Effect of static electric field on the HF band structure of a guanine stack

Effect of Static Electric Field on the HFBand Structure of a Guanine Stack

A. MARTINEZ,1 F. BOGAR,2,3 P. OTTO,1 J. LADIK1,2

1Chair for Theoretical Chemistry, Friedrich-Alexander-University, Erlangen-Nurnberg,Egerlandstrasse 3, D-91058 Erlangen, Germany2Laboratory of the National Foundation for Cancer Research at the Chair for Theoretical Chemistry,D-91058 Erlangen, Germany3Protein Chemistry Research Group of Hungarian Academy of Sciences, University of Szeged,Dom ter 8, H-6720 Szeged, Hungary

Received 5 August 2002; accepted 10 October 2003Published online 17 December 2003 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.10839

ABSTRACT: Putting dipoles in different arrangements in the planes of the guaninemolecules in a stack in DNA B conformation, in a model calculation the effect of theelectric field on the ab initio Hartree–Fock band structure was studied. The calculationswere performed at the stacking distances 3.12, 3.36, and 3.60 Å, respectively. Theelectric fields of the dipoles decreased the very broad conduction band of a guaninestack, caused some shifts in the band structure, but hardly influenced the fundamentalgap. The computed effective masses at the physically interesting band edges correspondto the bandwidth changes due to the electric field. The electric field significantlyinfluences the deformation potentials at the band edges and with it the transportproperties of the guanine stack. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 99:833–840, 2004

Key words: effect of electric field on the band structure of a G stack; static electricfield generated by dipoles; effect of external dipoles on DNA; different positions ofexternal dipoles around a G stack

Introduction

D NA in the cell is surrounded by water andother molecules (including nucleohistone

and other proteins with which its interacts in a

nucloesome) that have nonnegligible dipoles. Thismeans that an electric field acts in the cell on DNA.This field is at least partially time dependent due tothe chemical reactions that take place in the neigh-borhood of DNA and the motion of the water mol-ecules. These reactions, however, are not periodic;therefore, the usual methods for the treatment ofthe effect of a periodic electric field on DNA [likecoupled Hartree–Fock (HF) crystal orbital (CO)]

Correspondence to: J. Ladik; e-mail: [email protected]

International Journal of Quantum Chemistry, Vol 99, 833–840 (2004)© 2003 Wiley Periodicals, Inc.

theory [1] cannot be applied in this case even in thecase of a periodic base stack.

On the other hand, the potential of a static elec-tric field due to dipoles is easy to introduce into theHF CO theory [2]. Therefore, to have a feeling forthe effect of the electric field caused by static di-poles we performed calculations by putting dipolesin the neighborhood of a guanine stack. It should beemphasized that this model investigation has notthe aim to treat the effect of the water structurearound DNA but only to see in a simple modelcalculation the magnitude of change of the bandstructure due to the electric field generated by staticdipoles located in the neighborhood of a guaninestack in positions that do not cause steric hin-drances.

We have put the point changes in the form ofelectric dipoles (to preserve electric neutrality) inthe plane of a stacked base (or base pairs) and rotatethem by 36° going from one stacked base to thenext. In this way one does not destroy the periodicsymmetry of a DNA helix and the usual HF crystalorbital formalism can be applied for a base stackjust putting the potentials of the dipoles in theHamiltonian.

As a model calculation we have taken a guaninestack (guanine molecules have the largest overlapin a DNA stack [3]) in the DNA B geometry andperformed the band structure calculation. Besidesthe case of no extra change, in three such positionsof a 5-Å long dipole that does not cause sterichindrances with a poly(G-C) double helix (for de-tails see below).

For the purpose of a subsequent mobility calcu-lation a deformation potential computation hasbeen performed [4]. The calculations were done atthe equilibrium stacking distance of a0 � 3.36 Å andalso at the stacking distances of a� � a0 � 0.07a0

(3.12 and 3.60 Å). At this point it should be men-tioned that this model calculation is the first step totake into account the effect of the water structuresurrounding DNA. The effects of transient chemicalreactions in the neighborhood of DNA should bethe subsequent step. In both cases even if one dealswith a periodic nucleotide base stack one has to usethe coupled HF equations (to describe the timedependence) and the theory of disordered systems(to handle the aperiodicity introduced both by thewater structure and by the nonperiodic occurrenceof chemical reactions around DNA). At first ap-proximation one could perform these calculationson finite clusters to simplify the formalism.

