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.     

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.     

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.     

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.     

Membrane catalysis: synchronous multielectron reactions at the interface between two liquid phases. Bioenergetic mechanisms

Kinetics of multi-electron reactions at the interface between two immiscible liquids are considered. Calculations of the energy of solvent reorganization, of the work required to bring reactants and reaction products together, and of the electrostatic contributions to the Gibbs free energy of the reaction during electron transfer between reactants which are in different dielectric media are reported. Conditions under which the free energy of activation of the interfacial reaction of electron transfer decreases are established. The influence of the distance between reactants and of the dielectric permittivity of the non-aqueous phase on the solvent reorganization energy value is studied. Conditions under which multielectron reactions at the interface proceed are discussed. The biophysics and biochemistry of photosynthesis and respiration are considered as examples of multielectron processes.     

The role of solvent polarity in the free energy of transfer of amino acid side chains from water to organic solvents

The transfer free energies of amino acid side chains from water to N-methylacetamide have been determined and compared with those obtained from other model systems. Although the process of transfer from water to N-methylacetamide represents transfer from a lower dielectric phase to a higher dielectric phase, the transfer free energies of most of the amino acid side chains are nearly the same as those obtained from the water to ethanol system. Among the apolar side chains studied, only the transfer free energies of methionine and the aromatic side chains are apparently influenced to some extent by the polarity of the organic solvent phase. The transfer free energies of the neutral polar side chains also exhibit significant dependence on solvent polarity. The van't Hoff plots for most of the apolar side chains exhibit nonlinear curves, indicating that the enthalpy of transfer from water to N-methylacetamide is temperature-dependent. It is suggested that to assess the contribution of the hydrophobic free energy to the stability of globular proteins, it is probably not necessary to account for variation in the internal environment of the protein.     

Intramolecular dielectric screening in proteins

This paper investigates the microscopic mechanisms of charge screening by proteins. For this purpose, we introduce the generalized susceptibility of a protein in response to a point charge, which is a scalar quantity dependent on position within the protein. The contribution to the susceptibility from atomic polarizabilities, associated with electronic degrees of freedom, is found to be highly uniform. By contrast, that from dynamic dipolar relaxation, associated with nuclear degrees of freedom, varies greatly between different regions of the protein. We investigate the possible r?le of this variation in the activity of proteins that interact functionally with charged species, and we formulate and test the hypothesis that this variation is correlated to functional activity. Model calculations give encouraging support to this hypothesis. The protein's dielectric properties are represented by a standard model in which electronic relaxation is described by a set of atomic polarizabilities, and dipolar relaxation is treated as a perturbation to normal mode dynamics. The model yields the desired susceptibility in closed form. Its obvious limitations are discussed. It is applied to several test systems, and is compared to various continuum models. Four model alpha-helices are considered, three of which play a r?le in vivo in the binding of charged ligands. We show that the intramolecular screening, and its spatial variation, can indeed play a part in this binding. The electron transfer between ferri- and ferrocytochrome c is considered. The dielectric relaxation of each molecule, associated respectively with its oxidation or its reduction, is known to be directly related to the activation free energy for the electron transfer reaction. Our analysis of the dielectric susceptibility will thus permit an estimate of this activation free energy. We show that the relaxation of the atomic positions ("dipolar relaxation") contributes 1 kcal/mol to this activation free energy, and that the molecule achieves this low value by providing a low dipolar susceptibility throughout its central part. In this case, the spatial variation of the susceptibility has a clear functional r?le.     

How proteins trigger excitation energy transfer in the FMO complex of green sulfur bacteria

A simple electrostatic method for the calculation of optical transition energies of pigments in protein environments is presented and applied to the Fenna-Matthews-Olson (FMO) complex of Prosthecochloris aestuarii and Chlorobium tepidum. The method, for the first time, allows us to reach agreement between experimental optical spectra and calculations based on transition energies of pigments that are calculated in large part independently, rather than fitted to the spectra. In this way it becomes possible to understand the molecular mechanism allowing the protein to trigger excitation energy transfer reactions. The relative shift in excitation energies of the seven bacteriochlorophyll-a pigments of the FMO complex of P. aestuarii and C. tepidum are obtained from calculations of electrochromic shifts due to charged amino acids, assuming a standard protonation pattern of the protein, and by taking into account the three different ligand types of the pigments. The calculations provide an explanation of some of the earlier results for the transition energies obtained from fits of optical spectra. In addition, those earlier fits are verified here by using a more advanced theory of optical spectra, a genetic algorithm, and excitonic couplings obtained from electrostatic calculations that take into account the influence of the dielectric protein environment. The two independent calculations of site energies strongly favor one of the two possible orientations of the FMO trimer relative to the photosynthetic membrane, which were identified by electron microscopic studies and linear dichroism experiments. Efficient transfer of excitation energy to the reaction center requires bacteriochlorophylls 3 and 4 to be the linker pigments. The temporal and spatial transfer of excitation energy through the FMO complex is calculated to proceed along two branches, with transfer times that differ by an order of magnitude.     

