# American Institute of Mathematical Sciences

October  2021, 14(10): 3459-3478. doi: 10.3934/dcdss.2021018

## Computational and numerical simulations for the deoxyribonucleic acid (DNA) model

 1 School of Management & Economics, Jiangsu University of Science and Technology, 212003 Zhenjiang, China 2 Department of Basic Science, Higher Technological Institute 10th of Ramadan City, El Sharqia, 44634, Egypt 3 Institute of Space Sciences, Magurele-Bucharest, Romania 4 Department of Mathematics, Faculty of Science, Jiangsu University, 212013, China 5 Department of Mathematics, Obour High Institute For Engineering and Technology, 11828, Cairo, Egypt 6 Mathematics Department, Faculty of Science, Taif University P.O.Box 11099, Taif 21944, Saudi Arabia

* Corresponding author email: mostafa.khater2024@yahoo.com

Received  October 2019 Revised  January 2020 Published  October 2021 Early access  March 2021

In this research paper, the modified Khater method, the Adomian decomposition method, and B-spline techniques (cubic, quintic, and septic) are applied to the deoxyribonucleic acid (DNA) model to get the analytical, semi-analytical, and numerical solutions. These solutions comprise much information about the dynamical behavior of the homogenous long elastic rods with a circular section. These rods constitute a pair of the polynucleotide rods of the DNA molecule which are plugged by an elastic diaphragm that demonstrates the hydrogen bond's role in this communication. The stability property is checked for some solutions to show more effective and powerful of obtained solutions. Based on the role of analytical and semi-analytical techniques in the motivation of the numerical techniques to be more accurate, the B-spline numerical techniques are applied by using the obtained exact solutions on the DNA model to show which one of them is more accurate than other, to explain more of the dynamic behavior of the homogenous long elastic rods, and to show the coincidence between the different types of obtained solutions. The obtained solutions verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows the power and effectiveness of them for applying to many different forms of the nonlinear evolution equations with an integer and fractional order.

