a Xi :nlin/0307031 2 [nlin.PS] 25 No 2003
E ec o base-pai inhomogenei ies on cha ge
anspo along DNA media ed by wis and
adial pola ons
F Palme o, JFR A chilla
ETS Ingenie ´ıa In o m´a ica. Uni e sidad de Se illa.
A da Reina Me cedes s/n, 41012-Se illa, Spain
D Hennig
F eie Uni e si ¨a Be lin,
Fachbe eich Physik, A nimallee 14, 14195-Be lin (Ge many)
FR Rome o
Facul ad de F´ısica. Uni e sidad de Se illa.
A da Reina Me cedes s/n, 41012-Se illa, Spain
15 Jul 2003
Abs ac
Some ecen esul s o a h ee–dimensional, semi–classical, igh –
binding model o DNA show ha he e a e wo ypes o pola ons,
named adial and wis pola ons, ha can anspo cha ge along he
DNA molecule. Howe e , he exis ence o wo ypes o base pai s in
eal DNA, makes i c ucial o ind ou i cha ge anspo also exis
in DNA chains wi h di e en base pai s. In his pape we add ess
his p oblem in i s simple case, an homogeneous chain excep o a
single di e en base pai , wha we call a base-pai inhomogenei y,
and i s e ec on cha ge anspo . Radial pola ons expe ience ei he
e lec ion o apping. Howe e , wis pola ons a e good candida es
o cha ge anspo along eal DNA. This anspo is also e y obus
wi h espec o weak pa ame ic and diagonal diso de .
Keywo ds:DNA, pola ons, cha ge anspo
PACS: 87.-15. ,63.20.K ,63.20.Ry,87.10.+e
1
1 In oduc ion
Cha ge anspo in biomolecules is o ele an in many biological unc ions.
Mo eo e , elec onic anspo plays a well known and undamen al ole in
many o biological p ocesses, such as ne ous conduc ion, pho osyn hesis, cel-
lula espi a ion and edox eac ions. Recen ly, DNA–media ed cha ge ans-
po has ocused much a en ion wi h expe imen s ha ha e been conduc ed
o de e mine i s conduc i e p ope ies, and heo e ical s udies ha e add essed
he possibili y o cha ge anspo and i s e iciency [1]. The impo ance o
cha ge anspo h ough DNA is signi ican because some mu a ions in li -
ing sys ems and adical mig a ions a e c i ical issues in ca cinogenesis s udies
and may yield insigh in o damage p e en ion o epai p ocesses [2, 3, 4].
Mo eo e , cha ge anspo has been demons a ed o p oceed wi hin HeLa
cell nuclei [5] as well as in he nucleosome co e pa icles [6], and can p o ide
a p ac ical me hod o gene ic sc eening o known gene sequences and an al-
e na i e me hod o hyb idiza ion based a ays [7]. Ma e ial scien is s ha e
hough ha DNA has a undamen al physical in e es o he de elopmen
o DNA–based molecula echnologies, as i possesses ideal s uc u al and
molecula ecogni ion p ope ies o use in sel assembling nanode ices wi h
a de ini e molecula a chi ec u e [8].
Pho ochemical elec on ans e expe imen s ha e been ca ied ou wi h
em osecond esolu ion, and ha e shown, wi h he p ecise con ol o he DNA
sequences, a as ime scale (≤5 ps) p ocess, a ibu ed o cha ge mobili y
along an unpe u bed double s and, and a slowe p ocess (≤75 ps), ha
hey belie e e lec s cha ge hopping be ween pe u bed domains along he
DNA s and [9].
Expe imen al esul s ha e explained ha DNA ac s as a linea chain wi h
o e lapping πo bi als loca ed a he s acked base pai s and i s conduc i i y
is e y sensi i e o dis up ion, caused by base–pai mis akes o in e ac ions
wi h p o eins, in he base–pai s ack. Depending on he base pai sequences
he ans e can be slowed o inhibi ed [10].
