Fractional nonlinear dynamics of DNA breathing

Accepted Manuscript

Fractional nonlinear dynamics of DNA breathing Alain Mvogo, Germain H. Ben-Bolie, Timoleon C. Kofane´ ´ PII: DOI: Reference:

S1007-5704(16)30525-1 10.1016/j.cnsns.2016.12.031 CNSNS 4071

To appear in:

Communications in Nonlinear Science and Numerical Simulation

Received date: Revised date: Accepted date:

16 August 2016 21 November 2016 29 December 2016

Please cite this article as: Alain Mvogo, Germain H. Ben-Bolie, Timoleon C. Kofane, ´ ´ Fractional nonlinear dynamics of DNA breathing, Communications in Nonlinear Science and Numerical Simulation (2016), doi: 10.1016/j.cnsns.2016.12.031

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ACCEPTED MANUSCRIPT Highlights

• The classical Lagrangian and Hamiltonian formulations of the DNA molecular chain is reviewed and extended to include coordinates with time derivative of fractional order. • The analytical procedure for obtaining breather solutions is performed through the application of a fractional perturbation technique.

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• The system exhibits analytical and numerical fractional breather-like modes with high amplitude and velocity when the fractional order decreases.

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ACCEPTED MANUSCRIPT Fractional nonlinear dynamics of DNA breathing Alain Mvogo,1, 2, ∗ Germain H. Ben-Bolie,3, 2 and Timol´ eon C. Kofan´ e4, 2 1

Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon 2 Centre d’Excellence Africain en Technologies de l’Information et de la (CETIC), University of Yaounde I, Cameroon 3 Laboratory of Nuclear Physics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, University of Yaounde, Cameroon 4 Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon (Dated: December 30, 2016)

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The classical Lagrangian and Hamiltonian formulations for the nonlinear dynamics of a homogeneous Peyrard-Bishop DNA molecular chain is reviewed and extended to include coordinates with time derivative of fractional order γ (0 < 2γ < 2). We obtain the equations of motion depending on γ. The analytical procedure for obtaining nonlinear waves solutions is performed through the application of a powerful fractional perturbation technique. The results show that both the amplitude and the velocity of waves increase when γ decreases. Accordingly, for low values of γ, the system exhibits highly localized waves with high amplitude and velocity. The numerical results agree with the theoretical ones and show that the system can support fractional breather-like modes. Keywords fractional derivatives, Peyrard-Bishop model, fractional perturbation technique, fractional breather modes. PACS numbers:

INTRODUCTION

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The DNA is undoubtedly one of the most important biological molecule. Its nonlinear dynamics has been extensively studied due to its biologically functionalities in protection, transcription, translation and replication of the genetic code, and proteins recognition [1]. On the basis of mechanical point of view, attempts to study these processes theoretically have lead to several approximate models. Englander et al. [2] suggested the first nonlinear DNA model. The investigators demonstrated that the nonlinear effects promote concentration of vibrational energy in localized soliton-like excitations [2]. Thereafter, simplified models have been proposed to describe the angular distortions of DNA [3, 4]. Through the micromanipulation experiments of real DNA molecules, the evidence was brought to rather show the great importance of radial displacements of bases during the processes of replication and transcription [5–8]. The Peyrard–Bishop (PB) DNA model [5], which has successfully been used to analyze experiments on short DNA sequences [11], has gained popularity in that direction. Although significant progress have been made to describe DNA with nonlinear models, it remains really difficult to relate all its characteristics to a specific mathematical model. For realistic modeling, one improves the existing models, while taking into account the effects that were previously neglected or unknown. This is done in keeping with the progress in the development of pertinent mathematical and numerical methods. Recent researches motivated the establishment of strategies taking advantage

∗ Electronic

address: mvogal [email protected]

