DNA as a molecular wire

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DNA as a molecular wire Article in Superlattices and Microstructures · July 2000

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Superlattices and Microstructures, Vol. 28, No. 4, 2000 doi:10.1006/spmi.2000.0915 Available online at http://www.idealibrary.com on

DNA as a molecular wire Y URI A. B ERLIN , A LEXANDER L. B URIN , M ARK A. R ATNER† Department of Chemistry, Center for Nanofabrication and Molecular Self-Assembly and Materials Research Center, Northwestern University, 2145 N Sheridan Road, Evanston, Illinois 60208-3113, U.S.A. (Received 26 July 2000) Physical mechanisms that might assure the functioning of DNA as a molecular wire are considered on the basis of recent progress in understanding long-range charge transfer in this biologically important molecule. Our analysis shows that DNA behaves as an insulator at low bias, while beyond the threshold the current sharply increases. Such behaviour concurs with recent experimental observations and is explained by the decrease of the energy gap between the HOMO of guanine bases and the Fermi level of the contact with the voltage applied across the individual DNA molecule. We propose a model for the hole injection in DNA, which is based on the dynamic control of this process by internal motions of base pairs in the stack. The temperature dependence of the voltage gap obtained within this model is found to be in reasonable agreement with the available experimental data. For systems, where charge transfer is controlled by changes in the relative orientation of the donor and acceptor and where the equilibrium states are optimally overlapped, the model predicts the decrease of the tunneling transfer rate with temperature. We also demonstrate that depending on the structure of the stack, hole transport along DNA wires above the voltage threshold can proceed via two different mechanisms. In the case of duplex DNA oligomers with stacked adenine–thymine and guanine–cytosine pairs migration of injected holes can be viewed as a series of short-range hops between energetically appropriate guanine bases. By contrast, in double-stranded poly(guanine)–poly(cytosine) the band-like motion of holes through bases dominates. c 2000 Academic Press

Key words: molecular wires, tunneling, charge transfer in DNA, configurational dynamics.

1. Introduction Deoxyribonucleic acid (DNA) is a macromolecule consisting of a double helix with an aromatic π-stack core where the basis of the pyrimidine deoxynucleotides (thymine, T; cytosine, C) and purine deoxynucleotides (adenine, A; guanine, G) participate in Watson–Crick base pairing (AT; CG). This unique molecular structure allows DNA to fulfill important biological tasks including coding, storage and propagation of genetic information. However, the current interest in DNA is not restricted to its role in biology. In particular, the advent of molecular electronics has stimulated interest in the possibility to exploit this molecule in functional mesoscopic electronic devices [1–9] and in molecular computing [10, 11]. † Address correspondence to any author. E-mail: [email protected], [email protected] or [email protected]

