Toshiko Ichiye

394 Ichiye

Therefore, ∆G° is related to the redox potentials E °i of the donor and acceptor via the Nernst equation,

where n is the number of electrons transferred and F 23.06 kcal/(mol V) is Faraday’s constant. Thus, an understanding of the redox potentials of electron transfer proteins in an electron transport chain is essential, because the redox potentials determine the direction of favorable flow of the electrons. One caution: There may be differences between the redox potentials of the isolated reactants and those of the reactants in an electron transfer complex [2].

The electron transfer rates in biological systems differ from those between small transition metal complexes in solution because the electron transfer is generally long-

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range, often greater than 10 A [1]. For long-range transfer (the nonadiabatic limit), the rate constant is

where HAB is the electronic (or tunneling) matrix element, λ is the environmental reorganization energy, and ∆G‡ is the activation energy for the reaction. HAB is a measure of the electronic coupling between the reactants and the products at the transition state, and λ is a measure of the energy required to change the environment from the equilibrium state around the reactants to what would be the equilibrium state around the products while the reactants remain in their initial state. According to Marcus theory [3], ∆G‡ is given by

where ∆G° is the free energy described above. Thus, understanding λ is important, because it entails understanding how the protein keeps the activation energy ∆G‡ low enough to promote the long-range transfer.

The calculation of E°i and λ from computational methods is the focus here. Generally, the energetics of these quantities are separated into contributions from the inner and outer shells. For transfer between small molecules, the inner shell generally is defined as the entire solutes A and D, and the outer shell is generally defined as only the solvent. However, in a more practical approach for proteins, the inner shell is defined as only the redox site, which consists of the metal plus its ligands no further than atoms of the side chains that are directly coordinated to the metal, and the outer shell is defined as the rest of the protein plus the surrounding solvent. Thus

where IP, the ionization potential, is the negative of the Franck–Condon energy required to add an electron to the species i [4]; ∆SHE, the standard hydrogen electrode correction, is 4.43 V [5]; ∆Giin, the inner shell relaxation energy, is the change in energy for species i when the metal ligand geometry (i.e., the internal coordinates) has the oxidized versus reduced values; and ∆G iout, the solvent or outer shell relaxation energy, is the change upon reduction of the solvation polarization energy [3] of species i. In the second equality, ∆G sitei is the change upon reduction in the intrinsic energy of the redox site i. Similarly,

where λin, the inner shell reorganization energy, is the energy required to change the metal ligand geometry of the reactants from the initial to the final values while the charges of the reactants remain in the initial state and λout, the outer shell reorganization energy, is the energy required to change the solvent polarization from an equilibrium configuration around the reactants to one around the products, again while the charges of the reactants remain in the initial state.

The simplest approach to calculating both ∆Gi and λ is to assume that the energetics of the redox site and the outer shell can be determined independently (i.e., the energies are uncoupled). Thus, as a first approximation, the change in the energy of the redox site can be calculated from quantum mechanical calculations of analogs, and the change in the outer shell energy can be calculated from classical calculations. The coupling between the redox site and outer shell energies is through the potential energy parameters, especially the partial charges, of the redox site used in the classical calculations. Because environmental effects due to the protein and/or solvent may influence the electronic structure, a higher order approximation is to use mixed quantum–classical methods (see Chapter 11) such as are used in a study of electronic tunneling pathways in ruthenium-modified myoglobin [6]; however, such calculations are not yet routine.

III.CALCULATION TECHNIQUES FOR ELECTRON TRANSFER PROTEINS

Computer simulations of electron transfer proteins often entail a variety of calculation techniques: electronic structure calculations, molecular mechanics, and electrostatic calculations. In this section, general considerations for calculations of metalloproteins are outlined; in subsequent sections, details for studying specific redox properties are given. Quantum chemistry electronic structure calculations of the redox site are important in the calculation of the energetics of the redox site and in obtaining parameters and are discussed in Sections III.A and III.B. Both molecular mechanics and electrostatic calculations of the protein are important in understanding the outer shell energetics and are discussed in Section III.C, with a focus on molecular mechanics.

A.Quantum Chemistry of the Redox Site

Quantum mechanical electronic structure calculations of the redox site are often necessary for even classical calculations of electron transfer proteins, such as when parameters are lacking for a metal redox site. Although semiempirical methods have been used for these types of studies, here the focus is on ab initio methods. Currently, density functional theory (DFT) methods are the ab initio quality electronic structure calculations of choice for metal sites because Hartree–Fock (HF) methods are insufficient and the inclusion of configuration interactions (CI), which are important for these systems, with high level basis sets leads to intractable calculations (i.e., the computational dependence on the number of electrons n is n 3 for DFT, n 4 for MP2, and n 5 for CI techniques). In addition, DFT gives excellent results for transition metals, including bond dissociation energies, bond lengths, conformational analysis, ionization potentials, and electron affinities [7,8]. An excellent review covers the application of these methods to transition metal complexes, including redox sites within proteins and analogs of protein redox sites [9].

