Fluctuations in Biological and Bioinspired Electron-Transfer Reactions

Abstract

Central to theories of electron transfer (ET) is the idea that nuclear motion generates a transition state that enables electron flow to proceed, but nuclear motion also induces fluctuations in the donor-acceptor (DA) electronic coupling that is the rate-limiting parameter for nonadiabatic ET. The interplay between the DA energy gap and DA coupling fluctuations is particularly noteworthy in biological ET, where flexible protein and mobile water bridges take center stage. Here, we discuss the critical timescales at play for ET reactions in fluctuating media, highlighting issues of the Condon approximation, average medium versus fluctuation-controlled electron tunneling, gated and solvent relaxation controlled electron transfer, and the influence of inelastic tunneling on electronic coupling pathway interferences. Taken together, one may use this framework to establish principles to describe how macromolecular structure and structural fluctuations influence ET reactions. This framework deepens our understanding of ET chemistry in fluctuating media. Moreover, it provides a unifying perspective for biophysical charge-transfer processes and helps to frame new questions associated with energy harvesting and transduction in fluctuating media.

1. INTRODUCTION

Is the reactivity of biomolecules dictated by their average molecular structure, by routine thermal excursions from the average, or by rare large-scale conformational changes? Addressing these questions is central to establishing a description of bioenergetics at the molecular scale. Here, we address these and closely related issues in the framework of biological and bioinspired electron transfer (ET). We review how a new generation of theory and experiments is unveiling the role played by protein and solution structure, fluctuations, and relaxation in ET kinetics and mechanisms.

Early formulations of nonadiabatic ET reactions were based on Fermi’s golden rule and emphasized the probability of encountering configurations with quasi-degenerate donor and acceptor states (through the Franck-Condon factor), and the propensity for electron tunneling in these configurations (through the tunneling matrix element). Recent studies have focused on how structural fluctuations [those generating donor-acceptor (DA) degeneracy and others] may influence the transfer rate, revealing a richness of mechanistic behavior.

The familiar Fermi’s golden-rule ET rate constant is $k_{\mathit{\text{ET}}}=\frac{2\pi }{\hslash }{T}_{\mathit{\text{DA}}}^2(F.C.)$, where T_DA is the DA electronic coupling interaction and (F.C.) is the Franck-Condon factor (1–7). The description of ET in terms of a single rate constant suggests that the ET kinetics is single exponential and it is appropriate if 1/k_ET is slow compared with timescales of fluctuations that may modulate the reaction rate. Such fluctuations could include conformational interconversions and solvent relaxation. These molecular fluctuations may affect both the DA energy gap and the DA tunneling matrix element, and the fluctuations may lead to regimes that require modification of the golden-rule rate constant, even in the limit of single exponential kinetics.

2. COUPLING FLUCTUATION EFFECTS IN ELECTRON TRANSFER

In the following, we discuss generalizations of the nonadiabatic ET rate expression when fluctuations of the molecular structure and of the solvent modulate T_DA with a fluctuation timescale that is fast compared with 1/k_ET. When the coupling fluctuations are slow on the timescale of nuclear motion through the crossing region of the donor and acceptor potential surfaces (validating the Condon approximation), $T_{\mathit{\text{DA}}}^2$ in the nonadiabatic rate expression should be replaced by its thermal average $\langle T_{\mathit{\text{DA}}}^2\rangle$. If the coupling fluctuations are fast on the timescale of nuclear motion through the crossing region, the tunneling electron may exchange energy with the nuclear degrees of freedom of the bridge, enabling inelastic tunneling. This effect has a well-understood influence on the Franck-Condon factor and can also change T_DA significantly, as inelastic markers left on tunneling pathways change the overall amplitude for tunneling (vide infra).

We now present a framework for exploring how bridge fluctuations influence ET kinetics. We use a time-dependent version of Fermi’s golden rule with a semiclassical approximation to explore the relevant timescales, both in general and for some specific systems. It is helpful to express the nonadiabatic ET rate constant as the Fourier transform of time-dependent correlation functions (see 8, 9). We consider an ET system described by the Hamiltonian Ĥ = Ĥ^D + Ĥ^A + Ĥ^B + V̂, where Ĥ^D and Ĥ^A are electronic-vibrational (vibronic) Hamiltonians for the donor (D) and acceptor (A) diabatic potential surfaces (Figure 1), Ĥ^B is the vibronic bridge Hamiltonian, and V̂ couples donor (acceptor) and bridge electronic states. In particular, ${\hat{H}}^D=U_D^{\text{min}}|D\rangle \langle D|+{\hat{H}}^{D(\mathit{\text{vi}})}\text{and}{\hat{H}}^A=U_A^{\text{min}}|A\rangle \langle A|+{\hat{H}}^{A(\mathit{\text{vi}})}$, where Ĥ^D(vi) and Ĥ^A(vi) are the vibrational Hamiltonians for the U_D and U_A harmonic diabatic surfaces (Figure 1), and Ĥ^B = Ĥ^B(el) + Ĥ^B(vi) + Ĥ^B(e−v), where Ĥ^B(el) and Ĥ^B(vi) are the electronic and vibrational bridge Hamiltonians, respectively. Ĥ^B(e−v) describes the bridge electronic-vibrational interactions. In this model, the nonadiabatic ET rate constant is

$$k_{\mathit{\text{ET}}}({\omega }_{\mathit{\text{DA}}})=\frac{1}{{\hslash }^2}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\mathit{\text{dte}}}^{i{\omega }_{\mathit{\text{DA}}}t}{C}_{T_{\mathit{\text{DA}}}}(t)C_{\mathit{\text{FC}}}(t),}$$ (1)

where ${\omega }_{\mathit{\text{DA}}}=(U_D^{\text{min}}-U_A^{\text{min}})/\hslash$ is the frequency corresponding to the energy gap (Figure 1), C_FC(t) is the time-dependent Franck-Condon factor,

$$C_{\mathit{\text{FC}}}(t)={\langle e^{i{\hat{H}}^{D(\mathit{\text{vi}})}t/\hslash }{e}^{-i{\hat{H}}^{A(\mathit{\text{vi}})}t/\hslash }\rangle }_D,$$ (2)

and

$$C_{T_{\mathit{\text{DA}}}}(t)={\langle {\hat{T}}_{\mathit{\text{DA}}}(t){\hat{T}}_{\mathit{\text{DA}}}(0)\rangle }_B={\langle e^{i{\hat{H}}^{B(\mathit{\text{vi}})}t/\hslash }{\hat{T}}_{\mathit{\text{DA}}}(0)e^{-i{\hat{H}}^{B(\mathit{\text{vi}})}t/\hslash }{\hat{T}}_{\mathit{\text{DA}}}(0)\rangle }_B$$ (3)

is the correlation function of the vibronic DA electronic coupling (tunneling matrix element). 〈…〉_D and 〈…〉_B denote thermal averages over the Ĥ^D(vi) and Ĥ^B(vi) vibrational eigenstates, respectively. The vibronic tunneling matrix element to second order in V̂ is

$${\hat{T}}_{\mathit{\text{DA}}}(0)=\hat{V}{(E_{\mathit{\text{tun}}}-{\hat{H}}^B)}^{-1}\hat{V},$$ (4)

where E_tun is the total electronic-vibrational energy of the system when the electron is in the initial donor electronic state (10–12).

Figure 1.

Donor and acceptor diabatic potential energy curves as a function of the reaction coordinate Q, which is assumed to be a single normal mode ({r⃑_i} denote atom coordinates). The diagram shows the crossing region, activation energy, energy gap $U_D^{\text{min}}-U_A^{\text{min}}$, and reorganization energy λ = k(Q_D − Q_A)²/2, associated with the donor and acceptor potentials (k is the curvature of the surfaces).

