Energy localization and shape transformations in semiflexible polymer rings

PHYSICAL REVIEW E 93, 062227 (2016) Ene gy localiza ion and shape ans o ma ions in semi lexible polyme ings Yu. B. Gaididei,1J. F. R. A chilla,2V. J. S ´ anchez-Mo cillo,3and C. Go ia4 1Bogolyubo Ins i u e o Theo e ical Physics, Me ologichna S ee 14 B, 03143 Kie , Uk aine 2G upo de F´ ısica No Lineal, Uni e sidad de Se illa, ETSI In o m´ a ica, A. Reina Me cedes s/n, E-41012 Se illa, Spain 3Ins i u o de In es igaci´ on pa a la Ges i´ on In eg ada de las Zonas Cos e as, Uni e sidad Poli ´ ecnica de Valencia, Pa anim 1, E-46730 G ao de Gandia, Spain 4Depa men o Applied Ma hema ics and S a is ics, Uni e si y o he Basque Coun y, E-48080 Bilbao, Spain (Recei ed 30 Oc obe 2015; e ised manusc ip ecei ed 20 Ap il 2016; published 30 June 2016) Shape ans o ma ions in d i en and damped molecula chains a e conside ed. Closed chains o weakly coupled molecula subuni s unde he ac ion o spa ially homogeneous ime-pe iodic ex e nal ield a e s udied. The coupling be ween he in e nal exci a ions and he bending deg ees o eedom o he chain modi ies he local bending igidi y o he chain. In he absence o d i ing he a ay akes a ci cula shape. When he ene gy pumped in o he sys em exceeds some c i ical alue he chain unde goes a nonequilib ium phase ansi ion: The ci cula shape o he agg ega e becomes uns able and he chain akes he shape o an ellipse o , in gene al, o a polygon. The exci a ion ene gy dis ibu ion becomes spa ially nonuni o m: I localizes in such places whe e he chain is mo e la . The weak in e ac ion o he chain wi h a la su ace es ic s he dynamics o a la mani old. DOI: 10.1103/PhysRe E.93.062227 I. INTRODUCTION Nonlinea localiza ion phenomena a e widely ecognized as key o unde s anding he exci a ion dynamics in many physical and echnological aspec s such as ligh p opaga ion, cha ge, and ene gy anspo in condensed-ma e physics, o dynamics o mic omechanical oscilla o a ays [1–4]. Recen ad ances in manu ac u ing o mic o- and nano-elec omechanical sys- ems [5] ha e made i possible o ab ica e ex e nally ac ua ed esonan nanos uc u es and s udy in insic localized s a e o ma ion in d i en mic o-mechanical can ile e a ays [1–4]. Elec ic ac ua ion is he mos common ac ua ion me hod because o i s simplici y and high e iciency. Howe e , o he ac ua ion me hods including he mal mechanical s esses, magne ic ields, and op ical exci a ion a e also used [6,7]. I is well known ha in conse a i e sys ems o nonlinea oscilla o s he modula ional ins abili y o band edge plane wa es leads o c ea ion o spa ially localized s a es. Howe e , as i was shown qui e ecen ly [8] in damped and d i en la ices in insic localized s a es appea ia a new ins abili y mechanism, di e en om he modula ional ins abili y. New aspec s in nonlinea ene gy localiza ion appea in sys ems wi h complica ed geome y. Nonlinea whispe ing galle y modes o a nonlinea Maxwell equa ion in a mic odisk we e in es iga ed in [9], and he exci a ion o whispe ing-galle y- ype elec omagne ic modes by a mo ing luxon in an annula Josephson junc ion was ound in [10]. Localiza ion o linea and nonlinea exci a ions in pa abolically cu ed wa eguides was s udied in [11]. A cu ed chain o nonlinea oscilla o s was conside ed in [12] and i was shown ha he in e play o cu a u e and nonlinea i y leads o a symme y b eaking when an asymme ic s a iona y s a e becomes ene ge ically mo e a o able han a symme ic s a iona y s a e. The in e ac ion o classical anha monic localized modes wi h geome y was conside ed in Re s. [13–17]. Ano he ecen example o localiza ion in cu ed geome ies was epo ed in [18,19], whe e spa ial ins abili ies o a ci cula ing o coupled pendula pa ame ically d i en by a e ical ha monic o ce a e discussed. No mal oscilla ion modes [19] (b ea hing, dipole, quad upole) and localized pa e ns o di e en ypes [18] (b ea he s and kinks) a e p edic ed and obse ed in such mechanical sys em. The analogy be ween he conside ed disc e e mechanical sys em and a gas bubble ca i a ing unde he ac ion o an acous ic ield was also es ablished. The bulk o heo e ical esul s has been achie ed o a ays o nonlinea oscilla o s wi h ixed geome y: linea chains and wo-dimensional la ices. Un il ecen ly he e ha e been ew heo e ical and nume ical s udies o nonlinea exci a ions in sys ems wi h lexible geome y. Many ypes o