Electron Hole Transport in DNA - The Journal of Physical Chemistry B

Messer, A.; Carpenter, K.; Forzley, K.; Buchanan, J.; Yang, S.; Razskazovskii, Y.; Cai, Z.; Sevilla, M. D. J. Phys. Chem. B 2000, 104, 1128−1136. [A...
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10398

J. Phys. Chem. B 2001, 105, 10398-10406

Electron Hole Transport in DNA Johan Olofsson and Sven Larsson* Department of Physical Chemistry, Chalmers UniVersity of Technology, S-41296 Go¨ teborg, Sweden ReceiVed: March 20, 2001; In Final Form: June 1, 2001

Electron hole transport along DNA chains is studied theoretically. Couplings between the DNA bases adenine, cytosine, guanine, and thymine, placed as in DNA in a single strand or diagonally in a double strand, are calculated using ab initio methods. Coupling as well as reorganization energies, which are also calculated, show a great variation between bases. To decide whether a periodic DNA strand has localized or delocalized electrons, a novel theoretical model that uses reorganization energy and coupling as input, is used. The model suggests that electron holes on infinite one-dimensional chains of periodic DNA localize if the ratio of reorganization energy to coupling is larger than about four. The hole is weakly trapped on an infinite guanine strand (G)x whereas it is delocalized in thymine (T)x and cytosine (C)x strands. Some mixed DNA strands also show a delocalized behavior, but most are localized to a single G in agreement with EPR measurements. Rates of stepwise hole transfer between G sites in mixed chains are calculated and show agreement with experimental data. In electron-transfer steps between localized sites, the tunneling rate decreases exponentially with distance with a quite large β factor in agreement with earlier theoretical work.

1. Introduction Ever since the discovery of the repetitive structure of DNA, it has been hypothesized that in DNA the unoccupied π* orbitals of the bases form a conduction band.1-3 This possibility, which concerns primarily periodic DNA,3,4 has recently been supported by experiments.5,6 EPR experiments on aperiodic DNA suggest, however, localized behavior in the ground states of the ionized systems.7 A number of other experiments, using fluorescence quenching techniques, have been interpreted to mean that the electron hole tunnels over large distances in a single step with an exceptionally low decrease (β = 0.1 Å-1) of rate with distance,8-10 according to:

k ∼ exp(-βR)

(1)

The low β is not confirmed by other electron transfer (ET) experiments on similar systems, and other interpretations have been suggested.11,12 Theoretical calculation of the rate of single ET steps in DNA shows that the rate decreases fast with the number of intervening bases, as in the case of proteins.13 Interpretation of other results obtained by photoinduced ET14-17 has led to a multistep model where the electron hole (or charge) leaps between guanine bases.18-31 Such a model implies a high rate of transport if the number of non-G bases between the guanines is small. In particular Ratner et al. have shown that the distance dependence in that case is 1/N, where N is the number of guanine units.27,28 If there is a great number of nonguanine units in sequence, the transfer probability through this part of the chain decreases exponentially, since the transfer mechanism is tunneling. There is good reason to believe that the rate of tunneling in DNA is normal, as in, for example, proteins, where it decreases exponentially with distance and, therefore, is restricted to short distances. Since tunneling depends in a detailed way on electronic structure, the electrons of the bridge have to be included in the * To whom correspondence should be addressed. Telephone: +46-031 7723058, Fax: +46-(0)-31 7723858, E-mail: [email protected]

calculation of the coupling32 that is used in the Marcus model.33 In the present case we will use an approximate form of this tunneling theory, called ‘superexchange’,30,34 that is applicable in multistep cases of ET between DNA bases. Provided that the intervening molecules are the same, the coupling decreases exponentially but slower than in the case of open space. The same behavior is obtained as in the Gamov tunneling model.35 The bases (or in other cases solvent molecules) are promoting rather than hindering ET. If the intervening medium is nonuniform, which is the rule in a biological system, ET may still be approximately exponential, with β ranging between 0.7 and 1.5. In a very inhomogeneous system, there may be ‘pathways’ or structures that are more favorable than others, with nonexponential behavior resulting.36 Transport of positive holes may be viewed as tunneling between guanine bases22,27-30 which are the easiest to oxidize.37 Schuster et al.19,23,26 and Giese et al.24,25,29,38 have estimated rates of ET between such sites. More recently, direct measurement of hole transport in DNA has been performed using photophysical methods.15-17 To our knowledge there is no sufficiently accurate calculation that shows whether the electron hole is really delocalized in periodic DNA. Such a calculation requires that full geometry optimization is performed and compared for the localized and delocalized forms. A simulation based on calculated couplings and reorganization energies may be used for the infinite system.39 Experimentally the recent work of Fink et al. suggests that conductivity in DNA is possible without activation energy,5 whereas the work of Porath et al. suggests that DNA is semiconducting with a rather wide band gap.6 In both cases, however, the authors argue convincingly that transport occurs in a band and not by stepwise hopping. In the present paper, results will be presented using a novel theoretical model that shows that infinite Tx and Cx have delocalized holes whereas Gx and Ax have weakly trapped holes. In the calculation of rate of ET, the reorganization energy (λ) is in a sense the inverse of vibrational overlap.30 λ varies from system to system and has to be calculated using accurate

10.1021/jp011052t CCC: $20.00 © 2001 American Chemical Society Published on Web 09/29/2001

Electron Hole Transport in DNA

J. Phys. Chem. B, Vol. 105, No. 42, 2001 10399

Figure 1. DNA bases.

