research article equilibrium configurations of the...

Research ArticleEquilibrium Configurations of the Noncircular Cross-SectionElastic Rod Model with the Elliptic KB Method

Yongzhao Wang12 Qichang Zhang1 and Wei Wang1

1Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control School of Mechanical EngineeringTianjin University Tianjin 300072 China2School of Mathematics and Statistics Xuchang University Xuchang 461000 China

Correspondence should be addressed to Qichang Zhang qzhangtjueducn

Received 3 November 2014 Accepted 21 February 2015

Academic Editor Lakshmanan Shanmugam

Copyright copy 2015 Yongzhao Wang et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The mechanical deformation of DNA is very important in many biological processes In this paper we consider the reducedKirchhoff equations of the noncircular cross-section elastic rod characterized by the inequality of the bending rigidities One familyof exact solutions is obtained in terms of rational expressions for classical Jacobi elliptic functions The present solutions allow theinvestigation of the dynamical behavior of the system in response to changes in physical parameters that concern asymmetry Theeffects of the factor on the DNA conformation are discussed A qualitative analysis is also conducted to provide valuable insightinto the topological configuration of DNA segments

1 Introduction

DNA is a long polymer made of millions (or even hundredsof millions) of nucleotides arranged in two complementarystrands forming a double helix Genetic information in livingcells is carried in the linear sequence of nucleotides in DNAConformational features and mechanical properties of DNAin vivo (such as supercoil formation and bendtwist rigidity)play an important role in its packing gene expressionprotein synthesis [1 2] protein transport [3] and so forthas misfolding of DNA has become the major cause of manyillnesses such as paroxysmal nocturnal hemoglobinuria(PNH) disease [4] Thus it is necessary to understand thebasic mechanisms of DNA folding that leads to new waysfor preventing such diseases In recent years geometricalconfiguration of a DNA chain has attracted considerableattentions

The elastic properties of ds-DNA molecules are believedto play an important role in many biological functions [56] Over the past two decades the elastic properties havebeen extensively studied with the development of single-molecule manipulation techniques [7ndash11] and experimentcapabilities Analytical models based on classical elasticity

theory [12ndash14] have no spatialtemporal limitations and havewidely been used to study the DNA configurations Theelastic rod model to research the flexible structures is toassume that they are made of an elastic material obeyingthe appropriate laws of elasticity The well-known Kirchhoffmodels for rods are widely used to describe the stationarystates of elastic filaments within the approximation of linearelasticity theory through a system of six coupled ordinarydifferential equations [12] In 1859 Kirchhoff discovered thatthe equations that describe the thin elastic rod in equilibriumare mathematically identical to those used to describe thedynamics of the heavy top Shi and Hearst [15] derived atime-independent one-dimensional nonlinear Schrodingerequation for the stationary state configurations of supercoiledDNA Xue et al [16] extended the Schrodinger equation to fitthe noncircular Kirchhoff elastic rod by using the complexrigidity Wang et al [17 18] rebuilt the initial Kirchhoffequations in a complex style to suit the character of obviousasymmetry and the periodically varying bending coefficientswhich is embodied on the cross-section by consideringthe mathematical background of DNA double helix andintroduced a complex form variable solution of the torqueto obtain a simplified second ordinary differential equation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 351362 7 pageshttpdxdoiorg1011552015351362

2 Mathematical Problems in Engineering

with single variable However in above work the complexexpression of 119872

3according to the complex normal form

method is not accurate we will correct this error in thefollowing section

In recent years the analysis of static and dynamicconfigurations of elastic rod has drawn great attentions Inthis paper we will consider the revised reduced Kirchhoffequations of the noncircular cross-section elastic rod char-acterized by the inequality of the bending rigidities It iscrucial to find the exact or approximate solutions for therevised simplified second ordinary differential equation inorder to investigate the configurations of DNA segmentsReference [19] applied the enhanced cubication method todevelop approximate solutions for the most common nonlin-ear oscillators and leads to amplitude-time response curvesand angular frequency values Reference [20] developed anonlinear transformation approach to obtain the equivalentrepresentation form of conservative two-degree-of-freedomnonlinear oscillators Lai and Chow [21] used Jacobi ellipticKrylovndashBogoliubov (KB)method to find two families of exactsolutions for oscillators with quadratic damping and mixed-parity nonlinearityMotivated by the above literatures reviewthis paper focuses on accurate solutions for the reducedKirchhoff equations and undertakes a qualitative analysis ofthe topological configuration of DNA segments

The paper is divided into four parts In the next sectionthe reduced form of Kirchhoff rsquos equations is revised In thethird section the periodic solutions of the equations arefound and the effects of anisotropic on configuration of DNAare discussed Finally some conclusions are drawn and thepaper is closed

2 The Reduced Form of Kirchhoff Equations

As a coarse-grained description aDNA can be approximatelyregarded as a thin flexible and inextensible rod or string[12 15 16] The classical theory of elasticity describes thegeometry of an elastic rod in terms of its center line R =

