Extensive simulations of cytochrome c in solution are performed to address the apparent contradiction between large reorganization energies of protein electron transfer typically reported by atomistic simulations and much smaller values produced by protein electrochemistry. The two sets of data are reconciled by deriving the activation barrier for electrochemical reaction in terms of an effective reorganization energy composed of half the Stokes shift (characterizing the medium polarization in response to electron transfer) and the variance reorganization energy (characterizing the breadth of electrostatic fluctuations). This effective reorganization energy is much smaller than each of the two components contributing to it and is fully consistent with electrochemical measurements. Calculations in the range of temperatures between 280 and 360 K combine long, classical molecular dynamics simulations with quantum calculations of the protein active site. The results agree with the Arrhenius plots for the reaction rates and with cyclic voltammetry of cytochrome c immobilized on self-assembled monolayers. Small effective reorganization energy, and the resulting small activation barrier, is a general phenomenology of protein electron transfer allowing fast electron transport within biological energy chains.

A combination of experimental data and theoretical analysis provides evidence of a bell-shaped kinetics of electron transfer in the Arrhenius coordinates ln k vs 1/T. This kinetic law is a temperature analogue of the familiar Marcus bell-shaped dependence based on ln k vs the reaction free energy. These results were obtained for reactions of intramolecular charge shift between the donor and acceptor separated by a rigid spacer studied experimentally by Miller and co-workers. The non-Arrhenius kinetic law is a direct consequence of the solvent reorganization energy and reaction driving force changing approximately as hyperbolic functions with temperature. The reorganization energy decreases and the driving force increases when temperature is increased. The point of equality between them marks the maximum of the activationless reaction rate. Reaching the consistency between the kinetic and thermodynamic experimental data requires the non-Gaussian statistics of the donor–acceptor energy gap described by the Q-model of electron transfer. The theoretical formalism combines the vibrational envelope of quantum vibronic transitions with the Q-model describing the classical component of the Franck–Condon factor and a microscopic solvation model of the solvent reorganization energy and the reaction free energy.

We present a model of the dynamical transition of atomic displacements in proteins. Increased mean-square displacement at higher temperatures is caused by the softening of the force constant for atomic/molecular displacements by electrostatic and van der Waals forces from the protein–water thermal bath. Displacement softening passes through a nonergodic dynamical transition when the relaxation time of the force–force correlation function enters, with increasing temperature, the instrumental observation window. Two crossover temperatures are identified. The lower crossover, presently connected to the glass transition, is related to the dynamical unfreezing of rotations of water molecules within nanodomains polarized by charged surface residues of the protein. The higher crossover temperature, usually assigned to the dynamical transition, marks the onset of water translations. All crossovers are ergodicity breaking transitions depending on the corresponding observation windows. Allowing stretched exponential relaxation of the protein–water thermal bath significantly improves the theory–experiment agreement when applied to solid protein samples studied by Mössbauer spectroscopy.

The Marcus theory of electron transfer predicts a bell-shaped dependence of the reaction rate on the reaction free energy. The top of the “inverted parabola” corresponds to zero activation barrier when the electron-transfer reorganization energy and the reaction free energy add up to zero. Although this point has traditionally been reached by altering the chemical structures of donors and acceptors, the theory suggests that it can also be reached by varying other parameters of the system including temperature. We find here dramatic evidence of this phenomenon from experiments on a fullerene–porphyrin dyad. Following photoinduced electron transfer, the rate of charge recombination shows a bell-shaped dependence on the inverse temperature, first increasing with cooling and then decreasing at still lower temperatures. This non-Arrhenius rate law is a result of a strong, approximately hyperbolic temperature variation of the reorganization energy and the reaction free energy. Our results provide potentially the cleanest confirmation of the Marcus energy gap law so far since no modification of the chemical structure is involved.

We report 11 μs of molecular dynamics simulations of the electron-transfer reaction between primary and secondary quinone cofactors in the bacterial reaction center. The main question addressed here is the mechanistic reason for unidirectional electron transfer between chemically identical cofactors. We find that electron is trapped at the secondary quinone by wetting of the protein pocket following electron transfer on the time-scale shorter than the backward transition. This mechanism provides effective rectification of the electron transport, making the reaction center a molecular diode operating by cyclic charge-induced electrowetting.

