a Xi :nlin/0308003 1 [nlin.PS] 1 Aug 2003
Cha ge anspo in poly(dG)-poly(dC)
and poly(dA)-poly(dT) DNA polyme s
D. Hennig1, E.B. S a iko 2, J.F.R. A chilla3and F. Palme o3
1F eie Uni e si ¨a Be lin, Fachbe eich Physik, Ins i u ¨u Theo e ische Physik
A nimallee 14, 14195 Be lin, Ge many
2Ka olinska Ins i u e, Cen e o S uc u al Biochemis y NOVUM
S - 14157 Huddinge, Sweden
3G oup o Nonlinea Physics, Depa amen o de F´
isica Aplicada I
ETSI In o m´a ica, A da Reina Me cedes, s/n. 41012 - Se illa, Spain
Augus 1, 2003
Abs ac
We in es iga e he cha ge anspo in syn he ic DNA polyme s buil up
om single ypes o base pai s. In he con ex o a pola on-like model,
o which an elec onic igh -binding sys em and bond ib a ions o he
double helix a e coupled, we p esen es ima es o he elec on- ib a ion
coupling s eng hs u ilizing a quan um-chemical p ocedu e. Subsequen
s udies conce ning he mobili y o pola on solu ions, ep esen ing he s a e
o a localized cha ge in unison wi h i s associa ed helix de o ma ion, show
ha he sys em o poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA poly-
me s, espec i ely possess quan i a i ely dis inc anspo p ope ies.
While he o me suppo s unidi ec ionally mo ing elec on b ea he s a -
ibu ed o highly e icien long- ange conduc i i y he b ea he mobili y
in he la e case is compa a i ely es ained inhibi ing cha ge anspo .
Ou esul s a e in ag eemen wi h ecen expe imen al esul s demon-
s a ing ha poly(dG)-poly(dC) DNA molecules ac s as a semiconduc ing
nanowi e and exhibi s be e conduc ance han poly(dA)-poly(dT) ones.
PACS numbe s: 87.-15. , 63.20.K , 63.20.Ry
1 In oduc ion
Pa icula ly wi h iew o possible applica ions in molecula elec onics based
on bioma e ials elec onic anspo (ET) h ough DNA has ecen ly become
o in ensi ied in e es [1],[2]. Al hough he deba e whe he DNA cons i u es a
conduc o is s ill ongoing, he e exis al eady s ong expe imen al e idence ha
DNA o ms an e ec i ely one-dimensional molecula wi e [3]. Among se e al
heo e ical a emp s o desc ibe he cha ge anspo mechanism in DNA he
pola on app oach has u ned ou la ely o be a p omising candida e o modeling
cons uc i e in e play be ween he cha ge ca ying sys em and ib a ional de-
g ees o eedom o he DNA conspi ing o es ablish cohe en ET [4]-[8]. Recen
1
expe imen s a e in suppo o he pola on mechanism o ET in DNA polyme s
[9]. The p esen s udy deals wi h he heo e ical desc ip ion o ET in syn he -
ically p oduced DNA polyme s consis ing o a single ype o base pai s, i.e.
ei he poly(dG)-poly(dC) o poly(dA)-poly(dT) DNA polyme s [10]. U ilizing
a nonlinea app oach based on he concep o b ea he and pola on solu ions we
explo e whe he conduc i i y depends on he ype o he DNA polyme which
migh be o in e es o he design o syn he ic molecula wi es. Mo eo e , we
amelio a e p eceding s udies o ET in DNA [11],[12] in he sense ha , ins ead
o adjus ing he coupling pa ame e s, we use now c edible es ima es o hem
de i ed wi h he help o quan um-chemical me hods.
2 Model o pola on-like cha ge anspo in DNA
Ou model o cha ge anspo in DNA is based on he inding ha he cha ge
mig a ion p ocess is dominan ly in luenced by he ans e se ib a ions o he
bases ela i e o each o he in adial di ec ion wi hin a base pai plane [13].
In ac , he impac o o he ib a ional deg ees o eedom (e.g. wis mo ions,
helical pi ch changes and longi udinal acous ic phonons along he s ands which
a e signi ican ly es ained by he backbone igidi y) can be expec ed o be
negligible wi h espec o ET in DNA. Hence, he mo ion can be iewed as
con ined o he base pai planes [14].
