Scaling is a term borrowed from engineering, which, as its name suggests, refers to adaptations of a functional system to operate at different production scales. This is achieved by increasing the number of functional units, increasing the size of a functional unit, increasing flow through the functional unit or by redesigning the system altogether. Biological systems take advantage of all four of these mechanisms when adapting to serve organisms of different sizes.
Allometry is the study of size and its consequences, and the scaling of biological functional systems can be studied and described mathematically by using allometric equations. The term allometry is derived from the Greek alloios, meaning different, and is used to distinguish allometric scaling from isometric scaling.
Isometric scaling applies to figures whose proportions remain the same at all sizes (i.e. geometrical figures), whereas allometric scaling applies to figures whose proportions change as a function of size.
P is the metabolic rate, W is the body weight in kilogram. The constant b is referred to as the exponent, which describes how the parameter scales over different values of body weight.
It is important to realize that exponents are not constants and have no physiological meaning by themselves. They simply provide a means of describing, in mathematical language, the effect of size (W) on a given parameter (P). If P and W increased in direct proportion, the exponent would be 1.
Blood volume, for example, increases in direct proportion to body mass and therefore scales with an exponent of 1; this is an isometric relationship. Haemoglobin concentration is constant across species and the exponent is therefore 0. Skeletal mass increases out of proportion with body mass in order to withstand the increased static load, locomotive stresses and impact forces generated and received by larger animals. In this case, the exponent is greater than 1.
The value of the exponent for whole body metabolic rate was originally calculated by Max Kleiber to be 0.74 (Kleiber, 1932. A few years later, Brody et al. published their famous mouse to elephant curve and calculated the exponent to be 0.734 (Brody, 1945). A value of 0.75 is now accepted because it is easier to use, and the difference from 0.734 is considered to be statistically negligible (Schmidt-Nielsen, 1984). However, it should be noted that exponents in the range 0.6–0.8 have been reported for metabolic rate (Agutter and Wheatley, 2004). A value of 0.75 means that the whole body metabolic rate increases as body weight increases, but to a lesser extent than would be expected of a simple proportional relationship. It follows on from this that the specific metabolic rate (the metabolic rate per unit mass) decreases as animals get larger (the exponent is −0.25); the metabolic rate of 1 g blue whale tissue is 1000 times less than that of 1 g shrew tissue (Kirkwood, 1983).