Evidence against universal metabolic allometry


†Correspondence: E-mail: folmer.bokma@oulu.fi


  • 1A long-standing theoretical problem in biology is the relation between organism size M and metabolic rate Q.
  • 2This long history of research into such a basic pattern is due to the particular shape of the relationship: it is generally observed that the relation is well described as Q ∝ Mb where b is most often between 0·5 and 1.
  • 3Numerous analyses have focused on determining b using interspecific data, in particular to test whether b = 2/3 or b = 3/4.
  • 4Here a large amount of intraspecific data on fish are analysed, which show that b ≠2/3, b ≠ 3/4, and especially that there is no single, universal value of b.


Despite numerous established relations between body size and metabolic rate within, as well as between, species from various taxa (e.g. Peters 1986), there is still no consensus on what shapes the relation. An early theory suggested it is shaped by the geometry of organisms maintaining stable body temperature. In Euclidean geometry, the surface area of an object of a specific shape increases as the 2/3 power of its volume. If the rate of heat loss is assumed to be proportional to the surface area of the body, the power required to keep such an object at constant temperature increases with the 2/3 power of its volume. Initially this was found to be quite close to the observed pattern, but as empirical deviations from b = 2/3 accumulated it became replaced by b ≈ 0·72–0·73, which was eventually replaced by b = 3/4 (see Dodds, Rothman & Weitz 2001 and references therein).

Although many accepted b = 3/4 as a ubiquitous rule, the value remained purely empirical: there was no theoretical explanation except that it was observed that if organism geometry had a fourth dimension, metabolic rate would scale as body size to the power 3/4 (Blum 1977). Only recently a serious potential fourth dimension was proposed. With the discovery of fractal geometry in biology, it was noted that the internal transport networks most organisms use (e.g. lungs, trachea, the blood vascular network) appear to have a fractal dimension. It was suggested that if organisms use such networks to minimize the cost of transporting substances inside the body, then metabolic rate scales as the 3/4 power of body size (West, Brown & Enquist 1997, 1999). This theory, which elegantly relies on the intuitive supposition that organisms have evolved so as to maximize internal energetic efficiency, has been argued to be the reason for various relationships associated with scaling in plants as well as animals (Enquist, Brown & West 1998; Gillooly et al. 2001; West, Brown & Enquist 2001).

There is little evidence against 3/4-power scaling though some have questioned its generality (Whitfield 2001). It has been shown that the individual processes that make up the whole-organism metabolic rate (e.g. ventilation, cardiac work, circulation) have different scaling characteristics (Darveau et al. 2002; Weibel 2002; see also Reich 2001). Also it was shown that in birds the observed relations are not statistically significantly different from the 2/3 scaling relationship (Dodds et al. 2001), and that mammal species below and above the modal body size of a taxon appear to have different scaling relations (Lovegrove 2000; Dodds et al. 2001).

However, most evidence for (and against) universal 3/4-power scaling of metabolic rate comes from analyses of interspecific data that may be affected by ignoring of species’ phylogenetic non-independence (Clarke & Johnston 1999; Symonds & Elgar 2002). Thus, the variability of estimates of b in literature may be due to (i) measurement error, (ii) neglect of the historical relationships of species and (iii) true variation between species (Lovegrove 2000). The phylogenetic uncertainty in interspecific analyses can be considerable, especially if one takes into account that it is unknown whether metabolic rate evolves in a punctuated manner or gradually over time (Lovegrove 2000; Bokma 2002). It is therefore more informative to investigate intraspecific allometry, avoiding assumptions on tempo and mode of metabolic rate evolution. It has been suggested that b differs between interspecific and intraspecific relationships, but if the relationship between body size and metabolic rate is shaped by geometrical factors – whether this geometry is Euclidean or fractal does not matter – metabolic rate should increase with body size within species in the same way as between species (West et al. 2001).


