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

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 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.

Algebraic Morphology of DNA--RNA Transcription and Regulation

“…The later group is found as governing the structure of many transcription factors and is associated to the link found in [7, Figure 2]. The card seq for π 3 is [7,14,89,264,1987,11086,93086 Table 2 lists a few selected transcription factors, their card seq and the corresponding singular surfaces, if any As announced, in the selected transcription factors, there exists a correlation between the lack of 'syntactical freedom', or the presence of a singular surface in the character variety, with an identified disease.…”

“…In this case this surface seems to possess a regulatory action that may be lost in the latter case. In [7,Table 3], the potential diseases associated to a non-free group structure are mentioned. In the present section, one explores in Table 2 the case of a 3-letter sequence of a TF and in Table 3 the case of 3-letter sequence of a miRNA.…”

“…This contrasts with some sequences encountered in another context. In [7], we found two cases of two-base sequences whose corresponding Groebner basis contains the surfaces…”

“…Recently, we also introduced concepts for representing the groups f p over the Lie group SL(2, C). The SL(2, C) character variety of f p and its Groebner basis are topological ingredients, they feature algebraic geometric properties of the group f p under question [6,7]. For the definition of the Groebner basis of an ideal containing multivariate polynomial rings, the reader may consult Reference [8].…”

Algebraic Morphology of DNA–RNA Transcription and Regulation

“…Recently, we also introduced concepts for representing the groups f p over the Lie group SL(2, C). The SL(2, C) character variety of f p and its Groebner basis are topological ingredients, they feature algebraic geometric properties of the group f p under question [12][13][14]. A reminder of the theory is given in the next section.…”