Hiroaki Yamada and Kazumoto Iguchi- Some Effective Tight-Binding Models for Electrons in DNA Conduction:A Review

8/3/2019 Hiroaki Yamada and Kazumoto Iguchi- Some Effective Tight-Binding Models for Electrons in DNA Conduction:A Review

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arXiv:1003.5604v1[cond-mat.dis-nn]29Mar2010

Some Effective Tight-Binding Models for Electrons in DNA Conduction:A Review

Hiroaki YamadaYPRL, Aoyama 5-7-14-205, Niigata 950-2002, JAPAN

Kazumoto IguchiKIRL, 70-3 Shinhari, Hari, Anan, Tokushima 774-0003, Japan

(Dated: March 30, 2010)

Quantum transport for DNA conduction has widely studied with interest in application as acandidate in making nanowires as well as interest in the scientific mechanism. In this paper, wereview recent works with concerning the electronic states and the conduction/transfer in DNApolymers. We have mainly investigated the energy band structure and the correlation effects oflocalization property in the two- and three-chain systems (ladder model) with long-range correlationas a simple model for electronic property in a double strand of DNA by using the tight-binding model.In addition, we investigated the localization properties of electronic states in several actual DNAsequences such as bacteriophages of Escherichia coli, human-chromosome 22, compared with thoseof the artificial disordered sequences with correlation. The charge transfer properties for poly(dA)-poly(dT) and poly(dG)-poly(dC) DNA polymers are also presented in terms of localization lengthswithin the frameworks of the polaron models due to the coupling between the charge carriers andthe lattice vibrations of the double strand of DNA.

PACS numbers:

I. INTRODUCTION

Recent interests on semiconducting DNA polymershave been stimulated by successful demonstrations of thenanoscale fabrication of DNA, where current-voltage (I-V) measurements for poly(dA)-poly(dT) and poly(dG)-poly(dC) DNA polymers have been done [14]. For suchartificial periodic DNA systems, the energy band struc-ture is a useful starting point in order to interpret theexperimental results such as semiconductivity and themetal-insulator transition [5].

On the other hand, Tran et al. measured conductivity

along the double helix of lambda phage DNA (DNA)at microwave frequencies, using the lyophilized DNA inand also without a buffer [6]. The conductivity is stronglytemperature dependent around room temperature with acrossover to a weakly temperature dependent conductiv-ity at low temperatures [2]. Yu and Song showed that theDNA can be consistently modeled by considering thatelectrons may hop through the variable range hopping forconduction without invoking additional ionic conductionmechanisms, and that electron localization is enhancedby strong thermal structural fluctuations in DNA [7].Indeed, the sequence of base pairs(bp) of the DNAis inhomogeneous as in disordered material systems.

Moreover, charge migration in DNA have been mainlyaddressed in order to clarify the mechanism of damagerepair which are essential to maintain the integrity ofthe molecule. The precise understanding of the DNA-mediated charge migration would be important on thedescriptions of damage recognition process and protein

Electronic address: [email protected];

URL: http://www.uranus.dti.ne.jp/~hyamada/

binding, or on the engineering biological processes [53].The stacked array of DNA bp provides an extended pathto a long-range charge transfer although the dynami-cal motions or the energetic sequence-dependent hetero-geneities are expected to reduce the long-range migra-tion. Photoexcitation experiments have unveiled thatcharge excitations can be transfered between metalloin-tercalators through the guanine highest occupied molec-ular orbitals of the DNA bridge[8]. The subsequentlylow-temperature experiments showed that the radiation-induced conductivity is related to the mobile charge car-riers, migrating within frozen water layers surrounding

the DNA helix, rather than through the base-pair core.Those experiments are summarized as follows: (i)

Band gap reduction of a double strand of DNA, (ii)Transition from the tunneling hopping to the band hop-ping, (iii) Anomalously strong temperature dependenceof band gap, (iv) Highly nonlinear temperature depen-dence of the DC conductivity, (v) Low conductivity ofDNA with a complicated sequence such as DNA andhigh conductivity of DNA with a simple sequence suchas poly(dG)-poly(dC) and poly(dA)-poly(dT). These re-sults suggest that the anomalously strong temperaturedependence on the physical quantities is attributed tothe self-organised extrinsic superconductive characterof DNA due to the formation of donors and accepters.Here we would like to note the following: Usually, asin solid state physics, the extrinsic semiconductor is re-alized when external impurities are introduced in puresubstrate materials. Such impurities produce the extrin-sic semiconductive nature. However, in real DNA helices,there are no such impurities from outside; but there ex-ist already a complicated arrangement of bases of adenine(A), guanine(G), cytosine (C) and thymine (T) inside ofthe DNA, where among the bases each one of bases mayregard other bases as impurities. Hence, a kind of self-
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organized extrinsic semiconductive nature may appear.However, the electronic transport properties in DNA

are still controversial mainly due to the complexity of theexperimental environment and the molecule itself. Al-though theoretical explanations for the phenomena havebeen tried by some standard pictures used in solid statephysics such as polarons, solitons, electrons and holes,the situation is still far from unifying the theoretical

scheme.Each DNA sequence is packed in a chromosome, vary-

ing in length from 105bp in yeast to 109bp in human.In general, the length of a mutation is relatively short(10bp) as compared to the length of a gene (103106bp).Because the mutation rate is very law the mechanicaland thermodynamic characteristics are maintained forthe mutation [9]. In Sect.6, we give a brief discussionabout the mutation as the proton transfer between thenormal and tautomer states.

The Watson-Crick(W-C) base-pair sequence is essen-tial for DNA to fulfill its function as a carrier of thegenetic code. The specific characteristics is also used

extensively even in various fields such as anthropologyand criminal probe. As observed in the power spectrum,the mutual information analysis and the Zipf analysisof the DNA base sequences such as human chromosome22(HCh-22), the long-range correlation exists in the to-tal sequence, as well as the short-range periodicity. Inthis article, we mainly discuss the relationship betweenthe correlation in the DNA base sequences and the elec-tronic transport/transfer.

The electrical transport of DNA is closely relatedto the density of itinerant electrons because of thestrong electron-lattice interaction. Resistivities of twotypical DNA molecules, such as poly(dG)-poly(dC) and

DNA, are calculated. At the half-filling state, thePeierls phase transition takes place and the poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers ex-hibit a large resistivity. When the density of itinerantelectrons departs far from the half-filling state, theresistivity of poly(dG)-poly(dC) becomes small. For theDNA, there is no Peierls phase transition due to theaperiodicity of its base pair arrangement. The resistiv-ity of poly(dG)-poly(dC) decreases as the length of themolecular chain is increasing, while that of DNA in-creases as the length is increasing.

In Sect.3, we introduce the Huckel model of DNAmolecules to treat with the electrons. Moreover, inSect.4, we give the effective polaron model including the

electron-lattice coupling dynamics. In Sect.5, we presentcorrelation effects of localization in the ladder models andthe formation of localized polarons (Holsteins polaronsor solitons) due to the coupling between the charge car-riers and the lattice vibrations of the double strand ofDNA. These polarons in DNA act as donors and accep-tors and exhibit an extrinsic semiconductor character ofDNA. The results are discussed in the context of experi-mental observations.

The present review article essentially follows the line

in Refs. [5, 1115]. In particular, it is written with aview of the localization and/or delocalization problemin the quasi-one-dimensional tight-binding models withdisorder. In appendices we give some calculations andexplanations related to main text.

II. CORRELATED DNA SEQUENCES

As is well known, DNA has the specific binding proper-ties, i.e., only A-T and G -C pairs are possible, where thebases of nucleotide are A, T, G, C. The backbones of thebases, sugar and phosphate groups, ensure the mechani-cal stability of the double helix and protect the base pairs.Since the phosphate groups are negatively charged, thetopology of the duplex is conserved only if it is immersedinto an aqueous solution containing counterions such asN a+ and Mg+ that neutralize the phosphate groups.

The clustering of similar nucleotides can be clarified bystudying the properties of the cluster size distributionson the various real DNA sequences, ranging from the

viral to higher eukaryotic sequences. It is shown thatthe distribution function P(S) about the number S ofthe consecutive C-G or A-T clusters becomes P(S) exp{S} [16, 17]. The values of the scaling exponent of CG are much larger than of AT. The maximum valueof the A-T cluster size is found to be much larger thanthat of the C-G cluster size, which implies the existenceof large A-T clusters.

Moreover, it has been found that the base sequences ofvarious genes exhibit a long-range correlation, character-ized by the power spectrum S(f) f (0.1 < < 0.8)in the low frequency limit (f


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namic limit.Accordingly, it is very interesting to compare the lo-

calization nature of the electronic states in the real DNAsequence with those in the artificial disordered sequencewith long-range correlation. Recently, Krokhin et al.have reported that much longer localization length hasbeen observed in the exon regions than in the intronregions for practically all the allowed energies and for

all randomly selected DNA sequences[20]. Through thestatistical correlations in the nucleotide sequence, theysuggest that the persistent difference of the localizationproperty is related to the qualitatively different informa-tions stored by exons and introns.

In the Sect.5, we numerically give localization nature ofthe electronic states in some real DNA sequences such asbacteriophages of Escherichia coli (E.coli) and HCh-22,and so on in ladder models [13, 21]. We also investigatethe correlation effect on the localization property of theone-electronic states in the disordered ladder models witha long-range structural correlation that is generated bythe modified Bernoulli map. Obviously, the correlation in

the DNA sequences affects not only the electronic con-duction but also the twist vibration of the backbones.However, such effects can be approximately ignored inthe scope of this article.