Mo eo e , he obus , malleable one–dimensional s uc u e o DNA can
be used o design elec onic de ices based on bioma e ials [11, 12, 13, 14, 15].
The s uc u e o he ben double helix λ-DNA can be modelled by a ne -
wo k o oscilla o s aking in o accoun de o ma ions o he hyd ogen bonds
wi hin a base pai and wis mo ions be ween consecu i e base pai s. The
h ee–dimensional semi–classical igh –binding model o DNA ha was i s
p oposed in Re . [16, 17] makes also he assump ion ha he elec on mo-
ion is p edominan ly in luenced by ib a ional modes o he double helix.
The nonlinea in e ac ion be ween he elec on and he ib a ional modes is
esponsible o he o ma ion o pola ons o elec on- ib on b ea he s. These
2
a e localized exci a ion pa e ns ha can be s a ic, p oducing cha ge local-
iza ion, o mobile, b inging abou cha ge anspo . I has been conside ed
some o he models wi h inhomogenei ies [18].
A a ian o his model has been p oposed in Re . [19], which li s he
es ic ion o he p e ious one, ha he pe u ba ion o he spa ial a i-
ables we e small enough o pe o m a linea app oxima ion in he dynamical
equa ions. Two ypes o pola ons appea depending on he pa ame e α,
ha desc ibes he coupling be ween he ans e in eg al and he dis ance
be ween nucleo ides, and he o m o ac i a ion: adial pola ons and wis
pola ons. Fo adial pola ons, he de o ma ions associa ed by he elec onic
cha ge a ec mainly o he adial a iables, and o wis pola ons, o he
angula ones. In he las case, he mo ing pola on is slowe han in he p e i-
ous one, bu i is mo e obus wi h espec o he in oduc ion o pa ame ic
diso de . See e e ence abo e o de ails and Re . [20] o wis pola ons in
o he model.
In his pape we conside he mo emen o pola ons along DNA chains
made ou o di e en base pai s. The mos simple one consis s o homo-
geneous DNA chains excep o a single di e en base pai , which we call
base-pai inhomogenei y.
We ha e ound ha when cha ge anspo is media ed by means o a wis
pola on, his mo ing ca ie can anspo cha ge along a base-pai inhomo-
genei y in a e y e icien way. Mo eo e , cha ge anspo pe sis s unde
he in oduc ion o a high deg ee o pa ame ic diso de , ha ep esen s he
inhe en diso de in he su ounding medium and he inhomogeneous dis i-
bu ion o coun e ions along he DNA duplex [21]. On he o he hand, adial
pola ons a e ei he e lec ed o apped by he base-pai inhomogenei y, mak-
ing hem poo candida es o cha ge anspo along he e ogeneous DNA a
leas in he amewo k o he igh -binding models.
2 DNA model
2.1 Desc ip ion
We conside a a ian [19] o he model Hamil onian o cha ge anspo
along DNA ha was i s p oposed in [16]. This a ian consis s o con-
side ing he ull dynamical equa ions o he sys em ins ead o pe o ming
linea app oxima ions based on he assump ion ha he de o ma ions o he
molecule we e small.
These models we e designed o implemen he basic cha ac e is ics o he
DNA double helix s uc u e needed o a p ope desc ip ion o he cha ge
3
n
l0+dn n−1
nθ0+φn
θ0
R0
l0
Figu e 1: Ske ch o he model. Filled ci cles ep esen bases. The a iables
used in he ex a e displayed.