As one has to expect, the presence of the electricfield changes significantly the bandwidths and po-sitions of the bands at the different stacking dis-tances. This causes also significant changes in thedeformation potentials. Therefore, according to thismodel calculation even the transport properties ofDNA can change significantly due to the presenceof an electric field.

Subsequently, after a brief discussion of themethods applied we present and discuss the resultsobtained.

Methods

As mentioned above the standard HF COmethod [2] for helical systems [5] was applied, add-ing to the one-electron part of the Hamiltonian thepotential of a dipole. Because this method was de-scribed in numerous articles, we do not repeat thedescription of the formalism here (see [4–6]). Thecomputations were performed with the aid of theErlangen program package with the slight modifi-cation necessary to introduce a static electric field.

In Figure 1 the choice of the dipoles is shown.Dip.1 (radial dipole) is in the C6ON3 line of aguanine molecule. The negative point charge is lo-cated away from N3 along this line. The length ofthe dipole is 5 Å. In Dip.2 the positions of thenegative and positive elementary charges are inter-changed. Dip.3 (tangential dipole) is perpendicularto the dipoles 1 and 2, respectively, with the nega-tive point charge nearer to the N11H2 group ofguanine. This dipole intersects the C6ON3 line inthe middle point of dipoles 1 and 2, respectivelyand has also a length of 5 Å. All the dipoles are inthe plane of guanine. At the subsequent bases of thestack the dipoles are rotated by 36° together withthe DNA helix. Therefore, the inhomogeneous fieldof the dipoles has no z-components (z is the direc-tion of the main axis of the double helix). Conse-quently, they do not influence the periodicity of theguanine stack and therefore the formalism has notto be changed.

The geometry of the guanine molecules as wellthat of the helix was taken from the HyperChemdatabase [6]. The dipole of the guanine molecule isnot given here, because in the applied formalism nodipole-dipole interactions are calculated. Instead ofthis the static periodic field was introduced to theHamiltonian (see above).

The calculations were performed using the dou-ble-� basis of Gianolo and Clementi [7]. Nearest-

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neighbors’ interactions approximation was used,which in the case of the guanine stack due to therelatively large stacking distance gives a HF bandstructure of acceptable quality. (In the case of sec-ond-neighbors’ interactions the HF band structurehardly changes [8]). This is, of course, not so inevery system but due to the rather large stacking

distance in the stack (according to detailed investi-gation [9]) this statement is valid in our case.) Thenumber of k points in the half of the Brillouin zonewas 15. Besides the equilibrium stacking distance(3.36 Å) all calculations were repeated at 3.12 and3.60 Å, respectively (a0 � 0.07a0). Taking the alge-braic average of the shift of the upper limit of thevalence band and that the lower limit of the con-duction band (�W) we could calculate the deforma-tion potential

12��W1��a1�

a � ��W2��a2�a � . (1)

Here, �W1 and �W2 are the shifts of the band edgesin the case of compression and dilatation, respec-tively (�a1 � �0.24 Å, �a2 � 0.24 Å), and a0 is theequilibrium stacking distance of 3.36 Å. If one usesfor �ai (i � 1, 2) the shifts of the upper limits of thevalence bands, one obtains the deformation poten-tial of the holes; if the shifts of the lower limits ofthe conduction bands are taken into account, oneobtains the deformation potential of the electrons(see Table IV below).

For the determination of the effective masses, theexpression

1m* �

1�2

�2E�k2 (2)

was used. The second derivatives were calculatednumerically fitting first the dispersion curves of thevalence and conduction bands, respectively, by afourth-order polynome and taking both right andleft from the band edges four points. The secondderivatives were taken then at the upper edge of thevalence band and at the lower edge of the conduc-tion band, respectively (see Table III below). The

TABLE I ______________________________________HF band structure of a guanine stack without extracharges at different stacking distances (eV-s).

