On the action of cytochrome c: correlating geometry changes upon

Gautam Basu, Akio Kitao, Atsuo Kuki, and Nobuhiro Go. The Journal of Physical Chemistry B 1998 102 (11), 2085-2094. Abstract | Full Text HTML | PDF | ...
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J. Phys. Chem. 1983, 87, 1683-1694

ARTICLES On the Action of Cytochrome c: Correlating Geometry Changes upon Oxidation with Activation Energies of Electron Transfer Antonle K. Churg, Robert M. Welss, Arleh Warshel,’ Deparfment of Chemistfy, University of Southern California, Los Angeles, California 90007

and Tsunehiro Takano Institute for Proteln Research, Osaka University, Osaka 565, Japan (Received September 13, 1982; In Final Form: December 2, 1982)

A microscopic treatment of the energeticsof electron transfer reactions in proteins and aqueous environments is presented. Using this treatment we correlate the observed X-ray structural changes between reduced and oxidized cytochrome c and the function of the protein as an electron transfer catalyst. The correlation is based on estimating the reorganization energy, a , which determines the activation free energy of electron transfer for a fixed distance between the donor and acceptor. It is shown that, for electron exchange reactions, CY can be estimated from electrostaticconsiderations and from the X-ray structures of the reduced and oxidized forms of the protein without the self-energy (the folding energy) of the protein. The catalytic advantage of the cytochrome is assessed by comparing a for electron exchange between ferrous and ferric heme prosthetic groups in their respective cytochrome environments with that in aqueous solution. The computed values of CY for electron exchange at a distance of 8 A are 6.4 kcal/mol in the protein system and 32 kcal/mol in water, indicating that, compared to a polar solvent, the protein undergoes a small dielectric relaxation upon oxidation of the heme. Calculated energy surfaces for the relaxation as a function of protein conformation are presented. These are the first energy surfaces computed for an entire enzyme reaction, and they demonstrate the consistency of the X-ray structures with the reaction energetics.

I. Introduction Electron transfer chains play a central role in biological energy transduction. A unique opportunity to examine quantitatively molecular factors that control biological electron transfer processes is presented by the recent X-ray studies of Takano and Dickerson’ which compared the structures of reduced and oxidized cytochrome c. The X-ray results define small but experimentally significant differences between the static time-averaged conformations of ferri- and ferrocytochrome in the crystal. To understand how these differences in equilibrium structure bear on the catalytic function of the protein, one must examine the structures in the framework of a theory of electron transfer rates. At first it is not obvious what the equilibrium structures taken a t face value can tell us about the rate of electron transfer. However, the theory of electron transfer reaction rates2” estabishes a formal relationship between the activation free energy for electron transfer and a parameter called the “solvent reorganization energy”. This parameter is defined as the free energy released when the solent reorients from the equilibrium configurations around the reactants to those around the products, while the solute charges are fixed at the values of the products. (1) (a) T h o , T.; Dickerson, R. E.Proc. Nutl. Acud. Sci. U.S.A. 1980, 77,6371-5. (b) J.Mol. Biol. 1981,153, 79-94. ( c ) Ibid. 1981,153,95-115. (2) (a) Marcus, R. A. J.Chem. Phys. 1956,24,96678. (b) Ibid. 1956, 24, 979-89. (3) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155-96. (4) Levich, V. G. Adu. Electrochem. Electrochem. Eng. 1966, 4, 249-371. ( 5 ) Hopfield, J. J. Proc. Nutl. Acud. Sci. U.S.A. 1974, 71, 3640-4.

0022-365418312087-1683$0 1.5010

Viewing the cytochrome as the solvent environment of the heme, one sees that the change in equilibrium protein structure when the heme changes from ferrous to ferric is literally a “solvent reorganization”. The change in conformation can, therefore, be related to the electron transfer rate, given a relation between the solvent reorganization energy and the activation free energy for electron transfer. The important role of the solvent in the energetics of electron transfer reactions can readily be appreciated. Transfer of an electron from one molecule to another entails a redistribution of charge, and inherently at least one of the reactants or products has a net charge. The energy of the solute-olvent system depends on the solute charge distribution. Polar solvents, especially, will adopt equilibrium configurations to lower the electrostatic energy of the system. On the other hand, electron transfer can only occur if the solvent adopts configurations such that the total energy of the system is the same for the initial and final solute charge distributions. Except in unusual cases, the equilibrium configurations do not satisfy this conditions. Thus, the electron transfer rates depend on the probability or frequency of reaching the solvent configurations which do satisfy the electron transfer condition. This probability can be expressed as (exp[-Ag’/kTI) where Ag* is the contribution of the solvent rearrangement to the total activation energy of the reaction. According to Marcus’ t h e ~ r y , which ~ - ~ treats the solvent as a dielectric continuum, this activation free energy is given by Ag’ = (a

+ AG0)’/4a

where AGO is the overall free energy change and a is the 0 1983 American Chemical Society

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The Journal of Physical Chemistry, Vol. 87,No. 10, 1983

solvent reorganization energy. Although the quantitative validity of eq 1 for real solvents requires further examination (see ref 6 and section I1 of this paper) it can be used as a qualitative estimate of Ag’. Equation 1 tells us that, for a given AGO, the crucial parameter that determines the activation free energy (and the reaction rate) is the reorganization free energy. Thus, understanding the catalytic action of cytochromes requires understanding of the factors that detemine their reorganization energy. Fortunately, although the reorganization energy is not accessible to direct experimental measurements, it is directly related to the X-ray data which describe the change in equilibrium configuration of the cytochrome when the heme switches its oxidation state. The correlation between these structural changes and the reorganization energy is independent of the specific intermolecular contacts between the donor and acceptor molecules (which play an important role in optimizing the electron transfer rate). (Admittedly, the structure of an individual cytochrome may be altered on formation of a complex with its donor or acceptor. However, the quantity relevant to the reorganization energy is the change in structure upon electron transfer. Since only the X-ray structures of noncomplexed cytochromes are available, we assume that the effect of changes in structure of the isolated molecule are qualitatively similar to those occurring in the complex. This assumption is, in fact, also implicit in the formal theory of “outer-sphere” or “nonadiabatic” electron transfer.) Equation 1was derived for an ideal solvent, treated as a dielectric continuum. In this approximation the free energy (apart from an additive constant) is given by the electrostatic energy of the system, which is given as a quadratic function of the field from the charges of the donor and acceptor molecules. In this case it is possible to show2S3that the free energy can be expressed as a parabolic function of a single formal parameter and that the activation free energy is given by eq 1. However, for discrete systems such as proteins or “real” polar solvents eq 1 might not be rigorously valid.6 In order to treat discrete systems one should convert the macroscopic continuum concept to a microscopic picture. It is necessary to consider the multidimensional coordinate space of the system, to define a unique one-dimensional reaction coordinate, and to obtain the free energy curves for the reactants and products states as a function of this reaction coordinate. The height of the intersection of such curves is %*. If the free energy curves are not parabolic in shape, Q*will not be related to the reorganization free energy by eq I. This work starts by rederiving the theory of electron transfer for microscopic systems and real solvents. The derivation concentrates on defining the reaction coordinate and clarifying the relation between the dynamical fluctuations of the system and &*. We present a rigorous semiclassical prescription for the evaluation of the intersection of the free energy curves and determination of &’. This prescription, which was used in preliminary studies,6 is presented here not for practical calculation but as a conceptual framework to clarify the approximations involved in estimating &*from the time average equilibrium coordinates, given by the X-ray data. In the practical calculation of &*we use two independent approaches. (1) In the first approach it is assumed that eq 1 is a reasonable approximation for Ag‘ and the reorganization energy is evaluated by converting the observed geometry changes (taken at their face value) into a change in electrostatic (6) Warshel, A. J . Phys. Chem. 1982, 86, 2218-24.

