Charge transport in a DNA model with solvent interaction
The charge transport in the modified DNA model is studied by taking into account the factor of solvent and the effect of coupling motions of nucleotides. 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. By applying the continuum approximation, we show that the original differential-difference equation for the DNA dynamics can be reduced to a set of three coupled nonlinear equations. The linear stability analysis of wave solutions of the coupled systems is performed and the growth rate of instability is found numerically. We also investigate the impact of solvent interaction. The solvent factor introduces a new behavior to the wave patterns, modifying also the intrinsic properties of localized structures. In the numerical simulations, we show that the solitons exists when taking into account the effect of solvent and confirms an highest propagation of localized structures in the systems. The effect of solvent forces introduces a robustness behavior to the formed patterns, reinforcing the idea that the information in the DNA model is confined and concentrated to specific regions for efficiency. We also show that the localized structures can be disappeared with the highest value of solvent factor and thereafter the information within the molecule is not perceptible or not transmitted to another sites.
KeywordsCharge transport Peyrard–Bishop model Solvent interaction
Charge transfer through biomolecular systems is one of the most promising ongoing investigations in biophysics and nanotechnology. So, understanding the charge-transport mechanisms is essential for the development of molecular electronic devices. In this context, the study of different modes of charge transfer, both theoretical and experimental, is devoted to illumination of the mechanisms of charge transport [1, 2, 3]. In this context, Cherstvy et al.  studied the non-linear effect in charge transport. Sun-Yong et al.  studied the effect of the base pair on the charge transport in double-strand DNA. The authors proved that the charge transport decreases when the base pairs are opened. Dirk et al.  proved that the soliton is responsible for the energy transfer and localization. Toko et al.  investigated the propagation of localized structures in a DNA model, which takes into account helicity and solvent interaction by using the Peyrard and Bishop model. The authors showed that the solvent interaction term modulates and increases the width of the pulses . In recent years, localized and nonlinear excitations [8, 9, 10, 11, 12, 13, 14, 15, 16], (solitons, discrete breathers, intrinsic localized modes) have been drawing increasing attention and are widely believed to be responsible for several effects in molecular chains, such as charge and thermal conductivity, energy transfer, and localization. A particular interesting discrete system that support solitons and localized modes is desoxyribonucleic acid, or DNA. In this system, localization of energy has been suggested as a precursor of the transcription bubble , and moving localized oscillations as a method of transport of information along the double strand .
In recent years, due to experiments on single molecules of DNA, models with two and more degrees of freedom have been introduced with insistence on the radial and torsional aspects. We take as an example Barbi, Cocco and Peyrard , and later improved by Cocco and Monasson [13, 14]. In recent research, localized structure waves are paid too much attention while studying DNA dynamics in the presence of some perturbations. The impact of damping and thermal fluctuations on lattice soliton patterns due to the presence of thermal noise were studied by Arévalo et al. [15, 16] and Ekobena et al. , who demonstrated a gradual increase energy soliton pattern due to the presence of thermal noise in the bi-exciton molecular chain. Kalosakas et al.  shows that the thermal fluctuations do not destroy completely the soliton localization in biological molecule. Many authors, for example Samora-Sillero et al. , have shown that the most standard mechanism through which bright solitons or solitary wave structures appear is through the activation of modulational instability (MI) of plane waves. Fialko and Laklno [20, 21]has studied the transport of charge and hole along the short DNA molecule by using the Peyrard–Bishop–Holstein (PBH) model [22, 23]. These authors investigate the impact of long-range transfer of charges through the DNA molecule.
Then, Kornyshev et al.  studied the process of denaturation in a DNA model by nonlinear effects of torsional deformations. Hidayat et al.  investigated the impact of the viscosity in the process of denaturation. The result obtained showed how at a certain temperature the increase of the viscosity coefficient will decrease the melting temperature. The way that the solvent factor affects the transport of charges via MI, in the DNA model with the vibrational and rotational coupled motions, is the main focus of the present work. This work is organized as follows. In Section 2, we propose the model and we derive the equations of motion. In Section 3, linear analysis is also studied in this part and predictions for some localized structure formation. The validity of this analysis is proved by numerical simulations in Section 4 where we will point out the effect of solvent interaction, which will lead to a conclusion in Section 5.
2 Model and equations of motion
3 MI analysis and DNA wave patterns
4 Numerical analysis of MI and DNA wave patterns
We discuss solvent interactions and the possibility of generation of soliton-like excitations along the DNA model, based on a set of coupled nonlinear equations. The linear stability has been studied under the continuum approximation and the emergence of localized structures in DNA model have been displayed. From numerical methods, we have plotted the region of stability/instability due to the increase of solvent interaction. We also showed that our model could be subjected to MI, as indicated by the numerical representation of the growth rate of instability. Our analytical predictions have been verified numerically, where the pattern of charges has been displayed. In this case, increasing the solvent factor, the domains of instability increases and prevents charge spreading. The potential barrier brought by the term fsD destroys the H-bonds and blocks charge spreading. We also show that the solvent factor does not completely destroy the emergence of localized modes but prevents the propagation of the flow of information in the cells. We note that the charge patterns are sensitive to solvent interactions as they get more localized with increasing fs and ls. Further increasing fs, has revealed another interesting feature, as patterns of charge can be localized over a long time for highest values of ls. The solvent molecules can collide and bring additional charges in the patterns. This explains the highest density of charges observed in Fig. 6. Taking into account the solvent factor in our model, the localized structures become robust with high values of ls. The spectrum of behaviors is displayed in Fig. 6. We can see that, the localized modes persists with robustness behavior a much better candidate of non-linear modes responsible for a locally open state where biological functioning takes place. The efficiency of transport of charges is affected by stretching of the molecule. The density of loads vanishes and the efficiency drops steeply. Thereafter, the life-time of charges decreases with an increase of fs reinforces the spatial confinement of the charge carrier in specific domains. Increasing fs and ls in the model also revealed that due to the environment effect of the charges, we observe that the bubbles form spatiotemporal “hot-spots”, which inhibit charge propagation along the strands and enhance its confinement. Thereafter, the flux of charges is concentrated for a long times in specific sites. Consequently, the transfer of information along the molecule is also blocked. This explains some chromosome diseases or mutations due to an accumulation of mutations that increase its proliferation capabilities, the instability of its genome, and its ability to escape systems that eliminate abnormally proliferating cells. Our results obtained suggest that it is possible to reduce some chromosome diseases by including the solvent factor interactions and χ in DNA model. Taking into account the effect of solvent factor and the coupling constant, the density of charges can vanish, and only the small quantity of charges can migrate in the DNA model. We can conclude that, the solvent factor can be facilitators or inhibitors charge transport in the modified DNA model and the good conduction of current is also depend on the flux of charges carrier. However, theoretical investigations of charge transport and localization in DNA are complicated, not only due to the intrinsic disorder caused by the different nucleotides and dynamics of bases present in DNA but also because of the softness of the biomolecule. However, in this present study, other factors are not considered, such as viscosity and diffusion effect. It is important to consider their influence in the charge transport model, which is work that we are currently doing.
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Conflict of interests
The authors declare that they have no competing financial interests. There are no competing interests related to this work. There are no known conflicts of interest associated with this work.
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