The Hamil onian o he ET along a s and in DNA comp ises wo pa s
H=Hel +H ib ,(1)
wi h Hel is he pa which desc ibes he ET o e he base pai s and H ib ep-
esen he dynamics o adial ib a ions o he base pai s . The elec onic pa
is gi en by a igh -binding sys em
Hel =X
n
En|cn|2−Vn n−1c∗
ncn−1+cnc∗
n−1.(2)
The index ndeno es he si e o he n− h base on a s and and |cn|2de e mines
he p obabili y o ind he elec on (cha ge) esiding a his si e. Enis he
local elec onic ene gy and Vnn−1is he ans e ma ix elemen media ing he
anspo o he elec on along he s acked base pai s. We make he usual
assump ion ha ET akes place only along he base pai sequence on a s and
excluding in e -s and ET.
The ib onic pa o he Hamil onian H ib models dynamical changes o he
adial equilib ium posi ions o he bases. Supposing ha hese adial ib a ions
can be ea ed classically and ha monically we ep esen H ib as
H ib =1
2X
n1
M(p
n)2+MΩ2
2
n.(3)
The adial coo dina es nquan i y he adial displacemen s o he base uni s
om hei equilib ium posi ions along he line b idging wo bases o a base pai
wi hin he base pai plane. Mdeno es he educed mass and Ω is he ha monic
equency. Due o he cons ain en o ced by he suga -phospha e backbone
adial displacemen s lead no only o changes o he equilib ium leng hs o he
2
n
l0+dn n−1
θ0
θ0
R0
l0
Figu e 1: Ske ch o he s uc u e o he DNA model. The bases a e ep esen ed
by bulli s and he geome ical pa ame e s R0,l0,θ0, nand dn n−1a e indica ed.
hyd ogen bonds (b idging wo bases o a base pai ) bu a e also connec ed wi h
a change o he h ee-dimensional dis ance be ween wo consecu i e bases on
a s and en ailing de o ma ions o he co esponding co alen bonds. In an
expansion up o i s o de a ound he equilib ium posi ions [15] his s acking-
dis ance is de e mined by
dn n−1=R0
l0
( 1 −cos θ0) ( n+ n−1).(4)
The geome ical pa ame e s R0,θ0and l0de e mine he equilib ium alue o
he adius, he wis angle be ween wo adjacen base pai s, and he equilib-
ium leng h o he co alen bond linking wo consecu i e bases along a s and,
espec i ely. The la e is gi en by
l0=qa2+ 4R2
0sin2(θ0/2) ,(5)
wi h abeing he dis ance be ween neighbo ing base pai s measu ed along he
o ien a ion o he helix axis. A ske ch o he s uc u e o he DNA model and he
designa ion o he geome ical pa ame e s R0,l0,θ0, nand dn n−1is p esen ed
in Figu e 1.
Wi h espec o he in e ac ion be ween he elec onic and he ib a ional
deg ees o eedom a iable, cnand n, espec i ely, i is assumed ha he elec-
onic pa ame e s Enand Vn n−1a e modi ied by displacemen s o he bases
wi hin he base pai s. As a quan um-chemical compu a ion o he geome y
dependence o he elec onic pa ame e s Enand Vn n−1 e eals hei mos sig-
ni ican modula ion o igina es om adial dis o ions o he helix which a e
ela ed wi h hyd ogen and co alen bond de o ma ions, espec i ely. Howe e ,
he in luence o small angle de o ma ions on he alues o he elec onic pa am-
e e s can be disca ded (see also u he below).
3
The modula ion o he on-si e elec onic ene gy E0by he adial ib a ions
o he base pai s is exp essed as
En=E0+k n.(6)
On he o he hand he ac ual cha ge occupa ion has i s impac on he local
adial dis o ion o he helix. Fu he mo e, he ans e ma ix elemen s Vn n−1
a e assumed o depend on he h ee-dimensional s acking-dis ance be ween wo
consecu i e bases on a s and as ollows
Vn n−1=V0(1 −α dn n−1).(7)
The quan i y α egula es how s ong Vn n−1is in luenced by he dis ance.
As ypical pa ame e s o DNA molecules one inds [13],[15]: a= 3.4˚
A,
R0≈10˚
A,θ0= 36◦,E0= 0.1eV , Ω = 6.252 ×1012 s−1,V0≃0.1eV and
M= 4.982 ×10−25kg.
A e scaling he ime as →Ω one passes o he dimensionless quan i ies:
˜ n=sMΩ2
V0
n,˜
k=k
pMΩ2
V0
,˜
E0=E0
V0
,(8)
˜α=sV0
MΩ2
α , ˜
R0=sMΩ2
V0
R0,(9)
and o he sake o con enience he ildes a e omi ed in he ollowing. The
alues o he scaled pa ame e s a e ob ained as R0= 34.862 and l0= 24.590.