Churg et

al.

interaction between the protein and the heme. (2) The second approach evaluates &*in a more direct way. The reaction coordinate is approximated by the vector connecting the reduced and oxidized equilibrium structure. This reaction coordinate is used for locating the intersection of the reactants and products potential surfaces, where the height of this intersection is used as an estimate of Ag*. It is found that the two independent approaches give very similar results. This finding provides a consistency check on our approximations and on the validity of Marcus’ theory for electron transfer in proteins. The results of the calculations are used to assess the catalytic advantage of the protein. This is done by comparing the calculated reorganization energy for electron exchange between two cytochromes to the corresponding reorganization energy for outer-shell electron transfer reaction in aqueous solutions. It is found that the protein reorganization energy is much smaller than the corresponding reorganization energy of polar solvents. This indicates that the catalysis of electron transfer reaction by cytochromes is due, at least in part, to restriction of the reorientation of their dipoles. Section 1I.l.a treats the statistical problems of defining the activation free energy parameters and the reaction coordinate for a discrete solvent, sections II.1.b and 1I.l.c give practical prescription for estimating the reorganization energy, and section 11.2 summarizes our computational methods. Section 111.1 uses the X-ray structures of the reduced and oxidized cytochrome c and evaluates the activation free energy for electron exchange between two cytochromes. Section 111.2 assesses the catalytic advantage of the protein by calculating the reorganization energy for the corresponding outer-sphere electron transfer reaction in aqueous solutions. The results are discussed in section IV.

11. Theory and Methods ( 1 ) Microscopic Theory of Electron Transfer. ( a ) General Considerations. The theory of electron transfer reactions was originally derived by Marcus2p3and by Levich4using a continuum model for the solvent and treating the solute molecules as charged spheres. However, for proteins (and real polar solvents) microscopic models are needed. Thus we present a somewhat different treatment based on semiclassical considerations.6 We derive the rate starting, as did Levich, from time-dependent quantum mechanics. However, we obtain the rate in terms of an activation free energy (as did Marcus) as opposed to the activation (potential) energy of Levich. The derivation in this section is aimed at clarifying the microscopic relationship between the rate and the activation free energy. In particular we try to consider the relation between the dynamical fluctuations of the system and the thermodynamic parameters of the rate expression. The prescription for the rigorous semiclassical treatment is presented here not for the purpose of a practical numerical calculation, but as a justification for approximating the activation free energy by Marcus’ relation eq 1. The reader who is not interested in this somewhat formal consideration can start from eq 1and then skip this section and proceed to section II.lb which discusses the practical evaluation of eq 1. The rate of electron transfer between a donor, D,, and an acceptor, A2, is given by4

,& = JP(R)k(R) d3R where d3R is the volume element of the system, P(R) is

Geometry Changes on Oxidation of Cytochrome c

the probability that D1 and A2 will be at a given distance and orientation, R, and k(R) is the rate of electron transfer at a fixed R. P(R) reflects the work required to bring the reactants to a given R. Because the function P(R)k(R) has a sharp maximum at some value of R, say R,, the rate constant is k N P(R,)k(R,). Accordingly, the total activation energy is the sum of two components: the free energy of association at separation R, (-kbT In P(R,) + constant), and the activation free energy for reaction at fixed R, (-kbT In k(R,)). Since P(R,) is not directly related to the observed structural changes upon electron transfer we focus in this work on the fadors that determine k(R) for electron exchange at fixed R. The factors that determine k(R) in eq 2 can be best understood from Figure 1. The figure pertains to an electron transfer between D1 and A2 held at a fixed spearation in a polar solvent. The reaction involves two electronic states of the entire system: the reactant state (D1A2+)and the product state (D1+A2),where the superscript "+" indicates that the corresponding molecule is oxidized. The curves VI and VD describe the total potential energies for the nuclei in thest two states (the Born-Oppenheimer potential surfaces) as functions of the spatial coordinates of the solvent molecules, r. These potential surfaces include the internal energy involved in forming the donor and acceptor states in vacuo and the electrostatic interaction between the solvent dipoles and the donor and acceptor as well as the interaction among the solvent dipoles themselves. An hypothetical example is sketched at the bottom of Figure 1. When the solvent dipoles are oriented toward A2 the potential of D1A2+is lower than that of Dl+A2,and when they are oreinted toward D1 the potential of D1+A2is lower than that of D1A2+.For certain configurations of solvent dipoles, the potential energy of the system is the same for both D1A2+and D1+A2(as will be shown below, the electron transfer reaction occurs at these configurations). Many configurations of solvent dipoles are associated with each value of the potential, while the number of dipole configurations yielding the same value of the potential may not be constant over the space of the solvent coordinates. Thus, a microscopic model which simulates electron transfer reactions in solution VI and VI, are very complicated multidimensional surfaces. This situation is to be contrasted with dielectric continuum models such as the polaron approximation of Levich, where the degrees of freedom of the solvent can be represented by a single formal coordinate. The simplest way to consider the multidimensional case is to evaluate the potential difference (VII - VI), as a function of time rather than coordinates. This an be obtained by following the classical trajectories that describe the fluctuations, r(t), of the solvent molecules (and therefore VI(r(t)) and VII(r(t))as a function of time? If we assume that the reaction is initiated at a time t = 0 where the system is in the reactant state, at the initial stage of the reaction the solvent dipoles will follow trajectories that fluctuate around an average orientation and reach, from time to time, orientations where the difference between the potential VI and VI, is small. At these configurations the system has a significant probability for crossing from the reactant state (I) to the product state (11) (such a surface crossing process corresponds to an electron transfer from D1to A2). In the case of weak electronic interaction between the donor and acceptor (i.e., in socalled "outer-sphere" or "nonadiabatic" electron transfer reaction where R > 4 A) the probability for surface crossing and electron transfer is significant only for those classes of solvent configurations r* that give VI = V,. The probability for a reaction is determined by the

