IC/2009/034

Available at: http://publications.ictp.it

United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

DISCRETE INSTABILITY IN THE DNA DOUBLE HELIX

Conrad Bertrand Tabi1 Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, Alidou Mohamadou2 Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon and Condensed Matter Laboratory, Department of Physics, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon and Timol´eon Cr´epin Kofan´e3 Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.

MIRAMARE – TRIESTE June 2009

1

[email protected] [email protected] 3 [email protected] 2

Abstract Modulational instability (MI) is explored in the framework of the base-rotor model of DNA dynamics. We show in fact that, the helicoidal coupling introduced in the spin model of DNA reduces the system to a modified discrete sine-Gordon (sG) equation. The MI criterion is thus modified and displays interesting features because of the helicoidal coupling. This is confirmed in the numerical analysis where a critical value of the helicoidal coupling constant is derived. In the simulations, we have found that a train of pulses are generated when the lattice is subjected to MI, in agreement with analytical results obtained in a modified discrete sG equation. Also, the competitive effects of the harmonic longitudinal and helicoidal constants on the dynamics of the system are notably pointed out. In the same way, it is shown that MI can lead to energy localization which is high for some values of the helicoidal coupling constant.

1

Leading paragraph It is well known that the modulated signals are very important and widely used in engineering since they can be transmitted along much longer distance than nonmodulated waves. As a whole, those waves have the form of soliton-like objects and, one of the ways to produce them is modulational instability. Modulational instability has also been shown to be a pathway to energy localization in biomolecules and in discrete systems, in general. It is a result of the interplay between nonlinearity and dispersion and arises in continuous as well as in discrete systems. Different available methods are used to study the condition for the formation of modulated wave, and the one addressed here is the so-called Discrete multiple scaling method. The base-rotor model, where we introduce the helicoidal coupling, is addressed and our results show that, the interplay between stacking and helicoidal couplings can give rise to highly localized structures with a consequence on energy localization.

1

Introduction

The great interest aroused by solitons and localized structures formation in biomolecules and in DNA in these last years is proof of the fundamental role they play in human life. In fact, DNA molecules are the support for genetic information, and knowledge of their sequence is essential from the biological and medical point of view. Furthermore, the leading phenomena of transcription and replication confer to DNA mechanical features that are still not yet fully understood by physicists and biophysicists alike. From the physical point of view, the DNA molecule is nothing but a system consisting of many interacting atoms organized in a special way in space. It was shown by Franklin and Gosling [1], Watson and Crick [2], Wilkins et al. [3] and Crick and Watson [4] that: (i) under usual external conditions (temperature, pH, etc.) the molecule has the double helix form and (ii) the helix is not a static structure. On the contrary, DNA is a very flexible molecule. On this background, following the pioneering work of Englander et al. [5], a number of mathematical models of the DNA double chain have been proposed over the years, focussing on different aspects of the DNA molecule and on different biological, physical, and chemical processes in which DNA is involved. Those models include the plane-base rotor model by Yomosa [6, 7], later improved by Takeno and Homma [8] and the Peyrard-Bishop (PB) model [10], later improved by Dauxois [11] and Barbi [12]. In the first model attention has been paid to the degree of freedom characterizing base rotation in the plane perpendicular to the helical axis along the backbone structure. DNA dynamics, in the above models has been shown to be governed by the sine-Gordon equation, while the second model has been used to study the DNA thermal denaturation (or melting) and the corresponding degree of freedom were related to straight or radial separation of the two strands which are wound together in the DNA double helix. Together with the Yakushevich model [13, 14, 15, 16] (which is concerned with rotational and torsional degrees of freedom of the molecule), the PB model has been widely used in the 2

literature [17, 18, 19, 20]. Most of the studies fulfilled in the framework of the above-mentioned applied theoretical models refer to the formation of bubbles and localized structures. These bubbles has been shown to play a vital role in DNA processes such as replication, transcription, recombination, and reparation, or the DNA transcription to several types of RNA including those involved in the protein synthesis. The use of localized structures in explaining these phenomena sets the problem of their creation and stability. In the past years, it was found by many investigators that this is a rather complicated and substantial problem itself. In the framework of the PB model, Daumont et al. [21], and more recently, Tabi et al. [20] have shown that the discreteness of the system causes the instability of the extended solitons. They tend to self-modulate evolving to localized soliton-like modes that interact nonelastically and grow the largest ones at the expenses of the smallest [19, 20, 22, 23]. This phenomenon, known as modulational instability (MI) is the outcome of the interplay between nonlinearity and dispersive/diffraction effects. MI arises in continuous as well as in discrete systems but, as a whole, the analysis of such nonlinear systems is performed by first deriving from the original model a simpler limit equation which usually results to the nonlinear Schr¨odinger (NLS) equation [20, 21, 24]. This universal limit is obtained by a very general approach called multiscale perturbation analysis [20, 21, 24]. In the case of discrete systems where nonlinear localized modes (also called discrete breathers) are known to exist [25], MI is usually described in a semi-discrete multiple scale analysis where the envelope of the carrier wave is treated as a continuous function. Recently, Meir et al. [26] have reported the first experimental observation of MI in physical systems. Their studies reveal a diversity of behaviors forbidden in continuous media. Leon and Manna [27] have proposed a theoretical approach to study discrete instability in nonlinear lattices. They show that discrete multiple scale analysis for boundary value problems in nonlinear discrete systems leads to a first order, strictly MI (disappearing in the continuous envelope limit) above a threshold amplitude for wavenumbers, beyond the zero of group velocity dispersion. This approach has been applied to the electrical lattice [28] with a good agreement with the experimental results. In connection with the studies of these authors, we will also apply the discrete multiple scale analysis to the DNA lattice model and study the resulting MI as well as soliton formation through MI. Therefore, the model we consider here is inspired from the Zhang model [29] where kink and anti-kink solitons are known. That model, after some mathematical simplifications, reduces to the known sine-Gordon equation. With in mind the fact that the B-DNA is mainly helicoidal, we introduce the helicoidal coupling as done by Gaeta [16], Dauxois [30] and Tabi et al. [20]. The impact of the helicoidal coupling constant, on the formation of localized structures, is investigated. The present discrete multiple scale analysis relies on the definition of a large grid scale via the comparison of the magnitude of the related difference operator, and on the expansion of the wavenumber in powers of frequency variations due to nonlinearity. The rest of the paper is therefore organized as follows: in Section 2, we discuss the model and we derive the equations that describe the dynamics of the 3

hydrogen bonds. In Section 3, a brief presentation of the multiple scale method is made and the envelope equation is derived. In Section 4, after deriving the condition for obtaining localized structures in the model understudy, numerical investigations are performed in order to confirm analytical predictions. The last section is dedicated to some concluding remarks.

2

Model and dynamical equations

In this work, we consider the so-called B-form of the DNA molecule as presented in Fig.1(a). S and S 0 represent the complementary strands in the double helical structure. Each arrow in the figure stands for the hydrogen bonding effects that take place in the molecule between the complementary bases. The z-axis is chosen along the helical axis of DNA. In Figs.1(b) and (c), we show the horizontal projection of the nth base pair in the xy and xz planes, respectively. In these figures, Qn and Q0n denote the tips of the nth bases belonging to the strands S and S 0 . Pn and Pn0 represent the points where the bases in the nth base pair are attached to the strands S and S 0 , respectively. As a whole, modelling the dynamics of the DNA molecule should take into account at least three interactions: the hydrogen bonding effect which, in fact, models the interactions taking place between two bases in a pair. From a heuristic point of view, this energy depends on the distance between two bases in a pair. Thus from Fig.1(b), the square of the distance between the edges of the arrows (Qn Q0n )2 is written as [9, 29] (Qn Q0n )2 = 2 + 4r2 + (zn − zn0 )2 + 2(zn − zn0 )(cos θn − cos θn0 ) − 4r[sin θn cos φn + sin θn0 cos φ0n ] + 2[sin θn sin θn0

(1)

× (cos φn cos φ0n + sin φn sin φ0n ) − cos θn cos θn0 ] where r is the radius of the circle depicted in Fig.1(b). The base-base interaction energy can be understood in a clearer and more transparent way by introducing Sn = (Snx , Sny , Snz ) and 0

0

0

0

Sn = (Snx , Sny , Snz ) given by [9, 29] Snx = sin θn cos φn , 0

0

0

Sn x = sin θn cos φn ,

Sny = sin θn sin φn , 0

0

0

Sn y = sin θn sin φn ,

Snz = cos θn 0

0

Sn z = cos θn

(2)

for the strands S and S 0 , respectively. In terms of this, Eq.(1) can be rewritten as [9, 29] 0 0 0 0 (Qn Q0n )2 = 2 + 4r2 + 2 Snx Snx + Sny Sny − Snz Snz − 4r Snx + Snx

(3)

In the above equation (3), the longitudinal compression along the direction of the helical axis has been neglected and we have assumed zn = zn0 [9]. So, in what follows, we consider the anisotropic Heisenberg model of DNA shown in Fig. 2 [29]. In the case of spin chains, each arrow represents a group of atom at that lattice point. In the present model, arrows are marked anti-parallel as in an antiferromagnetic model. Here the z -direction (i.e. the direction of the helix axis) is chosen as the easy axis of magnetization in the spin chain. The spin-spin interaction takes into account

4

the stacking interaction between nearest neighbors (the nth pair interacts with both the (n + 1)th and (n − 1)th pairs). Another effect, newly introduced in this model, is due to water filaments that link units at different sites. In particular, they have a good probability to form between nucleotides which are a half turn of the helix apart on different chains, i.e., which are near to each other in space due to the double helix geometry; these water filaments-mediated interactions are therefore also called helicoidal interactions ( the nth pair interact with both the (n + h)th and (n − h)th pairs) [20, 30, 31] with a pitch h that could be equal to 4 [20, 30, 31, 32] or to 5 [33, 34]. Since the helicoidal pitch is 11, we will use in the rest of the paper h = 5. Further, in this study, we assume the two strands to be the same and could, therefore, shear the same parameters. With these considerations, the Hamiltonian of the Heisenberg model for an anisotropically chain of coupled spin model and antiferromagnetic rung coupling (between bases belonging to the complementary strands, i.e. interstrand interaction) is written as X 0y 0 0x 0 y x z + Sny Sn+1 ) − K1 f Snz Sn+1 − Jf (Snx Sn+1 + Sny Sn+1 ) H= [−Jf (Snx Sn+1 n 0

0

0

0

0

y z x z + Sny Sn+h ) − K2 gSnz Sn+h − K1 f Snz Sn+1 − Jg(Snx Sn+h

−

0x Jg(Snx Sn+h

+

0y ) Sny Sn+h

−

0z

0z K2 gSnz Sn+h

+

0 η(Snx Snx

+

(4)

0 Sny Sny )

0z

+ µSnz Sn + A(Snz )2 + A0 (Sn )2 ], where J corresponds to the intrastrand interaction constant or the stacking energy between the nth base and its neighbors, in the plane normal to the helical axis in the strands S and S 0 . When K1 and K2 are not equal to J, they introduce anisotropy in the intrastrand and in the helicoidal interactions. µ and η represent measures of the interstrand interaction or hydrogen bonding energy, between the bases of similar sites in both the strands along the direction of the helical axis and in a plane normal to it respectively. Here we have assumed that there exist an average almost uniform interactions A that assume positive values, which are the uniaxial anisotropy coefficients leading to rotation of the bases in a plane normal to the helical axis. The dimensionless parameters f and g indicate the intrastrand stacking energy and the helicoidal stacking energy strength, respectively. 0

0

It is then possible to rewrite the Hamiltonian (4) in terms of the variables (θn , φn ) and (θn , φn ) as follows: H=

X

[−Jf sin θn sin θn+1 cos(φn+1 − φn ) − K1 f cos θn cos θn+1

n 0

0

0

0

0

0

0

0

− Jf sin θn sin θn+1 cos(φn+1 − φn ) − K1 f cos θn cos θn+1 0

(5)

− Jg sin θn sin θn+h cos(φn+h − φn ) − K2 g cos θn cos θn+h 0

0

0

− Jg sin θn sin θn+h cos(φn+h − φn ) − K2 g cos θn cos θn+h 0

0

0

0

+ η sin θn sin θn cos(φn − φn ) + µ cos θn cos θn + A cos2 θn + A cos2 θn ] The quasi-spin model thus introduced implies that, the dynamics of bases in DNA can be described 5

by the following equations of motion [9, 29] θ˙n =

−1 ∂H ˙0 1 ∂H 1 ∂H −1 ∂H 0 , φ˙ n = , θn = , φ˙ n = sin θn ∂φn sin θn ∂θn sin θn0 ∂φ0n sin θn0 ∂θn0

(6)

where the overdot represents the time derivative. When the anisotropy energy A is much larger than the other parameters (i.e, A J, K1 , K2 , η, µ) [9], substituting Eq.(6) into the Hamiltonian (5), the equations of motions become [9] 0 0 φ˙ n = 2A cos θn

φ˙ n = 2A cos θn ,

(7)

By introducing the above terms into Eq(5), we get the following Hamiltonian H=

XI 0 [ (φ˙ 2n + φ˙ n2 ) − Jf sin θn sin θn+1 cos(φn+1 − φn ) − K1 f cos θn cos θn+1 2 n 0

0

0

0

0

0

0

0

− Jf sin θn sin θn+1 cos(φn+1 − φn ) − K1 f cos θn cos θn+1 0

(8)

− Jg sin θn sin θn+h cos(φn+h − φn ) − K2 g cos θn cos θn+h 0

0

0

− Jg sin θn sin θn+h cos(φn+h − φn ) − K2 g cos θn cos θn+h 0

0

0

+ η sin θn sin θn cos(φn − φn ) + µ cos θn cos θn ] where I =

1 2A

is the moment of inertia of the bases around the axes at Pn (Pn0 ). The above

Hamiltonian can be rewritten under the absolute minima of the potential as [9, 29] X I 02 0 0 2 ˙ ˙ (φ + φn ) + Jf [2 − cos(φn+1 − φn ) − cos(φn+1 − φn )] H= 2 n n X 0 0 0 + Jg[2 − cos(φn+h − φn ) − cos(φn+h − φn )] − η[1 − cos(φn − φn )]

