Classical and quantum nonlinear localized excitations in discrete systems

arXiv:nlin/0504021v2 [nlin.PS] 14 Apr 2005 Classical and quantum nonlinear localized excitations in discrete systems FR Romero,∗JFR Archilla, F Palmero, B S´anchez-Rey, A ´ Alvarez, J Cuevas, and JM Romero Grupo de F´ısica No Lineal. Departamento de F´ısica at´ ømica, molecular y nuclear y Departamento de F´ısica Aplicada I. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012-Sevilla (Spain) (Dated: April 10, 2005) Discrete breathers, or intrinsic localized modes, are spatially localized, time–periodic, nonlinear excitations that can exist and propagate in systems of coupled dynamical units. Recently, some experiments show the sighting of a form of discrete breather that exist at the atomic scale in a magnetic solid. Other observations of breathers refer to systems such as Josephson–junction arrays, photonic crystals and optical-switching waveguide arrays. All these observations underscore their importance in physical phenomena at all scales. The authors review some of their latest theoretical contributions in the field of classical and quantum breathers, with possible applications to these widely different physical systems and to many other such as DNA, proteins, quantum dots, quantum computing, etc. PACS numbers: 63.20.Pw, 63.20.Ry, 63.50.+x, 66.90.+r, 87.10.+e Keywords: Discrete breathers, Nonlinear excitations, Nonlinear dynamics, Klein–Gordon lattices, Polarons, DNA. I. INTRODUCTION Nonlinear physics of discrete systems has attained an enormous development in the previous years. A landmark is the prove of the existence of spatially localized, time–periodics oscillations in discrete nonlinear systems, by MacKay and Aubry [46]. They are now termed discrete breathers (DB), nonlinear localized excitations, or intrinsic localized modes, and ever since there are many experimental data and theoretical results published about these important new phenomenon in physics. They appear in lattices of oscillators interacting through nonlinear forces. The central class of system for which the concept of DB was developed is classical, autonomous spatially discrete Hamiltonian or reversible systems. A type of systems of this class is called Klein–Gordon lattices. They are Hamiltonian systems consisting of one degree of freedom anharmonic Hamiltonian oscillators coupled weakly in a lattice, that is, the frequency of oscillation of each oscillator varies non–trivially with the amplitude. Most studies on breathers have been done utilizing this type of models, with applications to biomolecules, such as DNA, solids, etc. Other classes of systems where breathers occur are: autonomous forced damped systems, and time–periodically forced systems. Considering applications to molecular crystals, quantum effects must be taken into account and the concept of quantum discrete breather should be developed. This review summarize some of the latest theoretical contributions in the field of localized nonlinear excitations carried out by the Nonlinear Physics Group of the Sevilla University. The structure of this work has the following scheme: •Section II: Breathers in Klein–Gordon lattices with competing short and long–range interactions –A: DNA model with dipole–dipole coupling –B: Numerical methods for obtaining breathers and determining their stability –C: Bifurcations •Section III: Moving breathers in homogeneous DNA chains –A: Moving breathers in DNA chains without curvature –B: Moving breathers in bent DNA chains –C: Parameters values of DNA •Section IV: Interaction of moving breathers with mass impurities ∗Electronic address: [email protected] 2 •Section V: Interaction of moving breathers with vacancies •Section VI: Small and large amplitude breathers in Fermi-Pasta-Ulam lattices •Section VII: Dark breathers in Klein–Gordon lattices •Section VIII: Polarons in biomolecules •Section IX: Quantum breathers –A: Quantum breathers in a translational invariant lattice –B: Trapping in lattices with broken translational symmetry •Section X: Thermal evolution of enzyme-created oscillating bubbles in DNA II. BREATHERS IN KLEIN–GORDON LATTICES WITH COMPETING SHORT AND LONG–RANGE INTERACTIONS Most of the studies about breathers in Klein–Gordon lattices refer to models which include only short–range interacting forces between the oscillators. This section reviews some of the results relative to breathers solutions in Klein–Gordon chains which include also long–range interactions. A one–dimensional Klein–Gordon lattice is a chain of identical oscillators with an on–site potential V, each one of them coupled with its nearest neighbours through a parameter C≪1. The Hamiltonian of a chain of Noscillators is: H= N X n=1 1 2˙u2 n+V(un) + 1 2C(un−un−1)2,(1) where unrepresents the coordinates of the oscillators referred to their equilibrium positions, and V(un) represents the on–site potential. In addition to the short–range interaction coupling, other types of forces can also exist. For example, if each oscillator has an intrinsic dipole moment, there exist also long–range forces due to the dipole–dipole coupling, and consequently the Hamiltonian must be modified. The existence of these two competing forces could alter substantially the properties of the breathers solutions. The study is presented in the next subsections, selecting a DNA model which has also into account the existence of dipole–dipole coupling. A. DNA model with dipole–dipole coupling In DNA there exists different kinds of interactions between the main atomic groups. One of them is the stacking interaction between neighbour 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 [35, 51]. The dipole–dipole interactions between different base pairs become critical 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, 6, 23]. The Peyrard-Bishop DNA model [53] was initially introduced considering only the stacking interactions between base pairs. It can be modified by adding an energy term that takes into account the dipole–dipole forces. The Hamiltonian can be written as: H= N X n=1   1 2m˙u2 n+V(un) + 1 2C(un+1 −un)2+1 2X p6=n J |p|un+pun .