PART I Chapter 8 Conclusions. With evolution all capsid structures are topologically related

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68 PART I Chapter 8 Conclusions With evolution all capsid structures are topologically related We have seen how three concepts dominate the scene one begins with the snub dodecahedron forming the

This we describe with two snub dodecahedra of different chirality that interpenetrate to the great rhombicosidodecahedron.

1 68 PART I Chapter 8 Conclusions With evolution all capsid structures are topologically related We have seen how three concepts dominate the scene one begins with the snub dodecahedron forming the smallest of all viruses - the Panicum Mosaic Virus of 159 Å. This is continued within T=1 when one molecule forms the strange symmetry with the Adeno-Associated Virus (AAV-2). This we describe with two snub dodecahedra of different chirality that interpenetrate to the great rhombicosidodecahedron. Which is what happens in Pseudo T=3, T=3, T=4 and many are the capsid structures that fit into this bilateral concept. We demonstrate now the transition from the bilateral concept to the second - the morphotropic concept. Also in a topological way as shown in fig The disk concept the third - is described in Bilateral and morphotropic The Cowpea Chlorotic in Fig represents the bilateral evolution. The Cowpea mosaic in represents the beginning of the morphotropic evolution. There are several ways to describe a morphotropic structure block is a popular word they are simple and big and lay the path to the real big ones. We also show the topological relationship between the two cowpeas in Fig8.1.1 Cowpea Chlorotic 2 Cowpea Mosaic Mosaic drawn as Chlorotic with topological similarity

unit has an advanced shape - allowing for the many variations in bilateral symmetry there are in Pseudo T=3 and T=3.

6 how Cowpea Chlorotic evolves to BPMV, Polio type 3 and Echovirus type 12

6 Cowpea Chlorotic BPMV Polio type 3 Echovirus type 12

2 69 Fig Cowpea Chlorotic The asymmetric units of the two cowpeas 5 Cowpea Mosaic We notice that the chlorotic asymmetric unit has an advanced shape - allowing for the many variations in bilateral symmetry there are in Pseudo T=3 and T=3. As examples we show below in fig how Cowpea Chlorotic evolves to BPMV, Polio type 3 and Echovirus type 12, all Pseudo T=3. fig Cowpea Chlorotic BPMV Polio type 3 Echovirus type 12 The Cowpea mosaic as shown in is of simpler shape as we see below size and simplicity is taken over to fulfill demand for evolution. We demonstrate this below with asymmetric units of the Cowpea mosaic structure, the Semliki structure, the Simian structure, the PRD and the PBCV-1 structures in fig This is the morphotropic concept. The explanation of the word and the historical background is given in Appendix. Fig Cowpea mosaic asymm unit Semliki asymm. unit The disk and structure

70 Simian asymm. unit, one disk PRD asymm.

2 The disk Of outstanding importance is the third

dictated by symmetry around the 3 or 2 fold axes in a

Right through bilateral symmetry of small capsids,

block structures to capsids of any size.

With a series of pictures we show below some examples

3 70 Simian asymm. unit, one disk PRD asymm. unit, 4 disks PBCV-1 The Rossmann disk Fig The disk Of outstanding importance is the third concept; the disk concept, which with its structure dictated by symmetry around the 3 or 2 fold axes in a growing asymmetric unit, builds the complete capsid. Right through bilateral symmetry of small capsids, and via the two Cowpeas, over to the morphotropic block structures to capsids of any size. The disks may have shapes of a triangle, a pentagon, a hexagon, or simply a circle. The sizes vary between Å. With a series of pictures we show below some examples in figs Fig BPMV 2 Human Rhinovirus 3 Echovirus 12, note the rods 4 Swollen disk of BMV 5 Disk of Bacteriophage Ga

4 71 6 BMV virus 7 Its disk 8 Norfalk (disk derived from relation with BMV) 9 Two fold symmetry, Nudarella, T=4 10 Human Hepatitis T=4 11 Semliki T=4 12 Simian,T=7, no symmetry 13 HK 97 T=7, no symmetry 14 Human Adenovirus,T=13 The asymmetric 15Rice dwarf,t=13, The asymmetric unit contains 4 different disks unit contains 4 different triangular(tilings) disks

72 16 PBCV-1 PseudoT =169d, The Rossmann disk of ordinary size.

