Proc. Nat. Acad. Sci. USA Vol. 71, No. 9, pp. 3640-3644, September 1974

Electron Transfer Between Biological by Thermally Activated Tunneling (photosynthesis/oxidative phosphorylation/cytochromes) J. J. HOPFIELD Department of Physics, Princeton University, Princeton, New Jersey 08540; and Bell Laboratories, Murray Hill, New Jersey 07974 Contributed by J. J. Hopfield, June 24, 1974

ABSTRACT A theory of electron transfer between two We bypass possible Winfield-like complications, and assume fixed sites by tunneling is developed. Vibronic coupling in that there are no other electron states available at low enough the individual molecules produces an activation energy to transfer at high temperatures, and temperature-inde- energies to be thermally accessible. Section I shows that pendent tunneling (when energetically allowed) at low transfer between two fixed sites, in suitable approximation, is temperature. The model is compared with known results mathematically isomorphic with the conceptually simpler on electron transfer in Chromatium and in Rhodopseudo- problem of excitation transfer by the F6rster (7, 8) (dipolar) monas spheroides. It quantitatively interprets these results, with parameters whose scale is verified by com- mechanism. In Section II, the simplest possible model of the parison with optical absorption spectra. According to this coupling of electronic states to molecular thermal motions description, the separation between linking sites. for is developed and used to calculate the temperature-dependent electron transfer is 8-10 A in Chromatium, far smaller electron transfer rate. The model is compared with experi- than earlier estimates. mental results in Section III. The transfer of an electron from one to another is an I. The two-site tunneling description of electron transfer essential part of oxidative phosphorylation and photosyn- thesis. The reversible oxidation of the of cytochrome c We consider the problem of the transfer of an electron be- by cytochrome oxidase is a specific example of such a process, tween two sites a and b, with the electron initially in a wave one of several electron transfers in the sequence of reactions function p0a localized around site a. The final state will have resulting in oxidative phosphorylation. The overall effective- the electron in (pb, localized around b. spa and ptb weakly over- ness of such processes as photosynthesis or oxidative phos- lap, as sketched in Fig. 1. Because of the overlap between phorylation depends both on there being a large electron these wave functions, there is a matrix element Tab of the transfer rate for desired transfers and a small rate for in- Hamiltonian between these two one-particle states. The appropriate transfers. A particular cytochrome (or -sulfur meaning of Tab can be seen from the special case of sites a protein) seems to have, as its sole chemical function, the and b being equivalent, in which case the overlap generates a ability to exchange electrons with two other molecules A and splitting 2Tab between the bonding and anti-bonding states B, which (apparently) cannot directly exchange electrons. (

