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.

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).

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.

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.

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).

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).

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.

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.

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 .

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 .

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.