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What makes structure and shape of capsid or crystal Sten Andersson Sandforsk, Institute of Sandvik S-38074 Löttorp, Sweden www.sandforsk.se 12 June 2006 Abstract The inner wavefuctions of the harmonic oscillator give the morphologies or structures for various capsids. For example n=4 gives the structure of the capsid that desribes the Herpes virus (T=16) and also the Human hepatitis virus as described with the new polyhedron Tri-pent-hex I. The structures of crystal and capsid are compared. As several shapes of these structures are derived with the harmonic oscillator the vibrations of particles, or building block units, are discussed as important for structure building operations. In an appendix all the classic polyhedra(+ two new) of five fold symmetry are directly compared with important capsid morphologies. Introduction In this article we discuss capsids or capsid crystals that have five fold symmetry, and crystals. Used alone the word ‘crystal’ means something having crystallographic symmetry. Inorganic solid state materials of crystallographic symmetry have been of great importance in the developments in technology. In biology there is one non-living material of special importance, the virus capsid of non-crystallographic symmetry. The construction ‘non-crystallographic symmetry’, used by The international Union of Crystallography in for example The International Tables for Crystallography to describe the groups

!

53m and

!

532, is not a good formulation. That construction is in the case of virus, or quasi crystals too general, if not just wrong. We shall call them for structures or crystals that have five fold, or pentagonal symmetry. Because that is what they are. Both kinds of developments of these two groups continue steadily. Evolution changes capsid to the demand of biology. As mankind changes technology to the demand of society.

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As understood the two groups build structures, with atoms or molecules in one group that interact to what are often called building block units, which in turn interact to form a desired material. In the other group protein units, often also called building blocks, interact and form structures with properties demanded by evolution and biology. Using the pentagonal symmetry codes we described a number of capsid structures as derived with mathematics from spherical space and confirmed our general observation from solid state science that complex structures are built from parts of simple ones. We did that in two steps–the smaller capsids are described by asymmetric units or proteins that are repeated with five fold symmetry to form capsid structures. Then the complex capsids, ‘the big ones’ are obtained by extending the regions between 5 fold axes with material from a normal virus. Example is Blue tongue as filled with hexagon structure material from the cosackie structure type(ref 1). The expansion occurs with ordinary translation and the Blue tongue structure of five fold symmetry has local parts of crystallographic symmetry. Now to the question: What makes structure and shape of capsid or crystal? What is it that these two groups of phenomena have in common? Beside the obvious and important fact that both are structures. Let us for a moment forget about chemistry, physics, biology and just do some relationship between the geometry of capsids, mathemathics and structures. Some mathematics of crystallographic symmetry, five fold symmetry and capsids. In spherical space with radial periodicity there is the body centered cubic packing, or also called the rhombic dodecahedral, a common form of steel(one of the best packings ever):

!

Sum[e"(x+y"n)2

+ e"(x"y"n)2

+ e"(y+z"n)2

+ e"(y" z"n)2

+ e"(z+x"n)2

+ e"(z"x"n)2

,

{n, m, - m, " 2}]" 5.25 = 0

1

which for three different m give Fig 1.

m=2 m=4 m=6 Fig 1 The rhombic dodecahedra with their faces

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In spherical space with radial periodicity there is the dodecahedral packing of bodies as in eq 2 and fig 2. The difference between the equations is only in τ but indeed this formulation brings out five fold periodicity. For lower m there are capsid structures with m=1 for the dodecahedron, m=2 the icosidodecahedron. m=3 is easily related to the (small) rhombicosidodecahedron. Higher m(80) led to a description of the alloy quasi crystal structure(ref 2).

!

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

+ e"(x"# y"n)2

+ e"(y+# z"n)2

+ e"(y"# z"n)2

+ e"(z+# x"n)2

+ e"(z"# x"n)2

,

{n, m# , - m# , "#}]" const = 0

2

m=1 m=2 m=3 Fig 2. Dodecahedral packing of bodies after eq 2. Going up in exponent to 4 in eq 2 there are more capsid structures. We shall now use the mathematics of the harmonic oscillator to shortly describe crystallographic symmetry and then in more detail the five fold symmetry and structure of capsids. The Schrödinger solution to the harmonic oscillator is a symmetrical body that contains a periodic structure. Which really is the inside of the wave function that constitutes the Schrödinger solution(ref 3). We give below the wave function for a primitive cubic system for a complete calculation(ref 4).

