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Home , Icosidodecahedron, Rhombicosidodecahedron, Truncated dodecahedron, Truncated icosahedron

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Mathematics, mixed polyhedra and virus structures

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

Abstract We propose the new concept of mixed polyhedra. With Hardy deviations, and using only two simple symmetry codes we describe the pentagonal virus structures. A vast amount of hypothetical capsid structures can be derived.

Introduction

We continue to describe how viruses are built in terms of natural science and mathematics. We tried first to make the concept grow – it was soon exhausted. We tried established concepts like repeated structure building blocks - asymmetric units – or simply using nets. These approaches work but are not good enough.

Our choice of method became abstract but truly exact - the practical use of symmetry codes from the mathematics of the pentagonal space will now be explored. We start with a simple example.

The pentagonal structure in fig 1a after eq 1 we have described earlier as a new polyhedron (ref 1,2). But as the has two different corners we now draw two different structures in fig 1b. One describes the and the other the .

Such mixed polyhedra are a direct continuation of the greek concept. And indeed this gives an accurate description to Bacteriophage GA (ref 4b) in fig 1c.

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

m=3, p=2, cons =6.55 !

Fig 1a Two particles b Mixed polyhedron c Bacteriophage GA 2

Classification of polyhedra

Solids change in a uniform way.

These solids are the , the , and the icosidodecahedron, all shown in fig 2.

Fig 2a b c

Dodecahedron Icosahedron Icosidodecahedron

Solids change volume in a non-uniform way.

These solids are the , the , the rhombicosidodecahedron, the dodecahedra and the great rhombicosidodecahedron. These solids change volume in a non-uniform way with Hardy deviations (ref 1) as identical polygons need not be in contact. If the in the truncated icosahedron change size the become trigonal in symmetry. Similarly the can change size in the truncated dodecahedron. We simply say that the Hardy deviations can be described as expanding, or contraction, and combined with rotation. These are very useful operations in the description of many virus .

Fig 3a b c Truncated dodecahedron Truncated icosahedron Rhombicosidodecahedron

Fig 3d e f The two snub dodecahedra The great rhombicosidodecahedron

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If the two snub dodecahedra are added the great rhombicosidodecahedron (to right) is obtained. So the great rhombicosidodecahedron is a mixed polyhedron as well. If the pentagons in one snub are rotated the rhombicosidodecahedron is obtained (no picture).

The structure codes

We give below the two important structure codes in exponential form that give the shapes for the dodecahedron and the icosahedron in eqs 2 and 3(ref 1).

(x+" y) (x#" y) (y+" z) (y#" z) (z+" x) (z#" x) e + e + e + e + e + e dodecahedron 2

(x+" 2 y) (-x+" 2 y) (y+" 2 z) (-y+" 2 z) (z+" 2 x) (-z+" 2 x) e + e + e + e + e + e + icosahedron 3 ! ("(x+ y+z)) ("(x# y-z)) ("(-x# y+z)) ("(-x+ y-z)) e + e + e + e

We shall use the structure codes we find for the polyhedra in fig 3. We find we can describe the simplest to the very complicated virus structures using these codes given in ! fig 4.

Icosahedron Dodecahedron Icosidodecahedron Fig 4a b c

Fig 4d e f g h Truncated dod Truncated icos Rombicosidodecahedron The two snub dodecahedra

The five symmetry codes in figs 4 d-h contain possible Hardy deviations. The truncated dodecahedron code in fig 4d can change the distance between triangles, which means contraction or . Four of them can be transformed into each via Hardy rotaion. The pentagons in the code in fig 4i can rotate as described towards the structures of the rombicosidodecahedron and the truncated icosahedron.

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Fig 4i

In the extension this means that all virus capsid structures of pentagonal symmetry can be described by the truncated dodecahedron and the snub dodecahedron as given in fig 4j. With proper Hardy deviations.

Fig 4j The two fundamental symmetry codes for pentagonal symmetry. The Hardy deviations are obvious.

The mixed polyhedra and virus structures

BPMV and the great rhombicosidodecahedron

We explained the principle of mixed polyhedra in the introduction and continue here with the great rhombicosidodecahedron that gives the BPMV virus.

