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Donor-Acceptor model of electron transfer through proteins
J. theor. Biol. (1978) 73, 29-50
Donor-Acceptor
Model of Electron Transfer Through Proteins
V. N. KHARKYANEN, E. G. PETROV AND I. I. UKRAINSKII Institute of Theoretical Physics, 252130 Kiev, U.S.S.R. (Received 11 August 1977) It is shown that the interaction of proteins with a polar solvent leads to the formation of electron donor groups in the proteins. The highest occupied level of these groups in situated near the bottom of the conduction energy band of protein. The transfer of an electron from the donor group through the conduction band of protein to the spatially removed acceptor group is considered including the possible relaxation processes.
1. Electronic States in Polypeptides In 1941, Szent-Gykgyi put forward the hypothesis that the transformations occurring during biochemical reactions at any part of a protein macromolecule are transferred to its other parts due to electron migration along a polypeptide chain. Szent-GyGrgyi (1941, 1957, 1968), Ryll (1948) and some other authors (see references cited by Gutman & Lyons, 1967) emphasized the fact that the ideas on electrons moving along protein are important for understanding the mechanism of many biochemical processes. For example, the scheme of enzymic redox reactions proposed by Cope (1964a,b) shows that the rate of radical formation is limited by the velocity of electron motion through protein. It is now clear that to elucidate the electron transfer mechanism and the accompanying electron-conformation transformations in biosystems is one of the most important problems in bioenergetics. A study of the nature of charge separation and electron transport can give a deeper insight into molecular machinery, thus making it possible to simulate bioprocesses. The concepts of physics, in particular, very fruitful in these applications.
those of solid-state
determine
helicity
physics, appear to be
For example, the Davydov (1968) exciton theory for molecular crystals was used to study the migration of optical excitations over chlorophyle molecules in chloroplasts of green plants and in photosynthetic bacteria. This theory has also been successfully applied to the degree of protein
in studies of hypochromic
effect
29
0022~5193/78/07o6-0029$02.00/0
0 1978 Academic Press Inc. (London) Ltd.
30
V.
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(Volkenstein, 1975). Davydov (1973, 1974, 1975, 1977) developed the exciton theory of the deformation of a polypeptide chain and proposed, on its basis, the soliton model of muscle contraction. Although the migration of uncharged excitations over protein appears to be an established fact and it plays an important physiological role in certain cases, the electron migration over protein remains questionable as yet. The experimental verification of the Szent-Gyorgyi hypothesis performed by Eley with co-workers (Eley, Parfitt, Perry & Taysum, 1953; Cardew & Eley, 1959; Eley & Spivey, 1960; Eley, 1962) and by Rosenberg (1962u) has shown that the current excitations of protein are separated from its ground state by the energy gap AE w 2.3- 3.5 eV. Theoretical calculations of polypeptide energy band structure have given a still greater value of the gap. Thus, at physiological temperatures T w 300 “K (N O-025 eV) the protein is a good insulator. This fact forced many authors to conclude (Blumenfeld, 1974; Volkenstein, 1975) that electrons cannot migrate through a polypeptide chain in biological reactions. (Most biological reactions occur in the dark, therefore the optical excitation of an electron in the protein conduction band should be excluded.) In order to elucidate what objections are usually raised against a possible physiological role of protein electroconductivity, we shall consider the experimental data on electronic conductivity of proteins and the quantum mechanical calculations of the energy band structure. In the above experiments the D.C. conductivity was evaluated with the formula IS = a,(T)*exp (-AE/2kT), where a,(T) is th equantity weakly depending on temperature T. This formula, however, does not give a correct value of the energy gap for current excitations in a protein macromolecule, since the experiments were made using pellets. The value of AE characterizes the potential barriers for the electron transfer between protein macromolecules (see, e.g. Ladik, 1972). As a matter of fact these experiments can allow us only to conclude that the energy gap between a conduction and an upper filled electron energy band does not exceed the measured value of AE (provided that the impurity contributions into conductivity are negligible). The experiments of Eley (1963) and Rosenberg (1962a,b) on hydrated proteins have shown that the conductivity increases exponentially (because the effective value of AE decreases) with increasing hydration. Rosenberg associates this fact with decreasing barrier of the electron transition between macromolecules, the decrease in BE by 0.4 eV being su.lBcient to increase the electro-conductivity by a factor of lo8 at T = 300 “K. There are no reliable experimental data on the value of E for proteins, since the contribution of intermolecular electron transitions and that of impurities have not been evaluated. In order to determine the electronic conductivity of protein, more direct experimentation is necessary.
ELECTRON
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The electron energy bands of polypeptide chains were tist calculated by Laki (1942), Evans & Gergely (1949), and later refined by Suard, Bertheir & Pullman (1961) and Ladik (1963, 1964). The calculations were made for the a-helical structure of a polypeptide chain stabilized by hydrogen bonds I I between plane peptide groups H-N-C&O. The allowed energy band widths obtained (0.1-O-2 eV) are due to the weak overlapping of n-electron orbitals of neighbouring peptide groups, and the energy gap (up to 5 eV) is connected with the,n-electron states in a single peptide (amide) group. The lowest vacant band originates from a n*-antibonding orbital delocalized over peptide N-C=0 group. The above calculations were evidence that the protein electronic conductivity at physiological temperatures is negligibly small. However, Brillouin (1962) supposed that due to the presence of the H-C-R
groups in proteins,
the radicals R can appear as impurities, whose energy levels fill up the space between the highest occupied and the lowest vacant electron zones. According to Brillouin, the energy bands are also formed by electron states of the o-type, this resulting in seven filled and seven vacant allowed energy bands (see, e.g. Ladik, 1972). The calculations of the electronic structure of proteins taking into account the radicals R have not been performed. However, several detailed calculations on 7~ and a-bands for polypeptide chains of poly+alanine and polyglycine have been made since 1970 (see papers by Fujita, Imamura (1970), Imamura & Fujita (1974) Duke, Eilers & O’Leary (1975) and the references listed in these papers). These calculations have given the energy gap to be more than 10 eV. This value is two-three times greater than that obtained earlier in the n-electronic approximation. Moreover, this value is not in agreement with the experimental energies of first optical transitions in the protein chain which have been established to be 5.3 and 6.7 eV (see e.g. Volkenstein, 1975). This fact indicates that the approximations used for describing the excited states of polypeptide chains are insufficient. One of the limitations of the calculations mentioned above seems to be the employment of a minimal atomic basis set which includes only the Slater orbitals of valent and inner atomic shells, more exactly, the 16 24 a-&z functions for the atoms of the second row (N, C, 0) and the 1s orbital for the hydrogen atom. It is known, however, (see, e.g. Frish, 1963) that in optical spectra of the second row elements, the Rydberg states (shells next following the valent ones) play an important role. In particular, in the cases of carbon, nitrogen and oxygen atoms which form the valent core of a polypeptide chain, the optical transition between terms of the type
32
V. N. KHARKYANEN
ET AL.
(Is)~(~s)~(~P)” + (1~)~(24~(3p”-~3s corresponds to an energy of about 7 eV. Even this value is less than the gap between the upper filled and the lowest vacant energy band obtained in pointed above papers. Thus, taking into account the 3s and 3p states is rather important for a correct description of excited states of molecules and polymers consisting of the second row elements (an exception should be made for the conjugated systems). Such calculations involving 3s and 3p states are now performed only for a number of small molecular systems (see Buenker & Peyerimhoff, 1975a,b; 1976). Of these systems ethylene C,H, and ethane C,H, are of great interest for our purposes. A very interesting result is that the lowest optical single excitation of ethylene is connected with the transition 71-+ 3s rather than the transition between z-electron levels, rc + rc* (see Fig. 1). In ethane the vertical transition 0(2x, 2p) -+ 3s corresponds to an energy of 9.16 eV, while the transition 0(2s, 2p) + 0*(2s, 2p]) lies in the continuous spectrum. “&(///%,:i’::;; :; :‘: ‘;:
(a)
I
,,;
,i
,:
-+Vacuumt level
(6)
FIG. 1. The scheme of the lowest optical transitions in ethane (a) and ethylene (b).
Using these data we construct a qualitative picture of the electron levels of a polypeptide chain. First of all we answer the question of why more rigorous calculations of the polypeptide energy band structure with allowance c- and a-electrons result in bad agreement with experiment. One of the reasons for this disagreement is connected with the absence of 3s and 3p orbitals for carbon, oxygen and nitrogen atoms in the basis sets used. Indeed, an optical transition of the n + 3s type in polypeptides should have an energy lower than that of ethylene owing to the splitting of atomic 3s, 3p levels into allowed energy bands, i.e. AE(rc + 3s) < 7 eV. The states of these bands would probably be mixed to a great extent with non-bonding MO’s of the n-type of the peptide groups 0=X-N, since the polypeptide
ELECTRON
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33
symmetry group does not contain elements allowing to perform a rigorous o--71separation of electron states. As to the electron structure of the polypeptide ground state, i.e. the structure of valence energy bands, the calculations in the minimal atomic basis sets seem to have correctly revealed its properties. According to Fujita & Imamura (1970) the upper filled energy band of polyglycine originates from the n-states of lone pairs of N-C&O groups and bonding z-electronic states slightly mixed with bonding o-states. The optical bands 5.3 eV and 6.7 eV of proteins can be connected with n + (3s, n*) and 71--f (3s, x*) transitions. The n -+ D* (2s,2p) and 0(2s, 2~7) + 0*(2x, 2~) transitions appear to be situated as in the case with ethylene and ethane in the continuous spectrum. We can thus suggest that the energy bands of polypeptide chains have the structure given in Fig. 2. Taking into account the diffusive nature of 33, 3p orbitals of C, N, 0 atoms and using the well-known Slater rules for qualitative evaluations we can obtain that the bottom of the lower vacant band corresponds to energies of about N 1 eV (neglecting the core electron polarization effects) and the width of this band is of the order of a few electron-volts. Let us compare the energy band structure of a polypeptide chain, given in Fig. 2, with the results of n-electron methods often used for the treatments of optical experiments. According to 7r-electronic approximation the lowest vacant and highest occupied energy bands are of the n-type whose small
Conduction band
J
Valent energy band n,a(2p) u (2s,2p)
FIG. 2. The scheme of the upper filled and vacant states of electrons of the covalent core of a polypeptide chain. 2
34
V.
N.
