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Electronic decoherence for electron transfer in blue copper proteins
7 September 2001
Chemical Physics Letters 345 (2001) 159±165
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Electronic decoherence for electron transfer in blue copper proteins Daren M. Lockwood, Yuen-Kit Cheng 1, Peter J. Rossky * Department of Chemistry and Biochemistry, Institute for Theoretical Chemistry, University of Texas at Austin, Austin, TX 78712-1167, USA Received 18 May 2001
Abstract We present a molecular dynamics investigation of the electronic decoherence rate for electron transfer (ET) in a solvated protein molecule. We ®nd that decoherence occurs on an ultrafast time scale of 2.4 fs, considerably faster than ¯uctuations in the electronic coupling. Both protein and solvent dynamics play important roles. Solvent alone would give rise to a decoherence time of 3.0 fs, as compared to 4.1 fs from the protein matrix alone. This implies that both solvation and protein dynamics can strongly aect both the rate and mechanism of ET. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Electron transfer (ET) reactions are of interest in a diverse set of contexts and have been the subject of extensive experimental and theoretical research [1]. ET in molecular wires is of interest in the ®eld of molecular electronics [2], while ET reactions in proteins are biologically important and make up vital components of metabolic pathways [3]. The eects of nuclear degrees of freedom on ET rates have traditionally been taken into account by introducing a constant averaged electronic coupling [1,4], in which case the ET rate can be factored into a purely nuclear component, the Franck±Condon (FC) factors, and a purely
*
Corresponding author. Fax: +1-512-471-1624. E-mail address: [email protected] (P.J. Rossky). 1 Present address: Department of Chemistry, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
electronic component. The FC factors can be conveniently approximated in condensed phase environments at high temperatures by a semiclassical Marcus expression [1,4]. However, at low temperatures [4], and in cases where so-called bridging electronic states become signi®cantly populated [2,5], ET rates in general require treatment of the dynamics of electronic coherence. Decoherence in general, corresponding to decay of o-diagonal elements of quantum mechanical density matrices [6±8], has been the subject of intense interest in the theoretical physics community, in relation to such issues as the classical limit of quantum mechanics [8]. Only recently, however, have methods been developed which enable quantitative investigation of the eect on electronic dynamics for realistic chemical systems [7,9,10]. Electronic decoherence is a characteristic of the electronic reduced density matrix obtained by integrating over the nuclear degrees of freedom [7]. The decay of o-diagonal elements of this
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reduced density matrix results when nuclear trajectories on alternative electronic surfaces diverge from one another. This is expected to occur rapidly in condensed phase environments [9,11]. Accurate prediction of the rates of electronic processes thus requires an explicit treatment of electronic coherence. The rate of electronic coherence loss plays a clear role in modulating both the rate and mechanism of ET [2,5]. For the high electronic decoherence rates anticipated in condensed phase environments, direct tunneling between two tightly bound electronic states becomes less ecient as the decoherence rate increases. This can be understood based on either FC factors [4,9], or within the framework of open systems theory (and, in the limit of in®nite decoherence rates, of the Zeno eect) [6]. Sequential ET over bridge sites is favored when rapid coherence loss is associated with each ET step, especially if the energy gaps between donor, bridge, and acceptor sites are small [2,5,12]. It is thus of considerable interest to identify features of the condensed phase environment that control the duration of electronic coherence. 2. ET in ruthenated azurin Pseudomonas aeruginosa azurin is a well-characterized electron-transport protein [13±15]. This small blue copper protein contains a single copper redox center and a surface histidine group to which ruthenium complexes may be attached, as described by Gray and Winkler [13,16]. These complexes act as electron donors and acceptors, and permit one to photo-induce ET over controlled distances in natural structures. Here, we consider the speci®c case of direct ET between metal ions in ruthenated azurin. The electronic state is taken to be a linear combination of two tightly bound electronic states, an initial (donor) electronic state j1i and a ®nal (acceptor) electronic state j2i, throughout its evolution. The ET rate is given by the Golden Rule [1]. Expressed in a time dependent representation, the thermal rate constant expression can be written as [10]
k1!2 h
2
Z
1 1
dthijV12 eiH2 t=h V21 e
iH1 t= h
jii
:
1
T
Here, jii is the initial nuclear state, and the brackets h iT denote a thermal average over alternative initial nuclear states. The integrand in Eq. (1) is given by the weighted time dependent overlap of two nuclear states. e iH1 t=h jii yields the nuclear state obtained by propagating jii for a time t on the initial electronic surface j1i. The electronic coupling element V21 promotes the timeevolved nuclear state e iH1 t=h jii to surface j2i. The second nuclear state is obtained by promoting hij to surface j2i through the action of V12 , followed by propagation for a time t on surface j2i [9]. In Eq. (1), we have written the Golden Rule rate expression in its most general form. It is worth noting that if one assumes that an eective averaged electronic coupling can be factored out, the rate expression can be expressed as a product of a nuclear (FC) factor and an electronic coupling factor [9], which is the form most commonly used to discuss ET rates [1]. Alternatively, if