electronic structures and long-range electron transfer through dna...

Electronic Structures and Long-RangeElectron Transfer Through DNA Molecules

YUAN-JIE YE,∗ YAN JIANG†

Department of Protein Engineering, Institute of Biophysics, Chinese Academy of Sciences, 15 DatunRoad, Chaoyang District, Beijing 100101, China

Received 19 July 1999; accepted 30 December 1999

ABSTRACT: Quantum chemical calculation on an entire molecule of segments ofnative DNA was performed in an ab initio scheme with a simulated aqueous solutionenvironment by overlapping dimer approximation and negative factor counting method.The hopping conductivity was worked out by random walk theory and compared withrecent experiment. We conclude that electronic transport in native DNA molecules shouldbe caused by hopping among different bases as well as phosphates and sugar rings. Blochtype transport through the delocalized molecular orbitals on the whole molecular systemalso takes part in the electronic transport, but should be much weaker than hopping. Thecomplementary strand of the double helix could raise the hopping conductivity for morethan 2 orders of magnitudes, while the phosphate and sugar ring backbone could increasethe hopping conductivity through the base stacks for about 1 order of magnitude. DNAcould transport electrons easily through the base stacks of its double helix but not itssingle strand. Therefore, the dominate factor that influences the electronic transfer throughDNA molecules is the π stack itself instead of the backbone. The final conclusion is thatDNA can function as a molecular wire in its double helix form with the conditions that itshould be doped, the transfer should be a multistep hopping process, and the time periodof the transfer should be comparable with that of an elementary chemical reaction.c© 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 112–130, 2000

Key words: DNA; hopping conductivity; electron transfer; charge-transfer; ab initio

Correspondence to: Y.-J. Ye.∗Present address: Baker Laboratory of Chemistry and Chemical

Biology, Cornell University, Ithaca, NY 14853-1301.†Present address: Department of Chemistry, Brown University,

Providence, RI 02912.

International Journal of Quantum Chemistry, Vol. 78, 112–130 (2000)c© 2000 John Wiley & Sons, Inc.

STRUCTURES AND TRANSFER THROUGH DNA MOLECULES

Introduction

I n a previous article [1] it was reported that DNAdouble helices can be good amorphous conduc-

tors when doped. Their hopping conductivity isgreater than proteins. The question remains is howdo the phosphate backbone and aqueous solutionenvironments influence the conductivity.

Recently, Barton and co-workers [2 – 9] reporteda series of experimental results that showed thatelectron can be quickly transported though DNAdouble helices over distances greater than 30 Å. Thisimplies that DNA is not only a carrier of geneticinformation, but also a pathway that is conduciveto charge transport. The mechanism of such elec-tronic transfer, however, remains unknown [10 – 14].The experiments herein support the conclusionspresented in the abovementioned article [1]. A com-parison of the theoretical results with those of exper-iments is also presented in this article to confirm thehopping mechanism of long-range electronic trans-fer through DNA helices.

The experiments of Barton’s group stimulatedgreat debates in recent years because they are con-troverted by both experiments and theory [15 – 25].There seems to be a paradox in the long-rangeelectronic transfer through DNA double helices.Some experiments show distance-dependent trans-fer, whereas others show sequence-dependent phe-nomena. All these results show the complexity ofliving materials and indicate that the long-rangeelectronic transfer is conditional. Therefore, it is nec-essary to work out the electronic structures of anentire molecular system of segments of DNA toreveal the conditions that make DNA moleculestransfer electronic charges distance dependently orindependently; that is, in which cases DNA cantransfer electrons through long distances and un-der what conditions it cannot transfer electroniccharges.

The electronic structures of DNA have been stud-ied in the decades [26 – 32] since the double helicalstructure of DNA was proposed by Watson andCrick [33]. Owing to developments in crystallineexperiments, some three-dimensional structures ofDNA molecules and DNA–protein complexes havebeen reported [34 – 42]. Some of these structuresprovide well determined geometries that can beused directly to perform quantum chemical calcu-lations to obtain their electronic structures. Thesenative DNA segments that really exist in the worldhave shown much more variety in their three-

dimensional conformations that could influence theelectronic structures and transport properties thathave been found in calculations on native pro-teins [43 – 46]. Therefore, quantum chemical calcu-lations on entire molecular systems with their welldefined three-dimensional conformations that aredetermined by X-ray diffraction on single crystalsbring us to investigate the properties of DNA mole-cules in more detail.

It is important to perform quantum chemical cal-culations on an entire system of biomacromoleculesbecause the biomacromolecules act in global formswith their substrates, for example, in the case ofDNA–protein interactions [47, 48]. The electronicstructures and hopping conductivity of native pro-teins have been reported in previous articles [43 – 46,49 – 54]. Herein, we completely report the first ex-ample of electronic structures and hopping conduc-tivity in a segment of native DNA molecule. Basedon the results, we performed further calculations onthe double helices of the base stacks to investigatetheir electronic transport properties.

A segment of native DNA (operator), which con-trols the genetic expression in living cells, is taken asthe first example to be investigated by performingquantum chemical calculation on its entire mole-cular system. The overlapping dimer approxima-tion and negative factor counting (NFC) method[49 – 58], which have been used in calculations onthe electronic structures of native proteins [49 – 54],are applied to the calculation on the segment of na-tive DNA. The details of the approximation withthe simulation of the aqueous solution environmentfor the DNA case will be presented in the next sec-tion. The results of the electronic structures andhopping conductivity of the segment are shown inthe third and fourth sections. The results for basestacks in double and single strand forms withoutbackbones are described in the same sections. Inthe last section, discussion and conclusions are pre-sented.

Approximations and Methods

The coordinates of operator [39], a segment ofnative DNA molecule that dominates genetic ex-pression in living cells, was obtained from theBrookhaven National Laboratory Protein Data Bank(pdb1trr.ent). There are four single chains that con-sist of 16 bases in each segment in B conformation.The sequence is 5′-AGCGTACTAGTACGCT-3′ foreach chain. Only one segment, chain C in the data

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 113

YE AND JIANG

set, is used for the calculation. There was no hy-drogen atom in the original data set. All of the co-ordinates of hydrogen atoms are added to the dataset theoretically. The number of atoms in the DNAsegment is 507, including hydrogen atoms. WhenClementi’s minimal basis set is applied to the quan-tum chemical calculation on the entire segment,1867 basis functions should be used.

