The treatment of electrostatics and dielectric effects in molecular mechanics calculations necessary for redox property calculations can be divided into two issues: electronic polarization contributions to the dielectric response and reorientational polarization contributions to the dielectric response. Without reorientation, the electronic polarization contribution to ε is 2 for the types of atoms found in biological systems. The reorientational contribution is due to the reorientation of polar groups by charges. In the protein, the reorientation is restricted by the bonding between the polar groups, whereas in water the reorientation is enhanced owing to cooperative effects of the freely rotating solvent molecules.

Electronic polarization is included in simulations in two different ways. In many of the potential energy functions used for biological molecules such as the CHARMM and AMBER potentials and also those used for water in simulations of biological systems such as SPC [39], SPC/E [40], and TIP3P and TIP4P [41], electronic polarization effects are included implicitly in the potential energy parameters. In other words, these potential energy functions have been parametrized to give good structure and energetics in an ‘‘average’’ environment without including electronic polarization explicitly [42]. Thus, the approximation will tend to break down in mixed environments. For instance, although electronic polarization plays a relatively small role around singly charged ions, it plays a significant role for charges of magnitude greater than 2 [43], so it may be an issue for metal centers. The PDLD method [44,45] has electronically polarizable dipoles representing the polar groups of the protein surrounded by a grid of Langevin dipoles representing the solvent. However, some other features of the PDLD approach that warrant caution are that the Langevin dipoles, although including some reorientational dielectric effects, do not accurately reflect the structure of water because of the cubic grid to which they are confined and that charged side chains are often neutralized, which is reasonable only for those at the surface far from the point of interest. Several of the standard water models have been modified to include polarizability [46], but as yet they have not been integrated with a polarizable potential function for proteins.

The reorientational polarization contribution to the dielectric response comes from including the interactions of all polar groups, including those of both the protein and the solvent, in the calculation of the electrostatic component. As generally recommended for simulations of proteins, electron transfer proteins must be simulated with explicit water and counterions. However, if the interaction energy of the redox site with the rest of the protein needed for calculating redox properties is to be calculated explicitly from the simulation, then the use of methods such as droplet boundary conditions and/or spherical continuous or discontinuous cutoffs becomes questionable in calculating the energetics of a protein in solution because the contribution of the Born solvation energy of an ion in water is significant even at large distances [47].

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For instance, the contribution of water beyond 12 A from a singly charged ion is 13.7 kcal/mol to the solvation free energy or 27.3 kcal/mol to the solvation energy of that ion. The optimal treatment is to use Ewald sums, and the development of fast methods for biological systems is a valuable addition (see Chapter 4). However, proper account must be made for the finite size of the system in free energy calculations [48].

IV. REDOX POTENTIALS

Calculations of redox potentials may have two somewhat different purposes: (1) to calculate the redox potential of a given protein or (2) to calculate the differences in redox

potential between two proteins, most often the wild-type and a mutant. The calculation of the absolute redox potential Ei (or ∆Gi) of a given protein is important for determining the relative importance of various contributions to the redox potential. Electronic structure calculations are necessary to obtain the change upon reduction in the intrinsic energy of the redox site and are discussed in Section IV.A. Both molecular mechanics and electrostatic energy calculations are useful for calculating the change upon reduction of the energy of the protein and solvent (the outer shell) and are discussed in Section IV.B. Although calculation of differences in redox potentials between two proteins ∆Ei (or ∆∆Gi) would seem to necessitate calculation of ∆Gi, the assumption that most of the contributions to the redox potential will remain constant allows the use of much simpler calculation techniques and are discussed in Section V.

