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J. theor. Biol. (2002) 215, 5766 doi:10.1006/jtbi.2001.2494, available online at http://www.idealibrary.com on

Summary: J. theor. Biol. (2002) 215, 57­66
doi:10.1006/jtbi.2001.2494, available online at http://www.idealibrary.com on
Heterochronic Detection Through a Function for the Ontogenetic Variation
of Bone Shape
J. Cubo*w, D. Azagraz, A. Casinosy and J. Castanetn
Equipe Formations Squelettiques, CNRS UMR 8570-2, Pl. Jussieu ­ Case, 7077 ­ 75251, Paris Cedex
05, France zDepartamento de Analisis Matematico, Facultad de Ciencias Matematicas, Universidad
Complutense de Madrid, 28040 Madrid, Spain and yDepartament de Biologia Animal, Universitat de
Barcelona, Av. Diagonal, 645, 08028 Barcelona, Spain
(Received on 23 July 2001, Accepted in revised form on 2 November 2001)
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 s (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 sđt̃; the time tm at which periosteal growth rate first
equalizes endochondral growth rate was taken as the timing parameter to be compared


Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid


Collections: Mathematics