J. Phys. Chem. 1994,98, 3379-3388
3379
Electron Transfer in Proteins: A Novel Approach for the Description of Donor-Acceptor Coup1ing Spiros S. Skourtis,* Jeffrey J. Regan, and Jod Nelson Onuchic’ Department of Physics, University of California, San Diego, La Jolla, California 92093-031 9 Received: October 5, 1993; In Final Form: January 3, 1994”
We propose a nonperturbative Green’s function approach for analyzing the effects of conformation on electrontransfer matrix elements in proteins. This approach allows the analysis (at all levels of detail) of the contributions of different protein structural elements to the electronic coupling between donor and acceptor. It also describes propagation and interference effects semianalytically. We define electronic contact maps and electronic propagation maps which can be used to interpret the effect of the protein’s conformation on the total donoracceptor coupling.
Introduction Electron-transfer (ET) reactions play a significant role in biology. They constitute important steps in processes such as photosynthesis and oxidative phosphorylation. Biological ET reactions are also very interesting from a physicakhemical point of view because they involve the tunneling of an electron over a long distance (several angstroms). The electron tunnels from a localized donor state (D) to a localized acceptor state (A), both of which are embedded in protein or protein and cofactors.1s2 This medium enhances the rate of ET by providing virtual orbitals for the electron. The tunneling mechanism is often called superexchange.193-5 Biological ET reactions are usually (for possible exceptions see refs 6 and 7) nonadiabaticcll with the rate given by rate = x2*e D 2 ( ~ c )
CD
is the protein-mediated electronic matrix element between D and Aa3J2 FC describes the Franck-Condon factors between the D and A vibronic manifolds.8JOJ3-16 An important question in biological ET is how the protein-DA complex controls v“d,. The wealth of experiments on protein ET systems17J*presents the opportunity for detailed theoretical work to address this question. The most useful theory for v”d, has been the pathway model.19 It describes v”d, in terms of electronic coupling routes through the protein which connect D to A (ET pathways). These routes are sequences of covalent bonds, hydrogen bonds, and a few through-space jumps, that provide most of the electronic coupling between D and A. The pathway model has successfully predicted rate ratios for several protein ET reactions.20.2’ It also relates the structure of the protein to the DA matrix element. Its success has stimulated interest in the computation of ET matrix elements in artificial and biological systems. Currently there are several different approaches for computing ET matrix elements.22 Also, there have been several theoretical papers that use the Green’s function formalism to describe electronic tunneling.23 One of the current arguments in biological ET is whether the protein acts as an inhomogenous or a homogeneous electronic coupling medium. According to the simplest pathway picture, the protein is a highly inhomogeneous coupling medium that depends on the details of the protein’s structure. If only a few pathways (or pathway families) contribute, the ET protein can
* Authors to whom correspondence should be addressed. After 1/2/94, S. S. Skourtiswill be at the Department of Chemistry,University of Rochester, NY 14621. * Abstract published in Advance ACS Abstracts, March 1, 1994.
be thought of as an electronic circuit specially engineered to enhance the electroniccontacts between D and A, and is amenable to specific structural control (along the pathways). ET proteins would then be considered as specialized “enzymes”that “catalyze” specific ET reactions. The opposite point of view considers the protein as a highly homogeneous coupling medium.16 Due to the homogeneity, the effective matrix element is largely independent of structural details. In this situation the protein aids ET only in the sense that its electronic conductivity is better than that of free space and it is also sufficient for achieving the desired ET rate. Many globular proteins (Le. not just ET proteins) can do this job as long as the reorganization energy and energy gap between D and A are favorable. The question of whether ET proteins act as specific ET “enzymes” or as nonspecific ET media can only be resolved through the combination of experiment and detailed theory. First, it is necessary to study ET rates in many different biological and artificial systems. At the same time, one needs to understand how to measure the degree of homogeneity and inhomogeneity in the electronic contacts of a protein and, more generally, how to relate the protein’s structure to its function as an electrontunneling carrier. To this end, it is necessary to develop methods that allow one to understand in detail the ways in which different structural elements of the protein interact to influence the DA coupling. In this paper we propose a novel Green’s function approach for addressing the question of structure-function relationships for protein ET matrix elements. The approach is very general (Le. it is independent of the Hamiltonian used todescribe the protein), and it is nonperturbative. It allows one to break down the contributions to the effective matrix element of the different structural components of a protein, at any level of detail, and to describe propagation and interference effects semianalytically. It also leads to the concepts of electronic contact and electronic propagation maps (denoted by ECM and EPM, respectively). These maps describe the “electronic-coupling structure” of a protein. The ECM’s reflect the effect of the protein’s conformation on its electronic coupling Hamiltonian (they are thus analogous to the contact maps and Ramachandran plots used to describethe folded conformationof a protein). The E P M s reflect the effect of the protein’s conformation on the electronic propagation from D to A (i.e. on the DA matrix element). Such maps can be used to explore the question of structure-function relationships in ET proteins much in the same way that X-ray structures are used to interpret steric effects in enzyme function.
0022-3654/94/2098-3379%04.50/0 0 1994 American Chemical Society
3380 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
Skourtis et al.
I
“Streched”protein chain
+ Figure 1. An ET complex (top) identifying the bridge and D, A. The bridge is the tunneling medium between D and A. It may consist of protein, or protein and other molecules (e.g. waters). The bridge d and (I sites are bridge orbitals that directly couple to D and A (coupling strengths vm, v . ~ ) . The bottom drawing is a schematic representation of eq 2. In the two-state limit, the influence of the bridge on V$, (Etun)is described by G.d(Elun).
