%0 Journal Article %T Quantum Information in the Protein Codes, $3$-manifolds and the Kummer Surface %T Information quantique dans les codes des protéines, 3-variétés et la surface de Kummer %+ Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174) (FEMTO-ST) %A Fang, Fang %A Planat, Michel %A Aschheim, Raymond %A Amaral, Marcelo, M %A Irwin, Klee %Z https://doi.org/10.3390/sym13071146 %< avec comité de lecture %@ 2073-8994 %J Symmetry %I MDPI %V 13 %P 1146 %8 2021-06-16 %D 2021 %R 10.20944/preprints202103.0612.v1 %K protein structure %K DNA genetic code %K finite groups %K 3-manifolds %K Kummer surface %K cartographic group %K informationally complete characters %Z PACS: 02.20.-a, 03.65.Fd, 82.39.Rt, 87.10.-e, 87.14.-g, 87.14.EeMSC codes: 20C15, 92D20, 20E45, 14H52, 14J28, 58D19, 57M27 %Z Life Sciences [q-bio]/Other [q-bio.OT] %Z Physics [physics]/Quantum Physics [quant-ph] %Z Mathematics [math]/Group Theory [math.GR]Journal articles %X 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. %G English %2 https://cnrs.hal.science/hal-03184138v2/document %2 https://cnrs.hal.science/hal-03184138v2/file/ProteinSecondary.pdf %L hal-03184138 %U https://cnrs.hal.science/hal-03184138 %~ CNRS %~ UNIV-FCOMTE %~ UNIV-BM %~ ENSMM %~ FEMTO-ST %~ INSMI %~ UNIV-BM-THESE %~ GIP-BE