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Feigenbaum Cascade of Discrete Breathers in a Model of DNA P. Maniadis, B. S. Alexandrov, A. R. Bishop, and K. Ø. Rasmussen Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545. (Dated: January 13, 2011) We demonstrate that period-doubled discrete breathers appear from the anti-continuum limit of the driven Peyrard-Bishop-Dauxois model of DNA. These novel breathers result from a stability overlap between sub-harmonic solutions of the driven Morse oscillator. Sub-harmonic breathers exist whenevera stability overlap is present within the Feigenbaum cascade to chaos and therefore an entire cascade of such breathers exists. This phenomenon is present in any driven lattice where theon-sitepotentialadmitssub-harmonicsolutions. InDNAthesebreathersmayhaveramifications for cellular gene expression. 1 1 Introduction.— Discrete breathers are spatially local- ear oscillators where multi-stability occurs between var- 0 ized,temporallyperiodicexcitationsinnonlinearlattices ious stages of the Feigenbaum period doubling cascade. 2 [1]. While discretebreatherssharemany traitswithsoli- Although, our study focuses on the PDB model the ob- n tonsthey standoutbecause oftheir localizationbrought served phenomena are rather general and not limited to a aboutbyadelicatesensitivitytolatticediscreteness. Dis- this specific system of nonlinear oscillators. J cretebreathers,havebeenubiquitouslystudiedinawide DNA model.—We consider the PBD model in the fol- 1 variety of physical systems and have been the subject lowing form [15]: 1 of intense theoretical and numerical scrutiny [1–5]. Al- ] though, dissipation and driving are typically key exper- my¨n = − U′(yn)−W′(yn+1,yn)−W′(yn,yn−1) S imental features, most of the theoretical and numerical − mγy˙ +F cosΩt, (1) P n 0 studies have excluded such effects. Only quite recently . n havetheeffects ofdissipationanddrivingbeenexplicitly where the Morse potential i nl considered in numerical studies of discrete breathers [6]. U(yn)=D[exp(−ayn)−1]2 Biomolecules represent a striking example of systems [ where the localization of discrete breathers has long represents the hydrogen bonding of the complementary 2 been considered important, and where dissipation and bases. Similarly, v driving cannot be ignored. DNA represents a spe- 5 k 6 cific biomolecular system where localization of energy W(yn,yn−1)= 2 (cid:16)1+ρe−β(yn+yn−1)(cid:17)(yn−yn−1)2 5 in terms of strand-separation dynamics is emerging as 2 a governing regulatory factor [7]. The Peyrard-Bishop- represents the stacking energy between consecutive base . 2 Dauxois (PBD) model of double-stranded DNA [8, 9] is pairs. The parameters D, a, and k in principle de- 1 arguablythemostsuccessfulmodelfordescribingthislo- pend on the type of the base pair (A-T or G-C), but 0 cal pairing/unpairing (breathing) dynamics, since it re- forsimplicity we willconsiderhomogeneousDNA inthis 1 producesawidevarietyofexperimentsrelatedtostrand- study. Thetermmγy˙ isthedragcausedbythesolvent, : n v separation dynamics [10]. Of particular importance is while F cosΩt is the (terahertz) drive [19]. In this sys- 0 Xi the role this model has played in demonstrating strong tem, Eq. (1), the linear resonance frequency is given as correlations between regulatory activity, such as protein Ω2 =2Da2/m. r 0 a binding and transcription, and the equilibrium propen- In order to understand how period-doubled breathers sity of double-stranded DNA for local strand separation oscillating at half the frequency, Ω/2, of the drive can [11–14]. Recently, this model was augmented to include appear at the anti-continuum limit, we first study the a monochromaticdrive in the terahertz frequency range, nonlinear response of a single (k = 