Synchronized Oscillations in Double-Helix B-DNA Molecules with Mirror-Symmetric Codons

A fully analytical treatment of the base-pair and codon dynamics in double-stranded DNA molecules is introduced, by means of a realistic treatment which considers different mass values for G, A, T, and C nucleotides and takes into account the intrinsic three-dimensional, helicoidal geometry of DNA in terms of a Hamitonian in cylindrical coordinates. Within the framework of the Peyrard-Dauxois-Bishop model we consider the coupling between stretching and stacking radial oscillations as well as the twisting motion of each base pair around the helix axis. By comparing the linearized dynamical equations for the angular and radial variables when going from the bp local scale to the longer triplet codon scale, we report an underlying hierarchical symmetry. The existence of synchronized collective oscillations of the base-pairs and their related codon triplet units are disclosed from the study of their coupled dynamical equations. The possible biological role of these correlated, long-range oscillation effects in double standed DNA molecules containing mirror-symmetric codons of the form XXX, XX’X, X’XX’, YXY, and XYX is discussed in terms of the dynamical equations solutions and their related dispersion relations.