The network of hydrogen bonds characteristic of bulk water is significantly disturbed at the protein–water interface, where local fields induce mutually frustrated dipolar domains with potentially novel structure and dynamics. Here the dipolar susceptibility of hydration shells of lysozyme is studied by molecular dynamics simulations in a broad range of temperatures, 140–300 K. The real part of the susceptibility passes through a broad maximum as a function of temperature. The maximum shifts to higher temperatures with increasing frequency of the dielectric experiment. This phenomenology is consistent with that reported for bulk relaxor ferroelectrics, where it is related to the formation of dipolar nanodomains. Nanodomains in the hydration shell extend 12–15 Å from the protein surface into the bulk. Their dynamics are significantly slower than the dynamics of bulk water. The domains dynamically freeze into a ferroelectric glass below 160 K, at which point the Arrhenius plot of the dipolar relaxation time becomes significantly steeper.

An exact, closed-form solution is obtained for the line shape function of an optical transition with the transition frequency depending linearly plus quadratically on a Gaussian coordinate of the thermal bath. The dynamical modulation of the line shape involves two parameters corresponding to the linear and quadratic components of the transition frequency. The increase of the second component results in a non-Gaussian line shape that splits into two Lorenzian lines in the limit of fast modulation.

Signatures of nonlinear and non-Gaussian dynamics in time-resolved linear and nonlinear (correlation) 2D spectra are analyzed in a model considering a linear plus quadratic dependence of the spectroscopic transition frequency on a Gaussian nuclear coordinate of the thermal bath (quadratic coupling). This new model is contrasted to the commonly assumed linear dependence of the transition frequency on the medium nuclear coordinates (linear coupling). The linear coupling model predicts equality between the Stokes shift and equilibrium correlation functions of the transition frequency and time-independent spectral width. Both predictions are often violated, and we are asking here the question of whether a nonlinear solvent response and/or non-Gaussian dynamics are required to explain these observations. We find that correlation functions of spectroscopic observables calculated in the quadratic coupling model depend on the chromophore’s electronic state and the spectral width gains time dependence, all in violation of the predictions of the linear coupling models. Lineshape functions of 2D spectra are derived assuming Ornstein–Uhlenbeck dynamics of the bath nuclear modes. The model predicts asymmetry of 2D correlation plots and bending of the center line. The latter is often used to extract two-point correlation functions from 2D spectra. The dynamics of the transition frequency are non-Gaussian. However, the effect of non-Gaussian dynamics is limited to the third-order (skewness) time correlation function, without affecting the time correlation functions of higher order. The theory is tested against molecular dynamics simulations of a model polar–polarizable chromophore dissolved in a force field water.

Proteins function by sampling conformational sub-states within a given fold. How this configurational flexibility and the associated protein dynamics affect the rates of chemical reactions are open questions. The difficulty in exploring this issue arises in part from the need to identify the relevant nuclear modes affecting the reaction rate for each characteristic time-scale of the reaction. Proteins as reaction media display a hierarchy of such nuclear modes, of increasingly collective character, that produce both a broad spectrum of static fluctuations and a broad spectrum of relaxation times. In order to understand the effect of protein dynamics on reaction rates, we have chosen to study a sub-nanosecond electron transfer reaction between the bacteriopheophytin and primary quinone cofactors of the photosynthetic bacterial reaction center. We show that dynamics affects the activation barrier of the reaction through a dynamical restriction of the configurational space sampled by the protein–water solvent on the reaction time-scale. The modes which become dynamically arrested on the reaction time-scale of hundreds of picoseconds are related to elastic motions of the protein that are strongly coupled to the hydration layer of water. Several mechanistic consequences for protein electron transfer emerge from this picture. Importantly, energy parameters used to define the activation barrier of electron transfer reactions lose their direct connection to equilibrium thermodynamics and become dependent in a very direct way on the relative magnitudes of the reaction and nuclear reorganization time-scales. As a result, the energetics of protein electron transfer need to be defined on each specific reaction time-scale. This perspective offers a mechanism to optimize protein electron transfer by tuning the reaction rate to the relaxation spectrum of the reaction coordinate.

