research article travelling wave solutions of nonlinear ...such as bell-shaped solitary waves and...
TRANSCRIPT
-
Research ArticleTravelling Wave Solutions of Nonlinear Dynamical Equations ina Double-Chain Model of DNA
Zheng-yong Ouyang1 and Shan Zheng2
1 Department of Mathematics, Foshan University, Foshan Guangdong 528000, China2Guangzhou Maritime College, Guangzhou Guangdong 510725, China
Correspondence should be addressed to Zheng-yong Ouyang; zyouyang [email protected]
Received 6 January 2014; Revised 24 February 2014; Accepted 10 March 2014; Published 3 April 2014
Academic Editor: Yonghui Xia
Copyright Β© 2014 Z.-y. Ouyang and S. Zheng. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
We consider the nonlinear dynamics in a double-chain model of DNA which consists of two long elastic homogeneous strandsconnectedwith each other by an elasticmembrane. By using themethodof dynamical systems, the bounded travelingwave solutionssuch as bell-shaped solitary waves and periodic waves for the coupled nonlinear dynamical equations of DNA model are obtainedand simulated numerically. For the same wave speed, bell-shaped solitary waves of different heights are found to coexist.
1. Introduction
In 1953, Waston and Crick discovered the structure ofdeoxyribonucleic acid (DNA) double helix [1], which openedthe area of molecular biology. The significance of the doublehelix model not only means the proven structure of DNA,but, more importantly, also helps to reveal the replicationmechanism of DNA. To further study the characteristicsof DNA, nonlinear science has been used to deal withDNA system, because its properties can be investigated bynonlinear models that combine the methods of physics withbiological tools [2]. The soliton theory especially whichpertains to nonlinear science has been widely applied in thestudy of DNA [3β6].
To study DNA structure from the view of nonlinearscience, it is necessary to look for the right nonlinearmathematical models. A number of researchers have tried toestablish mathematical models to describe DNA system. Atthe beginning, it is difficult to use a specific mathematicalmodel to simulate DNA system due to its complex structureand the presence of various movements [7]. So some simpli-fiedmodels were used to study only some internal movementin DNA [8β12]. Further, two movements which made themain contribution to the DNA denaturation process weretaken into account together in one model. For example, atwo-dimensional discrete model was used to describe the
two movements in DNA [13]. Another new double-chainmodel of DNA that consists of two long elastic homogeneousstrands was given to describe transverse movements alongthe hydrogen bond and longitudinal movements along thebackbone direction [14, 15].
In [14, 15], the nonlinear dynamical equations werederived and nonlinear dynamics of DNA were studied. Theexact solutions of the general nonlinear dynamic systemin the double-chain model of DNA were obtained anddiscussed numerically and analytically under some specialapproximate conditions. The nonlinear dynamical equationsfor the double-chain model were given as follows:
π’π‘π‘β π2
1π’π₯π₯
= π1π’ + πΎ1π’V + π
1π’3+ π½1π’V3,
Vπ‘π‘β π2
2Vπ₯π₯
= π2V + πΎ2π’2+ π2π’2V + π½
2V3 + π0,
(1)
where π’ denotes the difference of the longitudinal displace-ments of the bottom and top strands, that is, the displace-ments of the bases from their equilibrium positions alongthe direction of the phosphodiester bridge that connects thetwo bases of the same strands; V represents the difference ofthe transverse displacements of the bottom and top strands,that is, the displacements of the bases from their equilibriumpositions along the direction of the hydrogen bond that
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 317543, 5 pageshttp://dx.doi.org/10.1155/2014/317543
-
2 Abstract and Applied Analysis
connects the two bases of the base pair. The constants π1, π2,
π1, π2, πΎ1, πΎ2, π1, π2, π½1, π½2, and π
0in (1) are written as
π1= Β±β
π
π
, π2= Β±β
πΉ
π
, π1=
β2π
ππβ
(β β π0) ,
π2=
β2π
ππ
, πΎ1= 2πΎ2=
2β2ππ0
ππβ2
, π1= π2=
β2ππ0
ππβ3,
π½1= π½2=
4ππ0
ππβ3, π
0=
β2π (β β π0)
ππβ
,
(2)
where π,π,π,πΉ, π, π0, and β are, respectively, themass density,
the area of transverse cross-section, the Young modulus,the tension density of the strand, the rigidity of the elasticmembrane, the distance of the two strands, and the height ofthe membrane in the equilibrium position.
