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Calculation of hopping conductivity in aperiodic nucleotide base stacks
PERGAMON
Solid State Communications 112 (1999) 139–144 www.elsevier.com/locate/ssc
Calculation of hopping conductivity in aperiodic nucleotide base stacks Y.-J. Ye a,b,*, R.-S. Chen a,b, A. Martinez a, P. Otto a, J. Ladik a a
Chair for Theoretical Chemistry and Laboratory of the National Foundation for Cancer Research, University Erlangen–Nu¨rnberg, Egerlandstr. 3, D-91058 Erlangen, Germany b Department of Protein Engineering, Institute of Biophysics, Chinese Academy of Sciences, 15 Datun Road, Chawang District, Beijing 100101, People’s Republic of China Received 20 April 1999; accepted 30 June 1999 by A. Zawadowski
Abstract The electronic density of states (DOS) of aperiodic nucleotide base and base pair stacks were calculated previously by the negative factor counting (NFC) procedure. Applying the inverse iteration method, the localized electronic wave functions of the first 100 filled levels were determined. As a third step the primary hopping frequencies between the localized electronic wave functions (at different sites) were computed assuming interactions via acoustic phonons. Finally using the hopping frequencies as input of a random walk theory of Lax and coworkers the complex, frequency-dependent hopping conductivities s (v ) were determined. This procedure was performed for two different 100 base or base pair long sequences in the stack and for a 200 units long segment for a single stack. The influence of the application of a better basis set and that of correlation effects were also investigated. The results show an increase of s (v ) as compared to the ones of different protein chains and at v 1010 s21 they are close to 1 V 21 cm 21 in the case of 100 base pairs in the stacks. Further, the application of the better (double z ) basis and of correlation corrections of the level schemes increase s (v ). One can conclude that in aperiodic DNA there is hopping hole conduction (if its interaction with nucleoproteins generates holes via charge transfer) and its value is about 1 V 21 cm 21 at high frequencies. This result agrees well with the available experimental data. q 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: A. Disordered systems; A. Organic crystals; A. Polymers, elastomers, and plastics; D. Electronic transport
1. Introduction The hopping conductivity in aperiodic nucleotide base stacks was calculated in four steps: (1) The electronic density of states (DOS) of the stacks was calculated with ab initio negative factor counting (NFC) method in its matrix block form [1–5]. (2) Using the inverse iteration method [6] the Anderson localization of the different energy levels of these aperiodic systems was computed. It was found that the wave functions belonging to the different n p * Corresponding author. Department of Protein Engineering, Institute of Biophysics, Chinese Academy of Sciences, 15 Datun Road, Chawang District, Beijing 100101, People’s Republic of China. Tel.: 1 86-10-2020077; fax: 1 86-10-2027837.
(highest filled) energy levels of the bases were localized on one or two bases (base pairs) in the stack [7]. According to previous calculations on periodic polynucleotides (base stacks with the sugar-phosphate backbone of DNA) [8– 10] have shown that the highest filled (n p) and lowest unfilled (np 1 1) bands come from the base stacks and only the np 2 1 and np 1 2 bands, respectively, bands originate from the backbone. At the same time it was found that the band structure of the polynucleotides corresponds in a very good approximation to the superposition of the bands of the stacks and of the backbone, respectively [8–10]. Therefore, the band structure of the periodic stacks and the level distributions of the aperiodic ones describe the corresponding strands of DNA. Further, the water structure around DNA [11] hardly influences the band structure of a
0038-1098/99/$ - see front matter q 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00312-9
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periodic cytosine (C) stack [12], and the alternating positive and negative charges on the different constituents (net negative charges on the bases, positive ones on the sugar groups [8–10,13], again negative ones on the phosphates and positive K 1 counterions) screen each other out. Therefore, the results obtained for the aperiodic base stacks (DOS histograms) have to be quite similar to those which one would have obtained for native DNA in its B conformation [14]. As step (3) the hopping frequencies (primary jump rates) were calculated using the simple expression given in the book of Mott and Davis [15]. (4) As the last step using the random walk theory of Lax and coworkers [16–19] the complex frequency-dependent hopping conductivities of two different sequences of base and base pair stacks were calculated.