of fractional calculus in the modeling, the study and the control of many phenomena [12–16]. All physical phenomena occur in the form of fractional-order differential equations. Specifically, integer-order differential equations constitute a special case of fractional-order differential equations. Although it has a long history, the applications of fractional calculus to physics and engineering are just a recent focus of interest [17, 18]. In the last decades, fractional equations have played an important role in diverse topics such as fluids and plasma physics [19, 20], fractal theory [21], viscoelasticy [22], electrodynamics [23], optics [24, 25], thermodynamics [26], diffusive neural networks [27] and quantum theory [28–31]. The fact that many biological systems are systems with memory is now confirmed by many researches. Consequently, fractional dynamics can emerge as the concept of adopting fractional calculus in the study of dynamical systems by tacking advantage of the long memory properties of the fractional operators. DNA includes the history of evolution towards the particular species and the instructions for the growth of each individual during its lifetime. This has been motivated recently by Machado et al. [14] who addressed the DNA code analysis in the perspective of system dynamics and fractional calculus. In many cases, memory obeys the power law and the corresponding systems could be described by fractional differential equations. To the best of our knowledge, all above mentioned works [2–8] and many other nonlinear DNA models do not take into account memory effects or non-local effects that are important in DNA [32]. The advantage of modeling DNA using fractional derivatives is the non-local property, and this means that the next state of the system relies not only upon its present state but also upon all of its historical states. The nonlocality appears within the

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I.

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MODEL AND EQUATIONS OF MOTION

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II.

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The PB model [5] considers a simplified geometry for the DNA chain, which consists of a sequence of base pairs. Each base pair includes two degrees of freedom, un and vn corresponding to the displacements of the bases from their equilibrium positions along the direction of the hydrogen bonds that connect two bases in a pair. In order to take into account the memory effects in such a system, the Lagrangian depending on the fractional time derivative of coordinates can be written as follows

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X n 1 h dγ un dγ vn i 1 m ( γ )2 + ( γ )2 − C(un − un−1 )2 2 dt dt 2 n o 1 − C(vn − vn−1 )2 − V (un , vn ) , 2 (1)

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L=

where D is the dissociation energy and a the parameter homogeneous to the inverse of a length, which sets the spatial scale of the potential. dγ The derivative γ which is a fractional power of the dt d ordinary time derivative operator dt is formulated in terms of the left Riemann–Liouville fractional derivative. The parameter γ is the fractional order derivative such that 0 < γ < 1. The representation of this fractional derivative is given by [40] 1 dγ d q(t) = dtγ Γ(1 − γ) dt

0

t

0

0

(t − t )−γ q(t0 )dt ,

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where

Z

Γ(z) =

Z

+∞

tz−1 e−t dt,

where the sum is over all the base pairs of the molecule. The memory effects work through the fractional order parameter γ. The parameter m is a common mass used for all the nucleotides in a strand, and C the same coupling constant along each strand. The intrapair potential is the Morse potential h i2 V (un , vn ) = D e−a(un −vn ) − 1 ,

(2)

(3)

(4)

0

is the gamma function. The coordinate q can be replaced by x or y, respectively. To analyze the motions of the two strands, it is convenient to use the coordinates un and vn for in-of-phase and out-of-phase motions. They are defined as

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fractional derivatives. This has been illustrated clearly in [33–35]. In this work, we show that the nonlocality in DNA can lead to fractional equations for the in-ofphase and out-of-phase motions in DNA. The fractional Lagrangian of DNA model is constructed by replacing the classical derivatives with the fractional ones. The fractional Euler–Lagrange equations are obtained as a result of a fractional variational procedure [33–39]. The dynamics of the system is investigated by applying in the semidiscrete approximation, a fractional perturbation technique for deriving breather-like mode solutions. Results show that the breather parameters, such as its amplitude and velocity, are deeply influenced by the fractional order. According to the system parameters, the system exhibits highly localized waves with high velocity when the fractional order γ decreases. These waves can provide a better representation of the open states of DNA. The rest of the paper is organized as follows. In Sec. II, we propose the model and derive the fractional discrete equations of motion. In Sec. III, we use a fractional perturbation technique in the semi-discrete approximation, and show that the out-of-phase dynamical equation can be reduced to the NLS equation where the dispersion and the nonlinearity coefficients depend on the fractional order parameter. In Sec. IV, our numerical analysis are performed and we bring out of the impact of the fractional order on the dynamics of breather-like modes in the system. In Sec. V, we close the paper with the summary and concluding remarks.

