More than a quarter billion people today are infected with the hepatitis B virus (HBV), the World Health Organization estimates, and more than 850,000 of them die every year as a result. Although an effective and inexpensive vaccine can prevent infections, the virus, a major culprit in liver disease, is still easily passed from infected mothers to their newborns at birth, and the medical community remains strongly interested in finding better ways to combat HBV and its chronic effects. It was therefore notable last month when Reidun Twarock, a mathematician at the University of York in England, together with Peter Stockley, a professor of biological chemistry at the University of Leeds, and their respective colleagues, published their insights into how HBV assembles itself. That knowledge, they hoped, might eventually be turned against the virus.

Their accomplishment has gained further attention because only this past February the teams also announced a similar discovery about the self-assembly of a virus related to the common cold. In fact, in recent years, Twarock, Stockley and other mathematicians have helped reveal the assembly secrets of a variety of viruses, even though that problem had seemed forbiddingly difficult not long before.

Their success represents a triumph in applying mathematical principles to the understanding of biological entities. It may also eventually help to revolutionize the prevention and treatment of viral diseases in general by opening up a new, potentially safer way to develop vaccines and antivirals.

A Geodesic Insight

In 1962, the biologist-chemist duo Donald Caspar and Aaron Klug published a seminal paper on the structural organization of viruses. Among a series of sketches, models and X-ray diffraction patterns that the paper featured was a photograph of a building designed by Richard Buckminster Fuller, the inventor and architect: It was a geodesic dome, the design for which Fuller would become famous. And it was, in part, the lattice structure of the geodesic dome, a convex polyhedron assembled from hexagons and pentagons, themselves divided into triangles, that would inspire Caspar and Klug’s theory.

At the same time that Fuller was promoting the advantages of his domes—namely, that their structure made them more stable and efficient than other shapes—Caspar and Klug were trying to solve a structural problem in virology that had already attracted some of the field’s greats, not least among them James Watson, Francis Crick and Rosalind Franklin. Viruses consist of a short string of DNA or RNA packaged in a protein shell called a capsid, which protects the genomic material and facilitates its insertion into a host cell. Of course, the genomic material has to encode for the formation of such a capsid, and longer strands of DNA or RNA require larger capsids to shield them. It didn’t seem possible that strands as short as those found in viruses could achieve this.

Then, in 1956, three years after their work on DNA’s double helix, Watson and Crick came up with a plausible explanation. A viral genome could include instructions for only a limited number of distinct capsid proteins, which meant that in all likelihood viral capsids were symmetric: The genomic material needed to describe only some small subsection of the capsid and then give orders for it to be repeated in a symmetric pattern. Experiments using X-ray diffraction and electron microscopes revealed that this was indeed the case, making it apparent that viruses were predominantly either helical or icosahedral in shape. The former were rod-shaped structures that resembled an ear of corn, the latter polyhedra that approximated the sphere, consisting of 20 triangular faces glued together.