A. Calculation of the Energy Change of the Redox Site

The change upon reduction in the intrinsic energy of the redox site (∆G isite) is composed of the negative of the Frank–Condon ionization potential (IP) and the energy change due to changes in the internal coordinates of the redox site (∆G iin) [Eq. (5)]. Thus, it is the difference in the absolute energy of the oxidized species in its equilibrium (or relaxed) conformation versus the reduced species in its equilibrium (or relaxed) conformation, which can be calculated via quantum chemistry calculations of the oxidized and reduced species that include geometric optimization in the presence of the reaction field due to the environment. These absolute energies are often given in atomic units (au), where 1 au 27.21 eV.

The preferred method for calculating ∆G sitei is to use DFT, for the reasons just described. The difference is significant for the [1Fe] analog in vacuum, because the DFT calculation gives a value of 1.79 eV whereas the HF calculation gives a value of 2.13 eV. The HF calculation is clearly sensitive to the lack of CI, because calculations using an effective core potential on the Fe give values of 0.251 and 1.18 eV at the HF and MP2 (second-order Møller–Plesset) levels, respectively. The values are also sensitive to the use of geometric optimization, lowering ∆G isite by 0.150–0.130 eV for the [2Fe-2S] site relative to geometries of the oxidized and reduced species [11]. Assuming a single geometry for both oxidation states can lead to considerable errors in ∆G sitei , because ∆Giin is about 0.5 eV in the HF calculation of the [1Fe] site. The effects of the environment on ∆G sitei alone are variable, less than 0.1 eV in the Mn-SOD site but 0.2–0.3 eV in the [2Fe2S] clusters.

B. Calculation of the Energy Changes of the Protein

The calculation of ∆G iout, the energy change upon reduction of the energy of the outer shell—the rest of the protein plus solvent—along with the quantum chemical value for ∆G sitei allows comparisons to experimental redox potentials through Eq. (5). In addition, it allows the decomposition of the various factors contributing to ∆Giout. Calculation of ∆Giout from molecular mechanical techniques entails a free energy simulation between the oxidized and reduced states (see Chapter 9). However, it is useful to perform molecular dynamical simulations or even energy minimizations of the oxidized and reduced states prior to such a calculation because information can be gained about the structures of both states when there is an experimental structure of only one state. In addition, simulations of both states, especially when higher net charges concentrated at the redox site are in-

volved, are important in assessing the simulation condition for free energy simulations. Moreover, these simulations can then be used to calculate ∆E outi , which gives the enthalpic contribution minus the PV term to ∆G outi . The decomposition of the contributing factors to ∆Eiout is generally simpler than for ∆Giout. The calculation of ∆Eiout from molecular mechanics is described first, followed by the calculation of ∆G outi from free energy simulations and finally the calculation of ∆G iout from Poisson–Boltzmann calculations.

1. Molecular Mechanics Calculations of ∆Eouti

Calculation of ∆Eiout E iout,red E iout,oxd (where the subscripts red and oxd denote the reduced and oxidized states, respectively, of species i) from molecular mechanics entails calculating the total classical potential energy of the system when species i is in both the reduced and oxidized states, where the state is determined by the partial charges of the redox site. Since the total potential energy is dependent on the conditions (temperature, pressure, number of particles) of the system, it is thus necessary to have the same conditions in the oxidized and reduced states to calculate the total ∆E iout. This type of calculation is useful in determining various contributions to the energetics of a redox reaction such as the contributions of structural relaxation of the protein versus change in charge of the redox site. In addition, because the energies are simply sums of various terms, electrostatics versus van der Waals versus internal coordinate contributions and various components of the protein such as backbone versus side chain can be calculated by summing over only the appropriate terms. Thus, specific contributions may be focused on with the caveat that there may be compensating changes in other components.

