article: 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.
  Molluskan shells grow via an accretive process occurring at the shell margin by an organ called the mantle, which is a thin elastic membrane lining the inner surface of the shell. Over each increment, the mantle extends slightly beyond the calcified shell edge, while adhering to the rigid shell. The mantle then secretes matrix proteins, which, through biomineralization and calcification harden into a new layer of shell. Within this process is an interesting mechanical interaction, due to the fact that the form taken by the mantle along the growing shell front is fixed in the calcified edge, while the form of the calcified edge partly determines the shape of the mantle at the next growth increment [25]. As the mantle may have grown since the last shell secretion, its margin may be longer than the shell edge, and hence attachment to the shell may induce deformation of the mantle tissue that is then recorded and fixed in the shell shape upon secretion and calcification. We introduce our foundational notions of two distinct modes of growth. A process that creates new surface where none existed before is labelled as surface growth. In contrast, if growth takes place by deposition where material already exists, i. e. , over pre-existing volume, followed by elastic relaxation, the effect is to locally increase the material volume without adding new surface. For this reason we label it as volume growth. From a mechanical point of view, shell growth may thus be summarized by the steps illustrated schematically in Fig 1: (1) the mantle extends beyond the shell edge while also growing along the shell margin (volume growth), and (2) adhering to the rigid shell, creating an elastic deformation (morphoelasticity); in this deformed configuration, (3) new shell material is secreted (surface growth of the shell), and thus (4) a new layer of shell appears in the shape of the deformed mantle, which undergoes biomineralization and calcification, and the process repeats. The same basic process occurs in all shell-building mollusks, and yet produces a hugely diverse output of shell shapes and ornamentations. A general goal is to produce a mathematical and computational framework to explore this diversity: in particular how mechanical properties of the mantle, growth rates, and geometry conspire to produce the beautiful and varied outcomes observed in this phylum. However, a complete mathematical description is inherently challenging, as it links complex shell geometry (helicospiral, e. g.), elements of both surface and volume growth, nonlinear elastic deformations, and calcification. Previous work by some of the authors has approached this problem in a setting of one-dimensional elasticity, treating the interaction between the mantle margin and shell as a rod on an evolving elastic foundation. Here, our objective is to develop an algorithmic approach and computational tools to model the problem in a setting of three-dimensional nonlinear morphoelasticity. However, the mathematical details of such a computation are quite complex; indeed the combination of surface and volume growth is itself a significant challenge in biomechanics, with evolving reference configurations and multiple growth tensors. Here, we have the added complexity of the distinction between the growing shell and the growing mantle, as well as the additional process of shell calcification. Hence, a proper description involves the delicate treatment of surfaces evolving due to combined mechanisms of growth, mechanical forces, and a calcification front that plays the role of a moving boundary condition. To simplify the description, in this paper we formulate a mathematical description of the process that treats the mantle and shell edge as a single elastic object undergoing surface growth, volume growth, and calcification. The mathematical details are provided in the following subsection (which may be skipped by the reader whose interest lies primarily in the outcome of applying the growth models). In short, our algorithmic approach is to execute the process shown in Fig 1. In step (1) of the figure, extension of the mantle edge beyond the shell edge is modelled as surface growth. Volume growth is assumed to occur only in the direction parallel to the shell edge, and produces an excess of length of mantle relative to the calcified shell edge. Upon attachment, the calcified shell edge acts as a boundary condition constraining one edge of the mantle margin in step (2). Elastic deformation is determined via mechanical equilibrium within the framework of finite elasticity (and computed via the finite element method). This is the morphoelastic volume growth step that causes out-of-plane deformation of the mantle. Subsequent secretion of new shell material then occurs over the extended and deformed mantle in step (3) and the calcification front advances. Configurations of both the mantle and the shell are updated in step (4), and the process is repeated. As discussed further below, there are variations possible on exactly how this process is implemented in a computational setting. Our objective in this paper is not to exhaustively explore all possibilities, but rather to demonstrate the general validity of the algorithm and examine some basic properties of the shell patterns that emerge as output.