Donor-Acceptor Coupling in the Two-State Limit In the following discussion we often refer to the tunneling medium between D and A as the ”bridge”. The bridge may consist of protein, protein and water, or protein and cofactors (Figure 1). It excludes those states that are the D and A of the E T reaction. In the limits where the effective two-state picture of E T is ~alid,24*~5 the effective tunneling matrix element between D and A (in eq 1) is given by
(Figure 1). G ( E ) = l/(E-Hbridgc)is the bridge Green’s function (Hbidge is the bridge Hamiltonian), and Etunis the tunneling energy
of the transferring electron (for a precise definition of E,,, see refs 24 and 25). la) and Id) denote the bridge sites that couple to D and A (a site is an atomic or a covalent-bond orbital). In this work they are referred to as the bridge donor and acceptor. VA. and VDd denote the electronic couplings between these states and D and A. In the two-state limit, the tunneling matrix elements Gad(&) give the dependence of the DA coupling on the bridge. We concentrate on these matrix elements.
Effect of Through-Space Couplings on the Donor-Acceptor Matrix Element
The aim is to understand the effect of through-space (ThSp) electronic couplings on G,&??tun).By through-space couplings we mean all the noncovalent couplings of the bridge, such as hydrogen bonds and through-space jumps. These couplings connect different bridge sites. Some of them are between the protein chain and bridge molecules (e.g. waters). Others are between sites of the protein chain. The latter can be roughly described as short range and long range. Short-range couplings connect Drotein sites that are verv close to each other alone the chain. iong-range couplings conAect sites that are far fromiach other. Such sites interact because the folded conformation of the protein brings them close in space. Long-range couplings allow the tunneling electron to take shortcuts, thus avoiding large parts
Ca
Ca
U
P
Figure 2. Range of ThSp coupling defined with respect to the protein
chain. Short-rangecouplingsconnect sites that are close along the chain. Long-range couplings connect distant sites along the chain, and they reflect the folded conformation of the protein. A folded protein chain is shown, where A,,, and A, denote amino acid side chains, a and j3 denote protein sites connected by a ThSp coupling, and Sogdenotw the coupling’s strength. of the covalent chain. Generally, the “structure” of all ThSp couplings reflects the bridge’s conformation. For example, the long-range couplings reflect the folded conformation of the protein (Figure 2). Suppose we know the total Hamiltonian of the system (Le. including all ThSp couplings). One can compute each of the God(&,,,) in eq 2 using the total Hamiltonian. This method takes into account all electronic couplings, but it provides limited intuition about the effects of the ThSp couplings (and thus of the bridge’s conformation) on the G.d(Et,,,,). To analyze such effects, we propose an approach that computes each God(Etu,)including all electronic couplings, but it adds the ThSp couplings in steps. At each step, only couplings of similar importance to Gd(Etun) are added. To identify couplings of a given importance, the protein is asked. That is, we do not use our own intuition to decide which couplings are of a given importance. (The extent to which the protein can be trusted depends on the Hamiltonian used to describe it.) Thedefinition of importance, the method of asking the protein, and the details of this approach are described in the following sections. In this iterative approach, the starting Hamiltonian may correspond to (i) the covalently linked protein chain (backbone + amino acid chains), (ii) the covalently linked protein chain with water molecules and any other molecules that comprise the bridge, or (iii) the system of (ii) with few or with many ThSp couplings already included (e.g. a single ET pathway or a family of E T pathways). Generally, the choice of the initial system depends on the particular question to be addressed. The stepwise addition of ThSp couplings involves only those couplings that are not included in the initial Hamiltonian.
Defining the Importance of ThSp Couplings In the above approach we use the concepts of individual and collective importance of ThSp couplings. For a bridge with n ThSp couplings, the individual importance to the electronic propagation of an additional ThSp coupling is given by
Electron Transfer in Proteins: A Novel Approach
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3381
n*+l)(Etun) is the exact scattering matrix:
G2(E,,,) is the electronic propagation before the addition of the is the electronic propagation after the coupling. GF1)(EIun) addition of the coupling. Similarly, the collective importance of m additional ThSp couplings is given by
T("+l) a@
where Glmd+"')(E,,,)is the electronic propagation after all m ThSp couplings have been added. As an example, assume that the initial system is the covalent protein chain (no ThSp couplings). Then, the individual importance of a ThSp coupling is given by
=
1 -[sa@ 1-@
p+1) = 1 @a -11s-
- Sagpa x ~ $ 1
(12)
-sap,,x G
(13)
@a
3
The function @(E,,,) appearing in the denominators above is given by
+ s,,G$+ saps,, x (G;~G# - G~JG:)
=s,,G~
GL? denotes the electronic propagation along the covalent chain, i.e. GL? G$")(E,,,) = (all(E,,,-H(~))ld), whereH(mV)is the covalent Hamiltonian. G!J is the electronic propagation along the covalent chain and the ThSp coupling. G$)(EIun)= (all/ (E,,, - H(l))ld),where H c l ) = H(m) 1ThSp. Similarly, the importance of m ThSp couplings to the covalent propagation is given by
+
where G$)(EtUn)describes the electronic propagation along the covalent chain and the m ThSp couplings. G$')(Etun)= (all/ (E,,, - H(m))ld),where H(m) = Hcmv) mThSp. The importance parameters defined above can be used to categorize the ThSp couplings of a bridge. For example, couplings with 1R$)1
