B ea he s in a sys em wi h helici y and dipole in e ac ion
B. Sa
´nchez-Rey, J. F. R. A chilla, and F. Palme o
Depa amen o de Fı
´sica Aplicada I, Uni e sidad de Se illa, A enida Reina Me cedes s/n, 41012-Se illa, Spain
F. R. Rome o
Facul ad de Fı
´sica, Uni e sidad de Se illa, A enida Reina Me cedes s/n, 41012-Se illa, Spain
共Recei ed 31 Janua y 2002; published 3 July 2002兲
Recen pape s ha ha e s udied a ian s o he Pey a d-Bishop model o DNA, ha e aken in o accoun he
long ange in e ac ion due o he dipole momen s o he hyd ogen bonds be ween base pai s. In hese models
he helici y o he double s and is no conside ed. In his pape we ha e pe o med an analysis o he in luence
o he helici y on he p ope ies o s a ic and mo ing b ea he s in a Klein-Go don chain wi h dipole-dipole
in e ac ion. I has been ound ha he helici y enla ges he ange o exis ence and s abili y o s a ic b ea he s,
al hough his e ec is small o a ypical helical s uc u e o DNA. Howe e , he e ec o he o ien a ion o he
dipole momen s is conside ably highe wi h anscenden al consequences o he exis ence o mobile b ea he s.
DOI: 10.1103/PhysRe E.66.017601 PACS numbe 共s兲: 63.20.Pw, 63.20.Ry, 63.50.⫹x, 87.10.⫹e
I. INTRODUCTION
A g ea deal o a en ion has been paid o he in e play
be ween geome y and nonlinea i y in loca ing p oblems in
ecen yea s. The ela ionship be ween geome y and nonlin-
ea i y has an impo an ole in he unc ions o some biomol-
ecules, such as DNA, whe e he localiza ion o ene gy has
been pu o wa d as a p ecu so y mechanism o he ansc ip-
ion bubble 关1兴, and mo ing localized exci a ions as a
me hod o anspo ing in o ma ion along he double s and
关2兴.
The ac ha hyd ogen bonds ha link each pai o bases
in DNA ha e a ini e dipole momen , has b ough abou he
in oduc ion o models 关3–6兴wi h long ange dipole-dipole
in e ac ion. Apa om i s heo e ical in e es , his in e ac ion
becomes ele an when he seconda y s uc u e o DNA is
conside ed. The shape o he molecule can in luence he lo-
caliza ion and anspo p ope ies o ene gy, which is
hough o play a biological unc ion 关7兴. Some o hese mod-
els 关3兴s udy he e ec s o he cu a u e in a chain o non-
linea oscilla o s using he disc e e nonlinea Sch o
¨dinge
equa ion. O he models conside Klein-Go don sys ems o
s udy kinks 关4兴, b ea he s in cu ed chains 关5兴o b ea he s
wi h wo compe ing in e ac ions 关6兴. Howe e , all hese
models wi h long ange in e ac ion ail o ake in o accoun
he peculia helicoidal s uc u e o he DNA chain, al hough
his has been conside ed in some models 关8兴wi hou he
dipole in e ac ion.
In his pape , we s udy he e ec o helici y on he p op-
e ies o b ea he s in a Klein-Go don model wi h dipole-
dipole in e ac ion. These pe iodic nonlinea localized oscil-
la ions in disc e e sys ems a e e y localized exci a ions ha
appea as a consequence o he nonlinea i y and disc e eness
o he sys em 关9兴. They a e specially sui able o biomol-
ecules when conside ing exci a ions ha in ol e a ew uni s,
ha is, a om he con inuous limi . They can be s a ic bu ,
unde ce ain condi ions, also mo e and anspo ene gy
along he sys em 关10兴.
We ha e ound ha he in oduc ion o helici y enhances
he s abili y o s a ic b ea he s, al hough his e ec is ela-
i ely small o he ypical helicoidal s uc u e o he DNA.
On he o he hand, he p o ile o he s a ic b ea he s and he
p ope ies o mo ing ones a e s ongly dependen on he
ela i e o ien a ion be ween he dipole momen s.
