Breather trapping and breather transmission in a DNA model with an interface

Breather trapping and breather transmission in a DNA model with an interface. A Alvarez and FR Romero∗ Grupo de F´ısica No Lineal. ´ Area de F´ısica Te´ ørica. Facultad de F´ısica. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012-Sevilla (Spain) JFR Archilla and J Cuevas Grupo de F´ısica No Lineal. Departamento de Fisica Aplicada I. ETSI Inform´atica. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012-Sevilla (Spain) PV Larsen Department of Mathematics, Technical University of Denmark. DK-2800 Kgs. Lyngby (Denmark) (Dated: March 24, 2006) We study the dynamics of moving discrete breathers in an interfaced piecewise DNA molecule. This is a DNA chain in which all the base pairs are identical and there exists an interface such that the base pairs dipole moments at each side are oriented in opposite directions. The Hamiltonian of the Peyrard–Bishop model is augmented with a term that includes the dipole–dipole coupling between base pairs. Numerical simulations show the existence of two dynamical regimes. If the translational kinetic energy of a moving breather launched towards the interface is below a critical value, it is trapped in a region around the interface collecting vibrational energy. For an energy larger than the critical value, the breather is transmitted and continues travelling along the double strand with lower velocity. Reflection phenomena never occur. The same study has been carried out when a single dipole is oriented in opposite direction to the other ones. When moving breathers collide with the single inverted dipole, the same effects appear. These results emphasize the importance of this simple type of local inhomogeneity as it creates a mechanism for the trapping of energy. Finally, the simulations show that, under favorable conditions, several launched moving breathers can be trapped successively at the interface region producing an accumulation of vibrational energy. Moreover, an additional colliding moving breather can produce a saturation of energy and a moving breather with all the accumulated energy is transmitted to the chain. PACS numbers: 63.20.Pw, 63.20.Ry, 63.50.+x, 66.90.+r, 87.10.+e Keywords: Moving breathers, breathers collisions, Klein–Gordon lattices, dipole–dipole interaction, DNA dynamics I. INTRODUCTION Nonlinear physics of DNA has experienced an enormous development in the previous years. There are many experimental data and theoretical results published about the nonlinear properties of DNA (for a review see, e.g., [1]. The DNA molecule is a discrete system consisting of many atoms having a quasi-one-dimensional structure. It can be considered as a complex dynamical system, and, in order to investigate some aspects of the dynamics and the thermodynamics of DNA, several mathematical models have been proposed. Among them, it is worth remarking the Peyrard–Bishop model [2] introduced for the study of DNA thermal denaturation. This model, and some variations of it, has also been used extensively for the study of the dynamical properties of DNA. In DNA there exist different kinds of interactions between the main atomic groups. One of them is the stacking interaction between neighbouring bases along the DNA axis, these are short–range forces which stabilize the DNA structure and hold one base over the next one forming a stack of bases. There exist also long–range forces, due to the finite dipole moments of the hydrogen bonds within the nucleotides. The existence of these forces is corroborated both by theoretical and experimental studies. Recently, quantum chemical calculations, using the second Moler-Plesset perturbation method, have determined bonds lengths and bonds angles involved in hydrogen bonds between the bases of DNA base pairs, as well as dipole moments based at the MP2/6-31G(d) and MP2/6-31G(d,p) basis sets [3]. The dipole moments calculated for the adenine-thymine base pair at equilibrium are, respectively, 1.44 D and 1.29 D. The values calculated for the guanine-cytosine base pair at equilibrium are, respectively, 5.88 D and 5.79 D. All these ∗Electronic address: [email protected] values are within the range of the parameters values for the dipole moments considered in this paper. Some experiments carried out on the dynamics of the B-A transition of DNA double helices, through the analysis of the transients observed under electric field pulses, show that dipolar strectching is the main driving force for the B-A reaction [4]. The dipole–dipole interactions between different base pairs should never be ignored and they may play a crucial role for the dynamical properties of DNA, for example when the geometry of the double strand of DNA is taken into account, as the distance between base pairs and, therefore, the intensity of the coupling between them, depends on the shape of the molecule [5–11]. Most of the studies have been done considering DNA homogeneous models. Nevertheless, the DNA molecule is essentially an inhomogeneous system, and this inhomogeneity is characterized by the existence of