Dynamical transition, hydrophobic interface, and the temperature dependence of electrostatic fluctuations in proteins

Reorganization Energies, Entropies, and Free Energy Surfaces for Electron Transfer

Reorganization energies for an intramolecular self-exchange electron-transfer reaction are calculated by quantum-classical molecular dynamics simulations in four solvents with varying polarity and at temperatures ranging from 250 to 350 K. The reorganization free energies for polar solvents decrease systematically with increasing temperature, indicating that they include substantial contributions from entropy changes. The variances of the energy gap between the reactant and product states have a major component that is relatively insensitive to temperature. Explanations are suggested for these observations, which appear to necessitate rethinking the free energy functions of a distributed coordinate that frequently are used in discussions of reaction dynamics.

Protein electron transfer: is biology (thermo)dynamic?

Simple physical mechanisms are behind the flow of energy in all forms of life. Energy comes to living systems through electrons occupying high-energy states, either from food (respiratory chains) or from light (photosynthesis). This energy is transformed into the cross-membrane proton-motive force that eventually drives all biochemistry of the cell. Life's ability to transfer electrons over large distances with nearly zero loss of free energy is puzzling and has not been accomplished in synthetic systems. The focus of this review is on how this energetic efficiency is realized. General physical mechanisms and interactions that allow proteins to fold into compact water-soluble structures are also responsible for a rugged landscape of energy states and a broad distribution of relaxation times. Specific to a protein as a fluctuating thermal bath is the protein-water interface, which is heterogeneous both dynamically and structurally. The spectrum of interfacial fluctuations is a consequence of protein's elastic flexibility combined with a high density of surface charges polarizing water dipoles into surface nanodomains. Electrostatics is critical to the protein function and the relevant questions are: (i) What is the spectrum of interfacial electrostatic fluctuations? (ii) Does the interfacial biological water produce electrostatic signatures specific to proteins? (iii) How is protein-mediated chemistry affected by electrostatics? These questions connect the fluctuation spectrum to the dynamical control of chemical reactivity, i.e. the dependence of the activation free energy of the reaction on the dynamics of the bath. Ergodicity is often broken in protein-driven reactions and thermodynamic free energies become irrelevant. Continuous ergodicity breaking in a dense spectrum of relaxation times requires using dynamically restricted ensembles to calculate statistical averages. When applied to the calculation of the rates, this formalism leads to the nonergodic activated kinetics, which extends the transition-state theory to dynamically dispersive media. Releasing the grip of thermodynamics in kinetic calculations through nonergodicity provides the mechanism for an efficient optimization between reaction rates and the spectrum of relaxation times of the protein-water thermal bath. Bath dynamics, it appears, play as important role as the free energy in optimizing biology's performance.

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.

Protein electron transfer: Dynamics and statistics

Electron transfer between redox proteins participating in energy chains of biology is required to proceed with high energetic efficiency, minimizing losses of redox energy to heat. Within the standard models of electron transfer, this requirement, combined with the need for unidirectional (preferably activationless) transitions, is translated into the need to minimize the reorganization energy of electron transfer. This design program is, however, unrealistic for proteins whose active sites are typically positioned close to the polar and flexible protein-water interface to allow inter-protein electron tunneling. The high flexibility of the interfacial region makes both the hydration water and the surface protein layer act as highly polar solvents. The reorganization energy, as measured by fluctuations, is not minimized, but rather maximized in this region. Natural systems in fact utilize the broad breadth of interfacial electrostatic fluctuations, but in the ways not anticipated by the standard models based on equilibrium thermodynamics. The combination of the broad spectrum of static fluctuations with their dispersive dynamics offers the mechanism of dynamical freezing (ergodicity breaking) of subsets of nuclear modes on the time of reaction/residence of the electron at a redox cofactor. The separation of time-scales of nuclear modes coupled to electron transfer allows dynamical freezing. In particular, the separation between the relaxation time of electro-elastic fluctuations of the interface and the time of conformational transitions of the protein caused by changing redox state results in dynamical freezing of the latter for sufficiently fast electron transfer. The observable consequence of this dynamical freezing is significantly different reorganization energies describing the curvature at the bottom of electron-transfer free energy surfaces (large) and the distance between their minima (Stokes shift, small). The ratio of the two reorganization energies establishes the parameter by which the energetic efficiency of protein electron transfer is increased relative to the standard expectations, thus minimizing losses of energy to heat. Energetically efficient electron transfer occurs in a chain of conformationally quenched cofactors and is characterized by flattened free energy surfaces, reminiscent of the flat and rugged landscape at the stability basin of a folded protein.

