The medium reorganization energy for the charge transfer reactions in proteins

A low static dielectric permittivity of proteins causes the low reorganization energies for the charge transfer reactions inside them. This reorganization energy does not depend on the pre-existing intraprotein electric field. The charge transferred inside the protein interacts with its aqueous surroundings; for many globular proteins, the effect of this surroundings on the reorganization energy is comparable with the effect of reorganization of the protein itself while for the charge transfer in the middle of membrane the aqueous phase plays a minor role. Reorganization energy depends strongly on the system considered, and hence there is no sense to speak on the "protein reorganization energy" as some permanent characteristic parameter. We employed a simple algorithm for calculation of the medium reorganization energy using the numerical solution of the Poisson-Boltzmann equation. Namely, the reaction field energy was computed in two versions - all media having optical dielectric permittivity, and all the media with the static one; the difference of these two quantities gives the reorganization energy. We have calculated reorganization energies for electron transfer in cytochrome c, various ammine-ruthenated cytochromes c, azurin, ferredoxin, cytochrome c oxidase, complex of methylamine dehydrogenase with amicyanin, and for proton transfer in α-chymotrypsin. It is shown that calculation of the medium reorganization energy can be a useful tool in analysis of the mechanisms of the charge transfer reactions in proteins.     

Effective activation energy of enzymatic and nonenzymatic reactions. Evolution-imposed requirements to enzyme structure

The difference of the activation energies in a protein globule and water has been treated in terms of the theory of an elementary act of charge transfer reaction with regards to the energy spent on the transfer of charged reactants from water into the protein. The protein was treated as a structureless dielectric with a given optical and static dielectric constants surrounded by the aqueous phase. Reactions of different types (charge exchange between reactants, charge separation, neutralization, etc.) have been analyzed both under prevalence of purely electrostatic effects and under considerable nonelectrostatic contributions to the activation energies. It is shown that for all one-electron and most multi-electron reactions involving two reaction centres the energy spent for charged reactant transfer from water into protein is greater than the concomitant activation energy gain. The same effect takes place in a number of cases for multi-centre processes as well. To overcome the entropy hindrances, the reactants and catalysts must combine into multiparticle complexes, i.e. form microscopic regions of low dielectric constant. This results in increased effective activation energy as compared to reactions in water. It has been hypothesized that in order to make up for this loss the evolution has selected the proteins which are characterized by considerable intraglobular permanent electric fields. The presence in proteins of high concentrations of strongly polar peptide groups renders them advantageous in this respect over other polymers that are less polar.     

Role of the protein's low dielectric constant in the functioning of the photosynthetic reaction center

The high efficiency of the energy storage in the photosynthetic reaction center (RC) is determined by a successful competition of electron transfer from bacteriopheophytin to quinone, as compared to backward recombination of the primary charge-separated state. This relationship is caused by a fine matching of the reorganization energy and the free energy gap making the forward processes activationless, and hence very fast, and mismatching of these two quantities for the backreaction, therefore retarding it strongly. In this study, we show that this matching is due to a low dielectric constant of the RC's protein core because a low dielectric affects strongly electrostatic polarization components of both the reorganization energy and the equilibrium free energy of reaction. If the protein and membrane were replaced by a homogeneous medium with a high dielectric constant, the effective energy storage would be impractical.     

Effects of freezing out protein vibrational modes on electron transfer kinetics in bacterial reaction centers

We propose a model that some vibrational modes of the protein in bacterial photosynthetic reaction centers may be frozen at low temperatures. The freezing of the protein-environmental motion can affect the electron transfer rate through changes in the reorganization energy and the free energy gap. We offer a qualitative explanation of the different kinetics of the ET processes in reaction centers which are cooled in the dark and cooled under illumination.     

Converting structural changes upon oxidation of cytochrome c to electrostatic reorganization energy

The observed X-ray structural differences between reduced and oxidized cytochrome c are converted to electrostatic energy. This conversion is used to estimate the protein reorganization energy which determines the protein contribution to the activation barrier for the electron transfer reaction. It is shown that the reorganization energy of cytochrome c is much smaller than the corresponding energy for electron transfer in water and that this is consistent with the role for cytochromes as electron transfer catalysts.     

Electron transfer in proteins: in search of preferential pathways

Electron migration between and within proteins is one of the most prevalent forms of biological energy conversion processes. Electron transfer reactions take place between active centers such as transition metal ions or organic cofactors over considerable distances at fast rates and with remarkable specificity. The electron transfer is attained through weak electronic interaction between the active sites, so that considerable research efforts are centered on resolving the factors that control the rates of long-distance electron transfer reactions in proteins. These factors include (in addition to the distance and nature of the microenvironment separating the reactants) thermodynamic driving force and the configurational changes required upon reaction. Several of these aspects are addressed in this review, which is based primarily on recent work performed by the authors on model systems of blue copper-containing proteins. These proteins serve almost exclusively in electron transfer reactions, and as it turns out, their metal coordination sites are endowed with properties uniquely optimized for their function.     