0) oscillator. For and was suggested to represent a simplified model for periodic driving the single oscillator supports periodic DNA dynamics in the presence of terahertz radiation solutions with the period (T = 2π/Ω) of the drive, and, [15]. In this recent work [15] the existence of discrete due to the softness of the Morse potential, this oscillator breathers oscillating at half the frequency of the drive also supports solutions with period nT, where n is any was numerically observed, and it was argued that the positiveinteger. Thesesolutionsaremostsimplystudied existence and spontaneous generation of such breathing at zero dissipation (γ = 0); we will then examine the statesmayhavesignificantramificationsforcellulargene modifications resulting when γ 6= 0. A Newton method expression: Thisargumenthasreceivedsomeexperimen- [2] was used to track the solutions with periods T and tal support [16, 17]. 2T. TheNewtonprocedurewasinitiatedforaverysmall Here,weinvestigatetheseperiod-doubledbreatherex- drivingamplitude (F ≃0). The responseofthe system, 0 citations further, and show that they appear naturally namely the amplitude of the period T and 2T solutions, from the anti-continuum limit [18] in systems of nonlin- was followed as |F | was increased. 0 2 0.7 γ=0 00..65 TP2 u u CP2 sBP3 u γ=0.01 γ=0.1 0.8 a ement 1 ude y ( Å )0 000...243 u BPss4BP1 s CP1 u γ=0.001 ment ( Å )00..46 Displac−01 4 Time t2 L0att2ic0e10 0 Amplit -00..011TP2 u BP4 us uTP1 s BP1 u BP4 s BPs2 Displace0.2 -0.2 u CP2 sBP3 u u BP4 s 0 chaos chaos -0.3 -0.2 -0.1 0 0.1 4 8 12 16 20 24 −0.20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Force F ( nN ) Time t ( ps ) 0 FIG. 1: (Color online) Nonlinear response manifold of a single oscillator driven at Ω =7.0 rad/ps (< Ω0). The NLRM 0.5 b for the period-T solutions (solid black line) and the period- ent 0.5 0.4 m dpmoaeurntbtslseodcfosntohnleeucttNioeLdnRstoM(dBafsPohr4e)dtshoreeludptieloirnnioesd).-aT4rThees(shstooalwbidnilibttyolugoeefthltihenerevwsaeirtgih-- Å ) 0.3 Displace−0.05 Time t Lattice100 oussolutions isindicated bythelettersplaced inthevicinity nt ( 0.2 6 4 2 0 20 e of the lines (’s’ for stable and ’u’ for unstable). Important m e turning points (TP), bifurcation points (BP), and crossing plac 0.1 points(CP)areshownbycircles. Finally,thegreyareasrep- Dis 0 resent the chaotic regions that result from the Feigenbaum period-doublingcascades. Theinset shows thebehaviornear −0.1 CP1 in the presence of dissipation. γ = 0 (dashed red line), γ =0.001ps−1, γ =0.01ps−1, and γ =0.1ps−1 −0.2 0 1 2 3 4 5 6 Time t ( ps ) Softperiod-doubled breathers: Ω<Ω . —Intheabove FIG. 2: Illustration of period-doubled breathers. Shown are 0 fashion we constructed the nonlinear response manifolds the central site y0 of the breather (solid red lines) together (NLRM) [20] depicted for Ω = 7 rad/ps in Fig. 1. The with the closest neighbor sites y1 (dashed blue lines), and y2 (dashed-dotted green lines). For clarity, panel a) shows NLRM for the oscillations with period T is shown by 10y1 and 10y2, respectively. Panel a) shows the period- the solid black line, while the dashed red line represents doubledbreathercorrespondingtothestabilityoverlapshown the NLRM for the period-doubled (2T) solutions. Fi- in Fig. 1 (Ω = 7 rad/ps and γ = 0) for F0 = 7.06 pN and nally, the blue lines segments (connected to BP ) rep- k = 0.02eV˚A−2. Similarly, panel b) shows a period-doubled 4 resent parts of the NLRM for the period 4T solutions. breatherforΩ=5.35rad/ps,γ =1ps−1,F0 =115.6pN,and ThestabilityofthesolutionscanbeobtainedbyFloquet k =0.0065eV˚A−2. The insets shows the full spatio-temporal evolution of theperiod-doubled breathers. stability analysis [2] and is indicated by the letters ’s’ (stable)and’u’(unstable)placedadjacenttotherespec- tive