In this article, the overlapping dimer approxi-mation is applied to build up the Fock matrix ofthe whole system as has been done in calculationson native proteins. In this way, the whole sys-tem is partitioned into dimers in which two bases,sugar rings, and phosphates are included. Somepseudoatoms are put at the ends of the dimers tosimulate the chemical environment of their nearestneighbor units. Figure 1 shows the details of theconstruction of the dimers in the case of the DNAmolecule. Gazdy et al. [55] have proven that theresults obtained from the Fock matrix of a whole

system constructed by local self-consistent calcula-tion on these dimers are similar to those calculatedby a self-consistent field (SCF) on the entire sys-tem.

It is necessary to simulate the aqueous solutionenvironment in the calculation because the data setwas obtained from X-ray diffraction on a singlecrystal in aqueous solution and because, under ordi-nary conditions, native DNA molecules are mainlysurrounded by water molecules. The phosphatebackbone of the native DNA molecule has one neg-ative charge on every unit. It will abstract positivecharges around it when it is in aqueous solution.Therefore, in the calculation the aqueous solutionenvironment is simulated by putting point chargesaround the phosphate backbone (see Fig. 2).

All of the dimers are calculated by a SCF quan-tum chemical method with an ab initio scheme.After all dimers are worked out, we can constructthe Fock (and overlap) matrix F (and S) of the whole

FIGURE 1. Construction of the overlapping dimers of the segment of native DNA. (a) The dimer at the beginning ofthe segment. There is no phosphate group at the beginning unit because it was not determined experimentally [39].(b) The dimers inside the segment. (c) The dimer at the end of the segment. The pseudoatoms are added at the ends ofeach dimer to simulate the chemical environment of the adjacent units of the dimer. They are included in the SCFcalculations on the dimers, but are not included in the construction of the Fock matrix for the whole segment.

114 VOL. 78, NO. 2


FIGURE 2. Simulation of the aqueous solutionenvironment surrounding the phosphates. Point chargesare placed along the direction of the P—O bonds. Thedistances between the point charges and the oxygenatoms are the same as have been applied in calculationson native proteins [51].

segment of DNA molecule by the formulas

F(S)ij (n, n) = 1

m

m∑k= 1

F(d)ij (n, n, k), m = 1, 2, . . . , (1)

F(S)ij (n, n′) = F(d)

ij (n, n′), n 6= n′, (2)

where F(S)ij (n, n′) are the matrix elements of the Fock

matrix of the whole segment, and F(d)ij (n, n, k) and

F(d)ij (n, n′) are matrix elements of the dimers. The n

and n′ indicate the numbering of the units and kindicates the dimers in which the nth unit is con-tained. In this way the elements of the off-diagonalblocks of the matrices are taken directly from thoseof the dimers, and the diagonal blocks are takenfrom the averages of the diagonal blocks that cor-respond to different neighboring dimers.

The Fock (and overlap) matrix of the entire sys-tem constructed in this way has the form

F(1, 1) F(1, 2) 0 0F(2, 1) F(2, 2) F(2, 3)

0. . . . . . . . .

. . . . . . . . . 0. . . . . . F(N − 1, N)

0 0 F(N, N− 1) F(N, N)

(3)

in which the F(n, n′)s stand for the matrix blocks thatbelong to units n and n′. Its eigenvalues can be cal-culated by the NFC method [56 – 58] and the eigen-vectors (wave functions) corresponding to them can

be solved by inverse iteration [59]. The programENFC [49, 60], in which the NFC method is a simplecase, is used to solve this eigenvalue problem.

The same approximation used in the previousarticle [1] is applied to further calculations on thedouble helix of the base stacks of the operator andthe helices that were used in the recent experimentof Barton et al. For the sake of clarity, we briefly re-peat the formulas used in the previous article.

First of all, the 16 basic dimers, which consist oftwo bases (base pairs), are calculated by an ab ini-tio quantum chemical method. The geometry of thebases (base pairs) in these dimers is taken from thesame data set of the crystalline structure of the op-erator. The orientation of the bases (base pairs) istaken according to the standard structure of B-formDNA [34]; that is, the distance between two bases(base pairs) is 3.36 Å and the rotation angle betweenthe two bases (base pairs) is 36◦. Then, after the 16basic dimers have been worked out, the Fock (andoverlap) matrix F (and S) of the whole molecularsystem of a DNA segment can be obtained. The firststep is to perform a rotation on the matrix F (and S)of each dimer according to

F′ = RFR−1 (4)

in which

R = R1 ⊕ R2 ⊕ · · · ⊕ RN (5)

and

N = n1 + n2, (6)

where N is the total number of atoms of the dimer inwhich the number of atoms in the first base pair is n1

and in the second is n2, and Ri is the rotation matrixof the ith atom in the dimer. The Fock (and overlap)matrix F (and S) of the whole molecular system of aDNA segmant can be constructed from the rotatedmatrices of the dimers by Eqs. (1) and (2).

The hopping conductivity in the segment can becalculated by the same formulas used for nativeproteins, in which random walk theory is applied.(Random walk theory has been widely appliedin condensed matter physics [61 – 67], especiallyto solve the problem of electronic charge transferthrough amorphous conductors [62 – 66].) The de-tails have been reported in previous articles [52, 53].The only difference between the current work andprevious works is that the unit that consists of themolecule is a nucleotide instead of an amino acidresidue. The formulas are briefly described as fol-lows.


YE AND JIANG

The hopping conductivity can be determined bythe Einstein relationship

σ (ω) = nVe2

kBTD(ω) (7)

in which nV is the number density of charge carri-ers in the volume of a DNA segment, e is the chargeof an electron, kB is the Boltzmann constant, and Tis the absolute temperature. The diffusion constantD(ω) can be calculated by solving the master equa-tion∂P[X(n, j), t|X(n0, i0), 0]

∂t= −0X(n,j)P

[X(n, j), t

∣∣X(n0, i0), 0]

+∑

X(n′ ,j′) 6= X(n,j)

hX(n′ ,j′)→X(n,j)P[X(n′, j′), t

∣∣X(n0, i0), 0].

