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Category: GoV

GoV 2 : Viruses and quasi-crystals

If you look around for mathematical theories of the structure of viruses, you quickly end up with the work of Raidun Twarock and her group at the University of York.



We’ve seen her proposal to extend the Caspar-Klug classification of viruses. Her novel idea to distribute proteins on the viral capsid along Penrose-like tilings shouldn’t be taken too literally. The inherent aperiodic nature of Penrose tiles doesn’t go together well with perfect tilings of the sphere.

Instead, the observation that these capsid tilings resemble somewhat Penrose tilings is a side-effect of another great idea of the York group. Recently, they borrowed techniques from the theory of quasicrystals to gain insight in the inner structure of viruses, in particular on the interaction of the capsid with the genome.

By the crystallographic restriction theorem no $3$-dimensional lattice can have icosahedral symmetry. But, we can construct aperiodic structures (quasicrystals) which have local icosahedral structure, much like Penrose tilings have local $D_5$-symmetry

This is best explained by de Bruijn‘s theory of pentagrids (more on that another time). Here I’ll just mention the representation-theoretic idea.

The isometry group of the standard $5$-dimensional lattice $\mathbb{Z}^5$ is the group of all signed permutation $5 \times 5$ matrices $B_5$ (Young’s hyperoctahedral group). There are two distinct conjugacy classes of subgroups in $B_5$ isomorphic to $D_5$, one such subgroup generated by the permutation matrices
\[
x= \begin{bmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 \end{bmatrix} \qquad \text{and} \qquad
y = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \end{bmatrix} \]
The traces of $x,x^2$ and $y$, together with the character table of $D_5$ tell us that this $5$-dimensional $D_5$-representation splits as the direct sum of the trivial representation and of the two irreducible $2$-dimensional representations.
\[
\mathbb{R}^5 = A \simeq T \oplus W_1 \oplus W_2 \]
with $T = \mathbb{R} d$, $W_1 = \mathbb{R} u_1 + \mathbb{R} u_2$ and $W_2 = \mathbb{R} w_1 + \mathbb{R} w_2$ where
\[
\begin{cases}
(1,1,1,1,1)=d \\
(1,c_1,c_2,c_3,c_4)= u_1 \\
(0,s_1,s_2,s_3,s_4) = u_2 \\
(1,c_2,c_4,c1,c3)= w_1 \\
(0,s_2,s_4,s_1,s_3)= w_2
\end{cases}
\]
and $c_j=cos(2\pi j/5)$ and $s_j=sin(2 \pi/5)$. We have a $D_5$-projection
\[
\pi : A \rightarrow W_1 \quad (y_0,\dots,y_4) \mapsto \sum_{i=0}^4 y_i(c_i u_1+s_i u_2) \]
The projection maps the vertices of the $5$-dimensional hypercube to a planar configuration with $D_5$-symmetry.



de Bruijn’s results say that if we take suitable ‘windows’ of lattice-points in $\mathbb{Z}^5$ and project them via the $D_5$-equivariant map $\pi$ onto the plane, then the images of these lattice points become the vertices of a rhombic Penrose tiling (and we get all such tilings by choosing our window carefully).



This explains why Penrose tilings have a local $D_5$-symmetry. I’ll try to come back to de Bruijn’s papers in future posts.

But, let’s go back to viruses and the work of Twarock’s group using methods from quasicrystals. Such aperiodic structures with a local icosahedral symmetry can be constructed along similar lines. This time one starts with the standard $6$-dimensional lattice $\mathbb{Z^6}$ with isometry group $B_6$ (signed $6 \times 6$ permutation matrices).

This group has three conjugacy classes of subgroups isomorphic to $A_5$, but for only one of them this $6$-dimensional representation decomposes as the direct sum of the two irreducible $3$-dimensional representations of $A_5$ (the decompositions in the two other cases contain an irreducible of dimension $4$ or $5$ together with trivial factor(s)). A representant of the crystallographic relevant case is given by the signed permutation matrices
\[
x= \begin{bmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1
\end{bmatrix} \qquad \text{and} \qquad y=
\begin{bmatrix}
0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & -1 & 0 & 0
\end{bmatrix} \]

Again, using suitable windows of $\mathbb{Z}^6$-lattice points and using the $A_5$-equivariant projection to one of the two $3$-dimensional components, one obtains quasicrystals with local $A_5$-symmetry.