Measurements of fish metabolic rate were downloaded from the online database at http://www.fishbase.org. Of all available data on metabolic rate (n = 6555 measurements from 313 species), those obtained from measurements under stress were excluded. Also 1158 measurements obtained from active animals were excluded. Of the remaining 3575, another 3 were excluded as they did not specify the mass of the fish measured. The 3572 measurements that were finally selected for analysis came from 113 species in 217 series, and ranged in temperature from −1·5 °C to 38 °C and in mass from 0·01 g to 10·4 kg.

The relation between metabolic rate and body size within species was determined by least squares linear regression of the natural logarithm of oxygen consumption on the natural logarithm of body mass. Regression coefficients and intercepts were calculated for all combinations of temperature and salinity for which there were at least three measurements (after averaging measurements at identical body size) at different body size. This was done because temperature affects metabolic rate (Gillooly et al. 2001) and also salinity might do so. The remaining data yielded a total of 198 regressions based on a total of 1738 data points for further analysis. For some species more than one regression was available. For example, the Carp Bream Abramis brama is represented twice, with a series of 3 measurements at 18 °C and a series of 32 measurements at 20 °C. Variance in the 198 estimates of b may be due to chance alone, or to true variation between species. Equality of regression coefficients was assessed by analysis of covariance (ancova, Zar 1996, p. 362).

The data contain a number of measurements at constant body mass and constant conditions, which allow estimation of measurement error. Measurement error was defined as the standard deviation of repeated measurements SDe. One hundred and fourteen such series of repeats (containing 421 measurements) were used to determine the best linear relation between average (Q) and standard deviation of repeated measurements as: SDe = exp(0·94 ln Q − 1·72). This relation between SDe and Q was used to obtain Monte Carlo estimates of statistical power for the 198 regressions. For all mass data supporting a certain observed regression coefficient, Q was calculated as Mb + e, where e is measurement error, sampled from a normal distribution with zero mean and standard deviation SDe. From those simulated data regression coefficients were estimated (50 repeats for every regression). These estimates are on average b, but deviate more or less from this expectation depending on the number and spread of the mass data. Their variance indicates the statistical power of the observed regression. This procedure may be somewhat imprecise, as it fails to partition error variance into for example within- and between-observer components, but it gives a relatively good picture of which are the most informative estimates of b.


intraspecific allometry

The regressions yielded a variety of body size exponents, as expected in a large set of estimates, especially as some were based on small sample sizes. Indeed, the variance of estimates decreases with increasing sample size (Fig. 1a) and with increasing body size range (Fig. 1b). The most informative estimates of body size exponents (with least sampling variance in the simulations) are from large samples spanning a large body size range. Even these estimates deviate considerably from b = 2/3 and b = 3/4 (Fig. 1c). The most powerful sample is one containing 37 measurements of Sea Trout Salmo trutta trutta, over a body size range from 0·1 to 600 g. The corresponding estimate of the body size exponent of metabolic rate is 0·86 (r2 = 0·997) which is highly significantly different from both 2/3 and 3/4 (P < 0·001).

Figure 1.

Estimates of the body size exponent of metabolic rate plotted against (a) the number of measurements, (b) the difference between the largest and smallest body sizes in the sample, and (c) a Monte Carlo estimate of statistical power which unites sample size and body size range. Horizontal lines indicate b = 2/3 and b = 3/4. The best estimates (the rightmost in all panels) place the exponent in between 0·5 and 1.

The weighted average of estimated body size exponents was calculated, with the weighting factor the inverse of the standard deviation of simulated estimates. The average slope is 0·715, which is between the ‘traditional’ hypothesized values.

Estimates of the body size exponent were tested for significant departure from the hypothesized values 2/3 and 3/4 using t-tests. When compared against a theoretical value of 3/4, 21 out of 198 estimates are significant at the 5% level after sequential Bonferroni correction. When compared against the theoretical value 2/3, 16 out of 198 estimates remain significant after sequential Bonferroni correction. The data thus clearly reject both hypothesized body size exponents.

As some of the data do not agree with the hypothesized exponents, it is possible that there is another exponent, perhaps between 2/3 and 3/4, or that there is no universal scaling relation. In the latter case the individual estimates come from populations with different body size exponents b. ancova revealed that the 198 estimates are very unlikely to come from populations with identical b (F197,1342 = 7·33, P << 0·001). If the analysis of covariance is restricted to the 25 most powerful estimates (thus excluding the estimates from small samples and small body size ranges), this conclusion remains (F24,466 = 15·96, P << 0·001).