anspo dynamics. The s a ing poin is he wis -opening model [22, 23,
24] ha has aken in o accoun he helicoidal s uc u e o DNA and he
o sional de o ma ions induced by he opening o he base pai s. The bases
a e conside ed as single nonde o mable objec s. The helicoidal s uc u e
o DNA is desc ibed in a cylind ical e e ence sys em and he n– h base
pai has wo deg ees o eedom, namely ( n, φn), whe e n ep esen s he
adial displacemen o he base pai om he equilib ium alue R0, and φn
ep esen s he angula de ia ion om equilib ium angles wi h espec o a
ixed ex e nal e e ence ame. As we a e in e es ed in base pai ib a ions
and no in acous ic mo ions, he cen e o mass o each base pai is ixed,
i.e., he wo bases in a base pai a e cons ained o mo e symme ically
wi h espec o he molecule axis. Mo eo e , he dis ances be ween wo
neighbo ing base pai planes will be ea ed as ixed, because in he axial
di ec ion DNA is less de o mable han wi hin he base pai planes [22].
A ske ch o he helicoidal s uc u e o he DNA model is shown in Fig. 1.
The ans e sal displacemen s o he base pai s a e de o ma ions o he
H-bonds and he angula wis and he adial ib a ional mo ion e ol e in-
dependen ly on wo di e en ime scales, hen hey can be conside ed as
decoupled deg ees o eedom in he ha monic app oxima ion o he no mal
4
mode ib a ions [25].
2.2 Hamil onian
The Hamil onian o he elec on anspo along a s and o DNA is gi en by
b
H=b
Hel +b
H ad +b
H wis , whe e b
Hel co esponds o he pa ela ed wi h he
pa icle cha ge anspo o e he base pai s, b
H ad desc ibes he dynamics o
he H-bond ib a ions and b
H wis is he pa co esponding o he dynamics
o he ela i e wis angle be ween wo consecu i e base pai s. This elec onic
pa is desc ibed by a igh –binding sys em o he o m:
b
Hel =X
n
En|nihn| − Vn−1,n|n−1ihn| − Vn+1,n|n+ 1ihn|,(1)
whe e |ni ep esen s a localized s a e o he cha ge ca ie a he n h base pai .
The quan i ies {Vn,n−1}a e he nea es –neighbo ans e in eg als along base
pai s, and {En}a e he ene gy on–si e ma ix elemen s. A gene al elec onic
s a e is gi en by |Ψi=Pncn( )|ni, whe e cn( ) is he p obabili y ampli ude
o inding he cha ged pa icle in he s a e |ni. The ime e olu ion o he
{cn( )}is ob ained om he Sch ¨odinge equa ion i~(∂Ψ/∂ ) = b
Hel|Ψi
The nucleo ides a e la ge molecules and hey mo e much mo e slowly han
a cha ged pa icle, hen he la ice oscilla o s can be desc ibed classically and
b
H ad and b
H wis a e, de ac o, classical Hamil onians. Fo homogeneous chains
hey a e gi en by (omi ing he ha o e hem):
H ad =X
n1
2M(p
n)2+MΩ2
,n
2 2
n,(2)
H wis =X
n1
2J(pφ
n)2+JΩ2
φ
2(φn−φn−1)2,(3)
whe e p
nand pφ
na e he conjuga e momen a o he adial and angula co-
o dina es, espec i ely. In hese exp essions Mis he mass o each base pai
(M= 2m,being man a e age es ima ion o he nucleo ide mass), J=M R2
0
is he momen o ine ia o each base pai and Ω ,n is he linea adial e-
quency which is p opo ional o he s eng h o he hyd ogen bonds. Indeed,
o an homogeneous chain all hese equencies a e equals, ha is, Ω2
,n =bnΩ2
wi h bn= 1 ∀n.
In his pape we a e in e es ed in he s udy o cha ge anspo along
DNA s ands whe e all he base pai s a e o he same ype, say A-T (o
C-G), excep only one o hem which is o a di e en ype, say C-G (o A-
T). We can suppose ha he a io be ween he elas ic cons an s o bonds
5

in a C-G base pai and an A-T base pai is 3/2 because he i s in ol es
h ee hyd ogen bonds and he second wo o hem. Then, as in Re . [26]
we ake bn= 0.8 o an A-T base pai , and bn= 1.2 o a C-G base pai .
