesting areas of study. To provide a specific context for this point, consider the data for resting metabolic rate in reptiles and mammals (Fig. 1a). To physiologists the 10–12-fold difference in mean resting metabolic rate between a typical reptile and mammal is important, reflecting predominantly the energetic cost of an endotherm metabolism (for which the underlying physiology is now well understood: Else & Hulbert 1987; Else, Turner & Hulbert 2004). Also of interest is the heterogeneity in the scaling exponent between different vertebrate classes: in mammals and birds overall b ∼ 0·67 and is statistically quite distinct from the prediction of the MTE which is b ∼ 0·75 (Dodds, Rothman & Weitz 2001; White & Seymour 2003, 2004; White, Phillips & Seymour 2006). At the level of class (all mammals or all birds) this is a robust result which cannot be explained away on the basis of vagaries of taxonomic sampling (as has been argued by West & Brown 2004). Savage et al. (2004b) argued that b ∼ 0·67 was the result of an uneven distribution of mass values; they pooled data into 52 mass bins and proposed a value of b close to the MTE prediction of b ∼ 0·75. Two major sources of variance in any overall relationship between resting metabolic rate and mass in mammals are the heterogeneity in scaling relationships with taxonomic order (Kozlowski & Konarzewski 2005) and variation in body temperature (despite the narrow range of body temperatures in mammals: White & Seymour 2003; White et al. 2006). These sources of error, however, affect all such broad compilations; they apply equally to those groups where the fitted exponent matches the MTE prediction as to those where it does not. The major mass-related error in measures of resting metabolic rate in mammals and birds probably comes from the influence of circadian rhythms (Aschoff 1982), which makes it more likely with decreasing size that any measured metabolic rate will not be representative, and hence that the fitted relationship overestimates the true slope. Recent debates have also considered the separate issue of measurement error, and Farrell-Gray & Gotelli (2005) have shown from a careful meta-analysis that errors in body mass estimates may be more pervasive than previously thought: whilst the body mass of any individual may be measured precisely, the error comes from selection of an inappropriate value to represent a given species. The critical point is thus whether the data for mammals and birds are an interesting deviation from the central tendency described by the MTE equation, or a fundamental challenge to the theory itself. It is intriguing that the two groups whose scaling most differs from the scaling relationship predicted by the MTE are both endotherms; perhaps in birds and mammals considerations of heat flow override the scaling dictated by fractal-like supply networks. A recent study of metabolic rate in insects suggests that here also resting metabolic rate exhibits b ∼ 0·67, perhaps reflecting a very different system for oxygen delivery to the tissues (Niven & Scharlemann 2005), although an earlier study of a more comprehensive data set had indicated a steeper slope for arthropods (Addo-Bediako, Chown & Gaston 2002). More recently the applicability of a universal b ∼ 0·75 has been questioned for terrestrial plants (Li, Han & Wu 2005; Reich et al. 2006), a comprehensive analysis by Glazier (2005) has revealed considerable heterogeneity in the value of b across a wide range of animal groups, and Makarieva, Gorshkov & Li (2003, 2005) have examined the scaling of metabolic rate from the perspective of a minimum cost for maintaining living tissue. The question of the scaling of metabolic rate is far from resolved. The sequence of points made by Gillooly et al. (2006) centre on two key issues, namely the relevance of a simple Boltzmann correction to complex physiology and the difference between within-species and between-species scaling relationships. Rather than reply to their critique point-by-point, I will discuss these two issues, identifying areas of agreement, disagreement and misinterpretation. All discussions of the thermal physiology of organisms acknowledge their debt to the Stefan–Boltzmann equation and the Arrhenius concept of activation energy. Despite its intuitive appeal, however, there are many reasons why classical statistical thermodynamics can be extrapolated to whole-organism physiology only with great care. Although originally developed for bimolecular reactions in the gas phase under ideal conditions, classical statistical thermodynamics provides a powerful explanation for the reaction rate of simple processes in dilute solutions under equilibrium conditions. The key problem, long recognized by physiologists, is that organisms are complex, non-equilibrium systems and the cell contents are far from ideal dilute solutions. Most important, however, is that statistical thermodynamics is based on the assumption that all the entities under consideration are identical, and that the only change in the system is its temperature. Neither assumption is valid for organismal physiology. In particular, organisms that have evolved to live at different temperatures have enzymes with different kinetic behaviour, mitochondria with different composition, structure and kinetics, and a different intracellular milieu. Clearly an organism is very different from the simple system to which classical statistical thermodynamics applies. Nevertheless it is well established empirically that resting metabolic rate increases monotonically with temperature. Gillooly et al. (2001, 2006) argue that a strong statistical fit to a simple Arrhenius model indicates that predictions of the MTE are well supported. Clarke & Johnston (1999) showed clearly that in a carefully selected data set for teleost fish the relationship between resting metabolic rate and temperature was described equally parsimoniously by any one of three statistical models (Arrhenius, exponential, double-logarithmic). As commented by Gillooly et al. (2006) we did suggest that in the absence of any other evidence we would use the Arrhenius model; however, this was a choice of statistical model, not of underlying mechanism. The difference in viewpoint here is that Gillooly et