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Electron transfer between [4Fe–4S] clusters
Chemical Physics Letters 495 (2010) 131–134
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Electron transfer between [4Fe–4S] clusters Alexander A. Voityuk * Institució Catalana de Recerca i Estudis Avançats, Barcelona and Institut de Química Computational, Universitat de Girona, 17071 Girona, Spain
a r t i c l e
i n f o
Article history: Received 4 February 2010 In final form 19 June 2010 Available online 25 June 2010
a b s t r a c t Iron–sulfur clusters [4Fe–4S] are major components in biological electron transport (ET). Using the DFT/ B3LYP method, we calculate electronic coupling for low- and high-potential ET between [4Fe–4S] clusters and explore its sensitivity to structural parameters of the system and external electric field. As an example, we consider a model of the bacterial respiratory complex I and estimate the role of neighboring amino acids in facilitating the ET process between the clusters. Our results suggest that the superexchange mechanism rather than hole hopping should be operative in ET between [4Fe–4S] redox centers. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Fifty years ago, Mortenson et al. discovered the first iron–sulfur protein, ferredoxin from Clostridium pasteurianum [1]. This protein contains two iron–sulfur clusters [4Fe–4S] that mediate electron transport within the protein. Nowadays, [4Fe–4S] and other iron– sulfur clusters are known to be major redox-active centers in biological electron transfer (ET) including the photosynthetic and respiratory electron transport chains [2–4]. In particular, two or multiple [4Fe–4S] centers mediate ET in hydrogenases [5]. In the cubane-shaped [4Fe–4S] core, each iron atom is connected with three l3-sulfur ions. Three different redox states, [4Fe–4S]n+, with n = 1, 2, and 3, have been identified in proteins. Usually the [4Fe– 4S]n+ core is coordinated by four thiolate groups of cysteine resulting in [Fe4S4(SR)4]n4. For brevity, the last complex is commonly referred to as [4Fe–4S]n+. In some systems, cysteine is replaced by other amino acids, e.g. histidine. The redox couple [4Fe–4S]2+/ [4Fe–4S]1+ and [4Fe–4S]3+/[4Fe–4S]2+ are operative in low- and high-potential ET systems, respectively. To describe the electronic structure of [4Fe–4S]2+, the cluster can be seen as made up of two high-spin [2Fe–2S]+ units (S = 9/2). Because these units are coupled antiferromagnetically, the ground electronic state of [4Fe–4S]2+ has a total spin of zero [6]. The redox potential of [4Fe–4S] is sensitive to electrostatic effects of the protein matrix [7–9]. In this Letter, we describe a DFT study of ET between [Fe4S4(SR)4] clusters embedded in a protein as shown in Fig. 1. Based on the electronic coupling values calculated for different mutual arrangements of the clusters, we consider the distance dependence of the low- and high-potential ET rates. It will be shown that the coupling does not depend significantly on the orientation of the redox centers and external electric field. As an example, we consider ET in the bacterial respiratory complex I
and estimate contribution of neighboring amino acids in the superexchange-mediated coupling. 2. Computational details Computer simulations provide valuable information about the dependence of chemical transformation on structural and electronic features of reaction centers and its surroundings. Theoretical and computational approaches to modeling of enzymatic redox reactions are considered in several reviews [10–12]. The rate of nonadiabatic ET
kET
" # 2p jVj2 ðDG þ kÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ¼ h 4kkB T 4pkkB T
ð1Þ
is determined by three key parameters – the electronic coupling V, the driving force DG, and the reorganization energy k [13]. Because our system consists of two identical redox centers, the driving force for ET in both 2+/1+ (low-potential) and 3+/2+ (high-potential) redox couples is assumed to be zero (DG = 0). In this case, the activa2 tion barrier for the ET is expressed as DG# ¼ ðDGþkÞ ¼ 4k. In proteins, k 4k is usually assumed to be in the range from 0.5 to 1 eV [11,12,14]. At k room temperature, the factor exp½ 4kT changes from 7 103 (k = 0.5) to 5 105 (k = 1.0). Thus, depending on the k parameter used in Eq. (1), kET varies within two orders of magnitude. If DG = 0, the rate constant at 298 K can be approximately estimated as
k0 ðs1 Þ 1:7 1016 V 2 k1=2 expð10kÞ
ð2Þ
In Eq. (2), V and k are in eV. 2.1. Electronic coupling
* Fax: +34 972418356. E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.06.057
Because accurate values of V can hardly be derived from experiment data, quantum chemical methods are invoked to compute
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A.A. Voityuk / Chemical Physics Letters 495 (2010) 131–134
2.2. Structures
Fig. 1. Large model includes two [4Fe–4S] clusters and 5 amino-acid residues.
