Protein Electron Transfer in Solution, Protein Powders, and Electrode Confinement

Abstract

Transport of electrons in biological electron transport chains involves interfaces and confinement, while measurements of rates of protein electron transfer and computer simulations are mostly conducted in solutions. Here, we employ molecular dynamics simulations of the azurin protein to compare activation parameters of electron transfer in solution with the corresponding parameters in confinement. We study the redox chemistry of azurin in protein powders and in confinement between two gold slabs, modeling conditions of an electrochemical experiment. The polarizability of the active site of the protein is included in the calculations and is found to be the main factor in lowering the reaction activation barrier. It couples to strong and inhomogeneous electric fields in confinement and protein powders, resulting in an observable reorganization energy of ∼0.1–0.2 eV, comparable to electrochemical measurements. Corresponding low activation barriers, ∼2k _B T, yield a weak temperature effect on the reaction rate. No dynamical medium control of protein half-reactions is found in our calculations, producing rate constants exponentially decaying with the distance to the electrode.

Introduction

Natural photosynthesis and respiratory chains of biology rely on fast charge separation to avoid wasteful recombination pathways. − The challenge to understand physical mechanisms of enzymatic catalysis and overall efficiency of charge transport is not limited to natural energy chains, but extends to a recently posed puzzle on an unusually high conductivity of proteins lacking redox cofactors − and high conductivity of cytochrome nanowires enabling extracellular electron transport in anaerobic bacterial respiration. , Protein conductivity was shown to be sensitive to enzymatic function. , This preliminary observation, if confirmed, opens the possibility of fluorescent-free single-molecule monitoring , of biological activity by electroanalytical techniques. ,

Despite the obvious importance of protein conductivity, the field has still not matured, and even the basic mechanism of charge conductivity is debated. Two alternative mechanisms , are considered − : (i) hopping (incoherent) conductivity between localized (or polaronic for DNA) states and (ii) superexchange (coherent) conductivity through closely located protein cofactors. Conduction through aromatic residues decays with the protein length according to Ohm’s law, thus suggesting hopping as the conductivity mechanism, also confirmed by simulations. Nevertheless, a number of measurements, mostly in protein junctions, indicate a low temperature dependence of protein conductivity, inconsistent with the typically anticipated activation barriers deduced from solution experiments. ,− Either the activation free energy must be much lower than suggested by solution measurements or quantum coherent transport, at least through some parts of the protein, has to be involved. At the same time, conductance through multiheme cytochrome nanowires is Arrhenius-type, − suggesting a hopping mechanism of conductivity.

A resolution of the current inconsistency between measured protein conductivities and solution kinetic measurements can be sought in the special properties of interfaces where conductivity is measured. In fact, thin-film protein electrochemistry has long reported reorganization energies of electron transfer for azurin in the range of 0.2–0.3 eV, , far below the magnitudes of 0.8 eV reported from solution experiments. Another globular protein, cytochrome c, demonstrates reorganization energies from 0.58 ± 0.04 to 0.22 eV depending on the protocol of protein immobilization on the surface of a self-assembled monolayer (SAM). Similar reorganization energy magnitudes were reported for electron transport through living bacterial biofilms facilitated by intercytochrome electron hops in multiheme cytochrome nanowires.

A number of very recent measurements have suggested that interfaces can significantly lower the activation barrier for electron transfer or a half-reaction. − As such, measurements for ferrocene attached to flexible DNA linkers are particularly revealing. The reorganization energy of ≃0.84 eV for ferrocene in solution is reduced to nearly zero, < 0.02 eV, in a sub-10 nm nanogap between a planar substrate and the AFM tip. , A significant reduction of the activation barrier has been consistently reported in recent interfacial measurements and simulations. − ,

At least two physical effects are expected to change the reaction energetics at the interface: (i) heterogeneity of the reaction conditions contributing to the distribution of microstates sampled by the reacting system and (ii) confinement of the water solvent with dramatic alterations to the water electrostatics. − This paper studies these two medium effects on the energetics of protein electron transfer. We report simulations of the half-reaction of azurin protein in the protein powder and in confinement between two electrodes, and compare them to simulations in solution as reported in our previous study. We find that it is mostly a strong and inhomogeneous electric field at the interface that reduces the reaction activation barrier (option (i)) by coupling to the electronic polarizability of the protein active site. Effects of the interfacial electric field modifying electrochemical rates were considered in the past, but not in terms of coupling to the polarizability of the protein active site.

The protein powder was modeled with four azurin proteins in the simulation box (Figure a), solvated with different numbers N _w of TIP3P water molecules. Three configurations (labeled as W1, W2, and W3) were considered to reproduce typical values of the hydration level h (water mass/protein mass) in protein powders. The hydration levels h = 0.51, 1.13, and 3.25 were achieved with N _w = 1586 (W1), 3523 (W2), and 10138 (W3). To further explore the effect of the interface on the activation barrier of a redox reaction, we also produced simulation trajectories of azurin confined between two gold plates separated by d ≃ 4.4 nm. As we report below, this environment is very distinct from bulk water in polarity and effectively traps the protein both translationally and rotationally, over the length of the simulation trajectory, within the confined volume. Nevertheless, the activation barrier of a half-reaction is not strongly affected by the interfacial structure of confined water (option (ii) above).

1.