Whether DFT or HF methods are used, there are several issues regarding the details

of the calculation that will affect various aspects of the quality of the calculation; these issues are mentioned here and are discussed more fully with respect to the specific calculation later. Comparisons throughout this chapter refer to unrestricted HF calculations of the [1Fe] or rubredoxin-type site by our group, which use a full core on the iron and geometry optimization unless otherwise specified, and to spin-unrestricted DFT calculations by Noodleman and coworkers [9], which use the local density approximation for exchange and correlation with nonlocal corrections as a perturbation or as part of the SCF procedure, frozen core orbitals, and experimental geometries unless otherwise specified.

One issue concerns whether or not the geometry of the redox site is optimized. For the iron-sulfur sites, geometry optimization has given longer FeES bond lengths than are

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found in experiment—both an HF calculation of a [1Fe] site (0.1–0.2 A) [10] and a DFT

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calculation of a [2Fe-2S] site (0.5–0.2 A) [11]. For the blue copper sites, geometry optimi-

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zation has given metal–ligand distances that differ by up to 0.2 A from experimental averages. However, the latter come from crystal structures of the blue copper proteins, which are at much lower resolution than analog structures and have a large range of metal–

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ligand bonds (0.2–0.5 A). Another approach is to use experimental geometries from high resolution crystal structures of analogs of the redox site in the proper oxidation state. Less preferably crystal structures of the protein may be used, although they are lower resolution and may exist in only a single oxidation state [12]. The use of analogs is a good approximation for iron-sulfur proteins, because the redox site structures are relatively invariant even for proteins with very different folds and/or redox potentials [13]. However, care must be taken for blue copper proteins, because the redox site structure can vary significantly in different proteins [4].

Another issue is whether the calculation is of the redox site in vacuum or with environmental effects. If the energetics of the metal site are independent of its environment, the calculation of an analog in vacuum is a good approximation. The influence of the protein on the electronic structure of the redox site apparently is rather small for Fe-S proteins, from both experimental [13] and theoretical [10] studies. However, the protein plays a much larger role in the blue copper proteins, as is seen in both experiment [4] and theory [14]. A presumably more accurate but more computationally intensive process entails adding a reaction field that represents the environment. In one approach, a dielectric continuum reaction field (possibly including fixed charges and a low dielectric region representing a protein environment) is incorporated into the self-consistent field (SCF) of the entire system to give a self-consistent reaction field. Examples include DFT calculations of the manganese superoxide dismutase (Mn-SOD) active site using oxidized geometries only in aqueous solution [12] and the [2Fe-2S] site in two different proteins [11]. Other approaches include using QM/MM methods (see Chapter 11).

B. Potential Energy Parameters

The first major obstacle in studying electron transfer and/or metalloproteins is often the lack of potential energy parameters for metal sites in proteins. Although parameters for hemes existed in some of the earliest parameter sets because of the numerous studies of myoglobin [15], hemoglobin [16], and cytochrome [17], there is a dearth of parameters for other metal sites. Parameters for iron-sulfur sites have been recently developed [18–21] based on spectroscopic data for the force constants, crystallographic data for the equilibrium values, and quantum mechanical calculations for the partial charges and for the van der Waals parameters (see Chapter 2). Parameters for other sites have also been developed [22–25].

The force constants and equilibrium values for the internal coordinates should be

obtained from experimental data given the computational intensity of obtaining electronic structure calculations of sufficient quality for these quantities. If possible, the equilibrium values should be obtained from analogs of the redox site, because generally the structures are at much higher resolution than those of proteins and it is possible that the protein strains the analog away from its equilibrium internal coordinates.