The se o coupled equa ions o mo ion ead as
i τ ˙cn= (E0+k n)cn
−(1 −α dn+1,n)cn+1 −(1 −α dn n−1)cn−1(10)
¨ n=− n−k|cn|2−R0
l0
(1 −cos θ0)
×α[c∗
n+1cn+cn+1c∗
n] + [c∗
ncn−1+cnc∗
n−1](11)
and he a io τ=~Ω /V0= 0.0411 de e mines he ime scale sepa a ion be-
ween he slow elec on mo ion and he as bond ib a ions. (No ice ha any
E0cn e m on he .h.s. o Eq. (10) can be elimina ed by a gauge ans o ma ion
cn→exp(−iE0 /τ)cn.)
Con a y o p e ious s udies we use o ou compu a ions c edible alues o
he elec on-mode coupling s eng hs kand αas a esul o ou a quan um-
chemical compu a ional p ocedu e.
In his wo k, we pe o m quan um-chemical calcula ions on symme ical
homodime s consis ing o wo nucleoside Wa son-C ick base pai s (adenosine-
hymidine (AT) and guanosine-cy idine (GC) base pai s eps (BPS)) s acked
o e each o he o mimic he con en ional A-DNA and B-DNA con o ma ions
( o a schema ic iew see [16]). Taking in o accoun DNA backbone a leas in
o m o in ac suga moie ies, ins ead o subs i u ing i by p o ons o me hyl
g oups, is necessa y o he consis ency o he calcula ions [17] and o co ec ly
desc ibe cha ge ans e h ough DNA duplexes [18]. We use semiempi ical
all- alence-elec on PM3 Hamil onian [19] wi hin he MOPAC7 e sion o CI
4
(con igu a ion in e ac ion) app oxima ion, as desc ibed in de ail in [20]. We
chose he e PM3-CI me hod, bu no an ab ini io one, since
a) The molecula agmen s in ol ed a e e y la ge;
b) Simila ab ini io calcula ions e en o smalle segmen s expe ience di icul ies
wi h SCF con e gence [21];
c) The wo k [17] used simila semiempi ical Hamil onian (AM1) o analogous
molecula agmen s;
d) PM3-CI app oxima ion is good a desc ibing exci ed s a es o nucleoside base
pai s eps [20];
e) CI app oxima ion is indispensable when cha ged s a es o nucleic acid bases
a e conside ed using semiempi ical quan um chemis y (c ., e.g., [22] and e e -
ences he ein).
He e he e ec s o small, bu non-negligible (up o 0.1 ), adial s e ching
and comp essing o Wa son-C ick hyd ogen bonds (WC H-bonds) on he MO
ene gies o he BPS unde s udy ha e been es ima ed. Speci ically, we pe u bed
in he abo e sense he equilib ium leng hs o he WC H-bonds in only one o he
base pai s in all he BPS in ol ed and moni o ed he esul ing changes in he
ene gies o he highes occupied molecula o bi al and o he occupied molecula
o bi al nex o he highes one (HOMO and HOMO-1, espec i ely), as com-
pa ed wi h he equilib ium alues o hese ene gies. Acco ding o he Koopmans
heo em well known in quan um chemis y (c ., e.g., [21] and e e ences he ein),
he HOMO ene gy is app oxima ely equal o molecula ionisa ion po en ial and
in he igh -binding app oxima ion could be iewed as he si e ene gy, whe eas
he di e ence be ween he HOMO and HOMO-1 ene gies is app oxima ely wice
he hopping in eg al. Wi h his in mind, we we e able o es ima e kand αpa-
ame e s o ou model by calcula ing linea eg ession o he co esponding si e
ene gy and hopping in eg al changes, espec i ely, on o WC H-bond dis ance
pe u ba ions. We we e able o a i e a a e y igh linea co ela ion be ween
he o me and he la e ones. In e es ingly, we ailed o e eal any co ela ion
be ween he igh -binding Hamil onian pa ame e changes and BPS wis an-
gle/helical pi ch pe u ba ions (’helical pi ch’ is he dis ance ain Eq. (5)). The
pe u ba ions in he wo la e pa ame e s we e also ela i ely small (up o 10
deg ees and 0.1 , espec i ely). Fo he coupling pa ame e s o he poly(dA)-
poly(dT) DNA polyme we ob ain: k= 0.0778917 eV/ ˚
Aand α= 0.053835 ˚
A−1.
