Symmetry of helicoidal biopolymers in the frameworks of algebraic geometry: α-helix and DNA structures

The chain of algebraic geometry and topology constructions is mapped on a structural level that allows one to single out a special class of discrete helicoidal structures. A structure that belongs to this class is locally periodic, topologically stable in three-dimensional Euclidean space and corresponds to the bifurcation domain. Singular points of its bounding minimal surface are related by transformations determined by symmetries of the second coordination sphere of the eight-dimensional crystallographic lattice E₈. These points represent cluster vertices, whose helicoid joining determines the topology and structural parameters of linear biopolymers. In particular, structural parameters of the α-helix are determined by the seven-vertex face-to-face joining of tetrahedra with the E₈ non-integer helical axis 40/11 having a rotation angle of 99°, and the development of its surface coincides with the cylindrical development of the α-helix. Also, packing models have been created which determine the topology of the A, B and Z forms of DNA.