Figure 2. Marcus model. Precursor (left) and successor (right) potential energy surfaces for the system (B1 + B2)+.

ab initio methods.40 Coupling (HAD) is half the electronic factor (∆) at the ET transition state (HAD ) ∆/2).41 ∆ is simply taken as the energy difference between the two electronic states involved at the avoided crossing for ET (plus and minus combinations of precursor and successor states). ∆ may be calculated using ab initio methods. In the present paper, we have calculated both λ and ∆ and from these numbers derived the ET rate using well-known theory.44 Voityuk et al.42 have recently carried out the coupling with a similar method as is used here.43 Transport of electrons through DNA is fast if the ET steps between guanines are short. An artificial, periodic chain should give a very high rate of transfer and possibly appear metallic,2-6,45 even if electron holes are weakly trapped along such a periodic chain. If the sequence is aperiodic, high transfer rate is possible if there are short distances between all guanines, according to the model of Ratner et al.27,28 Recent experiments, suggesting that even aperiodic λ-DNA is conducting5,46-48 could possibly be explained that way. At this point there may still be disagreement, however.49

Figure 3. (a) Reorganization energy (λ1+) for removing one electron from the neutral molecule. (b) Reorganization energy (λ2+) for adding one electron to the positive ion.

removing one electron from B1 and adding it to B2+ is obtained by first decreasing the number of electrons by one in B1 and calculating the energy at the optimized equilibrium geometry of neutral B1. The structure is subsequently reoptimized for B1+. The energy gain is equal to the reorganization energy λ1+ (Figure 3a). The reorganization energy for increasing by one the number of electrons on B2+ is obtained by calculating the total energy of neutral B2 in the geometry optimized for B2+. The difference between this energy and the optimized one for neutral B2 is equal to λ2+ (Figure 3b). Finally the total reorganization energy is obtained as λ ) λ1+ + λ2+. Structural changes are mainly in the CC bond lengths. Calculations have shown that the change in CH bond length can be neglected and usually also the bond angle changes.40 If there are no other important types of geometry modifications, it is possible to obtain the reorganization energy alternatively from the following approximate equation:

λ)

1

∑ k∆Ri2 i ) CC 2

(3)

bonds

2. Theory We are concerned with ET between a neutral (B1) and a positively charged (B2+) DNA base (Figure 1). Since the structural differences between B1 and B1+ and between B2 and B2+ are quite small, one may assume that ET takes place in the parabolic region near the equilibrium geometries on the total energy potential energy surface of (B1+B2)+ (Marcus model,33 Figure 2). Assuming the same force constants, the activation energy is easily calculated with the help of two parameters, λ and ∆G°.

Ea )

λ ∆G° 2 14 λ

(

)

(2)

∆G° is measured by the net free energy decrease of the system during a single ET step. The reorganization energy λ1+ for

where the force constant k ) 750 Nm-1 corresponds to a stretch frequency of 1456 cm-1 and ∆Ri is the change of bond length when one electron is added or removed. Contributions from CO and CN bond length changes may be taken into account in a similar way. We have assumed that electron transport primarily takes place through the DNA core of conjugated bases. In saturated structures the change in bond lengths at ionization are considerably larger than in a π system. There is a very high reorganization energy that leads to deep traps, preventing ET. In particular, semiconductivity is not possible through the sugar and phosphate part of DNA even if the electrons could be injected in the unoccupied MOs. ET by tunneling, on the other hand, is possible in a saturated system but with a higher β than in a conjugated chain.50 We have therefore included one or two bases (Figure

