539.12With the help of energy analysis suggested by McLaughlin and Scott for the sine-Gordon equation, evolution of kink velocity modeling the propagation of a local conformational perturbation along the DNA molecule under the simultaneous action of dissipation effects and special nonstationary external fields is investigated. For a harmonically time-dependent external force, the kink velocity is characterized by oscillations about a rather monotonically decreasing trend expressed by an explicit analytical formula. The trend velocity coincides with the results of calculations of the kink velocity averaged over the period on different time intervals. Similar results are obtained for a nonstationary force in the form of a step function. A numerical analysis, in particular, demonstrates that the trend velocity changes its behavior for certain values of rectangular pulse duration.Nonlinear perturbations propagating along the DNA molecule are important for biological activity of the molecule. In [1] it was demonstrated that the kink motion is modeled by the sine-Gordon equation, and the effect of the medium is modeled by two additional terms that consider the effects of dissipation and nonstationary external force.We now write down the sine-Gordon equation with the DNA parameters and additional terms in designations of [2, 3]:Here ( , ) z t φ = φ is the angular shift of bases from the equilibrium position at time t at the point whose coordinate along the DNA molecule is equal to z ; I is the moment of inertia of the bases; K′ is the constant characterizing the rigidity of the sugar-phosphate DNA chain; а is the distance between the nearest bases in the DNA; V is the energy required to break the hydrogen bonds inside of pairs; β is the dissipation coefficient; ( ) F t is the moment of generalized nonstationary external force (below we use the term force) whose value is naturally considered to be bounded in time, that is, the external force amplitude isEquation (1) is reduced to the well-known sine-Gordon equation forwhich (for example, see [4,5]) is integrated by the inverse scattering transform method and has soliton solutions. In particular, a single-soliton solution (kink) of sine-Gordon equation (3) has the form Tomsk State University,
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