In the high-temperature (classical) limit for the donor and acceptor vibrational degrees of freedom,

$$C_{\mathit{\text{FC}}}(t)\approx e^{-i\lambda t/\hslash }{e}^{-t^2/2{\tau }_{\mathit{\text{FC}}}^2},$$ (5)

where λ is the reorganization energy (Figure 1), and ${\tau }_{\mathit{\text{FC}}}=\hslash /\sqrt{2\lambda k_{B}T}$ is often called the Franck-Condon time (5, 7). The Franck-Condon time can be interpreted as the characteristic time for the system to move out of the DA diabatic surface crossing region (Figure 1) due to thermal fluctuations of the DA energy gap, i.e.,

$${\tau }_{\mathit{\text{FC}}}=\hslash /{\sigma }_{\mathrm{\Delta }U,}\text{where}{\sigma }_{\mathrm{\Delta }U}^2=\langle {({\hat{H}}^{A(\mathit{\text{vi}})}-{\hat{H}}^{D(\mathit{\text{vi}})})}^2\rangle -{\langle ({\hat{H}}^{A(\mathit{\text{vi}})}-{\hat{H}}^{D(\mathit{\text{vi}})})\rangle }^2=2\lambda k_{B}T.$$

In the classical limit for the bridge vibrational degrees of freedom, the electronic coupling depends parametrically on the atom trajectories, and the correlation function is

$$C_{T_{\mathit{\text{DA}}}}(t)=\langle T_{\mathit{\text{DA}}}(t)T_{\mathit{\text{DA}}}(0)\rangle ,$$ (6)

where 〈.…〉 denotes a classical thermal average. In practice, T_DA(t) is computed along classical molecular dynamics (MD) trajectories, i.e., T_DA(t) = T_DA[{r⃑₁(t), .… r⃑_N(t)}], where r⃑_i(t) is the position of atom i, and 〈‥〉 is the time (trajectory) average. A perturbative expression for T_DA(t) in a nonorthogonal donor, acceptor, and bridge (i_B, j_B) basis is

$$T_{\mathit{\text{DA}}}(t)={\displaystyle \sum _{i_B}{\displaystyle \sum _{j_B}(E_{\mathit{\text{tun}}}^{(\mathit{\text{el}})}{S}_{D,i_B}-V_{D,i_B})G_{i_B,j_B}(E_{\mathit{\text{tun}}}^{(\mathit{\text{el}})})(E_{\mathit{\text{tun}}}^{(\mathit{\text{el}})}{S}_{j_B,A}-V_{j_B,A}),}}$$ (7)

where $G_{i_B,j_B}(E_{\mathit{\text{tun}}}^{(\mathit{\text{el}})})={\{{(E_{\mathit{\text{tun}}}^{(\mathit{\text{el}})}{\tilde{S}}^{B(\mathit{\text{el}})}-{\tilde{H}}^{B(\mathit{\text{el}})})}^{-1}\}}_{i_B,j_B}$ is the matrix element of the electronic bridge Green’s function; S_ij and V_ij are donor- (acceptor-) to-bridge overlap and electronic Hamiltonian matrix elements, respectively; and S̃^B(el) and H̃^B(el) are bridge overlap and electronic Hamiltonian matrices (13). $E_{\mathit{\text{tun}}}^{(\mathit{\text{el}})}$ is the tunneling energy of the electron. All quantities on the right-hand side of Equation 7, including the tunneling energy, should be evaluated at time t (14, 15).

T_DA(t) is routinely computed using a variety of approaches, including Green’s function and energy-splitting analysis (11), generalized Mulliken-Hush analysis (16), and tunneling current methods (17). The timescale of T_DA(t) fluctuations is characterized by the coherence time (τ_coh) (8, 18, 19). τ_coh is the decay time of the C_TDA(t) correlation function; it measures how rapidly fluctuations of the electronic coupling randomize T_DA(t) values, e.g.,

$$C_{T_{\mathit{\text{DA}}}}(t)\approx {\sigma }_{T_{\mathit{\text{DA}}}}^2{e}^{-t^2/2{\tau }_{\mathit{\text{coh}}}^2+{\langle T_{\mathit{\text{DA}}}\rangle }^2,}$$ (8)

where ${\sigma }_{T_{\mathit{\text{DA}}}}=\sqrt{\langle T_{\mathit{\text{DA}}}^2\rangle -{\langle T_{\mathit{\text{DA}}}\rangle }^2}$. For small times compared with τ_coh, C_TDA(t) approaches the value $\langle T_{\mathit{\text{DA}}}^2\rangle$; for long times compared with τ_coh, C_TDA(t) approaches 〈T_DA〉². The strength of T_DA(t) fluctuations is characterized by

$$R_{\mathit{\text{coh}}}=\frac{{\langle T_{\mathit{\text{DA}}}\rangle }^2}{\langle T_{\mathit{\text{DA}}}^2\rangle }=\frac{1}{1+({\sigma }_{T_{\mathit{\text{DA}}}}^2/{\langle T_{\mathit{\text{DA}}}\rangle }^2)},$$ (9)

which is known as the coherence parameter (20). When coupling fluctuations are large compared with the average coupling $({\sigma }_{T_{\mathit{\text{DA}}}}^2\gg {\langle T_{\mathit{\text{DA}}}\rangle }^2)$, the coherence parameter is small, R_coh ≪ 1, and for small fluctuations $({\sigma }_{T_{\mathit{\text{DA}}}}^2\ll {\langle T_{\mathit{\text{DA}}}\rangle }^2)$, R_coh ≈ 1.

The parameters R_coh, τ_coh, and τ_FC are useful to describe distinct nonadiabatic ET rate regimes (see 8, 18, 19, 21). We focus next on single exponential ET kinetics. In this regime, we have τ_FC, τ_coh ≪ 1/k_ET [if any of the timescales is longer than the ET time, the kinetics may be multi-exponential or gated (22, 23)].

3. THE SLOW COUPLING–FLUCTUATION REGIME (τ_coh > τ_FC)

In this regime, T_DA(t) does not fluctuate while the donor and acceptor electronic states are in resonance. As a result, each DA energy crossing event is subject to an effectively constant coupling, equal to the value of T_DA(t) at the time of the crossing (i.e., the Franck-Condon approximation is valid). In Equation 1, we can thus replace C_TDA(t) by $C_{T_{\mathit{\text{DA}}}}(0)=\langle T_{\mathit{\text{DA}}}^2\rangle$, and the rate is proportional to the product of $\langle T_{\mathit{\text{DA}}}^2\rangle$ and the Franck-Condon weighted density of states, ${\rho }_{\mathit{\text{FC}}}={(4\pi \lambda k_{B}T)}^{-1/2}{e}^{-{([U_D^{\text{min}}-U_A^{\text{min}}]-\lambda )}^2/4\lambda k_{B}T}$ [assuming the classical high-temperature expression for C_FC(t) in Equation 5]. Here, one finds two limits: the weak T_DA fluctuation regime (R_coh ≈ 1) and the strong T_DA fluctuation regime (R_coh ≪ 1).

In the limit of weak T_DA fluctuations, bridge fluctuations do not change T_DA significantly. Hence, $\langle T_{\mathit{\text{DA}}}^2\rangle \approx {\langle T_{\mathit{\text{DA}}}\rangle }^2$, and the rate is

$$k_{\mathit{\text{ET}}}=\frac{2\pi }{\hslash }{\langle T_{\mathit{\text{DA}}}\rangle }^2{\rho }_{\mathit{\text{FC}}}.$$ (10)

This limit applies to a broad range of ET model systems with the donor and acceptor linked by rigid bridges. Paddon-Row and coworkers (24, 25), Miller and coworkers (26), Newton and coworkers (27), and others (6) have investigated ET kinetics in chemical systems with rigid organic bridges (Figure 2). Through the use of beautifully designed molecular structures and many different kinetic probes, investigators have shown that nonadiabatic ET in such systems proceeds via bridge-mediated electron or hole tunneling. Furthermore, the tunneling propensity is described by T_DA values that correspond to equilibrium bridge structures (see 4, 27, 28 for reviews of these studies). In these systems, thermal fluctuations of the bridge structure do not cause significant changes in T_DA (i.e., ${\sigma }_{T_{\mathit{\text{DA}}}}^2\ll {\langle T_{\mathit{\text{DA}}}\rangle }^2$).