2. Some computational details As was shown previously [20], if one performs the negative factor counting (NFC) procedure on Fock and overlap matrices constructed from the corresponding matrices of overlapping dimers the results for the DOS-s are in a very good approximation, the same as that when one would have constructed directly the full I and S matrices and would have applied for these matrices the NFC method. In applying this procedure we have performed the ab initio Hartree–Fock (HF) calculations mostly using Clementi’s minimal basis, but in some cases also his double z one [21]. For a single strand with a sequence which is a part of a human oncogene [22], the energy levels of the overlapping dimers were also corrected for correlation. For this purpose the inverse Dyson equation was applied in the diagonal approximation for the self-energy [23,24]
at every tenth size a positively charged arginine residue. Besides one has in nucleohistone protonated lysin. We have assumed 100 holes in 100 units long single base stacks and 200 holes in 100 units long base pair stacks. Of course, these holes cannot come only from that segment of nucleohistone, which corresponds to 100 units. Charge transfer occurs also above and below the chosen 100 units, which means that one has 100 or 200 holes, respectively, on the 100 or 200 highest originally filled levels. For the hopping frequencies we have applied the simple expression given by Mott and Davis [15] X i j n n 0 DEij cr cs kxr uxs l exp 2 h n;i! n 0 ;j nphonon : kB T r[n;s[n 2 Here n phonon is the acoustic phonon frequency for which we have taken following Ref. [14] 10 12 s 21, DEij is the energy difference of the level i on the unit n and of the jth one on n 0 , kB the Boltzmann constant and T is the absolute temperature. The level differences DEij were determined by studying the grid points of the NFC procedure. Eq. (2) is a rather crude approximation for the hopping frequencies. It would be more correct to express them with electron–phonon interaction matrix element as X el h n;i! n 0 ;j kcn;i uH^ el–ph uwac ph l t
X r[n;s[n 0
0
j n n 2 ac ^ el c i r cs kxr uxs l kwph uH el–ph ucn 0 ;j l
8 > < e 2 DEij kB T > : 1
if DEij . 0
:
if DEij # 0
1
3
The diagonal elements of the self-energy were approximated by many-body perturbation theory in second order using the Moeller–Plesset partitioning [25] and taking into account also relaxation effects [26]. The non-linear inverse Dyson equation was solved in an iterative way [27,28] (see also Refs. [29,30]). The DOS curves of single- and double-stranded stacks were obtained for 100 bases or base pairs, respectively, and in one case for 200 units long segments using the oncogene sequence and a random sequence generated by a Monte Carlo program. The obtained DOS curves will be published in detail elsewhere [30]. Using the inverse iteration method the wave functions of the 100 filled levels of the base stacks and 200 filled levels of the base pair stacks were determined. They were all localized on one or two base (pairs). We have looked after the filled levels because there is a strong charge transfer in a nucleoprotein. DNA with its negatively charged phosphate groups acts as an electron donor and the most frequent protein in a nucleoprotein, nucleohistone contains at least
Here H^ el–ph is the electron–phonon interaction operator (in the stacks the phonons describe the vibrations of the moleel cules with respect to each other), Cel n;i and Cn 0 ;j , respectively, describe the one-electron wave functions at site n belonging to the level i and at site n 0 of the level j, respectively. To find out the vibrational wave functions wac ph of the normal modes belonging to these vibrations, as well as to ^ ac calculate the matrix elements kCel n;i uHuwph l requires a separate detailed calculation. We plan to do this in the future. One should mention that in principle in Eqs. (2) and (3) one should sum over all phonon frequencies taking into account also all longitudinal phonons and other vibrations. However, in the case of hopping across a base stack the vibration of the bases relative to each other is obviously the dominant one. This argument is supported by the fact that in our previous hopping conductivity calculations on proteins [30–32] the obtained log s v–log v curves (s is the hopping conductivity and v the frequency) were in an order of magnitude agreement with those obtained for
vi vi;i 1HF i;i 1 S vi i;i 1HF i 1 S vi i;i :
Y.-J. Ye et al. / Solid State Communications 112 (1999) 139–144
141
Fig. 1. Logarithm of the absolute values of the complex hopping conductivities (us (v )u) as a function of the frequency: (a) (—) double strand (D) with sequence 1 (sequence 1) in the valence (v.) bands region taking into account 200 levels (Dseq1v200); (b) (_ · _ · _) (Dseq1v100).
different 3D inorganic chalcogenides [15] (despite that in proteins the hopping occurs first of all via phonons which are corresponding to the vibrations of the side chains relative to each other). Further, in proteins it is much more probable than in base stacks that longitudinal phonons play also a role in the hopping. Using Einstein’s relation, we can write
s v
nv e2 D v: kB T
4
Here nv is the number density of free charge carriers: for instance in the case of a single stranded chain with 100 bases 3 [14]. The denominator is the volume of nv 100=32121 A the single chain with 100 bases in the case of the oncogene sequence, D(v ) the frequency-dependent (v ) diffusion constant of the electrons. Using the random walk theory of Lax and coworkers [16– 19], again the same formulae as those in the case of proteins [30–32] were applied for the hopping conductivities (for the details see Refs. [30,32]). There are more refined methods for the calculation of hopping conductivities (see for instance Ref. [33]). These theories, however, usually require special restrictions on the
disordered polymers which cannot be fulfilled in the case of ab initio calculations of complicated biopolymers (see also Ref. [32]).
