Abstract
This article discusses dynamics of the fractal double-chain deoxyribonucleic acid model. This structure contains two long elastic homogeneous strands that serve as two polynucleotide chains of deoxyribonucleic acid molecules, bounded by an elastic membrane indicating hydrogen bonds between the base pairs of two chains. The semi-inverse variational principle and auxiliary equation method are employed to extricate soliton solutions. The collection of retrieved exact solutions includes bright, dark, periodic, and other solitons. The constraint conditions emerge naturally which ensure the presence of these solutions. Additionally, 2D and 3D graphs showing the impact of fractals on solutions are included. These plots use appropriate parameter values. Furthermore, sensitivity analysis of the considered model is also acknowledged. The outcomes reveal that these techniques are reliable, effective, and applicable to various biological systems.
1. Introduction
Deoxyribonucleic acid (DNA) is an interesting nonlinear model of biological sciences [1, 2]. Since it is needed for protein-coding, inheritance, and genetic instruction manual for life, it contains instructions for cell growth, reproduction, and death of a human. DNA molecules are the foundation of life so their dynamics are one of the interesting problems in biophysics. Researchers have been studying this structure during the last decades [3, 4]. The study of DNA mechanism predicts the presence of significant nonlinear structures. It has been established that localized waves are caused by nonlinearity and these waves are fascinating because they can transmit power without causing power loss [5–7].
Nonlinear partial differential equations (NLPDEs) have been considered for studying several nonlinear physical phenomena. Many physicists and mathematicians have worked hard to develop further precise alternatives to NLPDEs for a better understanding of these processes. Therefore, exact solutions of NLPDEs are essential for exploring physical explanations and qualitative aspects of different mechanisms [8–18]. These solutions demonstrate the dynamics of several nonlinear complex models symbolically and physically. Numerous methods were implemented to attain exact and wave solutions of the nonlinear governing model [19–26].
This paper introduces the fractal double-chain DNA model to scrutinize the double-helix structure. Fractal calculus has been a flourishing subject of biology, mathematics, and physics because it deals with the modeling of distinct nonlinear procedures [27–34]. Since the fractal model covers many powerful properties, which the traditional system fails to explain. The field of biophysics greatly benefits from a unique class of solitary wave solutions referred to as solitons of proposed problems [3]. Wave packets known as solitons propagate at a constant pace and maintain their shape despite nonlinearities and dispersion [35–43]. Semi-inverse scheme and auxiliary equation method (AEM) are two effective techniques implemented to derive a set of solitons in this manuscript.
The Ritz-like approach linked with the variational principle termed as He’s semi-inverse variational method [44] is applied to attain the bright solitons of fractal DNA model which may aid biologists to comprehend its physical significance. An effective and straightforward algebraic method for finding soliton solutions is the semi-inverse scheme [45]. Many authors contributed to develop this technique to analyze fractal models in distinct scientific fields [33, 46, 47]. Another method adopted here is AEM that retrieves dark, periodic, bright, and other shaped solitons. This reliable strategy is employed to obtain dual-mode solutions of various equations found in literature [48, 49]. It is the generalization of many existing techniques. By using various values of the parameters and fractal dimension, the nonlinear dynamics of DNA strands can be addressed. The sensitivity analysis assesses how different uncertainties affect the overall level of uncertainty in a mathematical model. Specific boundaries that are dependent on one or more parameters have been applied using this technique.
The rest of the article is organize as follows: The governing model is included in Section 2. In section 3, soliton solutions are extracted along with geometrical analysis by employing semi-inverse method. Section 4 comprises solitons obtained via AEM with graphs. Section 5 of the report discusses the findings. Section 6 provides a sensitivity analysis of the suggested system. The article’s conclusion is provided in Section 7.
2. Governing System
Consider the following two general nonlinear dynamical equations which describe double-chain model of DNA:
where is the difference between the top and bottom strands’ longitudinal displacements, i.e., the deviations of the bases from their equilibrium positions along the direction of the phosphodiester bridge, which joins the two bases of the same strands, is the difference between the bottom and top strands’ transverse displacements, i.e., the bases displacement from its equilibrium point with the pathway of hydrogen bond which joins two bases of base pair where
where , , , and represent the tension density, cross-sectional area, Young’s modulus, and the mass density of each strand, is the distance between the strands, while is the stiffness and is the membrane height in positive equilibrium. The difference between the longitudinal displacements of the bottom and top strands is in equation (1), as opposed to , which represents the difference between the transverse displacements of the lower and higher strands.
Noq using a transformation:
where and are constants, to simplify Equation (1) into the system of equations as follows:
Comparing Equations (4) and (5), we infer that and . So, Equation (5) can be written as where
3. Mathematical Analysis
The wave transformation reduces Equation (7) to the following ODE:
According to [50, 51], a fractal DNA model can be written as
where and are the fractal dimensional value and derivative, respectively, stated as
The variational principle [44] can be used to produce the following trial-functional:
The variational formulation of Equation (10) is given as
where is the kinetic energy and is the potential energy.
The above equations are the Lagrangian and Hamiltonian. Using the two scale transformation,
Equation. (13) can be written as
3.1. Soliton Solutions of Fractal Model
Using Ritz technique, one can construct the solitary wave solution as
where and are constants to be further calculated. Putting Equation (17) into Equation (16), we have
Setting stationary with respect to and , it results,
From Equations (19) and (20), we have
Now, Equation (17) can be described as
Inserting the value of in Equation (4), then, we have where .