3.12 Å 3.36 Å 3.60 Å

CB min 3.01 2.57 2.36CB width 1.63 1.75 1.79VB max �7.26 �7.76 �8.04VB width 0.70 0.48 0.33Gap 10.27 10.34 10.40

FIGURE 1. Guanine molecule of the stack and theposition of the different dipoles. Dip.1: Radial dipole oflength of 5 Å; the negative charge (at point A) is nearerto the guanine molecule at a distance of 10 Å from N3

along the N3OC6 line; the positive charge is at point B.Dip.2: Same radial dipole as dipole 1 but the negativeand positive charges are interchanged. Dip.3: Dipole of5-Å length, perpendicular to Dips.1 and 2 and inter-sects these dipoles at their midpoint. The negativecharge (in point C) is nearer to N11 of guanine, whilethe positive one is farther away (point D). All three di-poles are in the plane of the guanine molecule.

STATIC ELECTRIC FIELD AND HF BAND STRUCTURE

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 835

computation of the m* values were done at theequilibrium stacking distance a0 � 3.36 Å.

Results and Discussion

In Table I and Figure 2 the HF band structure of aguanine stack is presented in the absence of any ad-ditional charges. The stacking distances used were (asmentioned before) 3.12, 3.36, and 3.60 Å, respectively.

As one can see the conduction band is unusuallybroad (1.75 eV at 3.36 Å stacking distance) and itswidth changes insignificantly by going from a �3.36–3.60 Å. At the moment we cannot offer anexplanation of this anomalous behavior. One has tosee whether the same thing happens by using abetter basis set or introducing correlation correc-tions of the band structure at the MP2 level. It isunusual that at compression to a � 3.12 Å this bandbecomes somewhat narrower. (The valence band-width decreases monotonically with the increase ofa.) The limits of both of the valence and conductionbands, respectively, are shifted downward with in-

creasing a [in the inset of Fig. 2 (�) represents 3.12,(0) 3.36, and (�) 3.60 Å, respectively]. This indicatessomewhat repulsive stacking energies. The mini-mal gap value at (k � 0) is insignificantly changingwith the stacking distance because the band edge ofthe CB and VB bands, respectively, are shifteddownward nearly by the same amount. (This is trueboth for compression and dilatation.)

Turning to the cases of Dip.1 and Dip.2, respec-tively (see Table II and Figs. 3 and 4), we can seethat the conduction bandwidth becomes muchsmaller (1.04 eV at 3.36 Å) for Dip.1 than in the fieldfree case. It changes in the same anomalous way asin the dipole free case (the bandwidth is decreasingwith decreasing a), while in the case of the exchangeof charges (Dip.2) the widths behave as one wouldexpect (1.90 eV at a � 3.12 Å and 1.68 eV at a � 3.60Å). The same is true in both cases for the valencebandwidths. The upper edge of the valence band isshifted by about 0.5 eV in the case of Dip.1 while atDip.2 one can observe an opposite trend (upwardshift). The lower edge of the conduction band isshifted downward by about 0.6 eV at Dip.1 and by

FIGURE 2. HF double-� band structure of a guanine stack without extra charges. The conduction (CB) and valence(VB) band dispersions are shown at 3.12 Å (�), 3.36 Å (0), and 3.60 Å (�) stacking distances (in eV-s).

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about the same amount upward at Dip.2. Theseshifts obviously depend on the fact that in Dip.1 anegative point charge and at Dip.2 a positive pointcharge is closer to the guanine molecule. As in thecase of the changes of the stacking distance in theabsence of charges, the presence of the dipoleshardly changes the fundamental gap.

Finally, in Figure 5 (Dip.3, the so-called tangentialdipole) the width of the conduction band hardly

changes with a while the width of the valence bandchanges strongly (by 0.4 eV) between a � 3.12 and3.60 Å, respectively. Interestingly, the widths of theconduction bands are also here (similarly to the casesof no dipole and Dip.2) broad (values between 1.50 eVat a � 3.12 Å and 1.60 eV at a � 3.60 Å).

The conduction band is shifted in this case byabout 0.2 eV downward and the valence band byabout 0.10 eV upward as compared to the dipole

TABLE II ______________________________________________________________________________________________HF band structure of a guanine stack with different dipoles and stacking distances (eV-s).

VB max VB width CB min CB width Gap

Dip.1 3.12Å �7.88 0.59 2.43 0.90 10.313.36Å �8.48 0.41 1.92 1.04 10.403.60Å �8.80 0.28 1.64 1.11 10.44

Dip.2 3.12Å �6.62 0.82 3.46 1.90 10.083.36Å �7.04 0.57 3.10 1.65 10.153.60Å �7.28 0.38 2.95 1.68 10.23

Dip.3 3.12Å �7.15 0.72 2.80 1.50 9.943.36Å �7.65 0.50 2.38 1.59 10.033.60Å �7.93 0.34 2.16 1.60 10.09

FIGURE 3. Same band structures adding Dip.1 (see Fig. 1 for its definition).



free case. Therefore, the gap is by about 0.3 eVsmaller than in the case of the absence of dipolesbut changes only a little with the stacking distance.