3. Results and discussion In the calculations, we have used the 3D structures of the different chains [14]. Only those hoppings were included, which happen between first and second neighbor units. The volume of the double stranded (base pair) segment (with 100 ˚ 3. units) was calculated to be 64242 A In Fig. 1 the absolute value us (v )u as a function of v is presented for a double stranded DNA in a double log scale taking into account 200 levels or 100 levels, respectively, in the valence bands region using the oncogene sequence (sequence 1). In Fig. 2 results are shown for the same region of a single stack with 200 units and 200 and 100 levels, respectively, using sequence 1. In Fig. 3 us (v )u is given as a function of v for a single base stack with sequence 1 applying two different basis sets and in the case of double z basis (using the formalism to correct the energy levels of the dimers for correlation) also by taking into account
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Fig. 2. log us (v )u of 200 units (L) for the valence (v.) bands region of single (S) stranded DNA with sequence 1 of 200 units taking into account: (a) (—) 200 levels (LSseq1v200); (b) (- - -) 100 levels (LSseq1v100); (c) (- · - · -) for comparison of the results for a single chain with 100 units and 100 levels in the valence band region (Sseq1v100) are also shown.
Fig. 3. log us (v )u for a single (s) stack with sequence 1 and 100 levels in the valence (v.) bands region (Sseq1v100) with a minimal basis (- - -), with a double z basis (- · - · -) and with double z basis and correlation (—).
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correlation. One can see that going from the minimal basis to the double z one and introducing correlation corrections us (v )u increases at its saturation value by more than three orders of magnitude. Looking at the us vu 2 v curves one sees from Fig. 1 that the base pair stack with 200 filled levels has the largest value (us vu < 5 × 1021 V21 cm21 ) at v 5 × 1010 s21 (for sequence 1). One can wonder what is the biological relevance of us (v )u at the large v v $ 1010 s21 : One should point out that the time scale of the elementary steps of many biochemical reactions is in the picosecond range. Therefore, s (v ) with v between 10 10 and 10 12 s 21 is of biological interest. us (v )u has values of about 10 25 V 21 cm 21 at v 5 × 1010 s21 in the single stacks with 100 units (using both sequences) and 100 levels. Namely, in this case having 100 levels originating from the HOMO-s of the single bases, the Boltzmann factors become again smaller due to the larger DEij level spacings. us (v )u at the measured largest v [34] < 5 × 107 s21 have similar values (between 5 × 10 26 and 10 25 V 21 cm 21) as the single base stack in the valence bands regions (100 levels). On the other hand the base pair stack in the valence bands regions (200 levels taken into account again) has a much larger theoretical us (v )u value of <10 24 V 21 cm 21 at the mentioned v . At lower v (,5 × 103 s21 ) our us (v )u values are substantially larger 1027 2 5 × 1027 V21 cm21 than the experimentally found value of ,5 × 10210 V21 cm21 : This measurement, however, is 36 years old and one cannot find in the literature any other s (v ) values for DNA. Further, the experimental s values, especially at lower v -s, are not reliable. Namely the samples, because of the difficulties in their purification and characterization, are not very well defined. In the case of 200 units of a single base stack (Fig. 2) with 200 levels in the v. bands region us (v )u is about 10 22 V 21 cm 21 again at v 5 × 1010 s21 (after having a saddle region between v 105 and 10 9 s 21 with a us (v )u value of 5 × 1027 V21 cm21 ). If we take in the 200 units case only 100 levels, us (v )u decreases also to the above given value of 5 × 1027 V21 cm21 . Finally we have calculated us (v )u also with Clementi’s double z basis [21] for a single stack with sequence 1 and 100 levels in the v. bands region. In this case in the base dimer calculations we have corrected also the energy levels for correlation using the inverse Dyson equation with an MP2 self-energy. As one can see from Fig. 3 for the minimal basis the saturation value of us (v )u <7 × 1027 V21 cm21 occurs already at v 105 s21 . For the double z basis the saturation value of us (v )u is about an order of magnitude larger <6 × 1026 V21 cm21 and occurs only at s 1012 s: Finally the double z curve with correlation corrections has a saturation value of us vu < 2 × 1023 V21 cm21 : Thus the better basis increases us (v )u by one order of magnitude and the correlation effects by an additional 2.5 orders of magnitude. These results can be again easily inter-
143
preted if we take into account that both the better basis and the correlation effects [35] decrease the level spacings DEij increasing in this way the Boltzmann factors and with them the hopping frequencies.