R(119904) = (119909(119904) 119910(119904) 119911(119904)) three-dimensional curve parameter-ized by its arc-length 119904 In presence of externalmomentm andexternal load f which are distributed along the central axis R(as show in Figure 1) the static Kirchhoff equations in bodyfixed frame are as follows

119889F119889119904

+ f = 0

119889M119889119904

+ e3times F +m = 0

(1)

where F and M denote the elastic force and momentrespectively

As shown in [17 18] the complex vector basesD119894are used

to substitute the real form vectors e1 e2 e3

D119886=e1minus 119894e2

2

D119886=e1+ 119894e2

2

D3= e3

(2)

P0

r

P

PL

r + drP998400

dr

119813 + d119813

119820+ d119820

0

119839ds119846ds

minus119813

minus119820

Figure 1The deformed state loaded by forces andmoments per unitlength [17]

Thus the complex Kirchhoff equations in the case of zeroexternal momentm can be written out in terms ofD

119894 such as

1198651015840

119886+ 119894 (1198651198861205963minus 1198653120596119886) + 119891119886= 0

1198721015840

119886+ 119894 (119872

1198861205963minus1198723120596119886+ 119865119886) = 0

1198651015840

3+

119894

2(119865119886120596119886minus 1198653120596119886) + 1198913= 0

1198721015840

3+

119894

2(119872119886120596119886minus119872119886120596119886) = 0

(3)

where 119909119886= 1199091+ 1198941199092and 119909

119886denotes the complex conjugate

vector of 119909119886with 119909

119886 119909119886 1199093 being the projection on each

complex axis 119872119886= 1198721+ 1198941198722= 119860120596

1+ 119894119861120596

2 1198723= 119862120596

3

where 1198721and 119872

2are the bending moments and 119872

3is the

twisting component along the rodIn [17 18] a complex expression of119872

3is brought which

is analogous to the complex normal form method Howeverthis expression is inaccurate we revised the expression asfollows

1198723=120572

2(119872119886120596119886+119872119886120596119886) (4)

where 120572 is per unit length scale In this case both sides of (4)are consistent for the dimension and it is easy to find a newway to improve the reduced form while 120572 is a constant or afunction of arc-length

That produces the expressions of 1205961and 120596

2by solving (3)

and (4)

1205961=

(119860 minus 119861)1198723plusmn radic(119860 minus 119861)

21198723

2minus 41198601198611205722 (119872

1015840

3)2

21198601205721198721015840

3

1205962

1205962=radic(119860 minus 119861)1198723 ∓

radic(119860 minus 119861)21198723

2minus 41198601198611205722 (119872

1015840

3)2

2120572 (119860 minus 119861) 119861

(5)

Mathematical Problems in Engineering 3

Following with [17] the reduced form of Kirchhoffequations can be expressed as

11988921198723

1198891199042minus(119860 minus 119861)

2+ 81205722(119860 + 119861)119867

812057221198601198611198723

+(119860 minus 119861)

2+ (119860 + 119861)119862

21205721198601198611198621198723

2

minus4119860119861 minus 3 (119860 + 119861)119862

211986011986111986221198723

3+2120572

119862(1198891198723

119889119904)

2

= 0

(6)

where119867 is the Hamiltonian of systemWe simplify (6) by first making all the variables dimen-

sionless that is defining

119904 =119904

120572

1198723=1198723

119864119868

119860 =119860

119864119868

119861 =119861

119864119868

119862 =119862

119864119868

119867 =1198671205722

119864119868

(7)

where 119864 is the Youngrsquos modulus and 119868 denotes the moment ofinertia

The radios 119901 = 119860119862 and 119902 = 119861119862 can also be introduced(6) can be reduced to a more contracted form

11988921198723

1198891199042

minus(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

1198723

+(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

1198723

2

minus4119901119902 minus 3 (119901 + 119902)

21199011199021198622

1198723

3

+2

119862

(1198891198723

119889119904)

2

= 0

(8)

3 Periodic Solutions

Modeling periodical configurations in various practical prob-lems has attracted research interests of scientists from widespread areas including physicists chemists applied math-ematicians engineers and biologists Periodical configura-tions modeled have included morphologies of calcites silica-barium carbonate ropes polyethylene glycol the microstruc-ture of rods cables and ribbon general polymer helices [22]and of course those occurring in proteins and DNA [23]Theaim of this section is to find periodical solutions of (8) Interms of the phase plane the periodic solutions correspondto the formation of closed trajectories

Following with the line of [21] the use of Jacobi ellipticfunctions and the KB approximation scheme namely theelliptic KB method was proposed An exact solution of (8)is assumed in the form

1198723(119904) =

119886 cn (120596119904119898)

1 + 119887 cn (120596119904119898) (9)

where 119886 119887 120575 and 119898 are parameters to be determined Theparameter119898 is equal to 1198962 with 119896 being the elliptic modulus

Substituting (9) into (8) and setting the coefficients ofcn119894(120596119904119898) (119894 = 0 1 2 3 4) to zero we can easily obtain

2 (1 minus 119898) 1198861205962[119886

119862

minus 119887] = 0

(119901 minus 119902)2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ (1 + 21198872minus 2119898 minus 2119887

2119898)1205962= 0

minus 3119886119887(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 1198862[(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