(2) 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 bases. The on–site potential is the Morse potential, i.e., D(e−bun−1)2, represents the interaction energy due to the hydrogen bonds within the base 3 pairs, where Dis the well depth and represents the dissociation energy of a base pair, and b−1is related to the width of the well. The stacking energy is 1 2C(un+1 −un)2, where Cis the stacking coupling constant [53] . The last term of the Hamiltonian is the long-range dipole–dipole interaction term, where J=q2/4πε0d3is the dipole long-range interaction coupling constant [21], with qbeing the charge transfer due to the formation of the hydrogen bonds, and dthe distance between base pairs, which is supposed to be constant. It is possible to consider D= 1/2, b= 1, m= 1 without loss of generality. B. Numerical methods for obtaining breathers and determining their stability A static breather (SB) can be obtained by solving the full dynamical equations, which can be obtained applying Hamilton’s equations to (2). This aim can be achieved using common methods based on the anticontinuous limit [8, 46, 47, 48]. The implementation of these methods basically consists in calculating the orbit of an isolated oscillator at fixed frequency ωb, and using this solution as a seed to solve the complete dynamical equations by means of a Newton–Raphson continuation method. A SB can be obtained for a given values of the dipole–dipole coupling parameter, J, and the stacking coupling parameter, C. This solution can be continued by varying Cand maintaining Jconstant or vice versa. The linear stability of a breather can be studied by performing a Floquet analysis [10]. To this end, it is necessary to consider the evolution of a perturbation ξthrough the dynamical equations: ¨ ξn+V′′(un)ξn+C(2ξn−ξn+1 −ξn−1) + X p6=n J pξn+p= 0.(3) The functions Ω(0) ≡({ξn(0)},{˙ ξn(0)}) must be integrated until t=T, giving Ω(T)≡({ξn(T)},{˙ ξn(T)}). There exists a matrix of dimension 2Ncalled the monodromy, which is defined by Ω(T) = MΩ(0). The spectrum of this matrix has the following property: if λis an eigenvalue, then λ∗, 1/λ and 1/λ∗are also eigenvalues. Therefore, stability implies that all the eigenvalues have modulus unity [20]. The procedure of continuation of SBs can be done as long as an integer multiple of their frequency does not resonate with any of the frequencies of the linear modes (or phonons). This resonance provides an analytical expression for the upper boundary delimiting the existence of SBs: C= (ωb/ωo)2−1/4 + 3ζ(3)J/8. C. Bifurcations The study of the bifurcation loci of SBs is interesting for three different reasons: a) It gives the range of existence of the SBs; b) It gives the regions in the parameter space where the breathers are stable (and, therefore, physically observable in real systems) or unstable; c) Finally, but of paramount importance, it gives the values of the parameters where the breathers are movable. Two types of bifurcations appear as Cand Jare varying: 1) Stability bifurcations: they occur when a breather changes its stability. The existence of this type of bifurcation is a necessary condition for the existence of movable breathers (see Section III). 2) Breather extinctions: they occur when a breather is not continuable any longer. When only stacking interaction is considered, only stability bifurcations occur and moving breathers are possible. However, with only dipole-dipole interactions there are no stability bifurcations and, therefore, moving breathers do not exist. When both interactions exist simultaneously, the analysis of 1-site (site-centered) and 2-site (bond-centered) breathers, permits to establish the following staments: A) 1–site breathers can be moved only for J < Jc1. B) 2–site breathers are movable for J < Jc2being Jc2> Jc1. They are also movable for J∈[Jc2, Jo). Therefore, Jo establishes the maximum value of J(and also the minimum value of C) for which breathers can be movable, i.e., there are no moving breathers for C < Joand for J > Jo. This is the reason why Jois called the mobility limit [21]. The bifurcation loci for the stability bifurcations are shown in Fig. 1(a). Also, in Fig. 1(b) the dependence of the mobility limit with respect to the breather frequency is shown. This curve fits very well to the relation Jo=Aωr b, where A=0.1921 ±0.0002 and r=2.377 ±0.005. III. MOVING BREATHERS IN HOMOGENEOUS DNA CHAINS This section is dedicated to the study of travelling breathers, or moving breathers, in DNA chains described by the Peyrard–Bishop modified model. The first subsection presents this study for straight line DNA chains, or chains 4 (a) (b) 0 0.03 0.06 0.09 0.12 0.11 0.112 0.114 0.116 0.118 0.12 0.122 0.124 0.126 0.128 0.13 First exchange of stability J C Jo Jc1 1−site 2−site 0 0.03 0.06 0.09 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 Second exchange of stability J C Jc2 Jc1 1−site 2−site 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.08 0.1 0.12 0.14 0.16 0.18 ωb/ωo Jo FIG. 1: (a) Bifurcation loci for the first and the second exchange of stability in 1-site and 2-site breathers. The values corresponding to ωb= 0.8 are: Jc1= 0.076, Jc2= 0.085, Jo= 0.11263. (b) Dependence of Jowith respect to the breather frequency. without curvature. The next subsection considers DNA chains which have been bent and adapted to a parabola. A. Moving breathers in DNA chains without curvature Contrary to static breathers, moving breathers are not exact solutions of the dynamical equations of the model and they have a finite life due to the effects of phonon radiation. A moving breather is obtained by ‘kicking’ a static breather, i.e., perturbing its velocity components. However, not all breathers can be moved, and not every kick can move a breather. Therefore, it is important to know the conditions