III. HOMO-LUMO GAPS

Each compositional unit of a DNA polymer is complexalthough the DNA polymers can be regarded as quasi-one-dimensional systems. (See Fig.1.) In this section, wegive a brief review on the Huckel theory in order to ana-lyze the relationship between electronic structures of the

separate nucleotide groups and of their infinite periodicchains. Ladik tentatively concluded that electrons hopmainly between the bases along the helical axis of DNA,so that it is enough to take into account the overlap inte-grals between the orbitals of the adjacent base pairs inthe DNA duplexes[53]. These studies have revealed thatDNA polymers are insulators with an extremely largeband gap Eg (about 10-16 eV) and narrow widths of thevalence and conduction bands (about 0.3-0.8 eV). This isa consequence of the orbital mixing between the highestoccupied molecular orbital(HOMO) and the lowest unoc-cupied molecular orbital (LUMO) in the base groups ofnucleotide in the DNA. We can control the charge injec-tion into DNA and the conduction in DNA, by adjustingthe HOMO-LUMO of the polymer and the Fermi energyof electric leads, respectively.

Brumel et al. found that semiconduction in DNA isvery important for such systems since the activation en-ergies of nucleotides are lower than those of nucleosides[22]. This was the first suggestion that propagation alongthe sugar and phosphate groups play a significant role inthe transport properties of DNA as well as propagationalong the base-stacking of the nucleotide groups in thecenter of the DNA molecule.

FIG. 1: The single bases of A, G, C, T. Here dots () mean-electrons.

We revisit the problem of the electronic properties ofindividual molecules of DNA, in order to know electrontransport in the double strand of DNA as a mother ma-terial for the single and double strand of DNA, by takinginto account only electrons in the system[5]. To doso, we review the theory of electrons in DNA, usingthe Huckel approximation for electrons in both thesugar-phosphate backbone chain and the stacking ofthe nitrogenous bases of nucleotide.

A. Huckel Model

Applying the basic knowledge of quantum chemistry tobiomolecules of nitrogenous bases, we can find the totalnumber oforbitals and electrons in the bases. It issummarized in Table 1.

Let us consider the famous Huckel model in quan-tum chemistry [2227]. This model concerns only theorbitals in the system. In this context, this theoryis closely related to the so-called tight-binding modelin solid state physics, which concerns very localized or-bitals at atomic sites such as the Wanniers wavefunction.Therefore, this approach has been extensively applied tomany polymer systems such as polyacetylene with a great

success. In the so-called Huckel approximation we adoptthe orthogonality condition for the overlap integrals

Srr = 1, Srs = 0 (r = s), (1)where Srs denotes overlap integrals between atomic or-bitals at rth and sth sites. We assume the special formof the resonance integrals

hrs =1

2KSrs(hrr + hss), (2)


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Base A G C T S Ph|N| 5 5 3 5 - -|C| 5 5 4 2 5 0|O| 0 1 1 2 4 4|P| - - - - 0 1

Total 10 11 8 9 9 5-orbitals 10 11 8 8 4 5-electrons 12 14 10 10 8 8

TABLE I: The total numbers |N|, |C|, |O| and |P| of N, C, Oand P atoms and of-orbitals and -electrons in the bases ofA, G, C and T, and the sugar(S) and phosphate(Ph) groups,respectively. Here there are 12, 14, 10 and 10 -electrons forthe 10, 11, 8, 8 -orbitals in the A, G, C and T base molecules,respectively, while there are 8 and 8 -electrons for the 4 and5 -orbitals in the sugar and phosphate groups, respectively.Here we note that the carbon atom ofCH3 in the T moleculedoes not have any -electron since it forms the sp3hybridorbitals. This provides 8 -electrons for the T molecule.

with K = 1.75 and the overlap integrals Srs (r = s) are

not necessary to be diagonal, and otherwise. Usually theon-site (r = s) resonance integrals are called the Coulombintegrals denoted by r, while the off-site (r = s) inte-grals are called the resonance integrals denoted by rssuch that r = hrr, rs = hrs (r = s). Here we wouldlike to emphasize the following. In the sense of the Huckeltheory, the parameters are taken empirically. This meansthat the parameters are adjustable and feasible to giveconsistent results with the experimental results or the abinitio calculation results. Therefore, the exact values ofthe parameters are neither so important nor should betaken so seriously in this framework. This is becauseonce one can obtain much more precise values for the

Huckel parameters, one can provide the more plausibleresults from the Huckel theory. Although many effortsof the ab initio calculations have been done for DNAsystems, unfortunately at this moment there seem to bevery few first principle calculations for such parametersin the DNA systems to fill out this gap. Nevertheless,we must assign some values for the Huckel parametersin order to calculate the electronic properties of DNA inthe framework of the Huckel theory. So, we look back tothe original method about time when the Huckel theorywas invented.

For this purpose to use the standard Huckel theory, letus adopt some simple formulae for the Huckel parame-ters, which are defined as follows: Let X and Y be two

different atoms. Denote by X the Coulomb integral atthe X atom and by XY the resonance integral betweenthe X and Y atoms:

X = + aX, XY = lXY. (3)

Here aX and XY are the empirical parameters that aresupposed to be adopted from experimental data. And the

parameters and are important. These can be thoughtof the fundamental parameters in our problem of biopoly-mers. Conventionally we take as the Coulomb integralfor the 2pxorbital of carbon and as the resonanceintegral between the 2pxorbitals of carbon, such that = C 0, = CC 1. This means that the energylevel of a carbon atom is taken as the zero level, and theenergy is measured in units of the resonance integral be-tween carbon atoms. We note that the empirical valuesobtained from experiments are usually given by

6.30 6.61eV, 2.93 2.95eV. (4)

Since the biomolecules consist of the atoms C, N, O, P, letus find the plausible values of the Huckel parameters forthem to apply the Huckel model to biomolecules of DNA.The Paulings electronegativity for carbon (C), nitrogen(N), oxygen (O) and phosphorus (P) are the following:

C 2.55, N 3.0, O 3.5, P 2.1. (5)

Then we can defined the Huckel parameters for carbon,using the Sandorfys formula[28], Mullikens formula[29]and Streitwiesers formula[30], we find C = , N = + 0.7, O = + 1.53, P = 0.724. and we ob-tain O = + 2.53, N = + 0.276. and CC = ,C=C = 1.1, C

N = C

N = 0.8, C=N = 1.1,

CO = 0.9, C=O = 1.7. Here X(X) denotes theCoulomb integral of atomic state X with one(two) elec-tron(s) occupied. See appendix A for the formula. Andif one can get more accurate values from the ab initiocalculations, then we can always replace the Coulombintegrals by the new set of values.

B. HOMO-LUMO of Biomolecules

For example, considering the topology of the hoppingof electrons on the -orbitals, we now find the followingHuckel matrices HA for A:


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HA =

N CN 0 0 0 C=N 0 0 0 0CN C C=N 0 0 0 0 0 0 0

0 C=N N CN 0 0 0 0 0 00 0 CN C C=C 0 0 0 CN 00 0 0 C=C C CC CN 0 0 0

C=N 0 0 0 CC C 0 0 0 CN

0 0 0 0 CN 0 N C=N 0 00 0 0 0 0 0 C=N C CN 00 0 0 CN 0 0 0 CN N 00 0 0 0 0 CN 0 0 0 N

,

Paulis exclusion principle tells us that each state with anenergy level is occupied by a pair of electrons with spinup and down. So, electrons occupy the energy levels inthe spectrum from the bottom at low temperature. Sincethe lower energy levels with one half of the total numberofelectrons can be occupied by the electrons, thereappears an energy separation between the occupied andthe unoccupied states, which is called the energy gap.

The energy levels of the HOMOs and LUMOs are givenby H = 0.888, L = 0.789 for A, where the energy ismeasured in units of . We can do the same calcula-tion for G,C,T, respectively. Defining the energy gapbetween the LUMO and HOMO, = L H. Thenwe obtain the result as, A = 1.677, G = 1.555,C = 1.535, T = 1.713. (See Fig.2(a).) The to-tal energy Etot = 2

j=occ.states j of the electrons

of A, G, C, T are Etot(A) = 2 0.74, Etot(G) = 26.47,Etot(C) = 19.05, Etot(T) = 21.14, respectively. There-fore,

Etot(C) > Etot(A) > Etot(T) > Etot(G).

This shows that since the lower the ground state energythe more stable the system, the most stable molecule isG while the most unstable molecule is C.

HOMOs and LUMOs of sugar and phosphate maybe required when we consider charge conduction of theDNA polymer because the sugar and phosphate groupconstituting the backbones of DNA polymer also haveelectrons. We give the results for the single sugar-phosphate in Fig.2(b). Moreover, the results based onthe Huckel approximation have been extended to the sin-gle nucleotide systems such as A, G, C, T with the singlesugar-phosphate group, and the system of a single strandof DNA with an infinite repetition of a nucleotide groupsuch as A, G, C and T, respectively. See Ref.([5]) forthe more details. This reorganization, which is difficultto calculate due to the complexity of the combined sys-tem, may lead to a smaller HOMO-LUMO gap and widerband widths than for the bare molecule.

When the system of the nucleotide bases such as A, G,C, T exists as an individual molecule, there is always anenergy gap between the LUMO and HOMO states, wherethe order of the gap is about several eV. This means that

FIG. 2: (a) The spectrum of the -orbitals of A, G, C, T. (b)The spectrum of the -orbitals of a sugar-phosphate group.PO4 stands for the phosphate group where the electron hop-ping between the oxygen sites is taken into account, whilePO4 stands for the the phosphate group where the electronhopping between the oxygen sites is forbidden. SP stands forthe sugar-phosphate group where the electron hopping be-tween the oxygen sites is taken into account. The energiesare measured in units of ||. Here dots () mean -electronsand the level with four dots means the double degeneracy of

the level.

the nucleotides of A, G, C, T have the semiconductingcharacter in its nature. If the stacking of the baseis perfect, then there are two channels for electronhopping: One is the channel through the base stackingand the other is the one through the backbone chain ofthe sugar-phosphate. In this limit, the localized states ofthe original nucleotide bases become extended such that

the levels form the energy bands.