The partial charges for the metal sites are difficult to obtain via experiment, so quantum mechanical calculations provide the best source. Because the electrostatics of the redox site are potentially a crucial factor in calculating electron transfer properties, it is important to understand the sensitivity of the results to the values of the partial charges. The energetics of fixed structures are relatively insensitive to the values of the partial charges as long as the total charge is correct [26,27]. However, the partial charges become important for energy minimization and molecular dynamics simulations because perturbations of the protein may occur [28]. Either DFT or more traditional ab initio calculations can be used to provide wave functions, which are then used to generate single-point ‘‘partial charges.’’ Because Mulliken population analyses tend to be dependent on the basis set, the preferred method of generating the partial charges is to fit the molecular electrostatic potential determined by the nuclear interactions and the electron charge density while constraining the total charge and desired higher moments to match those determined from the electronic wave function using programs such as CHELPG [29]. However, difficulties may be encountered in obtaining good values for a coordinated metal site, because such fitting methods are relatively insensitive to the values of the partial charges of buried atoms such as the metal and can thus be more conformation-dependent than the Mulliken population analysis indicates [10]. Thus, Mulliken charges may be a better indicator of the conformational dependence of the partial charges, whereas fitted charges may be a better indicator of the magnitude of the partial charges. To obtain the best charges, fitted charges should be determined for conformations with a high degree of symmetry, which are likely to be the most accurate. Mulliken population analyses should then be compared for the high symmetry and any other desired conformation to determine if the desired conformation is likely to have the same charge distribution. The details of the electronic structure calculation may also have a significant effect on the values of the partial charges. For instance, the use of an effective core versus a full core on the metal in unrestricted HF calculations of the [1Fe] site lead to 10–15% changes in the CHELPG charge of the Fe [10]. Moreover, the use of geometry for the wrong oxidation state in the same study can lead to 5–10% changes in the CHELPG charge of the Fe. This study also indicated that overestimation of the bond lengths tends to reduce the CHELPG charge on the Fe but increase the Mulliken charge. For partial charges, the treatment of the environment becomes an issue, because the condensed phase environment tends to enhance polarization so that approaches such as using a dielectric continuum to represent either a solvent or protein environment should be considered. The effect of adding environmental effects via a dielectric continuum plus protein charges in a DFT calculation indicates less than a 1% change in charge on the Fe in the [2Fe-2S] site [11], but up to an almost 20% change in CHELPG charge on the Mn of the Mn-SOD site [12]. In determining partial charges of other organic molecules, the tendency for HF to overestimate polarization for the molecule in vacuum had been viewed as an approximate means of determining the increase in polarization as the molecule is transferred to the condensed phase [30]. Experimental verification is essential, because in studies of the [1Fe] site of rubredoxin, DFT calculations [27] and unrestricted HF calculations [10] differ by almost 50% on the Fe even though the identical fitting method was used to obtain the charges.

The van der Waals parameters of the metal in a liganded site may be significantly

different from that of the free ion in solution. Experimental determination of these parameters has not been forthcoming, so quantum chemical calculations again provide the best means. In cases in which there are several ligands such as when there is octahedral coordination, the metal may be inaccessible due to the van der Waals spheres of the ligands, so it is not necessary to determine separate van der Waals parameters for the metal as has been assumed for the Fe in the heme group in the CHARMM potential [31,32]. However, for the tetrahedral coordination found in the Fe-S sites, the blue copper sites, and the zinc sites, the metal is very accessible, and thus good metal van der Waals parameters are essential. Although DFT methods are generally preferable for metal sites, they are less well developed for treating the interactions between molecules, so HF methods have been used [21,33].

C.Molecular Mechanics and Electrostatics Calculations of the Protein

Molecular mechanics and electrostatics calculations have both played an important role in studying electron transfer proteins. Molecular mechanics calculations of these proteins use the same techniques (molecular dynamics, energy minimization) as for other proteins, although special consideration must be made in simulation conditions.

The treatment of long-range electrostatics and dielectric effects is an important issue in calculations of the outer shell energetics because of the importance of the electrostatic contribution to the redox properties. Two basic approaches have been used in atomistic studies of biological molecules: (1) treating the protein as a collection of charges embedded in a low dielectric continuum and the solvent as a high dielectric continuum, possibly with an ionic strength associated with it, and solving the Poisson or Poisson–Boltzmann equation as appropriate and (2) treating both the protein and solvent atoms explicitly as a collection of partial charges, with both allowed to reorient to mimic the orientational dielectric response, in a standard molecular mechanics calculation. The molecular mechanics techniques discussed here are energy minimization and molecular dynamics in standard implementations such as in CHARMM and DISCOVER; a review of the Protein Dipoles Langevin Dipoles (PDLD) method in the program POLARIS as applied to iron-sulfur proteins was published in 1996 [34].

Poisson and Poisson–Boltzmann calculations involve calculating the electrostatics due to partial charges of atoms in the protein, generally assuming that the protein itself has a low dielectric constant εp and the surrounding solvent has a high dielectric constant εw ( 80 for water) such as in the programs DelPhi [35,36], UHBD [37,38], and MEAD. Poisson–Boltzmann calculations include nonzero salt concentration, whereas Poisson calculations are for the special case of zero salt concentration. Among the strengths of this approach are that the long-range contributions (i.e., the Born solvation energy) and salt effects are calculated accurately. One of the weaknesses is that a single εp is assumed for the entire protein, whereas dielectric relaxation is a molecular phenomenon so the dielectric response varies within a protein. It is not even clear if there is a good average value for proteins, and values ranging from εp 2 to εp 10 are used. Another potential problem arises if specific interactions of the solvent with the redox site are important, as appears to be the case for rubredoxin from calculations [19] and resonance Raman experiments (J. Sanders-Loehr, personal communication) and also for ferredoxin from resonance Raman experiments [13]. This is an important consideration in general, because the redox sites are close to the surface in many electron transfer proteins. In addition, most Poisson-type calculations do not allow relaxation of the protein or dynamic effects.