KHARKYANEN
ET
AL.
widths are connected with the weak overlapping of rc-electron orbitals of O=C-N groups in the a-helical sections of a protein chain. When the helicity is disturbed n-electrons are localized in the peptide groups, since the H-C-R
fragments of the chain separating the peptide groups do not
contain rc-electrons. This circumstance is of great importance, because the n-electron protein system breaks at these sites and the rotation about ordinary C-C and C-N bonds is responsible for the conformational flexibility of a protein chain. As a result, the electron transport along the n-electronic system of proteins is impossible. Therefore, if the electron is forced into the vacant energy band thermally or optically, its motion is possible only within a-helical fragment of the chain. The same is true of a hole, e.g. in the presence of a strong electron acceptor, in the occupied band. According to the scheme proposed here which allows for the mixture of 3s- and rc*-states and the formation of a spread conduction band, the occurrence of an electron in a vacant band can lead to its migration along the chain involving also nonhelical sites. This may be responsible also for the optical excitation migration in proteins observed in the known experiments by Lautsch, Paceday, Sommer, Julius & Bodefeld (1959). We shall argue here that under certain circumstances the lowest 3s, rc* vacant energy band of a polypeptide chain can effect electron transfer from electron donor to acceptor in dark reactions, although the conduction band itself is separated from the upper filled band by an energy gap of about 5-8 eV, and the thermal excitation of an electron to this band from the occupied ones is not practically realizable. In other words, the insulator properties of a polypeptide chain and its conformational flexibility are not an impediment for electron transfer from donor to acceptor through protein macromolecules. 2. The Mechanism of Electron Donor-Acceptor Transfer along a Polypeptide Chain The energy scheme represented in Fig. 2 should be completed by electronic levels of protein R-groups which, according to Brillouin’s assumption (Brillouin, 1962), can be situated between the upper filled and the lowest vacant electronic energy bands of a polypeptide chain. However, Brillouin did not consider which are the R-groups that could have the electron energy levels close to the conduction band of the peptide chain core. We wish to emphasize that two of twenty R-groups of amino-acid residues, namely -CH,---COOand -CH2-CH,-COO(residues of asparagic and glutamic acids, respectively) have negatively charged groups -COOat
ELECTRON
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physiological values of pH = 6-8. Besides that, at the same pH the unbound -COOgroup of the C-end of the chain also possesses a negative charge. The problem arises as to what energy level corresponds an extra electron in the -COOgroup. To elucidate this question, we turn to experimental data on the electronic properties of similar molecular fragments and polymers. According to the measurements of the electron affinity of radical OH, the energy of the extra electron in OH- is equal to 2.2 eV, while in the Hz0 molecule the highest occupied level possesses an energy of 12.7 eV (the ionization potential of the water molecule). Thus, the breaking of a proton away from a molecule results in an increase in the energy of one of the electrons by about 10 eV. Almost the same situation occurs also in the case of the -COOH group. The electron affinity of polymers with saturated bonds in the main chain (polystyrene, polymethylmethacrylate) amounts to 3-4 eV (Duke & Fabish, 1976). The results obtained by Duke & Fabish (1976) show also that in polymer systems with saturated bonds where n-electronic states are absent, the injected electrons can move along the chain. Thus, the energy of an electron in the -COOgroup is close to that of the conduction band of the chain. Solvation of the -COOgroup does not change essentially this situation. Indeed, since -COOH group transforms into hydrated ions -COOand H+ at pH = 6-7 (see e.g. Lehninger, 1972), the energy difference in this process, AE = pH In 10Tat T = 300°K w 0.025 eV should not exceed 0.3 eV. The gain in proton energy due to hydration is equal to about 10 eV (Klotz, 1962). The loss in electron energy when a proton is removed is, according to the above data, 10-12 eV. Accordingly, the solvation energy of the -COOgroup and hence electron level lowering is less than 1 eV. Thus, if protein has a hydrated -COOgroup, the extra electron of this group has the energy close to the edge of the conduction band, and consequently, is partly delocalized due to overlapping of its wave function with 34 rc* orbitals of a polypeptide chain core. If there is an appropriately situated acceptor moiety which possesses an empty orbital with energy below the donor level of the -COOgroup, then the electron can be transferred along the polypeptide chain to the acceptor. We note, however, that in viva (T = 300” K, pH = 6-8) -COOgroups are not free as a rule. They form, for example, salt bridges (or hydrogen bonds) with NH: and NH+ groups of amino acid residues:
I -COO-.
. .H+-NH,
uncl -COO-.
. .H+-N<.
36
V.
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KHARKYANEN
ET
AL.
In this case a salt bridge can be realized both between two fragments of a chain (stabilization of the tertiary structure of proteins) and between fragments of two different chains (stabilization of the quartic structure of oligoproteins). Owing to the conformational transformation of proteins there may occur a rearrangement of salt bridges, resulting in rupture of individual bridges. When the salt bridge breaks, the -COOgroup of a corresponding amino acid residue becomes free and the excess electron becomes highly energetic. The energy increase of an excess electron is compensated by the energy of the hydration of the NH: group or by the hydration energy of a proton. Now we consider the model of electron donor-acceptor transfer through a polypeptide chain. We denote by D- the electron donor group attached to the chain core. As we noted above, at pH = 6-8 the C-end of the chain and the residues of asparagic and glutamic acids can play the role of this group. Let us denote by H’Bthe group connected with the donor moiety D- by a salt bridge. The N-end of the chain, the positively charged groups of amino acids can appear as H’Bgroups. Since the break-up of the salt bridge requires the free energy about 0.1-0.2 eV at pH = 6-8 and kT = 300” K the thermodynamic equilibrium shifts towards the formation of the salt bridges, but the thermodynamic probability of the salt bridge breaking differs from zero. When the salt bridge breaks up the energy of the excess electron of the -COOgroup rises to the bottom of the protein conduction band (the electron states in this band can be of a polaron type, in general case, since the conformational change of protein can assist the electron migration along the chain). When the energy of the extra electron in the -COOgroup becomes higher than the bottom of a conduction band, the relaxation to the lowest level of the band may occur owing to the interaction with thermostat. In this case the lowering of electron energy should not exceed the energy cost of the rupture of the salt bridge, i.e. the value of 0.142 eV since otherwise the thermodynamic equilibrium will be displaced towards breaking of the salt bridge. Generally speaking the interaction of donor orbital of -COOgroup with the conduction band states leads to splitting a local level off the band (if the donor level falls into the band) or to the repulsion of the initial donor level from the band (if the donor level is near the bottom of the conduction band). In both cases the conduction band states mix with the donor orbital of the -COOgroup. The interaction of the local level with zone levels is considered in the Appendix. When the electron donor group is not bound with cation group by the salt bridge, the energy of its excess electron is near the protein conduction band and the electron can be excited thermally into the conduction band.
ELECTRON
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Then, during the time z(I) [see formulae (12) and (13) in Appendix] between the levels of the protein vacant energy band Ek and the electron donor level EC’) the Boltzmann distribution of electrons is established. If we now join the acceptor moiety to the protein chain, the electron transfers to the acceptor during some time z % r(l) [see formulae (16) and (23) in Appendix]. In the case when the electron donor and electron acceptor moieties are joined to the protein chain but the pH of the medium is low so that the equilibrium of the reaction -COO- +H+ +r -COOH corresponds to the formation of the -COOH groups the probability of electron donor-acceptor transfer is essentially reduced since all the electrons on the -COOH group possess the low energies far from the vacant energy band of protein. If we increase the value of pH, the above reaction equilibrium will be shifted towards the formation of -COOgroups and hydrated protons and hence to the possibility of electron transfer to the acceptor through the protein chain. The -COOgroup itself becomes an electron acceptor when its excess electron is taken off. In the absence of exogenous donors of electrons, the unsaturated valency of the -COO* group can form a probably weak bond with other protein groups or solutes possessing unsaturated valencies, e.g. : -c/O-%c-
&O--s
10 o/ 10 ’ The formation of these bonds lowers the energy of the system and consequently decreases the activation energy of electron transfer. Besides this, the -COO* group can be reduced by exogenous donors. In this case, which probably is more realistic, the -COOgroup functions as a mediator of electron transfer from donors into the conduction band of protein. We have considered the -COOgroup as an example of protein fragments essential for electron transport processes. Other protein groups can also appear as electron donors or electron transfer mediators (ETM) in phenomena of electron donor-acceptor transfer through proteins. Thus, we mention that amido-, amino- and SH-groups of proteins appear as electron donors with respect to acceptor S-S bonds in biochemical reactions (see, e.g. Szent-Gyorgyi, 1968). Recently Atanasov et al. (1977) have shown that histidine residues A10 and GHl of myoglobin take part in the reaction of electron transfer from myoglobin to ferricytocbrome C. In our opinion it may occur that histidine appears as ETM in this process. Generally speaking, we suggest that electron donor or ETM groups in proteins appear due to the interaction with polar solvent, i.e. due to solvation of cations splitting off the groups: -A--A\ 4-f -AB it -A+ B+, &BQ+B
38
V.
N.
KHARKYANEN
where A- and --Aare donor metallic or organic cations.
ET
groups
AL.
B+ and B++ are solvated
(ETM),
Figure 3 gives the scheme of energy levels of the system of donor + acceptor + polypeptide chain. This scheme reflects the main features of the electron transport model proposed. The mathematical aspects of the model are discussed in Appendix. The kinetics of the electron transfer from donor to acceptor is described by the expressions (14), (15) and (22). The presence of a conduction band located not far from the donor and acceptor levels
A=0
(a)
(b)
FIG. 3. The energy diagram of the electron-proton states of the donor + acceptor -Ipolypeptide chain system (a) and the energy diagram of the electronic states of the same system (b). EC”) is the electron energy in the donor with a proton attached to the donor, E(‘) is the electron energy on the donor with a proton passed into the medium and hydrated, IP is the electron energy in the acceptor, E. is the energy of the electron at the centre of the conduction band, EcH+, is the energy of the proton hydration. The dotted line in Fig 3b shows the increase in energy of the electron when the proton moves to a polar medium. The energy distances indicate the order of the corresponding values.
provides the effective electron transfer from donor to acceptor. At the same time, the conduction band is separated from the valent ones by the significant energy gap (see Fig. 2). So, in the process of electron donor-acceptor transfer we deal with an impurity conductivity of proteins. The appearance of the impurity-namely, of the donor group with a highenergy electron-is a result of interaction with polar solvent. We may say that the polar solvent plays the role of temperatures exciting the electron of a donor group up to the high energy level. The role of real temperature is now reduced to the excitation of an electron from the highly energetic donor
ELECTRON
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39
level into the protein conduction band. If the energy gap between the donor level and the bottom edge of the conduction band is about 0.1-0.3 eV, then room temperature can provide the effective electron transport in dark reactions. During the electron transfer from donor to acceptor the total energy of the system is lowered by a value equal to the difference of electron energies of donor and acceptor levels. This energy can be consumed for the transformations of the protein conformation, the formation of new bonds, heat, etc.
3. Conclusion
The model of electron donor-acceptor transfer through proteins presented in this paper shows that despite the insulator properties of proteins, electrons can be transferred through polypeptide chain in dark reactions. The essential point is that in the electron transport model proposed here, the electron is not excited from the valent energy bands of protein into the conduction band. The highly energetic electron of the donor group is transferred, but not the electron of the covalent core of the polypeptide chain, as was assumed earlier. For the polypeptide chain core this electron is excessive and therefore it can move in the conduction band of proteins. In this process we deal with the impurity conductivity of proteins. Due to the significant contribution of the Rydberg (3s) atomic states in the formation of the conduction band in proteins, the excess electron mobility in fact weakly depends on protein conformations. We suppose also that the formation of the electron donor group is a result of the interaction of a protein macromolecule with a polar solvent. So, the donor group should locate in the hydrophilic part of proteins. We wish to note that the model of electron transfer proposed in the present paper gives a new interpretation of Szent-Gyorgyi’s ideas on the role of protein as electron conductors. Our interpretation of Szent-GyBrgyi’s hypothesis is argued by a number of facts of common knowledge from the fields of molecular physics and chemistry. This model can be checked in experiments on the synthesized polypeptide chains. For example, if we put the polyglycine chain through the artificial membrane, on one side of which there is water and on the other there is a non-polar liquid (Fig. 4), and place the C-end of the chain in the aquomedium, then attach the acceptor (such an acceptor can be, e.g. heme, heme + 0,) to the chain part in a non-polar medium. Next changing pH of the aqueous solution from the low (when -COOH group is stable) to the physiological values (when the -COOH group dissociates in hydrated fragments -COOand H+) we should expect the electron transfer from the
40
V.