(9)

n

While rewriting the Hamiltonian in the above form, we have restricted the bases to be rotating in the plane which is normal to the helical axis. In other words, we have restricted our problem 0

to a plane-base rotor model [9] by assuming θn = θn = π/2. It should be stressed that, the above Hamiltonian describes the dynamics of two coupled homogeneous ferromagnetic spin systems (in the XY -spin model). The first term of Hamiltonian (9) corresponds to the kinetic energies of the rotational motion of the nth base pair. The second term corresponds to the stacking interaction of adjacent bases. The third term is new and characterizes the helicoidal structure of the double helix. In this frame, we know that the pitch of the double helix ranged from 8 to 10 bases in the B-DNA. The last term corresponds to the on-site potential which models the interaction of the hydrogen bonds that connect two bases in a pair. The equations of the motions of the base pairs can then be derived from the Hamiltonian (9) as follows: I φ¨n = Jf [sin(φn+1 − φn ) − sin(φn − φn−1 )] 0

0

0

(10a)

+ Jg[sin(φn+h − φn ) − sin(φn − φn−h )] + η sin(φn − φn ) 0

0

0

0

0

0

0

I φ¨n = Jf [sin(φn+1 − φn ) − sin(φn − φn−1 )] 0

+ Jg[sin(φn+h − φn ) − sin(φn − φn−h )] + η sin(φn − φn ) 6

(10b)

On the basis of the above equations, while describing the dominant rotational motion of the bases, all the other small motions of the bases are ignored. In so doing, the difference in angular rotation of bases with respect to neighboring bases along the two strands will be small. We then 0

0

assume that sin(φn±a − φn ) ≈ (φn±a − φn ) and we set Ξn = φn + φn and ψn = φn − φn . The above equations become ¨ n = Jf (Ξn+1 + Ξn−1 − 2Ξn ) + Jg(Ξn+h + Ξn−h − 2Ξn ) IΞ I ψ¨n = Jf (ψn+1 + ψn−1 − 2ψn ) − Jg(ψn+h + ψn−h + 2ψn )

(11a) (11b)

+ 2η sin(ψn ) After rescaling the time as t →

p J/It and choosing η = −J/2, we get the following set of

uncoupled equations d2 Ξn = f (Ξn+1 + Ξn−1 − 2Ξn ) + g(Ξn+h + Ξn−h − 2Ξn ) dt2

(12a)

d2 ψn = f (ψn+1 + ψn−1 − 2ψn ) − g(ψn+h + ψn−h + 2ψn ) − sin(ψn ) dt2

(12b)

The equation (12b) is known as the modified discrete sG equation. The solution Ξn is a linear wave while the solution ψn represents a nonlinear solitonic wave [33, 34]. Keeping only linear terms in Eqs.(12), we can easily obtain their linear solutions assuming they have the form of plane waves Ξn = Aei(qdn−ωΞ t) + c.c.,

ψn = Bei(qdn−ωψ t) + c.c.

(13)

where A and B are constant amplitudes, qd the wave vector and ωΞ and ωψ the frequencies of the in-phase and the out-of-phase motions, respectively. Substituting (13) into the harmonic limit of (12), the corresponding frequencies, e.g., the frequencies of the in-phase and the out-of phase motions, usually called acoustic and optical are [33, 34] qd hqd ) + 4g sin2 ( ) 2 2 qd hqd = 1 + 4f sin2 ( ) + 4g cos2 ( ) 2 2

2 ωΞ2 = ωac = 4f sin2 (

(14)

2 ωψ2 = ωop

(15)

their different features are shown in Fig.3. In Fig.3(a), we have assumed f = 1 and g = 0. Since the main purpose of this work is to bring out the impact of the helicoidal coupling on the bearing of localized structures, the different features of the dispersion relation are going to be discussed. In fact, recently, Zdravkovi`c [33] and Tabi [34] have shown that the presence of the helicoidal coupling in the helicoidal Peyrard-Bishop model can give rise to highly localized structures. For this to be possible, they introduced the resonant mode and stated that this was possible by setting ωop = ωac . When the condition for resonance is fulfilled, there is at least one crossing point between the optical and acoustic dispersion curves [33, 34].

7

Coming back to the model under our study, the resonant mode in ensured if [33, 34] 1 1 2 hqd . ωac = ωop , i.e, + = sin 2 8g 2

(16)

This means that g = 0.25 = gcr ,

and,

qd =

π . h

(17)

According to Fig.3(b), when g = 0.2 < gcr , there is no crossing point. When g = gcr = 0.25, there is one crossing point at qd = π/5 [see Fig.3(c)]. For g = 0.3 > gcr , there are two crossing point [see Fig.3(d)]. All the curves have been plotted for f = 1. For the two last cases, there is resonance and the formation of highly localized structures is expected in the framework of the modified discrete sG equation (12b).

3

The multiple scales analysis

3.1

Discrete multiple scaling

Equation (12b) forms a system of nonlinear ODEs which cannot be solved exactly. Thus, there exist several techniques to convert them into more integrable systems [12, 20, 24, 34, 35]. In this frame, we apply the method developed by Leon and Manna [36, 27] known as the multiple scale analysis. The reductive perturbation method (or multiple scales analysis) allows the deduction of simplified equations from a basic model without losing its characteristic features. The method consists essentially in an asymptotic analysis of a perturbation series, based on the existence of different scales. More specially, the method generates a hierarchy of (small) scales for the space and time variations of the envelopes of a fundamental (linear) plane wave and all the overtones. Moreover, the scale is directly related to the small amplitude of the wave itself. The scaling of variables is performed via a Taylor expansion of the frequency ω(q0 ) in powers of a small deviation of the wave number q0 . This deviation from the linear dispersion relation is, of course, generated by the nonlinearity. There are actually three different approaches to multiple scales analysis for a discrete evolution. The first is obviously to go to the continuum limit right in the starting system, for which discreteness effects are wiped out. The second is the semi-discrete approach which consists in having a discrete carrier wave modulated by a continuous envelope. In the latter case, some discreteness aspects are preserved, in particular, the resulting MI may depend on the carrier frequency. The third stems from the adiabatic approximation, but the approach requires one to use the rotating wave approximation to artificially eliminate the overtones. The price to pay is that the predictions, for example the MI, are not trustworthy for large time [20, 24]. In order to study MI on the helicoidal base-rotor system, we perform multiple scale analysis of the discrete evolution equation (12b). The physical problem we are concerned with is the following: the first particle of the chain (say n = 0) is given oscillation (or submitted to an external force) at frequency Ω. In a linear chain, this oscillation would propagate without distortion as 8

the plane wave exp[i(Ωt + qnd)], where d is the lattice spacing. However, the nonlinearity induces some deviations from the value Ω, namely, the wave propagates with actual frequency ω and wave number q0 that is defined as [36] ω = Ω + λ, where

1 vg

=

∂Ω ∂q

q0 = q +

is the group velocity and 2cg =

∂2q ∂2Ω

λ + 2 cg λ2 + ... vg

(18)

is the group velocity dispersion.

For notation simplicity, we shall assume here that cg = 1, which does not reduce the generality of our task. Let us consider the wave packet given by the Fourier transform [36] Z i(ωt+q0 dn) ˆ ψn (t) = dω ψ(ω)e

(19)

With the use of Eqs.(18), Eq.(19) can be expanded as [36] Z iλ(t+ nd ) iλ2 2 nd i(Ωt+qnd) vg ˆ dλψ(λ)e e ψn (t) = e

(20)

By means of the following change of independent variables [36, 27] τn = (t + nd/vg ),

ξn = 2 n

(21)

we obtain the following expression for the wave packet [36, 27] ψn (t) = A(n, t)u(ξn , τn )

(22)

where A(n, t) = e

i(qnd+Ωt)

Z ,

u(ξn , τn ) =

i(λτn +λ ˆ dλψ(λ)e

2 ξ d) n

(23)

Eq.(23) has a clear physical meaning: one considers long distance (−2 ) effects in the retarded time to give the input disturbance enough time to reach the observed lattice point. In such a situation, the lattice is excited at one end. This corresponds, in other words, to a boundary value problem. The quantity u(ξn , τn ) defines the slow modulation. In order to keep discreteness in space variable for the envelope u(ξn , τn ), one fixes the small parameters as [36, 27] 2 = 1/N

(24)

and, for any given n, we shall consider only the set of points ..., n − N, n, n + N, ... of a large grid indexed by the slow variable m, that is, ....(n − N ) → (m − 1), n → m, (n + N ) → (m + 1)....

(25)

As a consequence, we can index the variable ξn by m in the new grid; we can call m a given point n and m + j the points n + jN for all j [36, 27]. To simplify the notation, we shall be using everywhere u(ξj , τj ) = u ˜j ,

u(ξj , τn ) = uj 9

(26)

for a given n and all j (note that u ˜n = un ). Hence, we are interested in expressing everything in terms of un = u(m, τ ) defined as [36, 27] un−N = um−1 ,

un = um ,

un+N = um+1 .

(27)

The problem is now to express the various different operators occurring in nonlinear evolutions for the product A(n, t)um in terms of different operators for um . The traditional approach to multiple scaling for continuous media originates from water waves theory for which the physical problem is usually that of the evolution of an initial disturbance (e.g. of the surface). In this case, the observer has to follow the deformation at the (linear) group velocity. This operation corresponds to making, in the general Fourier transform solution, the expansion of ω(q0 ) around small deviations of q0 from the linear dispersion law. We now have to obtain the analogous relations for the product ψn (t) = A(n, t)u(ξn , τn ) = An un , appearing in definition (19). The quantities ψn+1 + ψn−1 − 2ψn and ψn+h + ψn−h + 2ψn are factorized as follows [36]: ψn+1 + ψn−1 − 2ψn = [An+1 − 2An + An−1 ]um + [An+1 − An−1 ]( +

d ∂um ) vg ∂τ

d ∂ 2 um 2 2 [An+1 + An−1 ]( )2 + [An+1 − An−1 ][um+1 − um−1 ] 2 vg ∂τ 2 2

(28a)

+ ϑ(3 ) ψn+h + ψn−h + 2ψn = [An+h + 2An + An−h ]um + [An+1 − An−1 ]( +

hd ∂um ) vg ∂τ

2 hd ∂ 2 um 2 [An+h + An−h ]( )2 + [An+h − An−h ][um+h − um−h ] 2 vg ∂τ 2 2

(28b)

+ ϑ(3 ) The formula (28) constitutes our basic tool for deriving reduced models in what follows.

3.2

Evolution of the envelope

We seek a solution of the above equation (12b) in the form of a Fourier expansion in harmonics of the fundamental exp[i(qnd − Ωt)], where the Fourier components are developed in a Taylor series in powers of the small parameter , measuring the amplitude of the initial wave [36] ψn (t) =

∞ X p=1

p

p X

A(l) (n, t)u(l) p (m, τ )

(29)

l=−p

with ∗ A(l) (n, t) = exp(il(Ωt + qnd)) and u(−l) = (u(l) p p ) .

Inserting the above expression in (12b), after expanding the term in sinus until the third order,

10

we get the following system ∞ X p=1

−f

p

p X

l=−p p ∞ X X p

p=1

+

(l) 2 (l) (l) [2 ∂τ τ u(l) p (m, τ ) + 2ilΩ∂τ up (m, τ ) − (lΩ) up (m, τ )]A (n, t)

2

[(eilqd + e−ilqd − 2)u(l) p (m, τ ) + (

l=−p

(eilkd + e−ilkd )(

d )(eilqd − e−ilqd )∂τ u(l) p (m, τ ) vg

d 2 ) ∂τ τ u(l) p (m, τ ) vg

2 2 (l) (l) + (eilkd − e−ilkd )(u(l) p (m + 1, τ ) − up (m − 1, τ ))]A (n, t) 2 p ∞ X X hd ilhqd p )(e − e−ilhqd )∂τ u(l) +g [(eilhqd + e−ilhqd + 2)u(l) p (m, τ ) p (m, τ ) + ( vg p=1

2

(30)

l=−p

2 ilhqd hd 2 ) ∂τ τ u(l) (e − e−ilhqd ) p (m, τ ) + 2 vg 2 p ∞ X X (l) (l) (l) p (l) × (up (m + h, τ ) − up (m − h, τ ))]A (n, t) + [u(l) p (m, τ )A (n, t)]

+

(eilhqd + e−ilhqd )(

p=1 ∞ X

1 − [ 6

p=1

p

p X

l=−p

(l) 3 u(l) p (m, τ )A (n, t)] = 0,

l=−p

we obtain that, the coefficients of the constant term give, at different orders of :

(0)

u1 = 0; 2 :

(0)

u2 = 0; 3 :

(0)

u3 = 0

(31)

The coefficients of A(1) , at different order of , give: d[f sin(qd) − gh sin(hqd)] , Ω (1) d hd ∂ 2 u1 3 : [1 − f ( )2 cos(qd) + g( )2 cos(hqd)] vg vg ∂τ 2

2 : vg =

− if

(1) sin(qd)[u1 (m

+ 1) −

(1) u1 (m

(32)

− 1)]

1 (1) (1) (1) (1) + ig sin(hqd)[u1 (m + h) − u1 (m − h)] − |u1 |2 u1 = 0 2 One should nevertheless stress that, the order leads to the already found optical expression (15) of the dispersion relation Ω = ωψ . The last equation in 3 , of the above set, writes i iP ∂ 2 χm (χm+1 − χm−1 ) − (χm+h − χm−h ) + Q − γ|χm |2 χm = 0 2 2 ∂τ 2

(33)

g sin(hqd) −1 , γ= f sin(qd) 4f sin(qd) 1 d 2 hd Q= [f ( ) cos(qd) − g( )2 cos(hqd) − 1] 2f sin(qd) vg vg

(34)

with P =

11

From relation (29), the approximate solution ψn (t) of Eq.(12b) can be written as ψn (t) = χm (τ )ei(qnd−Ωt) + ϑ(2 ),

(35)

(1)

where χm (τ ) = u1 (m, τ ). Note that the higher order correction to ψn (t) are either explicitly (l)

expressed in terms of χ through expression up (m, τ ) or by linear inhomogeneous differential equations. This means that the theory is self-consistent and, in particular, that the overtones are by no means neglected.