4. Conclusions One would expect that if one would perform the calculation of a base pair stack of 200 units and would take into account all the 400 levels which come from the HOMO-s of the single bases, the hopping conductivity would reach a saturation value possibly somewhere above 1 V 21 cm 21. Such a value most probably will be obtained also when for 100 base pairs with 200 units correlation corrections could be introduced. However, in reality in DNA after 100 units most probably there is an impurity or a conformational disturbance, which would make the results for 200 or more units unrealistic. Of course it is necessary to introduce besides a better basis (which already has been done) correlation corrections for the base pair stacks, to reach a better-established conclusion about the hopping conductivity of DNA. Nevertheless, from the results already obtained we can conclude that us (v )u is obviously larger than in proteins and its value is around 1 V 21 cm 21. Therefore, most probably it plays an important role in the charge, energy and signal transport along a DNA double helix with obvious biological consequences. Finally it should be mentioned that recently Barton and coworkers [36–38] have shown that there is without doubt a hole conduction across the DNA base pairs for a distance of ˚ (though the conduction may persist for substantially 37 A ˚ [39]. Of course, the problem of larger distances of ,200 A charge transport across a nucleotide base stack is a complicated one. Some authors think that above 150 K the main mechanism is hopping which could be interrupted only by scavengers [40] (like 5-bromo-6-hydroxy-5,6-dihydrothimine (TOHBr), 5-bromocytosine (CBr), etc.). On the other hand we have found for proteins [32] that conformational changes can decrease very strongly the hopping distance. Other authors [41] have concluded using time and frequency resolved fluorescence spectrum measurements of electron acceptors that for smaller distances coherent charge transport (tunneling) and for larger distances incoherent hopping transport is dominant. This conduction can be interrupted by kinks or strand breaks [36–38]. Our previous results on hopping conductivity in proteins have shown that this is influenced first of all by the conformations of the proteins and not by their sequences [32]. Obviously, there is a similar situation in aperiodic DNA. As mentioned above there are still no (high) frequency conductivity measurements on DNA. Therefore, we could not find any experimental data with which we could directly compare our theoretical results.
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In our opinion even in the electronic ground state of DNA besides tunneling and hopping simultaneously other mechanisms, like sequentially mixed tunneling and hopping, charge transport along the sugar phosphate chain with surrounding water structure (which would be mostly ionic or protonic though it can be coupled to the motion of the electrons along the double strand) etc. should be considered. Therefore, further experimental and theoretical studies are necessary to understand at least the dominant factors of the charge transport of DNA. Acknowledgements We would like to express our gratitude to Prof J.K. Barton for putting at our disposal her results before publication. Further, we are indebted to Mrs Y. Jiang and to Prof A.K. Bakhshi and to Dr F. Bogar for their help and for the fruitful discussions. We are grateful to the “National Natural Scientific Foundation of China”, the Alexander-von-Humboldt Foundation, the German Academic Exchange (DAAD) and the “Deutsche Forschungsgemeinschaft” for the financial support which made it possible to perform this joint research in Erlangen. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
P. Dean, J.L. Martin, Proc. R. Soc. Ser. A 259 (1960) 409. P. Dean, Rev. Mod. Phys. 44 (1972) 127. M. Seel, Chem. Phys. 43 (1979) 103. R.S. Day, F. Martino, Chem. Phys. Lett. 84 (1981) 86. Y.-J. Ye, J. Math. Chem. 14 (1993) 121. J.H. Ilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. Y.-J. Ye, R.-S. Chen, A. Martinez, P. Otto, J. Ladik, Physica B, submitted for publication. J. Ladik, S. Suhai, Phys. Lett. A 77 (1980) 25. P. Otto, E. Clementi, J. Ladik, J. Chem. Phys. 78 (1983) 4547. E. Clementi, G. Corongiu, Int. J. Quant. Chem. QBS9 (1982) 213. G. Corongiu, E. Clementi, Biopolymers 20 (1981) 551.