biomolecules as polyme s o DNA chains belong o his ca ego y. Con o ma ional dynamics wi h accoun o coupling be ween he in e nal and mechanical deg ees o eedom was s udied in [20]. I was ound ha he p esence o nonlinea exci a ions may cause he buckling and collapse ins abili ies o an ini ially s aigh chain. These ins abili ies emain la en in a s aigh in ini ely long chain, because he bending o such a chain would equi e an in ini e ene gy. The ole o he cha ge-cu a u e in e ac ion on he o ma ion o he g ound s a e o closed semi lexible molecula chains was s udied in [21–23]. I was shown ha he coupling be ween cha ge ca ie s and he bending deg ees o eedom o he chain modi ies he local bending igidi y o he semi lexible chain. Due o he in e ac ion be ween cha ge ca ie s and he bending deg ees o eedom he ci cula shape o he agg ega e may become uns able and he chain akes he shape o an ellipse o , in gene al, o a polygon. The e, he polygon s uc u e is a esul o he sel -consis en in e ac ion be ween cha ge ca ie s and bending deg ees o eedom: ex ema o he cu a u e and o he cha ge densi y co ela e: In he case o he so ening cha ge-cu a u e in e ac ion maxima o cu a u e and cha ge densi y coincide, while in he case o he ha dening in e ac ion he minima o he cu a u e coincide wi h he maxima o he cha ge densi y. These esul s we e ob ained by assuming ha he cha ge ca ie dynamics is cohe en and he o al cha ge is an in eg al o mo ion. In his wo k we a e in e es ed in nonequilib ium shape ans o ma ions which may occu in closed ilamen s pos- sessing wo kinds o deg ees o eedom: high equency 2470-0045/2016/93(6)/062227(9) 062227-1 ©2016 Ame ican Physical Socie y YU. B. GAIDIDEI e al. PHYSICAL REVIEW E 93, 062227 (2016) elec omagne ically ac i e deg ees o eedom (in wha ol- lows we will call hem exci ons) and low equency modes which a e nonlinea ly coupled wi h exci ons. The e a e many examples o such sys ems. P obably, he bes known model o exci a ions in a biological chain is he Da ydo -Sco model o p o eins. See Re . [24] o a ecen e iew. In his model he so-called Amide I exci a ion consis s o he s e ching ib a ion o he C=0 bond o he pep ide g oups, which a e linked by hyd ogen bonds. The pe iodic o cing o he exci on modes can be p o ided by elec omagne ic wa es. The Amide I ib a ion in p o eins has a equency o 1665 cm−1o abou 50 THz, i.e., be ween he nea and mid-in a ed spec um. The e is abundan bibliog aphy dealing wi h he in e ac ion o in a ed adia ion wi h p o eins and in pa icula wi h he Amide I modes, including con o ma ional changes and pho oini ia ed dynamics. See, o example, [25,26] and e e ences he ein. The in e ac ion o he exci on wi h he bending deg ees o eedom is also conside ed in a e y simila model o Amide I exci a ions in c ys alline ace anilide [27]. Li e ime o he exci a ions is s ill a ma e o esea ch and deba e. The li e ime o Amide I exci a ions in he ace anilide c ys al has been expe imen ally de e mined in 2 ps bu i was also shown ha he exci a ion ene gy is no dissipa ed un il 35 ps sugges ing ha he Amide I exci a ion can be ans o med in o ano he mo e long-li ed exci a ion [28]. Howe e , Amide I exci a ions in a p o ein [29] su i e much mo e, up o 500 ps. Theo e ical calcula ions in he Da ydo -Sco model show ha he exci on can a el along he p o ein in a ew picoseconds ime [24]. Ano he example a e DNA minici cles [30,31] and ci - cula plasmids adso bed in a mica su ace [32] wi h hei a -in a ed-ac i e in e base hyd ogen-bond b ea hing modes (cha ac e is ic equency is ∼100 cm−1[33] and ypical li e ime is ∼10 ps [34]) which a e nonlinea ly coupled wi h o sional-acous ic modes [35]. An e ec i e con ol o he shape o he sys em by isible ligh is achie ed by inco po a - ing dye monome s in o liquid-elas ic elas ome s [36,37]. The liquid-elas ic elas ome s a e cha ac e ized by a s ong coupling be ween he o ien a ional o de and mechanical s ain. Unde he ac ion o linea pola ized ligh nema ic s ips doped wi h azo-dyes con ollably bend as monome s pho oisome ize be ween hei ans- and cis-s a es and educe he deg ee o he o ien a ional o de in he elas ome [38]. The aim o he pape is o s udy shape ans o ma ions in d i en and damped molecula chains. A gene ic model o closed chain o weakly coupled molecula subuni s unde he ac ion o a spa ially homogeneous ime-pe iodic ex e nal elec ic ield is s udied. In con