The Mathematics of Growing a Shell

Kaushik Bhattacharya is an engineer and materials scientist, but he has always been fascinated by seashells. Part of the draw is the "beautiful" idea that even simple organisms like mollusks can construct wonderfully complex structures over and again, even in differing environments. He marvels at nature's engineering capabilities.

"Manufacturing materials is a nasty, difficult process," Bhattacharya says. "How does nature do it so nicely in such simple organisms?"

Bhattacharya, Caltech's Howell N. Tyson, Sr., Professor of Mechanics and Materials Science, and postdoctoral scholar Huan Liu have now figured out a mathematical way to explain the puzzling reproducibility of mollusk shells—the animals' ability to make roughly the same shell, regardless of the challenges that an individual mollusk might face.

This mathematical method can reliably reproduce the characteristic shells of almost any class of mollusk—from the coiled shell of a nautilus to the fanlike shell of a scallop—beginning with nothing more than three elements: the shape at the leading edge of a developing shell (what the researchers call the "curve" element); a scaling factor that reflects how fast a shell expands; and the angle of the shell's rotation, whether it is making the tight spirals of a tower snail or the wide swooping whorls of a nautilus, for example.

The mollusk-shell growth process is both passive, meaning it happens without additional control, and energy efficient—and the scientists now hope to emulate this ability to make other materials with complex shapes, such as specialized manifolds for jet engines.

Bhattacharya and Liu, a Cecil and Sally Drinkward Postdoctoral Fellow at Caltech, describe their findings in a paper published in the journal Proceedings of the National Academy of Sciences on October 23.

About a year ago, Bhattacharya mentioned his interest in molluscan shells to Liu, noting that shell growth had to be governed by "some simple rule which is done repeatedly," he recalls. "It can't be that the mollusk is sensing the whole shape and then adding to it in a clever way."

By studying a variety of real seashells, including the conelike shells of some sea snails and the hinged two-part shells of bivalves, Liu saw that the growth lines of each shell had the same shape as the one before it. Like tree rings, these growth lines are laid down sequentially over time. As a mollusk grows, its mantle secretes material that the organism then makes into its shell, creating ridges or bands with each new growth.

Liu and Bhattacharya quickly realized that the growing edge of the first shell that forms in the mollusk's embryo—what is known as a protoconch—repeated one set of actions over and over. The growth lines only differed by a scaling factor, or scalar, and a turn, or rotation.

This suggested that a concept from mathematics called a Lie group (pronounced "lee") would be a natural way of describing seashell growth. (In this context, "group" is a collection of actions that can be combined or undone.) In a Lie group, those actions change smoothly rather than in fixed steps, and this characteristic makes them ideal for describing a process in which the same small operation is executed repeatedly. By treating the shell's growth as a Lie group that acts on the shell's growing edge, the researchers could work out a growth law—a mathematical formula that explains how the growing edge transforms as the shell grows. Using the law, they were able to accurately build up the full spiral shape of a shell by repeating the process over time.

"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.

Changing the values of two of the three elements in the growth law produces markedly different shells. The starting curve at the growing edge of the shell sets the shape of the shell's opening, while the scaling factor, which the researchers call lambda (λ) determines how fast the shell expands. By changing those two factors in the model, Bhattacharya and Liu were able to generate the shape of nearly all known molluscan shells.

"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."

The scientists also showed that this whole-shell description is also mathematically equivalent to a local growth law at the growing edge. New material is added only along that curve, and the relative rate of material addition depends on the local geometry at that curve, such as how sharply it curves on the shell surface. From the beginning of development, once the protoconch is established, each step of growth determines the next, which means the continued expansion of the shell is a passive process. The mollusk does not need additional energy or activity to direct the construction of the shell; it simply builds upon itself.

The value of lambda—the scaling factor—also turned out to be biologically meaningful. 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. And each of the five shell-bearing classes of mollusks (Gastropoda, Bivalvia, Cephalopoda, Monoplacophora, and Scaphopoda) occupies its own narrow range of lambda values.

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.

The paper is titled "Local growth laws determine global shape of molluscan shells." The work was supported by the Drinkward Postdoctoral Fellowship and the Howell N. Tyson, Sr., Professorship at Caltech.