1 on the Inside Structures of Virus Capsids

On the inside structures of virus capsids. (25 nov 2009)

Sten Andersson Sandforsk, Sandvik, Södra Långgatan 27, S-38074 Löttorp, Sweden

‘The growth cycle of poliovirus is extremely brief and extraordinarily efficient, the entire process is complete within 8 h, and yields in excess of 100,000 particles per cell are not uncommon’ (ref 1).

Abstract The insides of capsids-the genome region-of the Pariacoto insect virus (ref 2), the Noda fish virus(ref 3) and the AAV2 virus(ref 4) have structures that are analyzed with our exponential mathematics. We conclude that these three different viruses have structures in the genome regions we describe mathematically as the inside geometrical part of the functions of the outside shapes of capsid or spike structures. A simple example is the Pariacoto virus structure we show has a direct mathematical connection between spikes and RNA duplex.

Introduction

We show some polyhedra to use, partly in Bulatov fashion(ref 5) in d, i-l.

Fig 1a Dodecahedron b Icosidodecahedron c Truncated icosahedron d pentakisdodecahedron

e Tri-pent-hex II f Rhombicosidodecahedron g Great rhombicosidodecahedon

i Third stellation j Small stellated dodecahedron k Small icosihemidodecahedron l Triacontahedron of icosahedron

Mathematics to use (ref 6-12)

Equation of dodecahedral symmetry.

p "(x+# y"n)p "(x"# y"n) "(y+# z"n)p "(y"# z"n)p "(z+# x"n)p "(z"# x"n)p Sum[e + e + e + e + e + e , Eq 1 {n, m#, - m#, " #}]" const = 0

Structures after eq 1.

Fig 2a m=2, p=2,const=6.41 b m=2, p=4, const=6.45 c n,3,-3,-1, p=2 Icosidodecahedron Truncated icosahedron Triacontahedron const=5.315.

d Inside of icosidodecahedron e Inside of truncated icosahedron. const= 6.45 and 6.35 resp.

f m=3 p=2, const= 6.55 g Tri-pent-hex II

Equation of icosahedral symmetry

"(x+# 2 y"n)6 "(-x+# 2 y"n)6 "(y+# 2 z"n)6 "(-y+# 2 z"n)6 "(z+# 2 x"n)6 "(-z+# 2 x"n)6 Sum[e + e + e + e + e + e +

"(# (x+ y+z)"n)6 "(# (x" y-z)"n)6 "(# (-x" y+z)"n)6 "(# (-x+ y-z)"n)6 e + e + e + e , Eq 2 {n, 2#, - 2#, " #}]" const = 0

The polyhedron is the great rhombicosidodecahedron given below in fig 3a-d. !

Fig 3a After eq 2, Const=11.15. b Great rhombicosidodecahedron in green

c Rhombicosidodecahedron in yellow d The two chiral snub dodecahedra

At least four different structures are easily derived from the inside and shown in fig 4a-d

Fig 4a Icosidodecahedron b Trunc triacontahedron c Trunc icosahedron d Rhombicosidodecahedron

Virus structures to discuss

Asymmetric units

Fig 5a Nodavirus b Pariacoto c Poliovirus type 1 d AAV2 Max diam. 380 Å 360 Å 324 Å 295 Å

The Pariacoto insect virus, the Noda fish virus and the Poliovirus have similar capsid structures as seen in fig 5a-c. The spike structures from the truncated icosahedron (eq 1) are almost identical. The AAV2 structure from Böttcher (ref 4) in 5d is similar – it stimulated us to derive the mathematics for left and right in life (ref 6).

We describe the inside structures and start with Pariacoto. The dodecahedron in 6a and b is built of 30 copies of the RNA duplex fragment, which account for 35 % of the total RNA in the capsid. Mathematically this is shown with eq 1 to be the inside structure of a truncated icosahedron(the spike structure of the capsid) in fig 2e. We use a non convex polyhedron-the third stellation of the icosahedron-after Bulatov(ref 5) in fig 6c to locate 60 more copies of duplex RNA fragments that is required. In fig 6a,b,d this is done with red lines. The blue lines give the dodecahedron with the 30 copies of RNA fragments.

