Bacteriophage MS2 is a viral particle whose symmetrical capsid consists of 180 copies of asymmetrical coat proteins with triangulation number T = 3. The mathematical theorems in this study show that the phage particles in three-dimension (3D) might be an icosahedron, a dodecahedron, or a pentakis dodecahedron. A particle with 180 coat protein subunits and T = 3 requires some geometrical adaptations to form a stable regular polyhedron, such as an icosahedron or a dodecahedron. However, with mathematical reasons electron micrographs of the phage MS2 show that 180 coat proteins are packed in an icosahedron. The mathematical analysis of electron micrographs in this study may be a useful tool for surveying the platonic solid structure of a phage or virus particle before performing 3D reconstruction.
Many viruses and phages are icosahedral particles. However, spherical viruses may be polyhedral, forming dodecahedra or pentakis dodecahedra [
When an icosahedron is transferred to a flat sheet [
The coat protein of MS2 forms a shell that protects the phage nucleic acid and acts as a translational repressor [
The overall shape of MS2 is spherical, but is difficult to see the three-dimensional (3D) figure in electron microscopy (EM) two-dimensional (2D) images before 3D reconstruction. Therefore, this study also introduces a mathematical method to predict particle solid 3D figures with 2D EM images before performing 3D reconstruction.
The 3D structure of MS2 should be an isohedron or a regular polyhedron if the particles need to pack the asymmetrical proteins to make a stable symmetrical structure. The capsid of phage MS2 contains 180 identical copies of a coat protein with a T = 3 isohedral (such as icosahedral) shell [
Theorem 1. If a polyhedron has 180 subunits, where 180/n-polygons have n sides, n should be a positive integer and equal or less than 5.
Proof. A polyhedron is constructed with n-side polygons, the polyhedron has 180/n faces and the number of edges should be
By the Euler theorem,
A vertex of solid angle needs at least trimer, then
Thus, the maximum of n is 5, where the polygons are pentagons, quadrilaterals, or triangles.
Therefore, a polyhedron with 180 structure protein subunits should not be constructed with 30 hexagons.
Theorem 2. If a polyhedron with 180 subunits is constructed with 36 pentagons, the polyhedron should not be an isohedron or a regular polyhedron.
Proof. If 36 pentagons can construct an isohedron or a regular polyhedron, the number of faces is 36. If two faces form dihedral angles, then the number of edges = 36 × 5 ÷ 2 = 90.
By the Euler theorem,
If vertices are constructed with x n-polymers and y m-polymers, where x, y, n, and m are positive numbers, and
Since
the maximum of n is 3.
Thus,
The shell of an isohedron or a regular polyhedron consists of m-polymer units, and y m-polymers consist of 36 faces.
That is
The solution of 3 simultaneous equations
is
This is a contradiction with positive integer m.
Therefore, this theorem suggests that 36 pentagons cannot form an isohedron or a regular polyhedron.
Theorem 3. If a polyhedron with 180 subunits is constructed with an odd number of quadrilaterals, the polyhedron should not be an isohedron or a regular polyhedron.
Proof. Suppose that an isohedron or a regular polyhedron can be constructed with an odd number, 2n + 1, of quadrilaterals.
The number of faces is then 2n + 1, and the number of edges
By the Euler theorem,
The number of vertices is odd, which is a contradiction because an isohedron or a regular polyhedron is symmetrical, and the number of verticies should be even.
Thus, an odd number (such as 45) of quadrilaterals cannot form an isohedron or a regular polyhedron.
According to Theorems 1, 2, and 3, 45 quadrilaterals, 36 pentagons, or 30 hexagons (the polygons with more than 3 edges) cannot form an isohedron or a regular polyhedron with 180 subunits. Only 60 triangles can form a regular polyhedron for packing 180 subunits.
Lemma 4. If a convex solid angle is constructed with n equilateral triangles, n should be 3, 4, or 5.
Proof. In 3D models, a convex solid angle has at least by 3 faces.
For a solid angle, the total angle with 3 equilateral triangular angles is 180˚. The total angle with 4 equilateral triangular angles is 240˚. The total angle with 5 equilateral triangular angles is 300˚. The total angle with 6 equilateral triangular angles is 360˚, which is a cyclic angle.