as o he p e ious s udies [21–23], whe e he shape o ma ion o he molecula chain was due o a cha ge-bending in e ac ion, and he e o e ela ed o an equilib ium phase ansi ion, he e we discuss he shape ans o ma ion o he molecula chain as a nonequilib ium phase ansi ion which occu s due o he ene gy pumping in he sys em. The pape is o ganized as ollows. In Sec. II we desc ibe a model. Sec ion III p esen s he s a iona y analy ical solu ions and discusses he s abili y issues. In Sec. IV we p esen he esul s o nume ical simula ions, demons a ing he exci a ion o di e en shape modes depending on he pa ame e s. In Sec. Vwe p opose an analy ical app oach o he p oblem based on he Gale kin decomposi ion me hod, and compa e he analy ical esul s wi h he esul s ob ained di ec ly by nume ical simula ions. Finally, Sec. VI p esen s some concluding ema ks. II. THE MODEL We conside a simple phenomenological model o a polyme ing consis ing o pa icles, labeled by an index n, and loca ed a he poin s n={xn,y n,z n}(n=1...N). We a e in e es ed in he case when he a ay ep esen s a closed chain and so we impose he pe iodici y (closu e) condi ion on he coo dina es n= n+N. Each uni nis connec ed wi h i s wo neighbo s n+1 and n−1 by elas ic bonds. We will assume ha he chain is inex ensible: | n− n+1|=a( he bond leng h ais a cons an which we in wha ollows pu equal o 1). The change o he angle be ween he bond ec o s n+1=( n+1− n) and n=( n− n−1) is con olled by he bending po en ial which we ake in he o m, Ub=K 2 n κ2 n,(1) whe e Kis he elas ic modulus o he bending igidi y (sp ing cons an ) o he chain, κn≡| n+1− n|(2) de e mines he cu a u e o he chain a he poin n.Byusing he pa ame iza ion, n=(cos θn,sin θnsin φn,sin θncos φn),(3) he local cu a u e can be p esen ed in he o m, κn≈(θn+1−θn)2+sin θn(φn+1−φn)2.(4) The angles θnand φnsa is y he ela ion, θn+N=2π+θn,φ n+N=φn,(5) which s ems om he pe iodici y (closu e) condi ion. The bending igidi y o he chain Kcan be exp essed as K=lpkBT, (6) whe e lpis he pe sis ence leng h in uni s o chain pe iod, T is he empe a u e, and kBis he Bol zmann cons an . We conside he si ua ion when he chain adhe es o some su ace. The in e ac ion be ween he chain and he su ace ends o o ien all bond ec o s pa allel o he su ace: easy- su ace aniso opy in e ac ion. I is assumed weak enough in o de no o a ec he in e nal and bending dynamics o he chain. The su ace o adhe ence is pa allel o a x−yplane and he easy-su ace in e ac ion has he o m, Uw=w 2 n (ˆz· n)2 ≡w 2 n sin2θncos2φn,(7) whe e ˆz=(0,0,1) is a uni ec o along he zaxis, and he pa ame e wgi es he in ensi y o he su ace-chain in e ac ion. We assume ha each pa icle ep esen s a complex sub- uni o he chain which addi ionally o i s posi ion n( ), ca ies a high- equency in e nal exci a ion which can be cha ac e ized by a complex ampli ude n( ). Examples o 062227-2 ENERGY LOCALIZATION AND SHAPE TRANSFORMATIONS . . . PHYSICAL REVIEW E 93, 062227 (2016) such in e nal modes a e Amide I ib a ions in p o eins o base-pai ib a ions in DNA [39] and op ical exci a ions o dye-doped liquid-c ys al elas ome s [40], as commen ed in he in oduc ion. The Hamil onian o he high equency exci a ions (exci ons) has he o m, H=ω n|n|2+1 2J n|n+1−n|2,(8) whe e ωis he ene gy o he exci a ion ( he Planck cons an is se equal o 1), and he pa ame e Jcha ac e izes esonance coupling be ween subuni s. We will assume ha he p esence o he high- equency exci a ion a he si e nmodi ies locally he bending igidi y o , in o he wo ds, he e is a coupling be ween exci ons and bending deg ee o eedom o he o m, Uex−b=χ n|n|2κ2 n,(9) whe e he pa ame e χcha ac e izes he s eng h o he coupling (see Appendix o de ails). In wha ollows we es ic ou sel es o he case o ha dening exci on-cu a u e coupling: χ>0. Ene gy is pumped in o he sys em by exci ing i wi h a ime-pe iodic ex e nal o ce o ampli ude and equency . The co esponding in e ac ion ene gy is gi en by U =−  n (nei +c.c.).