The co esponding alues o he poly(dG)-poly(dC) DNA polyme a e de e -
mined as k=−0.090325 eV/ ˚
Aand α= 0.383333 ˚
A−1.
Tha we a e equipped wi h he quan um-chemical es ima es o he coupling
pa ame e s αand kis a de ini e s ep o wa d and dis inguishes he p esen
s udy om p e ious ones [11],[12]. We emphasize ha he quan um-chemical
es ima es o he coupling pa ame e s di e signi ican ly om he ones used o
he model s udy in [11],[12] whe e hese pa ame e s, in lack o eliable alues o
hem, we e ea ed as adjus able. As a consequence he e is a di e ence be ween
he ET scena io desc ibed in [11],[12] and he one we a e going o illus a e in
he ollowing.
3 S a iona y localized elec on- ib on s a es
Caused by he nonlinea in e play be ween he elec onic and he ib a ional
deg ees o eedom o he helix he o ma ion o pola onic elec on- ib a ion
5

Figu e 2: The spa ial pa e n o he pola onic elec on- ib a ion compound.
(a) The elec onic pa . Full (dashed) line: poly(dA)-poly(dT) (poly(dG)-
poly(dC)) DNA polyme . (b) Uni less adial de o ma ion pa e n. Assignmen
o line ypes as in (a).
compounds is possible [4]-[8],[11],[12]. We cons uc such localized s a iona y
solu ions o he coupled sys em (10),(11) wi h he help o he nonlinea map
app oach explained in de ail in [23]. In Figu e 2 we depic he p o iles o he
(s anding) pola on s a es o he poly(dA)-poly(dT) and he poly(dG)-poly(dC)
DNA polyme , espec i ely. In bo h cases he pola ons a e o ai ly la ge ex en-
sion (wid h). Rega dless o he DNA polyme ype he elec onic wa e unc ion
is localized a la ice si e and he en elope o he ampli udes decays mono oni-
cally and exponen ially wi h g owing dis ance om his cen al si e (base pai ).
Howe e , he elec onic wa e unc ion o he poly(dG)-poly(dC) DNA polyme
is s onge localized han he one o i s poly(dA)-poly(dT) coun e pa .
Acco dingly, he a ibu ed adial displacemen pa e ns a e exponen ially
localized a he cen al la ice si e. Conce ning he esul ing s a ic adial helix
de o ma ions we ind ha he e is a d as ic di e ence be ween he poly(dA)-
poly(dT) and he poly(dG)-poly(dC) DNA polyme s. In he o me case he
o e all non-posi i e adial ampli udes imply ha he H-b idges expe ience con-
ac ions. In con as , in he la e case he H-b idges ge s e ched. Ne e he-
less, hese de o ma ions a e a he weak, i.e. on he o de o 1.5 10−3˚
A.
4 Cha ge anspo
We s udy now he ET achie ed by mo ing pola ons. In ac , since he con-
s uc ed pola on solu ions a e o ai ly la ge ex ension ( he hal -wid h in ol es
.100 la ice si es) we can expec hem o be mobile. In o de o ac i a e po-
la on mo ion we used he disc e e g adien me hod [24] o ob ain sui able ini ial
pe u ba ions o he momen um coo dina es p
nwhich ini ia e cohe en mo ion
o he pola on compound.
We conside i s he case o he poly(dG)-poly(dC) DNA polyme . The
p opaga ion ea u es a e illus a ed in Figu e 3 whe e he spa io- empo al e o-
lu ion o he elec onic and he ib a ional adial b ea he a e shown. Fo a
DNA la ice consis ing o 300 si es (base pai s) he se o coupled equa ions
(10),(11) was in eg a ed using a ou h-o de Runge-Ku a me hod and pe i-
odic bounda y condi ions we e imposed. Main enance o he no m conse a ion
Pn|cn( )|2= 1 se ed o assu e accu a e compu a ions.
6
Figu e 3: B ea he mo ion along he DNA o he poly(dG)-poly(dC) DNA
polyme .(a) The elec onic b ea he . (b) The ib a ional b ea he . The adial
de o ma ions n( ) a e gi en in dimensionless uni s.
7
Figu e 4: Time e olu ion o he i s momen um o he elec onic occupa ion
p obabili y. Full (dashed) line: poly(dA)-poly(dT) (poly(dG)-poly(dC)) DNA
polyme .
