theoretical investigation of semiconductive properties in proteins. ii. the possibility of charge...

Theoretical Investigation of Semiconductive Properties in Proteins. 11. The Possibility of

Charge Transfer Between Proteins and Different Acceptor Molecules

SANDOR SUHAI,* Lehrstuhl fur Theoretische Chemie der Universitat Gottingen, 34 Gottingen, Federal Republic of Germany;

and THOMAS C. COLLINS and JANOS LADIK, Lehrstuhl fiir Theoretische Chemie der Friedrich-Alexander- Universitat Erlangen- Nurnberg, 852 Erlangen, and Laboratory of the National Foundation

for Cancer Research at the Chair of Theoretical Chemistry of the University Erlangen-Nurnberg, Federal Republic of Germany

Synopsis

The results of theoretical investigations are reported concerning the possibility of impurity-type charge carrier production in proteins. The energy band structures of the periodic protein model polyglycine calculated with the aid of the ab initio Hartree-Fock crystal orbital method and corrected afterwards for long-range correlation effects are compared with the empty levels of glyoxal, methyl-glyoxal, acrolein, and croton-aldehyde, suggested recently by Albert Szent-Gyorgyi as possible acceptors against proteins. The comparison with previous supermolecule calculations shows that appreciable charge transfer can be expected to glyoxal, methyl-glyoxal, and acrolein from the polypeptides, while croton-aldehyde is probably less efficient in this relation.

INTRODUCTION

Since the early suggestions of Szent-Gyorgyi' and Laki2 concerning the possible biological effect of semiconductive properties of proteins, many experimental and theoretical papers have appeared to explain these phe- nomena. It was verified from experiments that the conduction in proteins is of the electronic type.3 Theoretical investigations a t the same time suggested different pathways for electronic delocalization in these systems. Previous r-electron calculations established the existence of energy bands with a bandwidth of some tenths of an eV, corresponding to delocalization along the hydrogen bonds.P7 All-valence electronSl0 and ab initio crystal orbital calculations (unpublished results), on the other hand, showed that the inclusion of the interaction between different peptide units in the direction of the main polypeptide chain results in bandwidths of -1-2 eV.

* Present address: Lehrstuhl f i r Theoretische Chemie der Friedrich-Alexander Universitiit

+ Humboldt Professor on leave of absence from AFOSR, Washington D.C. 20332. Erlangen-Nurnberg, 852 Erlangen, Federal Republic of Germany.

Biopolymers, Vol. 18,899-908 (1979) 0 1979 John Wiley & Sons, Inc. 0006-3525/79/0018-0899$01.00

900 SUHAI, COLLINS, AND LADIK

Furthermore, the theoretical investigations of the electron and hole mo- bilities and free paths in a two-dimensional polyglycine structure have shown that the direction of the main chains is preferred for electrical conduction.’l In this case the values of the above-mentioned transport properties were comparable with the corresponding ones in conventional semiconductors.

The main problem in the semiconduction of proteins seems to be the origin of the charge carriers. The experimentally observed activation energies in proteins measured in the case of dc conduction lie in the region of 5-6 eV,3 being clearly too large to produce an appreciable number of electron-hole pairs and thus to facilitate intrinsic conduction. The ab initio energy band structures obtained recently for some periodic protein models (unpublished results) exhibit forbidden band gaps of -10 eV, somewhat larger than estimated previously on the basis of approximate calcula- t i o n ~ . ~ - ~ ~ In summary, it seems very probable that proteins are extrinsic semiconductors containing a certain amount of impurity (donor or acceptor) centers.

