Biological charge transfer via flickering resonance

Significance

Electron transport through DNA plays a central role in nucleic acid damage and repair, and it is usually modeled using a carrier tunneling mechanism (at short distances) and a hopping mechanism (at longer distances). We find that fluctuations into transient geometries that bring multiple bases into electronic degeneracy may support band-like transport during the resonance lifetimes over a distance of ≲15 Å, obviating the need to invoke electron tunneling at short distances. This line of research may help to reveal mechanisms of charge transport in multiheme proteins, bacterial nanowires, and synthetic nanowires, and may also assist in framing the mechanisms of coherent multipigment excitonic transport in light-harvesting proteins.

Abstract

Biological electron-transfer (ET) reactions are typically described in the framework of coherent two-state electron tunneling or multistep hopping. However, these ET reactions may involve multiple redox cofactors in van der Waals contact with each other and with vibronic broadenings on the same scale as the energy gaps among the species. In this regime, fluctuations of the molecular structures and of the medium can produce transient energy level matching among multiple electronic states. This transient degeneracy, or flickering electronic resonance among states, is found to support coherent (ballistic) charge transfer. Importantly, ET rates arising from a flickering resonance (FR) mechanism will decay exponentially with distance because the probability of energy matching multiple states is multiplicative. The distance dependence of FR transport thus mimics the exponential decay that is usually associated with electron tunneling, although FR transport involves real carrier population on the bridge and is not a tunneling phenomenon. Likely candidates for FR transport are macromolecules with ET groups in van der Waals contact: DNA, bacterial nanowires, multiheme proteins, strongly coupled porphyrin arrays, and proteins with closely packed redox-active residues. The theory developed here is used to analyze DNA charge-transfer kinetics, and we find that charge-transfer distances up to three to four bases may be accounted for with this mechanism. Thus, the observed rapid (exponential) distance dependence of DNA ET rates over distances of ≲15 Å does not necessarily prove a tunneling mechanism.

Chemical structure and, importantly, structural fluctuations determine the mechanism and kinetics of charge transfer. Redox energy fluctuations are of particular significance when transport barrier heights and the energy fluctuations are of similar magnitude. Indeed, the sensitivity of biological electron-transfer (ET) rates to conformational fluctuations and consequent (transient) delocalization is the topic of intense interest (1–3). Resonant enhancement of biological ET rates is consistent with a growing body of physical and structural data found in DNA ET through stacked nucleobases (4), extended delocalized structures of bacterial photosynthesis (including the special pair, bridging chlorophyll and pheophytin) (5), the polaronic states of oxidized porphyrin arrays up to seven porphyrin diameters in spatial extent (6), micrometer-scale bacterial nanowires (7, 8), multiheme oxidoreductases (9, 10), amino acid side chains in ribonucleotide reductase (11), engineered protein-based hopping-chains (12), and centimeter-scale charge-transport chains in filamentous bacteria (13). Here, we describe a transient or flickering resonance (FR) mechanism for ET. The FR mechanism arises when thermal fluctuations produce geometries that enable charge delocalization across the entire structure by bringing the donor (D), bridge (B), and acceptor (A) levels into energetic degeneracy. An electron or hole that is spatially localized at a starting D may move ballistically (i.e., with near constant velocity) through these fleeting structures to A. Because the probability of bringing multiple sites into energetic degeneracy is multiplicative and decays approximately exponentially with the number of sites (i.e., with distance), this transport mechanism could be mistaken for electron tunneling because of its distance dependence.

ET Kinetics

The analysis in this paper concerns multisite charge transfer. Conventional biological ET theory is based on two-state dynamics (2, 14–16), with D and A brought into electronic resonance by medium reorganization. At resonance, the electron propagates from D to A either by tunneling (through space or via a bridge, B) i.e., nonadiabatic ET, or by strong electronic mixing among D, B, and A (adiabatic ET). Here, we explore a limit where the intrinsic energy fluctuations and/or medium reorganization create multistate electronic degeneracy and we demonstrate that charge transport in the degenerate state results in exponential distance dependence of the ET rate in both adiabatic and nonadiabatic regimes.

Two-State ET.

To set the stage for the generalized flickering resonance (FR) mechanism, we explain this framework in the context of two-state D-to-A ET where the D and A energies are modulated by independent harmonic fluctuations. In this case, the FR picture is the standard picture of ET, where transfer takes place in thermally populated D–A resonances, within a redox energy matching window equal to the D-to-A coupling. To frame this discussion, we adopt the simplest kinetic model for two-state ET

$${\mathbf{D}}^{-}\mathbf{A}\underset{k_{off}}{\overset{k_{on}}{\rightleftarrows }}{\mathbf{R}}_2^{\mathrm{‡}}\left({\mathbf{D}}^{-}\mathbf{A}\right)\stackrel{1/{\tau }_{trans}}{\to }{\mathbf{R}}_2^{\mathrm{‡}}\left(\mathbf{D}{\mathbf{A}}^{-}\right)\stackrel{1/{\tau }_{trap}}{\to }\mathbf{D}{\mathbf{A}}^{-}.$$ [1]

