STABILITY OF LOCALIZED STANDING WAVES IN THE PEYRARD–BISHOP–DOKSUA DNA MODEL - Goloskubova, N. S.; Mikhlin, Yu. V. (Computer Science and Applied Mathematics, 2018)
Many recent works are devoted to study of wave processes in the DNA molecule, which are determined both by interest in the important biological problems, and the development of modern nano-technologies. One of the most successful models of the DNA molecule dynamics is the model developed by Peyrard, Bishop and Dauxois (the PBD model) which is considered in this paper. In the PBD model two strands of the DNA are represented by linked chains of hard disks, where the bonds within base pairs are described by the Morse potential and the interaction between opposite discs of different chains is represented by the anharmonic potential. In this model, the transfer of the DNA duplex in space as a whole is not considered, and only a divergence of different chains is studied. The stability of localized standing waves in the PBD model is investigated here by using the numerical-analytical procedure which is based on the well-known Lyapunov stability criterion. Localized standing waves are excited by either the initial displacement describing the divergence of the chosen opposite discs of two chains, or the corresponding initial velocity. Then we use a comparison of the current values of the kinetic energies of neighboring elements with initial kinetic energy of the excited node. Instability of localized standing wave is fixed if more than ten percent of the initial kinetic energy is transferred to neighboring discs. Calculations are realized in nodes of some mesh in chosen domain of the system parameters. These calculations are stopped when boundaries of the stability/instability regions are stabilized in the chosen mesh. This is a principal criterion of the choice of the calculation time. As a result, boundaries of stability/instability regions in some places of the system parameters are obtained for a case of two connected chains consisting of nine discs each. As an illustration, the calculations of the localized standing waves in such chains are given for the values of parameters taken in regions of the wave stability or instability. We can see, in particular, that a decrease of the parameter of interaction between pairs of bases along chain leads to increase of the localized standing wave stability.