## Biological Scaling

Not only do the structure of organisms and their dimensions change in a regular way with size, but many fundamental physiological, ecological, and evolutionary factors also scale in predictable ways. If the relationship is linear or geometric with body mass with a slope of ~1, the scaling is termed isometric (iso = same, metric = measure); the gut capacity of animals is an example of a trait that scales isometrically with body mass. Many relationships scale nonlinearly with body mass, with slopes less than or greater than 1; these are known as allometric relationships (allos = different, metron = measure; Figure 1), a term coined by Julian Huxley and Georges Teissier in 1936. Allometric relationships were first noted in the late 1890s by Eugene Dubois and Louis Lapicque working independently on the relationship between brain and body mass, and have been extensively explored in paleontology, physiological ecology, and other disciplines.

Among traits that scale allometrically are many fundamental physiological processes such as metabolic rate, fecundity, home range, and cost of locomotion. Allometric or isometric relationships with body size are often formulated as power functions:

 Y= 1 1 1 1 y aX1 Figure 1 Difference between isometric and allometric scaling in arithmetic space. Such relationships are typically logarithmically transformed to obtain a linear slope for ease in computation and take the form log Y = log a + b log X. where Y is the variable of interest, M is body size, b is the slope of the relationship (representing how the variable of interest changes with differences in body size), and represents a taxon-specific constant, sometimes referred to as the normalization or proportionality constant (the intercept at unity body mass when M = 1). Power laws are often logarithmically transformed such that log Y = log a + b log M because the exponent becomes the slope of a straight line, facilitating computations and comparisons. The intercept often varies in a regular way among groups; marsupials, for example, have a metabolic rate 30% lower than other mammals, which is reflected in the value of their normalization constant. These body size relationships allow comparisons within and among species at different taxonomic levels and also allow reasonably accurate predictions of many biological rates and times. Often what appear to be significant differences among organisms are a simple consequence of scaling effects. True deviations from predicted values can provide important insights into evolutionary history and adaptation. Considerable research has gone into formulating and comparing allo-metric relationships for a whole variety of traits and taxa. 