From this we can conclude that the presence of astatic electric field (depending on the orientation ofthe dipoles) can in some cases (Dip.1) significantlychange the width of the conduction band and withit the mobility. It can cause also some shifts of thevalence and conduction bands but has hardly anyeffect on the fundamental gap.

Looking at Table III of the effective masses cal-culated with the aid of Eq. (2) at the physicallyinteresting band edges one finds that in the case ofthe conduction band they correspond by and largeto the bandwidths. m* is at the conduction band1.09 m in the absence of dipoles and 1.67 m in thecase of Dip.1, which causes the largest decrease ofthe bandwidth. The substantially narrower valencebands have m* values between �1.93 and �2.69 m(here, m � the free electron mass).

Finally one can see from Table IV that the defor-mation potentials (which serve in first approximationfor the calculation mobilities of the charge carriers [8])

calculated with the aid of Eq. (1) are for the holesalways by 1 eV larger than the corresponding valuesfor the electrons. On the other hand, in the radial caseof Dip.1 the deformation potential is larger; in theother radial case (with opposite charges), Dip.2, it issmaller (for both holes and electrons) than in thedipole free case. Dip.3 (tangential dipole) hardlychanges the deformation potentials as compared tothe charge free case both for holes and electrons.

Conclusion

One can conclude that if one investigates the effectof a static inhomogenous field caused by dipoles indifferent arrangements on the band structure of aguanine stack in the DNA B geometry one finds that(1) the width of the broad conduction band can bestrongly influenced (depending on the position of thedipole) and (2) with this the effective mass and defor-mation potential can change also significantly.

In reality the electronic structure of DNA is firstinfluenced by time-dependent transient electric fields


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due to the reactions of molecules having dipoles.To handle this problem no appropriate methodsare available. At present, one could try as firstapproximation to take finite aperiodic clusters ofDNA and investigate the effect of a time-dependent field on the electronic structure of thecluster using the coupled HF method. Gatheringin this way enough experience one hopes withtime to be able to develop more serious methods

to handle the problem both for periodic and non-periodic longer DNA stacks.

Finally, it should be pointed out that the studyof the effects of electric fields on nucleotide basestacks will lead not only to the better understand-ing of their transport properties but most proba-bly will provide also further insight in the mech-anisms of its biologic functioning (according toprevious calculations the DNA–protein interac-tion depends also on their transport properties[10]).


TABLE III _____________________________________Effective electronic mass at the band edges (in freeelectron mass units) of a guanine stack withdifferent dipoles near the stack.

Dip.1 Dip.2 Dip.3 No charge

CB lower edge(k � 0) 1.67 0.87 1.23 1.09

VB upper edge(k � �/a) �2.69 �1.93 �2.11 �2.21

TABLE IV _____________________________________Deformation potential of a guanine stack withdifferent dipoles (see Fig. 1) near the stack (eV).

Dip.1 Dip.2 Dip.3 No charge

�e 5.51 3.53 4.44 4.57�h 6.44 4.60 5.46 5.50



ACKNOWLEDGMENTS

The authors express their gratitude to the Chairfor Theoretical Chemistry of FAU for providing thenecessary CPU time. J. L. thanks Prof. E. Brandasand Prof. J. Maruani for inviting him for the IVthCongress of the International Society of TheoreticalChemical Physics. The material presented in thisarticle forms a part of his lecture there.

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4. Shockley, W. Electrons and Holes in Semiconductors; VanNostrand, D.: New York, 1950; p 278.

5. Blumen, A.; Merkel, C. Phys Stat Solidi B 1977, 83, 425;Ladik, J. In: Nicolaides, C. A.; Beck, D. R., eds. Excited Statesin Quantum Chemistry; Reidel: Dordrecht, The Netherlands,1979; p 495.

6. Nucleic Acid Database, HyperChem; Hypercube, Inc.: Erlan-gen, Germany, 2003.

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Effect of static electric field on the HF band structure of a guanine stack