Moving breathers in a bent DNA model

Moving breathers in a bent DNA model J Cuevas a,1, F Palmero a, JFR Archilla a, FR Romero b aEscuela T´ecnica Superior de Ingenier´ıa Inform´atica. Departamento de Fisica Aplicada I. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012-Sevilla (Spain) bFacultad de Fisica. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012-Sevilla (Spain) Abstract We study the properties of moving breathers in a bent DNA model with short range interaction, due to the stacking of the base pairs, and long range interaction, due to the finite dipole moment of the bonds within each base pair. We show that the movement of a breather is hindered by the bending of the chain analogously to a particle in a potential barrier. Key words: Discrete breathers, Mobile breathers, Intrinsic localized modes, DNA PACS: 63.20.Pw, 63.20.Ry, 63.50.+x, 66.90.+r, 87.10.+e 1 Introduction Discrete breathers are localised oscillations that appear in nonlinear discrete systems. Their existence and stability, under some rather relaxed conditions, was proven by MacKay and Aubry and, ever since, they have been widely studied [1–3]. These localized excitations, under certain conditions, can move and transport energy along the system and they are usually called moving breathers [4–8]. A particularly interesting discrete system is the Deoxyribonucleic acid, or DNA. In this system, localization of energy has been suggested as a precursor of the transcription bubble [9–11], and moving localized oscillations as a method of transport of information along the double strand [12]. 1Corresponding author. E-mail: jcuev[email protected] Preprint submitted to Physics Letters A 27 March 2015 In order to explain some aspects of DNA dynamics, a great number of mathematical models have been proposed [13]. An interesting approach has been followed by Peyrard and Bishop (PB model) [9], who proposed a model in order to study the dynamics and the thermodinamics of base pairs opening in DNA denaturation and transcription. In this model, the double strand is equivalent to a Klein-Gordon chain, the variables are the distances between nucleotides within each base pair, and only short range interactions due to the stacking coupling are considered. Other DNA models ignore this kind of interaction and only consider long range interactions, whose origin lies in the dipole moments that characterize the hydrogen bonds between the nucleotides [14,15]. Nevertheless, we have shown that the existence of stacking interaction is a necessary condition to obtain moving breathers in these last kind of models [16]. In the framework of the Peyrard-Bishop model, the bending of the molecule is not relevant, because only nearest neighbour interaction is considered. However, when long range interactions are taken into account, the bending becomes relevant, and it can modify the dynamics of the system. Bending of DNA has been studied as an inhomogeneity in a chain with only nearest neighbour interaction [17,18], and long range interaction has been taken into account in homogeneous bent DNA models using the nonlinear Schr¨odinger equation (NLS) [19,20]. The relationship between static breathers and bending has also been studied in the framework of the Klein–Gordon equation [21,22]. In this paper, we study the properties of moving breathers in a bent DNA chain with short and long range interactions in a modified Peyrard-Bishop model [16]. The origin of the interactions lies, respectively, in the stacking of the base pairs and the finite dipole moment of the hydrogen bonds within each base pair. We have found that the bending of the chain can drastically change the properties of moving breathers. They can cross the bent region or be reflected, a behaviour which resembles the movement of a particle in a potential barrier. A similar phenomenon has been recently observed in a two–dimensional FPU model for rigid biopolymers [23,24]. Another important result is that the moving breather behaves as a quasiparticle with constant mass provided that the phonon radiation is small enough. Under these conditions, we have developed a method for calculating the instantaneous translational kinetic energy of the moving breather. 2 2 The model We consider a modification of the Peyrard-Bishop model, which consists in the addition of an energy term to the Hamiltonian, that takes into account the long range interaction due to the dipole–dipole forces [16]. The Peyrard–Bishop model has been studied assuming that the double strand has a planar geometry, i.e., the DNA chain lies in a plane. In order to study the influence on the dynamics of a different geometry, we suppose that the plane has been bent and adapted to a parabola of curvature κ(that is, the location of the n-th base pair is determined by the equation yn=κx2 n/2) [22]. All the dipole moments are perpendicular to the parabola and parallel among them. Thus, the hamiltonian of the system can be written as: H=T+UBP +UST +UDD (1) where Tis the kinetic energy: T=1 2m∑ n ˙u2 n,(2) being unthe transverse stretching of the hydrogen bonds connecting the two bases and mthe reduced mass of a pair of nucleotides. The term UBP represents the