Helical substructures of 4D constructions that determine the structure of α-helices.

In a 4D polytope {3, 3, 5}, a 40-vertex toroidal helix is selected that unites the vertices of two orbits of the axis 20/9 with the angle of rotation 9 × 360°/20 = 162°. Symmetrization of this helix allows one to select in the 3D spherical space a helix {40/11} with the angle of rotation of 99°. Its mapping into the 3D Euclidean space E³ determines the helix {40/11}, which coincides with the helix of atoms C_α in the α-helix. A tube polytope with the symmetry group ±[O×D₂₀] contains a toroidal helix {40/11}, constructed of 40 prismatic cells. The symmetry of the polytope, as well as the partition it induces on the lateral face of the prismatic cell, allow one to find additional vertices that do not belong to the polytope. Putting the vertices of the helix {40/11} in correspondence with the atoms C_α and the additional vertices with the atoms O, C', N, H, determines the peptide plane of the α-helix; its multiplication by the axis 40/11 leads to a polytope model of the α-helix. A radial contraction of the polytope model, with subsequent mapping into E³, leads to its densely packed structural realization - the α-helix that is universal in proteins. A polytope with the group of symmetry ±[O×D₂₀] arises in the family of tube polytopes with the starting group ±1/2[O×C_2n] at n = 5. Along with the axis 40/11 of a single α-helix, the screw axes of this family of polytopes determine the axes 7/2, 11/3, 15/4, 18/5 realized as the axes of the α-helices included in superhelices.