Solitons in DNA and Biological Implications

Modelling the deoxyribonucleic acid (DNA) dynamics and understanding the action of all the various factors involved remain the major challenge in nonlinear physics, biology and even in chemistry. This vivid interest arises of course not only from the biological relevance of DNA but also from its physical properties which can now be probed through single-molecule micromanipulation experiments. Many studies have clearly pointed out that DNA must be considered as a dynamical object whose nonlinear distortions could play a major role in its functions. DNA is a complex system which has many nearly isoenergetic ground states and may therefore be considered as a ﳿuctuating aperiodic system.
Nonlinear interactions can give rise to very stable excitations, called solitons, which can travel without changing their shape. Nonlinear interactions between atoms in DNA can give rise to intrinsically localized breather-like vibration modes. These modes can occur as a result of modulational instability of continuum-like nonlinear modes, which are created by energy exchange mechanisms between nonlinear excitations. The phenomenon of modulational insta- bility can be deﴀned as a self-induced break-up of an initially homogeneous wave during its evolution in a nonlinear medium. Furthermore, modulational instability has been suggested to be responsible for energy localization mechanism, leading to the formation of large-amplitude nonlinear excitations in hydrogen-bonded crystals or DNA molecules. So, it is important and worthwhile to study the properties of modulational instability in nonlinear media such as DNA. Therefore, the ﴀrst part of this book is dedicated to the study of modulational instability in some models of DNA nonlinear dynamics. That part is divided into three chapters. The ﴀrst chapter deals with MI and patterns formation in the Peyrard-Bishop model. The impact of viscosity, helicity as well as long-range interactions, on nonlinear dynamics will be investigated. In Chapter II, anharmonic models of DNA such as the Dauxois-Peyrard-Bishop and the twist- opening models are concerned. We bring out the impact of anharmonicity on the one hand, and the impact of the coupling between the twist angle and the radial displacement on the other hand. In Chapter III, we study MI in models which take into account solvent interactions and the coupling between charge and displacement. We show that, solvent interactions make the amplitude of the waves and the density of energy tunable, which means that, by playing on the value of the solvent interaction constant, it is possible to get high-amplitude waves and high energy density as well.
Because of the close relationship between MI and generation of solitons, the investigation of exact solutions, in particular solitons, for nonlinear mathematical physics is an interesting subject. Then, the second part of this work is dedicated to the study of soliton solutions and nonlinear excitations for DNA dynamics.
In Chapter IV, we investigate analytical solutions for the Peyrad-Bishop model. We pay attention to the impact of viscosity. In that chapter, we also study the impact of the strong nonlinear coupling between the upper and the lower cut-oﴁ modes of the optical band. In Chapter V, we investigate soliton excitations and bubble formation in anharmonic models of DNA, namely the Dauxois-Peyrard-Bishop model. The book ends with Chapter VI which is based on the study of soliton-like excitations in coupled models of DNA. In this respect, soliton solutions for the twist-opening model are investigated as well as formation of nonlinear localized modes in the two-component helicoidal model. For the last case, the resonance mode is investigated and the form of the eyelike bubble observed within transcription is simulated.
Our study ends with a general conclusion summarizing the most important results obtained and listing some other problems encountered. We also present the outlooks related to this work