J. Phys. Chem. 1995, 99, 2946-2948
2946
Nonperturbative Calculation of Electronic Coupling for Electron Transfer Reaction in Proteins Akira Okada* and Toshiaki Kakitani Department of Physics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan
Junichiro Inoue Department of Applied Physics, Faculty of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan Received: September 7, 1994; In Final Form: January 4, 1 9 9 9
Electronic couplings for electron transfer reaction in myoglobin whose His-48 is ruthenated are calculated nonperturbatively by applying a recursion method. An extended Huckel method is used for electronic calculations. A new index that measures the importance of each amino acid for electron transfer pathways is introduced. It is found that a few amino acids play a key role, amino acids without which the electronic couplings is decreased drastically. It is also found that the electronic coupling - is increased by some important amino acids around the key amino acids in this myoglobin derivative.
Introduction Electron transfer reactions in proteins play a major role in the energy conversion of biological systems. In recent years there has been considerable experimental research on electron transfers in As a result, it is made clear that these electron transfers can occur rapidly over a long ranges4 Theoretically, the rate constant for such electron transfers can be expressed as the product of the square of an electronic coupling matrix element (HDA,abbreviated as electronic couplings hereafter) and a nuclear Franck-Condon f a ~ t o r .Elec~ tronic couplings for electron transfers in small systems have been studied by using ab initio and by using more approximate methods.1°-12 For proteins, electronic couplings have been calculated by the one-electron approximation. Beratan, Betts, Onuchic, and co-workers calculated electronic couplings by way of pathway analysis with certain empirical decay factors.13-20 Goldman calculated electronic couplings by applying a renormalized perturbation expansion with certain empirical reasonace integrals and Coulomb integrals.21 Siddarth and Marcus calculated electronic couplings by an extended Huckel theory for the selected important amino acids by their artificial intelligence search m e t h ~ d . ~ *Their - ~ ~ search method is based on a perturbation theory, but electronic couplings are calculated nonperhubatively with no adjustable parameters for amino acids selected. In these theoretical studies of electronic couplings for electron transfer reactions in proteins, perturbation theories have been used somewhere in the calculational procedure, because these calculations are required to treat a large matrix that has several thousand dimensions. But, there is a suggestion that in these electron transfer reactions electronic couplings would not be adequately obtained by perturbational methods, because many intermediary amino acids involved must work coherently or c ~ o p e r a t i v e l y . Kuki ~ ~ ~ ~and ~ Wolynes calculated electronic couplings nonperturbatively by the path integral method using a pseudopotential for the protein.26 After this work Gruschus and Kuki calculated electronic couplings nonperturbatively by applying their inhomogeneous aperiodic lattice theory.27 In this study, we calculate electronic couplings for electron transfer by using the extended Huckel method and explore the @Abstractpublished in Advance ACS Absrracrs, February 1, 1995.
method to evaluate HDAby a nonperturbational calculation. For this purpose the recursion method?* which is a powerful tool for calculation of large sparse matrices, is used. A new index that measures the importance of each amino acid for electron transfer is introduced. Calculations are made for the zinc1 ruthenium-modified myoglobin, which was well studied experimentally by Gray et al.29 By the analysis using this index, it is found that the importance ranges widely among many amino acids and that a relatively small number of amino acids play a key role in the electron transfer pathway, without which the electronic coupling is decreased drastically.