In [16], introducing the transformation V = ππ’ + π (π andπ are constants) in (1) yields
π’π‘π‘β π2
1π’π₯π₯
= π’3(π1+ π½1π2) + π’2(2π½1ππ + ππΎ
1)
+ π’ (π1+ ππΎ1+ π½1π2) ,
π’π‘π‘β π2
2π’π₯π₯
= π’3(π2+ π½2π2) + π’2(
πΎ2
π
+
π2π
π
+ 3π½2ππ)
+ π’ (π2+ 3π½2π2) +
π2π
π
+
π½2π3
π
+
π0
π
.
(3)
Comparing (3) and using the physical quantities (2), it followsthat π = β/β2 and πΉ = π; (3) can be written as
π’π‘π‘β π2
1π’π₯π₯
= π΄π’3+ π΅π’2+ πΆπ’ + π·, (4)
where π΄ = ((β2πΌ/β3) + (4π2πΌ/β3)), π΅ = 6β2ππΌ/β2, πΆ =((β2πΌ/π
0) + (6πΌ/β)), andπ· = 0 with πΌ = ππ
0/ππ.
Reference [16] studied the traveling wave solutions of(4) and obtained one particular family of solitary kink-type solutions for different values of Riccati parameter of(4). Unfortunately the results are not complete, because (4)admits tanh-type, periodic-type, and bell-type solutions; [16]only presented the tanh-type solutions.
In this paper, we use the bifurcationmethod of dynamicalsystems [17β20] to study bell-type and periodic-type solu-tions of (6), which can improve the results in [16]. We lookfor solutions of (4) in the form of π’(π₯, π‘) = π(π) with π =(π₯/π1) β β2π‘. Substituting π’(π₯, π‘) = π((π₯/π
1) β β2π‘) into (4),
we have
πππ
= π΄π3+ π΅π2+ πΆπ, (5)
which is equivalent to the following two-dimensional system:ππ
ππ
= π¦,
ππ¦
ππ
= π΄π3+ π΅π2+ πΆπ. (6)
Obviously, system (6) has the first integral
π»(π, π¦) =
1
4
π΄π4+
1
3
π΅π3+
1
2
πΆπ2β
1
2
π¦2= π, (7)
whereπ is the Hamiltonian constant.
In fact, system (6) is a planar dynamical system whosephase orbits defined by the vector field of (6) determinetraveling wave solutions, so we first need to investigate somedifferent phase portraits of system (6) in different parameterregions. Then, by using the theory of dynamical system, wewill identify some different wave profiles and compute exactrepresentations for solitary wave solutions and periodic wavesolutions.
This paper is organized as follows. In Section 2, we drawbifurcation phase portraits of system (6), where explicitparametric conditions will be derived. In Section 3, we giveexact representations of solitary wave and periodic wavesolutions of (6) in explicit form. A short conclusion will begiven in Section 4.
2. Properties of Singular Points andBifurcation Phase Portraits
Firstly, we compute the equilibrium points of system (6)under different parametric conditions. It is clear that neitherthe parameters π΅ nor πΆ takes value zero from the forms ofparameters π΄, π΅, and πΆ; then π΄ equals zero if π = Β±β2/2.So we give equilibrium points in different parametric cases asfollows.
(1) When π΄ = 0, there are two equilibrium points (0, 0)and (βπΆ/π΅, 0).
(2) When π΄ ΜΈ= 0 and πΏ1> 0, there are three equilibrium
points (0, 0), (πβ, 0), and (π
+, 0) of (6) on the π-axis,
where πΏ1
= π΅2β 4π΄πΆ, π
β= (βπ΅ β βπΏ
1)/2π΄, and
π+= (βπ΅ + βπΏ
1)/2π΄.
Define π0(π) = π΄π
3+ π΅π2+ πΆπ; then the type of
equilibrium point (π, 0) can be decided by πΒ±
= βπ
0(π),
which are eigenvalues of linearized system of (6) at point(π, 0).
Secondly, we compute the intersection points of phaseportraits of (6) on the π-axis when the orbits pass throughthe equilibrium point (0, 0), namely, when the Hamiltonianconstantπ = β(0, 0) = 0. Let π(π) = (1/2)π΄π4 + (2/3)π΅π3 +πΆπ2, πΏ2= (4/9)π΅
2β 2π΄πΆ, π
0= β3π΄/2π΅, π
1= ((β2/3)π΅ β
βπΏ2)/π΄, and π
2= ((β2/3)π΅+βπΏ
2)/π΄. Whenπ΄ = 0, there are
a double zero point 0 and a zero point π0of π(π) on the π-
axis. When π΄ ΜΈ= 0 and πΏ2> 0, there are a double zero point 0
and two zero points π1and π
2of π(π) on the π-axis. In order
to draw bifurcation phase portraits of system (6), we need todiscuss the sign of parameters π΄, π΅, and πΆ. The intersectionpoints of phase portraits of (6) on the π-axis are given asfollows.