Figure 2.

(a) A schematic diagram and example molecular structure for a linear line-of-sight arrangement of donor-bridge-acceptor units in a supermolecule. (b) A schematic diagram and example molecular structure for a cleft geometry in which the line of sight between the acceptor and donor is through a gap rather than through a saturated bridge unit.

In the case of near-zero T_DA values for the equilibrium bridge structure, T_DA fluctuations can induce large changes in T_DA compared with the average, i.e., ${\sigma }_{T_{\mathit{\text{DA}}}}^2\gg {\langle T_{\mathit{\text{DA}}}\rangle }^2(R_{\mathit{\text{coh}}}\ll 1)$. In this regime, bridge structural fluctuations enhance the coupling and determine the observed ET rate constant,

$$k_{\mathit{\text{ET}}}=\frac{2\pi }{\hslash }\langle T_{\mathit{\text{DA}}}^2\rangle {\rho }_{\mathit{\text{FC}}}.$$ (11)

Here, one expects that the ET rate can increase with increasing temperature because the system accesses bridge conformations with larger couplings (9, 23). Whereas the regime of large T_DA fluctuations has wide applicability in long-distance biological ET, the studies of Zimmt and Waldeck show that the fluctuation-dominated regime can also be relevant to small chemical ET systems (vide infra). Using synthetic donor-bridge-acceptor structures where a cleft (i.e., a C-clamp) forms between donor and acceptor, they (30, 31) showed that ET occurs by electron (hole) tunneling through a solvent in the cleft (Figure 2). Their studies reveal that the electronic coupling could not be described using a static perspective but must be described by a temperature-dependent dynamical perspective (32). By design, these C-clamp structures have a zero average electronic coupling because of symmetry, so configurational fluctuations (donor-solvent-acceptor) are essential for the enhancement of the DA coupling; such systems provide an important testing ground for understanding the fluctuation-controlled transfer regime (33). A recent extension of this idea to water-soluble cleft molecules (see Figure 3) promises to provide a way to investigate electron tunneling through water molecules.

Figure 3.

Molecular structures showing C-clamp molecules with a pyrene acceptor and a dimethylaniline donor. The D-SSS-A supermolecule has a well-defined cleft, whereas the D-SRR-A supermolecule does not.

Expanding C_TDA(t) to second order in time, it is possible to write the ET rate in powers of (τ_FC/τ_coh): C_TDA(t) = C_TDA(0) + Ċ_TDA(0)t + C̈_TDA(0)t²/2. With $C_{\mathit{\text{FC}}}(t)\propto e^{-i\lambda t/\hslash }{e}^{-t^2/2{\tau }_{\mathit{\text{FC}}}^2}\text{and}{C}_{T_{\mathit{\text{DA}}}}(t)\approx {\sigma }_{T_{\mathit{\text{DA}}}}^2{e}^{-t^2/2{\tau }_{\mathit{\text{coh}}}^2}+{\langle T_{\mathit{\text{DA}}}\rangle }^2$, one finds

$$k_{\mathit{\text{ET}}}=\langle T_{\mathit{\text{DA}}}^2\rangle {\rho }_{\mathit{\text{FC}}}[1+\delta k],$$ (12)

where δk is the non-Condon correction to the rate equation of the slow fluctuation regime (Equation 11), to lowest order (second order) in (τ_FC/τ_coh) (8). Here, δk depends on R_coh and $U_D^{\text{min}}-U_A^{\text{min}}$ [in addition to (τ_FC/τ_coh)²], and it can be computed using MD simulations.

Numerous studies of ET reactions have examined the fluctuation dependence of the DA coupling (14, 15, 21, 28, 33–40), with the aim of identifying bridge conformations that enhance this coupling. Three studies (18, 19, 21) characterize known ET reactions in terms of R_coh, τ_coh, and δk (using MD simulations and semiempirical electronic structure analysis) to understand how these parameters relate to bridge structure and dynamics. Comparison of the results (18, 19, 21) is instructive because Reference 18 explores short-range solvent-mediated ET in the C-clamped molecules above, Reference 19 explores long-distance ET in the protein azurin, and Reference 21 studies ET in the photosynthetic reaction BPh⁻ → Q_A in the bacterial photosynthetic reaction center. In these three cases, the computed non-Condon correction values are similar (δk ≤ 1%), which means that the ET rates for all these systems are well described using Equation 11. The coherence parameters vary from R_coh ≤ 0.1 for References 18 and 19 (strong T_DA fluctuations) to 0.4 for Reference 21 (moderate T_DA fluctuations). Surprisingly, the τ_coh values are similar for all ET reactions in these studies (τ_coh is a few tens to 100 fs). In all cases, the tunneling matrix element fluctuations are slow (τ_coh > τ_FC), and this observation may be understood using simple arguments. For ET reactions with λ ≈ 1 eV, the room-temperature Franck-Condon time is expected to be very short, ${\tau }_{\mathit{\text{FC}}}=\hslash /\sqrt{2\lambda k_{B}T}~\text{fs}$ [for azurin τ_FC ≈ 3 – 4 fs, as computed using MD simulations (41), and for the C-clamp molecules (Figure 2), a range of τ_FC values that depend on the solvent are possible and have magnitudes of a few femtoseconds (18)]. Constrained MD simulations for azurin (19) show that the fastest bridge motions that determine τ_coh are largely valence angle vibrations with periods of a few tens of femtoseconds, much longer than the typical τ_FC times discussed above. Such vibrations have similar periods for a wide variety of chemical bridges, suggesting that the slow fluctuation regime is general for ET tunneling reactions at room temperature in polar media, with ~1-eV reorganization energies, normal (unstrained) chemical bridges, and Gibbs free energy gaps that do not place the ET reaction in the highly activated regime. Deviations from the regime of slow matrix element fluctuations could arise for systems with longer Franck-Condon times (see 42, 43 for deviations from the formula ${\tau }_{\mathit{\text{FC}}}=\hslash /\sqrt{2\lambda k_{B}T}$) and are more likely for low-bridge-gap tunneling systems, where the tunneling matrix element is sensitive to bridge deformations (11, 19, 44).

3.1. The Characteristic Length r_crit

It is interesting to examine why short-distance (R_DA ~ 7 Å) solvent-mediated ET in the C-clamp molecules (18) shows equally strong tunneling matrix element fluctuations as are found in long-distance (R_DA ~ 17–24 Å) protein-mediated ET in azurin (19) (R_coh ≤ 0.1 in both cases). Recent studies (45) suggest that there is a critical (medium-dependent) distance beyond which DA coupling fluctuations dominate the electron tunneling. Indeed, Reference 45 poses a statistical question regarding coupling: Is there a distance r_crit that characterizes, on average, a transition between 〈T_DA〉² and ${\sigma }_{T_{\mathit{\text{DA}}}}^2$ dominated kinetics? An answer to this question emerges from an electronic coupling analysis for MD sampled geometries of Ru-modified heme and blue-copper proteins (46), as well as for water-mediated self-exchange between cytochrome b₅ molecules. (47) For the protein-mediated tunneling systems, the transition between the 〈T_DA〉²-dominated regime and the ${\sigma }_{T_{\mathit{\text{DA}}}}^2$-dominated regime occurs at distances of ~6–7 Å, about the size of an amino acid residue (see Figure 4a). For the water-mediated self-exchange reaction, the transition to a fluctuation-dominated coupling regime occurs at distances about the size of a water molecule, ~2–3 Å (Figure 4a). Importantly, these critical distances for protein and water tunneling media coincide with the computed mechanical (κ_ij) and electron-tunneling (χ_ij) spatial correlation functions (45):

