Kaushik Bhattacharya, Caltech’s Howell N. Tyson, Sr., Professor of Mechanics and Materials Science, and postdoctoral scholar Huan Liu, a Cecil and Sally Drinkward Postdoctoral Fellow, describe the work in a paper published in the Proceedings of the National Academy of Sciences on October 23. Bhattacharya is an engineer and materials scientist by training, but he has long been drawn to seashells. Part of the appeal, he says, is the idea that even simple organisms can construct wonderfully complex structures over and over again, in differing environments. “Manufacturing materials is a nasty, difficult process,” he says. “How does nature do it so nicely in such simple organisms?”

The starting point for the collaboration was a hunch about how such a complicated object could emerge without any global plan. About a year ago, Bhattacharya mentioned his interest in molluscan shells to Liu and suggested that shell growth must be governed by “some simple rule which is done repeatedly.” As he recalls, “It can’t be that the mollusk is sensing the whole shape and then adding to it in a clever way.” In other words, the animal was unlikely to be monitoring the overall geometry of its shell and adjusting each new increment of material accordingly. Something far more local and mechanical had to be at work.

To find that rule, Liu studied a variety of real seashells, including the conelike shells of some sea snails and the hinged two-part shells of bivalves. The key evidence came from growth lines, the ridges or bands laid down sequentially as a mollusk grows, much like the rings of a tree. As the animal develops, its mantle secretes material that the organism then incorporates into its shell, producing a new growth line with each round of construction. Liu noticed something striking: on each shell, every growth line had the same shape as the one before it.

That observation pointed directly to the earliest stage of shell formation. The growing edge of the first shell that forms in the mollusk’s embryo, a structure known as the protoconch, appeared to repeat one set of actions over and over. Successive growth lines differed from one another only by a scaling factor, a single number that enlarges or shrinks the shape, and by a turn, or rotation. The entire diversity of adult shell forms, it seemed, could flow from a single curve being copied, scaled, and rotated again and again.

Mathematicians have a natural language for exactly this kind of process: the Lie group, pronounced “lee.” In this context, a group is a collection of actions that can be combined or undone, and in a Lie group those actions change smoothly rather than in fixed steps. That smoothness makes Lie groups ideal for describing a process in which the same small operation is executed repeatedly. By treating shell growth as a Lie group acting on the shell’s growing edge, the researchers could derive a growth law, a mathematical formula that specifies how the growing edge transforms as the shell grows. Applying the law repeatedly over time, they were able to build up the full spiral shape of a shell accurately. “We found that the wide variety of molluscan shells can be described by a simple family of Lie groups that only contain three elements—one curve, one scalar, and one rotation,” Liu says.

The three elements each carry a distinct geometric role. The curve is the shape at the leading edge of the developing shell, and it sets the shape of the shell’s opening. The scaling factor, which the researchers call lambda (λ), determines how fast the shell expands as it grows. The rotation angle governs how each new increment is turned relative to the previous one, producing everything from the tight spirals of a tower snail to the broad whorls of a nautilus. By changing the values of just two of these elements, the curve and the scaling factor, Bhattacharya and Liu were able to generate the shapes of nearly all known molluscan shells, from coiled cephalopod forms to the flattened fans of scallops.

Perhaps the most striking property of the model is its robustness. “What is remarkable about all of this is it’s very robust,” Bhattacharya says. “You can change things around, like the growth rate, or you can add a little defect, and still the overall shape of the shell remains very repeatable.” That robustness helps explain the long-standing puzzle of why mollusks raised under different conditions still produce shells that look so recognizably alike. Small perturbations in the growth process do not accumulate into distorted forms; the structure of the growth law itself absorbs them and preserves the overall geometry.

The researchers also showed that their whole-shell description is mathematically equivalent to a local growth law acting at the growing edge. New material is added only along that curve, and the relative rate of material addition depends on the local geometry there, such as how sharply the edge curves on the shell surface. Once the protoconch is established at the beginning of development, each step of growth determines the next, which means the continued expansion of the shell is a passive process. The mollusk does not need to expend additional energy or direct activity to steer the construction of its shell; it simply builds upon itself, with the local rule carrying the global form forward automatically.

The scaling factor lambda also turned out to carry biological meaning. Shells of mollusks within the same class on the phylogenetic tree, even some that look quite different from one another, have nearly the same lambda. Moreover, each of the five shell-bearing classes of mollusks, Gastropoda, Bivalvia, Cephalopoda, Monoplacophora, and Scaphopoda, occupies its own narrow range of lambda values. The single number that controls how fast a shell expands appears to be a signature shared across evolutionary lineages, linking shell geometry to molluscan ancestry in a quantitatively precise way.

Beyond explaining one of nature’s oldest engineering feats, the work points toward practical applications. Bhattacharya, who is interested in engineering materials in more efficient ways, says the results suggest the possibility of gentler and less energy-hungry methods of manufacturing complex parts. Because molluscan growth is both passive and energy efficient, adding material only locally and letting simple rules generate the final form, the same principle could inspire new approaches to fabricating shaped components, such as the specialized manifolds used in jet engines. The scientists now hope to emulate the mollusk’s strategy to make other materials with complex shapes. The paper, titled “Local growth laws determine global shape of molluscan shells,” was supported by the Drinkward Postdoctoral Fellowship and the Howell N. Tyson, Sr., Professorship at Caltech.

Subject of Research: A Lie-group-based mathematical growth law explaining the reproducible shapes of molluscan shells

Article Title: The mathematics of growing a shell

Article References: The mathematics of growing a shell. Original publication (Bibliographic details unavailable.)

Image Credits: AI Generated

DOI: Not provided

Keywords: mollusks, seashells, Lie groups, growth law, morphogenesis, mathematical biology, Caltech, protoconch, biomineralization, scaling factor, PNAS, materials science