Shells: vaults, viruses and spheres

Inside many of your cells float tiny hollow barrels called vaults. Around the genes of many viruses sits a shell shaped like a twenty-sided die. The two look nothing alike, but they are built the same way: from many copies of one protein, fitted together by symmetry.

This page looks at why nature builds that way, the math that decides which shells are possible, and a 2026 study that found vaults made by viruses of bacteria.

Many copies of one piece

A small virus carries very few genes. In 1956 Francis Crick and James Watson pointed out that such a virus can’t hold the instructions for a big shell made of many different proteins. It is far cheaper to write the recipe for one small protein and make lots of copies.

Then comes a second idea. If every copy is the same and each one meets its neighbors in the same way, the copies must sit in a symmetric pattern, like identical tiles. Crick and Watson predicted that small round viruses would have the symmetry of one of the regular solids. Most turned out to have the symmetry of the icosahedron.

Nobody has to build these shells. The pieces build themselves. Their shapes and sticky patches make them snap together in only one way, much as salt crystals grow out of salt water. This is called self-assembly. In 1955 Heinz Fraenkel-Conrat and Robley Williams pulled tobacco mosaic virus apart into its protein and its RNA, mixed them again in a test tube, and got working virus back.

Virus shells: the twenty-sided ball

An icosahedron has 20 triangle faces, 12 corners and 30 edges. Spin it on an axis through two opposite corners and it looks the same 5 times in one full turn, so that is a 5-fold axis. There are 6 of those. An axis through the centers of two opposite faces is 3-fold, and there are 10. An axis through the middles of two opposite edges is 2-fold, and there are 15.

Count every turn that leaves it looking the same, including doing nothing: $1 + 6 \times 4 + 10 \times 2 + 15 \times 1 = 60$. Each turn can carry one protein onto the place of another. So 60 is the largest number of copies that can sit in exactly the same surroundings on an icosahedron: three on each face. That is the simplest virus shell. The tiny satellite tobacco necrosis virus has one.

A bigger virus needs a bigger shell, which means more than 60 copies. In 1962 Donald Caspar and Aaron Klug found the way. Proteins can bend a little, so they can sit in places that are nearly the same. They called this quasi-equivalence. Most of the proteins gather in groups of six (hexamers), which lie flat like the cells of a honeycomb. To curve a flat honeycomb into a closed ball you need exactly 12 groups of five (pentamers), one at each corner of the icosahedron.

Where the pentamers go is set by two whole numbers. Start at one pentamer, walk $h$ steps along a row of the honeycomb, turn 60°, and walk $k$ more steps. You land on the next pentamer. That walk is one edge of a triangle face, and the shell has

$$T = h^2 + hk + k^2$$

times as many proteins as the simplest shell: $60T$ in all, made of 12 pentamers and $10(T-1)$ hexamers. (Check: $12 \times 5 + 10(T-1) \times 6 = 60T$.) $T$ is called the triangulation number.

Real shells bend the rules a little. In phage HK97 and herpes virus, one of the 12 corners holds a doorway (the portal) that the DNA goes in by, instead of a pentamer.

A number puzzle hiding in the shells. The list of possible values starts 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, … Numbers like 2, 5, 6 and 8 never appear. Why? $h^2 + hk + k^2$ is the squared length of the walk on the honeycomb grid, so the possible $T$ are exactly the squared distances between points of a hexagonal grid. Mathematicians treat those grid points as numbers, the Eisenstein integers $a + b\omega$ with $\omega = \tfrac{-1 + i\sqrt{3}}{2}$, and call the squared length the norm. A whole number is a norm exactly when every prime in it that leaves remainder 2 on division by 3 (2, 5, 11, 17, …) appears an even number of times. So 5 is out, $4 = 2 \times 2$ is in, and $7 = 4 + 2 + 1$ is in.

When $h$ and $k$ are different and neither is 0, as for $T = 7$, the walk can turn left or right. The shell comes in two mirror-image forms, like a left and a right hand.

Vaults: barrels inside the cell

In 1986 Nancy Kedersha and Leonard Rome found a new particle in rat liver cells. Under the electron microscope its arches reminded them of the vaulted ceilings of a cathedral, so they called it a vault. Vaults turned up in humans and in many other animals and some single-celled organisms. Yeast and fruit flies don’t have them.

