Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed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 (doi: 10.20944/preprints202103.0612.v1). 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 (doi: 10.20944/preprints202103.0612.v1).

## 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 Grothendieck's cartographic group are singled out as tentative models of such secondary structures. For the quaternary structure, there are links to the Kummer surface.

## Keywords

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

## Subject

LIFE SCIENCES, Biochemistry

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