N.
KHARKYANEN
ET
AL.
FIG. 4. The scheme of the experiment to study the possibility of the electron donoracceptor transfer through polypeptide chain due to the change of pH in aqueous solution.
-COO-
group to the acceptor, and from exogenous donors to the -COO
group.
Besides -COOgroup acting as the electron donor, a compound which can be excited by light in such a way that the energy of one of its electrons becomes comparable with the energy in the vacant band of the polypeptide chain can be used. In this case the compound acts as a photodonor of electrons. After illumination the compound is first oxidized with the reduction of the acceptor during the time zl, then it is reduced again in z2 9 21 time, removing the electron from the acceptor (Fig. 5). Polvoeotide
FPolypeptide i
conduction I-\
,------”
FValent
chain
band
-
------------.
energy
bond
,-
I-bhD
EPh D
of the chom --_--.
-.--
5. The scheme of the experiment to detect the electron transfer from a photodonor optically excited (PhD) to the acceptor one. E P,,Dis the PhD ground state energy, EP~D is the PhD excited state energy. Under the influence of light with frequency w1 = l/fi (E&D - EphD) the electron enters the conduction band from where it relaxes to the acceptor for the time zI. The inverse transfer is produced during time za and followed by the light emission with the frequency aa, (ta > tl). FIG. moiety
ELECTRON
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PROTEINS
The experimental study of the kinetics of the electron donor-acceptor transfer through the polypeptide, e.g. the measurement of the dependence of the electron transfer rate on the distance between the donor and the acceptor along the chain, on temperature and pH of medium, is of a great importance in elucidating the role of electron conductivity of proteins. The mechanism of electron transport through protein macromolecules described above very likely plays a certain role in a number of biochemical reactions involving electron transfer processes. This may be true for such processes as electron transfer in electron transport chains, electron transfer from cytochrome to photocentre, formation of transmembrane potentials, and so on. Electron transfer through proteins can also be responsible for allosteric interactions in enzymes, origination of gating currents in native membranes and mechanisms of opening of membrane channels. These applications of donor-acceptor electron transfer through proteins deserve special consideration, which is now in progress.
B. P., POSTNIKOVA, Molek. Biol. 11, 537.
ATANASOV,
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Chimiu
Ziirich 13, 129.
A. L. (1972). Biochemistry, Ch. 4. New York: Worth Publ. Lrrsnrrz, I. M. (1947~). Zh. Eksperim, Theor. Fiz. 17, 1017. Lrrsnrrz, I. M. (1947b). Zh. Eksperim. Theor. Fiz. 17, 1076. Lrrsnrrz, I. M. (1948).Zh. Eksperim. Theor. Fiz. 18, 293. Ovcmov, A. A., UKRAINSKII, I. I. & KV~NTZEL, G. F. (1972).Uspehi Fizich. Nauk. 108,81. pEm&mu, S. V. & JATZENKO, A. A. (1967). Zh. Eksperim. Theor. Fiz. 53, 1327. ~ELETMINSKII. S. V. & PETRov. E. G. (1968).Preprint ZTP-68-20. Kiev: Institutefor Theor. Physics.Acad. Sci. UkrainianS.S.B. ’ ROSENBERG, B. (1962a). J. Chem. Phys. 36, 816. LEHNINGER,
ROSENBERG, B. (19626). Nature 193, 364. R~L, N. V. (1948).Uspehi Fizich. Nauk. 35, 186. Sum, M., BERTHEIR, G. & PULLMAN, B. (1961).Biochim. SZENT-GY~RC+~I, A. (1941~).Science 93, 609. SZENT-GY~~RGYI, A. (1941b).Nature 148, 157. SZENT-GY~RGYI, A. (1957). Bioenergetics (Augenstein,L.
Press.
SZENT-GY~~RG~I, A. (1968). VOLKENSTEIN, M. V. (1975).
biophys. Actu. 52, 254.
G. ed.) New York: Acadmic
Bioelectronics New York, London: AcademicPress. Molecular Biophysics. Moscow:Nauka.
APPENDIX
The Electron Donor-Acceptor Transfer through a Polymer Chain with Account for Relaxation Processes We consider a regular chain (or quasi-one-dimensional crystal) with the electron donor group D- and electron acceptor group A attached to the n,th and n,th sites of the chain, respectively [see Fig. 6(a)]. Let us denote by E,, the energy of an electron in the lowest vacant state of any site of the chain, then, E, and E, are the energies of an excesselectron in the donor and acceptor groups respectively (E, - E,), L is a hopping integral for neighbouring sites of the chain which are separated by the distance a, Tr is the electron resonance integral between the donor group D- and the n,th site of the chain, T2 is the same integral for acceptor group and n,th site of the chain.
ELECTRON
TRANSFER
IN
. . . 0 “‘rp . . . & 8
43
PROTEINS
“’ p . . . 8D
A
(a)
1 a ttti
_/.-.-.-. -.-. 42
E(3)
I I + $1)
E(2)
(b)
FIG. 6. The polymeric chain with donor and acceptor (a). Two mechanisms of the electron donor-acceptor transfer through the chain (b). The thermal excitation of an electron from the donor into the conduction band (straight upward arrows); the electron relaxes to the ground level of the acceptor /W (the straight arrows directed downwards). If the thermal excitation is inefficient, the main contribution to the probability of the electron transfer can be made by the second mechanism connected with the electron transport via the direct interaction VIZ from the donor level E (I) to the vibronic sublevel of the acceptor Ec3). As a result of oscilatory relaxation, the electron is transferred to the level Ec2) (dotted and wavy arrows). Both the mechanisms lead to the losing in electron energy by the value E(l) - E@).
In order to consider the electron relaxation to the equilibrium state in the system of the polymer chain - donor + acceptor (or the electron donoracceptor transfer) we need the electron eigen states in this system. We determine them using the well-known results of impurity centres in onedimensional crystals [see papers by Lifshitz (1947a,b, 1948), Koster & Slater (1957), Ovchinnikov, Ukrainskii & Kventzel (1972) and references cited in these papers]. We shall use also the tight binding approximation and suppose that the impurities-donor and acceptor-can be described by the shortrange potential. If the direct interaction between the donor and the acceptor groups is absent, the energies of electron local levels E(p) (/A = 1,2) which lie below the bottom of the conduction band of the chain, Eo-2jLI > E@), can be found from the equation where Iv,kl” R, = Ej-E(@) - 2 ___ k &--E’“”
Ek = E,+2L
cos ak,
W9
44
V. N. KHARKYANEN
ET AL.
k is the wave number, N is the total number of the chain sites. As it can be seen from expressions (Al), (A2) the value of IJZ, which can be written ast:
-(E,-E'"')+J(E,-E'"))'-4LZ, 2L
,4 , < ,,@3) P
if E") < E,-2jLI, characterizes the effective interaction of the donor group with the acceptor group for a given value of the energy E”‘. It is essential to note that the value of I’,, decreases markedly (by exponential law) with increasing number of unit cells In, --n,l-between the donor and acceptor groups. From the calculations, e.g. by Duke, Eilers & O’Leary (1975), it follows that the width of conduction band in polyglycine (41LI in our notations) is of several electron-volts. So, the value of hoping integral, L, is about 1 eV. The same value can be taken for the vacant band arising from 3s states. Assuming that IL1 N 1 eV, 17’rl N O-5 eV, lTzl N 0.1 eV, E, - EC") = 2lLl+O*2 eV, l~~-n~) = 10, we obtain from expression (A3) for I’,,: IV,,1 N 10m4 eV. The smallness of the value of I’,, points to the fact that the energies Ecp)can be found with a good accuracy by putting equal to zero the values of Rjj. From the equations Rjj = 0 (j = 0 for the donor and j = 2 for the acceptor) we obtain the following expression for EC")
(E,-2)L(
> E@')
From this equation we can find the energy of an electron in the donor group, E(l), and in the acceptor group, EC'). The wave functions +r and tj2 which correspond to these levels, are localized near the donor and the acceptor. The larger the energetic distance of a local level from the edge of the conduction band, i.e. the value IA',,-2(LI -E(1,2)l, the stronger the localization of the wave functions. We now consider the electron donor-acceptor transfer. The system Hamiltonian has the form H, = c E,A:A,+E’1t4:A, +E(2)A:A2, (A4) where E,, E(l), EC') are the introduced above energies of electron in the conduction band, in the donor and acceptor local levels, Akf, A:, A: (& A,, A,) are the creation (destruction) operators of electron on the corresponding state. tWe obtain the expression for VI, from equation (A2) integrating over k.
ELECTRON
TRANSFER
IN
PROTEINS
45
We choose the interaction between our system of the chain + donor + acceptor and the thermostat in the usual form: V=
C
C
j=l,Z
(K:kAi+Ak+?CljA:Aj)(Bj,+Bi+,),
vk
645) Here ICYis the interaction parameter characterizing the electron transfer from the donor (j = 1) or from the acceptor (j = 2) to the nth unit of the chain. This transfer is accompanied by an emission or absorption of the vth oscillation mode of the donor or the acceptor group; B& Bjy are the creation and destruction operators of vibron. The chain + donor + acceptor system can be considered as a subsystem with a finite number of degrees of freedom, placed into thermostat. The establishing of equilibrium in the subsystem is described [see, e.g. Peletminskii & Jatzenko (1967), Peletminskii & Petrov (1968)] by timedependence of the matrix elements yl,(t) of the density matrix of the subsystem p(t), yr,(t) = (Zlp(t)lm > = ,S,p(t)$j,,.In our case land m correspond to the states with the energies &, E (l) , EC” and fI,,, are the following operators : /3kk’= A:Ak, = ~~k, 9jk = Af A, = ~~, ~jj = AfAj
= ~~, 912 = A:A2 = p,:.
Using the definitions of yl,,, and rr,(t), expressions (A4) and (A5) and the equations for yl,(t), obtained by Peletminskii & Petrov (1968), we obtain fl
Yjj(t)
=
-
2
I~k[~jj(t)njv-Ykk(t)(l+nj,)l,
it
Ykkct)
=
-
c vi
lyk[ykk(t)(l
+
njv)-yjj(t>njvl,
where Ijk E ~~~IIc~~I~~(E(‘)-E~-~o~,), and njv = [exp (BttOj,)-l]-’ is the Bose distribution function for the oscillators with the frequency wjV at temperature T, /I = (k,T)-‘, tz and k, are the Plank and Boltzmann constants, respectively. It is a very difficult problem to solve the set of A’+2 equations (A6) for a general case. We suppose that the interaction of the chain with the thermostat establishes the Boltzmann distribution of electron over the levels Ek for the time rg, but the establishing of the electron distribution between levels E(lB2) and zone levels Ek is realized during time r(l), r(‘) $ rB. If r(l), rc2) $ zs, we can describe the establishing of a statistical equilibrium in the
46
V.