Azurin in powder (a) and in confinement between two gold electrodes (b). The Cu atom of the active site is rendered gold. Four proteins are placed in a simulation box with TIP3P N _w water molecules to model protein powder with hydration levels of h = 0.51, 1.13, and 3.25 (water mass/protein mass). Shown in (a) is the configuration with h = 3.25 and N _w = 10138 (W3 configuration). A single azurin protein is placed between two 16.6 × 16.7 nm gold slabs separated by d ≃ 4.4 nm and solvated with N _w = 37 470 TIP3P water molecules.

The rate constant of long-ranged electron transfer is traditionally viewed as decaying exponentially with the distance between the donor and acceptor , or, alternatively, with the distance R between a reacting molecule and the electrode

$$k_{\mathrm{ET}}=k_{\mathrm{NA}}\propto e^{-\gamma R}$$ 1

This is the result of an exponential distance decay of the electron tunneling probability entering the nonadiabatic (NA) rate constant, with the exponential decay parameter γ somewhat varying with the intervening medium. , However, a number of measurements of electrochemical rates − have indicated the appearance of a plateau of the rate constant when the distance between the electrode and the protein attached to a self-assembled monolayer (SAM) coating the electrode was decreased (Figure ). The lack of the rate’s distance dependence in the plateau region, and its dependence on solution viscosity in this regime, was attributed to the crossover from nonadiabatic electronic transitions, with an exponential rate falloff (eq ), to the medium dynamics control of the reaction rate when the rate constant pre-exponential factor is determined by the medium’s friction at the top of the activation barrier. Given that the tunneling probability is generic for many ET reactions, the key parameter specifying the crossover is the effective relaxation time of the medium τ_eff, affected by the dynamics of medium polarization (Stokes shift dynamics) and the dynamics of the donor–acceptor distance.

2.

Dependence of the standard electrochemical rate of the half-reaction to azurin depends on the distance to the electrode. The data (points) combine the new measurements (black) with previously reported results: red, blue, and magenta. The transformation from the number of methylene groups n in the self-assembled monolayer to R is according to the equation: R = 1.9 + 1.12 × n. The dashed line is the fit of new data to the nonadiabatic reaction rate constant (eqs and ) with the fitting parameters: Δ₀ = 6 × 10^–6 eV and γ = 0.95 Å^–1. The dotted blue line is the fit of blue points to the equation k _ET = A exp­[−bR]/(1 + B exp­[−bR]) to guide the eye. The experimental data in this plot are adapted with permission from Figure of Szuster, J.; Murgida, D. H. New experimental evidence challenges the long-held paradigm of dynamical control in electrochemical protein electron transfer. Electrochim. Acta 2025, 540, 147240. Copyright (2025), with permission from Elsevier.

Fitting electrochemical kinetic data for cytochrome c covalently attached to SAMs required an unreasonably long relaxation time τ_eff ≃ 400 ns, much longer than any relevant time scales acquired from molecular dynamics (MD) simulations. Fits of kinetic data were improved by assuming a very loose attachment of the protein to the electrode surface, with a root-mean-squared displacement of 2–3 Å. Such values are not found in our simulations of powders and confinement, presented below. It seems that the theory-experiment discord was resolved by recent experimental results by Murgida and co-workers, who pointed out that previous measurements suffered from systemic errors arising from uncompensated resistance of the electrochemical cell. According to new measurements, the plateau region disappears, and the rate constant remains exponential down to R ≃ 6 Å (Figure ). Our calculations presented here confirm this outcome by demonstrating no plateau of the rate constant for the parameters relevant to the half-reaction of protein electron transfer.

While the appearance of the rate constant plateau in electrochemical measurements seems unlikely, the typical values of the donor–acceptor electronic coupling and the effective relaxation time τ_eff in the protein environment allow the dynamically controlled regime for intraprotein electron transfer between donor and acceptor cofactors at donor–acceptor distances below R* ≃ 12 Å for azurin, and below R* ≃ 11 Å between cofactors in a photosynthetic heliobacterial reaction center. Currently, there is no direct experimental evidence of the dynamic control of intraprotein electron transfer. Indirect evidence comes from observations of a strong effect of water deuteration on protein charge conductivity, , explained by the effect of deuteration on the relaxation time affecting the rate constant in the plateau region. If calculations of the distance-dependent rates of protein electron transfer are supported by measurements, then most intraprotein electron-transfer hops should occur in the regime of dynamical control of the reaction rate.

Simulations reported here ask the question of whether conditions of protein powders and confinement can dramatically change the energetics of protein electron transfer. We contrast azurin in solution with the same protein in the protein powder and electrode confinement. These simulations can be viewed as mimicking conditions of experiments performed with proteins immobilized at interfaces. However, there is no direct match between the simulation protocols used here and the conditions of thin-film protein electrochemistry. Periodic boundary conditions imposed by MD simulation protocols require using two electrodes with the protein solution confined between them (Figure b). This configuration is thus more relevant to the experimental conditions in electrode junctions. The gold slabs used to cap the protein solution do not carry electrical charges, and hence do not provide a fair model of the metal electrodes. The effects of constraining proteins to powders and electrode confinement on the energetics of charge transport are the only questions that can be addressed in this simulation setup.

As we discuss below, solution, powder, and electrode confinement produce comparable effects on the activation barrier of the half-reaction when the active site is nonpolarizable. The strength of the electric field and the spread of its magnitudes, however, generally increase at the interface. The interaction of the medium electric field with the polarizability of the active site is what separates these different media. We indeed find that interfaces are effective in lowering the barrier of the half-reaction due to the coupling of the interfacial electric field to the active-site polarizability.