In light of the above considerations, special attention has to be paid to Szent-Gyorgyi’s recent proposaP that unsaturated ketones or dicarbonyls may act as electron acceptors if they interact with proteins. According to his theory, charge transfer (CT) between these molecules and the proteins makes the latter ones conductors, and the conduction properties of proteins play an important role in the regulation of cell duplication.13-15

The proper quantitative description of the CT process between proteins and different possible acceptor molecules is rather difficult. A recent ab initio molecular orbital study of the glyoxal-formamide CT system16 has shown that quantitatively reliable results can be obtained only if one treats the interacting molecules as a supermolecule with a wave function delocalized over both partners. (Similar observations were also made earlier on the basis of semiempirical cal~u1ations.l~) Since such a treatment would be a formidable task in the case of proteins, we have to look for some qualitative features characterizing the glyoxal-formamide CT system that could be transferred to the case of proteins. Since formamide is similar in its structure to the building elements of the proteins chains, while glyoxal can be taken as a representative of the acceptor molecules proposed by S~ent-Gyorgyi,’~-’~ we can hope to obtain some information about the role of impurities in proteins.

The above-cited supermolecule calculations16 have shown that appreciable CT is possible also in the ground state even if the distance in energy between the highest filled donor level and the lowest unfilled acceptor level is relatively large (many eVs in this case; see Results and Discussion). This is in contradiction with the generally accepted opinion that “strong” (ground state) CT is only possible if this energy difference is very small. The amount of the transferred charge is, of course, very sensitive to the relative position of the partner molecules. According to our experience, the optimum configuration from the point of view of CT is when the planes

SEMICONDUCTIVE PROPERTIES IN PROTEINS. I1 901

of the two molecules are parallel. This situation can easily occur also in most protein structures.

Intuitively, one has the feeling that the strength of the CT should be related to the position of the empty acceptor levels relative to the donor- filled levels. This expectation was confirmed by our comparison of the wave functions of the glyoxal-formamide complex at different intermolecular distances. We found that in configurations with stronger CT, the relative weights of the lowest, originally empty, levels of the acceptor were substantially larger in the filled levels of the interacting complex than in configurations with less transferred charge. Considering this phenomenon from a perturbation theoretical point of view, we can see that to increase the strength of CT, the above mixing has to be enlarged. This necessitates the lowering of the energy difference between the levels in question. Thus special attention has to be paid to the determination of the unfilled levels of the molecules supposed to be acceptors against a given donor.

METHOD OF CALCULATION

The energy bands of polyglycine were calculated using the ab initio LCAO Hartree-Fock crystal orbital method.18 The atomic orbitals were taken as linear combinations (contractions) of normalized spherical Gaussians of the form @i(r ) = ( 2 ~ i / n ) ~ / ~ exp(-&). The coefficients dli in the contractions X l ( r ) = Z&@i(r), as well as the exponents qi, were taken from Ref. 19 for the first row atoms and from Ref. 20 for hydrogen. The 16 Gaussians on the atoms C, N, and 0 were contracted in this way to five atomic orbitals and 3 contracted primitive Gaussians represented the hydrogen 1s orbital (minimal basis). The only necessary further input of these calculations was the geometrical structure of the polypeptide chain. The atomic coordinates corresponding to a parallel-chain pleated-sheet conformation were taken from the work of Pauling and Corey.zl

Our earlier calculationszz have shown that in biological macromolecules whose dielectric constant differs considerably from unity, the long-range electron correlation effects along the macromolecular chains may substantially influence the positions, as well as the widths, of the physically most interesting bands. Applying the electronic polaron mode123-z5 with the approximations used before,zz we calculated the crystal momentum- dependent energy shifts for the conduction and valence bands of polyglycine according to the expression

E&(k) stands for the Hartree-Fock one-electron energy in the conduction and valence bands, respectively, t;Gn*+' is the theoretically obtained first singlet-singlet exciton energy (assumed to be k independent), and finally t, is the high-frequency dielectric constant of the system.

The ab initio molecular orbital calculations of the single molecules have


been performed with the same basis set as specified above. In connection with this basis set, it should be mentioned that though it is a minimal basis, as our comparative calculations show, its orbital exponents and contraction coefficients have been determined in such a way that they reproduce quite well the HOMO and LEMO levels of a double {-type extended basis.26 The chemical formulas of the four molecules (glyoxal, methyl-glyoxal, acrolein, and croton-aldehyde) suggested by S z e n t - G ~ o r g y i l ~ - ~ ~ as probably good acceptors against proteins are given in Fig. 1. The corresponding bond lengths and bond angles were taken from Ref. 27.

