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

• Published in: International journal of molecular sciences Vol. 23; no. 21; p. 13290
• Main Authors: Planat, Michel, Amaral, Marcelo M., Fang, Fang, Chester, David, Aschheim, Raymond, Irwin, Klee
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.11.2022; MDPI
• Subjects: Algebra; algebraic surfaces; Amino acids; DNA conformations; DNA structure; Eigenvalues; free groups; Gene regulation; Generators; Group theory; infinite groups; Nucleotide sequence; Quantum computing; Subgroups; Telomeres; Topology; Transcription factors
• ISSN: 1422-0067; 1661-6596; 1422-0067
• DOI: 10.3390/ijms232113290

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.