(8)

Here P[X(n, j), t|X(n0, i0), 0] represents the probabil-ity that a carrier arrives at center X(n, j) at time twhen it was at center X(n0, i0) at time t = 0, and

0X(n,j) =∑

X(n′ ,j′) 6= X(n,j)

hX(n,j)→X(n′ ,j′). (9)

In the master equation, the hopping frequencyhX(n,i)→X(n′ ,j) can be calculated from the electronicenergy levels and corresponding wave functions ob-tained by the quantum chemical calculations on thewhole molecular system of the DNA segment by theformula

hX(n,i)→X(n′ ,j) =

νphonon

( ∑r∈ns∈n′

C(i)r C( j)

s⟨φr(n)

∣∣φs(n′)⟩)2

× exp(−1Eij

kBT

), 1Eij > 0,

νphonon

( ∑r∈ns∈n′

C(i)r C( j)

s⟨φr(n)

∣∣φs(n′)⟩)2

,

1Eij ≤ 0,(10)

where C(i)r and C( j)

s are the linear coefficients of theith and jth molecular orbitals of the whole DNAsegment, φr(n) and φs(n′) are the basis functions ofthe molecular orbitals, r and s represent the num-bering of the basis functions, and n and n′ representthe numbering of the nucleotide. The difference be-tween the jth and the ith energy levels is 1Eij =Ej−Ei. The acoustic phonon frequency is νphonon andis taken as 1012 s−1 as in Ref. [52]. The center of theith molecular orbital in the nth nucleotide is definedas

X(n, i) =∑

A∈n wA(n, i)XA(n)∑A∈n wA(n, i)

, (11)

where

wA(n, i) =∑r∈A

C2r (n, i) (12)

in which A is an atom of the base (base pair).Equation (10) was first presented by Ladik et

al. [68]. Note that the summation runs only overthe basis functions of the nth and n′th nucleotide.This means that the wave functions are truncatedand the sum will be nonzero. (It will be zero whenthe summation runs over all nucleotides of the DNAmolecule because the wave functions are orthogonalfor the entire molecular system.) Further, it will haveexponential behavior because the basis functionsare represented by exponential functions. Therefore,Eq. (10) is a good approximation for calculating thehopping frequencies. For more details, see Mott andDavis [62].

The diffusion constant in Eq. (7) can be writtenin terms of the Laplace transform of the probabilityP[X(n, j), t|X(n0, i0), 0] of Eq. (8) as

D(ω) = −ω2

2d

∑X(n0,i0)

∑X(n,j)

[X(n, j)− X(n0, i0)

]2

× P̃[X(n, j), iω

∣∣X(n0, i0)]

f (EX(n0,i0)), (13)

where d is the dimensionality of the system, whichis taken as 1 in the calculations on DNA molecules,and P̃[X(n, j), iω|X(n0, i0)] is the Laplace transform ofthe probability P[X(n, j), t|X(n0, i0), 0],

P̃[X(n, j), iω

∣∣X(n0, i0)]

=∫ ∞

0eiωtP

[X(n, j), t

∣∣X(n0, i0), 0]

dt, (14)

and f (EX(n,i)) is the equilibrium distribution functionfor the localized carriers and it obeys the equation∑X(n′ ,j′) 6= X(n,j)

hX(n′ ,j′)→X(n,j) f (EX(n′ ,j′))

=∑

X(n′ ,j′) 6= X(n,j)

hX(n,j)→X(n′ ,j′) f (EX(n,j)) (15)

in which EX(n,j) is the energy level of the carrier atcenter X(n, j).

A formal solution of the Laplace transformationof probability, P̃[X(n, j), iω|X(n0, i0)], can be obtainedby the following methods. Performing the Laplacetransformation on both sides of Eq. (8) yields

iωP̃[X(n, j), iω

∣∣X(n0, i0)]− P

[X(n, j), 0

∣∣X(n0, i0), 0]

= −0X(n,j)P̃[X(n, j), iω

∣∣X(n0, i0)]

+∑

X(n′ ,j′) 6= X(n,j)

hX(n′ ,j′)→X(n,j)

× P̃[X(n′, j′), iω

∣∣X(n0, i0)]. (16)

116 VOL. 78, NO. 2


Obviously,

P[X(n, j), 0

∣∣X(n0, i0), 0] = δX(n,j),X(n0,i0). (17)

Therefore, the Laplace transformation of P[X(n, j),t|X(n0, i0), 0] fulfills the relation(

iω + 0X(n,j))̃P[X(n, j), iω

∣∣X(n0, i0)]

−∑

X(n′ ,j′) 6= X(n,j)

hX(n′ ,j′)→X(n,j)

× P̃[X(n′, j′), iω

∣∣X(n0, i0)]

= δX(n,j),X(n0,i0). (18)

Equation (18) can be written in matrix form as

(iωI−H)̃P(iω) = I (19)

in which the elements of the hopping matrix H aregiven by

HX(n,j),X(n,j) = −0X(n,j), (20)

HX(n,j),X(n′ ,j′) = hX(n′ ,j′)→X(n,j), X(n, j) 6= X(n′, j′)(21)

and I is the unit matrix. Therefore, a formal solutionfor P̃[X(n, j), iω|X(n0, i0)] can be written as

P̃[X(n, j), iω

∣∣X(n0, i0)] = [(iωI−H)−1]

X(n,j)X(n0,i0).

(22)

From Eq. (15) and the definition of the hoppingmatrix H, the equilibrium distribution function forthe localized carriers, f (EX(n,j)), can be obtained bysolving the eigenequation of the hopping matrix H,which is an asymmetric matrix. The solution is theright eigenvector that corresponds to the uniquezero eigenvalue of the matrix H (see Ref. [65] formore details).

Electronic Structures and Densityof States

DISTRIBUTION OF NET CHARGES

Table I shows the distribution of net charges foreach base and its groups. In the table group 1 con-sists of a phosphor atom and the two oxygen atomsthat are bonded only to the phosphor atom. Group 2consists of the sugar ring, including the oxygenatoms of O3′ and O5′, which are also bonded tothe phosphor atom. Group 3 consists of the bases.The same descriptions apply to the groups in Ta-bles III–V.