Citation: Raghda A. M. Attia, Dumitru Baleanu, Dianchen Lu, Mostafa M. A. Khater, El-Sayed Ahmed. Computational and numerical simulations for the deoxyribonucleic acid (DNA) model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3459-3478. doi: 10.3934/dcdss.2021018
##### References:
 [1] M. A. Abdelrahman, E. H. Zahran and M. M. Khater, The exp $(-\phi(\xi))$-Expansion method and its application for solving nonlinear evolution equations, International Journal of Modern Nonlinear Theory and Application, 4 (2015), 37. [2] R. A. Attia, D. Lu and M. M. A. Khater, Chaos and relativistic Energy-Momentum of the nonlinear time fractional duffing equation, Mathematical and Computational Applications, 24 (2019), 10-33.  doi: 10.3390/mca24010010. [3] M. D. Barkley and B. H. Zimm, Theory of twisting and bending of chain macromolecules, Analysis of the fluorescence depolarization of DNA, The Journal of Chemical Physics, 70 (1979), 2991-3007. [4] H. M. Baskonus and H. Bulut, Analytical studies on the (1+ 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation defined by seismic sea waves, Waves in Random and Complex Media, 25 (2015), 576-586.  doi: 10.1080/17455030.2015.1062577. [5] V. K. Fedyanin and L. V. Yakushevich, Scattering of neutrons and light by DNA solitons, Studia Biophysica, 103 (1985), 171-178. [6] S. Homma and S. Takeno, A coupled base-rotator model for structure and dynamics of DNA: Local fluctuations in helical twist angles and topological solitons, Progress of Theoretical Physics, 72 (1984), 679-693.  doi: 10.1143/PTP.72.679. [7] M. M. A. Khater, R. A. Attia and D. Lu, Explicit lump solitary wave of certain interesting (3+ 1)-Dimensional waves in physics via some recent traveling wave methods, Entropy, 21 (2019), 397-426.  doi: 10.3390/e21040397. [8] M. M. A. Khater, D. Lu and R. A. Attia, Lump soliton wave solutions for the (2+ 1)-dimensional Konopelchenko-Dubrovsky equation and KdV equation, Modern Physics Letters B, 33 (2019), 1950199, 20 pp. doi: 10.1142/s0217984919501999. [9] M. M. A. Khater, D. Lu and R. A. Attia, Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method, AIP Advances, 9 (2019), 025003. [10] M. M. A. Khater, R. Attia and D. Lu, Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions, Mathematical and Computational Applications, 24 (2019), 1-13.  doi: 10.3390/mca24010001. [11] M. M. A. Khater, R. A. Attia and D. Lu, Numerical solutions of nonlinear fractional Wu-Zhang system for water surface versus three approximate schemes, Journal of Ocean Engineering and Science, (2019). [12] K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two–variable system with noninteger–order derivative, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013145, 15 pp. doi: 10.1063/1.5086909. [13] A. A. Opanuga, E. A. Owoloko, O. O. Agboola and H. I. Okagbue, Application of homotopy perturbation and modified Adomian decomposition methods for higher order boundary value problems, (2017). [14] Z. Parsaeitabar and A. R. Nazemi, A third-degree B-spline collocation scheme for solving a class of the nonlinear Lane-Emden type equations, Iranian Journal of Mathematical Sciences and Informatics, 12 (2017), 15-34. [15] I. Sakalli, Analytical solutions in rotating linear dilaton black holes: Resonant frequencies, quantization, greybody factor and Hawking radiation, Physical Review D, 94 (2016), 084040, 12 pp. doi: 10.1103/physrevd.94.084040. [16] M. Turkyilmazoglu, Determination of the correct range of physical parameters in the approximate analytical solutions of nonlinear equations using the Adomian decomposition method, Mediterranean Journal of Mathematics, 13 (2016), 4019-4037.  doi: 10.1007/s00009-016-0730-8. [17] P. Wang, M. Zhao, X. Du and J. Liu, Analytical solution for the short-crested wave diffraction by an elliptical cylinder, European Journal of Mechanics-B/Fluids, 74 (2019), 399-409.  doi: 10.1016/j.euromechflu.2018.10.006. [18] L. V. Yakushevich, Nonlinear DNA dynamics: A new model, Physics Letters A, 136 (1989), 413-417. [19] M. Yaseen, M. Abbas, A. I. Ismail and T. Nazir, A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations, Applied Mathematics and Computation, 293 (2017), 311-319.  doi: 10.1016/j.amc.2016.08.028. [20] L. V. Yakushevich, Nonlinear DNA dynamics: A new model, Physics Letters A, 136 (1989), 413-417. [21] E. H. Zahran and M. M. Khater, Extended jacobian elliptic function expansion method and its applications in biology, Applied Mathematics, 6 (2015), 1174. [22] C. T. Zhang, Soliton excitations in deoxyribonucleic acid (DNA) double helices, Physical Review A, 35 (1987), 886-891.  doi: 10.1103/PhysRevA.35.886.