The elec onic componen se s o o mo e di ec ionally along he la ice in
unison wi h he ib a ional ampli ude pa e n. Howe e , he e emains a small
ampli ude ib a ional b ea he a he ini ial posi ion. In bo h cases he localized
s uc u es a e p ac ically p ese ed apa om he a ising ampli ude b ea hing
indica ing a pe iodic ene gy exchange be ween he elec onic and ib a ional
deg ees o eedom. Hence, long- ange ET is achie able. Wi h he ini ial injec-
ion o kine ic ene gy he ( ib a ional) sys em possesses now inc eased ene gy
con en so ha we obse e ib a ional b ea he s wi h ampli udes being la ge
han hose o hei s a ic equi alen s (c . Figu es 2 (b) and 3 (b)).
As he poly(dA)-poly(dT) DNA polyme is conce ned we ound ha i does
no exhibi such good conduc i i y as i s poly(dG)-poly(dC) coun e pa . The
di e en p opaga ion scena ios a e p ope ly illus a ed using he ime e olu-
ion o he i s momen um o he elec onic occupa ion p obabili y de ined
as ¯n( ) = Pnn|cn( )|2. One can clea ly obse e ha in he(dG)-(dC) case
elec on p opaga ion p oceeds un es ic edly and unidi ec ionally wi h uni o m
eloci y (see Figu e 4). Dis inc ly, he (dA)-(dT) elec on mo es i ine an ly in
a con ined egion, comp ising no mo e han i e bases, a ound he s a ing si e
es aining conduc i i y.
Rega ding he ene gy s o ing capaci y we moni o ed he no malized pa ic-
ipa ion numbe de ined as
p( ) = P( )
P(0) (12)
wi h
P( ) = 1
Pn|cn( )|4.(13)
Since he elec onic wa e unc ion is no malized he elec on b ea he is com-
ple ely con ined a a single si e i p= 1 and is uni o mly ex ended o e he
la ice i pis o he o de N, iz. he numbe o la ice si es. Hence, pmeasu es
how many si es a e exci ed o con ibu e o he elec onic b ea he pa e n.
8
Figu e 5: The no malized pa icipa ion numbe p( ). Full (dashed) line:
poly(dA)-poly(dT) (poly(dG)-poly(dC)) DNA polyme .
F om Figu e 5 we in e ha he (dG)-(dC) elec on b ea he ex ension pe -
o ms oscilla ions he maximal ampli udes o which co espond o sligh g ow h
o he spa ial wid h up o me ely ≃1.2 imes i s s a ing alue. In compa ison,
he wid h o he (dA)-(dT) elec onic ampli ude pa e n expe iences s onge
b oadening leading o a mo e ex ended elec on s a e.
Conside ing he ene gy exchange be ween he elec onic and ib a ional sub-
sys ems we obse e ha his p ocess e ol es wi h equency Ω . Hence, he
dynamical dis ibu ion be ween he elec onic and ib a ional ene gy is d i en
by he adial ha monic ib a ions. In pa icula , he amoun o kine ic en-
e gy lowing om he ib a ional sys em in o he elec onic subsys em (ini-
ially he elec onic kine ic ene gy is ze o) is go e ned by he s eng h o he
coupling α. Appa en ly, in he (dG)-(dC) case he ene gy sha ing is by a
mo e p onounced han in he (dA)-(dT) case due o he ac ha he coupling
s eng hs di e by almos an o de o magni ude, ha is α(dG)−(dC)= 0.110
and α(dA)−(dT )= 0.0154. This explains he la ge con en o kine ic ene gy
injec ed om he ib a ional deg ees o eedom in o he elec onic ones in-
ducing high elec on mobili y. Fu he mo e, he k’s ha e opposi e sign, i.e.
k(dA)−(dT )=−0.2591 and k(dG)−(dC)= 0.2234, which leads o he sign di e -
ence in he adial dis o ion pa e ns (dA)−(dT )
nand (dG)−(dC)
n.
An open ques ion which is subjec o u he esea ch is whe he he pola ons
su i e a ambien empe a u e. In spi e o he alues o he adial a iables
being small he pola on is a compound objec and he ac ha he ans e
in eg al elemen s Vn n−1≈0.1 eV a e la ge han kBT≈0.025 eV a T= 300 K,
sugges s ha hey would su i e. Equally he e ec o an elec ic ield du ing
he whole mo emen o he pola on is cu en ly in es iga ed.
In conclusion, we ha e ound ha conduc i i y in syn he ically p oduced
DNA molecules depends on he ype o he single base pai o which he poly-
me is buil o . While a pola on-like mechanism, elying on he nonlinea cou-
pling be ween he elec on ampli ude and adial ib a ions o he base pai s,
is esponsible o long- ange and s able ET in (dG)-(dC) DNA polyme s, he
9