different local ranges, or functional regions, with very specific sequences of base pairs and very specific functions. Numerical simulations show that discrete breathers (DBs), which are spatially localized nonlinear oscillations, can appear in models of crystals, biomolecules, and many others nonlinear discrete systems [12–14]. It is well known that DBs can be static, but they can also move under certain physical conditions [15–17], constituting a mechanism for the transport of energy and information along discrete systems. They are called moving breathers (MBs) and have been obtained in many different systems [18–20]. In the Peyrard–Bishop model, the existence of DBs has been demonstrated [21, 22], and DBs are thought to be the precursors of the bubbles that appear prior to the transcription processes in which large fluctuations of energy have been experimentally observed [23]. The study of MBs in Klein-Gordon models of DNA chains with the inclusion of long–range interactions was initiated in [24] using an augmented Peyrard–Bishop model. The system considered is homogeneous and without bending, that is, all the base pairs are identical, aligned in a straight line, and all the dipole moments are parallel along the same direction. The results show that MBs exist for a wide range of the parameter values, although the mobility is hindered when the intensity of the long–range interaction increases. Essentially, the effective mass of a breather increases as the intensity of the dipole–dipole interaction increases, and there exists a threshold of the dipole–dipole coupling constant above which the breather is not movable. The interaction of MBs with different kind of inhomogeneities in the DNA molecule may be also of great importance for the localization of energy [25, 26]. As a consequence of this interaction, MBs may be trapped in small regions, that is, vibrations remain affecting only to a small number of consecutive base pairs, producing an accumulation of energy, which can initiate denaturation or transcription bubbles. Also, trapping phenomena may be on the basis for the appearance of secondary breaks when DNA is irradiated with ionizing radiation [27]. Structural studies carried out with non coding sequences of DNA have shown that poly(dA.dT) sequences (called T-tracts) are abundant genomic DNA elements. Suter et al. [28]have shown that they exist as rigid DNA structures in nucleosome-free yeast promoters in vivo, that is, in vivo they are not folded in nucleosomes. Their data support that transcription activation depends on the length of the T-tracts and the same occurs when they are replaced by poly(dG.dC) sequences, another rigid structures. This fact suggests that the T-tracts operate not by recruitment of specific transcription factors, but rather by their intrinsic DNA structure, and that transcription activation must be intimately related with the dynamical properties of T-tracts. These considerations motivated us to study MBs in some types of poly(dA.dT) or poly(dG.dC) sequences. In this paper we study the effects of the collisions of MBs with local inhomogeneities in two different but related systems. The first system consists of an inhomogeneous DNA molecule formed by two consecutive homogeneous regions. All the base pairs are identical, but the dipoles of a region are oriented in opposite direction to the dipoles of the other one, as sketched in Fig. 1(a). One possibility is for a sequence being as ...AT/AT/AT/T A/T A/T A...,where AT and TA represent, respectively, the adenine-thymine base pair and the thymine-adenine base pair. The other possibility is for a sequence being as ...CG/CG/CG/GC/GC/GC...,whereCG and GC represent, respectively, the cytosine-guanine base pair and the guanine-cytosine base pair. The inhomogeneity is determined by the existence of an interface that separates these regions, that is, there exists a local inhomogeneity and the interface marks a discontinuity in the system. These kind of chains are called interfaced piecewise chains, they exist in real DNA and can also be synthesized in the laboratory [29, 30]. The second system consists in a DNA chain of identical base pairs with all the dipoles oriented in the same direction, except one of them which has an opposite orientation, that is, the local inhomogeneity is at the site of the inverted–dipole, as sketched in Fig. 1(b). In both systems, the simulations of MBs launched towards the inhomogeneity show that if the translational kinetic energy is below a critical value, which is determined by the values of the coupling parameters, the breather is trapped. For greater values, the breather is transmitted to the chain, losing part of its energy as phonon radiation. Moreover, reflection of the colliding breather never occurs. These results emphasizes the enormous importance of a change of the orientation of a single base pair in a homogeneous track of DNA chain, as it demonstrates that a very simple structural change creates a mechanisms for (a) (b) . . . . . . . . . . . . FIG. 1: Sketch of the dipole chains.