10400 J. Phys. Chem. B, Vol. 105, No. 42, 2001 1) in the calculations, but left out the sugars and phosphates. The relative position of the DNA bases and their orientation to each other are taken from HyperChem.51 The latter over-all structure was tested by molecular mechanics and found to be close to the optimized structures. The structures of each base and their positive ions are optimized individually using HartreeFock, MP2, and DFT (B3LYP) methods with up to 6 311-G(d) basis sets in the Gaussian-98 program.52 The optimized molecules are superposed and allowed to replace the corresponding HyperChem molecules in the right position and orientation. Geometry optimization is performed for the neutral molecule and the positive ion. The calculated geometries agree with experimental data and earlier calculations. The contribution to the reorganization energy from the region outside the ionized base is hard to estimate because of the heterogeneous character of this region, but this contribution has been considered very important in recent publications.53 It was even suggested that the main distance dependence of the rate is due to the distance dependence in the outer reorganization energy. The latter was calculated using global dielectric constants.53 We propose here that a more accurate way would be to calculate outer λ in the same way as inner λ, i.e., calculating bond length and bond order differences. This is done below for the phosphate and sugar groups and for the hydrogen bonds between the bases. The reorganization energy from the solvent has been ignored in the present work, since there is a large distance to the solvent through the sugar-phosphate or rather small space angle to the solvent of the grooves. The effective coupling HAD for ET between two bases, A and D, is defined as half of the energy splitting (∆/2) at an avoided crossing between the two states relevant for ET.41 If the potential energy surfaces have dimension 3M, where M is the number of nuclei, the seam of avoided crossing has dimension 3M - 1, in principle. In the present case, the coupling is determined by overlap of extended π-orbitals and cannot be expected to be very sensitive to the precise geometry of the transition state. In all cases the donor/acceptor MOs describing the electrons to be transferred are (φd + φa) and (φd - φa), with orbital energies + and -, respectively. In the present case φd and φa and their linear combinations are HOMO and HOMO-1 for the whole system. The coupling is thus half the HOMO-1 HOMO energy difference EN-1 - EN-1 at the avoided crossing where HOMO-1 and HOMO, respectively, are ionized. We use Koopmans’ theorem to estimate this total energy difference. Taking the difference in ionization energy for the neutral system we obtain: HOMO-1 HOMO ∆ ) EN-1 - EN-1 ≈ HOMO - HOMO-1 ) |+ - -| (4)

Even in the case of identical DNA bases, HOMO and HOMO-1 are mainly localized to a single base, since the bases are placed in DNA in such a way that there is no symmetry element. To find a transition state, the geometry is modified.40,41 To avoid a great number of heavy calculations, preliminary external charges are implemented that bring about equal localization with a help of charges in a very fast semiempirical calculation. The geometry of molecule and external charge is subsequently used in an ab initio calculation. The charge distribution remained so close to 50/50% that the final adjustment could be done in a few additional calculations. In a case like the present one with an open gap between two π systems, it is important to have a basis set that correctly describes the tail of the orbitals. In our experience, the approximations involved in Koopmans’ theorem appear to be

Olofsson and Larsson TABLE 1: Difference in Free Energy between the Guanine Molecule (donor) and the Different Bridge Molecules (adenine, cytosine, and thymine) ∆G (eV) EA-EG EC-EG ET-EG

0.47 0.65 0.62

less serious than inadequate basis sets. The ET rate constant for a single vibration of frequency νn is given as an Arrhenius equation.44

( )

kel ) νnκ exp -

Ea kBT

(5)

νn is the average nuclear attempt frequency which takes the system across the barrier, κ is the electronic factor, and Ea is the activation barrier. The structural dependence due to electron tunneling is included in the electronic factor:44

κ)

2[1 - exp(-νel/2νn)] 2 - exp(-νel/2νn)

(6)

νel may be called the “electronic frequency” and is given as

νel )

[ ]

∆2 π3 2h λkBT

1/2

(7)

In the case νel , νn we obtain that κ ) νel/νn and eq 5 may be written as

kel )

[ ] ( )

∆2 π3 2h λkBT

1/2

exp -

Ea kBT

(8)

The oxidation potentials of Seidel et al. using cyclic voltammetry and differential pulse polarography for the electrochemical measurements (Table 1)37 are used to calculate the change in free energy when one electron is removed from the DNA bases. The reported potentials are obtained using a hydrogen reference electrode (NHE) and are in good agreement with the potentials calculated from the Weller equation.54 3. Calculations and Results Reorganization Energy. The bond-length changes when one electron is removed from a base are given in Figure 4. The calculated reorganization energies for the hole state is given in Figure 5. In Figure 6 we obtained the reorganization energy from eq 3. The good agreement between Figure 5c and Figure 6 shows that the origin of the internal reorganization is mainly in the CC, CN, and CO bond length changes. Consequently, the reaction coordinate in the Marcus model is primarily the bond length change in the bases. The DFT method predicts considerably smaller total reorganization energies, λ1+ + λ2+, for the DNA bases than the Hartree-Fock and MP2 methods. For guanine and the basis set 6-311G(d) the difference is as large as 0.6 eV, using eq 3. Still the bond length changes (Figure 4) are in the same direction, but somewhat smaller by the DFT method. Since the reorganization energy contains the bond length change squared, the final λ is quite different between the methods. To try to determine which theoretical method is the best one to calculate reorganization energy, we used benzene (C6H6) as a ‘benchmark’ (Table 2). Remarkably enough, the MP2 predicts a considerably larger value of λ1+ for the best basis set. It is known that MP2 overcorrects the Hartree-Fock method for correlation effects

Electron Hole Transport in DNA

J. Phys. Chem. B, Vol. 105, No. 42, 2001 10401

Figure 4. Bond length changes when one electron is removed. Methods and basis sets (the lines are used to guide the eye): 9 MP2/6-311G(d), b HF/6-311G(d), and 2 DFT B3LYP/6-311G(d). (a) adenine, (b) cytosine, (c) guanine, and (d) thymine.

