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General polyhedra, virus structure and mutation. Sten Andersson Sandforsk, Institute of Sandvik, Södra Långgatan 27, S-38074 Löttorp, Sweden Abstract We describe how mutations can be organized as truncation and chiral operations in pentagonal space. This gives an unlimited number of possible variations and serves as a mathematical description of a new concept of polyhedra. The HK97 structure is n= 8 in the Blue tongue series. Introduction and general polyhedra We may consider two types of mutations in the evolution of virus structure. The first type has resulted in the differentiation into the various capsid polyhedral shapes. The second type has resulted in outside spike structures for recognition of receptors outside host cells. All mutations are geometrically connected, however, full filling the requirements of the pentagonal symmetry. Different shapes are therefore regarded here from such a geometric mutation perspective. From earlier work on biomathematics(1) and on description of virus structures (2,3) it became clear that Nature from evolution had developed an extended concept of polyhedra. We use truncation and chiral operations to describe this and arrive with the general polyhedra and give examples below. The Hardy deviation is an alternative description that also is used.

a b Fig 1 Hardy-deviated truncated icosahedra

c Pariacoto

d e f Hardy-deviated great rhombicosidodecahedra and snub dodecahedra are examples of general polyhedra and compared with two simple viruses in d – i above and below.

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g Hardy-deviated snub dodecahedra h Satellite Panicum Mosaic (i) Porcine Parvovirus [12]

Two kinds of truncations of Archimedes polyhedra are used in the description of variations in virus structures. One is the corner cutting and the other is the edge cutting. We show them in fig 2a-b. The simple deviations that occur at the truncations of the Platon polyhedra to the icosidodecahedron, the truncated dodecahedron, the rhombicosidodecahedron, and the truncated icosahedron are examples. Another example is the split up of the 6-fold and 10-fold axes into 2 interpenetrating 3-fold, and 2 interpenetrating 5-fold axes of rotation as shown in fig 1a-i. To this we add the very important variations in structure between the rhombicosidodecahedron and the two chiral forms of snub dodecahedron type as described with chiral motions below. These general polyhedra we describe as variations of the Archimedean polyhedra. They constitute the concept of a mathematical description of an unlimited number of mutations. Truncation and chiral motion in the bilateral case with three molecules in the asymmetric unit. From the two truncations in fig 2a and b we continue with the truncation in fig2c of the icosahedron(white) in green net. If the green pentagon in c is rotated with size change anti clock wise and slipping on the red pentagon we can derive the yellow net in d. This is the snub dodecahedral unit as shown above. By rotating the green net clockwise the blue net in 2e is obtained. These are really chiral motions of a virus particle under expansion. We have the great rhombicosidodecahedron drawn in white in 2f and we realize the green truncated icosahedron is the ‘mirror’ between the two chiral forms of the snub dodecahedron. In g there is the Pariacoto virus that also with the advanced spike structure (green truncated icosahedron) is described by the mentioned polyhedra.

Fig 2a

b

c

d

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e

f

g

We show in figs 2h and i a chiral mechanism for formation of structures that can be understood as mutations. In blue and yellow are chiral variants as above of Hardy deviated snub dodecahedra. In green there is the transition structure (truncated icosahedron) between the two forms.

h

i

Of course the structures above are also to be compared as complete polyhedral geometries. Below we show how this is developed to a mutation in black lines with the Black Beatle virus.

Mutation(in black) in Black Beatle

4 Four and seven molecules in the asymmetric unit. Nudaurelia and HK 97. For the Nudaurelia(ref 4) there are three different snub dodecahedral units as shown in figs3a-b. The drawing in white net in Fig 3c is constructed from combining the three proteins in the ASU and represent the surface called the Sinbid or Semliki surface (ref 2). We realize that small rotations in each one of the three snub dodecahedral units in white, black and blue, in different directions and combinations, would multiply into many varieties. So the chiral mechanism as described above gives a very great number of structures that mimic mutations. And always fulfilling conditions of symmetry. An example is given with a chiral motion in red in fig 3a how a chemical change in a protein molecule will spread over the entire capsid as a mutation.