N.
KHARKYANEN
ET
AL.
chain + donor + acceptor system at any time t >> zg assuming that there is a Boltzmann quasi-equilibrium between levels Ek, namely ykk(t) = y&t) ewPtEkmEo)= y&t) e-pZLcosak. (A7) Substituting the condition (A7) of the quasi-equilibrium into the set of equations (A6) and using the normalization condition Yll(o+Y22(0
+ T Ykk(Q = 1
we obtain the set of two equations a ~tYll(o
= -X1~~l+~l)Yll(~)+~lY22(t~-~,l~
(‘48) ;
Y22(0
=
+~z>r22(0-~21,
-X2Ccf2Yll(o+(l
where Xj = i
5 IJknj, = Bj[exp
(pAi)-
I]-‘?
Aj s E,-21LI
ep(Eo-Eco)
e@(b-E(~‘)
-E(j),
(A9)
is the constant not depending on temperature, I,,(x) is the Bessel function of the zeroth-order imaginary argument. If jL( N 1 eV, then at room temperature j&Cl % 1 and, hence, the Bessel function can be replaced by its asymptotic expression :
Let us solve the equation (A8). First of all we investigate the case when the acceptor is absent. In this case we obtain only the equation ;tYll(o
= -X1C(l+a,)Y,,(t)-a,l.
The solution of this equation with the initial conditions Yldt = rid N” Ydt = 0) = Yll(O> has the form
r1d0 = r11(0) - & [ As follows from equation
(All),
1I
e-.Yl(l+al)t ) c11 .
the equilibrium
(All) l+a, electron distribution
ELECTRON
TRANSFER
IN
47
PROTEINS
between levels of the system of chain + donor at t % zB is realized for the characteristic time z(r), and according to expressions (A9) and (AlO) 1 1+2JnB# l/N exp (PAI) - = x1(1 +a,) = B, 6412) z(1) exp U%> - 1 . If the level EC’) lies outside the edge of the band E,,-21L( by the value of O-2 eV or more, then at room temperature or lower, one in the denominator of expression (A12) the unity can be omitted so that -1 ’ ,,/m + exp (-/?A;) . G413) Tu) = Bl jy
1
At t g T(I) the values yll(t),
c ykk(t) which characterized the populations
of the donor level and conduc;ion band, are practically and coincide with equilibrium values
time-independent
Solving the set of equation (A8) we assume that initially
i.e. between sun-band donor level and the chain, the thermodynamic equilibrium is established and acceptor level is empty. Under these conditions the solution of the system (A8) has the form L k,-k,
y11(f)=(l+oll)~~~al+a2)
(-k,
1 Y22(t) = ~+a,);;?~,
+a,)
ekit+kl
ekzt) + ,+,“l,,
1
, 2
k,Ck, +x1(1 +a,) ek,t-
k, -k,
XlEl
k,Ckz+xlU+dJ
ekzt Xl%
+ i
a2
l+a,+a,
G414)
where
kl = 3{-Cxl(l+al)+~2(1+az)l 2
t- JC~l(l+cr,)-~z(l+a211~+4~l~(I~2~(2).
Taking into account the condition EC’) > Ec2) and the definition values xj, Olj, WC see that X~CQ % X~CZ,,CX~% ~1, ~1 % ~2. Therefore k1
N” -x2
2,
k2
,”
of the
-xla,
and hence Ik, 1 Q Ik21. From the comparison of the value k, with the value [d’)]-’ = x1(1 t-a,) M xlccl (the inequality olj % 1 denotes that in the equilibrium state the probability of finding the electron in the chain is
48
V.
N.
KHARKYANEN
ET
AL.
essentially less than the probability of its occurrence in the donor or acceptor) and the expression (A14) it follows that there is quasi-equilibrium between the donor level E(r) and zone levels Ek at any time t > r(l) and the final establishing of the equilibrium in whole, the system with the acceptor is realized for a time r = Ik,j-r. Therefore, at r > t % r(l) ala2 yll(t) ES(l+a,)(l+cc,+&je-; 1
ata2 YZZW
It + c(1-* l+a,+a, -
21
x2a2 cI1
1 +
xlal
e-2
a2
+
. (Al5)
1 +a, +a2 ( > The temperature dependence of the time pduring which the system finally reaches equilibrium has, according to the expressions for values xj, aj, the form 1 z = x2 z N B, exp [-/?(E,--2/LI -EC”]. (‘416) (1 +a,)(1
+a1
+a,)
Xl%
Thus, from (A6) it follows that the electron-donor-acceptor transfer is the more effective, the stronger the relaxation interaction of the acceptor with the chain characterized by the parameter B,. Besides, the transition efficiency increases with the temperature increase, since the probability of a thermal excitation of the electron from the local level EC’) into the conduction band rises with increase of T. It is essential that the value [equation (Al6)] does not include the distance in1 -n,/ between the donor and acceptor. However, it should be remembered that the Boltzmann equilibrium between the stationary levels of the chain Ek is assumed to be quickly established (during the time rs). The limiting stages of the electron transfer are the relaxation on the acceptor and the thermal excitation of electron from the donor into the conduction band. The transfer rate is independent of the way of preparation of the initial state. For example, we can take as an initial state the state when only the stationary level EC’) is populated. We may also prepare a non-stationary state. Whatever the way of preparation of the initial state, the information of initial states disappears during time r(l), since between donor level EC’) and levels Ek the Boltzmann distribution establishes before the electron falls on the acceptor level EC’) for the time z B r(l). At t % r the populations of all the levels achieve their equilibrium values, i.e. : yll(t B 7) 3 y;1 = y&t
9 7) = yi2 =
a1
l+a,+a,
a2
l+al+a,=e
e-fiEWZ-l
=
7
-/mvz- 1 ’
ELECTRON
TRANSFER
IN
PROTEINS
49
6417)
k
The expressions (A17) shows that the electron in the equilibrium state of the donor + acceptor -I- chain system will be delocalized mainly on the acceptor when
The considered mechanism of the electron donor-acceptor transfer according to expression (A16) is quite sensitive to the state of the donor level E(l). So, if in (A16) j&?& -2lLJ -E(l) % 1, the transfer time owing to the thermal excitation of an electron into conduction band is long, and the electron transfer proves to be more rapid due to the donor-acceptor interaction determined by the formulae (A3). In this case the problem is reduced to finding the time z” of the establishment of the electron equilibrium distribution between donor and acceptor when the electron via the interaction V,, between the donor and acceptor levels transfers to the vibronic sublevel of the acceptor E (3) which is isoenergetic to the donor level EC’) and then relaxes to the acceptor level E (‘) . Denoting the resonance interaction V,, between the donor level and the acceptor vibronic sublevel by IV, the electron populations of the levels E(l), Ec2(, Ec3) by yll, yzz, y3s respectively, and taking the relaxation interaction on the acceptor in the form VA = c 74&43% Y
we obtain the set of equations
+&4Q
= c G(?23@ $-?,A9 ”
50
V.
N.
KHARKYANEN
1 I
= Z(a)
Q =
eaR,
ET
g3)-~(2)
AL.
9
6420)
i(Q) = 211C &6(R
- 0,) = F[exp (,Q) - 11-r, Y F is the constant independent on temperature T. The set of four equations (A19) with the normalization condition yr, +y22+y33 = 1 is reduced to the set of three equations whose solution is characterized by three relaxation times. If the relaxation on the acceptor is faster than the resonance transition of electron from the level 1 to the level 3 (see Fig. 6), i.e. if IFI % IWJ, the limiting stage of the electron transfer is resonance transfer. In this case at any time t % T,,, where r, is the time of the equilibrium establishment on the acceptor (z- ’ = I(Q)[exp @?a)+ 11) between the acceptor levels EC’) and EC3), the Boltzmann distribution is established Y33(0 -
=
e-fin.
6421)
Y22(0
Using this equation and the initial condition yrr(O) = 1, ~~~(0) = 0, ~~~(0) = 0, yr3(0) = 0 at t 9 r,, we find the following solution of the system (A 19) yll(t> = ($n+2)-1[1+(eB*+1)
eP*(l-e-tf),
yzz(t) = (es”+2)-’ y33(t) = (ea”+2)-I(1 Imy,,
= - -e2w Z(l2)
e +J,
-e-:rl).
6422)
-/Jn -it e ’
where the time z” for the establishing of the final equilibrium system is given by the formula 1 4w2 -=F (1 +2em8”) $$. z”
state
in
the
w23)
At room and lower temperatures, if Q = E(l)- EC’) = 0.2 eV we have PCl P 1. Therefore, the value z”is practically constant for a wide temperature range, 2”- ’ = 4W2/F, but according to the determination of the value W = V,, [see equation (A3)] it exponentially decreases as the distance between donor and acceptors groups along the chain increases. The expressions (A16) and (A23) for relaxation time, allow one to investigate the picture of the electron donor-acceptor transfer through a regular polymer chain both at high and low temperatures.
Donor-Acceptor
Model of Electron Transfer Through Proteins
V. N. KHARKYANEN, E. G. PETROV AND I. I. UKRAINSKII Institute of Theoretical Physics, 252130 Kiev, U.S.S.R. (Received 11 August 1977) It is shown that the interaction of proteins with a polar solvent leads to the formation of electron donor groups in the proteins. The highest occupied level of these groups in situated near the bottom of the conduction energy band of protein. The transfer of an electron from the donor group through the conduction band of protein to the spatially removed acceptor group is considered including the possible relaxation processes.
1. Electronic States in Polypeptides In 1941, Szent-Gykgyi put forward the hypothesis that the transformations occurring during biochemical reactions at any part of a protein macromolecule are transferred to its other parts due to electron migration along a polypeptide chain. Szent-GyGrgyi (1941, 1957, 1968), Ryll (1948) and some other authors (see references cited by Gutman & Lyons, 1967) emphasized the fact that the ideas on electrons moving along protein are important for understanding the mechanism of many biochemical processes. For example, the scheme of enzymic redox reactions proposed by Cope (1964a,b) shows that the rate of radical formation is limited by the velocity of electron motion through protein. It is now clear that to elucidate the electron transfer mechanism and the accompanying electron-conformation transformations in biosystems is one of the most important problems in bioenergetics. A study of the nature of charge separation and electron transport can give a deeper insight into molecular machinery, thus making it possible to simulate bioprocesses. The concepts of physics, in particular, very fruitful in these applications.
those of solid-state
determine
helicity
physics, appear to be
For example, the Davydov (1968) exciton theory for molecular crystals was used to study the migration of optical excitations over chlorophyle molecules in chloroplasts of green plants and in photosynthetic bacteria. This theory has also been successfully applied to the degree of protein
in studies of hypochromic
effect
29
0022~5193/78/07o6-0029$02.00/0
0 1978 Academic Press Inc. (London) Ltd.
