Reflections on a Biometrics of Organismal Form

Tending to the Facial Surfaces of a Mathematical Biology Head-Scratcher: Why Does the Head of the Sea Turtle Natator depressus Resemble a Convex Zygomorphic Dodecahedron?

Two convex polyhedra that markedly resemble the head of the flatback sea turtle hatchling are identified. The first example is a zygomorphic tetragonal dodecahedron, while the other, an even better matching structure, is a related tetradecahedron, herein speculated to arise from this particular dodecahedron via known mechanisms gleaned from studies of the behavior of foams. A segmented, biomorphic, convex polyhedral model to address cephalic topology is thus presented stemming from solid geometry, anatomical observations, and a recently computed densest local packing arrangement of fifteen slightly oblate spheroids in which fourteen oblate spheroids surround a central such spheroid. This particular array of oblate spheroids shares salient structural features with the aforementioned dodecahedron. Successful testing of the model has been achieved by converting this array of fifteen oblate spheroids constructed with putty to the cephaloid dodecahedron in a process involving ventral elongation induced by stretching in the anterior direction along the anteroposterior axis (convergent extension). During convergent extension, the two left most anterolateral oblate spheroids that are in direct contact with the ventral spheroid of the array merge into a single lateral facet of the incipient dodecahedron, while the corresponding two right such oblate spheroids do the same. Thus, the fourteen outer oblate spheroids of the array give rise to the twelve facets of the finalized dodecahedron, while the central oblate spheroid remnant assumes an interior dodecahedral position. The hypothetical dodecahedron to tetradecahedron transformation entails the collapse of a tetravalent vertex (which is known to occur in foams as part of a T1 transition) followed by bilateral facet splitting. Remarkably, a model stipulating that convexity is to be retained in connection with this sequence of steps necessitates that the starting dodecahedral template undergoes modification to become a tetradecahedron in possession of precisely the highly ordered feature found at the top of the head of numerous specimens of the flatback sea turtle hatchling, namely, a fused medial pentagon-heptagon pair in the form of a pentagonal frontal scute and heptagonal frontoparietal scute. Such a possible new instance of geometric biomorphy, taken together with the correct anticipation of the cephalic pentagon-heptagon pair, might serve to instill further confidence in renewed efforts to shed light on morphogenesis with foam embryo models.

A New Method for Landmark-Based Studies of the Dynamic Stability of Growth, with Implications for Evolutionary Analyses

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 $$i=\sqrt{-1}$$ i = - 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.

Correlation of the human pubic symphysis surface with age-at-death: a novel quantitative method based on a bandpass filter

Age-at-death estimation from skeletal remains typically utilizes the roughness of pubic symphysis articular surfaces. This study presents a new quantitative method adapting a tool from geometric morphometrics, bandpass filtering of partial warp bending energy to extract only age-related changes of the surfaces. The study sample consisted of 440 surface-scanned symphyseal pubic bones from men between 14 and 82 years of age, which were landmarked with 102 fixed and surface semilandmarks. From the original sample, 371 specimens within Procrustes distance of 0.05 of the side-specific average were selected. For this subsample, age was correlated with total bending energy (calculated as summed squared partial warps amplitudes) for a wide range of plausible bandpass filters. For our subsample's 188 right-side surfaces, the correlation between age and bandpass filtered versions of bending energy peaks relatively sharply at r = -0.648 for ages up through 49 years against the first seven partial warp amplitudes only. The finding for left symphyses is similar. The results demonstrate that below the age 50, the symphyseal surface form changes most systematically related to age may be best detected by a lowpass-filtered version of bending energy: signals at the largest geometric scales of roughness rather than its full spectrum. Combining this method with information from other skeletal features could further improve age-at-death estimation based on the symphyseal pubic surface.

Centric Allometry: Studying Growth Using Landmark Data

The geometric morphometric (GMM) construction of Procrustes shape coordinates from a data set of homologous landmark configurations puts exact algebraic constraints on position, orientation, and geometric scale. While position as digitized is not ordinarily a biologically meaningful quantity, and orientation is relevant mainly when some organismal function interacts with a Cartesian positional gradient such as horizontality, size per se is a crucially important biometric concept, especially in contexts like growth, biomechanics, or bioenergetics. “Normalizing” or “standardizing” size (usually by dividing the square root of the summed squared distances from the centroid out of all the Cartesian coordinates specimen by specimen), while associated with the elegant symmetries of the Mardia–Dryden distribution in shape space, nevertheless can substantially impeach the validity of any organismal inferences that ensue. This paper adapts two variants of standard morphometric least-squares, principal components and uniform strains, to circumvent size standardization while still accommodating an analytic toolkit for studies of differential growth that supports landmark-by-landmark graphics and thin-plate splines. Standardization of position and orientation but not size yields the coordinates Franz Boas first discussed in 1905. In studies of growth, a first principal component of these coordinates often appears to involve most landmarks shifting almost directly away from their centroid, hence the proposed model’s name, “centric allometry.” There is also a joint standardization of shear and dilation resulting in a variant of standard GMM’s “nonaffine shape coordinates” where scale information is subsumed in the affine term. Studies of growth allometry should go better in the Boas system than in the Procrustes shape space that is the current conventional workbench for GMM analyses. I demonstrate two examples of this revised approach (one developmental, one phylogenetic) that retrieve all the findings of a conventional shape-space-based approach while focusing much more closely on the phenomenon of allometric growth per se. A three-part Appendix provides an overview of the algebra, highlighting both similarities to the Procrustes approach and contrasts with it.

Pathologies of Between-Groups Principal Components Analysis in Geometric Morphometrics

Good empirical applications of geometric morphometrics (GMM) typically involve several times more variables than specimens, a situation the statistician refers to as “high p/n,” where p is the count of variables and n the count of specimens. This note calls your attention to two predictable catastrophic failures of one particular multivariate statistical technique, between-groups principal components analysis (bgPCA), in this high-p/n setting. The more obvious pathology is this: when applied to the patternless (null) model of p identically distributed Gaussians over groups of the same size, both bgPCA and its algebraic equivalent, partial least squares (PLS) analysis against group, necessarily generate the appearance of huge equilateral group separations that are fictitious (absent from the statistical model). When specimen counts by group vary greatly or when any group includes fewer than about ten specimens, an even worse failure of the technique obtains: the smaller the group, the more likely a bgPCA is to fictitiously identify that group as the end-member of one of its derived axes. For these two reasons, when used in GMM and other high-p/n settings the bgPCA method very often leads to invalid or insecure biological inferences. This paper demonstrates and quantifies these and other pathological outcomes both for patternless models and for models with one or two valid factors, then offers suggestions for how GMM practitioners should protect themselves against the consequences for inference of these lamentably predictable misrepresentations. The bgPCA method should never be used unskeptically—it is always untrustworthy, never authoritative—and whenever it appears in partial support of any biological inference it must be accompanied by a wide range of diagnostic plots and other challenges, many of which are presented here for the first time.