In this $3$-dimensional case the replacements of the thick and thin rhombi are these four parallellepipeda, known as the Amman blocks



which must be stacked together obeying the gluing condition that dots of the same colour must be adjacent.

Has anyone looked at a possible connection between the four Amman blocks (which come in pairs) and the four (paired) nucleotides in DNA? Just an idle thought…

These blocks grow into quasicrystals with local icosahedral symmetry.



The faces on the boundary of such a sphere-like quasicrystal then look a lot like a Penrose tiling.

How can we connect these group and representation-theoretic ideas to the structure of viruses? Here’s another thought-provoking proposal coming from the York group.

Take the $A_5$ subgroup of the hyperoctahedral group in six dimensiona $B_6$ generated by the above two matrices (giving a good $A_5$-equivariant projection $\pi$ to three dimensional space) and consider an intermediate group
\[
A_5 \subsetneq G \subseteq B_6 \]
Take a point in $\mathbb{R}^6$ and look at its orbit under the isometries of $G$, then all these points have the same distance from the origin in $\mathbb{R}^6$. Now, project this orbit under $\pi$ to get a collection of points in $\mathbb{R}^3$.

As $\pi$ is only $A_5$-equivariant (and not $G$-equivariant) the image points may lie in different shells from the origin. We can try to relate these shells of points to observational data on the inner structures of viruses.

Here’s a pretty convincing instance of such a correlation, taken from the thesis by Emilio Zappa “New group theoretical methods for applications in virology and quasicrystals”.



This is the inner structure of the Hepatitis B virus, showing the envelope (purple), capsid protein (cream) and genome (light blue). The coloured dots are the image points in the different shells around the origin.

Do viruses invade us from the sixth dimension??

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GoV 1 : Geometry of viruses

As you may have guessed from the symmetries of Covid-19 post, I did spend some time lately catching up with the literature on the geometric structure and symmetries of viruses. It may be fun to run a little series on this.

A virus is a parasite, so it cannot reproduce on its own and needs to invade a host cell to replicate. All information needed for this replication process is stored in a fragile DNA or RNA string, the viral genome.

This genome needs to be protected by a coating made of proteins, the viral capsid. Most viruses have an additional fatty protection layer, the envelope, decorated by virus (glyco)proteins (such as the ‘spikes’ needed to infiltrate the host cell).



Most viruses are extremely small (between 20 and 200nm), our friend the corona-virus measures between 80 and 120nm. So, its genome is also pretty small (the corona genome has around 30.000 base pairs). To maximise its information, the volume of the protective capsid must be as large as possible, and must be formed by just a few different proteins (to free as much space in the code of the genome for other operations) and clusters of them are distributed over the polyhedral capsid, as symmetric as possible.

This insight led Watson and Crick, the discoverers of the structure of DNA, to the ‘genetic economy’-proposal that most sphere-like viruses will have an icosahedral capsid because the icosahedron is the Platonic solid with the largest volume and rotational symmetry group. They argued that the capsid is most likely constructed from a single subunit (capsomere), which is repeated many times to form the protein shell.



Little is known about capsid formation, that is the process in which the capsid proteins self-assemble into an icosahedral shape, nor about the precise interplay between the genome and the capsid proteins. If we would understand these two things better it might open new possibilities for anti-viral drugs, by either blocking the self-assembly process or by breaking the genome-capsid interaction.

A first proposal for the capsid structure was put forward by Caspar and Klug. Their quasi-equivalence principle asserts that each of the 20 triangular faces of the icosahedron is subdivided in 3 subunits, each consisting of at least one protein.

