Upon the Joyeux-Buyukdagli model of DNA, the helicoidal interactions are introduced, and their effects on the dynamical behaviors of the molecule investigated. A theoretical framework for the analysis is presented in an external force field, taking into account Stokes and hydrodynamics viscous forces. In the semi-discrete approximation, the dynamics of the molecule is found governed by the cubic complex Ginzburg-Landau (CGL) equation. By choosing an appropriate decoupling ansatz, the cubic CGL equation is transformed into a nonlinear differential equation whose analytical solitary wave-like solutions can be explored by means of the direct method, which is more tractable in case where the form of soliton solutions is known. Based on this, a dissipative bright-like soliton solution is obtained. Numerical experiments have been done, and relevant results were brought out, such as the quantitative and qualitative influences of the helical interactions on the parameters of the traveling bubble. The important role-played by these interactions in the DNA biological processes is brought out, showing that depending on the wave number, their effects can increase, decrease, or keep constant the bubble angular frequency, velocity, amplitude, and width, as well as the energy involved by enzymes in the initiation of DNA biological processes. This can prevent some coding or reading errors and resulting genetic damages. Analytical predictions and numerical experiments were in good agreement.

Transport memory effects on nonlinear wave propagation are addressed in a damped Peyrard-Bishop-Dauxois model of DNA dynamics. Under the continuum and overdamped limits, the multiple-scale expansion method is employed to show that an open-state configuration of the DNA molecule is described by a complex nonlinear Schrödinger equation. For the latter, solutions are proposed as bright solitons, which suitably represent the open-state configuration that takes place along the DNA molecule in the form of bubbles. A good agreement between numerical experiments and analytical predictions on the impact of memory effects on the angular frequency, velocity, width, and amplitude of the moving bubble is obtained. It also appears that memory effects can modify qualitatively and quantitatively the nonlinear dynamics of DNA, including the energy brought by enzymes for the initiation of the processes of replication and transcription.