The Journal of Physical Chemistty, Vol. 87, No. 10, 1983

J

Reacti;n

Coordinate

t

m DI

1685

A2

8.0

-

Figure 1. A schematic representation of the energetics of electron transfer reaction D1 A,+ D1+ A, in a polar solvent at a fixed distance between D, and A,. The upper diagram represents a twodimensional section through the multidimensional diabatic potential surfaces of the system. V I and VI,are, respectively, the Born-OpA,' solvent and D,' penheimer potential surfaces of the D, A, solvent systems. The reaction coordinate 8 is defined by the locus of points 6' corresponding to the classes of coordinates S' that give successive vaiues of (AV)'(i.e., ( V I , - VI)= d where C is a constant). The vector r(8) is the least energy set of position vectors along the reaction coordinate 8. The probability of electron transfer reaction is determined by the relative number of times that a trajectory of the solvent dipoles, which move on the multidimensional potential surface VI,reaches position vectors r * where VI = VII. This probability is expressed as exp{-Agl'(6*)lk,T) where the relative free energy AgI(B) and the corresponding one for fluctuations on VI, are described in the middle figure. The Ag's are defined by the relative probability that trajectories on either VI or VI,will pass through a given value of 8 and will have the corresponding value of AV (e.g., the relative probability of having AV = 0 defined Ag * by Ag * = -k,T In {P&AV=O)))). The curves gI and gIIwill not necessarily be the same as VI and V I I , particularly when the number of dipole configurations associated with AV changes with 8. As a result the activation free energy AgI* will differ from the activation potential A V l * . The solvent Configurations 6,' and BIl0 that give the maximum number of the same AVvalue are not necessarily identical with the minimua of V Iand V I ] . A specific hypotheticalexample is given in the lower part of the figure: The solvent dipoles for 8 = 0 stabilize the reactant state and those for 6 = T stabilizes the product state. The configurations for 6 = 8' stabilize equally the reactant and product states so that VI= V I , . Since there are twice as many configurations for 8' than for 8,' the activation free energy Ag,* involves an entropic contribution of k,T In 2. I t is also assumed that A,' interacts less strongly with the solvent dipoles than Dl+ and that part of this difference is compensated by a larger entropic contributiondue to solvent motions in Bll0 than in 8,'. Thus the free energy difference, AGO, between the product and reactant states is not equivalent to the corresponding difference in potential energy, AVO. The rate is determined by the probability that solvent fluctuations will reach 6'. This probability is determined by Ag* and not by the potential difference A V * . The reorganization energy, a , is given by V,,(8,') - V1(Bi') - AGO.

+

+

+

+

+

+

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The Journal of Physical Chemistty, Vol. 87, No. 10, 1983

product of (1)the probability of surface crossing at r*,and (2) the probability that the solvent trajectories will reach r*. The probability of surface crossing at r* can be evaluated semiclassically by integrating the transition probability along solvent trajectories.6 The resulting crossing probability can be approximated by the Landau-Zener which gives, for the one-dimensional harmonic case, the rate expression4 k(R) =(U2(R)/ h)(a/akbT)'/' exp(-AVI*(x') /kbV

(3)

where h is Planck's constant, k b is Boltzmann's constant, ) u(R) is the Hamiltonian matrix element which was estimated to decay exponentially with an h dependence of exp(-2.7(R -4.6));9 a is the reorganization energy (defined in Figure l), X is the one-dimensional reaction coordinate, and AVI*(X*)is the difference between VI(X*)and V1(Xt). Equation 3 expresses the rate as a product of the crossing probability at X*,where VI and VI1 intersect, and the relative probability, P(X*), of reaching the intersection point

Churg et al.

k(R) = (a2(R)/h)(7f/akbT)1/2eXp(-&*(8*)/kbT)

(8)

The activation free energy, &(e*), can be evaluated by a molecular dynamics approach6running trajectories of the solvent (or protein) system in state I and counting the number of times, n(O'),that the VI and VI, intersect relative to the number of times, n(OIo),that VI, - VI reach its most probable value (the value obtained most times along the trajectory). AgI*(O*)is given by

&I*(O*)

= -kbT In (n(d*)/n(OIo)]

(9)

Such studies, which involve an umbrella-samplingmethod

(where the system is driven to S* by a constraint potential (#D~A~+~HI#D~+A with a solute charge distribution of intermediate value

P ( X *) /P(XO ) = exp{-A VI' (X' ) /kbT)

(4)

In the multidimensional case6 the rate is obtained by substituting for the one-dimensional probability in eq 3, [exp{-AVI*(X')/kbT)], the corresponding multidimensional probability of reaching the intersection of VI and VII. (In the multidimensionalcase the activation energy, AVI*(X*), should be replaced by the proper activation free energy, Ag*(B*). This point can be seen clearly by inspecting the hypothetical case presented in Figure 1. In this case the multidimensional reaction coordinate of the solvent is represented by rotation of the solvent dipoles. Since there are two solvent configurations with VI = Vu and only one configuration minimizes VI the relative probability of reaching the intersection of VI and VI, is determined by [&*(e') = AVI(O*)- k b T In 21 rather than by AV(X*).) The multidimensional probability of reaching r* is treated formally in the following way: The relation

VII(ri)- VI(#) = C'

(5)

where C' is a particular constant, defines a hypersurface Si in the solvent configuration space (i.e., A' is the class of all r' that satisfy eq 5). The generalized reaction coordinate 8 is then defined as the locus of points perpendicular to the successive A' taken for a monotonic change in C'. Thus, for example, all the coordinates r' that satisfy the relation VI, = VI correspond to the same value of reaction coordinate f? = 8*. Similarly the coordinates rIthat give the most probable value of VI, - VI, in the reactant state, correspond to 8 = BIo (see Figure 1). The probability of reaching 8* is given by exp(-AVI*(S*) / k b q dS *

P(8*)/P(Bo) =

1exp{-VI(SO)

/kbTJ dSo

(6)

This probability can also be defined in terms of a free energy &*(8*) so that P(O')/P(4O)= exp(-&*(O')/kbT) (7) The probability of being at other points along the reaction coordinate can also be defined in terms of the corresponding Ag(O). The overall rate is given now by (7) Landau, L. Phys. 2.Sowjetunion 1932, 2, 46. (8)Zener, C . Proc. R. Soc. London, Ser. A 1932, 137, 363-40. (9)Warshel, A.; Schlosser, D. Proc. Notl. Acad. Sci. U.S.A. 1981, 78, 5564-8.

between those of the reactant and the products6),are beyond the scope of this work. However, for a continuum model of the solvent which implies that the orientation of the solvent dipoles is linearly related to the local field one obtains the Marcus' relation

&*(e*)

= (a

+ AG0)2/4a

(10)

where AGO is the free energy difference between the product and reactant states and a is the reorganization free energy a = ( v I I ( e I o ) - vI(eIo)) - AGO (11) The validity of this expression for real polar solvents or for proteins has never been demonstrated. For example, eq 11does not give the correct Ag' for the model presented in Figure 1. Thus we will consider eq 10 as an approximation and examine the corresponding calculated results by a direct evaluation of the energy along the reaction coordinate. This work will focus on the evaluation of a and &*from the X-ray geometry changes observed upon oxidation of cytochrome c. In order to assess the catalytic advantage of the protein we also calculate a for the corresponding reaction in aqueous solution. The qualitative difference between the solvent reorganization in the aqueous and cytochrome systems can be appreciated by comparing Figures 1 and 2. In the latter system, the configuration of dipoles in the immediate vicinity of the solutes (i.e., the heme prosthetic groups) is established by the protein structure; at larger distance from the solutes are dipoles of the water surrounding the protein. As will be shown here, the protein structure limits the extent of dipole reorientation in the entire system, and consequently the reorganization energy is smaller than that of the corresponding electron transfer process in homogeneous solution. Equation 8 gives the rate of electron transfer between perfectly rigid donor and acceptor molecules. When the intramolecular vibrations of the donor and acceptor are taken into account (see Appendix A) one obtains an intramolecular contribution to the reorganization energy. This gives a = a, +

aintra

(12)

where aintra is the intramolecular contribution of the donor and acceptor and a8is the contribution of the surrounding solvent (aBwill be called apin electron transfer in proteins and a, in electron transfer in water). The intramolecular contribution, cyintra, is only 1 kcal/mol in the present case. ( b ) Calculation of the Reorganization Energy i n Proteins. This section will describe our approach for evaluating the reorganization energy for electron transfer in cytochromes. As sketched in Figure 2, the proteins (with surrounding water) are formally treated as the solvent for the heme prosthetic groups. Thus, for electron transfer

The Journal of Physical Chemistv, Vol. 87, No.

