Geometric analysis and estimation of the growth rate gradient on gastropod shells

A Mathematical Model for Mollusc Shells Based on Parametric Surfaces and the Construction of Theoretical Morphospaces

In this study, we propose a mathematical model based on parametric surfaces for the shell morphology of the phylum Mollusca. Since David Raup’s pioneering works, many mathematical models have been proposed for different contexts to describe general shell morphology; however, there has been a gap in the practicality of models that allow the estimation of their parameter values in real specimens. Our model collects ideas from previous pioneering studies; it rests on the equation of the logarithmic spiral, uses a fixed coordinate system (coiling axis), and defines the position of the generating curve with a local moving system using the Frenet frame. However, it improves upon previous models by applying apex formation, rotations, and substantially different parameter definitions. Furthermore, the most conspicuous improvement is the development of a simple and standardized methodology to obtain the six theoretical parameters from shell images from different mollusc classes and to generate useful theoretical morphospaces. The model was applied to reproduce the shape of real mollusc-shell specimens from Gasteropoda, Cephaloda and Bivalvia, which represent important classes in geological time. We propose a specific methodology to obtain the parameters in four morphological groups: helicoidal, planispiral, conic, and valve-like shells, thereby demonstrating that the model offers an adequate representation of real shells. Finally, possible improvements to the model are discussed along with further work. Based on the above considerations, the capacity of the model to allow the construction of theoretical morphospaces, the methodology to estimate parameters and from the comparison between several existing models for shells, we believe that our model can contribute to future research on the development, diversity and evolutionary processes that generated the diversity in mollusc shells.

Growth and morphogenesis of the gastropod shell

Gastropod shell morphologies are famously diverse but generally share a common geometry, the logarithmic coil. Variations on this morphology have been modeled mathematically and computationally but the developmental biology of shell morphogenesis remains poorly understood. Here we characterize the organization and growth patterns of the shell-secreting epithelium of the larval shell of the basket whelk Tritia (also known as Ilyanassa). Despite the larval shell's relative simplicity, we find a surprisingly complex organization of the shell margin in terms of rows and zones of cells. We examined cell division patterns with EdU incorporation assays and found two growth zones within the shell margin. In the more anterior aperture growth zone, we find that inferred division angles are biased to lie parallel to the shell edge, and these divisions occur more on the margin's left side. In the more posterior mantle epithelium growth zone, inferred divisions are significantly biased to the right, relative to the anterior-posterior axis. These growth zones, and the left-right asymmetries in cleavage patterns they display, can explain the major modes of shell morphogenesis at the level of cellular behavior. In a gastropod with a different coiling geometry, Planorbella sp., we find similar shell margin organization and growth zones as Tritia, but different left-right asymmetries than we observed in the helically coiled shell of Tritia These results indicate that differential growth patterns in the mantle edge epithelium contribute to shell shape in gastropod shells and identify cellular mechanisms that may vary to generate shell diversity in evolution.

Integrating multilocus DNA data and 3D geometric morphometrics to elucidate species boundaries in the case of Pyrenaearia (Pulmonata: Hygromiidae)

To accurately delimit species the use of multiple character types is essential as all speciation processes are not equally reflected in different data (e.g. morphological, molecular or ecological characters). With the introduction of geometric morphometrics methods and advances in 3D technology, a comprehensive combination of molecular and morphological data has been enabled in groups where exhaustively quantifying and measuring morphological shape change was not possible before such as gastropod shells. In this study, we combined multilocus coalescent species delimitation methods with 3D geometric morphometrics of shell shape to delimit species within the land snail genus Pyrenaearia. A new taxonomic scheme was constructed for the genus identifying ten species. Two nominal species were synonymized and a hitherto unrecognized cryptic species was identified. Our findings support the importance of combining multiple lines of evidence as molecular and morphological data on their own do not yield the same information. Further, the integration of morphological and molecular data shows the importance of allometry in shell shape and suggests a combined effect of population history and selection in different environments on shells morphological variation. Our new taxonomy and phylogenetic reconstruction suggest that, besides the glacial cycles of the Pleistocene, passive dispersal and rock substrate complexity could also have been involved in the speciation of the genus.

A Method for Quantifying, Visualising, and Analysing Gastropod Shell Form

Quantitative analysis of organismal form is an important component for almost every branch of biology. Although generally considered an easily-measurable structure, the quantification of gastropod shell form is still a challenge because many shells lack homologous structures and have a spiral form that is difficult to capture with linear measurements. In view of this, we adopt the idea of theoretical modelling of shell form, in which the shell form is the product of aperture ontogeny profiles in terms of aperture growth trajectory that is quantified as curvature and torsion, and of aperture form that is represented by size and shape. We develop a workflow for the analysis of shell forms based on the aperture ontogeny profile, starting from the procedure of data preparation (retopologising the shell model), via data acquisition (calculation of aperture growth trajectory, aperture form and ontogeny axis), and data presentation (qualitative comparison between shell forms) and ending with data analysis (quantitative comparison between shell forms). We evaluate our methods on representative shells of the genera Opisthostoma and Plectostoma, which exhibit great variability in shell form. The outcome suggests that our method is a robust, reproducible, and versatile approach for the analysis of shell form. Finally, we propose several potential applications of our methods in functional morphology, theoretical modelling, taxonomy, and evolutionary biology.