Effect of base-pair inhomogeneities on charge transport along the DNA molecule, mediated by twist and radial polarons

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. 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