Tubular Breakdown of Electron Transfer in Proteins - Advances in

8 Tubular Breakdown of Electron Transfer in Proteins Downloaded by NANYANG TECH UNIV LIB on November 6, 2014 | http://pubs.acs.org Publication Date: April 17, 1998 | doi: 10.1021/ba-1998-0254.ch008

J. J. Regan, F. K. Chang, and J. N. Onuchic Department of Physics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0319

We propose and apply a new tubular approach to compute the electron tunneling coupling in a protein. This approach goes beyond the single -pathway view to incorporate multiple-path effects and expose how inter ferencearisingfrom the structure can determine the coupling. An appli cation to recent experiments in Ru-modified azurin is presented. The experimental data are used in a novel way to determine the proper effective electron tunneling energy to use in the model. The data are inter preted in terms of interfering tubes, and hydrogen bonds play a critical role in this interference. As tubes can be blocked or created by mutation, the theory suggests how experimental control of rates can be achieved.

jyiultiple-site experiments (J) have led to a new theoretical approach for electron tunneling (ET) beyond the single-pathway picture (2-4). This ap proach emphasizes tubes—tighdy grouped families of pathways—and looks for interactions between these families rather than focusing on individual paths. In some cases, for a given donor D and acceptor A, the electron transfer can be thought of as pathway-like," wherein the protein bridge can be physically reduced to a tube without changing the overall coupling. In other cases, the transfer is characterized by multiple tubes that can interfere with one another, and a single-path assumption will fail to identify all the structural elements that control the coupling. Reducing the protein to only the relevant parts (tubes) that mediate the tunneling matrix element is a useful tool for understanding E T in a biological medium. 4

The Electron Transfer Model The E T model used here arises from the standard non adiabatic expression for the rate constant for electron transfer k : ET

©1998 American Chemical Society

In Photochemistry and Radiation Chemistry; Wishart, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1998.

111

112

PHOTOCHEMISTRY AND RADIATION CHEMISTRY

= f

kT E

|r

D A

|

F.C.

2

(1)

where h is Planck's constant divided by 2ir, F.C. is the Franck-Condon density of nuclear states, and T is the tunneling matrix element, given by D A

Downloaded by NANYANG TECH UNIV LIB on November 6, 2014 | http://pubs.acs.org Publication Date: April 17, 1998 | doi: 10.1021/ba-1998-0254.ch008

^DA^tim) = Σ 3odGda (Etun) βαΑ d,a

G

= V^-fi ^tun -

(2)

( 3 ) n

where Ε tun is the energy of the tunneling electron, d and a are the bridge orbitals coupled to D & A, respectively, β'β are the couplings and G a the Green's function propagator between d and a. In the couplings discussed here, differences in T are expected to be much larger than differences in the F.C. term because the rates reported here are k with an optimized F.C. factor. Also, from experiment to experiment the local environment of D and A stays the same while the intervening medium changes. One therefore expects the relative rates to be determined by the tunneling matrix element T alone. All electronic properties of the protein are contained in f i , a single-electron tight-binding (5) Hamiltonian, representing the protein. A "state" in this system is an electron residing in a particular tight-binding site, and the only sites used are (1) the σ-bonding orbitals and lone-pair orbitals in the protein matrix, and (2) an orbital centered on both the C u and the Ru. These orbitals are simply localization sites and are not treated in any further detail; Η is just a large extended-Hiickel-like matrix, with a dimension equal to the number of orbitals recognized in the protein. An off-diagonal element in Η is the coupling between two states and is directly related to the probability that an electron will move between the two sites involved. Two of the sites in the protein are special in that they are associated with the D and A states, whereas the remaining states in the protein are collectively referred to as the bridge (H is partitioned into a D A subsystem and a bridge subsystem). The sum in equation 2 is over the bridge entrance and exit states, indexed by d and a, respectively. These are the states with direct coupling to D and/or A . When the energies of D and A are close to each other relative to their distance in energy from the closest bridge state, and when coupling to the bridge is small relative to this distance, then (6) the D A subsystem can be viewed as an effective two-state Hamiltonian with a coupling determined by virtual occupation of the bridge. The so-called tunneling energy, E n , is an energy parameter indicative of the energies of the D and A states. All bridge states have energy zero on the energy scale used here. The direct coupling between two orbitals which share the same atom (a covalent link) is taken as a constant 7 (a pathway-like Hamiltonian). In this model, a "hydrogen bond" (H bond) is an interaction between a a

D A

max

D A

tU



8.