From the table we can see that there is electronictransfer from the backbone to the bases. The num-ber of the electron transferred is about 0.22 per unit,

TABLE IThe net charge distribution.

Bases andpositions Net charge Group 1 Group 2 Group 3

AdenineUnit 1 −0.4801 — −0.2612 −0.2189Unit 6 −1.0077 −0.0072 −0.7714 −0.2292Unit 9 −1.0036 −0.0263 −0.7548 −0.2224Unit 12 −1.0001 −0.0069 −0.7612 −0.2320

CytosineUnit 3 −1.0278 −0.0125 −0.8062 −0.2091Unit 7 −0.9771 0.0198 −0.7720 −0.2250Unit 13 −1.0031 −0.0011 −0.8019 −0.2001Unit 15 −0.9952 0.0009 −0.7767 −0.2194

GuanineUnit 2 −1.0044 −0.0460 −0.7375 −0.2209Unit 4 −0.9719 0.0212 −0.7723 −0.2207Unit 10 −0.9922 −0.0367 −0.7508 −0.2048Unit 14 −0.9902 0.0006 −0.7780 −0.2129

ThymineUnit 5 −1.0138 −0.0142 −0.7886 −0.2109Unit 8 −1.0013 0.0117 −0.7836 −0.2294Unit 11 −1.0165 −0.0343 −0.7628 −0.2194Unit 16 −0.5148 0.0032 −0.2993 −0.2187

which is in good agreement with previous crystalorbital calculation results on periodic model sys-tems of DNA [26]. From the results we can see thatthe negative value of the net charge of a base isapproximately equals to the positive value of thenet charge of a hydrogen atom. Therefore, we canalso conclude that the phosphate and the sugar ringin a nucleotide may be replaced by a single hy-drogen atom to simplify the calculations on basestacks to investigate electronic transfer through theπ stacks.

BAND STRUCTURES

The calculated curves of the density of states(DOS) are shown in the Figure 3. Comparison withnative proteins [44, 51, 53] reveals that the curvesof proteins and DNA are similar. This means thatthe DOSs of living materials have a common fea-ture. However, there are some differences betweenthe curves of the two different kinds of living ma-terials. There is a clearly separated subband in theconduction band region and a small peak in the va-lence band region in the DNA case that consist ofthe frontier molecular orbitals localized at differentbases as will be shown in the Tables III and IV.


YE AND JIANG

FIGURE 3. The electronic density of states (DOS) curves of the segment of native DNA molecule (a) for both valenceand conduction band regions using a grid of 0.5 eV, (b) for the valence band region using a grid of 0.05 eV, and (c) forthe conduction band region using a grid of 0.05 eV. (The grids are not the scales shown on the abscissa, but the steplengths in the DOS histograms.)

The energy gap is estimated to be 10.807 eV. It istoo large to allow intrinsic conductivity in the nativeDNA molecule. The DNA molecule should be an in-sulator under ordinary conditions because thermalenergy is not sufficient to pump electrons from thefull filled valence band region to the empty conduc-

tion band region. The DNA molecule can transportelectrons only when it is doped, that is, when itis stimulated by photoexcitation such as radiationor bound by other molecules such as proteins andother chemicals. In the former case, the electrons inthe valence band region are pumped to the conduc-

118 VOL. 78, NO. 2


FIGURE 3. (Continued)

tion band region to create free charge carriers, whilein the latter case, biochemical reactions take elec-trons from the valence band region or put electronsin the conduction band region to create free chargecarriers.

Table II shows the rates of distributions of wavefunctions of different energy levels on the units ofthe molecule. The component of a wave functionlocalized at the nth unit, a(n), is estimated by theformula

a(n) =∑mn

j= 1 C2j (n)∑N

n= 1∑mn

j= 1 C2j (n)

(23)

in which mn is the number of basis functions in thenth unit and Cj(n) is the coefficient of the jth basisfunction in the nth unit. The total number of theunits is N. The same formula is applied to the com-ponents of each group. Units are neglected whenthe components of a wave function localized at itare less than 0.05. The energy bands are divided bythe numbering of the molecular orbitals to calcu-late the rates of distributions. From the table, onecan see clearly that the molecular orbitals are local-ized at one or two units at the edges of both bands.The more the molecular orbitals are inside thebands, the more delocalized they are. Therefore, themechanism of electronic transport through a native

TABLE IIThe distribution rates of energy levels.

Percentage of distributions (%)Numbering ofenergy levels 1–2 3–4 5–6 7–8 9–10

The conduction band

1826–1867 97.6 2.4 0.0 0.0 0.01746–1825 81.2 17.5 1.2 0.0 0.01666–1745 23.8 53.8 20.0 2.5 0.01586–1665 6.2 51.2 35.0 6.2 1.21506–1585 6.2 57.5 31.2 5.0 0.01426–1505 8.8 40.0 36.2 11.2 3.81346–1425 15.0 52.5 22.5 8.8 1.21266–1345 71.2 22.5 5.0 1.2 0.0

The valence band

1186–1265 80.0 20.0 0.0 0.0 0.01106–1185 15.0 56.2 25.0 3.8 0.01026–1105 21.2 47.5 25.0 6.2 0.0946–1025 22.5 42.5 28.8 5.0 1.2866–945 11.2 42.5 36.2 8.8 1.2786–865 6.2 50.0 30.0 11.2 2.5706–785 11.2 52.5 28.8 7.5 0.0626–705 50.0 35.0 12.5 2.5 0.0546–625 75.0 18.8 5.0 1.2 0.0466–545 98.8 1.2 0.0 0.0 0.0386–465 67.5 31.2 1.2 0.0 0.0


YE AND JIANG

TABLE IIIThe frontier orbitals of the valence bands region.