NLRMs. The stability of a given solution changes at the turning points (TP), bifurcation points (BP), or doubled) when Ω < 5.5 rad/ps (not shown in Fig. 1). at the crossing points (CP) of the NLRM. Since, the The ramifications of these stability overlaps for the con- Morsepotentialis asymmetricwithrespectto y =0,the struction of discrete breathers is now clear. In the anti- NLRMs lack symmetry with respect to a π-phase shift continuum limit (k = 0) of N driven oscillators, we can (±F0)ofthedrive. However,anequivalencebetweenthe constructsolutionswhereN−1oftheoscillatorsperform NLRM branchesforF0 and−F0 remains,asis indicated stable period-T oscillations, while the remaining oscilla- by our labeling of the TPs, BPs, and CPs. tor resides stably in the period-doubled state. Perform- From Fig. 1 it can be noticed that, for a region of ing standard numerical continuation [21] of this state to small positive force amplitudes, F , the period-T solu- finite values of the coupling, k, will retain the stabil- 0 tion is stable up to TP (F ≃ ±14 pN) and so is the ity of the solution and lead to a stable period-doubled 1 0 period-doubled solution between CP (F =0) and BP discrete breather solution. Examples of such period- 2 0 3 (F ≃ 21.5 pN). This means that in the region between doubled breathers arising in each of the two described 0 0 < F < 14 pN both solutions are stable. A similar stability overlapregionsareshownin Fig. 2. Panela)of 0 stability overlap can be observed for the regions F = 0 Fig. 2 shows the period-doubled breather in the stabil- 0 to TP (period-T, F < 0) and BP to BP (period- ity overlap region for F > 0. It is clearly seen that the 1 0 4 1 0 3 1.5 s Thescenariodescribedabovegenerallyholdsalsowhen s chaos u u s u dissipation is taken into account. The modifications to the NLRM causedby dissipationis thoroughlydiscussed s u for a different system of nonlinear equations in Ref. [6]. Å ) 1 s u The main effect of dissipation occurs at the crossing y ( 0 u s u points CP1 and CP2. For γ = 0 the period-T solutions de s areinphasewith the drivewhenthe responseamplitude plitu 0.5 u u y0 is below CP1, but it is inanti-phase (π-phase shifted) m chaos when y0 is above CP1. Similarly, the period-doubled so- A lutionsexperiencea π-phaseshiftatCP . The introduc- us 2 u tion of dissipation creates an additional phase shift be- 0 s u s s tween the drive and the response everywhere along the u s NLRM.Thisadditionalphaseshiftislargestcloseto the -6 -4 -2 0 2 4 6 crossingpoints and cause the solution to disappear close Force F ( nN ) 0 to the crossing points (small |F |). This effect is illus- 0 trated in the inset of Fig. 1 for the CP crossing point. FIG. 3: Nonlinear response manifold of a single oscillator 2 Itisclearthatforweakdissipationthedescribedstability driven at Ω =13.194 rad/ps (> Ω0). The NLRM for the period-T solution (solid black line) and the period-doubled scenario persists. However, as the dissipation increases solutions (dashedred line) are shown together with parts of the stability overlapregion is eventually lost and with it the NLRM for the period-4T (blue line segments) solutions. the stable period-doubled breathers. Thestabilityofthesolutionsisindicatedbythelettersplaced Hard period-doubled breathers: Ω > Ω . — Figure 0 inthevicinityofthelines(’s’forstableand’u’forunstable). 