4

Modulational instability analysis

4.1

Linear stability analysis

The continuous version of Eq.(33) is a well-known model for boundary value problems in optical fibers [37]. Its discrete version (without the second term) has been found by Leon and Manna using the same technique [27]. The found modified equation thus takes helicity into consideration. For instance, in our knowledge, the DNA dynamics is the only system to be described by such an equation. Thus, it has stationary solutions in the form [27] χm = Bei(λm−µτ )

(36)

This plane wave solution obeys the dispersion relation µ2 = −

1 [sin(λ) − P sin(hλ) + γ|B|2 ] Q

(37)

By replacing P and γ by their expressions, we get the above nonlinear dispersion relation in the form µ2 = −

1 g sin(hqd) |B|2 [sin(λ) − sin(hλ) − ] Q f sin(qd) 4f sin(qd)

(38)

As we are considering a boundary value problem, µ and B are the given frequency and amplitude of the modulation. Modulated waves are then expected in the DNA model under study if the right-hand side of Eq.(38) is negative. In this frame, there is no real solution λ if Q < 0,

2 . |B|2 > 4[f sin(qd) sin(λ) − g sin(hqd) sin(hλ)] = Bcr

(39)

The above relation represents the condition for a plane wave to be unstable in the modified discrete sG model. It is also a modification of the condition predicted by Leon and Manna [27] for a discrete sG model without the term of helicity. In the case of the simple discrete sG model (without helicity), obtained for g = 0, the highest value of the threshold amplitude is obtained for sin(λ) = −1 [27]. In the case where helicity is taken into account, for h = 5, right-hand side of Eq.() is positive for sin(λ) = sin(hλ) = 1, i.e, λ = π/2. This means that 2 . |B|2 > 4[f sin(qd) − g sin(hqd)] = Bcr

12

(40)

However, this does not guarantee that the right-hand side of Eq.() is the highest for sin(λ) = sin(hλ) = 1. For example, it is possible that the expression in brackets is higher if sin(λ) is a little bit smaller than 1 and sin(hλ) is negative. To clarify this issue, we have plotted in Fig.4 the functions sin(λ) and sin(hλ). The crossing point of the two red dashed lines indicates that the value which better satisfies our needs is λ = 3π/10. For this value, sin(λ) is 0.8 and sin(hλ) is -1. The corresponding threshold amplitude is therefore given by 4 2 |B|2 > [4f sin(qd) + 5g sin(hqd)] = Bcr . 5

(41)

The behaviors of the threshold amplitude will then be discussed with respect to the critical value gcr derived in the second section. Thus, our analysis will be based on two cases: when g < gcr and when g ≥ gcr . Therefore, the following manifestations of MI are observed: • For g = 0, the system is described by Fig.5(a). The instability region belong to the whole interval [0; π] and there is only one side band. This features confirm the results by Leon and Manna [27]. • For g < gcr , we first observe that helicity breaks the instability domain into satellite side bands [see Fig.5(a), where the region between the two red dashed lines is the region of instability]. Furthermore the instability region is reduced by the helicoidal structure of the molecule. Furthermore it is obvious that the amplitude have increased. • For g ≥ gcr , we have the features displayed in Figs.5(c) and (d) which confirms the fact that the region of instability is reduced by the helicoidal coupling. Also, for values of g greater or equal to gcr , it is possible to observe large oscillations of strands. The behaviors displayed by the model understudy show that, this model can support large amplitude oscillation depending on the value of the helicoidal coupling constant g. Such results prove that, the present model is much richer than the simple discrete sG model [29, 38, 39, 40], that only consider stacking interactions between neighboring pairs and which can be recovered here by letting g = 0. In the framework of the Peyrard-Bishop-Dauxois model [10, 11], it has also been shown that helicity can deeply modify the instability criterion (in comparison to the works of Peyrard and co-workers [21, 24]) and can give rise to important behaviors in the process of soliton bearing through MI [20].

4.2

Numerical analysis of MI

It is well known that one of the main effects of MI, which refers to the exponential growth of certain modulation sideband of nonlinear plane waves propagating in a dispersive medium as a result of the interplay between nonlinearity and dispersion effect, is the generation of localized structures. In this section, we make computer simulations of the instability of a plane wave in the modified discrete sG equation in order to bring out some features of MI in the DNA 13

model understudy. Comparisons are made between numerical results and analytical predictions performed in the previous sections and a particular attention will be paid to the consequence of the interplay between stacking and helicoidal effects in the bearing localized wave patterns. However, it seems important to stress that, the linear stability analysis is based on the linearization around the unperturbed carrier wave, which is valid only when the amplitude of the perturbation is small in comparison with that of the carrier wave. Clearly, the linear approximation should fail at large time scales as the amplitude of an unstable sideband grows exponentially. Furthermore, the linear stability analysis neglects additional combination waves generated through a wave-mixing process, which, albeit small at the initial stage, can become significant at large time scales if its wave number falls in an instability domain. Linear stability analysis therefore cannot tell us the long time evolution of a modulated nonlinear plane wave. In this frame, we have performed numerical simulations of the modified discrete sG equation (12b) using the standard fourth-order Runge-Kutta scheme, with an integration time scale ∆t = 5 × 10−3 . The initial condition in accordance with Eq.(35), is a modulated wave. The number of base pairs have been chosen so that we do not encounter wave reflection at the the end of the molecule and periodic boundary conditions have been used. As already announced, two cases have been considered: the case g < gcr and the case g ≥ gcr . For the two cases, we have chosen B = 2.5, = 0.01, qd = 0.58π and λ = 0.3π. When the nearest-neighbor effect is more important in the system, i.e. g < gcr , we have the features displayed in Fig.6. We see that the plane wave excitation breaks into trains of waves which have the appearance of multisoliton shapes with breathing motion. Attention is paid to the oscillations of the bases 400 [see Fig.6(a)] and 800 [see Fig.6(b)]. We observe that, the oscillations of the bases undergo slight modulation for n = 400 and n = 800 even if modulation becomes important for the second case. This confirms our analytical predictions and once more shows that MI is a crucial issue for soliton instability [11], and is also considered a precursor to soliton formation because, it typically occurs in the same parameter region as that where solitons are observed. When the effect of the helicoidal coupling becomes important, i.e. g ≥ gcr , one obtains the results depicted in Fig.7. Even in this case, the plane wave breaks into trains of soliton-like structures. But the amplitude of waves is higher than in the first case and oscillations are obvious. At base pair n = 400, one clearly observes that the wave pattern displayed is that of an extended wave that propagates with a breathing motion. At base pair n = 800, the soliton objects are well separated, even if their amplitude is lower than for n = 400. One could then relate these exact breathing motions and the increasing of the amplitude to the presence of helicoidal terms. The corresponding values of the helicoidal coupling parameter (g ≥ gcr ) cause better and more efficient modulation of waves. This simply means that, when such a condition is fulfilled, there is a better information transfer through the molecule. Also, such behaviors cannot be observed for g < gcr , which, once more, suggest the importance the helicoidal stacking interactions. In the same way, in order to substantiate the importance of the helicoidal coupling in the 14

model, we have investigated energy localization by using the following relation 1 1 En = ψ˙ n2 + f [(ψn+1 − ψn )2 + (ψn − ψn−1 )2 ] 2 2 1 + g[(ψn + ψn+h )2 + (ψn + ψn−h )2 ] + [1 − cos(ψn )] 2

(42)

derived from (9) and (12b). For g < gcr [Fig.8(a)] and g ≥ gcr [Fig.8(b)], energy is really localized in the structure. This particularly accounts for the robustness of energy localization is the available biological models [18, 19, 20, 21]. For each of the two cases considered here, different features are displayed. In the case g < gcr , we see that energy is sheared by the whole lattice and gives rise to slight radiation in its spectrum accompanied by small localized structures. On the other hand, the case g ≥ gcr undergoes large patterns which tend to localize on specific sites.

5

Conclusion

In the framework of the so-called dynamic plane-base rotor model, we have taken into account the helicity of the B-DNA to investigate the MI of a plane wave. The dynamical equation, which finally appeared in the form of a modified discrete sG equation, was derived from a suitable Hamiltonian in analogy with the Heisenberg model of anisotropically coupled spin chain or spin ladder with ferromagnetic legs and antiferromagnetic rung coupling [29], where helicoidal terms have been introduced. We have found that, helicity brings about a new term in the equation describing the nonlinear dynamics of the molecule. To study MI, we have first derived from the original mode a simpler limit equation which results to be the discrete nonlinear Schrodinger (DNLS) equation in discrete space (23) and with space and time exchanged. This universal limit is obtained by a very general approach called multiple scale perturbation analysis. The MI criterion has been reported to be a modification of the one by Leon and Manna [27], due to the presence of helicoidal terms. A critical value of the helicoidal coupling parameter has been derived and on the basis of the chosen parameter values, there has been established a relationship between MI and the critical value gcr of g. It has, in fact, been found that, for g < gcr , the system does not feel the presence of helicoidal term. When g ≥ gcr , the analytical MI analysis predicts large amplitude oscillations as already seen in the framework of the PBD model [33, 20, 34]. Numerical experiments have been carried out in order to confirm the analytical predictions. It has been observed that, in the case where g < gcr , there is MI since the initial modulated plane wave breaks into train of extended solitons and soliton-like objects. In this case, the oscillations are slight at base n = 400 and tend to be accentuated at base n = 800. When g ≥ gcr , the amplitude of the waves is higher than in the first case and, as n increases, the waves tend to break into more precise soliton objects. This can be analyzed as the action of the waves flowing in the molecule and which plays an important role in the conformational concerns taking place in the B-DNA molecule. On the other hand, the increase of the amplitude of the waves trains mainly describes the action of RNA polymerase which breaks progressively the hydrogen bonds 15

for the messenger RNA to come and copy the genetic code. We believe that the existence and the formation of solitons in the DNA molecule could be a proper candidate to explain how data are exchanged during basic biological phenomena such as transcription and replication. As it is well known, for the hydrogen bonds to be broken, there should be a concentration of the enzyme and of the energy brought through the hydrolysis of ATP. It should nevertheless be stressed that, while speaking of mechanical models of DNA, we exclude consideration of the all-important interactions between DNA and its environment. The latter includes at least the fluid in which the DNA is immersed, and interaction with this leads to energy exchanges; one should thus include in the equations describing DNA dynamics both dissipation terms and random terms due to interaction with molecules in the fluid (even if the helicoidal terms included here stand for the presence of water filaments within the molecular environment). It would then be of interest to bring out the effect of such factors that could interestingly modify the process of MI studied in the model under consideration. However, the results obtained in the present study once more show the importance of helicity in the bearing of high localized structures.

Acknowledgments Fruitful discussions with Prof. Pierre Ngassam of the Department of Biology and Animal Physiology of the University of Yaound´e I are acknowledged. Tabi thanks the Condensed Matter and Statistical Physics Section of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, for the invitation, where this work has been finalized.

16

Appendices A

The discrete multiple scaling

From Eq.(30), the equation which describes constant terms A(0) is given by 2 (0) 3 ∂ u1 ∂τ 2

+

2 (0) 4 ∂ u2 ∂τ 2

+

2 (0) 5 ∂ u3 ∂τ 2

(0)

− f (1 + 1 − 2)u1 + 3 f

d vg

2

(0) ∂ 2 u1 ∂τ 2

! A(0)

2 2 (0) ! 2 (0) 2 (0) ∂ u ∂ u hd ∂ u1 (0) 2 3 5 3 − 4 f A(0) + f − g(1 + 1 + 2)u − g 1 2 2 2 ∂τ ∂τ vg ∂τ ! 2 2 (0) ∂ u hd (0) (0) 2 + 2 g(1 + 1 + 2)u2 + 4 g + 3 g(1 + 1 + 2)u3 A(0) 2 vg ∂τ ! 2 2 (0) (0) 3 hd ∂ u (u ) (0) (0) (0) 3 + 5 g A(0) = 0 + u1 + 2 u2 + 3 u3 − 3 1 vg ∂τ 2 6 (0)

(0)

(43)

(0)

This allows to derive the expressions of u1 , u2 , and u3 at different order of (0)

(0)

: 0 + 4gu1 + u1 = 0 (0)

(0)

2 : 0 + 4gu2 + u2 = 0 2 2 (0) (0) (0) hd ∂ u1 (u1 )3 ∂ 2 u1 (0) (0) 3 + g + 4gu + u − =0 : 3 3 ∂τ 2 vg ∂τ 2 6 (0)

(0)

(0)

The above system yields u1 = 0, u2 = 0, and u3 = 0.