interaction energy due to the hydrogen bonds within each base pair: UBP =∑ n V(un),(3) where V(un) is the Morse potential, i.e., V(u) = D(e−b u −1)2, being Dthe well depth, which represents the dissociation energy of a base pair, and ba spatial scale factor. UST is the short range interaction term, representing the stacking energy between base pairs: UST =1 2k∑ n (un+1 −un)2,(4) 3 where kis the stacking coupling constant. UDD is the long range interaction term, due to the dipole–dipole interaction. It can be expressed as [16]: UDD =1 2∑ n,i J∗ niunui,(5) where, J∗ ni =       J∗ |rn−ri|3for i=n 0 for i=n. (6) The vector rndescribes the position of the n-th base pair. We assume that the chain is inextensible, so that the distance between neighbouring sites remains constant: |rn−rn+1| ≡ d. The coupling constant Jis related to the charge transfer due to the formation of the hydrogen bonds (q) and the distance between base pairs (d), in the following way: J∗=q2 4πεod3.(7) The Hamiltonian can be written as: H= N ∑ n=1 (1 2m˙u2 n+D(e−b un−1)2+1 2k(un+1 −un)2+1 2∑ i J∗ inuiun),(8) With an appropriate change of variables [16], the distance between neighbouring sites is 1 and the dynamical equations become: F({un})≡¨un+ (e−un−e−2un) + C(2un−un+1 −un−1) + ∑ i Jinui= 0,(9) where C=k/2Db2and Jin =J∗ in/2Db2. Furthermore, we define J=J∗/2Db2 as a new dimensionless dipole–dipole coupling constant. 4 3 Moving breathers and parameter values Our aim is to study the dynamics of a moving breather travelling through a bent chain. In order to obtain a moving breather, we perturb the velocity of a static breather far from the parabola vertex, launching it to the bent part. Static breathers have been calculated using common methods based on the anticontinuous limit [26–29]. Once a suitable static breather is obtained, it is made movable using the marginal mode method [7,8], which basically consists in adding to the velocities of the static breather a perturbation of magnitude λcolinear to the direction of a linear localized mode, and letting the system evolve in time. In this context, an useful concept for describing the moving breather dynamics is the effective mass [7,8], a measure of the moving breather inertia. If the kinetic energy added to the breather is E=λ2/2, it is found that the resulting translational velocity of the breather, v, is proportional to λ[7]. Consequently, moving breathers can be considered as a quasi-particle with mass m∗, which can be defined through the relation: 1 2m∗v2=1 2λ2=E. (10) A difficult issue is the choice of appropriate values of the parameters in order to fulfill two different requirements. On the one hand, there is only a small range of them that allows breather mobility. On the other hand, the parameters must be consistent with real DNA. The last aspect is controversial by itself as explained below. The requirement that the breathers can move is our first priority, in agreement with the experimentally observed “breathing” modes in DNA [30,31]. While there exists a general agreement in the literature about the order of magnitude of the parameters Dand b, there are not accurate data about the value of the elastic constant k, which oscillates between 0.01eV/˚ A2and 10eV/˚ A2[10,32]. In their original paper, Peyrard and Bishop [9] used a set of values D= 0.33 eV, b= 1.8˚ A−1and k= 3.0×10−3eV/˚ A2. This set of parameters gives a value of C= 0.014. Afterwards, Dauxois, Peyrard and Bishop used the values D= 0.04 eV, b= 4.45 ˚ A−1and k= 0.06 eV/˚ A2 [11], which imply that C= 0.038. Consequently, as Cmust be greater than 0.12 in order to obtain moving breathers [16], they would not exist in the Peyrard–Bishop model with these values of the parameters. For a given model, a tuning procedure must be achieved in order to correctly fit the parameters to experimental results. Even the experimental results are controversial as each one refers to different aspect of DNA dynamics (torsion, bending, stretching, ...) [33]. Some recent works propose a generalization of the PB model with different values of the elastic constant k. In [33], the 5 0 20 40 60 80 100 120 140 160 −15 −10 −5 0 5 10 XE Periods vo=0.0311 vo=0.0260 Fig. 1. Evolution of the breather energy centre (XE) for two different initial velocities. The curvature is κ= 2. selected parameter values gives a coupling parameter C= 0.63. In [34,32], normal modes corresponding to hydrogen bonds excitations are theoretically investigated, with a range of the coupling parameter C∈(0.06,0.31) that shows a good agreement with neutron scattering experiments. In this paper, we have chosen the breather frequency ωb= 0.8, so that the nonlinear effects are significant but not too strong, as the nonlinearities in DNA are thought to be weak. The stacking and dipole–dipole coupling parameters chosen are, respectively, C= 0.24 and J= 0.02, which provides with moving breathers with low phonon radiation for small enough values of λ[16]. We have considered different spatial configuration varying the parameter κ. 4 Numerical results In figure 1 we show the evolution of the energy centre [16] of a moving breather in a bent chain. If the added kinetic energy, E=λ2/2 is smaller than