3 shows the NLRM when the frequency of the drive Important turning points, bifurcation points, and crossing (Ω = 13.194 rad/ps) is larger than the linear resonance points are shown by circles. Finally, thegrey areas represent frequency of the on-site potential. In this case only a the chaotic regions that result from the Feigenbaum period- doublingcascades. Astabilityoverlapcanbeobservedinthe single, but larger, stability overlap between the period- vicinityofF0 =0. Theinsetshowsamagnifiedversionofthe T solution and the period-doubled solution exists in the area between thechaotic regions. vicinity of F = 0. Again, the period-doubled breathers 0 can be generatedin this overlapregion. It is noteworthy that for soft potentials, such as the Morse potential, or- breather,whichextendsover 5latticesites,performsos- dinary period-T breathers cannotexists above the linear cillationsatexactlyhalfthefrequencyofthebackground. Ω resonance frequency: This frequency region can only 0 The background is frequency-locked to the drive at fre- be accessed for hard potentials. quency Ω. Similarly, panel b) of Fig. 2 shows the period Twoexamplesoftheperiod-doubledbreathersforΩ= doubled breather that exists in the stability overlap re- 13.194rad/psaregiveninFig. 4. Panela)showsaperiod gion for F0 < 0. The breather is more localized and doubled-breather in a dissipation-less (γ = 0) system, its dynamical behavior is more complex. This breather andit is clearthat the breathersites oscillate athalf the roughlyfollowsthebackgroundperiod-Toscillationsand frequency of the background, which is frequency locked theperiod-doublingoccursasamodulationofthebreath- to the drive. Similarly, panel b) shows a period-doubled ing amplitude. For weak driving the dynamics must be breather at strong dissipation γ = 1ps−1. The phase very similar to the undriven case, as is seem in panel a). shift introduced by the dissipation is apparent. At stronger driving the dynamics is more dominated by Discussionandconclusions. —Wehavedemonstrated the drive and become more intricate, as in the case of the existence of a new kind of sub-harmonic discrete panel b). breathers in driven lattices of coupled nonlinear oscil- It should be noted that, in the absence of a stability lators. The phenomenon was illustrated within the overlap,unstable period-doubledbreatherscanof course drivenPBD model, whichconsistsof a lattice ofcoupled still be constructed. An interesting example of this oc- Morse oscillators. Specifically, we have shown that these cursforF <23.3nNwhereFig. 1showsthattheperiod- breathersappearnaturallyfromtheanti-continuumlimit 0 doubled solution is stable while the period-T solution is in a fashion similar to familiar discrete breathers. We unstable. Close to BP the period-T solution is very showthatthesub-harmonicbreathersarisefromastabil- 2 weaklyunstable(theFloquetexponentis∼1.04),sothat ity overlap between sub-harmonic solutions of the single a weakly unstable ’dark’ breather can be constructed in oscillator. This new kind of breathers exist even when this region by positioning N-1 of the oscillators in the the frequency of the drive is above the linear resonance period-doubledstateswhile the remainingoscillatorisin frequencyofthesofton-sitepotential. Ordinarybreather the period-T state. We have found that numerical con- solutions do not exist in such cases for soft potentials. tinuation of this state produces a weakly unstable ’dark’ Although, we focused here on the stability overlap breather that can be sustained for tens of periods. between the frequency locked solution and the period- 4 double-strandedDNA, andtherebycouldleadtomodifi- 0.25 a nt 0.5 cation in cellular gene expression by affecting transcrip- 0.2 me tionandotherprocesses. Sucheffectshaverecentlybeen e ac 0 experimentally observed [16, 17]. Future extensions of 0.15 Displ Time t Lattice 0 our approach will include effects of gene sequence and nt ( Å ) 0.1 −0.5 2 1 0 20 10 other disorder, as well as temperature. e m 0.05 ace Thisresearchwascarriedoutundertheauspicesofthe Displ 0 NationalNuclearSecurityAdministrationoftheU.S.De- −0.05 partment of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. −0.1 −0.15 0 0.5 1 1.5 2 2.5 Time t ( ps ) 0.25 [1] S. Flach and A. V. Gorbach, Phys.Rep. 467, 1 (2008). b 0.5 nt [2] S. Aubry,Physica D 103, 201 (1997). 