As indicated in the Introduction, the calculation of the empty levels of the acceptors requires great caution. For example, consider the CT process qualitatively in localized terms, i.e., starting with a zeroth-order wave function composed of the orbitals of the single molecules (taking, for in- stance, in the simplest case the highest filled-donor and lowest unfilled- acceptor levels). It is evident that for a physically correct description, we have to use such an empty orbital, which experiences the effect of the positive hole left behind on the donor site. This situation is analogous to the one arising in the course of atomic or molecular excitations. It is well known, on the other hand, that the unfilled orbitals obtained by the Har- tree-Fock (HF) method are related to an incorrect N particle potential, in which the effect of the removal of a particle is not taken into account. This problem can be solved in a relatively simple way by using the excitation Hamiltonian Pci) of the 6A8 method2%

(2)

where 13 is the original Fock operator and 6 is a projection operator pro- jecting onto the subspace of the virtual levels. The operator a (i) has the following form for a singlet-singlet excitation starting from the i th filled

pci, = p + 6A(i)6

glyoxal methyl-gboxol

II 0 OcroIeiO

crdon-aldehyde

Fig. 1. The chemical formulas of glyoxal, methyl-glyoxal, acrolein, and croton-aldehyde.


TABLE I Valence and Conduction Band Edges of Polyglycinea

Band Band Minimum Maximum Bandwidth

Valence band (HF) -11.252 -9.154 2.098 Conduction band (HF) 3.817 5.195 1.378 Valence band (HF + LRC) -10.373 -8.465 1.908 Conduction band (HF + LRC) 3.102 4.357 1.255

a Values calculated first with the ab initio Hartree-Fock (HF) crystal orbital method and corrected afterwards for long-range correlation (LRC) effects. All energies in eV.

It can be shown28 that the eigenfunctions of the modified Fock operator Fci) represent the excited states of the system experiencing the correct N - 1 particle potential. [The filled orbitals remain unchanged due to the presence of the operator 6 in Eq. (2).]

RESULTS AND DISCUSSION

In Table I we present the minima and maxima of the physically most interesting two bands of polyglycine calculated first with the ab initio H F crystal orbital method and corrected afterwards by the electronic polaron model to take into account the long-range correlation effect along the polypeptide chain. In the calculation of the long-range correlation cor- rection, we assumed no dispersion for the exciton energy

n*-n*+l texc

and used the first singlet-singlet excitation energy obtained by the 6A6 method for the segment of polyglycine containing two unit cells (see below). For the high-frequency dielectric constant, the value of tm = 3.5 has been used, as recommended in Ref. 29. It can be seen from Table I that the obtained valence and conduction bands are broad enough to make possible a large mobility transport. (They are of the same order of magnitude, but somewhat larger than the corresponding semiempirically calculated all- valence electron energy bands used in the transport calculations of Ref. 11.) The original HF bandwidths (2.098 and 1.378 eV for the valence and conduction bands, respectively) are reduced by -10% through the long-range correlation effect. (The new bandwidths are 1.908 and 1.255 eV, respectively.)

Table I also shows that the gap between the valence and conduction bands is very large in polyglycine. Though the HF value (12.971 eV) is somewhat reduced (to 11.567 eV) by the long-range correlation effect, it is still too large to permit charge-carrier generation through an intrinsic mechanism. In the case of the possibility of a “weak” charge transfer


TABLE I1 One-Electron Energy Levels of Some Simple Systems Containing the Peptide Groupa

Peptide System %*-l en* 6"*+1 %*+2

Formamide H F - 11.928 -11.654 5.441 10.555 6A6 -11.928 -11.654 0.455 2.386

Methylformamide H F -11.603 -1 1.489 5.492 10.549 6A6 -11.603 - 11.489 0.065 2.918

Double unit of H F -10.817 -10.495 5.596 6.313 polyglycine 6.46 -10.817 -10.495 -1.543 0.099

*The results are from the ab initio Hartree-Fock molecular orbital study (HF) and the excitation Hamiltonian of the 0.46 method. All energies are in eV. n* represents the highest filled orbital.