Geometry Changes on Oxidation of Cytochrome c

10, 1983 1687

charges, Q, and the protein charges, q, and V,, is the electrostatic interaction among the protein charges. Vind is the energy of the induced dipoles of the protein in the field of the heme and protein charges (which represents the microscopic dielectric effect of the protein), V, is the energy of the surrounding water molecules, and Vmd is the ocnformational energy of the system (bond stretching, angle bending, and torsional potential of the protein). V,, is the van der Waals energy of the system. On a microscopic level the electrostatic interactions VQ, and V,, are the same as in vacuo and they are given by

a;

VQ, + V,,

332 Z Q i q j / r i j+ 332C’qjqgrjf (16) ij

,,red I

W Figure 2. A schematic representation of the relation between structure and energetics in electron transfer from a donor, protein D,, to cytochrome c (A2). The figure uses the notation of Figure 1 and describes the changes in the dipoles of the proteins and the surrounding water molecule (indicated by enricled arrows) along the reaction coordinates. The illustrated protein dipoles of cytochrome c are as calculated for Figure 4, the tertiary structure of cytochrome c is as diagrammed by Adman.30 The figure illustrates that the reorientation of the dipoles of the proteins is much smaller than the corresponding reorientatlon of the solvent dipoles in an electron transfer reaction in polar solvents (Figure 1). Thus the reorganization energy, a,the activation energy, AV*, and the activation free energy, Ag*,are much smaller for electron transfer in proteins than in polar solvents.

between two cytochromes, the relevant quantity is the reorganization energy of the protein environement, a,,.In principle, in order to calculate the protein reorganization energy one should scan many protein configurations and find those that give most often the same value of Vu - VI. This set of coordinates will correspond to OIo in Figures 1 and 2 and the value of VII(OIo- VI(OIo)is a for an exchange reaction. However, here we introduce two assumptions: (i) The protein coordinates that give the most probable value of VI, - VI are also the most probable coordinates. (ii) The most probable protein coordinates are those obtained from the X-ray studies. With these assumptions, OIo correspondsto the vectors (rlred, r2OX), where Ired and roxare the equilibrium coordinates of the cytochrome around the reduced and oxidized heme, respectively, and the subscripts 1 and 2 indicate the donor and acceptor subsystems. Since A G O = 0 for electron exchange, eq 11 is recast as

where the subscripts I and I1 refer to the initial and final heme charge distributions. Expressing VI and VI, as explicit functions of the heme charges one obtains ap= V(Q10x,Q2red,rlred,r20x) - V(Qlred,Q20x,r~d,r20x)

(14) where Q is the vector of the partial atomic charges of the heme atoms. Evaluation of eq 14 requires an explicit expression for the potential V of the protein-heme system. This potential is given here (for fixed heme coordinates) by

where r is the vector of the protein coordinates, and Q is the vector of the partial atomic charges of the heme atoms. VQ, is the electrostatic interaction between the heme

jj‘

where the charges are in units of le1 (electronic charge), the energy is given in kcal/mol, and the distance in A. The explicit form of the other terms in eq 15 and their numerical evaluation will be considered in section 11.2. Substituting the V of eq 15 in eq 14, using the fact (see V , depend on Q (and below) that only VQ, + Vind neglecting the small contribution to CY from the interaction between q1 and Q2),we obtain

+

CYp

CY,,*=

=

CYp*

+

CYy*

AvQ,q[(Qlred_*Qlox);rl’ed] + A V Q , [ ( Q ~ ~ ~ Q P+~C)Y, ;’ ~(17) ~~~]

+

ay* = ~ ~ V i ~ d [ ( Q l ’ ~ d Q l ~ ~ )AVind[Q20;rl’~~]

Q2red;r20x] where a,* and aY* designate the contributions of the permanent and induced dipoles, respectively. CY,^ is an abbreviation for the quantity Vw(Q10X,Q2red,rlred,r20x) Vw(Q~ed,Q20x,rld,r20x) evaluated with the water molecules around the cytochromes in their initial equilibrium conformations; aPwincludes contributions from both the permanent and the induced water dipoles around the protein (to simplify the notation we do not divide the water contribution to upinto “permanent- and “induced dipole components). The evaluation of CY,^ is identical with that described in section 11.3 concerning electron transfer processes in water. The simple expressions of eq 17 are obtained because we are treating an exchange reaction and (the self-energy or folding the terms V,, + Vconf+ vVdw energy of the protein) do not change upon change of Q at fixed r. In order to simplify the discussion we will concentrate on a,,*and add the effect of CY?*that reduces cyp only at the final stage of discussing the numerical results. Using the explicit form of VQ (eq 16) one obtains (using the symmetry between r1 and r2) CY,,*=

C[Ui(rlred)QY- Ui(rlred)Q:ed+ Ui(rloX)QYdi

Uj(rloX)Qlox] + a,P (18)

where Viis the electrostatic potential from all the protein atoms at the location of the ith atom of the heme (i.e., Vi(r) = 332Cjqj/ri.).Equation 18 will be used in section 11.3 to evaluate t i e activation energy, Ag’. This estimate of Ag* will be verified by a direct calculation using the observed geometry changes of the protein to obtain an estimate of the transition-state geometry r*,and using the change in electrostatic energy upon change from rIoto r* as an estimate of the activation free energy. ( c ) The Reorganization Energy for Electron Exchange between Two Heme Groups in Aqueous Solution. The calculation of the reorganization energy in aqueous solution is more complicated than the corresponding calculation for cytochromes; in proteins, the equilibrium configurations are given from the X-ray results; in aqueous solutions the

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The Journal of Physical Chemistry, Vol. 87, No. 10, 1983

equilibrium configurations and the large fluctuation around them are not available experimentally and should be evaluated by computer simulation. Preliminary attempts to evaluate Ag* by molecular dynamics approches (using umbrella sampling methods to drive the system near the transition-state region) are described in ref 6. Such approaches are very time consuming and are entirely beyond the scope of this paper which deals mainly with Ag' in proteins. Instead we use the fixed center Langevin dipole (FCLD) modellOJ1which simulates average properties of the solvent. Using this model and the considerations given in Appendix B we obtain a , e -l/ZCp).[Di(Q") - Di(Q')]

(19)

1

where the pi are the solvent dipoles and Di are the fields from the heme charges on the ith water dipole (Di = xjQjrij/riF). This expression will be used in section 111. (2) Methods of Calculation. The vectors roxand rrd of the coordinates of reduced and oxidized cytochrome c are obtained as follows: The carbon and nitrogen atomic coordinates are those reported by Takano and Dickerson.' Hydrogens were added to the carbon-nitrogen atom skeleton using standard geometries.12 Hydrogens of serine oxygen and the water olecules in the X-ray data set were oriented to minimize their local electrostatic energy under the steric constraint of their neighboring atoms. The coordinates of the bulk water molecules are evaluated as described in Appendix B. The point charges of the protein atoms, q,are those used in ref 10. The partial atomic charges of the reduced and oxidized heme atoms, Q, were calculated by the QCFF/PI method extended to include hetero atoms.16 Two sets of Q differing by 50% (due to different CT charge density but summing to the same values) gave the same a pto within 0.5 kcal/mol. The various terms in the protein-heme potential surface (eq 15) are evaluated as follows: The electrostatic interactions VQqand Vqqare given by eq 16. The energy of the induced dipoles of the protein in the field of the heme and protein charges, Vin&is determined by the microscopic dielectric approachlo*"assigning an induced dipole, pj = -yjE,, to each protein atom, where the y j are the atomic polarizabilities and the Ej are the local fields due to the charges of the system and all the other induced dipoles (determined self-consistently). The interaction between induced dipoles on bonded atoms is not included since this interaction corresponds in our classical approximation to the interaction between two overlaping conducting spheres. With these induced dipoles we obtain

