Nonlinear excitations in DNA molecule

Abstract

Applied mathematics is one of the research fields that developed over the last few thousand years and still continues to develop. Mathematical models allow researchers to analyze a simplified structure of a biological system and predict its behavior. In fact, interdisciplinary research can offer answers to several unexplained phenomena and mathematical biology, in particular, allows the analysis of living organisms. Such analysis might involve the appearance, the development or even the death of the organisms, or simply explain the causes and the conditions in which a process takes place. Deoxyribonucleic acid (DNA) is one of a major focus for mathematical biologists, biophysicists, chemists as well as biologists. DNA is a long polymer consisting of two chains of bases, in which the genetic information is stored. A base from one chain has a corresponding base on the other chain which together forms a so-called base-pair. Our starting is focused on the nonlinear dynamics of the DNA molecule, particularly the study of its denaturation. We revisit one of the first nonlinear models of denaturation process of DNA, which neglects the inhomogeneities due to the base sequence and the asymmetry of the two strands: the Peyrard Bishop (PB) model. Using the semi-discrete method, we derive a nonlinear Schrödinger equation with higher-order term. Thereafter, the modulational instability (MI) is investigated through the linear stability analysis as well as the variational approach. The effect of the coupling between the base pairs is examined. Knowing that the PB model does not take into account the helical nature of the DNA molecule, we consider the Peyrard Bishop Dauxois model, which takes into account the double helix of DNA. We show using the reductive perturbation method that the dynamics of the system can be described by a set of coupled nonlinear Schrödinger equations. The relevant MI scenarios are explored and we note that the system is stable under the modulation for certain parameter values of the PBD model. We also point out the impact of the group velocity on the stability of the system understudy. The PBD model does not take into account the angle of torsion between pairs of adjacent bases and the effect of solvent. So, we reconsidered the PBD model improved by Marco Zoli. The generalized discrete nonlinear Schrödinger equation is then derived for the twisted DNA with solvent interaction. We present an analytical expression for the MI gain and show the effects of twist angle on MI gain spectra as well as on stability diagram. Numerical simulations are carried out to show the validity of the analytical approach. The impact of the twist angle is investigated and we obtain that the twist angle affects the dynamics of stable patterns gener-XVIII ated through the molecule. Energy localization in the framework of twisted DNA with solvent interaction is also studied. The rotational and torsional degrees of freedom of the DNA sequence are considered to play an important role for DNA transcription as witnesses rotational model of Yakushevich. Using the perturbation method, we derive from this model a discrete coupled nonlinear Schrödinger equation that allowed us to study the MI through a simple mode excitation as well as through a double mode excitation. We have focused our attention on the effect of the coupling parameters, including the torsion coefficients. Direct numerical simulations of the nonlinear Schrödinger equation have been carried out in order to verify the validity of our analytical results. We have also been interested to look for exact analytical solutions of the PB and PBD models. As regards the model of PB, the effect of harmonic coupling is highlighted, while in the PBD model, we threw our sights on the impact of the angle of torsion and the solvent coefficient. The stability of these solitons has been also investigated.