this parameter. We note that reliable estimation of electronic couplings remains a considerable challenge especially for systems containing transition metals. In the model under study, donor and acceptor have several nearly-degenerate states. Structural fluctuations of the redox centers and their environment may lead to reordering of the electronic states. Consequently, a two-state model that takes into account only one state of each cluster, cannot appropriately describe the electronic coupling in the system. As the reordering of the donor and acceptor levels occurs at the picoseconds time scale and thus is much faster than ET, averaging over several states should be used to estimate the matrix element [15]. The effective coupling squared V2 can be estimated as
V 2 ¼ N2 V 2rms
V rms
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N1 X N2 X u 1 ¼t V2 N1 N2 i¼1 j¼1 ij
ð3aÞ
ð3bÞ
In Eq. (3), N1 and N2 are the numbers of (nearly) degenerate diabatic states localized on the donor and acceptor sites, respectively. For each cluster, we considered 3 states of the lowest energy (N1 = N2 = 3). Eq. (3a) accounts for N2 pathways from each state of the donor to the acceptor. Two methods, the Generalized Mulliken–Hush (GMH)[16] and the Fragment Charge Difference (FCD) [17] were employed to calculate the coupling. The adiabatic states were approximated by Kohn–Sham orbitals stemming from the DFT/B3LYP calculation of systems containing two [Fe4S4(SR)4]2 clusters. DFT calculations provide reasonable estimates of ET couplings [18–21], Vrms for low-potential ET (between [4Fe–4S]2+ and [4Fe–4S]1+) was computed using beta-spin lowest unoccupied MOs (LUNOs), whereas the matrix element for high-potential ET (between [4Fe–4S]3+ and [4Fe–4S]2+) was derived using alpha-spin highest occupied MOs (HOMOs). The iron–sulfur clusters [4Fe–4S]n+ are characterized by a delicate energy balance among different spin states; many of these states are nearly degenerate and lie within several kT [6]. Because ET couplings between donor and acceptor levels is averaged (Eq. (3)), a plausible estimate of Vrms, can be obtained using the high-spin state of the system. For each redox site, we considered 3 electronic states of the lowest energy and averaged the couplings for 3 3 ET pathways connecting the corresponding diabatic states.
We consider several models. The largest model (147 atoms) shown in Fig. 1 merges two clusters and 5 amino-acid residues (Pro, His, Phe, Tyr, and Trp) that are involved in mediating ET coupling. The smallest system (56 atoms) includes only two [Fe4S4 (SR)4]2 complexes. In addition, we considered several structures generated from the large model by truncation of some residues. The structure of the systems is based on the experimental geometries [22,23]. To estimate the distance dependence of the coupling, a number of structures were generated, where the [4Fe–4S] units 0 are separated by the distance R ranging from 10.0 to 18.0 Å A. Several structures were obtained by mutual rotation of the clusters at the 0 fixed distance R = 11.6 A Å. DFT calculations were performed using the spin–unrestricted scheme and the standard B3LYP functional [24] that provides good estimates for electronic couplings [20]. For H, C, N, O and S, the 631G* basis was used; for Fe, we employed the LanL2DZ basis set, where inner electrons are substituted by effective core potentials and the valence orbitals are represented by double-Zeta quality functions [25]. The calculations were carried out with the program GAUSSIAN03 [26]. Despite much effort, we could not localize the appropriate antiferromagnetic states of the [4Fe–4S] dimer (in these states, each cluster is made up of two high-spin [2Fe–2S]+ units with S = 9/2 and has the total spin of 0). Because of that, we used the Kohn– Sham molecular orbitals obtained for high-spin states of the system. We note that averaging over several low-lying electronic states, Eq. (3), should improve the accuracy of the derived couplings.
3. Results and discussion First we consider the small model (reference system) consisting of two [Fe4S4(SR)4]2 clusters. Adiabatic states for low-potential ET are represented by beta LUMOs of the high-spin system, whereas the HOMOs of the dimer (alpha-spin eigenvalues and eigenvectors) are used to derive Vrms values for high-potential ET (3+/2+ redox couple). All these states are found to be quite localized on a single site (Fig. 2). In particular, LUMO, LUMO+2 and LUMO+5 of the dimer are localized on the first cluster, whereas LUMO+1, LUMO+3 and LUMO+4 – on the second center. These states lie within a small gap (0.3 eV). A very similar situation is found for HOMOs. 3.1. Distance dependence Table 1 lists the electronic coupling calculated for several structures, where the redox centers are separated by the distance R 0 ranging from 10 to 18 Å A. The GMH and FCD methods give very similar values of Vrms. As seen from Fig. 2, the electron density in [Fe4S4(SR)4] is delocalized over the cluster including thiolate groups. Relatively short contacts between these ligands facilitate the electronic coupling of the clusters. For instance, in the dimer 0 with R = 17 Å A, the distance0 between sulfur atoms of the nearest thiolate ligands RS is 11.1 Å A. As expected, the electronic coupling decays exponentially with the distance between the redox centers. For low-potential ET
logVðLPÞ ¼ 0:07 0:32R
ð4aÞ
and for high-potential ET
logVðHPÞ ¼ 1:73 0:42R where the electronic coupling V is in eV and R is in Å.
ð4bÞ
A.A. Voityuk / Chemical Physics Letters 495 (2010) 131–134
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Fig. 2. Kohn–Sham orbitals represent initial and final states for low- and high-potential ET between iron–sulfur clusters.
