Charge transfer in DNA: effective Hamiltonian approaches

Aperiodic crystals in biology

Biological systems display a broad palette of hierarchically ordered designs spanning over many orders of magnitude in size. Remarkably enough, periodic order, which profusely shows up in non-living ordered compounds, plays a quite subsidiary role in most biological structures, which can be appropriately described in terms of the more general aperiodic crystal notion instead. In this topical review I shall illustrate this issue by considering several representative examples, including botanical phyllotaxis, the geometry of cell patterns in tissues, the morphology of sea urchins, or the symmetry principles underlying virus architectures. In doing so, we will realize that albeit the currently adopted quasicrystal notion is not general enough to properly account for the rich structural features one usually finds in biological arrangements of matter, several mathematical tools and fundamental notions belonging to the aperiodic crystals science toolkit can provide a useful modeling framework to this end.

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 that 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 corresponding to the bp local scale with those of 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.

Base-Pairs' Correlated Oscillation Effects on the Charge Transfer in Double-Helix B-DNA Molecules

By introducing a suitable renormalization process, the charge carrier and phonon dynamics of a double-stranded helical DNA molecule are expressed in terms of an effective Hamiltonian describing a linear chain, where the renormalized transfer integrals explicitly depend on the relative orientations of the Watson-Crick base pairs, and the renormalized on-site energies are related to the electronic parameters of consecutive base pairs along the helix axis, as well as to the low-frequency phonons' dispersion relation. The existence of synchronized collective oscillations enhancing the π-π orbital overlapping among different base pairs is disclosed from the study of the obtained analytical dynamical equations. The role of these phonon-correlated, long-range oscillation effects on the charge transfer properties of double-stranded DNA homopolymers is discussed in terms of the resulting band structure.