Dependence of morphometric allometries on the growth kinetics of body parts.

As overall size varies, the sizes of body parts of many animals often appear to be related to each other by a power law, commonly called the allometric equation. Orderly scaling relationships among body parts are widespread in the animal world, but there is no general agreement about how these relationships come about. Presumably they depend on the patterns of growth of body parts, and simple analyses have shown that exponential growth can lead to size relationships that are well-described by the allometric equation. Exponential growth kinetics also allow for a simple biological interpretation of the coefficients of the power relationship. Nevertheless, many tissues do not grow with exponential kinetics, nor do they grow for the same period of time, and the consequences of more realistic growth patterns on the resulting allometric relationships of body parts are not well understood. In this paper I derive a set of allometric equations that assume different kinetics of growth: linear, exponential and sigmoidal. In these derivations I also include differences in development times as a variable, in addition to differences in the growth rates and initial sizes of the two structures whose allometric relationship is compared. I show how these equations can be used to deduce the effect of different causes of variation in absolute size on the resulting allometry. Variation in size can be due to variation in the duration of development, variation in growth rate or variation in initial size. I show that the meaning of the coefficients of the allometric equation depends on exactly how size variation comes about. I show that if two structures are assumed to grow with sigmoidal kinetics (logistic and Gompertz) the resulting allometric equations do not have a simple and intuitive structure and produce graphs that, over a sufficiently large range of sizes, can vary from linear, to sigmoidal to hump-shaped. Over a smaller range of absolute sizes, these sigmoid growth kinetics can produce nearly linear allometries in both the arithmetic and logarithmic domains. I will argue that although growth kinetics are likely to be sigmoidal in most cases, natural selection will restrict variation in absolute size and the parameters of growth kinetics to regions where the allometric relations are linear, or nearly so.