Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

DNA sequence and structure under the prism of group theory and algebraic surfaces

Version 1 : Received: 16 September 2022 / Approved: 19 September 2022 / Online: 19 September 2022 (05:34:50 CEST)

A peer-reviewed article of this Preprint also exists.

Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces. Int. J. Mol. Sci. 2022, 23, 13290. Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces. Int. J. Mol. Sci. 2022, 23, 13290.

Abstract

Taking a DNA-sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group $\pi$, one can discriminate two important families: (i) the cardinality structure for conjugacy classes of subgroups of $\pi$ is that of a free group on $1$ to $4$ bases and the DNA word, viewed as a substitution sequence, is aperiodic. (ii) The cardinality structure for conjugacy classes of subgroups of $\pi$ is not that of a free group, the sequence is generally not aperiodic and topological properties of $\pi$ have to be determined differently. The two cases rely on DNA conformations such as A-DNA, B-DNA, Z-DNA, G-quadruplexes, etc. We found a few salient results: Z-DNA, when involved in transcription, replication and regulation in a healthy situation implies (i). The sequence of telomeric repeats comprising $3$ distinct bases, most of the time, satisfies (i). For $2$-base sequences in the free case (i) or non free case (ii), the topology of $\pi$ may be found in terms of the $SL(2,\mathbb{C})$ character variety of $\pi$ and the attached algebraic surfaces. The linking of two unknotted curves --the Hopf link-- may occur in the topology of $\pi$ in cases of biological importance, in telomeres, G-quadruplexes, hairpins and junctions, a feature we already found in the context of models of topological quantum computing. For $3$- and $4$ base sequences, other knotting configurations are noticed and a building block of the topology is the $4$-punctured sphere. Our methods have the potential to discriminate potential diseases associated to the sequences.

Keywords

DNA conformations; transcription factors; telomeres; infinite groups; free groups; algebraic surfaces; aperiodicity; character varieties

Subject

Biology and Life Sciences, Biochemistry and Molecular Biology

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