Thus, we believe that our Huckel approach in this pa-per fills out the gap between the simple approach ofmathematical models for tight-binding calculations andthe approach of the quantum chemical models for ab ini-tio calculations from the first principle, where in the for-mer we assume one orbital with one electron at one nu-cleotide while in the latter we include all electrons andatoms in the system.


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IV. EFFECTIVE POLARON MODELS

First principle methods are powerful enough to under-stand the basic electronic states of the molecules. Onthe other hand, the complementary model-based Hamil-tonian approach is effective for understanding those ofpolymers as well. In this section, we give some effective1D models and the basic properties of polymers such as

trans-polyacetylene and DNA polymers. In the last sec-tion, we treated HOMOs and LUMOs and the occupationof orbitals by the electrons with spin. Hereafter, weomit spin of electrons when we consider the one-electronproblems, for the sake of simplicity in notation.

A. Tight-binding Models for Polymers

It is known that the electronic properties of planarconjugated systems are dominated by systems withone orbital per site. The electronic Hamiltonian for 2pzorbitals through a tight-binding model with the nearest

neighbors interactions only:

Hel =n

EnCnCn

n

Vnn+1(CnCn+1 + Cn+1C

n),(6)

where Cn and Cn are creation and annihilation operators

of an electron at the site n. The matrix elements are

obtained from the extended Huckel theory as given inthe previous section:

En = n (7)Vnm = (K/2)(n + m)Snm, (8)

where n is ionization energy (Coulomb integral) of thenth 2pz orbital, and Snm is overlap integral (resonanceintegral) between the nth and mth orbitals centered atneighbour sites given in Sect.3. We usually treat thecopolymer as a system with one orbital per site andrepresent the electronic Hamiltonian for the 2pz carbonand nitrogen orbitals through a tight-binding model withnearest neighbor interactions. DNA polymers are alsobelieved to form an effectively one-dimensional molecularwire, which is highly promising for diverse applications.Basically, the carriers mainly propagate along the aro-matic stacking of the strands (the interstrand cou-pling being much smaller), so that the one-dimensionaltight-binding chain model can be a good starting pointto minimally describe a DNA wire[10].

Bruinsma et al. also introduced the effective tight-binding model that describes the site energies of a carrierlocated at the nth molecule as,

H =n

EnCnCn

n

tDNA cos(n,n+1(t))(CnCn+1 + C

n+1Cn), (9)

where Cn(Cn) creation(annihilation) operator of the car-rier at the site n [7, 31, 32]. The carrier site energiesEn are chosen according to the ionization potentials ofrespective DNA bases as, A = 8.24eV, T = 9.14eV,C = 8.87eV, and G = 7.75eV, while the hopping in-tegral, simulating the stacking between adjacentnucleotides, is taken as tDNA = 0.1 0.4eV. n,n+1denotes the relative twist angle deviated from equilib-rium position between the nth and (n + 1)th moleculesdue to temperature T. Then we can estimate the or-der of the hopping integral. Let n,n+1 is an indepen-

dent random variable that follows Gaussian distributionwith n,n+1 = 0 and 2n,n+1 = kBT/I2I, where I isthe reduced moment of inertia for the relative rotationof the two adjacent bases and I is the oscillator fre-quency of the mode (I2I = 250K). Therefore the fluctu-ation of the Hopping term is about tDNAcos(n,n+1) tDNA(1 2n,n+1/2) 0.16eV at room temperature,which is much smaller than that ( 1.4eV) in the diag-onal part. Accordingly, as a simple approximation, wecan deal with the diagonal fluctuation with keeping the

hopping integral constant.

Roche investigated such a model for poly(GC) DNApolymer and DNA, with the on-site disorder arisingfrom the differences in ionization potentials of the basepairs, and with the bond disorder tDNA cos(n,n+1(t)) re-lated to the random twisting fluctuations of the nearest-neighbor bases along the strand[32]. While for poly(GC)the effect of disorder does not appear to be very dramatic,the situation changes when considering

DNA.

Actually, modern development of physico-chemical ex-perimental techniques enables us to measure directlythe DNA electrical transport phenomena even in singlemolecules. Moreover, several groups have recently per-formed numerical investigations of localization propertiesof the DNA electronic states based on the realistic DNAsequences.


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B. Twist Polaron Models for DNA Polymers

In this subsection we give an effective 1D model withrealistic parameters for the electronic conduction of thepoly(dA)-poly(dT) and poly(dG)-poly(dC) DNA poly-mers.

Specially, a distinctive feature of biological polymersis a complicated composition of their elementary sub-

units, and an apparent ability of their structures to sup-port long-living nonlinear excitations. In their polaron-like model, Hennig and coworkers studied the electronbreather propagation along DNA homopolynucleotideduplexes, i.e. in both poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers, and, for this purpose, they esti-mated the electron-vibration coupling strength in DNA,using semiempirical quantum chemistry[3335]. Changet al. have also considered a possible mechanism to ex-plain the phenomena of DNA charge transfer. The chargecoupling with DNA structural deformations can create apolaron and thus promote a localized state. As a re-sult, the moving electron breather may contribute to the

highly efficient long-range conductivity[36]. Recent ex-periments seem to support the polaron mechanism forthe electronic transport in DNA polymers.

The Hamiltonian for the electronic part in the DNAmodel is given by

Hel(t) =n

En(t)CnCn

n

Vnn+1(t)(CnCn+1 + Cn+1C

n),(10)

where Cn and Cn are creation and annihilation operatorsof an electron at the site n. The on-site energies En(t)are represented by

En(t) = E0 + krn(t), (11)

where E0 is a constant and rn denotes the structuralfluctuation caused by the coupling with the transver-sal Watson-Crick H-bonding stretching vibrations. Theschematic illustration is given in Fig.3. The transfer in-tegral Vnn+1 depends on the three-dimensional distancednn+1 between adjacent stacked base pairs, labeled by nand n + 1, along each strand. And it is expressed as

Vnn+1(t) = V0(1 dnn+1(t)) . (12)Parameters k and describe the strengths of interac-tion between the electronic and vibrational variables, re-spectively. The 3D displacements dnn+1 also give riseto variation of the distances between the neighboring

bases along each strand. The first order Taylor expansionaround the equilibrium positions is given by

dnn+1(t) =R00

(1 cos 0)(rn(t) + rn+1(t)). (13)

R0 represents the equilibrium radius of the helix, 0 is theequilibrium double-helical twist angle between base pairs,and 0 the equilibrium distance between bases along onestrand given by

0 = (a2 + 4R20 sin

2(0/2))1/2, (14)

parameter value

kAT 0.778917 eVA1

AT 0.053835 A1

kGC -0.090325 eVA1

GC 0.383333 A1

TABLE II: Basic parameters for DNA molecules. Thesubscripts, AT and GC, for k and denote for ones ofthe poly(dA)-poly(dT) and poly(dG)-poly(dC) DNA poly-mers, respectively. The other parameters are E0 = 0.1eV,V0 = 0.1eV, a = 3.4A, R0 = 10A and 0 = 36

.

with a being the distance between the neighboring basepairs in the direction of the helix axis. We adopt real-istic values of the parameters obtained from the semi-empirical quantum-chemical calculations. (See table 2.)Further, we consider {rn} as independent random vari-ables generated by a uniform distribution with the width(rn [W, W]). Accordingly, fluctuations in both theon-site energies and the off-diagonal parts in the Hamil-tonian (1) are mutually correlated because they are gen-erated by the same random sequence rn. (See Fig.2(c).)The typical value of W is W = 0.1[A], which approxi-mately corresponds to the variance in the hydrogen bondlengths in the Watson-Crick base pairs, as seen in the X-ray diffraction experiments [37].

A quasi-continuum spectrum is possessed of a wealthof dynamical modes. In principle, each of these can in-fluence the DNA charge transfer/transport. But, sincethere are more or less active modes, it is possible totake the whole manifold of the DNA molecular vibra-tions into two parts: vibrations which are most activeand a stochastic bath consisting of all the other ones.Computer simulations have pinpointed that the dynam-ical disorder is crucially significant for the DNA trans-fer/transport. Several attempts to formulate stochasti-cal models for the interplay of the former and the lat-ter have already appeared in the literature(see, for ex-amples). In this paper, we will deal with the polaron-like model of Hennig and coworkers as described in theworks[34, 35], where charge+breather propagation alongDNA homopolynucleotide duplexes [i.e. in the bothpoly(dG)-poly(dC) and poly(dA)-poly(dT) DNA poly-mers] has been studied.

C. Localization due to Static Disorder

In this subsection, we assume that the fluctuations of{rn} are frozen (quenched disorder) and independent fordifferent links; that is, we investigate localization prop-erties of the poly(dA)-poly(dT) and poly(dG)-poly(dC)DNA polymers[38].

The Schrodinger equation Hel| = E| is written in


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8

FIG. 3: Sketch of the structure of the DNA model. The basesare represented by bullets and the geometrical parametersR0, 0, 0, rn+1 and dnn+1 are indicated.

the transfer matrix form,n+1

n

=

EEnVnn+1

Vnn1Vnn+11 0

n

n1

, (15)

where n is the amplitude of the electronic wavefunction| = n n|n, where |n = Cn|0 at the base pairsite n. |0 is the Fermi vacuum. We use the localizationlength and/or Lyapunov exponent calculated by themapping (15) in order to characterize the exponential lo-calization of the wave function. Originally the Lyapunovexponent (the inverse localization length) is defined inthe thermodynamic limit (N

). However, here we

use the following definition of the Lyapunov exponent forthe electronic wave function with a large system size N[38, 39].

