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usual situation of having studies using all three different methods on the same system provided the opportunity for comparison. One point made was that each method has specific strengths, which are summarized in the review. MD methods have the advantages of allowing reorganization of the atoms, including dynamic information, and including explicit solvent, and FEMD has the added advantage that the connection to formal redox potentials is direct, because free energies rather than enthalpies are calculated. In addition, all three groups came to similar conclusions, although with different magnitudes of energetics, although the only common electrostatic contribution was the polar groups of the backbone and polar side chains.

V.DIFFERENCES IN REDOX POTENTIALS

Electron transfer proteins can modulate electron transfer processes by varying the outer shell energy. For instance, differences in redox potential of up to a few hundred millivolts are seen between homologous proteins with the same redox site, and even larger differences occur between nonhomologous electron transfer proteins with the same redox site [61]. Despite the rapidly growing number of crystal structures of electron transfer proteins, the structural origins of differences in redox potentials for a given redox site remain unclear. The differences may be due to intrinsic changes in the ionization potential of the redox site or to extrinsic changes in the energetics of the environment surrounding the redox site. By assuming that only the energetics of a few degrees of freedom lead to the redox potential differences, the calculation of differences in the energy change upon reduction for different proteins with the same redox site can be greatly simplified. Electronic structure calculations are necessary to examine differences that arise in the intrinsic energy of the redox site and are discussed in Section V.A. Both molecular mechanics and electrostatic energy calculations can be used for calculating the differences in the reduction energy of the outer shell and are discussed in Section V.B.

A.Calculation of Differences in the Energy Change of the Redox Site

Changes in the electronic structure of the redox site can lead to changes in the redox potential, mainly by changing the ionization potential or, more subtly, because geometry or charge distribution changes may alter the energetics of the environment surrounding the redox site. Three possible ways that the electronic structure of the redox site can be altered are by mutations of the ligands of the metal, by mutations that alter ligand geometry, and by substitution of the metal ion by another metal ion. The effects of these changes on the ionization potential and partial charges of the redox site can be determined simply by electronic structure calculations of the appropriate analogs of the wild-type and mutant using the methods of the previous section. Other mutations of the outer shell are thought not to influence the ionization potential but rather the electrostatic interaction energy between the redox site and the outer shell.

B. Calculation of Differences in the Energy Change of the Protein

Changes in the environment of the redox site can lead to changes in the redox potential via alteration of the interaction energy of the redox site with the outer shell. In many

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proteins such as the cytochromes [62] and the iron-sulfur proteins [63], the major way that nature influences the redox potentials is apparently through the protein environment of the cluster. One important contribution comes from the electrostatic environment created by the protein and solvent [34,64]. However, mutations involving surface charged residues often show very little effect [65,66] or unpredictable effects [67] on the redox potential. Moreover, our studies show surface charged residues are screened by solvent and counterions [19,68]. Buried charged side chains have been shown to be important in some proteins [69,70] but do not occur naturally in all proteins. Theoretical calculations indicate that a combination of the electrostatic contributions from the polar backbone, polar side chains, and solvent can impact the redox potential [68,71–73]. These polar interactions are complex because they depend on both orientation and distance and may vary dynamically. Moreover, large redox potential differences appear to be composed of many such small interactions as opposed to a few key interactions. Because of this, it has been almost impossible to sort out the contributions by simple inspection of structural data or by mutations guided by physical intuition alone. Therefore, molecular mechanics calculations provide a powerful means to sort out important effects in these molecules.

Calculation of how changes in the protein environment influence the redox potential depend in part on the magnitude of the changes involved. The rationale is that the overall backbone fold of the protein determines the general range of the potential and thus the general function, while specific sequences tune this potential (or are involved in binding specificity or are nonfunctional). Generally, most single-site mutations that affect neither the metal site nor the folding result in changes in redox potential of less than 100 mV. At one extreme are two proteins with the same redox site but completely different folds and redox potentials. For instance, the bacterial ferredoxins and the high potential ironsulfur proteins (HiPIPs) both have the [4Fe-4S] site, but the redox potential for the 2 /3 couple is so much lower for the HiPIPs that only the 1 /2 couple has been seen experimentally whereas the 2 /3 couple is seen experimentally for the ferredoxins. For such cases, the approach should be to perform calculations of each couple according to the methods of Section V.A. On the other hand, proteins with similar folds but different sequences can be compared by using much simpler strategies, which are discussed here.

1. Structure/Sequence–Function Analysis

The simplest approach to understanding differences in redox potentials between homologous proteins is to analyze the experimental structures of multiple homologous proteins plus additional sequences of multiple homologous proteins in conjunction with available experimental redox potentials, in what might be termed a structure/sequence–function analysis. This analysis is based on the idea that a redox potential difference between homologous proteins can be identified by differences in the electrostatic potential contribution of specific residues as calculated from experimental structures of representative proteins. Furthermore, similar redox potential differences can be introduced into a given protein by introducing the appropriate amino acid mutations identified from the homologous protein study. Both MD and FEMD provide means of testing the latter assumption prior to experimental studies. The major advantage of examining multiple experimental X-ray crystal or NMR solution structures rather than molecular dynamics simulations is that changes in energy are often small (less than 2 kcal/mol) and may involve structural

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changes that are small (distances less than 0.5 A). Thus, fluctuations in the simulations

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or inaccuracies due to the potential may be on the order of the changes that are seen in different experimental structures. However, a disadvantage of examining these experimental structures is that specific protein–solvent interactions and dynamic effects are difficult to examine.