(10) The equa ion o mo ion o he complex ampli ude nhas he o m, i˙ n=−iα n+ω n−J(n+1+n−1−2n) +χκ 2 nn+ e −i ,(11) whe e α−1gi es he li e ime o he exci on. Fo he sake o simplici y we will neglec ine ia e ec s in he dynamics o he mechanical subsys em and ake he equa ions o mo ion o he bending deg ees o eedom in he o m o o e damped Lag ange-Rayleigh equa ions, ∂F ∂˙ ξn=− ∂ ∂ξn (Ub+Uw+Uex−b), (12) ξn∈(θn,φn) whe e Fis a dissipa i e unc ion which is gi en by he exp ession, F=η1 2 n (˙ n+1−˙ n)2,(13) he pa ame e ηbeing he elaxa ion ime o he bending deg ees o eedom. The dissipa i e unc ion (13) desc ibes in e nal ic ion, which is due o i e e sible p ocesses aking place wi hin he sys em. The unc ion (13) is he disc e e e sion o he dissipa ion unc ion which is usually used in mac oscopic elas ici y heo y [41]. In e ms o he pa ame iza- ion (3) he dissipa ion unc ion (13) akes he o m, F=η1 2 n˙ θ2 n+sin2θn˙ φ2 n.(14) F om Eqs. (12) and (14) we ge η˙ θn=−(K+2χ|n−1|2)(θn−θn−1)+(K+2χ|n|2) ×(θn+1−θn+sin θncos θn(φn+1−φn)) −wsin2θncos2φn,(15) ηsin2θn˙ φn=−(K+2χ|n−1|2)sin 2θn−1(φn−φn−1) +(K+2χ|n|2)sin 2θn(φn+1−φn) +wsin2θnsin φncos φn.(16) To simpli y no a ions i is con enien o use a escaled complex ampli ude, ans e o a o a ing ame o e e ence, n=K χψne−i ,(17) and measu e all ele an a iables in e ms o he coupling s eng h χ, de ining ¯ =χ ,¯ω=ω/χ,¯ =/χ,¯ = /√χK,¯ J=J/χ,¯α=α/χ,¯η=ηχ/K, and ¯w=w/K. In he escaled a iables Eqs. (11), (15), and (16) ake he o m, i˙ ψn=−(δ+iα)ψn−J(ψn+1+ψn−1−2ψn)+κ2 nψn+ , (18) η˙ θn=−(1 +2|ψn−1|2)(θn−θn−1)+(1 +2|ψn|2) ×(θn+1−θn+sin θncos θn(φn+1−φn)) −wsin2θncos2φn,(19) ηsin2θn˙ φn=−(1 +2|ψn−1|2)sin 2θn−1(φn−φn−1) +(1 +2|ψn|2)sin 2θn(φn+1−φn) +wsin2θnsin φncos φn,(20) whe e “ba s” a e omi ed o simplici y and δ=−ωis a de uning equency. Ou analy ical app oach is based on he assump ion ha he elaxa ion ime ηis sho (η→0) and he e o e he bending deg ees o eedom a e sla ed o he exci on ones. In his case om Eqs. (19) and (20) we ge φn=π 2, (21) θn+1−θn=A( ) 1+2|ψn|2, whe e he unc ion A( ) is chosen in he o m, A( )=2πN  n=1 1 1+2|ψn|2−1 ,(22) o sa is y he pe iodic bounda y condi ions gi en by Eq. (5). By inse ing Eq. (21) in o Eq. (18), we ob ain ha he exci on dynamics is go e ned by he ollowing nonlinea in eg o-di e en ial equa ion, i˙ ψn=−(δ+iα)ψn−J(ψn+1+ψn−1−2ψn) +A2( ) (1 +2|ψn|2)2ψn+ . (23) 062227-3 YU. B. GAIDIDEI e al. PHYSICAL REVIEW E 93, 062227 (2016) No e ha acco ding o Eq. (21), he azimu al angle is ixed, so mo ion is con ined o a plane, as Fig. 2shows. III. SOLUTIONS AND STABILITY Equa ion (23) has a spa ially homogeneous solu ion, ψn( )=,  = δ−1 R2+iα,(24) when all subuni s oscilla e in phase and he chain has a ci cula shape wi h he cu a u e κn=2π/N ≡1/R. To in es iga e he s abili y o he spa ially homogeneous solu ion (24) (and he ci cula shape o he chain) we assume ha ψn( )=+ϕn( ), wi h N n=1ϕn( )=0, and linea ize Eqs. (22) and (23) wi h espec o ϕn( ). As a esul we ob ain i˙ϕn=− δ+1 R2+iαϕn−J(ϕn+1+ϕn−1−2ϕn) +2 R2 1 (1 +2||2)(ϕn−22ϕ∗ n).(25) The s abili y is analyzed by conside ing solu ions o he linea sys em (25)o he o m, ϕn( )=˜ϕexp i2πk Nn−iz ,(26) whe e zis a complex equency. No e ha , in he linea app oxima ion, he cu a u e o he chain is gi en by he exp ession, κn=1 R1−2ϕ ∗ n+∗ϕn 1+2||2,(27) which means ha in Eq. (25) he in ege kmus sa is y inequali y k⩾2 o ul ill he closu e condi ion (35); mo eo e , he exci a ion o he k h exci on mode co esponds o a k-gonal de o ma ion o he chain: ellip ical o k=2, iangula o k=3, e c. Inse ion o Eq. (26) in o Eq. (25) leads o zk=−iα ±δkδk+8 R2||2 1+2||2,(28) whe e δk=−ωkand ωk=ω+1 R2+4Jsin2πk N(29) is he equency o he k h exci on mode in he ci cula chain. Di ec inspec ion o Eq. (28) shows ha Im(zk)>0 and he spa ially homogeneous s a e (24) is uns able wi h espec o exci ing he k h exci on mode when ⩾ kand ⩽ωk(δk< 0), whe e 2 k=α2+δ−1 R22Nk,(30) Nk=−1 2 α2+δ2 k α2+δkδk+4 R2.(31) We ace ou ins abili y cu es o he spa ially homogeneous s a e (24)in(δ, ) space. Equa ion (30) de ines he k h mode s abili y cu es in he (δ, ) plane. These esul s demons a e −0.1 −0.05 0 0.05 0.1 0.1 0.2 0.3 0.4 J=0.2 δ k=2 k=4 0 0.2 0.4 0.6 0.1 0.2 0.3 0.4 δ J=1.2 k=2 k=3 k=4 FIG. 1. Ins abili y cu es in (δ, ) space a which he k h mode o he spa ially homogeneous solu ion des abilizes o k=2,3,4. The s eng h o esonance coupling J a ies in he panels as indica ed. We se weak losses, α=0.01 and η=0.01. In he unshaded egions, spa ially homogeneous s a es a e s able. ha o la ge esonance coupling be ween subuni s J, when J>J k=1 N2 4π1−α2R4 4 sin π Nsin π N(2k+1),(32) he egion o ins abili y o he homogeneous solu ion spli s in o sepa a e a eas (see he bo om panel in Fig. 1) whe e he exci on modes wi h di e en kg ow inde ini ely. Howe e , o J<J k he k- and (k+1) h a eas o ins abili