This study is the most comprehensive analysis to date of the body mass scaling of individual FMR. Our analysis accounts for nonindependence in the data arising from shared evolutionary history and looks at both mean scaling exponents and taxonomic heterogeneity in scaling exponents in a unified framework. Results confirmed our hypotheses that (i) taxonomic heterogeneity in scaling exponent is statistically meaningful (i.e. strongly supported by our AIC results) and substantial relative to the difference 3/4−2/3 and the difference between the mean slopes for birds (0·71) and mammals (0·64); and (ii) variation is most important at the order and species levels of taxonomy. Hence, taxonomic variation in scaling exponents easily exceeds differences among various theoretical predictions for average scaling exponent, seeming to diminish in importance debates about what is the ‘correct’ average scaling exponent, and what are the reasons for it, relative to the importance of explaining taxonomic variation in scaling exponents. In the following sections, we compare our average exponent results with the predictions of several theories, as well as, and more importantly in our view, comparing our results about variation in exponents to theory. We also examine the issue of curvature in plots of log metabolic rate vs. log body mass, because it pertains to the comparisons with theory. Results support the heat dissipation limit theory of Speakman & Król (2010) and the metabolic levels boundary hypothesis of Glazier (2010) more so than other theories.
Recent work examining species-averaged data detected significant convex curvature in log RMR vs. log body mass scatter plots for mammals (Kolokotrones et al. 2010; see also Hayssen & Lacy 1985); discrepancies among prior empirical studies of the scaling of mammalian RMR were explained as a result of curvature, with studies focusing on smaller body masses reporting slopes close to 2/3 and studies focusing on larger masses reporting slopes close to 3/4. Our FMR data for mammals also appear to show convex curvature (Fig. 1; Fig. S7 for significance), but a focus on smaller body masses cannot explain the fact that our mean slope for mammals was close to 2/3 because we did not focus on smaller body masses: the range of masses we used was similar to that of the large collections of Kolokotrones et al. (2010). Savage, Deeds & Fontana (2008) and Kolokotrones et al. (2010) offered refinements to the supply-network theory to explain observed curvature in their RMR plots. The theory of quantum metabolism also predicts curvature (Agutter & Tuszynski 2011). However, heat dissipation limit theory (Speakman & Król 2010) provides an alternative explanation for apparent curvature that seems better supported by the FMR data presented here. This theory suggests that the greater thermal conductivity of water compared to air leads to a greater capacity to dissipate heat and therefore a higher FMR in aquatic animals. Data for aquatic mammals should therefore exhibit the same slope but a higher intercept than terrestrial mammals on log FMR vs. log body mass plots (Speakman & Król 2010). Of the 56 mammalian individuals in our data set that are aquatic, 51 have a body mass >10 kg (Fig. S7). We tested the hypothesis that the apparent curvature in our mammalian data results from the presence of many large-bodied aquatic animals by fitting three models to our mammalian data: a linear model, a quadratic model and a linear model with different intercepts for aquatic and nonaquatic species. The latter model gave the best fit and had higher intercept for aquatic mammals than for nonaquatic ones (Fig. S7), supporting the heat dissipation theory explanation for apparent curvature. Curvature is not real, in the sense that it can be explained best by linear models with regression line elevations varying by group in a way consistent with the heat dissipation limit theory.
As the theory of West, Brown & Enquist (1997) was originally billed as a universal theory, one may expect its generalizations (Savage, Deeds & Fontana 2008; Kolokotrones et al. 2010) to also be universally applicable and to predict curvature for birds as well as mammals. Our avian data do not appear curved (Fig. 1; Fig. S7 for statistical tests). While potentially inconsistent with the models of Savage, Deeds & Fontana (2008) and Kolokotrones et al. (2010), this is consistent with the heat dissipation theory because aquatic birds are not so predominantly large as to cause curvature in scatter plots by having elevated FMR. We again fitted a linear model, a quadratic model and a linear model with different intercepts for aquatic and nonaquatic birds, repeating this for a variety of ways of categorizing birds as aquatic/nonaquatic (Fig. S7). In all cases, the two-intercept model was the best fit, and the intercept for aquatic birds was higher than that for nonaquatic. These arguments do not disqualify the theories of Savage, Deeds & Fontana (2008) and Kolokotrones et al. (2010) but they do suggest that researchers could usefully examine what predictions those theories make for heterogeneity of curvature across major taxa.