II. THE MODEL
The model is inspi ed by he p ima y s uc u e o DNA,
wi h dipole momen s pe pendicula o he helix axis, and
whe e he s e ching o he hyd ogen bonds wi hin base pai s
is desc ibed as a a ia ion o he dipole momen s. Mo e de-
ailed jus i ica ion o he model can be ound in 关6兴.
We deno e
␾
n he angle o he ndipole wi h espec o a
e e ence axis pe pendicula o he helix axis. Then, he
angle be ween he nea es neighbo ing dipoles is
␪
w⫽
␾
n
⫺
␾
n⫺1. We ha e conside ed his neighbo ing angle cons an
along he chain, and i will be called he wis ing angle. Thus,
␾
n⫹m⫺
␾
n⫽m
␪
w , and, he e o e, 2
␲
/
␪
w dipoles a e
needed o comple e a u n o sc ew. In DNA, o example,
he wis ing angle is 36° and a u n o sc ew equi es en
base pai s. Figu e 1 shows a ske ch o he model, whe e i
can be app ecia ed ha he sys em o dipoles ha e a helicoi-
dal s uc u e.
FIG. 1. Ske ch o he model a equilib ium. The a ows ep e-
sen he dipoles momen s, pe pendicula o he helix axis.
PHYSICAL REVIEW E 66, 017601 共2002兲
1063-651X/2002/66共1兲/017601共4兲/$20.00 ©2002 The Ame ican Physical Socie y66 017601-1
In he app op ia e dimensionless a iables, he Hamil-
onian o ou sys em becomes
H⫽兺
n⫽1
N
冉
1
2u
˙n
2⫹V共un兲⫹1
2J兺
m⫽n⫺N/2
n⫹N/2 unum
兩
n⫺m
兩
3
⫻cos关
␪
w共n⫺m兲兴
冊
,共1兲
whe e Nis he numbe o a iables. The a iables
兵
un
其
n⫽1
N,
whe e un⫾N⫽un, ep esen , in he con ex o he Pey a d-
Bishop model o DNA 关1兴, he ans e sal displacemen s o
he wo complemen a y nucleo ides in he n h pai wi h e-
spec o he molecula axis. In ou model, hey desc ibe he
s e ching o he dipoles wi h espec o hei equilib ium
leng hs. V(un) is he on-si e po en ial, which, in DNA mod-
els, desc ibes he hyd ogen bonds linking he wo bases, and
he pa ame e Jmeasu es he s eng h o he long ange
dipole-dipole in e ac ion. We ha e chosen he on-si e po en-
ial as he Mo se po en ial, gi en by
V共un兲⫽1
2共e⫺un⫺1兲2.共2兲
The eason o his, is ha i is a sui able po en ial o ep-
esen ing chemical bonds, being asymme ic, wi h a ha d
pa , modeling he epulsion be ween a oms o molecules,
and a so pa ha becomes la , modeling he b eakage o
he bond.
The dynamical equa ions become
u
¨n⫹V⬘共un兲⫹J兺
m⫽n⫺N/2
n⫹N/2 cos关
␪
w共n⫺m兲兴
兩
n⫺m
兩
3um⫽0, 共3兲
whe e n⫽1,...,N. To s udy he linea modes o he sys em
we eplace V⬘(un) in Eq. 共3兲wi h he linea e m un, which
implies ha he ime has been scaled so ha he linea e-
quency
␻
0⫽1. Conside ing solu ions o he o m un
⫽eiqn⫺iw he ollowing dispe sion ela ion is ob ained:
wk⫽
冑
1⫹2J兺
m⫽1
N/2 cos共m
␪
w兲
m3cos共mqk兲,共4兲
whe e qk⫽2
␲
k/N, wi h k⫽1,...,Ndue o he pe iodic
bounda y condi ions.
The a ia ion o he phonon band wi h he helici y is
shown in Fig. 2, whe e he equencies o he linea modes
a e ep esen ed as a unc ion o he wis ing angle,
␪
w , o a
ixed alue o he coupling pa ame e J⫽0.1. The e ec o
he wis ing is a na owing o he phonon band, which will
enhance he ange o exis ence and s abili y o he b ea he s.
This has been con i med nume ically.
III. BREATHER EXISTENCE AND STABILITY
We ha e s udied he exis ence and s abili y o b ea he s in
his model using he s anda d nume ical me hods desc ibed
in Re . 关11兴.