(a): the interfaced piecewise chain, (b): the single inverted–dipole chain localization and accumulation of energy. Moving breathers can also collide with a trapped breather. Our study of these collision processes shows that multiple scenarios are possible, one of them is the possibility of getting an accumulation of energy at the interface up to a saturation threshold value. All the accumulated energy can be carried away by a successful incoming breather. These phenomena suggest a way for the DNA molecule to accumulate enough energy to break the hydrogen bonds between the base pairs on opposing strands. Another important result concerning these systems, with a local inhomogeneity and with competing short and long–range interactions, is that there exists a relevant parameter determining the range of translational kinetic values for which MBs are trapped, this is the quotient between the dipole–dipole and the stacking coupling constants. The critical kinetic energy increases, or the size of the ”trapping window”, if this quotient increases. This paper is organized as follows: In section II, we introduce the interfaced piecewise DNA model and the single inverted dipole DNA model, which includes the long–range interactions and has into account their respective chain conformations. In section III, we investigate the characteristics of the linear modes of the first system, as they are important for determining possible behaviors of MBs when they collide with the interface. It is shown that the linear modes spectrum of this system is similar to the spectrum of a homogeneous DNA molecule with a single inverted dipole. In section IV, we generate numerically MBs which are launched towards the interface and we study the collision phenomena, exploring all the possible scenarios that appear when the translational kinetic energies of the MBs and the coupling parameters are varied. Thereafter, we show that the collisions of MBs in the case of a single inverted dipole DNA molecule, bring about similar effects to the interfaced case. Section V presents some interesting phenomena that can appear when moving breathers collide with a previous trapped breather. Finally, section VI summarizes our results and contains some conclusions. II. INTERFACED PIECEWISE DNA MODEL We consider a modification of the Peyrard-Bishop DNA model [2], with the addition of a dipole-dipole energy term. Thus, the Hamiltonian of the system can be written as [24] H= N  n=1 ⎛ ⎝ 1 2m˙u2 n+D(e−bun−1)2+1 2ε(un+1 −un)2+1 2μ m=n Jn,mumun⎞ ⎠(1) The term 1 2m˙u2 nrepresents the kinetic energy of the nucleotide of mass mat the nth site of the chain, and unis the variable representing the transverse stretching of the hydrogen bond connecting the base at the nth site. The Morse potential, i.e., D(e−bun−1)2, represents the interaction energy due to the hydrogen bonds within the base pairs, being Dthe well depth, which corresponds to the dissociation energy of a base pair, and b−1is related to the width of the well. The stacking energy is 1 2ε(un+1 −un)2,whereεis the stacking coupling constant. The last term of the Hamiltonian, i.e., 1 2μnm=nJn,mumun,is the long-range dipole–dipole interaction term, where μis the dipole–dipole coupling constant. The expression for this constant is μ=q2/4πε0d3[24], qbeing the charge transfer within the dipole and dthe distance between neighbouring base pairs, which is supposed to be constant. Finally, Jn,m is the dipole-dipole coupling factor given by Jn,m =αn,m |m−n|3,(2) where |m−n|is the normalized distance between base pairs, and αn,m takes the value 1 if the dipoles at nand m are parallel, and −1 if they are antiparallel. In order to keep the spatial homogeneity in a finite system with periodic boundary conditions [31], the number of particles affected by the long–range interaction must be limited. In consequence, if N is the number of base pairs, we suppose that the long–range interaction affects to (N−1)/2 neighboring base pairs, if N is odd, or (N−2)/2, if N is even, to each direction of a given site of the chain. In fact, we use periodic boundary conditions in a way that both the last dipole and the first one are pointing along the same direction, this implies that the interfaced piecewise chain must has the geometry of a M¨obius band, and in this way there exists only one interface in the chain. As an example, we can consider a small system with N= 8, with the dipoles located at n=+1,+2,+3,+4 pointing in one direction and the other ones located at n=−1,−2,−3,0 pointing in opposite direction. Then, the dipole–dipole coupling factors Jn,n+pcan be written in a matrix form as follows: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 010.1250 0.0370 0 0.0370 0.1250 1 10 10.1250 −0.0370 0 0.0370 0.1250 0.1250 1 0 1 −0.1250 −0.0370 0 0.0370 0.0370 0.1250 1 0 −1−0.1250 −0.0370 0 0−0.0370 −0.1250 −10 10.1250 0.0370 0.0370 0 −0.0370 −0.1250 1 0 1 0.1250 0.1250 0.0370 0 −0.0370 0.1250 1 0 1 10.1250 0.0370 0 0.0370 0.1250 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ We perform the following changes of variables: t→ω0t, un→bun,ε→ε mω2 0 ,μ→μ mω2 0 ,H→H 2D,D→1 2,(3) where ω0=2b2D/m is the frequency of an isolated oscillator in the harmonic limit. It becomes unity in the scaled system. With these changes, the dynamical equations become: ¨un+(e−un−e−2un)+ε(2un−un+1 −un−1)+μ m=n αn,m |m−n|3um=0.