Fig3a (mutation in red)

b

c

The patterns for viruses like Bacteriophage alfa-3 and Simian with four and five proteins in the ASU give similar pattern. With the HK97 with six proteins in the ASU we approach the real big ones and can discuss the manifolds of mutations of the big virus structures as shown below. This HK 97 virus structure (ref 5) is described as built of 5 snub dodecahedral units in five different nets in red blue, white, pink and yellow, and the truncated icosahedron in the sixth net in green. All in figs 4a and b. The truncated icosahedron is included in snub dodecahedral concept as the intermediate of zero chirality. Again the possible variations that keep symmetry are the rotations around one of the symmetry axes, and volume changes.

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Fig 4a

b

Chiral operations as demonstrated above with the snub dodecahedral case are becoming very rich when the number of molecules in the asymmetric unit is growing. We may have a good way for describing mutations but we are deep in trouble with the description of bigger capsids by using the asymmetric units in that way. The HK97 may be the watershed. The ASU:s for HK97 and Archaeal (ref 6) are shown in figs 5a and b. We show below a complete descripton of the structure of HK97 with the hexagon asymmetric unit as line shadowed in fig 5d. And we see empty hexagons just confirming HK97 is member number 8 in the Blue tongue series. We show the Archaeal virus, one of the biggest with n=10 in the same series in 5c(ref 6). So there are good structure descriptions. But if we do the detailed plot with the snub dodecahedral block unit for Archaeal as we have done for HK97 and Nudarelia, we realize the immense number of possible mutations for Archaeal. We believe we need both for a deeper understanding of the utterly advanced structure of the 13 different snub dodecahedral units that build the complete structure of 13 different nets. In order to understand the manifold of possible mutations it is enough to change one of the proteins in the HK97 hexagon, and one of the proteins in each of the five hexagons in Archeael. Doing that systematically we have different ways to estimate the manifolds of mutations. We have marked the changes in the ASU:s with black lines respectively in figs 5 a,b,c,d.

Archaeal ASU Fig 5a

HK97 ASU b

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Fig 5c Archaeal

d HK97

Small variations in structure of the proteins of the hexagon as the asymmetric unit we say give mutation. It is the same mechanism as given for HK97 in fig 4, but a lot simpler, and necessary when going to discuss mutations in the big virus structures. Acknowledgment. I thank Kåre Larsson. Without his help this article had not been written. For all the information used above we acknowledge the very remarkable Viper database: VIPERdb2: an enhanced and web API enabled relational database for structural virology. Mauricio Carrillo-Tripp, Craig M. Shepherd, Ian A. Borelli, Sangita Venkataraman, Gabriel Lander, Padmaja Natarajan, John E. Johnson, Charles L. Brooks, III and Vijay S. Reddy Nucleic Acid Research 37, D436-D442 (2009); doi: 10.1093/nar/gkn840 References 1 Sten Andersson, Kåre Larsson, Marcus Larsson, and Michael Jacob; BIOMATHEMATICS, Elsevier, Amsterdam, 1999 2 Sten Andersson, Zeit Anorg Allg Chem. 2008,634, 2161. The structure of virus capsids. Part I. 3 Sten Andersson, Zeit Anorg Allg Chem. 2008,634, 2504. The structure of virus capsids. Part II.

4 Helgstrand, C., Munshi, S., Johnson, J.E. & Liljas, L. Virology (2004) 318:192.

7 The Refined Structure of Nudaurelia Capensis Omega Virus Reveals Control Elements for a T = 4 Capsid Maturation 5 Wikoff, W.R., Liljas, L., Duda, R.L., Tsuruta, H., Hendrix, R.W. & Johnson, J.E. Science (2000) 289:2129. Topologically Linked Protein Rings in the Bacteriophage HK97 Capsid. 6 George Rice, Liang Tang, Kenneth Stedman, Francisco Roberto, Josh Spuhler, Eric Gillitzer, John E. Johnson, Trevor Douglas, and Mark Young Proc Natl Acad Sci U S A. 2004 May 18; 101(20): 7716–7720 The structure of a thermophilic archaeal virus shows a double-stranded DNA viral capsid type that spans all domains of life.