and also that very large basis sets are necessary in the MP2 method. On the other hand, the DFT method is possibly less accurate in the case of ions. Experimentally, the reorganization energy λ1+ is identical to the difference between vertical and adiabatic ionization potential. Our results agree quite well with experimental numbers55-57 (Table 3). The average error is minimized for the DFT method, but the trends from one base to the other are not completely correct with this method. The MP2 results, on the other hand, show the correct trends but are on the average 60% too large. This corresponds to bond length changes being on the average larger by about 25% than the experimental numbers. The reorganization energy due to sugar and phosphate was obtained using semiempirical methods from calculated bond length changes after ionization of the base. Very small values were obtained which can be explained as resulting from a change of the electric field. The hydrogen bonds between two bases were also studied. Using a semiempirical method, we first found that ionization from a base pair does not lead to redistribution of protons. The bond length changes in the hydrogen-oxygen and hydrogen-nitrogen bonds, although significantly larger than the change of CH bond lengths, are too small to give any significant contribution to the reorganization energy. The water solvent, finally, can be expected to give a nonnegligible contribution, despite the small space angle to groove. In view of other possible errors, we decided, however, to ignore this contribution for the time being. Coupling. The calculated electronic couplings in the DFT method [basis set 6-311G(d)] and the HF method [basis set 6-31G(d)] are given in Tables 4 and 5. The results from the

two different methods agree quite well. Our results also agree reasonably well with the calculations by Voityuk et al.42 who use a different basis set in the Hartree-Fock method. In both cases, surprisingly large values of the couplings are obtained. Semiempirical methods, using a simpler representation of the valence orbitals of the carbon atoms, give a much smaller coupling and cannot be used in the present case. 4. Coherent Electron Transport in a Conduction Band Aperiodic 1D systems are believed to be nonconducting if the diagonal disorder is greater than the bandwidth. A difference in oxidation potentials of the bases is a contribution to disorder in the Anderson localization model,58 applied to DNA. Thermal agitation gives another contribution, even if the bases are the same. Finally, disorder may be due to localization at absolute temperature T ) 0, which may occur even if the bases are the same in periodic DNA. The latter problem is not solved by Anderson localization and requires special attention.39 In a periodic, infinite system, the MOs are delocalized Bloch orbitals. Split of symmetry is normally not permitted. Consequently there may be a hidden solution of the Schro¨dinger equation with a lower total energy, corresponding to localized electron or hole, with local nuclear geometry resembling that of the negative or positive ion, respectively.39 The translational symmetry is lost. In a finite system with a hole state, a quantum chemical calculation can determine, in principle, whether the hole is localized or delocalized. In practice, most real systems are too large for accurate calculation. For example a system consisting of three bases is too large for an accurate calculation but too small to decide about localization.

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Figure 6. Reorganization energies for removing one electron, calculated using (eq 3). Methods and basis sets: 9 MP2/6-311G(d), b HF/ 6-311G(d), and 2 DFT B3LYP/6-311G(d).

TABLE 2: Reorganization Energy (eV) Calculated from the Total Energies for Removing One Electron from Benzene (compressed structure, two short, and four long bonds) HFa HFb HFc DFTa DFTb DFTc MP2a MP2b MP2c a

6-31G(d)

b

λ2+

λ1+

λTOT ) λ1++ λ2+

0.228 0.226 0.216 0.150 0.150 0.148 0.127 0.125 0.124

0.153 0.225 0.216 0.149 0.151 0.152 0.129 0.247 0.127

0.381 0.451 0.432 0.299 0.301 0.300 0.255 0.372 0.251

6-311G(d)

c

6-311+G.

For each base the bond lengths are determined by the π bond orders, which in turn are proportional to the electronic occupation (between 0 and 1) on that base. The occupation is equal to the coefficient squared in the total wave function of the system. At the same time, reorganization energy λ depends on bondlength change squared, i.e., bond order squared. Consequently, λ depends on occupation number squared or wave function coefficient to the fourth power.39 The diagonal matrix element may thus be written as:

Hij ) ∆µi - λ1i+ ‚Ci4

(9)

+

Figure 5. (a) Calculated reorganization energy λ1 for removing one electron from the neutral molecule. (b) Calculated reorganization energy λ2+ for adding one electron to the positive ion. (c) Calculated total reorganization energy λ ) λ1+ + λ2+. Methods and basis sets: 9 MP2/ 6-311G(d), b HF/6-311G(d), and 2 DFT B3LYP/6-311G(d).

A theory for localization in periodic solids, where the sites are molecular groups, has recently been worked out by one of us in collaboration with Klimkajns.39 Delocalization along the chain of molecules is favored by a large interaction (H12 ) ∆/2) between the molecules and counteracted by partial trapping of the electron on a single site. The model39 shows that one may understand localization or delocalization in an infinite crystal as an interaction between wave functions, where each wave function Φi corresponds to a hole on base i. If at a certain moment one particular base i is a positive ion, the energy of this site is lowered by the reorganization energy λ1i+, compared to all other wave functions. The final interaction is between wave functions, each corresponding to a particular base, where those with occupation have a modified diagonal matrix element.

where ∆µi is the oxidation potential of the site relative to the oxidation potential of G (Table 1), λ1i+ the reorganization energy of site i for removing an electron (Figure 5), and Ci the coefficient of the wave function at the site. The off-diagonal matrix element is the ordinary coupling Hij. The coefficients are determined from the following eigenvalue problem:

(

∆µ1 - λ11+C14 H21 .... 0

H12

0.... ∆µ2 - λ12+C24 H23... .... .... 0 0...