where j and k run over all the protein atoms and i runs over the heme atoms. The energy is given in kcal/mol, the distance in A, and the charges in atomic units. The energy of the surrounding water molecules, V,, is evaluated by the microscopic FCLD modellOJ1(cf. Appendix B) repar(10) Warshel, A.; Levitt, M. J . Mol. Biol. 1976, 103, 227-49. (11) Warshel, A. J.Phys. Chem. 1979,83, 1640-52. (12) Levitt, M. J . Mol. Biol. 1974, 82, 393-420. (13) Warahel, A.; Karplus, M. J. Am. Chem. SOC.1972, 94, 5612-25. (14) Warshel, A.; Levitt, M. Quantum Chemistry Program Exchange No. 247, Indiana University, 1974. (15) Warshel, A. In 'Modern Theoretical Chemistry";Segal, G., Ed.; Plenum: New York, 1977; Vol. 7, pp 133-71. (16) Warshel, A.; Lappicirella, A. J . Am. Chem. SOC.1981, 103, 4664-73. (17) Warshel, A. Acc. Chem. Res. 1981, 14, 284-90.

Churg et al.

ametrized to include both permanent moments (pa = 1.35 D) and polarizability (1.4 A3). The conformational energy, Vconf,and the van der Waals energy, Vvdw,may be represented by the standard force field approach'* using the potential functions and parameters described in ref 12. However, these terms are not calculated here since the contribution of vwdand Vvdwto the reorganization energy is rigorously zero (see section 1I.l.b). Furthermore, the value of these energy contributions is meaningless without convergent energy minimization. 111. Results and Discussion (1) Calculations of Ag' for Electron Exchange between Cytochromes. This section will use the observed structural differences between reduced and oxidized cytochrome c to estimate the activation free energy for electron exchange between two cytochromes immobilized with respect to translational motion. Calculations using two independent approaches will be presented in the next two sections; the first is based on evaluating the reorganization energy and the second on direct estimate of the potential surfaces for the reaction. (a) Evaluating Ag* from the Reorganization Energy. According to section I1 the activation free energy for electron exchange between two cytochromes can be approximated by Ag* = (ap+ aintra)/4.As indicated by eq 17, a p is the change in electrostatic energy for the process

-

cytl(rlred,hemered) + cytZ(r20x,hemeox) cytl(rlred,hemeox) + cyt2(r20x,hemered)(21) where rredand roxdenote the equilibrium conformations of the protein when the heme i s reduced and oxidized, respectively. The components of cyp due to the permanent dipoles of the protein and the surrounding water, CY,*,can be evaluated by eq 18 which can be recast as ap* = -CIUi(rlox)- Ui(rlred)](Q;oX - QYd) i

+ CY,^ =

-CAUjAQi

+ a,P

(22)

1

where Ui(rlox)or Ui(rIred)denotes the electrostatic potential Vi = xqj/rij !qj is the charge on the j t h protein atom and rij is the distance between heme and protein atoms when the protein is in the roxor rredconformation), Qioxor QYd refers to the partial electronic charge on the ith heme atom when the heme is oxidized or reduced. For convenience we abbreviate this expression in Viand Qi as -CiAUiAQi;in the present discussion AUi refers to changes in potential due only to neutral groups of the protein, i.e., the protein dipoles. The justification for neglecting the effect of the surface charges is given in Appendix C. The changes in potential, AUi, when the cytochrome goes from reduced to oxidized conformation are described in Figure 3. The calculated changes in partial atomic charges of the heme atoms, AQi, when the heme becomes oxidized are indicated in brackets in Figure 3. Only those porphyrin atoms which are in the electronic delocalization pathway exhibit significant AQi's. Thus the charges and potentials at atoms of the porphyrin skeleton alone are relevant to the evaluation of -CiAUiAQi. As can be seen from Figure 3, the AUi's are mainly negative while the AQi's sum to +1.0. To the extent that the protein relaxes around the positively charged ferric heme, negative values of AUi would be anticipated. However, the pattern of AUi's has a distinct gradient; the atom with the most positive calculated AUi is in the vicinity of Tyr-46 while the atoms with the most negative AUi's are in the region (18) Lifson, S.; Warshel, A. J . Chem. Phys. 1968, 49, 6116-24.

The Journal of Physical Chemistry, Vol. 87, No. 10, 1983

Geometry Changes on Oxidation of Cytochrome c

1689

AU

U - 3 . 2 (.OB) 6-3.8 C - 4.4 d -5.7 e-4.8 f -6.9 9-4.1 h -5.2

(am) (.071 )

(.OBB)

(.me) (.OM) (.041) (.OTO)

i - 2 . 9 (.om) j -0.9( k 3.6 (.010) / -3.5 (.042) m- I . I (.073) n 3.5 (.060) 0 + 3. I (-015) p + 1 . 1 (.oae) q + 1.0 (.ooo) r + 1.9( . 0 3 5 ) S +3.5 ( . 0 6 2 ) t +4.3 (.oeo) u +5.2 (.IOO) Y +3.2 (.oo8) w-3.3 (.019)

-

t

a,' )'

0.0 (.OlB) 0.0 (.035)

Fipre 3. The changes in the electrostatic potential, A q , AU,= (U,(rOx) - u,(P"),at the heme r a t o m locations, i , upon transition of the protein from reduced to oxidized conformation. The potential is defined as the energy in kcal/mol of a unit positive test charge. The changes in the heme partial atomic charges, AQ,, upon oxidation are given in brackets. The calculated changes in charge include the effect of polarization of the heme electrons by the electrostatic field of the protein (see eq 21 of ref 16). The protein reorganization energy is given approximately by -C,AU,AQ,.

defied by Tyr-67, Trp-59, and Asn-52. These two regions are, in fact, those for which the largest X-ray structural differences between oxidized and reduced cytochrome c were observed. In general, the dipoles nearest the prosthetic group have the largest effect on AU,; those associated with the region Gly-45-Trp-59 are shown in Figure 4. Only the dipole of Asn-52 changes direction by more than a few degrees. The calculated value of -CiAUiAQi (obtained directly from Figure 3) is 2.4 kcal/mol. In addition to this contribution, which reflects the rearrangement of the protein dipoles, we have to include in CY the contribution of the rearrangement of the surrounding water molecules, a,P, the relaxation of the hemes, aintra, and the reorientation of the induced dipoles of the protein, cyy*. a,P evaluated by using eq 19 and the procedure described in Appendix B is 3 kcal/mol. shea evaluated as described in Appendix 1is -1 kcal/mol. With these calculated values we obtain the total CY by CY = -CAU,"Qi CYwp CYintra + CYy* = i

+

+

6.4 kcal/mol

+ ay* (23)

since aY*is inherently negative the upper limit for CY is 6.4 kcal/mol, and the upper limit for the corresponding estimate of the activation energy is Ag' = a / 4 < 6.4/4 = 1.6 kcal/mol. As will be shown in section 111.2 this contribution of the reorganization free energy to Ag' is much smaller than the corresponding contribution in electron transfer in aqueous solutions. It should be emphasized that the calculated Ag' cannot be compared with the experimentally observed Ag' for electron exchange.lg In this case the work of association plays a crucial role since two cytochrome c molecules are not naturally optimized redox partners. ( b ) Direct Calculation of Ag'. The analysis presented in the previous section is based on eq 11which is rigorously (19)Gupta, R.K.; Koenig, S. H.; Mfield, A. G . J.Magn. Reson. 1972, 7, 66-73.

