Operational Point Homology by Cartesian Transformation to Standard Shape: Examples from Setal Positions in Phytoseiid Mites

Abstract

Hansell, R. I. C. (Department of Zoology, University of Toronto, Toronto, Canada M5S lAl), F. L. Bookstein (Center for Human Growth and Development, University of Michigan, Ann Arbor, Michigan 48109), and H. J. Rowell (Department of Zoology, University of Toronto, Toronto, Canada M5S lAl) 1980. Operational point homology by Cartesian transformation to standard shape: Examples from setal positions in phytoseiid mites. Syst. Zool., 29:43-49.-The geometric equivalence of loci on organisms is used as a basis for the operational definition of point homology. A mathematical and physical method of transforming the organism outline to a unit circle and of mapping the internal points are compared with reference to taxonomic problems in mesostigmatic mites. [Taxonomy, Homology, Shape transformation.] In the study of organisms, most of the attributes or character states we measure are associated with locations, or points, in or on the body. Comparing the attributes of two organisms depends on instructions about the correspondences of location, that is, on homology. In the modern literature, typified by Jardine and Jardine (1969; see also 1971), homology is taken to be identification of of things. We have argued elsewhere (Bookstein, 1978a: 120ff.) that whenever such a homology is unambiguously computed it reduces to a geometric homology function which declares equivalences of loci among organisms by reference to points only. Only this approach can avoid the severe logical problems and ambiguities involved in the definition of parts and their relationships. (For various approaches to this determination see, among others, Darwin, 1859; Jardine, 1967, 1969a; Owen, 1847; Sattler, 1967; or Sneath, 1969.) An interesting example of the problems inherent in determining the homology of internal points is a study of the phytoseiids by Rowell, Chant, and Hansell (1978). For a closely related family of mites, Ascidae, which possesses all the dorsal setae of the phytoseiids plus others, a standard system of nomenclature (Fig. 1) has now been accepted (Lindquist and Evans, 1965). It is an axiom of this nomenclature that relative setal position is genetically fixed and stable ontogenetically. The problem for acarologists, according to Rowell et al. (1978) is <to determine which setae have been lost in the evolution of the Phytoseiidae. Now for the phytoseiids, the insertions of setae on the exoskeleton have shown relatively constant within genera; and the presence or absence of setae, and their attributes, are important for the taxonomy of the group (Chant, 1965; Chant et al., 1978). In these taxonomic papers, setal homologies were determined by positional relations with other setae, with pores, and with the perimeters of the exoskeletal shield. But the clear definition of these positional relations is obscured by the different shapes of these mites. Rowell et al. (1978) removed this source of confusion by transforming the shape of the dorsal shield to a circle before determining the homologies of the setal insertions. Their method was a physical analogue of the technique of Cartesian transformation suggested by D'Arcy Thompson (1917). There has recently appeared in the biomathematical literature a purely geometric implementation of Thompson's method (Bookstein,