)( ) ()

0 0 .... ∆µN - λ1N+CN4

C1 C )E 2 ... CN

N is the number of bases included in the simulation.

C1 C2 ... CN

(10)

Electron Hole Transport in DNA

J. Phys. Chem. B, Vol. 105, No. 42, 2001 10403

TABLE 3: Calculated and Experimental Vertical (ve) or Adiabatic (ad) Ionization Potentials and λ1+ (eV) for the DNA Bases DFT/6 -311G(d) A C G T a

HF/6- 311G(d) +

experimenta

MP2/ 6-311 G(d) +

+

ad

ve

λ1

ad

ve

λ1

ad

ve

λ1

ad

ve

λ1+

7.78 8.19 7.34 8.37

7.99 8.30 7.72 8.58

0.21 0.11 0.37 0.21

6.84 7.08 6.24 7.44

7.22 7.30 6.78 7.84

0.38 0.23 0.54 0.40

8.00 8.72 7.76 8.58

8.36 9.11 8.48 9.05

0.36 0.39 0.72 0.36

8.26 8.68 7.77 8.87

8.44 8.94 8.24 9.14

0.18 0.26 0.47 0.27

ref 56, 57.

TABLE 4: Calculations of the Intrastrand Electronic Couplings in Single-Stranded DNA Performed for the Different Pairs of the DNA Basesa base pair

DFTb

HFc

AA AC AG AT CA CC CG CT GA GC GG GT TA TC TG TT

0.052 0.071 0.057 0.145 0.029 0.087 0.051 0.120 0.099 0.136 0.076 0.173 0.117 0.090 0.112 0.141

0.027 0.080 0.027 0.205 0.031 0.109 0.057 0.176 0.115 0.126 0.096 0.174 0.124 0.094 0.111 0.195

a The notation GC refers to 5′-GC-3′ orientation. b B3LYP/6311G(d). c 6-31G(d).

TABLE 5: Calculations of the Interstrand Electronic Couplings in Double-Stranded DNA Performed for the Different Combinations of the Bolded and Underlined DNA Basesa base pair

DFTb

HFc

AT/TA TA/AT AG/TC GA/CT AC/TG CA/GT CG/GC GC/CG CC/GG CC/GG CA/GT AC/TG GC/CG CG/GC AG/TC GA/CT TT/AA TT/AA AT/TA TA/AT

0.088 0.072 0.022 0.013 0.081 0.035 1.7 × 10-4 1.6 × 10-3 4.5 × 10-3 0.025 2.5 × 10-4 1.1 × 10-3 0.033 0.025 0.017 2.4 × 10-3 9.7 × 10-3 9.5 × 10-3 1.3 × 10-3 1.1 × 10-3

0.059 0.077 0.019 0.014 0.071 0.041 3.0 × 10-4 9.5 × 10-4 8.0 × 10-3 0.017 2.2 × 10-4 1.3 × 10-3 0.027 0.014 0.017 2.5 × 10-3 0.015 8.5 × 10-3 6.5 × 10-4 9.0 × 10-4

a The notation GC/CG refers to 5′-GC-3′ dimer hydrogen bonded to 3′-CG-5′ dimer. b B3LYP/6-311G(d). c 6-31G(d).

An iterative procedure is started by assuming that a hole is delocalized over the system (Ci ≈ 0). The first solution has larger coefficients for the site with the lowest oxidation potential. In the second iteration, the diagonal matrix elements are updated using the bond-length changes implied by previous eigenvector. This leads to nonzero contribution λ1i+Ci4 in the diagonal Hamiltonian matrix elements (eq 9). If the absolute value of the coupling Hij (Tables 4,5) is not large enough compared to

TABLE 6: Character of Hole State in Infinite Chainsa DNA Ax Cx Gx Tx (AC)x (AG)x (AT)x (CG)x (CT)x (GT)x (GAC)x (CAG)x (CAT)x (TAC)x (ACT)x (ATC)x (CTA)x (TCA)x (ACA)x (AGA)x (ATA)x (CAC)x (CGC)x (CTC)x (TGT)x (TAT)x (TCT)x (GAG)x (GCG)x (GTG)x

DFT

HF

A+

A+

deloc G+ deloc A+C AG+ deloc CG+ deloc G+T G+AC CAG+ CA+T TA+C A+CT A+TC CTA+ TCA+ A+CA AG+A A+TAb CA+C CG+C deloc TG+T deloc deloc GAG+ GCG+ GTG+

deloc G+ deloc A+C AG+ A+T CG+ deloc G+T G+AC CAG+ CA+T TA+C A+CT A+TC CTA+ TCA+ A+CA AG+A A+TAb CA+C CG+C CT+C TG+T TA+T deloc GAG+ GCG+ GTG+

MP2 A+ C+ G+ deloc A+C AG+ A+T CG+ CT+ G+T G+AC CAG+ CA+T TA+C A+CT A+TC CTA+ TCA+ A+CA AG+A A+TAb CA+C CG+C CT+C TG+T TA+T TCT+ GAG+ GCG+ GTG+

a Deloc means delocalized hole. If the hole is delocalized, the + denotes site where the hole is localized within the repeat unit. The reorganization energy λ1+ is calculated using the DFT, HF, and MP2 methods. b Hole almost delocalized over both A (coefficients 0.65 and deloc 0.56).

