Triangulation Number
Icosahedral virus capsids are typically assigned a triangulation number (T-number) to describe the relation between the number of pentagons and hexagons i.e. their quasi-symmetry in the capsid shell. The T-number idea was originally developed to explain the quasi-symmetry by Caspar and Klug in 1962.
For example, a purely dodecahedral virus has a T-number of 1 (usually written, T=1) and a truncated icosahedron is assigned T=3. The T-number is calculated by (1) applying a grid to the surface of the virus with coordinates h and k, (2) counting the number of steps between successive pentagons on the virus surface, (3) applying the formula:
- =
where and h and k are the distances between the successive pentagons on the virus surface for each axis (see figure on right). The larger the T-number the more hexagons are present relative to the pentagons.
| capsid parameters | hexagon/pentagon system | triangle system | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| (h,k) | T | # hex | Conway notation | image | geometric name | # tri | Conway notation | image | geometric name | |
| (1,0) | 1 | 0 | D | Dodecahedron | 20 | I | Icosahedron | |||
| (1,1) | 3 | 20 | tI dkD |
Truncated icosahedron | 60 | kD | Pentakis dodecahedron | |||
| (2,0) | 4 | 30 | cD=t5daD | Truncated rhombic triacontahedron | 80 | k5aD | Pentakis icosidodecahedron | |||
| (2,1) | 7 | 60 | dk5sD | Truncated pentagonal hexecontahedron | 140 | k5sD | Pentakis snub dodecahedron | |||
| (3,0) | 9 | 80 | dktI | Hexapentatruncated pentakis dodecahedron | 180 | ktI | Hexapentakis truncated icosahedron | |||
| (2,2) | 12 | 110 | dkt5daD | 240 | kt5daD | Hexapentakis truncated rhombic triacontahedron | ||||
| (3,1) | 13 | 120 | 260 | |||||||
| (4,0) | 16 | 150 | ccD | 320 | dccD | |||||
| (3,2) | 19 | 180 | 380 | |||||||
| (4,1) | 21 | 200 | dk5k6stI tk5sD |
420 | k5k6stI kdk5sD |
Hexapentakis snub truncated icosahedron | ||||
| (5,0) | 25 | 240 | 500 | |||||||
| (3,3) | 27 | 260 | tktI | 540 | kdktI | |||||
| (4,2) | 28 | |||||||||
| (5,1) | 31 | |||||||||
| (6,0) | 36 | 350 | tkt5daD | 720 | kdkt5daD | |||||
| (4,3) | 37 | |||||||||
| (5,2) | 39 | |||||||||
| (6,1) | 43 | |||||||||
| (4,4) | 48 | 470 | dadkt5daD | 960 | k5k6akdk5aD | |||||
| (6,2) | 48 | |||||||||
| (5,3) | 49 | |||||||||
| (5,4) | 61 | |||||||||
| (6,3) | 64 | |||||||||
| (5,5) | 75 | |||||||||
| (6,4) | 76 | |||||||||
| (6,5) | 91 | |||||||||
| (6,6) | 108 | |||||||||
| ... | ||||||||||
T-numbers can be represented in different ways, for example T=1 can only be represented as a icosahedron or a dodecahedron and, depending on the type of quasi-symmetry, T=3 can be presented as a truncated dodecahedron, an icosidodecahedron, or a truncated icosahedron and their respective duals a triakis icosahedron, a rhombic triacontahedron, or a pentakis dodecahedron.
Read more about this topic: Capsid
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