that a breather must fulfill in order to be movable and what the characteristics of the perturbations are. As indicated in [11, 17, 19], the following steps must be performed in order to obtain a moving breather: 1) To look for the existence of the two complementary stability bifurcations for the 1–site breather and the 2–site breather. Their bifurcation loci must have a region where they are fairly close. The static breather that can be moved should be obtained for values of the parameters close enough to these bifurcation loci. However, as shown in [21] it is only necessary that the static breather is “quasi-stable”, i.e., not very stable or not very unstable; 2) To perturb the breather with the velocity components of the marginal mode at the neighbouring bifurcation. Time evolution of a moving breather is shown in Fig. 2. An useful concept used for describing the breather dynamics is its effective mass [11, 17]. It is a measure of the breather inertia to external forces. The breather velocity must be perturbed in a direction colinear to the marginal mode in order to obtain moving breathers. This mode is the responsible for the stability bifurcation. If the normalized marginal mode is V, the perturbation added to the breather velocities, which are zero at t= 0, is given by λV, and λ is the magnitude of the perturbation. Thus, the kinetic energy added to the breather by the initial kick is 1/2λ2. It is found that the resulting translational velocity of the breather, v, is proportional to λ. Then, the concept of effective mass can be defined through the relation m∗v2/2 = λ2/2. Therefore, m∗= (λ/v)2. Consequently, moving breathers can be considered as a quasi-particle with mass m∗. The effective mass is a quantitative measure of the breather mobility. Larger mass indicates smaller mobility. Numerical studies show that, for high values of the dipole–dipole coupling, the predicted range of existence of moving breathers decreases and, in some cases, it is impossible for the breather to be moved even in the neighbourhood of the stability bifurcations. Another important result is that the dipole–dipole interaction affects to the mobility of the breathers. The study of the dependence of the breather effective mass, shows that the mobility decreases when the intensity of the dipole–dipole coupling increases (see Fig. 3). 5 FIG. 2: Evolution of a moving breather for C= 0.1244, J= 0.06 and an initial ‘kick’ λ= 0.10. The breather involves basically three sites that oscillate in phase. B. Moving breathers in bent DNA chains Suppose a DNA chain lying in a plane which 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) [5]. All the dipole moments are perpendicular to the plane and parallel among them. Thus, the modified Hamiltonian of Eq. (2) is: H= N X n=1   1 2m˙u2 n+V(un) + 1 2C(un+1 −un)2+1 2X p6=n J |~rn−~rp|3un+pun ,(4) where ~rn= (xn, yn), and it is assumed that the chain is inextensible, so that the distance between neighbouring sites remains constant: |~rn−~rn+1| ≡ d. Fig. 4(a) illustrates the evolution of the energy centre [21] 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. Fig. 4(b) shows that the critical energy increases monotonically with the curvature. Chain bending acts as a hindrance for the movement of breathers. This hindrance resembles to the experimented by a particle moving 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, i.e., the turning points for different values of E. Furthermore, if the breather has a constant effective mass m∗, this potential barrier can be obtained using the expression: Eb(x) = 1 2λ2[1 −(v(x)/vo)2],(5) where v(x) is the translational velocity and vois its value at t= 0. Fig. 5 show that there exists a good agreement between the barriers 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 6 0.2 0.3 0.4 20 25 30 35 40 45 J=0 Breather mass 0.2 0.3 0.4 45 50 55 60 65 70 75 80 85 90 95 J=0.03 C0.2 0.3 0.4 80 90 100 110 120 130 140 150 160 170 J=0.05 FIG. 3: Breather mass versus stacking coupling parameter Cfor dipole–dipole coupling parameter J= 0 (left), J= 0.03 (center) and J= 0.05 (right). Note that the maximum value of the mass occurs at the first bifurcation, and that the breather mass for J= 0.05 is always greater than the mass for J= 0. Note also the different scales at the Y-axis. (a) (b) 0 20 40 60 80 100 120 140 160 −15 −10 −5 0 5 10 XE Periods vo=0.0311 vo=0.0260 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. 4: (a)Evolution of the breather energy centre (XE) for two different initial velocities. The curvature is κ= 2. (b) Critical kinetic energy Ecversus curvature κ. 7 −1 −0.5 0 0.5 1 6 7 8 9 10 11 12 x 10−3 Eb XE FIG. 5: Potential barrier calculated finding the turning points (solid line) and using Eq. (5) 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. to the discreteness of the system [19]. This result confirms that, in this case, a moving breather behaves as a particle of constant effective mass m∗. It is worth remarking that a similar study has been performed in bent alpha–helix proteins [7]. C. Parameters values of DNA The results presented are valid for all chains with both short range and long range interactions given by equations (2) and (4). When they model a DNA chain, the following parameter values should be used [25]: D= 0.04 eV, b= 4.45 ˚ A−1,d= 3.4˚ A, J= 0.0031 eV/˚ A2,C= (0.01,10) eV/˚ A2,m= 300 amu and the time unit is 0.88 ps. For this set of values, the minimal kinetic energy for a moving breather to cross the bending point is small, and, in consequence, it seems plausible that most MBs will have enough energy to pass through bending points, although the ones with low energy will be reflected. IV. INTERACTION OF MOVING BREATHERS WITH MASS IMPURITIES Sometimes a chain can have a single point inhomogeneity, that is, the chain is homogeneous except at a single site where,for example, an impurity or a different type of particle is located. Moving breathers can be generated far apart the impurity and be launched towards it. This section presents some results relative to the effects of the collisions of moving breathers with an impurity in a Klein–Gordon chain. 