show all references

##### References:
 [1] M. A. Abdelrahman, E. H. Zahran and M. M. Khater, The exp $(-\phi(\xi))$-Expansion method and its application for solving nonlinear evolution equations, International Journal of Modern Nonlinear Theory and Application, 4 (2015), 37. [2] R. A. Attia, D. Lu and M. M. A. Khater, Chaos and relativistic Energy-Momentum of the nonlinear time fractional duffing equation, Mathematical and Computational Applications, 24 (2019), 10-33.  doi: 10.3390/mca24010010. [3] M. D. Barkley and B. H. Zimm, Theory of twisting and bending of chain macromolecules, Analysis of the fluorescence depolarization of DNA, The Journal of Chemical Physics, 70 (1979), 2991-3007. [4] H. M. Baskonus and H. Bulut, Analytical studies on the (1+ 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation defined by seismic sea waves, Waves in Random and Complex Media, 25 (2015), 576-586.  doi: 10.1080/17455030.2015.1062577. [5] V. K. Fedyanin and L. V. Yakushevich, Scattering of neutrons and light by DNA solitons, Studia Biophysica, 103 (1985), 171-178. [6] S. Homma and S. Takeno, A coupled base-rotator model for structure and dynamics of DNA: Local fluctuations in helical twist angles and topological solitons, Progress of Theoretical Physics, 72 (1984), 679-693.  doi: 10.1143/PTP.72.679. [7] M. M. A. Khater, R. A. Attia and D. Lu, Explicit lump solitary wave of certain interesting (3+ 1)-Dimensional waves in physics via some recent traveling wave methods, Entropy, 21 (2019), 397-426.  doi: 10.3390/e21040397. [8] M. M. A. Khater, D. Lu and R. A. Attia, Lump soliton wave solutions for the (2+ 1)-dimensional Konopelchenko-Dubrovsky equation and KdV equation, Modern Physics Letters B, 33 (2019), 1950199, 20 pp. doi: 10.1142/s0217984919501999. [9] M. M. A. Khater, D. Lu and R. A. Attia, Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method, AIP Advances, 9 (2019), 025003. [10] M. M. A. Khater, R. Attia and D. Lu, Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions, Mathematical and Computational Applications, 24 (2019), 1-13.  doi: 10.3390/mca24010001. [11] M. M. A. Khater, R. A. Attia and D. Lu, Numerical solutions of nonlinear fractional Wu-Zhang system for water surface versus three approximate schemes, Journal of Ocean Engineering and Science, (2019). [12] K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two–variable system with noninteger–order derivative, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013145, 15 pp. doi: 10.1063/1.5086909. [13] A. A. Opanuga, E. A. Owoloko, O. O. Agboola and H. I. Okagbue, Application of homotopy perturbation and modified Adomian decomposition methods for higher order boundary value problems, (2017). [14] Z. Parsaeitabar and A. R. Nazemi, A third-degree B-spline collocation scheme for solving a class of the nonlinear Lane-Emden type equations, Iranian Journal of Mathematical Sciences and Informatics, 12 (2017), 15-34. [15] I. Sakalli, Analytical solutions in rotating linear dilaton black holes: Resonant frequencies, quantization, greybody factor and Hawking radiation, Physical Review D, 94 (2016), 084040, 12 pp. doi: 10.1103/physrevd.94.084040. [16] M. Turkyilmazoglu, Determination of the correct range of physical parameters in the approximate analytical solutions of nonlinear equations using the Adomian decomposition method, Mediterranean Journal of Mathematics, 13 (2016), 4019-4037.  doi: 10.1007/s00009-016-0730-8. [17] P. Wang, M. Zhao, X. Du and J. Liu, Analytical solution for the short-crested wave diffraction by an elliptical cylinder, European Journal of Mechanics-B/Fluids, 74 (2019), 399-409.  