+ (2119898 minus 1)2

119862

1205962] = 0

minus 31198872(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 119886119887(119901 minus 119902)

2+ (119901 + 119902)

119901119902119862

minus 11988624119901119902 minus 3 (119901 + 119902)

21199011199021198622

+ (1198872minus 2119898 minus 2119887

2119898)1205962= 0

minus 1198873(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 1198861198872(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

minus 11988621198874119901119902 minus 3 (119901 + 119902)

21199011199021198622

minus 21198871198981205962minus 119886119898120596

2 2

119862

= 0

(10)

The initial condition is as follows

11987230= 1198723(0) =

119886

1 + 119887

1198723

1015840

(0) = 0

(11)

Solving the algebraic equations (10) and (11) the solutionsof 119886 119887119898 and 120596 can be straightforwardly determinedWe set119886 119887119898 120596119867 and119872

30as the unknowns the solutions can be


solved by the six equationsThe exact solution can be expressas follows

119886 = 119862

119887 = 1

120596 = radic3119901 + 3119902 minus 4119901119902

8119901119902

119898 =41199012+ 119901 (25 minus 36119902) + 119902 (25 + 4119902)

8 (119901 (3 minus 4119902) + 3119902)

119867 = (119888(minus (119901 minus 119902)2+ 1198882119901119902

sdot (minus

2 (119901 + (119901 minus 119902)2+ 119902)

1198882119901119902

+4119901119902 minus 3 (119901 + 119902)

21198882119901119902)))

sdot (8 (119901 + 119902))minus1

11987230=119862

2

(12)

The following conditions hold for the existence of a periodicalsolution in (8)

3119901 + 3119902 minus 4119901119902 gt 0

41199012+ 119901 (25 minus 36119902) + 119902 (25 + 4119902)

8 (119901 (3 minus 4119902) + 3119902)gt 1

(13)

In this case the exact solution can be also expressed as

1198723(119904) =

119886 dn (120596119904119898)

1 + 119887 dn (120596119904119898) (14)

where

120596 =

radic2 (119901 + (119901 minus 119902)2+ 119902) 119901119902 minus 7 (4119901119902 minus 3 (119901 + 119902)) 2119901119902

4radic2

119898 = minus4 (4119901119902 minus 3 (119901 + 119902))

sdot (119901119902(

2 (119901 + (119901 minus 119902)2+ 119902)

119901119902

minus7 (4119901119902 minus 3 (119901 + 119902))

2119901119902))

minus1

(15)

and 119886 119887 are the same as in (12)

According to the theories of ordinary differential equa-tions [24] significant information can be extracted fromphase plane analysis For this aim (8) can be expressed as

119889119909

119889119904= 119891 (119909 119910) = 119910

119889119910

119889119904= 119892 (119909 119910) =

(119901 minus 119902)2119862 + 8 (119901 + 119902)119867

8119901119902119862

119909

minus(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

1199092+4119901119902 minus 3 (119901 + 119902)

21199011199021198622

1199093minus

2

119862

1199102

(16)

There are three critical points

1199091= 0

1199092=

119888 (2 (119901 + 1199012+ 119902 minus 2119901119902 + 119902

2) minus 119875)

4 (minus3119902 + 119901 (minus3 + 4119902))

1199093=

119888 (2 (119901 + 1199012+ 119902 minus 2119901119902 + 119902

2) + 119875)

4 (minus3119902 + 119901 (minus3 + 4119902))

(17)

where

119875

= radic2

sdot (21199014+ 81199013(2 minus 3119902) + 119901

2(23 minus 56119902 + 60119902

2)

+ 1199022(23 + 2119902 (8 + 119902)) minus 2119901119902 (minus23 + 4119902 (7 + 3119902)))

12

(18)

It is clear that 119891(119909 minus119910) = minus119891(119909 119910) and 119892(119909 minus119910) =

119892(119909 119910) thus the system (16) is reversible [21] As discussedin [21] a closed orbit in phase space is formed by a twintrajectory which is shown in Figure 1 In this case the criticalpoint 119909

2is a centre the periodical solution is around the

point Figure 2 is in accordance with the resultWithout loss of generality we choose 119901 = 415 119902 =

1615 and 119888 = 1 By calculation (13) is held so there isa periodical solution as shown in Figure 2 The evolution of1198723(119904) is shown in Figure 3 which confirms the results in

Figure 2In (12) the frequency 120596 and elliptic modulus 119898 depend

only on the radios 119901 and 119902 not on 119862 it is clearly evident thatthe radios mainly reflect the period of the solution Sequencedependence and anisotropy of bending persistence length hasbeen widely noticed in the base-pair steps approaches inwhich relative rotation and displacement of every two seg-ments are defined trough six parameters slide shift rise tiltroll and twist [25] Looking atmicroscopic structure of DNAmacromolecule suggests that bending toward the groove iseasier than bending toward the backbone [26] which isconfirmed byMonte Carlo simulationsThus the asymmetrymay be an important factor in DNA reconfiguration process


03 035 04 045 05

minus015

minus01

minus005

0

005

01

015

y

x

Figure 2 Phase space of (16)