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xn =

un + vn un − vn √ and yn = √ . 2 2

(5)

Obviously, xn describes the movement of the center of mass of the nucleotide pair at the site n, while yn represents the stretching of the pair. According to Eq. (2) and Eq. (3), the fractional Lagrangian of the system can be rewritten as X n 1 h dγ un dγ vn i 1 m ( γ )2 + ( γ )2 − C(xn − xn−1 )2 L= 2 dt dt 2 n 2 o  √ 1 − C(yn − yn−1 )2 − D e−a 2yn − 1 . 2 (6) Since the Lagrangian in Eq. (1) and Eq. (6) is formulated only in the terms of the left RL fractional derivative, we assume the fractional Euler-Lagrange equation [38, 39] to be in the form:  ∂L ∂L d γ + −  dγ y  = 0, ∂yn dt n ∂ dtγ

(7)

Then, the equations of motion for un and vn after some algebras read (−1)γ

(−1)γ

d2γ xn C = − (xn+1 − 2xn + xn−1 ) , dt2γ m

(8)

d2γ yn C =− (yn+1 − 2yn + yn−1 ) 2γ dt m√  (9) 2 2aD −a√2yn  −a√2yn − e e −1 . m

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√ Yn = a 2yn ,

τ=



r

Da2 2γ t, m

Y (z, τ ) =

C k= 2 , 4a D

(10)

(12)

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where ωg2 = 1, α = − 32 and β = 76 . In the next sections, we relate the analysis of Eq. (13) to breather-like modes in DNA. We will begin by applying the multiple scale method in the semi-discrete approximation to Eq. (13). After, the numerical analysis will be performed on Eq. (9) in order to get an idea of the impact of the fractional order γ of the lattice on breather modes propagation. THE FRACTIONAL SEMI-DISCRETE APPROXIMATION

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In this section, we use the semi-discrete approximation to study the dynamics of DNA breathing. We first assume small amplitude oscillation of the nucleotide around the bottom of the Morse potential. This assumption allows to set the out-of-phase transverse motion in the form [9] Yn = εψn ,

ε  1.

(13)

Replacing Yn defines below into Eq. (13), we obtain (−1)γ

d2γ ψn = − k (ψn+1 − 2ψn + ψn−1 ) dt2γ + ωg2 (ψn + εαψn2 + ε2 βψn3 ).

(15)

j=1

d = D0 + εD1 + ε2 D2 + .... dτ

(11)

Since we need to perform analytical calculation, we expand the substrate potential up to the third order approximation, Eq. (12) becomes d2γ Yn (−1)γ = − k (Yn+1 − 2Yn + Yn−1 ) dτ 2γ + ωg2 (Yn + αYn2 + βYn3 ),

εj ψj (Z0 , Z1 , Z2 , ..., T0 , T1 , T2 , ...),

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d2γ Yn = − k (Yn+1 − 2Yn + Yn−1 ) dτ 2γ + e−Yn − e−2Yn .

∞ X

where Zj and Tj are independent variables. Using the ∂ , the time derivatives are expressed in notation Dj = ∂T j terms of fast and slow time scales as follows:

which transform Eq. (9) in the form (−1)γ

Equation (14) describes the fractional dynamics of the perturbed out-of-phase equation of motion. It has been shown that the multiple scale expansion in the semidiscrete approximation is more adapted to solve such equation [41]. This approximation is a perturbation technique in which the amplitude is treated in the continuum limit, while the carrier waves are kept discrete. The technique allows the study of the modulation of the wave. We proceed by making a change of variables according to the space and new time scales Zj = εj z and Tj = εj τ , respectively. The importance of the multiple scale expansion lays on finding a solution Y (z, τ ) depending on these new sets of variables as perturbation series of functions. We consider that

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For γ = 1, we get the classical equations for xn and yn . The first of the equations describes usual linear waves (phonons) while the second one contains nonlinear terms (see below) and describes nonlinear waves (breathers). So to investigate nonlinear waves in our fractional DNA model, we restrict our attention on the second fractional equation. In this work, the values of parameters used to perform the numerical analysis have been borrowed from the dynamical and denaturation properties of DNA. They are [9, 10]: D = 0.03 eV , a = 4.5 ˚ A−1 , m = 300 a.m.u, and C = 0.06 eV/˚ A2 . On the other hand, it is convenient for the analytical and numerical calculations to transform Eq. (9) into a dimensionless form by defining the dimensionless variables

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And similarly, the spatial derivatives can be expressed ∂ , by DZj = ∂Z j ∂ = DZ0 + εDZ1 + ε2 DZ2 + .... ∂z

(17)

We look for modulated wave solutions of Eq. (14) in the form ψn = F1n eiθn + εF2n e2iθn + c.c.,