(4) III. ANALYSIS OF THE LINEAR MODES The study of the linear modes (phonons) of a nonlinear discrete system gives a necessary information for predicting some properties of the MBs with a given frequency. For example, the frequencies of MBs must be not too close to the phonon band. Otherwise, they emit a large amount of phonon radiation simultaneously to their movement. Furthermore, as it was shown in [32], the analysis of linear modes allows to predict the scenario of the collisions of MBs with inhomogeneities without the necessity of performing numerical simulations. The dynamical equations of the interfaced piecewise system can be linearized if we assume that the amplitudes of the oscillations are small enough. Then, the linearized dynamical equations are: 0 0.05 0.1 0.15 0.2 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 μ ω 0 0.05 0.1 0.15 0. 2 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 μ ω Top mode Bottom mode Top mode Bottom mode FIG. 2: Left: Frequencies of the linear modes varying the coupling parameter μfor ε=0.129. The upper line represents the top mode and the lower line represents the bottom mode. Right: Top and bottom mode frequencies with respect to μ. Crosses and dots represent, respectively, analytical and numerical values. ¨un+un+ε(2un−un+1 −un−1)+μ p=0 Jn,n+pun+p=0.(5) The inhomogeneity of these equations determines the existence of two localized modes, one of them is above (top mode) and the other one is below (bottom mode) the extended modes (phonon band). We have calculated numerically the frequencies of the linear modes for different values of the parameters εand μ. Fig. 2(left) shows the dependence of the frequency spectrum with respect to the parameter μfor the fixed value of ε=0.129, which is an appropriate value to obtain MBs with low dispersion [9]. In this figure, it can be observed the existence of localized modes whose origin relies in the fact that the interface acts as an inhomogeneity in the lattice. The localized modes are two, one of them is above (top mode) and the other one is below (bottom mode) the extended modes (phonon band). The generic profiles of the localized modes are shown in Fig. 3. Their vibration patterns depend on the parameter μ. In particular, we have observed that there exists a critical value μcsuch that for μ<μ c, the top mode has a zig-zag vibration pattern, while the bottom mode vibrates in phase. For μ>μ c, the vibration patterns are interchanged. As it is shown below, μc≈ε(e.g. for ε=0.129, μc=0.123). This fact is related to the existence of a competition between the attractive (stacking) and the repulsive (dipole–dipole) interaction. The first one is dominant for εμc and vice versa. An approximate explicit expression that gives the dependence of the localized modes frequencies with respect to the coupling constant μcan be obtained considering that the long-range interaction is limited to nearest neighbour base pairs. A study of infinity-range interaction could be performed, but it would be limited to frequencies close to the phonon band [5, 31]. This approximation is justified as the interaction decays rapidly. We have checked that the results are practically coincident except for values of μfar from the ones considered in our study. The localized modes can be found using Green’s lattice function methods [21, 33, 34]. However, the same results can be obtained in a more straightforward way using the following ansatz: un=a0r|n|exp(i ωt)forn>0 un=a1r|n−1|exp(i ωt)forn<1,(6) where r is a spatial decay factor. The sign of r indicates the vibration pattern of the localized mode. If r>0, the particles vibrate in phase, and if r<0, the mode has a zig-zag vibration pattern. We have supposed that the dipoles at n≥1 are all antiparallel to the dipoles at n≤0, then, there exists only two neighboring dipoles which have a different orientation, and the interface is located between n=0andn=1. Asthe localized modes are centered between both (see Fig. 3), we assume that: −20 −15 −10 −5 0 5 10 15 20 −1 −0.5 0 0.5 1 n un(0) Top mode −20 −15 −10 −5 0 5 10 15 20 −0.2 0 0.2 0.4 0.6 0.8 n un(0) Bottom mode FIG. 3: Generic profiles of the top and bottom localized modes for the coupling values μ=0.05 and ε=0.129. a0=−a1(corresponds to the top mode) a0=a1(corresponds to the bottom mode) .(7) With the application of these ans¨atze we obtain that the frequencies of the top and bottom modes are, respectively: ω2 top =ω2 0+2μ2+2ε2+εμ ε+μ;ω2 bottom =ω2 0+2με−μ ε+μ.(8) The decay factors for the top and bottom modes are, respectively: rtop =μ−ε μ+ε;rbottom =−rtop.