0 5 10 15 20 25 30

032

034

03

036

038

042

04

05

044

046

048

s

M3

Figure 3 The evolution of1198723(119904)

In the following the simulations of DNA configurations aredone under different radios between 119901 and 119902

Experimental results show that the elastic thin rod modelconsidered DNA molecular internal structure and externalenvironment as a whole which significantly shows asymme-try As shown in [26] 119902119901 varies from 1 to 4 The effects ofthe asymmetry on the frequency 120596 and elliptic modulus 119898in (12) are studied in Figure 4 Figure 5 shows the period 119879

of the periodical solution changes with the asymmetry 119902119901

1 2 3 4

11

12

13

14

15

qp

m

120596

Figure 4 Variations of119898 and 120596 with respect to 119902119901

1 2 3 469

71

73

75

77

qp

T

Figure 5 Variations of the period 119879 with respect to 119902119901

The topological parameter twisting number 119879120596 is defined as

[12]

119879120596=

1

2120587int

119871

0

1205963(119904) 119889119904 (19)

Figure 6 shows the twisting number of the elastic rod per unitlength with respect to the asymmetry 119902119901 From Figures 4ndash6 we can find that DNA segment will undergo a series ofalteration with the change of the asymmetry which shows afairly well agreement with [27]


1 2 3 4

0066

0067

0068

0069

007

T120596

qp

Figure 6 Variations of 119879120596with respect to 119902119901

4 Conclusion

In the present paper we revised the reduced formofKirchhoffequations which characterizes the equilibrium configura-tions of DNA segments with the noncircular cross-sectionThe Jacobi ellipticKrylovndashBogoliubov (KB)method is used tofind one family of exact periodical solutions of the Kirchhoffequations The effect of the asymmetry on the equilibriumconfigurations of DNA is discussed the results show thatthe asymmetry is an important factor in the process ofDNA elastic rod reconfiguration which shows a fairly wellagreement with [27]

Finally it should be noted that the interfacial energymodel is only a coarse-grained model Many data of DNAis still measured in future experience such as the presenceof DNA segments environment The DNA configurationwith periodically varying bending rigidities will be alsoinvestigated in future work

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 Nature Sci-ence Foundation of China (no 11372210) Research Fundfor the Doctoral Program of Higher Education of China(no 20120032110010) and the Tianjin Research Programof Application Foundation and Advanced Technology (no12JCZDJC28000) This support is greatly appreciated

References

[1] K Luger A W Mader R K Richmond D F Sargent and T JRichmond ldquoCrystal structure of the nucleosome core particle at28 A resolutionrdquo Nature vol 389 no 6648 pp 251ndash260 1997

[2] S E Halford and J F Marko ldquoHow do site-specific DNA-binding proteins find their targetsrdquoNucleic Acids Research vol32 no 10 pp 3040ndash3052 2004

[3] J Elf G-W Li and X S Xie ldquoProbing transcription factordynamics at the single-molecule level in a living cellrdquo Sciencevol 316 no 5828 pp 1191ndash1194 2007

[4] J-I Nishimura Y Murakami and T Kinoshita ldquoParoxysmalnocturnal hemoglobinuria an acquired genetic diseaserdquo Amer-ican Journal of Hematology vol 62 no 3 pp 175ndash182 1999

[5] Z Bryant M D Stone J Gore S B Smith N R Cozzarelliand C Bustamante ldquoStructural transitions and elasticity fromtorque measurements on DNArdquo Nature vol 424 no 6946 pp338ndash341 2003

[6] J F Allemand D Bensimon R Lavery and V CroquetteldquoStretched and overwound DNA forms a Pauling-like structurewith exposed basesrdquo Proceedings of the National Academy ofSciences of theUnited States of America vol 95 no 24 pp 14152ndash14157 1998

[7] S B Smith Y Cui andC Bustamante ldquoOverstretching B-DNAthe elastic response of individual double-stranded and single-stranded DNA moleculesrdquo Science vol 271 no 5250 pp 795ndash799 1996

[8] T Lionnet S Joubaud R Lavery D Bensimon and V Cro-quette ldquoWringing out DNArdquo Physical Review Letters vol 96no 17 Article ID 178102 4 pages 2006

[9] C Bustamante J C Macosko and G J L Wuite ldquoGrabbingthe cat by the tail manipulating molecules one by onerdquo NatureReviews Molecular Cell Biology vol 1 no 2 pp 130ndash136 2000

[10] T R Strick V Croquette and D Bensimon ldquoSingle-moleculeanalysis of DNA uncoiling by a type II topoisomeraserdquo Naturevol 404 no 6780 pp 901ndash904 2000

[11] B Alberts D Bray J Lewis M Raff K Roberts and J DWatsonMolecular Biology of the Cell Garland Publishing NewYork NY USA 1994

[12] Y Z Liu Nonlinear Mechanics of Thin Elastic Rod TheoreticalBasis of Mechanical Model of DNA Tsinghua University PressBeijing China 2006

[13] X H Zhou ldquoA one-dimensional continuous model for carbonnanotubesrdquoThe European Physical Journal B-CondensedMatterand Complex Systems vol 85 pp 1ndash8 2012

