A coordinate-free approach to the analysis of growth patterns: models and theoretical considerations.

Developmental biology holds keys to our understanding of morphological pattern formation whether these patterns are expressed in the fossil record or among extant species. Though much is known about osseous growth at the cellular level (e.g. Hall, 1991), we have minimal understanding of the coordinated processes that combine to produce a complex, three-dimensional form. We have proposed a framework for the coordinate-free representation of form, a statistical method for comparing and modelling growth trajectories for complex morphologies, and a means for the eventual elucidation of the role of growth in the evolution of morphology. Our method uses the coordinate locations of biological landmarks to represent form as a matrix of all possible linear distances between landmarks, the form matrix. When two forms are expressed in this way, comparison of these forms is accomplished by computing the ratios of like linear distances, the form difference matrix. When the forms being compared are from a growth series, the matrix of ratios is called a growth matrix. Patterns of growth for two groups can be compared by computing the growth difference matrix. We applied growth difference matrix analysis to the study of sexual dimorphism of ontogeny in the M. fascicularis craniofacial skeleton. Growth matrices describing growth in male and female M. fascicularis were presented along with the growth difference matrix that describes sexual dimorphism of growth to underscore the detailed information available from this analytical technique. The method is quite general and can be applied to two- or three-dimensional data sets of landmark coordinates (cross-sectional or longitudinal) collected from almost any developing structure. The methods that we propose enable us to go beyond a mathematical summary of the comparison of forms and the comparison of growth patterns. We provide examples of how growth patterns might be used in the study of phylogenetic relationships. Our plans for use of this method in the study of evolutionary change assumes that morphological change in the craniofacial skeleton results from evolutionary change in developmental units (as defined by Atchley & Hall, 1991) that underlie morphological structure. We believe we have the basic tools to ultimately propose informed phylogenies based solely on developmental data. This task requires the identification of 'growth features' and the polarization of these features as primitive or derived. It is also advisable to determine a set of primitive growth features for the groups of interest. This will necessitate the inclusion of outgroups in our growth analysis.(ABSTRACT TRUNCATED AT 400 WORDS)