REGAN ET AL.

Tubular Breakdown of ET in Proteins

113

σ-bonding orbital (between a heavy atom and a hydrogen) and a lone-pair orbital on another heavy atom. If one arranges H so that the diagonal is ordered like the amino acid sequence, then Η-bonds (and through-space jumps) are far-off diagonal elements of H. Previous experience has shown that H-bonds are vital for mediating E T in proteins, and for strong Η-bonds, such as those involved in protein secondary structure, our current results indicate that their contribution to E T is comparable to that provided by covalent links. H-bond couplings are treated as distance-independent covalent links (providing a direct coupling of 7). Recent experimental results (7,8) support this hypothesis. The Hamiltonian used here has only covalent links and Η-bonds; no through-space jumps are important. The upshot of this is that the theory models the protein bridge with pre cisely one parameter: the ratio y/E . A l l the covalent bonds and Η-bonds in the bridge are treated as equivalent (i.e., they are all 7), meaning that the Hamiltonian matrix H used with E to compute G = l/(£ n - H) is just 7 times a sparse matrix of Fs. The ratio y/E appears in expansions of G matrix elements. This is a highly simplified picture of the protein. Despite its simplic ity, the H used here exhibits the primary features required for this problem. There is a rough exponential decay of coupling with distance (down a tube), the coupling sign changes (9) with each step from orbital to orbital (because 7 / E < 0), quantum interference effects (interfering tubes) arise that have a direct connection to the secondary and tertiary structure of the protein, and most important, the computed ratios of T are within an order of magnitude of experimental rate ratios (see the section " A n Example in Azurin"). More complicated Hamiltonians could and should be used, but they will only be understood in terms of the basic features already present in this Hamiltonian. tun

t u n

tU

tun

tun

D A

Gda as a Sum of Pathways. A pathway is a specific sequence of bridge orbitals, starting at a site d (which is directly coupled to the donor D), and ending at a site a (directly coupled to A)—for example, the - N - C - C - N - C - C - bond sequence of a protein backbone could be a seg ment of a pathway. The determination of the relevance of a particular pathway is discussed below in the section "The Pathway Approximation"; here paths are simply defined and discussed in general terms. Consider a bridge with only two states, as in Figure 1, where the donor is coupled to only one of the bridge a

a

Figure 1. A simple two-state bridge, with its effects in G ^ expanded as a sum of pathways.


114


states and the acceptor is coupled to the other. Because of this, the sum in equation 2 has only one term, Τ (Ε^ ) = $ G (Ft n) β*Α· Recognizing the bridge as a two-state subsystem, for which the desired Green's function matrix element has a simple form, we have ΌΑ

G

da(£)

(E _ (


X 1

E l ) (

+

η

- - y i ~ (E-

E - E) 2

(Ε - EiKE -E ) 2

da

Dd

+

EJTE

[(£ - Ei){E

U

-

E) 2

- E )f 2

+

")

( 4 )

The Gda matrix element can be written as a sum of terms, each of which corre sponds to a specific sequence of states (a pathway). The terms in this sum are depicted in Figure 1. In this simple case of a two-state bridge, each term multiplies in another factor of y , representing a trip from a back to d and back to a again. More complicated bridges are harder to expand this way, but the same idea applies. This "sum of pathways" view shows the potential role of interference in this model. There will be interference effects buried in the calculation of an individual G } (as in this example), and there will be interfer ence effects in the T sum, where different G matrix elements between different bridge exit and entrance points are added. Although these two catego ries are useful, it is more useful to define two categories of interference effects that are closer to each other than these two extremes. 2

t

D A

a

da

The Pathway Tube. The first category of interference is called trivial interference, and this is the interference that arises from nearest-neighbor, next-nearest-neighbor, and backscatter effects in propagation down a simple structure, like an ideal linear alkane chain or a protein backbone, centered on a core pathway. The coupling provided by a protein backbone is a much stronger function of backbone length than it is of the types of residues encountered along the backbone. The amide hydrogens, the lone-pairs on the oxygens, even the residues themselves, all provide trivially different alternative pathways that interfere in a way that can easily be renormalized (10) into a much simpler set of states with the connectivity of a string of pearls. A n example of this is shown in Figure 2, which is discussed in the section "The Pathway Approximation." A pathway tube is the set of states one finds by first identifying a core pathway (between some d and a) that never visits the same state twice, then adding to this set all nearest neighbors of the core states, then again adding the neighbors of these extra states. This catches all hanging orbitals off the core pathway, and this subset of the bridge is called a pathway tube. To find multiple tubes, a generalization of this method is used that differentiates between differ ent tubes. Such tubes are shown in Figure 6. A tube is a centrally useful concept from an experimental point of view in that it can sometimes be blocked, or created, via molecular replacement.