$${\kappa }_{\mathit{\text{ij}}}=\frac{\langle ({\stackrel{\rightharpoonup }{r}}_i-\langle {\stackrel{\rightharpoonup }{r}}_i\rangle )({\stackrel{\rightharpoonup }{r}}_j-\langle {\stackrel{\rightharpoonup }{r}}_j\rangle )\rangle }{\sqrt{\langle {({\stackrel{\rightharpoonup }{r}}_i-\langle {\stackrel{\rightharpoonup }{r}}_i\rangle )}^2\rangle \langle {({\stackrel{\rightharpoonup }{r}}_j-\langle {\stackrel{\rightharpoonup }{r}}_j\rangle )}^2\rangle }},{\chi }_{\mathit{\text{ij}}}=\frac{\langle (G_{\mathit{\text{ii}}}^2-\langle G_{\mathit{\text{ii}}}^2\rangle )(G_{\mathit{\text{ij}}}^2-\langle G_{\mathit{\text{ij}}}^2\rangle )\rangle }{\sqrt{\langle {(G_{\mathit{\text{ii}}}^2-\langle G_{\mathit{\text{ii}}}^2\rangle )}^2\rangle \langle {(G_{\mathit{\text{ij}}}^2-\langle G_{\mathit{\text{ij}}}^2\rangle )}^2\rangle }}.$$ (13)

Figure 4.

(a) Dependence of ln(R_coh) on the donor-acceptor (DA) distance R_DA. The vertical lines are error bars, and the horizontal line denotes the value of R_coh where ${\langle T_{\mathit{\text{DA}}}\rangle }^2={\sigma }_{T_{\mathit{\text{DA}}}}^2,R_{\mathit{\text{coh}}}=0.5$. (b) The scatter function $S_{R_{\mathit{\text{DA}}}}(X)=\sqrt{\text{var}[X]R_{\mathit{\text{DA}}}}/\mathit{\text{avg}}[X]R_{\mathit{\text{DA}}},\text{where}X=\langle T_{\mathit{\text{DA}}}^2\rangle$ (Equation 14), or $X={\langle T_{\mathit{\text{DA}}}\rangle }^2,{\sigma }_{T_{\mathit{\text{DA}}}}^2$, is plotted for different R_DA distances. An S_RDA of the scale of unity indicates that the specific protein structure largely determines the observable rate, whereas S_RDA ≪ 1 means that an effective tunneling barrier could be used to describe the tunneling medium (45).

Here, r⃑_i represent bridging atom positions, and G_ij is the bridge electronic Green’s function between two-center bond orbitals i and j (Equation 7). κ_ij describes the loss of correlation among atomic motions that arises from thermal fluctuations of the bridge structure, and χ_ij describes the loss of correlation among electron-tunneling propagations through the bridge that involves different tunneling distances. The loss of tunneling correlation is also a function of the bridge structural fluctuations. For both water and protein media, the κ_ij and χ_ij decay (correlation) lengths coincide with r_crit for which, on average, ${\langle T_{\mathit{\text{DA}}}\rangle }^2={\sigma }_{T_{\mathit{\text{DA}}}}^2$. Thus, the onset of a fluctuation-dominated regime is related to the loss of mechanical and tunneling-propagation correlation in the bridge. For water-mediated tunneling, mechanical and tunneling correlation is lost, on average, for distances greater than the size of a water molecule, whereas for proteins, this distance is larger on average (45). The water results suggest that, for solvent-mediated ET in general, the fluctuation-dominated regime will occur at distances greater than the size of the solvent molecules. This analysis explains the results that find ET in C-clamp molecules (R_DA ~ 7 Å) and in azurin (R_DA ~ 17–24 Å) to be fluctuation dominated (18, 19). Even though the tunneling distances are different in the two systems, they are each larger than the critical distance for the respective tunneling media. It would be interesting to repeat the above analysis for the BPh⁻ → Q_A ET reaction, where R_coh = 0.4 (21).

In Gray & Winkler’s (46) tunneling-limited Ru-modified protein ET experiments, all derivatives with anomalously slow rates for their DA distance (in the heme proteins myoglobin, cytochrome c, and cytochrome b₅₆₂) are in the weak fluctuation regime $({\langle T_{\mathit{\text{DA}}}\rangle }^2<{\sigma }_{T_{\mathit{\text{DA}}}}^2)$ and have only a few coupling pathways, which are axial to the heme donor. Protein derivatives with fast and average rates (for their distance) fall in the strong fluctuation regime $({\langle T_{\mathit{\text{DA}}}\rangle }^2>{\sigma }_{T_{\mathit{\text{DA}}}}^2)$ and have tunneling matrix elements that are determined by interactions among many alternative pathway structures, rather than by a few dominant paths (48). The above observations suggest a useful hierarchy for categorizing protein structural effects on ET. At a first coarse level, rates drop rapidly as a function of distance because of the underlying tunneling mechanism. At a second level, rates through β-strands are more rapid for their distance than for α-helices at the same distance as a consequence of tunneling pathway length effects (49). At a third level, edge-coupled hemes and chlorophylls tend to sample many interfering pathways and will have fluctuation-dominated and multipathway-mediated tunneling. As we discuss below, fluctuation-dominated coupling need not wash out the influence of the protein’s fold on the ET rate; rather, it seems likely that it may remove the possibility of a single weak link causing dramatic slowing (for a given distance) (50).

3.2. Fluctuations and Structure

A natural question that arises from an analysis of how fluctuations impact ET kinetics is whether fluctuations wash out the influence of medium structure on the ET rates. This question has been addressed within the database of protein-mediated ET (in the case of Ru proteins) and water-mediated ET (for cytochrome b₅ self-exchange) (45). If tunneling medium fluctuations wash out structural features, the tunneling medium could be described by an effective structureless tunneling barrier. To address this key issue, which has deep connections to the role of evolution in determining tunneling pathway structure (48, 51–54), we computed the metric (45)

$$S{[\langle T_{\mathit{\text{DA}}}^2\rangle ]}_{R_{\mathit{\text{DA}}}}=\frac{\sqrt{\mathit{\text{var}}{[\langle T_{\mathit{\text{DA}}}^2\rangle ]}_{R_{\mathit{\text{DA}}}}}}{\mathit{\text{avg}}{[\langle T_{\mathit{\text{DA}}}^2\rangle ]}_{R_{\mathit{\text{DA}}}}},$$ (14)

where the variance (var) and the average (avg) are computed for all distinct ET species with the same transfer distance. A value of S_RDA ≪ 1 at a given R_DA means that an average barrier model that reproduces $\mathit{\text{avg}}{[\langle T_{\mathit{\text{DA}}}^2\rangle ]}_{R_{\mathit{\text{DA}}}}$ gives, to a good approximation, $\langle T_{\mathit{\text{DA}}}^2\rangle$ for all ET species at R_DA. A value of S_RDA ≥ 1 means that the $\langle T_{\mathit{\text{DA}}}^2\rangle$ for any single ET species is not representative of the $\langle T_{\mathit{\text{DA}}}^2\rangle$ for the other ET species at that distance (Figure 5).

Figure 5.

Schematic diagram explaining how donor-acceptor (DA) coupling fluctuations could wash out DA coupling structural differences with increasing DA distance. The diagram shows possible T_DA probability densities for two pairs of different electron-transfer (ET) species, each pair having the same average values of R_DA. σ₁ and σ₂ represent the root-mean-squared coupling fluctuations σ_TDA of each species in the pair. (a) For R_DA < r_crit, coupling fluctuations are small and do not wash out structural differences in $\langle T_{\mathit{\text{DA}}}^2\rangle$, i.e., $S_{R_{\mathit{\text{DA}}}<r_{\mathit{\text{crit}}}}=({\langle T_{\mathit{\text{DA}}}^2\rangle }_2-{\langle T_{\mathit{\text{DA}}}^2\rangle }_1)/({\langle T_{\mathit{\text{DA}}}^2\rangle }_2+{\langle T_{\mathit{\text{DA}}}^2\rangle }_1)~1$. (b) For R_DA > r_crit, the increase in coupling fluctuations could wash out structural differences in $\langle T_{\mathit{\text{DA}}}^2\rangle$, i.e., S_RDA>rcrit ≪ 1, leading to an average barrier limit. Surprisingly, our simulations do not observe this second regime (b) even though coupling fluctuations are large (45).