(18)

with θn = q ∗ n − ωτ , where q ∗ is the dimensionless wave vector, ω is the frequency and c.c. stands for the complex conjugate terms. In the rest of the work, we omit the asterisk in q ∗ , for simplicity. F1 , and F2 are unknown functions that must be determined. The derivative of a fractional order should be subject to a rule that is generalization of the Leibniz rule [42] ∞   X Dτν f g =

   Γ(ν + 1) Dτν−k f Dτk g Γ(k + 1)Γ(ν − k + 1) k=0 (19) to the case of differentiation and integration of fractional order [40]. Dν is the Riemann-Liouville derivative and Dk is the derivative of integer order k. The fractional derivative of function ψn can be therefore expressed as follows: Dτν ψn

(14)

(16)

=

h   Γ(ν + 1) Dν−k eiθn Dk F1n Γ(k + 1)Γ(ν − k + 1) k=0    i + ε Dν−k e2iθn Dk F2n + ... , ∞ X

(20)

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d where Dν ≡ dτ ν (ν = 2) is the Riemann-Liouville derivak tive and D is derivative of integer order k. The equality in Eq. (6) allows one to use the Liouville representation of the fractional derivative applied to the exponential function:

To order [ε2 , exp[iθn ]], introducing ξ1 = Z1 − υg T1 and τ2 = T2 , we get the NLS equation governing the slow envelope evolution

i (21)

∂F1 ε2 ∂ 2 F1 ∂F1 + ..., ± ε2 + ∂Z1 ∂Z2 2 ∂Z12 ∂F2 ε2 ∂ 2 F2 ∂F2 =F2 ± ε + .... ± ε2 + ∂Z1 ∂Z2 2 ∂Z12

F2(n±1)

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F1(n±1) =F1 ± ε

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Collecting all the above informations in Eq. (14), order [ε0 , exp[iθn ]] is associated with the linear approximation and after some algebras leads to the dispersion relation

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ω 2γ = ωg2 + 4k sin2 (q/2).

(23)

To order [ε1 , exp[iθn ]], we get the equation ∂F1 ∂F1 + υg = 0, ∂T1 ∂Z1

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(24)

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where υg is the group velocity given by υg =

k sin(q) . ω 2γ−1

P =

1 2ω 2γ−1

h

k cos(q) −

F2 = δF12 ,

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i Γ(2γ + 1) ω (2γ−2) υg2 , (29) Γ(3)Γ(2γ − 1)

and Q=−

n o 2 ω (2αδ + 3β) . g 2γ−1 1

2ω

(30)

The coefficients P and Q represent the dispersion and the nonlinearity coefficients, respectively. For P Q > 0, Eq. (28) has an envelope soliton solution describing a small-amplitude breather. This solution is ξ − v τ  h v i 1 e 2 e F1 = A sech exp i (ξ1 − vc τ2 ) , Le 2P

(31)

where A is the amplitude of the envelope and Le is its width. The expression ve2 − 2ve vc > 0, where ve and vc are the velocities of the envelope and the carrier waves, respectively. A and Le are given respectively by

A=

s

ve2 − 2ve vc , 2P Q

Le = p

2P . ve2 − 2ve vc

(32)

The analysis reveals that we have a group of parameters coming from the mathematical procedure: ε, ve , and vc . The velocities ve , and vc are included in the solution of the NLS equation. On the other hand, ε does not have any physical meaning. This is nothing but a “working” parameter, helping us to distinguish big and small terms in the series expansion Eq. (18). Hence, one would expect that DNA dynamics does not depend on it. In other words, ε exists in the derivations but is not expected to determine the final solution Yn (t). A careful investigation of all the formulaes shows that only two mathematical parameters are relevant and they are: εve , and εvc . We can therefore introduce parameters Ve and η defined as

(25)

To order [ε1 , exp[2iθn ]], we obtain the expression for the second harmonic given by

(28)

with

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To obtain the solution of Eq. (14), the new set of variables Zj and Tj should be considered from the physical line and allows us to impose the condition that for small values of ε, the asymptotical expansion must converge uniformly. The procedure will consist of replacing the form of the solution Eq. (18) and its derivatives in the different terms of Eq. (14). We then group terms in the same power of ε, which leads us to a system of equations. Each of those equations will correspond to each approximation for specific harmonics. Nonlinear terms in Eq. (14) incite one to predict that through frequencies superpositions, the first harmonics of the wave will contain terms in e±2iθn as well as terms without any exponential dependence. Since the envelope function varies slowly in space and time, we use the continuum approximation for F1,n , F0,n and F2,n in a multiple scale expansion such that they can be obtained at the order ε2 by a Taylor expansion

∂F1 ∂ 2 F1 2 + Q|F1 | F1 = 0, +P ∂τ2 ∂ξ12

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dγ iωt e = (iω)γ eiωt . dtγ

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Ue = εve ,

η=

vc . ve

(33)

This allows us to reduce the number of parameters to be studied in this work. This, practically, means that η remains the single mathematical parameter. εA and L/ε can therefore be written as

with αωg2 δ= , (2ω)2γ − ϕ

ϕ=

ωg2

2

+ 4k sin (q).