Energy level Positions and Components

(eV) bases Total Group 1 Group 2 Group 3

(1) −8.9808 4 Guanine 0.985 0.000 0.001 0.984(4) −9.2323 10 Guanine 0.970 0.000 0.001 0.969(5) −9.5462 12 Adenine 0.062 0.000 0.001 0.062

13 Cytosine 0.899 0.000 0.009 0.889(8) −9.7735 15 Cytosine 0.994 0.000 0.013 0.981(9) −9.8309 1 Adenine 0.985 0.000 0.008 0.977

(12) −10.2748 6 Adenine 0.968 0.000 0.006 0.962(13) −10.5181 16 Thymine 0.990 0.000 0.040 0.950(16) −10.6705 11 Thymine 0.993 0.000 0.073 0.920(17) −11.0389 3 Cytosine 0.060 0.001 0.058 0.001

4 Guanine 0.940 0.923 0.012 0.004(40) −11.5435 10 Guanine 0.103 0.002 0.100 0.001

11 Thymine 0.895 0.892 0.002 0.000(41) −11.5778 8 Thymine 0.195 0.001 0.001 0.193

9 Adenine 0.756 0.007 0.080 0.669(50) −11.7068 1 Adenine 0.997 0.000 0.004 0.993

DNA molecule should be dominated by hoppingamong different localized molecular orbitals insteadof Bloch type transport through delocalized orbitals.

Some frontier molecular orbitals [highest occu-pied and lowest unoccupied (HOMO and LUMO)]of the whole molecular system are shown in the Ta-bles III and IV. From the tables one can see that all of

the frontier molecular orbitals are localized at differ-ent units. They are localized at different bases thanthe phosphates mixed with bases and sugar rings inthe valence band region, and also at different basesthan sugar rings mixed with phosphates in the con-duction band region. The order of the types of basesat which the molecular orbitals localized is G > C >

TABLE IVThe frontier orbitals of the conduction bands region.



(1∗) 1.8264 5 Thymine 0.986 0.000 0.014 0.972(4∗) 1.9651 11 Thymine 0.993 0.000 0.013 0.980(5∗) 1.9740 3 Cytosine 0.985 0.000 0.028 0.958(8∗) 2.1146 13 Cytosine 0.962 0.000 0.021 0.941(9∗) 2.2957 6 Adenine 0.990 0.000 0.010 0.979

(12∗) 2.5738 1 Adenine 0.959 0.000 0.010 0.949(13∗) 2.7635 14 Guanine 0.986 0.000 0.025 0.961(16∗) 2.9712 2 Guanine 0.294 0.000 0.008 0.286

4 Guanine 0.686 0.000 0.016 0.671(17∗) 3.1805 9 Adenine 0.959 0.000 0.025 0.934(40∗) 5.4551 4 Guanine 0.968 0.000 0.076 0.892(41∗) 7.3185 10 Guanine 0.059 0.006 0.049 0.004

11 Thymine 0.908 0.025 0.254 0.629(50∗) 7.7548 10 Guanine 0.674 0.043 0.598 0.033

11 Thymine 0.268 0.115 0.116 0.036

120 VOL. 78, NO. 2


TABLE VThe delocalized molecular orbitals in the bands.



(185∗) 11.6116 3 Cytosine 0.109 0.006 0.081 0.0224 Guanine 0.126 0.009 0.097 0.0205 Thymine 0.117 0.020 0.088 0.0106 Adenine 0.061 0.028 0.030 0.0047 Cytosine 0.061 0.002 0.036 0.024

10 Guanine 0.066 0.011 0.043 0.01211 Thymine 0.116 0.017 0.047 0.05212 Adenine 0.075 0.005 0.021 0.04813 Cytosine 0.062 0.003 0.047 0.01114 Guanine 0.085 0.011 0.014 0.060

(455) −19.2686 1 Adenine 0.060 — 0.028 0.0322 Guanine 0.065 0.010 0.050 0.0053 Cytosine 0.056 0.004 0.014 0.0385 Thymine 0.160 0.038 0.099 0.0236 Adenine 0.143 0.034 0.055 0.0537 Cytosine 0.055 0.029 0.022 0.0048 Thymine 0.058 0.003 0.023 0.0329 Adenine 0.153 0.002 0.052 0.099

10 Guanine 0.160 0.003 0.128 0.02911 Thymine 0.053 0.018 0.034 0.001

A > T according to the energy levels of the orbitalsfrom high to low in the valence band region and isT < C < A < G according to the levels from lowto high in the conduction band region. The ordersare also in good agreement with the results obtainedby crystal orbitals. However, the widths of the sub-bands are much wider than in results of previousworks [26 – 32]. This means that the energy levels ofthe HOMOs and LUMOs that are localized at dif-ferent kinds of bases consist of fine subbands in theDOS curves.

Note that the energy differences between the sub-bands of different bases are smaller than the widthsof the subbands themselves. The largest differencebetween the energy levels of the bases that have thesame type is 0.4439 eV (10.24 kcal/mol) in the va-lence band region and 0.2781 eV (6.413 kcal/mol)in the conduction band region. These differencesstrongly influence the calculations on hopping con-ductivity because calculation of the elementary hop-ping frequencies is based on them [see Eq. (10)].These amounts also are important in biolochem-ical reactions when a high energy bond of anATP molecule supplies about 9 kcal/mol, that is,0.4 eV [69].

The delocalized wave functions in the middle ofboth valence and conduction bands are shown inthe Table V. From the table one can see that thesemolecular orbitals hardly distribute at all over thewhole molecular system of the segment. This showsthat the Bloch type electronic transport still exists inthe case when the deep levels in the bands take partin electronic transport.

To sum up the foregoing analysis, the conclusionto be drawn from these tables is that the elec-tronic transport in native DNA molecules shouldbe caused by hopping among different bases aswell as between phosphates and sugar rings. Blochtype transport through the delocalized molecularorbitals on the whole molecular system also takespart in the electronic transport, but should be muchweaker than hopping.