(E, N) = 1(E, N) =ln(|N|2 + |N1|2)

2N. (16)

We use the appropriate initial conditions 0 = 1 = 1,and for large N(>> ) the localization length and theLyapunov exponent are independent of the boundarycondition. The energy-dependent transmission coeffi-cient T(E, N) of the system between metallic electrodesis given as T(E, N) = exp(2N) and is related to Lan-dauer resistance via

= (1

T)/T in units of the quan-

tum resistance h/2e2( 13[km]) [40, 41].We have numerically investigated the localization

properties of electronic states in the adiabatic polaronmodel of poly(dG)-poly(dC) and poly(dA)-poly(dT)DNA polymers with the realistic parameters obtained us-ing the semi-empirical quantum-chemical calculations.

We compare the localization properties of thepoly(dG)-poly(dC), poly(dA)-poly(dT) DNA polymersand the mixed cases. Figure 5(a) and (b) show the local-ization length and the Lyapunov exponent in the three

0.102

0.101

0.100

0.099

0.098

Vnn+1

200150100500

n

G-CA-T

(b)

0.110

0.105

0.100

0.095

0.090

En

20015010050

G-CA-T

n

(a)

0.1015

0.1010

0.1005

0.1000

0.0995

0.0990

0.0985

Vnn+1

0.1050.1000.095En

A-TG-C

(c)

FIG. 4: (a)The on-site energy EneV and (b)transfer integral

Vnn+1eV as a function of the base pair site n. The parametricplot En versus Vnn+1 is shown in (c). W = 0.1 and the otherparameters are given in text. The unit of the energy andthe spatial length are eV and the number of nucleotide basepair(bp), respectively, throughout the present paper.

1.8x10-3

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

L

yapunovexponent

0.20.10.0-0.1Energy

A-TG-Cmixed

(b)4000

3000

2000

1000

0

Localizationlength

0.30.20.10.0-0.1Energy

A-TG-Cmixed

W=0.1,N=2^22(a)

FIG. 5: Comparison (a)localization length and (b)Lyapunovexponent in poly(dG)-poly(dC), poly(dA)-poly(dT) and themixed DNA polymers. W = 0.1, 0 = 36

and N = 222.


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9

12x103

10

8

6

4

2

Localizatonlength

0.30.20.10.0-0.1 Energy

N=2^222^192^16

A-T, W=0.1(a)

6000

4000

2000

0

0.30.20.10.0-0.1Energy

G-C,W=0.1

2^222^192^16

(b)

FIG. 6: The localization length as a function of the energyfor several system sizes N(= 216, 219, 222) in the (a)poly(dA)-poly(dT) DNA polymers, (b)poly(dG)-poly(dC) DNA poly-mers. W = 0.1 and 0 = 36

.

types of polymers with W = 0.1. In low energy regions,the localization length in the poly(dG)-poly(dC) DNApolymer is larger than that in the poly(dA)-poly(dT)DNA polymer. The system size dependence of the lo-calization length is given in Fig.6 in relation to the reso-nance energy. Indeed, the localization length of the DNApolymers is larger than > 2000[bp] in almost all theenergy bands for all models; it is much larger than thesystem size of the oligomer used in the experiments. Asis seen in Fig.6, the smaller the size of the system themore complex the resonance peaks become.

D. Quantum Diffusion in the Fluctuating

Environment

In this subsection we numerically investigate quan-

tum diffusion of electrons in the Hennig model ofpoly(dG)-poly(dC) and poly(dA)-poly(dT) with a dy-namical disorder[15]. We assume that the diffusion iscaused by a colored noise associated with the stochas-tic dynamics of distances r(t) between two Watson-Crick

base pair partners: r(t)r(t ) = r0 exp(|t t |/).These fluctuations can be regarded as a stochastic pro-cess at high temperature, with phonon modes being ran-domly excited[42, 43]. In the model, the characteristicdecay time of correlation can control the spread ofthe electronic wavepacket. Interestingly, the white-noiselimit 0 can in effect correspond to a sort of mo-tional narrowing regime(see, examples), because we findthat such a regime causes ballistic propagation of thewavepacket through homogeneous DNA duplexes. Still,in the adiabatic limit , DNA electronic statesshould be strongly affected by localization. The ampli-tude r0 of random fluctuations within the base pairs (thefluctuation in the distance between two bases in a basepair) and the correlation time , are very critical param-eters for the diffusive properties of the wavepackets. Itis interesting to find the ballistic behavior in the whitenoise limit vs the localization in the adiabatic limit, sincethere is a number of experimental works observing ballis-

tic conductance of DNA in water solutions, which is alsotemperature-independent. Zalinge et al. have tried toexplain the latter effect, using a kind of acoustic phononmotions in DNA duplexes, which seems to be plausible,but not the only possible physical reason[44, 45]. Wewill propose an alternative explanation for the observedtemperature-independent conductance, based upon ournumerical results.

Generally, in the case of quantum diffusion the tempo-ral evolution of the electronic state vector | is describedby the time-dependent Schrodinger equation i|t =Hel(t)|, which then becomes

ieffnt

= En(t)n Vnn+1(t)n+1 Vn1n(t)n1,(17)

where n = n| and the effective Planck constanteff = 0.53. We redefined the scaled dimensionless vari-

ables En(t) and Vnn+1(t) in Eq.(1) such thatEnV0

En,Vnn+1V0

Vnn+1. We used mainly the 4th order Runge-Kutta-Gill method in the numerical simulation for thetime evolution with time step t = 0.01. In some cases,we confirmed ourselves that the accuracy in the obtainedresults is in accord with the one in the results that aregained by the help of the 6th order symplectic integratorthat is the higher order unitary integration.

40

30

20

10

0

DiffusionrateD(t)

108642 1

r n0 =135

(b) G-C40

30

20

10

0

DiffusionrateD(t)

108642 1

r n0 =1

35

(a) A-T

FIG. 7: Diffusion rate D(t) as a function of 1 for severalfluctuation strengths rn0 = 1, 3, 5 at (a)A-T model and (b)G-C model, respectively

600

500

400

300

200

100

0

MSD

m(t)

108642time t

(b) tau=0.01300

200

100

0

MSD

m(t)

2520151050time t

(a) tau=1.0

A-TG-Cmixed

FIG. 8: Short-time behavior ofm(t) of A-T, G-C, and mixedmodels with rn0 = 1.0 at (a) = 1 and (b) = 0.01.

It has been demonstrated that the motional narrow-ing affects the localization in the poly(dG)-poly(dC) and


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10

poly(dA)-poly(dT) DNA polymers. In either case of themodel DNA polymers, the temperature-dependence be-comes virtually suppressed when the motion of the wavepacket is characterized by the ballistic propagation.

We have also investigated the temporal diffusion ratein almost all the diffusive ranges. (See Fig.7.) We foundthat the diffusion rate of the A-T model is larger thanthat of the G-C model at comparatively low tempera-

tures. Interestingly enough, for relatively high tempera-tures in the diffusive range of the wavepacket motion thedifference between the two DNA systems gets smaller.

Figure 8 shows the short-time behavior in the casesof = 1 and = 0.01 depicted on a larger time scale.It follows that in the short-time behavior the (dG)15 (dC)15 case is more diffusive than its (dA)15 (dT)15counterpart within the range from which the spread ofthe wavepacket is

m 15.

We used periodic sequences, which means that E0and V0 are constant values as the static parts of theon-site and hopping terms for poly(dG)-poly(dC) andpoly(dA)-poly(dT) DNA polymers, respectively. This

includes the mixed model as well. Then the motionalnarrowing for dynamical disorder makes time-evolutionof the wavepacket ballistic. However, we remark that themotional narrowing strongly localizes the wavepacket ifwe use a disordered sequence for the static parts of Enand/or Vnn+1.

E. Quantum Diffusion Coupled with Vibrational

Modes

Let us denote rn(t) by the stretching vibration of W-C bonds at the n site as before. The external harmonic

perturbation is equivalent to the coupling with quantumlinear oscillators or with phonon modes in solid statephysics [46, 47]. We replace the disordered fluctuationrn(t) with the harmonic time-dependent one such as

rn(t) = rn0

Mi=1

i cos(it + (i)n ), (18)

where M and i are the number of the frequency compo-nent and the strength of the perturbation, respectively.

The initial phases {(i)n } [0, 2] at each site n are ran-domly chosen. In the numerical calculation of this sec-tion, we take i =

M, for simplicity, and take incom-

mensurate numbers of i O(1) as the frequency set.In the limit of M , the time-dependent perturba-tion approaches the stochastic fluctuation as discussed inthe last subsection. In particular, we can regard the ap-proximation as an electronic system coupled with highlyexcited quantum harmonic oscillators. One of the ad-vantage of this model is that although the number of au-tonomous modes is limited due to the computer power,we can freely control the number M of frequency com-ponents in the harmonic perturbation. This is also a

simple model to investigate electronic diffusion coupledwith lattice vibrations.

In Fig.9(a), the MSD is shown for the poly(dG)-poly(dC), the poly(dA)-poly(dT) and the mixed DNApolymers. We find that all the cases exhibit the normaldiffusion of electron without any stochastic perturbation.As shown in Fig.9(b), the diffusion rate decreases as theperturbation strength increases. As a result the coupled

motion of charges and the lattice breathers connectedwith the localized structural vibrations may contributeto the highly efficient long-range conductivity. (See ap-pendix D.)

3000

2500

2000

1500

1000

500

0

MSD

m(t)

403020100

t

A-TG-Cmixed

(a) 2500

2000

1500

1000

500

0

MSD

m(t)

403020100 t

(b)

FIG. 9: (a) MSD m(t) in A-T, G-C, and mixed models per-turbed by M = 1, = 0.5. (b) MSD m(t) of A-T modelperturbed by M = 1 for = 0.5, 0.7, 1.0, 2.0 from above. Weset rn0 = 1.