8 (a) (b) FIG. 6: (a) Frequencies of the linear modes versus the parameter α. At α=αres and α=αc, two different bifurcations occur, being the first one due to the resonance between the impurity mode and the breather. (b) Different regimes in the interaction of a moving breather with an impurity. Consider a Klein–Gordon chain with nearest neighbours attractive interactions, with Hamiltonian given by: H= N X n=1 1 2˙u2 n+Vn(un) + 1 2C(un−un−1)2,(6) where Vn(un) = Dn(e−un−1)2is a substrate potential at the n-th site. The inhomogeneity is introduced assuming a different well depth at a single site, i.e., Dn=Do(1 + αδn,0), where α∈[−1,∞) is a parameter which tunes the magnitude of the inhomogeneity. The particle located at n= 0 is an impurity. The dynamical equations can be linearized if the amplitudes of the oscillations are small enough. These equations have N−1 non-localized solutions (linear extended modes) and one localized solution, (linear impurity mode). Their frequencies, ωEand ωL, are given, respectively, by: ω(q, α) = rω2 o+ 4Csin2q(α) 2, ω2 L=ω2 o+ 2C+ sign(α)pα2ω4 o+ 4C2,(7) where q∈(0, π] if α < 0 and q∈[0, π) if α > 0. Fig. 6(a) shows the dependence on α. Numerical simulations show that, depending of the critical values of the parameter α, there exist four different regimes in the moving breather–impurity interaction [24]: 1) Barrier. The impurity acts as a potential barrier. It occurs either with α > 0 or α∈(−1, α1) with α1<0. If α&0, the breather can pass through the impurity provided the translational velocity is high enough [23]. Excitation. The impurity is excited and the breather is reflected. This case appears for α∈(α1, α2), an example can be seen in Fig. 7(a). 3) Trapping. The breather is trapped by the impurity. It occurs in the interval α∈(α2, α3). When the moving breather is close to the impurity, it becomes trapped while its center oscillates between the neighbouring sites, as Fig. 7(b) shows. 4) Well. The impurity acts as a potential well. It occurs for α∈(α3,0) and consists of an acceleration of the breather as it approaches to the impurity, and a deceleration after the breather has passed through the impurity. It is observed that the breather bifurcates with the zero solution at α=αres. That is, for αsmaller than this value, no impurity breather exists. At α=αres, the frequency of the impurity mode coincides with the moving breather frequency, i.e., in (7), ωL=ωb. The scenario for the trapped breathers when α < 0 is the following: the impurity mode has q= 0, and also all the particles of the impurity breather vibrate in phase; this vibration pattern indicates that the impurity breather bifurcates from the impurity mode and it will be the only localized mode that exists when the impurity is excited for α > αres. Thus, when the moving breather reaches the impurity, it can excite the impurity mode. For α < αres, the moving breather is always reflected. In addition, the impurity breather does not exist. Therefore, there might be a connection between both facts, i.e., the existence of the impurity breather seems to be a necessary condition in order 9 (a) (b) FIG. 7: (a) Interaction of a breather with an impurity for α=−0.52, which corresponds to the impurity excitation case. (b) Evolution of the moving breather for α=−0.3, which corresponds to the trapping case. The moving breather becomes trapped by the impurity; afterwards, the breather emits phonon radiation and its energy centre oscillates between the sites adjacent to the impurity. to obtain a trapped breather. If α > 0, the impurity mode has q=πbut the impurity breather’s sites vibrate again in phase, that is, the impurity breather does not bifurcate from the impurity mode. There are two different localized excitations: the tails of the (linear) impurity mode and the impurity breather. Thus, if the moving breather reaches the impurity site, it will excite these localized excitations. Therefore, we conjecture that the existence of both linear localized entities at the same time may be the reason why the impurity is unable to trap the breather when α > 0. All these facts allows the formulation of the following trapping hypothesis: The existence of an impurity breather for a given value of αis a necessary condition for the existence of trapped breathers. However, if there exists an impurity mode with a vibration pattern different from the impurity breather one’s, the trapped breather does not to exist. V. INTERACTION OF MOVING BREATHERS WITH VACANCIES Another type of local inhomogeneity that can exists in a homogeneous chain is a vacancy, that is, the absence of a particle at a single site. A moving breather can be launched towards the vacancy and the effects of the collision should be analyzed. Consider a Hamiltonian Frenkel–Kontorova model with an anharmonic interaction potential [16]: H=X n 1 2˙x2 n+1 4π2[1 −cos(2πx)] + C1 2b2[exp(−b(xn+1 −xn−1)) −1]2,(8) where {xn}are the absolute coordinates of the particles. The choice of a periodic potential makes easy to represent a vacancy. Thus, if the vacancy is located at site nv(see Fig. 8), the displacements of the particles with respect to their equilibrium positions are:    un=xn−nL n < nv un=xn−(n+ 1)L n > nv. (9) It is possible to generate a moving breather travelling towards the vacancy at the site nv, and simulate the collision with this type of point inhomogeneity. The initial perturbation, {Vn}has been chosen as V=λ(. . . , 0,−1/√2,0,1/√2,0,...), where the nonzero values correspond to the neighboring sites of the initial center of the breather. 