Theory and Method Formulation of HDA by the partitioning technique is given by30
where ID)is the donor state, IA) is the acceptor state, and E is the transfemng electron's energy. The indices D, A and M indicate the donor, acceptor, and medium, respectively. VDAis the coupling between the donor and acceptor. VDMand V m are defined in the same way. The Green function &(E) is defined as
where HMis the Hamiltonian of the medium. In the long-range electron transfer, the direct electronic coupling between the donor state and the acceptor state is much smaller than the indirect coupling. Then, we neglect the former and write HDA as
(3) The calculation of HDA was made nonperturbatively by applying the recursion method, which is a powerful tool for calculation of large sparse matrices. We now shortly recall the recursion method.28 In this method, the Hamiltonian matrix is tridiagonalized, and a diagonal element of the Green function is expressed in the continued fraction of the elements of the tridiagonalized Hamiltonian matrix. In the present study, the
0022-365419512099-2946$09.00/0 0 1995 American Chemical Society
Letters
J. Phys. Chem., Vol. 99, No. 10, 1995 2947
extended Huckel method was used to construct an effective oneelectron Hamiltonian matrix, and off-diagonal elements of an overlap integral were neglected. By this method, we can calculate the right-hand side of eq 3 nonperturbatively. More details of the calculation method will be published elsewhere. In the investigation of the electron pathway by the path integral method using a pseudopotential for an excess electron, Kuki and Wolynes derived a formula26
0
6
0
2
Number of Amino Acids 14 26 38
60
84
10
12
h
(4) where V(r)is a potential at point r and e w ( r ) is the density of electron pathways in a kink at the point r. Equation 4 states that a functional derivative of In HDA by V(r) measures the importance at the position I in the electron transfer pathways. In their formulation of HDA using the one-electron pseudopotential, the hole transfer effect was not taken into account. In analogy with eq 4,we introduce an index
I-
4
6
8
R ( 4 Figure 1. Calculated electronic couplings HDAby varying the value of R. The number written above the figure is the number of amino acids which are included in the range defined by R. A dashed line indicates the converging value of HDA.
TABLE 1: Indices of Amino Acids for Electron Transfer Reaction in ("&Ru(His-48)Mb(ZnP) (Values Calculated for R = 8 A) where Vi is a potential at amino acid i. If we replace Vi in this formula by V(r),the right side of eq 5 equals the left side of eq 4. Therefore, we can consider that the value of Zi measures the importance of the amino acid i in the electron transfer pathways. Furthermore, HDAin eq 5 is formulated by using the extended Huckel molecular orbital theory, so it includes both effects of electron transfer and hole transfer.
Results and Discussion
In this study we consider the zinchthenium-modified myoglobin which was well studied experimentally by Gray et al.29 Calculation was made for HDA of sperm whale myoglobin, whose heme was replaced by Zn mesoporphyrin and whose His48 was ruthenated ((NH3>5Ru(Hi~-48)Mb(ZnP)). In the present study, the donor is the triplet Zn mesoporphyrin (3ZnP*) and the acceptor is the ruthenium complex ((NH3)5Ru3+). Accordingly, the donor state was taken as the HOMO of the Zn mesoporphyrin and the acceptor state as the LUMO of the ruthenium complex. We used the value of the energy of the HOMO of the Zn mesoporphyrin as the value of E. The value of E was -11.36 eV, and that was calculated by the extended Huckel theory. The parameters in the extended Huckel theory were taken from refs 31 and 32. The protein coordinates were obtained from the Brookhaven protein data bank (code number 1MBC),33and the coordinates of protons were produced by using BIOGRAF.34 The ruthenium complex coordinates were obtained from the Cambridge crystallographic data file.35 The zinchthenium-modified myoglobin coordinates were produced from these data by computer manipulation. Water molecules were removed. First, we confined the protein medium to those amino acids which are involved or crossed by the region that is made up by two spheres and one cylinder of radius R. Two spheres are the spheres whose centers are the Zn site of the donor and the Ru site of the acceptor, and the axis of cylinder is the line connecting these two sites. We calculated HDAby varying the value of R. The result is plotted in Figure 1. From this, we find that HDAconverges to 4.6 x eV for R 2 8 A. This fact indicates that those amino acids which locate outside the spheres and cylinder of radius larger than 8 A make little or no contribution to the electron coupling factor.