8.

REGAN ET AL.


•

C)

·

γ

γ

Gb α

115


θ 0

α

·

γ

θ

γ

θ

α

α

^

θ

α

γ

Θ α

0

Figure 2. This figure shows how alkane would be converted to a Hamiltonian used in this model (a —>b), and then how this periodic Hamiltonian could be further reduced (b —* c)to result in the "string of pearls" Η below it. The new energies and couplings in this chain have the effects of the side groups renormalized into them.

The Sum of Tubes. The second category of interference is that of interfering tubes. Once tubes have been defined, one can ask how they inter fere. Whereas the "trivial" interference of the previous section is buried in the calculation of G for a single tube, this kind of interference can be thought of as sitting at the coarse level of the T sum, or just below it. Sometimes there will be only one tube between a given bridge entrance (d) and exit (a) pair, in which case each G represents a single tube, and the tube interference is explicitly the T sum. Sometimes, however, there will be multiple tubes per G matrix element, when more than one tube share the same D - A pair, and this is the level "just below" the T sum. d a

D A

d a

D A

d a

D A

The Pathway Approximation.

The pathways approach (11) pro

vides a way to find the virtual route through the protein matrix that contributes the most to the electronic coupling, and easily estimates the coupling provided by this route by turning the Green's function calculation into a simple scalar product along the path rather than an inversion of H. If a single tube dominates, then this product of decay factors is a good estimate of the coupling, as described below.



116


Renormalization. Any bridge can be ' renormalized" using matrix parti tioning (12). The idea is to simply divide up the bridge into the states one wants to keep (one will always want to keep the states at the bridge exit and entrance points, for example), and the states one wants to eliminate. These latter states are removed from the bridge, but their effect on the calculation of any matrix elements between the remaining states is perfectly accounted for by "renormalizing" the energies of these remaining states and the coupling between them. In general, one might want to eliminate states to clarify struc tural elements in the protein (e.g., reduce a residue to a single state), or to simplify later calculations, as discussed later. For example, consider the alkane molecule of Figure 2a. Simple systems like these are frequently used in the E T literature (13-15). The σ-bonding orbitals between the C - C and C - H pairs have been identified with energies a and ah respectively. Three classes of coupling have also been identified: 7 , between C - C and C - C bonds; 7h, between C - H and C - H bonds; and 7 , between C - C and C - H bonds. Consider the four-orbital subsystem outlined in Figure 2a, wherein the two C - H bonding orbitals physically located between the two C - C bonding orbitals are coupled to each other and to the C - C orbitals, but are otherwise isolated. From the point of view of the rest of the system, only the C - C bonding orbitals are visible, and one can define new energies and couplings for these orbitals which will include the effects of the C - H bonds implicitly. This permits one to reduce the system as shown in Figure 2b-2c. The original Hamiltonian, c

C

X

H =

7χ

7χ

7c

Ύχ

«h

7h

7χ

Ύχ

7h

as a function of E for the 122, 124, and 126 couplings, for two different choices of weights. In both cases, as E gets further from the bridge energy, T drops off as we would expect from G = 1/(E - H). In this single-electron model (no shielding), G's dependence on E may be too strong, but the general trend should be that as expected for hole tunneling. The highest bridge eigenvalue is at - 2 7 (7 < 0); if we attempt D A

D A

1

Α

t u n

t u n

D A

t i m


8.

REGAN ET AL.


125

to use an E at this value or lower, we violate the two-state model. This is not a problem because, as described above, E is chosen to provide the appropriate distance decay down a covalent chain for a given choice of D and A, and therefore it always falls outside the "bridge band" (the range of eigenvalues of the bridge). These weights can be given reasonable values. In detailed studies of blue copper proteins, Solomon (32, 34, 35) analyzed the electronic structure of the metal and its ligands in an effort to better understand ligand-to-metal charge transfer interactions that, among other things, give these proteins their striking blue color. We only need a rough estimate of these couplings, as our model is only meant to be accurate to an order of magnitude in the bridging matrix element. Solomon found that the strongest coupling is to the SG sulfur in 112, and the weakest is to the S D sulfur in 121. If the absolute coupling to 112 is 1, then the absolute coupling to 121 is roughly 0.1. Any interaction with the lone-pair orbitals on 4 5 : 0 is likely to be no more than the 121 coupling, so it is also set to 0.1. The coupling to the N's of 46 and 117 is, on the other hand, somewhat stronger; we will take it to be about 0.25 on this absolute scale. This ordering of CYS > HIS > M E T will be referred to in what follows as the "rational" set of weights. The 112 interaction is clearly the strongest, and it will be seen to dominate the couplings in the experiments considered here, but as a rule all bridge entrances must be taken into account. Also, although not done here, it would be possible to position an acceptor so that C u coupling to 112 would be unimportant compared to one of the other bridge entrances. t u n