!

" =

N e#1/2(x

2+ y

2+ z

2)H[n,(x)] H[n,(y)] H[n,(z)]

3

4

Fig 3a

!

" 2=.001, n=3 b n=20 We also give bodies from cubic close packing, as well as from simple hexagonal below in fig 3b-c.

Fig 3b-c. Cubic close packing and simple hexagonal packing. And with five fold symmetry in eq 4 we do analogous calculations, and for various quantum numbers we get typical capsid structures in fig 4 we shall analyze below:

!

" =

N e#1/2[(2$

2+ 2)[x

2+ y

2+ z

2]

H[n,($ x + y )] H[n,(x + $ y)] H[n,(y + $ z)] H[n,(-$ x + y )] H[n,(x -$ y)] H[n,(y -$ z)]

4

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n=1 n=2 n=3 n=4 Fig 4 Five fold symmetry and the harmonic oscillator for various quantum numbers The finite periodicity in spherical space is obtained via the exponential scale mathematics of summations of Gauss distributions. The harmonic oscillator is composed of a Gauss distribution function(damping), and a product of its n (quantum numbers) derivatives. Which makes the two approaches very similar in mathematical properties. The main difference is that the Schrödinger equation has a physical contribution in its description of the vibrations of a particle. The virus capside and the harmonic oscillator We have derived relationships between the shape of virus capsids, mathematics of polyhedra and finite periodic functions in spherical space(refs 1,2). Using the mathematics for the harmonic oscillator we report here below on some remarkable agreements between the shapes of the oscillators, the insides of the wave functions, and the shapes of virus capsids With the dodecahedral symmetry code in the oscillator as in eq 4 we find that outside shape and very much the inside periodic structure grow complicated with increasing quantum number. We are then limited only to lower numbers. We see from above that in the cubic case increased quantum number does not change the shape. Which justifies the use of lower quantum numbers below. The periodic change of shape, as described before(ref 2) is a property of the five fold periodicity. To this we may add that the remarkable result is a justification in itself. For n=1 in equation 4 there is fig 5a and the dodecahedral symmetry code delivers the dual form, an icsahedron for n=1 and

!

" 2=. 01. Which is in a very good agreement with the capsid of Bacteriophage Alpha 3 in b. The extra smaller bodies in fig 5a describe the dual dodecahedron which also is seen in the capsid in the 3 fold positions. A square around the 2 fold axis in the capsid shows also the presence of the rhombicosidodecahedron which is shown in fig 5c.

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Fig 5a, n=1 b Bact. Alpha 3 c rhombicosidodecahedron For n =2 there is the icosidodecahedron in fig 6a which is in perfect agreement with the Bacteriopage phi 174 in fig 6b. Which also contains an elongated square which agrees with the smaller extra bodies in fig 6a. This is a distortion of the rhombicosi dodecahedron which in undistorted form is present in 6c in alfalfa mosaic virus. Note that the same type of distortion is present in the calculated form in fig 6a, as well as in the observed capsid in fig 6b.

Fig 6a n=2, b Bact phi 174 T=3 c alfalfa mosaic n=3 gives the truncated icosahedron in fig 7a which here also serves as a model for coxsackievirus B3 and several others(ref 1).

Fig 7a n=3 b Coxsackievirus T=3

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The famous and complicated Herpes virus with T=16 in fig 8b has a morphology of extraordinary similarity to the calculated oscillator for n=4 in fig 8a, as well as the new polyhedron Tri-pent-hex I(ref 1) in fig 8.c We like to add here that for the earlier described Tri-pent-hex I and Tri-pent-hex II the hexagons in Tri-pent-hex I are centered in two fold positions, while in Tri-pent-hex II they are in three fold positions(as in the truncated icosahedron). This new polyhedron in fig 8c is in very remarkable agreement with the virus capsid structure for Human hepatitis shown in fig 8d. Note that the hexagons and the pentagons in Human hepatitis are unfilled contrary to the case for Herpes, and the calculated oscillator.

