# Triangulation number of the SARS CoV-19 virus

Is the triangulation number of the SARS CoV-19 virus capsid, in the sense of the Caspar-Klug theory, known? In case the Caspar-Klug theory does not apply to it, is it known what is its tiling, in the sense of the viral tiling theory of Twarock (and coauthors)? I am a mathematician, trying to learn the geometric and group-theoretic aspects of virus structures.

Coronaviruses, like many viruses with a lipid membrane, are pleomorphic. The individual particles don't have a consistent shape. Here is a cryo-electron micrograph of SARS-CoV-1:

It might not be obvious, but each particle seen here is unique, they are not different rotations of identical copies. Since there is no consistent symmetry, there can be no triangulation number.

The image is from the following reference: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1563832/

• Thank you very much for your explanation. It is very helpful for me!
– user59912
May 19, 2020 at 11:14

After reading just about every paper on SARS-CoV-2 structure I could find, I think the answer is a bit more nuanced. The coronavirus capsid is probably somewhat regular, but not perfectly regular. Calling it "pleomorphic" is probably more confusing than descriptive here - it's not that the virus can be produced in several distinct forms by cells, but rather it can rearrange itself (eg under mechanical forces), and it can also be damaged during preparation of samples for microscopy which can create the appearance of having different forms. Carefully collected SEM images show a very uniform and narrow distribution of particle sizes and shapes.

There are a couple of bits of information which may help:

1. A broken open virus releases an apparently helical strand of RNA+N protein which is about 15nm in diameter and very long, and
2. There are some regular arrangements of N proteins visible inside intact particles by cryo-EM tomography, typically a hexagon plus one protein in the middle (these are also 15nm in diameter) (Source: Yao 2020, Molecular Architecture of the SARS-CoV-2 Virus)

The most likely way to combine those two observations is that the strand folds itself into a hollow sphere, perhaps in a way similar to examples from other enveloped viruses (eg PDB ID 3J3Q). Looking at a snapshot at any point in time, the strand winds around such that most of the copies of N form a lattice, although the lattice may be distorted, and probably has a few places where the spacing changes and/or there are defects in the lattice; but also this may not be a fixed arrangement over time.