Other empirical studies have found no or limited evidence of curvature in some data sets (Capellini, Venditti & Barton 2010; Isaac & Carbone 2010), and a recent study suggested curvature is specific only to certain mammalian clades (Müller et al. 2012). If some groups within each data set, such as aquatic representatives in mammalian and bird data sets, are more able to dissipate heat than others, one may expect heterogeneous curvature results for different data sets according to whether better heat dissipators are larger or smaller than other organisms considered in the particular data set, or distributed evenly across body masses. Ehnes, Rall & Brose (2011) found curvature in basal rate data for soil invertebrates, and some studies have shown that intraspecific scaling can be nonlinear for various ectotherms (Glazier 2005; Killen et al. 2007; Moran & Wells 2007; Streicher, Cox & Birchard 2012); these results are interesting but not directly relevant to heat dissipation theory, which applies to endotherms.
Heterogeneity in slopes and comparison with theory
In agreement with previous studies (e.g. Capellini, Venditti & Barton 2010), we found variability in b. Isaac & Carbone (2010) showed that for species-averaged basal rates, the mean slope 3/4 was well supported, but that taxonomic variability around that mean was sufficiently great that, for instance, ‘extreme’ values outside the range 0·5–1 should not be unexpected even for whole orders. Our conclusions are analogous: our order-level random-effect standard deviation was 0·0871, compared with 0·105 for RMR across metazoa in Isaac & Carbone (2010). This means that, for individual FMR data, our model predicts that 5% of bird orders will have slopes outside the range 0·54–0·88 and 5% of mammal orders should have slopes outside the range 0·47–0·81. These values are quantiles for normal distributions with means 0·71 and 0·64 and standard deviations 0·0871.
Riek's (2008) is the only other analysis of individual FMR data we are aware of, but that study is limited to arguing for the importance of including a random effect of study in models (which we did). We find it counter-intuitive to model only the random effects of study while ignoring the pseudo-replication resulting from shared evolutionary history. Our results show that taxonomic heterogeneity of slope, particularly at the order level, is at least as important as heterogeneity related to study effects (Table 2).
Theories exist that try to explain variation in the exponent, b. These include the theories of Savage, Deeds & Fontana (2008) and Kolokotrones et al. (2010), the metabolic-level boundaries hypothesis (Glazier 2005, 2010), the cell metabolism hypothesis (Kozłstrokowski, Konarzewski & Gawelczyk 2003) and the quantum metabolism theory reviewed by Agutter & Tuszynski (2011). Many theories fit at least some aspects of empirical data, so it is hard to resoundingly disprove any of them. Nevertheless, our heterogeneity-of-slope results do partly support some theories and partly contradict others. For example, several theories predict that b will take a value between 2/3 and 1 (Kozłstrokowski, Konarzewski & Gawelczyk 2003; Glazier 2005; van der Meer 2006). These theories are not entirely consistent with our data, since, taking order-level slopes to be normally distributed with standard deviation 0·0871 and mean 0·71 for birds and 0·64 for mammals, as estimated by our model, and then using quantiles, we find that 31% of bird orders and 62% of mammal orders are predicted to have slopes less than 2/3. The quantum metabolism theory predicts that 1/2<b<1. Only 5% of mammal orders and 1% of bird orders are expected by our results to have slope <1/2, so 1/2 may be a sensible choice if a lower bound is needed. Figure 2 shows order-level slope estimates provided by our best-fitting model, as well as model-average order-level slopes. The fact that few orders have confidence intervals in Fig. 2 that fall entirely below 2/3 should not be interpreted as contradicting our assessment that 31% of bird orders and 62% of mammal orders are predicted to have slopes <2/3. While the statistical methods of this study are not designed to provide great confidence about the scaling exponent for any particular order, they do strongly support the presence of substantial variation among order-level scaling exponents, both among orders for which data were included and, by inference, orders yet to be sampled. So we can say with great confidence that a substantial fraction of orders have scaling exponents below 2/3, even though we can only confidently identify a few specific orders with slope below that value.