We report atomistic molecular dynamics simulations (200 ns) of the first, rate-limiting electron transfer in the electron transport chain in a bacterial bc1 complex. The dynamics of the energy gap between the donor and acceptor states include slow components, on the time-scale of tens of nanoseconds. These slow time-scales are related to large-scale elastic motions of the membrane-bound protein complex, which modulate both electrostatic and induction interactions of the electron with the protein–water–lipid thermal bath. The combined effect of these interactions is a high, ∼ 5 eV, reorganization energy of electron transfer as calculated from their variance. The reorganization energy does not reach equilibrium on the length of simulations and the system is nonergodic on this time-scale. To account for nonergodicity, two reorganization energies are required to describe the activation barrier, and their ratio is tuned by the relative time-scales of nuclear reorganization and of the reaction.Keywords: bioenergetics; non-Gaussian statistics; nonergodic chemical kinetics; protein dynamics; protein electron transfer;

We show that electrostatic fluctuations of the protein–water interface are globally non-Gaussian. The electrostatic component of the optical transition energy (energy gap) in a hydrated green fluorescent protein is studied here by classical molecular dynamics simulations. The distribution of the energy gap displays a high excess in the breadth of electrostatic fluctuations over the prediction of the Gaussian statistics. The energy gap dynamics include a nanosecond component. When simulations are repeated with frozen protein motions, the statistics shifts to the expectations of linear response and the slow dynamics disappear. We therefore suggest that both the non-Gaussian statistics and the nanosecond dynamics originate largely from global, low-frequency motions of the protein coupled to the interfacial water. The non-Gaussian statistics can be experimentally verified from the temperature dependence of the first two spectral moments measured at constant-volume conditions. Simulations at different temperatures are consistent with other indicators of the non-Gaussian statistics. In particular, the high-temperature part of the energy gap variance (second spectral moment) scales linearly with temperature and extrapolates to zero at a temperature characteristic of the protein glass transition. This result, violating the classical limit of the fluctuation–dissipation theorem, leads to a non-Boltzmann statistics of the energy gap and corresponding non-Arrhenius kinetics of radiationless electronic transitions, empirically described by the Vogel–Fulcher–Tammann law.

Theories of activated transitions traditionally separate the dynamics and statistics of the thermal bath in the reaction rate into the preexponential frequency factor for the dynamics and a Boltzmann factor for the statistics. When the reaction rate is comparable to relaxation frequencies of the medium, the statistics loses ergodicity and the activation barrier becomes dependent on the medium dynamics. This scenario is realized for mixed-valence self-exchange electron transfer at temperatures near the point of solvent crystallization. These complexes, studied by Kubiak and coworkers, display anti-Arrhenius temperature dependence on lowering temperature when approaching crystallization; that is, the reaction rate increases nonlinearly in Arrhenius coordinates. Accordingly, the solvent relaxation slows down following a power temperature law. With this functional form for the relaxation time, nonergodic reaction kinetics accounts well for the observations.Keywords: electron delocalization; nonergodic reorganization energy; Stokes shift dynamics;

Linear solvation theories are well established to describe electrostatic hydration of small solutes when the hydration free energy is dominated by the electrostatic free energy of the solute multipole. In contrast, hydration of nanometer solutes is driven by surface hydration. We address the question of whether the linear-response thermodynamics established for small multipolar solutes applies to surface hydration. To this end, molecular dynamics simulations are carried out on a model C180 solute that carries no global multipole, but the surface of which is decorated with radially pointing dipoles. Linear response is dramatically violated in this case. Further, two crossovers in the solvation thermodynamics are discovered as the surface polarity is increased. Both transformations produce strongly nonlinear solvation response. The second, more collective, crossover leads to a dramatic slowing down of the interfacial dynamics, reaching the time-scales of nanoseconds. Our picture offers the possibility of flipping water domains at interfaces of nanoparticles and biomolecules.Keywords: dynamical slowdown; hydration; non-Gaussian statistics; nonlinear solvation; solvation thermodynamics; surface transition;