Results

Reorganization Energy of Electron Transfer

We monitor the energy gap between the quantum-mechanical energy level of the active site of the protein and the highest occupied electronic level in the metal with the electrochemical potential μ_m. This vertical energy gap becomes the reaction coordinate X for electron transfer in electrochemistry. − The instantaneous value of the reaction coordinate can be written as follows

$$X^{\mathrm{pol}}=\pm {\lambda }^{\mathrm{St}}+\delta X^{\mathrm{pol}}$$ 2

where δ(···) denotes the deviation from the average values, and the first term in this equation defines the Stokes-shift reorganization energy of electron transfer. It is given by the difference of the average values of the reaction coordinate in two oxidation states of the protein

$${\lambda }^{\mathrm{St}}=\frac{1}{2}[{⟨X^{\mathrm{pol}}⟩}_{\mathrm{Ox}}-{⟨X^{\mathrm{pol}}⟩}_{\mathrm{Red}}]$$ 3

as reflected by ± sign in eq , where ‘+’ is assigned to i = 1 = Ox and ‘–’ is assigned to i = 2 = Red. Equation applies to the equilibrium electrode potential and needs to be modified by adding eφ to the right-hand side when the overpotential φ is applied to the electrode; e is the elementary charge.

The fluctuating energy gap of a polarizable active site of the protein X ^pol includes two components. The first part, X ^C, is the change in the Coulomb interaction energy of the active site with the environment when the electron is transferred from the metal to the electrode. Given that the electronic density is generally delocalized over a few atoms of the active site, this physics is captured by assigning atomic charge differences Δq _j to a subset of atoms N _a of the active site. Obviously, the partial atomic charges satisfy the sum rule of ∑ _j Δq _j = −e. The charges Δq _j interact with the medium through site electrostatic potentials ϕ _sj to yield the overall Coulomb energy contributing to the instantaneous energy gap

$$X^{\mathrm{C}}=\sum _{j=1}^{N_{\mathrm{a}}}\mathrm{\Delta }{q}_j{\varphi }_{sj}$$ 4

The electronic density of the active site is also deformed (polarized) by electrostatic interactions with the medium. The simplest representation of this physics is in terms of the vacuum polarizability of the protein’s active site, which alters due to charge transfer. Assuming that the change in polarizability between Red and Ox states is Δα _i = α _Red(q _i ) – α _Ox(q _i ), one accounts for polarizability effects through the second term in the following equation

$$X^{\mathrm{pol}}=X^C-\frac{1}{2}\mathbf{E}·\mathrm{\Delta }{\mathit{\alpha }}_i·\mathbf{E}$$ 5

The polarizabilities α _i (q _j ) (i, j = Ox, Red) are second-rank tensors contracted in the last term of eq with the vector of the electric field E produced by the medium at the active site of the protein. The difference in polarizabilities, Δα _i , carries the dependence on the oxidation state for which it is calculated (index i) through the equilibrium nuclear coordinates q _i of the active site optimized in the corresponding oxidation state. The polarizability is therefore calculated based on the optimized geometry in the Ox state for the forward transition, and with optimized geometry in the Red state for the backward transition. The polarizability difference is then calculated by preserving the optimized geometry in each state and performing single-point calculations of the polarizability upon changing the oxidation state. For instance, Δα _Ox = α _Red(q _Ox) – α _Ox(q _Ox) is calculated based on the equilibrated optimized geometry q _Ox on the active site in the Ox state (Supporting Information). The reason for the more complex mathematics involved in the second term in eq is that this term is a free energy of polarizing the electronic density of the active site, in a given oxidation state, in contrast to energies involved in defining the Coulomb energy gap X ^C.

The reaction coordinate, as defined by eq , yields the free energy surfaces of electron transfer intersecting at the tunneling configuration of zero energy gap, X = 0 (either X ^C or X ^pol). Electrode overpotential in the electrochemical experiment plays the role of the reaction Gibbs energy in solution reactions, vertically offsetting the free energies of electron transfer to produce the observable electrode current. The effect of the overpotential on the activation barrier is described by the Marcus equation when the free energy surfaces are parabolic, but requires calculations involving nonparabolic free energy surfaces when the picture of shifted Marcus parabolas fails. , We will not consider nonzero overpotentials here, and instead will focus on the question of how the medium affects the barrier of a symmetric, zero-overpotential half-reaction. As such, we will define the activation barrier through the observable (reaction) reorganization energy λ ^r according to the standard equation of Marcus theory

$$\mathrm{\Delta }{F}^{†}=\frac{1}{4}{\lambda }^r$$ 6

This result directly follows from the Marcus theory of electrochemical reaction at zero overpotential. We stress, however, that in the case of nonparabolic free energy surfaces, λ ^r in eq needs to be directly calculated from the crossing of two free-energy surfaces of electron transfer F _i (X) at the tunneling configuration X = 0.

The reorganization energy of electron transfer in the standard Marcus picture can be defined either from the first moments of the reaction coordinate (eq ) or from the second moments of X. The latter yields the variance reorganization energy, defined separately for each oxidation state

$${\lambda }_i=\frac{1}{2}\beta {⟨{(\delta X^{\mathrm{pol}})}^2⟩}_i$$ 7

where β = (k _B T)⁻¹ is the inverse temperature.