Ωφis he linea wis equency, and we ep esen by θn,n−1= (φn−φn−1),
he de ia ion o he ela i e angle be ween wo adjacen base pai s om i s
equilib ium alue θ0.
In gene al, he ioniza ion po en ial o di e en nucleo ides di e s an
amoun o 0.2-1.0 eV [27]. In ou model, his implies di e en alues o
he on si e ene gies E0
n o each base pai s. In his wo k, we ha e ocused in
geome ical e ec s due o he s e ching o he chain o e he cha ge, and we
ha e conside ed E0
nindependen on he ype o base.
The elec onic pa o he Hamil onian, b
Hel, has a dependence on he
s uc u e a iables nand φn h ough he dependence o he ma ix elemen s
Enand Vn,n−1on hem. The ene gy on–si e ma ix elemen s a e gi en by [28]
En=E0
n+k n, exp essing he modula ion o he on–si e elec onic ene gies
{E0
n}by he adial de o ma ions o he base pai s. The ans e ma ix
elemen s Vn,n−1, which a e esponsible o he anspo o he elec on along
he s acked base pai s, a e assumed o depend on he dis ances be ween wo
consecu i e bases along a s and dn,n−1as Vn,n−1=V0(1−αdn,n−1), whe e αis
a pa ame e ha desc ibes he in luence o he dis ance be ween nucleo ides,
and he la e is de e mined by
dn,n−1= [a2+ (R0+ n)2+ (R0+ n−1)2−
2(R0+ n)(R0+ n−1) cos(θ0+θn,n−1)]1/2−l0,(4)
wi h l0= (a2+ 4R2
0sin2(θ0/2))1/2,whe e ais he e ical dis ance be ween
wo consecu i e base pai s.
In his pape , as in Re . [19], we ha e no used he expansion o his ex-
p ession up o i s o de a ound he equilib ium posi ions, as was pe o med
in Re s. [16, 17]. This allow us o conside pa ame e s ha allow la ge
de o ma ions in he angula a iables.
Realis ic pa ame e s o he DNA a e gi en in Re s. [23, 29]. We ha e
conside ed: a= 3.4˚
A,m= 300 amu, R0= 10˚
A,Ω = 8 ×1012 s−1, Ωφ=
9×1011 s−1. Ab ini io calcula ions ind ha be ween adjacen nucleo ides,
he ans e in eg al is in o de o 0.1-0.4 eV [30] . Al hough his ans e
in eg al is di e en o each pai o di e en nucleo ides, we will conside he
same alue o all neighbo ing cases V0= 0.1 eV, a supposi ion widely used
and can be alid in o de o ep oduce ab ini io esul s and expe imen s [31].
We scale he ime acco ding o →Ω , and we in oduce he dimen-
sionless quan i ies: ˜ n= n(MΩ2
/V0)1/2,˜
kn=kn/(MΩ2
V0)1/2,˜
En=En/V0,
˜
Ω = Ωφ/Ω ,˜
V=V0/(JΩ2
), ˜α=α(V0/M Ω2
)1/2,˜
R0=R0(MΩ2
/V0)1/2.