V. TIGHT-BINDING MODELS FOR THE

LADDER SYSTEMS

In this section, to investigate the energy band struc-ture for periodic sequences and the localization proper-ties for the correlated disordered sequences, we intro-duce the ladder models of DNA polymers. Although inRefs.[11, 13, 21], we assumed the ladder structure con-sisting of sugar and phosphate groups, the model canbe applicable to the DNA-like substances such as theladder structure of bases pairs without sugar-phosphatebackbones. When we apply the model for the sugar andphosphate chains, the chain A and B are constructedby the repetition of the sugar-phosphate sites, and theinter-chain hopping Vn at the sugar sites come from thenucleotide base-pairs, i.e., A T or G C pairs. (SeeFig.10(a).)

A. The Ladder Models and the Dispersions

Since there is only one orbital per site, there aretotally 2N orbitals in the ladder model of the DNAdouble chain. Let us denote by An (

Bn ) the orbital

at site n in the chain A(B). By superposition of the

orbitals, the Schrodinger equation H| = E| be-


11/25

11

FIG. 10: Models of the double strand of DNA. (a) the two-chain model, (b) the three-chain model we adapted in themain text.

comes,

An+1,nAn+1 + An,n1

An1 + An,n

An + Vn

Bn = E

An ,

Bn+1,nBn+1 + Bn,n1

Bn1 + Bn,n

Bn + Vn

An = E

Bn ,

where An+1,n (Bn+1,n) means the hopping integral be-tween the nth and (n + 1)th sites and An,n (Bn,n) theon-site energy at site n in chain A (B), and Vn is the hop-ping integral from chain A(B) to chain B(A) at site n, re-spectively. Furthermore it can be rewritten in the matrixform, n+1 = Mnn, where n = (

An ,

An1,

Bn ,

Bn1)

.The transfer explicit matrix is given in appendix B. Gen-erally speaking, we would like to investigate the asymp-

totic behavior (N ) of the products of the matricesMd=2(N) = Nk=1Mk = MNMN1 M1.We consider the band structure for the periodic case by

setting An+1,n = Bn+1,n = a(b) at odd (even) site n andVn = v(0) at odd(even) sites, Ann = Bnn = (Ann =Bnn = ) for odd(even) site n, for simplicity.

A simple way to solve the above equations is to use theBloch theorem:

An+2 = ei2ksAn ,

Bn+2 = e

2iksBn , (19)

where s is the length between the adjacent base groupsand is assumed to be equivalent to the length betweenthe adjacent sugar-phosphate groups and the wavenum-

ber k is defined as 2s k 2s . (We take s = 1in the following calculations.) The Schrodinger equationbecomes

n+1 = Mn, (20)

M =

E (a + beik) 0 0(a + beik) E 0 v

0 0 E (a + beik)0 v (a + beik) E

, (21)

where n = (A2n+1, A2n,

B2n+1,

B2n)

t. Denote the deter-minant of M by D(M). Solving the equation D(M) = 0for E, we obtain the energy dispersion of the system.Here we show the simple case of = = 0,

E()+ (k) =

1

2

v

v2 + 4(a2 + b2) + 8ab cos2k

,(22)

E() (k) =

1

2

v

v2 + 4(a2 + b2) + 8ab cos2k

.(23)

E(+) (k) ( E

() (k) ) stand for the upper(lower) bands

in channel

, respectively[11]. The lowest and uppermiddle (the lower middle and highest) bands correspondto bonding (antibonding) states between the adjoint or-bitals in the interchains. The more detail of the energyband for general case is given in appendix C. In Fig.11(a)the energy band structure is given with varying the in-terchain hopping v. Figure 11(b) shows the cross-sectionview at v = 1. There is a band gap Eg(v) at the cen-ter in between the lower and upper middle bands in thespectrum for the whole range of v when = = 0.

Eg(v) = E(+) (/2) E()+ (/2) =

4(a b)2 + v2 v.(24)

The other band gap g(v),

g(v) = v +1

2

4(a b)2 + v2

4(a + b)2 + v2

,(25)

appears in between the lowest and the lower middle bands(the upper middle and highest bands) when v > vc 2ab/a

2

+ b2

(otherwise, it is negative and thereforesemimetallic). There is a transition from semiconduc-tor to semimetal as the electron hopping between thenitrogenous bases of nucleotide is increased.

The two-chain model can be easily extended to thethree-chain case. (See Fig.10(b).) Cn+1,n means the hop-ping integral between the nth and (n + 1)th sites andCn,n the on-site energy at site n in chain C, and Vn andUn are the hopping integral between the chains. TheSchrodinger equation H| = E| becomes,


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12

An+1,nAn+1 + An,n1

An1 + An,n

An + Vn

Cn = E

An ,

Cn+1,nCn+1 + Cn,n1

Cn1 + Cn,n

Cn + Vn

An + Un

Bn = E

Cn , (26)

Bn+1,nBn+1 + Bn,n1

Bn1 + Bn,n

Bn + Un

Cn = E

Bn ,

It can be also rewritten in the matrix form, n+1 =Md=3(n)n where n = (

An+1,

An ,

Cn+1,

Cn ,

Bn+1,

Bn )

t

and the explicit transfer matrix is given in the appendixC.

For simplicity, we set An+1,n = Bn+1,n = a andCn+1,n = c at site n, and Vn = Un. We can analyti-cally obtain the energy band structure for the three chainmodel by using the Bloch theorem.

An+1 = eiksAn ,

Cn+1 = e

iksCn , Bn+1 = e

iksBn . (27)

B. Localization in the Ladder Models

As was discussed in Sect. 2 and Sect. 3, the sequencesof the realistic DNA polymers are not periodic and ac-company a variety of disorder. Generally, the random-ness makes the electronic states localized because of thequasi-one-dimensional system and affects electronic con-duction and optical properties, and so on. In the presentsection, we investigate the correlation effect on the local-ization properties of the one electronic states in the dis-ordered, two-chain (ladder) and three-chain models witha long-range structural correlation as a simple model forthe electronic property in the DNA[14]. The relationship

between the correlation length in the DNA sequences andthe evolutionary process is suggested. Moreover, it isinteresting if the localization property is related to theevolutionary process.

Figure 12 shows the DOS as a function of energy forthe binary disordered systems. The sequence of the in-terchain hopping Vn takes an alternative value WGC orWAT. We find that some gaps observed in the periodiccases close due to the disorder corresponding to the base-pair sequences.

Figure 13(a) shows the energy dependence of the Lya-punov exponents (1 and 2) for some cases in the asym-metric modified Bernoulli system characterized by thebifurcation parameters B0 and B1 controlling the corre-lation. (See appendix G for modified Bernoulli system.)They are named as follows: Case(i):B0 = 1.0, B1 = 1.0,Case(ii):B0 = 1.0, B1 = 1.9 and Case(iii):B0 = 1.7, B1 =1.9. Apparently, the case(i) is more localized than thecases(ii) and (iii) in the vicinity of the band center|E| < 1. The comparison between the case(ii) and thecase(iii) shows the effect of asymmetry of the map on thelocalization. The ratio of the G-C pairs RGC 0.2 forthe case(ii), RGC 0.47 for the case(iii). In the energyregime |E| > 1, the Lyapunov exponent 2 in the case(ii)

FIG. 11: Energy bands of the decorated ladder model. (a)The energy bands as a function of V. (b) The snap shot ofthe energy bands when V = 1. k means the wave vector inunits of

ssuch that 1.0 k 1.0, V means the -electron

hopping integral between the inter chain sites, and E meansthe energy in units ofV = 1. Here we have taken the values = 0, = 0, a = 0.9, b = 1.2.

is smaller than the one in the case(iii) in spite of the samecorrelation strength B1. As a result, we find that in theDNA ladder model, correlation and asymmetry enhancethe localization length ( 1/2 ) of the electronic statesaround |E| < 1, although the largest Lyapunov expo-nents 1 do not change effectively at all. Figs. 13(b),(c) and (d) show the nonnegative Lyapunov exponentsin the real DNA sequences of (b) HCh-22, (c) bacterio-


13/25

13

phages of E. Coli and (d) histon proteins. In the case ofHCh-22, we used two sequences with N = 105, extractedfrom the original large DNA sequences. The result showsthat the Lyapunov exponents do not depend the detail ofthe difference in the sequences of HCh-22. Although theweak long-range correlation has been observed in HCh-22 as mentioned in the introduction, it does not affect onthe property of localization. The sequences we used are

almost symmetric (RGC 0.5).

-3 -2 -1 0 1 2 3

(a)

DOS(a.u.)

-3 -2 -1 0 1 2 3E

(b)

FIG. 12: DOS as a function of energy for the binary disor-dered cases in the two-chain model. (a)WAT = 2.0, WGC =1.0, a = 1.0, b = 1.0. (b)WAT = 2.0, WGC = 1.0, a = 1.0, b =0.5. The on-site energy is set st Ann = Bnn = 0.

In addition, in the numerical calculation we set the on-site energy as Ann = Bnn = Cnn = 0 for simplicity. Thesequence

{Cnn+1

}can be also generated by correspond-

ing to the base-pairs sequence. The localization prop-erties in the simple few-chain models with the on-sitedisorder have been extensively investigated [48]. Figure14 shows the energy dependence of the Lyapunov expo-nents in the three chain cases. We can observe that all theLyapunov exponents i(i d) are changed by the corre-lation. The least nonnegative Lyapunov exponent 3 isdiminished by the correlation. In particular, it is foundthat the correlation of the sequence enhances the localiza-tion length defined by using the least non-negative Lya-punov exponent. We can see that the localization lengthdiverges at the band center E = 0.

In Fig.15(a) the result for asymmetric modifiedBernoulli system is shown. Apparently, the correlationand/or asymmetry of the sequence effect a change in thesecond and third Lyapunov exponent. In contrast, al-though the global feature of 1 is almost unchanged, thelocal structure of the energy dependence is changed bythe change ofB0. Fig.15(b) shows the results in phage-fdand phage-186 in the three-chain model. It is found thatthe structure of the energy dependence around |E| < 2is different from that in the artificial sequence by themodified Bernoulli map.