The overall D-to-A electron or hole transfer rate is denoted $k_{ET}$. ${\mathbf{D}}^{-}$ is the initially prepared state with an electron or hole on D in a state that is not resonant (R) with A. ${\mathbf{R}}_2^{\mathrm{‡}}\left({\mathbf{D}}^{-}\mathbf{A}\right)$ and ${\mathbf{R}}_2^{\mathrm{‡}}\left(\mathbf{D}{\mathbf{A}}^{-}\right)$ denote the subensemble of structures with D and A in two-state resonance and the electron on D or A, respectively. $\mathbf{D}{\mathbf{A}}^{-}$ denotes the ET product with the electron or hole on the relaxed acceptor. $k_{on}$ is the rate to reach two-state resonance ${\mathbf{R}}_2^{\mathrm{‡}}\left({\mathbf{D}}^{-}\mathbf{A}\right)$, $1/{\tau }_{trans}$ is the inverse time for electron transfer at resonance, and $1/{\tau }_{trap}$ is the trapping rate (considered irreversible in the context of transition state theory). The overall ET rate in this kinetic scheme is

$$k_{ET}=\left(1/{\tau }_{\ast }\right)\exp \left[-\mathrm{\Delta }{G}_{act}/K_{B}T\right].$$ [2]

For nonadiabatic ET, $\mathrm{\Delta }{G}_{act}=\mathrm{\Delta }{G}_{act}^{NA}={\left(\mathrm{\Delta }{G}_{f,i}^{\left(0\right)}+\lambda \right)}^2/4\lambda$ and $1/{\tau }_{\ast }\mathrm{=}2\pi V^2/\hslash \sqrt{4\pi \lambda K_{B}T}$. For adiabatic ET, $\mathrm{\Delta }{G}_{act}=\mathrm{\Delta }{G}_{act}^{AD}$ and, if ET is limited by $k_{on}$, then $1/{\tau }_{\ast }\mathrm{=}\omega$ is the reaction coordinate motion attempt frequency. If ET is limited by solvent relaxation and/or by trapping, $1/{\tau }_{\ast }\mathrm{=}1/{\tau }_{trap}$ (e.g., ${\tau }_{trap}\approx {\tau }_L$ is the longitudinal relaxation time). Approximate expressions can be derived for $\mathrm{\Delta }{G}_{act}^{AD}$. For example, when $\mathrm{\Delta }{G}_{f,i}^{\left(0\right)}=0$, $\mathrm{\Delta }{G}_{act}^{AD}=\left(\lambda /4\right)-\left|V\right|$ (15). In our FR reformulation of two-state (D-to-A) ET,

$$k_{ET}=\left(1/\tau \right)P_{match}\left(2\right),$$ [3]

where $P_{match}\left(2\right)$ is the probability that the D and A energy levels differ by less than the D–A electronic coupling V. If we assume independent energy level fluctuations for D and A sites, as well as independent fluctuations of the coupling V, then $P_{match}\left(2\right)={\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}dV{P}_V\left(\mathit{V}\right)}{\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}d{E}_1{\rho }_1\left(E_1\right)}{\displaystyle {\int }_{E_1-\left|V\right|}^{E_1+\left|V\right|}d{E}_2{\rho }_2\left(E_2\right)}$, where ρ₁(E₂) and ρ₂(E₂) are the probability distribution functions for the electronic redox energies of the ET active states of sites 1 and 2. This formulation extends Hopfield’s description of ET rates in terms of Gaussian broadened electron removal and insertion spectral functions (16). In Eq. 3, $1/\tau$ is generally different from $1/{\tau }_{\ast }$ because $P_{match}\left(2\right)\ne \exp \left[-\mathrm{\Delta }{G}_{act}/K_{B}T\right]$. For example (SI Text, section S4), in the approximation of constant coupling $\left(P\left(V\right)=\delta \left(V-\left|V_0\right|\right)\right)$, the exact two-state matching probability is

$$P_{match}\left(2\right)=\left(\theta /\sqrt{2\pi }\right)\exp \left(-{\left(\mathrm{\Delta }{G}_{i,f}^{\left(0\right)}+\lambda -V_{\ast }\right)}^2/4\lambda K_{B}T\right),$$ [4]