17

(44)

The coefficient of A(1) are governed by the equation ! ! (1) (1) 2 (1) 2 (1) ∂ u ∂ u ∂u ∂u (1) (1) 2 1 A(1) + 2 A(1) 2 + 2iΩ 1 − Ω2 u1 + 2iΩ 2 − Ω2 u2 ∂τ 2 ∂τ 2 ∂τ 2 ∂τ ! (1) 2 (1) ∂ u ∂u (1) 3 + 3 A(1) + 2iΩ 3 − Ω2 u3 ∂τ 2 ∂τ ! (1) d ∂u (1) − f (eiqd + e−iqd − 2)u1 + ( )(eiqd − e−iqd ) 1 A(1) vg ∂τ ! 2 (1) 2 ∂ u 2 d 2 iqd (1) (1) 1 − f ( ) (e + e−iqd ) A(1) + (eiqd − e−iqd ) u1 (m + 1) − u1 (m − 1) 2 vg ∂τ 2 2 ! (1) d ∂u (1) − 2 f (eiqd + e−iqd − 2)u2 + ( )(eiqd − e−iqd ) 2 A(1) vg ∂τ ! 2 2 (1) 2 d ∂ u (1) (1) 2 − 2 f ( )2 (eiqd + e−iqd ) A(1) + (eiqd − e−iqd ) u2 (m + 1) − u2 (m − 1) 2 vg ∂τ 2 2 ! (1) d ∂u (1) A(1) − 3 f (eiqd + e−iqd − 2)u3 + ( )(eiqd − e−iqd ) 3 vg ∂τ ! 2 2 2 (1) d ∂ u (1) (1) 3 ( )2 (eiqd + e−iqd ) + (eiqd − e−iqd ) u3 (m + 1) − u3 (m − 1) − 3 f A(1) 2 vg ∂τ 2 2 ! (1) hd ihqd (1) ihqd −ihqd −ihqd ∂u1 + g (e +e + 2)u1 + ( )(e −e ) A(1) vg ∂τ ! 2 (1) 2 ihqd 2 hd 2 ihqd (1) (1) −ihqd ∂ u1 −ihqd +e ) ( ) (e + (e −e ) u1 (m + h) − u1 (m − h) A(1) + g 2 2 vg ∂τ 2 ! (1) hd ihqd (1) −ihqd ∂u2 2 ihqd −ihqd −e ) − g (e +e + 2)u2 + ( )(e A(1) vg ∂τ ! 2 (1) 2 hd 2 ihqd 2 ihqd (1) (1) −ihqd ∂ u2 2 −ihqd + g ( ) (e +e ) + (e −e ) u2 (m + h) − u2 (m − h) A(1) 2 2 vg ∂τ 2 ! (1) ∂u hd (1) − 3 f (eihqd + e−ihqd + 2)u3 + ( )(eihqd − e−ihqd ) 3 A(1) vg ∂τ ! 2 2 (1) 2 hd 2 ihqd ∂ u3 (1) (1) + 3 g ( ) (e + e−ihqd ) + (eihqd − e−ihqd ) u3 (m + h) − u3 (m − h) A(1) 2 2 vg ∂τ 2 (1) 2 (1) 3 (1) 3 1 (−1) (1) 2 + u1 + u2 + u3 − u1 (u1 ) A(1) = 0 2 (45)

18

At different order of , we have 2

: Ω = 1 + 4f sin

2

qd 2

2

+ 4g cos

hqd 2

(1) hd ∂u1 d sin(hqd) : 2i Ω − f ( ) sin(qd) + g vg vg ∂τ qd hqd (1) − Ω2 − 1 − 4f sin2 − 4g cos2 u2 2 2 2 (1) g hd 2 ihqd ∂ u1 f d 2 iqd −iqd −ihqd 3 ) + ( ) (e +e ) : 1 − ( ) (e + e 2 vg 2 vg ∂τ 2 (1) f iqd (1) e − e−iqd u1 (m + 1) − u1 (m − 1) − 2 (1) g ihqd (1) −ihqd + e −e u1 (m + h) − u1 (m − h) 2 (1) 1 (1) (1) d hd ∂u2 − |u1 |u1 + 2i Ω − f ( ) sin(qd) + g sin(hqd) 2 vg vg ∂τ qd hqd (1) − Ω2 − 1 − 4f sin2 − 4g cos2 u3 = 0 2 2 2

(46)

At order , we get the linear dispersion relation. At order 2 , we recover the linear dispersion relation Ω2 and the group velocity vg =

1 (f d sin(qd) − ghd sin(hqd)) Ω

(47)

(1)

At order 3 , we get the equation in u1 as follows ! 2 (1) hd g ∂ 2 u1 (eiqd + e−iqd ) + (eihqd + e−ihqd ) 2 vg ∂τ 2 f (1) (1) − (eiqd − e−iqd ) u1 (m + 1) − u1 (m − 1) 2 1 g ihqd (1) (1) (1) (1) + (e − e−ihqd ) u1 (m + h) − u1 (m − h) − |u1 |u1 = 0 2 2 f 1− 2

d vg

2

(48)

(1)

With u1 (m, τ ) = χm (τ ), the above equation reads i iP ∂ 2 χm (χm+1 − χm−1 ) − (χm+h − χm−h ) + Q − γ|χm |2 χm = 0 2 2 ∂τ 2

(49)

−1 g sin(hqd) , γ= f sin(qd) 4f sin(qd) 1 d hd Q= [f ( )2 cos(qd) − g( )2 cos(hqd) − 1] 2f sin(qd) vg vg

(50)

with P =

19

References [1] R. E. Franklin and R. G. Gosling, Nature (London) 171, 740 (1953). [2] J. D. Watson and F. H. C. Crick, Nature (London) 171, 737 (1953). [3] M. H. F. Wilkins, W. E. Seeds, A. R. Stokes and H. R. Wilson, Nature (London) 172, 759 (1953). [4] F. H. C. Crick and J. D. Watson, Proc. R. Soc. (London) A223, 80 (1954). [5] S. W. Englander et al., Proc. Nad. Acad. Sci. U.S.A. 77, 7222 (1980). [6] S. Yomosa, Phys. Rev. A 27, 2120 (1983). [7] S. Yomosa, Phys. Rev. A 30, 474 (1984). [8] S. Takeno and S. Homma, Prog. Theor.Phys. 70, 308 (1983). [9] S. Homma and S. Takeno, Prog. Theor.Phys. 72, 679 (1984). [10] M. Peyrard and A. R. Bishop, Phys. Rev. Lett. 62, 2755 (1989). [11] T. Dauxois, M. Peyrard and A.R. Bishop, Phys. Rev. E 47, R44 (1993). [12] M. Barbi, S. Cocco and M. Peyrard, Phys. Lett. A. 253, 358 (1999). [13] L. V. Yakushevich, Phys. Lett. A 136, 413 (1989). [14] L. V. Yakushevich, A. V. Savin and L. I. Manevitch, Phys. Rev. E 66, 016614 (2002). [15] L. V. Yakushevich, Nonlinear Physics of DNA, 2nd Ed. (Wiley, Chichester, 2004). [16] G. Gaeta, Phys. Lett. A 190 (1994) 301; J. Nonl. Math. Phys. 14, 57 (2007). [17] K. Forinash, T. Cretegny and M. Peyrard, Phys. Rev. E 55, 4740 (1997). [18] E. Zamora-Sillero, A. V. Shapovalov and F. J. Esteban, Phys. Rev. E 76, 066603 (2007). [19] C. B. Tabi, A. Mohamadou and T. C. Kofane, J. Comput. Theor. Nanosci. 5, 647 (2008). [20] C. B. Tabi, A. Mohamadou and T. C. Kofane, J. Phys.: Condens. Matter 20, 415104 (2008). [21] I. Daumont, T. Dauxois and M. Peyrard, Nonlinearity 10, 617 (1997). [22] J. Cuevas, J. F. R. Archilla, Yu. B. Gaididei, and F. R. Romero, Physica D 163, 106 (2002). [23] P. V. Larsen, P. L. Christiansen, O. Bang, and Yu. B. Gaididei, Phys. Rev. E 69, 026603 (2004). [24] Y. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992). [25] A. V. Gorbach and M. Johanson, Eur. Phys. J D 29, 77 (2004).

20

[26] J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel and J. S. Aitchison, Phys. Rev. Lett. 92, 163902 (2004). [27] J. Leon and M. Manna, Phys. Rev. Lett. 83, 2324 (1999). [28] P. Marqui´e, J. M. Bilbault and M. Remoissenet, Phys. Rev. E 51, 6127 (1995). [29] M. Daniel and V. Vasumathi, Physica D 231, 10 (2007). [30] T. Dauxois, Phys. Lett. A 159, 390 (1991). [31] S. Zdravkovi`c and M. V. Satari`c, Physica Scripta 64, 612 (2001). [32] S. Zdravkovi`c and M. V. Satari`c, Int. J. Mod. Phys. B 17, 5911 (2003). [33] S. Zdravkovi`c and M. V. Satari`c, Europhys. Lett. 78, 38004 (2007). [34] C. B. Tabi, A. Mohamadou and T. C. Kofane, Eur. Phys. J. D 50, 307 (2008). [35] M. Remoissenet, Phys. Rev. B 33, 2386 (1986). [36] J. Leon and M. Manna, J. Phys. A: Math. Gen. 32, 2845 (1999). [37] M. Wadati, T. Lizuka and T. Yajima, Physica (Amsterdan) 51D, 388 (1991). [38] M. Salerno and Y. S. Kivshar, Phys. Lett. A 193, 263 (1994). [39] M. Salerno, Phys. Rev. A 46, 6856 (1992). [40] G. Gaeta, Phys. Rev. E 74, 021921 (2006).

21

(a)

(b)

(c)

Figure 1: (a) A schematic structure of the B-form of DNA; (b) A horizontal projection of the nth base pair in the xy-plane; (c) A projection of the nth base pair in the xz-plane.

Figure 2: A schematic representation of DNA as an anisotropically coupled spin chain model or spin ladder.

22

2.5

2.5 Acoustic branch Optical branch

Acoustic branch Optical branch

1.5

1.5

ω

2

ω

2

1

1

0.5

0.5

0 0

0.2

0.4

0.6

0.8

0 0

1

qd/π

0.2

0.4

0.6

0.8

1

qd/π

(a)

2.5

(b)

3

Acoustic branch Optical branch

Acoustic branch Optical branch 2.5

2

2

ω

ω

1.5 1.5

1 1 0.5 0.5 0 0

0.2

0.4

0.6

0.8

qd/π

0 0

1

(c)

0.2

0.4

0.6 qd/π

0.8

1

(d)

Figure 3: Optical and acoustic frequencies as a function of the wave number qd and for different values of g and f = 1: Panel (a) shows the dispersion curves for g = 0. The curves are similar to those obtained for the discrete sG model. For g = 0.2 < gcr , we have the configuration of panel (b). The dispersion curves oscillate and there is no crossing point between the optical and the acoustic curves. When the condition () is fulfilled (g = gcr = 0.25), there is one crossing point and resonance is possible. Panel (d) shows the dispersion curves for g = 0.3 > gcr . In this case, there are two crossing points between the optical and the acoustic curves. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 sin(hλ) sin(λ)

−0.8 −1 0

0.2

0.4

λ/π

0.6

0.8

1

Figure 4: The panel shows the plot of the functions sin(λ) (green curve) and sin(hλ) (blue curve). The possible value of λ which leads to the highest value of the threshold amplitude Bcr is the one situated at the crossing point of the the two red dashed lines, λ = 3π/10.

23

2

2.5 Unstable

Unstable

1.8 1.6

2

Stable

Stable

1.4 1.5 cr

Bcr

1.2

B

1 Stable

0.8

1 Stable

0.6 0.4

Stable

Stable

0.5

0.2 0 0

0.2

0.4

0.6

0.8

0 0

1

qd/π

0.2

0.4

0.6

0.8

1

qd/π

(a)

2.5

(b)

2.5 Unstable Unstable

2

Stable

Stable

Stable

Stable

2

B

cr

1.5

Bcr

1.5

1

1 Stable

Stable

Stable

Stable

0.5

Stable

Stable

0.5

0 0

0.2

0.4

0.6

0.8

0 0

1

qd/π

0.2

0.4

0.6

0.8

1

qd/π

(c)

(d)

Figure 5: The panels show the threshold amplitude Bcr versus the wave number qd for different values of g with f = 1: (a) g = 0. the result is similar to the one obtained by Leon and Manna [27] for the discrete sG model. In the second panel, the threshold amplitude is plotted for g = 0.2 < gcr . The amplitude has increased and the region of instability has been reduced due to helicity. In panel (c), the threshold amplitude has been plotted for g = 0.25 = gcr and the last panel depicts the case g = 0.3 > gcr . −3

8

x 10

0.03

n = 800 n = 400

6

0.02

4

2 ψn/ 2 π

ψn/ 2 π

0.01

0

0

−2 −0.01

−4 −0.02

−6

−0.03 0

200

400

600 Time

800

1000

1200

(a)

−8 0

200

400

600 Time

800

1000

1200

(b)

Figure 6: Dynamics of modulated waves as a function of time (t.u.) showing the MI of slowly modulated plane waves for g < gcr (a) Soliton-like objects at the base pair 400; (b) Soliton-like objects at the base pair 800. The system undergoes slight oscillations. So, for the considered case, modulations are not efficient and cannot ensure better transmission of information within the double helical structure. 24

0.4

0.2

n = 400

n = 800

0.2

0.1

0.1

0.05 ψn/ 2 π

0.15

ψn/ 2 π

0.3

0

0

−0.1

−0.05

−0.2

−0.1

−0.3

−0.15

−0.4 0

200

400

600 Time

800

1000

1200

(a)

−0.2 0

200

400

600 Time

800

1000

1200

(b)

Figure 7: Dynamics of modulated waves as a function of time (t.u.) showing the MI of slowly modulated plane waves for g ≥ gcr (a) Soliton-like objects at the base pair 400; (b) Soliton-like objects at the base pair 800. Oscillations for this case are more efficient and clearly show that when g ≥ gcr , data are better transmitted within the DNA model. This bring out the importance of helicoidal models over the simple model that only take into consideration nearest-neighbor coupling.

−3

−3

x 10

800

x 10

800

1.5 7

700

700

600

600

6

5

1

Time

500

Time

500

400

4

400 3 0.5

300

300 2

200

200 1

100 50

100

150 n

200

250

300

100

0

(a)

50

100

150 n

200

250

300

0

(b)

Figure 8: Energy localization in the DNA double helix for (a) g < gcr ; (b) g ≥ gcr . The horizontal axis indicates the position of the base pairs and the vertical axis corresponds to time (time is going upward). In the first case we have En = 1.62 × 10−3 and En = 7.8 × 10−3 in the second, respectively. It is obvious that the case g ≥ gcr is responsible for high energy localization, as shown in (b).

