Mollusks are mathematicians

Three basic rules

Mollusks are ingenious architects, and the durable yet aesthetic shells are the houses they have built for themselves, protecting their soft flesh from wind, rain or predation. Many of these shells have strikingly complex shapes—logarithmic helixes decorated with fractal spines or other things—all of which almost perfectly obey mathematical rules. Of course, mollusks don't know much mathematics, and what surprised researchers was how these inferior beings could construct incredibly intricate structures so precisely.

For more than 100 years, scientists have realized that cells, tissues, and organs follow the same set of physical laws as everything else in the world. But most biologists in the 20th century focused their research on genes, wanting to understand how genetic codes guide the production of biological patterns and what they do. In recent decades, researchers have gradually begun to use physics-based mathematical models to solve problems related to biological forms. In this direction, our work over the past few years provides some interesting insights into how shells form.

Differential geometry is a discipline of mathematics that studies curves and surfaces, and through this discipline, we learned that mollusks can produce beautifully shaped shells by following only a few mathematical rules when building their homes. The interaction of these laws with the mechanical forces subjected to shell growth produces an infinite number of different shell shapes. Our findings help explain how many clades in the largest mollusk taxa, gastropods, independently evolved complex biological features such as spines. For these different organisms, there is no need to go through the same genetic variation to get similar decorative shapes, because the laws of physics determine many things.

The shells of mollusks are built from a mantle membrane: at the opening or opening of the shell, the thin and soft organ of the mantle secretes a substance rich in calcium carbonate layer by layer. To form spiral shells unique to gastropods such as snails, only three basic laws need to be followed.

The first rule is expansion: by evenly depositing more material than before, mollusks can continuously create slightly larger openings than before. In this process, the initial circular opening becomes a cone.

The second rule is rotation: by depositing more material on the side of the shell, the mollusk can rotate completely on the basis of the original shell mouth, and get a shell like a doughnut or ring.

The third rule is torsion: mollusks rotate the deposit sites of their shells. By expanding and rotating, you can get a flat spiral shell like the chambered Nautilus. Coupled with torsion, the shape of the shell becomes a non-planar helix described mathematically.

The formation of decorations

For some shell builders, the story ends here, and their dwellings are so neat and beautiful. For other builders, the dwelling needed more decoration. In order to understand how decorative structures such as spines are formed, we must consider the forces generated when the shell grows. In fact, the secretion process of shells is carried out in a special mechanical system. The mantle membrane is connected to the shell through the so-called generating region, which is composed of substances secreted by the mantle membrane that has not yet been calcified. It is the interaction between the mantle membrane and the shell that allows the shell to take on a variety of shapes.

Any mismatch between the shell mouth and the mantle membrane can cause compression of the mantle membrane tissue itself. Compared to the shell mouth, if the coat film is too small, it must be stretched to adhere to the shell mouth. Conversely, if the mantle membrane is too large, it will have to compress itself to fit the shell mouth. Therefore, if the generating zone is deformed due to these pressures, then the new shell-forming material secreted by the mantle membrane at this time will be permanently solidified on the shell according to this deformation, and further affect the next growth stage of the mantle membrane. Fundamentally, as long as the growth rate of the shell does not exactly coincide with the growth rate of the mollusk itself, deformation occurs, forming what we call ornamentation or shell ornamentation.

Thorns are the most prominent type of decoration on shells, generally protruding vertically outward relative to the shell mouth, usually protruding several centimeters from the surface of the shell. With the explosive growth of the mantle membrane, these protrusions also form periodically. In an outbreak of growth, the mantle membrane grows so fast that it exceeds the shell mouth and can no longer align with the shell mouth. At this time, the mantle membrane will bend slightly, and the shell-forming material it secretes will also bend with it. In the next burst of growth, the mantle membrane grows further and again over the shell opening, which in turn amplifies the bend. We deduce that it is the repeated growth process that interacts with mechanical forces to eventually form a series of spines on the shell, and the specific pattern of these spines is mainly determined by the explosive growth rate of the mantle membrane and the stiffness of the mantle membrane.

Mathematical models confirm growth patterns

To test this idea, we developed a mathematical model to describe the growth of the mantle membrane, in which the growth basis of the mantle membrane increases with each growth process. When we experimented with typical growth patterns and material properties, the model yielded a variety of spines, very similar to the shapes that people observed on real shells, thus validating our hypothesis.

Thorns are not the only kind of decoration that mollusks are likely to add to their shells. The extinct ammonite is a close relative of today's cephalopods (nautilus, octopus, etc.), and another shell ornament has been found on its shell fossils. Ammonites ruled the oceans for 335 million years and became extinct about 65 million years ago. In addition to having a planar logarithmic spiral shell, ammonites are most notable by having regular ribs parallel to the edges of the shell. The mechanism by which this shell ornament is produced may be the same as that of the spine, stemming from the interaction of the growth process with mechanical forces, except that the shape of the two is completely different. Although the forces act are the same, the magnitude of the forces is different from the geometric environment.

The shell mouth of ammonite is basically rounded. When the radius of the mantle membrane is larger than the radius of the shell at this time, the mantle membrane will be compressed, but this degree of compression is not enough to produce spines. In this case, the oppressed mantle membrane extends outward, increasing the radius of the shell that grows out of the back. But at the same time, the outward movement of the mantle membrane is resisted by the generating region that is calcifiing, which acts like a torsion spring to maintain the growth direction of the shell.

We suspect that the antagonism of these two forces forms an oscillatory system: the radius of the shell increases, reducing the degree of compression of the mantle membrane, but when the radius of the shell exceeds the mantle membrane, a tension is created on the latter; the "stretched" mantle membrane begins to grow inward to reduce the tension, and is again oppressed because it exceeds the radius of the shell. The mathematical description of this "morphological mechanical oscillator" confirms our hypothesis: during the growth of mollusks, increases in wavelength and amplitude produce regular ribs. In addition, the results predicted by these mathematical models dovetail very well with the known ammonite shapes.

At the same time, mathematical models also predict that the faster the radius of the shell mouth of the mollusk increases, the less obvious the ribs on the shell will be. This may explain why the more curved the shell, the more pronounced its ribs become. The relationship between the rate of shell expansion and rib formation also solves a long-standing mystery in the study of mollusk evolution from a mechanical and geometric point of view: at least 200 million years ago, the chambered nautilus and its close relative, the nautilus, have very smooth shells. Some observers believe that the taxon apparently did not evolve much since then. In fact, several nautilus living today are often considered "living fossils". However, biophysical growth models show that the smooth shells on the Nautilus are purely the result of rapid expansion of the shell mouth. The actual evolution of nautilus may have occurred far more than the shape of their shells.

Until now, we still have a lot to look into how molluscs can construct amazing dwellings. For example, about 90% of gastropods are "right-handed", that is, their shells are built in a clockwise rotational direction, and only 10% build shells from the left. Scientists are only just beginning to study the mechanisms behind the prevalence of this right-handed spiral. In addition, there are some delicate shell ornaments whose origins remain unexplained, like fractal spines in many species of the osteospiaceae. As a model organism that studies the shape of nature, there are many secrets about the shells of marine mollusks, and the discovery of mechanical control of shell development has added to the charm of shell research.

Translation/Shen Hua Source: Global Science Public Account