the correlation function is replaced by the classical correlation function for the coupling, one obtains a rate expression recently used by Daizadeh et al. [15]. It is also worth noting that here we focus on the ET rate between two localized electronic states, in which case the rate expression (1) can be derived either based on unitary time evolution of the entire system or based on an open systems treatment [17,18]. In the current study, an electron transport protein is solvated under ambient conditions. In this and similar cases, the nuclear states can be described by quasi-classical coherent states with small uncertainty in momentum [9], which can be approximated for short times by a product of frozen Gaussian wave packets [10]. The centers of these wave packets follow classical trajectories. The coupling elements may then be evaluated along the classical nuclear trajectory on the initial electronic surface [9] yielding Z 1
2 k1!2 h dt V12q c 0V21q c t ijeiH2 t=h e iH1 t=h ji T 1 Z 1 2 h dthV12q c 0V21q c thi2 tji1 tiiT ; 1
2
D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
where V21q c t is evaluated along the classical nuclear trajectory. hi2 tji1 ti is the overlap between wave packets propagated on the alternative electronic surfaces. The electronic decoherence rate is the rate at which this overlap decays, as nuclear trajectories on the two surfaces diverge from one another [7]. If this decay is more rapid than the ¯uctuations in V21q c and uncorrelated with V21q c t, 2 then a factor of hjV21q c j iT can be separated. The rate is then proportional to the integral of the decaying overlap. Higher decoherence rates then correspond directly to reduced ET rates, demonstrating the signi®cance. Rearranging Eq. (2), one sees that the scaled 2 transition rate h2 k1!2 =hjV21q c j iT is equal to the contribution from FC factors. As such, it reduces at high T to the usual high T Marcus expression, often implemented in the present context [4]. The high T result includes the eect of the classical nuclear density of states, and corresponds to electronic transitions only at nuclear con®gurations where electronic surfaces cross [4]. Eq. (2) is more generally valid, and, for very short decoherence times, can be a required form [4,9]. In particular, the rate given by Eq. (2) is explicitly dependent on the rate of the decaying overlap of wave packets on dierent electronic surfaces, an eect (electronic decoherence) that can be neglected for suciently high temperatures in the case of only two electronic states [4]. Representing the ET rate in the time domain provides a convenient route to evaluating nuclear contributions in a nearly classical framework. Classical molecular dynamics or Monte Carlo simulations can be used to obtain the ensemble of dierent nuclear con®gurations on the initial electronic surface. Further, in this work we use a short time approximation involving only the initial forces on nuclei; we do not require or make use of any explicitly dynamical information. If kn is the spatial width of the Gaussian wave packet associated with nucleus n, the characteristic quantum decoherence time over which the quantity hi2 tji1 ti decays is explicitly given by [4,9] * + X k2 2 2 n ~ F1n ~ ; 3 sD F2n h2 n 2 T
161
where ~ Fjn is the force on nucleus n if it is evolving on the electronic surface jji. The duration of electronic coherence naturally decreases as the dierences between forces on the electronic surfaces increase. A suitable value for kn in the case of nearly classical nuclei is given by [10] k2n
h2 ; 6Mn kB T
4
where Mn is the mass of nucleus n and T is the temperature. Alternative expressions for kn give decoherence rates diering by less than a factor of 2 for the case of relaxation of a solvated electron [7,9]. We note that in the limit of in®nite nuclear masses or high temperatures, the quantum decoherence time diverges, and it follows that this decoherence eect does not appear in mixed quantum±classical approximations with a purely classical bath [7,9]. We note that the process of electronic decoherence described here should not be confused with the pure dephasing introduced by classical baths; unlike electronic decoherence, pure electronic dephasing rates increase with temperature [4,9]. In this limit of high T or large masses, the classical Marcus ET expression, often invoked for solvent (`outer sphere') nuclear modes [1], is recovered [4]. The separate treatment of high frequency solute (`inner sphere') nuclear modes in standard ET formulations via quantum harmonic models [1,15] does, of course, account for the same quantum eects as decoherence [9], within the limitations of the harmonic model. However, a key result that will be clear from the analysis below of the present solvated protein system is that the identi®cation and characterization of those modes for which the quantum correction is important is far more complex than might be anticipated. 3. Simulation and results A 100 ps molecular dynamics simulation of Ru bpy2 (im)(His 83)-modi®ed azurin [16] in water was performed in the canonical ensemble employing periodic boundary conditions, at a temperature of 300 K, using the computer program TI N K E R [19,20]. Initial atomic coordinates
