Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

Planat, Michel; Amaral, Marcelo M.; Fang, Fang; Chester, David; Aschheim, Raymond; Irwin, Klee

doi:10.3390/cimb44040095

Cited by 7 publications

(25 citation statements)

References 36 publications

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“…According to reference [ 5 ], aperiodicity correlates to the syntactical freedom of ordering rules. This statement was checked in the realm of transcription factors (Section 4 in [ 4 ]). Let us introduce the concept of a general substitution rule in the context of free groups.…”

Section: Materials and Methodsmentioning

confidence: 99%

“…This approach was applied to binding motifs of transcription factors [ 4 ]. The binding motif rel in the finitely presented group

is split into appropriate segments so that

with the substitution rules

.…”

Section: Materials and Methodsmentioning

confidence: 99%

“…We coined the term ‘syntactical freedom’ for recognizing this property, with inspiration from an earlier work [ 5 ]. We also showed that such free groups have the distinctive property of generating an aperiodic substitution rule, providing a connection between (group) syntactical freedom and irrational numbers (Section 4 in [ 4 ]). For an infinite group, the representation cannot be based on characters but on the so-called character variety.…”

Section: Introductionmentioning

confidence: 95%

“…For modeling DNA in its various conformations taken in transcription factors, telomeres and other building blocks of molecular biology, we need infinite groups defined from a motif. A sequence of the DNA nucleotides serves as the generator of the group [ 4 ]. In this context, it has been found that a group that is not free is often the witness of a potential disease.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 , we briefly account for the many types of topologies that DNA can show, in terms of double strands or more strands. Then, we recall the mathematical concepts employed in our paper with some redundancy with earlier work [ 4 , 6 ].…”

Section: Introductionmentioning

confidence: 99%

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DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces

Planat

Amaral

Fang

et al. 2022

IJMS

Self Cite

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Taking a DNA sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group π, one can discriminate between two important families: (i) the cardinality structure for conjugacy classes of subgroups of π is that of a free group on one to four bases, and the DNA word, viewed as a substitution sequence, is aperiodic; (ii) the cardinality structure for conjugacy classes of subgroups of π is not that of a free group, the sequence is generally not aperiodic and topological properties of π 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 three distinct bases most of the time satisfies (i). For two-base sequences in the free case (i) or non-free case (ii), the topology of π may be found in terms of the SL(2,C) character variety of π and the attached algebraic surfaces. The linking of two unknotted curves—the Hopf link—may occur in the topology of π in cases of biological importance, in telomeres, G-quadruplexes, hairpins and junctions, a feature that we already found in the context of models of topological quantum computing. For three- and four-base sequences, other knotting configurations are noticed and a building block of the topology is the four-punctured sphere. Our methods have the potential to discriminate between potential diseases associated to the sequences.

show abstract

Section: Materials and Methodsmentioning

confidence: 99%

“…This approach was applied to binding motifs of transcription factors [ 4 ]. The binding motif rel in the finitely presented group

is split into appropriate segments so that

with the substitution rules

.…”

Section: Materials and Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces

Planat

Amaral

Fang

et al. 2022

IJMS

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Group Theory of Messenger RNA Metabolism and Disease

Planat,

Amaral,

Chester

et al. 2024

Gene Expr

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Our recent work has focused on the application of infinite group theory and related algebraic geometric tools in the context of transcription factors and microRNAs. We were able to differentiate between "healthy" nucleotide sequences and disrupted sequences that may be associated with various diseases. In this paper, we extend our efforts to the study of messenger RNA (mRNA) metabolism, showcasing the power of our approach. We investigate (1) mRNA translation in prokaryotes and eukaryotes, (2) polyadenylation in eukaryotes, which is crucial for nuclear export, translation, stability, and splicing of mRNA, (3) microRNAs involved in RNA silencing and post-transcriptional regulation of gene expression, and (4) identification of disrupted sequences that could lead to potential illnesses. To achieve this, we used: (a) infinite (finitely generated) groups f p , with generators representing the r + 1 distinct nucleotides and a relation between them [e.g., the consensus sequence in the mRNA translation (i), the poly (A) tail in item (ii), and the microRNA seed in item (iii)]; (b) aperiodicity theory, which connects healthy groups f p to free groups F r of rank r and their profinite completion ˆr F , and (c) the representation theory of groups f p over the space-time-spin group SL 2 (C), highlighting the role of surfaces with isolated singularities in the character variety. Our approach could potentially contribute to the understanding of the molecular mechanisms underlying various diseases and help develop new diagnostic or therapeutic strategies.

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Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

Amaral¹,

Fang²,

Chester³

et al. 2022

Preprint

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It is shown that the representation theory of some finitely presented groups thanks to their $SL_2(\mathbb{C})$ character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. We study the connection of $SL_2(\mathbb{C})$ character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link $H$, whose character variety is a Del Pezzo surface $f_H$ (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups as well as that of the fundamental group for the singular fibers $\tilde{E}_6$ and $\tilde{D}_4$ contain $f_H$. A surface birationally equivalent to a $K_3$ surface is another compound of their character varieties.

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Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

Cited by 7 publications

References 36 publications

DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces

DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces

Group Theory of Messenger RNA Metabolism and Disease

Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

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