We find that S_RDA is of order unity for all transfer distances (Figure 4b), indicating that the influence of structural differences between proteins on the mean-squared ET couplings is large. Even at large R_DA, where ${\sigma }_{T_{\mathit{\text{DA}}}}^2$ dominates, the diversity of values at any distance is not washed out by structural fluctuations. Thus, the protein structure defines the tunneling rates, even when fluctuations dominate the ensemble of coupling values. This seems to be the case even for water-mediated interprotein ET (45).

The fluctuation analysis described here has been used mostly to study small-molecule and protein ET, although there is a growing body of related work on nucleic acids (see 55, and references therein). It will be interesting to use the parameters defined above to characterize fluctuations and structural disorder in DNA ET reactions. The DNA ET mechanism may vary from tunneling to thermally activated hopping, so that a simple tunneling element analysis may not describe the kinetics (44, 56, 57). Moreover, the folded structure of DNA does not show the wide variability found in proteins, so it may be interesting to study the dependency of R_coh and S_RDA on sequence, rather than on distance. The coherence parameter approach was recently applied to DNA charge transfer (58, 59).

MOLECULAR C-CLAMPS: INTERPOSING A WATER MOLECULE THAT MEDIATES TUNNELING BETWEEN DONOR AND ACCEPTOR

Modern synthetic tools allow the synthesis of tailored cleft or C-clamp structures, in which the dimensions of the C-clamp define the tunneling medium. The structure D-SSS-A (see Figure 3) has the dimensions and shape that will permit one water molecule to bridge between the donor and acceptor, but one DMSO molecule is too large to fit. The observed ET rate for D-SSS-A in water is much larger than that in DMSO. For the structure D-SRR-A, in which the dominant tunneling paths are through bonds, the ET rates are about the same in the two solvents (29).

WHAT ARE THE CHARACTERISTICS OF PROTEINS WITH LARGE VERSUS SMALL FLUCTUATIONS IN THE DONOR-ACCEPTOR TUNNELING INTERACTION, T_DA?

Proteins are dynamical objects: Their geometries fluctuate so the interactions that they mediate between redox cofactors that bind to them also fluctuate. Figure 5 illustrates the small versus large DA fluctuation regimes. In the small fluctuation regime, the width of the T_DA distributions for pairs of cofactors bound at different sites with the same DA separation distances (R_DA) is narrow compared to the differences in the $\langle T_{\mathit{\text{DA}}}^2\rangle$ values. When fluctuations are large, the fluctuations may cause the differences in the mean-squared $\langle T_{\mathit{\text{DA}}}^2\rangle$ values to become quite similar.

4. THE FAST COUPLING–FLUCTUATION REGIME (τ_coh ≤ τ_FC)

In this regime, T_DA(t) fluctuates while the donor and acceptor are resonant, so the Franck-Condon approximation may not be valid. In this limit, the rate equation is more complex than suggested by Equation 11 because there is no clear separation between the C_FC(t) and C_TDA(t) timescales in Equation 1. Theoretical studies of how matrix element fluctuations influence ET rates demonstrate the existence of various kinetic regimes (8, 23, 60–68). In the fast fluctuation limit (τ_coh < τ_FC) and for activationless (or nearly activationless) ET, the rate decreases with decreasing τ_coh because fluctuations take the system out of the activationless regime. For large activation free energies, the fluctuations can enhance the rate. In general, this rate enhancement is more pronounced in the inverted region because of inelastic transitions, and the overall effect is to reduce the drop in rate with increasing energy gap (and to change the rate’s temperature dependence).

Inelastic tunneling introduces the exchange of energy between the tunneling electron and the bridge vibrations. If bridge vibrations are described quantum mechanically, the nonadiabatic rate expression is (8, 12, 65) k_ET = ∑_i P_νi ∑_f k_ET(ν_i → ν_f), where

$$k_{\mathit{\text{ET}}}({\nu }_i\to {\nu }_f)=(2\pi /\hslash )|\langle D;{\nu }_i|\hat{T}|A;{\nu }_f\rangle {|}^2\times {\rho }_{\mathit{\text{FC}}}({\nu }_i\to {\nu }_f),$$ (15)

assuming thermal equilibration of the vibrational degrees of freedom prior to ET. k_ET is a thermally weighted sum of individual rates k_ET(ν_i → ν_f), each associated with a ν_i-to-ν_f transition induced by the exchange of energy with the tunneling electron (ν_i and ν_f are initial and final bridge vibrational states, respectively). The statistical weights, P_νi, are Boltzmann populations of initial ν_i states. k_ET(ν_i → ν_f) in Equation 15 contains a vibronic tunneling matrix element 〈D; ν_i|T̂|A; ν_f〉 (T̂ is given in Equation 4) and a thermally weighted Franck-Condon factor ρ_FC(ν_i → ν_f). If the vibrational degrees of freedom associated with the donor and acceptor diabatic surfaces (Figure 1) are treated classically,

$${\rho }_{\mathit{\text{FC}}}({\nu }_i\to {\nu }_f)={(4\pi \lambda k_{B}T)}^{-1/2}\text{exp}[-{([U_D^{\text{min}}+{\epsilon }_{{\nu }_i}-(U_A^{\text{min}}+{\epsilon }_{{\nu }_f})]-\lambda )}^2/4\lambda k_{B}T],$$ (16)

where ε_νi and ε_νf are the energies of bridge vibrational states ν_i and ν_f, and λ is the reorganization energy of the donor and acceptor diabatic surfaces (Figure 1). Equation 16 shows that the energy gap for elastic tunneling (ε_νi = ε_νf) is $U_D^{\text{min}}-U_A^{\text{min}}$, regardless of the initial (final) bridge vibrational state. For inelastic tunneling (ε_νi ≠ ε_νf), the energy gap is shifted by an amount equal to the change in the bridge vibrational energy, given by $U_D^{\text{min}}+{\epsilon }_{{\nu }_i}-(U_A^{\text{min}}+{\epsilon }_{{\nu }_f})=U_D^{\text{min}}-U_A^{\text{min}}+({\epsilon }_{{\nu }_i}-{\epsilon }_{{\nu }_f})$, and it can have a strong effect on the ET rate. For an elastic ET reaction strongly in the inverted regime, the elastic ET rate is suppressed by the large activation energy; e.g., if $U_D^{\text{min}}=U_A^{\text{min}}+2\lambda \text{and}\lambda \gg k_{B}T,\text{then}{E}_{\mathit{\text{act}}}={(U_D^{\text{min}}-U_A^{\text{min}}-\lambda )}^2/4\lambda =\lambda /4\gg k_{B}T$. If a bridge vibration with frequency ℏω ≈ λ can interact with the tunneling electron, however, then inelastic tunneling is possible because the electron can deposit energy equal to λ in any state ν_i of the oscillator, exciting it to state ν_f with ε_νi − ε_νf = −ℏω = −λ. The corresponding rate k_ET(ν_i → ν_f) is activationless because its energy gap is reduced by λ, i.e., $E_{\mathit{\text{act}}}={([U_D^{\text{min}}+{\epsilon }_{{\nu }_i}-(U_A^{\text{min}}+{\epsilon }_{{\nu }_f})]-\lambda )}^2/4\lambda =0$. Therefore, inelastic tunneling rates in the inverted region are enhanced, and the overall rate does not drop by the amount predicted for pure elastic tunneling. The effect is stronger for large bridges where the bridge vibrational spectrum is very dense (65). Enhanced ET rates in the inverted regime have been reported for small-molecule ET (69) and for biological ET (70), although it is not clear that the enhancement arises from inelastic tunneling.

