Marcus treatment of endergonic reactions: A commentary

Two forms of the equation for expression of the rate constant for electron transfer through a Marcus-type treatment are discussed. In the first (exergonic) form, the Arrhenius exponential term was replaced by its classical Marcus term; in the second (endergonic) form, the forward rate constant was replaced by the reverse rate constant (the forward rate constant in the exergonic direction), which was expanded to an equivalent Marcus term and multiplied by the equilibrium constant. When the classical Marcus treatment was used, these two forms of the rate equation give identical curves relating the logarithm of the rate constant to the driving force. The Marcus term for the relation between activation free-energy, ΔG^#, reorganization energy, λ, and driving force, ΔG^o, derived from parabolas for the reactant and product states, was identical when starting from exergonic or endergonic parabolas. Moser and colleagues introduced a quantum mechanical correction factor to the Marcus term in order to fit experimental data. When the same correction factor was applied in the treatment for the endergonic direction by Page and colleagues, a different curve was obtained from that found with the exergonic form. We show that the difference resulted from an algebraic error in development of the endergonic equation.     

Marcus treatment of endergonic reactions: a commentary

Two forms of the equation for expression of the rate constant for electron transfer through a Marcus-type treatment are discussed. In the first (exergonic) form, the Arrhenius exponential term was replaced by its classical Marcus term; in the second (endergonic) form, the forward rate constant was replaced by the reverse rate constant (the forward rate constant in the exergonic direction), which was expanded to an equivalent Marcus term and multiplied by the equilibrium constant. When the classical Marcus treatment was used, these two forms of the rate equation give identical curves relating the logarithm of the rate constant to the driving force. The Marcus term for the relation between activation free-energy, DeltaG#, reorganization energy, lambda, and driving force, DeltaG(o), derived from parabolas for the reactant and product states, was identical when starting from exergonic or endergonic parabolas. Moser and colleagues introduced a quantum mechanical correction factor to the Marcus term in order to fit experimental data. When the same correction factor was applied in the treatment for the endergonic direction by Page and colleagues, a different curve was obtained from that found with the exergonic form. We show that the difference resulted from an algebraic error in development of the endergonic equation.     

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.     

Electron-transfer reaction between copper macrocyclic complex and ascorbate anion. Application of the Marcus theory

The kinetics of the reduction of a copper(II) complex with a tetrabenzo(b, f, j, n) (1, 5, 9, 13)tetraazacyclohexadecine (TAAB) by ascorbic acid has been investigated in aqueous buffer solutions of pH 2.8 up to 5.0. The polarographic half-wave potential of Cu(TAAB)²⁺ reduction in aqueous media and acetonitrile and the acid equilibrium constant of Cu(TAAB)²⁺ have been determined. The rate constant of the outer-sphere electron-transfer reaction is discussed in terms of the Marcus theory. The previously reported self-exchange kinetic parameters per one-electron couples HA^0/- and Cu(TAAB)^2+/+ have been confirmed, and estimates of some other parameters of the electron self-exchange reactions and the cross-reactions are reported here.     

Transition-state ensemble in enzyme catalysis: possibility, reality, or necessity?

Proteins are not rigid structures; they are dynamic entities, with numerous conformational isomers (substates). The dynamic nature of protein structures amplifies the structural variation of the transition state for chemical reactions performed by proteins. This suggests that utilizing a transition state ensemble to describe chemical reactions involving proteins may be a useful representation. Here we re-examine the nature of the transition state of protein chemical reactions (enzyme catalysis), considering both recent developments in chemical reaction theory (Marcus theory for SN2 reactions), and protein dynamics effects. The classical theory of chemical reactions relies on the assumption that a reaction must pass through an obligatory transition-state structure. The widely accepted view of enzymatic catalysis holds that there is tight binding of the substrate to the transition-state structure, lowering the activation energy. This picture, may, however, be oversimplified. The real meaning of a transition state is a surface, not a single saddle point on the potential energy surface. In a reaction with a "loose" transition-state structure, the entire transition-state region, rather than a single saddle point, contributes to reaction kinetics. Consequently, here we explore the validity of such a model, namely, the enzymatic modulation of the transition-state surface. We examine its utility in explaining enzyme catalysis. We analyse the possibility that instead of optimizing binding to a well-defined transition-state structure, enzymes are optimized by evolution to bind efficiently with a transition-state ensemble, with a broad range of activated conformations. For enzyme catalysis, the key issue is still transition state (ensemble) stabilization. The source of the catalytic power is the modulation of the transition state. However, our definition of the transition state is the entire transition-state surface rather just than a single well-defined structure. This view of the transition-state ensemble is consistent with the nature of the protein molecule, as embodied and depicted in the protein energy landscape of folding, and binding, funnels.     

Marcus rate theory applied to enzymatic proton transfer

The hydration of CO(2) and the dehydration of HCO(3)(-) catalyzed by the carbonic anhydrases is accompanied by the transfer of protons between solution and the zinc-bound water molecule in the active site. This transfer is facilitated by amino acid residues of the enzyme which act as intramolecular proton shuttles; variants of carbonic anhydrase lacking such shuttle residues are enhanced or rescued in catalysis by intermolecular proton transfer from donors such as imidazole in solution. The resulting rate constants for proton transfer when compared with the values of the pK(a) of the donor and acceptor give Bronsted plots of high curvature. These data are described by Marcus theory which shows an intrinsic barrier for proton transfer from 1 to 2 kcal/mol and work terms or thermodynamic contributions to the free energy of reaction from 4 to10 kcal/mol. The interpretation of these Marcus parameters is discussed in terms of the well-studied pathway of the catalysis and structure of the enzymes.     