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for the azurin molecule were extracted from the crystal structure deposited at the Brookhaven Protein Data Bank, while initial atomic coordinates for the water molecules were taken from a pre-equilibrated box of 6500 water molecules, with of an atom in the protein those waters within 3.5 A matrix removed. In those cases where two oppositely charged groups, whether amino acid side apart, the chain or metal ion, were less than 7.5 A two groups were considered to constitute a single neutral group (where eective distances were based on metal ion centers, centers of nitrogens in ammonium groups, and centers of carbons in carboxylate groups). The nearest water molecules to any remaining, `isolated', charged groups were replaced by a sodium or chloride ion, so as to achieve local charge neutrality. Prior to generating Newtonian trajectories, atoms in the protein matrix were subjected to energy minimization to remove large forces. The water molecules were modeled using the SPC potential [21], while parameters for atoms in the protein matrix were based on the CHARMM parameter set [22]. In order to calculate a characteristic decoherence time via Eqs. (3) and (4), one requires the dierences in forces on the nuclei corresponding to each electronic state. CHARMM-compatible parameters for the Cu2 ion and its ligands in type I blue copper proteins have been published by Ungar et al. [23]. To determine the charge distribution on the ligands for Cu1 , we followed [23]. Linear extrapolation between CHARMM parameters for the protonated and deprotonated forms of the ligands was used, such that ligands had the same net charges as obtained from semi-empirical electronic structure calculations. For the ruthenium complex, the geometry of the six rings liganded to the ruthenium ion, relative to one another and to the ruthenium ion, were constrained based on the crystal structure. Hence only assignment of Lennard±Jones parameters and partial charges for these atoms were required to model the complex. Lennard±Jones parameters for the ruthenium ion were taken from the UFF force ®eld [24], and the associated charge equilibration (Qeq) method [25±27] was used to determine the total charge on each of the ligands. The charge distributions for the histidine and imidazole ligands of the ruthe-
nium ion were modeled as for the Cu1 ligands [23]. For the two 2,20 -bipyridine molecules liganded to the ruthenium ion, charges were taken directly from the Qeq results, while Lennard±Jones parameters for pyridines were taken from the OPLS parameter set [28]. For the Ru2 case, the net charges on each of the six ligands were nearly the same, while the partial charge on the Ru2 ion was reduced from its formal charge of +2 to +0.56. For the Ru3 case, the ruthenium partial charge was about the same, with the dierence taking place primarily on the ligands. To calculate the force dierences, we consider not only the eect of charge redistribution accompanying ET, but also the change in covalent radii of the metal ions. These two eects are suf®cient for modeling the metal complexes. The UFF force ®eld supports this claim and models metal complexes well [24,29], and further, crystal structures for azurin are largely insensitive to effects other than the radii of the metal ions [30]. Based on data for hexamine ruthenium [31], we took the increase in ruthenium radius on going from the +3 to +2 oxidation state to be 0.04 A, while the bond force constants between ruthenium and nitrogen were taken from the UFF force ®eld [4]. The decrease in the covalent radius of the copper ion on going from its +1 to +2 oxidation based on crystallostate was taken to be 0.05 A, graphic data for azurin and model copper complexes [30,32]. The decoherence rate is in any case not found to be very sensitive to the changes in metal ion radii, as noted below. The dierences in forces on the nuclei between the two electronic states were calculated from 800 evenly spaced con®gurations taken from the last 80 ps of the trajectory. The force dierences and Eqs. (3) and (4) yield a calculated characteristic (average) electronic decoherence time of 2.4 fs. This value is based on an initial electronic state in which both metal ions have formal charges of +2. The thermodynamically favored direction for ET depends on the selection of ruthenium ligands [13]. However, test simulations indicate that the electronic decoherence rate is largely independent of the direction of ET, as would be expected from a linear response model. We note that the characteristic electronic decoherence time of 2.4 fs is
D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
indeed considerably shorter than the characteristic time scale for ¯uctuation of the electronic coupling [15]. Indeed, 2.4 fs is shorter than the periods of the highest frequency modes in water; for instance, it is about a factor of 5 smaller than the vibrational period of the O±H bond. Because Eq. (3) takes the form of a sum over nuclei, it is possible to divide the nuclei into categories, and carry out the important analysis of the contribution of a given subset of the nuclei to sD2 . Of particular interest is the magnitude of the contribution from the solvent, as compared to the protein matrix, since this bears on the question of how hydration of a protein aects the ET rate and on the potential role of protein design as well. The form of Eq. (3) makes clear the fact that the squared decoherence rate can be expressed as 2 2 sD2 sD;w sD;p ;
5
163
where sD;w is the decoherence time that would arise from water contributions only, and sD;p is the decoherence time that would arise from only the remaining atoms in the system. Thus, the separation desired is a natural one in the present formulation. We ®nd that solvent molecules alone are sucient to give rise to a decoherence time of 3.0 fs, as compared to a decoherence time of 4.1 fs from only the remaining nonsolvent atoms in the system. As such, solvent molecules play a large role in reducing the rate of direct electronic tunneling between the donor and acceptor sites. To understand the eect of distance from ET sites on the contributions to decoherence, we have also determined contributions to sD2 as a function of a cuto distance from the metal ions. The result is shown in Fig. 1. Close to the metal ions, most atoms are part of the protein matrix, and so
Fig. 1. Contributions to sD2 from atoms within a speci®ed cuto distance of the metal ion centers. The solid line denotes the result obtained by including all atom types. The + symbols denote the contribution from atoms in the solvent molecules, while the } symbols denote the contribution from the remaining atoms. The symbols denote the contribution from atoms in the solvent molecules which are closer to the center of the copper ion than to the center of the ruthenium ion.
