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.
Highlights
We found in a previous work that the approach of quantum computation based on magic states [1,2,3] may be used to explore the symmetries and the structure of the genetic code [4,5,6]
In our previous paper [5], it was shown that 7-fold symmetry may be mirrored in the finite group G7 = Z7 × 2O whose characters may be mapped to the amino acids of the genetic code
Since the group G7 is successful for encoding the genetic code and that, at the same time, it provides an assignment to the 20 amino acids through the corresponding characters, one can ask ourselves if G7 may be used to define a secondary structure in a protein
Summary
We found in a previous work that the approach of quantum computation based on magic states [1,2,3] may be used to explore the symmetries and the structure of the genetic code [4,5,6]. From several protein examples belonging to highly symmetric complexes, that the secondary code has to obey some structural algebraic constraints relying to free group theory. In both cases, a theory close to the observed patterns is based on the oriented hypercartographic group H2+ , a straightforward generalization of the cartographic group C2 introduced by A. One proposes a local mapping of the amino acids to a protein secondary structure with pseudo-helices, sheets and coils based on the characters of the group G7. Following our previous work in [4,5], we find that the nucleosome complex allows to define another group theoretical model of the genetic code based on the characters of the group G8.
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