# 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 $G_n:=\mathbb{Z}_n \rtimes 2O$ (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.
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https://hal-cnrs.archives-ouvertes.fr/hal-03184138
Contributor : Michel Planat <>
Submitted on : Friday, April 2, 2021 - 1:55:35 PM
Last modification on : Wednesday, April 7, 2021 - 3:11:26 AM

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### Citation

Fang Fang, Michel Planat, Raymond Aschheim, Marcelo Amaral, Klee Irwin. Quantum Information in the Protein Codes, $3$-manifolds and the Kummer Surface. 2021. ⟨hal-03184138⟩

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