  We detail the mathematical framework that is the foundation for the eventual computational treatment of the growth processes outlined above. This requires descriptions of surface and volume growth, elasticity and calcification. In our model, de novo shell material is configured by a combination of mechanisms among those introduced above: (a) surface growth (creation of new surface) by mantle extension along a unit vector, s1, which is tangential to the shell surface and perpendicular to the mantle margin, i. e. s1 denotes the general direction of shell growth; (b) growth in size of the mantle over its pre-existing extent (therefore “volume” growth) manifesting in its expansion along another unit vector s2, which is tangential to the shell surface and in the direction of the mantle margin and shell edge (since the mantle takes on the shape of the shell surface near its leading edge, the mantle margin tracks the shell edge); and (c) formation of crests and valleys nominally perpendicular to the undeformed shell surface, and along the unit vector s3. We have s3 = s1 × s2, and more specifically, si · sj = δij, where the triad {s1, s2, s3} changes along the curved shell surface (Fig 2). The third mechanism above arises from elastic bifurcations from a smooth surface, and post-bifurcation deformation driven by “excess” mantle growth relative to the previous shell increment. As explained in the Mollusk shell growth mechanics subsection of the Introduction, this is the morphoelastic mechanism, which is susceptible to a continuum mechanical treatment. It is key to development of the elaborate, antimarginal decorations of molluskan shells [20,21,29–32]. Although surface growth due to mantle extension perpendicular to the shell edge occurs only following volume growth along the shell edge, we have described the steps (a-c) in the order of surface growth, volume growth and morphoelastic deformation. This is for mathematical purposes only. In our model, these mechanisms are not separated in time. At the outset we remark that the need for precision in describing the array of configurations and mechanisms leads to complexity of notation. The first step toward an algorithmic implementation is a discretization of the continuous processes of surface growth, morphoelastic volume growth, and evolution of the calcification front. The time interval of interest t ∈ [0, T] is discretized by instants t0, t1, …, tN, into sub-intervals [t0, t1], …, [tN−1, tN], where t0 = 0 and tN = T. For simplicity, we also consider deposition times τ = t0, t1, …. In a time step Δt = tk+1 − tk, the leading surface of material secreted by the mantle, ω t t, advances by v1Δt s1. Fig 4 depicts the relevant geometry (generating curve, mantle front surface in reference and deformed configurations parameterized by deposition time) and the growth processes driving the mantle’s shape by surface growth and morphoelastic volume growth. Secretion of shell material and calcification are implied, but not shown. The preceding mathematical model is continuous in time. It describes the biological processes in the sequence of (1) the mantle’s surface growth (extension), (2) morphoelastic volume growth, (3) shell growth by secretion on the mantle’s current configuration, followed by (4) calcification. However, the discrete model operates with time steps Δt = tk+1 − tk. While the time-continuous setting led to complex notation for evolving configurations, the time-discrete setting allows some simplifications in this regard. The above four processes are implemented in parallel over [tk, tk+1]. We note that the time discretization reflects a time-discontinuous process of growth and calcification.
  The influences of the surface and volume growth rates, and of the geometry via reference curvature, have been established. We now consider the combination of these effects, and their temporal and spatial variations, in two mechanisms that lead to more complex ornamentations.
