Symmetrical-geometry constructions defining helicoidal biostructures. The case of alpha-helix

Abstract: The chain of algebraic geometry constructions permits to transfer from the minimal surface with zero instability index, and from the lattice over the ring of cyclotomic integers to the tetra-block helix. The tetra-block is the 7-vertex joining of four tetrahedra sharing common faces; it is considered as a building unit for structures approximated by the chains of regular tetrahedra. The minimality condition of the 7 - vertex tetrablock as a building unit is the consequence of its unique mapping by the Klein's quartic (which is characterized by the minimal hyperbolic Schwartz triangle) into the minimal finite projective geometry. The topological stability of this helix provided by the pitch to radius ratio H/R of 2{\pi}/({\tau}+1) ({\tau} is the golden section) and by the local rotation axis order of 40/11=40exp(-H/R). These parameters determine the helix of C{\alpha} atoms inside the alpha - helix with the accuracy of up to 2%. They explain also the bonding relationship i -- i+4 between the i-th amide group and the (i+4)-th carbonil group of the residues in the peptide chain and the observed value of the average segment length of the alpha-helix which is equal to 11 residues. The tetra-block helix with the N, C{\alpha}, C', O, H atoms in the symmetrically selected positions, determines the structure of the alpha - helix. The proposed approach can display adequately the symmetry of the helicoidal biopolymers.