[14] X H Zhou ldquoSome notes on low-dimensional elastic theories ofbio- and nano-structuresrdquoModern Physics Letters B vol 24 no23 pp 2403ndash2412 2010

[15] Y Shi and J E Hearst ldquoThe Kirchhoff elastic rod the nonlinearSchrodinger equation and DNA supercoilingrdquo The Journal ofChemical Physics vol 101 no 6 pp 5186ndash5199 1994

[16] YXue Y-Z Liu andL-QChen ldquoTheSchrodinger equation fora Kirchhoff elastic rod with noncircular cross sectionrdquo ChinesePhysics vol 13 no 6 pp 794ndash797 2004

[17] W Wang Q-C Zhang and G Jin ldquoThe analytical reductionof the Kirchhoff thin elastic rod model with asymmetric crosssectionrdquo Acta Physica Sinica vol 61 no 6 Article ID 0646022012

[18] W Wang Q-C Zhang and Q-Z Xie ldquoAnalytical reductionof the non-circular Kirchhoff elastic rod model with theperiodically varying bending rigiditiesrdquo Physica Scripta vol 87no 4 Article ID 045402 6 pages 2013


[19] A Elıas-Zuniga and O Martınez-Romero ldquoAccurate solutionsof conservative nonlinear oscillators by the enhanced cubica-tion methodrdquoMathematical Problems in Engineering vol 2013Article ID 842423 9 pages 2013

[20] A Elias-ZunigaDO Trejo I F Real andOMartinez-RomeroldquoA transformation method for solving conservative nonlineartwo-degree-of-freedom systemsrdquo Mathematical Problems inEngineering vol 2014 Article ID 237234 14 pages 2014

[21] S K Lai and K W Chow ldquoExact solutions for oscillators withquadratic damping and mixed-parity nonlinearityrdquo PhysicaScripta vol 85 no 4 Article ID 045006 2012

[22] M Barros and A Ferrandez ldquoA conformal variational approachfor helices in naturerdquo Journal of Mathematical Physics vol 50no 10 Article ID 103529 20 pages 2009

[23] M Yavari ldquoB- to Z-DNA transition probed by the Feolirsquosformalism for a Kirchhoff modelrdquo International Journal ofModern Physics B vol 27 no 23 Article ID 1350121 2013

[24] E A Coddington and N Levinson Theory of Ordinary Differ-ential Equations Tata McGraw-Hill Education 1955

[25] W K Olson N L Marky R L Jernigan and V B ZhurkinldquoInfluence of fluctuations on DNA curvature a comparison offlexible and static wedge models of intrinsically bent DNArdquoJournal of Molecular Biology vol 232 no 2 pp 530ndash554 1993

[26] A Balaeff L Mahadevan and K Schulten ldquoModeling DNAloops using the theory of elasticityrdquo Physical Review E Statis-tical Nonlinear and Soft Matter Physics vol 73 no 3 ArticleID 031919 23 pages 2006

[27] F Mohammad-Rafiee and R Golestanian ldquoThe effect ofanisotropic bending elasticity on the structure of bent DNArdquoJournal of Physics Condensed Matter vol 17 no 14 pp S1165ndashS1170 2005

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with single variable However in above work the complexexpression of 119872

3according to the complex normal form

method is not accurate we will correct this error in thefollowing section

In recent years the analysis of static and dynamicconfigurations of elastic rod has drawn great attentions Inthis paper we will consider the revised reduced Kirchhoffequations of the noncircular cross-section elastic rod char-acterized by the inequality of the bending rigidities It iscrucial to find the exact or approximate solutions for therevised simplified second ordinary differential equation inorder to investigate the configurations of DNA segmentsReference [19] applied the enhanced cubication method todevelop approximate solutions for the most common nonlin-ear oscillators and leads to amplitude-time response curvesand angular frequency values Reference [20] developed anonlinear transformation approach to obtain the equivalentrepresentation form of conservative two-degree-of-freedomnonlinear oscillators Lai and Chow [21] used Jacobi ellipticKrylovndashBogoliubov (KB)method to find two families of exactsolutions for oscillators with quadratic damping and mixed-parity nonlinearityMotivated by the above literatures reviewthis paper focuses on accurate solutions for the reducedKirchhoff equations and undertakes a qualitative analysis ofthe topological configuration of DNA segments

The paper is divided into four parts In the next sectionthe reduced form of Kirchhoff rsquos equations is revised In thethird section the periodic solutions of the equations arefound and the effects of anisotropic on configuration of DNAare discussed Finally some conclusions are drawn and thepaper is closed

2 The Reduced Form of Kirchhoff Equations

As a coarse-grained description aDNA can be approximatelyregarded as a thin flexible and inextensible rod or string[12 15 16] The classical theory of elasticity describes thegeometry of an elastic rod in terms of its center line R =

R(119904) = (119909(119904) 119910(119904) 119911(119904)) three-dimensional curve parameter-ized by its arc-length 119904 In presence of externalmomentm andexternal load f which are distributed along the central axis R(as show in Figure 1) the static Kirchhoff equations in bodyfixed frame are as follows