25

Available at: http://publications.ictp.it

United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

DISCRETE INSTABILITY IN THE DNA DOUBLE HELIX

Conrad Bertrand Tabi1 Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, Alidou Mohamadou2 Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon and Condensed Matter Laboratory, Department of Physics, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon and Timol´eon Cr´epin Kofan´e3 Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.

MIRAMARE – TRIESTE June 2009

1

[email protected] [email protected] 3 [email protected] 2

Abstract Modulational instability (MI) is explored in the framework of the base-rotor model of DNA dynamics. We show in fact that, the helicoidal coupling introduced in the spin model of DNA reduces the system to a modified discrete sine-Gordon (sG) equation. The MI criterion is thus modified and displays interesting features because of the helicoidal coupling. This is confirmed in the numerical analysis where a critical value of the helicoidal coupling constant is derived. In the simulations, we have found that a train of pulses are generated when the lattice is subjected to MI, in agreement with analytical results obtained in a modified discrete sG equation. Also, the competitive effects of the harmonic longitudinal and helicoidal constants on the dynamics of the system are notably pointed out. In the same way, it is shown that MI can lead to energy localization which is high for some values of the helicoidal coupling constant.

1

Leading paragraph It is well known that the modulated signals are very important and widely used in engineering since they can be transmitted along much longer distance than nonmodulated waves. As a whole, those waves have the form of soliton-like objects and, one of the ways to produce them is modulational instability. Modulational instability has also been shown to be a pathway to energy localization in biomolecules and in discrete systems, in general. It is a result of the interplay between nonlinearity and dispersion and arises in continuous as well as in discrete systems. Different available methods are used to study the condition for the formation of modulated wave, and the one addressed here is the so-called Discrete multiple scaling method. The base-rotor model, where we introduce the helicoidal coupling, is addressed and our results show that, the interplay between stacking and helicoidal couplings can give rise to highly localized structures with a consequence on energy localization.

1

Introduction

The great interest aroused by solitons and localized structures formation in biomolecules and in DNA in these last years is proof of the fundamental role they play in human life. In fact, DNA molecules are the support for genetic information, and knowledge of their sequence is essential from the biological and medical point of view. Furthermore, the leading phenomena of transcription and replication confer to DNA mechanical features that are still not yet fully understood by physicists and biophysicists alike. From the physical point of view, the DNA molecule is nothing but a system consisting of many interacting atoms organized in a special way in space. It was shown by Franklin and Gosling [1], Watson and Crick [2], Wilkins et al. [3] and Crick and Watson [4] that: (i) under usual external conditions (temperature, pH, etc.) the molecule has the double helix form and (ii) the helix is not a static structure. On the contrary, DNA is a very flexible molecule. On this background, following the pioneering work of Englander et al. [5], a number of mathematical models of the DNA double chain have been proposed over the years, focussing on different aspects of the DNA molecule and on different biological, physical, and chemical processes in which DNA is involved. Those models include the plane-base rotor model by Yomosa [6, 7], later improved by Takeno and Homma [8] and the Peyrard-Bishop (PB) model [10], later improved by Dauxois [11] and Barbi [12]. In the first model attention has been paid to the degree of freedom characterizing base rotation in the plane perpendicular to the helical axis along the backbone structure. DNA dynamics, in the above models has been shown to be governed by the sine-Gordon equation, while the second model has been used to study the DNA thermal denaturation (or melting) and the corresponding degree of freedom were related to straight or radial separation of the two strands which are wound together in the DNA double helix. Together with the Yakushevich model [13, 14, 15, 16] (which is concerned with rotational and torsional degrees of freedom of the molecule), the PB model has been widely used in the 2

literature [17, 18, 19, 20]. Most of the studies fulfilled in the framework of the above-mentioned applied theoretical models refer to the formation of bubbles and localized structures. These bubbles has been shown to play a vital role in DNA processes such as replication, transcription, recombination, and reparation, or the DNA transcription to several types of RNA including those involved in the protein synthesis. The use of localized structures in explaining these phenomena sets the problem of their creation and stability. In the past years, it was found by many investigators that this is a rather complicated and substantial problem itself. In the framework of the PB model, Daumont et al. [21], and more recently, Tabi et al. [20] have shown that the discreteness of the system causes the instability of the extended solitons. They tend to self-modulate evolving to localized soliton-like modes that interact nonelastically and grow the largest ones at the expenses of the smallest [19, 20, 22, 23]. This phenomenon, known as modulational instability (MI) is the outcome of the interplay between nonlinearity and dispersive/diffraction effects. MI arises in continuous as well as in discrete systems but, as a whole, the analysis of such nonlinear systems is performed by first deriving from the original model a simpler limit equation which usually results to the nonlinear Schr¨odinger (NLS) equation [20, 21, 24]. This universal limit is obtained by a very general approach called multiscale perturbation analysis [20, 21, 24]. In the case of discrete systems where nonlinear localized modes (also called discrete breathers) are known to exist [25], MI is usually described in a semi-discrete multiple scale analysis where the envelope of the carrier wave is treated as a continuous function. Recently, Meir et al. [26] have reported the first experimental observation of MI in physical systems. Their studies reveal a diversity of behaviors forbidden in continuous media. Leon and Manna [27] have proposed a theoretical approach to study discrete instability in nonlinear lattices. They show that discrete multiple scale analysis for boundary value problems in nonlinear discrete systems leads to a first order, strictly MI (disappearing in the continuous envelope limit) above a threshold amplitude for wavenumbers, beyond the zero of group velocity dispersion. This approach has been applied to the electrical lattice [28] with a good agreement with the experimental results. In connection with the studies of these authors, we will also apply the discrete multiple scale analysis to the DNA lattice model and study the resulting MI as well as soliton formation through MI. Therefore, the model we consider here is inspired from the Zhang model [29] where kink and anti-kink solitons are known. That model, after some mathematical simplifications, reduces to the known sine-Gordon equation. With in mind the fact that the B-DNA is mainly helicoidal, we introduce the helicoidal coupling as done by Gaeta [16], Dauxois [30] and Tabi et al. [20]. The impact of the helicoidal coupling constant, on the formation of localized structures, is investigated. The present discrete multiple scale analysis relies on the definition of a large grid scale via the comparison of the magnitude of the related difference operator, and on the expansion of the wavenumber in powers of frequency variations due to nonlinearity. The rest of the paper is therefore organized as follows: in Section 2, we discuss the model and we derive the equations that describe the dynamics of the 3

hydrogen bonds. In Section 3, a brief presentation of the multiple scale method is made and the envelope equation is derived. In Section 4, after deriving the condition for obtaining localized structures in the model understudy, numerical investigations are performed in order to confirm analytical predictions. The last section is dedicated to some concluding remarks.

2

Model and dynamical equations

In this work, we consider the so-called B-form of the DNA molecule as presented in Fig.1(a). S and S 0 represent the complementary strands in the double helical structure. Each arrow in the figure stands for the hydrogen bonding effects that take place in the molecule between the complementary bases. The z-axis is chosen along the helical axis of DNA. In Figs.1(b) and (c), we show the horizontal projection of the nth base pair in the xy and xz planes, respectively. In these figures, Qn and Q0n denote the tips of the nth bases belonging to the strands S and S 0 . Pn and Pn0 represent the points where the bases in the nth base pair are attached to the strands S and S 0 , respectively. As a whole, modelling the dynamics of the DNA molecule should take into account at least three interactions: the hydrogen bonding effect which, in fact, models the interactions taking place between two bases in a pair. From a heuristic point of view, this energy depends on the distance between two bases in a pair. Thus from Fig.1(b), the square of the distance between the edges of the arrows (Qn Q0n )2 is written as [9, 29] (Qn Q0n )2 = 2 + 4r2 + (zn − zn0 )2 + 2(zn − zn0 )(cos θn − cos θn0 ) − 4r[sin θn cos φn + sin θn0 cos φ0n ] + 2[sin θn sin θn0

(1)

× (cos φn cos φ0n + sin φn sin φ0n ) − cos θn cos θn0 ] where r is the radius of the circle depicted in Fig.1(b). The base-base interaction energy can be understood in a clearer and more transparent way by introducing Sn = (Snx , Sny , Snz ) and 0

0

0

0

Sn = (Snx , Sny , Snz ) given by [9, 29] Snx = sin θn cos φn , 0

0

0

Sn x = sin θn cos φn ,

Sny = sin θn sin φn , 0

0

0

Sn y = sin θn sin φn ,

Snz = cos θn 0

0

Sn z = cos θn

(2)

for the strands S and S 0 , respectively. In terms of this, Eq.(1) can be rewritten as [9, 29] 0 0 0 0 (Qn Q0n )2 = 2 + 4r2 + 2 Snx Snx + Sny Sny − Snz Snz − 4r Snx + Snx

(3)

In the above equation (3), the longitudinal compression along the direction of the helical axis has been neglected and we have assumed zn = zn0 [9]. So, in what follows, we consider the anisotropic Heisenberg model of DNA shown in Fig. 2 [29]. In the case of spin chains, each arrow represents a group of atom at that lattice point. In the present model, arrows are marked anti-parallel as in an antiferromagnetic model. Here the z -direction (i.e. the direction of the helix axis) is chosen as the easy axis of magnetization in the spin chain. The spin-spin interaction takes into account

4

the stacking interaction between nearest neighbors (the nth pair interacts with both the (n + 1)th and (n − 1)th pairs). Another effect, newly introduced in this model, is due to water filaments that link units at different sites. In particular, they have a good probability to form between nucleotides which are a half turn of the helix apart on different chains, i.e., which are near to each other in space due to the double helix geometry; these water filaments-mediated interactions are therefore also called helicoidal interactions ( the nth pair interact with both the (n + h)th and (n − h)th pairs) [20, 30, 31] with a pitch h that could be equal to 4 [20, 30, 31, 32] or to 5 [33, 34]. Since the helicoidal pitch is 11, we will use in the rest of the paper h = 5. Further, in this study, we assume the two strands to be the same and could, therefore, shear the same parameters. With these considerations, the Hamiltonian of the Heisenberg model for an anisotropically chain of coupled spin model and antiferromagnetic rung coupling (between bases belonging to the complementary strands, i.e. interstrand interaction) is written as X 0y 0 0x 0 y x z + Sny Sn+1 ) − K1 f Snz Sn+1 − Jf (Snx Sn+1 + Sny Sn+1 ) H= [−Jf (Snx Sn+1 n 0

0

0

0

0

y z x z + Sny Sn+h ) − K2 gSnz Sn+h − K1 f Snz Sn+1 − Jg(Snx Sn+h

−

0x Jg(Snx Sn+h

+

0y ) Sny Sn+h

−

0z

0z K2 gSnz Sn+h

+

0 η(Snx Snx

+

(4)

0 Sny Sny )

0z

+ µSnz Sn + A(Snz )2 + A0 (Sn )2 ], where J corresponds to the intrastrand interaction constant or the stacking energy between the nth base and its neighbors, in the plane normal to the helical axis in the strands S and S 0 . When K1 and K2 are not equal to J, they introduce anisotropy in the intrastrand and in the helicoidal interactions. µ and η represent measures of the interstrand interaction or hydrogen bonding energy, between the bases of similar sites in both the strands along the direction of the helical axis and in a plane normal to it respectively. Here we have assumed that there exist an average almost uniform interactions A that assume positive values, which are the uniaxial anisotropy coefficients leading to rotation of the bases in a plane normal to the helical axis. The dimensionless parameters f and g indicate the intrastrand stacking energy and the helicoidal stacking energy strength, respectively. 0

0

It is then possible to rewrite the Hamiltonian (4) in terms of the variables (θn , φn ) and (θn , φn ) as follows: H=

X

[−Jf sin θn sin θn+1 cos(φn+1 − φn ) − K1 f cos θn cos θn+1

n 0

0

0

0

0

0

0

0

− Jf sin θn sin θn+1 cos(φn+1 − φn ) − K1 f cos θn cos θn+1 0

(5)

− Jg sin θn sin θn+h cos(φn+h − φn ) − K2 g cos θn cos θn+h 0

0

0

− Jg sin θn sin θn+h cos(φn+h − φn ) − K2 g cos θn cos θn+h 0

0

0

0

+ η sin θn sin θn cos(φn − φn ) + µ cos θn cos θn + A cos2 θn + A cos2 θn ] The quasi-spin model thus introduced implies that, the dynamics of bases in DNA can be described 5

by the following equations of motion [9, 29] θ˙n =

−1 ∂H ˙0 1 ∂H 1 ∂H −1 ∂H 0 , φ˙ n = , θn = , φ˙ n = sin θn ∂φn sin θn ∂θn sin θn0 ∂φ0n sin θn0 ∂θn0

(6)

where the overdot represents the time derivative. When the anisotropy energy A is much larger than the other parameters (i.e, A J, K1 , K2 , η, µ) [9], substituting Eq.(6) into the Hamiltonian (5), the equations of motions become [9] 0 0 φ˙ n = 2A cos θn

φ˙ n = 2A cos θn ,

(7)

By introducing the above terms into Eq(5), we get the following Hamiltonian H=

XI 0 [ (φ˙ 2n + φ˙ n2 ) − Jf sin θn sin θn+1 cos(φn+1 − φn ) − K1 f cos θn cos θn+1 2 n 0

0

0

0

0

0

0

0

− Jf sin θn sin θn+1 cos(φn+1 − φn ) − K1 f cos θn cos θn+1 0

(8)

− Jg sin θn sin θn+h cos(φn+h − φn ) − K2 g cos θn cos θn+h 0

0

0

− Jg sin θn sin θn+h cos(φn+h − φn ) − K2 g cos θn cos θn+h 0

0

0

+ η sin θn sin θn cos(φn − φn ) + µ cos θn cos θn ] where I =

1 2A

is the moment of inertia of the bases around the axes at Pn (Pn0 ). The above

Hamiltonian can be rewritten under the absolute minima of the potential as [9, 29] X I 02 0 0 2 ˙ ˙ (φ + φn ) + Jf [2 − cos(φn+1 − φn ) − cos(φn+1 − φn )] H= 2 n n X 0 0 0 + Jg[2 − cos(φn+h − φn ) − cos(φn+h − φn )] − η[1 − cos(φn − φn )]