Influence of the solvent structure on the electrostatic interactions in proteins

The proper estimation of the influence of the many-body dynamic solvent microstructure on a pairwise electrostatic interaction (PEI) at the protein-solvent interface is very important for solving many biophysical problems. In this work, the PEI energy was calculated for a system that models the interface between a protein and an aqueous solvent. The concept of nonlocal electrostatics for interfacial electrochemical systems was used to evaluate the contribution of a solvent orientational polarization, correlated by the network of hydrogen bonds, into the PEI energy in proteins. The analytical expression for this energy was obtained in the form of Coulomb's law with an effective distance-dependent dielectric function. The asymptotic and numerical analysis carried out for this function revealed several features of dielectric heterogeneity at the protein-solvent interface. For charges located in close proximity to this interface, the values of the dielectric function for the short-distance electrostatic interactions were found to be remarkably smaller than those determined by the classical model, in which the solvent was considered as the uniform dielectric medium of high dielectric constant. Our results have shown that taking into consideration the dynamic solvent microstructure remarkably increases the value of the PEI energy at the protein-solvent interface.     

Correlation of electron transfer rate in photosynthetic reaction centers with intraprotein dielectric properties

A number of the electrogenic reactions in photosystem I, photosystem II, and bacterial reaction centers (RC) were comparatively analyzed, and the variation of the dielectric permittivity (epsilon) in the vicinity of electron carriers along the membrane normal was calculated. The value of epsilon was minimal at the core of the complexes and gradually increased towards the periphery. We found that the rate of electron transfer (ET) correlated with the value of the dielectric permittivity: the fastest primary ET reactions occur in the low-polarity core of the complexes within the picosecond time range, whereas slower secondary reactions take place at the high-polarity periphery of the complexes within micro- to millisecond time range. The observed correlation was quantitatively interpreted in the framework of the Marcus theory. We calculated the reorganization energy of ET carriers using their van der Waals volumes and experimentally determined epsilon values. The electronic coupling was calculated by the empirical Moser-Dutton rule for the distance-dependent electron tunneling rate in nonadiabatic ET reactions. We concluded that the local dielectric permittivity inferred from the electrometric measurements could be quantitatively used to estimate the rate constant of ET reactions in membrane proteins with resolved atomic structure with the accuracy of less than one order of magnitude.     

Voltammetric probes of cytochrome electroreactivity: the effect of the protein matrix on outer-sphere reorganization energy and electronic coupling probed through comparisons with the behavior of porphyrin complexes

Using surface-modified electrodes composed of omega-hydroxyalkanethiols, an experimentally based value for the inner-sphere reorganization energy of the bis(imidazole)iron porphyrin system has been obtained by examining the solvent dependence of the reorganization energy of bis(N-methylimidazole)meso-tetraphenyl iron porphyrin. The value obtained (0.41 +/- 0.06 eV) is remarkably similar to values we have recently reported for the reorganization energy of cytochrome b(5) (0.43 +/- 0.02 eV) and cytochrome c (0.58 +/- 0.06 eV). This strongly suggests that the protein matrix mimics the behavior of a low dielectric solvent and effectively shields the heme from the solvent. The effect of the orientation of the heme relative to the electrode was also explored by sytematically varying the steric bulk of the axial ligands. On the basis of a good linear correlation between the electronic coupling and the cosine of the angle between the heme plane and the surface of the electrode, it is suggested that a parallel orientation of the heme yields a maximum in the electronic coupling. Relevance to interheme protein electron transfer is discussed.     

Electron transfer between horse heart and Candida krusei cytochromes c in the free and bound states

Electron transfer between horse heart and Candida krusei cytochromes c in the free and phosvitin-bound states was examined by difference spectrum and stopped-flow methods. The difference spectra in the wavelength range of 540-560 nm demonstrated that electrons are exchangeable between the cytochromes c of the two species. The equilibrium constants of the electron transfer reaction for the free and phosvitin-bound forms, estimated from these difference spectra, were close to unity at 20 degrees C in 20 mM Tris-HCl buffer (pH 7.4). The electron transfer rate for free cytochrome c was (2-3).10(4) M-1.s-1 under the same conditions. The transfer rate for the bound form increased with increase in the binding ratio at ratios below half the maximum, and was almost constant at higher ratios up to the maximum. The maximum electron exchange rate was about 2.10(6) M-1.s-1, which is 60-70 times that for the free form at a given concentration of cytochrome c. The activation energy of the reaction for the bound cytochrome c was equal to that for the free form, being about 10 kcal/mol. The dependence of the exchange rate on temperature, cytochrome c concentration and solvent viscosity suggests that enhancement of the electron transfer rate between cytochromes c on binding to phosvitin is due to increase in the collision frequency between cytochromes c concentrated on the phosvitin molecule.     