Table 1 Distance dependence of the direct electronic coupling (in eV) between [4Fe–4S] clusters for low- and high-potential ET. R
Low-potential ET 0
(Å A) 10 11 12 13 14 15 16 17 18
FCD
High-potential ET GMH
3
1.01 10 3.70 104 1.56 104 7.73 105 3.18 105 1.25 105 7.48 106 4.66 106 3.09 106
FCD 3
1.01 10 3.66 104 1.62 104 8.49 105 3.17 105 1.22 105 7.42 106 5.35 106 3.23 106
GMH 3
5.05 10 1.36 103 3.89 104 1.45 104 5.58 105 2.33 105 9.72 106 4.20 106 1.87 106
4.84 103 1.33 103 3.58 104 1.34 104 5.39 105 2.33 105 9.62 106 4.02 106 1.76 106
volved in ET (for instance, in p stacks). If there are several nearly degenerate electronic states, the average matrix element should be less sensitive to the mutual arrangement of the donor and acceptor sites (provided the distance between donor and acceptor does not change). Fig. 3 shows the results obtained 0 for different conformations of the [4Fe–4S] dimer with R = 11.6 A Å, where the mutual orientation of the redox centers is changed by rotation of one cluster around its center). As seen, the coupling V2St calculated using only two electronic states is much more sensitive than Vrms (Eq. (3b)). For instance, by passing between conformations 4 and 5, V2St becomes stronger by a factor of 60 (Fig. 3) while the Vrms value increases only by a factor of 1.5.
3.3. Is the electronic coupling sensitive to the electrostatic interaction with surroundings? To explore this point, the dimer was calculated applying electric field F of variable strength in two directions (parallel and perpendicular to ~ R). As seen from Fig. 4, the electric field does not significantly affect the electronic coupling. It should be noted that the ET driving force DG will change by 0.62 eV (2 orders of magnitude in the ET rate!) if the electric field of 0.001 au is applied along ~ R. Thus, unlike DG, the electronic coupling appears to be quite insensitive
Fig. 3. Dependence of the root-mean-square coupling, Vrms, and the two-state coupling V2St on the orientation of iron–sulfur clusters (the distance between the 0 clusters in all configurations is 11.6 A Å). The line shows the coupling value in the reference system.
3.2. Conformational effects ET couplings can be very sensitive to mutual orientation of the donor and acceptor sites as demonstrated for DNA [27,28], proteins and polypeptides [29,30], and organic materials [31]. Strong conformational effects on the coupling are expected for systems, where only two states (one of donor and one of acceptor) are in-
Fig. 4. Effect of the external electric field F on the electronic coupling between iron– sulfur clusters (F in au, V in eV). Two directions of the electric field are considered: parallel (j) and perpendicular (4) to the vector ~ R connecting the cluster centers. The line shows the coupling when F = 0.
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A.A. Voityuk / Chemical Physics Letters 495 (2010) 131–134
Table 2 Electronic coupling (in eV) for ET between [4Fe–4S] clusters (D and A) in the bacterial respiratory complex I.
a
Model
FCD
GMH
D—A D-Pro-A D-Tyr-A D-Trp-A D-His-Phe-A D-Pro-A D-Pro-Tyr-A D-Pro-Trp-A D-Pro-His-Phe-A D-Pro-Tyr-Trp-His-Phe-Aa
0.058 104 0.665 104 0.054 104 0.054 104 0.066 104 0.756 104 0.750 104 0.665 104 0.704 104 0.756 104
0.059 104 0.664 104 0.053 104 0.054 104 0.066 104 0.766 104 0.767 104 0.664 104 0.726 104 0.766 104
The structure is shown in Fig. 1.
to the electrostatic potential created by protein environment on the redox centers. Superexchange interactions are ubiquitous for ET in proteins, where long-range ET occurs, being mediated by the off-resonance superexchange electronic coupling with the molecular fragments between the donor and acceptor sites [12,32]. Fig. 1 shows the surroundings of the redox centers that belong to N5 and N6a units of the bacterial respiratory complex I [23] (the distance between the 0 [4Fe–4S] centers is 16.8 Å A). The main ET chain of the protein complex includes seven iron–sulfur clusters [23]. ET between clusters of N5 and N6a is expected to be the rate-determining step because of the longest distance between these redox centers. Table 2 lists electronic couplings for high-potential ET between the clusters calculated for different models. As seen, the residue Pro gives the largest contribution to the superexchange interaction increasing the coupling by an order of magnitude as compared to the direct donor–acceptor coupling. Significantly smaller contributions are found for other surroundings. Note that including an additional bridging unit may increase or decrease the coupling (Table 2) because of the positive or negative interference of the superexchange pathways. According to our calculations, the electronic states of the bridge units in the system (Fig. 1) are at least 2 eV higher in energy than the donor and acceptor states. It means that the amino-acid residues cannot serve as intermediate states in the ET between the redox centers. The probability of thermal activation of the