(27)

εA = A0 = Ue

r

1 − 2η , 2P Q

Le 2P √ =L= . (34) ε Ue 1 − 2η

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Ve = vg + Ue =

Ω . Θ

(35)

An expression for the velocity Ue = εve can consequently be written as Pq h Ue = 1−η

s

1+

i 2(1 − η)(ω − qυg ) −1 . 2 Pq

(36)

This allows us to delete the most troublesome dimensionless parameter ε. Then the final expression of Yn (t) is

where Θ and Ω are explicitly now given by

Ue Ue Θ= q+ , Ω= ω+ (υg + ηUe ). 2P 2P

(38)

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Furthermore, from Eq. (38), the wave amplitude will be given by   Ym = 2A0 1 + A0 δ .

(39)

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Finally, the coherence of the soliton solution of the model will be effective if the number of the nucleotide pairs covered by a single soliton is known. Assuming that the wave covers an integer number of N nucleotides [43, 44], we also redefine the dimensionless wavenumber q of the carrier component of the soliton such that 2π . N

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(40)

We now discuss on the impact of the fractional order γ on the breather parameters. For this, we plot first in Fig.1 the product P Q for different values of q (respectively N ) in the interval γ ∈ [0.1, 1]. In Fig. 1(a), we plot P Q for N = 1, 5, 6, 7. We observe that the product P Q is negative for all γ chosen in the interval [0.1, 1]. The same result has been also obtained for all the values of N > 7. In this case, the system is modulationally stable and Eq. (28) has an envelope-hole (or dark) soliton solution which does not correspond to the small-amplitude limit of breather modes [45]. In Fig. 1(b), we can see that for N = 2, 3, the product P Q is positive for all γ ∈ [0.4, 1]. In this case, the system is modulationally

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NUMERICAL RESULTS

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hn − V τ i e Yn (τ ) =2A0 sech L n hn − V τ i e (37) × cos(Θn − Ωτ ) + A0 sech L o × δ cos[2(Θn − Ωτ )] ,

unstable and can support breather mode solutions. We plot in Fig. 1(c) the product P Q for N = 4. We can see that the product P Q is negative for γ ∈ [0.1, 0.5[ and positive for γ ∈]0.55, 1]. Therefore the model under study can support breather mode solutions for wave numbers π q = π, q = 2π 3 and q = 2 . We plot in Figs. 2, 3 and 4 the breather parameters as a function of the fractional order γ for N = 2, 3, 4, respectively. In Figs. 2(a) and 2(b), we observe that for N = 2 both the amplitude and the velocity of the wave decrease as the parameter γ increases. However, we remark in Fig. 2(b) that the wave can be see to be static since its velocity remains very weak. In principle, stationary envelope solitons can be viewed as discrete breather. In Figs. 3(a), 3(b) and Figs. 4(a), 4(b), we chose N = 3, 4. We observe that both the amplitude of the wave and its velocity increase [this can be seen from Fig. 3(a) and 4(a), Fig. 3(b) and 4(b)] as the fractional order γ decreases. In the next section to confirm the correctives of the analytical solution, we perform the numerical simulations of de discrete equation of motion Eq. (9).