0749–6036/00/100241 + 12

$35.00/0

c 2000 Academic Press

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A key structural feature, which makes DNA a promising candidate for applications in nanometer-scale electronics, is the array of π-stacked base pairs. The striking resemblance of the base pair stack to conductive one-dimensional aromatic crystals prompted the proposal that the stack of base pairs in the interior of the double helix can provide a one-dimensional pathway for charge migration [12]. Although first posed in the 1960s [12, 13], the question of whether DNA is able or not to conduct electrical charges is still debated. Most experimental studies [14–43] addressing this fundamental question invoke the methodology widely employed for probing the distance dependence of electron transfer processes. In typical experiments a hole donor and acceptor are intercalated in DNA [15, 19, 22, 32, 34] or chemically attached to a well-defined oligonucleotide [18, 23, 24, 35, 36, 38]. The efficiency of hole transfer from the donor to the acceptor bridged by the base pair sequence is then determined by measuring, for instance, the quenching of the fluorescence of the donor [17, 18, 22, 26, 32, 35, 37] for bridges of different lengths or the damage yield at certain sites along the bridge [21, 23, 29, 33, 38, 40, 41]. The observed far-reaching translocation of charge [15, 23, 29, 33, 37, 38, 41] (up to ca. 200 Å) and a weak distance dependence of charge transfer deduced from these experiments have triggered a discussion on the hypothetical behavior of DNA as a molecular wire [27, 31, 44–50]. Significant progress in experimental technique for probing electrical conduction through individual molecular objects [51–53] allows the direct verification of this hypothesis by measuring a current I as a function of the potential U applied across DNA molecules. Fink and Schönenberger [7] were the first to perform such experiments for a single DNA rope, which consists of a few molecules, by employing a modified low-electron point source microscope. As follows from the linear I –U curves measured in the range ±20 mV under highvacuum conditions, the upper value for the resistance of the 600-nm-long DNA rope at room temperature is between 2.5 and 3.3 M. From these studies, it is concluded that the charge transport mechanism must be of electronic nature and should be distinct from the mechanism of the radiation-induced conductivity of hydrated DNA at low temperatures [20]. More recently Porath et al. [9] extended the voltage range up to 4 V and studied the current passing through a double-stranded poly(G)–poly(C) DNA oligomer of length 10.4 nm suspended between two platinum nanoelectrodes. They found that the measured I values do not exceed 1 pA below a threshold voltage of a few volts. This shows that the system under investigation behaves as an insulator at low bias. However, beyond the threshold the current sharply increases indicating that DNA can transport charge carriers. The observed voltage gap and its widening with increase of temperature is concluded to be incompatible with the available models of charge transport through DNA. Being partially motivated by experiments mentioned above, the present work has two main objectives. The first is to consider physical mechanisms that might assure the functioning of DNA as a molecular wire. We demonstrate that this can be accomplished on the basis of theoretical advances in understanding long-range charge transfer in this molecule [54–59]. Our analysis shows that electric current caused by the motion of holes can occur above the voltage threshold due to two different mechanisms. In the case of duplex DNA oligomers with stacked AT and GC pairs the migration of injected holes can be viewed as a series of shortrange hops between energetically appropriate G bases. In contrast, in double-stranded poly(G)–poly(C) the band-like motion of holes dominates. The latter mechanism implies that holes are delocalized over distances that are larger than the mean plane-to-plane distance between bases in the stack. Theoretically one can expect that the band-like motion will be affected by rearrangements of molecular units inside DNA, especially by rotations of base pairs around the stack axis [60]. However, this possibility has not been investigated theoretically in sufficient details. To our knowledge, the results published in the literature are restricted to the cases, where a hole added to the stack of base pairs is able either to undergo band-like motion perturbed by molecular rearrangement [54, 59] or to distort the stack structure with the subsequent formation of a polaron [41, 61]. Another aspect of the problem, which, to our knowledge, has not been theoretically studied yet, is the possible effect of molecular rearrangements on the position of the HOMO level of poly(G)–poly(C) wire. If the effect is significant, one can expect that dynamic disorder associated with these rearrangements becomes

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important for the efficiency of hole injection in DNA. Therefore, our second objective is the formulation of a theoretical model for hole injection controlled by internal reorientation dynamics within the stack. The model suggests that changes in the relative orientation or positions of two adjacent G bases slow down the injection rate as the applied voltage approaches a threshold value. This resembles the situation common for gated electron transfer reactions [62] and for unistep transitions coupled to hierarchical dynamics of the glassy and protein environment [63], where configuration changes are assumed to ‘turn on’ green or red traffic lights for the migration along the coordinate governing the rate process (the so-called ‘traffic light’ process) [64]. We demonstrate that for a molecular wire composed of the poly(G)–poly(C) duplex our model yields nonlinear I –U curves with the low-temperature voltage gap about 2.4 eV, in agreement with experimental observations [9]. Furthermore, the calculations performed enable us to conclude that for electric conductivity controlled by internal reorientation dynamics of stacked GC pairs the voltage gap becomes larger as temperature increases. This trend was indeed observed in experiments of Porath et al. [9], but has remained unexplained. For systems, where bridge-mediated charge transfer between donor and acceptor is controlled by changes in the relative orientation of mediators and where the equilibrium state is optimally overlapped, our theoretical treatment predicts a decrease of the tunneling transfer rate with temperature. This supports an earlier idea [60] concerning the effect of twisting motion on the conducting properties of polymers and on the variation of these properties with temperature.