hopping process is negligible (1035). Let us estimate the ET rate between the clusters using Eq. (2). For the effective coupling squared, V 2 ¼ 3 V 2rms with Vrms = 0.756104 eV (Table 2) and the reorganization energy k=1 eV, one obtains kET = 1.3104 s1. For smaller values of k, kET is higher (e. g. kET = 3.2105 s1 when k = 0.7 eV). Experimentally, the lowest limit of the ET rate is 102 s1 [33,34]. Wittekindt et al. calculated the electronic coupling between clusters using a semiempirical tight-binding model [35]. They estimated the matrix element for the bottleneck step to be 3109 eV, which gives kET 2104 s1 (at k = 0.7 eV). Because the estimated rate is 6 orders of magnitude smaller than the lowest limit of the rate, it was suggested that aromatic amino acids act as stepping stones in the ET process. Using some (not properly justified) values for the energetics of hole states on aromatic amino-acid residues in the system, they derived the ET rate for the multistep hopping, 103 s1, in agreement with the observed data. According to our DFT calculations, however, the electronic coupling between the clusters in the system (Fig. 1) is considerably stronger (Table 2) and no intermediate states (stepping stones) are required. Moreover, the states with a hole on the bridge units are considerably higher (P2 eV) than the cluster levels and cannot be thermally
populated. Thus, the hole hoping in the considered system appears to be neither required nor allowed. 4. Conclusions Using DFT calculations of several models consisting of two complexes [Fe4S4(SR)4] and closest amino-acid residues we have estimated the electronic coupling between the iron–sulfur clusters [4Fe–4S]. The matrix element Vrms is obtained by averaging the coupling values computed for nearly degenerate electronic states of the redox centers. The distance dependence of the matrix element has been found for the low- and high-potential ET systems. The effective coupling is shown to be not very sensitive to the external electric field and mutual orientation of the clusters. In the redox system modeling the rate-limiting ET step in the bacterial respiratory complex I, the superexchange interaction with surroundings increases the coupling value by an order of magnitude. Our results suggest that the superexchange mechanism rather than hole hopping should be operative in ET between iron–sulfur clusters in proteins. Acknowledgment The work was supported by the Spanish Ministerio de Educación y Ciencia (Project No. CTQ2009-12346). References [1] L.E. Mortenson, R.C. Valentine, J.E. Carnahan, Biochem. Biophys. Res. Commun. 7 (1962) 448. [2] R. Cammack, Adv. Inorg. Chem. 38 (1992) 281. [3] W. Lubitz, E. Reijerse, M. van Gastel, Chem. Rev. 107 (2007) 4331. [4] D.C. Johnson, D.R. Dean, A.D. Smith, M.K. Johnson, Ann. Rev. Biochem. 74 (2005) 247. [5] B.J. Lemon, J.W. Peters, in: A. Messerschmidt, R. Huber, T. Poulos, K. Wieghardt (Eds.), Handbook of Metalloproteins, Wiley, Weinheim, Germany, 2001. [6] L. Noodleman, C.J. Peng, D.A. Case, J.-M. Mouesca, Coord. Chem. Rev. 144 (1995) 199. [7] J.-M. Mouesca, J.L. Chen, L. Noodleman, D. Bashford, D.A. Case, J. Am. Chem. Soc. 116 (1994) 11898. [8] E.I. Solomon, B. Hedman, K.O. Hodgson, A. Dey, R.K. Szilagyi, Coord. Chem. Rev. 249 (2005) 97. [9] A. Dey et al., E. I. Science 318 (2007) 1464. [10] S. Shaik, S. Cohen, Y. Wang, H. Chen, D. Kumar, W. Thiel, Chem. Rev. 110 (2010) 949. [11] A. Warshel, Ann. Rev. Biophys. Biomol. Struct. 32 (2003) 425. [12] H.B. Gray, J.R. Winkler, Q. Rev. Biophys. 36 (2003) 341. [13] R.A. Marcus, N. Sutin, Biochim. Biophys. Acta 811 (1985) 265. [14] I.V. Kurnikov, A.K. Charnley, D.N. Beratan, J. Phys. Chem. B 105 (2001) 5359. [15] A. Troisi, A. Nitzan, M.A. Ratner, J. Chem. Phys. 119 (2003) 5782. [16] R.J. Cave, M.D. Newton, J. Chem. Phys. 106 (1997) 9213. [17] A.A. Voityuk, N. Rösch, J. Chem. Phys. 117 (2001) 5607. [18] K. Senthilkumar et al., J. Am. Chem. Soc. 127 (2005) 14894. [19] J. Huang, M. Kertesz, M. J. Chem. Phys. 122 (2005) 234707. [20] M. Félix, A.A. Voityuk, J. Phys. Chem. A 112 (2008) 9043. [21] T. Kubar, P.B. Woiczikowski, G. Cuniberti, M. Elstner, J. Phys. Chem. B 112 (2008) 7937. [22] I. Bertini, A. Donaire, B.A. Feinberg, C. Luchinat, M. Piccioli, H. Yuan, Eur. J. Biochem. 232 (1995) 192. [23] L.A. Sazanov, P. Hinchliffe, Science 311 (2006) 1430. [24] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [25] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [26] M.J. Frisch et al., GAUSSIAN 03, Gaussian, Inc., Pittsburgh, PA, 2003. [27] A.A. Voityuk, K. Siriwong, N. Rösch, Phys. Chem. Chem. Phys. 3 (2001) 5421. [28] A. Troisi, G. Orlandi, J. Phys. Chem. B 106 (2002) 2093. [29] I.A. Balabin, D.N. Beratan, S.S. Skourtis, Phys. Rev. Lett. 101 (2008) 158102. [30] D.N. Beratan et al., Acc. Chem. Res. 42 (2009) 1669. [31] J.L. Bredas, D. Beljonne, V. Coropceanu, J. Cornil, Chem. Rev. 104 (2004) 4971. [32] D.N. Beratan, J.N. Onuchic, J.R. Winkler, H.B. Gray, Science 258 (1992) 1740. [33] A.D. Vinogradov, Biochim. Biophys. Acta 1364 (1998) 169. [34] M.S. Sharpley, R.I. Shannon, F. Draghi, J. Hirst, Biochemistry 45 (2006) 241. [35] C. Wittekindt, M. Schwarz, T. Friedrich, T. Koslowski, J. Am. Chem. Soc. 131 (2009) 8134.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Electron transfer between [4Fe–4S] clusters Alexander A. Voityuk * Institució Catalana de Recerca i Estudis Avançats, Barcelona and Institut de Química Computational, Universitat de Girona, 17071 Girona, Spain
