Abstract
The dynamics of the Peyrard-Bishop model for vibrational motion of DNA dynamics, which has been extended by taking into account the rotational motion for the nucleotides (Silva et al., J. Biol. Phys. 34, 511–519, 2018) is studied. We report on the presence of the modulational instability (MI) of a plane wave for charge migration in DNA and the generation of soliton-like excitations in DNA nucleotides. We show that the original differential-difference equation for the DNA dynamics can be reduced in the continuum approximation to a set of three coupled nonlinear equations. The linear stability analysis of continuous wave solutions of the coupled systems is performed and the growth rate of instability is found numerically. Numerical simulations show the validity of the analytical approach with the generation of wave packets provided that the wave numbers fall in the instability domain.
Highlights
The development of the theory of the modulation instability (MI) as a well-known nonlinear phenomenon has attracted considerable attention and started almost simultaneously and occurred in parallel in hydrodynamics by Benjamin and Feir [1, 2] and in electrodynamics by Ostrovskii et al [3, 4]
Nonlinear interactions between atoms in DNA can give rise to intrinsically localized breather-like vibration modes [18]. Such localized modes, being large amplitude vibrations of a few (2 or 3) particles, can facilitate the disruption of base pairs and in this way initiate conformational transitions in DNA. These modes can occur as a result of the modulational instability of continuum-like nonlinear modes [19], which is created by energy exchange mechanisms between the nonlinear excitations
According to results obtained by many authors about the modulational instability, it is known that linear stability analysis can determine the instability field in space parameter e r=8 r=1 r=5
Summary
The development of the theory of the modulation instability (MI) as a well-known nonlinear phenomenon has attracted considerable attention and started almost simultaneously and occurred in parallel in hydrodynamics by Benjamin and Feir [1, 2] and in electrodynamics by Ostrovskii et al [3, 4]. A model which has two degrees of freedom per base pair: one radial variable related to the opening of the hydrogen bonds and an angular one related to the twisting of each base-pair responsible for the helicoidal structure of the molecule, has been built by Barbi et al [15], and later improved by Cocco and Monasson [16] Such a model provides an extension of the Peyrard-Bishop approach towards a more realistic description of biological processes. Such localized modes, being large amplitude vibrations of a few (2 or 3) particles, can facilitate the disruption of base pairs and in this way initiate conformational transitions in DNA These modes can occur as a result of the modulational instability of continuum-like nonlinear modes [19], which is created by energy exchange mechanisms between the nonlinear excitations.
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