R 9an (Pb

FIG. 1. The wave functions

be infinitely sharp. The single atom state characterizing an ex- x -O citation on atom a is given a spectral shape Sa(E) characterized FIG. 2. The description of an electron removal process by a by a weighted optical emission spectrum at energy E of the configuration-coordinate diagram. The two curves represent the transition )a/ i*&a, with an appropriate normalization. This total energy of the system as a function of the coordinate for the emission spectrum includes all effects of the interaction of two states with and without the electron. Da(E) is the thermal the electronic excitation on atom a with its environment. probability distribution of the vertical separation between these Similarly, the excitation of b is characterized by its ab- states. sorption spectrum Sb'(E), including all effects of the motions of atoms in the transition Vb - b. The rate of excitation systems, and lack some of the complexity of the electro- transfer from a to b by the F6rster mechanism can then be chemical problem. At the same time, Eq. 2 will permit exten- written (7, 8) sions beyond Gaussian and high-temperature spectral func- tions. The treatment of our simpler problem is modeled on the Wab = Sa(E)Sb'(E)dE. [1] usual description of tunneling through an insulating barrier (2T/h)JUabI2f between two metals (10). While Uab is essentially temperature-independent, Wab and II. Results from a symmetric model of D(E) Wba are both temperature-dependent due to the temperature We next describe the simplest available model of D(E), dependences of the spectra involved in the overlap integral. which can be based on the analog to a symmetric configura- A precise parallel exists for the transfer of an electron tion-coordinate description (11) of optical emission spectra. from site a to b, including the effects of vibronic coupling. Fig. 2 gives the essence of the physical description of this The analog to the emission spectrum Sa(E) of fa' i/ia is configuration-coordinate description. The upper curve de- the electron removal spectral distribution Da(E). In the scribes the energy as a function of a vibrational coordinate x presence of the coupling between the electronic state S0a and with the electron in state (pa, and the lower curve the energy the nuclear motions, the removal of an electron (which can as a function of the same coordinate in the absence of an be thought of as being destroyed or transferred to a fictional electron in Spa. In the symmetric model, the curvature ka of the state of zero energy) is characterized by a distribution of ground and excited states are the same, and there is no energies Da(E). Da(E) is broad for exactly the same reasons entropy of electron transfer. At temperature T, the classical of atomic position readjustment that make Sa(E) broad. probability distribution of being at x if the electron is present Similarly, there is an electron insertion spectrum Db'(E) is that describes the distribution of energy changes that result from the insertion of an electron (from the fictional state at / ka 1/2 P x) = I (X - Xa)2/2KT). zero energy) into electronic state 'pb. The rate of electron 2TrKT/)exp(-ka [3] transfer can then be written in exact analogy to Eq. 1 as The electronic removal is a vertical transition between the = [2] two energy curves. Given P(x), the energy distribution Wab (2Tr/h)ITabj2 Da(E)Db'(E)dE. Da(E) is This equation can also be directly calculated from the usual Da(E) = (2TK~a 12 (E Ea +1:2kaxa2)2 [4] quantum mechanical expression for first-order transition rates. Wab is in this problem always due to tunneling, although Eq. 4 is based on a classical probability distribution, valid the usual temperature dependence of Da(E) and Db'(E) will when KT is greater than the vibrational energies hw = KTa make this tunneling rate temperature-dependent. of the relevant vibrational coordinates. The single most Eq. 2, with D(E) approximated by a high temperature important effect of quantum mechanical corrections is to form of Eq. 4, represents a special case of the general theory product a zero-point width (11) to the distribution Eq. 3. (9) of electron transfer in solution electrochemistry. The The modification of Eq. 4 valid (with restrictions-see next suppositions (a) of fixed, well-separated sites, (b) of inde- paragraph) to lower temperatures is pendent atomic motions interacting with the electron at each site, and (c) no important effect of atomic motions on Da(E) = (1/2T0a2)1/2 exp -((E - Ea + 1/2kaxa2)2/2aa2) {5] are to transfer between distant TabI particularly appropriate Ca2 = coth sites embedded in a more or less rigid matrix. These approxi- kaxa2(KTa/2) Ta/2T mations are relevant to many cases of transfer in biological which reduces to Eq. 4 at high temperature. Downloaded by guest on September 29, 2021 3642 Biophysics: Hopfield Proc. Nat. Acad. Sci. USA 71 (1974)

TEMPERATURE ("KELVIN)