119889F119889119904

+ f = 0

119889M119889119904

+ e3times F +m = 0

(1)

where F and M denote the elastic force and momentrespectively

As shown in [17 18] the complex vector basesD119894are used

to substitute the real form vectors e1 e2 e3

D119886=e1minus 119894e2

2

D119886=e1+ 119894e2

2

D3= e3

(2)

P0

r

P

PL

r + drP998400

dr

119813 + d119813

119820+ d119820

0

119839ds119846ds

minus119813

minus119820

Figure 1The deformed state loaded by forces andmoments per unitlength [17]

Thus the complex Kirchhoff equations in the case of zeroexternal momentm can be written out in terms ofD

119894 such as

1198651015840

119886+ 119894 (1198651198861205963minus 1198653120596119886) + 119891119886= 0

1198721015840

119886+ 119894 (119872

1198861205963minus1198723120596119886+ 119865119886) = 0

1198651015840

3+

119894

2(119865119886120596119886minus 1198653120596119886) + 1198913= 0

1198721015840

3+

119894

2(119872119886120596119886minus119872119886120596119886) = 0

(3)

where 119909119886= 1199091+ 1198941199092and 119909

119886denotes the complex conjugate

vector of 119909119886with 119909

119886 119909119886 1199093 being the projection on each

complex axis 119872119886= 1198721+ 1198941198722= 119860120596

1+ 119894119861120596

2 1198723= 119862120596

3

where 1198721and 119872

2are the bending moments and 119872

3is the

twisting component along the rodIn [17 18] a complex expression of119872

3is brought which

is analogous to the complex normal form method Howeverthis expression is inaccurate we revised the expression asfollows

1198723=120572

2(119872119886120596119886+119872119886120596119886) (4)

where 120572 is per unit length scale In this case both sides of (4)are consistent for the dimension and it is easy to find a newway to improve the reduced form while 120572 is a constant or afunction of arc-length

That produces the expressions of 1205961and 120596

2by solving (3)

and (4)

1205961=

(119860 minus 119861)1198723plusmn radic(119860 minus 119861)

21198723

2minus 41198601198611205722 (119872

1015840

3)2

21198601205721198721015840

3

1205962

1205962=radic(119860 minus 119861)1198723 ∓

radic(119860 minus 119861)21198723

2minus 41198601198611205722 (119872

1015840

3)2

2120572 (119860 minus 119861) 119861

(5)



11988921198723

1198891199042minus(119860 minus 119861)

2+ 81205722(119860 + 119861)119867

812057221198601198611198723

+(119860 minus 119861)

2+ (119860 + 119861)119862

21205721198601198611198621198723

2

minus4119860119861 minus 3 (119860 + 119861)119862

211986011986111986221198723

3+2120572

119862(1198891198723

119889119904)

2

= 0

(6)



119904 =119904

120572

1198723=1198723

119864119868

119860 =119860

119864119868

119861 =119861

119864119868

119862 =119862

119864119868

119867 =1198671205722

119864119868

(7)



11988921198723

1198891199042

minus(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

1198723

+(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

1198723

2

minus4119901119902 minus 3 (119901 + 119902)

21199011199021198622

1198723

3

+2

119862

(1198891198723

119889119904)

2

= 0

(8)




1198723(119904) =

119886 cn (120596119904119898)

1 + 119887 cn (120596119904119898) (9)



2 (1 minus 119898) 1198861205962[119886

119862

minus 119887] = 0

(119901 minus 119902)2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ (1 + 21198872minus 2119898 minus 2119887

2119898)1205962= 0

minus 3119886119887(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 1198862[(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

+ (2119898 minus 1)2

119862

1205962] = 0

minus 31198872(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 119886119887(119901 minus 119902)

2+ (119901 + 119902)

119901119902119862

minus 11988624119901119902 minus 3 (119901 + 119902)

21199011199021198622

+ (1198872minus 2119898 minus 2119887

2119898)1205962= 0

minus 1198873(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 1198861198872(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

minus 11988621198874119901119902 minus 3 (119901 + 119902)

21199011199021198622

minus 21198871198981205962minus 119886119898120596

2 2

119862

= 0

(10)


11987230= 1198723(0) =

119886

1 + 119887

1198723

1015840

(0) = 0

(11)





119886 = 119862

119887 = 1

120596 = radic3119901 + 3119902 minus 4119901119902

8119901119902

119898 =41199012+ 119901 (25 minus 36119902) + 119902 (25 + 4119902)

8 (119901 (3 minus 4119902) + 3119902)

119867 = (119888(minus (119901 minus 119902)2+ 1198882119901119902

sdot (minus

2 (119901 + (119901 minus 119902)2+ 119902)

1198882119901119902

+4119901119902 minus 3 (119901 + 119902)

21198882119901119902)))

sdot (8 (119901 + 119902))minus1

11987230=119862

2

(12)


3119901 + 3119902 minus 4119901119902 gt 0

41199012+ 119901 (25 minus 36119902) + 119902 (25 + 4119902)

8 (119901 (3 minus 4119902) + 3119902)gt 1

(13)