Reorganization energies in the Ox and Red states of azurin, listed in Table , were calculated from variances of the energy gap in two oxidation states. The combination of the Stokes-shift and variance reorganization energies provides an estimate of the reaction reorganization energy when the free-energy surfaces can be approximated by parabolas near their corresponding minima. By adopting the effective curvature of two parabolas through the mean reorganization energy λ = (λ_Ox + λ_Red)/2, one can estimate the reaction reorganization energy as follows

$${\lambda }^r={({\lambda }^{\mathrm{St}})}^2/\lambda$$ 8

As mentioned above, direct calculations of the activation barrier are required to estimate λ ^r in eq from the activation barrier when free-energy surfaces are strongly nonparabolic.

1. Reorganization Energies (eV) for Different Configurations of the Protein in Solution, Powder, and Electrode Confinement.

prot.	λSt	λOx	λRed	λSt	λOx	λRed
	nonpolarizable			polarizable
	solution
λ(L)	1.17	1.16	1.17	1.04	1.16	3.12
λ(∞)	0.99	1.35	1.36	0.851	1.35	3.31
	protein powder
⟨λ⟩p	1.053	1.052	1.549	0.846	1.165	3.288
λ(∞)	0.768	1.337	1.834	0.561	1.450	3.573
	electrode confinement
λ(L)	1.37	1.60	1.55	1.14	1.61	8.74
λ(∞)	1.25	1.72	1.67	1.02	1.73	8.86

^aCalculated from MD in the simulation box of size L.

^bCorrected for finite-size effects to produce λ­(∞) and λ^St(∞).

^cCalculated from the distribution P̅(X) (eq ) averaged over the protein molecules in the powder.

Energetics of Electron Transfer

Electrochemical measurements consistently report a λ ^r of ≃0.2 – 0.3 eV for azurin, in contrast to a λ ^r of ≃0.8 eV in solution. Standard protocols ,, of molecular dynamics (MD) simulations of protein solutions focus on sampling the Coulomb energy gap X ^C (eq ) and typically yield λ ^r ≃ 1.5 eV, exceeding both the solution and electrochemical measurements. To account for the discrepancy with measurements, the polarizability of the active site was introduced to protein electron transfer. , While this formalism agrees with experimental reorganization energies of cytochrome c, calculations based on sums-over-states consistently underestimate polarizability, leading to reorganization energies of azurin exceeding electrochemical reports. We therefore re-evaluated here the polarizabilities of the Ox and Red states by allowing the geometry of the active site to relax in each oxidation state. The second-rank polarizability tensor was calculated from the linear dependence of the electronic induced dipole moment on the strength of the applied electric field (Supporting Information). As we show below, this procedure provides a more consistent account of the active-site polarizability, bringing the calculations in better accord with measurements.

Solution

Simulations of azurin in solution (see Supporting Information for the simulation protocol) were used to produce the free-energy surfaces of electron transfer from normalized distributions P _i (X) of the reaction coordinate according to the following equation

$$F_i(X)=-k_{\mathrm{B}}T\ln P_i(X)$$ 9

Figure a compares F _i (X) for polarizable and nonpolarizable active sites of azurin. The free energy surfaces cross at X = 0, and the reaction free energy is zero at zero electrode overpotential. The vertical distance between the crossing point and the free-energy minimum is used to determine the activation barrier and the reaction reorganization energy, λ ^r , according to eq (Table ). An estimate of λ ^r through eq is accurate only if the free-energy surfaces can be approximated by parabolas with similar curvatures. The deviation between the direct calculation of λ ^r and eq quantifies the asymmetry between F _i (X) and deviations from the parabolic shapes.

3.

Free energy surfaces of the half-redox reaction for azurin in solution (a), in powder (b), and in electrode confinement (c). The green and blue lines refer to a nonpolarizable active site (X ^C reaction coordinate), while red and orange lines refer to the polarizable active site (X ^pol reaction coordinate). The activation barrier ΔF ^† = F _i (0) – F _i,min is used to determine the reaction reorganization energy according to eq . In the case of protein powder, the free-energy surfaces refer to averages over four proteins in powder F̅ _i (X) as defined by eq .

2. Reorganization Energy λ ^r (eV) for Different Configurations of the Protein in Solution, Powder, and Electrode Confinement.

configuration	eq	eq	corr.	eq	eq	corr.
	nonpolarizable			polarizable
solution	0.92	1.18	0.57	0.56	0.51	0.25
protein powder	0.82	0.85	0.30	0.39	0.32	0.10
electrode confinement	0.99	1.19	0.74	0.30	0.25	0.16

^aCalculated from the activation barrier as λ ^r = 4ΔF ^† according to eq .

^bCalculated from eq by using λ­(L) without applying finite-size corrections.

^cCorrected as f _e λ^St(∞)²/λ­(∞) using λ­(∞) and λ^St(∞) (Table ), followed by multiplying with the correction factor f _e = 0.8 accounting for the solvent polarizability.

The free-energy surfaces are calculated without system-size corrections, and they are compared in Table (column 3) to eq with no such corrections from the results listed in Table . However, MD results for half-reactions involve significant system size effects , because altering the total charge of the simulation box in a half-reaction also alters its interaction with the uniform background charge appearing in the standard implementation of Ewald sums used to calculate Coulomb interaction in MD simulations. The finite-size correction term, scaling as 1/L with the box size L, is added to the variance reorganization energy λ _i and is subtracted from the Stokes-shift reorganization energy λ^St. The finite-size correction thus widens the gap between λ _i and λ^St. While eq gives results comparable to direct calculations from the crossing of free-energy surfaces, the use of size-corrected reorganization energies λ _i (∞) and λ^St(∞) in eq produces lower values (Table ).