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D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
contributions from such atoms are dominant at short distance. However, water molecules more distant from the metal ions contribute very signi®cantly to the squared decoherence rate, and the water contribution is strikingly long-ranged in character. To better understand the origin of this remarkable eectiveness of the water molecules in causing coherence loss, we calculated the contribution to sD2 arising solely from hydrogen atoms in water molecules. We ®nd that about 90% of the water contribution can be attributed to the hydrogen atoms. This importance of hydrogen atoms can easily be understood by examining the form of Eqs. (3) and (4), where the contributions of atoms with smaller mass are weighted more highly. As such, the eciency with which water induces decoherence can be understood as arising from the high density of polar hydrogen atoms. We note that the long range of the water contribution results from the contribution of this high density and eectiveness, combined with the increasing volume of water associated with a given radial distance. The range is not a peculiarity of the protein system, but manifests itself comparably for atomic ions in solution [4]. Although the density of polar protons is considerably lower in the protein than in the aqueous solvent, the importance of polar hydrogens leads to the corresponding result that about 75% of the protein matrix contributions (neglecting changes in metal ion covalent radii) also arise from hydrogen atoms. Also shown in Fig. 1 is the contribution of water molecules which are closer to the copper ion than the ruthenium ion. Clearly, the location of the ruthenium complex on the surface of the protein leads to a greater eect from nearby water molecules. It must be noted, however, that even the water molecules closest to the copper ion alone are sucient to give rise to a decoherence time of about 5.5 fs, while the protein matrix alone would give a decoherence time of about 4.1 fs. Hence, neglecting the Ru solvation would still give a 3.3 fs decoherence time. As such, the design of this protein, in which the copper ion is buried well within the protein matrix, is not sucient to prevent a very high rate of electronic coherence loss
due to the solvent molecules, nor is the decoherence time much longer even in the absence of solvent molecules. We brie¯y return to the issue of the role of distribution of charge over the ligands and of changes in covalent radii in determining the decoherence time. To test the former, we evaluated the decoherence time that would be obtained if charge transfer occurred strictly between metal ion centers. The decoherence time so obtained was 2.3 fs, as opposed to the value of 2.4 fs obtained with the correct distribution of charge over the ligands. To test the latter issue, we calculated the decoherence time that would be obtained if the ion covalent radii remained constant in the ET process. The decoherence time so obtained was 2.5 fs. We conclude that the decoherence rate is largely insensitive to such details, and can be obtained here by considering only the change in electrostatic forces if a point charge of )1 is transferred between the metal centers. 4. Conclusions For the example of direct ET in the blue copper protein ruthenated azurin, we have shown that electronic coherence decay due to coupling of the electronic state to the nuclear state of the surroundings is extremely rapid. The origins of this decay lie comparably in the quantum dynamics of the solvent and the protein matrix and the contributions are remarkably long ranged. One concludes that the quantum character of the dynamics of these `outer sphere' modes is an important feature for ET in proteins. Of special importance is the fact that the small characteristic electronic decoherence time of 2.4 fs can preferentially promote sequential ET over bridge sites. In addition, modi®cation of the ligands of ruthenium [13] and copper [33] can reduce the thermodynamic driving force which is also expected to enhance the role of sequential transfer [5,12]. To quantify this competition between sequential and direct transfer, it will be necessary to quantitatively explore the characteristics of electronic coherence associated with those individual ET steps which compose alternative transfer pathways [33] in azurin.
D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
Acknowledgements This work was supported by a grant from the Robert A. Welch Foundation. Additional support was provided by the National Science Foundation. We thank J.W. Ponder for use of the software package TI N K E R , available at http://dasher.wustl. edu/tinker. References [1] P.F. Barbara, T.J. Meyer, M.A. Ratner, J. Phys. Chem. 100 (1996) 13148. [2] W.B. Davis, W.A. Svec, M.A. Ratner, M.R. Wasielewski, Nature 396 (1998) 60. [3] D. Voet, J.G. Voet, Biochemistry, second ed., Wiley, New York, 1995. [4] D.M. Lockwood, H. Hwang, P.J. Rossky, Chem. Phys. 268 (2001) 285. [5] W.B. Davis, M.R. Wasielewski, M.A. Ratner, V. Mujica, A. Nitzan, J. Phys. Chem. A. 101 (1997) 6158. [6] D. Gilulini, et al., Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin, 1996. [7] O.V. Prezhdo, P.J. Rossky, Phys. Rev. Lett. 81 (1998) 5294. [8] W.P. Schleich, Nature 403 (2000) 256. [9] O.V. Prezhdo, P.J. Rossky, J. Chem. Phys. 107 (1997) 5863. [10] E. Neria, A. Nitzan, J. Chem. Phys. 99 (1993) 1109. [11] K. Wynne, R.M. Hochstrasser, Adv. Chem. Phys. 107 (1999) 263. [12] L.D. Zusman, D.N. Beratan, J. Chem. Phys. 110 (1999) 10468. [13] H.B. Gray, J.R. Winkler, Annu. Rev. Biochem. 65 (1996) 537.
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[14] O. Farver, I. Pecht, Adv. Chem. Phys. 107 (1999) 555. [15] I. Daizadeh, E.S. Medvedev, A.A. Stuchebrukhov, Proc. Natl. Acad. Sci. USA 94 (1997) 3703. [16] J.R. Winkler, H.B. Gray, Chem. Rev. 92 (1992) 369. [17] G.C. Shatz, M.A. Ratner, Quantum Mechanics in Chemistry, Prentice-Hall, New Jersey, 1993. [18] K. Blum, Density Matrix Theory and Applications, Plenum Press, New York, 1996. [19] R.V. Pappu, R.K. Hart, J.W. Ponder, J. Phys. Chem. B. 102 (1998) 9725. [20] J.W. Ponder, TI N K E R : Software Tools for Molecular Design, Version 3.7, Washington University School of Medicine, 1999. [21] H.J.C. Berendsen, J.P.M. Postma, W.F. Van Gunsteren, J. Hermans, in: G. Pullman (Ed.), Intermolecular Forces, Reidel, Dordrecht, 1981, p. 331. [22] A.D. MacKerell, et al., J. Phys. Chem. B. 102 (1998) 3586. [23] L.W. Ungar, N.F. Scherer, G.A. Voth, Biophys. J. 72 (1997) 5. [24] A.K. Rappe, C.J. Casewit, K.S. Colwell, W.A. Goddard III, W.M. Ski, J. Am. Chem. Soc. 114 (1992) 10024. [25] A.K. Rappe, W.A. Goddard III, J. Phys. Chem. 95 (1991) 3358. [26] D.M. Lockwood, S.B. Schougaard, R.E. Westacott, J.T. McDevitt, unpublished. [27] Cerius2 , Version 3.8, Molecular Simulations Inc., USA, 1998. [28] W.L. Jorgensen, N.A. McDonald, J. Mol. Struct. 424 (1998) 145. [29] A.K. Rappe, K.S. Colwell, C.J. Casewit, Inorg. Chem. 32 (1993) 3438. [30] W.E.B. Shepard, B.F. Andersen, D.A. Lewandoski, G.E. Norris, E.N. Baker, J. Am. Chem. Soc. 112 (1990) 7817. [31] M.J. Ondrechen, M.A. Ratner, D.E. Ellis, J. Am. Chem. Soc. 103 (1981) 1656. [32] J.M. Guss, P.R. Harrowell, M. Murata, V.A. Norris, H.C. Freeman, J. Mol. Biol. 192 (1986) 361. [33] J.J. Regan, et al., Inorg. Chem. Acta 275±276 (1998) 470.