The data reject not only both hypothesized body size exponents b = 2/3 and b = 3/4 but – more importantly – also the hypothesized existence of universal scaling of metabolic rate. That result does not rely on the estimates of b from small samples, which may be affected by the use of least squares regression where the use of reduced major axis regression or other statistics (Dodds et al. 2001) might have been more appropriate. The differences between these alternatives are negligible in the statistically powerful samples if the correlation coefficients are high. In fact, the correlation coefficients in the 25 most powerful samples are high, and these samples also strongly reject the existence of universal scaling.

The finding that b ≠ 2/3 is in distinction with the conclusion of Dodds et al. (2001) who carefully re-analysed available data on bird metabolic rate and found that the data do not reject b = 2/3. The traditional theoretical explanation for b = 2/3 is that heat leaves the body through the surface of the body, which in Euclidean geometry scales with the 2/3 power of body volume. It is tempting to argue that changes in shape during individual development cause deviations from intraspecific 2/3-power scaling, or that not all heat is lost through the body surface. However, it can be shown largely on theoretical grounds that the scaling of basal metabolic rate is not caused by scaling of the requirements to maintain stable body temperature. If basal metabolic rate is set by the requirement to maintain stable body temperature, organisms would decrease metabolic rate as far as permitted by the environmental circumstances. The contrary is observed in homeotherms, which do not decrease metabolic rate below a certain level as environmental temperature rises. This level is defined as basal metabolic rate, and its existence proves that there is a limit below which metabolic rate cannot go, and that this limit is not set by requirements to maintain stable body temperature. It is this limit that scales. Thus, basal metabolic rate does not scale due to the geometry of heat loss. Fish are not homeotherms like birds and show rather different overall physiology, yet unless metabolic rate scales in fish for different reasons from those in homeotherms, the argument still holds that there is little reason to expect b = 2/3.

It is therefore more interesting that the data also reject b = 3/4, the prediction of much recent physically inclined theoretical work (West et al. 1997, 1999). If the scaling of metabolic rate is determined by the fractal dimension of branching networks as suggested, deviations from b = 3/4 may be due to non-fractal branching. Biological networks are not true fractals that break an organism into smaller but self-similar structures. Biological networks are instead what Bejan (1997) appropriately called ‘constructals’: structures that are built up of small units that together form larger, more or less self-similar structures. As compared with adult individuals from different species, especially in individuals of increasing size of the same species, that building process may be dominantly present.

So far, however, the data support theories that explicitly allow for differences in scaling relations (Kozlowski & Weiner 1997; Darveau et al. 2002; Weibel 2002). Kozlowski & Weiner's (1997) as well as Darveau et al.'s (2002) theories however, ‘predict’ the scaling of maximum and basal metabolic rate from observed scaling relationships of the individual processes that make up those overall rates. They thus fail to explain why metabolic rate would scale, and why the individual processes would scale as they do. That is a serious shortcoming, as it is not unknown that just like organisms (Hammond & Diamond 1997) non-living systems such as car engines and electrical amplifiers cannot increase their maximum power without increasing power at stationary performance, which suggests a universal, perhaps thermodynamic (Gnaiger 1983, 1987), reason for some sort of scaling. Therefore theories explaining allometric scaling should not only explain the expected value of b, but also where the variation around it stems from, and why metabolic rate scales in the first place.

We are just starting to understand the basic rules that appear to govern much of biology's seemingly dazzling complexity. It cannot be overemphasized how important those unifying developments are in the light of the biological tradition of emphasizing differences. It appears, however, that with respect to the scaling of metabolic rate there just are more differences than present unifying theories allow for.


I want to thank Esa Hohtola, Mikko Mönkkönen and two reviewers for stimulating criticism. This study was supported by project 50412 of the National Academy of Finland. All data used in the analysis are available from the online database at http://www.fishbase.org, as well as from the author on request.