Abstract
A matrix manipulation new to the quantitative study of develomental stability reveals unexpected morphometric patterns in a classic data set of landmark-based calvarial growth. There are implications for evolutionary studies. Among organismal biology’s fundamental postulates is the assumption that most aspects of any higher animal’s growth trajectories are dynamically stable, resilient against the types of small but functionally pertinent transient perturbations that may have originated in genotype, morphogenesis, or ecophenotypy. We need an operationalization of this axiom for landmark data sets arising from longitudinal data designs. The present paper introduces a multivariate approach toward that goal: a method for identification and interpretation of patterns of dynamical stability in longitudinally collected landmark data. The new method is based in an application of eigenanalysis unfamiliar to most organismal biologists: analysis of a covariance matrix of Boas coordinates (Procrustes coordinates without the size standardization) against their changes over time. These eigenanalyses may yield complex eigenvalues and eigenvectors (terms involving i=sqrt{-1}); the paper carefully explains how these are to be scattered, gridded, and interpreted by their real and imaginary canonical vectors. For the Vilmann neurocranial octagons, the classic morphometric data set used as the running example here, there result new empirical findings that offer a pattern analysis of the ways perturbations of growth are attenuated or otherwise modified over the course of developmental time. The main finding, dominance of a generalized version of dynamical stability (negative autoregressions, as announced by the negative real parts of their eigenvalues, often combined with shearing and rotation in a helpful canonical plane), is surprising in its strength and consistency. A closing discussion explores some implications of this novel pattern analysis of growth regulation. It differs in many respects from the usual way covariance matrices are wielded in geometric morphometrics, differences relevant to a variety of study designs for comparisons of development across species.
Highlights
At present the literature of organismal growth studies does not offer an appropriate biomathematical toolkit for studying growth stability more than one variable at a time: the kind of analysis that is necessary if we are to understand the variation of configurations of skeletal landmarks over postnatal development
Discussions of canalization in today’s evo-devo literature that interpret it as reduction of variance (e.g., Gonzalez and Barbeito-Andrés, 2021) typically adhere to Conrad Waddington’s original trope of 1942, where the representation of the “constancy of the wild type” is by some hypothetical quantity unambiguously occupying the vertical axis of his famous diagram of developmental
In a different disciplinary context, until the advent of paleogenomics most of “multivariate palaeobiology” (Reyment, 1991) was based on an intuition about which traits are dynamically stable enough over normal growth to serve as numerical characters in a systematic analysis of forms of indeterminate age
Summary
At present the literature of organismal growth studies does not offer an appropriate biomathematical toolkit for studying growth stability more than one variable at a time: the kind of analysis that is necessary if we are to understand the variation of configurations of skeletal landmarks over postnatal development. There is following the pioneering, much-cited work of Potthoff and Roy (1964), a literature of technical biometrics centered on data series of otherwise unrestricted structure that follow specimens from more than one group over developmental time Nothing in this tradition, to my knowledge, takes advantage of the possibility that the physical dimension of time and the conceptual domain of the variables might be structured in a useful way. 5 returns to the conceptual level of discussion, suggesting relationships of this newly borrowed matrix method with topics such as knockout study design, paleostudies of selection gradients, and other matters of current evo-devo interest Following this discussion is an algebraic Appendix that engages with the pedagogical challenge here, the need to introduce our community to the general mathematical strategy of which principal component analysis is, alas, too specialized a special case
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