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The above notations are possible if 0 ≤ η < 0.5. The requirements for the coherent mode can be fulfilled if [43, 44]

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In section III, simplifications have been brought to Eq. (9) using a semi-discrete technique to make it more mathematically manipulable. This technique only gives information about the dynamical equation and cannot tell us anything on the longtime evolution of breather-like modes of the system. In our recent paper, we proposed both the semi and the linearly implicit Riesz fractional finite-difference schemes to solve efficiently the complex fractional Ginzburg-Landau equation derived from a diffusive neural network with non local effects [27]. In this paper, we deal with numerical analysis in a discrete system. Then, we integrate Eq. (20) through the fractional fourth-order Runge–Kutta computational technique for a lattice of 300 cells (base pairs). To obtain such a fractional numerical technique, algorithms can be found in Refs. [46, 47]. The differential operators are taken in the Riemann–Liouville sense and the initial conditions are specified according to Caputo’s suggestion, thus allowing for interpretation in a physically meaningful way. It is well dγ known that Dγ ≡ γ has an m-dimensional kernel, and dt therefore we certainly need to specify m initial conditions in order to obtain a unique solution of the straightforward form of a fractional differential equation, viz Dγ y(t) = f (t, y(t)),

(41)

with some given function f . Now, according to the standard mathematical theory [40], the initial conditions corresponding to (11) must be of the form

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k = 0, 1, 2, ..., m,

(42)

with given values bk . In practical applications, these values are frequently not available, and it may not even be clear what their physical meaning is. Therefore Caputo [49] has suggested that one should incorporate the classical derivatives of the function y, as they are commonly used in initial value problems with integer-order equations, into the fractional-order equation, giving Dγ (y − Tm−1 [y])(t) = f (t, y(t)),

(43)

V.

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where Tm−1 [y] is the Taylor polynomial of order m − 1 for y. Then, one can specify the initial conditions in the classical form. The parameters used to perform the numerical analysis have been borrowed from the dynamical and denaturation properties of DNA. They are: D = 0.03 eV , a = 4.5 ˚ A−1 , m = 300 a.m.u, and the coupling parameter C = 0.06 eV/˚ A2 , which are known in DNA-like models [9, 10]. While performing the simulations, we consider the DNA chain with 300 base pairs and put the breather center in the site with the number n = 50. In Figure 5, we have represented the evolution of the solution at different time [τ = 0, τ = 40 and τ = 80] according to the values of γ: γ = 0.55 and γ = 0.1. We can see in Figure 5(a) that the solution is well a localized modulated solution, and its propagates structurally stable. The modulated solution involving in the system appears in the form of a breather-like coherent structure. In Figure 5(b), we observe that for γ = 0.1, the amplitude of the breather increases and it propagates more rapidly as compared with the case where γ = 0.55. The obtained numerical results are in perfect accord with the analytical ones.

To achieve our goal, the Lagrangian of the system has been reformulated to include coordinates with the Riemann-Liouville derivatives of fractional order. We have used a new perturbation method in the semidiscrete approximation to study the dynamics of breather-like modes in DNA. The out-of-phase motion equation has been reduced to the NLS equation where the coefficients depend not only on the wave number and the system parameters, but also on the fractional order. Regions with different behaviors concerning the modulational instability of a plane wave have been predicted. For P Q > 0, we have derived the analytical soliton solution describing the propagation of nonlinear waves in the system. We have found as compared to the classical model [5] that the decreasing of the fractional order γ leads to highly localized waves with high amplitude and velocity that can propagate in DNA. Therefore, in the fractional model, we have breathers of small amplitude and breathers of high amplitude. The first type describes information transfer in DNA, and the second type belonging to the resonant mode describes the open states of DNA [50]. We have found also that the fractional DNA modeling can give the opportunity to describe particulary the DNA at the scale of a few of base pairs, which is now a challenge topic in nonlinear science [32]. The numerical results have agreed the analytical ones. This work provides some new mathematical methods in the area of soliton theory. In a forthcoming paper, the results will be discussed in the framework of the Peyrard–Bishop–Dauxois model [8, 51].

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γ−k

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CONCLUSION

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In this paper, we have investigated the coherent dynamics of breather-like modes in a fractional DNA model.

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Acknowledgments

References

[1] C. R. Calladine, H. R. Drew, R. F. Luisi, and A. A. Travers, Understanding DNA (Amsterdam: Elsevier 1997). [2] S. W. Englander et al., Proc. Natl. Acad. Sci. U.S.A. 77, 7222 (1980). [3] S. Yomosa, Phys. Rev. A 27, 2120 (1983). [4] S. Homma and S. Takeno, Prog. Theor. Phys. 70, 308 (1983).

A. Mvogo acknowledges fruitful discussions with Pr. Sylvie Paycha (from the Department of Mathematics, University of Potsdam, Germany).