Flgure 4. The dipoles associated with residues Gly-45-Trp-59 of cytochrome c . Each residue is electroneutral and can be assigned two-point dipoles positioned at their respective centers of charge: the peptide dipole (N-H-C-0) and the dipole moment of the remaining atoms. In the figure dipoles of the same residue are connected; the convention is (0-= )(+ -). The heme plane is at 45' to the plane of the drawing. The reorientation of Asn-52 is indicated by a dashed arc.

-

valid only if the protein can be considered as a polar solvent which behaves like a dielectric continuum and satisfies eq 10. Thus it is important to examine the result obtained in the previous section by an independent evaluation of Ag'. Fortunately, since we deal with small geometry changes upon electron transfer it is possible to estimate Ag* by direct evaluation of the coordinates where VI = VII.In principle, we should search for all the points S*where VI = Vm However, as in the evaluation of aPwe will consider the contribution from only one set of coordinates: the coordinates along the least energy pathway between the reactants and products. That is, we replace the Ag curve of Figure 1by the potential VI along the least energy pathway. Finding the least energy pathway requires energy minimization using the adiabatic mapping approach'O and might involve serious convergence problems. Here we take a much simpler approach, taking the X-ray coordinates at their face value and defining the coordinates of the system along the reaction coordinate by a simple interpolation: rl(6) = rlred 6(rlox- rlred) (24) rz(6)= r 2 O X 8(rZred - rzox)

+ +

where the coefficient 6 assumes values 0 < 6 < 1. The reaction free energy is estimated by the electrostatic contribution (CiUi(r(6))Qi)to the potential surface. This approximation is justified by the following argument: The correct variation of the reaction free energy along the real reaction pathway (not the interpolated one) is expected to involve only small contributions from strain energy terms (nonbonded repulsion, bond stretching, etc.) since these interactions involve strong forces which relax along least energy pathways. Thus the main contribution is expected to come from electrostatic energy. Since the electrostatic contributions are not sensitive to small geometry changes they can be estimated along the interpolated reaction coordinate rather than along the real reaction coordinate. Note that, on the other hand, strain energy contributions might give unstable and meaningless results along the interpolated reaction coordinate. The calculations are summarized in Figure 5. Parts a and b of Figure 5 represent the contribution of the cytochrome to CiUiQifor reduced and oxidized heme, re-

1690

The Journal of Physical Chemistty, Vol. 87, No. 10, 1983

Churg et al.

T

I

-r;ed

Protein conformation-

T

I

-rpK Flgure 6. The reorganization energy, cyw@), for electron exchange between ferrous and ferric heme in water for several distances, R , along the axis normal to the heme. The calculations are based on representing the average orientation of the solvent dipoles by the fixed center Langevin model and using eq 30.

,red (C)

-

rPK

Protein conformotion -+

I

I

Reaction coordinates

-

Flgure 5. A direct interpolation of the potential surface for electron transfer reaction using the observed X-ray structure. The changes in the potential of the system are represented by the changes in the protein-heme electrostatic potential. I t is assumed that all other contributions except this of the induced dipoles (V,, V,,J cancel out (this is rigorously correct for the equilibrium coordinates of the reactants and products in an exchange reaction). The effect of the induced dipoles reduces the calculated changes of VQ,qby about half. The reaction coordinates are determined by the interpolation formula rl(t9) = rlrd 0(rlox - rlrd), r,(@ = r2OX %(rid - r?) where the subscripts 1 and 2 refer to molecules “1” and “2”, % is a number between 0 and 1, and rOx and are the corresponding observed X-ray coordinates. Since the geometry changes between roxand rrd are small it is reasonable to take 8 as the generalized reaction coordinate of section 11.1. The upper figure (a) presents the variation of the electrostatic interaction between the charges of the reduoed heme and the permanent dipoles of the protein ( V,,,) as the protein c o n f m t l o n is changed linearly from that observed for reduced cytochrome c (fd) to that observed for the oxidized cytochrome @Ox)). The middle figure (b) shows the variation of the electrostatic interaction between the protein dipoles and the charges of the oxidized heme. The lower flgwe (c) presents the potential surfaces of the electron exchange reactlon where V , is the protein-heme electrostatic potential when the charges Q are those of the reactants (Did A?) and V I , is the protein-heme electrostatic potential when the charges Q are those of the products (DloXf A,’“).

+

+

+

+

spectively. The electrostatic interaction between the neutral (reduced) heme and the protein dipoles is seen to be independent of the protein conformation, whereas the oxidized heme (with charge 1+)is more stable in the oxidized Conformation than in the reduced conformation by 2.5 kcal/mol. Clearly if the oxidized heme were calculated to be less stable in the oxidized conformation the model would be invalidated. On the other hand, the calculation suggests that CiUjQialso defines the qualitative features of the free energy surfaces for electron exchange at a fixed heme-heme distance. The curves for CiViQiof a pair of cytochromes are drawn in Figure 5c. Tracing the VI curve from left to right describes the electrostatic energy of the reactants as a function of protein conformation: One heme is reduced and its protein goes from reduced toward oxidized conformation while the other is oxidized and its protein goes from oxidized toward reduced conformation. The curve of VI, corresponds to the products in which the heme charges have been reversed but the change in protein

conformation proceeds in the same dirrection; i.e., the right end of the VI, curve corresponds to ferric heme with protein in oxidized conformation plus ferrous heme with its protein in the reduced conformation. The calculation presented in Figure 5c gives the potential, due to the permanent dipoles of the protein, for the entire electron transfer reaction. (Contributions from the induced dipoles of the protein and from dipoles of the surrounding water molecules are not included.) The activation energy is given by the difference between the electrostatic energy a t r* (the intersection of VI and VII) and that at the minimum at (rIred,r2Ox); accordingly, the value of &*is 0.62 kcal/mol, in agreement with the value 0.60 kcal/mol obtained by taking the contribution to the reorganization energy from the permanent dipoles (-EiAUiAQiof the previous section) and applying eq 10. (The value 0.6 kcal/mol should not be confused with the value 1.6 kcal/mol obtained in section III.l.a, which includes the water contribution.) The good agreement between the protein contribution to &*obtained by eq 10 and by the direct calculations presented in Figure 6c is due to the fact that CjViQiis a parabolic function along the interpolated reaction coordinate. The fact that the crystallographic data define parabolic curves for the electrostatic potential suggests that major trends in the energetics of electron transfer in proteins can be predicted by harmonic approximations as in the case of electron transfer in polar solutions. (c) Estimating the Reliability of the Calculations. In assessing computer modeling like the one reported here it is important to estimate the error associated with the calculations. In contrast to regular energy calculations in proteins, which might involve very significant convergence problems, the protein conformational energy cancels out in the present case. Thus we only have to estimate the error associated with the electrostatic calculations. As mentioned in the Method section, we checked the sensitivity of the calculation to the distribution of the heme charges and obtained 0.5 kcal/mol difference in (Y up to 50% differences in the Q of specific atoms. The sensitivity to the X-ray coordinates was examined by comparing calculations using the two crystallographically independent ferricytochromes in the tetragonal crystal. This gave electrostatic energies equal within 0.5 kcal/mol whereas the difference between oxidized and reduced structures as given in Figure 6b is -3 kcal/mol. More importantly, the main point of the paper is not what is the exact value of &*but that Ag* is small. This conclusion will stay valid even if the error in cy is as large as (Y itself, which is probably an upper limit for the error bar since rox- rred can be considered to be at the limit of the X-ray precision. (2)Calculations of Ag* for Electron Exchange in Water. The catalytic role of the protein must be assessed by