[12] P. Otto, J. Ladik, G. Corongiu, S. Suhai, W. Fo¨rner, J. Chem. Phys. 77 (1981) 5026. [13] Y.-J. Ye, unpublished results. [14] R.A. Dickerson, H.R. Drew, B.N. Conner, R.M. Wing, A.V. Fratini, M.L. Kopka, Science 216 (1982) 475. [15] N.F. Mott, E.A. Davis, Electronic Processes in Non-Crystalline Materials, Clarendon Press, Oxford, 1971, p. 215 and Fig. 7.15. [16] T. Odagaki, M. Lax, Phys. Rev. B 26 (1982) 6480. [17] H. Cher, M. Lax, Phys. Rev. B 7 (1973) 4491. [18] T. Odagaki, M. Lax, Phys. Rev. B 24 (1981) 5284. [19] T. Odagaki, M. Lax, R.S. Day, Phys. Rev. B 30 (1984) 6911. [20] B. Gazdy, M. Seel, J. Ladik, Chem. Phys. 86 (1984) 41. [21] L. Gianolo, R. Pavoni, E. Clementi, Gazz. Chim. Ital. 180 (1978) 108. [22] G.I. Bell, R. Piclet, J. Writter, Nucl. Acid Res. 8 (1980) 4091. [23] L.S. Cederbaum, W. Domcke, Adv. Chem. Phys. 36 (1977) 205. [24] C.M. Liegener, Chem. Phys. 133 (1989) 173. [25] C. Moeller, M.S. Plesset, Phys. Rev. 46 (1934) 618. [26] N.A. Ostlund, A. Szabo, Modern Quantum Chemistry, Macmillan, New York, 1982, p. 398. [27] I. Palmer, J. Ladik, J. Comp. Chem. 15 (1994) 814. [28] F. Boga`r, J. Ladik, to be published. [29] J. Ladik, Phys. Rep. (1999) in press. [30] Y.-J. Ye, J. Ladik, Phys. Rev. B 48 (1993) 5120. [31] Y.-J. Ye, J. Ladik, Phys. Rev. B 51 (1995) 13091. [32] Y.-J. Ye, J. Ladik, Int. J. Quant. Chem. 52 (1994) 491. [33] J.W. Haus, K.W. Kehr, Phys. Rep. 150 (1987) 263. [34] C.T. O’Konski, Rev. Mod. Phys. 35 (1963) 721. [35] J. Ladik, Quantum Theory of Polymers as Solids, Plenum Press, New York, 1988, chap. V. [36] D.B. Hall, R.E. Holmkin, J.K. Barton, Nature 382 (1996) 731. [37] D.B. Hall, J.K. Barton, J. Am. Chem. Soc. 119 (1997) 5045. [38] M.R. Arkin, E.D.A. Stemp, S.C. Pulver, J.K. Barton, Chem. Biol. 4 (1997) 369. [39] J.K. Barton, Lecture at the 25th Anniversary Conference of the National Foundation of Cancer Research, Washington, 1998. [40] Y. Razskazovskii, S.G. Swarts, J.M. Falcome, C. Taylor, M.D. Sevilla, J. Phys. Chem. B 101 (1997) 1460. [41] A. Okada, V. Chernyak, S. Mukamel, Phys. Chem. A 102 (1998) 1241.
Solid State Communications 112 (1999) 139–144 www.elsevier.com/locate/ssc
Calculation of hopping conductivity in aperiodic nucleotide base stacks Y.-J. Ye a,b,*, R.-S. Chen a,b, A. Martinez a, P. Otto a, J. Ladik a a
Chair for Theoretical Chemistry and Laboratory of the National Foundation for Cancer Research, University Erlangen–Nu¨rnberg, Egerlandstr. 3, D-91058 Erlangen, Germany b Department of Protein Engineering, Institute of Biophysics, Chinese Academy of Sciences, 15 Datun Road, Chawang District, Beijing 100101, People’s Republic of China Received 20 April 1999; accepted 30 June 1999 by A. Zawadowski
Abstract The electronic density of states (DOS) of aperiodic nucleotide base and base pair stacks were calculated previously by the negative factor counting (NFC) procedure. Applying the inverse iteration method, the localized electronic wave functions of the first 100 filled levels were determined. As a third step the primary hopping frequencies between the localized electronic wave functions (at different sites) were computed assuming interactions via acoustic phonons. Finally using the hopping frequencies as input of a random walk theory of Lax and coworkers the complex, frequency-dependent hopping conductivities s (v ) were determined. This procedure was performed for two different 100 base or base pair long sequences in the stack and for a 200 units long segment for a single stack. The influence of the application of a better basis set and that of correlation effects were also investigated. The results show an increase of s (v ) as compared to the ones of different protein chains and at v 1010 s21 they are close to 1 V 21 cm 21 in the case of 100 base pairs in the stacks. Further, the application of the better (double z ) basis and of correlation corrections of the level schemes increase s (v ). One can conclude that in aperiodic DNA there is hopping hole conduction (if its interaction with nucleoproteins generates holes via charge transfer) and its value is about 1 V 21 cm 21 at high frequencies. This result agrees well with the available experimental data. q 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: A. Disordered systems; A. Organic crystals; A. Polymers, elastomers, and plastics; D. Electronic transport
1. Introduction The hopping conductivity in aperiodic nucleotide base stacks was calculated in four steps: (1) The electronic density of states (DOS) of the stacks was calculated with ab initio negative factor counting (NFC) method in its matrix block form [1–5]. (2) Using the inverse iteration method [6] the Anderson localization of the different energy levels of these aperiodic systems was computed. It was found that the wave functions belonging to the different n p * Corresponding author. Department of Protein Engineering, Institute of Biophysics, Chinese Academy of Sciences, 15 Datun Road, Chawang District, Beijing 100101, People’s Republic of China. Tel.: 1 86-10-2020077; fax: 1 86-10-2027837.