The force fields employed in MD simulations are nonpolarizable and do not provide a complete account of the electronic polarization of the medium. Since only nuclear degrees of freedom participate in solvent reorganization of electron transfer, the corresponding polarization fluctuations are screened by fast electronic polarization, following adiabatic changes in nuclear configurations. This effect is accounted for through the Pekar factor in standard theories of polarons in crystals , and in theories of electron transfer. , The effect of electronic polarization is more complex in molecular theories of liquids, but can be empirically accounted for by multiplying reorganization energies from MD simulations by a factor, f _e ≃ 0.8. Accounting for the system-size and polarizability effects noticeably lowers the reaction reorganization energy. For instance, it drops from λ ^r ≃ 0.92 to 0.57 eV for the nonpolarizable azurin and from 0.56 to 0.25 eV for polarizable azurin in solution. Likewise, the drop is from λ ^r ≃ 0.82 to 0.30 eV for nonpolarizable azurin in the powder considered below, and from 0.39 eV to merely 0.10 eV for polarizable azurin in the same powder configuration.

The effect of active-site polarizability on the reorganization energies is more significant in the Red state than in the Ox state. This is related to different magnitudes of polarizability changes in the forward and backward half-reactions as quantified by the isotropic trace, Δα _i = (1/3)­Tr­[Δα _i ]: Δα_Ox = 22.9 ų and Δα_Red = – 75.5 ų (see Supporting Information for polarizability tensors). The larger change in the polarizability in the Red state (total protein charge, Q = −3) combines with a stronger electric field at the active site (Figure ) to produce an overall much stronger effect on the free-energy surfaces and the corresponding reorganization energies.

4.

Active site of azurin (a) with electric field, E, from the protein–water medium calculated at the Cu atom of the active site. Normalized distributions of electric-field magnitudes in solution (b), powder (c), and in confinement (d) in Ox (blue) and Red (green) states of azurin.

Protein Powder

The free-energy surfaces for proteins in powders require a more careful definition, as half-reactions can occur to multiple proteins with heterogeneous electrostatics. This complication applies to most conditions of electrochemical interfacial reactions, and our simulations of protein powders provide us with a glimpse of the significance of medium heterogeneity in protein electrochemistry.

The statistical definition of the free-energy surfaces for powders includes the average of individual distributions P _i (X) for separate redox proteins

$$e^{-\beta {\stackrel{-}{F}}_i(X)}=\stackrel{-}{P}(X)=N_{\mathrm{p}}^{-1}\sum _{m=1}^{N_{\mathrm{p}}}{P}_i^{(m)}(X)$$ 10

where in our simulations the number of proteins in the powder N _p = 4, and P _i (X) are distributions of the energy gap for individual proteins. This definition of the free-energy surface assumes a certain separation of time scales: the half-reaction is assumed to be slower than the time of transitions between configurations specific to individual proteins, as characterized by the transition time, τ_p. If this is not the case, then the heterogeneous average over individual proteins should apply to the population dynamics of the electron density in the Red or Ox state with the rate constant k _p. The overall decay of the electron population comes as the average over exponential decay channels for individual proteins, similarly to that in eq : ⟨exp­[−k _p t]⟩_p. This limiting case implies that there are proteins in the powder with large values of the variance reorganization energies, λ _i, that will have a low activation barrier (eq ) and correspondingly fast reaction rates. Conductivity of protein powders will always be channeled through those active configurations when k _pτ_p ≫ 1. The latter situation is consistent with electrode-junction experiments, while slow half-reactions, from submilliseconds to seconds, are characteristic of thin-film electrochemistry (Figure ).

Assuming the separation of time scales (slow half-reactions, k _pτ_p ≪ 1), the distributions of the Coulomb energy gap, X ^C, are heterogeneous between four proteins in the simulation box (Figure a for the Red state and Figure S4a for the Ox state). One obtains an increased reorganization energy, ⟨λ⟩_p, calculated from the variance of P̅(X) in eq , as a consequence of this heterogeneity (Table ). A larger reorganization energy reflects shallower free-energy surfaces F- _i (X) calculated from eq (Figure c). However, the distribution of the electric field at azurin’s active site is by far more heterogeneous than that of the Coulomb energy gap X ^C (Figures b and S4b), leading to a greater spread of distributions, P _i (X), calculated for polarizable active sites (Figure b). As a result, the free-energy surfaces F̅ _i (X) are much shallower than individual F _i (X) (Figure d), and the variance reorganization, ⟨λ⟩ _p , calculated from P̅(X), is substantially increased (Table ). Overall, a heterogeneous distribution of the electric field in the protein powder helps to significantly lower the activation barrier for a half-reaction.

5.

Normalized distribution (dashed lines) of the energy gap, X ^C, (a) and X ^pol (b) for four proteins in the simulation box with N _w = 1586 water molecules (W1 configuration) in azurin’s Red state. Panels (c, d) show the corresponding free-energy surfaces according to eq . The black solid lines refer to average distributions, P-(X), and free energies F-(X), calculated according to eq .

