408	Ichiye
Poxd	∆Goxd→red
	→ Pred
∆Goxd→red* ↓	↓ ∆Gred→red*

P*oxd → P*red

∆Goxd*→red*

Scheme 2

where P and P* denote the native and mutated protein, respectively, and oxd and red denote the oxidized and reduced forms, respectively. The free energy difference is given by

∆∆G ∆Goxd*→red* ∆Goxd→red	(12)
∆∆G ∆Gred→red* ∆Goxd→oxd*	(13)

where Eq. (12) is simply the definition of ∆∆G as the difference between the reduction free energies of the wild-type and mutant, and Eq. (13), which comes from the thermodynamic cycle, gives ∆∆G as the difference between the free energies of mutation of the oxidized and reduced states. Calculation of ∆∆G by Eqs. (12) and (13) will be referred to as reduction and mutation, respectively, free energy calculations.

VI. ELECTRON TRANSFER RATES

The environmental (i.e., solvent and/or protein) free energy curves for electron transfer reactions can be generated from histograms of the polarization energies, as in the works of Warshel and coworkers [79,80].

A. Theory

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Figure 1, where R represents the reactants D and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P′ is the Franck–Condon state on the P-surface. The activation free energy, ∆G‡, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. For self-exchange transfer reactions, we need only λ to calculate ∆G‡, because ∆G° 0. Moreover, we can write

	λ ∆G(rD,red, qD,oxd; rA,oxd, qA,red) ∆G(rD,oxd, qD,oxd; rA,red, qA,red)	(14)
	∆G(rD,red, qD,oxd; rA,oxd, qA,red) ∆G(rD,red, qD,red; rA,oxd, qA,oxd)

where the difference lies in the second term. Here, rD and rA represent the nuclear configurations of all atoms near D and A, respectively; qD and qA represent the charged states (via the partial charges of the appropriate atoms) of D and A, respectively; and oxd and red denote oxidized and reduced, respectively. Thus, λ is the energy of the Franck–Condon transition from the R (i.e., qD qD,red and qA qA,oxd) to the P (i.e., qD qD,oxd and qAqA,red) surface at X R0. Note that by examining λ, one does not have to explicitly define the polarization coordinate.

Simulations of Electron Transfer Proteins

409

Figure 1 Reaction coordinate diagram for electron transfer reactions.

More generally, the connection between the free energy surface and the simulation data can be made by the relation [81]

∆G(X) kB T ln[P(X)/P(Xmin)]

(15)

where P(X) is the probability of a value of X, kB is Boltzmann’s constant, and T is the temperature. P(X) can be calculated by making a histogram of X obtained from a simulation. The activation free energy can also be calculated from the ratio of the probability of being at the transition state (X 0) versus the probability of being in the equilibrium state (X Xmin),

∆G‡ kB T ln[P(0)/P(Xmin)]

(16)

This definition of ∆G‡ is thus dependent on the exact definition of X. Here the polarization coordinate X is defined to be the difference in the energy between when the excess electron is on A [curve ∆GP(X) of Fig. 1] and when the excess electron is on D [curve ∆GR(X) of Fig. 1] [79], i.e.,

	X ∆V V(rD, qD,oxd; rA, qA,red) V(rD, qD,red; rA, qA,oxd)	(17)
	∆G(rD, qD,oxd; rA, qA,red) ∆G(rD, qD,red; rA, qA,oxd)

where the second equality holds because there is no entropy change upon making a vertical transition between the two curves of Figure 1, plus some small terms corresponding to the change in geometry of the redox site (i.e., the potential energy parameters) on going from the oxidized to the reduced state and vice versa. Therefore, at the left-hand minimum, R0, Xmin λ. For a self-exchange reaction, the transition state corresponds to when the energy of the excess electron on either protein is the same; i.e., ∆V 0.

410

Ichiye

If the distribution of X is assumed to be a Gaussian about Xmin, the minimum of either the leftor the right-hand side, the activation free energy [Eq. (4)] becomes [82]

∆G‡ kB T λ2/2σ2

(18)

where σ2 (X Xmin)2, i.e., the mean-square fluctuations of X, which can also easily be calculated from the simulation. This will be referred to as the Gaussian fluctuation approximation.

As indicated above, the free energy curve predicted by simulation is obtained by making a histogram of values of X in the trajectory [Eq. (15)]. However, because thermal fluctuations in a molecular dynamics simulation generally are not sufficient to allow sampling far away from P(Xmin), two methods are used to extend the free energy surface. First, non-Boltzmann or ‘‘umbrella’’ sampling [79,83,84] can be used to obtain the free energy surface away from X Xmin (see Chapter 10). Here, the nuclear rearrangement due to intermediate values of charge on the two redox centers is considered, meaning that the charge density of the transferring electron is (1 z)e on one center and ze on the other. More specifically, for a protein, all parameters (partial charges and equilibrium bond lengths and angles) of the redox site should be scaled according to

Uoxd,z (1 z)Uoxd zUred	(19a)
and
Ured,z (1 z)Ured zUoxd	(19b)
The expression for the probability with X ∆V is
P(∆V ) c (z)eβz∆VP (z)(∆V )	(20)

where c (z) is a normalization constant and P (z) indicates the probability when the system has the charge distribution characterized by z. The second method is actually a special case of umbrella sampling when z 1, so that c (1) c (0) [81,85]. Thus, the left-hand side of the P curve, ∆G P, is given by ∆G R ∆V and the right-hand side of the R curve is given by ∆G P ∆V (Fig. 1).

B. Application

Free energy curves for the self-exchange reaction between two rubredoxins (Rd1 and Rd2) were generated from MD simulations [86,87].

Rd12 Rd21 → Rd11 Rd22

(R3)

The study of self-transfer reactions is a great simplification for theoretical studies, although not for experimental studies, and is thus a useful starting point. The free energy for electron transfer reactions in a variety of small molecule systems has been studied using MD methods [79–81,84,85,88–90]. The applications to proteins have been more limited. Several applications have been made by the Warshel group, including studies of the crystal structures of oxidized and reduced cytochrome c [91]. The simulations used to evaluate ∆G ‡ for the self-exchange reaction of Rd were actually separate simulations of Rd in the oxidized and reduced form [19], which implies that the two rubredoxins are separated by a distance great enough that the polarization of solvent around one does not affect the other, i.e., the infinite separation limit. Thus, the polarization of the entire system is estimated by summing that of separate simulations of the protein in the oxidized and reduced forms,

Simulations of Electron Transfer Proteins	411
∆V V1(r1, q1,oxd) V2(r2, q2,red) V1(r1, q1,red) V2(r2, q2,oxd)	(21)

The first and third terms on the right-hand side of Eq. (21) can be evaluated from the reduced simulation by using the oxidized and reduced partial charges, respectively, and the second and fourth terms can be evaluated from the oxidized simulation by using the reduced and oxidized partial charges, respectively. Using Eqs. (17) and (21), the solvent reorganization energy becomes

λ Xmin V1(r1,red, q1,oxd) V2

(r2,oxd, q2,red)

(22)

V1(r1,red, q1,red) V2(r2,oxd , q2,oxd)

Thus, the z 0 surface (and equivalently the z 1 surface) is generated from the original oxidized and reduced simulations. Additional simulations were performed to sample values of the polarization coordinate away from the minima, using parameters scaled according to Eq. (19).

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