t u n

One set of Tf) curves (gray) in Figure 7 was generated with all weights set to 0.1, the smallest weight in the rational set. The second set of curves (black) was generated using the rational weights. Note that the E dependence line for 126 in Figure 7 shifts up more than the line for 122 does, with this general increase in coupling weights. This is because 126 has more paths going into it from the strong 112 coupling than 122 does (Figure 6). Because these tubes are all the same length, they interfere coherently (same sign and nearly the same magnitude), so any increase in the 112 coupling is bound to help 126 more than it will help 122. These paths traverse the ladder of Η-bonds between the two parallel β-strands seen in Figure 6. The same argument applies to 124, although the effect is not as strong because it has fewer additional paths than 126 does. Figure 8 shows how experimental data can be used to calibrate E . In each of the E T reactions considered here, D and A (and the F.C. factors) are assumed to be the same, so we expect that E will also be the same for all reactions. But what single value of E should be used with our highly renormalized and simplified bridge Hamiltonian? The experimental data determine it: the proper E to use is where the theoretical curves intersect the straight experimental lines in this figure: E — 2 . 0 5 7 . Recall that E is an effective energy parameter; this value for £ makes sense when used with the simple Hamiltonian employed in this model and should not be converted to a real energy (see the discussion at beginning of this section). A

t u n

t i m

t u n

t u n

t u n

t u n

t u n

t u n



126


Figure 8. Ratios of Tf> (functions ofE ) compared with ratios of expérimental hnax (straight lines). Theoretical curves cross experimental coupling-ratio lines at roughly the same place, suggesting — 2.05 γ as an appropriate value for E . Two different sets of donor couplings were used (see Figure 7). Three experimental ratios are shown even though only two ratios are uniquely determined. A

tun

tun

The tube results suggest that in the cases of 122,124, and 126, the principal coupling is provided by the tubes from 112 and, to a much lesser extent, 121 (because of weaker copper coupling), and that in any event, these tubes inter fere coherently. Because of the dominance of the 112 tubes, ratios of T ' s (for acceptors on the same β-strand) will not depend on the choice of weights so strongly that shifting the weights slightly will select a substantially different E . For the case with all weights the same, the dominant tube leaves the C u at 121 to feed directly into 122, and so on. For the rational set of weights, the main coupling is through 112. Because the paths down the 112 strand are just a constant three steps longer than those down the 121 strand, the ratios in one strand are similar to those in the other, so both sets of theoretical curves in Figure 8 (for each set of weights) cross experimental lines at roughly the same energy. As Figure 7 shows, the rational weights shift 126 up more than 124, which DA

t u n


8.

REGAN ET AL.

127


Table I. G (-2.057) (in units of -7-1), a = X:HIS:NE2-RU, for the full protein and the subset of the protein relevant to the β-strand experiments da

d

X = J22

112:CYS:SG-CU 121:MET:SD-CU 117:HIS:ND1-CU 46:HIS:ND1-CU 45:GLY:0-L02

-6.1e-05 -3.1e-04 -3.0e-06 l.le-06 2.1e-06

X = 124

X = 126

-5.3e-06 -1.4e-05 -1.5e-07 7.9e-08 1.5e-07

-3.1e-07 -6.6e-07 -7.0e-09 4.4e-09 7.8e-09

-5.5e-06 -1.4e-05 -1.6e-07 7.2e-08 1.4e-07

-3.3e-07 -7.0e-07 -7.4e-09 3.5e-09 6.8e-09


Full Protein

Subset of Protein 112:CYS:SG-CU 121:MET:SD-CU 117:HIS:ND1-CU 46:HIS:ND1-CU 45:GLY:0-L02