Electrode Confinement

In the last configuration studied here, azurin protein was placed between two Au(111) gold slabs separated by a distance of d ≃ 4.4 nm (see the Supporting Information for the simulation protocol). The direction perpendicular to the two plates is assigned to the z-axis of the laboratory frame. The distribution of azurin positions P(z) along the z-axis yields the potential of mean force (PMF), F(z) = −k _B T ln P(z) (Figure ). The position of azurin is defined as the z-coordinate of its copper atom relative to the topmost gold atom in the lower Au(111) slab. The PMF profiles are approximately parabolic, with the average distances to the slab of 2.9 (Ox) and 3.4 (Red) nm. The standard deviations from the minima are ≃ 0.02 nm. The protein is thus strongly constrained in its motion relative to the plates, resembling weak electrostatic or van der Waals attachment to a surface monolayer in thin-film electrochemistry. This confinement is also similar to what we find in protein powders since the corresponding PMFs are very close (filled vs open points in Figure ). As we discuss below, in addition to being positionally constrained, azurin is rotationally arrested in powders and in electrode confinement.

6.

Potential of mean force, F(z) = −k _B T ln P(z), for azurin in electrode confinement (filled points) and in the protein powder (open points). The free- energy functions along the z-axis perpendicular to the two gold slabs are calculated from the corresponding position probabilities, P(z). The curves in the powder (W1 configuration) are shifted horizontally to allow a comparison.

To summarize, calculations of reorganization energies based on the Coulomb energy gap X ^C produce consistent results in all three media with no dramatic effect of changing the protein environment on the activation barrier of the half-reaction (Tables and ). The reaction reorganization energies are 0.3–0.7 eV in all three media when the polarizability of the active site is neglected. They reduce to 0.1–0.25 eV for the polarizable active site. The latter range of values implies an activation barrier at the level of a few k _B T, and essentially no temperature dependence of the reaction rate. The lowest activation barriers are found for protein powders, where heterogeneously distributed electric fields at protein active sites allow parts of the ensemble with very low activation barriers. An exponential dependence of the rate constant on the activation barrier will predominantly select those low-barrier proteins for charge conductivity.

Dynamical Control of Redox Half-Reactions

Kinetic measurements of protein reaction rates in solution are typically analyzed in the framework of the Marcus–Levich theory of long-distance electron transfer. , It requires three parameters to determine the reaction rate: electronic coupling, V _e , reorganization energy of electron transfer, λ, and Gibbs energy of the reaction, ΔF ₀. In the case of electrode reactions, the reaction Gibbs energy is replaced with the electrode overpotential φ, ΔF ₀ = eφ that can be continuously swept in the electrochemical experiment.

The model of the electrochemical discharge adopted elsewhere and here is that the redox protein resides in a weak harmonic potential originating from interfacial interactions and keeping it at the equilibrium distance, R _e , from the electrode. Such weak harmonic PMFs are indeed realized in our simulations of proteins in powders and electrode confinement (Figure ). Electronic delocalization between the protein and the metal electrode is defined through the parameter, Δ = πV _μ ρ_μ, in terms of direct electronic coupling, V _μ, between the reactant and the Fermi level of the metal and the density of electronic conduction states at the Fermi level, ρ_μ. Since V _μ decays exponentially with the distance, the distance dependence of Δ can be written in the form

$$\mathrm{\Delta }(R)={\mathrm{\Delta }}_e{e}^{-\gamma \delta R}$$ 11

where δR = R – R _e is the deviation from the equilibrium distance R _e and γ ∼ 1 Å^–1 is the exponential falloff parameter describing the lowering of the electron tunneling probability with widening of the tunneling barrier.

Electronic delocalization leads to electronic transitions between the protein and the metal electrode, with the golden-rule rate constant calculated by summation over the Fermi distribution of conduction electrons ,

$$k_{\mathrm{NA}}=\frac{⟨\mathrm{\Delta }⟩}{\hslash }\mathrm{erfc}\left[\frac{{\lambda }^r+e\phi }{\sqrt{4{\lambda }^r{k}_{\mathrm{B}}T}}\right]$$ 12

where erfc­(x) is the complementary error function. The average electronic coupling, ⟨Δ⟩, in this equation comes from averaging of the distance-dependent Δ­(R) over thermally induced oscillations in the confining PMF, F(z) (Figure ). Assuming a harmonic PMF, one gets

$$⟨\mathrm{\Delta }⟩={\mathrm{\Delta }}_e{e}^{{\gamma }^2⟨\delta R^2⟩/2}={\mathrm{\Delta }}_0{e}^{-\gamma R_e}$$ 13

Thermal fluctuations of the metal-electrode distance thus lead to an enhancement of the effective electronic coupling accounted for in the parameter Δ₀ in eq .

Equations and still predict an exponential falloff of the reaction rate due to an exponential decay of ⟨Δ⟩ with the equilibrium distance to the electrode. A major modification of the rate expression is achieved by incorporating the medium (protein and water) dynamics, which compete with tunneling and thus reduce the reaction rate at the top of the activation barrier. − The rate constant accounting for the medium dynamics is obtained by dividing the nonadiabatic rate constant in eq by the correction term 1 + g to arrive at the following equation for the reaction rate constant

$$k_{\mathrm{ET}}={(1+g)}^{-1}{k}_{\mathrm{NA}}$$ 14

The crossover parameter, g ∝⟨Δ⟩τ_eff, is proportional to the product of ⟨Δ⟩ and an effective medium relaxation time τ_eff. Decreasing the distance to the electrode (increasing ⟨Δ⟩) or slowing the medium dynamics (increasing τ_eff) brings the reaction to the dynamically controlled regime at the crossover distance R = R* satisfying the condition g(R*) = 1. At R < R*, ⟨Δ⟩ cancels out from the nominator and denominator in eq , and the rate constant switches from the nonadiabatic, distance-dependent function to the limit of Kramers’ kinetics − k _ET ∝ τ_eff when the rate constant reaches a plateau as a function of R.