Additionally, we look another soliton solution in the form:
where and are constants to be further calculated. Substituting Equation (24) in Equation (16), we have
When we keep stationary with respect to and , it gives
From Equations (26) and (27), we have
Equation (24) becomes
Plugging the value of in Equation (4), we have
where .
4. Illustration of the AEM
The following statement illustrates the general NLPDE structure:
where is polynomial function of and its derivatives in relation to two independent variables and . Use the single variable conversion to reduce Equation (31) into ODE of the form:
Here, is a polynomial function with both linear and nonlinear terms and the superscripts of show its ordinary derivative with respect to . The algorithm of AEM suggests the initial solution of Equation (32) as satisfying the auxiliary equation
where , , , …, are coefficients to be evaluated such that . The value of is determined by balancing the highest order derivative and nonlinear term involved in Equation (9).
Now, putting Equation (33) into Equation (9) and performing few steps of algebra yields a system of algebraic equations in .
The family of solutions of Equation (34) can be obtained as follows:
Family 1. When and ,
Family 2. When and ,
Family 3. When and and ,
Family 4. When and and ,
Family 5. When and ,
Family 6. When and ,
Family 7. When ,
Family 8. When , and ,
Family 9. When and ,
Family 10. When ,
Family 11. When and ,
Family 12. When and ,
Family 13. When ,
Family 14. When ,
Family 15. When ,
Family 16. When ,
Family 17. When and ,
Family 18. When ,
4.1. Application of AEM
The balancing principle employed to Equation (9) yields the value of index . Hence, Equation (33) takes the form:
Now, invoking Equation (53) into Equation (9) gives a system of equations which is further evaluated via Maple, it generates where
Insertion of Equation (54) into Equation (53) results to
By substituting the solutions specified by Equation (34) into Equation (58), the solutions retrieved are
For Family 1, when and ,
For Family 2, when and ,
For Family 3, when , and ,
For Family 4, when , and ,
For Family 5, when and ,
For Family 6, when and ,
For Family 7, when ,
For Family 8, when , and ,
For Family 9, when and ,
For Family 12, when and ,
For Family 13, when ,
For Family 14, when ,
For Family 15, when ,
For Family 16, when ,
For Family 18, when ,
For Family 17, when and ,
5. Results and Discussion
This section covers the graphical interpretation of the results and the impact of the fractal parameter on them. Two powerful integration approaches, namely, semi-inverse scheme and AEM are used to extract soliton solutions of governing model. It has been established that the approaches currently provided for the double-chain DNA model, which were utilized to create closed-form exact results, are novel and distinct from those currently in use. The innovative solitonic solution structure and the new equations that yielded distinct types of solutions are the observable characteristics for finding solutions from the method outlined. Graphics that elaborate the various novel exact solitons in the forms of dynamics and nonlinear waves are presented for a physical description of the solutions that have been achieved. The semi-inverse principle offers bright soliton solutions Equations (22), (23), (29), and (30) of the aforesaid system. The physical significance of these solitons is shown in terms of modulus of by assigning particular values of free parameters. Equations (23) and (30) show the same graphical behavior with just translation given in Equation (4) as in the figures. In Figures 1 and 2, these solitons in the form of 3D plots for fractal dimension value and 2D graphs for are provided. Dual-wave periodic, dark, bright solitons are raised by executing AEM. The dynamics of DNA strands and are presented in Figures 3–12. 3D sketches for fractal value and 2D graphics for are provided. The fractal impact is displayed by the irregularity in the curves of solutions. A few representative solutions are graphically illustrated to consider the appropriate connotation of dual-wave behaviors of the DNA system. The propagation of solitons and collisions of dual-mode pulses are examined using graphs. It is important to note that the proposed schemes may be used to generate soliton solutions for any NLPDE.
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6. Sensitivity Analysis
The sensitive analysis of the formulated soliton solutions is demonstrated in this section. There are several research publications on the methods and applications of sensitivity analysis of parameter uncertainty to mathematical problems. The goal of the study is to identify and classify the different types of uncertainty that can affect how well a mathematical equation or framework performs in relation to its inputs. Results are presented based on various parametric values, and sensitivity is investigated by taking into account how a little change in input can significantly alter output. A detailed analysis of Equation (9) is introduced in Figures 13 and 14.
7. Conclusion
In this manuscript, semi-inverse method and AEM have been successfully applied to the double-chain DNA model that is one of the interesting models of current biophysics since it is related to an organism’s life. It is likely that the well-known cubic nonlinear Klein-Gordon equation is the linearly reduced model to (1). The fractal DNA system has a high influence because it is used to describe the nonlinear dynamics of DNA molecules. The dark, periodic, bright, and other soliton solutions are derived which may help biologists for physical simulation of suggested equations. It should be highlighted that our results are novel and different from those of earlier investigations [3, 9]. The semi-inverse scheme is a fascinating integration tool to deduce variational principles for various differential models, whereas AEM is compelling to derive a family of dual-wave solitons of any NLPDE. The relevant choices of parameters enable us to discuss fractal behavior of the system. Also, the nature of attained solutions is reviewed by their 3D and 2D graphics. By using various initial conditions, the system is subjected to sensitivity analysis, which is then visualized using graphs. The acquired effects might help spark original suggestions for future biological applications.
Data Availability
Data is available on request.
Conflicts of Interest
The authors have no conflict of interest regarding the publication of this paper.
Authors’ Contributions
All authors have equal contribution on this paper.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, for funding this project under grant number R.G.P. 2/29/43. Emad E. Mahmoud acknowledges the Taif University Researchers Supporting Project number TURSP-2020/20, Taif University, Taif, Saudi Arabia.