1.0

0.8

0.6

0.4

0.2

0.0

3210-1-2-3

HC22_1HC22_2(b)

1.0

0.8

0.6

0.4

0.2

0.0

3210-1-2-3

Case(1)Case(2)Case(3)

(a)

1.0

0.8

0.6

0.4

0.2

0.0

3210-1-2-3

Lambda186(c)

0.4

0.3

0.2

0.1

0.0

210-1-2

Early H1Late H1

(d)

FIG. 13: Lyapunov exponents (1, 2) as a function of en-ergy in the ladder model. (a) Modified Bernoulli model, wherecases(1):B0 = 1.0, B1 = 1.0, Case(2):B0 = 1.0, B1 = 1.9 andCase(3):B0 = 1.7, B1 = 1.9 are shown. (b)HC-22. (c) bac-teriophages of E. coli (phage-, phage-186). (d) early his-tone H1 and late histone H1. The on-site energy is set atAnn = Bnn = 0, a = 1.0, b = 0.5. The size of the se-quence is N = 105 for (a), N = 105 for (b), N = 48510for the phage- in (c), N = 30624 for the phage-186 in (c),N = 787 for the early histone H1 in (d), and N = 1182 forthe late histone H1 in (d).

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

i

-4 -2 0 2E

B=1.0B=1.8

FIG. 14: Lyapunov exponents i(i = 1, 2, 3) as a function of

energy in the correlated three-chain model. The parametersWAT = 2.0, WATAT = 1.0, a = 1.0, b = 0.5. The on-siteenergy is set at Ann = Bnn = Cnn = 0.

C. Polaron Models for a Double Strand of DNA

The overlap of the electronic orbitals along the stackedbase pairs provides a pathway for charge propagationover 50A and the reaction rate between electron donors


14/25

14

3

2

1

0

Lyapunovexponents

3210-1-2-3

case(1)case(2)

(a)

3

2

1

0

Lyapunovexponents

3210-1-2-3Energy

phage-sfdphage-186

(b)

FIG. 15: Lyapunov exponents (1, 2, 3) as a function of en-ergy in the three-chain model. (a) Modified Bernoulli model,(b) bacteriophages of E. coli (phage-fd, phage-186). The pa-rameters are same as ones in Fig.14 except for on-site energiesof C-chain (Cn,n = 0).

and acceptors does not decay exponentially with dis-tance. A multiple-step hopping mechanism was recently

proposed to explain the long-range charge transfer be-havior in DNA. In this theory, the single G-C base pairis considered as a hole donor due to its low ionizationpotential if compared to the one on the A-T base pairs.Besides the multiple-step hopping mechanism, polaronmotion has also been considered as a possible mecha-nism to explain the phenomena of DNA charge transfer.The charge coupling with the DNA structural deforma-

tions may create a polaron and cause a localized state.The polaron behaves as a Brownian particle such thatit collides with low energy excitations of its environmentthat acts as a heat bath. For the most physical systems,the acoustical and optical phonons are the main latticeexcitations as in the single-chain case given in AppendixD.

In this section, we present the formation of localizedpolarons due to the coupling between charge carriers andlattice vibrations of a double strand of DNA[12]. Thesepolarons in DNA act as donors and acceptors, which ex-hibit an extrinsic semiconductor character of DNA. Theresults are discussed in the context of experimental ob-

servations.Let xn and yn be configuration angles for rotation of

the nth base group along the course of the backbonechains A and B of DNA ladder, respectively. For thesake of simplicity, we assume that all masses Mn of thenucleotide groups and the bond lengths dn between SandBn groups are identical so that Mn = M and dn = d, giv-ing the moment of inertia In = I0 = M d

2. In this case,the lattice Hamiltonian Hph can be given by

Hph(xn, yn) =Nn=1

{I02

(x2n + y2n) +

A2

(x2n + y2n) +

K2

(xn+1 xn)2

+K

2(yn+1 yn)2 + S

2(xn yn)2} (28)

where A, K and S are the parameters for the rotationalenergy, the stacking energy and the bonding energy ofthe bases, respectively. We note here that various gen-eralizations and modifications of the model are straight-forwardly possible by adapting a different combination ofthe interactions and the hopping integrals. In the follow-

ing, we give result only for HOMO. We can obtain theresult for LUMO in simple replacements.

On the other hand, generalizing the idea of Holstein forthe one-dimensional molecular crystal to our case of thedouble strand of DNA, the tight bingding Hamiltonianfor electrons is given by

HHel =

Nn=1

{taA(n+1)aAn taA(n1)aAn + AH(xn)aAn aAn + vn(|xn yn|)aAn aBn }

+Nn=1

{taB(n+1)aBn taB(n1)aBn + BH(yn)aBn aBn + vn(|xn yn|)aBn aAn }


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15

For the case of LUMO, we can obtain the result by a bin HLel. Here a

An (b

An ) is the electron creation operator

in the HOMO (LUMO) at the nth nucleotide group ofP, S and Bn for the chain A.

AH(xn) and

BH(yn) are the

electron-lattice coupling Helph given by

AH(x) = AH+ FHx, (29)

BH(y) = BH

FHy, (30)

respectively. We can also obtain AL(xn), BL (yn) for

LUMO. Moreover, we assume the relation:

|L H| >> 4|t|, (31)since this condition is realized in most of the DNA sys-tems and guarantees the semiconductivity of the DNA.The total Hamiltonian Htot is given by

Htot = Hph + HHel + H

Lel, (32)

while the wavefunction | given by

| =

s=A,B

Nn=1

{snasn |0 + snbsn |0 + snasn |0 + snbsn |0}(33)

where an (bn) means the electron creation operator at

site n in chain A (B). An (Bn ) represents the HOMO

and An (Bn ) the LUMO, at the nth nucleotide site in the

chain A(B), respectively. Applying to the Schrodingerequation Htot| = Etot| we obtain the following equa-tions for the HOMO states of nucleotide groups in thedouble strand of DNA:

t{An+1 + An1} + AHAn + FHxnAn vnBn = EAn(34)

t{Bn+1 +

Bn1} +

BH

Bn FHyn

Bn vn

An = E

Bn(35)

where E EtotHph(xn, yn), vn v(|xnyn|). For theLUMO states of nucleotide groups in the double strandof DNA, the similar equations are given by replacing and H by and L, respectively. Moreover, according toargument of Holstein, we find the following coupled dis-cretized nonlinear Schrodinger equation(DNSE) describ-ing the electronic behavior under the lattice vibrationsin the ladder model of DNA. In Appendix E, we give thederivation of the coupled DNSEs. The single-chain ver-sion of the DNSE and some comments on the physicalmeanings are also given in Appendix D.

D. DC-Conductivity of the Double Strand of DNA

We exclusively focused on the low-energy transport,when the charge injection energies are small comparedwith the molecular bandgap of the isolated moleculewhich is of the order of 2 3eV. In the experiment of theconductance property of the DNA, temperature depen-dence is important. Finite temperature can also reducethe effective system size and leads to the changes in the

transport property. Moreover, the effects of the stackingenergy and of temperature can be considered by intro-ducing the fluctuation in the hopping energy such as theSu-Schriefer-Heegar model for polyacetilene [49, 50].

We consider DC-conductivity of periodic DNA se-quence based on the LUMO, HOMO band ofelectronsand small polaron [12]. It is known that DNA behavesn(p)

type extrinsic semiconductor, where the donors

(acceptors) for the LUMO (HOMO) band are positively(negatively) charged small polarons with the total num-ber Nd(Na) and the energy d = E

spL = Ec F2L/I020,

a = EspH = Ev + F

2H/I0

20, where Ev(Ec) is the valence

(conduction) band edge. Following the standard argu-ment, denote by nc and nD the numbers of electrons inthe conduction (LUMO) band and in the donor levels,respectively; and denote by pv and pA the numbers ofelectrons in the valence (HOMO) band and in the ac-ceptor levels, respectively. We now have the relation:nc + nd = Nd Na +pv +pa, where

nd =Nd

1

2e(d

) + 1

, pa =Na

1

2e(

d) + 1

, (36)

where 1/KBT, is chemical potential of the system.If we suppose d > > kBT, a >> kBT andnd


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16

FIG. 16: The DC conductivity is shown for the n-type (orp-type) extrinsic semiconductor where the energy gap Eg =0.33eV, the hopping energy for the HOMO (or LUMO) bandt = 0.2eV and the activation energy for the polaron hoppingof (a) Ea = 0.001t, (b) Ea = 0.03t and (c) Ea = 0.05t aretaken, respectively. Nc = Pv = 10

8 and n = ND NA = 103

are assumed.

FIG. 17: Energy gap Eg(T) of the ntype (or ptype) ex-trinsic semiconductor is shown for electron conduction (orhole conduction) where the energy gap Eg = 2eV, t = 0.1eV,Nc = Pv = 10

9 and n = 106 are used, respectively.

host material is a homogeneous crystal of Si or Ge withfour valence bonds, then the impurities are As atomswith five valence bonds and boron(B) atoms with threevalence bonds, which then provides an inhomogeneousmaterial with impurity levels. The As atoms play a roleof donors with positive charge such that the system be-comes an n-type semiconductor, while the B atoms playa role of acceptors with negative charge such that thesystem becoms a p-type semiconductor. Such the systemof n- or p-type semiconductor exhibits an extrinsic semi-conductor character. On the other hand, in our systemsof DNA semiconductors, there exist no such impurities

exerted from outside; but already there exist inside theDNA a complicated arrangement of bases of A, G, C andT, such that among the bases, each one of bases may re-gard other bases as impurities. Hence, a kind of extrinsicsemiconductive nature may appear as a result of self or-ganization of the base arrangement. This is the meanigof our terminology of extrinsic semiconductor for theDNA systems. Therefore, it is more preferable for us to

put self-organized in front of extrinsic semiconductorfor DNA such as self-organized extrinsic semiconduc-tor in order to represent the semiconducting characterof DNA.