-6.3e-05 -3.2e-04 -3.0e-06 l.le-06 2.2e-06

NOTE: The agreement between these two sets of results shows that the rest of the protein can be neglected in the coupling calculations.

itself shifts up more than 122, and this causes both the ratios 124/126 and 122/124 to drop (Figure 8). Unfortunately, the 122/124 ratio drops more than the 124/126 ratio because 124 is helped more relative to 122 than 126 is relative to 124. Thus, the theoretical 124/126 ratio stays higher than the theoretical 122/124 ratio, in contrast to the experimental results. This minor discrepancy is due to the the rough treatment of Η-bonds in this model—they are all treated in exactly the same way. The actual N - O bond length of the 112-121 H-bond feeding 122 is 0.2-0.3 A longer than the ones feeding 124 and 126, and the coupling to 122 should not improve as much as it does when the 112 weight is increased. If the lengths of these Η-bonds were incorporated into the model, it would resolve this problem (36). However, a detailed description of the tuning of Η-bonds is not the aim of this paper; rather, we seek a qualitative bridge model with only one adjustable parameter (y/E ). The full Gda matrix elements taken between bridge entrance and exit points for the full protein bridge are provided for E = -2.057 i upper half of Table I. From this table, and the expression for T in equation 2, we can immediately see which bridge entrances are important in which reactions, and what effects the β weights will have, because the weights multiply these num bers directly in the T sum. The largest bridge couplings to the β-strand acceptors (122, 124, 126) are of course via 112 and 121 (though the better weight at the 112:SG-Cu coupling will make the 112 tubes far more important). The numbers in the lower half of Table I represent the G a elements taken over a subset of the protein—just the part seen in Figure 6, plus a little more, totaling about 30% of the protein. These numbers have the same order of tun

nt

t u n

n

e

D A

D A

a


128


magnitude (and most important they have the same sign) as the numbers for the full protein, indicating that the subset of the protein used provides essentially all the coupling; the rest of the protein can be dismissed.

The Tubular Breakdown A straightforward goal of theories describing E T in proteins is to predict the value of the tunneling matrix element T . This discussion goes a step further, to define what it is about the structure of the protein that determines this value. This approach converts a protein, as represented by its atomic coordinates, to a very simple Hamiltonian that has just enough information to retain all essential features of the electron tunneling problem. The coupling derived from this model can be broken down into contributions from individual tubes, each of which is a a family of similar pathways through specific sequences of covalent or Η-bonds. The tubes encapsulate trivial interference effects and can expose crucial intertube interferences. The set of all tubes that are important to the coupling can be identified—and the rest of the protein can be dismissed. A purely quantum mechanical effect like tube interference is directly related to the secondary and tertiary structure of the protein, and Η-bonds play a central role in this effect, as they are the primary factor distinguishing a protein from what would otherwise be an effective (and uninteresting) one-dimensional chain. As proposed in earlier work, the treatment of H bonds on equal footing with covalent bonds leads to theoretical predictions that are consistent with rate measurements, indicating the critical role that Η-bonds can play in E T . This chapter discusses the particular case of β-strand tubes. However, tube analysis is generally applicable to other motifs in different proteins. Different patterns of tube interferences will exist for different motifs, but the framework for analysis will remain the same. The step from a tube analysis of a protein to experimental design is obvious, as one anticipates that there will be situations where a tube can be created or blocked by an appropriate mutation.


D A

Acknowledgments We thank David Beratan and Spiros Skourtis for helpful comments about this work. This research was supported by the National Science Foundation (NSF) (Grant MCB-93-16186) and the National Institutes of Health (GM48043). J. J. Regan was also supported by the Berkeley Program in Mathematics in Biology (NSF Grant DMS-94-06348).

References 1. Langen, R.; Chang, I. J.; Germanas, J. R.; Richards, J. H.; Winkler, J. R.; Gray, H . B. Science (Washington, D. C.) 1995, 268, 1733-1735.