2. Electric current through stacked base pairs 2.1. Background As in other molecular wires [65, 66], the current through the DNA molecule connecting two nanoelectrodes is determined by two processes, i.e. the injection of charge carriers onto the stack of base pairs and their transport along the stack. A central physical factor governing the efficiency of the former process is the location of the Fermi level, E F , of the metallic contact relative to the energy levels of the stack. Obviously the injection efficiency would be high if the Fermi level could align with one of the occupied or unoccupied molecular orbitals. However, it seems fairly certain that usually this is not the case. Instead, the Fermi level should fall in the HOMO–LUMO gap in order to preserve the charge neutrality of the molecule, and the potential energy barrier will arise at the metal–molecular junction (Fig. 1). This is an important theme in molecular electronics, clearly discussed in the contribution to this volume by the Purdue group. In the absence of the electric field, the height of the barrier 1 that must be overcome to generate an ‘electronic’ hole in the molecule with the ionization potential I can be approximated as [67] 1 = I − W,

(1)

where W is the work function of the metallic contact. The application of eqn (1) for the estimation of the 1 value requires information about energetics of nucleobases within the stack. The experimental data on one-electron redox potentials of nucleobases in solution [68, 69] show that the energy of the hole when residing on adenine (A), cytosine (C) or thymine (T) bases is higher than when on G by 0.5–0.7 eV. If the same trend is maintained in DNA, the hole injection will proceed by electron transfer from the G site of the stack to metal. According to ab initio calculations [70], the ionization potential of G base is equal to 7.75 eV, and hence eqn (1) yields 1 = 2.39 eV for DNA bridging two platinum contacts with the work function 5.36 eV [71]. Taking the typical barrier length to be equal to 5 Å [72], it can be verified that thermally activated generation of holes is precluded, while the probability of their injection due to electron tunneling, Ttun (E F ), is about 4 × 10−3 . This implies that at low voltages (with much less than ∼ 1 V being dropped across the molecule) the resistance R of the DNA wire evaluated from the Landauer formula R = h¯ π/(e2 Ttun (E F )) = 12.9(K)/Ttun (E F ) [73] is expected to be about 3.6 M

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Fig. 1. Schematic picture of energy levels at the metal–DNA contact. The metal work function W defines the position of the metal Fermi level E F . A hole is injected to the highest occupied molecular orbital (HOMO) of the DNA with the ionization potential I with respect to the vacuum level.

in reasonable agreement with the value measured by Fink and Schönenberger [7]. Thus we conclude that at low bias the metal–molecular junction makes the main contribution to the resistance of the DNA wire. The situation, however, becomes different at higher voltages, since the gap between the HOMO of G bases and the Fermi level of the contact decreases with U . In the vicinity of a certain voltage threshold, which corresponds to the energy crossing, the conduction of the DNA wire will be controlled by the ability of the base pair stack to transport a charge rather than by the efficiency of the injection process. Recent theoretical [54–59] and experimental [29, 33, 38, 43] studies of long-range charge transfer in DNA show that a hole moves along stacks of AT and GC base pairs undergoing sequential tunneling transitions between neighboring G sites separated by fragments containing A and/or T bases. The basis of this hopping mechanism is that a guanine cation cannot oxidize A (or T or C) because of the larger ionization potential of A compared with that of G, but can oxidize another G. According to standard electron transfer theory [74], each oxidation step proceeds with the rate khop , which exponentially decreases with the length, L AT , of the AT bridge between two adjacent G bases, i.e. khop = k0 exp(−β L AT ),

(2)

where k0 is the preexponential factor and β is the fall-off parameter. The direct consequence of eqn (2) is the effect of arrangement and number of G bases in the stack on the hole mobility µ and therefore on the conductance of DNA wires near the voltage threshold. As follows from our analysis [56, 58] of charge-transfer experiments [29, 33] performed for various base pair sequences, khop decreases by about a factor of 0.3 for each intervening AT base pair linked directly to the previous pair
AT
like AA TT or about an order of magnitude for cross linked pairs like TA . Therefore the highest mobility of holes along an AT-GC stack should be expected for base pair sequences with one repeating AT pair between G bases. For these regular sequences, the value of the hole drift mobility can be evaluated from the Einstein relation e ∼ e khop L 2 , µ= D= (3) AT kB T kB T where kB is the Boltzmann constant, T is temperature, and D is the diffusion coefficient of holes. To determine the mobility from eqn (3), we use the G–G0 hopping rate of about 109 s−1 roughly estimated by Jortner et al. [55] for the strand GTTGTTGTT. . . TTG. For a sequence with one AT pair between G bases, this value should be larger by a factor of three due to the dependence of khop on the number of intervening

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10−2

Hole mobility (cm2 V−1 s−1)

10−3 10−4

10−5

10−6

10−7

0

1

2

3

4

5

6

Number of repeating AT pairs between two G bases Fig. 2. Hole mobility along a DNA wire composed of N repeating AT pairs between two adjacent G bases on the number of AT base pairs within the bridge. The numerical results for stacks with strands G(TT) N G(TT) N G(TT) N . . . (TT) N G and G(TA) N G(TA) N G(TA) N . . . (TA) N G are shown by circles and squares, respectively (see text).