al. (2001) plot their data in Arrhenius form on the basis of the MTE model and are happy with the fit to data (though the variation in fitted slope for various groups is ∼15%). In contrast Clarke & Johnston (1999) set out with no a priori assumptions concerning underlying mechanisms, and simply compared the fit of different statistical models. A critical difference between the views of Gillooly et al. (2001) and Clarke (2004) concerns the extent to which the relationship between temperature and metabolic rate is influenced by the complexities of cellular physiology. Clarke (2004) emphasized the importance of feedbacks in metabolic control and of entropy in enzyme kinetics, but Gillooly et al. (2006) dismiss these effects as trivial. This misunderstands the role of entropic effects in the complex sequence of events involved in enzyme catalysis, events in which water plays an integral role. To see why entropy is important, it is necessary to consider the thermodynamics of enzyme catalysis. The basic theory of reaction kinetics was developed for simple dilute aqueous systems involving small molecules, where processes are typically dominated by enthalpic changes. The enzyme-mediated processes that characterize physiology are far more complex. They take place either in a complex gel-like cellular environment, or on a membrane. In particular, water is not only a solvent but is also a reactant or product in all the major classes of reaction (condensation, hydrolysis, hydrogenation, dehydrogenation). The most important role for water, however, comes through the weak-bond with such as have and this them the of and involves it in the or of many In entropy many of the between and These and processes are typically the in enzyme catalysis, and the overall kinetics using theory 1999). this the important of entropy to enzyme catalysis, determining the individual from and entropy is made by in water itself. has long been recognized as an important of the thermodynamics of enzyme kinetics and in evolutionary although the concept has been by 2002; but also by and al. emphasized important statistical in the for and argued that this on the of such The statistical point is but is now recognized as a real and its in the thermodynamics of in water is well 1999). of the thermodynamics of enzyme thus clearly the importance of and the role of there is more than to the of reaction rate in physiology. Gillooly et al. (2006) compare the temperature sensitivity of ATP in vitro with that of whole-organism metabolic rate. ATP for a considerable of whole-organism oxygen but physiologists have long recognized that mitochondria in vitro are in a very different from that in Most the are in that are not whereas in oxygen are very much et al. 2002). most critical for the developed by Gillooly et al. (2006) is that they have argued that isolated and in vitro are from of supply and thus from the predictions of the MTE equation, in that their model (West & Brown 2004). This is not to that in vitro cannot provide fundamental into physiology, only that to in is not at all because of the of feedbacks and level metabolic Indeed the strong dependence of ATP on temperature revealed by in vitro may be a fundamental on whole-organism ecology and history (Clarke 2003). this dependence is not a direct mechanistic relationship to temperature is indicated by the extent to which it is by & and also that the temperature sensitivity of ATP generation and is quite different & 1999). do not, however, make ATP ATP is with the key control being well known and long important are energy of the of ATP to that of and control by in the as ATP are ATP The of and hence the for oxygen as is by for ATP (Fig. In direct the MTE equation links metabolic rate and to for an organism whose mass constant there are no other variables in the This is thus a if temperature ATP generation and hence metabolic rate increase (Fig. and effects of temperature on metabolic rate. (a) A the rate of of and hence for is by the for the rate of ATP This is thus a (b) The Metabolic Theory of whereby the temperature of an organism its metabolic rate This is a supply The model of Clarke Here the temperature the level of the of processes resting metabolic and the for oxygen sets the level of metabolic rate. This is an effect of temperature on metabolic rate, and is a The of the are and processes are shown in Gillooly et al. (2006) argue that in organisms supply and are always in This is any organism must be to supply the ATP that circumstances As metabolic increases (for for or supply increases a is The point made by Clarke (2004) was and is in Fig. As temperature the rate of many processes not resting metabolic rate These and hence in the rate of many which may between processes if ATP the rate at which it can be The nature and extent of the of temperature on these different processes et al. Clarke & Fraser but the effect is an for which is as an increase in whole-organism oxygen The relationship between temperature and metabolic rate is thus complex and not It however, be described statistically and the ecological consequences of the observed empirical relationship be (Clarke 2003). Gillooly et al. (2006) the evolutionary (Clarke 2004; Clarke & Fraser with metabolic clearly did not for these two as equivalent misunderstands The concept of metabolic on the thermal physiology of fish by The specific was that organisms which had adapted over evolutionary time to live at temperatures would have a resting metabolic rate than from the metabolic relationship established for fish living at temperatures (for more see Clarke This would be equivalent to fish a value of the normalization constant b0 than other fish that fish would above the central tendency expressed by the MTE equation for all data clearly at in fish, does not Clarke & Johnston 1999). to the concept of was to the in the nature or of enzymes or & but these intracellular adjustments are as an of or acclimation to temperature in a The evolutionary is quite It that the resting metabolic rate of an organism is the result of a between resting and for with the level being set by It therefore why a of organisms such as fish different groups by different resting metabolic rates by in the value of the normalization constant (Fig. 