Complete Quantum Information in the DNA Genetic Code

Abstract

We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature $G=\mathbb{Z}_5 \rtimes H$ where $H=\mathbb{Z}_2 . S_4\cong 2O$ is isomorphic to the binary octahedral group $2O$ and $S_4$ is the symmetric group on four letters/bases. The second group has signature $G=\mathbb{Z}_5 \rtimes GL(2,3)$ and points out a threefold symmetry of base pairings. For those groups, the representations for the $22$ conjugacy classes of $G$ are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the $22$ characters of $G$ attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.

SL(2, C) Scheme Processing of Singularities in Quantum Computing and Genetics

“…Over the last few years, the authors of this article have found that some mathematical techniques employed for quantum information processing and quantum computing may also apply to biology at the genome scale. More precisely, group theory and representations of symmetries with characters of finite groups have been used for topological quantum computing (TQC) [19] or elementary particles [29], and the the encoding of proteins [30]. Methods for dealing with infinite groups and SL(2, C) representations of such groups in TQC papers [5,31] were similarly employed for transcription factors [6] and miRNAs [6].…”

SL(2, C) Scheme Processing of Singularities in Quantum Computing and Genetics

“…Over the last few years, the authors of this article have found that some mathematical techniques employed for quantum information processing and quantum computing may also apply to biology at the genome scale. More precisely, group theory and representations of symmetries with characters of finite groups have been used for topological quantum computing (TQC) [14] or elementary particles [38], and the encoding of proteins [39]. Methods for dealing with infinite groups and SL(2, C) representations of such groups in TQC papers [11,40] were similarly employed for transcription factors [12] and miRNAs [12].…”

Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

“…We made use of group theory for investigating transcription factors in genetics. Finite group theory plays a big role in the attempts to model the genetic code, see [25,26] and other references therein. Finitely generated groups (whose cardinality is infinite) are necessary to deal with the secondary structures of proteins [2].…”