The Journal of Physical Chemistry, Vol. 87, No. 10. 1983

Geometry Changes on Oxidation of Cytochrome c

comparing the quantity aPwith the corresponding reorganization energy for electron exchange between two hemes in water, a,, using eq 19. The calculated results are presented in Figure 6. a , is found to be an order of magnitude larger than cyp. At the heme-heme distance relevant to electron transfer between cytochromes, R 8 A, CY, 35 kcal/mol, whereas a p 5 kcal/mol. The experimentally observed rates for electron exchange between hemes in water are not directly relevant to a comparison of a, and ap. The reaction in water probably occurs by collision of the donor and acceptor within the same solvent cage (E 5 A); the shorter distance between reactants substantially enhances the coupling between donor and acceptor and accelerates the rate. It is interesting to note that the computed dependence of a, on the heme-heme separation differs from that estimated by the standard macroscopic treatment. That is, the a , derived with the dielectric continuum approximation is given (for electron exchange between two spherical reactants of radius a and separation R ) by3p4

-

-

-

-

a = 332(1/n2 - l/e)(l/a - 1/R)

(25)

where a is given in kcal/mol and R and a in A. n and e are the index of refraction and static dielectric constant of the medium. The distance dependence of CY, is

a,(R) = a,(m)[l - a / R ]

(26)

This functional form of a,(R) does not describe the results in Figure 6. In general, quantities such as the effective cavity radius of the solute and the effective distance between reactants are poorly defined by macroscopic approaches. Evidently, porphyrins, which have disk-shaped charge distributions, are inappropriately treated as spheres.

IV. Concluding Remarks This work correlates the observed structural differences between reduced and oxidized cytochrome c with the role of the protein as a catalyst for electron transfer reactions. I t is found that the reorganization energy i n the protein is much smaller t h a n the corresponding reorganization energy for electron transfer i n water. Apparently the structural rearrangements needed to reach the transition state are very small in the protein and therefore the rearrangement energy and the corresponding activation energy are small. The present analysis is based on a purely electrostatic model considering the protein as a solvent for its prosthetic group. This model is clearly valid of the reorganization energy, a,in an exchange reaction, since all other terms cancel out (see derivation of eq 17). However, there is not guarantee that the activation free energy is given by Ag’ = a / 4 in proteins (which do not necessarily behave like ideal polar solvents). This point was examined by evaluating the electrostatic contribution to the potential surface and obtaining Ag’ in a direct way. It was found that to a good approximation Ag’ is indeed given by a / 4 . This suggests the interesting possibility that the average response of the protein dipoles to changes in their local electrostatic field is similar to the average response of the dipoles of polar solvents, which change linearly with their local field. Note that when the average reorientation is given by A p = yE eq 10 is rigorously correct. The proposal that the protein helps to accelerate electron transfer by optimizing the reorganization energy is further supported by a recent study20that examines the rate of electron transfer from cytochrome c to the chlo(20)Warshel, A. In ‘Interaction between Iron and Proteins in Oxygen and Electron Transport”; Chien Ho., Ed.; Elsevier/North Holland New York, 1982;pp 111-5.

1691

rophyll dimer in bacterial photosynthesis. When the observed activation energy for this process and the calculated were used it was found that the contribution of aintra contribution of protein relaxation to a is -2 kcal/mol. This estimate is in good agreement with the present work which gave CiAUiAQi= 2.4 and a large contribution of the surroundung water, aP , 3. The contribution of the surrounding water is probably about half as large in the photosynthetic system since the donor is in water and the acceptor in the membrane. This work analyzed only one aspect of the relation between the structure of the protein and its role in electron transfer reactions. A more complete analysis should address the problems of efficiency of vectorial electron transfer. Recent considerations of these aspects indicated that the overall efficiency of electron transfer across a chain of proteins (DlA2A3--A,) is determined by the energetics of the intermediate steps.g The optimal energetics lead to a downhill transfer of the electron which involves a redox gradient < (Go)i)where the AGO for the forward reaction is negative. The redox gradient is related to the protein structure in two ways: (1)The dipoles of the proteins determine the solvation energy of the prosthetic groups that serve as electron carriers and thus determine their redox potentials. (2) The assembly of the ith protein and its partners in the chain (Ai-l and Ai) should follow the specific order that provides the optimal redox gradient. That is, the system should be assembled as Ai-lAiAi+l and not as AiAi-lAi+l if the redox potentials are > (Go)i> (Go)i+l).This specific assembly is determined by the complimentarity of the surface charges (and other groups), which lead to intermolecular “recognition”between the members of an electron transfer chain. For example, the positive surface charges of cytochrome c play a crucial role in determining its binding to various redox p a r t n e r ~ . ~ l Indeed - ~ ~ experimental results reviewed by SutinZ5indicate that the surface charges are a crucial factor in determining Ag’. Specific correlation of the structure of redox partners with their role in efficient electron transfer will greatly benefit from X-ray structures of donor-acceptor complexes. An interesting result emerges from inspection of Figure 3. The figure shows a nonuniform pattern of AU because the potential gradient across the heme is different in the reduced and oxidized conformations of the protein. This result indicates that the electric field across the heme is oriented differently in the reduced and oxidized cytochrome. It is intriguing to speculate that the electric field inside the protein guides electron flow to and from the heme, where the protein establishes a direction of “easy flow” which, at equilibrium, is opposite in the reduced and oxidized cytochrome c molecules. The capability of “guiding” electron flow might be enhanced by conformational changes when intermolecular contacts are formed between cytochrome c and its specific acceptor or donor.

-

Acknowledgment. This work was supported by Grant GM24492 from the National Institutes of Health to A.W. and by an EMBO Senior Fellowship to A.W. (21)(a) Holwerda, R.A.; Read, R. A.; Scott, R. A.; Wherland, S.; Gray, H. B.; Millett, F. J. Am. Chem. SOC. 1978,100,5028-33. (b) Holwerda, R.A.; Knaff, D. B.; Gray, H. B.; Clemer, J. D.; Crowley, R.; Smith, J. M.; Mauk, A. G. Ibid. 1980,102, 1142-5. (22)Ferguson-Miller, S.;Brautingan, D. L.; Chance, B.; Waring, A.; Margoliash, E. Biochemistry 1978,17,2246-9. (23)Koppenol, W. H.; Vroonland, C. A. J.; Braams, R. Biochim. Biophys. Acta 1978,503,499-508. (24)Butler, J.; Davies, D. M.; Sykes, A. G.; Koppenol, W. H.; Osheroff, N.; Nargoliash, E. J.Am. Chem. SOC. 1981,103, 469-71. (25)Sutin, N.Adu. Chem. Ser. 1977,No. 162, 156-72.