The Mo se po en ial is a so po en ial wi h he conse-
quence ha he equency o a b ea he has o be lowe han
he linea equency
␻
0⫽1. Thus, we ha e chosen,
␻
b
⫽0.8 so ha he nonlinea e ec will be signi ican bu on
he o he hand no o e ly s ong, as he nonlinea i ies in
DNA a e hough o be weak.
Fi s o all, he helici y in luences he b ea he p o ile. As
is shown in Fig. 3, o a ixed alue o he coupling pa am-
e e J, he inc ease o he wis ing angle p oduces a ansi ion
om a zigzag p o ile 共 he nea es neighbo oscilla ing in an-
iphase兲 o a bell p o ile 共all dipoles oscilla ing in phase兲.
This e ec ollows om he spa ial p o ile o he phonon
s a e wi h he lowes equency since he b ea he equency
is below he phonon band, and all he highe ha monics a e
way oo high o be ele an . Fo
␪
w⬍
␲
/2 he in e ac ion is
e ec i ely ‘‘an i e omagne ic,’’ which leads o s agge ed
phonons a he lowe band edges. In he same way o
␪
w
⬎
␲
/2 a ‘‘ e omagne ic’’ in e ac ion is p esen , which leads
o a nons agge ed phonon a he lowe band edge. The
b ea he bi u ca es om he lowe band edge phonons and
hus e ains he p ope y o he phonon s uc u e. Fo
␪
w
⫽
␲
/2 he sys em sepa a es in o wo nonin e ac ing subla -
ices: e en and odd si es. As a esul , in his case, he nea es
neighbo s a e a es and he odd si e subla ice emains un-
exci ed.
One-si e b ea he s a e s able a low coupling as was
p o ed by Aub y 关12兴. Fo any alue o he wis ing angle
␪
w⬍90° hey can be con inued om he an icon inuous
limi ill
␻
ben e s he phonon band. Jus be o e he b ea he
disappea s, i becomes uns able due o he occu ence o a
FIG. 2. E ec o helici y on he b ead h o he linea spec um.
The cu es ep esen he lowe and highe limi s o he phonon
band as a unc ion o he wis ing angle in deg ees o ixed cou-
pling pa ame e J⫽0.1. wis in dimensionless uni s.
FIG. 3. P o iles o he one-si e b ea he when wis ing is in-
c eased o ixed coupling J⫽0.1 and Mo se po en ial. Dimension-
less uni s.
BRIEF REPORTS PHYSICAL REVIEW E 66, 017601 共2002兲
017601-2
ha monic bi u ca ion in he e olu ion o he Floque eigen-
alues. The inc ease o he wis ing enhances he s abili y as
is shown in Fig. 4 共ci cles兲. This can be unde s ood i we
conside only he nea es neighbo in e ac ion 共NNI兲. Then
he in luence o helici y on he s abili y o he b ea he s
could be desc ibed by an e ec i e coupling Je ⫽Jcos
␪
w .
The one-si e b ea he wi hou wis ing loses i s s abili y o a
coupling alue o Jc
0. Wi h wis ing and only NNI his would
occu o Jc⫽Jc
0/cos
␪
w 共dash lines in Fig. 4兲, which concu s
wi h he nume ical esul s.
The wo-si e b ea he , which consis o wo neighbo ing
oscilla o s exci ed in phase, is also s able a low coupling.
This can be unde s ood in e ms o Aub y’s band heo y 关12兴.
When coupling is inc eased, a bubble o ins abili y appea s
due o K ein c unches be ween he phonon band eigen alues
and a localized eigen alue o he Floque ope a o . I we
con inue inc easing he coupling he double b ea he de i-
ni ely becomes uns able due o he occu ence o a subha -
monic bi u ca ion. Again, he e ec o he wis ing is o en-
la ge he ange o s abili y owa d highe alues o he
coupling pa ame e 共 ull ci cles in Fig. 4兲. This sugges s ha
wis ing migh be a way o con ol he s abili y o he b ea h-
e s in eal sys ems.