Chemical Physics Letters 345 (2001) 159±165
www.elsevier.com/locate/cplett
Electronic decoherence for electron transfer in blue copper proteins Daren M. Lockwood, Yuen-Kit Cheng 1, Peter J. Rossky * Department of Chemistry and Biochemistry, Institute for Theoretical Chemistry, University of Texas at Austin, Austin, TX 78712-1167, USA Received 18 May 2001
Abstract We present a molecular dynamics investigation of the electronic decoherence rate for electron transfer (ET) in a solvated protein molecule. We ®nd that decoherence occurs on an ultrafast time scale of 2.4 fs, considerably faster than ¯uctuations in the electronic coupling. Both protein and solvent dynamics play important roles. Solvent alone would give rise to a decoherence time of 3.0 fs, as compared to 4.1 fs from the protein matrix alone. This implies that both solvation and protein dynamics can strongly aect both the rate and mechanism of ET. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Electron transfer (ET) reactions are of interest in a diverse set of contexts and have been the subject of extensive experimental and theoretical research [1]. ET in molecular wires is of interest in the ®eld of molecular electronics [2], while ET reactions in proteins are biologically important and make up vital components of metabolic pathways [3]. The eects of nuclear degrees of freedom on ET rates have traditionally been taken into account by introducing a constant averaged electronic coupling [1,4], in which case the ET rate can be factored into a purely nuclear component, the Franck±Condon (FC) factors, and a purely
*
Corresponding author. Fax: +1-512-471-1624. E-mail address: [email protected] (P.J. Rossky). 1 Present address: Department of Chemistry, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
electronic component. The FC factors can be conveniently approximated in condensed phase environments at high temperatures by a semiclassical Marcus expression [1,4]. However, at low temperatures [4], and in cases where so-called bridging electronic states become signi®cantly populated [2,5], ET rates in general require treatment of the dynamics of electronic coherence. Decoherence in general, corresponding to decay of o-diagonal elements of quantum mechanical density matrices [6±8], has been the subject of intense interest in the theoretical physics community, in relation to such issues as the classical limit of quantum mechanics [8]. Only recently, however, have methods been developed which enable quantitative investigation of the eect on electronic dynamics for realistic chemical systems [7,9,10]. Electronic decoherence is a characteristic of the electronic reduced density matrix obtained by integrating over the nuclear degrees of freedom [7]. The decay of o-diagonal elements of this
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 8 5 5 - 7
160
D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
reduced density matrix results when nuclear trajectories on alternative electronic surfaces diverge from one another. This is expected to occur rapidly in condensed phase environments [9,11]. Accurate prediction of the rates of electronic processes thus requires an explicit treatment of electronic coherence. The rate of electronic coherence loss plays a clear role in modulating both the rate and mechanism of ET [2,5]. For the high electronic decoherence rates anticipated in condensed phase environments, direct tunneling between two tightly bound electronic states becomes less ecient as the decoherence rate increases. This can be understood based on either FC factors [4,9], or within the framework of open systems theory (and, in the limit of in®nite decoherence rates, of the Zeno eect) [6]. Sequential ET over bridge sites is favored when rapid coherence loss is associated with each ET step, especially if the energy gaps between donor, bridge, and acceptor sites are small [2,5,12]. It is thus of considerable interest to identify features of the condensed phase environment that control the duration of electronic coherence. 2. ET in ruthenated azurin Pseudomonas aeruginosa azurin is a well-characterized electron-transport protein [13±15]. This small blue copper protein contains a single copper redox center and a surface histidine group to which ruthenium complexes may be attached, as described by Gray and Winkler [13,16]. These complexes act as electron donors and acceptors, and permit one to photo-induce ET over controlled distances in natural structures. Here, we consider the speci®c case of direct ET between metal ions in ruthenated azurin. The electronic state is taken to be a linear combination of two tightly bound electronic states, an initial (donor) electronic state j1i and a ®nal (acceptor) electronic state j2i, throughout its evolution. The ET rate is given by the Golden Rule [1]. Expressed in a time dependent representation, the thermal rate constant expression can be written as [10]
k1!2 h
2
Z
1 1
dthijV12 eiH2 t=h V21 e
iH1 t= h
jii
:
1
T
Here, jii is the initial nuclear state, and the brackets h iT denote a thermal average over alternative initial nuclear states. The integrand in Eq. (1) is given by the weighted time dependent overlap of two nuclear states. e iH1 t=h jii yields the nuclear state obtained by propagating jii for a time t on the initial electronic surface j1i. The electronic coupling element V21 promotes the timeevolved nuclear state e iH1 t=h jii to surface j2i. The second nuclear state is obtained by promoting hij to surface j2i through the action of V12 , followed by propagation for a time t on surface j2i [9]. In Eq. (1), we have written the Golden Rule rate expression in its most general form. It is worth noting that if one assumes that an eective averaged electronic coupling can be factored out, the rate expression can be expressed as a product of a nuclear (FC) factor and an electronic coupling factor [9], which is the form most commonly used to discuss ET rates [1]. Alternatively, if the correlation function is replaced by the classical correlation function for the coupling, one obtains a rate expression recently used by Daizadeh et al. [15]. It is also worth noting that here we focus on the ET rate between two localized electronic states, in which case the rate expression (1) can be derived either based on unitary time evolution of the entire system or based on an open systems treatment [17,18]. In the current study, an electron transport protein is solvated under ambient conditions. In this and similar cases, the nuclear states can be described by quasi-classical coherent states with small uncertainty in momentum [9], which can be approximated for short times by a product of frozen Gaussian wave packets [10]. The centers of these wave packets follow classical trajectories. The coupling elements may then be evaluated along the classical nuclear trajectory on the initial electronic surface [9] yielding Z 1