Id

oo 4

1000 /T FIG. 3. The rate of photolysis-initiated transfer of an electron from cytochrome in Chromatium as a function of temperature. The experimental points are from deVault and Chance (5). The solid line is a plot of Eq. 8 with Ta = Tb = 350'K, E. - Eb = 0.05 eV, 1/2k4.2 = '/2kbXb2 = 0.5 eV, and ITab = 4 X 10-4 eV. The form of Db'(E) follows from an exactly parallel model, to water and its phases. Figure 3 shows the experimental except that P(x) is replaced by electron transfer rate and an approximate fit to the experi- mental data by use of Eq. 8. In our ignorance of the two sites (kb/2wrKT) /2 exp(-kbx2/2KT) [6] of transfer, it is pointless to differentiate between the two and the transition is in the opposite direction, whence sites, a and b, as far as vibronic parameters are concerned. We pick kaxa2 = kixb2 and Ta = Tb. Eq. 8, thus reduced, Db'(E) = (1/2-rob2) /2 exp(-(E + Eb - l/2kb4b2)2/2c-b2) contains four effective parameters, namely Ta, kaxa2 Tabf, [7] and (Ea - Eb). 4± = coth Tb/2T. The characteristic temperature Ta is 3500 70°. (About ab2 kbXb2(KTb/2) 1500 marks the turning point of the data between two regions From Eqs. 2,5, and 7 of temperature behavior, and the characteristic turning point involves Ta/2 in Eq. 5.) The other three parameters Wab = are not uniquely determined. Fortunately, Ea - Eb is well 22r/hITabI2(1/2Xra2)'/2 exp(-(Ea - Eb- A)2/2cT2) [8] limited by usual constraints on electron transfer. Ea is greater where than Eb, for the electron transfer takes place even at zero temperatures. But successive steps in electron transfer chains 02 = (kaXa2/2)KTa coth Ta/2T + (kbXb2/2)KTb coth Tb/2T normally have their standard potentials within about and 0.05 V (unless energy is being usefully extracted in the step). (Ea - Eb) occurs only in the exponential of Eq. 8. If it is A = '/2kaxa2 + 1/2kbxb2 Wba = Wab exp(- (Ea - Eb)/KT). given a "typical" value of 0.05 V (1.18 keal), then A is so in order to the data that the values of both A and are and large fit At high temperatures, quantum effects unimportant, are insensitive to whether (Ea - Eb) is in error by a involved in 8 are valid. Tab the approximations calculating Eq. factor of five. We thus obtain 1/2kaxa2 = 0.5 ± 0.1 eV (11.5 The process appears thermally activated, for at high tem- ± cx kcal) and ITabI = 4 X 10-4 0-6 eV (9.6 cal). The value of peratures, u2 T. At low temperatures, the use of this 1/2kaxa2 is sharply constrained by the high temperature expression is limited at best to the case Ea > Eb + KTa or b, is much less determined restrictions on the size of a. activation energy. ITa1,| definitely with further possible depending because it occurs only as a prefactor. Eq. 8 is a strong coupling result, also requiring '/2kaxa2/KTa The magnitude of Ta and '/2kaxa2 can be directly checked >>. on a semiquantitative basis by comparing the parameters just determined, appropriate to adding (or removing) an III. The scale of parameters electron to a cytochrome, to the parameters relevant to opti- The general scale of parameters for electron transfer can be cally exciting a similar heme electron without removing it. established by making a fit to an appropriate experiment. The general considerations that were used to generate the The transfer of an electron from a cytochrome to fill a hole shape of Da(E) in Section II are identical with those of the made available by a flash of light (the earliest stages of photo- configuration-coordinate description of the broadening of synthesis) has been studied in Chromatium by deVault and optical spectral lines. The only important difference between Chance (5). This transfer seems likely to come close to the optical excitation of electrons and the removal of electrons idealized problem the theory describes. It has simple kinetics, is that the removal of an electron is a somewhat larger per- has been studied over a wide range of temperatures, persists turbation, so the effective '/2kaXa2 for an optical transition of to very low temperatures, and does not appear closely coupled the heme is expected to be comparable to, but smaller than, Downloaded by guest on September 29, 2021 Proc. Nat. Acad. Sci. USA 71 (1974) Electron Transfer between Biological Molecules 3643 that for electron removal. Similar vibrations will be involved, rate of transfer would be almost temperature independent, so Ta should be essentially the same for the two cases. The and enhanced from the low-temperature ground state result full width at half maximum for an optical transition with a (above) by a factor of 1.7 X 107. In addition, the excited distortion parameter (1/2kaxa2)optical is state is less well bound, and the tunneling barrier appears less high. With the parameters previously used, the tunneling width(T) = 2.34[(KTa) (1/2kaXa2)optical coth (Ta/2T)]'/2. barrier would now be only 1 eV high, and the matrix element If we use the value of 1l2kaXa2 deduced from tunneling, the Tab would be raised a factor of 6 from its ground-state value. predicted linewidth as a function of temperature is 0.290 eV The total increase in the rate of transfer from the excited at 1000K and below, 0.337 eV at 200'K, and 0.388 eV at state compared to the ground state is a factor of 6 X 108, with 3000K. For a typical Soret transition at 430 nm, the cor- the dominant factor arising from the change in the exponen- responding full widths at half maximum are 43, 50, and 58 nm, tial of Eq. 8, which eliminates the usual Stokes shift sup- respectively. For comparison, the full width at half maximum pression of the transfer rate. This general