λ1i+ to keep the electron delocalized, the hole is becoming more and more localized on site i. When the iterative procedure converges, we may find either localization to a single site or complete delocalization over the chain. The transition from localized to delocalized, if by some means the coupling or reorganization energy can be decreased continuously, is a sharp transition.39 Calculations on a periodic, linear chain show that the transition occurs when λ is about four times the coupling matrix element. The results are shown in Table 6. Partial trapping is a cause of resistivity. If the Marcus barrier for ET is large, the carrier is trapped and conduction hindered, although a calculation on a symmetric system with translational symmetry gives perfectly delocalized bands. In our case of DNA bases we are close to the region between localization and delocalization. There are experimental data that suggest that conductivity is possible in aperiodic DNA (usually λ-DNA)6,46-48 or without activation energy.5 Even proximity-induced superconductivity seems to have been measured in aperiodic solids.48 Tran et al. used microwave frequencies in a study of λ-DNA (aperiodic) at different temperatures and found a strong temperature dependence consistent with an activation barrier.47 In this case,

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TABLE 7: Wave Function Coefficients for the CCCTAGGG Sequences and CTCGAGa DNA sequence

C

C

C

T

A

G

G

G

CCCTAGGGb CCCTAGGGc

0 0

0 0

0 0

0.001 0.004

0.009 0.028

0.108 0.221

0.988 0.951

0.106 0.213

DNA sequence

C

T

C

G

A

G

CTCGAGb CTCGAGc

0 0.001

0.009 0.016

0.099 0.133

0.994 0.989

0.048 0.069

0.007 0.018

a The DFT (B3LYP) method and 6-311G(d) was used to calculate the electronic coupling (∆). b The reorganization energy (λ) was calculated with the MP2 method and the basis set 6-311G(d). c The reorganization energy (λ) was calculated with the DFT (B3LYP) method and the basis set 6-311G(d).

activation is hardly attributable to a band gap, as in the case of the periodic DNA studied by Porath et al.6 but consistent with stepwise ET in the ‘conduction band’. 5. Stepwise Hopping: Calculation of Rate The localization model used above39 may be employed also for aperiodic DNA if the diagonal elements are modified with the respective reduction potentials. In all cases, when this model has been employed on an aperiodic solid, we have obtained agreement with the ESR results, for example to those of Debije et al.7 and Schieman et al. (Table 7).59 G+ radicals are easily formed and stable, due to low oxidation potential. (dA-dT)+ radicals are reduced by G. If G is absent a localized A+ ought to be possible, since A is also localized (Table 6). However, (AT)x is delocalized, at least by the DFT calculation. In dA-dT there is the possibility that the hole disappears in a faster time than the EPR time constant. The rate of transport between G and G+ is calculated using eqs 6-8. In both cases, we need information about coupling and reorganization energy for G. The coupling HGxG is obtained from a super exchange model:

HGxG )

HGxHxG Ex - EG

(11)

for the case of one intervening non-G base x. For the case of two intervening non-G bases, x and y, we obtain:

HGxyG )

HGxHxyHyG (Ex - EG)(Ey - EG)

(12)

with obvious extension to cases with a greater number of intervening non-G bases. The extended superexchange model, eq 12, leads to exponential decrease with distance, which is uniform if the bases are the same. The results are presented in Tables 8 and 9. Using the superexchange method to calculate the coupling (Table 10), we obtained β values in the range 0.7-1.7 Å-1, which agree with experimental numbers.60 Some examples of ET rate calculations are given in Tables 8 and 9. For a system where (AT)n forms a bridge between G and G+, we obtain EA - EG ) 0.47 eV and ET - EG ) 0.62 eV, which gives β ) 0.84 Å-1 for the (AT)n bridge, and for β ) 1.06 Å-1 for (CT)n. As a comparison, Meggers et al. obtain β ) 0.7 (0.1 Å-1 for the (AT)n bridge.22 For Tn, Lewis et al. obtain β ) 0.64 Å-1.61 We obtain β ) 0.87 Å-1 with the DFT data and β ) 0.68 Å-1 with the HF data. Using the HF data calculated by Voityuk et al.42 we obtain β ) 0.80 Å-1. In both cases ET is through the Tn bridge, which is directly connected to the G bases. Lewis et al. also found61 that there is no photoinduced ET in a An or Tn hairpin. This