2 k1!2 h dt V12q c 0V21q c t ijeiH2 t=h e iH1 t=h ji T 1 Z 1 2 h dthV12q c 0V21q c thi2 tji1 tiiT ; 1
2
D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
where V21q c t is evaluated along the classical nuclear trajectory. hi2 tji1 ti is the overlap between wave packets propagated on the alternative electronic surfaces. The electronic decoherence rate is the rate at which this overlap decays, as nuclear trajectories on the two surfaces diverge from one another [7]. If this decay is more rapid than the ¯uctuations in V21q c and uncorrelated with V21q c t, 2 then a factor of hjV21q c j iT can be separated. The rate is then proportional to the integral of the decaying overlap. Higher decoherence rates then correspond directly to reduced ET rates, demonstrating the signi®cance. Rearranging Eq. (2), one sees that the scaled 2 transition rate h2 k1!2 =hjV21q c j iT is equal to the contribution from FC factors. As such, it reduces at high T to the usual high T Marcus expression, often implemented in the present context [4]. The high T result includes the eect of the classical nuclear density of states, and corresponds to electronic transitions only at nuclear con®gurations where electronic surfaces cross [4]. Eq. (2) is more generally valid, and, for very short decoherence times, can be a required form [4,9]. In particular, the rate given by Eq. (2) is explicitly dependent on the rate of the decaying overlap of wave packets on dierent electronic surfaces, an eect (electronic decoherence) that can be neglected for suciently high temperatures in the case of only two electronic states [4]. Representing the ET rate in the time domain provides a convenient route to evaluating nuclear contributions in a nearly classical framework. Classical molecular dynamics or Monte Carlo simulations can be used to obtain the ensemble of dierent nuclear con®gurations on the initial electronic surface. Further, in this work we use a short time approximation involving only the initial forces on nuclei; we do not require or make use of any explicitly dynamical information. If kn is the spatial width of the Gaussian wave packet associated with nucleus n, the characteristic quantum decoherence time over which the quantity hi2 tji1 ti decays is explicitly given by [4,9] * + X k2 2 2 n ~ F1n ~ ; 3 sD F2n h2 n 2 T
161
where ~ Fjn is the force on nucleus n if it is evolving on the electronic surface jji. The duration of electronic coherence naturally decreases as the dierences between forces on the electronic surfaces increase. A suitable value for kn in the case of nearly classical nuclei is given by [10] k2n
h2 ; 6Mn kB T
4
where Mn is the mass of nucleus n and T is the temperature. Alternative expressions for kn give decoherence rates diering by less than a factor of 2 for the case of relaxation of a solvated electron [7,9]. We note that in the limit of in®nite nuclear masses or high temperatures, the quantum decoherence time diverges, and it follows that this decoherence eect does not appear in mixed quantum±classical approximations with a purely classical bath [7,9]. We note that the process of electronic decoherence described here should not be confused with the pure dephasing introduced by classical baths; unlike electronic decoherence, pure electronic dephasing rates increase with temperature [4,9]. In this limit of high T or large masses, the classical Marcus ET expression, often invoked for solvent (`outer sphere') nuclear modes [1], is recovered [4]. The separate treatment of high frequency solute (`inner sphere') nuclear modes in standard ET formulations via quantum harmonic models [1,15] does, of course, account for the same quantum eects as decoherence [9], within the limitations of the harmonic model. However, a key result that will be clear from the analysis below of the present solvated protein system is that the identi®cation and characterization of those modes for which the quantum correction is important is far more complex than might be anticipated. 3. Simulation and results A 100 ps molecular dynamics simulation of Ru bpy2 (im)(His 83)-modi®ed azurin [16] in water was performed in the canonical ensemble employing periodic boundary conditions, at a temperature of 300 K, using the computer program TI N K E R [19,20]. Initial atomic coordinates