(1) If π΄ = 0, there are two intersection points (0, 0) and(π0, 0) of (6) on the π-axis (see Figure 1).
(2) If π΄ > 0, there are three intersection points (0, 0),(π1, 0), and (π
2, 0) of (6) on the π-axis and π
1< π2
(see Figure 2).(3) If π΄ < 0, there are three intersection points (0, 0),
(π1, 0), and (π
2, 0) of (6) on the π-axis and π
2< π1
(see Figure 3).
-
Abstract and Applied Analysis 3
y
0 ππ0
(a) π΅ > 0, πΆ > 0
y
0 ππ0
(b) π΅ > 0, πΆ < 0
y
0 ππ0
(c) π΅ < 0, πΆ > 0
y
0 ππ0
(d) π΅ < 0, πΆ < 0
Figure 1: The bifurcation phase portraits of system (6) with π΄ = 0.
y
0 ππ1 π2
(a) π΅ > 0, πΆ > 0
y
0 ππ1 π2
(b) π΅ > 0, πΆ < 0
y
0 ππ1 π2
(c) π΅ < 0, πΆ > 0
y
0 ππ1 π2
(d) π΅ < 0, πΆ < 0
Figure 2: The bifurcation phase portraits of system (6) with π΄ > 0.
y
0 ππ1π2
(a) π΅ > 0, πΆ > 0
y
0 ππ1π2
(b) π΅ > 0, πΆ < 0
y
0 ππ1π2
(c) π΅ < 0, πΆ > 0
y
0 ππ1π2
(d) π΅ < 0, πΆ < 0
Figure 3: The bifurcation phase portraits of system (6) with π΄ < 0.
Finally, from the above analysis about system (6), we givetwelve different phase portraits of (6) which are shown inFigures 1β3.There exist homoclinic orbits and periodic orbitsin some phase portraits of Figures 1β3, which correspond tobell-type and periodic-type waves and are of special interestto us.
3. Exact Explicit Representations ofSolitary Wave Solutions
(1) The case π΄ = 0, π΅ > 0, and πΆ > 0 (see Figure 1(a)).
Corresponding to the homoclinic orbit of (6) passingthrough points (π
0, 0) and (0, 0) on theπβπ¦ plane, there exists
a smooth solitary wave solution. Now, (7) becomes
π¦2=
2
3
π΅π3+ πΆπ2
(π0β©½ π β©½ 0) . (8)
Choosing π(0) = π0, substituting (8) into the first equation of
(6), and integrating along the homoclinic orbit, we have
β«
π0
π
1
π β(2/3) π΅π + πΆ
ππ =π. (9)
Solving (9) yields
π’1= β
3πΆ
π΅ (cosh (βπΆ π) + 1)
. (10)
(2) The case π΄ = 0, π΅ < 0, and πΆ > 0 (see Figure 1(c)).
Similarly, in Figure 1(c) the homoclinic orbit has thefollowing expression:
π¦2=
2
3
π΅π3+ πΆπ2
(0 β©½ π β©½ π0) . (11)
Thus the corresponding smooth solitary wave solution isobtained as (10) and its profile is shown in Figure 4(b).
(3) The case π΄ > 0, π΅ > 0, and πΆ > 0 (see Figure 2(a)).
In this case, on the π β π¦ plane the homoclinic orbit of(6) passing through points (π
2, 0) and (0, 0) has the following
expression:
π¦2=
1
2
π΄π4+
2
3
π΅π3+ πΆπ2
(π2β©½ π β©½ 0) . (12)
-
4 Abstract and Applied Analysis
0
π
π
(a)
0
π
π
(b)
u2
u3
0
π
π
(c)
u2
u3
0
π
π
(d)
0
π
π
(e)
0
π
π
(f)
Figure 4: Smooth solitary wave and periodic wave solutions.