SEQUENCE-ENERGY “SPECTRUM” INNATIVE DNA

The results of the previous subsection indicatethat the energy levels of the HOMOs and LUMOslocalized at the bases that have different types con-sist of different fine subbands and form a spectrumwith the sequence of native DNA molecule. Table VI


YE AND JIANG

TABLE VIThe sequence–energy spectrum of the operator.a

|1E| HOMO Sequence LUMO |1E|(kcal/mol) (eV) (eV) and position (eV) (eV) (kcal/mol)

−9.8309 1 A 2.573815.25 0.661 0.388 8.95

−9.1694 2 G 2.962112.25 0.531 0.988 22.79

−9.7008 3 C 1.974016.60 0.720 0.997 23.00

−9.9808 4 G 2.971238.84 1.684 1.145 26.40

−10.6651 5 T 1.82649.00 0.390 0.469 10.82

−10.2748 6 A 2.295712.96 0.562 0.246 5.67

−9.7127 7 C 2.049720.98 0.910 0.091 2.09

−10.6226 8 T 1.959013.91 0.603 0.361 8.33

−10.0193 9 A 2.320318.15 0.787 0.529 12.20

−9.2323 10 G 2.849333.16 1.438 0.884 20.39

−10.6705 11 T 1.965115.58 0.676 0.384 8.86

−9.9949 12 A 2.349210.35 0.449 0.235 5.41

−9.5462 13 C 2.114611.59 0.503 0.649 14.96

−9.0434 14 G 2.763516.84 0.730 0.663 15.30

−9.7735 15 C 2.100217.17 0.745 0.268 6.18

−10.5181 16 T 1.8324

a The HOMOs and LUMOs are localized at each base and their energy levels are for the entire molecular system of the operator.

shows the spectrum of the segment of the operatorcalculated. The sequence and positions of bases areshown in the middle column and the differences be-tween the energy levels of the HOMOs and LUMOs,localized at the neighboring bases, are displayed oneach side.

From the table it can be seen that the energy lev-els of the HOMOs and LUMOs of each base differalong the sequence of the segment of DNA mole-cule. A sequence-energy “spectrum” that has finestructures results. The energy differences betweenthe HOMOs or LUMOs of the nearest neighborbases are large enough to influence the quantities ofhopping frequencies. Those adjacent bases that havethe same types also have quite different energy dif-

ferences at different positions in the DNA sequence.For example, there are three TAs that have HOMOenergy differences of 0.390, 0.603, and 0.676 eV (9.00,13.91, and 15.58 kcal/mol) and LUMO energy dif-ferences of 0.469, 0.361, and 0.384 eV (10.82, 8.33,and 8.86 kcal/mol), respectively. The same differ-ences can be found for the other cases. The largestenergy difference between neighboring bases is1.684 eV (38.84 kcal/mol) for the G4T5. Therefore,such an sequence–energy “spectrum” should existin native DNA molecules because the calculatedsegment really exists in the world and its three-dimensional conformation, which has been welldetermined experimentally, is used in the calcula-tions.

122 VOL. 78, NO. 2


FIGURE 4. The ac conductivity of the operator. The solid line (——–) represents the results calculated with abackbone, the dashed line (---------) represents results calculated without a backbone, and the dashed and dotted line(- · - · - · - · -) stands for the results obtained with the approximation presented in the Ref. [1]. (a) The real parts of theconductivities; (b) their imaginary parts; (c) their absolute value.

The same indications are found in the results ofcalculations on the double helices of base stacks inwhich the geometries of each kind of base are identi-cal. Therefore, the energy spectrum is caused mainlyby global interactions among the bases, and it alsoforms different hopping patterns to transfer chargesthrough different DNA molecules.

Hopping Conductivity

The hopping conductivity in the segment of na-tive DNA is shown in the Figure 4. In the calcula-tion, the volume of the segment was estimated to be6158 Å3, and it is comparable to that of the repres-


YE AND JIANG

FIGURE 4. (Continued)

sor (9224 Å3), which is the native protein moleculebound to it. Therefore, this segment that has a basicbiological function can be recognized as one of thesmallest segments of native DNA molecule to whichthe random walk theory can be applied.

Two additional calculations on the same se-quence were performed to investigate the influencesof the phosphate backbone and the three-dimen-sional conformations on the hopping conductivityin the DNA segment. One of the calculations wasperformed with the original coordinates of X-raydiffraction without the phosphate and sugar ringbackbone. The other was calculated with the samemethod presented in the previous article [1], that is,with the equations presented in the second sectionof this paper. The results are also displayed in Fig-ure 4.

Comparison with native proteins [43 – 46, 52, 53]indicates that the curves of the ac conductivity vs.frequency of both living materials are qualitativelysimilar. The largest value of hopping conductivityin the segment of native DNA is 10−4 �−1 cm−1

when the phosphate backbone is taken into ac-count. This is still much lower than that of nativeproteins, which means that the segment of nativeDNA cannot transport electrons easily in its singlestrand. The largest value of hopping conductivi-ties of the base stacks without a backbone is about10−5 �−1 cm−1, which is in 1 order of magnitudesmaller than with a backbone. This means that the

influence of the backbone will increase the hoppingconductivity by 1 order of magnitude.

Figure 5 shows the hopping conductivities ofsegments of native DNA molecules in their singleand double strand cases. Figure 5(a) shows that thelargest value of the hopping conductivity of theoperator is above 10−3 �−1 cm−1 and is larger thanthat of single strand by more than 2 orders of magni-tude. Therefore, we assume that the complementarystrand of DNA molecules helps transfer electronsthrough the base stacks. The results of the calcula-tions on the sequence used in the experiment [3, 4]are presented in the Figure 5(b), which shows thatthe largest value of that sequence of 28 base pairs(5′-CGCGATATGGGCGCATTAACCAGAATTC-3′)is about 1.57 × 10−2 �−1 cm−1. The rate of chargetransfer has been reported as about 150 ps throughthe 28 base pairs [4]. Using the formula

σ = elτSU

, (24)

in which e is the electronic charge, l and S are thelength and cross section of the DNA molecule, re-spectively, τ is the time period to transfer a chargethrough the DNA molecule, and U is the voltagedrop, which was estimated for this DNA segmentto be 0.8 V (see Ref. [4]), the conductivity of theDNA molecule can be estimated to be about 6.72 ×10−2 �−1 cm−1. The theoretical value of hoppingconductivity is on the same order of magnitudeas the experimental value. This result supports the

124 VOL. 78, NO. 2


FIGURE 5. The absolute values of ac conductivities of the double helices and single strands of different sequences.(a) The operator. The solid line (——–) represents the results of the double helix and the dashed line (---------) stands forthe single strand. (b) The sequences used in the experiments. The solid line (——–) represents the results of the doublehelix of 28 base pairs, the dashed line (---------) stands for the results of its single strand, the dashed and dotted line(- · - · - · - · -) represents the results of the double helix of 16 base pairs, and the dotted line (· · · · · · · ·) stands for theresults of its single strand.

conclusions in the previous paper that DNA doublehelices can function as molecular wires to transportcharges.