6
2.3 Dynamical equa ions
Using he expec a ion alue o he elec onic con ibu ion o he Hamil o-
nian, he new classical Hamil onian H=hφ|b
H|φi/V0is gi en in he scaled
a iables (omi ing he ildes) by
H=X
nn1
2( ˙ 2
n+bn 2
n) + R2
0
2[˙
φ2
n+ Ω2(φn−φn−1)2] +
(E0
n+k n)|cn|2−(1 −αdn,n−1)(c∗
ncn−1+cnc∗
n−1)o.(5)
The Sch ¨odinge equa ion and he Hamil onian equa ions lead o he scaled
dynamical equa ions o he sys em:
iτ˙cn= (E0
n+k n)cn
−(1 −α dn+1,n)cn+1 −(1 −α dn n−1)cn−1,(6)
¨ n=−bn n−k|cn|2
−α∂dn,n−1
∂ n
(c∗
ncn−1+cnc∗
n−1) + ∂dn+1,n
∂ n
(c∗
n+1cn+cn+1c∗
n),(7)
¨
φn=−Ω2(2φn−φn−1−φn+1)
−α V ∂dn,n−1
∂φn
(c∗
ncn−1+cnc∗
n−1) + ∂dn+1,n
∂φn
(c∗
n+1cn+cn+1c∗
n)(8)
whe e he quan i y τ=~Ω /V0measu es he ime scale sepa a ion be ween
he as elec on mo ion and he slow bond ib a ions. In he o de ed case,
wi h En=E0,∀n, wi h he limi o α= 0 he se o coupled equa ions
ep esen s he Hols ein sys em, widely used in s udies o pola on dynamics
in one-dimensional la ices. Also, o α=k= 0, and andom E0
n, he
Ande son model is ob ained.
The scaled pa ame e s ake he alues τ= 0.053, Ω2= 0.013, V=
2.5×10−4,R0= 63.1 and l0= 44.5. We ix he alue k= 1 and conside he
pa ame e αas adjus able.
3 Cha ge anspo media ed by mobile po-
la ons
3.1 Mobile pola ons
The p incipal in e es o his pape consis s in he s udy o cha ge anspo
along he double s and by mo ing pola ons in p esence o a base-pai inho-
mogenei y. We need i s o ob ain localized s a iona y solu ions o Eqs. (6-8).
7
In Appendix A, we p esen he p ocedu e ha we ha e ollowed o ob aining
hese s a iona y solu ions applied o di e en cases: homogeneous chains,and
inhomogeneous chains. The pola on mo ion can be ac i a ed unde ce ain
condi ions. A sys ema ic me hod o do his, known as he pinning mode-
me hod [32], consis s o pe u bing he (ze o) eloci ies o he g ound s a e
wi h localized, spa ially an isymme ic modes ob ained in he icini y o a
bi u ca ion. This me hod leads o mo ing en i ies wi h e y low adia ion,
bu has he incon enience o being applicable only in he neighbo hood o
ce ain alues o he pa ame e s. An al e na i e is he disc e e g adien
me hod [33], pe u bing he (ze o) eloci ies o he s a iona y s a e {˙ n(0)},
{˙
φn(0)}in a di ec ion pa allel o he ec o s (∇ )n= ( n+1 − n−1) and/o
(∇φ)n= (φn+1 −φn−1). Al hough his me hod does no gua an ee mobili y,
i ne e heless p o es o be success ul in a wide pa ame e ange. We will
deno e he ene gy o he pe u ba ion as ∆K=λ2/2, whe e λis he modulus
o he ec o used o pe u b he sys em (pa allel o ∇ nand/o ∇φn). This
ene gy gi es us he di e ence o he ene gy be ween he mo ing pola on and
he s a ic one, and i is usually called he ac i a ion ene gy.
3.2 Homogeneous chains
In his sec ion we ecall he basics esul s ob ained in Re . [19], see his e -
e ence o de ails. Fo homogeneous chains in absence o diagonal diso de ,
E0
n=E0,∀n, and i he pa ame e αis small enough, i is no possible o
mo e he pola ons by pe u bing he angula a iables. Mobili y can be ac-
complished only by pe u bing h ough he adial a iables. Howe e , o
la ge alues o he pa ame e α(α&0.01), he pola ons can only be mo ed
by pe u bing he angula a iables, i.e., pe u ba ions o he adial a iables
canno ac i a e mobili y. Ne e heless, o in e media e alues o he pa am-
e e α(0.005 .α.0.01), he pola ons can become mobile by pe u bing
any se o a iables, he adial o he angula ones. The mo emen is a he
di e en , when he pola on p opaga ion is ac i a ed by means o only adial
pe u ba ions, i s cha ac e is ics a e simila o he adial mo abili y egi-
men, and likewise when i is ac i a ed by angula pe u ba ions. A de ailed
analysis o hese di e en egimes can be ound in he e e ence abo e. In
gene al, adial mo abili y equi es less ene gy and has highe eloci y han
he angula one. Also, he limi s o hese egimes a e no exac , depending
on he kine ic ene gy o he pe u ba ion.