Correlation of electron transfer rate in photosynthetic reaction centers with intraprotein dielectric properties

A number of the electrogenic reactions in photosystem I, photosystem II, and bacterial reaction centers (RC) were comparatively analyzed, and the variation of the dielectric permittivity (ε) in the vicinity of electron carriers along the membrane normal was calculated. The value of ε was minimal at the core of the complexes and gradually increased towards the periphery. We found that the rate of electron transfer (ET) correlated with the value of the dielectric permittivity: the fastest primary ET reactions occur in the low-polarity core of the complexes within the picosecond time range, whereas slower secondary reactions take place at the high-polarity periphery of the complexes within micro- to millisecond time range. The observed correlation was quantitatively interpreted in the framework of the Marcus theory. We calculated the reorganization energy of ET carriers using their van der Waals volumes and experimentally determined ε values. The electronic coupling was calculated by the empirical Moser-Dutton rule for the distance-dependent electron tunneling rate in nonadiabatic ET reactions. We concluded that the local dielectric permittivity inferred from the electrometric measurements could be quantitatively used to estimate the rate constant of ET reactions in membrane proteins with resolved atomic structure with the accuracy of less than one order of magnitude.     

Effects of medium polarization and pre-existing field on activation energy of enzymatic charge-transfer reactions

The highly organized spatial structure of proteins' polar groups results in the existence of a permanent intraprotein electric field and in protein's weak dielectric response, i.e. its low dielectric constant. The first factor affects equilibrium free energy gap of a charge-transfer reaction, the second (medium polarization effect) influences both equilibrium and non-equilibrium (reorganization) energies, decreasing the latter substantially. In the framework of the rigorous 'fixed-charge-density' formalism, the medium polarization component of the reaction activation energy has been calculated, both for the activation energy of the elementary act proper, and the effective activation energy accounting for the charges' transfer from water into a low-dielectric structureless medium. In all typical cases of reactions, the energy spent for charge transfer from water into structureless 'protein' is larger than the gain in activation energy due to the protein's low reorganization energy. Therefore, the low dielectric constant of proteins is not sufficient to ensure their high catalytic activity, and an additional effect of the pre-existing intraprotein electric field, compensating for an excessive charging energy, is necessary. Only a combined action of low reorganization energy and pre-existing electric field provides proteins with their high catalytic activity. The dependence of activation energy on the globule geometry has been analyzed. It is shown that, for each reaction, an optimum set of geometric parameters exists. For five hydrolytic enzymes, the optimum globule radii have been calculated using the experimental geometry of their active sites. The calculated radii agree satisfactorily with the real sizes of these macromolecules, both by absolute and by relative values.     

Conformational mobility of proteins and activation energy of enzymatic charge transfer reactions: an analysis in dielectric formalism

In calculating the medium reorganization energy and the activation energy of charge transfer enzymatic reactions, an allowance is made for the enhanced conformational mobility of the protein external region. The two-layer model is proposed, the outer layer having a higher static dielectric constant. The calculations show that the higher mobility in the outer layer causes some quantitative rather than qualitative changes. The main result obtained earlier is confirmed: the reorganization energy for charge transfer reaction in protein globule is much lower than in water and for this reason the activation energy also decreases. The higher dielectric constant of the outer layer somewhat favours the introduction of charge into active site and hence favours the natural selection of proteins as enzymes. This effect cannot exclude the necessity of other factors stabilizing ionic forms inside the protein globule. Freezing of conformational mobility (say, at low temperatures) hinders the charge transfer process as a consequence of the difficulty in equalizing the initial and final energy levels.     

Medium reorganization energy and enzymatic reaction activation energy

Reorganization and activation energies for charge transfer reactions occurring inside a dielectric sphere have been calculated by solving the problem of polar medium reorganization within and outside a dielectric sphere placed in another infinite dielectric. The dielectric sphere is assumed to simulate a protein globule, i.e. an enzyme molecule. It has been shown that for some reaction types the activation energy tends to decrease as the globule radius increases and that for each of the reaction types considered there is an optimal globule radius an increase of which does not bring about any tangible activation energy reduction. The calculated optimal radii for different processes are in good agreement with the increasing molecular sizes in the series: ribonuclease less than or equal to lysozyme less than serine proteinases approximately equal to cysteine proteinases less than NAD-dependent dehydrogenases. The calculated radii are usually about 1.5 to 1.7 times (and molecular masses about 4-5 times) smaller than the experimental ones. The reasons for this discrepancy are discussed and it has been suggested that the approximate nature of the treatment of a protein globule as a structureless dielectric is the main reason. It is shown that charge transfer at an acute angle to the globule surface is the optimum process. For endoergonic reaction stages it is the net charge transfer towards the periphery and for exoergonic ones that in the reverse direction which are advantageous. These conclusions are consistent with the data about the structure of the above-mentioned enzymes.     