Hydrophobicity of proteins and interfaces: insights from density fluctuations

Macroscopic characterizations of hydrophobicity (e.g., contact angle measurements) do not extend to the surfaces of proteins and nanoparticles. Molecular measures of hydrophobicity of such surfaces need to account for the behavior of hydration water. Theory and state-of-the-art simulations suggest that water density fluctuations provide such a measure; fluctuations are enhanced near hydrophobic surfaces and quenched with increasing surface hydrophilicity. Fluctuations affect conformational equilibria and dynamics of molecules at interfaces. Enhanced fluctuations are reflected in enhanced cavity formation, more favorable binding of hydrophobic solutes, increased compressibility of hydration water, and enhanced water-water correlations at hydrophobic surfaces. These density fluctuation-based measures can be used to develop practical methods to map the hydrophobicity/philicity of heterogeneous surfaces including those of proteins. They highlight that the hydrophobicity of a group is context dependent and is significantly affected by its environment (e.g., chemistry and topography) and especially by confinement. The ability to include information about hydration water in mapping hydrophobicity is expected to significantly impact our understanding of protein-protein interactions as well as improve drug design and discovery methods and bioseparation processes.

Comparing the temperature dependence of photosynthetic electron transfer in Chloroflexus aurantiacus and Rhodobactor sphaeroides reaction centers

The process of electron transfer from the special pair, P, to the primary electron donor, H(A), in quinone-depleted reaction centers (RCs) of Chloroflexus (Cf.) aurantiacus has been investigated over the temperature range from 10 to 295 K using time-resolved pump-probe spectroscopic techniques. The kinetics of the electron transfer reaction, P* → P(+)H(A)(-), was found to be nonexponential, and the degree of nonexponentiality increased strongly as temperature decreased. The temperature-dependent behavior of electron transfer in Cf. aurantiacus RCs was compared with that of the purple bacterium Rhodobacter (Rb.) sphaeroides . Distinct transitions were found in the temperature-dependent kinetics of both Cf. aurantiacus and Rb. sphaeroides RCs, at around 220 and 160 K, respectively. Structural differences between these two RCs, which may be associated with those differences, are discussed. It is suggested that weaker protein-cofactor hydrogen bonding, stronger electrostatic interactions at the protein surface, and larger solvent interactions likely contribute to the higher transition temperature in Cf. aurantiacus RCs temperature-dependent kinetics compared with that of Rb. sphaeroides RCs. The reaction-diffusion model provides an accurate description for the room-temperature electron transfer kinetics in Cf. aurantiacus RCs with no free parameters, using coupling and reorganization energy values previously determined for Rb. sphaeroides , along with an experimental measure of protein conformational diffusion dynamics and an experimental literature value of the free energy gap between P* and P(+)H(A)(-).

Electrostatics of the protein-water interface and the dynamical transition in proteins

Atomic displacements of hydrated proteins are dominated by phonon vibrations at low temperatures and by dissipative large-amplitude motions at high temperatures. A crossover between the two regimes is known as a dynamical transition. Recent experiments indicate a connection between the dynamical transition and the dielectric response of the hydrated protein. We analyze two mechanisms of the coupling between the protein atomic motions and the protein-water interface. The first mechanism considers viscoelastic changes in the global shape of the protein plasticized by its coupling to the hydration shell. The second mechanism involves modulations of the local motions of partial charges inside the protein by electrostatic fluctuations. The model is used to analyze mean-square displacements of iron of metmyoglobin reported by Mössbauer spectroscopy. We show that high displacement of heme iron at physiological temperatures is dominated by electrostatic fluctuations. Two onsets, one arising from the viscoelastic response and the second from electrostatic fluctuations, are seen in the temperature dependence of the mean-square displacements when the corresponding relaxation times enter the instrumental resolution window.