We report numerical simulations of three hydrated heme proteins, myoglobin, cytochrome c, and cytochrome B562. The properties of interest are the dynamics and statistics of the electric field and electrostatic potential at heme’s iron, as well as their separation into the protein and water components. We find that the electric field produced by both the protein and the hydration water relaxes on the time scale of 3–6 ns, and the relaxation time of the electrostatic potential is close to 1 ns. The slow dynamics of the electrostatic observables is accompanied by their large variances. For the electrostatic potential, a large amplitude of its fluctuations leads to a gigantic reorganization energy of a half redox reaction changing the redox state of the protein. Both a large magnitude and a slow relaxation time of the electric field fluctuations are required to explain the onset of large mean-square displacements of iron at the point of protein’s dynamical transition. These requirements are met by the simulations which are used to explain the temperature dependence of heme iron displacements measured by Mössbauer spectroscopy. All three phenomena, (i) nanosecond dynamics, (ii) protein dynamical transition and a large high-temperature excess of atomic mean-square displacements, and (iii) the gigantic reorganization energy, are explained here by one physical mechanism. This mechanism involves two components: nanosecond motions of the protein surface residues and polarization of the interfacial water by the protein charges. Global nanosecond conformations of the protein move the surface water. Since water is polarized, these movements create large-amplitude electrostatic fluctuations, sufficient to modify displacements of groups inside the protein and yield reorganization energies of protein electron transfer far exceeding those found for small molecules. Water follows adiabatically the protein motions. Therefore, the relaxation times of the protein and its hydration layer are close, leading to matching temperatures of the dynamical transition for the two components.

Despite its diversity, life universally relies on a simple basic mechanism of energy transfer in its energy chains—hopping electron transport between centers of electron localization on hydrated proteins and redox cofactors. Since many such hops connect the point of energy input with a catalytic site where energy is stored in chemical bonds, the question of energy losses in (nearly activationless) electron hops, i.e., energetic efficiency, becomes central for the understanding of the energetics of life. We show here that standard considerations based on rules of Gibbs thermodynamics are not sufficient, and the dynamics of the protein and the protein–water interface need to be involved. The rate of electronic transitions is primarily sensitive to the electrostatic potential at the center of electron localization. Numerical simulations show that the statistics of the electrostatic potential produced by hydration water are strongly non-Gaussian, with the breadth of the electrostatic noise far exceeding the expectations of the linear response. This phenomenon, which dramatically alters the energetic balance of a charge-transfer chain, is attributed to the formation of ferroelectric domains in the protein's hydration shell. These dynamically emerging and dissipating domains make the shell enveloping the protein highly polar, as gauged by the variance of the shell dipole which correlates with the variance of the protein dipole. The Stokes-shift dynamics of redox-active proteins are dominated by a slow component with the relaxation time of 100–500 ps. This slow relaxation mode is frozen on the time-scale of fast reactions, such as bacterial charge separation, resulting in a dramatically reduced reorganization free energy of fast electronic transitions. The electron transfer activation barrier becomes a function of the corresponding rate, self-consistently calculated from a non-ergodic version of the transition-state theory. The peculiar structure of the protein–water interface thus provides natural systems with two “non's”—non-Gaussian statistics and non-ergodic kinetics—to tune the efficiency of the redox energy transfer. Both act to reduce the amount of free energy released as heat in electronic transitions. These mechanisms are shown to increase the energetic efficiency of protein electron transfer by up to an order of magnitude compared to the “standard picture” based on canonical free energies and the linear response approximation. In other words, the protein–water tandem allows both the formation of a ferroelectric mesophase in the hydration shell and an efficient control of the energetics by manipulating the relaxation times.

Numerical simulations of hydrated proteins show that protein hydration shells are polarized into a ferroelectric layer with large values of the average dipole moment magnitude and the dipole moment variance. The emergence of the new polarized mesophase dramatically alters the statistics of electrostatic fluctuations at the protein−water interface. The linear response relation between the average electrostatic potential and its variance breaks down, with the breadth of the electrostatic fluctuations far exceeding the expectations of the linear response theories. The dynamics of these non-Gaussian electrostatic fluctuations are dominated by a slow (≃1 ns) component that freezes in at the temperature of the dynamical transition of proteins. The ferroelectric shell propagates 3−5 water diameters into the bulk.

We report the results of extensive numerical simulations and theoretical calculations of electronic transitions in the reaction center of Rhodobacter sphaeroides photosynthetic bacterium. The energetics and kinetics of five electronic transitions related to the kinetic scheme of primary charge separation have been analyzed and compared to experimental observations. Nonergodic formulation of the reaction kinetics is required for the calculation of the rates due to a severe breakdown of the system ergodicity on the time scale of primary charge separation, with the consequent inapplicability of the standard canonical prescription to calculate the activation barrier. Common to all reactions studied is a significant excess of the charge-transfer reorganization energy from the width of the energy gap fluctuations over that from the Stokes shift of the transition. This property of the hydrated proteins, breaking the linear response of the thermal bath, allows the reaction center to significantly reduce the reaction free energy of near-activationless electron hops and thus raise the overall energetic efficiency of the biological charge-transfer chain. The increase of the rate of primary charge separation with cooling is explained in terms of the temperature variation of induction solvation, which dominates the average donor−acceptor energy gap for all electronic transitions in the reaction center. It is also suggested that the experimentally observed break in the Arrhenius slope of the primary recombination rate, occurring near the temperature of the dynamical transition in proteins, can be traced back to a significant drop of the solvent reorganization energy close to that temperature.