We would like to note that in the ddimensional dis-ordered systems, the activated hopping between local-ized states, i.e., the variable range hopping, could be adominant mechanism for the dcconductivity, and thetemperature dependence is governed by

= 0e(T0T )

dd+1

, (40)

where T0 = 8Wa/kB. Here W is the energy difference

between the two sites, is Lyapunov exponent, and a thedistance between the nearest neighboring bases. It is rea-sonable to expect that the experiments of charge trans-port in DNA suggest such temperature-dependence e

T0/T at relatively low temperature due to the

1d aperiodic base-pair sequence. The main contribu-tion was given by the interaction with water moleculesand not with counterions. Further, polaron formationwas not hindered by the charge-solvent coupling. Andthe interaction rather increased the binding energy (self-localization) of the polaron by around 0.5eV, which ismuch larger than relevant temperature scales.

VI. CONDUCTION AND PROTON TRANSFER

Gene mutations sometimes cause human disorders suchas cancer. Simultaneously, gene mutation can be a driv-ing force for processes of biodiversity and evolution[51].When a system contains hydrogen bonds, proton mo-tion also needs to be considered. Instead of oscillatorymotions in a single-well potential, protons can tunnelfrom one side to another in a double-well potential ofthe hydrogen bond. This proton tunneling causes aninterstrand charge hopping. And it could generate thetautomeric base pairs and destroy the fidelity of the W-C base pairs.

Lowdin proposed that proton tunneling contributes tothe formation of rare tautomeric of DNA W-C base pairswhose accumulation could result in DNA damages, pointmutations, and even tumor growth[52]. The argument re-lies on the assumption that the rare tautomers are morestable, and once the intermolecular proton transfer oc-curs, lifetime of the tautomers is significantly larger thantime of DNA repair. Indeed, the results from the quan-tum chemical and statistical mechanical calculations in-dicate that at room temperature at least the GC tau-


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17

tomers (G C) would have a sufficient lifetime tocause the DNA damage through mismatches of the W-Cbase pairs.

Ladik speculated that semiconductivity of DNA mightbe related to the origin of cancer due to transmutationof genes[53]. Very recently, Shi et al. have also reportedcorrelation between the cancereous mutation hot spotsand the charge transport along DNA sequences[54].

The proton transfer causes fluctuation of the poten-tial and more or less affects on the localization/transportproperty of the charged carriers. At the same time, theproton transfer suffers from the lattice vibration and theelectron transport along the base sequences of DNA. Ac-cordingly, we should treat the coupling between the pro-ton transfer and the phononic and electronic states whenwe investigate the stability of the tutomatic states(theexcited states). Recent theoretical studies have shownthat charged protonated base pairs display smaller acti-vation barriers, which make the proton transfer and thetunneling more facile.

Chang et al.[36] have given the effective 1D modelHamiltonian H

tot= H

elph(t) + H

+ H

el, where

H =

n Hn represents Hamiltonian describing theproton transfers. Here Hamiltonian Helph includes theelectron and phonon modes and the coupling. For theproton motion in hydrogen bonds, they used a two-levelsystem in order to describe the tunneling behavior atmolecular site n

Hn =1

2(pzn + tpxn), (41)

where zn and xn are Pauli matrices assigned at site n, p

the energy bias between the two localized proton statesand tp the tunneling matrix element between the states.They have modeled the coupling between the protons in

the hydrogen bond and the charges in the DNA strandas

Hel =n

n(zn 1)CnCn (42)

where n is the coupling intensity. When the proton isin the lower energy state or there is no charge around thehydrogen bond (CnCn = 0), the coupling vanishes.

Indeed, the normal G-C base pair is in the lower energystate and its tautomeric form is in an excited state. Whenp >> tp, the probability of having a tautomeric formis extremely small. However, the cation of a G-C basepair has almost the same energy as that of its tautomericform p tp. In this case, the proton state becomesdelocalized in hydrogen bonds.

The symmetry of potential well and the hight of barrieressentially affects on the tunneling probability betweenthe potential wells. The probabilistic amplitude of pro-ton could influence the transport of the charged carriersthough the Coulomb interaction. In addition, the two-level approximation will be broken down if the chaoticmotion occurs in the dynamics due to the coupling withlattice vibrations [55].

VII. SUMMARY AND DISCUSSION

We briefly reviewed our recent works with concern-ing the electronic states and the conduction/transfer inDNA polymers. In Sect.3, based on the Huckel model,we have discussed the basics of quantum chemistry wherethe electronic states of atoms in biochemistry such asC, N, O, P and the electronic configurations of the ni-

trogenous bases, the sugar-phosphate groups, the nu-cleotides and the nucleotide bases are summarized, re-spectively. In Sect.4, based on the tight-binding approx-imation of the DNA sequences, we have investigated thelocalization properties of electronic states and quantumdiffusion, using a stochastic-bond-vibration approach forthe poly(dG)-poly(dC), the poly(dA)-poly(dT) and themixed DNA polymers within the frameworks of the po-laron models. At that time, we assumed that the dynam-ical disorder is caused by the DNA vibrational modes,which are caused by a noise associated with stochasticdynamics of the distances r(t) between the two Watson-Crick base pair partners. In Sect.5, we have mainly in-

vestigated the energy band structure and the localiza-tion property of electrons in the disordered two- andthree-chain systems (ladder model) with long-range cor-relation as a simple model for electronic property in adouble strand of DNA by using the tight-binding model.In addition, we investigated the localization properties ofelectronic states in several actual DNA sequences such asbacteriophages of Escherichia coli, human chromosome22, compared with those of the artificial disordered se-quences. In Sect.2 and 4, we gave a brief explanationfor DNA bases sequences and gene mutations, respec-tively, which are related to the DNA carrier conductionphenomena [5659]. The relationship between the long-range correlations and the coherent charge transfer in thesubstitutional DNA sequences has been studied, based onthe transfer matrix approaches[60, 64].

Recently, the tight-binding models for the DNA poly-mers have been extended to the decorated ladder modelsand the damaged DNA models by some groups. Further-more, the I-V characteristics of the ladder models cou-pled with environments has been investigated as simplemodels for DNA polymers [6163].

Appendix A: Huckel Parameters

Sandorfys formula[28], Streitwiesers formula[30], andMullikens formula[29] are given by following relations,

X = +X C

C 4.1, (A1)

X = X + , (A2)

CX =SCXSCC

, (A3)

respectively. See Ref.[5] for the details.


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Appendix B: Transfer Matrix Method

Let us define the four-dimensional column vector inthe main text as n (An , An1, Bn , Bn1)t. Then, it

can be rewritten in the following form: n+1 = Mnn,where

Mn =

An VnUn Bn

. (B1)

Mn is the 4 4 transfer matrix with the 2 2 matrices:

An

EAn,nAn+1,n

An,n1An+1,n1 0

, Bn

EBn,nBn+1,n

Bn,n1Bn+1,n1 0

, (B2)

Un Un

An+1,n0

0 0

, Vn

VnBn+1,n

0

0 0

. (B3)

According to the sequence ofN segments, we have to takethe matrix product M(N) MNMN1 M1, which isalso a 4 4 matrix. When the double chain system isperiodic, we adopt the Bloch theorem:

An+N = An ,

Bn+N =

Bn , (B4)

where = eikN and k is the wavevector. Then, we find a4 4 determinant D() that is a fourth order polynomialof :

D() det[M(N) I4]= 4 a13 + a22 a3 + a4 = 0. (B5)

It provides the wave vector k in the system such thatk = 2j/N for j = N/2, , N/2. Here, if four rootsare written as 1, 2, 3, and 4, then the following con-ditions hold

a1 trM(N) =4i=1

i, a2 4

i


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The density of states (DOS) D(E) is calculated for eachchannel , respectively:

dk =1

Nd cos1[x(E)/2] (B16)

= 1N

x

4 x2

dE = D(E)dE. (B17)

Therefore, the total DOS is given as the sum of D+(E)and D(E):

D(E) = D+(E) + D(E), (B18)

where D(E) [D+(E)] means the DOS contributed fromthe bonding (antibonding) channel (+), respectively.It agrees with the result on the tight-binding model forthe ladder structure. Physically speaking, the (+)channel means the bonding (antibonding) states betweentwo parallel strands of the DNA.

Appendix C: Transfer Matrices for Ladder Systems

In this appendix, we give the explicit expression ofthe energy band for the decolated ladder model given inFig.10(a). In the unit cell, the period is taken as N = 2and it contains four orbitals. We apply the result inAppendix B for the case. Let the transfer matrix methodbe

M =

A VU B

. (C1)

M is a 4 4 transfer matrix with 2 2 matrices:

A = An+1An

(E)(E)ab ab EaEa ba

= B,(C2)

V = An+1Vn

Eab v 0 va 0

= U. (C3)

Let us calculate trM and tr(M2). We find

trM = tr(A) + tr(B) = 2P 2R (C4)tr(M2) = tr(A2) + tr(B2) + tr(U V) + tr(V U)

= 2(P + Q)2 + 2(R2 2), (C5)

where

P =(E )(E )

ab, Q =

(E )vab

, R =a2 + b2

ab.(C6)

The discriminant D is given by D = 2tr(M2

) (trM)2

+8 = 4Q2. As a result, we obtain

2cos2k =1

2(trM

D) (C7)

= P R Q (C8)=

(E )(E )ab

a2 + b2

ab (E )v

ab(C9)

Solving the above for E, we can obtain the energy bands.

E()+ =

1

2

( + ) v

( )2 + v2 + 4(a2 + b2) + 2( + )v + 4v + 8ab cos2k

(C10)

E() =

1

2

( + ) + v +

( )2 + v2 + 4(a2 + b2) 2( + )v 4v + 8ab cos2k

(C11)

In Fig.1(a), the energy band structure for a = b = 1when = 1, = 0 is given with varying the interchainhopping v. Figure 1(b) shows the cross-section view atv = 1.