For a convex solid angle constructed with n equilateral triangles, n should be 3, 4, or 5. The angle of a vertex equal to or greater than 360˚ is flat or concave.
Theorem 5. If a polyhedron is constructed with 60 identical equilateral triangles, the polyhedron is not an isohedron or a regular polyhedron.
Proof. If a polyhedron is constructed with 60 identical equilateral triangles, as in
Lemma 4 shows that vertex Q is not a convex solid angle. Instead, vertex Q may belong to the following:
1) Vertex Q is a concave solid angle.
This hypothesis is not true because the polyhedron is convex.
2) Point Q is at the center of hexagon
Vertex O and pentagon PQRST form a regular pentagonal cone. A dihedral angle forms between triangles OPQ and OQR on line
3) Vertex Q locates on a dihedral angle.
Vertex
Theorem 6. A polyhedron becomes an icosahedron when the solid angles of the hexamers of the polyhedron spread to be flat and the connecting lines between the vertices of 2 nearby pentagonal cones become the ridges of dihedral angles.
Proof. As
Spreading and flattening the solid angles of the hexamers in the pentagonal cone
Therefore, a polyhedron becomes an icosahedron.
Theorem 7. A polyhedron becomes a dodecahedron when the solid angles of pentamers of the polyhedron spread out and become flat.
Proof. In
and
Vertex Q becomes a convex solid angle and the polyhedron becomes a pentakis dodecahedron.
When the edge lines at vertex O decrease to allow the solid angle O to disappear and become flat, the pentagonal cone OPQRST flattens to become a pentagon PQRST.
Finally, the polyhedron becomes a dodecahedron.
Theorems 1, 2, and 3 show that 45 quadrilaterals, 36 pentagons, or polygons with 6 or more edges cannot form an isohedron or a regular polyhedron with 180 subunits. Only 60 triangles can form a regular polyhedron for packing 180 subunits. In the MS2 particle, the way to pack 180 coat proteins is to form an isohedron or a regular polyhedron with 60 identical triangles (Lemma 4). The 3D polyhedron constructed with 60 identical equilateral triangles in
The 3D structure of MS2 [
An array of hexamers is the basic unit for generating an icosahedron [
For MS2 capsids with the principal of quasi-equivalence [
This study hypothesizes that regular Platonic solids allow a single type of asymmetric subunit to assemble into a well-defined spherical structure [
With electron micrographs [
The 5 regular Platonic solids are tetrahedron, octahedron, cube, dodecahedron, and icosahedron. The icosahedron has a common symmetry with the dodecahedron, and the octahedron is similar to the cube [
The mathematical reason from Theorem 6 and 7 shows that the T = 3 phage MS2 particles may be icosahedral, dodecahedral, or pentakis dodecahedral particles. The projections of icosahedral particles in phage MS2 EM images exhibit unequilateral hexagons in twofold views, regular hexagons in threefold views, and regular decagons in fivefold views (
regular decagon. In threefold views, 3 models have different projections: regular hexagons form from icosahedra, hexagon-like dodecagons form from dodecahedra, and regular decagons form from pentakis dodecahedra. In twofold views, the projections of icosahedra and dodecahedra are unequilateral hexagons (
Although the 3D structure reconstructed from the EM images shows the Platonic solids of the particles, the wrong order of 3D reconstruction might yield a false solid figure [
Symmetry | Asymmetry | |||
---|---|---|---|---|
Twofold | Threefold | Fivefold | ||
Icosahedron | Unequilateral hexagon | Regular hexagon | Regular decagon | -- |
Dodecahedron | Unequilateral hexagon | Hexagon-like dodecagon | Regular decagon | (two forms) |
Pentakis dodecahedron | regular decagon | Regular decagon | Regular decagon | -- |
primary survey of the Platonic solid of the particle, the 3D reconstruction can be performed confidently with a right symmetrical order [
In conclusion, a viral particle with 180 coat protein subunits and T = 3 requires some geometrical adaptation to form a stable regular polyhedron, such as an icosahedron or a dodecahedron. The mathematical analysis of MS2 particles reveals the EM projections of icosahedral particles. The MS2 particles are confirmed to be icosahedra.
We gratefully acknowledge Mr. Jaw-Fu Gau for his EM assistance.