Non-Gaussian Lineshapes and Dynamics of Time-Resolved Linear and Nonlinear (Correlation) Spectra

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

Termination of Biological Function at Low Temperatures: Glass or Structural Transition?

Energy of life is produced by electron transfer in energy chains of respiration or photosynthesis. A small input of free energy available to biology puts significant restrictions on how much free energy can be lost in each electron-transfer reaction. We advocate the view that breaking ergodicity, leading to violation of the fluctuation–dissipation theorem (FDT), is how proteins achieve high reaction rates without sacrificing the reaction free energy. Here we show that a significant level of nonergodicity, represented by a large extent of the configurational temperature over the kinetic temperature, is maintained in the entire physiological range for the cytochrome <i>c</i> electron transfer protein. The protein returns to the state consistent with the FDT below the crossover temperature close to the temperature of the protein glass transition. This crossover leads to a sharp increase in the activation barrier of electron transfer and is displayed by a kink in the Arrhenius plot for the reaction rate constant

Theory of Protein Charge Transfer: Electron Transfer between Tryptophan Residue and Active Site of Azurin

One reaction step in the conductivity relay of azurin, electron transfer between the Cu-based active site and the tryptophan residue, is studied theoretically and by classical molecular dynamics simulations. Oxidation of tryptophan results in electrowetting of this residue. This structural change makes the free energy surfaces of electron transfer nonparabolic as described by the Q-model of electron transfer. We analyze the medium dynamical effect on protein electron transfer produced by coupled Stokes-shift dynamics and the dynamics of the donor–acceptor distance modulating electron tunneling. The equilibrium donor–acceptor distance falls in the plateau region of the rate constant, where it is determined by the protein–water dynamics, and the probability of electron tunneling does not affect the rate. The crossover distance found here puts most intraprotein electron-transfer reactions under the umbrella of dynamical control. The crossover between the medium-controlled and tunneling-controlled kinetics is combined with the effect of the protein–water medium on the activation barrier to formulate principles of tunability of protein-based charge-transfer chains. The main principle in optimizing the activation barrier is the departure from the Gaussian-Gibbsian statistics of fluctuations promoting activated transitions. This is achieved either by incomplete (nonergodic) sampling, breaking the link between the Stokes-shift and variance reorganization energies, or through wetting-induced structural changes of the enzyme’s active site

Solvent-Induced Shift of Spectral Lines in Polar–Polarizable Solvents

Solvent-induced shift of optical transition lines is traditionally described by the Lippert–McRae equation given in terms of the Onsager theory for dipole solvation. It splits the overall shift into the equilibrium solvation by induced dipoles and the reaction field by the permanent dipoles in equilibrium with the chromophore in the ground state. We have reconsidered this classical problem from the perspective of microscopic solvation theories. A microscopic solvation functional is derived, and continuum solvation is consistently introduced by taking the limit of zero wavevector in the reciprocal-space solvation susceptibility functions. We show that the phenomenological expression for the reaction field of permanent dipoles in the Lippert–McRae equation is not consistent with the microscopic theory. The main deficiency of the Lippert–McRae equation is the use of additivity of the response by permanent and induced dipoles of the liquid. An alternative closed-form equation for the spectral shift is derived. Its continuum limit allows a new, nonadditive functionality for the solvent-induced shift in terms of the high-frequency and static dielectric constants. The main qualitative outcome of the theory is a significantly weaker dependence of the spectral shift on the polarizability of the solvent than predicted by the Lippert–McRae formula

Non-Gaussian Statistics and Nanosecond Dynamics of Electrostatic Fluctuations Affecting Optical Transitions in Proteins

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

Dipolar Nanodomains in Protein Hydration Shells

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

Complex Dynamics of Water in Protein Confinement

This paper studies single-molecule and collective dynamics of water confined in protein powders by means of molecular dynamics simulations. The single-particle dynamics show a modest retardation compared to the bulk but become highly stretched in the powder, with the stretching exponent of ≃0.2. The collective dynamics of the total water dipole are affected by intermolecular correlations inside water and by cross-correlations between the water and the protein. The dielectric spectrum of water in the powder has two nearly equal-amplitude peaks: a Debye peak with ≃16 ps relaxation time and a highly stretched peak with the relaxation time of ≃13 ns and a stretching exponent of ≃0.12. The slower relaxation component is not seen in the single-molecule correlation functions and can be assigned to elastic protein motions displacing water in the powder. The loss spectrum of the intermediate scattering function reported by neutron-scattering experiments is also highly stretched, with the high-frequency wing scaling according to a power law. Translational dynamics can become much slower in the powder than in the bulk but are overshadowed by the rotational loss in the overall loss spectrum of neutron scattering

Theory and Electrochemistry of Cytochrome <i>c</i>

Extensive simulations of cytochrome <i>c</i> 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 <i>c</i> 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

Coulomb Soup of Bioenergetics: Electron Transfer in a Bacterial <i>bc</i>₁ Complex

We report atomistic molecular dynamics simulations (200 ns) of the first, rate-limiting electron transfer in the electron transport chain in a bacterial <i>bc</i>₁ 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

Half Reactions with Multiple Redox States Do Not Follow the Standard Theory: A Computational Study of Electrochemistry of C₆₀

The standard theory of electron transfer advanced by Marcus predicts that the solvent reorganization energy of electron transfer does not depend on the redox state of the reactant. For instance, it should be the same in the reduced and oxidized states of a half reaction. This theory prediction is verifiable by measuring activation barriers of electron transfer reactions involving multiple oxidation states. We use here the opportunity offered by electrochemistry of C₆₀, which allows charges from 0 to −4 in a sequence of reduction half reactions. We find that the activation barrier does change with altering redox state of a fullerene, which can be experimentally verified by measuring Arrhenius slopes of corresponding reaction rates. This outcome is connected to the alteration of the molecular polarizability caused by electronic transitions. Classical molecular dynamics simulations of a fullerene in water are combined here with the analytical Q-model of electron transfer involving polarizable molecules. The main outcome of the study is that altering molecular polarizability makes the reorganization energy and the reaction activation barrier depend on the redox state of the reactant