1198723(119904) =

119886 dn (120596119904119898)

1 + 119887 dn (120596119904119898) (14)

where

120596 =


4radic2

119898 = minus4 (4119901119902 minus 3 (119901 + 119902))

sdot (119901119902(

2 (119901 + (119901 minus 119902)2+ 119902)

119901119902

minus7 (4119901119902 minus 3 (119901 + 119902))

2119901119902))

minus1

(15)



119889119909

119889119904= 119891 (119909 119910) = 119910

119889119910

119889119904= 119892 (119909 119910) =

(119901 minus 119902)2119862 + 8 (119901 + 119902)119867

8119901119902119862

119909

minus(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

1199092+4119901119902 minus 3 (119901 + 119902)

21199011199021198622

1199093minus

2

119862

1199102

(16)


1199091= 0

1199092=

119888 (2 (119901 + 1199012+ 119902 minus 2119901119902 + 119902

2) minus 119875)

4 (minus3119902 + 119901 (minus3 + 4119902))

1199093=

119888 (2 (119901 + 1199012+ 119902 minus 2119901119902 + 119902

2) + 119875)

4 (minus3119902 + 119901 (minus3 + 4119902))

(17)

where

119875

= radic2

sdot (21199014+ 81199013(2 minus 3119902) + 119901

2(23 minus 56119902 + 60119902

2)

+ 1199022(23 + 2119902 (8 + 119902)) minus 2119901119902 (minus23 + 4119902 (7 + 3119902)))

12

(18)









03 035 04 045 05

minus015

minus01

minus005

0

005

01

015

y

x


0 5 10 15 20 25 30

032

034

03

036

038

042

04

05

044

046

048

s

M3





1 2 3 4

11

12

13

14

15

qp

m

120596


1 2 3 469

71

73

75

77

qp

T



[12]

119879120596=

1

2120587int

119871

0

1205963(119904) 119889119904 (19)



1 2 3 4

0066

0067

0068

0069

007

T120596

qp


4 Conclusion





Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11988921198723

1198891199042minus(119860 minus 119861)

2+ 81205722(119860 + 119861)119867

812057221198601198611198723

+(119860 minus 119861)

2+ (119860 + 119861)119862

21205721198601198611198621198723

2

minus4119860119861 minus 3 (119860 + 119861)119862

211986011986111986221198723

3+2120572

119862(1198891198723

119889119904)

2

= 0

(6)



119904 =119904

120572

1198723=1198723

119864119868

119860 =119860

119864119868

119861 =119861

119864119868

119862 =119862

119864119868

119867 =1198671205722

119864119868

(7)



11988921198723

1198891199042

minus(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

1198723

+(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

1198723

2

minus4119901119902 minus 3 (119901 + 119902)

21199011199021198622

1198723

3

+2

119862

(1198891198723

119889119904)

2

= 0

(8)




1198723(119904) =

119886 cn (120596119904119898)

1 + 119887 cn (120596119904119898) (9)



2 (1 minus 119898) 1198861205962[119886

119862

minus 119887] = 0

(119901 minus 119902)2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ (1 + 21198872minus 2119898 minus 2119887

2119898)1205962= 0

minus 3119886119887(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 1198862[(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

+ (2119898 minus 1)2

119862

1205962] = 0

minus 31198872(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 119886119887(119901 minus 119902)

2+ (119901 + 119902)

119901119902119862

minus 11988624119901119902 minus 3 (119901 + 119902)

21199011199021198622

+ (1198872minus 2119898 minus 2119887

2119898)1205962= 0

minus 1198873(119901 minus 119902)

2119862 + 8 (119901 + 119902)119867

8119901119902119862

+ 1198861198872(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

minus 11988621198874119901119902 minus 3 (119901 + 119902)

21199011199021198622

minus 21198871198981205962minus 119886119898120596

2 2

119862

= 0

(10)


11987230= 1198723(0) =

119886

1 + 119887

1198723

1015840

(0) = 0

(11)





119886 = 119862

119887 = 1

120596 = radic3119901 + 3119902 minus 4119901119902

8119901119902

119898 =41199012+ 119901 (25 minus 36119902) + 119902 (25 + 4119902)

8 (119901 (3 minus 4119902) + 3119902)

119867 = (119888(minus (119901 minus 119902)2+ 1198882119901119902

sdot (minus

2 (119901 + (119901 minus 119902)2+ 119902)

1198882119901119902

+4119901119902 minus 3 (119901 + 119902)

21198882119901119902)))

sdot (8 (119901 + 119902))minus1

11987230=119862

2

(12)


3119901 + 3119902 minus 4119901119902 gt 0

41199012+ 119901 (25 minus 36119902) + 119902 (25 + 4119902)

8 (119901 (3 minus 4119902) + 3119902)gt 1

(13)


1198723(119904) =

119886 dn (120596119904119898)

1 + 119887 dn (120596119904119898) (14)

where

120596 =


4radic2

119898 = minus4 (4119901119902 minus 3 (119901 + 119902))

sdot (119901119902(

2 (119901 + (119901 minus 119902)2+ 119902)