AT pairs specified above. Taking now the mean plane-to-plane distance between stacked bases to be equal to 3.4 Å [26, 32], we conclude that the hole mobility calculated from eqn (3) for holes moving along the sequence with one repeating AT pairs between G bases is expected to be 2 × 10−3 cm2 V−1 s−1 at room temperature. Similar estimations made for other regular sequences show that µ indeed decreases with the number of intervening AT pairs and depends on the particular base pairs stacked on the same strand (Fig. 2). It is worth noting that the breaking of the positional order in the base pair sequences can dramatically reduce the mobility of holes undergoing hopping motion along the stack. There are two possible reasons for this effect. One reason lies in the fact that the hole mobility along the stack with irregular positions of G bases is determined by the time it takes for a charge carrier to jump through the longest AT fragments. Another reason is the existence of thermodynamic traps in sequences where several adjacent G bases are stacked on the same strand [29, 70]. The origin of such traps can be understood within the tight-binding approach, which allows the calculation of the ionization potential of stacking G bases using the Hückel Hamiltonian Hˆ =

N X

εk n k − t

k=1

n k = ck+ ck .

N −1 X

+ (ck+ ck+1 + ck+1 ck ),

k=1

(4)

Here cn+ and cn are the creation and annihilation operators for a hole at the nth site, respectively, t is the transfer integral, and εn is the energy of a hole at the nth site (for a single site ε1 corresponds to the ionization potential I1 of a single G base). The solution of the Schrödinger equation with the Hamiltonian (4) enables one to estimate the ionization potential for any number, NG , of stacked G in terms of the lowest energy needed to create one hole. The results of our tight-binding calculations with parameters I1 = 7.75 eV and

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Ionization potential

8

7.5

7

6.5 0

1

2

3

4

5

Number of stacked G bases Fig. 3. Ionization potential versus the number of G bases stacked on the same strand. The squares are ab initio calculations [70], and the triangles are the results of tight-binding calculations.

t = 0.4 eV are shown in Fig. 3. As can be seen from the numerical results obtained, the ionization potential of the stacked G bases decreases with NG in good agreement with the results of ab initio studies [70]. Accordingly, the energy of hole located, for instance, on GGG is lower than the energy of G+ by about 0.7 eV, and hence the triple GGG serves as a sink for moving hole in accord with experimental findings [29]. It is interesting that in the limit of a large number of stacked G bases the ionization potential tends to the value I1 − 2t = 6.95 eV which corresponds to the I value for the poly(G)–poly(C) duplex. The difference in ionization potentials for the individual G and for G bases stacked at the same strand can be explained by the formation of the band with width 2t due to ππ-interactions between neighboring GC pairs. By virtue of eqn (1), this implies that for poly(G)–poly(C) oligomers, the energy barrier, 1, for hole injection becomes smaller in comparison with irregular DNA stacks consisting of both AT and GC pairs. The lowering of the potential barrier, in turn, will reduce the voltage threshold, Uc , for poly(G)–poly(C) duplex as compared to irregular DNA. To estimate Uc , we exploit the simplest model possible, which suggests that the potentials at the injecting and collecting contacts vary with the applied voltage as W − U/2 and W + U/2, respectively. If, in addition, one assumes that the applied field does not affect the electronic structure of stacked base pairs, the HOMO of the molecular wire is expected to cross the Fermi level at the voltage Uc = 2(I − W )/e = 21/e.