We now explore how tunneling interactions, 〈D; ν_i|T̂|A; ν_f〉 in Equation 4, change when the tunneling is inelastic. We consider a bridge with two parallel electronic pathways (bridge orbitals B1 and B2 both coupled to the D and A states; see Figure 6), where each bridge orbital energy is modulated by a pathway-specific bridge vibrational mode (y₁ for B1 and y₂ for B2). Assuming linear electronic-vibrational coupling, we have, in Equation 4, Ĥ^B = Ĥ^B(el) + Ĥ^B(vi) + Ĥ^B(e−v), where

$${\hat{H}}^{B(\mathit{\text{el}})}=U_{B1}^{\text{min}}|B1\rangle \langle B1|+U_{B2}^{\text{min}}|B2\rangle \langle B2|,$$ (17)

$${\hat{H}}^{B(\mathit{\text{vi}})}={\displaystyle \sum _n\hslash {\omega }_1(n+1/2)|n(y_1)\rangle }\langle n(y_1)|+{\displaystyle \sum _m\hslash {\omega }_2(m+1/2)|m(y_2)\rangle \langle m(y_2)|,}$$ (18)

and

$${\hat{H}}^{B(e-v)}=-F_1{\hat{y}}_1|B1\rangle \langle B1|-F_2{\hat{y}}_2|B2\rangle \langle B2|.$$ (19)

Figure 6.

(a) A donor-bridge-acceptor supermolecule that might be used to realize a molecular double-slit experiment. (b) The vibrations of the bridge could act to measure which pathway the electron follows as it tunnels from the donor (D) to the acceptor (A), if some of the atoms in one pathway are isotopically substituted in order to create pathway-localized vibrations.

The tunneling process |D; ν_i〉 → |A; ν_f〉, where |ν_i〉 = |n(y₁)〉|m(y₂)〉 and |ν_f〉 = |n′(y₁)〉|m′(y₂)〉, is now described by the vibronic tunneling matrix element 〈D; n(y₁)m(y₂)|T̂|A; n′(y₁)m′(y₂)〉. Using Equation 4, one can show that the exchange of energy between the electron and the vibrations identifies the electronic path traversed by the electron. The reason is that, in this model, the vibrations are pathway specific (12, 71). Any elastic tunneling matrix element is the sum over the parallel pathway contributions (here, B1 and B2), which interfere coherently, e.g., 〈D; 0(y₁), 0(y₂)|T̂|A; 0(y₁), 0(y₁)〉 = Path(B1) + Path(B2). An inelastic matrix element, such as 〈D; 1(y₁), 0(y₂)|T̂|A; 0(y₁), 0(y₂)〉, comprises only Path(B1) (12, 71). Therefore, inelastic tunneling can erase the interference among coupling pathways; it is possible that such effects are relevant to protein ET (17). Recent studies have shown that, in a system in which elastic tunneling pathways interfere destructively and ET is symmetry forbidden, the enhancement of inelastic tunneling induced by exciting bridge vibrational modes with infrared irradiation can switch on ET (72). The field of inelastic tunneling ET reactions is closely related to inelastic tunneling spectroscopy (e.g., see 73, 74; reviewed in 75, 76).

A UNIMOLECULAR WHICH-WAY INTERFEROMETER

Leaving spatially localized vibrational energy behind on (or picking it up from) a tunneling pathway records the route that the electron traversed. Knowing this route with certainty removes all coupling pathways that do not include that specific coupling step (i.e., the group in which the vibrational energy was deposited) from the pathway sum (12). Figure 6 illustrates the concept of a molecular which-way interferometer. The key element is that spatially localized and distinguishable normal modes couple to bridge sites. Experiments to test these ideas are employing infrared excitation of bridge vibrational modes in a three-color experiment (77).

5. SOLVENT CONTROL OF ELECTRON TRANSFER

In the above sections, we explore nonadiabatic ET, in which the rate is limited by electron tunneling and tunneling fluctuations. It is well-known that for adiabatic ET, the rate-limiting step may be solvent relaxation, and the solvent fluctuation timescale becomes particularly important (1–7). The importance of solvent dynamics in the intermediate to nonadiabatic limits has not been well appreciated, however. Recent studies of the transition from solvent polarization control to tunneling control of ET, using the U-shaped molecules shown in Figure 7, are elucidating the relevant timescale considerations for such ET processes.

Figure 7.

Molecular structures for the three donor-bridge-acceptor supermolecules, which have different pendant groups in the line of sight between the naphthalenic donor and the dicyanoethylene acceptor units, that were studied by Paddon-Row, Waldeck, and coworkers.

For the U-shaped molecules shown in Figure 7, T_DA is smaller than k_BT, but not that much smaller, and it is possible to observe a change in ET mechanism by changing the solvent friction. Three different limiting regimes, or mechanisms, are observed in ET reactions: nonadiabatic ET, adiabatic ET, and solvent-controlled ET. Criteria for transitions between the adiabatic and nonadiabatic regime are related to the nature of the nuclear motion through the crossing (e.g., ballistic or overdamped) and can be formulated in terms of the Landau-Zener curve-crossing problem (1–7). In the nonadiabatic case (discussed in the previous sections), the electronic coupling is very weak in the sense that the system has to move through the curve-crossing region many times before the electronic state changes from the donor to the acceptor (see Figure 1). In the discussion that follows, we use Equation 11 for the nonadiabatic rate constant and denote the nonadiabatic rate k_NA rather than k_ET. Furthermore, we assume that in addition to the classical mode that modulates the DA energy gap (U_D and U_A in Figure 1), there is a high-frequency quantum mode that also modulates the gap (with reorganization energy λ̃ and frequency ω̃ = 2πν̃). In this case the Franck-Condon factor has both classical mode and quantum mode contributions, and the nonadiabatic rate constant becomes

$$k_{\mathit{\text{NA}}}=\frac{4{\pi }^2}{h}\langle T_{\mathit{\text{DA}}}^2\rangle \frac{1}{\sqrt{4\lambda \pi k_{B}T}}{\displaystyle \sum _{n=0}^{\infty }\text{exp}(-S)}\left(\frac{S^n}{n!}\right)\text{exp}\left[-\frac{{(\mathrm{\Delta }G+\lambda +nh\tilde{\nu })}^2}{4\lambda k_{B}T}\right],$$ (20)

where λ is the reorganization energy of the classical mode of Figure 1 $(\mathrm{\Delta }G=U_A^{\text{min}}-U_D^{\text{min}})$, and S = λ̃/hν̃ is the Huang-Rhys factor associated with the reorganization energy and frequency of the quantum mode (1–7). Equation 20 assumes that the initial vibrational state of the quantum mode is the ground state (i.e., hν̃ ≫ k_BT). The Franck-Condon factor in Equation 20 is similar to ρ_FC(ν_i → ν_f) of Equation 16. Both Franck-Condon factors describe the exchange of energy between the electron and quantum vibrations of the system. In Equation 16 the vibrations modulate the DA coupling, whereas in Equation 20, they modulate the DA energy gap (8, 65).

In the adiabatic case of large coupling, the reaction proceeds by nuclear motion through the transition state on a single electronic potential energy surface (the lower adiabatic energy surface arising from the coupling T_DA between U_D and U_A in Figure 1). The effect of T_DA on the rate constant is manifest only through its role in determining the adiabatic surface energy barrier, which is less than E_act of Figure 1. In the solvent-controlled limit, the electronic coupling may still be small; however, the rate constant is affected by frictional coupling. In this case, the characteristic time spent in the curve-crossing region is long enough that the electronic state changes from D to A for nearly every approach to the DA crossing, even though the coupling is weak. Hence the reaction appears adiabatic in the sense that the rate is limited by nuclear dynamics rather than by the electron-tunneling probability.