The extension to the decolated three chains (d = 3) isstraightforward. Here we give the transfer matrix for a

simple system of coupled three chains. n+1 = Mnnwhere n = (An+1, An ,

Cn+1,

Cn ,

Bn+1,

Bn )

t. 66 trans-

fer matrix with the 2 2 submatrices is given in blocktridiagonal form:

Mn =

An V

An 0

VCn Cn UCn

0 UBn Bn

, (C12)

where

Cn

ECn,nCn+1,n

Cn,n1Cn+1,n1 0

, VAn

VnAn+1,n 00 0

, VCn

VnCn+1,n 00 0

,

UCn UnCn+1,n 0

0 0

, UBn

UnBn+1,n 00 0

.


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FIG. 18: Energy bands of the decorated ladder model. (a)The energy bands as a function of V. (b) The snap shot ofthe energy bands when V = 1. k means the wave vector inunits of

ssuch that 1.0 k 1.0, V means the -electron

hopping integral between the inter chain sites, and E meansthe energy in units of V = 1. Here we have taken the values = 1.0, = 0, a = b = 1.

For simplicity, we set An+1,n = Bn+1,n = a, Cn+1,n = c,Ann = Bnn = , Cnn = , and Vn = Un = v at site n.Then

M =

Ea 1 va 0 0 01 0 0 0 0 0

vc 0 Ec 1 vc 00 0 1 0 0 00 0 va 0 Ea 10 0 0 0 1 0

. (C13)

Even for general multichain models, some useful for-mula exist in order to obtain the eigenvalues of blocktridiagonal matrices and the determinants of the corre-

sponding block tridiagonal matrices [6567].

Appendix D: Nonlinear Schrodinger Equation

In this appendix, we derive the discrete nonlinearSchrodinger equation for 1D electronic system coupledwith lattice oscillations. The relative motions betweentwo different base pairs can be represented by an acous-

tical phonon mode and the vibrational motion inside abase pair by optical phonons. These modes represent thelattice distortions such as sliding, twisting or bending.The Hamiltonian that describes these modes is given by

Hph =n

{ p2n

2M+

1

2M2s(un+1 un)2} +

n

{ P2n

2M+

1

2M 2ov

2n}. (D1)


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idndt

= t(n+1(t) + n1(t)) + a|n(t)|2n(t) + Enn(t). (D7)

The time-independent version is given as

En = t(n+1 + n1) + |n|2n + Enn, (D8), which appears in the Holstein polaron model on the lat-tice. The DNSEs (40) and (41) show more various prop-erties due to the inhomogeneity of En and the strength for nonlinearity, etc [70, 71].

For example, localization-delocalization transitiontakes place, depending on the coupling strength and theinitial state. The DNSE has been studied in the contextof delocalization due to nonlinearity. Indeed, Eq.(D7)describes the 1D disordered waveguide lattice, whichis called the Gross-Pitaevsky (GP) equation on a dis-cretized lattice [72]. When the absolute value of the non-

linearity parameter a is greater than some critical valuec, the excitation is self-trapped. It was mathematically

proved that the DNLS has quasi-periodic self-trapped so-lutions called the discrete breathers[73, 74]. On the other

hand, it is found that at moderate strength of nonlinear-ity the spreading of the wavepacket algebraically growsas

n)2 t( 0.2) [75].

Appendix E: Coupled Nonlinear SchrodingerEquations

In this appendix, we derive the coupled nonlinearSchrodinger equations in order to describe the effect offormation of a double strand of DNA on the polarons in

the HOMO band, following the Holsteins argument[12].The energy of HOMO band is expressed as,

EH({xn, yn}) = Hph({xn, yn}) +Nn=1

H(xn)|An |2 + H(yn)|Bn |2

Nn=1

t

(An+1 + An1)

Bn + (

Bn+1 +

Bn1)

An

Nn=1

vn(An

Bn +

Bn

An ). (E1)

Differentiating the energy with respect to xp, yp, respec-tively, we approximately obtain the most contributed co-ordinates xp, yp by

EHxp

= 0, EHyp = 0;

xp = |Ap |2 (Ap Bp + Bp Ap ), (E2)yp = |Bp |2 (Ap Bp + Bp Ap ), (E3)

where = FH/I020, = /I0

20. In the derivation,

we have assumed vp = v |xp yp|, SABH

xp= 0,

SABHyp

= 0, where SABH = (Ap

Bp +

Bp

Ap ). Note that

|An |2(|Bn |2) means the frontier orbital density atp-th nu-cleotide group in chain A(B) and SABH means the over-lapping integral of the frontier orbitals for electrons inthe HOMO at p-th nucleotide group between the chainsA and B. Substituting the expressions into Eq.(34) and(35), we obtain the following coupled DNSEs:

t{An+1 + An1} + ABH An vABn Bn = EAn (E4)t{Bn+1 + Bn1} + BAH Bn vABn An = EBn (E5)

where

ABHn = AH FH|An |2 + FHSABH , (E6)

BAHn = BH FH|Bn |2 FHSABH , (E7)

vABn = v FH(|An |2 + |Bn |2). (E8)

By the same argument, similar equations can be obtainedfor the LUMO band case as well, just by replacing theorbitals of electrons in HOMO bands with the frontierorbitals for holes in the LUMO bands, respectively. Inthe limit v 0, 0, it becomes the decoupled DNSEin Eq.(84) without randomness.

Appendix F: Lyapunov Exponents and Multichannel

Conductance

The definition for energy dependence of the Lyapunovexponents is given by,

i = limN

1

2Nlog i(Md(N)

Md(N)), (F1)


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23

where i(. . . ) denotes the i-th eigenvalue of thematrix Md(N)

Md(N) [38]. As the t ransfer m a-trix Td(N) is symplectic, the eigenvalues of theMd(N)

Md(N) have reciprocal symmetry around theunity as e1 ,...,eded,...,e1 , where 1 2 . . . d 0. The d denotes the number of channels; i.e.d = 2 in the two-chain model and d = 3 in the three-chainmodel.

The Lyapunov exponent is related to the DOS (E)as an analogue of the Thouless relation (the generalizedThouless relation) [68]:

di

i(E)

ln |E E |(E)dE . (F2)

Accordingly, we can see that the singularity in the largestLyapunov exponent is strongly related to the singularityin the DOS.

Generally speaking, in the quasi-one-dimensional chainwith the hopping disorder, the singularity in the DOS,the localization length and the conductance at the bandcenter depend on the parity, the bipartiteness and theboundary condition. Since discussions on the details isout of scope of this paper, we give simple comments.Note that the parity effects appear in the odd numberchains with the hopping randomness. In the odd numberchain with the hopping randomness, only one mode atE = 0 is remained as an extended state, i.e. d = 0,while other exponents are positive, d1 > > 1 > 0.The behavior is seen in Fig.14 in the main text. Then,non-localized states with = 0 determine the conduc-tance. Although we have ignored the bipartite structurein the three-chain models for simplicity, if we introduce

bipartiteness in the intrachain hopping integral Vn(= Un)another delocalized state due to chiral symmetry appearsat E = 0.

Furthermore, we find that in the thermodynamic limit(n ), the largest channel-dependent localizationlength d = 1/d determines the exponential decay in theLandauer conductance g(n) that is measured in units ofe2/h at zero temperature and serves as the localizationlength of the total system of the coupled chains [40, 41].It is given as

g(n) = 2d

i=1

1

cosh2ni(n) 1

exp(

2n

d(n)

), (F3)

for n , where n denotes the system size along thechain [39]. Recently, electron transport for molecularwires between two metallic electrorodes has been alsoinvestigated by several techniques.If we impose a voltage difference V over the molecularstructure, we have one end point with energy EF + eVand the other still EF. Only electrons in that range con-tribute to the conductance. Therefore, the I V charac-

teristics can be evaluated as

I(EF) =V e

dE( f

E)i

Ti(E) (F4)

V e

ij

Tij(EF), (F5)

where f(E) 1/(eE/kBT

+ 1) means the Fermi distribu-tion function and Tij |tij |2 is the squared transmissionamplitudes between the i-th and j-th channels. At thezero-temperature limit, all quantities can be evaluatedby the transmittion matrix at E = EF.

Appendix G: Modified Bernoulli Map

The correlated binary sequence {Vn} and/or {Cnn+1}of the hopping integrals can be generated by the modifiedBernoulli map [76, 77]:

Xn+1 =

Xn + 2B01XB0n (Xn I0)Xn 2B11(1 Xn)B1 (Xn I1), (G1)

where I0 = [0, 1/2), I1 = [1/2, 1). B0 and B1 arethe bifurcation parameters that control correlation inthe sequence. We set 1 < B0 < B1 < 2 for sim-plicity. The asymmetry of the map (B0 = B1) cor-responds to the asymmetric property in the distribu-tion for a real sequence in the double helix DNA, wherethe number of the A-T pairs is not equal to that ofG-C pairs; they are different from random binary se-quences with equal weight. We introduce an indicatorRGC for the rate of the G-C pairs in the sequences such

as RGC = (NG + NC)/(NG + NC + NA + NT), whereNG, NC, NA and NT denote the numbers of symbols G,C, A and T in the base sequence, respectively.

In the case of B0 = B1( B), depending on the valueB, the correlation function C(n)( Vn0Vn0+n) (n0 = 1,n is even ) decays following inverse power-law as C(n) n

2BB1 for large n (3/2 < B < 2). The power spectrum

becomes S(f) f 2B3B1 for small f. We focus on theGaussian and non-Gaussian stationary regions (1 < B

Documents

Hiroaki Yamada and Kazumoto Iguchi- Some Effective Tight-Binding Models for Electrons in DNA Conduction:A Review