a r t i c l e
i n f o
Article history: Received 4 February 2010 In final form 19 June 2010 Available online 25 June 2010
a b s t r a c t Iron–sulfur clusters [4Fe–4S] are major components in biological electron transport (ET). Using the DFT/ B3LYP method, we calculate electronic coupling for low- and high-potential ET between [4Fe–4S] clusters and explore its sensitivity to structural parameters of the system and external electric field. As an example, we consider a model of the bacterial respiratory complex I and estimate the role of neighboring amino acids in facilitating the ET process between the clusters. Our results suggest that the superexchange mechanism rather than hole hopping should be operative in ET between [4Fe–4S] redox centers. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Fifty years ago, Mortenson et al. discovered the first iron–sulfur protein, ferredoxin from Clostridium pasteurianum [1]. This protein contains two iron–sulfur clusters [4Fe–4S] that mediate electron transport within the protein. Nowadays, [4Fe–4S] and other iron– sulfur clusters are known to be major redox-active centers in biological electron transfer (ET) including the photosynthetic and respiratory electron transport chains [2–4]. In particular, two or multiple [4Fe–4S] centers mediate ET in hydrogenases [5]. In the cubane-shaped [4Fe–4S] core, each iron atom is connected with three l3-sulfur ions. Three different redox states, [4Fe–4S]n+, with n = 1, 2, and 3, have been identified in proteins. Usually the [4Fe– 4S]n+ core is coordinated by four thiolate groups of cysteine resulting in [Fe4S4(SR)4]n4. For brevity, the last complex is commonly referred to as [4Fe–4S]n+. In some systems, cysteine is replaced by other amino acids, e.g. histidine. The redox couple [4Fe–4S]2+/ [4Fe–4S]1+ and [4Fe–4S]3+/[4Fe–4S]2+ are operative in low- and high-potential ET systems, respectively. To describe the electronic structure of [4Fe–4S]2+, the cluster can be seen as made up of two high-spin [2Fe–2S]+ units (S = 9/2). Because these units are coupled antiferromagnetically, the ground electronic state of [4Fe–4S]2+ has a total spin of zero [6]. The redox potential of [4Fe–4S] is sensitive to electrostatic effects of the protein matrix [7–9]. In this Letter, we describe a DFT study of ET between [Fe4S4(SR)4] clusters embedded in a protein as shown in Fig. 1. Based on the electronic coupling values calculated for different mutual arrangements of the clusters, we consider the distance dependence of the low- and high-potential ET rates. It will be shown that the coupling does not depend significantly on the orientation of the redox centers and external electric field. As an example, we consider ET in the bacterial respiratory complex I
and estimate contribution of neighboring amino acids in the superexchange-mediated coupling. 2. Computational details Computer simulations provide valuable information about the dependence of chemical transformation on structural and electronic features of reaction centers and its surroundings. Theoretical and computational approaches to modeling of enzymatic redox reactions are considered in several reviews [10–12]. The rate of nonadiabatic ET
kET
" # 2p jVj2 ðDG þ kÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ¼ h 4kkB T 4pkkB T
ð1Þ
is determined by three key parameters – the electronic coupling V, the driving force DG, and the reorganization energy k [13]. Because our system consists of two identical redox centers, the driving force for ET in both 2+/1+ (low-potential) and 3+/2+ (high-potential) redox couples is assumed to be zero (DG = 0). In this case, the activa2 tion barrier for the ET is expressed as DG# ¼ ðDGþkÞ ¼ 4k. In proteins, k 4k is usually assumed to be in the range from 0.5 to 1 eV [11,12,14]. At k room temperature, the factor exp½ 4kT changes from 7 103 (k = 0.5) to 5 105 (k = 1.0). Thus, depending on the k parameter used in Eq. (1), kET varies within two orders of magnitude. If DG = 0, the rate constant at 298 K can be approximately estimated as
k0 ðs1 Þ 1:7 1016 V 2 k1=2 expð10kÞ
ð2Þ
In Eq. (2), V and k are in eV. 2.1. Electronic coupling
* Fax: +34 972418356. E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.06.057
Because accurate values of V can hardly be derived from experiment data, quantum chemical methods are invoked to compute
132
A.A. Voityuk / Chemical Physics Letters 495 (2010) 131–134
2.2. Structures
Fig. 1. Large model includes two [4Fe–4S] clusters and 5 amino-acid residues.