may be explained by the greater oxidation potential for A, and T. Fukui et al. in a different experiment used An as a bridge and found β ) 1.47. Our data (Table 10) is 1.3 for DFT and 1.68 for HF. 6. Discussion and Conclusions We have carried out accurate ab initio calculations to determine the relevant parameters of coupling (∆/2) and reorganization energy (λ) in order to decide theoretically about the transport mechanism of electrons in DNA. Oxidation potentials of the bases were taken from experiments.37 It was somewhat surprising to find that the coupling appears to be large enough compared to reorganization energy to have delocalized π* orbitals in some cases of periodic DNA such as Cx, Tx, and (AT)x (Table 7). In these cases semiconductivity should occur, where the only activation is to lift the electron to the conduction band. On the other hand, there are still large uncertainties, particularly in the calculation of the reorganization energy. Furthermore, a significant contribution from the water solvent, neglected here, would make all systems localized. In aperiodic DNA it is more reasonable to have stepwise ET between bases of the same kind. Since guanine has the lowest oxidation potential, ET in DNA can normally be understood as stepwise hopping between G sites. Each step is a tunneling step. Alternatively, the mechanism may be called thermally induced tunneling, vibration assisted tunneling, or phonon-assisted polaron-hopping. In all cases, we are dealing with ET according to the Marcus model,33 or if vibrational quantization is important, according to the Bixon-Jortner model30 (see ref 44 for further references). We have found that the couplings (Hxy ) ∆/2) between adjacent bases are of magnitude 0.03-0.17 eV. The deviations between the different methods is, however, rather large, which suggests that the methods to calculate couplings should be improved. Voityuk et al. have used a slightly different basis set at the Hartree-Fock level.42 Their calculated couplings are consistent with ours both regarding intrastrand and interstrand couplings. Not surprisingly, the latter are in both cases smaller than the coupling between the neighboring bases of the same strand. The textbook model suggests that conductivity occurs if the Fermi level passes through an energy band and semiconductivity if the Fermi level is in a sufficiently narrow energy gap. In reality, this model has to be considerably extended.63-65 First of all, electrons may be photoexcited or injected into the conduction band, in which case the problem of activation across the gap disappears. On the other hand, electrons may be trapped in the conduction band so that significant activation energy is needed to transport electrons by stepwise hopping. In the Marcus model, the applied field leads to lowering of the right-hand parabola compared to the left one (Figure 2).64 To get the maximum conductivity, this lowering, from one base to the next, has to be equal to λ. De Pablo et al. have found a very high resistance for λ-DNA with mixed bases,49 which is in agreement with our view. On the other hand, as long as the activation barrier is smaller than the vibrational energy (corresponding to CC stretch) very fast ET will occur, which may be evident as a quite high conductivity. Delocalization in infinite systems is often discussed in terms of the Anderson58 or Mott-Hubbard models.63 The first model is a simplified model where the physical problem concerns what happens in an ordered, delocalized system, if some ‘disorder’ occurs, for example, aperiodicity or thermal fluctuations. In our case, the problem we pose is a different one: whether the system is localized or delocalized without disorder. The ratio between reorganization energy and coupling is here decisive.

Electron Hole Transport in DNA

J. Phys. Chem. B, Vol. 105, No. 42, 2001 10405

TABLE 8: Electronic Coupling Obtained from a Superexchange Model for the Case of One or None Intervening Non-G Base x (Eq 11) and Hole-hopping Rate Constants at 298 K (Eq 5) G-A-G+ G-C-G+ G-T-G+ A-A+ C-C+ G-G+ T-T+ a

HGx (eV)

HxG (eV)

Ex - EGa (eV)

HGxG (eV)

k (s-1)

t ) 1/k (s)

0.057 0.136 0.112

0.099 0.051 0.173

0.47 0.65 0.62

1.2 × 10-2 1.1 × 10-2 3.1 × 10-2

1.0 × 106 8.3 × 105 6.8 × 106 1.0 × 1010 1.3 × 109 3.1 × 107 7.1 × 109

9.7 × 10-7 1.2 × 10-6 1.5 × 10-7 9.8 × 10-11 7.9 × 10-10 3.2 × 10-8 1.4 × 10-10

Difference in free energy between the guanine molecule (donor) and the different bridge molecules (adenine, cytosine and thymine).

TABLE 9: Electronic Coupling Obtained from a Superexchange Model for the Case of Two Intervening Non-G Basesa HGx (eV) +

G-A-A-G 5′-G-A-C-G+-3′ 3′-G-A-C-G+-5′ 5′-G-A-T-G+-3′ 3′-G-A-T-G+-5′ G-C-C-G+ 5′-G-C-T-G+-3′ 3′-G-C-T-G+-5′ G-T-T-G+

0.057 0.099 0.057 0.099 0.057 0.136 0.136 0.051 0.112

Hxy (eV) 0.052 0.071 0.029 0.145 0.117 0.087 0.12 0.090 0.141

HyG (eV)

Ex - EGb (eV)