Choosing π(0) = π2, substituting (12) into the first equation
of (6), and integrating along the homoclinic orbit, we have
β«
π2
π
1
π β(1/2) π΄π 2+ (2/3) π΅π + πΆ
ππ =π. (13)
Solving (13), it follows that
π’2=
4πΆπΌ1
πΌ2
1πββπΆ|π|
+ ((4/9) π΅2β 2π΄πΆ) π
βπΆ|π|β (4/3) π΅πΌ
1
, (14)
where πΌ1
= (2(β2π΅2+ 9π΄πΆ + π΅β4π΅
2β 18π΄πΆ))/(β6π΅ +
3β4π΅2β 18π΄πΆ).The profiles of solutions π’
1and π’2both have
figure caption Figure 4(a).
(4) The case π΄ > 0, π΅ < 0, and πΆ > 0 (see Figure 2(c)).
In this case, on the π β π¦ plane the homoclinic orbit of(6) passing through points (π
1, 0) and (0, 0) has the following
expression:
π¦2=
1
2
π΄π4+
2
3
π΅π3+ πΆπ2
(0 β©½ π β©½ π1) . (15)
Choosing π(0) = π1, substituting (15) into the first equation
of (6), and integrating along the homoclinic orbit, we have
β«
π1
π
1
π β(1/2) π΄π 2+ (2/3) π΅π + πΆ
ππ =π. (16)
Solving (16), we get
π’3=
4πΆπΌ2
πΌ2
2πβπΆ|π|
+ ((4/9) π΅2β 2π΄πΆ) π
ββπΆ|π|β (4/3) π΅πΌ
2
, (17)
where πΌ2
= (2(2π΅2
β 9π΄πΆ + π΅β4π΅2β 18π΄πΆ))/(6π΅ +
3β4π΅2β 18π΄πΆ). The profile of π’
3is shown in Figure 4(d).
(5) The case π΄ < 0, π΅ > 0, and πΆ > 0 and π΄ < 0, π΅ < 0,and πΆ > 0 (see Figures 3(a) and 3(c)).
In this case, there exist two homoclinic orbits and theirexpressions are (12) and (15), respectively. Similarly we getthe corresponding solitary wave solutions as π’
2and π’3, which
coexist for the same speed.The plane graphs of the coexistingsolitary wave solutions are shown in Figure 4(c), whenπ΄ < 0,π΅ > 0, and πΆ > 0, and Figure 4(d), when π΄ < 0, π΅ < 0, andπΆ > 0.
(6) The case π΄ < 0, π΅ > 0, and πΆ < 0 (see Figure 3(b)).
In this case, on theπβπ¦plane the orbit of (6)which passesthrough points (π
1, 0) and (π
2, 0) is a periodic orbit; it has the
following expression:
π¦2=
1
2
π΄π4+
2
3
π΅π3+ πΆπ2, (0 < π
2β©½ π β©½ π
1) . (18)
Choosing π(0) = π1, substituting (18) into the first equation
of (6), and integrating along the periodic orbit, we have
β«
π1
π
1
π β(1/2) π΄π 2+ (2/3) π΅π + πΆ
ππ =π. (19)
Solving (19) we obtain the following periodic wave solution:
π’4=
6πΆ
β4π΅2β 18π΄πΆ cos (ββπΆπ) β 2π΅
. (20)
The profile of π’4is shown in Figure 4(e).
-
Abstract and Applied Analysis 5
(7) The case π΄ < 0, π΅ < 0, and πΆ < 0 (see Figure 3(d)).
Similarly, corresponding to the periodic orbit of (6)which passes through points (π
1, 0) and (π
2, 0), there exists
a periodic wave solution. Now, the expression of the periodicorbit is
π¦2=
1
2
π΄π4+
2
3
π΅π3+ πΆπ2, (π
2β©½ π β©½ π
1< 0) . (21)
Choosing π(0) = π2, substituting (21) into the first equation
of (6), and integrating along the periodic orbit, we have
β«
π
π2
1
π β(1/2) π΄π 2+ (2/3) π΅π + πΆ
ππ =π. (22)
Solving (22), it follows that
π’5=
6πΆ
ββ4π΅2β 18π΄πΆ cos (ββπΆπ) β 2π΅
. (23)
The profile of π’5is shown in Figure 4(f).