Figure 2. Estimates of slope by order for (a) birds and (b) mammals. Filled circles and horizontal lines mark the best model's random-effects estimates together with their 95% confidence intervals, offset by the best model's fixed-effects estimates. Vertical lines mark the model-averaged fixed-effects estimate. Crosses mark model-averaged values per order, computed by summing model-averaged fixed-effect slopes and model-averaged conditional means of the random effect of order on slope. Models without a random effect of order on slope were treated as having a conditional mean of zero. As far as we are aware, it is not possible to compute model-averaged confidence intervals on predictions that include random effects, so the crosses are not accompanied by confidence intervals.
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The theories of Savage, Deeds & Fontana (2008) and Kolokotrones et al. (2010) predict that orders with smaller average body size will have shallower slope (i.e. smaller scaling exponent). We assessed this by fitting four linear regression models. Response variables were order-level slopes for birds or mammals (Fig. 2), and predictor variables were one of two measures of average order body size (making four possible combinations for four models). Average order body sizes were either the mean of the -transformed body masses of individuals in our data set in the order, or the mean of the -transformed body masses of the species in our data set in the order, where species log body mass was the mean of the logs of the individuals in the species. In no case was a regression trend visible; all P-values were >0·05. Therefore, our data provide no support for the idea that orders with smaller body size have shallower slope.
The metabolic levels boundary hypothesis (Glazier 2005, 2008, 2010) predicts that orders of higher ‘metabolic level’ should also have shallower log FMR vs. log body mass slope. The technique used by Isaac & Carbone (2010) to test the hypothesis is unfortunately flawed, because their estimates of metabolic level are not independent of body size. The output of our statistical model can be used to test the metabolic levels boundary hypothesis because it provides a measure of metabolic level that is not confounded by body size, as follows. Order-level slopes and average order body sizes were computed as in the prior paragraph, using both methods reported there for computing average order body sizes. Order-level intercepts were computed analogously to order-level slopes (Fig. 2), using model averaging. Order-level slopes and intercepts together allow the identification of an order-level regression line for log FMR vs. log body size. The metabolic level for an order was defined as the height of this line at the average order body mass minus the height of the class-level regression line at the same body mass; the class-level regression line was determined by the model-averaged fixed-effects slope and intercept for the class to which an order belongs (Aves or Mammalia). Testing for a negative correlation between order metabolic level and order slope gave significant results for birds (Pearson R = −0·591 or −0·584, P = 0·010 or 0·011, respectively, for a one-sided test using the two ways of measuring average order body size) and a nonsignificant but still negative correlation for mammals (Pearson R = −0·079 or −0·071, P = 0·399 or 0·409). Thus, our results provide some support for the metabolic levels boundary hypothesis. Poorly represented orders are expected to be affected by statistical ‘shrinkage’ (Isaac & Carbone 2010), which may have reduced the strength of the effect seen here. In all cases, correlation coefficients were stronger and P-values lower when orders were excluded that had fewer than 10 individuals in our data set. The metabolic levels boundary hypothesis predicts clearly that there should be a negative relationship between metabolic level and slope, b, for data on resting or basal metabolic rates, but it also predicts a positive relationship for data measured during intensive exercise, and for the intermediate case of FMR, Glazier (2010) says ‘... a negative correlation between b and L [metabolic level] should also be seen in field animals and those engaged in minimal (routine) activities, as long as maintenance costs remain a large proportion of the energy budget.’ So we add the caveat that our results support the theory if FMR can be seen as routine activity as suggested by Glazier (2010), maintenance costs can be considered a large proportion of the energy budget in the field, and hence, the theory is interpreted to predict a negative correlation between b and L for FMR.