30
V.
N.
KHARKYANEN
ET
AL.
(Volkenstein, 1975). Davydov (1973, 1974, 1975, 1977) developed the exciton theory of the deformation of a polypeptide chain and proposed, on its basis, the soliton model of muscle contraction. Although the migration of uncharged excitations over protein appears to be an established fact and it plays an important physiological role in certain cases, the electron migration over protein remains questionable as yet. The experimental verification of the Szent-Gyorgyi hypothesis performed by Eley with co-workers (Eley, Parfitt, Perry & Taysum, 1953; Cardew & Eley, 1959; Eley & Spivey, 1960; Eley, 1962) and by Rosenberg (1962u) has shown that the current excitations of protein are separated from its ground state by the energy gap AE w 2.3- 3.5 eV. Theoretical calculations of polypeptide energy band structure have given a still greater value of the gap. Thus, at physiological temperatures T w 300 “K (N O-025 eV) the protein is a good insulator. This fact forced many authors to conclude (Blumenfeld, 1974; Volkenstein, 1975) that electrons cannot migrate through a polypeptide chain in biological reactions. (Most biological reactions occur in the dark, therefore the optical excitation of an electron in the protein conduction band should be excluded.) In order to elucidate what objections are usually raised against a possible physiological role of protein electroconductivity, we shall consider the experimental data on electronic conductivity of proteins and the quantum mechanical calculations of the energy band structure. In the above experiments the D.C. conductivity was evaluated with the formula IS = a,(T)*exp (-AE/2kT), where a,(T) is th equantity weakly depending on temperature T. This formula, however, does not give a correct value of the energy gap for current excitations in a protein macromolecule, since the experiments were made using pellets. The value of AE characterizes the potential barriers for the electron transfer between protein macromolecules (see, e.g. Ladik, 1972). As a matter of fact these experiments can allow us only to conclude that the energy gap between a conduction and an upper filled electron energy band does not exceed the measured value of AE (provided that the impurity contributions into conductivity are negligible). The experiments of Eley (1963) and Rosenberg (1962a,b) on hydrated proteins have shown that the conductivity increases exponentially (because the effective value of AE decreases) with increasing hydration. Rosenberg associates this fact with decreasing barrier of the electron transition between macromolecules, the decrease in BE by 0.4 eV being su.lBcient to increase the electro-conductivity by a factor of lo8 at T = 300 “K. There are no reliable experimental data on the value of E for proteins, since the contribution of intermolecular electron transitions and that of impurities have not been evaluated. In order to determine the electronic conductivity of protein, more direct experimentation is necessary.
ELECTRON
TRANSFER
IN
PROTEINS
31
The electron energy bands of polypeptide chains were tist calculated by Laki (1942), Evans & Gergely (1949), and later refined by Suard, Bertheir & Pullman (1961) and Ladik (1963, 1964). The calculations were made for the a-helical structure of a polypeptide chain stabilized by hydrogen bonds I I between plane peptide groups H-N-C&O. The allowed energy band widths obtained (0.1-O-2 eV) are due to the weak overlapping of n-electron orbitals of neighbouring peptide groups, and the energy gap (up to 5 eV) is connected with the,n-electron states in a single peptide (amide) group. The lowest vacant band originates from a n*-antibonding orbital delocalized over peptide N-C=0 group. The above calculations were evidence that the protein electronic conductivity at physiological temperatures is negligibly small. However, Brillouin (1962) supposed that due to the presence of the H-C-R
groups in proteins,
the radicals R can appear as impurities, whose energy levels fill up the space between the highest occupied and the lowest vacant electron zones. According to Brillouin, the energy bands are also formed by electron states of the o-type, this resulting in seven filled and seven vacant allowed energy bands (see, e.g. Ladik, 1972). The calculations of the electronic structure of proteins taking into account the radicals R have not been performed. However, several detailed calculations on 7~ and a-bands for polypeptide chains of poly+alanine and polyglycine have been made since 1970 (see papers by Fujita, Imamura (1970), Imamura & Fujita (1974) Duke, Eilers & O’Leary (1975) and the references listed in these papers). These calculations have given the energy gap to be more than 10 eV. This value is two-three times greater than that obtained earlier in the n-electronic approximation. Moreover, this value is not in agreement with the experimental energies of first optical transitions in the protein chain which have been established to be 5.3 and 6.7 eV (see e.g. Volkenstein, 1975). This fact indicates that the approximations used for describing the excited states of polypeptide chains are insufficient. One of the limitations of the calculations mentioned above seems to be the employment of a minimal atomic basis set which includes only the Slater orbitals of valent and inner atomic shells, more exactly, the 16 24 a-&z functions for the atoms of the second row (N, C, 0) and the 1s orbital for the hydrogen atom. It is known, however, (see, e.g. Frish, 1963) that in optical spectra of the second row elements, the Rydberg states (shells next following the valent ones) play an important role. In particular, in the cases of carbon, nitrogen and oxygen atoms which form the valent core of a polypeptide chain, the optical transition between terms of the type
32
V. N. KHARKYANEN
ET AL.
(Is)~(~s)~(~P)” + (1~)~(24~(3p”-~3s corresponds to an energy of about 7 eV. Even this value is less than the gap between the upper filled and the lowest vacant energy band obtained in pointed above papers. Thus, taking into account the 3s and 3p states is rather important for a correct description of excited states of molecules and polymers consisting of the second row elements (an exception should be made for the conjugated systems). Such calculations involving 3s and 3p states are now performed only for a number of small molecular systems (see Buenker & Peyerimhoff, 1975a,b; 1976). Of these systems ethylene C,H, and ethane C,H, are of great interest for our purposes. A very interesting result is that the lowest optical single excitation of ethylene is connected with the transition 71-+ 3s rather than the transition between z-electron levels, rc + rc* (see Fig. 1). In ethane the vertical transition 0(2x, 2p) -+ 3s corresponds to an energy of 9.16 eV, while the transition 0(2s, 2p) + 0*(2s, 2p]) lies in the continuous spectrum. “&(///%,:i’::;; :; :‘: ‘;:
(a)
I
,,;
,i
,:
-+Vacuumt level
(6)
FIG. 1. The scheme of the lowest optical transitions in ethane (a) and ethylene (b).
Using these data we construct a qualitative picture of the electron levels of a polypeptide chain. First of all we answer the question of why more rigorous calculations of the polypeptide energy band structure with allowance c- and a-electrons result in bad agreement with experiment. One of the reasons for this disagreement is connected with the absence of 3s and 3p orbitals for carbon, oxygen and nitrogen atoms in the basis sets used. Indeed, an optical transition of the n + 3s type in polypeptides should have an energy lower than that of ethylene owing to the splitting of atomic 3s, 3p levels into allowed energy bands, i.e. AE(rc + 3s) < 7 eV. The states of these bands would probably be mixed to a great extent with non-bonding MO’s of the n-type of the peptide groups 0=X-N, since the polypeptide
ELECTRON
TRANSFER
IN PROTEINS
33
symmetry group does not contain elements allowing to perform a rigorous o--71separation of electron states. As to the electron structure of the polypeptide ground state, i.e. the structure of valence energy bands, the calculations in the minimal atomic basis sets seem to have correctly revealed its properties. According to Fujita & Imamura (1970) the upper filled energy band of polyglycine originates from the n-states of lone pairs of N-C&O groups and bonding z-electronic states slightly mixed with bonding o-states. The optical bands 5.3 eV and 6.7 eV of proteins can be connected with n + (3s, n*) and 71--f (3s, x*) transitions. The n -+ D* (2s,2p) and 0(2s, 2~7) + 0*(2x, 2~) transitions appear to be situated as in the case with ethylene and ethane in the continuous spectrum. We can thus suggest that the energy bands of polypeptide chains have the structure given in Fig. 2. Taking into account the diffusive nature of 33, 3p orbitals of C, N, 0 atoms and using the well-known Slater rules for qualitative evaluations we can obtain that the bottom of the lower vacant band corresponds to energies of about N 1 eV (neglecting the core electron polarization effects) and the width of this band is of the order of a few electron-volts. Let us compare the energy band structure of a polypeptide chain, given in Fig. 2, with the results of n-electron methods often used for the treatments of optical experiments. According to 7r-electronic approximation the lowest vacant and highest occupied energy bands are of the n-type whose small
Conduction band
J
Valent energy band n,a(2p) u (2s,2p)
FIG. 2. The scheme of the upper filled and vacant states of electrons of the covalent core of a polypeptide chain. 2
34
V.
N.
KHARKYANEN
ET
AL.
widths are connected with the weak overlapping of rc-electron orbitals of O=C-N groups in the a-helical sections of a protein chain. When the helicity is disturbed n-electrons are localized in the peptide groups, since the H-C-R
fragments of the chain separating the peptide groups do not
contain rc-electrons. This circumstance is of great importance, because the n-electron protein system breaks at these sites and the rotation about ordinary C-C and C-N bonds is responsible for the conformational flexibility of a protein chain. As a result, the electron transport along the n-electronic system of proteins is impossible. Therefore, if the electron is forced into the vacant energy band thermally or optically, its motion is possible only within a-helical fragment of the chain. The same is true of a hole, e.g. in the presence of a strong electron acceptor, in the occupied band. According to the scheme proposed here which allows for the mixture of 3s- and rc*-states and the formation of a spread conduction band, the occurrence of an electron in a vacant band can lead to its migration along the chain involving also nonhelical sites. This may be responsible also for the optical excitation migration in proteins observed in the known experiments by Lautsch, Paceday, Sommer, Julius & Bodefeld (1959). We shall argue here that under certain circumstances the lowest 3s, rc* vacant energy band of a polypeptide chain can effect electron transfer from electron donor to acceptor in dark reactions, although the conduction band itself is separated from the upper filled band by an energy gap of about 5-8 eV, and the thermal excitation of an electron to this band from the occupied ones is not practically realizable. In other words, the insulator properties of a polypeptide chain and its conformational flexibility are not an impediment for electron transfer from donor to acceptor through protein macromolecules. 2. The Mechanism of Electron Donor-Acceptor Transfer along a Polypeptide Chain The energy scheme represented in Fig. 2 should be completed by electronic levels of protein R-groups which, according to Brillouin’s assumption (Brillouin, 1962), can be situated between the upper filled and the lowest vacant electronic energy bands of a polypeptide chain. However, Brillouin did not consider which are the R-groups that could have the electron energy levels close to the conduction band of the peptide chain core. We wish to emphasize that two of twenty R-groups of amino-acid residues, namely -CH,---COOand -CH2-CH,-COO(residues of asparagic and glutamic acids, respectively) have negatively charged groups -COOat
ELECTRON
TRANSFER
IN
PROTEINS
35
physiological values of pH = 6-8. Besides that, at the same pH the unbound -COOgroup of the C-end of the chain also possesses a negative charge. The problem arises as to what energy level corresponds an extra electron in the -COOgroup. To elucidate this question, we turn to experimental data on the electronic properties of similar molecular fragments and polymers. According to the measurements of the electron affinity of radical OH, the energy of the extra electron in OH- is equal to 2.2 eV, while in the Hz0 molecule the highest occupied level possesses an energy of 12.7 eV (the ionization potential of the water molecule). Thus, the breaking of a proton away from a molecule results in an increase in the energy of one of the electrons by about 10 eV. Almost the same situation occurs also in the case of the -COOH group. The electron affinity of polymers with saturated bonds in the main chain (polystyrene, polymethylmethacrylate) amounts to 3-4 eV (Duke & Fabish, 1976). The results obtained by Duke & Fabish (1976) show also that in polymer systems with saturated bonds where n-electronic states are absent, the injected electrons can move along the chain. Thus, the energy of an electron in the -COOgroup is close to that of the conduction band of the chain. Solvation of the -COOgroup does not change essentially this situation. Indeed, since -COOH group transforms into hydrated ions -COOand H+ at pH = 6-7 (see e.g. Lehninger, 1972), the energy difference in this process, AE = pH In 10Tat T = 300°K w 0.025 eV should not exceed 0.3 eV. The gain in proton energy due to hydration is equal to about 10 eV (Klotz, 1962). The loss in electron energy when a proton is removed is, according to the above data, 10-12 eV. Accordingly, the solvation energy of the -COOgroup and hence electron level lowering is less than 1 eV. Thus, if protein has a hydrated -COOgroup, the extra electron of this group has the energy close to the edge of the conduction band, and consequently, is partly delocalized due to overlapping of its wave function with 34 rc* orbitals of a polypeptide chain core. If there is an appropriately situated acceptor moiety which possesses an empty orbital with energy below the donor level of the -COOgroup, then the electron can be transferred along the polypeptide chain to the acceptor. We note, however, that in viva (T = 300” K, pH = 6-8) -COOgroups are not free as a rule. They form, for example, salt bridges (or hydrogen bonds) with NH: and NH+ groups of amino acid residues:
I -COO-.