where $\theta =2\left|V_0\right|/\sqrt{2\lambda K_{B}T}$, $\lambda ={\lambda }_D+{\lambda }_A$ is the total reorganization energy, and ${\mathit{V}}_{\ast }\mathrm{\in }\left(-\left|V_0\right|,+\left|V_0\right|\right)$. Eq. 4 is valid in both the adiabatic $\left(\theta \ge 1\right)$ and nonadiabatic $\left(\theta \ll 1\right)$ regimes. From Eq. 4, we see that $\tau$ in Eq. 3 is related to ${\tau }_{\ast }$ in Eq. 2 by the exact relation $1/{\tau }_{\ast }\mathrm{=}\left(\theta /\sqrt{2\pi }\right)\times 1/\tau$ in any regime. In the nonadiabatic regime we have $\theta \ll 1$ and $\tau \ll {\tau }_{\ast }$. Further, in the nonadiabatic regime, $P_{match}\left(2\right)\approx P_{match}^{NA}\left(2\right)$, where $P_{match}^{NA}\left(2\right)=\left(\theta /\sqrt{2\pi }\right)\exp \left(-\mathrm{\Delta }{G}_{act}^{NA}/K_{B}T\right)$ and $\mathrm{\Delta }{G}_{act}^{NA}={\left(\mathrm{\Delta }{G}_{i,f}^{\left(0\right)}+\lambda \right)}^2/4\lambda$ (SI Text, section S4). Thus, we can write the nonadiabatic rate either as $k_{ET}^{NA}=\left(1/{\tau }_{\ast }\right)\exp \left(-\mathrm{\Delta }{G}_{act}^{NA}/K_{B}T\right)$ with $1/{\tau }_{\ast }\mathrm{=}2\pi V_0^2/\hslash \sqrt{4\pi \lambda K_{B}T}$ or $k_{ET}^{NA}=\left(1/\tau \right)P_{match}^{NA}\left(2\right)$, where, using $1/{\tau }_{\ast }\mathrm{=}\left(\theta /\sqrt{2\pi }\right)\times 1/\tau$, we find $\tau \mathrm{=}{\tau }_{Rabi}/{\pi }^2$ (${\tau }_{Rabi}\equiv \mathrm{h}/(2\left|{\mathrm{V}}_0\right|)$ is the Rabi time). Moreover, $P_{match}^{NA}\left(2\right)=2\left|V_{DA}\right|\times FC$, where FC is the well-known Franck–Condon factor: $FC=\langle \delta \left(E_{final}-E_{in}\right)\rangle ={\left(\sqrt{4\pi \lambda K_{B}T}\right)}^{-1}\exp \left(-\mathrm{\Delta }{G}_{act}^{NA}/K_{B}T\right)$ (SI Text, section S4). In the adiabatic limit we have $\theta \mathrm{\approx }1$ and $\tau \mathrm{\le }{\tau }_{\ast }$. Importantly, when $\mathrm{\Delta }{G}_{i,f}^{\left(0\right)}=0$, $P_{match}\left(2\right)\approx \left(1/3\right)\exp \left(-\mathrm{\Delta }{G}_{act}^{AD}/K_{B}T\right)$, where $\mathrm{\Delta }{G}_{act}^{AD}=\left(\lambda /4\right)-\left|{\mathrm{V}}_0\right|$ and $\tau ={\tau }_{\ast }/3$ (SI Text, section S4). As such, we have two equivalent strategies to formulate the two-state ET rate (Eqs. 2 and 3). An advantage of using the matching probability (Eq. 3) in formulating the ET rate is that it proves a computationally accessible means to estimate ET rates without first assuming a limiting coupling regime (i.e., adiabatic or nonadiabatic), and it enables generalization to multistate resonance.

A Generalized ET Rate for Multistate Resonance.

For two-states, the FR rate is identical to two-state ET rates (adiabatic or nonadiabatic, depending on the magnitude of V_DA). For an N-state system, the FR mechanism defines a particular ET channel (other channels could well coexist). The FR kinetic model and the rate associated with FR are formulated in analogy to the above approach with the kinetic scheme

$${\mathbf{D}}^{-}{\mathbf{B}}_{N-2}\mathbf{A}\underset{k_{off}}{\overset{k_{on}}{\rightleftarrows }}{\mathbf{R}}_N^{\mathrm{‡}}\left({\mathbf{D}}^{-}{\mathbf{B}}_{N-2}\mathbf{A}\right)\stackrel{1/{\tau }_{trans}}{\to }{\mathbf{R}}_N^{\mathrm{‡}}\left(\mathbf{D}{\mathbf{B}}_{N-2}{\mathbf{A}}^{-}\right)\stackrel{1/{\tau }_{trap}}{\to }\mathbf{D}{\mathbf{B}}_{N-2}{\mathbf{A}}^{-}.$$ [5]

R^‡_N(D⁻B_N−2A) and R^‡_N(DB_N−2A⁻) denote the subensemble of structures with all N units (including D and A) in degeneracy and the electron on D or A, respectively. $\mathbf{D}{\mathbf{B}}_{N-2}{\mathbf{A}}^{-}$ is the product with the trapped electron on A, and P_match is the probability of matching all N states.