y o e lap and k-gonal and (k+1)-gonal p o iles can be simul aneously s able o he same pa ame e alues. The op panel o Fig. 1 p esen s he si ua ion when he homogeneous s a e is uns able simul aneously wi h espec o he modes k=2 and k=3. IV. NUMERICAL RESULTS To ind he s uc u es which appea as a esul o such an ins abili y we sol ed nume ically Eqs. (18)–(20) o a chain o N=36 elemen s and di e en pa ame e se s. The s a ing 062227-4 ENERGY LOCALIZATION AND SHAPE TRANSFORMATIONS . . . PHYSICAL REVIEW E 93, 062227 (2016) con igu a ion co esponded o he exci on subsys em in he g ound s a e: ψn(0) =0 and ini ially, all he chain poin s we e placed a (almos ) symme ic poin s on he ci cle o an app op ia e adius ( he symme y was b oken by changing he local cu a u e o he chain by 0.1%). The simula ions we e pe o med o wo ini ial seeds: θn=2π Nn+10−3sin 2πk Nn, φn=π 2−10−3cos 2πk Nn,k=2,3.(33) Small losses a e conside ed in all he cases, α=η=0.01. The coo dina es o he subuni s can be p esen ed in he o m, xn( )= n  m=1 cos θm( )−1 N N  m=1 (N−m−1) cos θm( ), yn( )= n  m=1 sin θm( )sinφm( ) −1 N N  m=1 (N−m−1) sin θm( )sinφm( ), zn( )= n  m=1 sin θm( ) cos φm( ) −1 N N  m=1 (N−m−1) sin θm( ) cos φm( ),(34) whe e he las e ms in he exp essions o xn,yn, and zn ix he cen e o mass o he chain a he coo dina e o igin. In e ms o he pa ame iza ion (34) he closu e condi ion eads N  m=1 cos θm( )= N  m=1 sin θm( )sinφm( ) = N  m=1 sin θm( ) cos φm( )=0.(35) The esul s o he ull scale 3D simula ions a e shown in Figs. 2and 3. I is seen ha while he ini ial s a e o he chain has a 3D shape, he s a iona y s a e has a la shape pa allel o he x-yplane. We checked ha he shape con e ges o a wo-dimensional x-yp o ile e en o a a he weak easy-su ace aniso opy pa ame e : w∼10−4. The uppe ow o Fig. 3shows he elipselike (k=2), iangula (k=3), and e agonal (k=4) dis ibu ions o he chain subuni s ob ained om nume ical simula ions. The lowe ow shows he co esponding ene gy and cu a u e dis ibu ion along he chain. The elipselike and iangula modes ha e been ob ained o he same o cing, coupling, and de uning pa ame e s, bu di e en ini ial seeds, Eq. (33) wi h k=2 and 3, espec i ely. This clea ly indica es he exis ence o mul is abili y p edic ed by he p e ious analysis, i.e., mul iple k-gons, wi h di e en alues o k, can be simul aneously s able o he same pa ame e alues. The ene gy and cu a u e dis ibu ions also demons a e ha he cu e is mo e la whe e he exci a ion densi y is maximal. Such a beha io is gene ic. The pa ame e s −5 0 5−5 0 5 −5 0 5 y x z× 104 −5 0 5−5 0 5 −5 0 5 z× 104 x y FIG. 2. The op panel shows he ini ial 3D shape o he chain, and he bo om panel shows he s a iona y shape which is achie ed by he chain in he p esence o he same d i ing as in Fig. 3. used o ob aining he e agonal shape we e di e en om he ellipselike and iangula shapes (see cap ion). We conclude ha he polygon s uc u e is a esul o he sel - consis en in e ac ion be ween exci a ions and bending deg ees o eedom; ex ema o he cu a u e and o he exci a ion densi y co ela e: In he case unde conside a ion when he exci on-bending in e ac ion locally ha dens he chain s i ness (χ>0) he minima o he cu a u e coincide wi h he maxima o he exci a ion densi y. V. MINIMAL MODEL. GALERKIN APPROACH To gain some insigh in o he mechanism o shape ans- o ma ions we will use a Gale kin app oach by expanding he complex ampli ude ψn( ) in o Fou ie modes, ψn= N 2  k=−N 2+1 Fk( )exp i2πkn N.(36) 062227-5 YU. B. GAIDIDEI e al. PHYSICAL REVIEW E 93, 062227 (2016) FIG. 3. (Uppe ow) Mode shapes o index k=2 (le ), k=3 (cen e ), and k=4 ( igh ), ob ained in he a ea o pa ame e s whe e he exci on modes a e linea ly uns able. Modes k=2 (elipselike) and k=3 ( iangula ) coexis and ha e been ob ained o =0.01,J =0.2,δ =0.03. Mode k=4 ( e agonal) was ob ained o =0.2,J =0.6,δ =0.25. (Lowe ow) Densi y o exci a ion ene gy dis ibu ion |ψn|2(dashed line, escaled by a ac o 10−1) and cu a u e a ia ion κn(solid line) along he chain. No e ha F1( )=0in(36) because he ha monics wi h k=1 canno con ibu e o he expansion (36) due o he closu e condi ion, Eq. (35). Le s us in oduce an ac ion unc ional S=e2α Ld , whe e Lis a Lag angian unc ion o he o m, L= ni 2(˙ ψnψ∗ n−c.c.)+δ|ψn|2−J|ψn+1−ψn|2 −2π2 n 1 1+2|ψn|2−1.