Perturbed soliton excitations in the DNA double helix

We study the nonlinear dynamics of the inhomogeneous DNA double-helical chain using the dynamic plane-base rotator model by considering angular rotation of bases in a plane normal to the helical axis. The DNA dynamics in this case is found to be governed by a perturbed sine-Gordon equation, while taking into account the interstrand hydrogen bonding energy, between bases, and the intrastrand inhomogeneous stacking energy and by making an analogy with the Heisenberg model of the Hamiltonian of an inhomogeneous anisotropic spin ladder with ferromagnetic legs and antiferromagnetic rung coupling. In the homogeneous limit the dynamics is governed by the kink-antikink soliton of the sine-Gordon equation which represents the formation of an open state configuration in the DNA double helix. The effect of inhomogeneity in the stacking energy in the form of localized and periodic variations on the formation of open states in DNA is studied under perturbation. The perturbed soliton is obtained using a multiple-scale soliton perturbation theory by solving the associated linear eigenvalue problem and by constructing the complete set of eigenfunctions. The inhomogeneity in stacking energy is found to modulate the width and speed of the soliton depending on the nature of the inhomogeneity. Also it introduces fluctuations in the form of a train of pulses or periodic oscillations in the open state configuration.