A computational framework for the morpho-elastic development of molluskan shells by surface and volume growth

Mollusk shells are an ideal model system for understanding the morpho-elastic basis of morphological evolution of invertebrates’ exoskeletons. During the formation of the shell, the mantle tissue secretes proteins and minerals that calcify to form a new incremental layer of the exoskeleton. Most of the existing literature on the morphology of mollusks is descriptive. The mathematical understanding of the underlying coupling between pre-existing shell morphology, de novo surface deposition and morpho-elastic volume growth is at a nascent stage, primarily limited to reduced geometric representations. Here, we propose a general, three-dimensional computational framework coupling pre-existing morphology, incremental surface growth by accretion, and morpho-elastic volume growth. We exercise this framework by applying it to explain the stepwise morphogenesis of seashells during growth: new material surfaces are laid down by accretive growth on the mantle whose form is determined by its morpho-elastic growth. Calcification of the newest surfaces extends the shell as well as creates a new scaffold that constrains the next growth step. We study the effects of surface and volumetric growth rates, and of previously deposited shell geometries on the resulting modes of mantle deformation, and therefore of the developing shell’s morphology. Connections are made to a range of complex shells ornamentations.Author Summary: Molluska are the second most diversified phylum of the animal kingdom, and their evolutionary success can be partly attributed to the hard shell that provides both protection and support to the soft body. The distinctive anatomical features of these hard shells are their rich pigmentation patterns and complex structural ornamentations. While the pigmentation patterns are primarily of biochemical origin, the ornamentations result from mechanical deformation of the mantle due to growth induced forces. This mechanical basis of “growth and form” has been previously investigated using simplified morpho-mechanical models, but restricted to reduced geometric representations. Here we propose a three-dimensional computational framework coupling morphology, incremental surface growth by accretion, and morpho-elastic volume growth, to enable an improved representation of the growth and structural parameters controlling the evolution of these ornamentations. We study the effects of growth rates, and of previously deposited shell geometries on the resulting modes of mantle deformation, and present a “phase diagram” of morphogenesis in molluskan shells. Our main motivation for focusing on generic physical processes involved in development is that they may shape living beings in a predictive way and partly determine the spectrum of forms that have been and could have been generated during evolution.