The FR channel ET rate is

$$k_{ET}^{FR}=\left(1/{\tau }_{\ast }\right)\exp \left(-\mathrm{\Delta }{G}_{act\left(N\right)}/K_{B}T\right)\text{or}{k}_{ET}^{FR}=\left(1/\tau \right)P_{match}\left(N\right).$$ [6]

ΔG_act(N) is the activation free energy to reach N-state D–B–A resonance and ${\tau }_{\ast }$ is the rate-limiting time scale associated with the kinetics ($k_{on}^{-1}$, ${\tau }_{trans}$, or ${\tau }_{trap}$ for the N-state system). In [5], ${\tau }_{trans}$ is interpreted as the D-to-A transport time for N-state D–B–A resonance with a finite lifetime. The minimum value of ${\tau }_{trans}$ is obtained in the infinite FR lifetime limit, which gives ${\tau }_{trans}^{\mathit{min}}\approx {\tau }_{trans}^{bal}\approx {\mathrm{R}}_{\mathbf{D}\mathbf{A}}ħ/(2{\mathrm{r}}_{\mathrm{n}\mathrm{n}}{\mathrm{V}}_{\mathrm{n}\mathrm{n}})$ [using a mean carrier velocity $\langle v\rangle ={\left(1/\hslash \left[\mathrm{\partial }E\left(k\right)/\mathrm{\partial }k\right]\right)|}_{E_{Fermi}}=2r_{nn}\left|V_{nn}\right|/\hslash$, where r_nn is the nearest-neighbor distance (17)]. For a nearest-neighbor coupling of 0.1 eV and a 3.4-Å separation among neighbors, the carrier velocity is ∼1 Å/fs which places ${\tau }_{trans}^{\mathit{min}}$ in [5] in the range of tens of femtoseconds for a few redox sites. The ${\tau }_{trans}$ step in [5] generates transient carrier population on the bridge independent of the process that ultimately limits the overall rate of the FR channel ([5]).

As in the two-state limit, we could develop a multistate description where $k_{ET}^{FR}=\left(1/{\tau }_{\ast }\right)\exp \left(-\mathrm{\Delta }{G}_{act\left(N\right)}/K_{B}T\right)$. However, the $P_{match}\left(N\right)$ formulation is particularly convenient in terms of computation and interpretation because of its explicit treatment of the influence of nearest-neighbor coupling on the energy-matching window (Fig. S1). For an N-state system with uncorrelated redox energy fluctuations, the probability of simultaneously matching all levels, summed over all matching energies, is

$$\begin{array}{l}{P}_{match}\left(N\right)={\displaystyle \underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}dV{P}_V\left(\mathit{V}\right)}{\displaystyle \underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}d{E}_1{\rho }_1\left(E_1\right)}\\ \underset{{\epsilon }_1-\left|V\right|}{\overset{{\epsilon }_1+\left|V\right|}{{\displaystyle \int }}}d{E}_2{\rho }_2\left(E_2\right)\dots {\displaystyle \underset{\mathit{max}\left\{{\epsilon }_1,{\epsilon }_2,\dots {\epsilon }_{N-1}\right\}-\left|V\right|}{\overset{min\left\{{\epsilon }_1,{\epsilon }_2,\dots {\epsilon }_{N-1}\right\}+\left|V\right|}{\int }}d{E}_N{\rho }_N\left(E_N\right)}.\end{array}$$ [7]

The upper limit of integration for the Nth site takes the minimum value of all site energies for sites 1 to N − 1 plus |V|, and the lower limit of the Nth site integral takes the maximum value of all site energies for sites 1 to N − 1 minus |V|. In this study, the matching probabilities are calculated using numerical integration for fewer than six sites and are computed using Monte Carlo methods for larger systems (18). We can derive exact expressions for $P_{match}\left(N\right)$ or $\mathrm{\Delta }{G}_{act\left(N\right)}$ that are valid for arbitrary intersite couplings and can also derive corresponding relations between ${\tau }_{\ast }$ and $\tau$ (SI Text, section S4). In general, $P_{match}\left(N\right)=f\times \exp \left(-\mathrm{\Delta }{G}_{act\left(N\right)}\left(\left\{E_i\right\},\left\{V_{rms\left(i\right)}\right\},\left\{{\lambda }_i\right\}\right)/K_{B}T\right)$, where, approximately, $f\mathrm{\propto }{\theta }_i^{N-1}$, with ${\theta }_i=V_{rms\left(i\right)}/{\sigma }_{E\left(i\right)}$, where ${\sigma }_{E\left(i\right)}=\sqrt{2{\lambda }_i{K}_{B}T}$ is the rms fluctuation of the ith site redox energy (and λ_i is the corresponding redox site reorganization energy), and $V_{\left(i\right)}^{rms}$ is the ith nearest-neighbor coupling. ${\theta }_i$ is interpreted as a local (nearest-neighbor) adiababticity parameter, ${\theta }_i={\tau }_{FC\left(i\right)}/{\tau }_{Rabi\left(i\right)}$. The FC time ${\tau }_{FC\left(i\right)}=h/\sqrt{2{\lambda }_i{K}_{B}T}$ is the time associated with the loss of the nearest-neighbor resonance arising from energy fluctuations of the ith site. ${\tau }_{Rabi\left(i\right)}=h/2V_{\left(i\right)}^{rms}$ is the Rabi time associated with resonance between sites i and i + 1. Therefore, $f\mathrm{\propto }{\theta }_i^{N-1}$ is a generalized N-state resonance adiabaticity parameter (SI Text, section S4). Even if nearest-neighbor resonances are quasiadiabatic with ${\theta }_i$ slightly less than unity, the N-state resonance becomes nonadiabatic as N grows because $f\mathrm{\propto }{\theta }_i^{N-1}$. Thus, the effective lifetime of the N-state resonance decreases with increasing N. In Eq. 6, $\left(\tau /{\tau }_{\ast }\right)\propto f$ and, in a generalized nonadiabatic limit defined by $f\ll 1$, we have ${\tau }_{\ast }\mathrm{\gg }\tau$. If the overall rate is limited by carrier transmission through the N-state resonance, ${\tau }_{\ast }$ in Eq. 6 is interpreted as ${\tau }_{trans}$, where ${\tau }_{trans}\mathrm{\approx }{\tau }_{trans}^{bal}/f$ and ${\tau }_{trans}^{bal}$ is in the range of tens of femtoseconds for a few redox sites. That is, the nonadiabatic electron transmission time through the FR bridge is the band transport time increased by a nonadiabatic factor associated with degeneracy breaking.