(37) No e ha minimizing he ac ion, δS/δψ∗ n=0, leads o he e olu ion equa ion Eq. (23). By inse ing Eq. (36) in o Eq. (37), expanding he La- g ange unc ion (37) in e ms o he ampli udes Fk(k= 0) up o he second o de , and ca ying ou summa ion o e n, one can ob ain an e ec i e Lag angian unc ion in he o m, L=i 2 k (˙ FkF∗ k−c.c)+δ−1 R2|F0|2 + k=0δk+1 R2|Fk|2−2 R2 1 1+2|F0|2 × k=0F2 k−FkF−k∗2−F2 0F∗ kF∗ −k + (F0+F∗ 0).(38) F om he ac ion S he e olu ion equa ions o he complex ampli udes Fkcan be ob ained in he o m, i˙ =− δ−1 R2+iα +4 R2 1 (1 +2|F0|2)2 × k=0Fk 2−F∗ kF∗ −kF2 0F0 +FkF−k(1 +|F0|2)F∗ 0− , (39) i˙ Fk=− δk+1 R2+iα Fk+2 R2 1 1+2|F0|2 × k=0Fk−2F2 0F∗ −k,k=2,3,... (40) Fo each k he se o Eqs. (39) and (40) has wo kinds o s a iona y solu ions: (1) Fk=0 o k= 0 and F0≡is gi en by Eq. (24). In his s a e he exci a ion ene gy is homogeneously dis ibu ed along he ci cula chain. The s abili y o his s a e is discussed in he p e ious sec ion (see Fig. 1). 062227-6 ENERGY LOCALIZATION AND SHAPE TRANSFORMATIONS . . . PHYSICAL REVIEW E 93, 062227 (2016) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.1 0.2 0.3 0.4 0.5 c Mode ampli udes F0 F2 FIG. 4. Bi u ca ion diag am o he lowes ha monic modes |F0| and |F2|o he exci on wa e unc ion as a unc ion o he no malized pumping s eng h / c,whe e cis he c i ical pump alue a which he ins abili y appea s ( 2in he case shown). The heo e ical solu ions (lines) we e ob ained in he amewo k o he minimal model, while nume ical esul s (symbols) co espond o ull scale nume ical simula ions. Pa ame e s a e N=36, J=0.2, α=0.01, η=0.01, and δ=0.03. (2) Fk= 0. In his s a e he chain is k-gonally de o med and he exci a ion ene gy is concen a ed in he la pa s o he chain [as i is seen om Eq. (27) a maximum o he cu a u e co esponds o a minimum o he exci a ion densi y and ice e sa]. The s abili y egions in he (δ, ) pa ame e space o such kind o s a iona y s a e a e p esen ed as shaded a eas in Fig. 1. The s a iona y s a e has a ema kable ea u e which dis inguishes i om he i s one. In con as o he case o he ci cula chain when he ampli ude o he spa ially homogeneous componen |F0|is a linea unc ion o he pump in ensi y [see Eq. (24)] in he s a iona y s a e o he second kind his componen does no depend on , |F0|2=−1 2 α2+δ2 k α2+δkδk+4 R2,(41) and coincides wi h he h eshold alue Nka which he spa ially homogeneous exci a ion dis ibu ion becomes uns able wi h espec o exci a ion o he k h exci on mode. A his poin a Hop bi u ca ion occu s and he ampli ude o he k h mode o → ke ol es as Fk∼ 2− 2 k,(42) wi h kgi en by Eqs. (30) and (31). This is illus a ed in Fig. 4whe e he ampli ude o he spa ially homogeneous componen |F0|and he ampli ude o he i s non anishing Fou ie mode |F2|a e p esen ed as a unc ion o ex e nal o ce .Fo < 2 he ampli ude F0∼Fwhile F2=0. A = 2 he Hop bi u ca ion akes place and he beha io o he Fou ie ha monics quali a i ely changes: F0 emains cons an and he ampli ude o he second ha monics g ows as Eq. (42). To e i y hese esul s we ca ied ull scale nume ical simula ions o he same se o pa ame e s. These esul s a e shown in Fig. 4wi h symbols. F om his analysis, one can conclude ha he minimal model gi es a easonable desc ip ion o he dynamics o he sys em. VI. CONCLUSIONS AND DISCUSSION In his pape , we ha e s udied he dynamics o closed chains o weakly coupled molecula subuni s unde he ac ion o a spa ially homogeneous ime-pe iodic ex e nal ield. Chains become la because o a weak in e ac ion wi h a la su ace. We in es iga ed he ole o he exci on-cu a u e in e ac ion on he o ma ion o he shape and ene gy dis ibu ion o closed semi lexible molecula chains. The epo ed shape ans o ma ion o he molecula chain is he esul o a nonequilib ium phase ansi ion, which is media ed by an ex e nal d i ing (ene gy pumping) in he sys em. We ha e ound ha in he absence o d i ing he a ay akes a s able ci cula shape. When he d i ing in ensi y exceeds some c i ical le el he ci cula shape o he chain becomes uns able and he chain akes a polygonal shape. In he case unde conside a ion, when he exci a ion locally ha dens he bending igidi y o he chain, he exci a ion ene gy is localized in such places whe e he chain is mo e la . The ansi ion o he polygonal shape is due o a Hop bi u ca ion. ACKNOWLEDGMENTS Y.B.G. acknowledges pa ial inancial suppo om a special p og am o he Na ional Academy o Sciences o Uk aine, and is hank ul o he Depa men o Applied Ma hema ics and Compu e Science and he Depa men o Physics, Technical Uni e si y o Denma k as well as he Uni e si y o Se ille o hospi ali y. J.F.R.A acknowledges G an No. 2011/FQM-280 om CEICE, Jun a de Andaluc´ ıa Spain. J.F.R.A. and V.J.S.