this parameter. We note that reliable estimation of electronic couplings remains a considerable challenge especially for systems containing transition metals. In the model under study, donor and acceptor have several nearly-degenerate states. Structural fluctuations of the redox centers and their environment may lead to reordering of the electronic states. Consequently, a two-state model that takes into account only one state of each cluster, cannot appropriately describe the electronic coupling in the system. As the reordering of the donor and acceptor levels occurs at the picoseconds time scale and thus is much faster than ET, averaging over several states should be used to estimate the matrix element [15]. The effective coupling squared V2 can be estimated as
V 2 ¼ N2 V 2rms
V rms
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N1 X N2 X u 1 ¼t V2 N1 N2 i¼1 j¼1 ij
ð3aÞ
ð3bÞ
In Eq. (3), N1 and N2 are the numbers of (nearly) degenerate diabatic states localized on the donor and acceptor sites, respectively. For each cluster, we considered 3 states of the lowest energy (N1 = N2 = 3). Eq. (3a) accounts for N2 pathways from each state of the donor to the acceptor. Two methods, the Generalized Mulliken–Hush (GMH)[16] and the Fragment Charge Difference (FCD) [17] were employed to calculate the coupling. The adiabatic states were approximated by Kohn–Sham orbitals stemming from the DFT/B3LYP calculation of systems containing two [Fe4S4(SR)4]2 clusters. DFT calculations provide reasonable estimates of ET couplings [18–21], Vrms for low-potential ET (between [4Fe–4S]2+ and [4Fe–4S]1+) was computed using beta-spin lowest unoccupied MOs (LUNOs), whereas the matrix element for high-potential ET (between [4Fe–4S]3+ and [4Fe–4S]2+) was derived using alpha-spin highest occupied MOs (HOMOs). The iron–sulfur clusters [4Fe–4S]n+ are characterized by a delicate energy balance among different spin states; many of these states are nearly degenerate and lie within several kT [6]. Because ET couplings between donor and acceptor levels is averaged (Eq. (3)), a plausible estimate of Vrms, can be obtained using the high-spin state of the system. For each redox site, we considered 3 electronic states of the lowest energy and averaged the couplings for 3 3 ET pathways connecting the corresponding diabatic states.
We consider several models. The largest model (147 atoms) shown in Fig. 1 merges two clusters and 5 amino-acid residues (Pro, His, Phe, Tyr, and Trp) that are involved in mediating ET coupling. The smallest system (56 atoms) includes only two [Fe4S4 (SR)4]2 complexes. In addition, we considered several structures generated from the large model by truncation of some residues. The structure of the systems is based on the experimental geometries [22,23]. To estimate the distance dependence of the coupling, a number of structures were generated, where the [4Fe–4S] units 0 are separated by the distance R ranging from 10.0 to 18.0 Å A. Several structures were obtained by mutual rotation of the clusters at the 0 fixed distance R = 11.6 A Å. DFT calculations were performed using the spin–unrestricted scheme and the standard B3LYP functional [24] that provides good estimates for electronic couplings [20]. For H, C, N, O and S, the 631G* basis was used; for Fe, we employed the LanL2DZ basis set, where inner electrons are substituted by effective core potentials and the valence orbitals are represented by double-Zeta quality functions [25]. The calculations were carried out with the program GAUSSIAN03 [26]. Despite much effort, we could not localize the appropriate antiferromagnetic states of the [4Fe–4S] dimer (in these states, each cluster is made up of two high-spin [2Fe–2S]+ units with S = 9/2 and has the total spin of 0). Because of that, we used the Kohn– Sham molecular orbitals obtained for high-spin states of the system. We note that averaging over several low-lying electronic states, Eq. (3), should improve the accuracy of the derived couplings.
3. Results and discussion First we consider the small model (reference system) consisting of two [Fe4S4(SR)4]2 clusters. Adiabatic states for low-potential ET are represented by beta LUMOs of the high-spin system, whereas the HOMOs of the dimer (alpha-spin eigenvalues and eigenvectors) are used to derive Vrms values for high-potential ET (3+/2+ redox couple). All these states are found to be quite localized on a single site (Fig. 2). In particular, LUMO, LUMO+2 and LUMO+5 of the dimer are localized on the first cluster, whereas LUMO+1, LUMO+3 and LUMO+4 – on the second center. These states lie within a small gap (0.3 eV). A very similar situation is found for HOMOs. 3.1. Distance dependence Table 1 lists the electronic coupling calculated for several structures, where the redox centers are separated by the distance R 0 ranging from 10 to 18 Å A. The GMH and FCD methods give very similar values of Vrms. As seen from Fig. 2, the electron density in [Fe4S4(SR)4] is delocalized over the cluster including thiolate groups. Relatively short contacts between these ligands facilitate the electronic coupling of the clusters. For instance, in the dimer 0 with R = 17 Å A, the distance0 between sulfur atoms of the nearest thiolate ligands RS is 11.1 Å A. As expected, the electronic coupling decays exponentially with the distance between the redox centers. For low-potential ET
logVðLPÞ ¼ 0:07 0:32R
ð4aÞ
and for high-potential ET
logVðHPÞ ¼ 1:73 0:42R where the electronic coupling V is in eV and R is in Å.
ð4bÞ
A.A. Voityuk / Chemical Physics Letters 495 (2010) 131–134
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Fig. 2. Kohn–Sham orbitals represent initial and final states for low- and high-potential ET between iron–sulfur clusters.