The effective relaxation time is a combined effect of the dynamics of the energy gap X(t) and the dynamics of the distance to the electrode R(t). The latter affects τ_eff through the time τ_γ required for the protein to diffuse, with the diffusion constant D, through the distance of tunneling decay γ^–1

$${\tau }_{\gamma }={({\gamma }^{2}D)}^{-1}$$ 15

Assuming that the energy-gap dynamics are characterized by the relaxation time τ_X, the dynamical crossover parameter is determined by the following equation

$$g={\tau }_{\mathrm{eff}}\frac{⟨\mathrm{\Delta }⟩}{\hslash }$$ 16

with

$${\tau }_{\mathrm{eff}}=2{\tau }_{\gamma }{e}^{{\gamma }^2⟨\delta R^2⟩}h({\tau }_{\mathrm{X}},{\tau }_{\gamma })$$ 17

$$h({\tau }_{\mathrm{X}},{\tau }_{\gamma })=1-\frac{x_0}{\sqrt{x_0^2+4{\tau }_{\mathrm{X}}/{\tau }_{\gamma }}}$$ 18

and $x_0=({\lambda }^r+e\phi )/\sqrt{2{\lambda }^r{k}_{\mathrm{B}}T}$ (φ = 0 is assumed here).

Equation was derived in ref for single-exponential decays of the time autocorrelation functions of X(t) and R(t). However, both the Stokes-shift , (X(t)) and distance (R(t)) dynamics are multiexponential in our simulations, and an extension of the theoretical framework to multiple relaxation times is required.

All steps of the formalism leading to eqs – can be retraced, when the equilibrium ensemble averages are augmented with averages over the relaxation times τ _X and τ _R representing single-exponential relaxations in multiexponential correlation functions. This extension is the formalism of Markovian embedding. , The dynamics of two coordinates Y (t) = X(t), R(t) are described by two time autocorrelation functions C _Y(t) = ⟨δY (t)­δY (0)⟩, where δY (t) = Y (t) – ⟨Y ⟩ _i . The corresponding normalized time correlation functions S _Y(t) = C _Y(t)/C _Y(0) are fitted to sums of decaying exponents with relaxation times τ_X,j and τ_R,j

$$\begin{array}{lll}& S_{\mathrm{X}}(t)=\sum _j{a}_j{\mathrm{e}}^{-t/{\tau }_{\mathrm{X},j}},& \sum _j{a}_j=1\\ & S_{\mathrm{R}}(t)=\sum _k{b}_k{\mathrm{e}}^{-t/{\tau }_{\mathrm{R},k}},& \sum _k{b}_k=1\end{array}$$ 19

The relation for an effective relaxation time thus involves the average over multiple relaxation times, assigned weights a _j and b _k, in the corresponding multiexponential representations

$${\tau }_{\mathrm{eff}}=2{\mathrm{e}}^{{\gamma }^2⟨\delta R^2⟩}\sum _{j,k}{a}_j{b}_k{\tau }_{\gamma ,k}h({\tau }_{\mathrm{X},j},{\tau }_{\gamma ,k})$$ 20

where τ_γ,k = τ_R,k /(γ²⟨(δR)²⟩.

The dynamics of X(t) are consistent between the three media studied here, producing average (integrated) relaxation times τ_X in the range of 10–100 ps (Tables S6, S7, and S9). The dynamics are also linear, showing essentially no distinction between normalized time correlation functions in the Ox and Red states, as shown in Figure , which compares the results for the Coulomb component X ^C(t) (eq ) with the energy-gap dynamics of polarizable azurin (eq ). We find a significant slowing down when the active site is polarizable, reflected in a much slower long-time relaxation tail. For instance, the relaxation time in solution grows from about 0.05 ns to roughly 2 ns in the Ox state, while increasing from about 0.01 ns to over 35 ns for electrode confinement in the Red state. In the powder systems, particularly at intermediate (W2) and high (W3) hydration, τ_X shifts from subnanoseconds to tens of nanoseconds.

7.

Normalized time autocorrelation functions of nonpolarizable (green and blue) and polarizable (orange and red) azurin in solution (a), in protein powder (b), and in electrode confinement (c). Black, dotted lines indicate multiexponential fits (see Supporting Information).

Based on eq , both the slow dynamics of the electric field E and the slow rotations of the anisotropic polarizability tensor Δα can potentially contribute to much slower energy-gap dynamics when polarizability is included. We find that rotations of the protein, occurring on a time scale of ≃15 ns in solution, are fully arrested in powders and in confinement. The electric field is thus the only medium mode responsible for the slow relaxation tails of S _X(t). A correlation between τ_X and τ_E is, however, hard to establish (Table ). We also note that the orientational arrest of the protein in powders and confinement found from MD is recorded on a simulation time of ≤1 μs. In comparison, proteins immobilized on SAMs in thin-film electrochemical experiments show distributions of orientations occurring on a millisecond time scale. These slow dynamics are obviously not captured by MD simulations.

3. Average Relaxation Times of the Energy Gap τ_X (ns) for the Nonpolarizable (np) and Polarizable (pol) Active Sites .

	Ox			Red
system	τX	τX	τE	τX	τX	τE
solution	0.048	1.19	8.5	0.017	1.43	15.0
powder (W2)	0.080	7.29	1.1	0.078	16.57	6.2
confined	0.007	3.74	3.6	0.011	35.28	11.5

^aAlso shown are relaxation times τ_E (ns) of the electric field at the active site.

