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

International audienceTaking a DNA sequence, a word with letters/bases A, T, G and C, asthe relation between the generators of an infinite group p, one candiscriminate between two important families: (i) the cardinalitystructure for conjugacy classes of subgroups of p is that of a freegroup on one to four bases, and the DNA word, viewed as asubstitution sequence, is aperiodic; (ii) the cardinality structurefor conjugacy classes of subgroups of p is not that of a freegroup, the sequence is generally not aperiodic and topologicalproperties of p have to be determined differently. The two casesrely on DNA conformations such as A-DNA, B-DNA, Z-DNA,G-quadruplexes, etc. We found a few salient results: Z-DNA, wheninvolved in transcription, replication and regulation in a healthysituation, implies (i). The sequence of telomeric repeatscomprising three distinct bases most of the time satisfies (i). Fortwo-base sequences in the free case (i) or non-free case (ii), thetopology of p may be found in terms of the SL(2,C) charactervariety of p and the attached algebraic surfaces. The linking oftwo unknotted curves—the Hopf link—may occur in the topology of pin cases of biological importance, in telomeres, G-quadruplexes,hairpins and junctions, a feature that we already found in thecontext of models of topological quantum computing. For three- andfour-base sequences, other knotting