Heterochronic detection through a function for the ontogenetic variation of bone shape

Abstract

Heterochrony, evolutionary modifications in the rates and/or the timing of development, is widely recognized as an important agent of evolutionary change. In this paper, we are concerned with the detection of this evolutionary mechanism through the analysis of long bone growth. For this, we provide a function sigma (t) for the ontogenetic variation of bone shape by taking the ratio of two Gompertz curves explaining, respectively, the relative contribution to long bone growth of (a) endochondral ossification and (b) periosteal ossification. The significance of the fitting of this function to empirical data was tested in Anas platyrhynchos (Anseriformes). In this function sigma(t), the time t(m) at which periosteal growth rate first equalizes endochondral growth rate was taken as the timing parameter to be compared between taxa. On the other hand, the maximum rate of ontogenetic change in bone shape (maximum slope, beta) from hatching to t(m) was taken as the rate parameter to be compared. Comparisons of these parameters between the plesiomorphic condition and the derived character state would provide evidence for hypornorphosis (earlier occurrence of t(m)), hypermorphosis (delayed occurrence of t(m)), deceleration (smaller beta) or acceleration (higher beta).

Regarding the phylogenetic context, the ancestral condition for the character of interest should be estimated to polarize the direction of the heterochronic change. We have quantified the influence of the phylogenetic history on the variation of adult bone shape in a sample of 13 species of Anseriformes and 17 species from other neornithine orders of birds by using permutational phylogenetic regressions. Phylogenetic effects are significant, and this fact allows the optimization of bone shape onto a phylogenetic tree of Anseriformes to estimate the ancestral condition for Anas platyrhynchos.