Heat dissipation limit theory does not make explicit statements about taxonomic variation in b, but the derivation of the theory in Speakman & Król (2010) suggests ways it might be amplified to explain variation; an expanded theory could be tested against our results. The theory assumes that heat dissipation, and therefore metabolic rate, is proportional to , where d is the depth of an insulating layer (feathers, blubber or fur), k is the thermal conductivity of that layer, A is the surface area of the organism and and are the ambient and core body temperatures, respectively. Using empirical data and theory to write each of these components as a power function of animal mass, Speakman & Król (2010) conclude that metabolic rate should be proportional to . However, the component allometries, , , , , are probably subject to taxonomic heterogeneity in exponents, which would ramify through the formula to produce taxonomic heterogeneity in the scaling of metabolic rate. Assembling the appropriate data on insulating-layer depths, thermal conductivities, etc., would allow future workers to test the theory. Presumably, supply-network theories could be tested in an analogous way by examining aspects of the circulatory systems of different orders of mammals or birds, but these measurements seem harder to get than the measurements needed to test the theory of Speakman & Król (2010).
Another likely rewarding avenue for future research is carrying out an analysis similar to ours but for BMR or RMR, and making comparisons with theory and between FMR and RMR results. Recent years have seen an increasing interest in the ecological and evolutionary causes and consequences of intraspecific variation in RMR (e.g. Clarke & Johnston 1999; Glazier 2005; Burton et al. 2011; White, Schimpf & Seymour in press). Much data on individual resting rates are scattered in the literature, or have been partially collected, but to our knowledge, no comprehensive collection of individual measurements of RMR and body size for birds and mammals has been assembled. Most published BMR data sets contain species-averaged data. For instance, Isaac & Carbone (2010) carried out an analysis like ours on a large collection of species-averaged data. White, Phillips & Seymour (2006) present some individual data, but their values appear to be averages for mammal and bird species. Ehnes, Rall & Brose (2011), Riveros & Enquist (2011) and much work of Glazier (2005) have examined individual-level data sets, but some of those works focus on clades other than birds and mammals, and the collections examined for birds and mammals are not comprehensive. White, Schimpf & Seymour (In press) studied a collection of individual measurements, but it was not intended to be a comprehensive collection, as they had different research goals. Clarke & Johnston (1999) provide a large data set of individual-level measurement for fish. Burton et al. (2011) review intraspecific variation in resting rates, including information pertinent to birds and mammals, but do not provide or analyse a comprehensive database.
Comparisons between BMR and RMR scaling and the scaling of other types of metabolic rate, including FMR, have been made by many authors, including Nagy (2005), White & Seymour (2005) and others. Glazier has examined the topic in depth, and his metabolic levels boundary hypothesis offers explanations of differences (Glazier 2010). But compiling a comprehensive database and comparing RMR and FMR data using unified statistical models, such as ours, that simultaneously take into account central tendency scaling exponents, taxonomic variation in exponents and evolutionary nonindependence of data can probably improve understanding of the differences between RMR and FMR scaling and help develop theoretical explanations such as the metabolic levels boundary hypothesis. Scaling exponents of metabolic rate are predicted by the metabolic levels boundary hypothesis to be influenced both by volume-related constraints on energy use and production, which scale with exponent 1, and by surface-area-related constraints on fluxes of resources and waste products, which scale with exponent 2/3. At very low metabolic levels (e.g. rates measured during dormancy), surface-area constraints are not predicted to be important, so the metabolic levels boundary hypothesis predicts rates at that level will scale with exponent 1. The same scaling is predicted at very high metabolic levels (maximal metabolic rate, measured during strenuous exercise), because surface-area constraints are temporarily avoided through physiological mechanisms such as stored energy in muscle tissues and temporary tolerance to waste build-ups. At some intermediate metabolic level, surface-area constraints dominate. Therefore, the metabolic levels boundary hypothesis predicts that as metabolic level increases from minimal, through resting rates and field rates, to maximal, scaling exponents will decline from 1 to 2/3 and then will climb back to 1 again. There appears to be some variation and uncertainty in the precise level at which the minimum of 2/3 is achieved, and Glazier (2010) identifies the question of how metabolic level precisely affects scaling exponents as one of several main area the metabolic levels boundary hypothesis could be developed in future work (Glazier 2010, his point three on p. 125). A comprehensive and unified analysis of both RMR and BMR (and possibly other levels if sufficient data can be compiled) using appropriate statistical methods seems likely to help illuminate this and other aspects of our understanding of the true variety of metabolic scaling relationships and the reasons for this variety.