(5)

Experimentally the energy crossing manifests itself as the voltage gap in the observed I –U curves. Equation (5) is applicable to the evaluation of this observable only in the low-temperature limit, where the molecular motion in the wire is frozen and does not perturb the alignment of base pairs optimal for the injection process. Using the ionization potential of the poly(G)–poly(C) duplex calculated above, we conclude that in the case of platinum electrodes the voltage gap for the poly(G)–poly(C) molecular wire is above 3 V. This

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crude estimate is consistent with the experimental value reported by Porath et al. [9] for the poly(G)–poly(C) duplex between two platinum contacts at cryogenic temperatures. 2.2. Dynamic model for poly(G)–poly(C) duplex To describe the injection of holes along the poly(G) strand at finite temperatures, the dynamics of the stack has to be taken into account, since different types of internal molecular motion observed in DNA on various timescales [75] can influence the coupling between the nearest-neighboring GC pairs in the poly(G)–poly(C) molecular wire. Particular dynamic modes that can affect the coupling include bending, flipping of bases [76] and especially their twisting [70]. To avoid a choice of the specific dynamic mode, which is currently not justified experimentally, it is convenient to invoke a method commonly used for the description of the transition rate coupled to rearrangements of the molecular environment [77]. We introduce the configuration coordinate ϕi which determines the relative arrangement of adjacent ith and (i + 1)th G bases. Then the potential energy associated with the configuration degree of freedom near the equilibrium position ϕi = 0 can be approximated by 1 u i = Fϕi2 (6) 2 with F being the stiffness constant. The effect of molecular motion on the coupling within the stack is modeled by the dependence of the transfer integral, ti,i+1 , for sites i and i + 1 on the coordinate ϕi . In analogy with eqn (6), the explicit form of this dependence can be derived by expanding ti,i+1 (ϕi ) as a power series in ϕi . If we restrict ourselves to the terms of the second order in ϕi , the result reads ti,i+1 (ϕi ) = t (1 − Aϕi − Bϕi2 ).

(7)

Such dependence on the configuration coordinate occurs very often, particularly for π-type chromophores [60]. Note that the coupling constants A and B in eqn (7) depend on the position of the maximum for the transfer integral. If ti,i+1 (ϕi ) reaches the maximum value at ϕi = 0 (i.e. at the equilibrium position of the configuration coordinate), then A = 0 and B > 0. This is particularly true for twisting motion as evident from ab initio calculations of Sugiyama and Saito for G bases [70] and from earlier studies [78]. Once ti,i+1 (ϕi ) is specified, the derivation of the temperature dependence of the mean transfer integral hti,i+1 i is straightforward. All we need to do is to average eqn (7) over the Boltzmann distribution  1/2   F Fϕ 2 wB = exp − . (8) 2πkB T 2kB T As a result, we get   kB T hti j i = t (1 − Bhϕ 2 i) ≈ t 1 − B . (9) F Thus, the transfer integral becomes smaller as temperature increases, since molecular rearrangements within the stack reduce π π-interaction between adjacent G bases. Consequently, the ionization potential of the poly(G)–poly(C) duplex, which falls off with t, will increase, and the voltage gap will widen with T . 2.3. Temperature dependence of the voltage gap The substitution of eqn (9) into (5) makes evident that the voltage gap Uc should exhibit a linear increase with temperature, with slope and intercept given by dUc kB B ≈ 4t , (10) dT eF Uc (0) = 2(I1 − W − 2t)/e. (11)