(9) ¿From these expressions, it is easy to obtain that μc=ε. The discrepancies from this value and the numerical one rely on our assumption that long-range interactions are limited to nearest neighbours. The values of the parameters for which the breather frequency ωb=ωbottom are related to the scattering properties, see Ref. [32]. However, these values are excluded in our study, because as we require the existence of moving breathers, we need that μ<M and ε>M,withM≈0.2ω2.4 b[24]. In other words, ε>μis a necessary condition for the existence of moving breathers. The equality ωb=ωbottom implies that μ=[(2ε−δ)+√42+δ2−8εδ]/2, with δ≡ω2 b−ω2 0<0. Thus, it is straightforward to demonstrate that μmust be greater than in order to get breather resonance with the bottom mode frequency. This region is inaccessible for the moving breathers, as it is shown above. The analytical and numerical results that give the dependence of the frequencies of the localized modes with respect to μare represented in Fig. 2(right). As can be appreciated, there is an excellent agreement between the two approaches. The single inverted dipole system also has two linear localized modes, but now they are localized at the inverted dipole site. The frequencies of these modes are: ω2 inv =ω2 0+2ε±2ε2−4εμ −14μ2 ε2+6εμ −15μ2.(10) IV. EVOLUTION OF MOVING DISCRETE BREATHERS IN THE INTERFACED PIECEWISE DNA CHAIN. The dynamics of MBs can be strongly affected by the existence of the interface. It is expected that MBs differing only in their translational kinetic energies can produce different effects when they interact with the interface. In order to study these effects, we have simulated many collisions processes launching MBs towards the interface with different translational kinetic energies. MBs can be generated numerically by adding to the velocities of a static breather a perturbation of magnitude λ collinear to the pinning mode [15, 16]. The pinning mode is an anti–symmetric linear localized mode, which may appear in the set of linear perturbations of the system when the coupling is strong enough. This perturbation breaks down the translational symmetry of the system and the breather moves with a translational kinetic energy given by K=λ2/2. The time evolution of the breathers have been studied using a Calvo’s 5th order symplectic algorithm [35]. Recent quantum–mechanical calculations Ref. [36] establishes that in DNA the order of magnitude of the parameter μis 0.002, therefore, as we are interested in applications to DNA, we have limited our study to this order. MBs are not exact solutions of the dynamical equations and some phonon radiation is emitted simultaneously to the movement of breathers. This radiation is minimized if the breather frequency is not close to the phonon band. In all the simulations, the breather frequency has been fixed to the value of ωb=0.8, which is an appropriate value. For smaller values of ωb, the breather gets more and more sharply localized and highly pinned to the lattice, and for ωb0.67 its movement is not possible. Also, phonon radiation increases when the value of the translational kinetic energy increases and some effects due to the size of the system can appear. For that reason the calculations have been tested using different systems sizes, to make sure that the results are not modified by boundary effects (most of the simulations have been done with N>200). The values of the parameter λhave been limited to the interval (0.01,0.3), as for this interval the phonon radiation is small enough. Finally, the first group of simulations have been done with the value of the stacking coupling constant ε=0.129. We have performed a large number of simulations and, essentially, we have found that there exist only two different dynamical regimes, which can be characterized as follows: Trapping regime : If the translational kinetic energy of a MB is smaller than a critical value, i.e. λ<λ c, the breather is trapped at the discontinuity region, that is, only some particles around the interface remain oscillating. The critical translational kinetic energy depends on the values of the parameters μ,  and ωb. An example of the time evolution of a MB with λsmaller than the critical value, is observed in Fig. 4(a). However, not all the moving breather energy is trapped after the collision because some energy is transferred to the chain as phonon radiation. To see this, we have studied the evolution of the ”central energy”, i.e. the energy of some few particles around the interface, which is represented in Fig. 5(a). Before the collision the central energy is zero, then it increases quickly taking the value of the incident breather energy, and a small decay of this energy occurs which is due to phonon radiation. This can be appreciated in the central zoom figure of Fig. 5(a). The analysis of the Fourier spectrum of this trapped breather, carried out after the initial decay of the central energy and at an early stage of the evolution, shows a frequency of value 0.8, which is the internal frequency ωbof the launched breather, and a frequency of value 0.02, which corresponds to an oscillating movement of the breather around the interface. This type of movement can be appreciated in Fig. 4(a). The absolute values of the Fourier components calculated soon after the collision of the breather with the interface are represented in Fig. 6, and the central energy of this breather