The dynamics of the energy gap become nonlinear when polarizability is included. The relaxation time τ_X depends on the oxidation state, which is a general property of Stokes-shift dynamics involving a changing polarizability. We find up to an order-of-magnitude ratio of τ_X in the Red and Ox states (Table ). When the reaction is in the dynamical regime, this high ratio helps to channel electrons in a specific direction in intraprotein charge transport. However, we do not find the dynamic regime feasible for electrochemical electron transfer.

Slower energy-gap dynamics of polarizable azurin are still insufficient to reach g _i > 1 for the dynamic crossover parameter. We used recent data for azurin electrochemical rates (dashed line in Figure ) to extract the electronic coupling Δ₀ = 6 × 10^–6 eV and the tunneling decay parameter γ = 0.95 Å^–1 from fitting the measured rates to eq . These parameters, along with exponential relaxation times and relative weights of decaying exponents fitting the time correlation functions (Supporting Information), were used in eq to calculate the effective relaxation time and the dynamical crossover parameter (eq ). We find (eq ) τ_eff ≃ 0.8 – 1.8 ns in the protein powder and τ_eff ≃ 0.1 – 0.2 ns in the electrode confinement for polarizable azurin. These values are significantly lower than the τ_eff ≃ 0.4 μs required to fit experimental data. Indeed, when combined with the fitted value of ⟨Δ⟩, these values of τ_eff yield the dynamic crossover parameters below unity in the entire range of distances relevant for thin-film protein electrochemistry , (Figure ). The correction factor (1 + g)⁻¹ in eq can be neglected, and we conclude that the dynamical regime of electron transfer should not be observed in electrochemical measurements, provided the conditions of thin-film electrochemistry are consistent with confinement effects studied by MD simulations here. The rate constant should not reach a plateau as a function of the electrode monolayer thickness and instead is expected to show an exponential distance decay. However, similar parameters relevant for intraprotein electron transfer between the donor and acceptor cofactors allow the dynamical regime of electron transfer to occur at donor–acceptor distances below R* ≃ 10 – 14 Å. ,

8.

g _i calculated for Ox (blue, i = 1) and Red (green, i = 2) states of polarizable azurin in powder (W1 configuration) and in electrode confinement (Conf.); λ ^r = 0.3 eV, Δ₀ = 6 × 10^–6 eV, and γ = 0.95 Å^–1 are adopted in the calculations. The exponential relaxation times and fractional amplitudes a _j , b _k in eq are taken from exponential fits of corresponding time correlation functions, as explained in the Supporting Information.

Discussion

This study addresses three major problems that persisted in experimental and theoretical literature on protein electron transfer: (i) the distinction between activation barriers of protein electron transfer reported from solution and electrochemical measurements, (ii) a disconnect between reports of the plateau region in electrochemical kinetic measurements and theories of solvent dynamical control of electron transfer, and (iii) low activation barriers and apparent lack of temperature dependence for electron transport through protein monolayers and multilayers in solid-state metal/protein/metal junctions. Temperature-independent electron transport over the length of tens of nanometers is inconsistent with either tunneling or large-barrier hopping. , Issues (i) and (iii) are connected and can potentially be attributed to special properties of interfaces, allowing significantly lower activation barriers. Issue (ii) has recently been resolved by recognizing that previously reported plateau regions result from deficiencies in experimental protocols. Consistent with new experimental data, the theory and simulations presented here do not allow solvent dynamical control or plateau regions in thin-film electrochemistry.

Proteins in powders and in water confinement reside in media significantly different in many physical properties from bulk solutions. One might anticipate that the special structure of interfacial water might contribute to lower barriers for protein electron transfer. Despite these expectations, our simulations have shown that interfacial water structure only mildly affects the activation barrier of a half-electrode reaction (Table ). The main factor lowering the activation energy of electron transfer is the active-site polarizability, depending on the oxidation state. Special properties of the interface in accelerating electron transfer are, therefore, not related to a specific structure of interfacial water but to strong and inhomogeneous interfacial electric fields coupled to the active-site polarizability. The protein powder is particularly conducive to accelerating redox reactions, since the field distribution is highly heterogeneous. Accounting for active-site polarizability brings activation barriers that are in accord with measurements reported by thin-film electrochemistry. The activation barriers become low, ∼2k _B T. This is comparable to the activation barrier of ∼k _B T reported from measurements of conductivity in bacteriorhodopsin junctions and can potentially explain the small temperature effects on hopping rates measured in electrode confinement.

Slow dynamics of interfacial electric fields make the energy-gap dynamics (Stokes-shift dynamics) significantly slower than what follows solely from Coulomb interactions of the transferred electron with medium polarization. Nevertheless, even these slow dynamics are insufficient to produce the dynamical crossover parameter above the threshold value g = 1 (eq ) to turn the reaction into the dynamically controlled regime. We find no plateau for the half-reaction rate constant as a function of the distance from the electrode in all environments studied here.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c12456.

• (i) Simulation protocols of azurin protein in solution, powders, and in electrode confinement, (ii) analysis of the energy-gap statistics produced by MD simulations and calculations of finite-size corrections, (iii) fitting of time correlation functions describing protein and water dynamics affecting protein half reactions, (iv) details of quantum calculations of polarizability of azurin’s active site in two oxidation states (PDF)

The authors declare no competing financial interest.