[5] M. Peyrard and A. R. Bishop, Phys. Rev. Lett. 62, 2755 (1989). [6] T. Dauxois, Phys. Lett. A. l59, 390 (l991). [7] T. Dauxois, M. Peyrard, and A. R. Bishop, Phys. Rev. E 47, R44 (1993). [8] T. Dauxois and M. Peyrard, Phys. Rev. E 51, 4027 (1995). [9] T. Dauxois, M. Peyrard, and A. R. Bishop, Physica D 66,35 (1993). [10] M. Peyrard, Nonlinearity 17, R1 (2004).

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(1975). [46] I. Petr´ aˇs, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation Series: Nonlinear Physical Science, (Springer, HEP. 2011). [47] I. Podlubny, T. Skovranek, and B. M. Vinagre, Matrix Approach to Discretization of ODEs and PDEs of Arbitrary Real Order (2008), MathWorks, Inc. Matlab Central File Exchange, URL: www.mathworks.com/matlabcentral/fileexchange/22071. [48] B. F. Putnam, L. L. Van Zandt, E. W. Prohofsky, K. C. Lu, and W. N. Mei, Biophys. J. 35, 271 (1981). [49] M. Caputo, Geophys. J. Roy. Astronom. Soc. 13, 529 (1967). [50] M. B. Hakim, S. M. Lindsay, and J. Powell, Biopolymers 23, 1185 (1984). [51] T. Dauxois, Phys. Lett. A 59, 390 (l991).

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[11] A. Campa and A. Giansanti, Phys. Rev. E 58, 3585 (1998). [12] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis 5, 367 (2002). [13] R. Caponetto, R. Dongola, L. Fortuna, I. Petr´ as, Fractional Order system: modeling and control applications. World Scientific Series on Nonlinear Science (Series A 72) (2010). [14] J. A. Tenreiro Machado, A. C. Costa, M. D. Quelhas, Commun. Nonlinear Sci. Numer. Simul. 16, 2963 (2011). [15] T.T. Hartley, C.F. Lorenzo, H.K. Qammer. Chaos in a fractional order Chua’s system. IEEE Transactions Circuits and Systems I, 42(8), 485-490 (1995). [16] I. Grigorenko, E. Grigorenko. Chaotic Dynamics of the Fractional Lorenz System. Phy. Rev. Lett. 91, 485 034101 (2003). [17] I. Podlubny. Fractional differential equations. New York: Academic Press. (1999). [18] R. Hilfer, editor. Applications of fractional calculus in physics. New Jersey: World Scientific; (2001). [19] B. A. Carreras, V. E. Lynch, and G. M. Zaslavsky, Phys. Plasmas 8, 5096 (2001). [20] V. E. Tarasov, Phys. Plasmas 12, 082106 (2005). [21] A. Carpinteri and F. Mainardi, Fractals and fractional calculus in continuum mechanics (Springer, 1997). [22] V. V. Novikov and K. V. Voitsekhovskii, J. App. Mech. Tech. Phys. 41, 149 (2000). [23] N. Engheta, IEEE Ant. Prop. Mag. 39, 35 (1997). [24] J. C. Gutierrez-Vega, Opt. Lett. 32, 1521 (2007). [25] J. C. Gutierrez-Vega, Opt. Express 15, 6300 (2007). [26] A. Gaies and A. El-Akrmi, Phys. Scr. 70, 7 (2004). [27] A. Mvogo, A. Tambue, G. H. Ben-Bolie, and T.C. Kofane, Commun. Nonlinear Sci. Numer. Simulat. 39, 396 (2016). [28] N. Laskin, Phys. Rev. E 66, 056108 (2002). [29] A. Mvogo, G. H. Ben-Bolie, and T. C. Kofane, Phys. Lett. A 378, 2509 (2014). [30] N. Laskin, Phys. Lett. A 268, 298 (2000). [31] A. Iomin, Phys. Rev. E 80, 022103 (2009). [32] M. Peyrard, S. Cuesta-Lopez, and G. James, Nonlinearity 21, T91 (2008). [33] D. Baleanu, Commun Nonlinear Sci Numer Simulat 14, 2520–2523 (2009). [34] D. Baleanu, S.I. Muslih, and E. M. Rabei, Nonlinear Dyn. 53, 67–74 (2008). [35] D. Baleanu and S.I. Muslih, Phys. Script. 72, 119-121 (2005). [36] F. Riewe, Phys. Rev. E 53, 1890 (1996). [37] F. Riewe, Phys. Rev. E 55, 3581 (1997). [38] O. P. Agrawal, J. Math. Anal. Appl. 272, 368 (2002). [39] E. M. Rabei et al., J. Math. Anal. Appl. 327, 891 (2007). [40] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993). [41] M. Remoissenet, Phys. Rev. B. 33, 2386 (1986). [42] J. Liouville, Memoire sur le calcul des differentielles a ` indices quelconques. J l’Ecole R Polytech [Extrait (complet) du Tome 13, section 21, p.71-162. (see p.118)] (1832). [43] S. Zdravkovic and M. V. Sataric, Phys. Lett. A 373, 126 (2008). [44] S. Zdravkovic, J. Nonl. Math. Phys. 18, 463 (2011). [45] M. Remoissenet and M. Peyrard, Phys. Rev. B 29, 3153