  Mechanics has been recognized as a framework for explaining biological growth and form since at least the appearance of D’Arcy Thompson’s work [27]. However, a large part of the literature on morphological aspects of growth since the 1970s, such as that assembled by Meinhardt [38] and others, has focused on applying analytic or semi-analytic generating curves to the forms of shells, horns and antlers. The coupling of three-dimensional form to material forces and displacements, one aspect of which is morphoelasticity, has remained a more difficult problem. The difficulty stems from the complexity attained by the coupled equations, especially where nonlinear elasticity appears, and has been very well laid out in [20] and [21]. Consequently, it is only with the marriage of mathematics and numerical methods that general, three-dimensional, initial and boundary value problems have been solved [33,39]. The literature on computational treatments of biological growth also has, in our eyes, suffered a limitation: problems addressable by the model of inhomogeneous, volume growth, i. e. , morphoelasticity, have formed the mainstay of this body of work. Effective as this treatment has been in explaining tumor growth [40–42], aspects of cardiovascular systems, and the folding of soft, layered structures during morphogenesis [43], it cannot be elegantly extended to accretive, surface growth. For such problems, the morphoelastic treatment is restricted to representing advancing fronts by a thickening surface layer. Under its effect, the front’s motion is an emergent phenomenon that is controlled by local, pointwise, volume growth. Neither the elaborate, generated surfaces, nor their accompanying elastic fields can be represented by such an application of volume growth with its foundations in local volume changes, rather than de novo deposition of material. Against this backdrop, we have presented, to the best of our knowledge, the first combined computational framework for accretive, surface growth and local, morphoelastic, volume growth. The mathematical basis for this framework in terms of generating surfaces, evolving reference configurations and moving fronts has been crucial because it has provided a rigorous foundation on which to elucidate the discretized, space-time formulation, as well as the finite element framework. The discretized space-time is a faithful reflection of the coupled processes of accretive surface growth and morphoelastic volume growth. There are alternatives to the finite element framework, however. Variants of level-set, phase field and immersed boundary methods would allow propagation of surface growth and the calcification front by fractions of an element width. Arbitrary changes in the propagation directions s1 and vc could also be easily represented. We do not, however, see that this restriction to propagation by integral element widths presents a fundamental limitation in the shell morphologies and ornamentations that can be represented by the approach presented here. The advantages listed above for immersed boundary type methods could be approached by nonuniform element sizes in the advancing surface, and stepped fronts approximating changing directions on average. Most crucially, our work has identified the prominent role played by geometry in controlling mantle deformation under the driver of morphoelastic growth. The active mantle width, which is a direct outcome of the surface growth rate, has a very visible influence on the mode number, mode shape and, as we have demonstrated, on the appearance of hierarchical ornamentation. The curvature of reference surfaces is the other parameter by which geometry acts directly to control the locations of crests and valleys. The evolving reference configurations, as each generation of active mantle is enslaved to the reference curvature of the previous generation, present a pathway for coupling of morphoelastic volume growth with surface growth and curvature. Fig 15 is a “phase diagram” illustrating salient aspects of our studies over this parameter space. We have not attempted to compile computational demonstrations that match molluskan morphologies with high fidelity. While, in our opinion, most of our computations match well with features of actual molluskan morphologies, others such as in Fig 10c and 10d are less satisfying, especially in representation of spiny outgrowths (see the following paragraph in this regard). It is also true, however, that more complexity could be introduced to the model. Contact mechanics is one such addition, which is on our critical path, but must await a future communication. Another is the spatial variation of material properties, which we have already addressed before (see following paragraph). While a detailed tuning over such effects may add insight to mechanisms, several trends are visible in what we have explored here. We have experimented with some variations on our basic themes; variations that typically are not describable with analytic forms, but are ubiquitous in nature, thus making them obvious candidates for computational exploration. This is the rational for investigations of temporal and spatial variations in growth. We note that previous work by some of the authors [36] already has highlighted the role of a spatial variation in material properties in shaping the sharp spines seen on the shells of bivalves, cephalopods and gastropods. The present communication adds to this emerging picture of the influence of physics, by shifting the focus to geometry, which, driven by morphoelastic growth, acts through the mechanism of surface growth and a parametric dependence on curvature. Of particular interest is the further coupling of this framework with pattern formation by a range of reaction-transport equations. This would make accessible well-studied developmental milestones such as the patterning and morphogenesis of limbs and digits. Other problems in morphogenesis, such as the formation of skeletons are also within reach of our computational framework. Finally, we note that we have modelled antimarginal ornamentation events via bursts of growth. It is more likely that the growth rate does not change significantly at the location of ornamentations, but rather that the thickness of the mantle decreases, creating an increase in the length of the active mantle strip and a corresponding decrease in stiffness without requiring a strong increase in material. This effect would be interesting to incorporate in future studies, as the decrease in stiffness would further amplify the amplitudes of transverse deformation, and also may be non-uniform along the length of the mantle strip.
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.
keywords: medicine and health sciences
  classical mechanics
  engineering and technology
  applied mathematics
  mechanical stress
  geometry
  finite element analysis
  leading edges
  physiological processes
  developmental biology
  mathematics
  damage mechanics
  morphogenesis
  deformation
  aerospace engineering
  physics
  curvature
  physiology
  secretion
  biology and life sciences
  calcification
  physical sciences
title: A computational framework for the morpho-elastic development of molluskan shells by surface and volume growth