. .H+-NH,
uncl -COO-.
. .H+-N<.
36
V.
N.
KHARKYANEN
ET
AL.
In this case a salt bridge can be realized both between two fragments of a chain (stabilization of the tertiary structure of proteins) and between fragments of two different chains (stabilization of the quartic structure of oligoproteins). Owing to the conformational transformation of proteins there may occur a rearrangement of salt bridges, resulting in rupture of individual bridges. When the salt bridge breaks, the -COOgroup of a corresponding amino acid residue becomes free and the excess electron becomes highly energetic. The energy increase of an excess electron is compensated by the energy of the hydration of the NH: group or by the hydration energy of a proton. Now we consider the model of electron donor-acceptor transfer through a polypeptide chain. We denote by D- the electron donor group attached to the chain core. As we noted above, at pH = 6-8 the C-end of the chain and the residues of asparagic and glutamic acids can play the role of this group. Let us denote by H’Bthe group connected with the donor moiety D- by a salt bridge. The N-end of the chain, the positively charged groups of amino acids can appear as H’Bgroups. Since the break-up of the salt bridge requires the free energy about 0.1-0.2 eV at pH = 6-8 and kT = 300” K the thermodynamic equilibrium shifts towards the formation of the salt bridges, but the thermodynamic probability of the salt bridge breaking differs from zero. When the salt bridge breaks up the energy of the excess electron of the -COOgroup rises to the bottom of the protein conduction band (the electron states in this band can be of a polaron type, in general case, since the conformational change of protein can assist the electron migration along the chain). When the energy of the extra electron in the -COOgroup becomes higher than the bottom of a conduction band, the relaxation to the lowest level of the band may occur owing to the interaction with thermostat. In this case the lowering of electron energy should not exceed the energy cost of the rupture of the salt bridge, i.e. the value of 0.142 eV since otherwise the thermodynamic equilibrium will be displaced towards breaking of the salt bridge. Generally speaking the interaction of donor orbital of -COOgroup with the conduction band states leads to splitting a local level off the band (if the donor level falls into the band) or to the repulsion of the initial donor level from the band (if the donor level is near the bottom of the conduction band). In both cases the conduction band states mix with the donor orbital of the -COOgroup. The interaction of the local level with zone levels is considered in the Appendix. When the electron donor group is not bound with cation group by the salt bridge, the energy of its excess electron is near the protein conduction band and the electron can be excited thermally into the conduction band.
ELECTRON
TRANSFER
IN
PROTEINS
37
Then, during the time z(I) [see formulae (12) and (13) in Appendix] between the levels of the protein vacant energy band Ek and the electron donor level EC’) the Boltzmann distribution of electrons is established. If we now join the acceptor moiety to the protein chain, the electron transfers to the acceptor during some time z % r(l) [see formulae (16) and (23) in Appendix]. In the case when the electron donor and electron acceptor moieties are joined to the protein chain but the pH of the medium is low so that the equilibrium of the reaction -COO- +H+ +r -COOH corresponds to the formation of the -COOH groups the probability of electron donor-acceptor transfer is essentially reduced since all the electrons on the -COOH group possess the low energies far from the vacant energy band of protein. If we increase the value of pH, the above reaction equilibrium will be shifted towards the formation of -COOgroups and hydrated protons and hence to the possibility of electron transfer to the acceptor through the protein chain. The -COOgroup itself becomes an electron acceptor when its excess electron is taken off. In the absence of exogenous donors of electrons, the unsaturated valency of the -COO* group can form a probably weak bond with other protein groups or solutes possessing unsaturated valencies, e.g. : -c/O-%c-
&O--s
10 o/ 10 ’ The formation of these bonds lowers the energy of the system and consequently decreases the activation energy of electron transfer. Besides this, the -COO* group can be reduced by exogenous donors. In this case, which probably is more realistic, the -COOgroup functions as a mediator of electron transfer from donors into the conduction band of protein. We have considered the -COOgroup as an example of protein fragments essential for electron transport processes. Other protein groups can also appear as electron donors or electron transfer mediators (ETM) in phenomena of electron donor-acceptor transfer through proteins. Thus, we mention that amido-, amino- and SH-groups of proteins appear as electron donors with respect to acceptor S-S bonds in biochemical reactions (see, e.g. Szent-Gyorgyi, 1968). Recently Atanasov et al. (1977) have shown that histidine residues A10 and GHl of myoglobin take part in the reaction of electron transfer from myoglobin to ferricytocbrome C. In our opinion it may occur that histidine appears as ETM in this process. Generally speaking, we suggest that electron donor or ETM groups in proteins appear due to the interaction with polar solvent, i.e. due to solvation of cations splitting off the groups: -A--A\ 4-f -AB it -A+ B+, &BQ+B
38
V.
N.
KHARKYANEN
where A- and --Aare donor metallic or organic cations.
ET
groups
AL.
B+ and B++ are solvated
(ETM),
Figure 3 gives the scheme of energy levels of the system of donor + acceptor + polypeptide chain. This scheme reflects the main features of the electron transport model proposed. The mathematical aspects of the model are discussed in Appendix. The kinetics of the electron transfer from donor to acceptor is described by the expressions (14), (15) and (22). The presence of a conduction band located not far from the donor and acceptor levels
A=0
(a)
(b)
FIG. 3. The energy diagram of the electron-proton states of the donor + acceptor -Ipolypeptide chain system (a) and the energy diagram of the electronic states of the same system (b). EC”) is the electron energy in the donor with a proton attached to the donor, E(‘) is the electron energy on the donor with a proton passed into the medium and hydrated, IP is the electron energy in the acceptor, E. is the energy of the electron at the centre of the conduction band, EcH+, is the energy of the proton hydration. The dotted line in Fig 3b shows the increase in energy of the electron when the proton moves to a polar medium. The energy distances indicate the order of the corresponding values.
provides the effective electron transfer from donor to acceptor. At the same time, the conduction band is separated from the valent ones by the significant energy gap (see Fig. 2). So, in the process of electron donor-acceptor transfer we deal with an impurity conductivity of proteins. The appearance of the impurity-namely, of the donor group with a highenergy electron-is a result of interaction with polar solvent. We may say that the polar solvent plays the role of temperatures exciting the electron of a donor group up to the high energy level. The role of real temperature is now reduced to the excitation of an electron from the highly energetic donor
ELECTRON
TRANSFER
IN
PROTEINS
39
level into the protein conduction band. If the energy gap between the donor level and the bottom edge of the conduction band is about 0.1-0.3 eV, then room temperature can provide the effective electron transport in dark reactions. During the electron transfer from donor to acceptor the total energy of the system is lowered by a value equal to the difference of electron energies of donor and acceptor levels. This energy can be consumed for the transformations of the protein conformation, the formation of new bonds, heat, etc.
3. Conclusion
The model of electron donor-acceptor transfer through proteins presented in this paper shows that despite the insulator properties of proteins, electrons can be transferred through polypeptide chain in dark reactions. The essential point is that in the electron transport model proposed here, the electron is not excited from the valent energy bands of protein into the conduction band. The highly energetic electron of the donor group is transferred, but not the electron of the covalent core of the polypeptide chain, as was assumed earlier. For the polypeptide chain core this electron is excessive and therefore it can move in the conduction band of proteins. In this process we deal with the impurity conductivity of proteins. Due to the significant contribution of the Rydberg (3s) atomic states in the formation of the conduction band in proteins, the excess electron mobility in fact weakly depends on protein conformations. We suppose also that the formation of the electron donor group is a result of the interaction of a protein macromolecule with a polar solvent. So, the donor group should locate in the hydrophilic part of proteins. We wish to note that the model of electron transfer proposed in the present paper gives a new interpretation of Szent-Gyorgyi’s ideas on the role of protein as electron conductors. Our interpretation of Szent-GyBrgyi’s hypothesis is argued by a number of facts of common knowledge from the fields of molecular physics and chemistry. This model can be checked in experiments on the synthesized polypeptide chains. For example, if we put the polyglycine chain through the artificial membrane, on one side of which there is water and on the other there is a non-polar liquid (Fig. 4), and place the C-end of the chain in the aquomedium, then attach the acceptor (such an acceptor can be, e.g. heme, heme + 0,) to the chain part in a non-polar medium. Next changing pH of the aqueous solution from the low (when -COOH group is stable) to the physiological values (when the -COOH group dissociates in hydrated fragments -COOand H+) we should expect the electron transfer from the
40
V.
N.
KHARKYANEN
ET
AL.
FIG. 4. The scheme of the experiment to study the possibility of the electron donoracceptor transfer through polypeptide chain due to the change of pH in aqueous solution.
-COO-
group to the acceptor, and from exogenous donors to the -COO
group.
Besides -COOgroup acting as the electron donor, a compound which can be excited by light in such a way that the energy of one of its electrons becomes comparable with the energy in the vacant band of the polypeptide chain can be used. In this case the compound acts as a photodonor of electrons. After illumination the compound is first oxidized with the reduction of the acceptor during the time zl, then it is reduced again in z2 9 21 time, removing the electron from the acceptor (Fig. 5). Polvoeotide
FPolypeptide i
conduction I-\
,------”
FValent
chain
band
-
------------.
energy
bond
,-
I-bhD
EPh D
of the chom --_--.