(highest filled) energy levels of the bases were localized on one or two bases (base pairs) in the stack [7]. According to previous calculations on periodic polynucleotides (base stacks with the sugar-phosphate backbone of DNA) [8– 10] have shown that the highest filled (n p) and lowest unfilled (np 1 1) bands come from the base stacks and only the np 2 1 and np 1 2 bands, respectively, bands originate from the backbone. At the same time it was found that the band structure of the polynucleotides corresponds in a very good approximation to the superposition of the bands of the stacks and of the backbone, respectively [8–10]. Therefore, the band structure of the periodic stacks and the level distributions of the aperiodic ones describe the corresponding strands of DNA. Further, the water structure around DNA [11] hardly influences the band structure of a
0038-1098/99/$ - see front matter q 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00312-9
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periodic cytosine (C) stack [12], and the alternating positive and negative charges on the different constituents (net negative charges on the bases, positive ones on the sugar groups [8–10,13], again negative ones on the phosphates and positive K 1 counterions) screen each other out. Therefore, the results obtained for the aperiodic base stacks (DOS histograms) have to be quite similar to those which one would have obtained for native DNA in its B conformation [14]. As step (3) the hopping frequencies (primary jump rates) were calculated using the simple expression given in the book of Mott and Davis [15]. (4) As the last step using the random walk theory of Lax and coworkers [16–19] the complex frequency-dependent hopping conductivities of two different sequences of base and base pair stacks were calculated.
2. Some computational details As was shown previously [20], if one performs the negative factor counting (NFC) procedure on Fock and overlap matrices constructed from the corresponding matrices of overlapping dimers the results for the DOS-s are in a very good approximation, the same as that when one would have constructed directly the full I and S matrices and would have applied for these matrices the NFC method. In applying this procedure we have performed the ab initio Hartree–Fock (HF) calculations mostly using Clementi’s minimal basis, but in some cases also his double z one [21]. For a single strand with a sequence which is a part of a human oncogene [22], the energy levels of the overlapping dimers were also corrected for correlation. For this purpose the inverse Dyson equation was applied in the diagonal approximation for the self-energy [23,24]
at every tenth size a positively charged arginine residue. Besides one has in nucleohistone protonated lysin. We have assumed 100 holes in 100 units long single base stacks and 200 holes in 100 units long base pair stacks. Of course, these holes cannot come only from that segment of nucleohistone, which corresponds to 100 units. Charge transfer occurs also above and below the chosen 100 units, which means that one has 100 or 200 holes, respectively, on the 100 or 200 highest originally filled levels. For the hopping frequencies we have applied the simple expression given by Mott and Davis [15] X i j n n 0 DEij cr cs kxr uxs l exp 2 h n;i! n 0 ;j nphonon : kB T r[n;s[n 2 Here n phonon is the acoustic phonon frequency for which we have taken following Ref. [14] 10 12 s 21, DEij is the energy difference of the level i on the unit n and of the jth one on n 0 , kB the Boltzmann constant and T is the absolute temperature. The level differences DEij were determined by studying the grid points of the NFC procedure. Eq. (2) is a rather crude approximation for the hopping frequencies. It would be more correct to express them with electron–phonon interaction matrix element as X el h n;i! n 0 ;j kcn;i uH^ el–ph uwac ph l t
X r[n;s[n 0
0
j n n 2 ac ^ el c i r cs kxr uxs l kwph uH el–ph ucn 0 ;j l
8 > < e 2 DEij kB T > : 1
if DEij . 0
:
if DEij # 0
1
3
The diagonal elements of the self-energy were approximated by many-body perturbation theory in second order using the Moeller–Plesset partitioning [25] and taking into account also relaxation effects [26]. The non-linear inverse Dyson equation was solved in an iterative way [27,28] (see also Refs. [29,30]). The DOS curves of single- and double-stranded stacks were obtained for 100 bases or base pairs, respectively, and in one case for 200 units long segments using the oncogene sequence and a random sequence generated by a Monte Carlo program. The obtained DOS curves will be published in detail elsewhere [30]. Using the inverse iteration method the wave functions of the 100 filled levels of the base stacks and 200 filled levels of the base pair stacks were determined. They were all localized on one or two base (pairs). We have looked after the filled levels because there is a strong charge transfer in a nucleoprotein. DNA with its negatively charged phosphate groups acts as an electron donor and the most frequent protein in a nucleoprotein, nucleohistone contains at least
Here H^ el–ph is the electron–phonon interaction operator (in the stacks the phonons describe the vibrations of the moleel cules with respect to each other), Cel n;i and Cn 0 ;j , respectively, describe the one-electron wave functions at site n belonging to the level i and at site n 0 of the level j, respectively. To find out the vibrational wave functions wac ph of the normal modes belonging to these vibrations, as well as to ^ ac calculate the matrix elements kCel n;i uHuwph l requires a separate detailed calculation. We plan to do this in the future. One should mention that in principle in Eqs. (2) and (3) one should sum over all phonon frequencies taking into account also all longitudinal phonons and other vibrations. However, in the case of hopping across a base stack the vibration of the bases relative to each other is obviously the dominant one. This argument is supported by the fact that in our previous hopping conductivity calculations on proteins [30–32] the obtained log s v–log v curves (s is the hopping conductivity and v the frequency) were in an order of magnitude agreement with those obtained for
vi vi;i 1HF i;i 1 S vi i;i 1HF i 1 S vi i;i :
Y.-J. Ye et al. / Solid State Communications 112 (1999) 139–144
141
Fig. 1. Logarithm of the absolute values of the complex hopping conductivities (us (v )u) as a function of the frequency: (a) (—) double strand (D) with sequence 1 (sequence 1) in the valence (v.) bands region taking into account 200 levels (Dseq1v200); (b) (_ · _ · _) (Dseq1v100).