16 −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Imaginary part Real part −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 Imaginary part Real part FIG. 13: Evolution of the Floquet multipliers with cubic on–site potential and attractive interaction for: (left) a breather with ε= 0.1; (right) a dark breather with ε= 0.004. The breather frequency is in both cases ωb= 0.8. −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 Imaginary part Real part −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 Imaginary part Real part FIG. 14: Floquet eigenvalues for a Morse on–site potential and repulsive coupling. Left: ε= 0.024, the system is still stable. Right: ε= 0.035, the system becomes unstable due to oscillatory and subharmonic bifurcations. VIII. POLARONS IN BIOMOLECULES The study of charge transport (CT) in biomolecules, and particularly in DNA, is of utmost importance. Electronic transport plays a very important role in many biological processes, such as DNA repair after damage by radiation or chemical agents, photosynthesis, nevous conduction, etc. Also, because the quasi–one–dimensional structure of DNA and other biomolecules could be used to design electronic devices based on biomaterials (for a detailed bibliographical information, see i.e. [38]). When an electron moves along the double strand of DNA it is accompanied by a local deformation of the molecule, forming a polaron or an electron–vibron breather. The polaronic character of the CT along DNA has been studied by means of different models, some of then based on the Peyrard–Bishop DNA model [53], in which the only variables are the distances between bases within each base pair. have permitted to show that the polaronic charge transport 17 0 0.5 1 |λ| −3 −2 −1 0 1 2 3 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 θ E FIG. 15: Band structure for a quartic hard on–site potential with attractive interaction at ε= 0.041 and ωb= 1.2. −10 −5 0 5 10 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Sites −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 Imaginary part Real part FIG. 16: Left: Floquet multipliers with a quartic hard on–site potential and attractive interaction at ε= 0.041 and ωb= 1.2. A harmonic and a small oscillatory instabilities appear. Right: the velocities components (the position ones are zero) of the eigenvalue corresponding to the harmonic instability. along DNA can be mediated by the coupling of the charge carrying unit with the hydrogen–bond deformations, In particular, it has been shown in some simple, but powerful models based on the proposed by Peyrard–Bishop [53], that the polaronic CRT along DNA can be mediated by coupling of the charge carrying unit with the hydrogen– bond deformations, and that this transport survives to a small amount of parametric and structural disorder [37], or by means of the coupling of this charge unit with the twist deformation of neighbor base pairs [52]. In this last case, the moving polaron is slower than in the previous one, but it is more robust under the introduction of parametric disorder. These differents regimes apperars for different values of a parameter α, the couples the transference integral with the deformations of the hydrogen bonds, a parameter which at present there is not any realible experimental data. Consider a three-dimensional semi–classical tight–binding model for synthetically produced DNA polymers built up 18 rn l0+dn n−1 nθ0+φn θ0 R0 l0 FIG. 17: Sketch of the structure of the DNA model. The bases are represented by bullets. from single type of base pairs, i.e. either poly(dG)-poly(dC) or poly(dA)-poly(dT) DNA polymers [54]. Utilizing a nonlinear approach based on the concept of breather and polaron solutions it is possible to explore if the conductivity depends on the type of the DNA polymer. If they have quantitatively distinct transpor properties or not might be of interest for the design of synthetic molecular wires. Moreover, it has been possible to ameliorate preceding studies of electronic transfer in DNA [37], [36] in the sense that, instead of adjusting the coupling parameters, plausible estimates for them, derived with the help of quantum-chemical methods, has been used. The model for CT in DNA is based on the finding that the charge migration process is dominantly influenced by the transverse vibrations of the bases relative to each other in radial direction within a base pair plane. The impact of other vibrational degrees of freedom are expected to be negligible with respect to charge transport in DNA. Then, the motion can be viewed as confined to the base pair planes [61]. The Hamiltonian for CT in along a strand in DNA has two parts H=Hel +Hvib, with Hel describing the CT over the base pairs and Hvib the dynamics of radial vibrations of the base pairs or the vibronic part. A sketch of the structure of the DNA model is shown in Fig. 17. Supposing that vibrational dynamics can be described classically, it is easy to formulate the dynamical equations of the system (a detailed description of procedure can be found in [52]).It is possible to use some typical parameter values for DNA molecules and compute plausible values for the electron–coupling strengths as a result of quantum–chemical computational procedure [38]. The nonlinear interplay between the electronic and the vibrational degrees of freedom, makes possible the formation of polaronic electron–vibration compounds. Also, there exist some stationary solutions that can be calculated with the help of the nonlinear map approach explained in detail in [42]. In general, the polarons are of fairly large extension (width). Regardless of the DNA polymer type, the electronic wave function is localized at a lattice site and the envelope of the amplitudes decays monotonically and exponentially as distance grows from this central site (base pair). However, the electronic wave function of the poly(dG)-poly(dC) DNA polymer is stronger localized than the one of its poly(dA)-poly(dT) counterpart. As polarons solutions are fairly large extended (the half-width involves .100 lattice sites) it is plausible to think that the can be mobile. This polaron motion can be activated using the discrete gradient method