Zusman (78, 79) generalized the rate constant expression for ET, k_ET, to describe a transition between the normal nonadiabatic limit k_NA and a solvent-controlled limit k_SC, namely

$$\frac{1}{k_{\mathit{\text{ET}}}}=\frac{1}{k_{\mathit{\text{SC}}}}+\frac{1}{k_{\mathit{\text{NA}}}},$$ (21)

which shows that the measured ET rate k_ET can be limited by either the electronic motion (where k_NA is small) or the nuclear motion (when k_SC is small). The slower process is rate controlling. In the classical limit he found

$$k_{\mathit{\text{SC}}}=\frac{1}{{\tau }_S}\sqrt{\frac{\lambda }{{\pi }^3{k}_{B}T}}\text{sin}\left(\pi \sqrt{\frac{\mathrm{\Delta }{G}^{\ne }}{\lambda }}\right)\text{exp}(-\frac{\mathrm{\Delta }{G}^{\ne }}{\lambda }),$$ (22)

in which the ET rate is proportional to the solvation rate, 1/τ_s, where τ_s is the solvation time. It is the characteristic time that it takes for the solvent to respond to a change in the charge distribution of a solute and has been quantified in a number of different ways (vide infra). In Equation 22, ΔG^≠ = (ΔG + λ)²/4λ (in the notation of Figure 1, $\mathrm{\Delta }G\to U_A^{\text{min}}-U_D^{\text{min}}\text{and}\mathrm{\Delta }{G}^{\ne }\to E_{\mathit{\text{act}}}$). Because the solvation time τ_s increases dramatically with decreasing temperature, especially in viscous solvents, the solvation time becomes more important as the temperature is lowered. Zusman used the inequality

$$\frac{{\pi }^2|T_{\mathit{\text{DA}}}{|}^2{\tau }_S}{\hslash {\lambda }_0}\text{exp}(-S)\gg \left(\text{sin}\frac{\pi }{2}\left(\frac{\mathrm{\Delta }G}{\lambda }+1\right)\right)$$ (23)

to assess when the solvent friction should be important for the ET rate. In the context of our previous discussion of fluctuating DA couplings, |T_DA| in Equation 23 should be interpreted as $T_{\mathit{\text{DA}}}^{\mathit{\text{rms}}}=\sqrt{\langle T_{\mathit{\text{DA}}}^2\rangle }$. For small driving forces, |ΔG| ≪ λ, this criterion is $\frac{{\pi }^2{\tau }_S{|T_{\mathit{\text{DA}}}^{\mathit{\text{eff}}}|}^2}{\hslash \lambda }\gg 1,\text{where}|T_{\mathit{\text{DA}}}^{\mathit{\text{eff}}}|=|T_{\mathit{\text{DA}}}|\text{exp}(-S/2)$. Hence, the solvent-controlled regime may be interpreted to arise from a solvent-driven change of adiabaticity, characterized by an adiabaticity parameter g, where (80)

$$g=\frac{|T_{\mathit{\text{DA}}}^{\mathit{\text{eff}}}{|}^2{\pi }^2{\tau }_S}{\hslash \lambda }.$$ (24)

When g ≫ 1, the reaction is solvent controlled, and when g ≪ 1, it is nonadiabatic.

In addition to Zusman’s model, there have been several theoretical approaches to the problem of solvent-controlled ET and to the transition from nonadiabatic to adiabatic ET [e.g., for reviews, see Calef & Wolynes (81), Morillo & Cukier (82) Hynes (83), Sumi and Marcus (84), Onuchic and coworkers (86), Sparpaglione & Mukamel (87, 88), Rips & Jortner (89), and Jortner & Bixon (4)]. Sumi & Marcus (84) considered the combined effects of intramolecular vibrations and diffusive solvent orientational motions on ET. They described the reaction as proceeding along a two-dimensional effective potential energy surface, V(q, X). The coordinate X corresponds to the solvent polarization (the polarization response of the solvent to changes of the charge distribution), and q is an intramolecular vibrational coordinate, which includes the fast nuclear motions typical of an ET reaction in the nonadiabatic or adiabatic limit. To find the reaction rate, they solved the Fokker-Planck equation for diffusive motion along X and treated the motion along q through a rate constant k(X) that depends on the fast motions in the normal way (e.g., Equation 1) and depends parametrically on X (84, 85).

It was found that the electron transfer rate of the U-shaped molecules in Figure 7 in fast solvents (90, 91) could be modeled by using Equation 21 with an internal (quantum mode) reorganization energy λ̃ of 0.65 eV and an effective quantum mode frequency of 1600 cm⁻¹. Comparison with solvation models indicated that the solvent (classical mode) reorganization energy λ was between 1.2 eV and 1.4 eV for molecules 1 and 2 in Figure 7 in NMA (N-methylacetamide) and NMP (N-methylpropionamide) (vide infra). The ratio λ̃/λ is thus approximately 0.5, which places these reactions in the narrow reaction window limit of Sumi & Marcus (84). This limit is also one in which Zusman’s predictions (Equations 21 and 22) should apply. Sumi & Marcus pointed out the nonexponential character in the narrow reaction window limit; however, Zusman’s treatment does not address this feature.

By studying chemical systems in different solvents with a range of solvent relaxation times, Waldeck and coworkers (29, 85, 92, 93) explored the limit in which the ET time τ_ET = 1/k_ET is controlled by τ_s, which is commonly modeled by a characteristic dielectric relaxation (Debye) time (79) or the characteristic time measured in a dynamic Stokes shift experiment (94). They reported a transition from the nonadiabatic ET regime to the solvent-controlled limit and showed how it depends on the solvent relaxation time and the electronic coupling strength. The molecules shown in Figure 7 (molecules 1, 2, and 3) have empirically determined electronic couplings in the range of tens to hundreds of wave numbers (depending on the pendant group that is placed in the cleft). In solvents with rapid polarization fluctuations, as measured by their dynamic Stokes shift relaxation times, the ET followed a first-order (i.e., exponential) rate and proceeded in the nonadiabatic limit. However, in slow solvents (e.g., NMP), the rate displayed a nonexponential behavior that was controlled by the solvent relaxation rate.

Because the rate was nonexponential, Waldeck and coworkers (93, 95) defined a rate constant k_ET from the inverse of the correlation time for the decay of the reactant state (i.e., the time integral of the temporal profile of the initially excited state), so it was possible to plot the ln(k_ET T^1/2) versus the inverse of the temperature T (see Figure 8). The rate constants for all three compounds were similar at low temperatures and deviate from one another at higher temperatures. The deviation occurred first for compound 1 (at 260 K to 270 K; NMP solvation time of approximately 240 ps) and then for compounds 2 and 3 at higher temperature (above 310 K; NMP solvation time of approximately 55 ps). Thus, the trend in characteristic times for the different solutes correlated with the change in electronic coupling |T_DA| (see Table 1) that was determined for these three molecules in aromatic solvents and acetonitrile, which have a rapid solvation response (τ_s is small).

Figure 8.

Experimental electron transfer rate constants for compounds 1 (gold squares), 2 (purple triangles), and 3 (black diamonds) in NMP plotted versus the inverse of the temperature. Figure adapted with permission from Reference 93.

Table 1.

Fitting parameters for compound 1, 2, and 3 in N-methylpropionamide (NMP) at 295K^a

Compound	|TDA| (cm−1)	λ (eV)	ΔG (eV)
1	90	1.24	−0.35
2	273	1.59	−0.57
3	147	1.50	−0.52

^aSee Reference 29.