When only the energetics of the protein are of interest, a simple approximation is to calculate the energetics of a single structure of the oxidized state and a single structure of the reduced state. The structures may simply be from experimental crystal or NMR studies of the protein in both states as in a study of cytochrome c [26]. If structures do not exist for both states, the structures of each state may be obtained by energy minimization using the potential energy parameters of that state, as in studies of rubredoxin [18,49]; however, the structures must each be carefully minimized to a local minimum. The structures can also be obtained from average structures from molecular dynamics simulations of each state. It is important that if the structures are from energy minimizations or molecular dynamics calculations that have used electrostatic cutoffs, the calculation of the energy itself should not use cutoffs. A more accurate calculation is to calculate the average energies from the molecular dynamics simulations. If the energetics of the solvent are also important, then it is crucial to do molecular dynamics simulations of each state with periodic boundary conditions so that either the pressure or the volume can be fixed.

The calculation of ∆E iout requires calculations of the total energies of each state, not just the interaction energies of the redox site with its environment. This means that interactions between atoms of the environment (protein and solvent) must also be included; the protein contribution may be over 20 kcal/mol [50]. One simplification is to examine only the energetics due to the protein itself (i.e., redox site with protein atoms and between protein atoms), which is useful when the solvent has not been treated by way of molecular dynamics. This restricts the analysis to the backbone plus the nonpolar, polar, and buried charged side chains, because it is necessary to include solvent and counterions to calculate the surface charged side chain contributions correctly. The contribution of the charged side chains at the surface is actually quite small owing to cancellation by the solvent and counterions. Interestingly, calculations of structures from molecular dynamics of rubredoxin [19] indicate that most of the large positive electrostatic potential from polar

because proteins are so flexible. Thus, an important simplification for many problems is to assume that when the relaxation energy ∆E r [Eq. (7)] is mainly due to the electrostatic contribution, i.e.,

where ∆E is the electrostatic energy change when the environment relaxes after the charge change and ∆E nonel is the nonelectrostatic energy change when the environment relaxes after the charge change. Of course, ∆E q [Eq. (7)] is purely electrostatic.

2. Free Energy Simulations of ∆G outi

The free energy changes of the outer shell upon reduction, ∆G iout, are important, because the Nernst equation relates the redox potential to ∆G. Free energy simulation methods are discussed in Chapter 9. Here, the free energy change of interest is for the reaction

When applying free energy methods, the length of simulations and λ increment needed for the reduction free energy calculation are of some concern, because changes in charge upon reduction result in large changes in energy. However, these changes do not generally involve large changes in structure or configurations; i.e., the primary structural change is the reorientation of dipoles [51]. Indeed, fast convergence (40 ps) was found for the calculation of the free energy difference of hydration between uncharged Ne and Na in water using the slow growth (SG) method [52]. This is particularly encouraging, because the change from a neutral species to a singly charged species, which involves breaking the symmetry of the disordered environment around the neutral species to impose an orientation of the solvent dipoles, is a greater perturbation than increasing the charge for charged species, which simply involves increasing the existing orientation. Both thermodynamic perturbation (TP) and thermodynamic integration (TI) can be used. A possible advantage of the TI method is that the contributions to the total ∆∆G are additive and thus decomposable, although the validity of this decomposition is controversial [53–55].

3. Poisson–Boltzmann Calculations of ∆Eouti

Calculations of ∆G iout by Poisson–Boltzmann methods have also been carried in which ∆G iout is assumed to be the sum of (1) the electrostatic interactions of the heme charges with the protein charges, screened by the polarizability of the water and protein, and (2) the reaction field energy from the polarization of solvent and protein [56]. Thus, both dielectric screening and entropic contributions are included in an approximate fashion, although no relaxation effects are included, because only a single structure is used. For the four hemes in the photosynthetic reaction center of Rhodopseudomonas viridis [56], major contributing factors include the axial ligands and the propionic acids. The reaction field energy is similar for all four hemes, indicating that this is not likely to be a major contributor to free energy differences between sites. However, the calculations for the protein assumed that the charge change was localized on the iron, which is likely to influence the results, especially for the highly delocalized heme system.

4. Summary

Recently, calculations of the photosynthetic reaction center by three different groups have been reviewed [57], which used DelPhi [58], PDLD [59], and CHARMM [60]. The un-