Table 1 Distance dependence of the direct electronic coupling (in eV) between [4Fe–4S] clusters for low- and high-potential ET. R
Low-potential ET 0
(Å A) 10 11 12 13 14 15 16 17 18
FCD
High-potential ET GMH
3
1.01 10 3.70 104 1.56 104 7.73 105 3.18 105 1.25 105 7.48 106 4.66 106 3.09 106
FCD 3
1.01 10 3.66 104 1.62 104 8.49 105 3.17 105 1.22 105 7.42 106 5.35 106 3.23 106
GMH 3
5.05 10 1.36 103 3.89 104 1.45 104 5.58 105 2.33 105 9.72 106 4.20 106 1.87 106
4.84 103 1.33 103 3.58 104 1.34 104 5.39 105 2.33 105 9.62 106 4.02 106 1.76 106
volved in ET (for instance, in p stacks). If there are several nearly degenerate electronic states, the average matrix element should be less sensitive to the mutual arrangement of the donor and acceptor sites (provided the distance between donor and acceptor does not change). Fig. 3 shows the results obtained 0 for different conformations of the [4Fe–4S] dimer with R = 11.6 A Å, where the mutual orientation of the redox centers is changed by rotation of one cluster around its center). As seen, the coupling V2St calculated using only two electronic states is much more sensitive than Vrms (Eq. (3b)). For instance, by passing between conformations 4 and 5, V2St becomes stronger by a factor of 60 (Fig. 3) while the Vrms value increases only by a factor of 1.5.
3.3. Is the electronic coupling sensitive to the electrostatic interaction with surroundings? To explore this point, the dimer was calculated applying electric field F of variable strength in two directions (parallel and perpendicular to ~ R). As seen from Fig. 4, the electric field does not significantly affect the electronic coupling. It should be noted that the ET driving force DG will change by 0.62 eV (2 orders of magnitude in the ET rate!) if the electric field of 0.001 au is applied along ~ R. Thus, unlike DG, the electronic coupling appears to be quite insensitive
Fig. 3. Dependence of the root-mean-square coupling, Vrms, and the two-state coupling V2St on the orientation of iron–sulfur clusters (the distance between the 0 clusters in all configurations is 11.6 A Å). The line shows the coupling value in the reference system.
3.2. Conformational effects ET couplings can be very sensitive to mutual orientation of the donor and acceptor sites as demonstrated for DNA [27,28], proteins and polypeptides [29,30], and organic materials [31]. Strong conformational effects on the coupling are expected for systems, where only two states (one of donor and one of acceptor) are in-
Fig. 4. Effect of the external electric field F on the electronic coupling between iron– sulfur clusters (F in au, V in eV). Two directions of the electric field are considered: parallel (j) and perpendicular (4) to the vector ~ R connecting the cluster centers. The line shows the coupling when F = 0.
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Table 2 Electronic coupling (in eV) for ET between [4Fe–4S] clusters (D and A) in the bacterial respiratory complex I.
a
Model
FCD
GMH
D—A D-Pro-A D-Tyr-A D-Trp-A D-His-Phe-A D-Pro-A D-Pro-Tyr-A D-Pro-Trp-A D-Pro-His-Phe-A D-Pro-Tyr-Trp-His-Phe-Aa
0.058 104 0.665 104 0.054 104 0.054 104 0.066 104 0.756 104 0.750 104 0.665 104 0.704 104 0.756 104
0.059 104 0.664 104 0.053 104 0.054 104 0.066 104 0.766 104 0.767 104 0.664 104 0.726 104 0.766 104
The structure is shown in Fig. 1.
to the electrostatic potential created by protein environment on the redox centers. Superexchange interactions are ubiquitous for ET in proteins, where long-range ET occurs, being mediated by the off-resonance superexchange electronic coupling with the molecular fragments between the donor and acceptor sites [12,32]. Fig. 1 shows the surroundings of the redox centers that belong to N5 and N6a units of the bacterial respiratory complex I [23] (the distance between the 0 [4Fe–4S] centers is 16.8 Å A). The main ET chain of the protein complex includes seven iron–sulfur clusters [23]. ET between clusters of N5 and N6a is expected to be the rate-determining step because of the longest distance between these redox centers. Table 2 lists electronic couplings for high-potential ET between the clusters calculated for different models. As seen, the residue Pro gives the largest contribution to the superexchange interaction increasing the coupling by an order of magnitude as compared to the direct donor–acceptor coupling. Significantly smaller contributions are found for other surroundings. Note that including an additional bridging unit may increase or decrease the coupling (Table 2) because of the positive or negative interference of the superexchange pathways. According to our calculations, the electronic states of the bridge units in the system (Fig. 1) are at least 2 eV higher in energy than the donor and acceptor states. It means that the amino-acid residues cannot serve as intermediate states in the ET between the redox centers. The probability of thermal activation of the hopping process is negligible (1035). Let us estimate the ET rate between the clusters using Eq. (2). For the effective coupling squared, V 2 ¼ 3 V 