Hierarchical mechanical patterns in morphogenesis: from mollusc shells to plants, fungi and animals

Hierarchical patterns, made up of subunits of different sizes intercalated with each other, are present in diverse organisms spanning the plant, fungi, and animal kingdoms. Despite these structures appearing in different kingdoms, at different scales, at different levels of biological organization, and involving different developmental mechanisms, their sequential development follows a generic principle of recursive subdivision of space, associated with domain growth and an irreversibility condition. To investigate the morphogenesis of such hierarchical patterns, we develop a theoretical framework, based on morphoelasticity, the biomechanics of growth. In our model, the hierarchical pattern is modelled as a sum of Gaussians, each representing a subunit. The size and spacing of these Gaussians is then determined by minimizing the mechanical energy of the growing system. Our framework is simple enough to be analytically tractable, enabling us to identify the mechanisms necessary for hierarchical pattern formation and providing new insight into the developmental process involved. Our work is specifically motivated by and applied to the context of mollusc shells, in which the hierarchical ridge pattern is in some species dilated to form beautifully exuberant shell edges, which may take the form of a row of needle-like or fractal-like spines, nevertheless maintaining the hierarchical form.