We ha e no conside ed he wo-si e b ea he in an iphase
because i coincides wi h he one-si e b ea he wi h zigzag
p o ile, i.e., he New on me hod con e ges o he same solu-
ion i we s a a he an icon inuous limi wi h one nonlinea
oscilla o o wi h wo nea es neighbo oscilla o s in an-
iphase.
A a he di e en si ua ion is he one wi h
␪
w⬎90°. Fi s ,
he one-si e b ea he is always s able un il i disappea s. Sec-
ond, he wo-si e b ea he is uns able a low coupling bu
becomes s able jus be o e i s ex inc ion. This beha io has
impo an consequences o he mobili y o hese b ea he s as
shown in he ollowing sec ion.
Fo he sake o ho oughness, we ha e also s udied he
e ec o wis ing wi h a ha d
␾
4po en ial V(un)⫽un
2
⫹1/4un
4, and a b ea he equency
␻
b⫽1.2. Quali a i ely he
esul s a e simila excep o he ac ha b ea he s wi h
␪
w⬍90° and b ea he s wi h
␪
w⬎90° exchange hei p op-
e ies.
IV. MOBILE BREATHERS
S a ic b ea he s unde ce ain condi ions can be mo ed.
The s anda d me hod o mo e a b ea he consis s in pe u b-
ing i s eloci y wi h spa ially an isymme ic ec o , called
he ma ginal mode 关10兴. Typically, his me hod wo ks wi hin
a ce ain ange o pa ame e s nea an exchange s abili y bi-
u ca ion. This occu s when a one-si e b ea he becomes un-
s able and a wo-si e b ea he does he opposi e a a nea by
poin .
We ha e looked o mobile b ea he s in ou sys em bo h
wi h a ha d
␾
4po en ial and wi h a Mo se po en ial, bu we
ha e only had success wi h Mo se po en ial and ‘‘ e omag-
ne ic’’ in e ac ion, i.e.,
␪
w⬎90o. In his pa icula case, we
ound a simila si ua ion o a s abili y exchange and we we e
able o mo e he b ea he pe u bing i wi h he uns able
localized mode o he wo-si e b ea he . This is an in e es ing
esul because his con igu a ion is equi alen o a chain o
an ipa allel dipoles wis ed
␲
⫺
␪
w⬍
␲
/2. In ac , we can
only expec pa allel dipoles in syn he ic DNA.
A use ul concep o desc ibing he b ea he mo emen is
i s e ec i e mass. I he no m o he pe u ba ion eloci y is
␭, he kine ic ene gy added o he b ea he by he pe u ba-
ion is E⫽␭2/2. The esul ing ansla ional eloci y o he
b ea he , , is ound o be p opo ional o ␭关10兴. Thus, mo -
ing b ea he s can be conside ed as a quasipa icle wi h a
mass o m*, which can be de ined h ough he ela ion
m* 2/2⫽␭2/2.
We ha e s udied he dependence o he e ec i e mass,
m*, wi h he coupling J. Figu e 5 shows he esul o an i-
pa allel dipoles (
␪
w⫽180°). Two di e en beha io s we e
ob ained depending on he ini ial condi ions. I we pe u b
he wo-si e b ea he , we obse e ha i s e ec i e mass in-
c eases mono onically wi h he coupling 共 ull squa es in Fig.
5兲. This e lec s he ac ha he wo-si e b ea he becomes
s able wi h inc easing coupling. Bu i a s a ic one-si e
b ea he is chosen as he ini ial con igu a ion, a minimum o
m*appea s showing he exis ence o an op imal alue o he
coupling o mo e his b ea he 共see blank squa es in Fig. 5兲.
We belie e ha his minimum exp esses a balance be ween
he wo opposi e e ec s p oduced by an inc ease o he cou-
pling on he s abili y o he one-si e and wo-si e b ea he s.
Simila esul s a e ob ained o o he alues
␪
w⬎
␲
/2.
FIG. 4. Range o s abili y o he one-si e 共ci cles兲and wo-si e
共 ull ci cles兲b ea he s as a unc ion o he wis ing angle in deg ees.
Jcis he maximum alue o he coupling pa ame e o which he
b ea he is s able. The dashed lines ep esen he alues calcula ed
wi hin he NNI app oxima ion. Jcis in dimensionless uni s.
