Transverse interaction in DNA molecule
Introduction
There are a couple of models describing DNA molecule (Gaeta et al., 1994, Yakushevich, 1998). According to most of them solitonic waves move through the chain. Namely, if DNA is imagined as a series of particles and springs then any perturbation on a certain particle will affect the neighbouring springs and nucleotides and, consequently, propagate along the molecule. Of course, such propagation is nothing but the wave. The wave can be either a breather, which is a modulated solitonic wave (Peyrard and Bishop, 1989, Dauxois, 1991, Forinash et al., 1997, Barbi et al., 1999, Zdravković and Satarić, 2006, Tabi et al., 2008), or a kink-soliton (Englander et al., 1980, Yakushevich, 1989, Yakushevich et al., 2002, Gaeta, 2006, Gaeta, 2007, Daniel and Vasumathi, 2007, Daniel and Vasumathi, 2008, Vasumathi and Daniel, 2008, Tabi et al., 2009a). Results of some numerical calculations bring about a conclusion that the envelope soliton may exist in DNA (Fei et al., 1994, Miles, 1981). Recently, we gave a couple of experiment proposals whose goal should be to study the nature of such wave (Zdravković and Satarić, 2008, Zdravković and Satarić, 2009b, Zdravković and Satarić, 2010).
It is well known that a DNA molecule consists of two compatible chains. In this paper we are concerned with the transverse interaction between the two strands of the molecule. This interaction is very often modelled by the Morse potential. This means that this paper may represent a contribution to any model of DNA using the Morse potential. As an example we very briefly describe the well-known Peyrard–Bishop–Dauxois (PBD) model (Peyrard and Bishop, 1989, Dauxois, 1991) in Section 2. Note that the name of the model is not unique as in some papers a similar model introduced in Dauxois et al. (1993) is also called as the PBD model.
A key problem in this issue is the influence of inhomogeneity on DNA dynamics. In Section 3 we study under what conditions the solitonic wave could be stable. In other words, we are looking for a combination of the two relevant parameters, which brings about a constant shape of the bottom of the Morse potential well. For this to be clear some knowledge of DNA dynamics based on the idea of the solitonic wave is important and this is explained in Section 2.
We close the paper with some concluding remarks in Section 4.
Section snippets
The Peyrard–Bishop–Dauxois model
According to the PBD model the DNA chain is treated as a perfectly homogenous periodic structure and only transversal motions of nucleotides are taken into consideration. This means that a common mass m is used for all the nucleotides as well as the values of a couple of parameters describing geometry and chemical bonds in DNA.
If the displacements of the nucleotides at the site n from their equilibrium positions are un and for the two strands then the Hamiltonian for the DNA chain is (
Transverse interaction
It was mentioned above that the DNA molecule consists of two compatible chains, which is a well known fact. Chemical bonds between neighbouring nucleotides belonging to the same strands are strong covalent bonds. On the other hand, nucleotides at a certain site n, belonging to the different strands, are connected through weak hydrogen interaction. This is where nonlinear effects are coming from and this interaction can be described by the Morse potential energy (Yakushevich, 1998, Peyrard and
Conclusion and discussions
In this paper we studied the influence of inhomogeneity on the solitonic wave. We stated that the soliton was not too much sensitive as, otherwise, it would be very unstable. Hence, according to this statement, we offered expected values for a′, given by Eqs. (13), (17). We showed that the bottoms of the Morse potentials VM can be very similar for both the AT and the CG pairs. If so then the solitonic wave is stable, i.e. it is very slightly affected by inhomogeneity. This can happen even if
Acknowledgement
This research was supported by funds from Serbian Ministry of Sciences, grants III45010, OI171009 and 142034G.
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