(9)

n

While rewriting the Hamiltonian in the above form, we have restricted the bases to be rotating in the plane which is normal to the helical axis. In other words, we have restricted our problem 0

to a plane-base rotor model [9] by assuming θn = θn = π/2. It should be stressed that, the above Hamiltonian describes the dynamics of two coupled homogeneous ferromagnetic spin systems (in the XY -spin model). The first term of Hamiltonian (9) corresponds to the kinetic energies of the rotational motion of the nth base pair. The second term corresponds to the stacking interaction of adjacent bases. The third term is new and characterizes the helicoidal structure of the double helix. In this frame, we know that the pitch of the double helix ranged from 8 to 10 bases in the B-DNA. The last term corresponds to the on-site potential which models the interaction of the hydrogen bonds that connect two bases in a pair. The equations of the motions of the base pairs can then be derived from the Hamiltonian (9) as follows: I φ¨n = Jf [sin(φn+1 − φn ) − sin(φn − φn−1 )] 0

0

0

(10a)

+ Jg[sin(φn+h − φn ) − sin(φn − φn−h )] + η sin(φn − φn ) 0

0

0

0

0

0

0

I φ¨n = Jf [sin(φn+1 − φn ) − sin(φn − φn−1 )] 0

+ Jg[sin(φn+h − φn ) − sin(φn − φn−h )] + η sin(φn − φn ) 6

(10b)

On the basis of the above equations, while describing the dominant rotational motion of the bases, all the other small motions of the bases are ignored. In so doing, the difference in angular rotation of bases with respect to neighboring bases along the two strands will be small. We then 0

0

assume that sin(φn±a − φn ) ≈ (φn±a − φn ) and we set Ξn = φn + φn and ψn = φn − φn . The above equations become ¨ n = Jf (Ξn+1 + Ξn−1 − 2Ξn ) + Jg(Ξn+h + Ξn−h − 2Ξn ) IΞ I ψ¨n = Jf (ψn+1 + ψn−1 − 2ψn ) − Jg(ψn+h + ψn−h + 2ψn )

(11a) (11b)

+ 2η sin(ψn ) After rescaling the time as t →

p J/It and choosing η = −J/2, we get the following set of

uncoupled equations d2 Ξn = f (Ξn+1 + Ξn−1 − 2Ξn ) + g(Ξn+h + Ξn−h − 2Ξn ) dt2

(12a)

d2 ψn = f (ψn+1 + ψn−1 − 2ψn ) − g(ψn+h + ψn−h + 2ψn ) − sin(ψn ) dt2

(12b)

The equation (12b) is known as the modified discrete sG equation. The solution Ξn is a linear wave while the solution ψn represents a nonlinear solitonic wave [33, 34]. Keeping only linear terms in Eqs.(12), we can easily obtain their linear solutions assuming they have the form of plane waves Ξn = Aei(qdn−ωΞ t) + c.c.,

ψn = Bei(qdn−ωψ t) + c.c.

(13)

where A and B are constant amplitudes, qd the wave vector and ωΞ and ωψ the frequencies of the in-phase and the out-of-phase motions, respectively. Substituting (13) into the harmonic limit of (12), the corresponding frequencies, e.g., the frequencies of the in-phase and the out-of phase motions, usually called acoustic and optical are [33, 34] qd hqd ) + 4g sin2 ( ) 2 2 qd hqd = 1 + 4f sin2 ( ) + 4g cos2 ( ) 2 2

2 ωΞ2 = ωac = 4f sin2 (

(14)

2 ωψ2 = ωop

(15)

their different features are shown in Fig.3. In Fig.3(a), we have assumed f = 1 and g = 0. Since the main purpose of this work is to bring out the impact of the helicoidal coupling on the bearing of localized structures, the different features of the dispersion relation are going to be discussed. In fact, recently, Zdravkovi`c [33] and Tabi [34] have shown that the presence of the helicoidal coupling in the helicoidal Peyrard-Bishop model can give rise to highly localized structures. For this to be possible, they introduced the resonant mode and stated that this was possible by setting ωop = ωac . When the condition for resonance is fulfilled, there is at least one crossing point between the optical and acoustic dispersion curves [33, 34].

7

Coming back to the model under our study, the resonant mode in ensured if [33, 34] 1 1 2 hqd . ωac = ωop , i.e, + = sin 2 8g 2

(16)

This means that g = 0.25 = gcr ,

and,

qd =

π . h

(17)

According to Fig.3(b), when g = 0.2 < gcr , there is no crossing point. When g = gcr = 0.25, there is one crossing point at qd = π/5 [see Fig.3(c)]. For g = 0.3 > gcr , there are two crossing point [see Fig.3(d)]. All the curves have been plotted for f = 1. For the two last cases, there is resonance and the formation of highly localized structures is expected in the framework of the modified discrete sG equation (12b).

3

The multiple scales analysis

3.1

Discrete multiple scaling

Equation (12b) forms a system of nonlinear ODEs which cannot be solved exactly. Thus, there exist several techniques to convert them into more integrable systems [12, 20, 24, 34, 35]. In this frame, we apply the method developed by Leon and Manna [36, 27] known as the multiple scale analysis. The reductive perturbation method (or multiple scales analysis) allows the deduction of simplified equations from a basic model without losing its characteristic features. The method consists essentially in an asymptotic analysis of a perturbation series, based on the existence of different scales. More specially, the method generates a hierarchy of (small) scales for the space and time variations of the envelopes of a fundamental (linear) plane wave and all the overtones. Moreover, the scale is directly related to the small amplitude of the wave itself. The scaling of variables is performed via a Taylor expansion of the frequency ω(q0 ) in powers of a small deviation of the wave number q0 . This deviation from the linear dispersion relation is, of course, generated by the nonlinearity. There are actually three different approaches to multiple scales analysis for a discrete evolution. The first is obviously to go to the continuum limit right in the starting system, for which discreteness effects are wiped out. The second is the semi-discrete approach which consists in having a discrete carrier wave modulated by a continuous envelope. In the latter case, some discreteness aspects are preserved, in particular, the resulting MI may depend on the carrier frequency. The third stems from the adiabatic approximation, but the approach requires one to use the rotating wave approximation to artificially eliminate the overtones. The price to pay is that the predictions, for example the MI, are not trustworthy for large time [20, 24]. In order to study MI on the helicoidal base-rotor system, we perform multiple scale analysis of the discrete evolution equation (12b). The physical problem we are concerned with is the following: the first particle of the chain (say n = 0) is given oscillation (or submitted to an external force) at frequency Ω. In a linear chain, this oscillation would propagate without distortion as 8

the plane wave exp[i(Ωt + qnd)], where d is the lattice spacing. However, the nonlinearity induces some deviations from the value Ω, namely, the wave propagates with actual frequency ω and wave number q0 that is defined as [36] ω = Ω + λ, where

1 vg

=

∂Ω ∂q

q0 = q +

is the group velocity and 2cg =

∂2q ∂2Ω

λ + 2 cg λ2 + ... vg

(18)

is the group velocity dispersion.

For notation simplicity, we shall assume here that cg = 1, which does not reduce the generality of our task. Let us consider the wave packet given by the Fourier transform [36] Z i(ωt+q0 dn) ˆ ψn (t) = dω ψ(ω)e

(19)

With the use of Eqs.(18), Eq.(19) can be expanded as [36] Z iλ(t+ nd ) iλ2 2 nd i(Ωt+qnd) vg ˆ dλψ(λ)e e ψn (t) = e

(20)

By means of the following change of independent variables [36, 27] τn = (t + nd/vg ),

ξn = 2 n

(21)

we obtain the following expression for the wave packet [36, 27] ψn (t) = A(n, t)u(ξn , τn )

(22)

where A(n, t) = e

i(qnd+Ωt)

Z ,

u(ξn , τn ) =

i(λτn +λ ˆ dλψ(λ)e

2 ξ d) n

(23)

Eq.(23) has a clear physical meaning: one considers long distance (−2 ) effects in the retarded time to give the input disturbance enough time to reach the observed lattice point. In such a situation, the lattice is excited at one end. This corresponds, in other words, to a boundary value problem. The quantity u(ξn , τn ) defines the slow modulation. In order to keep discreteness in space variable for the envelope u(ξn , τn ), one fixes the small parameters as [36, 27] 2 = 1/N

(24)

and, for any given n, we shall consider only the set of points ..., n − N, n, n + N, ... of a large grid indexed by the slow variable m, that is, ....(n − N ) → (m − 1), n → m, (n + N ) → (m + 1)....

(25)

As a consequence, we can index the variable ξn by m in the new grid; we can call m a given point n and m + j the points n + jN for all j [36, 27]. To simplify the notation, we shall be using everywhere u(ξj , τj ) = u ˜j ,

u(ξj , τn ) = uj 9

(26)

for a given n and all j (note that u ˜n = un ). Hence, we are interested in expressing everything in terms of un = u(m, τ ) defined as [36, 27] un−N = um−1 ,

un = um ,

un+N = um+1 .

(27)

The problem is now to express the various different operators occurring in nonlinear evolutions for the product A(n, t)um in terms of different operators for um . The traditional approach to multiple scaling for continuous media originates from water waves theory for which the physical problem is usually that of the evolution of an initial disturbance (e.g. of the surface). In this case, the observer has to follow the deformation at the (linear) group velocity. This operation corresponds to making, in the general Fourier transform solution, the expansion of ω(q0 ) around small deviations of q0 from the linear dispersion law. We now have to obtain the analogous relations for the product ψn (t) = A(n, t)u(ξn , τn ) = An un , appearing in definition (19). The quantities ψn+1 + ψn−1 − 2ψn and ψn+h + ψn−h + 2ψn are factorized as follows [36]: ψn+1 + ψn−1 − 2ψn = [An+1 − 2An + An−1 ]um + [An+1 − An−1 ]( +

d ∂um ) vg ∂τ

d ∂ 2 um 2 2 [An+1 + An−1 ]( )2 + [An+1 − An−1 ][um+1 − um−1 ] 2 vg ∂τ 2 2

(28a)

+ ϑ(3 ) ψn+h + ψn−h + 2ψn = [An+h + 2An + An−h ]um + [An+1 − An−1 ]( +

hd ∂um ) vg ∂τ

2 hd ∂ 2 um 2 [An+h + An−h ]( )2 + [An+h − An−h ][um+h − um−h ] 2 vg ∂τ 2 2

(28b)

+ ϑ(3 ) The formula (28) constitutes our basic tool for deriving reduced models in what follows.

3.2

Evolution of the envelope

We seek a solution of the above equation (12b) in the form of a Fourier expansion in harmonics of the fundamental exp[i(qnd − Ωt)], where the Fourier components are developed in a Taylor series in powers of the small parameter , measuring the amplitude of the initial wave [36] ψn (t) =

∞ X p=1

p

p X

A(l) (n, t)u(l) p (m, τ )

(29)

l=−p

with ∗ A(l) (n, t) = exp(il(Ωt + qnd)) and u(−l) = (u(l) p p ) .

Inserting the above expression in (12b), after expanding the term in sinus until the third order,

10

we get the following system ∞ X p=1

−f

p

p X

l=−p p ∞ X X p

p=1

+

(l) 2 (l) (l) [2 ∂τ τ u(l) p (m, τ ) + 2ilΩ∂τ up (m, τ ) − (lΩ) up (m, τ )]A (n, t)

2

[(eilqd + e−ilqd − 2)u(l) p (m, τ ) + (

l=−p

(eilkd + e−ilkd )(

d )(eilqd − e−ilqd )∂τ u(l) p (m, τ ) vg

d 2 ) ∂τ τ u(l) p (m, τ ) vg

2 2 (l) (l) + (eilkd − e−ilkd )(u(l) p (m + 1, τ ) − up (m − 1, τ ))]A (n, t) 2 p ∞ X X hd ilhqd p )(e − e−ilhqd )∂τ u(l) +g [(eilhqd + e−ilhqd + 2)u(l) p (m, τ ) p (m, τ ) + ( vg p=1

2

(30)

l=−p

2 ilhqd hd 2 ) ∂τ τ u(l) (e − e−ilhqd ) p (m, τ ) + 2 vg 2 p ∞ X X (l) (l) (l) p (l) × (up (m + h, τ ) − up (m − h, τ ))]A (n, t) + [u(l) p (m, τ )A (n, t)]

+

(eilhqd + e−ilhqd )(

p=1 ∞ X

1 − [ 6

p=1

p

p X

l=−p

(l) 3 u(l) p (m, τ )A (n, t)] = 0,

l=−p

we obtain that, the coefficients of the constant term give, at different orders of :

(0)

u1 = 0; 2 :

(0)

u2 = 0; 3 :

(0)

u3 = 0

(31)

The coefficients of A(1) , at different order of , give: d[f sin(qd) − gh sin(hqd)] , Ω (1) d hd ∂ 2 u1 3 : [1 − f ( )2 cos(qd) + g( )2 cos(hqd)] vg vg ∂τ 2

2 : vg =

− if

(1) sin(qd)[u1 (m

+ 1) −

(1) u1 (m

(32)

− 1)]

1 (1) (1) (1) (1) + ig sin(hqd)[u1 (m + h) − u1 (m − h)] − |u1 |2 u1 = 0 2 One should nevertheless stress that, the order leads to the already found optical expression (15) of the dispersion relation Ω = ωψ . The last equation in 3 , of the above set, writes i iP ∂ 2 χm (χm+1 − χm−1 ) − (χm+h − χm−h ) + Q − γ|χm |2 χm = 0 2 2 ∂τ 2

(33)

g sin(hqd) −1 , γ= f sin(qd) 4f sin(qd) 1 d 2 hd Q= [f ( ) cos(qd) − g( )2 cos(hqd) − 1] 2f sin(qd) vg vg

(34)

with P =

11

From relation (29), the approximate solution ψn (t) of Eq.(12b) can be written as ψn (t) = χm (τ )ei(qnd−Ωt) + ϑ(2 ),

(35)

(1)

where χm (τ ) = u1 (m, τ ). Note that the higher order correction to ψn (t) are either explicitly (l)

expressed in terms of χ through expression up (m, τ ) or by linear inhomogeneous differential equations. This means that the theory is self-consistent and, in particular, that the overtones are by no means neglected.

