Working Paper Article Version 2 This version is not peer-reviewed

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

Version 1 : Received: 24 March 2021 / Approved: 24 March 2021 / Online: 24 March 2021 (17:48:44 CET)
Version 2 : Received: 20 April 2021 / Approved: 21 April 2021 / Online: 21 April 2021 (08:36:37 CEST)

How to cite: Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Quantum Information in the Protein Codes, $3$-manifolds and the Kummer Surface. Preprints 2021, 2021030612 Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Quantum Information in the Protein Codes, $3$-manifolds and the Kummer Surface. Preprints 2021, 2021030612

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 $G_n:=\mathbb{Z}_n \rtimes 2O$ (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 $\alpha$ helices, $\beta$ 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.

Subject Areas

protein structure; DNA genetic code; informationally complete characters; finite groups; $3$-manifolds; Kummer surface; cartographic group

Comments (1)

Comment 1
Received: 21 April 2021
Commenter: Michel Planat
Commenter's Conflict of Interests: Author
Comment: The paper was improved following the reports by referees in a first submission to the journal Symmetry.
In addition, the group G of section 2.2 was incorrectly called the "cartographic group" but it is known as the "oriented hypercartographic group".
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