A pragmatic approach to structure based calculation of coupled proton and electron transfer in proteins

The coupled motion of electrons and protons occurs in many proteins. Using appropriate tools for calculation, the three-dimensional protein structure can show how each protein modulates the observed electron and proton transfer reactions. Some of the assumptions and limitations involved in calculations that rely on continuum electrostatics to calculate the energy of charges in proteins are outlined. Approaches that mix molecular mechanics and continuum electrostatics are described. Three examples of the analysis of reactions in photosynthetic reaction centers are given: comparison of the electrochemistry of hemes in different sites; analysis of the role of the protein in stabilizing the early charge separated state in photosynthesis; and calculation of the proton uptake and protein motion coupled to the electron transfer from the primary (Q(A)) to secondary (Q(B)) quinone. Different mechanisms for stabilizing intra-protein charged cofactors are highlighted in each reaction.     

Protein contributions to redox potentials of homologous rubredoxins: an energy minimization study.

The energetic contributions of the protein to the redox potential in an iron-sulfur protein are studied via energy minimization, comparing homologous rubredoxins from Clostridium pasteurianum, Desulfovibrio gigas, Desulfovibrio vulgaris, and Pyrococcus furiosus. The reduction reaction was divided into 1) the change in the redox site charge without allowing the protein to respond and 2) the relaxation of the protein in response to the new charge state, focusing on the latter. The energy minimizations predict structural relaxation near the redox site that agrees well with that in crystal structures of oxidized and reduced P. furiosus rubredoxin, but underpredicts it far from the redox site. However, the relaxation energies from the energy-minimized structures agree well with those from the crystal structures, because the polar groups near the redox site are the main determinants and the charged groups are all located at the surface and thus are screened dielectrically. Relaxation energies are necessary for good agreement with experimentally observed differences in reduction energies between C. pasteurianum and the other three rubredoxins. Overall, the relaxation energy is large (over 500 mV) from both the energy-minimized and the crystal structures. In addition, the range in the relaxation energy for the different rubredoxins is large (300 mV), because even though the structural perturbations of the polar groups are small, they are very near the redox site. Thus the relaxation energy is an important factor to consider in reduction energetics.     

Theoretical studies of enzymic reactions: Dielectric,electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme

A general method for detailed study of enzymic reactions is presented. The method considers the complete enzyme-substrate complex together with the surrounding solvent and evaluates all the different quantum mechanical and classical energy factors that can affect the reaction pathway. These factors include the quantum mechanical energies associated with bond cleavage and charge redistribution of the substrate and the classical energies of steric and electrostatic interactions between the substrate and the enzyme. The electrostatic polarization of the enzyme atoms and the orientation of the dipoles of the surrounding water molecules is simulated by a microscopic dielectric model. The solvation energy resulting from this polarization is considerable and must be included in any realistic calculation of chemical reactions involving anything more than an isolated molecule in vacuo. Without it, acidic groups can never become ionized and the charge distribution on the substrate will not be reasonable. The same dielectric model can also be used to study the reaction of the substrate in solution. In this way the reaction in solution can be compared with the enzymic reaction.In this paper we study the stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme. It is found that electrostatic stabilization is an important factor in increasing the rate of the reaction step that leads to the formation of the carbonium ion intermediate. Steric factors, such as the strain of the substrate on binding to lysozyme, do not seem to contribute significantly.     