Redox-promoting protein motions in rubredoxin

Proteins are dynamic objects, constantly undergoing conformational fluctuations, yet the linkage between internal protein motion and function is widely debated. This study reports on the characterization of temperature-activated collective and individual atomic motions of oxidized rubredoxin, a small 53 residue protein from thermophilic Pyrococcus furiosus (RdPf). Computational modeling allows detailed investigations of protein motions as a function of temperature, and neutron scattering experiments are used to compare to computational results. Just above the dynamical transition temperature which marks the onset of significant anharmonic motions of the protein, the computational simulations show both a significant reorientation of the average electrostatic force experienced by the coordinated Fe(3+) ion and a dramatic rise in its strength. At higher temperatures, additional anharmonic modes become activated and dominate the electrostatic fluctuations experienced by the ion. At 360 K, close to the optimal growth temperature of P. furiosus, simulations show that three anharmonic modes including motions of two conserved residues located at the protein active site (Ile7 and Ile40) give rise to the majority of the electrostatic fluctuations experienced by the Fe(3+) ion. The motions of these residues undergo displacements which may facilitate solvent access to the ion.

Probing the effect of water-water interactions on enzyme activity with salt gradients: a case-study using ribonuclease t1

Water molecules interact with one another via hydrogen bonds. Experimental and theoretical evidence indicates that these hydrogen bonds occur in two modalities--high- and low-angle hydrogen bonding--and that the addition of various solutes to water affects only the number of water molecules participating in a specific type of hydrogen bond interactions, not the nature of the water-water interactions. In this work, we have investigated the effect of each of these hydrogen bonding types upon the activity of the enzyme ribonuclease t1. This was done through perturbation of the water hydrogen bonding distribution by using various salts. Our results indicate that various salts differ in their ability to reduce the enzymatic activity of ribonuclease t1, and this ability is well correlated with the ability of each salt to promote high-angle hydrogen bonding in water. By applying the two-phase model of liquid water (i.e., liquid water being modeled as an equilibrium existing between two phases, LD and HD water), we demonstrate that our results are compatible with the assumption that increasing the population of high-angle hydrogen bonds among water molecules stabilizes the more compact, less active conformations of the enzyme. This indicates that the structures that proteins adopt in water solution depend upon the nature of interactions between water molecules.

Terahertz response of dipolar impurities in polar liquids: on anomalous dielectric absorption of protein solutions

A theory of radiation absorption by dielectric mixtures is presented. The coarse-grained formulation is based on the wave-vector-dependent correlation functions of molecular dipoles of the host polar liquid and a density structure factor of the solutes. A nonlinear dependence of the dielectric absorption coefficient on the solute concentration is predicted and originates from the mutual polarization of the liquid surrounding the solutes by the collective field of the solute dipoles aligned along the radiation field. The theory is applied to terahertz absorption of hydrated saccharides and proteins. While the theory gives an excellent account of the observations for saccharides, without additional assumptions and fitting parameters, experimental absorption coefficient of protein solutions significantly exceeds theoretical calculations with dipole moment of the bare protein assigned to the solute and shows a peak against the protein concentration. A substantial polarization of protein's hydration shell, resulting in a net dipole moment, is required to explain the disagreement between theory and experiment. When the correlation function of the total dipole moment of the protein with its hydration shell from numerical simulations is used in the analytical model, an absorption peak, qualitatively similar to that seen in experiment, is obtained. The existence and position of the peak are sensitive to the specifics of the protein-protein interactions. Numerical testing of the theory requires the combination of dielectric and small-angle scattering measurements. The calculations confirm that "elastic ferroelectric bag" of water shells observed in previous numerical simulations is required to explain terahertz dielectric measurements.

Fluctuations in biological and bioinspired electron-transfer reactions

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.

Energetics of bacterial photosynthesis

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.

Nonergodic activated kinetics in polar media

A theoretical formulation is developed for the activated kinetics when some subset of nuclear modes of the thermal bath is slower than the reaction and ergodicity of the thermal bath is not maintained. Nonergodic free energy profiles along the reaction coordinate are constructed by using restricted canonical ensembles with the phase space available to the system found by solving a self-consistent kinetic equation. The resulting activation barrier incorporates not only thermodynamic parameters but also dynamical information from the time autocorrelation function of the solute-solvent interaction energy. The theory is applied to the reactions of solvolysis and charge transfer in polar media.