1692

The Journal of Physical Chemistry, Vol. 87, No. 10, 1983

Appendix A. Intramolecular Contribution to the Rate of Electron Transfer Reaction Equation 9 gives the rate of electron transfer between perfectly rigid donor and acceptor molecules. When quantum mechanical tunneling between the vibronic states of the solutes is taken into account, the rate is given by6

k = (a2/h)(T/akbn-'" m,m'

Cm,J2 exp(-&'m,,/kbq exp{-Em/kb?7 /&ib

x (AI)

where m and m' denote the set of vibrational quantum numbers of the normal modes of the solutes in the initial and final state, respectively. Ag',,, is given by

& ,'

= [AGO

+ C1 hvl(m/ - mi) + aI2/4a

(A2)

where a is the solvent reorganization of free energy, AGO is the equilibrium free energy change for the reaction, and ul is the frequency of the Ith normal mode. C,, is the Franck-Condon overlap between vibrational state m of the reactants and vibrational state m' of the products; E , is the vibrational energy of reactant vibrational state m relative to the zero-point energy ( E , = ClhvlmJ,and z,b is the vibrational partition function. The Franck-Condon factors, C,,, can be evaluated according to the procedure described in ref 26. The crucial parameters are the origin shifts, A,, which give the projections of the geometry changes upon electron transfer along the normal mode vectors of the molecule. This work deals with electron transfer between hemes and the corresponding calculated origin shifts are given in Table I of ref 27. For the electron exchange reaction between hemes a t room temperature, tunneling corrections due to modes with hul > kbT are negligible since AGO = 0 and only the ml = 0 levels of these modes are populated. However, there is an a contribution due to the solute geometry changes associated with the low-frequency modes (hul C kbT). In principle, these contributions can be evaluated numerically from eq A1 by summing overa 11 m and m'. However, since the low-frequency modes can be treated classically it is possible to obtain their contribution to a by the same approach used in evaluating the solvent contribution. The resulting inis evaluated from the tramolecular contribution, aintra, realtion V I p=

where the ul are the vibrational frequencies in cm-', r f ) are the dimensionless normal mode coordinates of state I, and Al are the dimensionless origin shifts. (The dimensionless variables related to the corresponding dimensioned variables q and A by rf = y-'I2q and A = yW1I2Awhere y = 2h/4ir2cv) aintra

= \~IIIIintra(qI=O)- VIintra(q'=O)

=

C '/zhCDlAf (A41 1

From the calculation of Warshel and W e i s P it is found = 1 kcal/mol. Thus, the intramolecular virbathat aintra tional effects are quite small in the present case. However, it is important to keep in mind a point which is often overlooked: when IAGOI is much larger than a (e.g., for a very exothermic reaction), the classical Ag* becomes very (26) Warshel, A,; Dauber, P. J. Chem. Phys. 1976,66, 5477-88. (27) Warshel, A.; Weiss, R. M. In "Frontiers of Biological Energetics"; Dutton, P. L.; Leigh, J. S.; Scarpa, A,; Ed.; Academic Press: New York, 1978, Vol. 1. pp 30-6.

Churg et

al.

large but (in contrast to the prediction based on the classical Ag*) the rate does not become very slow since an alternative quantum channel of crossing from reactant vibrational states with low quantum numbers to excited vibrational states of the products becomes acce~sible.~

Appendix B. Calculations of Electron Transfer in Aqueous Solution Our microscopic calculation of the reorganization energy in aqueous solution is based on using the FCLD model.l0J1 The FCLD model places the solvent molecules on a grid around the solute molecules and simulates the average alignment of the solvent dipoles by Langevin-type dipoles. In order to evaluate energies of excited states (e.g., the enrgy of V , in Figure 1 when VI is the ground state of the solutesolvent system) we add an induced dipole P., = yE to the permanent dipole and obtain PI

= PI + YE,

031)

where y is the electronic polarizability of a water molecule (taken here as y = 1.4 A3). The average projection of the ith permanent dipole in the direction of the local field at its site is given by

P I = PO(COth (X,) -~ / ~ , X I = CpOIEil/kbT

~

~

1

/

1

~

1

1

032)

where po is realted to the permanent dipole moment of water solvent, C reflects the resistance to orientation of the solvent dipole (due, for example, to hydrogen bonds), and E, is the local field at the ith dipole, given by

E, = D,

+ E,' + E,,

(B3)

where D, is the field from the solute charges at the ith solvent dipole (D, = ~,QJrll/rzJ3) and E, and E, are the fields due to the permanent and induced dipoles of the solvent a t the ith dipole. That is

E,I = C-W5*r1f/r,c3)

(B4)

I#,

For ground states (when the solvent permanent dipoles are at the equilibrium configuration with respect to their local field) we evaluate the local field as

E, = D,/t(r,) + E,'

(B5)

where t ( r ) is a microscopic dielectric function. By iteratively evaluating the field of eq B3 it was foundlo that t ( r ) behaves as 1 + r. The ground-state solvation free energy of charges Q' is given by

GsoI(Q1) = -CP,'.D,(Q') - '/2CP1.E,' - '/zCrlEJ*

(B6)

where the first term is the interaction between the permanent dipoles and the solute charges; the second term is the dipole-dipole interaction between permanent dipoles; the factor of in the second term arises because each dipole-dipole interaction is counted twice in this form of the expansion. The last term is the energy of the induced dipoles. This expression was used to fit the calculated and observed solvation energies of different ions by selecting the best value for boand C in the expression for the Langevin dipole; this gave po = 1.52 D and C = 0.5. The free energy of the excited state formed upon changing the solute charges from Q' to Q" while holding the solvent permanent dipoles in their ground-state orientation is given by G8,~*(Q1',~')= -CP,'.D,(Q")- '/,CP,.E,' + C ' / ~ Y ~ E , ( Q " , P ' )(B7) ~~ 1

I

Geometry Changes on Oxidation of Cytochrome c and the solvent reorganization energy a, is given by i

Oi(Q')) + 1/2Cr(lEi(Q",C1')12 - lEi(Q1,p1)12) (B8) i

It is found numerically that to a good approximation the second term which represents the effect of the induced dipoles is given by the negative half of the first term. This can be rationalized by noting that the induced dipoles correspond in the continuum dielectric model to a dielectric constant of 2, which screen the change in interaction between the permanent charge distribution (Q and p) of the system. Thus we approximate a by aw

-'/2CC1;"[Di(Q1') - Di(Q')l

(B9)

1

Appendix C. The Contribution of Surface Charges to the Reorganization Energy This Appendix concerns the evaluation of a,P of eq 17 when the protein has unbalanced surface charges. To estimate the contribution of surface charges, q,, of cytochrome c to the reorganization energy we use a two-step procedure. In the first step, the unbalanced surface charges (e.g., q8 = 1+ for lysine) are held fixed in their positions, r:, in the reactant state and the reorganization energy due to the surrounding water and surface charges is evaluated. In the second step the surface charges are allowed to relax to their positions in the product state,r;, and the correspondingchange in the reorganization energy is evaluated. Accordingly, the components of the reorganization energy due to the water surrounding the protein in the presence of the surface charges and the surface charges can be expressed as %q,

- a,(q,(r:)) + aqs(qs(rsl)) + Aaw,q,(rs+,I1)

(Cl)

where the first two terms arise in step 1 due, respectively, to the water in the presence of the surface charges and to the surface charges. The third term arises in step 2 of the calculation. It will be shown that aq8(q&r,')) is identically zero, AawJr8Lr:I) is small (