Symmetry types, systems and their multiplicity in the structure of adenovirus capsid. I. Symmetry networks and general symmetry motifs

Symmetry types, systems and their multiplicity in the structure of adenovirus capsid. II. Rotational facet groups of five-, three- and two-fold symmetry axes

The icosahedral adenovirus capsid has three rotational symmetry axes of different types. The six five-fold, ten three-fold and the fifteen two-fold axes have two superficial points each, altogether 62. The axes determine the number and location of the identical rotational facet groups and that during the different rotational phases which other regular facets and with what multiplicity shall be covered by them. The number of rotational facets of the five-, three- and two-fold rotational symmetry axes is 4, 6.66 and 10, respectively. In all the three cases, there are two kinds of possible arrangements of the facets. During the rotation--when the facets of the facet group placed on one by one to the neighbouring identical facet groups--at the five-fold axes, the facets of the rotational facet group get into cover position 12 times with all the 20 regular capsid facets, 20 times at the three-fold axes, and 30 times at the two-fold axes in a way that a different facet combination (facet hit) falls to every facet, and the original symmetry is not disturbed. After all, this means 240, 400 and 600 facet combinations, i.e. multiplicity in case of five-, three- and two-fold symmetry axes respectively, and these numbers correspond with that of the theoretically possible variations. The same results can be calculated by multiplying the number of real rotations of the capsid bringing the body into itself, i.e. the number 60 with the number of facets contributing to the five-, three- and two-fold rotational phases. The other way of the determination of multiplicity takes into account that all the facet groups of the capsid rotate simultaneously during all the rotational phases, and this multiplies the number of multiplicity with the number of the rotational types five-, three- and two-fold which result in one and the same multiplicity number in the case of five-, three- and two-fold symmetry, alike 1200. Perpendicular to the five-fold symmetry axes with the line of intersection drawn horizontally in the middle along the 6 geodetic ribbon like motifs a regular decagonal intersection forms and the capsid can be cut into two equal parts, in which the polypeptides show a 72 degree rotation from each other, but with a proper rotation the polypeptides get into a congruent position, which means 300 or 600 specific facet combinations. The capsid similar to the icosahedron has also 15 virtual mirror planes which divide the capsid into two, identically arranged halves, forming six right angle triangles on each facet, altogether 120 smaller rectangular so-called Mobius-triangles on the surface. In the three-fold symmetry axis of the facets, these triangles in two separate groups of three can be rotated symmetrically with 120 degrees according to the orientation of the polypeptide subunits in a way that the hexon and other polypeptides too nearly cover each other. Consequently, the adenovirus capsid is a symmetrically arranged body in which several various symmetry types and symmetry systems can be found and their structural symmetry elements exist simultaneously and covering each other. The icosahedral symmetry types and systems are valid and functional simultaneously and in parallel with great multiplicity, but the existence of more than 1500 structural elements in several depth levels, their order of location and distribution make the symmetry of the capsid richer and more complex.