Below, we describe a simple analytical approximation to Eq. 7 for a bridge with uncorrelated fluctuations, equal average site energies, and site energy fluctuations larger than coupling energy fluctuations (nonadiabatic limit). Importantly, (i) $P_{match}\left(N\right)$ decays exponentially with N; (ii) the distance dependence of $P_{match}\left(N\right)$ coincides with the slope of the hole transfer rates and yields in the experiments of Lewis et al. (19) and Giese et al. (20), for reasonable values of the hole transfer rate parameters; and (iii) fits to experimental ET rates and yields require $\tau \approx 1-15\text{fs}$. To identify the rate-limiting step associated with the kinetic scheme of [5], it is necessary to consider ${\tau }_{\ast }$, which is longer than $\tau$. Different possibilities for the rate-limiting steps are discussed below.

The Multisite Matching Probability

As a starting point, we assume statistically independent fluctuations for the multiple sites in the ET system. Standard deviations (σ_E, SD) for DNA base energies estimated from quantum mechanical (QM) analysis of molecular dynamics (MD) sampled structures indicate σ_E ≈ 0.2 eV without solvent (21–23); inclusion of solvent interactions increases these values to ∼0.3–0.5 eV (24, 25). Fig. S1 shows multisite energy-matching probabilities for a four-site linear system with Gaussian fluctuations of equal mean energies (σ_E = 0.16 eV), and with nearest-neighbor couplings fixed at |V_nn| = 10⁻¹, 10⁻², or 10⁻³ eV. Fig. S1 indicates that the four-site energy matching probability drops several orders of magnitude for each order of magnitude decrease in |V_nn|, because smaller values of |V_nn| require closer site energy matching to establish FR.

The probability of matching N fluctuating site energies will decrease in a multiplicative fashion, with one multiplier per site. The energy-matching probability thus drops exponentially with distance. The data in Fig. S2 show the exponential decay of energy level-matching probabilities for a homogeneous chain (solid line) and an inhomogeneous chain (dashed line). For the homogeneous chain, all sites have the same mean energy values and σ_E = 0.6 eV; each nearest-neighbor coupling has zero mean and V_rms = 0.1 eV (the SD of the coupling magnitudes). For the inhomogeneous chains, D and A have the same mean energy values, whereas the bridging site energies are offset by 1 eV. The distance between neighboring sites is assumed to be 3.4 Å. In the case where all of the sites have the same mean energy (Fig. S2, solid line), the exponential decay constant for the matching probability as a function of distance is ∼0.6 Å⁻¹. In the inhomogeneous case (Fig. S2, dashed line), the exponential decay constant is ∼0.7 Å⁻¹. Thus, the average decay exponent increases as the energy offsets grow among the sites. The matching probability is several-fold smaller for the inhomogeneous chains compared with the homogeneous chains, although the decay of the matching probabilities with distance is not very different (because the bridging spectral functions are energy matched to one another in both cases).

The simple exponential distance dependence of the energy matching probabilities found in Fig. S2 is readily understood using an approximate model. Consider a chain with site energies that are described as independent Gaussian random variables with zero means and identical nonzero SDs, σ_E. The joint probability density function for the N-site redox energies is ρ(E₁) × ρ(E₂) × … × ρ(E_N), where $\rho \left(E\right)=\left(1/\sqrt{2\pi {\sigma }_E^2}\right)e^{-E^2/2{\sigma }_E^2}$. An upper bound to the energy-matching probability is approximately $P_{match}\left(N\right)\approx {\left(2{\displaystyle {\int }_0^{V_{rms}}dE\rho \left(E\right)}\right)}^N$. Because $V_{rms}/{\sigma }_E\approx 1/6$ in the example above, we can expand the exponential functions in terms of ${\left(E/{\sigma }_E\right)}^2$. Retaining the zeroth-order term, we find $P_{match}\left(N\right)\approx {\left(\sqrt{2/\pi }\left(V_{rms}/{\sigma }_E\right)\right)}^N$ or

$$P_{match}\left(N\right)\approx e^{-\mathrm{\Phi R}},\mathrm{\Phi }=\left(1/\mathrm{\Delta }R\right)\ln \left[\sqrt{\pi /2}\left({\sigma }_E/V_{rms}\right)\right]$$ [8]

