# Allometry and the Surface Rule of System Theory in Biology

Let us now proceed to the third model which is the so-called principle of allometry. As is well known, many phenomena of metabolism, and of biochemistry, morphogenesis, evolution, etc., follow a simple equation:

i.e., if a .variable y is plotted logarithmically against another variable x, a straight line results. There are so many cases where this equation applies that examples are unnecessary. Therefore let us look instead at fundamentals. The so-called allometric equation is, in fact, the simplest possible law of relative growth, the term taken in the broadest sense; i.e., increase of one variable, y, with respect to another variable x. We see this immediately by writing the equation in a somewhat different form:

As can easily be seen, the allometric equation is a solution of this function which states that the ratio of the relative increase of variable y to that of x is constant. We arrive at the allometric relation in a simple way by considering that any relative growth —only presupposed it is continuous—can generally be expressed by:

R. G. R. (y,x) = F,                               (7.3)

where F is some undefined function of the variables concerned. The simplest hypothesis is that F be a constant, a, and this is the principle of allometry.

However, it is well known that historically the principle of allometry came into physiology in a way very different from the derivation given. It appeared in a much more special form when Sarrus and Rameaux found around 1840 that metabolic rate in animals of different body weight does not increase in proportion to weight, but rather in proportion to surface. This is the origin of the famous surface law of metabolism or law of Rubner, and it is worthwhile to take a look at Rubner’s original data of about 1880 (Table 7.1). In dogs of varying weight, metabolic rate decreases if calculated per unit of weight; it remains approximately constant per unit surface, with a daily rate of about 1000 kcal. per square meter. As is well known, the so-called surface law has caused an enormous debate and literature. In fact, Rubner’s law is a very special case of the allometric function, y representing basal metabolic rate, x body weight, and the exponent a amounting to 2/3.

I believe that the general derivation just mentioned puts the surface law into correct perspective. Endless discussions of some 80 years are overcome when we consider it a special case of al- lometry, and take the allometric equation for what it really is: a highly simplified, approximate formula which applies to an astonishingly broad range of phenomena, but is neither a dogma nor an explanation for everything. Then we shall expect all sorts of allometric relationships of metabolic measures and body size—with a certain preponderance of surface or 2/3-power functions, considering the fact that many metabolic processes are controlled by surfaces. This is precisely what we find (Table 7.2). In pther words, 2/3 is not a magic number; nor is there anything sacred about the 3/4 power which more recently (Brody, 1945; Kleiber, 1961) has been preferred to the classical surface law. Even the expression: Gesetz der fortschreitenden Stoffwechselreduktion (Lehmann, 1956)— law of progressive reduction of metabolic rate—is not in place because there are metabolic processes which do not regress with increasing size.

Furthermore, from this it follows that the dependence of metabolic rates on body size is not invariable as was presupposed by the surface law. It rather can vary, and indeed does vary, especially as a function of (1) the organism or tissue in question; (2) physiological conditions; and (3) experimental factors.

As to the variation of metabolic rate depending on the organism or tissue concerned, I shall give later on examples with respect to total metabolism.

Differences in size dependence of Qo2 in various tissues are shown in Figure 7.3. A similar example is presented in Table 7.3 with respect to comparison of intra- and interspecific allometries. Variations of size-dependence of metabolic rate with physiological conditions are demonstrated by data obtained in our laboratory in an important aspect which has been little investigated. The size-dependence of metabolism as expressed in the allometry exponent a varies, depending on whether basal metabolic rate (B.M.R.), resting metabolism, or metabolism in muscular activity is measured. Figure 7.4 shows such variation in rats, comparing basal and nonbasal metabolic rates. Figure 7.5 gives a more extensive comparison in mice, including different degrees of muscular activity. These data confirm Locker’s statement (1961a) that with increasing intensity of metabolic rate, a tends to decrease. Variations in the slope of the regression lines are also found in invertebrates when metabolic rates of fasting and nonfasting animals are compared (FIG. 7.6). Variations of a with experimental conditions deserve much more attention than usually given. Often the attitude is taken as if Qo2 were a constant characteristic of the tissue under consideration. This is by no means the case. Variations appear, for example., with different bases of reference, such as fresh weight, dry weight, N-content, etc. (Locker, 1961 b). The simplest demonstration is change of the medium. Not only—as every experimenter knows—does the absolute magnitude of Qp2 vary greatly depending, e.g., on whether saline or medium with metabolites is used; the same is true of size dependence or the parameter a (FIG. 7.7). Locker’s rule, as mentioned previously, again is verified; its confirmations by the experiments summarized in Figures 7.4, 7.5 and 7.7 are particularly impressive because they were obtained independent of and prior to statement of the rule. The variation of Qo2 in different media indicates that different partial processes in respiration are measured.

Table 7.2

Equations relating quantitative properties with body weights among mammals.