-M. acknowledge inancial suppo om P ojec No. FIS2015-65998-C2-2-P om MINECO, Spain. APPENDIX The aim o his Appendix is o de i e an explici o m o he exci on-bending in e ac ion o a gene ic 3D ilamen . To desc ibe he ilamen lexibili y we use a disc e e wo mlike chain model. In he ame o his model he ilamen is conside ed as a chain o igid links, leng h a, wi h e ices loca ed a he poin s n=(xn,yn,zn),(n=1,...N). We a e in e es ed in he case when he chain is closed and so we impose he pe iodici y condi ion on he coo dina es: n+N= n.We assume ha each e ex ep esen s a complex subuni o he polyme which addi ionally o i s posi ion n( ) ca ies an in e nal elec omagne ically ac i e deg ee o eedom which can be cha ac e ized by he complex ampli ude n( ). We conside a ilamen whe e elec omagne ically ac i e subuni s a e coupled o bending deg ees o eedom. This coupling o igina es om he change in he in e ac ion ene gy (i.e., he an de Waals in e ac ion, iso opic pa o mul ipole-mul ipole in e ac ions, he in e molecula exchange in e ac ion, e c.) o he subuni wi h all neighbo ing subuni s in i s ansi ion o he exci ed s a e, and can be w i en as ollows: Eeb = n,n V(| n− n|)|ψn|2,(A1) 062227-7 YU. B. GAIDIDEI e al. PHYSICAL REVIEW E 93, 062227 (2016) whe e V(| n− n|) is he change o he in e ac ion be ween he n h subuni and he n h subuni when he o me occu s in he exci ed s a e. In he nex -nea es neighbo app oxima ion he in e ac ion be ween he subuni s has he o m, Eex−b= N  n=1{V(| n− n+1|)+V(| n− n−1|) +V(| n− n+2|)+V(| n− n−2|)}|n|2 = N  n=12V(a)+Va4−κ2 n+1 +Va4−κ2 n−1|n|2,(A2) whe e he de ini ion o he local cu a u e κn(2) is used. By assuming ha κ2 n1 and neglec ing dispe sion e ec s in he exci on-bending in e ac ion, i.e., 1 2(κ2 n+1+κ2 n−1)≈κ2 n, one can ob ain app oxima ely ha Eex−b= n m=±1,±2 V(ma)|n|2+Uex−b, Uex−b=χ n κ2 n|n|2.(A3) He e he pa ame e , χ=−adV( ) d  =2a ,(A4) cha ac e izes he in ensi y o he exci on-bending in e ac ion. [1] P. L. Ch is iansen, M. P. Sø ensen, and A. C. Sco (eds.), Nonlinea Science a he Dawn o he 21s Cen u y (Sp inge , Be lin, 2000). [2] L. V´ azquez, R. S. Mackay, and M. P. Zo zano (eds.), Localiza ion and Ene gy T ans e in Nonlinea Sys ems (Wo ld Scien i ic, Singapo e, 2003). [3] C. M. Soukoulis, Pho onic C ys als and Ligh Localiza ion in he 21s Cen u y, Na o Science Se ies, Vol. 563 (Kluwe Academic Publishe s, Do d ech , 2001). [4] M. Sa o, B. E. Hubba d, and A. J. Sie e s, Colloquium: Nonlin- ea ene gy localiza ion and i s manipula ion in mic omechanical oscilla o a ays, Re . Mod. Phys. 78,137 (2003). [5] H. G. C aighead, Nanoelec omechanical sys ems, Science 290, 1532 (2000). [6] R. H. Blick, A. E be, L. Pescini, A. K aus, D. V. Scheible, F. W. Beil, E. Hoehbe ge , A. Hoe ne , J. Ki schbaum, H. Lo enz, and J. P. Ko haus, Nanos uc u ed silicon o s udying undamen al aspec s o nanomechanics, J. Phys.: Condens. Ma e 14,R905 (2002). [7] B. Ilic, S. K ylo , K. Aubin, R. Reichenbach, and H. G. C aig- head, Op ical exci a ion o nanoelec omechanical oscilla o s, Appl. Phys. Le . 86,193114 (2005). [8] P. Maniadis and S. Flach, Mechanism o disc e e b ea he exci- a ion in d i en mic o-mechanical can ile e a ays, Eu ophys. Le . 74,452 (2006). [9] T. Ha ayama, P. Da is, and K. S. Ikeda, Nonlinea Whispe ing Galle y Modes, Phys.Re .Le .82,3803 (1999). [10] A. Wall a , A. V. Us ino , V. V. Ku in, I. A. She eshe sky, and N. K. Vdo iche a, Whispe ing Vo ices, Phys. Re . Le . 84, 151 (2000). [11] Yu. B. Gaididei, P. L. Ch is iansen, P G Ke ekidis, H. B¨ u ne , and A. R. Bishop, Localiza ion o nonlinea exci a ions in cu ed wa eguides, New J. Phys. 7,52 (2005). [12] Yu. B. Gaididei, S. F. Mingalee , and P. L. Ch is iansen, Cu a u e-induced symme y b eaking in nonlinea Sch ¨ odinge models, Phys. Re . E 62,R53 (2000). [13] J. F. R. A chilla, P. L. Ch is iansen, S. F. Mingalee , and Yu. B. Gaididei, Nume ical s udy o b ea he s in a ben chain o oscilla o s wi h long- ange in e ac ion, J. Phys. A: Ma h. Gen. 34,6363 (2001). [14] J. F. R. A chilla, P. L. Ch is iansen, and Yu. B. Gaididei, In e play o nonlinea i y and geome y in a DNA- ela ed, Klein-Go don model wi h long- ange dipole-dipole in e ac ion, Phys.Re .E65,016609 (2001). [15] J. F. R. A chilla, Yu. B. Gaididei, P. L. Ch is iansen, and J. Cue as, S a iona y and mo ing b ea he s in a simpli ied model o cu ed alpha-helix p o eins, J. Phys. A: Ma h. Gen. 35,8885 (2002). [16] P. V. La sen, P. L. Ch is iansen, O. Bang, J. F. R. A chilla, and Yu. B. Gaididei, Bubble gene a ion in a wis ed and ben DNA-like model, Phys.Re .E70,036609 (2004). [17] P. V. La sen, P. L. Ch is iansen, O. Bang, J. F. R. A chilla, and Yu. B. Gaididei, Ene gy unneling in a ben chain o Mo se oscilla o s wi h long- ange coupling, Phys. Re . E 69,026603 (2004). [18] V. J. S´ anchez-Mo cillo, N. Jim´ enez, J. Chaline, A. Bouakaz, and S. Dos San os, Spa io- empo al dynamics in a ing o coupled pendula: Analogy wi h bubbles, in Localized Exci a ions in Nonlinea Complex Sys ems,edi edbyR.Ca e e o-Gonzalez e al. (Sp inge , Cham, 2014), pp. 251–262. [19] J. Chaline, N. Jim´ enez, A. Meh em, A. Bouakaz, S. Dos San os, and V. J. S´ anchez-Mo cillo, Mac oscopic acous o-mechanical analogy o a mic obubble, J. Acous . Soc. Am. 138,3600 (2015). [20] S. F. Mingalee , Yu. B. Gaididei, P. L. Ch is iansen, and Y. S. Ki sha , Nonlinea i y-induced con o ma ional ins abili y and dynamics o biopolyme s, Eu ophys. Le . 59,403 (2002). [21] Yu. B. Gaididei, P. L. Ch is iansen, and W. J. Zak zewski, Con- o ma ional ans o ma ions induced by he cha ge-cu a u e in e ac ion: Mean- ield app oach, Phys.Re .E74,021914 (2006). [22] Yu. B. Gaididei, C. Go ia, P. L. Ch is iansen, and M. P. Sø ensen, Con o ma ional ans o ma ions induced by he cha ge-cu a u e in e ac ion a ini e empe a u e, Phys. Re . E78,051908 (2008). [23] Yu. B. Gaididei, C. Go ia, and P. L. Ch is iansen, Lange in dynamics o con o ma ional ans o ma ions induced by he cha ge–cu a u e in e ac ion, J. Biol. Phys. 35,103 (2009). [24] L. C uzei o, The VES hypo hesis and p o ein con o ma ional changes, Z. Phys. Chem. 230,743 (2016). 062227-8 ENERGY LOCALIZATION AND SHAPE TRANSFORMATIONS . . . PHYSICAL REVIEW E 93, 062227 (2016) [25] Z. Ganim, H. S. Chung, A. W. Smi h, L. P. De lo es, K. C. Jones, and A. Tokmako .Amide I wo-dimensional in a ed spec oscopy o p o eins.Acc. Chem. Res. 41,432 (2008). [26] A. Ba h and C. Zsche p, Wha ib a ions ell us abou p o eins, Q. Re . Biophys. 35,369 (2002). [27] L. C uzei o, The amide I band o c ys alline ace anilide: Old da a unde new ligh , in Quodons in Mica, Sp inge Se . Ma e . Sci., Vol. 221, edi ed by J. F. R. A chilla e al. (Sp inge , Be lin, 2015), pp. 401–424. [28] J. Edle and P. Hamm, Sel - apping o he amide I band in a pep ide model c ys al, J. Chem. Phys. 117,2415 (2002). [29] A. Xie, L. an de Mee , and R. H. Aus in, Exci ed-s a e li e imes o a -in a ed collec i e modes in p o eins, J. Biol. Phys. 28, 147 (2002). [30] A. D. Ba es and A. Maxwell, DNA Topology (Ox o d Uni e si y P ess, Ox o d, 2005). [31] A. D. Ba es, A. Noy, M. M. Pipe akis, S. A. Ha is, and A. Maxwell, Small DNA ci cles as p obes o DNA opology, Biochem. Soc. T ans. 41,565 (2013). [32] G. Wi z, K. Rechendo , J. Adamcik, and G. Die le , Con o - ma ion o Ci cula DNA in Two Dimensions, Phys.Re .Le . 101,148103 (2008). [33] Y. Z. Chen and E. W. P oho sky, Sequence and empe a u e dependence o he in e base hyd ogen-bond b ea hing modes in B-DNA polyme s: Compa ison wi h low- equency Raman peaks and hei ole in helix mel ing, Biopolyme s 35,573 (1995). [34] C. C. Shih and S. Geo ghiou, Ha monic analysis o DNA dynamics in a iscous medium, J. Biomol. S uc . Dyn. 17, 921 (2000). [35] V. L. Golo, Th ee-wa e in e ac ion be ween in e s and modes o he DNA, J. Exp. Theo . Phys. 101,372 (2005). [36] S. J. Wol man, G. D. Jay, and G. P. C aw o d, Liquid-c ys al ma e ials ind a new o de in biomedical applica ions, Na . Ma e . 6,929 (2007). [37] Y. Ji, J. E. Ma shall, and E. M. Te en je , Nanopa icle- liquid c ys alline elas ome composi es, Polyme s 4,316 (2012). [38] M. Camacho-Lopez, H. Finkelmann, P. Pal y-Muho ay, and M. Shelley, Fas liquid-c ys al elas ome swims in o he da k, Na . Ma e . 3,307 (2004). [39] M. Pey a d, Nonlinea Exci a ions in Biomolecules (Sp inge , Be lin, 1995). [40] E. M. Te en je , M. Cla ke, S. Ho a, and M. Wa ne , Liq- uid c ys alline elas ome s: Dynamics and elaxa ion s uc u e, Philos. T ans. R. Soc. London A 361,1(2003). [41] L. D. Landau and E. M. Li shi z, Theo y o Elas ici y (Pe gamon, Ox o d, 1986). 062227-9