different 3D inorganic chalcogenides [15] (despite that in proteins the hopping occurs first of all via phonons which are corresponding to the vibrations of the side chains relative to each other). Further, in proteins it is much more probable than in base stacks that longitudinal phonons play also a role in the hopping. Using Einstein’s relation, we can write
s v
nv e2 D v: kB T
4
Here nv is the number density of free charge carriers: for instance in the case of a single stranded chain with 100 bases 3 [14]. The denominator is the volume of nv 100=32121 A the single chain with 100 bases in the case of the oncogene sequence, D(v ) the frequency-dependent (v ) diffusion constant of the electrons. Using the random walk theory of Lax and coworkers [16– 19], again the same formulae as those in the case of proteins [30–32] were applied for the hopping conductivities (for the details see Refs. [30,32]). There are more refined methods for the calculation of hopping conductivities (see for instance Ref. [33]). These theories, however, usually require special restrictions on the
disordered polymers which cannot be fulfilled in the case of ab initio calculations of complicated biopolymers (see also Ref. [32]).
3. Results and discussion In the calculations, we have used the 3D structures of the different chains [14]. Only those hoppings were included, which happen between first and second neighbor units. The volume of the double stranded (base pair) segment (with 100 ˚ 3. units) was calculated to be 64242 A In Fig. 1 the absolute value us (v )u as a function of v is presented for a double stranded DNA in a double log scale taking into account 200 levels or 100 levels, respectively, in the valence bands region using the oncogene sequence (sequence 1). In Fig. 2 results are shown for the same region of a single stack with 200 units and 200 and 100 levels, respectively, using sequence 1. In Fig. 3 us (v )u is given as a function of v for a single base stack with sequence 1 applying two different basis sets and in the case of double z basis (using the formalism to correct the energy levels of the dimers for correlation) also by taking into account
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Fig. 2. log us (v )u of 200 units (L) for the valence (v.) bands region of single (S) stranded DNA with sequence 1 of 200 units taking into account: (a) (—) 200 levels (LSseq1v200); (b) (- - -) 100 levels (LSseq1v100); (c) (- · - · -) for comparison of the results for a single chain with 100 units and 100 levels in the valence band region (Sseq1v100) are also shown.
Fig. 3. log us (v )u for a single (s) stack with sequence 1 and 100 levels in the valence (v.) bands region (Sseq1v100) with a minimal basis (- - -), with a double z basis (- · - · -) and with double z basis and correlation (—).