The hopping conductivity of another sequence,which also has 16 base pairs (5′-ACGGGCATGCGT-

TCGT-3′) [3], is also displayed in Figure 5(b). Com-parison with the curve of absolute value of theoperator [see Fig. 5(a)] shows that the conductiv-ity of native DNA molecules is sequence-dependentbecause the segments of different sequences have


YE AND JIANG

quite different conductivities although they havethe same lengths. Putting Figure 5(a) and (b) to-gether shows that the curve of conductivity of the28 base pairs lies between those curves of DNAsegments comparised of 16 base pairs. This resultshows that the electronic transport through nativeDNA molecules is distance-independent.

Table VII displays the conductivity of the seg-ment of 28 base pairs for which the theoreticalresults were worked out by different methods. All ofthe results are on the same order of magnitude as theexperimental value. The better the applied basis setis the more exact is the result obtained. The resultsworked out by the density functional theory (DFT)methods in which correlation correction have beenincluded are also in agreement with the experimen-tal results.

There are numerous recent experimental studiesthat seem to conflict with the Barton rates. However,the conductivity of a DNA molecule can exponen-tially decay when the electronic charge is transferedby an elementary hopping process, which is thesuperexchange of electronic charge between a donorand an acceptor, such as the results reported byLewis et al. [17]. In this case, the master equation (8)has two state behavior and can be solved analyti-cally as was done by Ye and Scheraga [71]. In thiscase, from Eqs. (7)–(15), the hopping conductivitycan be expressed as follows: Consider the simplestcase in which only two energy levels, Ed and Ea,are involved. Represent the hopping frequency fromthe Ed to the Ea as hd→a and that from Ea to Ed as ha→d,respectively. Then the master equations are

∂P[X(Ed), t|X(Ea), 0]∂t

= −hd→aP[X(Ed), t

∣∣X(Ea), 0]

+ ha→dP[X(Ea), t

∣∣X(Ea), 0]

(25)

and∂P[X(Ea), t|X(Ed), 0]

∂t= +hd→aP

[X(Ed), t

∣∣X(Ed), 0]

− ha→dP[X(Ea), t

∣∣X(Ed), 0]

(26)

and the hopping matrix H is

H =(−hd→a ha→d

hd→a −ha→d

). (27)

This can be simplified as

H =(−a b

a −b

). (28)

Therefore,

iωI−H =(

iω + a −b−a iω + b

). (29)

Its inverse matrix is

(iωI−H)−1

= 1

iω+a

[1+ ab

(iω+a)(iω+b)−ab

] b(iω+a)(iω+b)−ab

a(iω+a)(iω+b)−ab

1iω+b

[1+ ab

(iω+a)(iω+b)−ab

] .

(30)

Therefore, the Laplace transformations of P[X(Ed),t|X(Ea), 0] and P[X(Ea), t|X(Ed), 0] are

P̃[X(Ed), iω

∣∣X(Ea)]

= ha→d

(iω + hd→a)(iω + ha→d)− hd→a × ha→d(31)

and

P̃[X(Ea), iω

∣∣X(Ed)]

= hd→a

(iω + hd→a)(iω + ha→d)− hd→a × ha→d. (32)

The equilibrium distribution coefficients can be ob-tained from the right eigenvector that correspondsto the zero eigenvalue of the matrix H in the Eq. (27).

TABLE VIIThe hopping conductivity of the seguence of 28 base pairs.

Conductivity Transfer rateMethods (×10−2 �−1 cm−1) (×109 s−1)

HF, minimal basis set 1.57 1.56HF, DZ basis set 4.04 4.00DFT, LYP functional, DZ basis set 9.30 9.20DFT, PW91 functional, DZ basis set 3.77 3.74Experimental value [4] 6.72 6.67

Note: Clementi’s basis sets [70] are applied to all calculations.

126 VOL. 78, NO. 2


They are

f[X(Ea)

] = hd→a

hd→a + ha→d(33)

and

f[X(Ed)

] = ha→d

hd→a + ha→d. (34)

Substituting into Eqs. (13) and (7) yields

σ (ω) = nVe2l2

kT· hd→a × ha→d

hd→a + ha→d· ω

2 + iω(hd→a + ha→d)ω2 + (hd→a + ha→d)2 ,

(35)

in which l is the distance between the donor and theacceptor, nV is the number of effective charge carri-ers in unit volume, and ω is the external alternativeelectric field that is supplied by the change of atomicpositions of the molecular system during the bio-chemical reaction, for example, by vibration of thedonor and the acceptor. In the case of two state hop-ping, the charge carrier number is 1. Therefore,

nV = 1lS

, (36)

where S is the cross section of the DNA segment.Substituting it into Eq. (24) yields∣∣σ (ω)

∣∣ = elτSU

= e2l2

lSkT· hd→a × ha→d

hd→a + ha→d

·∣∣∣∣ω2 + iω(hd→a + ha→d)ω2 + (hd→a + ha→d)2

∣∣∣∣. (37)

Therefore, the rate 1/τ can be expressed as

1τ= eU


hd→a + ha→d·∣∣∣∣ω2 + iω(hd→a + ha→d)ω2 + (hd→a + ha→d)2

∣∣∣∣, (38)

that is,

1τ= eU


hd→a + ha→d· 1/√

1+[

(hd→a + ha→d)ω

]2

.