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for the azurin molecule were extracted from the crystal structure deposited at the Brookhaven Protein Data Bank, while initial atomic coordinates for the water molecules were taken from a pre-equilibrated box of 6500 water molecules, with of an atom in the protein those waters within 3.5 A matrix removed. In those cases where two oppositely charged groups, whether amino acid side apart, the chain or metal ion, were less than 7.5 A two groups were considered to constitute a single neutral group (where eective distances were based on metal ion centers, centers of nitrogens in ammonium groups, and centers of carbons in carboxylate groups). The nearest water molecules to any remaining, `isolated', charged groups were replaced by a sodium or chloride ion, so as to achieve local charge neutrality. Prior to generating Newtonian trajectories, atoms in the protein matrix were subjected to energy minimization to remove large forces. The water molecules were modeled using the SPC potential [21], while parameters for atoms in the protein matrix were based on the CHARMM parameter set [22]. In order to calculate a characteristic decoherence time via Eqs. (3) and (4), one requires the dierences in forces on the nuclei corresponding to each electronic state. CHARMM-compatible parameters for the Cu2 ion and its ligands in type I blue copper proteins have been published by Ungar et al. [23]. To determine the charge distribution on the ligands for Cu1 , we followed [23]. Linear extrapolation between CHARMM parameters for the protonated and deprotonated forms of the ligands was used, such that ligands had the same net charges as obtained from semi-empirical electronic structure calculations. For the ruthenium complex, the geometry of the six rings liganded to the ruthenium ion, relative to one another and to the ruthenium ion, were constrained based on the crystal structure. Hence only assignment of Lennard±Jones parameters and partial charges for these atoms were required to model the complex. Lennard±Jones parameters for the ruthenium ion were taken from the UFF force ®eld [24], and the associated charge equilibration (Qeq) method [25±27] was used to determine the total charge on each of the ligands. The charge distributions for the histidine and imidazole ligands of the ruthe-
nium ion were modeled as for the Cu1 ligands [23]. For the two 2,20 -bipyridine molecules liganded to the ruthenium ion, charges were taken directly from the Qeq results, while Lennard±Jones parameters for pyridines were taken from the OPLS parameter set [28]. For the Ru2 case, the net charges on each of the six ligands were nearly the same, while the partial charge on the Ru2 ion was reduced from its formal charge of +2 to +0.56. For the Ru3 case, the ruthenium partial charge was about the same, with the dierence taking place primarily on the ligands. To calculate the force dierences, we consider not only the eect of charge redistribution accompanying ET, but also the change in covalent radii of the metal ions. These two eects are suf®cient for modeling the metal complexes. The UFF force ®eld supports this claim and models metal complexes well [24,29], and further, crystal structures for azurin are largely insensitive to effects other than the radii of the metal ions [30]. Based on data for hexamine ruthenium [31], we took the increase in ruthenium radius on going from the +3 to +2 oxidation state to be 0.04 A, while the bond force constants between ruthenium and nitrogen were taken from the UFF force ®eld [4]. The decrease in the covalent radius of the copper ion on going from its +1 to +2 oxidation based on crystallostate was taken to be 0.05 A, graphic data for azurin and model copper complexes [30,32]. The decoherence rate is in any case not found to be very sensitive to the changes in metal ion radii, as noted below. The dierences in forces on the nuclei between the two electronic states were calculated from 800 evenly spaced con®gurations taken from the last 80 ps of the trajectory. The force dierences and Eqs. (3) and (4) yield a calculated characteristic (average) electronic decoherence time of 2.4 fs. This value is based on an initial electronic state in which both metal ions have formal charges of +2. The thermodynamically favored direction for ET depends on the selection of ruthenium ligands [13]. However, test simulations indicate that the electronic decoherence rate is largely independent of the direction of ET, as would be expected from a linear response model. We note that the characteristic electronic decoherence time of 2.4 fs is
D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
indeed considerably shorter than the characteristic time scale for ¯uctuation of the electronic coupling [15]. Indeed, 2.4 fs is shorter than the periods of the highest frequency modes in water; for instance, it is about a factor of 5 smaller than the vibrational period of the O±H bond. Because Eq. (3) takes the form of a sum over nuclei, it is possible to divide the nuclei into categories, and carry out the important analysis of the contribution of a given subset of the nuclei to sD2 . Of particular interest is the magnitude of the contribution from the solvent, as compared to the protein matrix, since this bears on the question of how hydration of a protein aects the ET rate and on the potential role of protein design as well. The form of Eq. (3) makes clear the fact that the squared decoherence rate can be expressed as 2 2 sD2 sD;w sD;p ;
5
163
where sD;w is the decoherence time that would arise from water contributions only, and sD;p is the decoherence time that would arise from only the remaining atoms in the system. Thus, the separation desired is a natural one in the present formulation. We ®nd that solvent molecules alone are sucient to give rise to a decoherence time of 3.0 fs, as compared to a decoherence time of 4.1 fs from only the remaining nonsolvent atoms in the system. As such, solvent molecules play a large role in reducing the rate of direct electronic tunneling between the donor and acceptor sites. To understand the eect of distance from ET sites on the contributions to decoherence, we have also determined contributions to sD2 as a function of a cuto distance from the metal ions. The result is shown in Fig. 1. Close to the metal ions, most atoms are part of the protein matrix, and so
Fig. 1. Contributions to sD2 from atoms within a speci®ed cuto distance of the metal ion centers. The solid line denotes the result obtained by including all atom types. The + symbols denote the contribution from atoms in the solvent molecules, while the } symbols denote the contribution from the remaining atoms. The symbols denote the contribution from atoms in the solvent molecules which are closer to the center of the copper ion than to the center of the ruthenium ion.