Fig 8 a n=4 b Herpes T=16

Fig 8c Tri-pent-hex I d Human hepatitis T=4 (rotated)

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Discussion There are great differences in the structural sciences of capsid and crystal. There are also similarities like in the crystal building principal that complicated structures are constructed from simple ones. Crystals have group symmetry(two,three,four,or six fold symmetry), while capsids have five fold symmetry. Crystal structures and shapes are prepared, changed and manipulated by man in laboratory. Capsid structures and shapes are prepared, changed and manipulated by biology and evolution. Both crystal and capsid are doing jobs that are dependent on their structures. Why five fold for the capsid? We know it is best, as selected by evolution, but why is it so? The obvious alternative is the ordinary polyhedra like the cube, the octahedron, the tetrahedron, the cuboctahedron, the truncated cube, the truncated tetrahedron, the truncated octahedron, the rhombicuboctahedron, the truncated cuboctahedron and the snub cube. A unified mathematical description would need several different symmetry codes, meaning different periodicities. The five fold symmetry, with its nine polyhedra(+ two new), gives a mathematical description with only a couple of finite periodicities or symmetry codes, allowing for simple relationships between several important morphologies for capsid structures. The structures of these polyhedra allow for periodicity as shown with our mathematics. As if periodicity is a part of evolution. Or even a tool for evolution. The relationship between the icosahedron and the cuboctahedron is well known and illuminating – they have the same number of corners and the icosahedron has almost 10% smaller volume. This means much better packing which contributes to strength. This must be a very heavy reason in favor for the five fold symmetry of capsids. The icosahedral symmetry offers a simple form of finite periodicity, to build a frame strong enough to hold for the internal pressure needed for the ejection of the genome. Finally the snub cube and the snub dodecahedron are two related polyhedra as their names indicate. The relatively smaller snub cube as present around sodium in Pauling’s

!

NaZn13

is a very rare, or the only example in inorganic solid state science. Its rotational relationship with the rhombicuboctahedron is to our knowledge unknown in solid state.

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The corresponding relationship between the snub dodecahedron and the rhombicosidodecahedron on the other hand is important and common for capsid morphology as we have discussed(ref 1). How are structures with five fold symmetry coded into the genome? The answer is obvious – the vibrations. The particles building an icosahedron must have different, or harder oscillations as compared with the particles building a cuboctahedron. And we are then back to the production of the organic subunits – which is coded in a genome to give an ultimate structure with vibrations that favor five fold symmetry. This must be strongly related or connected to the forces that hold these subunits together. The subunits contain long tails which are entangled by the oscillations and hold different subunits together like burdocks. The thorns of the burdocks are then given a real physical description to the weak chemical forces often called dispersion forces. It is the oscillatory motions of entanglements which in interaction with electrostatics give such forces also called hydrophobic forces. Indeed an entanglement may be described as a quantized harmonic oscillator. A strong development in this field has been carried out by Ninham, Boström, Israelachvili, and several others(ref 5-6). Such hydrophobic interactions are beautifully used by Willem K. Kegel and Paul van der Schoot in their Competing Hydrophobic and Screened-Coulomb Interactions in Hepatitis B Virus Capsid Assembly(ref 7). We said above there is a corresponding tool in inorganic solid state to produce building block units that build complex structure. We said that 'The electronic shape of an atom or a molecule determine their vibrations which in their turn determine the structure of the solid it may form, and coupled quantized harmonic oscillators describe periodicity of crystals.' (ref 4) We extend this saying that atoms or ions form building blocks that vibrate in an entity to form complex structures. There is no genetic code for the simple chemist to make his compounds, at the best there might be a good recipe in his lab-journal. We should start to talk about an electronic code, following the explanation of the giant structure of NaCd2 recently carried out by Daniel C Fredrickson, Stephen Lee and Roald Hoffmann(Partly done in Sven Lidin’s laboratory in Stockholm): Interpenetrating Covalent and Ionic Sublattices in Intermetallics: the NaCd2 Structure.(In print) It was found that the driving force for the geometry is electronic and that the structure among other things has a covalent region interpenetrating an ionic, and that these two regions were separated by a minimal surface. As the driving force for evolution is mutation indeed we have a common physical denominator between crystals and capsids– the electron in its motion! Conclusion We have simple exponential mathematics that using finite periodicities in spherical, or elliptic space, describes the fundamental crystal structures, as well as the important part of different capsid structures. We have the quantized harmonic oscillator that gives the same result.