2rms with Vrms = 0.756104 eV (Table 2) and the reorganization energy k=1 eV, one obtains kET = 1.3104 s1. For smaller values of k, kET is higher (e. g. kET = 3.2105 s1 when k = 0.7 eV). Experimentally, the lowest limit of the ET rate is 102 s1 [33,34]. Wittekindt et al. calculated the electronic coupling between clusters using a semiempirical tight-binding model [35]. They estimated the matrix element for the bottleneck step to be 3109 eV, which gives kET 2104 s1 (at k = 0.7 eV). Because the estimated rate is 6 orders of magnitude smaller than the lowest limit of the rate, it was suggested that aromatic amino acids act as stepping stones in the ET process. Using some (not properly justified) values for the energetics of hole states on aromatic amino-acid residues in the system, they derived the ET rate for the multistep hopping, 103 s1, in agreement with the observed data. According to our DFT calculations, however, the electronic coupling between the clusters in the system (Fig. 1) is considerably stronger (Table 2) and no intermediate states (stepping stones) are required. Moreover, the states with a hole on the bridge units are considerably higher (P2 eV) than the cluster levels and cannot be thermally
populated. Thus, the hole hoping in the considered system appears to be neither required nor allowed. 4. Conclusions Using DFT calculations of several models consisting of two complexes [Fe4S4(SR)4] and closest amino-acid residues we have estimated the electronic coupling between the iron–sulfur clusters [4Fe–4S]. The matrix element Vrms is obtained by averaging the coupling values computed for nearly degenerate electronic states of the redox centers. The distance dependence of the matrix element has been found for the low- and high-potential ET systems. The effective coupling is shown to be not very sensitive to the external electric field and mutual orientation of the clusters. In the redox system modeling the rate-limiting ET step in the bacterial respiratory complex I, the superexchange interaction with surroundings increases the coupling value by an order of magnitude. Our results suggest that the superexchange mechanism rather than hole hopping should be operative in ET between iron–sulfur clusters in proteins. Acknowledgment The work was supported by the Spanish Ministerio de Educación y Ciencia (Project No. CTQ2009-12346). References [1] L.E. Mortenson, R.C. Valentine, J.E. Carnahan, Biochem. Biophys. Res. Commun. 7 (1962) 448. [2] R. Cammack, Adv. Inorg. Chem. 38 (1992) 281. [3] W. Lubitz, E. Reijerse, M. van Gastel, Chem. Rev. 107 (2007) 4331. [4] D.C. Johnson, D.R. Dean, A.D. Smith, M.K. Johnson, Ann. Rev. Biochem. 74 (2005) 247. [5] B.J. Lemon, J.W. Peters, in: A. Messerschmidt, R. Huber, T. Poulos, K. Wieghardt (Eds.), Handbook of Metalloproteins, Wiley, Weinheim, Germany, 2001. [6] L. Noodleman, C.J. Peng, D.A. Case, J.-M. Mouesca, Coord. Chem. Rev. 144 (1995) 199. [7] J.-M. Mouesca, J.L. Chen, L. Noodleman, D. Bashford, D.A. Case, J. Am. Chem. Soc. 116 (1994) 11898. [8] E.I. Solomon, B. Hedman, K.O. Hodgson, A. Dey, R.K. Szilagyi, Coord. Chem. Rev. 249 (2005) 97. [9] A. Dey et al., E. I. Science 318 (2007) 1464. [10] S. Shaik, S. Cohen, Y. Wang, H. Chen, D. Kumar, W. Thiel, Chem. Rev. 110 (2010) 949. [11] A. Warshel, Ann. Rev. Biophys. Biomol. Struct. 32 (2003) 425. [12] H.B. Gray, J.R. Winkler, Q. Rev. Biophys. 36 (2003) 341. [13] R.A. Marcus, N. Sutin, Biochim. Biophys. Acta 811 (1985) 265. [14] I.V. Kurnikov, A.K. Charnley, D.N. Beratan, J. Phys. Chem. B 105 (2001) 5359. [15] A. Troisi, A. Nitzan, M.A. Ratner, J. Chem. Phys. 119 (2003) 5782. [16] R.J. Cave, M.D. Newton, J. Chem. Phys. 106 (1997) 9213. [17] A.A. Voityuk, N. Rösch, J. Chem. Phys. 117 (2001) 5607. [18] K. Senthilkumar et al., J. Am. Chem. Soc. 127 (2005) 14894. [19] J. Huang, M. Kertesz, M. J. Chem. Phys. 122 (2005) 234707. [20] M. Félix, A.A. Voityuk, J. Phys. Chem. A 112 (2008) 9043. [21] T. Kubar, P.B. Woiczikowski, G. Cuniberti, M. Elstner, J. Phys. Chem. B 112 (2008) 7937. [22] I. Bertini, A. Donaire, B.A. Feinberg, C. Luchinat, M. Piccioli, H. Yuan, Eur. J. Biochem. 232 (1995) 192. [23] L.A. Sazanov, P. Hinchliffe, Science 311 (2006) 1430. [24] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [25] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [26] M.J. Frisch et al., GAUSSIAN 03, Gaussian, Inc., Pittsburgh, PA, 2003. [27] A.A. Voityuk, K. Siriwong, N. Rösch, Phys. Chem. Chem. Phys. 3 (2001) 5421. [28] A. Troisi, G. Orlandi, J. Phys. Chem. B 106 (2002) 2093. [29] I.A. Balabin, D.N. Beratan, S.S. Skourtis, Phys. Rev. Lett. 101 (2008) 158102. [30] D.N. Beratan et al., Acc. Chem. Res. 42 (2009) 1669. [31] J.L. Bredas, D. Beljonne, V. Coropceanu, J. Cornil, Chem. Rev. 104 (2004) 4971. [32] D.N. Beratan, J.N. Onuchic, J.R. Winkler, H.B. Gray, Science 258 (1992) 1740. [33] A.D. Vinogradov, Biochim. Biophys. Acta 1364 (1998) 169. [34] M.S. Sharpley, R.I. Shannon, F. Draghi, J. Hirst, Biochemistry 45 (2006) 241. [35] C. Wittekindt, M. Schwarz, T. Friedrich, T. Koslowski, J. Am. Chem. Soc. 131 (2009) 8134.