4

Modulational instability analysis

4.1

Linear stability analysis

The continuous version of Eq.(33) is a well-known model for boundary value problems in optical fibers [37]. Its discrete version (without the second term) has been found by Leon and Manna using the same technique [27]. The found modified equation thus takes helicity into consideration. For instance, in our knowledge, the DNA dynamics is the only system to be described by such an equation. Thus, it has stationary solutions in the form [27] χm = Bei(λm−µτ )

(36)

This plane wave solution obeys the dispersion relation µ2 = −

1 [sin(λ) − P sin(hλ) + γ|B|2 ] Q

(37)

By replacing P and γ by their expressions, we get the above nonlinear dispersion relation in the form µ2 = −

1 g sin(hqd) |B|2 [sin(λ) − sin(hλ) − ] Q f sin(qd) 4f sin(qd)

(38)

As we are considering a boundary value problem, µ and B are the given frequency and amplitude of the modulation. Modulated waves are then expected in the DNA model under study if the right-hand side of Eq.(38) is negative. In this frame, there is no real solution λ if Q < 0,

2 . |B|2 > 4[f sin(qd) sin(λ) − g sin(hqd) sin(hλ)] = Bcr

(39)

The above relation represents the condition for a plane wave to be unstable in the modified discrete sG model. It is also a modification of the condition predicted by Leon and Manna [27] for a discrete sG model without the term of helicity. In the case of the simple discrete sG model (without helicity), obtained for g = 0, the highest value of the threshold amplitude is obtained for sin(λ) = −1 [27]. In the case where helicity is taken into account, for h = 5, right-hand side of Eq.() is positive for sin(λ) = sin(hλ) = 1, i.e, λ = π/2. This means that 2 . |B|2 > 4[f sin(qd) − g sin(hqd)] = Bcr

12

(40)

However, this does not guarantee that the right-hand side of Eq.() is the highest for sin(λ) = sin(hλ) = 1. For example, it is possible that the expression in brackets is higher if sin(λ) is a little bit smaller than 1 and sin(hλ) is negative. To clarify this issue, we have plotted in Fig.4 the functions sin(λ) and sin(hλ). The crossing point of the two red dashed lines indicates that the value which better satisfies our needs is λ = 3π/10. For this value, sin(λ) is 0.8 and sin(hλ) is -1. The corresponding threshold amplitude is therefore given by 4 2 |B|2 > [4f sin(qd) + 5g sin(hqd)] = Bcr . 5

(41)

The behaviors of the threshold amplitude will then be discussed with respect to the critical value gcr derived in the second section. Thus, our analysis will be based on two cases: when g < gcr and when g ≥ gcr . Therefore, the following manifestations of MI are observed: • For g = 0, the system is described by Fig.5(a). The instability region belong to the whole interval [0; π] and there is only one side band. This features confirm the results by Leon and Manna [27]. • For g < gcr , we first observe that helicity breaks the instability domain into satellite side bands [see Fig.5(a), where the region between the two red dashed lines is the region of instability]. Furthermore the instability region is reduced by the helicoidal structure of the molecule. Furthermore it is obvious that the amplitude have increased. • For g ≥ gcr , we have the features displayed in Figs.5(c) and (d) which confirms the fact that the region of instability is reduced by the helicoidal coupling. Also, for values of g greater or equal to gcr , it is possible to observe large oscillations of strands. The behaviors displayed by the model understudy show that, this model can support large amplitude oscillation depending on the value of the helicoidal coupling constant g. Such results prove that, the present model is much richer than the simple discrete sG model [29, 38, 39, 40], that only consider stacking interactions between neighboring pairs and which can be recovered here by letting g = 0. In the framework of the Peyrard-Bishop-Dauxois model [10, 11], it has also been shown that helicity can deeply modify the instability criterion (in comparison to the works of Peyrard and co-workers [21, 24]) and can give rise to important behaviors in the process of soliton bearing through MI [20].

4.2

Numerical analysis of MI

It is well known that one of the main effects of MI, which refers to the exponential growth of certain modulation sideband of nonlinear plane waves propagating in a dispersive medium as a result of the interplay between nonlinearity and dispersion effect, is the generation of localized structures. In this section, we make computer simulations of the instability of a plane wave in the modified discrete sG equation in order to bring out some features of MI in the DNA 13

model understudy. Comparisons are made between numerical results and analytical predictions performed in the previous sections and a particular attention will be paid to the consequence of the interplay between stacking and helicoidal effects in the bearing localized wave patterns. However, it seems important to stress that, the linear stability analysis is based on the linearization around the unperturbed carrier wave, which is valid only when the amplitude of the perturbation is small in comparison with that of the carrier wave. Clearly, the linear approximation should fail at large time scales as the amplitude of an unstable sideband grows exponentially. Furthermore, the linear stability analysis neglects additional combination waves generated through a wave-mixing process, which, albeit small at the initial stage, can become significant at large time scales if its wave number falls in an instability domain. Linear stability analysis therefore cannot tell us the long time evolution of a modulated nonlinear plane wave. In this frame, we have performed numerical simulations of the modified discrete sG equation (12b) using the standard fourth-order Runge-Kutta scheme, with an integration time scale ∆t = 5 × 10−3 . The initial condition in accordance with Eq.(35), is a modulated wave. The number of base pairs have been chosen so that we do not encounter wave reflection at the the end of the molecule and periodic boundary conditions have been used. As already announced, two cases have been considered: the case g < gcr and the case g ≥ gcr . For the two cases, we have chosen B = 2.5, = 0.01, qd = 0.58π and λ = 0.3π. When the nearest-neighbor effect is more important in the system, i.e. g < gcr , we have the features displayed in Fig.6. We see that the plane wave excitation breaks into trains of waves which have the appearance of multisoliton shapes with breathing motion. Attention is paid to the oscillations of the bases 400 [see Fig.6(a)] and 800 [see Fig.6(b)]. We observe that, the oscillations of the bases undergo slight modulation for n = 400 and n = 800 even if modulation becomes important for the second case. This confirms our analytical predictions and once more shows that MI is a crucial issue for soliton instability [11], and is also considered a precursor to soliton formation because, it typically occurs in the same parameter region as that where solitons are observed. When the effect of the helicoidal coupling becomes important, i.e. g ≥ gcr , one obtains the results depicted in Fig.7. Even in this case, the plane wave breaks into trains of soliton-like structures. But the amplitude of waves is higher than in the first case and oscillations are obvious. At base pair n = 400, one clearly observes that the wave pattern displayed is that of an extended wave that propagates with a breathing motion. At base pair n = 800, the soliton objects are well separated, even if their amplitude is lower than for n = 400. One could then relate these exact breathing motions and the increasing of the amplitude to the presence of helicoidal terms. The corresponding values of the helicoidal coupling parameter (g ≥ gcr ) cause better and more efficient modulation of waves. This simply means that, when such a condition is fulfilled, there is a better information transfer through the molecule. Also, such behaviors cannot be observed for g < gcr , which, once more, suggest the importance the helicoidal stacking interactions. In the same way, in order to substantiate the importance of the helicoidal coupling in the 14

model, we have investigated energy localization by using the following relation 1 1 En = ψ˙ n2 + f [(ψn+1 − ψn )2 + (ψn − ψn−1 )2 ] 2 2 1 + g[(ψn + ψn+h )2 + (ψn + ψn−h )2 ] + [1 − cos(ψn )] 2

(42)

derived from (9) and (12b). For g < gcr [Fig.8(a)] and g ≥ gcr [Fig.8(b)], energy is really localized in the structure. This particularly accounts for the robustness of energy localization is the available biological models [18, 19, 20, 21]. For each of the two cases considered here, different features are displayed. In the case g < gcr , we see that energy is sheared by the whole lattice and gives rise to slight radiation in its spectrum accompanied by small localized structures. On the other hand, the case g ≥ gcr undergoes large patterns which tend to localize on specific sites.

5

Conclusion

In the framework of the so-called dynamic plane-base rotor model, we have taken into account the helicity of the B-DNA to investigate the MI of a plane wave. The dynamical equation, which finally appeared in the form of a modified discrete sG equation, was derived from a suitable Hamiltonian in analogy with the Heisenberg model of anisotropically coupled spin chain or spin ladder with ferromagnetic legs and antiferromagnetic rung coupling [29], where helicoidal terms have been introduced. We have found that, helicity brings about a new term in the equation describing the nonlinear dynamics of the molecule. To study MI, we have first derived from the original mode a simpler limit equation which results to be the discrete nonlinear Schrodinger (DNLS) equation in discrete space (23) and with space and time exchanged. This universal limit is obtained by a very general approach called multiple scale perturbation analysis. The MI criterion has been reported to be a modification of the one by Leon and Manna [27], due to the presence of helicoidal terms. A critical value of the helicoidal coupling parameter has been derived and on the basis of the chosen parameter values, there has been established a relationship between MI and the critical value gcr of g. It has, in fact, been found that, for g < gcr , the system does not feel the presence of helicoidal term. When g ≥ gcr , the analytical MI analysis predicts large amplitude oscillations as already seen in the framework of the PBD model [33, 20, 34]. Numerical experiments have been carried out in order to confirm the analytical predictions. It has been observed that, in the case where g < gcr , there is MI since the initial modulated plane wave breaks into train of extended solitons and soliton-like objects. In this case, the oscillations are slight at base n = 400 and tend to be accentuated at base n = 800. When g ≥ gcr , the amplitude of the waves is higher than in the first case and, as n increases, the waves tend to break into more precise soliton objects. This can be analyzed as the action of the waves flowing in the molecule and which plays an important role in the conformational concerns taking place in the B-DNA molecule. On the other hand, the increase of the amplitude of the waves trains mainly describes the action of RNA polymerase which breaks progressively the hydrogen bonds 15

for the messenger RNA to come and copy the genetic code. We believe that the existence and the formation of solitons in the DNA molecule could be a proper candidate to explain how data are exchanged during basic biological phenomena such as transcription and replication. As it is well known, for the hydrogen bonds to be broken, there should be a concentration of the enzyme and of the energy brought through the hydrolysis of ATP. It should nevertheless be stressed that, while speaking of mechanical models of DNA, we exclude consideration of the all-important interactions between DNA and its environment. The latter includes at least the fluid in which the DNA is immersed, and interaction with this leads to energy exchanges; one should thus include in the equations describing DNA dynamics both dissipation terms and random terms due to interaction with molecules in the fluid (even if the helicoidal terms included here stand for the presence of water filaments within the molecular environment). It would then be of interest to bring out the effect of such factors that could interestingly modify the process of MI studied in the model under consideration. However, the results obtained in the present study once more show the importance of helicity in the bearing of high localized structures.

Acknowledgments Fruitful discussions with Prof. Pierre Ngassam of the Department of Biology and Animal Physiology of the University of Yaound´e I are acknowledged. Tabi thanks the Condensed Matter and Statistical Physics Section of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, for the invitation, where this work has been finalized.

16

Appendices A

The discrete multiple scaling

From Eq.(30), the equation which describes constant terms A(0) is given by 2 (0) 3 ∂ u1 ∂τ 2

+

2 (0) 4 ∂ u2 ∂τ 2

+

2 (0) 5 ∂ u3 ∂τ 2

(0)

− f (1 + 1 − 2)u1 + 3 f

d vg

2

(0) ∂ 2 u1 ∂τ 2

! A(0)

2 2 (0) ! 2 (0) 2 (0) ∂ u ∂ u hd ∂ u1 (0) 2 3 5 3 − 4 f A(0) + f − g(1 + 1 + 2)u − g 1 2 2 2 ∂τ ∂τ vg ∂τ ! 2 2 (0) ∂ u hd (0) (0) 2 + 2 g(1 + 1 + 2)u2 + 4 g + 3 g(1 + 1 + 2)u3 A(0) 2 vg ∂τ ! 2 2 (0) (0) 3 hd ∂ u (u ) (0) (0) (0) 3 + 5 g A(0) = 0 + u1 + 2 u2 + 3 u3 − 3 1 vg ∂τ 2 6 (0)

(0)

(43)

(0)

This allows to derive the expressions of u1 , u2 , and u3 at different order of (0)

(0)

: 0 + 4gu1 + u1 = 0 (0)

(0)

2 : 0 + 4gu2 + u2 = 0 2 2 (0) (0) (0) hd ∂ u1 (u1 )3 ∂ 2 u1 (0) (0) 3 + g + 4gu + u − =0 : 3 3 ∂τ 2 vg ∂τ 2 6 (0)

(0)

(0)

The above system yields u1 = 0, u2 = 0, and u3 = 0.

