Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface

Abstract

Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The sequence unfolds in the 3-dimensional space through secondary (local foldings), tertiary (bonds) and quaternary (disjoint multiple) structures. The mere existence of the genetic code for the 20 letters of the linear chain could be predicted with the (informationally complete) irreducible characters of the finite group Gn:=Zn⋊2O (with n=5 or 7 and 2O the binary octahedral group) in our previous two papers. It turns out that some quaternary structures of protein complexes display n-fold symmetries. We propose an approach of secondary structures based on free group theory. Our results are compared to other approaches of predicting secondary structures of proteins in terms of α helices, β sheets and coils, or more refined techniques. It is shown that the secondary structure of proteins shows similarities to the structure of some hyperbolic 3-manifolds. The hyperbolic 3-manifold of smallest volume—Gieseking manifold—some other 3 manifolds and the oriented hypercartographic group are singled out as tentative models of such secondary structures. For the quaternary structure, there are links to the Kummer surface.

Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems

“…After we received an invitation to contribute to the present special issue of Sci "Mathematics and poetry, with a view towards machine learning" we thought that our current group theoretical approach of protein language [3] could be converted into an understanding of the poetic language, as well as an understanding of some musical structures.…”

“…A protein is a long polymeric linear chain encoded with 20 letters (the 20 amino acids). The surjective mapping of the 4 3 = 64 codons to the 20 amino acids is the DNA genetic code. It can be given a mathematical theory with appropriate finite groups [2,3].…”

“…The surjective mapping of the 4 3 = 64 codons to the 20 amino acids is the DNA genetic code. It can be given a mathematical theory with appropriate finite groups [2,3]. In addition, a protein folds in the three dimensional space with structural elements such as coils, α-helices and β-sheets, or other arrangements that determine its biological function.…”

“…In Section 3, we look at the secondary structures of two proteins. We take as examples the spike protein of the SARS-Cov-2 virus and a glycoprotein playing a role in the immune system (see [3] for our earlier work). In Section 4, the secondary structures are the musical forms of western music in the classical age and twentieth century music.…”

Beyond a Quartic Polynomial Modeling of the DNA Double-Helix Genetic Code

“…In a recent contribution Planat et al [23] defined a model for the structure of a nucleosome as secondary unit consisting of 8 histone proteins with a DNA double helix wound around. The assigned group was ( )…”