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2. Beratan, D. N.; Onuchic, J. N . Hopfield, J. J. J. Chem. Phys. 1987, 86, 4488-4498. 3. Beratan, D. N . Betts, J. N . Onuchic, J. N. Science (Washington, D. C.) 1991, 252, 1285. 4. Onuchic, J. N.; Beratan, D. N.; Winkler, J. R.; Gray, H . B. Annu. Rev. Biophys. Biomol. Struct. 1992, 21, 349-377. 5. Ashcroft, N . W.; Mermin, N. D. Solid State Physics; W.B. Saunders: New York, 1976. 6. Skourtis, S. S.; Onuchic, J. N. Chem. Phys. Lett. 1993, 209, 171-177. 7. Turro, N. J.; Murphy, C. J.; Arkin, Μ. Α.; Jenkins, Y.; Ghatlia, N. D.; Bossmann, S. H . ; Barton, J. K. Science (Washington, D. C.) 1993, 262, 1025-1029. 8. de Rege, P. J. F. Williams, S. Α.; Therien, M. J. Science (Washington, D. C.) 1995, 269, 1409, 1413. 9. Beratan, D. N. J. Am. Chem. Soc. 1986, 108, 4321-4326. 10. Onuchic, J. N . Beratan, D. J. Chem. Phys. 1990, 92, 722-733. 11. Beratan, D.N.;Onuchic, J. N . Photosyn. Res. 1989, 22, 173. 12. Löwdin, P. O. J. Math. Phys. 1962, 3, 969. 13. Liang, C. Newton, M . D. J. Phys. Chem. 1992, 96, 2855-2866. 14. Curtiss, L. Α.; Naleway, C. Α.; Miller, J. R. J. Phys. Chem. 1993, 97, 4050-4058. 15. Paddon-Row, M . N. J. Am. Chem. Soc. 1994, 116, 5328-5333. 16. Gama, A. A. S. da J. Theor. Biol 1990, 142, 251-260. 17. Onuchic, J. N . Andrade, P. C. P.; Beratan, D. N . J. Chem. Phys. 1991, 95, 1131. 18. Adman, E. T.; Jensen, L. H . Isr. J. Chem. 1981, 21, 8-120. 19. Gray, H . B. Chem. Soc. Rev. 1986, 15, 17-30. 20. Sykes, A. G. Advances Inorg. Chem. 1991, 36, 377-408. 21. Nar, H . ; Messerschmidt, Α.; Huber, R.; Van de Kamp, M.; Canters, G. W. J. Mol. Biol. 1991, 218, 427-447. 22. Van de Kamp, M.; Floris, R.; Hall, F. C.; Canters, G. W. J. Am. Chem. Soc. 1990, 112, 907-908. 23. Van de Kamp, M.; Silvestrini, M . C.; Brunori, M.; Beeumen, J. van; Hall, F. C.; Canters, G. W. Eur. J. Biochem. 1990, 194, 109-118. 24. Mikkelsen, Κ. V.; Skov, L. K.; Nar, H.; Farver, O. Proc. Natl. Acad. Sci. U.S.A. 1993, 90, 5443-5445. 25. Farver, O.; Pecht, I. J. Am. Chem. Soc. 1992, 114, 5764-5767. 26. Farver, O.; Skov, L. K. Nar, H.; Van de Kamp, M . Canters, G. W. Pecht, I. Eur. J. Biochem. 1992, 210, 399-403. 27. Farver, O.; Skov, L. K.; Pascher, T.; Karlsson, B. G.; Nordling, M.; Lundberg, L. G.; Vänngard, T.; Pecht, I. Biochemistry 1993, 32, 7317-7322. 28. Broo, Α.; Larsson, S. J. Phys. Chem. 1991, 95, 4925-4928. 29. Regan, J. J. Bilio, A. J. Di; Langen, R. Skov, L. K. Winkler, J. R. Gray, H . B. Onuchic, J. N . Chem. Biol. 1995, 2, 489-496. 30. Nar,H.;Messerschmidt, A. Huber, R. Van de Kamp, M.; Canters, G. W. J. Mol. Biol. 1991, 221, 765-772. 31. Regan, J. J. Risser, S. M . Beratan, D. N . Onuchic, J. N. J. Phys. Chem. 1993, 97, 13083-13088. 32. Solomon, E. I.; Lowery, M . D. Science (Washington, D. C.) 1993, 259, 1575-1581. 33. From the coordinates of chain "A," one of four azurin molecules in the unit cell of the Brookhaven Protein Data Bank entry "4AZU" (21, 30). Molecular replace mentand energy minimization were performed with Enzymix (36). 34. Lowery, M . D . Solomon, Ε. I. Inorg. Chim. Acta 1992, 200, 233-243. 35. Solomon, Ε. I.; Baldwin, M . J. Lowery, M . D. Chem. Rev. 1992, 92, 521-542. 36. Lee, F. S.; Chu, Ζ. T.; Warshel, Α. J. Comput. Chem. 1993, 14, 161-185. ;

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Tubular Breakdown of Electron Transfer in Proteins - Advances in

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