Fig6a b c d Pariacoto insect virus b RNA duplex c Third_stellation_of_icosahedron, d Blue rods as in b. Red rods hypothetical rest RNA

A capping of the Bulatov polyhedron gives fig 6e. The pariacoto capsid structure is also described with eq 2, or icosahedral geometry. In fig 4d there is an inside structure of the rhombicosidodecahedron with deviating symmetry as in fig 6h here. The globules in 6b are described as organized as a truncation in red of the small stellated dodecahedron in figs 6g-h. All polyhedra in 6c-h after Bulatov,

Fig 6e f g h Capping of third_stellation_ The small stellated dodecahedron Truncation to give red net in 6b of_icosahedron in e gives the triacotrahedron in f

The MGNNV virus

The white net in 7b connects the black circles in fish noda virus in 7a and is described by the new Tri-pent-hex II polyhedron in 8a and from fig 1f. Pictures are given in fig 2f-g as calculated from eq 1. Example of this structure type is given by the native calicivirus structure (ref 13) in fig 8b-c. Blue circles in 8c are black holes in fig 7a. The black holes are a part of the white solid circle (representing a sphere of course) just inside the two capsids layers. This white sphere contains also holes in some of the hexagons and pentagons (special positions marked in red in 8b) of the white net construction. If the spikes in the capsid structure are paired they describe these special positions of the inner white sphere. So this ‘inner capsid structure’ of holes is really an icosidododecahedron.

Fig 7 a Noda fish virus(Tang). b The net, inside out 6

Fig 8 a Tri-pent-hex II b Native calicivirus c Blue circles are black holes in fig 7a

A way to make a simple net is using the hexagons and pentagons as done in fig 9a. The polyhedron is the truncated icosahedron, which has 90 edges, but also 60 corners and 12 + 20 faces. In fig 9b we have marked the faces and the corners and we have 92 places with solid electron densities. We propose the duplex RNA fragments are anchored at the 92 positions and are orientated towards the center of the capsid. Before the center there is the inside of the truncated icosahedron, the icosidodechedron, calculated as given in fig 2e. The relation between the truncated icosahedron with its 92 position (60 corners and 20+12 faces, shown in 9b) and the icosidodechedron also with 92 positions (60 edges and 20+12 faces) is shown in fig 9c.

Fig 9 a b c The truncated icosahedron,60 corners+ Icosidodecahedron inside 12+20 faces=92 possible RNA fragments trunc icosahedron

White circles in 9c, originally at centers of edges, have moved slightly to give regular structure. Interesting here is to describe the MGNNV structure as a capsid from fig 5a of the rhombicosidodecahedron structure type, here in fig 9d. Its geometry is from fig 3c here given in 9 e, with the inside in 4 c, which is this very truncated icosahedron. Here in 9f and g.

Fig 9d e f g,9a repeated h Icosidodecahedron inside trunc icosahedron

From g we do the inside of a truncated icosahedron as above.

The AAV2 virus

We have earlier (ref 6) described the relationship between the great rhombicosidodecahedron with its two chiral snub dodecahedra (here in fig 3d) and the AAV2 capsid structure as also shown here in fig 10 a,b,c.

Fig 10 a b c AAV2 virus Great rhombicosidodecahedron Its inside The two chiral snub dodecahedra

After Böttcher (ref 4) there are globules that build the AAV2 inside net in fig 11a. The pattern given we describe with the icosidodecahedron in fig 11b. The globules are supposed to be in intimate contact with the genome.