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D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
contributions from such atoms are dominant at short distance. However, water molecules more distant from the metal ions contribute very signi®cantly to the squared decoherence rate, and the water contribution is strikingly long-ranged in character. To better understand the origin of this remarkable eectiveness of the water molecules in causing coherence loss, we calculated the contribution to sD2 arising solely from hydrogen atoms in water molecules. We ®nd that about 90% of the water contribution can be attributed to the hydrogen atoms. This importance of hydrogen atoms can easily be understood by examining the form of Eqs. (3) and (4), where the contributions of atoms with smaller mass are weighted more highly. As such, the eciency with which water induces decoherence can be understood as arising from the high density of polar hydrogen atoms. We note that the long range of the water contribution results from the contribution of this high density and eectiveness, combined with the increasing volume of water associated with a given radial distance. The range is not a peculiarity of the protein system, but manifests itself comparably for atomic ions in solution [4]. Although the density of polar protons is considerably lower in the protein than in the aqueous solvent, the importance of polar hydrogens leads to the corresponding result that about 75% of the protein matrix contributions (neglecting changes in metal ion covalent radii) also arise from hydrogen atoms. Also shown in Fig. 1 is the contribution of water molecules which are closer to the copper ion than the ruthenium ion. Clearly, the location of the ruthenium complex on the surface of the protein leads to a greater eect from nearby water molecules. It must be noted, however, that even the water molecules closest to the copper ion alone are sucient to give rise to a decoherence time of about 5.5 fs, while the protein matrix alone would give a decoherence time of about 4.1 fs. Hence, neglecting the Ru solvation would still give a 3.3 fs decoherence time. As such, the design of this protein, in which the copper ion is buried well within the protein matrix, is not sucient to prevent a very high rate of electronic coherence loss
due to the solvent molecules, nor is the decoherence time much longer even in the absence of solvent molecules. We brie¯y return to the issue of the role of distribution of charge over the ligands and of changes in covalent radii in determining the decoherence time. To test the former, we evaluated the decoherence time that would be obtained if charge transfer occurred strictly between metal ion centers. The decoherence time so obtained was 2.3 fs, as opposed to the value of 2.4 fs obtained with the correct distribution of charge over the ligands. To test the latter issue, we calculated the decoherence time that would be obtained if the ion covalent radii remained constant in the ET process. The decoherence time so obtained was 2.5 fs. We conclude that the decoherence rate is largely insensitive to such details, and can be obtained here by considering only the change in electrostatic forces if a point charge of )1 is transferred between the metal centers. 4. Conclusions For the example of direct ET in the blue copper protein ruthenated azurin, we have shown that electronic coherence decay due to coupling of the electronic state to the nuclear state of the surroundings is extremely rapid. The origins of this decay lie comparably in the quantum dynamics of the solvent and the protein matrix and the contributions are remarkably long ranged. One concludes that the quantum character of the dynamics of these `outer sphere' modes is an important feature for ET in proteins. Of special importance is the fact that the small characteristic electronic decoherence time of 2.4 fs can preferentially promote sequential ET over bridge sites. In addition, modi®cation of the ligands of ruthenium [13] and copper [33] can reduce the thermodynamic driving force which is also expected to enhance the role of sequential transfer [5,12]. To quantify this competition between sequential and direct transfer, it will be necessary to quantitatively explore the characteristics of electronic coherence associated with those individual ET steps which compose alternative transfer pathways [33] in azurin.
D.M. Lockwood et al. / Chemical Physics Letters 345 (2001) 159±165
Acknowledgements This work was supported by a grant from the Robert A. Welch Foundation. Additional support was provided by the National Science Foundation. We thank J.W. Ponder for use of the software package TI N K E R , available at http://dasher.wustl. edu/tinker. References [1] P.F. Barbara, T.J. Meyer, M.A. Ratner, J. Phys. Chem. 100 (1996) 13148. [2] W.B. Davis, W.A. Svec, M.A. Ratner, M.R. Wasielewski, Nature 396 (1998) 60. [3] D. Voet, J.G. Voet, Biochemistry, second ed., Wiley, New York, 1995. [4] D.M. Lockwood, H. Hwang, P.J. Rossky, Chem. Phys. 268 (2001) 285. [5] W.B. Davis, M.R. Wasielewski, M.A. Ratner, V. Mujica, A. Nitzan, J. Phys. Chem. A. 101 (1997) 6158. [6] D. Gilulini, et al., Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin, 1996. [7] O.V. Prezhdo, P.J. Rossky, Phys. Rev. Lett. 81 (1998) 5294. [8] W.P. Schleich, Nature 403 (2000) 256. [9] O.V. Prezhdo, P.J. Rossky, J. Chem. Phys. 107 (1997) 5863. [10] E. Neria, A. Nitzan, J. Chem. Phys. 99 (1993) 1109. [11] K. Wynne, R.M. Hochstrasser, Adv. Chem. Phys. 107 (1999) 263. [12] L.D. Zusman, D.N. Beratan, J. Chem. Phys. 110 (1999) 10468. [13] H.B. Gray, J.R. Winkler, Annu. Rev. Biochem. 65 (1996) 537.
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