0.099 0.051 0.136 0.112 0.173 0.051 0.112 0.173 0.173

0.47 0.47 0.47 0.47 0.47 0.65 0.65 0.65 0.62

Ey - EGb (eV)

HGxyG (eV)

k (s-1)

t ) 1/k (s)

0.47 0.65 0.65 0.62 0.62 0.65 0.62 0.62 0.62

1.3 × 1.2 × 10-3 7.3 × 10-4 5.5 × 10-3 3.9 × 10-3 1.4 × 10-3 4.5 × 10-3 2.0 × 10-3 7.1 × 10-3

1.3 × 104 1.0 × 104 3.9 × 103 2.2 × 105 1.1 × 105 1.5 × 104 1.5 × 105 2.8 × 104 3.7 × 105

8.0 × 10-5 1.0 × 10-4 2.6 × 10-4 4.5 × 10-6 8.9 × 10-6 6.7 × 10-5 6.6 × 10-6 3.5 × 10-5 2.7 × 10-6

10-3

a x and y (eq 12) and hole-hopping rate constants for electron transfer in DNA at 298 K (eq 5). b Difference in free energy between the guanine molecule (donor) and the different bridge molecules (adenine, cytosine, and thymine).

TABLE 10: β Values for Tunneling between Two Guanine Molecules through Different Bridging DNA Bases G-(A)x-G+ G-(C)x-G+ G-(T)x-G+ G-(AT)-G+ G-(CT)-G+

a

βa

βb

1.30 1.18 0.87 0.84 1.06

1.68 0.86 0.68 0.72 0.94

DFT B3LYP/6-311G(d). b HF/6-31G(d).

The Mott-Hubbard model63 concerns the localization problem, but the coupling is contrasted with the interelectronic repulsion U. In many cases, the local value of U is on the order of 10 eV, while the reorganization energy of the site is small. These parameters have, in fact, nothing to do with each other. Furthermore, the expectation value of Σ1/rij is not very different for different types of systems. For these reasons, use of the Mott-Hubbard model in connection with traditional quantum chemical models, which of course include U automatically, is not warranted. Even parametrized, quantum-chemical models such as the Hu¨ckel or ligand-field models include electronic repulsion implicitly in the parameters. The method used here is a simulation of accurate methods and therefore also includes the interelectronic repulsion. The model is, in fact, an extension to the infinite case of an accepted model by Hush that has been highly successful in determining localization properties in finite systems.65 The Mott-Hubbard model, on the other hand, refers back to valence bond models, but the important overlap integrals have been omitted. Obviously, the latter model cannot account for the great difference in behavior that occurs, for example, in DNA. The calculated results show that Cx and Tx stacks are, in fact, delocalized. Ladik has emphasized the possibility that a DNA strand could act as a wire. In his model, he used the bandwidth, which is closely related to our coupling, and a diffusion coefficient.3,4,66 It is unclear which property corresponds to reorganization energy in the Ladik models. Clearly, a very narrow band leads to a small velocity of purely electronic wave packets formed from the orbitals of the band. However, this applies, at best, to a completely delocalized metal with singleatom sites. In a system like DNA, coupling (or bandwidth) has

to be contrasted with reorganization energy. The localization model presented here shows a sharp metal-insulator transition in a periodic crystal. The conclusion that hopping is relayed between guanine units is not surprising and leads to fast electron transport through DNA if the guanine units are not too distant. The total distance dependence after many leaps in sequence is not exponential, as is clearly shown by Ratner et al.27-28 Theory is unusually consistent and appears to be in unanimous accord among scientists. There is probably no need for any new paradigm.67 Localization of spins is commonly found by EPR.7 As we have seen, it is not inconsistent with fast ET in DNA. The disagreement may be whether this stepwise hopping rate is sufficient to explain very fast electron transport and conductivity. Other interesting questions may be whether the current is proportional to the voltage or whether there are coulomb blockades. The conclusions of the present paper are not completely quantitative, because of remaining problems in the calculation of the reorganization energy. It is clear that a crucial point is the speed and magnitude of reorganization in the grooves. Some other theoretical and experimental factors have not been considered in detail in the present paper. If proton transfer contributes to trapping of the electron, as recently discussed by Debije et al.7 and Heller et al.,68 there would be important modifications in the present model. Furthermore, we have not yet considered in any detail the many possible mechanisms by which the electron or hole is created. It also remains to treat the case of negative charge carriers (electron transfer as opposed to hole transfer). Acknowledgment. We are grateful for support from NFR, the Swedish Natural Science Research Council. We have benefited from discussions with Prof. B. Norde´n, Dr. Per Lincoln and Dr. Tommi Ratilainen. References and Notes (1) Laki, K. Stud. Inst. Med. Chem. UniVersity Szeged 1942, 2, 43. (2) Eley, D. D.; Spivey, D. I. Faraday Soc. Trans. 1962, 58, 411415. (3) Ladik, J. Acta Phys. Acad. Sci. Hung. 1960, 11, 239; 1963, 15, 287; Nature 1964, 202, 1208. Hoffman, T. A.; Ladik, J. AdV. Chem. Phys. 1964, 7, 87-158.

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