17

(44)

The coefficient of A(1) are governed by the equation ! ! (1) (1) 2 (1) 2 (1) ∂ u ∂ u ∂u ∂u (1) (1) 2 1 A(1) + 2 A(1) 2 + 2iΩ 1 − Ω2 u1 + 2iΩ 2 − Ω2 u2 ∂τ 2 ∂τ 2 ∂τ 2 ∂τ ! (1) 2 (1) ∂ u ∂u (1) 3 + 3 A(1) + 2iΩ 3 − Ω2 u3 ∂τ 2 ∂τ ! (1) d ∂u (1) − f (eiqd + e−iqd − 2)u1 + ( )(eiqd − e−iqd ) 1 A(1) vg ∂τ ! 2 (1) 2 ∂ u 2 d 2 iqd (1) (1) 1 − f ( ) (e + e−iqd ) A(1) + (eiqd − e−iqd ) u1 (m + 1) − u1 (m − 1) 2 vg ∂τ 2 2 ! (1) d ∂u (1) − 2 f (eiqd + e−iqd − 2)u2 + ( )(eiqd − e−iqd ) 2 A(1) vg ∂τ ! 2 2 (1) 2 d ∂ u (1) (1) 2 − 2 f ( )2 (eiqd + e−iqd ) A(1) + (eiqd − e−iqd ) u2 (m + 1) − u2 (m − 1) 2 vg ∂τ 2 2 ! (1) d ∂u (1) A(1) − 3 f (eiqd + e−iqd − 2)u3 + ( )(eiqd − e−iqd ) 3 vg ∂τ ! 2 2 2 (1) d ∂ u (1) (1) 3 ( )2 (eiqd + e−iqd ) + (eiqd − e−iqd ) u3 (m + 1) − u3 (m − 1) − 3 f A(1) 2 vg ∂τ 2 2 ! (1) hd ihqd (1) ihqd −ihqd −ihqd ∂u1 + g (e +e + 2)u1 + ( )(e −e ) A(1) vg ∂τ ! 2 (1) 2 ihqd 2 hd 2 ihqd (1) (1) −ihqd ∂ u1 −ihqd +e ) ( ) (e + (e −e ) u1 (m + h) − u1 (m − h) A(1) + g 2 2 vg ∂τ 2 ! (1) hd ihqd (1) −ihqd ∂u2 2 ihqd −ihqd −e ) − g (e +e + 2)u2 + ( )(e A(1) vg ∂τ ! 2 (1) 2 hd 2 ihqd 2 ihqd (1) (1) −ihqd ∂ u2 2 −ihqd + g ( ) (e +e ) + (e −e ) u2 (m + h) − u2 (m − h) A(1) 2 2 vg ∂τ 2 ! (1) ∂u hd (1) − 3 f (eihqd + e−ihqd + 2)u3 + ( )(eihqd − e−ihqd ) 3 A(1) vg ∂τ ! 2 2 (1) 2 hd 2 ihqd ∂ u3 (1) (1) + 3 g ( ) (e + e−ihqd ) + (eihqd − e−ihqd ) u3 (m + h) − u3 (m − h) A(1) 2 2 vg ∂τ 2 (1) 2 (1) 3 (1) 3 1 (−1) (1) 2 + u1 + u2 + u3 − u1 (u1 ) A(1) = 0 2 (45)

18

At different order of , we have 2

: Ω = 1 + 4f sin

2

qd 2

2

+ 4g cos

hqd 2

(1) hd ∂u1 d sin(hqd) : 2i Ω − f ( ) sin(qd) + g vg vg ∂τ qd hqd (1) − Ω2 − 1 − 4f sin2 − 4g cos2 u2 2 2 2 (1) g hd 2 ihqd ∂ u1 f d 2 iqd −iqd −ihqd 3 ) + ( ) (e +e ) : 1 − ( ) (e + e 2 vg 2 vg ∂τ 2 (1) f iqd (1) e − e−iqd u1 (m + 1) − u1 (m − 1) − 2 (1) g ihqd (1) −ihqd + e −e u1 (m + h) − u1 (m − h) 2 (1) 1 (1) (1) d hd ∂u2 − |u1 |u1 + 2i Ω − f ( ) sin(qd) + g sin(hqd) 2 vg vg ∂τ qd hqd (1) − Ω2 − 1 − 4f sin2 − 4g cos2 u3 = 0 2 2 2

(46)

At order , we get the linear dispersion relation. At order 2 , we recover the linear dispersion relation Ω2 and the group velocity vg =

1 (f d sin(qd) − ghd sin(hqd)) Ω

(47)

(1)

At order 3 , we get the equation in u1 as follows ! 2 (1) hd g ∂ 2 u1 (eiqd + e−iqd ) + (eihqd + e−ihqd ) 2 vg ∂τ 2 f (1) (1) − (eiqd − e−iqd ) u1 (m + 1) − u1 (m − 1) 2 1 g ihqd (1) (1) (1) (1) + (e − e−ihqd ) u1 (m + h) − u1 (m − h) − |u1 |u1 = 0 2 2 f 1− 2

d vg

2

(48)

(1)

With u1 (m, τ ) = χm (τ ), the above equation reads i iP ∂ 2 χm (χm+1 − χm−1 ) − (χm+h − χm−h ) + Q − γ|χm |2 χm = 0 2 2 ∂τ 2

(49)

−1 g sin(hqd) , γ= f sin(qd) 4f sin(qd) 1 d hd Q= [f ( )2 cos(qd) − g( )2 cos(hqd) − 1] 2f sin(qd) vg vg

(50)

with P =

19

References [1] R. E. Franklin and R. G. Gosling, Nature (London) 171, 740 (1953). [2] J. D. Watson and F. H. C. Crick, Nature (London) 171, 737 (1953). [3] M. H. F. Wilkins, W. E. Seeds, A. R. Stokes and H. R. Wilson, Nature (London) 172, 759 (1953). [4] F. H. C. Crick and J. D. Watson, Proc. R. Soc. (London) A223, 80 (1954). [5] S. W. Englander et al., Proc. Nad. Acad. Sci. U.S.A. 77, 7222 (1980). [6] S. Yomosa, Phys. Rev. A 27, 2120 (1983). [7] S. Yomosa, Phys. Rev. A 30, 474 (1984). [8] S. Takeno and S. Homma, Prog. Theor.Phys. 70, 308 (1983). [9] S. Homma and S. Takeno, Prog. Theor.Phys. 72, 679 (1984). [10] M. Peyrard and A. R. Bishop, Phys. Rev. Lett. 62, 2755 (1989). [11] T. Dauxois, M. Peyrard and A.R. Bishop, Phys. Rev. E 47, R44 (1993). [12] M. Barbi, S. Cocco and M. Peyrard, Phys. Lett. A. 253, 358 (1999). [13] L. V. Yakushevich, Phys. Lett. A 136, 413 (1989). [14] L. V. Yakushevich, A. V. Savin and L. I. Manevitch, Phys. Rev. E 66, 016614 (2002). [15] L. V. Yakushevich, Nonlinear Physics of DNA, 2nd Ed. (Wiley, Chichester, 2004). [16] G. Gaeta, Phys. Lett. A 190 (1994) 301; J. Nonl. Math. Phys. 14, 57 (2007). [17] K. Forinash, T. Cretegny and M. Peyrard, Phys. Rev. E 55, 4740 (1997). [18] E. Zamora-Sillero, A. V. Shapovalov and F. J. Esteban, Phys. Rev. E 76, 066603 (2007). [19] C. B. Tabi, A. Mohamadou and T. C. Kofane, J. Comput. Theor. Nanosci. 5, 647 (2008). [20] C. B. Tabi, A. Mohamadou and T. C. Kofane, J. Phys.: Condens. Matter 20, 415104 (2008). [21] I. Daumont, T. Dauxois and M. Peyrard, Nonlinearity 10, 617 (1997). [22] J. Cuevas, J. F. R. Archilla, Yu. B. Gaididei, and F. R. Romero, Physica D 163, 106 (2002). [23] P. V. Larsen, P. L. Christiansen, O. Bang, and Yu. B. Gaididei, Phys. Rev. E 69, 026603 (2004). [24] Y. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992). [25] A. V. Gorbach and M. Johanson, Eur. Phys. J D 29, 77 (2004).

20

[26] J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel and J. S. Aitchison, Phys. Rev. Lett. 92, 163902 (2004). [27] J. Leon and M. Manna, Phys. Rev. Lett. 83, 2324 (1999). [28] P. Marqui´e, J. M. Bilbault and M. Remoissenet, Phys. Rev. E 51, 6127 (1995). [29] M. Daniel and V. Vasumathi, Physica D 231, 10 (2007). [30] T. Dauxois, Phys. Lett. A 159, 390 (1991). [31] S. Zdravkovi`c and M. V. Satari`c, Physica Scripta 64, 612 (2001). [32] S. Zdravkovi`c and M. V. Satari`c, Int. J. Mod. Phys. B 17, 5911 (2003). [33] S. Zdravkovi`c and M. V. Satari`c, Europhys. Lett. 78, 38004 (2007). [34] C. B. Tabi, A. Mohamadou and T. C. Kofane, Eur. Phys. J. D 50, 307 (2008). [35] M. Remoissenet, Phys. Rev. B 33, 2386 (1986). [36] J. Leon and M. Manna, J. Phys. A: Math. Gen. 32, 2845 (1999). [37] M. Wadati, T. Lizuka and T. Yajima, Physica (Amsterdan) 51D, 388 (1991). [38] M. Salerno and Y. S. Kivshar, Phys. Lett. A 193, 263 (1994). [39] M. Salerno, Phys. Rev. A 46, 6856 (1992). [40] G. Gaeta, Phys. Rev. E 74, 021921 (2006).

21

(a)

(b)

(c)

Figure 1: (a) A schematic structure of the B-form of DNA; (b) A horizontal projection of the nth base pair in the xy-plane; (c) A projection of the nth base pair in the xz-plane.

Figure 2: A schematic representation of DNA as an anisotropically coupled spin chain model or spin ladder.

22

2.5

2.5 Acoustic branch Optical branch

Acoustic branch Optical branch

1.5

1.5

ω

2

ω

2

1

1

0.5

0.5

0 0

0.2

0.4

0.6

0.8

0 0

1

qd/π

0.2

0.4

0.6

0.8

1

qd/π

(a)

2.5

(b)

3

Acoustic branch Optical branch

Acoustic branch Optical branch 2.5

2

2

ω

ω

1.5 1.5

1 1 0.5 0.5 0 0

0.2

0.4

0.6

0.8

qd/π

0 0

1

(c)

0.2

0.4

0.6 qd/π

0.8

1

(d)

Figure 3: Optical and acoustic frequencies as a function of the wave number qd and for different values of g and f = 1: Panel (a) shows the dispersion curves for g = 0. The curves are similar to those obtained for the discrete sG model. For g = 0.2 < gcr , we have the configuration of panel (b). The dispersion curves oscillate and there is no crossing point between the optical and the acoustic curves. When the condition () is fulfilled (g = gcr = 0.25), there is one crossing point and resonance is possible. Panel (d) shows the dispersion curves for g = 0.3 > gcr . In this case, there are two crossing points between the optical and the acoustic curves. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 sin(hλ) sin(λ)

−0.8 −1 0

0.2

0.4

λ/π

0.6

0.8

1

Figure 4: The panel shows the plot of the functions sin(λ) (green curve) and sin(hλ) (blue curve). The possible value of λ which leads to the highest value of the threshold amplitude Bcr is the one situated at the crossing point of the the two red dashed lines, λ = 3π/10.

23

2

2.5 Unstable

Unstable

1.8 1.6

2

Stable

Stable

1.4 1.5 cr

Bcr

1.2

B

1 Stable

0.8

1 Stable

0.6 0.4

Stable

Stable

0.5

0.2 0 0

0.2

0.4

0.6

0.8

0 0

1

qd/π

0.2

0.4

0.6

0.8

1

qd/π

(a)

2.5

(b)

2.5 Unstable Unstable

2

Stable

Stable

Stable

Stable

2

B

cr

1.5

Bcr

1.5

1

1 Stable

Stable

Stable

Stable

0.5

Stable

Stable

0.5

0 0

0.2

0.4

0.6

0.8

0 0

1

qd/π

0.2

0.4

0.6

0.8

1

qd/π

(c)

(d)

Figure 5: The panels show the threshold amplitude Bcr versus the wave number qd for different values of g with f = 1: (a) g = 0. the result is similar to the one obtained by Leon and Manna [27] for the discrete sG model. In the second panel, the threshold amplitude is plotted for g = 0.2 < gcr . The amplitude has increased and the region of instability has been reduced due to helicity. In panel (c), the threshold amplitude has been plotted for g = 0.25 = gcr and the last panel depicts the case g = 0.3 > gcr . −3

8

x 10

0.03

n = 800 n = 400

6

0.02

4

2 ψn/ 2 π

ψn/ 2 π

0.01

0

0

−2 −0.01

−4 −0.02

−6

−0.03 0

200

400

600 Time

800

1000

1200

(a)

−8 0

200

400

600 Time

800

1000

1200

(b)

Figure 6: Dynamics of modulated waves as a function of time (t.u.) showing the MI of slowly modulated plane waves for g < gcr (a) Soliton-like objects at the base pair 400; (b) Soliton-like objects at the base pair 800. The system undergoes slight oscillations. So, for the considered case, modulations are not efficient and cannot ensure better transmission of information within the double helical structure. 24

0.4

0.2

n = 400

n = 800

0.2

0.1

0.1

0.05 ψn/ 2 π

0.15

ψn/ 2 π

0.3

0

0

−0.1

−0.05

−0.2

−0.1

−0.3

−0.15

−0.4 0

200

400

600 Time

800

1000

1200

(a)

−0.2 0

200

400

600 Time

800

1000

1200

(b)

Figure 7: Dynamics of modulated waves as a function of time (t.u.) showing the MI of slowly modulated plane waves for g ≥ gcr (a) Soliton-like objects at the base pair 400; (b) Soliton-like objects at the base pair 800. Oscillations for this case are more efficient and clearly show that when g ≥ gcr , data are better transmitted within the DNA model. This bring out the importance of helicoidal models over the simple model that only take into consideration nearest-neighbor coupling.

−3

−3

x 10

800

x 10

800

1.5 7

700

700

600

600

6

5

1

Time

500

Time

500

400

4

400 3 0.5

300

300 2

200

200 1

100 50

100

150 n

200

250

300

100

0

(a)

50

100

150 n

200

250

300

0

(b)

Figure 8: Energy localization in the DNA double helix for (a) g < gcr ; (b) g ≥ gcr . The horizontal axis indicates the position of the base pairs and the vertical axis corresponds to time (time is going upward). In the first case we have En = 1.62 × 10−3 and En = 7.8 × 10−3 in the second, respectively. It is obvious that the case g ≥ gcr is responsible for high energy localization, as shown in (b).

25