FIG. 5. Dependence o he e ec i e mass o he mobile b ea h-
e s wi h he coupling pa ame e o
␪
w⫽180°. The blank squa es
co espond o mobile b ea he s ob ained om s a ic one-si e b ea h-
e s. Full squa es a e ob ained om s a ic wo-si e b ea he s.
BRIEF REPORTS PHYSICAL REVIEW E 66, 017601 共2002兲
017601-3
V. CONCLUSIONS
We ha e conside ed a sys em o oscilla ing dipoles wi h
helicoidal s uc u e in o de o s udy he e ec o helici y on
he exis ence and p ope ies o b ea he s. This s udy is mo-
i a ed by he helicoidal s uc u e o DNA, and he ac ha
i can be desc ibed by a educed dynamics whe e he only
deg ees o eedom a e he s e chings o he hyd ogen bonds
be ween base pai s, which ha e a ini e dipole momen . In
ou model, he helici y p oduces a na owing o he phonon
band, and an enla gemen o he ange o exis ence and s a-
bili y o he b ea he s, al hough his e ec is small o a
ypical helicoidal s uc u e o DNA.
The e ec o he o ien a ion o he dipole momen s, i.e., i
he wis ing angle is g ea e o no han 90°, is howe e
conside ably highe . In pa icula , we ha e only ound mo-
bile b ea he s wi h a Mo se po en ial and
␪
w⬎
␲
/2. Unde -
s anding he necessa y condi ions o mo e a b ea he is s ill
an open ques ion oday.
ACKNOWLEDGMENTS
This wo k was suppo ed by he Eu opean Union unde
he RTN p ojec , LOCNET, HPRN-CT-1999-00163. We ac-
knowledge Jesu
´s Cue as o his use ul commen s.
关1兴M. Pey a d and A.R. Bishop, Phys. Re . Le . 62, 2755 共1989兲.
关2兴M. Sale no and Yu. Ki sha , Phys. Le . A 193, 263 共1994兲.
关3兴P.L. Ch is iansen, YB. Gaididei, and S.F. Mingalee , J. Phys.:
Condens. Ma e 13, 1181 共2001兲; Yu.B. Gaididei, S.F. Minga-
lee , and P.L. Ch is iansen, Phys. Re . E 62, R53 共2000兲; S.F.
Mingalee , P.L. Ch is iansen, Yu.B. Gaididei, M. Johansson,
and K.O”. Rasmussen, J. Biol. Phys. 25,41共1999兲.
关4兴S.F. Mingalee , YB. Gaididei, E. Maje niko a, and S.
Shpy ko, Phys. Re . E 61, 4454 共2000兲.
关5兴J.F.R. A chilla, P.L. Ch is iansen, S.F. Mingalee , and Yu.B.
Gaididei, J. Phys. A 34, 6363 共2001兲; J.F.R. A chilla, P.L.
Ch is iansen, and Yu.B. Gaididei Phys. Re . E 65, 16 609
共2002兲.
关6兴J. Cue as, J. F. R. A chilla, Yu. B. Gaididei, and F. R. Rome o,
Physica D 共 o be published兲.
关7兴C. Calladine and H. D ew, Unde s anding DNA 共Academic
P ess, London, 1992兲.
关8兴M. Ba bi, S. Cocco, and M. Pey a d, Phys. Le . A 253, 358
共1999兲. M. Ba bi, S. Cocco, M. Pey a d, and S. Ru o, J. Mol.
Biol. 24,97共1999兲; G. Gae a, C. Reisss, M. Pey a d, and T.
Dauxois, Ri . Nuo o Cimen o 17,1共1994兲.
关9兴R.S. MacKay and S. Aub y, Nonlinea i y 7, 1623 共1994兲;S.
Flach and C.R. Willis, Phys. Rep. 295, 181 共1998兲.
关10兴Ding Chen, S. Aub y, and G.P. Tsi onis, Phys. Re . Le . 77,
4776 共1996兲; S. Aub y and T. C e egny, Physica D 119,34
共1998兲.
关11兴J.L. Ma in and S. Aub y, Nonlinea i y 9, 1501 共1996兲.
关12兴S. Aub y, Physica D 103, 201 共1997兲.
BRIEF REPORTS PHYSICAL REVIEW E 66, 017601 共2002兲
017601-4