Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, Wisconsin, United States of America.
Mathematical Institute, University of Oxford, Oxford, United Kingdom.
PLoS Comput Biol. 2019 Jul 29;15(7):e1007213. doi: 10.1371/journal.pcbi.1007213. eCollection 2019 Jul.
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.
软体动物壳是理解无脊椎动物外骨骼形态演变的形态和弹性基础的理想模型系统。在壳的形成过程中，套膜组织分泌蛋白质和矿物质，这些物质会钙化形成外骨骼的新的增量层。关于软体动物形态的大部分现有文献都是描述性的。对现有壳形态、新表面沉积和形态弹性体积生长之间的基本耦合的数学理解处于起步阶段，主要限于简化的几何表示。在这里，我们提出了一个通用的、三维的计算框架，将现有形态、增生表面生长和形态弹性体积生长结合起来。我们通过将其应用于解释贝壳在生长过程中的逐步形态发生来检验这个框架：通过套膜的增生生长来沉积新的材料表面，其形状由其形态弹性生长决定。最新表面的钙化不仅扩展了贝壳，还创建了一个新的支架，约束了下一步的生长。我们研究了表面和体积生长速率以及先前沉积的壳几何形状对套膜变形的影响，因此也对正在发育的壳的形态产生影响。与一系列复杂的贝壳装饰建立了联系。