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These two kinds of mathematics have two things in common. They are of finite periodicities and they contain the same symmetry codes. It is obvious that the mathematical symmetry codes give pictures in excellent agreement with the observed structures. In practice it is the genetic code that build the capsids, while it is an electronic code that builds the solid materials. And the two codes contain symmetry and periodicity. The symmetry code in the capsid case is built into the genetic coding for the production of organic subunits. These gigantic organic molecules must from their structure have vibrational properties that form the five fold symmetry when they are bonded together via entanglements, and form the capsid structure The doping of silicon is interference in the 3D structure. Similarly the mutation is an interference in the molecular and 3D structure that constitute the capsid Appendix The five fold symmetry. The virus capsids and polyhedra For all the informations used below we acknowledge the Viper database: C. M. Shepherd, I. A. Borelli, G. Lander, P. Natarajan, V. Siddavanahalli, C. Bajaj, J. E. Johnson, C. L. Brooks, III, and V. S. Reddy (2006). VIPERdb: a relational database for structural virology. Nucl. Acids Res. 34 (Database Issue): D386-D389 We show the relation of typical capsid structures to the regular polyhedra(ref 8). (+ two new, ref 1) This is made clear by always using the polyhedral corners in the description of a capsid structure and shape.

Human papilloma Desmodium yellow Icosahedron(corners 5 fold axes) and dodecahedron(corners 3 fold axes)

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Bacteriophage phi 174 Icosidodecahedron (corners 2 fold axes)

Satellite tobacco necrosis Truncated dodecahedron

Coxsackie virus Truncated icosahedron

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Alfalfa mosaic virus (Small) rhombicosidodecahedron

Bacteriophage GA Great rhombicosidodecahedron or truncated icosi-dodecahedron

Simian virus (7d lattice from Viper shows Snub dodecahedron position of 5 fold axes) We also show the AAV-2 virus, a T=1, and one of the smallest capsids, and having the morphology of the snub dodecahedron. From the picture it is easy to realize the rotation transformaion to the morphology of the (small) rhombicosidodecahedron.

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AAV-2

Herpes Tri-pent-hex I

Bacteriophage Q beta Tri-pent-hex II References Before we continue we reveal the use of a very remarkable database: For all their informations used above we acknowledge the Viper data base:

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C. M. Shepherd, I. A. Borelli, G. Lander, P. Natarajan, V. Siddavanahalli, C. Bajaj, J. E. Johnson, C. L. Brooks, III, and V. S. Reddy (2006). VIPERdb: a relational database for structural virology. Nucl. Acids Res. 34 (Database Issue): D386-D389

1 S. Andersson www.sandforsk.se, Virus structure in spherical space. Polyhedra, and beyond polyhedra, April, 2006 2 S.Andersson, Crystal structure and elliptic periodicity. Cubes and dodecahedra Solid State Science, In print. 3 L.Pauling and E.B. Wilson, INTRODUCTION TO QUANTUM MECHANICS, McGraw-Hill, 1968. 4 S. Andersson www.sandforsk.se, Crystal structures, the quantized harmonic oscillator and the exponential scale, October 2005. 5 J Mahanty and B. W. Ninham Dispersion forces Academic Press 1976 6 M.Boström, M., D. R. M. Williams, and B. W. Ninham. 2003. Specific ion effects: the role of co-ions in biology. Europhys. Lett. 63:610–615. 7 Willem K. Kegel and Paul van der Schoot, 2004. Competing Hydrophobic and Screened-Coulomb Interactions in Hepatitis B Virus Capsid Assembly. Biophys J. June; 86(6): 3905–3913. 8 H.M. Cundy and A.P. Rollet. Mathematical Models. Oxford University Press 1961.