-.--
5. The scheme of the experiment to detect the electron transfer from a photodonor optically excited (PhD) to the acceptor one. E P,,Dis the PhD ground state energy, EP~D is the PhD excited state energy. Under the influence of light with frequency w1 = l/fi (E&D - EphD) the electron enters the conduction band from where it relaxes to the acceptor for the time zI. The inverse transfer is produced during time za and followed by the light emission with the frequency aa, (ta > tl). FIG. moiety
ELECTRON
TRANSFER
IN
41
PROTEINS
The experimental study of the kinetics of the electron donor-acceptor transfer through the polypeptide, e.g. the measurement of the dependence of the electron transfer rate on the distance between the donor and the acceptor along the chain, on temperature and pH of medium, is of a great importance in elucidating the role of electron conductivity of proteins. The mechanism of electron transport through protein macromolecules described above very likely plays a certain role in a number of biochemical reactions involving electron transfer processes. This may be true for such processes as electron transfer in electron transport chains, electron transfer from cytochrome to photocentre, formation of transmembrane potentials, and so on. Electron transfer through proteins can also be responsible for allosteric interactions in enzymes, origination of gating currents in native membranes and mechanisms of opening of membrane channels. These applications of donor-acceptor electron transfer through proteins deserve special consideration, which is now in progress.
B. P., POSTNIKOVA, Molek. Biol. 11, 537.
ATANASOV,
REFERENCES G. B., SADYKOV, Yu. CH., and
VOLKENSTEIN,
M.
V. (1977).
BLUMENFELD, L. A. (1974). Problems of Biological Physics. Moscow: Nauka. BRILLOUIN, L. (1962). In Horizons in Biochemistry (Kasha M. & B. Pullman eds.) New York, London: Academic Press. BIJENKER, R. J., & PEYERIMHOFF, S. D. (1975a). In Advances in Quantum Chemistry. (P.-P. Lowdin, ed.). Vol. 9, New York: Academic Press. BUENKER, R. J. & PEYERIMHOFF, S. D. (19753). Chem. Phys. 8, 56. BUENKER, R. J. & PEYERIMHOFF, S. D. (1976). Chem. Phys. 9, 75. CARDEW, M. H. & ELEY, D. D. (1959). Dis. Faraday Sot. 27, 115. COPE, F. W. (1964a). Proc. natn. Acad. Sci. U.S.A. 51, 809. COPE, F. W. (1964b). J. them. Phys. 40, 2653. DAVYDOV, A. S. (1968). Theory of Molecular Excitons. Moscow: Nauka. DAVYDOV, A. S. (1973). J. theor. Biol. 38, 559. DAVYDOV, A. S. (1974). Stud. biophys. 47, 221. DAVYDOV, A. S. (1975). Ukrain. Fizich. Journ. 20, 179. DAVYDOV, A. S. (1977). Studia biophysics, 62, 1. DUKE, B. J., EILERS, J. E. & O’LEARY, B. (1975). Chem. Phys. Lett. 32, 602. DUKE, C. B. & FABISH, T. J. (1976). Phys. Rev. Lett. 37, 1075. ELEY, D. D., PARFITT, G. P., PERRY, M. J. & TAYSUM, D. H. (1953). Trans. Faraday. Sot. 49, 79. ELEY, D. D. & SPIVEY, D. J. (1960). Trans. Farad. Sot. 56, 1432. ELEY, D. D. (1962). In Horizons in Biochemistry (Kasha M. & B. Pullman eds). New York, London : Academic Press. ELEY, D. D. (1963). May & Baker Lab. Bull. 5, 55. EVANS, M. G. & GERGELY, J. (1949). Bioch. biophys. Acta. 52, 254. FRI~H, S. E. (1963). Optical Spectra of Atoms. Moscow, Leningrad: Fizmatgiz. FUJITA, IX & IMAMURA, A. (1970). .T. them. Phys. 53, 4555. GIJTMANN, F. & LYONS, L. E. (1967). Organic Semiconductors. New York, London, Sydney: John Wiley & Sons. IMAMURA, A. & FUJITA, H. (1974). J. them. Phys. 61, 115.
42
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N.
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ET
AL.
KOSTER, G. F. & SLATER, J. C. (1957). Phys. Rev. %, 1167. I(LOTZ, I. M. (1962). In Horizons in Biochemistry (KashaM.
York, London:AcademicPress.
& B, Pullmaneds).New
J. (1963). Actu Phys. Acad. Sci. Hung. 15, 287. LADE, J. (1964).Nature202, 1208. LADIK,J. (1972).Quantenbiochemie fiir Chemiker und Biologer. Budapest:Akad. Kiado. LAKI, K. (1942).Studies Inst. Medic. Chem. Univers. Szeged. 2, 43. LADIK,
LATJTSZH,
W.,
PACEDAY,
R.,
~OMMER,
I., JULIUS,
H. J. & B~DEFELD,
N.
(1959).
Chimiu
Ziirich 13, 129.
A. L. (1972). Biochemistry, Ch. 4. New York: Worth Publ. Lrrsnrrz, I. M. (1947~). Zh. Eksperim, Theor. Fiz. 17, 1017. Lrrsnrrz, I. M. (1947b). Zh. Eksperim. Theor. Fiz. 17, 1076. Lrrsnrrz, I. M. (1948).Zh. Eksperim. Theor. Fiz. 18, 293. Ovcmov, A. A., UKRAINSKII, I. I. & KV~NTZEL, G. F. (1972).Uspehi Fizich. Nauk. 108,81. pEm&mu, S. V. & JATZENKO, A. A. (1967). Zh. Eksperim. Theor. Fiz. 53, 1327. ~ELETMINSKII. S. V. & PETRov. E. G. (1968).Preprint ZTP-68-20. Kiev: Institutefor Theor. Physics.Acad. Sci. UkrainianS.S.B. ’ ROSENBERG, B. (1962a). J. Chem. Phys. 36, 816. LEHNINGER,
ROSENBERG, B. (19626). Nature 193, 364. R~L, N. V. (1948).Uspehi Fizich. Nauk. 35, 186. Sum, M., BERTHEIR, G. & PULLMAN, B. (1961).Biochim. SZENT-GY~RC+~I, A. (1941~).Science 93, 609. SZENT-GY~~RGYI, A. (1941b).Nature 148, 157. SZENT-GY~RGYI, A. (1957). Bioenergetics (Augenstein,L.
Press.
SZENT-GY~~RG~I, A. (1968). VOLKENSTEIN, M. V. (1975).
biophys. Actu. 52, 254.
G. ed.) New York: Acadmic
Bioelectronics New York, London: AcademicPress. Molecular Biophysics. Moscow:Nauka.
APPENDIX
The Electron Donor-Acceptor Transfer through a Polymer Chain with Account for Relaxation Processes We consider a regular chain (or quasi-one-dimensional crystal) with the electron donor group D- and electron acceptor group A attached to the n,th and n,th sites of the chain, respectively [see Fig. 6(a)]. Let us denote by E,, the energy of an electron in the lowest vacant state of any site of the chain, then, E, and E, are the energies of an excesselectron in the donor and acceptor groups respectively (E, - E,), L is a hopping integral for neighbouring sites of the chain which are separated by the distance a, Tr is the electron resonance integral between the donor group D- and the n,th site of the chain, T2 is the same integral for acceptor group and n,th site of the chain.
ELECTRON
TRANSFER
IN
. . . 0 “‘rp . . . & 8
43
PROTEINS
“’ p . . . 8D
A
(a)
1 a ttti
_/.-.-.-. -.-. 42
E(3)
I I + $1)
E(2)
(b)
FIG. 6. The polymeric chain with donor and acceptor (a). Two mechanisms of the electron donor-acceptor transfer through the chain (b). The thermal excitation of an electron from the donor into the conduction band (straight upward arrows); the electron relaxes to the ground level of the acceptor /W (the straight arrows directed downwards). If the thermal excitation is inefficient, the main contribution to the probability of the electron transfer can be made by the second mechanism connected with the electron transport via the direct interaction VIZ from the donor level E (I) to the vibronic sublevel of the acceptor Ec3). As a result of oscilatory relaxation, the electron is transferred to the level Ec2) (dotted and wavy arrows). Both the mechanisms lead to the losing in electron energy by the value E(l) - E@).
In order to consider the electron relaxation to the equilibrium state in the system of the polymer chain - donor + acceptor (or the electron donoracceptor transfer) we need the electron eigen states in this system. We determine them using the well-known results of impurity centres in onedimensional crystals [see papers by Lifshitz (1947a,b, 1948), Koster & Slater (1957), Ovchinnikov, Ukrainskii & Kventzel (1972) and references cited in these papers]. We shall use also the tight binding approximation and suppose that the impurities-donor and acceptor-can be described by the shortrange potential. If the direct interaction between the donor and the acceptor groups is absent, the energies of electron local levels E(p) (/A = 1,2) which lie below the bottom of the conduction band of the chain, Eo-2jLI > E@), can be found from the equation where Iv,kl” R, = Ej-E(@) - 2 ___ k &--E’“”
Ek = E,+2L
cos ak,
W9
44
V. N. KHARKYANEN
ET AL.
k is the wave number, N is the total number of the chain sites. As it can be seen from expressions (Al), (A2) the value of IJZ, which can be written ast:
-(E,-E'"')+J(E,-E'"))'-4LZ, 2L
,4 , < ,,@3) P
if E") < E,-2jLI, characterizes the effective interaction of the donor group with the acceptor group for a given value of the energy E”‘. It is essential to note that the value of I’,, decreases markedly (by exponential law) with increasing number of unit cells In, --n,l-between the donor and acceptor groups. From the calculations, e.g. by Duke, Eilers & O’Leary (1975), it follows that the width of conduction band in polyglycine (41LI in our notations) is of several electron-volts. So, the value of hoping integral, L, is about 1 eV. The same value can be taken for the vacant band arising from 3s states. Assuming that IL1 N 1 eV, 17’rl N O-5 eV, lTzl N 0.1 eV, E, - EC") = 2lLl+O*2 eV, l~~-n~) = 10, we obtain from expression (A3) for I’,,: IV,,1 N 10m4 eV. The smallness of the value of I’,, points to the fact that the energies Ecp)can be found with a good accuracy by putting equal to zero the values of Rjj. From the equations Rjj = 0 (j = 0 for the donor and j = 2 for the acceptor) we obtain the following expression for EC")
(E,-2)L(
> E@')
From this equation we can find the energy of an electron in the donor group, E(l), and in the acceptor group, EC'). The wave functions +r and tj2 which correspond to these levels, are localized near the donor and the acceptor. The larger the energetic distance of a local level from the edge of the conduction band, i.e. the value IA',,-2(LI -E(1,2)l, the stronger the localization of the wave functions. We now consider the electron donor-acceptor transfer. The system Hamiltonian has the form H, = c E,A:A,+E’1t4:A, +E(2)A:A2, (A4) where E,, E(l), EC') are the introduced above energies of electron in the conduction band, in the donor and acceptor local levels, Akf, A:, A: (& A,, A,) are the creation (destruction) operators of electron on the corresponding state. tWe obtain the expression for VI, from equation (A2) integrating over k.