for a distance ΔR between adjacent sites [corrections to this result are found by retaining higher-order terms in ${\left(E/{\sigma }_E\right)}^2$]. For the inhomogeneous case of Fig. S2, a similar argument leads to $P_{match}\left(N\right)\approx e^{-E_B^2/2{\sigma }_E^2}{e}^{-\mathrm{\Phi R}}$, where E_B is the mean energy gap between the D/A states and the N sites of the bridge. Using the values of Fig. S2 (typical for DNA), i.e., V_rms = 0.1 eV and σ_E = 0.6 eV, we find that Φ ≈ 0.6 Å⁻¹ for a 3.4-Å spacing between bases. This is an excellent approximation to the numerical values found in Fig. S2. For the inhomogeneous system in Fig. S2, the decay of the D-to-A rate for the superexchange mechanism would be proportional to the square of the B-mediated D–A tunneling matrix element $V_{DA}^2\propto e^{-\beta R_{dA}}$, where β ≈ (2/3.4 Å) ln[(ΔE_B/V_rms)] in the perturbative limit (1, 2, 26, 27). Substituting V_rms = 0.1 eV and ΔE_B = 1 eV gives β ≈ 1.4 Å⁻¹. As such, for the parameters used in the illustration, the FR mechanism has much softer distance dependence than the superexchange rate. This simple analytical argument is strictly valid for $V_{rms}/{\sigma }_E<1$ (the generalized nonadiabatic limit described above), and it can be used to assess the relative distance dependence of superexchange compared with FR as a function of V_rms, σ_E, and ΔE_B. An approximately exponential distance decay of the matching probability is also observed in molecular dynamics simulations and is discussed below. This simple analysis thus captures the source of the matching probability distance dependence. The crucial point from the above analysis is that FR transport requires all N − 2 bridging level energies to match each other, and the D and A energies; this matching probability drops exponentially with distance.

Correlated vs. Uncorrelated Site Energy Fluctuations

The illustrative FR computations above assume that site energy fluctuations for each site are uncorrelated Gaussians with ${\sigma }_{E_i}\approx \sqrt{2{\lambda }_i{K}_{B}T}$ (15, 16). Numerical simulations of Elstner demonstrate that the site energy fluctuations of neighboring bases in DNA are strongly correlated (25, 28). Mixed quantum/classical simulations indicate approximately Gaussian distributed site energy distributions (25, 29, 30) and nearest-neighbor electronic coupling distributions (25, 28, 31–34). However, the site energy fluctuation correlations break the simple connections between ${\sigma }_E$, $\lambda$, and $K_{B}T$, and produce values of λ ≈ 1.2 eV. Below, we compute the level-matching probabilities P_match (N) using molecular dynamics simulations that include correlated energy level fluctuations.

The FR Model for DNA Charge Transport Kinetics

Limitations of Current Models.

A large body of experimental data indicates two regimes of DNA charge transfer: a regime with strong (exponential) dependence of the rate/yield on distances up to ∼3–7 bp and a much weaker dependence for longer distances (35). Theoretical models have focused on describing short-distance transport with a tunneling model and longer-distance transport with a multistep or multirange hopping approach (36, 37). This framework has challenges associated with understanding transient B population in an apparent tunneling regime and with describing the transition between strong and weak distance-dependent regimes (SI Text, section S1 and refs. 4, 19, 21, 22, 30, and 38–41).

FR Descriptions of Charge Shift and Charge Separation in DNA.

The highest occupied molecular orbital (HOMO) energies of DNA bases are often used to estimate site energies in simple transport models (42, 43). The site energies of purines are higher than those of pyrimidines, so hole transfer is dominated by the purines. It is found that thermal fluctuations of the DNA bases produce distributions of base (gas phase) HOMO energies that are found to be approximately Gaussian distributed (25). Quantum mechanics/molecular mechanics (QM/MM) simulations that include solvent interactions further broaden the site energy distributions, producing increased probabilities of level matching even for sites with different mean energies (i.e., with different redox potentials). DNA sampling and energy computation protocols based on a model AT sequence, including correlated site energy fluctuations, are described in SI Text, section S2 (21, 22, 25, 28, 29, 44, 45), and these data are required for the FR kinetic analysis discussed below.

Charge Shift in the FR Framework.

We used the MD-derived parameters in the FR model to describe hole transfer and trapping in DNA systems studied by Giese et al. (20). As indicated in Fig. S3, a hole is injected onto a G site via photoexcitation of a 4′-acylated nucleotide. From there, the hole can either be lost to solvent in a chemical reaction that converts the G to an 8-oxoG, or it can propagate across an AT bridge to a lower-energy GGG state and exit to solvent from that unit (Fig. S3). The location of the 8-oxoG is then revealed through strand cleavage with piperidine, and the chemical yield ratio P_GGG/P_G for the cleavage products at the injection site G or the destination site GGG is measured using gel electrophoresis (20). Thus, the P_GGG/P_G ratio provides a measure of the G→GGG charge-transfer rate compared with the G→8-oxoG oxidation rate. The distance dependence of the P_GGG/P_G ratio is studied by varying the number of (AT) base pairs from 1 to 16 between the G and GGG traps.