a critical value Ec, the breather rebounds, but, if E > Ec, the breather passes through the bending point. Figure 2 shows that the critical energy increase monotonically with the curvature. Bending acts as a hindrance for the movement of breathers. This hindrance reminds to the experimented by a particle in a potential barrier. In this case, breather can be consider as a quasi-particle and this barrier can be calculated by finding the points where breathers rebound (or turning points) for different 6 0 2 4 6 8 10 12 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 κ Ec Fig. 2. Critical kinetic energy Ecversus curvature κ. values of E. Furthermore, if the breather has a constant mass m∗, this potential barrier can be obtained using the expression: Eb=1 2λ2(1−(v vo)) (11) where vis the translational velocity and vois its value at t= 0. In figure 3 can be observed a good agreement between the barrier calculated using both methods for a given value of κ. The barrier calculated by the second method exhibits an irregular shape, whose origin lies in the non-uniform behaviour of the translational velocity due to the discreteness of the system [35]. This result confirms that, in this case, a moving breather behaves as a particle of constant mass m∗. For the parameters used in our study, we have observed that the effect of the phonon radiation is not negligible if E&0.03, and the properties of the moving breathers change. Thus, in order to have a good agreement between both methods of calculating the potential barrier, the curvature of the parabola must be κ.4 (see figure 2). Otherwise, the value of the energy necessary for a moving breather to cross the bending point will be so high that the phonon radiation will alter the movement. 7 −1 −0.5 0 0.5 1 6 7 8 9 10 11 12 x 10−3 Eb XE Fig. 3. Potential barrier calculated finding the turning points (solid line) and using Eq. (11) for E= 0.0162 (dashed lined) and E= 0.0200 (dotted line). The curvature is κ= 2 and the critical energy is Ec≈0.0126. The zero value of the energy centre (XE) represents to the bending point. 5 Conclusions In this paper, we have studied the behaviour of moving breathers in a bent DNA chain. The movement of a breather is qualitatively similar to the experienced by a particle in a potential barrier. For a given initial velocity, breathers can cross the bending point if the curvature is lower than a critical value. Otherwise, the breather rebounds. This fact suggest that a slight conformational changes in the DNA molecule can modify the dynamic of nonlinear excitations. Another result is that the breather effective mass remains constant when the breather cross the bending point as long as the curvature is small enough. This fact provides with a method to calculate the instantaneous translational energy, which is given by Etrans =m∗v2/2, being vthe translational velocity of the breather. In a recent work, it is proposed a method to calculate the translational energy in a FPU chain based in the asymmetry between the difference between the maxima of the potential and kinetic energies and the total energy of the moving breather [24]. This analysis can be performed because the maximum kinetic and potential energies coincide with the total energy of the static breather, as the interaction potential is spatially symmetric. However, this method cannot be applied in our Klein–Gordon chain because the on–site potential is not spatially symmetric. A further study could consider 8 a spatially symmetric potential, as the double-well one, in order to find out whether both methods are equivalent. Acknowledgments This work has been supported by the European Commission under the RTN project LOCNET, HPRN-CT-1999-00163. JC acknowledges an FPDI grant from ‘La Junta de Andaluc´ıa’. References [1] RS MacKay and S Aubry. Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity, 7:1623–1643, 1994. [2] S Aubry. Breathers in nonlinear lattices: Existence, linear stability and quantization. Physica D, 103:201–250, 1997. [3] S Flach and CR Willis. Discrete breathers. Physics Reports, 295:181–264, 1998. [4] JL Mar´ın, JC Eilbeck, and FM Russell. Localized moving breathers in a 2-D hexagonal lattice. Phys. Lett. A, 248:225–229, 1998. [5] JL Marin, JC Eilbeck, and FM Russel. 2-d breathers and applications. In PL Christiansen and MP Soerensen, editors, Nonlinear Science at the dawn of the 21st century, pages 293–306. Springer, 2000. [6] JL Marin, FM Russell, and JC Eilbeck. Breathers in cuprate-like lattices. Phys. Lett. A, 281:21–25, 2001. [7] Ding Chen, S Aubry, and GP Tsironis. Breather mobility in discrete ϕ4lattices. Physical Review Letters, 77:4776, 1996. [8] S Aubry and T Cretegny. Mobility and reactivity of discrete breathers. Physica D, 119:34, 1998. [9] M Peyrard and AR Bishop. Statistical mechanics of a nonlinear model for dna denaturation. Physical Review Letters, 62:2755, 1989. [10] T Dauxois, M Peyrard, and CR Willis. Localized breather-like solution in a discrete Klein-Gordon model and application to DNA. Physica D, 57:267, 1992. [11] T Dauxois, M Peyrard, and AR Bishop. Dynamics and thermodynamics of a nonlinear model for DNA denaturation. Phys Rev E, 47:684, 1993. 9