(39)

Equation (39) is an exact representation of the elec-tronic transfer rate of the superexchange process inwhich only two electronic states are involved. It canbe simplified to obtain the empirical formula usedin the literature. When ω is larger than the elemen-tary hopping frequencies hd→a and ha→d, Eq. (39) canbe simplified as

1τ= eU


hd→a + ha→d. (40)

In the case that the electronic energy level of thedonor is higher than that of the acceptor, Ed > Ea,

that is, 1Ead = Ed − Ea > 0, substituting Eq. (10)into (40) yields

1τ= eU

kT· νphonon exp(−1Ead/kT)

1+ exp(−1Ead/kT)

·(∑

r∈ds∈a

C(i)r C(j)

s⟨φr(d)

∣∣φs(a)⟩)2

. (41)

The asymptotic behavior of the last term in Eq. (41)is an exponential function of the distance betweenthe donor and the acceptor when Slater type basisfunctions are applied [72]. Therefore, Eq. (41) can besimplified as

1τ≈ eU

kT· νphonon exp (−1Ead/kT)

1+ exp (−1Ead/kT)· exp(−βR). (42)

In the case that the electronic energy level of thedonor is lower than that of the acceptor, 1Eda =Ea − Ed > 0, we obtain

1τ≈ eU

kT· νphonon exp (−1Eda/kT)

1+ exp (−1Eda/kT)· exp(−βR). (43)

From Eqs. (42) and (43), it is obvious that the trans-fer rate is distance-dependent in the superexchangecases. When the temperature is taken as a constant,they become the empirical equation

1τ= A exp(−βR). (44)

The foregoing analysis indicates that the physicalmeaning of the parameter β in Eq. (44) is the inter-action between the electronic charge transferred andthe atomic groups of the donor and the acceptor. Asthe value of the β parameter increases, the interac-tion grows stronger.

Discussion and Conclusions

The results in the preceding sections show theelectronic structures of a segment of native DNAmolecule in more detail than before. The results in-dicate that the electronic transport in native DNAmolecules should be caused by hopping amongdifferent bases as well as phosphates and sugarrings when the DNA molecules are doped. Blochtype transport through the delocalized molecularorbitals on the whole molecular system also takespart in the electronic transport, but should be muchweaker than hopping. The results of the calculationson the ac conductivity also show that single strandsof native DNA molecules do not transport electrons


YE AND JIANG

easily under ordinary conditions in living cells with-out their complementary strands or binding withother molecules.

The atoms that are involved in the biochemi-cal reactions change their positions in a very shorttime period [73 – 78]. This time period correspondsto the high-frequency range (in the magnitude ofpicoseconds [74]). The change of the positions ofthe atoms supplies the external electric field onthe macromolecule and is irreversible. Therefore, ahigh-frequency electric current can be used as an ap-proximation to describe such processes. This is thereason why our interests are focused on the high-frequency range of ac conductivity.

There are several factors that influence the hop-ping conductivities through DNA molecules. Theresults presented in the preceding sections showthat the first important factor is the presence of thecomplementary strand of DNA molecules, whichhelps the DNA molecules transfer charges throughtheir double helices instead of single strands. An-other important factor is the sequences of the DNAmolecule. The different sequences cause differenthopping conductivities. The hopping conductivitiesof DNA molecules are sensitive to their sequencesinstead of their lengths. All these statements arein good agreement with recent experiments thatshowed that interference on the double helicesstrongly influences the electronic transfer throughDNA molecules [6, 9]. The phosphate and sugarring backbone also influence the hopping conduc-tivities of DNA molecules by increasing their con-ductivities. However, it is the π stack itself that isthe dominate factor of charge transfer through DNAmolecules.

The results in the preceding section indicate alsothat the electrons that are localized at the back-bone of the segment of native DNA molecule havemuch lower energy levels than those localized atthe bases. The highest energy level of the electronsat the backbone is 2.058 eV (47.46 kcal/mol) lowerthan the highest energy level of the electrons atthe bases. Therefore, the dominate factor that influ-ences hopping conductivity in the DNA moleculeshould be the π stack itself instead of the backbone.Thus the phosphate and sugar ring backbone can beneglected in the calculation on the hopping conduc-tivities through DNA molecules.

The theoretical result of the conformation-energy“spectrum” of native DNA molecules might be thereason why charge transfer through DNA moleculesis sequence-dependent and distance-independent.It should be pointed out that the sequence-energy

“spectrum” was detected experimentally by deter-mining the fine structures of the photoenergy spec-trum of the smallest segment that has palindromicstructure because the coordinates of atoms in thesegment are taken from a well determined data setof X-ray diffraction on a single crystal that really ex-ists in the world and the calculations are performedin an ab initio scheme.

Further, more exact results are obtained when abetter basis set is applied to the quantum chem-ical calculation on the DNA double helices. Theresults of DFT methods, in which many body effects(correlation corrections) were included, are also inagreement with the experimental value. Therefore,the final conclusion is that DNA double helices canfunction as molecular wires in biological processes.

However, the results presented in this paper re-veal that the long-range electronic transfer throughDNA molecules is conditional. One of the neces-sary conditions is that the DNA molecule be doped.Another condition is that the transfer be a multi-hopping instead of a single superexchange process.The third condition is that DNA molecules trans-fer only those electronic charges that are involvedin biochemical reactions that take a very short timeperiod in their elementary steps.

ACKNOWLEDGMENT

The authors express their gratitude for finan-cial support from the National Natural ScientificFoundation of China (project number 29473146),the Chinese High Technology Development Projects(project number 863-306-ZD-01-4), the ChineseHigh Performance Computing Foundation (projectnumber 96108), and the Chinese Pangdeng Project,as well as partial support from the Deutsche For-schungsgemeinschaft (project number 446-CHV-113/50/0) and for the use of the Dawn-1000computer at the National Research Center for Intel-lectual Computing System of the Chinese Academyof Sciences and the IBM/SP2 computer at the ChinaEducation and Research Network Center, as wellas the Power-Challenge/R10000 computer at theComputer Network Information Center of the Chi-nese Academy of Sciences. Y.-J. Ye also thanks Dr.C. L. Lawson of Brookhaven National Laboratoryfor supplying the complete data set of the trp-repressor/operator complex, and Ling-Ling Shenand Dr. Fei-Wu Chen for their kind help during thecalculations.

128 VOL. 78, NO. 2


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