Most viruses have much more than 60 proteins in their capsid, so Caspar and Klug introduced their $T$-number giving the number of proteins per subunit. One superimposes the triangulation of the icosahedron with the hexagonal plane lattice, then $T$ is the number of sub-triangles of these hexagons contained in each subunit. For $T = 7$ we have the following situation



Folding back the triangulation to form the icosahedron one then obtains a tiling consisting of hexagons (the green regions) and pentagons (the blue regions)



It turned out that many viruses with icosahedral symmetry consist of subunits having a different number of proteins, such as dimers (2 proteins), trimers (3 proteins), or pentamers (5 proteins) and these self-organise around a 2, 3, or 5-fold rotational axis of the icosahedron.

This led Reidun Twarock around 2000 to propose her virus tiling theory. This is a generalisation of the Caspar-Klug theory in which one superimposese the triangulation of the icosahedron with other tilings of the plane, consisting of two or more non-congruent tiles. Here an example which looks a bit like the aperiodic Penrose tilings of the plane.



Here’s a recent Quanta-Magazine article on Twarock’s work and potential consequences: The illuminating geometry of viruses.

And here’s an LMS Popular Lecture, from 2008, by Raidun Twarock herself: “Know your enemy – viruses under the mathematical microscope”.

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The symmetries of Covid-19

A natural question these days might be: “what are the rotational symmetries of the Covid-19 virus?”



Most illustrations show a highly symmetric object, suggesting it might have icosahedral symmetry. In fact, many viruses do have icosahedral symmetry as a result of the ‘genetic economy principle’ proposed by Watson and Crick in 1956, resulting in the Caspar-Klug classification of viral capsids.

But then, perhaps this icosahedral illusion is a result of design decisions illustrators made turning scientific data into pictures. Veronica Falconieri Hays wrote a beautiful article describing the effort going into this: How I built a 3d-model of the coronavirus for Scientific American. Here’s her final picture



And yes, icosahedral symmetry was one of her design decisions:

The M proteins form pairs, and it is estimated that there are 16–25 M proteins per spike on the surface of the virus. I ended up modeling 10 M protein pairs (so 20 M proteins) per spike in my model. Some researchers hypothesize that the M proteins form a lattice within the envelope (interacting with an underlying lattice of N proteins; see below). I decided to use an icosahedral sphere to create a regular distribution of the M protein dimers to hint at this hypothesis.

The spikes (or S-proteins) are the tentacles in these pictures, and one of the few hard figures on Corona is that ‘on average’ there are 74 of them.

This fact is enough to rule out icosahedral symmetry.

If the icosahedral rotation group (of order $60$, isomorphic to $A_5$) acts on the $74$ spikes, then each orbit consists of $60$ spikes unless that spike lies on a twofold, threefold or fivefold rotation axis, in which cases the number of spikes in its orbit are respectively $30$, $20$ or $12$. So, we can’t get a total number of $74$ spikes!

However, just looking at the number of spikes we cannot rule out octahedral symmetry!

The octahedral rotation group (of order $24$, isomorphic to $S_4$) will have orbits of size $24$ unless the spike lies on a twofold, threefold or fourfold rotation axis, giving orbits of size $12$, $8$ and $6$ respectively (the midpoints of edges, the vertices and the midpoints of faces of the octahedron), and

\[
74 = 24+24+12+8+6 \]

The most symmetric arrangement of spikes would be to subdivide each of the $8$ triangular faces of the octahedron into $6$ triangles with vertices the midpoint of the face, a vertex and a midpoint of an edge, and then to position the spikes on the axis through the vertices and midpoints of these smaller triangles.

Googling around I found very few references to symmetries of Covid-19, probably because it has an helical RNA-coil, which seems not to go well with Caspar-Klug type polyhedral viral capsids.

Here’s an exception: A structural model for the Coronavirus nucleocapsid by Federico Coscio, Alejandro D. Nadra, and Diego U. Ferreiro.



They propose a truncated octahedron as capsid (in transparent brown) with interior a continuous coil packing of blue and cyan helices. The virus membrane with the spikes and M proteins is drawn in blue.

If you have better info or references on the (conjectural) symmetries of Covid-19, please leave a comment.

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