effect in Marcus of the Soret band of typical six-coordinated iron in oxyhemo- theory has recently been noted (16). globin at room temperature (12) is about-32 nm, and it Experiments perhaps related to his calculation have been sharpens (13) about 10% on going to 210'K. As anticipated, carried out on the bacterium Rhodopseudomonas spheroides the scale of the temperature variation is similar to that inter- and on photosynthetic reaction centers taken from them. preted from the electron transfer data. The optical linewidth The rate of electron transfer from an excited state of molecule is similar in scale but somewhat smaller than the electron a to molecule b is >1.4 X 1011 sec-' (17). The rate of electron transfer linewidth, also as expected. The optical transition transfer from b back to the ground state of a below 80°K is has been calculated for a typical Soret TrT* transition, while 30 sec-' (6). This -enormous difference in rates may be an the electron to be added is placed in an "iron" orbital. How- indication of the strong effect of the energy difference on the ever, the nuclear magnetic resonance spectra (14) show that transfer rate. [The failure of the reaction b -o a to be ther- the highest energy orbital on ferric iron, the orbital from mally activated (18) can be accommodated within the general which the electron is being transferred, is widely delocalized tunneling framework, but involves details too complicated to on the heme, and the 7r-T* optical transition is strongly treat here.] mixed with iron d-states. Since the orbitals involved in elec- IV. Discussion tron transfer and in the optical excitation are similar mixtures, Our conclusions on the nature and range of the electron there is no need to distinguish between them in their general transfer process are totally different from the interpretations properties. that have previously been used in analyzing the same experi- The magnitude of the tunneling matrix element can be ments. We believe that previous errors of principle and of used to construct an approximate distance between the donor emphasis are responsible for the divergent conclusions. These and acceptor. Tab will fall approximately exponentially with models and their problems are summarized for comparison. separation R, with a characteristic length determined by the barrier height. An upper limits to the barrier height is about (a) "Low Temperature" Tunneling Description. These half the a a* band-gap of the surrounding material, yield- calculations (Eq. 1 of ref. 5 or Eq. 4 of ref. 6) are based ing a height of about 3 eV. That in hemoglobin are not on the penetration of a square barrier by a particle that is readily photooxidized in the Soret band suggests that the otherwise free. In order to make a comparison with a problem barrier is not unusually small. A 2-eV barrier height is taken of transfer between two localized sites, an effective collision as a reasonable estimate. For two carbon atoms in a r- frequency was introduced in an ad hoc fashion. It was guessed bonding configuration, then that this frequency factor should be constant and about 1015 sec1-. No discussion of the physics of how that frequency Tab - 2.7 exp(-0.72 R) factor came about was given (5, 6). The approach omits the where Tab is in electron volts and R in Angstroms. The pre- Franck-Condon factors that must always be present. The factor is evaluated by getting the correct r resonance integral present paper is equivalent to constructing a detailed quan- (15) at a normal bond length.* If two large aromatic groups tum mechanical description of this frequency factor. Because of Na and Nb atoms are in contact through one "edge" atom of vibronic coupling and Stokes shifts, this frequency factor on each, Tab will be multiplied by a normalization factor is many orders of magnitude smaller than previously assumed, (NaNb) - 1/2, and R then measures the separation between the and is temperature dependent. The estimates of 30 X for the edge atoms. Based on such an edge-to-edge contact between transfer distance (5, 6), therefore, lack a legitimate theoretical two such r-systems with Na - Nb 20, the Tab of 4 X 10-4 basis. eV corresponds to a separation of 8.0 A between the two atoms (b) "Barrier Fluctuation" Description of Temperature- through which the transfer takes place. Dependent Tunneling. Two descriptions of temperature- The scale of parameters shows that within a fixed geome- dependent transfer rates have been given (5). In these descrip- try, the rate of electron transfer will be greatly enhanced in tions, the effective distance an electron must tunnel is modu- transfer from an excited state. In the example just examined, lated by thermal fluctuations, and tunneling is easier at high the rate of electron transfer Wab at low temperatures is 260 temperatures. One of these descriptions was rejected by its sec1-. In an excited state higher by 1.0 eV instead of by 0.05 authors on the basis of unreasonable parameters required, eV but with no other factor changed, the argument of the while the other seemed to its authors reasonable. Both de- exponential in Eq. 8 would vanish. For the excited state, the scriptions make the same error as in (a) of completely failing to come to terms with the frequency factor. * We have used a resonance integral of 1.0 eV as a compromise Extensive studies have been made of tunneling through an between various views. insulating barrier between two metals. In this case, the tun- Downloaded by guest on September 29, 2021 3644 Biophysics: Hopfield Proc. Nat. Acad. Sci. USA 71 (1974)