1b). The was first in Clarke but only given the by Clarke The basis is the influence of temperature on cellular processes with evolutionary involving mass (Clarke & Johnston 1999). It is in its basis to the allometric of et al. and et al. this in a scaling Gillooly et al. (2006) this as and to The basic is clearly out in the model (Fig. and some broad predictions do (for that organisms with more will be by resting metabolic such as observed in et al. but the model is It cannot predict the resting metabolic rate of a particular species, any more than the MTE A difference is that whereas the MTE a central tendency for all the evolutionary is with both the overall relationship between resting metabolic rate and temperature, and the of variation It cannot make predictions because of the complexity of physiology and the nature of the but it does explain this complexity It is thus quite different in from the which this diversity within a models of the relationship between resting metabolic rate and temperature in (a) The The plot the metabolic relationship for two living at a temperature, one at a temperature and two at a temperature. The lines represent the acute effect of temperature on each species, an but of The represent the resting metabolic rate for each species at its living temperature. The relationship is different in and here also in from the within-species relationship that in this the difference in slopes has been for It is also different in being a statistical of individual evolutionary from Clarke & Fraser (b) The Metabolic Theory of The plot the within-species relationships for species, as for Fig. but with metabolic rate for body mass to the These species have been shown as individual which the fitting of the normalization b0, of the MTE equation to individual species. The relationship is thus different in and here also different in being a statistical of individual evolutionary that here also the difference in slopes has been for A critical of the application of the MTE equation is the taxonomic level at which the normalization b0, is fitted to empirical data. West et al. (1997, 1999) fitted the equation to the of from unicellular algae to trees, and from to whales, a central scaling relationship for all living The acknowledge that the fit variance from a variety of sources some as Gillooly et al. this overall MTE relationship a broad for all of it that many and physiologists as important, for the in resting metabolic rate between endotherms and the normalization constant at the level of class (Fig. this but at the of both in of b0 and the scaling at the level of order (Fig. the MTE equation is fit to individual species to allow for evolutionary (as proposed by Savage et al. and Gillooly et al. 2006), each species the of a evolutionary All is and the MTE to the evolutionary (Fig. The evolutionary adjustment is of only because of the applicability of the MTE to ecology or physiology thus concerns the taxonomic level at which it is At the level the MTE equation broadly but important evolutionary at is and evolutionary heterogeneity in both the normalization constant and the scaling exponent is Clarke (2004) argued that an assumption of the MTE was that interspecific and intraspecific scaling be identical, because the same Savage et al. (2004b) and Gillooly et al. (2006) both argue that evolutionary adjustment of metabolic rate place through variation in the normalization constant b0, which is the that many (e.g. et al. 2002; Clarke have argued the of the a range of data than Clarke & Johnston a different mass correction and a different temperature Gillooly et al. (2006) make a at in teleost fish, within-species and between-species scaling of resting metabolic rate with temperature are statistically all the Metabolic Theory of Ecology us to at we the It also produces such as the scaling relationship based on organisms with highly systems to apply also to organisms with mitochondria and isolated reaction It is in these areas that the theory will be most et al. 2005). The theory also sets a challenge to the more complex views of thermal physiology by et al. et al. and Clarke In particular it is not at all why the slope of relationship between resting metabolic rate and temperature in organisms is similar to that by simple Gillooly et al. (2001, 2006) this as an and direct control of metabolic rate by temperature. with the of organisms would argue that the complex involved mean that the relationship is more and complex, involving evolutionary at all (Clarke 2004; et al. 2002). This complexity would mean that the relationship between metabolic rate and temperature can be only a statistical description of a of separate evolutionary (albeit with a strong Clarke 2004). The critique that the mechanistic basis of the MTE is when compared with what we of organismal physiology is not to Clarke & (2004) argue that a Boltzmann temperature correction can be only as an of a much more relationship between metabolism and & (2004) argue that the complex in which organisms energy and means as there are no to simple models that are and & (2004) argue that the data do not provide links to and first that would the formulation of the MTE to all these is that the relationship between metabolism all and temperature is complex, and hence at present any overall description of this relationship must be statistical in the the challenge to physiologists from the MTE is to explain why the relationship between metabolic rate and temperature is what it Metabolism comprises a complex of some by some dominated by processes such as and involving which is temperature all of which take place in a complex, highly gel-like It would that be from the simple dilute systems of statistical thermodynamics and the relationship between whole-organism metabolic rate and temperature is a simple which a simple explanation in of a the lies in the central importance of water to enzyme kinetics and the temperature dependence of with rather than simple As Brown et al. and Gillooly et al. (2006) the complex links between temperature and whole-organism metabolic rate are one of the areas of physiology where is I not I West and Brown for many discussions of metabolism and Gillooly also an of critique. I also for a critical and Chown for