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correlation. One can see that going from the minimal basis to the double z one and introducing correlation corrections us (v )u increases at its saturation value by more than three orders of magnitude. Looking at the us vu 2 v curves one sees from Fig. 1 that the base pair stack with 200 filled levels has the largest value (us vu < 5 × 1021 V21 cm21 ) at v 5 × 1010 s21 (for sequence 1). One can wonder what is the biological relevance of us (v )u at the large v v $ 1010 s21 : One should point out that the time scale of the elementary steps of many biochemical reactions is in the picosecond range. Therefore, s (v ) with v between 10 10 and 10 12 s 21 is of biological interest. us (v )u has values of about 10 25 V 21 cm 21 at v 5 × 1010 s21 in the single stacks with 100 units (using both sequences) and 100 levels. Namely, in this case having 100 levels originating from the HOMO-s of the single bases, the Boltzmann factors become again smaller due to the larger DEij level spacings. us (v )u at the measured largest v [34] < 5 × 107 s21 have similar values (between 5 × 10 26 and 10 25 V 21 cm 21) as the single base stack in the valence bands regions (100 levels). On the other hand the base pair stack in the valence bands regions (200 levels taken into account again) has a much larger theoretical us (v )u value of <10 24 V 21 cm 21 at the mentioned v . At lower v (,5 × 103 s21 ) our us (v )u values are substantially larger 1027 2 5 × 1027 V21 cm21 than the experimentally found value of ,5 × 10210 V21 cm21 : This measurement, however, is 36 years old and one cannot find in the literature any other s (v ) values for DNA. Further, the experimental s values, especially at lower v -s, are not reliable. Namely the samples, because of the difficulties in their purification and characterization, are not very well defined. In the case of 200 units of a single base stack (Fig. 2) with 200 levels in the v. bands region us (v )u is about 10 22 V 21 cm 21 again at v 5 × 1010 s21 (after having a saddle region between v 105 and 10 9 s 21 with a us (v )u value of 5 × 1027 V21 cm21 ). If we take in the 200 units case only 100 levels, us (v )u decreases also to the above given value of 5 × 1027 V21 cm21 . Finally we have calculated us (v )u also with Clementi’s double z basis [21] for a single stack with sequence 1 and 100 levels in the v. bands region. In this case in the base dimer calculations we have corrected also the energy levels for correlation using the inverse Dyson equation with an MP2 self-energy. As one can see from Fig. 3 for the minimal basis the saturation value of us (v )u <7 × 1027 V21 cm21 occurs already at v 105 s21 . For the double z basis the saturation value of us (v )u is about an order of magnitude larger <6 × 1026 V21 cm21 and occurs only at s 1012 s: Finally the double z curve with correlation corrections has a saturation value of us vu < 2 × 1023 V21 cm21 : Thus the better basis increases us (v )u by one order of magnitude and the correlation effects by an additional 2.5 orders of magnitude. These results can be again easily inter-
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preted if we take into account that both the better basis and the correlation effects [35] decrease the level spacings DEij increasing in this way the Boltzmann factors and with them the hopping frequencies.
4. Conclusions One would expect that if one would perform the calculation of a base pair stack of 200 units and would take into account all the 400 levels which come from the HOMO-s of the single bases, the hopping conductivity would reach a saturation value possibly somewhere above 1 V 21 cm 21. Such a value most probably will be obtained also when for 100 base pairs with 200 units correlation corrections could be introduced. However, in reality in DNA after 100 units most probably there is an impurity or a conformational disturbance, which would make the results for 200 or more units unrealistic. Of course it is necessary to introduce besides a better basis (which already has been done) correlation corrections for the base pair stacks, to reach a better-established conclusion about the hopping conductivity of DNA. Nevertheless, from the results already obtained we can conclude that us (v )u is obviously larger than in proteins and its value is around 1 V 21 cm 21. Therefore, most probably it plays an important role in the charge, energy and signal transport along a DNA double helix with obvious biological consequences. Finally it should be mentioned that recently Barton and coworkers [36–38] have shown that there is without doubt a hole conduction across the DNA base pairs for a distance of ˚ (though the conduction may persist for substantially 37 A ˚ [39]. Of course, the problem of larger distances of ,200 A charge transport across a nucleotide base stack is a complicated one. Some authors think that above 150 K the main mechanism is hopping which could be interrupted only by scavengers [40] (like 5-bromo-6-hydroxy-5,6-dihydrothimine (TOHBr), 5-bromocytosine (CBr), etc.). On the other hand we have found for proteins [32] that conformational changes can decrease very strongly the hopping distance. Other authors [41] have concluded using time and frequency resolved fluorescence spectrum measurements of electron acceptors that for smaller distances coherent charge transport (tunneling) and for larger distances incoherent hopping transport is dominant. This conduction can be interrupted by kinks or strand breaks [36–38]. Our previous results on hopping conductivity in proteins have shown that this is influenced first of all by the conformations of the proteins and not by their sequences [32]. Obviously, there is a similar situation in aperiodic DNA. As mentioned above there are still no (high) frequency conductivity measurements on DNA. Therefore, we could not find any experimental data with which we could directly compare our theoretical results.
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In our opinion even in the electronic ground state of DNA besides tunneling and hopping simultaneously other mechanisms, like sequentially mixed tunneling and hopping, charge transport along the sugar phosphate chain with surrounding water structure (which would be mostly ionic or protonic though it can be coupled to the motion of the electrons along the double strand) etc. should be considered. Therefore, further experimental and theoretical studies are necessary to understand at least the dominant factors of the charge transport of DNA. Acknowledgements We would like to express our gratitude to Prof J.K. Barton for putting at our disposal her results before publication. Further, we are indebted to Mrs Y. Jiang and to Prof A.K. Bakhshi and to Dr F. Bogar for their help and for the fruitful discussions. We are grateful to the “National Natural Scientific Foundation of China”, the Alexander-von-Humboldt Foundation, the German Academic Exchange (DAAD) and the “Deutsche Forschungsgemeinschaft” for the financial support which made it possible to perform this joint research in Erlangen. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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