An evaluation of Poisson-Boltzmann electrostatic free energy calculations through comparison with experimental mutagenesis data

For systems involving highly and oppositely charged proteins, electrostatic forces dominate association and contribute to biomolecular complex stability. Using experimental or theoretical alanine-scanning mutagenesis, it is possible to elucidate the contribution of individual ionizable amino acids to protein association. We evaluated our electrostatic free energy calculations by comparing calculated and experimental data for alanine mutants of five protein complexes. We calculated Poisson-Boltzmann electrostatic free energies based on a thermodynamic cycle, which incorporates association in a reference (Coulombic) and solvated (solution) state, as well as solvation effects. We observe that Coulombic and solvation free energy values correlate with experimental data in highly and oppositely charged systems, but not in systems comprised of similarly charged proteins. We also observe that correlation between solution and experimental free energies is dependent on dielectric coefficient selection for the protein interior. Free energy correlations improve as protein dielectric coefficient increases, suggesting that the protein interior experiences moderate dielectric screening, despite being shielded from solvent. We propose that higher dielectric coefficients may be necessary to more accurately predict protein-protein association. Additionally, our data suggest that Coulombic potential calculations alone may be sufficient to predict relative binding of protein mutants.     

The enhancement of electrostriction caused by lowering the solvent dielectric constant leads to the decrease of activation energy in trypsin catalysis.

The contribution of electrostriction of the solvent to the stabilization of the negatively charged tetrahedral transition state of a trypsin-catalyzed reaction was probed by means of kinetic studies involving high-pressure and solvent dielectric constant. A good correlation was observed between the increased catalytic efficiency of trypsin and the decreased solvent dielectric constant. When the dielectric constant of the solvents was lowered by 4.68 units, the loss of activation energy and that of free energy of activation were 2.26 kJ/mol and 3.09 kJ/mol, respectively. The activation volume for k(cat) decreased significantly as the dielectric constant of the solvent decreased, indicating that the degree of electrostriction of the solvent around the charged tetrahedral transition state has been enhanced. These observations demonstrate that the increase in the catalytic efficiency of the trypsin reaction with decreasing dielectric constant resulted from the stabilization of electrostatic energy for the formation of an oxyanion hole, and this stabilization was caused by the increase of electrostricted water around the charged tetrahedral transition state. Therefore, we conclude that control of the solvent dielectric constant can stabilize the tetrahedral transition state, and this lowers the activation energy.     

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.     

Charges on proteins and distances of electron transfer in metalloprotein redox reactions

A modified form of the Debye-Marcus equation relating electron transfer rate constants to charges on proteins and distances of electron transfer has been applied to the reaction of chemically modified cytochrome f, in which positively charged amino groups are replaced with negatively charged carboxyl groups. The rate of electron transfer from reduced cytochrome f to ferricyanide decreased with increasing ionic strength when the native and singly substituted cytochrome f were used, although a sharp decrease was observed in the former case. When doubly or more than triply substituted cytochrome f was used, the rate of electron transfer was almost constant or increased with increasing ionic strength, respectively. The kinetic-ionic strength effects on this reaction can be well explained by the Debye-Marcus equation in which the charge and radius of the protein are treated as variable parameters. The results show the importance of local positive charges of about 2.0 on native cytochrome f and effective radius of about 11 A of cytochrome f for the electron transfer to ferricyanide. Since the net charge on the native cytochrome f is negative and the calculated radius of the protein is 22.8 A, the above results indicate that positive charges on the electron transfer site control the electrostatic interactions in this reaction. Previously reported data which had been analyzed by using the total net charge and full radius of the protein, were also well explained by the local charge and effective radius of the protein.(ABSTRACT TRUNCATED AT 250 WORDS)     

Electrochromic shift of chlorophyll absorption in photosystem I from Synechocystis sp. PCC 6803: a probe of optical and dielectric properties around the secondary electron acceptor

Nanosecond absorption dynamics at approximately 685 nm after excitation of photosystem I (PS I) from Synechocystis sp. PCC 6803 is consistent with electrochromic shift of absorption bands of the Chl a pigments in the vicinity of the secondary electron acceptor A(1). Based on experimental optical data and structure-based simulations, the effective local dielectric constant has been estimated to be between 3 and 20, which suggests that electron transfer in PS I is accompanied by considerable protein relaxation. Similar effective dielectric constant values have been previously observed for the bacterial photosynthetic reaction center and indicate that protein reorganization leading to effective charge screening may be a necessary structural property of proteins that facilitate the charge transfer function. The data presented here also argue against attributing redmost absorption in PS I to closely spaced antenna chlorophylls (Chls) A38 and A39, and suggest that optical transitions of these Chls, along with that of connecting chlorophyll (A40) lie in the range 680-695 nm.     