We report 11 μs of molecular dynamics simulations of the electron-transfer reaction between primary and secondary quinone cofactors in the bacterial reaction center. The main question addressed here is the mechanistic reason for unidirectional electron transfer between chemically identical cofactors. We find that electron is trapped at the secondary quinone by wetting of the protein pocket following electron transfer on the time-scale shorter than the backward transition. This mechanism provides effective rectification of the electron transport, making the reaction center a molecular diode operating by cyclic charge-induced electrowetting.

Proteins function by sampling conformational sub-states within a given fold. How this configurational flexibility and the associated protein dynamics affect the rates of chemical reactions are open questions. The difficulty in exploring this issue arises in part from the need to identify the relevant nuclear modes affecting the reaction rate for each characteristic time-scale of the reaction. Proteins as reaction media display a hierarchy of such nuclear modes, of increasingly collective character, that produce both a broad spectrum of static fluctuations and a broad spectrum of relaxation times. In order to understand the effect of protein dynamics on reaction rates, we have chosen to study a sub-nanosecond electron transfer reaction between the bacteriopheophytin and primary quinone cofactors of the photosynthetic bacterial reaction center. We show that dynamics affects the activation barrier of the reaction through a dynamical restriction of the configurational space sampled by the protein–water solvent on the reaction time-scale. The modes which become dynamically arrested on the reaction time-scale of hundreds of picoseconds are related to elastic motions of the protein that are strongly coupled to the hydration layer of water. Several mechanistic consequences for protein electron transfer emerge from this picture. Importantly, energy parameters used to define the activation barrier of electron transfer reactions lose their direct connection to equilibrium thermodynamics and become dependent in a very direct way on the relative magnitudes of the reaction and nuclear reorganization time-scales. As a result, the energetics of protein electron transfer need to be defined on each specific reaction time-scale. This perspective offers a mechanism to optimize protein electron transfer by tuning the reaction rate to the relaxation spectrum of the reaction coordinate.

Despite its diversity, life universally relies on a simple basic mechanism of energy transfer in its energy chains—hopping electron transport between centers of electron localization on hydrated proteins and redox cofactors. Since many such hops connect the point of energy input with a catalytic site where energy is stored in chemical bonds, the question of energy losses in (nearly activationless) electron hops, i.e., energetic efficiency, becomes central for the understanding of the energetics of life. We show here that standard considerations based on rules of Gibbs thermodynamics are not sufficient, and the dynamics of the protein and the protein–water interface need to be involved. The rate of electronic transitions is primarily sensitive to the electrostatic potential at the center of electron localization. Numerical simulations show that the statistics of the electrostatic potential produced by hydration water are strongly non-Gaussian, with the breadth of the electrostatic noise far exceeding the expectations of the linear response. This phenomenon, which dramatically alters the energetic balance of a charge-transfer chain, is attributed to the formation of ferroelectric domains in the protein's hydration shell. These dynamically emerging and dissipating domains make the shell enveloping the protein highly polar, as gauged by the variance of the shell dipole which correlates with the variance of the protein dipole. The Stokes-shift dynamics of redox-active proteins are dominated by a slow component with the relaxation time of 100–500 ps. This slow relaxation mode is frozen on the time-scale of fast reactions, such as bacterial charge separation, resulting in a dramatically reduced reorganization free energy of fast electronic transitions. The electron transfer activation barrier becomes a function of the corresponding rate, self-consistently calculated from a non-ergodic version of the transition-state theory. The peculiar structure of the protein–water interface thus provides natural systems with two “non's”—non-Gaussian statistics and non-ergodic kinetics—to tune the efficiency of the redox energy transfer. Both act to reduce the amount of free energy released as heat in electronic transitions. These mechanisms are shown to increase the energetic efficiency of protein electron transfer by up to an order of magnitude compared to the “standard picture” based on canonical free energies and the linear response approximation. In other words, the protein–water tandem allows both the formation of a ferroelectric mesophase in the hydration shell and an efficient control of the energetics by manipulating the relaxation times.