0.2 me 0.15 place 0 [3] D77.,C4h7e7n6,(S1.9A96u)b.ry,and G.P. Tsironis, Phys. Rev. Lett., ment ( Å ) 0.00.51 Dis−0.5 2 Time t1 L0attic2e0100 [[45]] MM(19..9JJ7oo)h.haannssssoonn,,S.anAdubSr.y,APuhbyrsy.,RNevo.nElin6e1a,ri5ty86410(2,010105).1 ace [6] P. Maniadis, S.Flach, Europhys.Lett. 74, 452 (2006). Displ 0 [7] B. S. Alexandrov, V. Gelev, S. W. Yoo, L. B. Alexan- drov, Y. Fukuyo, A. R. Bishop, K. Ø. Rasmussen, and −0.05 A. Usheva,Nucleic Acids Res. 6, 1790 (2010). −0.1 [8] M.Peyrard,andA.R.Bishop,Phys.Rev.Lett.62,2755, (1989). −0.15 0 0.5 1 1.5 2 2.5 [9] T. Dauxois,M.Peyrard,andA.R.Bishop,Phys.Rev.E Time t ( ps ) 47, R44 (1993). [10] B. S. Alexandrov, N. K. Voulgarakis, K. Ø. Rasmussen, FIG. 4: Illustration of period-doubled breathers. Shown are A. Usheva, A. R. Bishop, J. Phys. Condens. Matter 21, the central site y0 of the breather (solid red lines) together 034107 (2009), and references therein. with the closest neighbor sites y1 (dashed blue lines), and [11] C.H.Choi,G.Kalosakas,K.Ø.Rasmussen,M.Hiromura, y2 (dashed-dotted green lines). Panel a) shows the period- A. R. Bishop, A. Usheva, Nucleic Acids Res. 32, 1584 doubledbreathercorrespondingtothestabilityoverlapshown (2004). in Fig. 1 (Ω = 13.194 rad/ps and γ = 0) for F0 = 160 pN [12] C.H.Chu,Z.Rapti,V.Gelev,B.S.Alexandrov,E.Park, and k = 0.025eV˚A−2. Panel b) demonstrates the effects of J. Park,N.Horikoshi,A.Smerzi,K.Ø.Rasmussen,A.R. dissipation Ω=13.194 rad/ps, γ =1ps−1, F0 =209 pN, and Bishop, and A. Usheva,Biophys. J., 95, 597 (2008). k = 0.025eV˚A−2. The insets show the full spatio-temporal [13] B.S.Alexandrov,V.Gelev,Y.MonisovaY,L.B.Alexan- evolution of the period-doubled breathers. drov, A. R. Bishop, K. Ø. Rasmussen, and A. Usheva, Nucleic Acids Res. 37, 2405 (2009). [14] B.S.Alexandrov,V.Gelev,S.W.Yoo,A.R.Bishop,K.Ø. doubledsolution, it is clearthat in the presence ofa sta- Rasmussen, and A. Usheva, PLoS Comput Biol. 5(3), bility overlap the phenomena can occur between any of e1000313 (2009). the sub-harmonic solutions in the Feigenbaum cascade. [15] B.S.Alexandrov,V.Gelev,A.R.Bishop,A.Usheva,and K. Ø. Rasmussen, Phys.Lett. A 374, 1214 (2010). We have explicitly verified this by constructing stable [16] R. Shiurbaet al. Photoch. Photobio. Sci. 5, 799 (2006). periodic-nT (n = 3,4,5, and 6 ) breathers for small F 0 [17] J.Bock,Y.Fukuyo,S.Kang,M.E.Phipps,L.B.Alexan- and k, in the case where Ω > Ω . In this fashion a 0 drov, K. Ø. Rasmussen, A.R. Bishop, E.D. Rosen, J.S. cascade of sub-harmonic discrete breathers exists. The Martinez, H.-T. Chen, G. Rodriguez, B.S. Alexandrov, phenomenon is not specific to the system we have stud- and A.UshevaPLoS ONE5, e15806 (2010). ied: it is present in any driven lattice of nonlinear oscil- [18] R.S.MacKayandS.Aubry,Nonlinearity7,1623(1994). lators where the on-site potential admits sub-harmonic [19] In order to be consistent with prior work on DNA we use the following parameters unless different val- solutions. The sub-harmonic breathers also exist in the ues are explicitly given: D = 0.05eV, a = 4.2˚A−1, presence of dissipation, it is however important to note k=0.025eV˚A−2,β=0.35˚A−1,m=300amu,andρ=2. that in this case the cascade is truncated. With these parameters thelinear resonance frequency is The existence of the sub-harmonic breathers in DNA Ω0 = 2Da2/m=7.54 rad/ps, i.e.Ω0/2π=1.22 THz. is potentially very important because this suggests that [20] G. Koppidakis and S. Aubry, Phys. Rev. Lett. 84 3236 electromagneticterahertzradiationcanleadtothespon- (2000). taneous generation of stable local strand separation in [21] J. L. Marin and S. Aubry,Nonlinearity 9, 1501 (1994).