has the value 0.7975. The same calculations carried out after a time of 200 breather periods, shows a decrease of the Fourier component corresponding to ω=0.02, and a central energy of value 0.7943. A very slow–decaying process occurs and the oscillating trapped breather approaches to a static breather centered at the interface with frequency ωb=0.8. The energy of this existing static breather is 0.7646. Then, a necessary condition for the trapping effect is the existence of an inhomogeneity breather solution (IB) with the frequency of the MB and with a smaller energy. It consists of a static breather solution centered at the inhomogeneity to which the trapped breather decays. The trapping phenomena for small enough translational kinetic energies, and the oscillatory behavior as a whole of the trapped breather can be partially explained with the help of the collective coordinate method. In fact, the long–range interaction term of the Hamiltonian may be represented as follows: 1 2μ n m=n Jn,munum= n VEff nu2 n−1 4μ n m=n Jn,m(un−um)2,(11) where the first summation term in the rhs represents an effective on-site energy [37], with the on-site potential 500 1000 1500 2000 2500 3000 3500 (a) λ=0.05 500 1000 1500 2000 2500 3000 350 0 Time (b) λ=0.07 0 20 −20 n 0 20 −20 n FIG. 4: Two different regimes for the interaction of MBs with the interface. Displacement from the n-th equilibrium position versus time, for two different values of the translational kinetic energy: (a) λ<λ cand (b) λ>λ c,withμ=0.002, ε=0.129 and ωb=0.8. For these parameters values, λc=0.056. The interface is located between n=0andn=1. VEff n=1 2μ m=n Jn,m (12) and the second summation term represents an effective dispersion energy. In the case of a spatially homogeneous system the on-site potential does not depend on the index n, but for the interfaced piecewise system the potential VEff n,withμ=0.002, has the well profile shown in Fig. 7 (dots). It has a symmetric profile, and the depth is close to μ. The DNA chain with a point inhomogeneity given by a single inverted–dipole has an effective potential with the well profile shown in Fig. 7 (open circles). It is also a symmetric profile, and the depth is close to 2μ. The set of dynamical equations derived from the Hamiltonian (1) can be transformed, in the continuum limit, into the corresponding continuous partial differential equation. Within this context the collective coordinate method provides a good way to analyze the influence of a perturbation on a soliton (in our case we can consider the long range interaction term as a perturbation). The idea is the same as using the center of mass to analyze the behavior of a system of particles. The method introduces the center of the breather ”x” as a description variable, and the on-site potential affects as a potential energy of the form V(x)=1 2μ n VEff nu2 n−x.(13) If the breather is well localized, as occurs in our cases, V(x)≈VEff x. In other words, the translational movement of the breather center ”x” is affected by a symmetric potential well with the same form as the shown, for each case, in Fig. 7. If the translational kinetic energy of the breather is small enough, the phonon radiation emitted as it passes through the potential well prevents the transmission of the breather and it is trapped with an oscillating movement like a particle in a potential well. The remaining terms in Eq. (11) also contribute to the center of breather motion but they are not so significant. In fact, in the continuum approximation they lead to a space-dependent effective mass of excitation. 0 2000 4000 6000 0 0.2 0.4 0.6 0.8 (a) Time Central energy 2000 4000 6000 0.78 0.79 0.8 0.81 (zoom of (a)) 0 2000 400 0 0 0.2 0.4 0.6 0.8 (b) Time Central energy FIG. 5: Time evolution of the central energy corresponding to the collision processes: (a) in Fig. 4(a); (b) in Fig. 4(b). The central figure is a zoom of the left one 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1. 8 0 0.05 0.1 0.15 0.2 0.25 0.3 ω FIG. 6: Absolute values of the Fourier components of the trapped breather shown in Fig. 4(a) calculated soon after the collision Transmission regime : If the translational kinetic energy of a MB is bigger than the critical value, i.e. λ>λ c, the breather crosses the interface and continues its movement along the chain with smaller translational kinetic energy. In other words, the breather is transmitted as a consequence of the interaction with the interface, and a small part of its energy is transferred to the chain as phonon radiation. An example of a transmitted MB can be seen in Fig. 4(b), and the evolution of the corresponding central energy is shown in Fig. 5(b). The same qualitative results have been found in the case of the parallel-oriented dipole chain with a single inverted– dipole. In this case, the trapping site is centered at the inverted dipole. Both this system and the interfaced piecewise system have a symmetric effective on–site potential well due to the dipole coupling. Also, both of them have two linear localized modes. The nonexistence of an effective on–site potential