order residue (i) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
His Phe Glu Asp Phe Leu Thr Lys Lys Lys Thr Ala Ser
Arg Val Leu
His Phe
index I; (eV-l)
4.0 x lo-' 46 2.7 x lo-' 59 1.0 x lo-' 60 7.6 x 43 -7.3 x 49 5.7 x 39 4.3 x 63 4.0 x 3.7 x 47 3.7 x 42 67 3.5 x 3.4 x 57 58 3.3 x 45 -3.2 x 68 1.1 x 40 8.4 x 48 -4.7 x 33 4.7 x
64
order residue
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Lys Lys Leu Gly Met Leu Leu
His Ala Asp Ile Glu De His Ser Leu Leu Phe
(i)
index I; (eV-I)
62
3.6 x
50
3.2 x 61 1.9 x 65 1.6 x 10-3 5s -6.1 x 10-4 29 -3.3 x 10-4 32 -2.5 x 97 2.2 x 10-4 71 1.8 x 4.4 -1.7 x 10-4 99 9.2 x 10-5 54 -8.4 x 10-5 io7 -2.0 x 10-5 93 7.1 x 92 -1.3 x io4 2.4 x 10-7 89 -2.8 x 138 -5.2 x lo-'
Second, we calculated Zi for R = 8 8,(the range including 36 amino acids) by the approximated equation of eq 5 ,
1 ~ D I.=-' HDA AVi
A
where AHDAand AVi are the small changes of HDA and Vi, respectively. In the extended Huckel theory the valence orbital ionization energy H k of the atomic orbital k is the sum of kinetic and potential energy. If the value of Vi is increased by the constant value AVi, the value of Hkk increases by the constant value AV, for all the atomic orbital k belonging to the amino acid i. So we used the equation
where the atomic orbital k belongs to the amino acid i. The results for AVi = 0.01 eV are given in Table 1. We c o n f i e d that almost the same results for Zi are obtained for AV = 0.01, -0.01, and 0.001 eV. The values of Zi are written in the order of its absolute value. It is important to note that a few amino acids have large values of Zi and the other amino acids have relatively small but appreciable values of Zi.
Letters
2948 J. Phys. Chem., Vol. 99, No. 10, 1995 61
4
v
I
I
3.
II
J'
Nagoya University, HP in Department of Physics, Faculty of Science, Nagoya University, and HITAC-S-820180 at Computer Center, Institute for Molecular Science. This work was supported in part by Grants-in-Aid for Scientific Research on Priority Areas (No. 236,05235221; No. 229,06219213) from the Japanese Ministry of Education, Science and Culture.
References and Notes
0
2
4
6
8 1 0 12 14 16 18 2 0
22 24 26 2 8 30
Number of Amino Acids Figure 2. Calculated electronic couplings HDA as a function of the number of amino acids which are taken into account in the calculation in the order of its absolute value of the index 1,. A dashed line indicates the converging value of HDA.
Third, we calculated HDA as a function of number of amino acids which are taken into account in the calculation in the order of the absolute value of the index. The results are given in Figure 2. It is important to note that only the first two amino acids (His-64, Phe-46) make about 20% of the total electronic coupling. The first nine amino acids make about 100% of the total electronic coupling. Finally, we calculated HDAby considering all the amino acids except His-64 and Phe-46. As a result, we found that the value of HDAis reduced to 1/100 of the total electronic coupling. This result suggests that there are a relatively small number of key amino acids without which the electronic coupling is decreased drastically. By drawing the protein conformation by the NAMOD program36 (not shown), we find that His-64 and Phe-46 locate just on the line connecting the Zn mesoporphyrin and ruthenium complex.
Conclusion Electronic couplings for electron transfer reactions in the ruthenated derivative of myoglobin (NH3)5Ru(His-48)Mb(ZnP) were calculated by using the extended Huckel method. Nonperturbative calculations were made by applying the recursion method, which is a powerful tool for calculation of a large sparse matrix. A new index measuring the importance of amino acid for electron transfer pathways is introduced and calculated. As a result, it is found that a few amino acids which have the largest values of Zi play a key role in the electron tunneling pathway, without which the electronic coupling is decreased drastically in this myoglobin derivative. It is also found that the electronic coupling is increased by incorporating some important amino acids in addition to the key amino acids in this myoglobin derivative.
Acknowledgment. The calculations were done by using FACOM M-1800/20, W2600/10 in the Computation Center in
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