Direct voltammetric investigation of the electrochemical properties of human hemoglobin: relevance to physiological redox chemistry

Voltammetric measurements on solutions of human hemoglobin using gold electrodes modified with omega-hydroxyalkanethiols have yielded the first direct measure of the reorganization energy of the protein. The value obtained based on extrapolation of the experimentally measured currents, 0.76 eV, is independent of pH (i.e., over the physiologically relevant rage, pH 6.8-7.4) and is remarkably similar to values obtained for myoglobin. This result is perhaps surprising given the marked dependence of the measured reduction potential of hemoglobin on pH (i.e., the redox Bohr effect). Electron transfer rates from the electrode to hemoglobin were also measured. Using similarly measured heterogeneous electron-transfer rates for cytochrome b(5), it is possible to predict the magnitude of the homogeneous electron-transfer rate from cytochrome b(5) to methemoglobin using a formalism developed by Marcus. These predicted rates are in reasonable agreement with reported rates of this physiological reaction based on stopped-flow kinetics experiments. These results suggest that the intrinsic electroreactivity of these heme proteins is sufficient to account for physiologically observed rates. Residual differences between homogeneous phase kinetics and those predicted by heterogeneous phase reactions are suggested to be due to small reductions in the outer-sphere reorganization energy of both component proteins which arise due to solvent exclusion at the interface between the two proteins in complex.     

Lipid radicals--possible mediators of charge transfer and energy transformation (a hypothesis)

A number of enzymatic reactions with the participation of lipid radicals is discussed in the article. It is supposed that NADPH- and NADH-dependent formation of the lipid radicals has a functional importance. The uptake of oxygen by free radicals is considered as one of the reactions of radicals utilization. It is proposed that other reactions with participation of lipid radicals can take place in the membranes of microsomes and mitochondria: the reaction of electron transfer from flavoprotein to cytochrome P448 and the reaction of energy transfer which provide the coupling of oxidation and phosphorylation.     

Transesterification thio effects of phosphate diesters: free energy barriers and kinetic and equilibrium isotope effects from density-functional theory

Primary and secondary kinetic and equilibrium isotope effects are calculated with density-functional methods for the in-line dianionic methanolysis of the native (unsubstituted) and thio-substituted cyclic phosphates. These reactions represent reverse reaction models for RNA transesterification under alkaline conditions. The effect of solvent is treated with explicit (single and double) water molecules and self-consistently with an implicit (continuum) solvation model. Singly substituted reactions at the nonbridging O(P1) position and bridging O(2)('), O(3)('), and O(5)(') positions and a doubly substituted reaction at the O(P1) and O(P2) positions were considered. Aqueous free energy barriers are calculated, and the structures and bond orders of the rate-controlling transition states are characterized. The results are consistent with available experimental data and provide useful information for the interpretation of measured isotope and thio effects used to probe mechanism in phosphoryl transfer reactions catalyzed by enzymes and ribozymes.     

Application of Marcus theory of electron transfer for the reactions between HRP compound I and II and 2,4-disubstituted phenols

Reactions between horseradish peroxidase (HRP) compound I and II and some natural phenolic antioxidants were studied at pH 7. The bimolecular rate constants for these reactions were determined using a sequential mixing stopped-flow spectrometer. The rate constants for the reactions of compound I were found to be two orders of magnitude higher than those for compound II. The phenols under study showed a significant difference in their one-electron reduction potential values. As the rate constants also changed systematically with their one-electron potentials, the Marcus theory of electron transfer was applied to the above determined rate constants and the thermodynamic driving force (deltaG(o)), from which the reorganization energy (lambda) for the electron transfer from phenols to both compound I and II was estimated.     

Dielectric relaxation in proteins: the computational perspective

In photoexcitation and electron transfer, a new dipole or charge is introduced, and the structure is adjusted. This adjustment represents dielectric relaxation, which is the focus of this review. We concentrate on a few selected topics. We discuss linear response theory, as a unifying framework and a tool to describe non-equilibrium states. We review recent, molecular dynamics simulation studies that illustrate the calculation of dynamic and thermodynamic properties, such as Stokes shifts or reorganization free energies. We then turn to the macroscopic, continuum electrostatic view. We recall the physical definition of a dielectric constant and revisit the decomposition of the free energy into a reorganization and a static term. We review some illustrative continuum studies and discuss some difficulties that can arise with the continuum approach. In conclusion, we consider recent developments that will increase the accuracy and broaden the scope of all these methods.