kleiber's law

Because it was a bastard to write I though I'd share it...minus graphs

Kleiber’s Law Practical


Over time there has been debate as to whether the scaling exponent of the relationship between metabolic rate and body size is 0.67, or 0.75 as stated by Kleiber’s law (Symonds & Elgar, 2002). It was originally thought, by Max Rubner, that the relationship between body mass and metabolic rate was due to the fact that smaller animals have a larger body surface to body mass ratio than larger animals and was therefore related to heat loss (Schmidt-Nielsen, 1997). This has since been found to be untrue as the same relationship can be found in ectothermic organisms for which heat loss is determined by environment. The exponent of 0.75 b was discovered by Max Kleiber and shows that if the rate of oxygen consumption per unit body mass can be calculated a significant universal association between body size and oxygen consumption can be seen (Schmidt-Nielsen, 1997). An example of the importance of understanding body mass as a limiting factor on metabolic rate is the fact that resources needed by populations of organisms can be predicted using the equation M = kWb (Fielding & DeFoliart, 2008).

Although the 0.75 exponent has been widely accepted it has recently been found that the law is not universal. An alternative model, the metabolic-level boundaries hypothesis, predicts that the exponent b should be more closely related to metabolic level or the ecological lifestyle of a species, for example the resting metabolic rate for high energy species such as mammals and birds is generally 0.75 but is closer to 1.0 in low-energy species such as amphibians and fish (Glazier, 2007).

The correlation between metabolic rate and body mass can be seen within a single species as well as in differences between species; young animals will typically have a higher metabolic rate per unit mass than adults of the same species (Randall et al. 2001). As it has been found that larger animals have a lower metabolic rate per unit mass than smaller animals it can be assumed that metabolic rate does not increase accordingly to weight, therefore when testing the rate of oxygen consumption using goldfish Carrassius auratus of two different sizes the 0.75 exponent should be observed. If Kleiber’s law is correct the oxygen consumption of fish should increase with body mass, though the metabolic rate of small fish should be greater than that of big fish.

H1: There is a difference in the rate of oxygen consumption between big fish and small fish.

H0: There is no difference in the rate of oxygen consumption between big fish and small fish.


Working in a group of three, three glass jars were labelled; a control, one for a small fish and one for a big fish. The labelled jars were completely filled with de-chlorinated water into which one small and one big fish were carefully placed. The jars were closed to keep out residual air and the time at which each jar was closed once the fish was inside was recorded. The jars were then placed in an opaque plastic tub to reduce disturbance to the fish. The fish were sealed in the jars for approximately 30 minutes each, the exact time was recorded. Water from each jar was transferred to a labelled conical flask; the flasks were filled completely and stoppered. The mass of each fish was determined by weighing a half full beaker of water, adding the goldfish and then re-weighing the beaker. The two fish were then returned to the stock tank. The dissolved oxygen content of the water in the conical flasks was recorded using an oxygen meter; the rate of oxygen consumption was recorded and added to the class data set.


Results drawn from the whole class were used; it was found, as shown in figures 3 and 4, that the mean oxygen consumption of big fish was greater than that of small fish, and conversely the mean metabolic rate of small fish were greater than that of big fish. The slope shown in Figure 1 was calculated to determine the exponent that shows the change of metabolic rate with increase in body mass, it was found to be 0.831076. A Mann-Whitney U test was performed and it was found that big fish consume more oxygen than small fish.

(Mann-Whitney U test: U = 147.500, n1 = 21, n2 = 22, P = 0.042)


It can be concluded from the results that there is a difference in the rate of oxygen consumption between big fish and small fish. The results presented in figure 3 show that, as predicted, big fish consume more oxygen than small fish, and conversely figure 4 shows that the mean metabolic rate is higher in small fish than in big fish. The slope of approximately 0.83 that was calculated from figure 1 does, in fact, support Kleiber’s law; if an exponent closer to 1.0 had been found then it could be suggested that there is a metabolic rate directly proportional to body mass, which is known not to be the case (Schmidt-Nielsen, 1997). It can therefore be said that as organisms increase in mass their metabolic rate does not increase in an exactly proportional manner. The Mann-Whitney U test results confirm that the prediction was correct, and the significant result meant that the null hypothesis that there is no difference in the rate of oxygen consumption between big fish and small fish could be rejected (Mann-Whitney U test: U = 147.500, n1 = 21, n2 = 22, P = 0.042).

The fact that the exact 0.75 exponent was not observed is most likely due to inaccuracy when setting up the experiment, for example when ensuring that the jars were sealed with no residual air at the start of the experiment and also when the conical flasks were being stoppered. It is also possible that the recent metabolic-level boundaries hypothesis could be supported by the findings of this experiment; the theory hypothesises that the resting metabolic rates of low-energy species such as fish could be closer to 1.0 than 0.75 (Glazier, 2007). A potential confounding variable could be that there were some fish of the same size in both groups, though not applicable when the calculation the exponent is concerned. From the raw data in the class results there are several cases of this to be found, for example; a fish weighing 14.97g was classed as small and a fish weighing 14.8g was classed as big. To avoid inaccuracies in the data of future experiments it would be practical to weigh the fish before placing into categories rather than estimating size by sight.


Fielding, D.J., DeFoliart, L.S. 2008. Relationship of Metabolic Rate to Body Size in Orthoptera. Journal of Orthoptera Research. 17 (2) pp: 301-306.

Glazier, D.D. 2007. Ecology of Metabolic Scaling in Animals and Plants. Comparitive Biochemistry and Physiology – Part A: Molecular & Integratice Physiology. 148 (1) pp: S99.

Randall, D., Burggren, W., French, K. 2001. Eckert Animal Physiology. New York: W.H. Freeman & Company.

Schmidt-Nielsen, K. 1997. Animal Physiology: Adaptation and Environment. Cambridge: Cambridge University Press.

Symonds, M.R.E., Elgar, M.A. 2002. Phylogeny Affects Estimation of Metabolic Scaling in Mammals. Evolution. 56 (11) pp: 2330-2333



Mann-Whitney Test







Mean Rank

Sum of Ranks
















Test Statisticsa



Mann-Whitney U


Wilcoxon W




Asymp. Sig. (2-tailed)


a. Grouping Variable: group


H0: There is no difference in the rate of oxygen consumption between big fish and small fish.

α = 0.05


P (0.042) < α (0.05) = reject H0 = significant result


i’ve taken a chemistry course for a year and graduated, it was to chase an old dream. sometimes i wonder what the higher levels of biology are like, wish i had more time to study science but responsibilities get in the way. i’ve always dreamt of having an oversized sparrow for a pet, i hope one day sparrows will defy Kleiber and square-cube laws. gravity and the amount of oxygen available in the present day are also annoying factors. i’ll name my pet Yeyo, its diet will consist of cocaine and the flesh of my enemies.

for now i’ll just wait until some clever nutcase makes this possible. *winks at scientist friends*