119901119902

minus7 (4119901119902 minus 3 (119901 + 119902))

2119901119902))

minus1

(15)



119889119909

119889119904= 119891 (119909 119910) = 119910

119889119910

119889119904= 119892 (119909 119910) =

(119901 minus 119902)2119862 + 8 (119901 + 119902)119867

8119901119902119862

119909

minus(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

1199092+4119901119902 minus 3 (119901 + 119902)

21199011199021198622

1199093minus

2

119862

1199102

(16)


1199091= 0

1199092=

119888 (2 (119901 + 1199012+ 119902 minus 2119901119902 + 119902

2) minus 119875)

4 (minus3119902 + 119901 (minus3 + 4119902))

1199093=

119888 (2 (119901 + 1199012+ 119902 minus 2119901119902 + 119902

2) + 119875)

4 (minus3119902 + 119901 (minus3 + 4119902))

(17)

where

119875

= radic2

sdot (21199014+ 81199013(2 minus 3119902) + 119901

2(23 minus 56119902 + 60119902

2)

+ 1199022(23 + 2119902 (8 + 119902)) minus 2119901119902 (minus23 + 4119902 (7 + 3119902)))

12

(18)









03 035 04 045 05

minus015

minus01

minus005

0

005

01

015

y

x


0 5 10 15 20 25 30

032

034

03

036

038

042

04

05

044

046

048

s

M3





1 2 3 4

11

12

13

14

15

qp

m

120596


1 2 3 469

71

73

75

77

qp

T



[12]

119879120596=

1

2120587int

119871

0

1205963(119904) 119889119904 (19)



1 2 3 4

0066

0067

0068

0069

007

T120596

qp


4 Conclusion





Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886 = 119862

119887 = 1

120596 = radic3119901 + 3119902 minus 4119901119902

8119901119902

119898 =41199012+ 119901 (25 minus 36119902) + 119902 (25 + 4119902)

8 (119901 (3 minus 4119902) + 3119902)

119867 = (119888(minus (119901 minus 119902)2+ 1198882119901119902

sdot (minus

2 (119901 + (119901 minus 119902)2+ 119902)

1198882119901119902

+4119901119902 minus 3 (119901 + 119902)

21198882119901119902)))

sdot (8 (119901 + 119902))minus1

11987230=119862

2

(12)


3119901 + 3119902 minus 4119901119902 gt 0

41199012+ 119901 (25 minus 36119902) + 119902 (25 + 4119902)

8 (119901 (3 minus 4119902) + 3119902)gt 1

(13)


1198723(119904) =

119886 dn (120596119904119898)

1 + 119887 dn (120596119904119898) (14)

where

120596 =


4radic2

119898 = minus4 (4119901119902 minus 3 (119901 + 119902))

sdot (119901119902(

2 (119901 + (119901 minus 119902)2+ 119902)

119901119902

minus7 (4119901119902 minus 3 (119901 + 119902))

2119901119902))

minus1

(15)



119889119909

119889119904= 119891 (119909 119910) = 119910

119889119910

119889119904= 119892 (119909 119910) =

(119901 minus 119902)2119862 + 8 (119901 + 119902)119867

8119901119902119862

119909

minus(119901 minus 119902)

2+ (119901 + 119902)

2119901119902119862

1199092+4119901119902 minus 3 (119901 + 119902)

21199011199021198622

1199093minus

2

119862

1199102

(16)


1199091= 0

1199092=

119888 (2 (119901 + 1199012+ 119902 minus 2119901119902 + 119902

2) minus 119875)

4 (minus3119902 + 119901 (minus3 + 4119902))

1199093=

119888 (2 (119901 + 1199012+ 119902 minus 2119901119902 + 119902

2) + 119875)

4 (minus3119902 + 119901 (minus3 + 4119902))

(17)

where

119875

= radic2

sdot (21199014+ 81199013(2 minus 3119902) + 119901

2(23 minus 56119902 + 60119902

2)

+ 1199022(23 + 2119902 (8 + 119902)) minus 2119901119902 (minus23 + 4119902 (7 + 3119902)))

12

(18)









03 035 04 045 05

minus015

minus01

minus005

0

005

01

015

y

x


0 5 10 15 20 25 30

032

034

03

036

038

042

04

05

044

046

048

s

M3





1 2 3 4

11

12

13

14

15

qp

m

120596


1 2 3 469

71

73

75

77

qp

T



[12]

119879120596=

1

2120587int

119871

0

1205963(119904) 119889119904 (19)



1 2 3 4

0066

0067

0068

0069

007

T120596

qp


4 Conclusion





Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










03 035 04 045 05

minus015

minus01

minus005

0

005

01

015

y

x


0 5 10 15 20 25 30

032

034

03

036

038

042

04

05

044

046

048

s

M3





1 2 3 4

11

12

13

14

15

qp

m

120596


1 2 3 469

71

73

75

77

qp

T



[12]

119879120596=

1

2120587int

119871

0

1205963(119904) 119889119904 (19)



1 2 3 4

0066

0067

0068

0069

007

T120596

qp


4 Conclusion





Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4

0066

0067

0068

0069

007

T120596

qp


4 Conclusion





Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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