ELECTRON
TRANSFER
IN
PROTEINS
45
We choose the interaction between our system of the chain + donor + acceptor and the thermostat in the usual form: V=
C
C
j=l,Z
(K:kAi+Ak+?CljA:Aj)(Bj,+Bi+,),
vk
645) Here ICYis the interaction parameter characterizing the electron transfer from the donor (j = 1) or from the acceptor (j = 2) to the nth unit of the chain. This transfer is accompanied by an emission or absorption of the vth oscillation mode of the donor or the acceptor group; B& Bjy are the creation and destruction operators of vibron. The chain + donor + acceptor system can be considered as a subsystem with a finite number of degrees of freedom, placed into thermostat. The establishing of equilibrium in the subsystem is described [see, e.g. Peletminskii & Jatzenko (1967), Peletminskii & Petrov (1968)] by timedependence of the matrix elements yl,(t) of the density matrix of the subsystem p(t), yr,(t) = (Zlp(t)lm > = ,S,p(t)$j,,.In our case land m correspond to the states with the energies &, E (l) , EC” and fI,,, are the following operators : /3kk’= A:Ak, = ~~k, 9jk = Af A, = ~~, ~jj = AfAj
= ~~, 912 = A:A2 = p,:.
Using the definitions of yl,,, and rr,(t), expressions (A4) and (A5) and the equations for yl,(t), obtained by Peletminskii & Petrov (1968), we obtain fl
Yjj(t)
=
-
2
I~k[~jj(t)njv-Ykk(t)(l+nj,)l,
it
Ykkct)
=
-
c vi
lyk[ykk(t)(l
+
njv)-yjj(t>njvl,
where Ijk E ~~~IIc~~I~~(E(‘)-E~-~o~,), and njv = [exp (BttOj,)-l]-’ is the Bose distribution function for the oscillators with the frequency wjV at temperature T, /I = (k,T)-‘, tz and k, are the Plank and Boltzmann constants, respectively. It is a very difficult problem to solve the set of A’+2 equations (A6) for a general case. We suppose that the interaction of the chain with the thermostat establishes the Boltzmann distribution of electron over the levels Ek for the time rg, but the establishing of the electron distribution between levels E(lB2) and zone levels Ek is realized during time r(l), r(‘) $ rB. If r(l), rc2) $ zs, we can describe the establishing of a statistical equilibrium in the
46
V.
N.
KHARKYANEN
ET
AL.
chain + donor + acceptor system at any time t >> zg assuming that there is a Boltzmann quasi-equilibrium between levels Ek, namely ykk(t) = y&t) ewPtEkmEo)= y&t) e-pZLcosak. (A7) Substituting the condition (A7) of the quasi-equilibrium into the set of equations (A6) and using the normalization condition Yll(o+Y22(0
+ T Ykk(Q = 1
we obtain the set of two equations a ~tYll(o
= -X1~~l+~l)Yll(~)+~lY22(t~-~,l~
(‘48) ;
Y22(0
=
+~z>r22(0-~21,
-X2Ccf2Yll(o+(l
where Xj = i
5 IJknj, = Bj[exp
(pAi)-
I]-‘?
Aj s E,-21LI
ep(Eo-Eco)
e@(b-E(~‘)
-E(j),
(A9)
is the constant not depending on temperature, I,,(x) is the Bessel function of the zeroth-order imaginary argument. If jL( N 1 eV, then at room temperature j&Cl % 1 and, hence, the Bessel function can be replaced by its asymptotic expression :
Let us solve the equation (A8). First of all we investigate the case when the acceptor is absent. In this case we obtain only the equation ;tYll(o
= -X1C(l+a,)Y,,(t)-a,l.
The solution of this equation with the initial conditions Yldt = rid N” Ydt = 0) = Yll(O> has the form
r1d0 = r11(0) - & [ As follows from equation
(All),
1I
e-.Yl(l+al)t ) c11 .
the equilibrium
(All) l+a, electron distribution
ELECTRON
TRANSFER
IN
47
PROTEINS
between levels of the system of chain + donor at t % zB is realized for the characteristic time z(r), and according to expressions (A9) and (AlO) 1 1+2JnB# l/N exp (PAI) - = x1(1 +a,) = B, 6412) z(1) exp U%> - 1 . If the level EC’) lies outside the edge of the band E,,-21L( by the value of O-2 eV or more, then at room temperature or lower, one in the denominator of expression (A12) the unity can be omitted so that -1 ’ ,,/m + exp (-/?A;) . G413) Tu) = Bl jy
1
At t g T(I) the values yll(t),
c ykk(t) which characterized the populations
of the donor level and conduc;ion band, are practically and coincide with equilibrium values
time-independent
Solving the set of equation (A8) we assume that initially
i.e. between sun-band donor level and the chain, the thermodynamic equilibrium is established and acceptor level is empty. Under these conditions the solution of the system (A8) has the form L k,-k,
y11(f)=(l+oll)~~~al+a2)
(-k,
1 Y22(t) = ~+a,);;?~,
+a,)
ekit+kl
ekzt) + ,+,“l,,
1
, 2
k,Ck, +x1(1 +a,) ek,t-
k, -k,
XlEl
k,Ckz+xlU+dJ
ekzt Xl%
+ i
a2
l+a,+a,
G414)
where
kl = 3{-Cxl(l+al)+~2(1+az)l 2
t- JC~l(l+cr,)-~z(l+a211~+4~l~(I~2~(2).
Taking into account the condition EC’) > Ec2) and the definition values xj, Olj, WC see that X~CQ % X~CZ,,CX~% ~1, ~1 % ~2. Therefore k1
N” -x2
2,
k2
,”
of the
-xla,
and hence Ik, 1 Q Ik21. From the comparison of the value k, with the value [d’)]-’ = x1(1 t-a,) M xlccl (the inequality olj % 1 denotes that in the equilibrium state the probability of finding the electron in the chain is
48
V.
N.
KHARKYANEN
ET
AL.
essentially less than the probability of its occurrence in the donor or acceptor) and the expression (A14) it follows that there is quasi-equilibrium between the donor level E(r) and zone levels Ek at any time t > r(l) and the final establishing of the equilibrium in whole, the system with the acceptor is realized for a time r = Ik,j-r. Therefore, at r > t % r(l) ala2 yll(t) ES(l+a,)(l+cc,+&je-; 1
ata2 YZZW
It + c(1-* l+a,+a, -
21
x2a2 cI1
1 +
xlal
e-2
a2
+
. (Al5)
1 +a, +a2 ( > The temperature dependence of the time pduring which the system finally reaches equilibrium has, according to the expressions for values xj, aj, the form 1 z = x2 z N B, exp [-/?(E,--2/LI -EC”]. (‘416) (1 +a,)(1
+a1
+a,)
Xl%
Thus, from (A6) it follows that the electron-donor-acceptor transfer is the more effective, the stronger the relaxation interaction of the acceptor with the chain characterized by the parameter B,. Besides, the transition efficiency increases with the temperature increase, since the probability of a thermal excitation of the electron from the local level EC’) into the conduction band rises with increase of T. It is essential that the value [equation (Al6)] does not include the distance in1 -n,/ between the donor and acceptor. However, it should be remembered that the Boltzmann equilibrium between the stationary levels of the chain Ek is assumed to be quickly established (during the time rs). The limiting stages of the electron transfer are the relaxation on the acceptor and the thermal excitation of electron from the donor into the conduction band. The transfer rate is independent of the way of preparation of the initial state. For example, we can take as an initial state the state when only the stationary level EC’) is populated. We may also prepare a non-stationary state. Whatever the way of preparation of the initial state, the information of initial states disappears during time r(l), since between donor level EC’) and levels Ek the Boltzmann distribution establishes before the electron falls on the acceptor level EC’) for the time z B r(l). At t % r the populations of all the levels achieve their equilibrium values, i.e. : yll(t B 7) 3 y;1 = y&t
9 7) = yi2 =
a1
l+a,+a,
a2
l+al+a,=e
e-fiEWZ-l
=
7
-/mvz- 1 ’
ELECTRON
TRANSFER
IN
PROTEINS
49
6417)
k
The expressions (A17) shows that the electron in the equilibrium state of the donor + acceptor -I- chain system will be delocalized mainly on the acceptor when
The considered mechanism of the electron donor-acceptor transfer according to expression (A16) is quite sensitive to the state of the donor level E(l). So, if in (A16) j&?& -2lLJ -E(l) % 1, the transfer time owing to the thermal excitation of an electron into conduction band is long, and the electron transfer proves to be more rapid due to the donor-acceptor interaction determined by the formulae (A3). In this case the problem is reduced to finding the time z” of the establishment of the electron equilibrium distribution between donor and acceptor when the electron via the interaction V,, between the donor and acceptor levels transfers to the vibronic sublevel of the acceptor E (3) which is isoenergetic to the donor level EC’) and then relaxes to the acceptor level E (‘) . Denoting the resonance interaction V,, between the donor level and the acceptor vibronic sublevel by IV, the electron populations of the levels E(l), Ec2(, Ec3) by yll, yzz, y3s respectively, and taking the relaxation interaction on the acceptor in the form VA = c 74&43% Y
we obtain the set of equations
+&4Q
= c G(?23@ $-?,A9 ”
50
V.
N.
KHARKYANEN
1 I
= Z(a)
Q =
eaR,
ET
g3)-~(2)
AL.
9
6420)
i(Q) = 211C &6(R
- 0,) = F[exp (,Q) - 11-r, Y F is the constant independent on temperature T. The set of four equations (A19) with the normalization condition yr, +y22+y33 = 1 is reduced to the set of three equations whose solution is characterized by three relaxation times. If the relaxation on the acceptor is faster than the resonance transition of electron from the level 1 to the level 3 (see Fig. 6), i.e. if IFI % IWJ, the limiting stage of the electron transfer is resonance transfer. In this case at any time t % T,,, where r, is the time of the equilibrium establishment on the acceptor (z- ’ = I(Q)[exp @?a)+ 11) between the acceptor levels EC’) and EC3), the Boltzmann distribution is established Y33(0 -
=
e-fin.
6421)
Y22(0
Using this equation and the initial condition yrr(O) = 1, ~~~(0) = 0, ~~~(0) = 0, yr3(0) = 0 at t 9 r,, we find the following solution of the system (A 19) yll(t> = ($n+2)-1[1+(eB*+1)
eP*(l-e-tf),
yzz(t) = (es”+2)-’ y33(t) = (ea”+2)-I(1 Imy,,
= - -e2w Z(l2)
e +J,
-e-:rl).
6422)
-/Jn -it e ’
where the time z” for the establishing of the final equilibrium system is given by the formula 1 4w2 -=F (1 +2em8”) $$. z”
state
in
the
w23)
At room and lower temperatures, if Q = E(l)- EC’) = 0.2 eV we have PCl P 1. Therefore, the value z”is practically constant for a wide temperature range, 2”- ’ = 4W2/F, but according to the determination of the value W = V,, [see equation (A3)] it exponentially decreases as the distance between donor and acceptors groups along the chain increases. The expressions (A16) and (A23) for relaxation time, allow one to investigate the picture of the electron donor-acceptor transfer through a regular polymer chain both at high and low temperatures.