neling current is closely related to Eq. 2, with the functions times faster at room temperature than electron transfer be- Da(E) and Db'(E) replaced by their appropriate counterparts tween equivalent levels. These short-circuiting transfers must in metals. The tunneling matrix element is generally not be prevented (i.e., other things kept from being equal, as by appreciably temperature dependent. Though the analog is preventing approach) by structural and stereochemical con- not exact, it suggests that the temperature dependence of the siderations. When the range of electron transfer is too great, barrier is not the most likely source of temperature depen- it would be impossible to prevent these short-circuiting dence. transfers. (This rapid transfer between levels of considerably (c) Transfer by Thermal Excitation to a Free Electron State. different redox potentials may, however, be functionally use- The idea of this transfer description (5) is that the 3.3 kcal ful in photosynthesis for separating electrons and holes in spite of its free energy (0.14 eV) activation energy observed for electron transfer cost.) from cytochrome c represents the binding energy of an elec- This description of electron transfer by tunneling can in provide a framework for interpreting the function of struc- tron cytochrome c with respect to the conduction band of tural the surrounding material, and that thermally freed electrons features of electron transport molecules. To apply the react rapidly with the hole generated by the photon initiating model more generally and quantitatively it will be necessary to extend the transient process. The cytochrome then acts like a donor descriptions of D(E) to include the case in which in silicon. This description is unfortunately not internally there is an entropy change on electron transfer, a feature consistent. Donors of a depth of only 0.14 eV will be largely not included in the present description. ionized (18) at room temperature, with their electrons in the I thank R. G. Shulman, J. D. McElroy, and J. D. E. McIntyre conduction band (unless the cytochrome c concentration is for discussions of the problem. The work at Princeton was sup- greater than 1 mM). This model has also been used to suggest ported in part by NSF Grant GH40474. that the barrier height for tunneling is only 0.14 eV (with 1. Hodges, H. L., Holwerda, R. A. & Gray, H. B. (1974) J. concomitant huge distances in electron an Amer. Chem. Soc. 96, 3132-3137. possible transfer), 2. Marcus, R. A. (1964) Annu. Rev. Phys. Chem. 15, 155-196. interpretation we believe to be erroneous. 3. Newton, W. T. (1968) J. Chem. Educ. 45, 571-575. The experiments in Chromatium on which these various 4. Takano, T., Swanson, R., Kallai, 0. B. & Dickerson, R. E. interpretations of electron transfer have been based are all (1971) Cold Spring Harbor Symp. Quant. Biol. 36, 397-404. consistent with a single model containing four parameters, of 5. DeVault, D. & Chance, B. (1966) Biophys. J. 6, 825-847. 6. McElroy, J. D., Mauzerall, D. C. & Feher, G. (1974) Bio- which one is insensitively involved, and Qf which two others chim. Biophys. Acta 333, 261-278. can be semiquantitatively verified in optical absorption 7. Forster, T. (1946) Natur'wissenschaften 33, 166-182. studies. Enough information is available to evaluate the tun- 8. Dexter, D. L. (1953) J. Chem. Phys. 21, 836-850. neling parameter Tab of the theory and to estimate the separa- 9. Levich, V. G. (1965) "Present state of the theory of oxi- tion the sites on the donor and as dation-reduction in solution," in Advances in Electrochemistry between linking acceptor and Electrochemical Engineering, eds. Delahay, P. & Tobias, 8 A. (A barrier height of 1 eV would have increased this esti- C. W. (Interscience, New York), Vol. 4, pp. 249-372. mate by 2.5 A.) This distance is so much smaller than previous 10. Schrieffer, J. R. (1964) in Superconductivity (W. A. Ben- estimates (5, 6) (30-80 A) for such transfers that, if correct, jamin, Inc., New York), pp. 78-80. it must profoundly affect the view of the structural require- 11. Klick, C. C. & Schulman, J. H. (1957) Solid State Phys. 5, 97-172. ments for electron transfer. 12. Antonini, E. & Brunori, M. (1971) in Hemoglobin and Myo- A relatively short range of electron transfer is probably globin in their Interaction with Ligands (American Elsevier, imperative to the operation of electron transport molecules. New York), p. 18. They seem to be used to exchange electrons with another 13. Treu, J. (1973) "The Magnetic Circular Dichroism of molecule of similar standard redox If could Hemoglobin," Ph.D. Dissertation, Princeton University. potential. they 14. Shulman, R. G., Glarum, S. H. & Karplus, M. (1971) J. also exchange electrons with molecules with a very different Mol. Biol. 57, 93-115. redox potential, this would short-circuit the useful paths of 15. Salem, L. (1966) in The Molecular Orbital Theory of Con- the oxidative phosphorylation or photosynthetic electron jugated Systems (W. A. Benjamin, Inc., New York), pp. transport chains. The transfer mechanism described here has 109, 138, and 143. 16. Efrima, S. & Bixon, M. (1974) Chem. Phys. Lett. 25, 34-47. a propensity toward such short-circuits. It was shown in 17. Netzel, T. L., Rentzepis, P. M. & Leigh, J. (1973) Science Section III that, other things being equal, electron transfer 182, 238-241. between levels differing by one volt in redox is about 109 18. Parson, W. W. (1967) Biochim. Biophys. Acta 131, 154-172. Downloaded by guest on September 29, 2021