Fig 11 a AAV2 capsid b Icosidodecahedron in white. c The small icosihemidodecahedron. After Böttcher (ref 4) the difference map (blue structure) is superimposed to the map of the empty capsid (faint yellow structure). In fig 11c we describe the non-convex polyhedron, the small icosihemidodecahedron, as an obvious description from Bulatov’s picture. The number of vertices is 30, which are available for duplex RNA fragments. To this we add 12 centers of pentagons. 8

The icosahedral equation (Eq 2 above) gives fig 10 a, which also describes the capsid of AAV2. The mathematical inside of the function as shown in fig 4 a, and here in 10 b, gives also the Böttcher inside of the capsid – the icosidodecahedron.

Important structural relationships

The remarkable and simple relationships between the three virus structures as described below in fig 12 are direct and also via the dual forms:

The dual of the AAV-2 inside structure is the Pariacoto inside structure, which is directly related with the dual of the MGNNV inside structure, the pentakisdodecahedron. icosidodecahedron rhombic triacontahedron

dual rhombic dual icosidodecahedron dodecahedron added in blue triacontahedron Fig 12 AAV-2 Pariacoto

Pentakisdodecahedron Truncated icosahedron

dual truncated icosahedron dual pentakisdodecahedron

MGNNV

The frames of replication

We have constructed structures for insides of viruses using RNA duplex fragments as found for the pariacoto structure.

In the first case in fig13a the blue RNA dodecahedron with a structure of the third stellation of the icosahedron build the frame. 90 possible RNA duplex fragments.

In the second case in 13b a truncated icosahedron in white net with blue and yellow globules build a space. 92 possible RNA duplex fragments. 9

In the third case in 13c globules are organized in the corners of an icosidodecahedron with a structure between for the genome. 30 + 12(centers of pentagons) possible RNA duplex fragments.

Fig 13a Pariacoto virus b MGNNV virus c AAV 2

We say the genome is organized as a structure, or connected with a structure. When entering a cell, the capsids break to release the frames, which enter the ribosome where they are turned into RNA duplex strings. And RNA string replication results in more RNA fragments, frames, capsids and strings, and eventually the cell has transformed into a great number virus particles.

References

1 Geoffrey, BIOCHEMISTRY Zubay, Addison-Wesley, 1986. Page 1085.

2 Liang Tang, K. N. Johnson, L. A. Ball, T. Lin, M. Yeager, and J. E. Johnson. 2001. The structure of Pariacoto virus reveals a dodecahedral cage of duplex RNA. Nat. Struct. Biol. 8:77

3 Liang Tang,1 Chan-Shing Lin,2 Neel K. Krishna,1 Mark Yeager,3,4 Anette Schneemann,1 and John E. Johnson. Virus-Like Particles of a Fish Nodavirus Display a Capsid Subunit Domain Organization. Journal of Virology, June 2002, p. 6370-6375, Vol. 76, No. 12

4 B Böttcher. EMBL Electron microscopy and image processing of large biological complexes. Research Reports 2004, p 137 - 140.

5 Vladimir Bulatov, http://bulatov.org/polyhedra/

6 S. Andersson, Zeit Anorg Allg Chem. 2008,634, 2161. The structure of virus capsids. Part I.

7 S. Andersson, Zeit Anorg Allg Chem. 2008,634, 2504. The structure of virus capsids. Part II.

8 S. Andersson, Zeit Anorg Allg Chem. 2009, 635, 717. Virus Evolution and the Beginning. 10

9 S. Andersson, Zeit Anorg Allg Chem. 2009, 635, 725. Virus Structure. Stellations, Spikes, and Rods.

10 S. Andersson, Sandforsk.se 2009. Fundamental structure in virus evolution

11 S. Andersson, Sandforsk.se 2009 . General polyhedra, virus structure and mutation.

12 S Andersson, Sandforsk.se 2009 Mathematics and influenza virus capsids

13 Rong Chen , John D. Neill, Mary K. Estes, and B. V. Venkataram Prasad, PNAS , 2006, 103, 8048. X-ray structure of a native calicivirus: Structural insights into antigenic diversity and host specificity.