doi: 10.1016/j.euromechflu.2018.10.006. [18] L. V. Yakushevich, Nonlinear DNA dynamics: A new model, Physics Letters A, 136 (1989), 413-417. [19] M. Yaseen, M. Abbas, A. I. Ismail and T. Nazir, A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations, Applied Mathematics and Computation, 293 (2017), 311-319.  doi: 10.1016/j.amc.2016.08.028. [20] L. V. Yakushevich, Nonlinear DNA dynamics: A new model, Physics Letters A, 136 (1989), 413-417. [21] E. H. Zahran and M. M. Khater, Extended jacobian elliptic function expansion method and its applications in biology, Applied Mathematics, 6 (2015), 1174. [22] C. T. Zhang, Soliton excitations in deoxyribonucleic acid (DNA) double helices, Physical Review A, 35 (1987), 886-891.  doi: 10.1103/PhysRevA.35.886.
Periodic soliton wave of the longitudinal displacements of the top and down strands by using Eq. (2.6) in three(a), two(c)-dimensional and contour plot(b), when $\bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]}$
Periodic compacton traveling wave of the longitudinal displacements of the top and down strands by using Eq. (2.7) in three(a), two(c)-dimensional and contour plot(b), when $\bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]}$
Periodic kink traveling wave of the longitudinal displacements of the top and down strands by using Eq. (2.15) in three(a), two(c)-dimensional and contour plot(b), when $\bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]}$
Periodic cuspon traveling wave of the longitudinal displacements of the top and down strands by using Eq. (2.54) in three(a), two(c)-dimensional and contour plot(b), when $\bigg{[}\delta = 4,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]}$
Two dimensional plot of exact and approximate solution in combined, separate, and radar plots of Eqs. (2.53), (2.57)
Two dimensional plot of exact and numerical solution that obtained by cubic spline technique in combined, separate, and radar plots
Two dimensional plot of exact and numerical solution that obtained by quintic spline technique in combined, separate, and radar plots
Two dimensional plot of exact and numerical solution that obtained by septic spline technique in combined, separate, and radar plots
Analytical and semi-analytical solution of Eq. (2.1) at different point $\bigg{[}0.1\leq\xi\leq1\bigg{]}$
 Value of Values of Values of Absolute Value $\xi$ Approximate Solutions Exact Solutions of Error 0.1 0.0832639583 0.0832639583 3.85979 $\times 10^{-12}$ 0.2 0.1661133254 0.1661133244 9.8317$\times 10^{-10}$ 0.3 0.2481417478 0.2481417227 2.50479$\times 10^{-8}$ 0.4 0.3289591155 0.3289588670 2.48485$\times 10^{-7}$ 0.5 0.4081992403 0.4081977707 1.46967$\times 10^{-6}$ 0.6 0.4855272861 0.4855210208 6.26536$\times 10^{-6}$ 0.7 0.5606472102 0.5606259072 0.0000213030 0.8 0.6333096392 0.6332482704 0.0000613687 0.9 0.7033207446 0.7031650088 0.0001557360 1 0.7705527947 0.7701952621 0.0003575330
 Value of Values of Values of Absolute Value $\xi$ Approximate Solutions Exact Solutions of Error 0.1 0.0832639583 0.0832639583 3.85979 $\times 10^{-12}$ 0.2 0.1661133254 0.1661133244 9.8317$\times 10^{-10}$ 0.3 0.2481417478 0.2481417227 2.50479$\times 10^{-8}$ 0.4 0.3289591155 0.3289588670 2.48485$\times 10^{-7}$ 0.5 0.4081992403 0.4081977707 1.46967$\times 10^{-6}$ 0.6 0.4855272861 0.4855210208 6.26536$\times 10^{-6}$ 0.7 0.5606472102 0.5606259072 0.0000213030 0.8 0.6333096392 0.6332482704 0.0000613687 0.9 0.7033207446 0.7031650088 0.0001557360 1 0.7705527947 0.7701952621 0.0003575330
Analytical, numerical, and absolute values of error of obtained solutions of Eq. (2.1)
 Value of Values of Values of Absolute Values $\xi$ Approximate Solutions Exact Solutions of Error 0 0.0000000000 0.0000000000 0.0000000000 0.1 -0.0832541541 -0.0832639583 9.8042$\times 10^{-6}$ 0.2 -0.1660944493 -0.1661133244 0.0000188750 0.3 -0.2481152092 -0.2481417227 0.0000265135 0.4 -0.3289267807 -0.3289588670 0.0000320863 0.5 -0.4081627183 -0.4081977707 0.0000350524 0.6 -0.4854860388 -0.4855210208 0.0000349819 0.7 -0.5605943403 -0.5606259072 0.0000315669 0.8 -0.6332236466 -0.6332482704 0.0000246238 0.9 -0.7031509210 -0.7031650088 0.0000140878 1 -0.7701952621 -0.7701952621 1.11022$\times 10^{-16}$