4. Conclusion
In this paper, we employ the method of dynamical systemsto study the nonlinear dynamical equations of DNA model.The bifurcation phase portraits of the DNA under some para-metric conditions are drawn, and explicit exact expressions ofbell-shaped solitary waves and periodic waves are obtainedvia some special homoclinic and periodic orbits; their planargraphs are simulated. These solutions are new and differentfrom those in [14β16]; some existing results are improved.Furthermore, in general, the larger the wave speed, the higherthe wave crest. So for the same wave speed, it is interestingthat the bell-shaped solitary waves of different heights arefound to coexist.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (no. 11226303) and GuangdongProvince (no. 2013KJCX0189). The authors thank the editorsfor their hard working and also gratefully acknowledgehelpful comments and suggestions by reviewers.
References
[1] J. D. Watson and F. H. C. Crick, βMolecular structure of nucleicacids: a structure for deoxyribose nucleic acid,β Nature, vol. 171,no. 4356, pp. 737β738, 1953.
[2] M. Peyrard, S. C. LoΜpez, and G. James, βModelling DNA at themesoscale: a challenge for nonlinear science?βNonlinearity, vol.21, p. T91, 2008.
[3] T. Lipniacki, βChemically driven traveling waves in DNA,βPhysical Review E, vol. 60, p. 7253, 1999.
[4] S. Yomosa, βSolitary excitations in deoxyribonuclei acid (DNA)double helices,β Physical Review A, vol. 30, no. 1, pp. 474β480,1984.
[5] C. T. Zhang, βSoliton excitations in deoxyribonucleic acid(DNA)double helices,βPhysical ReviewA, vol. 35, no. 2, pp. 886β891, 1987.
[6] K. Forinash, βNonlinear dynamics in a double-chain model ofDNA,β Physical Review B, vol. 43, p. 10734, 1991.
[7] L. V. Yakushevich, Nonlinear Physics of DNA, John Wiley &Sons, Berlin, Germany, 2004.
[8] S. W. Englander, N. R. Kallenbanch, A. J. Heeger, J. A.Krumhansl, and S. Litwin, βNature of the open state in longpolynucleotide double helices: possibility of soliton excitations,βProceedings of the National Academy of Sciences of the UnitedStates of America, vol. 77, pp. 7222β7226, 1980.
[9] S. Yomosa, βSoliton excitations in deoxyribonucleic acid (DNA)double helices,β Physical Review A, vol. 27, no. 4, pp. 2120β2125,1983.
[10] S. Takeno and S. Homma, βA coupled base-rotator model forstructure and dynamics of DNAβlocal fluctuations in helicaltwist angles and topological solitons,β Progress of TheoreticalPhysics, vol. 72, no. 4, pp. 679β693, 1984.
[11] M. Peyrard and A. R. Bishop, βStatistical mechanics of a nonlin-ear model for DNA denaturation,β Physical Review Letters, vol.62, no. 23, pp. 2755β2758, 1989.
[12] T. Dauxois, M. Peyrard, and A. R. Bishop, βEntropy-drivenDNA denaturation,β Physical Review E, vol. 47, no. 1, pp. R44βR47, 1993.
[13] V. Muto, P. S. Lomdahl, and P. L. Christiansen, βTwo-dimensional discrete model for DNA dynamics: longitudinalwave propagation and denaturation,β Physical Review A, vol. 42,no. 12, pp. 7452β7458, 1990.
[14] D.X.Kong, S. Y. Lou, and J. Zeng, βNonlinear dynamics in a newdouble Chain-model of DNA,β Communications in TheoreticalPhysics, vol. 36, no. 6, pp. 737β742, 2001.
[15] X.-M. Qian and S.-Y. Lou, βExact solutions of nonlineardynamics equation in a new double-chain model of DNA,βCommunications in Theoretical Physics, vol. 39, no. 4, pp. 501β505, 2003.
[16] W. Alka, A. Goyal, and C. Nagaraja Kumar, βNonlinear dynam-ics of DNAβriccati generalized solitary wave solutions,β PhysicsLetters A: General, Atomic and Solid State Physics, vol. 375, no.3, pp. 480β483, 2011.
[17] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,Springer, Berlin, Germany, 1981.
[18] J. B. Li and Z. R. Liu, βSmooth and non-smooth traveling wavesin a nonlinearly dispersive equation,β Applied MathematicalModelling, vol. 25, pp. 41β56, 2000.
[19] J. B. Li and Z. R. Liu, βTraveling wave solutions for a class ofnonlinear dispersive equations,βChinese Annals ofMathematics,vol. 23, p. 397, 2002.
[20] Z. R. Liu and Z. Y. Ouyang, βA note on solitary waves formodified forms of Camassa-Holm and Degasperis-Procesiequations,β Physics Letters A, vol. 366, no. 4-5, pp. 377β381, 2007.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of