The first step in the analysis is to examine the energetics of experimental structures of multiple homologous proteins, preferably with measured redox potentials. An assumption that is generally made is that most of any redox potential differences between homologous proteins are not due to entropic effects, so that ∆G2 ∆G1 ∆∆G ∆∆E, where the subscripts refer to proteins 1 and 2. Furthermore, it is assumed that most of the differences lie in the electrostatic energy [18]. In most cases experimental structures are determined only for one oxidation state of the protein. An approximation to overcome this limitation is to assume that the structural relaxation of the homologous proteins is similar so that ∆∆Er 0, and the electrostatics of only a single state can be examined. Thus, using Eq. (7),

∆∆E ∆∆Eq neF(φ2 φ2)

(9)

where φi is the electrostatic potential of protein i, n is the number of electrons added, and e is the magnitude of an electron charge. If the change in charge is delocalized over several atoms, such as in the case of the Fe-S redox sites, a delocalized electrostatic potential can be defined as

	N		∆qi qj
			rij
φ	i redox j ≠i
	site atoms				(10)
		∆qi

i redox site atoms

where ∆qi is the change in charge of atom i of the redox site upon reduction, qj is the charge of any non-redox site atom j, and rij is the distance from atom i to atom j. The first sum is over all atoms of the redox site (i.e., those atoms that change charge upon reduction), and the second summation is over all atoms excluding the redox site. However, this definition is dependent on the partial charge parameters of the redox site, which are often uncertain or even unknown. Thus, a more approximate definition of φ, the electrostatic potential at a specific point 0, may be also be used:

φ	qj		(11)
	r0j
j≠redox

site atom

where r0j is the distance of the jth atom from the point 0 and the summation is over all atoms excluding the redox site. This relationship implies that all of the charge change is localized at a single point, which would be exact for a point charge description of a simple monatomic ion and is reasonable for the iron in the [1Fe] site. This latter definition of φ may be used as a parameter-independent estimate when there is uncertainty in the parameters. In either case, a dielectric constant of 1.0 with no cutoffs is used. The dielectric screening of the protein and solvent may be accounted for in an approximate manner using Poisson or Poisson–Boltzmann calculations as was done in a calculation of the differences between plastocyanin and rusticyanin [74]. However, it may be preferable not to, both

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for speed of calculation and because of the uncertainty in the local screening by the protein. If dielectric screening has not been included, caution should be exercised. In particular, the polar and charged side chain contributions should be calculated separately from each other, because the charged side chain contributions are likely to be greatly exaggerated. However, polar contributions have been the most important contribution in many of the cases studied [68,74].

An effective method for localizing causes of redox potentials is to plot the total backbone and side chain contributions to φ per residue for homologous proteins as functions of the residue number using a consensus sequence, with insertions treated by summing the contribution of the entire insertion as one ‘‘residue.’’ The results for homologous proteins should be examined for differences in the contributions to φ per residue that correlate with observed redox potential differences. These differences can then be correlated with any other sequence–redox potential data for proteins that lack crystal or NMR structures. In addition, any sequences of homologous proteins that lack both redox potentials and structures should be examined, because residues important in defining the redox potential are likely to have semi-sequence conservation of a few key amino acid types.

One example of a sequence determinant of redox potentials that has been identified in this manner is an Ala-to-Val mutation at residue 44, which causes a 50 mV decrease in redox potential (and vice versa) in the rubredoxins [68]. The mutation was identified because the sum of the backbone contributions to φ of residues 43 and 44 change by 40

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mV due to an 0.5 A backbone shift away from the redox site. This example points out the importance of examining the backbone contributions. The corresponding site-specific mutants have confirmed both the redox potential shift [75] and the structural shift [75].

A second example is that of an Ala-to-Cys mutation, which causes the formation of a rare SH S hydrogen bond between the cysteine and a redox site sulfur and a 50 mV decrease in redox potential (and vice versa) in the bacterial ferredoxins [73]. Here, the side chain contribution of the cysteine is significant; however, a backbone shift can also contribute depending on whether the nearby residues allow it to happen. Site-specific mutants have confirmed the redox potential shift [76,77] and the side chain conformation of cysteine but not the backbone shift in the case with crystal structures of both the native and mutant species [78]; the latter can be attributed to the specific sequence of the ferredoxin studied [73].

2. Molecular Dynamics Simulations

Molecular dynamics simulations are useful in understanding whether the electrostatic energy shifts seen between homologous proteins can be translated into site-specific mutations that shift the redox potential. The molecular dynamics simulations of the wild-type and proposed mutant can address whether the mutant will have the same structural and energetic shifts as occur between the homologous proteins with different redox potentials. The use of MD simulation allows for greater sampling of conformational space than energy minimization, thus enhancing the probability of properly modeling structural changes.

3. Free Energy Simulations

Free energy simulations are a useful means of quantitating whether the free energy and not simply the energy is shifting in the predicted manner for the mutant (see Chapter 9). The difference in the free energy changes upon reduction between a wild-type and a mutant, ∆∆G ∆G* ∆G, where the asterisk indicates the mutant, can be calculated in two ways via the thermodynamic cycle shown in Scheme 2,