to obtain suitable initial perturbations of the momentum coordinates which initiate coherent motion of the polaron compound. The electronic occupation probability is defined as ¯n(t) = Pnn|cn(t)|2, where the index ndenotes the site of the n−th base on a strand and cndetermines the probability to find the electron (charge) residing at this site. Results relative to the time evolution of the first momentum of the electronic occupation probability are shown in Fig. 18. Conductivity in synthetically produced DNA molecules depends on the type of the single base pair 19 of which the polymer is built of. While a polaron-like mechanism, relying on the nonlinear coupling between the electron amplitude and radial vibrations of the base pairs, is responsible for long-range and stable electronic transfer in (dG)-(dC) DNA polymers, the conductivity is comparatively weaker in the case of (dA)-(dT) DNA polymers. Especially when it comes to designing synthetic molecular wires these findings might be of interest. In fact, recent experiments suggest that electronic transfer through DNA molecules proceeds by polaron hopping [44]. Furthermore, these results comply with the findings of experiments which show also that poly(dG)-poly(dC) DNA polymers forms a better conductor than their poly(dA)-poly(dT) counterparts. FIG. 18: Time evolution of the first momentum of the electronic occupation probability. Full (dashed) line: poly(dA)-poly(dT) (poly(dG)-poly(dC)) DNA polymer. IX. QUANTUM BREATHERS At present, the phenomenon of localization of energy due to nonlinearity in discrete classical lattices is relatively well understood, but the understanding of the quantum equivalence of discrete breathers is very limited. There exist some theoretical results [28, 58] and some experimental observations of these states in different quantum systems, as mixtures of 4–methyl–pyridine [33], in Cu benzoate [9], and in doped alkalihalides [56]. Also, the study of discrete breathers in quantum lattices supporting a small number of quanta is not only interesting theoretically, this knowledge may be relevant to studies of quantum dots and quantum computing [45], and for studies of Bose–Einstein condensates trapped in optical traps [2]. This section focus the attention in one–dimensional lattices with a small number of quanta described by the the quantum version of the discrete nonlinear Schr¨odinger equation (QDNLS), also known as the Hubbard model. This is a particularly simple model for a lattice of coupled anharmonic oscillators, which has been used to describe the dynamics of a great variety of systems [57]. We present the results corresponding to a periodic lattice with fsites containing bosons [30]. Many of these results can be extended to a great variety of systems, for example, similar results have been obtained with a periodic lattice containing fermions, described by an attractive fermionic Hubbard model (FH) with two kinds of particles with opposite spins. This is a model of interest in connection with the theory of high-Tcsuperconductivity, and it can be used to describe bound states of electron and holes in some nanostructures as nanorings (excitons) [29]. The QDNLS Hamiltonian of the quantum system is ˆ H=− f X j=1 1 2γjb† jb† jbjbj+ǫjb† j(bj−1+bj+1),(24) 20 where b† jand bjare standard bosonic operators, γj/ǫjis the ratio of anharmonicity to nearest neighbor hopping energy, and periodic boundary conditions are imposed. As first step, only short range interactions are considered given by the hopping coefficients ǫj. This Hamiltonian conmutes with the number operator N=Pf j=1 b† jbj, and can be block-diagonalised easily when the number of bosons is small enough. Numerically exact solutions can be found restricting the study to small lattices with a small number of quanta. The simplest nontrivial case is for N= 2, where bound states corresponds to bound two-vibron states, as has been observed experimentally in several systems [39]. These results have been extended to more complicated situations, noting that many of them are valid for larger values of N. A. Quantum breathers in a translational invariant lattice In a homogeneous quantum lattice the parameters γjand ǫjare independent of j, i.e. γj=γand ǫj=ǫ. With periodic boundary conditions, it is possible to block–diagonalize the Hamiltonian operator using eigenfunctions of the translation operator ˆ T, defined as ˆ Tb† j=b† j+1 ˆ T. In each block, the eigenfunctions have a fixed value of the momentum k, with τ= exp(ik) being an eigenvalue of the translation operator [57]. In this way, it is possible to calculate the dispersion relation E(k) with a minimal computational effort. In general, as shown in 19, if parameter γis high enough, there exists an isolated ground state eigenvalue for each k which corresponds to a quantum breather. In this state, there is a high probability of finding the quanta on the same site, but due to the translational invariance of the system, there exists an equal probability of finding these quanta at any site of the system. In these cases, some analytical expressions can be obtained in some asymptotic limits [57]. −3 −2 −1 0 1 2 3 −6 −5 −4 −3 −2 −1 0 1 2 3 4 k E FIG. 19: Eigenvalues E(k). N= 2, f= 19 and γ= 4. For example, using the number–state–basis |ψn>= [na 1, na 2, ..., na f], where na irepresents the number of quanta at site i, in the case N= 2 and k= 0 the (unnormalized) ground state is given by |Ψi= [20 ...0] + [020 ...0] + ···+ [0 ...02] + O(γ−1). The coefficients aiof the first fterms are equal to unity and the rest are O(γ−1). 