(through light absorption) between the previously mentioned acceptor molecules and the polypeptide chain, a necessary condition is that the lowest empty states of the polypeptide should be higher than the corresponding levels of the acceptor molecules. The proper ab initio calculation of the exciton levels in polymers is, however, a rather intricate problem. As a first approximation, we calculated with the 6A6 method the excited levels of three molecules containing the peptide unit: formamide, methylformamide, and the previously mentioned double unit of polyglycine (Fig. 2).

For the bond lengths and bond angles, we have taken the same values as used in the polypeptide calculations.21 The investigation of these molecules may thus help in the approximate determination of the level distribution in the exciton bands of polypeptides. In Table I1 we show the

formmite methyl-iormmde

two urvt r e k of pdyglycne

Fig. 2. The chemical formulas of formamide, methylformamide, and a dipeptide structure that corresponds to the part of the polyglycine chain containing two peptide units. The ele- mentary cell of the infinite chain is shown with dotted lines.


TABLE I11 One-Electron Levels of the Suggested Acceptor Moleculesa

Acceptor Molecule %*-I en* %*-I h * + 2

Glyoxal HF -14.616 -12.343 1.023 6.477 6A0 -14.616 -12.343 -7.946 -2.476

Methyl-glyoxal H F -13.722 -1 1.683 1.225 5.857 6A6 -13.722 -11.683 -5.464 -1.135

Acrolein HF -11.223 -10.635 -1.488 0.935 6A6 -11.223 -10.635 -6.600 -6.077

Croton-aldehyde HF -12.060 -11.774 2.076 7.204 O A 6 -12.060 -11.774 -2.239 1.882

a Values calculated first with the ab initio Hartree-Fock (HF) molecular orbital method n* and corrected afterwards by an 6A6-type excitation Hamiltonian. All energies in eV.

denotes the highest filled level.

two highest filled (tn*-l, e n + ) and the two lowest unfilled (tn*+l, tn*+2)

one-electron levels of the above three molecules calculated with both the HF and 6A6 methods. In the course of the 6A6 calculations, we always assumed an excitation starting from the highest filled level. According to the OAO formalism,17 the eigenvalue differences €,*+I - en*, en*+2 - en*, etc., of the modified Fock operator fl(n*) give directly the excitation energies for the n* - n* + 1, n* - n* + 2, etc., transitions. Comparing the HF valence and conduction bands in Table I with the HF values of en* and tn*+l

in Table 11, we see that the position of the HF bands can be determined to an accuracy of -1 eV on the basis of the single-molecule levels. Applying similar considerations for the exciton bands, we can postulate that the first exciton band will be by 5-6 eV lower than the original conduction band (cf. the difference between the HF and 6A6 values of tn*+l in the third column of Table 11), i.e., in the region of -2 to -3 eV.

Turning to the acceptor molecules, in Table I11 the two highest filled and the two lowest unfilled levels are shown for glyoxal, methyl-glyoxal, acrolein, and croton-aldehyde calculated with the HF and 6A6 methods. (The hole was again placed on the highest filled level.) Looking a t the third column of Table 111, we can make two interesting observations. First, the four molecules can be divided into two very different parts according to their excitation energies: while for glyoxal, methyl-glyoxal, and acrolein the first calculated singlet-singlet excitation is a t 4 eV, in the case of croton-aldehyde it is only a t -9.5 eV. This is connected with the fact that the lowest unfilled levels of the first three molecules lie between 5.7 and 3.3 eV lower than the corresponding level of the fourth molecule. If our earlier considerations about the role of the acceptor-empty levels turn out to be right, this means that the first three molecules should be relatively much better acceptors in general than croton-aldehyde. The positions of the empty molecular levels calculated by the 6A6 method can be tested by comparing