 Value of Values of Values of Absolute Values $\xi$ Approximate Solutions Exact Solutions of Error 0 0.0000000000 0.0000000000 0.0000000000 0.1 -0.0832541541 -0.0832639583 9.8042$\times 10^{-6}$ 0.2 -0.1660944493 -0.1661133244 0.0000188750 0.3 -0.2481152092 -0.2481417227 0.0000265135 0.4 -0.3289267807 -0.3289588670 0.0000320863 0.5 -0.4081627183 -0.4081977707 0.0000350524 0.6 -0.4854860388 -0.4855210208 0.0000349819 0.7 -0.5605943403 -0.5606259072 0.0000315669 0.8 -0.6332236466 -0.6332482704 0.0000246238 0.9 -0.7031509210 -0.7031650088 0.0000140878 1 -0.7701952621 -0.7701952621 1.11022$\times 10^{-16}$
Analytical, numerical, absolute values of error of obtained solution of Eq. (2.1)
 Value of $\xi$ Approximate Exact Absolute Error 0 3.46945$\times 10^{-18}$ 0.0000000000 3.46945$\times 10^{-18}$ 0.1 -0.0832639540 -0.0832639583 4.22639$\times 10^{-9}$ 0.2 -0.1661133141 -0.1661133244 1.02421$\times 10^{-8}$ 0.3 -0.2481417081 -0.2481417227 1.46466$\times 10^{-8}$ 0.4 -0.3289588492 -0.3289588670 1.78158$\times 10^{-8}$ 0.5 -0.4081977514 -0.4081977707 1.92278$\times 10^{-8}$ 0.6 -0.4855210020 -0.4855210208 1.87683$\times 10^{-8}$ 0.7 -0.5606258910 -0.5606259072 1.62749$\times 10^{-8}$ 0.8 -0.6332482583 -0.6332482704 1.20805$\times 10^{-8}$ 0.9 -0.7031650035 -0.7031650088 5.25365$\times 10^{-9}$ 1 -0.7701952621 -0.7701952621 0.0000000000
 Value of $\xi$ Approximate Exact Absolute Error 0 3.46945$\times 10^{-18}$ 0.0000000000 3.46945$\times 10^{-18}$ 0.1 -0.0832639540 -0.0832639583 4.22639$\times 10^{-9}$ 0.2 -0.1661133141 -0.1661133244 1.02421$\times 10^{-8}$ 0.3 -0.2481417081 -0.2481417227 1.46466$\times 10^{-8}$ 0.4 -0.3289588492 -0.3289588670 1.78158$\times 10^{-8}$ 0.5 -0.4081977514 -0.4081977707 1.92278$\times 10^{-8}$ 0.6 -0.4855210020 -0.4855210208 1.87683$\times 10^{-8}$ 0.7 -0.5606258910 -0.5606259072 1.62749$\times 10^{-8}$ 0.8 -0.6332482583 -0.6332482704 1.20805$\times 10^{-8}$ 0.9 -0.7031650035 -0.7031650088 5.25365$\times 10^{-9}$ 1 -0.7701952621 -0.7701952621 0.0000000000
Analytical, numerical, absolute values of error of obtained solution of Eq. (2.1)
 Value of Values of Values of Absolute value $\xi$ Approximate Solution Exact solution of Error 0 0.0000000000 0.0000000000 0 0.1 -0.0832592356 -0.083263958 4.72267$\times 10^{-6}$ 0.2 -0.1661049996 -0.1661133244 8.32474$\times 10^{-6}$ 0.3 -0.2481357283 -0.2481417227 5.99441$\times 10^{-6}$ 0.4 -0.3289523831 -0.3289588670 6.48396$\times 10^{-6}$ 0.5 -0.4081921510 -0.4081977707 5.61967$\times 10^{-6}$ 0.6 -0.4855155144 -0.4855210208 5.50631$\times 10^{-6}$ 0.7 -0.5606213388 -0.5606259072 4.56842$\times 10^{-6}$ 0.8 -0.6332432277 -0.6332482704 5.04269$\times 10^{-6}$ 0.9 -0.7031623445 -0.7031650088 2.66428$\times 10^{-6}$ 1 -0.7701952621 -0.7701952621 1.11022$\times 10^{-16}$
 Value of Values of Values of Absolute value $\xi$ Approximate Solution Exact solution of Error 0 0.0000000000 0.0000000000 0 0.1 -0.0832592356 -0.083263958 4.72267$\times 10^{-6}$ 0.2 -0.1661049996 -0.1661133244 8.32474$\times 10^{-6}$ 0.3 -0.2481357283 -0.2481417227 5.99441$\times 10^{-6}$ 0.4 -0.3289523831 -0.3289588670 6.48396$\times 10^{-6}$ 0.5 -0.4081921510 -0.4081977707 5.61967$\times 10^{-6}$ 0.6 -0.4855155144 -0.4855210208 5.50631$\times 10^{-6}$ 0.7 -0.5606213388 -0.5606259072 4.56842$\times 10^{-6}$ 0.8 -0.6332432277 -0.6332482704 5.04269$\times 10^{-6}$ 0.9 -0.7031623445 -0.7031650088 2.66428$\times 10^{-6}$ 1 -0.7701952621 -0.7701952621 1.11022$\times 10^{-16}$
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