21 B. Trapping in lattices with broken translational symmetry When the translational invariance of a lattice is broken, the Hamiltonian operator does not commute with the translation operator and breathers can be localized around a particular point of the lattice. Perhaps the simplest way to do it is to consider non–flux boundary conditions, in order to simulate a finete–size chain. In this case, the ground state becomes localized around the middle of the chain but, when fis high enough, this effect vanishes. The existence of local inhomogeneities or impurities can also break the translational symmetry of the system, and this can be modeled by making some coefficients dependent on the site. This can occurs in system with twisted or bent geometries, as in models of globular proteins [31] or photonic crystals In these cases, the geometry effects can be considered introducing a long–range hopping term in the Hamiltonian operator. The breaking of translational symmetry of the system localizes the breather state around a particular site of the chain, as shown in Fig. 20. Note that the localization phenomena due to random variation of lattice parameters have been studied in harmonic models since the work of Anderson [4] but, in all cases, and due to anharmonic effects, there exist new localization effects, and, when it is present, the anharmonicity enhances the Anderson–like effect. 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 n |an|2 [4] 2 4 6 8 10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 n [31] FIG. 20: Square wave function amplitudes corresponding to localized components of the ground state. f= 11, N= 4 and γ= 2. Point impurity at the site ℓ= 6. (· · ·) homogeneous chain case, γim = 2, (- - -) γim = 2.1 , (—) γim = 2.5. X. THERMAL EVOLUTION OF ENZYME-CREATED OSCILLATING BUBBLES IN DNA This last section reviews some fundamental researches on the creation and subsequent evolution of breathers in DNA chains. For a double–stranded DNA molecule to replicate, the two strands of the double helix must be separated from each other, at least locally, with the breaking up of the hydrogen bonds. For long double–stranded DNA molecules, the rate of spontaneous strand separation is very low under physiological condition. Specific enzimes belonging to the large family of nucleic acid helicases (see [61]) (there are more than sixty different types), utilize the energy of ATP hydrolysis to power strand separation. The opening of a selected region of the twisted (Watson-Crick) DNA double helix, begins with a partial unwinding at an area called the replication fork and is observed as a bubble. Helicases are ubiquitous and versatile enzymes that, in conjunction with other components of the macromolecular machines, 22 carry out important biological processes such as DNA replication; DNA recombination; nucleotide excision repair (i.e. UV damage); RNA editing and splicing; the transfer of a single-stranded nucleic acid to other nucleic acid or a protein or its release into solution [62] and formation of T-loop telomeres [63]. Nucleic acid helicases have several structural varieties (monomeric, dimeric, trimeric, tetrameric and closed hexameric), but all of them use the hydrolysis of nucleoside triphosphate (NTP) to nucleoside diphosphate (NDP) as the preferred source of energy. Helicases have also a huge interest from a point of view of clinical disease. Actually, there are known up to five human monogenic hereditary disorders in which the mutated protein responsible for the disease carries an aberrant helicase activity: cockayne ’s syndrome, xeroderma pigmentosum, trichothiodystrophy, Bloom ’s syndrome, and Werner ’s syndrome. The study of the formation of DNA openings has been initiated recently utilizing models based on nonlinear lattice dynamics [1, 32]. With a view to vibrational excitations in DNA, the Peyrard–Bishop (PB) model [53] and its successors [14, 26] have been successfully applied to describe moving localized excitations (moving breathers), which reproduce typical features of the DNA opening dynamics such as the magnitude of the amplitudes and the time scale of the ”oscillating bubble” preceding full strand separation. The bubble formation process is initiated by structural deformations of selected regions of the parental DNA duplex that serves as a template for the replication. The process begins when the replication apparatus identifies the starting point and then gets combined to it. The replication apparatus is and assembled complex that forms the replication fork and opens it directionally. During this process the helical DNA is unwound and replicated. One of the components of that complex a type of helicase, which is unable to recognize the origin of the replication per se, and which requires the participation of specific proteins to lead it to the initiation site [27]. It is assumed that this enzyme operates in the way, that a local unwinding in combination with (rather small) stretchings of the H-bonds in this region occurs. A working model should demonstrate that there are indeed initial deformations that give rise to localized vibrations, constrained to a region of DNA, matching the properties of oscillating bubbles observed experimentally. These oscillating bubbles with their temporarily extended but yet unbroken H-bonds serve as the precursors to the replication bubble. Furthermore, it is important to know whether the stable radial and torsional breathers persist under imposed thermal perturbations. The research has been done using a nonlinear oscillator network model for the DNA double helix that is explained in detail in [1, 14, 18]. The equilibrium position of each base within the duplex configuration is described in a Cartesian coordinate system by x(0) n,i,y(0) n,i and z(0) n,i. The index pair (n, i) labels the n-th base on the i−th strand with i= 1,2 and 1 ≦n≦N, where Nis the number of base pairs considered. Displacements of the bases from their equilibrium positions are denoted by xn,i,yn,i and zn,i. The potential energy taking into account the interactions between the bases consists of four parts. The potential energy of the hydrogen bond within a base pair is modeled typically by a Morse potential Vn h=Dnhexp −α 2dn−1i2,(25) where the variables dndescribe dynamical deviations of the hydrogen bonds from their equilibrium lengths d0. The site-dependent depth of the Morse potential, Dn, depends on the number of involved hydrogen bonds for the two different pairings in DNA, namely the G-C and the A-T pairs. The former pair includes three hydrogen bonds while the latter includes only two. α−1is a measure of the potential–well width. The energies of the rather strong and rigid covalent bonds between the nucleotides nand n−1 on the i−th strand are modeled by harmonic potential terms Vn,i c=K 2l2 n,i ,(26) and ln,i describes the deviations from the equilibrium distance between two adjacent bases on the same strand. Kis the elasticity coefficient. Effects of stacking, which impede that, due to the backbone rigidity, one base slides over another [61] are incorporated in the following potential terms Vn,i s=S 2(dn,i −dn−1,i)2.