the calculated singlet excitation energies with the corresponding experimental ones. The study of the calculated molecular wave functions shows that in the case of glyoxal, methyl-glyoxal, and acrolein the theoretical values refer to an n + T* transition, while in croton-aldehyde the highest filled molecular level is of the T- type (T - T* transition). For the above excitation processes, transitions have been observed experimentally at 2.72 eV ( g l y ~ x a l ) , ~ ~ at 3.01 eV (acrolein), and between 5.9 and 6.5 eV (croton- aldehyde).30 Though the experimental excitation energy values do not agree very well with our theoretical ones (due to the approximate nature of the 8A8 method), for the n* + n* + 1 transitions they also show the same possible division of the molecules (glyoxal, methyl-glyoxal, and acrolein, on the one hand, and croton-aldehyde, on the other) as the theoretical excitation energies.

Second, we want to compare the tn*+l values of these acceptors with the position of the lowest excitonic levels of the polypeptide, on the one hand, and with the highest filled levels a t the top of the valence bands, on the other hand. The first comparison could show whether the intermolecular CT could be energetically favored at all. The second one could throw some light on the strength of the mixing between the empty acceptor states and the filled donor states in the course of the CT interaction, thus on the strength of CT itself. The difficulty with these comparisons is that our bA6 calculations always refer to situations in which the electron and the hole are situated on the same molecule. In the case of CT, we would be interested, however, in the interaction between the transferred electron on the acceptor and the remaining hole on the donor. It is evident that in the latter case the interaction would be weaker, and consequently, the empty acceptor levels would be shifted somewhat upwards from the pres- ently calculated positions. The correct quantitative treatment of this problem would require, however, that we write the wave function of the highest filled donor level into operator A. Calculating the matrix elements of the modified Fock operator in this way, we would have to calculate all those electron-electron repulsion integrals that contain two AOs from the donor and two from the acceptor. Since the computational difficulties of such a calculation would be comparable with those of the full supermolecule treatment, we can estimate at the moment only qualitatively the difference between the effect of the true operator a and that of the one used in our calculations. Inspection of the hole wave functions shows that they are delocalized in every case over the whole molecule, i.e., in a region of 4 A. Taking into account the r-l dependence of the Coulombic potential and having two molecules in parallel planes with a distance of 1.8-2 A, we judge the difference between the effects of the two hole potentials (once the hole is situated on the donor and once on the acceptor itself) to be about 3040%. We may thus expect empty acceptor levels between -4 and -5 eV in the case of glyoxal, methyl-glyoxal, and acrolein, and around -1 eV for croton-aldehyde.

Returning to the problem of comparing the level distributions in poly-


peptides and in our acceptors, we can say that croton-aldehyde should be a very weak acceptor against proteins if at all, since its empty level falls into the same region where the excited protein levels are also situated. Glyoxal, methyl-glyoxal, and acrolein, on the other hand, could act much better as electron acceptors in this relation, since their empty levels are estimated to be 3 eV lower than the corresponding protein levels. Finally, we should recall that earlier supermolecule calculations on the glyoxal-formamide CT complex16 resulted in a transferred charge of 0.2e at the intermolecular distance of 1.8 A. We can expect an even stronger CT in a similar polypeptide-acceptor configuration, since the top of the polypeptide valence band is nearly 3 eV higher than the highest filled formamide level. This means that its distance to the next empty acceptor level is only about 4 eV instead of 7 eV, the value for formamide.

We express our gratitude to Professor A. Szent-Gyorgyi for calling our attention to the problem and for numerous enlightening discussions. We are also very much indebted to the Alexander von Humboldt Foundation for giving a Senior US Scientist Award to one of us (T.C.C.). Furthermore, one of the authors (S.S.) expresses his special thanks to Professor W. A. Bingel for the hospitality and encouragement during his stay at Gottingen. We are further indebted to Dr. S. Abdulnur for calling our attention to newer geometrical data in the literature. The financial support of the Deutsche Forschungsgemeinschaft (Project No. La 371/3) and of the Fond der Chemischen Industrie are also gratefully acknowledged.

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Reinhold, New York.

Received January 10,1978 Accepted August 1,1978

theoretical investigation of semiconductive properties in proteins. ii. the possibility of charge...

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