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As has been found experimentally [9], the measured gap for the poly(G)–poly(C) duplex indeed increases with T following the linear law. Due to the wide scatter in the data, the results of measurement provide only order-of-magnitude evaluation of the experimental slope which gives dUc /dT ∼ 10−3 V K−1 . The estimated value of the slope can be compared with the prediction of eqn (10), assuming that twisting motion is the main factor affecting the value of the transfer integral. In this case t = 0.4 eV and B ≈ 7 according to ab initio calculations of Sugiyama and Saito [70]. Therefore, for F = 0.2 eV which corresponds to the typical barrier for internal rotations in organic molecules [79], eqn (10) yields dUc /dT ≈ 4 × 10−3 V K−1 in reasonable agreement with the experimental value. 2.4. Consequences for hole transfer studies The dynamic model of charge injection presented in Section 2.2 allows the useful generalization to the situation common for experimental studies of charge transfer in DNA [14–43]. In this case a hole is injected into the irregular DNA wire from a chemical donor rather than from a metallic electrode. Then the hole will undergo a tunneling transition from the chemical donor to the G base through the bridge of AT pairs with the rate, khop , given by eqn (2). The mechanism of this transition resembles superexchange transfer [74], but involves a band of bridge states instead of discrete energy levels of mediators. Besides, the dynamics of the bridge band can affect khop , much as the dynamics of poly(G)–poly(C) duplex affects hole transport along stacked GC pairs. To describe the coupling of hole transfer to the molecular rearrangements in the AT bridge, we introduce the probability distribution P(ϕ, t) that a hole will be on the donor at time t, if the elements of the AT bridge have configuration ϕ = ϕ1 , ϕ2 , . . . , ϕn . This distribution satisfies the multi-dimensional Smoluchowski equation [77] ∂P ∂2 P D ∂(u P) =D 2 + − khop (r ) exp(−βBϕ 2 a/2)P, (12) ∂t kB T ∂ϕ ∂ϕ Pn where D is the diffusion coefficient along the coordinates ϕ, u = 21 i=1 Fϕi2 is the potential associated with the configuration degree of freedom, and a is the mean plane-to-plane distance between stacked AT bases. Note that the stiffness constant F can differ from the same constant for the poly(G)–poly(C) molecular wire since the hole localized on the donor can affect the energetics of the DNA chain. In the simplest case of sufficiently long AT bridges, where the time evolution of the system along the coordinate ϕ is faster than the electron transfer, the approximate solution of eqn (12) is given by   u(ϕ) P(ϕ, t) = S(t) exp − . (13) kB T The function S(t) in eqn (13) can be specified by substituting eqn (13) into (12) and by subsequent integration of the result over the coordinates ϕ. This yields  n/2 d S(t) F = −khop (r ) S(t). (14) dt F + kB Tβ Ba Equation (14) implies that in the case under consideration the effective rate k for the unistep transfer of holes through the dynamic AT bridge varies with T as n/2  F k = khop (r ) . (15) F + kB Tβ Ba Unlike the vast majority of reactions known in chemical kinetics, the rate of this particular process decreases with temperature, which destroys the optimal overlap of bases needed for efficient charge transfer. The result

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obtained also clearly shows that the effective fall-off parameter βeff = −d(ln k)/d L AT becomes large as T increases. The corresponding temperature correction can be estimated from the expression   1 F + kB Tβa B ln . (16) βeff = β + 2a F As can be seen from eqn (16), the effect of temperature becomes significant, if kB Tβa B > F. According to our estimations, the latter inequality is satisfied for reasonable values of parameters involved. Therefore it will be interesting to test the predicted effect experimentally. Another result predicted by our model is the nonexponential decay of the survival probability for a hole initially localized on the donor. Such temporal behaviour follows from the averaging of eqn (12) over the configuration coordinate ϕ, see also [77]. The nonexponential decay will show up in transient absorption measurements and can help in understanding the appropriate experimental data [34].

3. Conclusions In summary, we have considered mechanisms of hole injection and transport in stacks of Watson–Crick base pairs, which make DNA oligomers promising candidates for application as a molecular wire. Our analysis has been focused on two types of stacks, which differ in the arrangement and number of guanine bases involved. The stacks of the first type consist of irregular sequences of pyrimidine and purine deoxynucleotides, while the second involves regular stacks containing only guanine–cytosine pairs. It has been shown that in spite of this structural distinction, the general feature of the injection process turns out to be common for both types of molecular wire: in both cases the motion of charge carriers in the metal–DNA junction proceeds via tunneling controlled by internal dynamics of the stack. A theoretical model proposed for this mechanism of hole injection predicts linear increase of the voltage gap with temperature and provides estimates for the slope and the intercept, which are in reasonable agreement with available experimental data. The extension of the model to the situation common for experimental studies of charge transfer in DNA has also been considered. It has been demonstrated that the structural differences manifest themselves in the mechanism of charge migration along the wire. For the irregular stacks, hole migration proceeds via hopping between energetically appropriate guanine bases, while in the regular stacks the band-like motion dominates. Using typical values of the hopping rate, the mobility of holes has been estimated for the periodic sequences with a certain number of adenine–thymine pairs between two guanines. Acknowledgements—This research is supported by funding from DoD/MURI and the Chemistry divisions of NSF and ONR. We are grateful to J Jortner, F D Lewis, M E Michel-Beyerle, and M R Wasielewski for helpful remarks.

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