A vault is a ribonucleoprotein: mostly protein, with a few short strands of RNA. It is huge for a molecule, about three times the mass of a ribosome, and about 67 nanometers long and 40 wide. Its shell is two cup-shaped halves joined rim to rim. Each half is a ring of 39 copies of the major vault protein (MVP), so the whole shell has 78. A few other proteins and the RNA sit inside.

What vaults do is still not well understood. Mice that can’t make them live normal lives in the lab.

Vaults from viruses of bacteria. A 2026 study by a life sciences team at Anthropic (a technical report, also sent to bioRxiv) searched about 15 billion protein sequences predicted from DNA. It found MVP-like proteins in 29 phyla (big groups) of bacteria and 4 of archaea. They are rare, in under 1% of the species whose genomes have been read, and several came from phages, the viruses that infect bacteria.

The team tried making 24 of these proteins in the lab bacterium E. coli. Seven built vault-like particles, and electron cryo-microscopy (cryo-EM) confirmed four. The best-studied one comes from a phage protein called L05. Its picture is sharp to about 3.2 Å, enough to trace each protein chain.

• It has 96 copies of the protein, 48 in each half, where the mammalian vault has 78.
• It is shorter and wider, about 54 by 47 nanometers, and it encloses about 25% more volume.
• Each protein has seven repeating segments in the wall where MVP has nine. Adding two more segments made a longer vault with about 20% more room inside.
• The wall has 48 ribs, but near the tip the chains end in a three-step pattern, so the cap repeats only every third copy (16-fold) and the very tip every sixth (8-fold).

Vault proteins now turn up in animals, bacteria and archaea, the three great branches of life. That hints the design may be very old. Phages also carry genes from one microbe to another, though, so the study doesn’t settle where vaults began.

A preprint is not the final word. bioRxiv is a server where scientists share reports before other experts have checked them (peer review). The numbers above come from the report itself and from news coverage of it. They may change when the work is reviewed and published.

Barrels versus balls

A vault and a virus shell use two different kinds of symmetry.

A vault half is a ring. Spin the ring by one copy and it looks the same, so the ring has an $n$-fold axis. Turn the whole vault upside down and the two halves swap. That flip can be made about $n$ different lines through the waist. Together these $2n$ moves are the symmetry group called $D_n$ (dihedral). For the human vault, $D_{39}$ has $2 \times 39 = 78$ moves, one for each copy of MVP. The phage vault’s wall is a ring of 48.

A ring can grow one copy at a time. Add a copy and the ring gets a little wider; any number fits. That is how nature got from 39 copies to 48. An icosahedral shell can’t do that. It must jump between the allowed values of $T$: 60, 180, 240, 420, 540, 720, 780 proteins, and nothing in between.

Other shells in nature

• Ferritin stores iron in almost every living thing. It is a hollow ball about 12 nanometers across, made of 24 proteins. It has the symmetry of a cube (octahedral symmetry), which has exactly 24 turns. One ferritin can hold up to about 4,500 iron atoms.
• Clathrin is a three-legged protein (a triskelion). The legs weave into cages of hexagons and pentagons around bits of cell membrane that the cell is pulling inside.
• Buckminsterfullerene, C₆₀, is a molecule of 60 carbon atoms found in 1985 by Harold Kroto, Robert Curl and Richard Smalley, who won the 1996 Nobel Prize in Chemistry. Its atoms sit at the corners of a truncated icosahedron: 12 pentagons and 20 hexagons, the same pattern as a classic soccer ball.
• Radiolarians are single-celled sea creatures with glassy skeletons. Ernst Haeckel drew hundreds of them, many with near-perfect symmetry, in his Art Forms in Nature (1904). One he named Circogonia icosahedra.
• Geodesic domes, made famous by Buckminster Fuller in the 1950s, build a near-sphere out of triangles. Caspar and Klug said Fuller’s domes helped them see how a virus shell could be put together. C₆₀ is named after him.

There is a rule behind the number 12 that keeps turning up. Any closed cage made only of pentagons and hexagons, with three edges meeting at each corner, has exactly 12 pentagons, however many hexagons it has. That follows from Euler’s formula for polyhedra, $V - E + F = 2$. A soccer ball, a C₆₀ molecule, a clathrin cage and the pentamers of a virus shell all obey it.