I an amoun o diso de in he on–si e ene gies E0
nis in oduced, wi h
andom alues |E0
n|<∆E, we ind ha mo ing pola ons exis below a c i i-
cal alue ∆Ec i . Beyond his alue, pola ons canno be mo ed. In gene al,
as shown in Fig. 2, co esponding o he mixed egime whe e pola on can be
8
Figu e 2: Veloci y o he pola on as a unc ion o he ene gy amoun ∆K
co esponding o he mixed egime when he pola on is ac i a ed by means
o angula pe u ba ions (α= 0.01). Ci cles and solid lines ep esen he
diso de ed case wi h ∆E= 0.05 and iangles and dashed line he o de ed
case.
ac i a ed (in absence o diso de ), by means o bo h angula and adial pe -
u ba ions, he mobili y induced by angula ac i a ion is mo e obus wi h
espec o pa ame ic diso de . I diso de is high enough, adial pe u ba-
ions ha could mo e pola on des oys i . Thus, i only can ac i a ed by
means o angula pe u ba ions. In his case, he mo emen is simila o
he o de ed case, he pola on has lowe eloci y, and he ac i a ion ene gy is
highe han in he adial mo abili y egime.
As shown in Fig. 2, i can be app ecia ed ha o a pola on in he mixed
egime, and i ∆Eis high enough, i is impossible o mo e i wi h adial
pe u ba ions, bu only wi h angula pe u ba ions. In his si ua ion, he
mo emen is e y simila o he o de ed case.
3.3 DNA chains wi h a base-pai inhomogenei y
The s udies o pola ons in homogeneous chains a e only applicable o syn-
he ic DNA made ou o a single ype o base pai . In eal DNA, he wo
di e en base pai s, A-T and G-C, combine in di e en ways cons i u ing
9
Figu e 7: P o iles o he g ound s a e in he homogeneous and o de ed chain.
(a) Wa e unc ion ampli udes |cn|2. (b) S a ic adial displacemen s n. (c)
S a ic wis s elonga ions θnn−1
16

Figu e 8: P o iles o he g ound s a e in he homogeneous and diso de ed
chain. (a) Wa e unc ion ampli udes |Cn|2. (b) S a ic adial displacemen s
n. (c) S a ic wis s elonga ions θnn−1
17
Figu e 9: P o iles o he g ound s a e in a C-G chain cen e ed in he inho-
mogenei y, an A-T base pai . (a) Wa e unc ion ampli udes |Cn|2. (b) S a ic
adial displacemen s n. (c) S a ic wis s elonga ions θn n−1
p e ious sec ion, his ac is de e minan in ela ion wi h he ansmission,
e lec ion o apping o a mo ing pola on by he inhomogenei y.
Acknowledgmen s
The au ho s acknowledge D . Jose Ma ´ıa Rome o, om he GFNL o he
Uni e si y o Se illa, o aluable sugges ions. They a e g a e ul o pa -
ial suppo unde he LOCNET EU ne wo k HPRN-CT-1999-00163. JFRA
acknowledges DH and he Ins i u ¨u Theo e ische Physik o hei wa m
hospi ali y. D.H. acknowledges suppo by he Deu sche Fo schungsgemein-
scha ia a Heisenbe g ellowship (He 3049/1-1).
Re e ences
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18
Figu e 10: P o iles o he g ound s a e in an A-T chain cen e ed in he
inhomogenei y, a C-G base pai .(a) Wa e unc ion ampli udes |Cn|2. (b)
S a ic adial displacemen s n. (c) S a ic wis s elonga ions θn n−1
19
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