Kinetics of reduction of high redox potential ferredoxins by the semiquinones of Clostridium pasteurianum flavodoxin and exogenous flavin mononucleotide. Electrostatic and redox potential effects

We have measured the ionic strength dependence of the rate constants for the electron-transfer reactions of flavin mononucleotide (FMN) and flavodoxin semiquinones with 10 high redox potential ferredoxins (HiPIP's). The rate constants were extrapolated to infinite ionic strength by using a theoretical model of electrostatic interactions developed in our laboratory. In all cases, the sign of the electrostatic interaction was the same as the protein net charge, but the magnitudes were much smaller. The results are consistent with a model in which the electrical charges are approximately uniformly distributed over the HiPIP surface and in which there are both short- and long-range electrostatic interactions. An electrostatic field calculation for Chromatium vinosum HiPIP is consistent with this. The presumed site of electron transfer includes that region of the protein surface to which the iron-sulfur cluster is nearest and appears to be relatively hydrophobic. The principal short-range electrostatic interaction would involve the negative charge on the iron-sulfur cluster. For some net negatively charged proteins, this effect is magnified, and for net positively charged HiPIP's, it is counterbalanced. The rate constants extrapolated to infinite ionic strength can be correlated with redox potential differences between the reactants, as has previously been shown for cytochrome-flavin semiquinone reactions. Both electrostatic and redox potential effects are magnified for the flavodoxin semiquinone as compared to the FMN semiquinone-HiPIP reactions. This was also observed previously for the flavin semiquinone-cytochrome reactions.(ABSTRACT TRUNCATED AT 250 WORDS)     

The origin and impact of bound water around intrinsically disordered proteins

Proteins and water couple dynamically over a wide range of time scales. Motivated by their central role in protein function, protein-water dynamics and thermodynamics have been extensively studied for structured proteins, where correspondence to structural features has been made. However, properties controlling intrinsically disordered protein (IDP)-water dynamics are not yet known. We report results of megahertz-to-terahertz dielectric spectroscopy and molecular dynamics simulations of a group of IDPs with varying charge content along with structured proteins of similar size. Hydration water around IDPs is found to exhibit more heterogeneous rotational and translational dynamics compared with water around structured proteins of similar size, yielding on average more restricted dynamics around individual residues of IDPs, charged or neutral, compared with structured proteins. The on-average slower water dynamics is found to arise from excess tightly bound water in the first hydration layer, which is related to greater exposure to charged groups. The more tightly bound water to IDPs correlates with the smaller hydration shell found experimentally, and affects entropy associated with protein-water interactions, the contribution of which we estimate based on the dielectric measurements and simulations. Water-IDP dynamic coupling at terahertz frequencies is characterized by the dielectric measurements and simulations.     

pH-dependent redox potential: how to use it correctly in the activation energy analysis

The activation barrier (the activation free energy) for the reaction's elementary act proper does not depend on the presence of reactants outside the reaction complex. The barrier is determined directly by the concentration-independent configurational free energy. In the case of redox reactants with pH-dependent redox potential, only the pH-independent quantity, the configurational redox potential enters immediately into expression for activation energy. Some typical cases of such reactions have been discussed (e.g., simultaneous proton and electron detachment, acid dissociation followed by oxidation, dissociation after oxidation, and others). For these mechanisms, the algorithms for calculation of the configurational redox potential from the experimentally determined redox potentials have been described both for the data related to a dissolved reactant or to a prosthetic group of an enzyme. Some examples of pH-dependent enzymatic redox reactions, in particular for the Rieske iron-sulfur protein, have been discussed.     

pH-dependent redox potential: how to use it correctly in the activation energy analysis

The activation barrier (the activation free energy) for the reaction's elementary act proper does not depend on the presence of reactants outside the reaction complex. The barrier is determined directly by the concentration-independent configurational free energy. In the case of redox reactants with pH-dependent redox potential, only the pH-independent quantity, the configurational redox potential enters immediately into expression for activation energy. Some typical cases of such reactions have been discussed (e.g., simultaneous proton and electron detachment, acid dissociation followed by oxidation, dissociation after oxidation, and others). For these mechanisms, the algorithms for calculation of the configurational redox potential from the experimentally determined redox potentials have been described both for the data related to a dissolved reactant or to a prosthetic group of an enzyme. Some examples of pH-dependent enzymatic redox reactions, in particular for the Rieske iron-sulfur protein, have been discussed.