The data in Figure 8 show that the ET rate of compound 2 is higher than that of compound 3 in NMP, and compound 1 has the lowest ET rate. This trend is reflected in the electronic coupling values extracted from a fit of the rate data in solvents with a fast solvation response by the semiclassical equation (Equation 20; see Table 1). The electronic coupling magnitude of compound 2 with a methoxy substituted pendant group is the largest among the molecules, and this large value can be rationalized in terms of that group’s electron affinity and ionization potential. The reorganization energy and Gibbs free energy parameters in Table 1 vary somewhat among the compounds, but the activation barrier for the reaction ΔG^≠ = (ΔG + λ)²/4λ is similar for the three structures at 295 K (from 0.160 eV to 0.164 eV). The similarity of the activation barriers (and energetic parameters) is consistent with the similar size, shape, and chemical structure of the molecules. This similarity is found even though the rate data appear to deviate substantially from one another as the temperature changes.

The self-consistency of this analysis can be assessed by considering the dependency of the rate constant on the solvation time. The different kinetic models predict that the ET rate constant is inversely proportional to the solvation time in the solvent-controlled regime, but that it becomes independent of solvent friction when the solvation time is short.

Figure 9 plots a reduced ET time ${\tau }_{\mathit{\text{ET}}}^{*}$ (Equation 25), defined in terms of Zusman’s timescale (78, 96, 97) as

$${\tau }_{\mathit{\text{ET}},Z}^{*}=\frac{\sqrt{{\pi }^3}}{\lambda \text{sin}\left(\pi \sqrt{\frac{\mathrm{\Delta }{G}^{\ne }}{\lambda }}\right)}{\tau }_S+\frac{\hslash }{\sqrt{\pi }}\frac{1}{|T_{\mathit{\text{DA}}}{|}^2}$$ (25)

versus the solvation time of NMP. For all of these molecules, a linear correlation exists between ${\tau }_{\mathit{\text{ET}}}^{*}$ and the solvation time at low temperature, and the ET times become independent of the solvation time at high temperatures. The Sparpaglione-Mukamel treatment (87, 88) is similar. Both models fail to predict the observed slopes quantitatively, although they otherwise provide an accurate description of the data. As discussed above, when g ≫ 1 the reaction is solvent controlled, and when g ≪ 1 it is nonadiabatic. This analysis predicts that the solvent-controlled regime is reached when τ_S ≫ 24 ps for compound 1, τ_S ≫ 2 ps for compound 2, and τ_S ≫ 6 ps for compound 3 in NMP. The experimental results (Figure 7) indicate that compounds 2 and 3 are in the solvent-controlled limit (coalescence of rates) when τ_S is near 56 ps, and for compound 1 the solvent-controlled limit is reached at around 240 ps. Thus, the trend in the rate data can be understood via the Zusman model.

Figure 9.

Plot of ${\tau }_{\mathit{\text{ET}}}^{*}$ (Equation 25) versus τ_S for compound 1 (gold squares), compound 2 (purple triangles), and compound 3 (black diamonds) in NMP. (a) The plot over the whole range of data. (b) Expansion of the plot in the high-temperature region 0 ≤ τ_S ≤ 60 ps (60 ps corresponds to the room temperature) for compounds 1, 2, and 3. Figure adapted with permission from Reference 95.

6. GATED ELECTRON TRANSFER VERSUS DYNAMICAL CONTROL

If any of the timescales discussed above is longer than the ET time, the ET kinetics may be multi-exponential or gated. Figure 10 shows rate constants for cytochrome c immobilized in two different ways in monolayer films (electrostatic adsorption and ligation of the iron heme) (98–100). Although the thickness dependencies of the rates are shifted from one another, determined by the immobilization, both tethering schemes show an exponential distance dependence for thick films and a weaker distance dependence for thin films. A number of studies (101–107) have found a change in the ET reaction mechanism with distance from an electrode for immobilized proteins. For cytochrome c immobilized by heme ligation, the tunneling regime is reached for long tethers (n ≥ 12) and is controlled by medium relaxation for short tethers (98, 108). The soft distance dependence of the rate and its sensitivity to the medium friction can be rationalized in different ways. For a gated mechanism, the conformation changes are slow with respect to the ET time and limit the rate for thin films; i.e., the system fluctuates to a geometry of large DA coupling. Alternatively, the experimental data can be explained by a change in the ET mechanism from nonadiabatic at long distances to solvent controlled at shorter distances.

Figure 10.

Protein immobilized on self-assembled monolayer (SAM) surfaces: (a) Cytochrome c is absorbed electrostatically to a SAM of carboxylic acid terminated thiols, and (b) cytochrome c is tethered to a SAM by a pyridyl group that replaces Met 80 as an axial ligand. (c) Plots of ln(k°) versus the number of methylene groups in alkyl SAMs on Au.

Yue and coworkers (99) distinguished tunneling control from relaxation control mechanisms by measuring the dependence of the ET rate on the overpotential. For electrostatic assemblies on silver, they found no dependence on the overpotential, indicating that ET is gated. For ligated-protein assemblies on gold electrodes, they found an overpotential dependence, suggesting that ET proceeds in an effective adiabatic/medium-controlled limit. Yue and coworkers (99) also performed a temperature-dependent study of protein ET and found that the activation barrier has contributions from both the ET reorganization energy and the medium friction’s apparent activation. These findings suggest that the so-called unusual distance dependence of the heterogeneous ET rate, reported for different proteins immobilized on metals, may be explained by the same underlying principles.

7. CONCLUSIONS AND PROSPECTS

Molecular fluctuations in ET systems not only enable access to the activated complex, but also cause DA electronic interactions to fluctuate. The magnitudes and timescales of these fluctuations can determine the reaction mechanism. The interplay among electronic coupling and nuclear fluctuations is particularly important in biological electron transfer because protein and water bridges dominate the electron-tunneling pathways. Simulations indicate that the Condon approximation appears to be valid for most of the proteins studied so far. For long-range ET, the rates tend to be controlled by fluctuations of the coupling away from its mean value. Indeed, because of destructive interferences among multiple paths, the mean coupling value tends to be quite small. Small-molecule studies show that solvent relaxation can change the adiabaticity of ET and, for highly sluggish solvents, limit the ET rate even for intermediate (approximately 100 cm⁻¹) couplings. Such effects are expected to be important for proteins in which a wide distribution of relaxation processes and timescales are present (see 109). Electrochemical studies of immobilized proteins are being used to reveal how the ET rate changes from the tunneling limit to relaxation control. Electron tunneling, gated ET, and solvent relaxation controlled ET are now well documented in both small-molecule and protein systems alike.

It is clear that inelastic tunneling, in addition to modifying the free energy dependence of ET kinetics, could have substantial effects on coupling pathway interferences. The quantitative theoretical framework that has emerged from these recent studies suggests that one may now be able to formulate general principles linking the structure, fluctuations, and function with ET kinetics. This framework also appears likely to frame new questions in molecular biophysical chemistry. For example, single-molecule studies of ET suggest the possibility of directly probing the role of fluctuations in ET reactions experimentally. Xie and colleagues (110) have reported measurements of ET in a flavoprotein, and Barbara and coworkers (111) have reported single-molecule measurements of ET at electrodes. Studies of this kind in systems with structural flexibility offer the opportunity to explore, in great detail, the role of fluctuations in bioenergetics. Indeed, Rubtsov and coworkers (77) just reported an ultrafast multipulse experiment that uses infrared excitation to perturb electron DA interactions during the course of an ET reaction.

FUTURE ISSUES

1. Experimental probes of pathway coherence, including inelastic labeling of paths, should be developed.

2. Experiments should be designed that are directly sensitive to coupling fluctuations.

3. Electron tunneling across organized water structures needs to be investigated.

4. Experiments should be designed that access multiple ET regimes in single systems: few-path tunneling, multipath tunneling, fluctuation-controlled tunneling, protein-gated tunneling, solvent relaxation limited transport, and carrier injection limited transport.

Glossary

ET

electron transfer

T_DA

tunneling matrix element

σ_TDA

variance of the tunneling matrix element

R_coh

coherence parameter

NMP

N-methylpropionamide