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
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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)
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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
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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.
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