The principle of similitude in biology From allometry to the formulation of dimensionally homogenous 'Laws'

Meaningful laws of nature must be independent of the units employed to measure the variables. The principle of similitude (Rayleigh, Nature 95:66-68, 1915) or dimensional homogeneity, states that only commensurable quantities (ones having the same dimension) may be compared, therefore, meaningful laws of nature must be homogeneous equations in their various units of measurement, a result which was formalized in the pi theorem (Vaschy, Ann Tel 19:25-28, 1892 ; Buckingham, Phys Rev 4:345-376, 1914). However, most relations in allometry do not satisfy this basic requirement, including the 3/4 Law (Kleiber, Hilgardia 6:315-351, 1932) that relates the basal metabolic rate and body mass, besides it is sometimes claimed to be the most fundamental biological rate (Brown et al., Ecol 85:1771-1789, 2004) and the closest to a law in life sciences (West and Brown, Phys Today 57:36-42, 2004). Using the pi theorem, here we show that it is possible to construct an unique homogeneous equation for the metabolic rates, in agreement with data in the literature. We find that the variations in the dependence of the metabolic rates on body mass are secondary, coming from variations in the allometric dependence of the heart frequencies. This includes not only different classes of animals (mammals, birds, invertebrates) but also differences during rest and exercise (basal and maximal metabolic rates). Our results demonstrate that most of the differences found in the allometric exponents (White et al., Ecol 88:315-323, 2007) are due to compare incommensurable quantities and that our dimensionally homogenous formula, unify these differences into a single formulation. We discuss the ecological implications of this new formulation in the context of the Malthusian's, Fenchel's, and the total energy consumed in a lifespan relations.