8.2.16 is now in full power with the giant structure of PBCV-1.

structures: Geometric evolution and structure relationship between different viruses follow the

In mathematics the bilateral symmetry is an exact mirror, in biology we follow Weyl(7) and say it is

5 72 16 PBCV-1 PseudoT =169d, The Rossmann disk of ordinary size. Capsid size 1900 Å The morphotropic principle operating with finite translation of a disk unit as in is now in full power with the giant structure of PBCV-1. The disk concept is obviously there to stay as a structure-building unit in the description of capsid geometry. 8.3 More about organization of structures We continue to shortly repeat the organization of virus structures: Geometric evolution and structure relationship between different viruses follow the Linnean approach. In mathematics the bilateral symmetry is an exact mirror, in biology we follow Weyl(7) and say it is not. We say biological structures in the first concept above are almost mirror images or have bilateral symmetry. Three simple series of capsid structures from the additions of hexagons, squares, and triangles to pentagons on a capsid surface. Block structures and morphotrophy, and finite translation in very big capsids. An organization of series is given below. Series 1. Addition of hexagons to a dodecahedron(pictures partly from Wikipedia) in figs Fig dodecahedron 2 trunc icosahedron 3 trunc triacontahedron(tt) 4 Simian polyhedron

6 73 The Blue Tongue series starts with fig and ends with fig Members beyond step=3(tt) with beginning of step=4(named Simian by us ) up to step=10 are all new polyhedra. The names given are the names of corresponding virus. Fig step=2 (Coxsackie) 7 step=3(semliki) 8 step=4 Simian 9 step=5 Cowpea mosaic 10 Step=6 Blue Tongue 11 step=7 Herpes 12 step 8 vacant 13 Step=9 Human Adenovirus Type 5 14 Step=10 Archaeal STIV Note that the net for Semliki as given above for step =3 is a simpler version from that given in

7 74 Series 2 Addition of hexagons to an icosidodecahedron in figs Fig icosidodecahedron 16 tri pent hex II (New polyhedron), 17 tri pent hex I (New polyhedron) 18 Bacteriophage alpha3 19 PBCV-1 20 Human hepatiti Series 3 Addition of hexagons to a rhombicosidodecahedron in figs Fig Rhombicosidodecahedron 21 Semliki polyhedron. New polyhedron

75 22 The phix174 virus 23 The Semliki virus

morphotropy, only some examples A simple

Blue Tongue structure, and its toplogical

8.4.1-2. Fig 8.4.1 Blue Tongue 2 T13 IBD The

8 75 22 The phix174 virus 23 The Semliki virus 8.4 Block structures, finite translation and morphotropy, only some examples A simple block of Alhambra tilings gives the Rice or Blue Tongue structure, and its toplogical relation with the T13 IBDV structure in figs Fig Blue Tongue 2 T13 IBD The Human Adenovirus is another example of a block structure in fig 8.4.3: Fig The commencement of local translation starts in figs

9 76 Fig Bacteriophage 5 Coxsackie 6Herpes 7 Human Adenovirus alpha3 And transferred to an icosahedron below: Fig The PBCV-1 structure is a beautiful example of local translation and morphotropy. Fig PBCV-1 structure

10 References Part I Before we continue we reveal the use of a very remarkable database: For all their information used above 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 1 S. Andersson, S.T. Hyde and H.G von Scnering, The intrinsic curvature of solids Z. Kristallogr. 210, 3 (1995). 2 S. Andersson, M. Jacob and S. Lidin, The exponential scale and crystal structures Z. Kristallogr. 210, 3 (1995). 3 S.Andersson, Crystal structure and elliptic periodicity. Cubes and dodecahedra. Solid State Science 8 (2006) The Structure and Evolution of the Major Capsid Protein of a Large, Lipid containing, DNA virus. Nandhagopal, N., Simpson, A., Gurnon, J.R., Yan, X., Baker, T.S., Graves, M.V., Van Etten, J.L. & Rossmann, M.G. Proc.Natl.Acad.Sci.USA (2002) 99: S. Andersson Virus structure in spherical space. Polyhedra, and beyond polyhedra, April, S. Andersson Morphotropic virus capsid structures, March, H Weyl, Symmetry, Princeton University Press, Page 4. 8 Andersson, S., and O Keeffe, M., Nature (1977). 9 O Keeffe, M., and Andersson, S., Acta Cryst. A (1977). 10 O Keeffe, M., Acta Cryst. A (1992). 11 Lidin, S., Jacob, M., and Andersson, S., J. Solid State Chem (1995). 12 S. Andersson, K. Larsson, M. Larsson and John C Fiala, Modeling of neuron organization in the CA1 region of rat brain with a helical dendrite structure. and

11 78 13 S. Andersson, K. Larsson,M. Larsson and John C Fiala, Modeling neuron organization: a case of rod packing. and 14 S. Andersson, K. Larsson,M. Larsson and John C Fiala, Modeling neuron organization: Rod packing in the CA1 region of brain of an adult rat. and 16 Andersson, S, Curved polyhedra, Z. f. Anorganische und Allgemeine Chemie. 631,499(2005).

Mathematics, mixed polyhedra and virus structures

1 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