Abstract
The developed apparatus of the “structural application” of algebraic geometry and topology makes it possible to determine topologically stable helicoidally-like packings of polyhedra (clusters). A packing found is limited by a minimal surface with zero instability index; this surface is set by the Weierstrass representation and corresponds to the bifurcation point. The symmetries of the packings under consideration are determined by four-dimensional polyhedra (polytopes) from a closed sequence, which begins with diamondlike polytope {240}. One example of these packings is a packing of tetrahedra, which arises as a result of the multiplication of a peculiar starting aggregation of tetrahedra by a fractional 40/11 axis with an angle of helical rotation of 99°. The arrangement of atoms in particular positions of this starting aggregation allows one to obtain a model of the α-helix. This apparatus makes it possible to determine a priori the symmetry parameters of DNA double helices.
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Original Russian Text © M.I. Samoylovich, A.L. Talis, 2013, published in Kristallografiya, 2013, Vol. 58, No. 5, pp. 639–651.
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Samoylovich, M.I., Talis, A.L. Structural regularities of helicoidally-like biopolymers in the framework of algebraic topology: II. α-Helix and DNA structures. Crystallogr. Rep. 58, 651–662 (2013). https://doi.org/10.1134/S106377451305009X
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DOI: https://doi.org/10.1134/S106377451305009X