Three spectral functions were introduced for the donor G, the bridge adenines (A), and the acceptor GGG (Fig. S4). The energy gap between the peaks of the spectral functions was set to ΔG + λ where ∆G is the free energy of charge shift between the sites and λ is the reorganization energy. ∆G is the charge shift reaction free energy for donor G to A or GGG. The free energy change for G^+•A to GA^+• ET is +0.35 eV (19) and from G^+•(GGG) to G(GGG)^+• is −0.7 eV (46). Typical λ values for DNA ET range from 0.4 eV (nearest-neighbor) to 1.2 eV (four B units between D and A) (45, 47). Smaller free energy differences between G and GGG states were reported as well (ca. 0.08 eV) (48). Decreased trapping free energies can also be accommodated in the FR model.

The FR probabilities for Giese’s structures are calculated using sampling from both MD and distribution functions using the energetics shown in Fig. S4 and a normally distributed energy matching window randomly selected from a Gaussian distribution with zero mean and SD of 0.1 eV, as described above with G–A couplings of 0.043 eV and A–A couplings of 0.08 eV (44). The FR charge-transfer rate is modeled with Eq. 6, where $\tau$ is an adjustable parameter used to fit the absolute values of the yields.

To compare our simulations with the experimental chemical yield ratios (P_GGG/P_G), we use the kinetic model (49) suggested by Jortner and coworkers. In this model, the water-trapping rate for the holes on GGG is assumed to be much larger than the back charge transfer (CT) rate, i.e., all of the holes that arrive at GGG are trapped. Therefore, the yield ratio is determined by the ratio of the charge transport rate and the trapping rate of the single G, $P_{GGG}/P_G\approx k_{CT}/k_r$. We set the trapping rate k_r to be ∼2.0 × 10⁹ s⁻¹, consistent with the earlier estimate of Jortner et al. (49). Giese and Spichty estimated the water-trapping rate to be as low as ∼6.0 × 10⁴ s⁻¹ (50), based on a tunneling rate through two B sites of 2.5 × 10⁶ s⁻¹ (51).

The computed and experimental chemical yield ratios are shown in Table 1. In the short-distance regime, the calculated chemical yield ratios have an exponential decay parameter of β ≈ 0.5 Å⁻¹, in good agreement with experimental data. As the chain length grows beyond 3–4 bp, the hole transfer presumably becomes dominated by incoherent (hopping) transport, and the FR mechanism is no longer relevant.

Table 1.

Computed P_match, experimental yield ratios, and fitted τk_r values (with k_r the water-trapping rate) for data of Giese et al. (20) (see Scheme S1)

Bridge units	Pmatch	PGGG/PG	τkr × 105
1	4.8 × 10−3	250	1.9
2	8.5 × 10−4	30	2.8
3	2.1 × 10−4	4	5.2
4	6.0 × 10−5	3.5	1.7

Application of the FR Model to DNA Hairpins.

We also used the coupling and site energy distribution functions described above in the FR model to explore charge transfer in stilbene-capped DNA hairpins studied by Lewis et al. (19, 41) (Fig. S5). In their experiments, hole transfer occurs between the photoexcited stilbene electron donor Sd and electron acceptor Sa separated by one to six AT bridging base pairs (19). In contrast to the Giese system, hole transfer between the stilbenes produces the charge-separated state Sa^−•/Sd^+•. The electron hole attraction in that state creates a D–A charge separation energy ramp (Fig. S6; cf. ref. 21), in addition to the redox potential difference between the stilbenes and adenines.

The mean energies of the spectral functions at each site are computed from the analysis of Grozema et al.: <E_i> = E_ion(A) − E_el.aff(Sa) − E_exc(Sa) − E_elst(A^+•Sa^−•) −ΔE_solv. Here, E_ion(A) is the gas-phase ionization potential of adenine in the DNA stack, E_el.aff (Sa) is the vacuum electron affinity of Sa, E_exc(Sa) is the optical excitation energy, and $E_{elst}\left(A^{+•}S{a}^{-•}\right)$ is the D/A Coulomb interaction in the charge-separated state (21). The SDs of these energy distribution functions are set to 0.6 eV, similar to the values from Elstner’s QM/MM analysis of DNA (25). If the excited state has exciplex character with partial delocalization onto the DNA, the site energies should be decreased.

Using the protocol described to treat site energy correlations, we calculated the matching probability for DNA hairpins (with one to three intervening ATs). The energy ramp produces large energy gaps between sites for systems with more than three AT bridging units, making the probability negligible to form a degenerate geometry at these larger distances. We therefore focus our study of the FR mechanism on Sa(A)_nSd (n = 1, 2, 3) DNA hairpins. In the energy matching probability analysis for the Sa(A)_nSd DNA, the nearest-neighbor coupling is taken to be normally distributed around zero with a SD of 0.1 eV (44). The FR charge-transfer rate is modeled with Eq. 6, where $\tau$ is an adjustable parameter used to fit the absolute values of the rates. The computed β-value is consistent with the measured kinetic data (Table 2).