Fig 5 The great rhombicosidodecahedron

Many different viruses (about 50, see Viper virus capsid) have structures that can be related with Hardy deviations. Biology makes the virus with two different proteins - red and green – obviously this is a picture of life and we give a mathematical version from eq 4 in fig 6a. 5

"(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 , 4 {n, 2#, - 2#, "#}] " const = 0

The two chiral forms of the snub dodecahedron in fig 6a join to describe the rhombicosidodecahedron of BPMV in fig 6 b. Simple Hardy deviations are obvious. !

Fig 6a const=11.15 p=6, b BPMV We continue with the analysis of the Pariacoto structure, for which we use the symmetry codes below:

Fig 7

The spike structure is given by equation 5.

p p p p "(x+# y"n) "(x"# y"n) "(y+# z"n) "(y"# z"n) "(z+# x"n)p "(z"# x"n)p Sum[e + e + e + e + e + e , 5 {n, m#, - m#, "#}]" const = 0

m=2,p=4. Const = 6.45 in original calc.

! The three structure codes are given yellow, green and red. 6

fig 8a Spike structure Pariacoto(4) b Structure codes c Calculated spike strucure

Simple Hardy translation making the pentagons change size in fig 8c and gives two sets of edges to the truncated icosahedron. All this has been thoroughly discussed earlier (ref 1).

Nudaurelia (ref 4a)

Fig 9a b c In yellow, green and red as in fig 10b.

We use the Nudaurelia structure to stress the similarity with BPMV. The yellow proteins in fig 10a and b are extra as represented by the code in fig 9a. Equation 6 with picture in 10c is also for the structures of Sinbid and Semliki.

"(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 , 6 {n, 2#, - 2#, "#}] = 10.1

!

Fig 10a b c

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Fig 10 d Semliki e Sinbid

PRD 1 (ref 7)

p p p p "(x+# y"n) "(x"# y"n) "(y+# z"n) "(y"# z"n) "(z+# x"n)p "(z"# x"n)p Sum[e + e + e + e + e + e , 7 {n, m#, - m#, "#}]" const = 0

!

Fig 11a b c d m=4,p=4, cons=6.5 e

Symmetry code in 11a-c.

The simple calculated structure after equation 7 is in perfect agreement with the observed PRD 1 in figs 11 d and e.

Bacteriophage alfa 3(ref 9)

The structure code is below in 12.

Fig 12 Bacteriophage alpha 3

Bacteriophage alpha 3 has an extra TD structure cod (see below) as compared with PRD. 8

Simian (ref 11)

This structure has been given a different description earlier (ref 1,2). Here we give each particle in the asymmetric unit, following from above, its own symmetry code with proper coloring. A simple description of a complicated structure.

Fig 13a

Fig 13b Simian c Asymmetric unit

KH 97 (ref 12)

In this complicated structure as shown in 14 b,c, there are six particles in the asymmetric unit in 14 d, and they give the coloring to their structure codes in fig 14 a.

Fig 14a

Fig 14b KH 97 c small holes d Asymmetric unit 9

KH 97 small holes

The KH 97 structure has spikes with small holes in the tips. We have derived the codes for these holes in fig 14e, and drawn them into fig 14c(above).

Fig 14e

Human rhinovirus

Viper pictures of Human rhinovirus in 15b-c with the tips of the spikes in red give one snub dodecahedrahon, while the roots of the spikes give a truncated icosahedron in yellow. Symmetry code in fig 15 a.

Fig 15a b c

Human hepatitis (ref 10)

The structure code is below in 16 a,b

Fig 16a b c d

Fig 16c Human hepatitis Net very similar to the simple part of the Herpes net. Pariacoto to right in d.

Human hepatitis in 16c has a beautiful structure with a simple structure code. We compare with Herpes as a mixed polyhedron with an extra icosidodecahedron in fig 18. We also compare with Pariaoto in d.

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Blue tongue

A series of important viruses were organized with a hexagonal net, the so called Blue Tongue series (ref 1). This manner of structure analysis can be given molecular resolution with the mixed polyhedral concept as shown with Blue Tongue itself in fig 17 a and b

Fig 17 a

Fig 17b Blue tongue Fig 18 b Herpes, also in Blue Tongue series.

Herpes ref 13

We continue with the Herpes structure. The nets in green, yellow and red are simplified as describing clusters of proteins. The structure has also been described as a Hermite in ref 2, with m=4. The nets in white, blue and black are connecting the blue – green net as described by the Blue Tongue series. The symmetry codes as in below.

blue yellow red white green black Fig 18 a

With the asymmetric unit we describe a bigger unit with protein particle resolution as shown in fig 19b. We have identified 10 codes in an Adeno mixed crystal as shown in Fig 19a. It is easy to do the rest but we ran short of colors.

