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 (Formula presented.), one can discriminate between two important families: (i) the cardinality structure for conjugacy classes of subgroups of (Formula presented.) 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 (Formula presented.) is not that of a free group, the sequence is generally not aperiodic and topological properties of (Formula presented.) 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 (Formula presented.) may be found in terms of the (Formula presented.) character variety of (Formula presented.) and the attached algebraic surfaces. The linking of two unknotted curves—the Hopf link—may occur in the topology of (Formula presented.) 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.