Table 2.

Computed P_match, experimental hole arrival rates of Lewis et al. (19), and fitted τ values

Bridge units	Pmatch	$k_{arrival}^{Expr}$, ns−1	τ, fs
1	7.6 × 10−3	580	13.1
2	2.1 × 10−4	27	7.8
3	5.3 × 10−5	3.4	15.6
4	1.3 × 10−5	1	13.0

Rate-Limiting Time Scales.

The distance dependence of the FR matching probability $P_{match}\left(N\right)$ is consistent with the measured distance dependence of the charge shift yields (Table 1) and charge separation rates (Table 2) at short distances. We have not assumed a particular ET regime (adiabatic or nonadiabatic), and it is remarkable that the slope of the distance dependence of DNA rates and yields is predicted successfully by the MD-derived $P_{match}\left(N\right)$. Our only fitting parameter is $\tau$ in Eq. 6. Recall that the time scale of the rate-limiting step (${\tau }_{\ast }$ in Eq. 6) in the generalized kinetic scheme of [5], is related to τ by $\left({\tau }_{\ast }/\tau \right)\propto {\displaystyle {\prod }_{i=1}^{N-1}{\theta }_i^{-1}}$ where ${\theta }_i\approx V_{rms}/{\sigma }_E$, regardless of the reaction adiabaticity. Assuming ${\theta }_i\approx 1/3\mathrm{to}1/6$ for the DNA systems studied here, ${\tau }_{\ast }$ is in the tens of femtoseconds to picoseconds range for the photoinduced charge separation experiments, and ${\tau }_{\ast }$ is in the hundreds of nanoseconds to tens of microseconds for the charge shift (yield) experiments. Which step(s) in [5] determine ${\tau }_{\ast }$? We do not expect ${\tau }_{trap}$ to depend strongly on distance, so it is not likely to be equal to ${\tau }_{\ast }$. ${\tau }_{trans}$ could be the origin of ${\tau }_{\ast }$ in the charge separation experiments because the fitted $\tau$ (Table 2) is as fast as tens of femtoseconds. However, we also expect $k_{on}$ to be distance dependent ([5]), because $\mathrm{\Delta }{G}_{act\left(N\right)}$ is distance dependent and so is the attempt frequency to form the N-state FR. Therefore, $k_{on}^{-1}$ could determine ${\tau }_{\ast }$ in both experiments, and it could be the rate-limiting step for FR (we expect $k_{on}$ to differ for charge shift and charge separation reactions). For ultrafast ET, it is possible that the time scale determining the ET rate prefactor will differ for different observation time windows, as ET active species can be depleted and replenished on multiple time scales.

In summary, the FR mechanism is compatible with the slopes of the distance dependences of the rates and yields in the charge separation and charge shift experiments at shorter distances. However, our analysis cannot resolve whether the FR channel is rate limited by the time scale of D–B–A transport in the N-state resonance conformation or by the activation rate to the N-state resonance conformation. Regardless of the rate-limiting kinetic step, the signatures of the FR mechanism are the exponential rate decay with distance and a transient carrier population on the bridge.

Conclusions

The FR probability drops approximately exponentially as a function of distance, so FR transport has a signature that can be mistaken for tunneling. This theoretical framework provides a scheme to quantify ideas of how stacking dynamics may influence the coherence of charge transfer. The FR mechanism is consistent with observed ET rates in hole transfer experiments of Giese et al. and Lewis et al. FR explains exponential distance decay of rates in the presence of transient D and A carrier populations that do not sum to unity as they must in the superexchange regime.

The redox energy-matching probability depends critically on nearest-neighbor couplings and energy gaps, as well as on their fluctuations. Level matching can be substantial, even when the D, B, and A sites are nonresonant on average. Fluctuations are expected to depend on temperature, structure, flexibility, and coupling pathway interferences. As such, one should be able to manipulate the energy-matching propensity. In particular, the FR model predicts approximately exponential distance dependence for the ET rate with decay exponent $\mathrm{\Phi }\propto \ln \left({\sigma }_E/V_{rms}\right)\propto \ln \left(\sqrt{2\lambda KT}/V_{rms}\right)$. This characteristic behavior provides a specific testable signature for the FR mechanism.

Disrupting energy level matching during electron transmission, combined with relaxation, can produce multirange hopping or mixed hopping and superexchange (30, 40). Future studies need to track the carrier dynamics that follows the disruption of resonance, to set more precise bounds on the admixture of coherent, incoherent, and mixed transport. The lessons of the present study are that FR (i) obviates the need to invoke tunneling in short distance DNA ET, (ii) rationalizes the observation of electron arrival at the A population without the concomitant reduction of the D population in a DNA hairpin that also displays steep distance-dependent ET rates, and (iii) suggests new approaches to controlling ET by varying site energy and coupling distributions. Building further on the isomorphism between energy and ET kinetics (16), it seems likely that the framework described here may be of use for analyzing coherences among excitonic states of artificial and natural light-harvesting complexes (52–59).