8

ACCEPTED MANUSCRIPT 0

9

0.14

−0.05

0.13

N=1 N=5 N=6 N=7

−0.1

0.12

PQ

0.11 −0.15

Ym

0.1

−0.2

0.09 0.08

−0.25 0.1

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

1

(a)

0.07 0.06

0.35 N=2 N=3

0.05

0.3

0.04 0.1

0.25

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

1

(a)

0.15

−18

4.6

0.1

4.4

0.05

4.2

0 0.1

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

1

(b)

4

−4

8

x 10

x 10

3.8 Ve

6 4

3.6 3.4 3.2

0

3

−2

2.8

−4 −6 −8 0.1

AN US

PQ

2

CR IP T

PQ

0.2

2.6 0.1

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

1

(c)

AC

CE

PT

ED

M

FIG. 1: Variations of the product P Q as a function γ. The parameter values are: The parameter values are: D = 0.03 eV , a = 4.5 ˚ A−1 , m = 300 a.m.u, and C = 0.06 eV/˚ A2 .

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

1

(b)

FIG. 2: Plot of the breather parameters as a function of the fractional order γ for N = 2. (a) The amplitude of the breather, (b) its velocity. Parameter values are the same as in Fig. 1.

ACCEPTED MANUSCRIPT 0.14

0.09

0.13

0.085

0.12

0.08

0.11

0.075 0.07 Ym

0.09

0.065

0.08

0.06

0.07

0.055

0.06

0.05

0.05

0.045

0.04 0.1

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

0.04 0.5

1

0.6

0.7

(a) 0.0247

0.029 0.028

0.0246 0.027 0.026

0.0245

Ve

Ve

0.025 0.0244

0.024 0.0243

0.022

AN US

0.023

γ

0.8

0.9

1

(a)

CR IP T

Ym

0.1

10

0.0242

0.021 0.02 0.1

0.2

0.3

0.4

0.5

γ

0.6

0.7

0.8

0.9

1

0.0241 0.5

(b)

AC

CE

PT

ED

M

FIG. 3: (Color online) Plot of the breather parameters as a function of the fractional order γ for N = 3. (a) The amplitude of the breather, (b) its velocity. The parameter values are the same as in Fig. 1.

0.6

0.7

γ

0.8

0.9

1

(b)

FIG. 4: Plot of the breather parameters as a function of the fractional order γ for N = 4. (a) The amplitude of the breather, (b) its velocity. The parameter values are the same as in Fig. 1.

ACCEPTED MANUSCRIPT

Yn (τ=0)

0.05 0 −0.05 −0.1 48

50

52

54

56

58 60 Base index (n)

62

64

66

68

Yn (τ=40)

0.1 0.05 0 −0.05 −0.1

50

55

60

65

Base index (n)

0 −0.05

45

50

55 Base index (n)

60

65

(a)

0

AN US

Yn (τ=0)

0.1

−0.1 −0.2 −0.3

46

48

50

52

54

56 58 Base index (n)

60

60 Base index (n)

65

62

64

Yn (τ=40)

0.1 0 −0.1

55

0.1 0 −0.1 55

60

65 Base index (n)

PT

−0.2

ED

Yn (τ=80)

0.2

70

70

CE

FIG. 5: Time behavior of propagating breather soliton in DNA for q = 2π (a) γ = 0.55 and (b) γ = 0.1. The pa3 rameter values are: The parameter values are: D = 0.03 eV , a = 4.5 ˚ A−1 , m = 300 a.m.u. and C = 0.06 eV/˚ A2 .

AC

66

M

−0.2 50

CR IP T

Yn (τ=80)

0.05

(b)

11