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Fig 19a red blue green purple yellow black light blue

green black grey

Fig 19 b

In the Blue Tongue series we show the two last members – m=9 and 10 below in 19c and d. But we now use the simplified nets as we started with Herpes. One code now represents three proteins from the asymmetric unit.

Fig 19c Human Adenovirus Type 5 Fig 19d Thermophilic archaeal virus

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Many are the hypothetical examples that can be found using the summation exponential mathematics. We show two below:

p p p p "(x+# y"n) "(x"# y"n) "(y+# z"n) "(y"# z"n) "(z+# x"n)p "(z"# x"n)p Sum[e + e + e + e + e + e , 8 {n, m#, - m#, "#}]" const = 0

!

Fig 18 m=4 p=2 cons 6.5 m=3, p=4 const 6.3

Conclusion

With the new concept mixed polyhedra we describe the known pentagonal virus capsid structures, and also a vast amount of hypothetical. This is done with the symmetry codes of only two structures, the truncated dodecahedron and the snub dodecahedron, and their Hardy deviations.

References 1 S. Andersson, Zeit Anorg Allg Chem. 2008,634, 2161. The structure of virus capsids. Part I.

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

3 Tang, L., Johnson, K.N., Ball, L.A., Lin, T., Yeager, M. & Johnson, J.E. The structure of pariacoto virus reveals a dodecahedral cage of duplex RNA.Nat.Struct.Biol. (2001) 8:77.

4 The Refined Structure of Nudaurelia Capensis Omega Virus Reveals Control Elements for a T = 4 Capsid Maturation Helgstrand, C., Munshi, S., Johnson, J.E. & Liljas, L. Virology (2004) 318:192.

5 Placement of the Structural Proteins in Sindbis virus Zhang, W., Mukhopadhyay, S., Pletnev, S.V., Baker, T.S., Kuhn, R.J. & Rossmann, M.G. J.VIROL. (2002) 76:11645.

6 Cryo-electron microscopy reveals the functional organization of an enveloped virus, Semliki Forest virus. Mancini, E.J., Clarke, M., Gowen, B.E., Rutten, T. & Fuller, S.D. Mol.Cell (2000) 5:255.

7 The X-Ray Crystal Structure of P3, the Major Coat Protein of the Lipid-Containing Bacteriophage Prd1, at 1.65 A Resolution. Benson, S., Bamford, J., Bamford, D. & Burnett, R. Acta Crystallogr.Sect.D (2002) 58:39.

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8 S. Andersson, Sandforsk.se 2009. On the inside structures of virus capsids.

9 Structural Studies of Bacteriophage alpha3 Assembly Bernal, R.A., Hafenstein, S., Olson, N.H., Bowman, V.D., Chipman, P.R., Baker, T.S., Fane, B.A. & Rossmann, M.G. J.Mol.Biol. (2003) 325:11.

10 Crystallization of Hepatitis B Virus Core Protein Shells: Determination of Cryoprotectant Conditions and Preliminary X-Ray Characterization Wynne, S.A., Leslie, A.G.W., Butler, P.J.G. & Crowther, R.A. Acta Crystallogr., Sect.D (1999) D55:557.

11 Structure of Simian Virus 40 at 3.8-A Resolution Liddington, R.C., Yan, Y., Moulai, J., Sahli, R., Benjamin, T.L. & Harrison, S.C. Nature (1991) 354:278.

12 The refined structure of a protein catenane: the HK97 bacteriophage capsid at 3.44 A resolution. Helgstrand, C, Wickoff WR, Duda RL, Hendrix RW, Johnson, J.E., Liljas, L. J Mol Biol. 2003, 334(5):885-99.

13 Alasdair Steven, Ph.D. Chief, Laboratory of Structural Biology Research, National Arthritis and Musculoskeletal and Skin Diseases (NIAMS) National Institutes of (NIH) 50 South Drive MSC 8025 Bethesda, Maryland 20892-8025

14. Fabry, C.M.S., Rosa-Calatrava, M., Conway, J.F., Zubieta, C., Cusack, S., Ruigrok, R.W.H. & Schoehn, G. Embo J. (2005) 24:1645.

15 The structure of a thermophilic archaeal virus shows a double-stranded DNA viral capsid type that spans all domains of life. 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