(27) The supposedly small deformations in longitudinal direction can be modeled by harmonic elasticity potential terms given by Vn,i l=C 2(zn,i −zn−1,i)2.(28) 23 FIG. 21: The initial radial distortions of the double helix. The inter-base distance dn(0) in ˚ A. The kinetic energy of a nucleotide is determined by En,i kin =1 2mp(x) n,i 2+p(y) n,i2+p(z) n,i2,(29) where mis the mass and p(x,y,z) n,i denotes the (x, y, z)−component of the momentum. The model Hamiltonian reads then as H=X i=1,2 N X n=1 En,i ,(30) with En,i =En,i kin +Vn h+Vn c+Vn,i s+Vn,i l,(31) and the summation in (30) is performed over all nucleotides and the two strands. Consider the initiation of the oscillating bubbles in DNA. The starting point is a DNA molecule for which a certain segment experiences initially angular and radial deformations due to the action of some helicase to which a region of the DNA is bound. In order to simulate the deforming action of enzymes, assume that initially a number of consecutive sites in the center of the DNA lattice (hereafter referred to as the central region) are exerted to forces acting in angular and radial direction such that in this region the molecule experiences twist reduction together with radial stretchings. These structural deformations can be extended over a region encompassing up to thirty base pairs and as can be demonstrated give rise to the formation of H-bridge breather solutions (extending over 15 −20 base pairs) reproducing the ”oscillating bubbles” observed for the DNA-opening process [13]. In Figs. 21 and 22 the localized initial distortions are shown. Both the angular and radial deformation patterns are bell-shaped, due to the fact that the enzymatic force is exerted locally, i.e. in the extreme case to a single base pair only for which the H-bond is deformed The first one being of non-positive amplitudes is linked with reduced twist while the latter one with non-negative amplitudes is associated with radial stretchings. The radial and angular deformation patterns are centered at the central lattice site (base pair) at which the H-bond stretching and twist angle reduction is at maximum. At either side of the central site the amplitudes approach progressively zero. The deformation energy amounts to 0.0362 eV. The associated set of coupled equations has been integrated with the help of a fourth-order Runge-Kutta method. For the simulation the DNA lattice consists of 799 sites and open boundary conditions are imposed. The same initial conditions are used both at zero and nonzero temperatures. The study of the temperature dependence of these oscillating bubbles has been done by means of bringing in contact the nonlinear oscillator network with a heat bath using the Nos-Hoover-method. It is demonstrated that the radial and 24 FIG. 22: The initial angular distortions of the double helix. torsional breathers sustain the impact of thermal perturbations, even at temperatures as high as room temperature. For T¿0 the breathers start to travel coherently towards an end of te DNA duplex, representing realistically the oscillating bubbles found in NAs i.e. they oscillate with periods in the range 0.3 -0.8 ps, are on the average extended over 10 -20 base pairs and possess maximal amplitudes of the order of 0.6 A. Quantitatively, the results regarding the temperature dependence of the breather evolution are suitably summarized by the time-evolution of the first momentum of the energy distribution defined as ¯n(t) = X i=1,2 N X n=1 (nc−n)En,i(t),(32) and the energy En,i is defined in Eq. (31) and ncis the site index corresponding to the center of the DNA lattice. This quantity describes the temporal behavior of the position of the center of a breather. Thus it represents a measure for the mobility of the breathers. Generally, the higher the temperature the larger the amplitude of the radial breather becomes and the faster the radial (and torsional) breather travels along the DNA lattice. This behavior is illustrated in Fig. 23. In summary, out of an initial non-equilibrium situation, for which the hydrogen bonds of an under-twisted segment of the DNA lattice have been stretched, breathers develop in the radial and angular displacement variables. For zero temperature the breathers remain standing in the initially excited region. However, for T > 0 the breathers start to travel coherently towards an end of the DNA duplex. The observed breathers represent realistically the oscillating bubbles found prior to complete unzipping of DNA, i.e. they oscillate with periods in the range 0.3−0.8ps, are on the average extended over 10 −20 base pairs and possess maximal amplitudes of the order of .0.6˚ A. These results demonstrate that the action of some enzyme, mimicked by localized radial and torsional distortions of the DNA equilibrium configuration, initiates in fact the production of oscillating bubbles in DNA. Moreover, these ”oscillating bubbles” sustain the impact of thermal perturbations. Acknowledgments We thank all the colleagues with whom we had very fruitful collaborations and discussions over the years. Without them this work would not have been possible. 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