Proton-coupled electron transfer at the Qo-site of the bc1 complex controls the rate of ubihydroquinone oxidation

The rate-limiting reaction of the bc(1) complex from Rhodobacter sphaeroides is transfer of the first electron from ubihydroquinone (quinol, QH(2)) to the [2Fe-2S] cluster of the Rieske iron-sulfur protein (ISP) at the Q(o)-site. Formation of the ES-complex requires participation of two substrates (S), QH(2) and ISP(ox). From the variation of rate with [S], the binding constants for both substrates involved in formation of the complex can be estimated. The configuration of the ES-complex likely involves the dissociated form of the oxidized ISP (ISP(ox)) docked at the b-interface on cyt b, in a complex in which N(epsilon) of His-161 (bovine sequence) forms a H-bond with the quinol -OH. A coupled proton and electron transfer occurs along this H-bond. This brief review discusses the information available on the nature of this reaction from kinetic, structural and mutagenesis studies. The rate is much slower than expected from the distance involved, likely because it is controlled by the low probability of finding the proton in the configuration required for electron transfer. A simplified treatment of the activation barrier is developed in terms of a probability function determined by the Br?nsted relationship, and a Marcus treatment of the electron transfer step. Incorporation of this relationship into a computer model allows exploration of the energy landscape. A set of parameters including reasonable values for activation energy, reorganization energy, distances between reactants, and driving forces, all consistent with experimental data, explains why the rate is slow, and accounts for the altered kinetics in mutant strains in which the driving force and energy profile are modified by changes in E(m) and/or pK of ISP or heme b(L).     

Two enzyme active transport in vitro with ph induced asymmetrical functional structures: I. The model and its analytical treatment

A class of systems is characterized by the asymmetrical distribution of a sink and a source reaction, the asymmetry of the global chemical equation (energy liberation) and by an asymmetrical one-wave space profile. These systems belong to the family of primary chemical cells and can deplete and enrich the media they separate. A “ one way ” transport-reaction chain is needed for specific “ real ” active transport. A two enzyme model of this class is described in which the spatial asymmetry is due to a (diffusive) pH gradient; this distribution of “ potential ” enzyme activities is called the “ functional structure ”. Equal potential enzyme activities and absence of reactive back action on local pH are assumed in the “ square model ” version of the pump. Analytical expressions of the enzymatic diffusion reactions are derived for zero and first order kinetics, i.e. in function of substrate concentrations. Tables of equations are presented. The intrinsic properties of the pump are characterized by (dimensionless) transport reaction parameters, (membrane composition); the “ potential ” activity is controlled by the pH gradient; the “ effective ” pumping is also a function of the substrate concentrations on the boundaries.     

Association Kinetics from Single Molecule Force Spectroscopy Measurements

Single molecule force spectroscopy is often used to study the dissociation of single molecules by applying mechanical force to the intermolecular bond. These measurements provide the kinetic parameters of dissociation. We present what to our knowledge is a new atomic force microscopy-based approach to obtain the activation energy of the association reaction and approximate grafting density of reactive receptors using the dependence of the probability to form molecular bonds on probe velocity when one of the interacting molecules is tethered by a flexible polymeric linker to the atomic force microscopy probe. Possible errors in the activation energy measured with this approach are considered and resulting corrections are included in the data analysis. This new approach uses the same experimental setup as traditional force spectroscopy measurements that quantify dissociation kinetics. We apply the developed methodology to measure the activation energy of biotin-streptavidin association (including a contribution from the steric factor) and obtain a value of 8 ± 1 kT. This value is consistent with the association rate measured previously in solution. Comparison with the solution-derived activation energy indicates that kinetics of biotin-streptavidin binding is mainly controlled by the reaction step.     

Reduction of intrinsic kinetic and thermodynamic barriers for enzyme-catalysed proton transfers from carbon acid substrates

Many enzymes catalyse the heterolytic abstraction of the alpha-proton from a carbon acid substrate. Gerlt and Gassman have applied Marcus formalism to such proton transfer reactions to argue that transition states for concerted general acid-general base catalysed enolization at enzyme active sites occur late on the reaction coordinate (J. Am. Chem. Soc. 115 (1993) 11552). We postulate that as an enzyme evolves, it may decrease deltaG++ for a proton transfer step associated with substrate enolization by following the path of steepest descent on the two-dimensional surface corresponding to deltaG++, as defined by Marcus formalism. We show that for an enzyme that has decreased deltaG++ following the path of steepest descent, the values of the intrinsic kinetic (deltaG++(int,E)) and thermodynamic (deltaG(E)0) barriers for proton transfer reactions on the enzyme may be predicted from the known values of deltaG++(int,N) and deltaG(N)0 for the corresponding non-enzymic reaction and the free energy of activation on the enzyme (deltaG++(E)). In addition, the enzymic transition state will occur later on the reaction coordinate than the corresponding non-enzymic transition state (i.e. x++(E)>x++(N)) if the condition (6 - square root 2)/82deltaG++(int,N).     

Cellular growth and division in the Gillespie algorithm

Recent experimental studies elucidating the importance of noise in gene regulation have ignited widespread interest in Gillespie's stochastic simulation technique for biochemical networks. We formulate modifications to the Gillespie algorithm which are necessary to correctly simulate chemical reactions with time-dependent reaction rates. We concentrate on time dependence of kinetic rates arising from the periodic process of growth and division of the cellular volume, and demonstrate that a careful re-derivation of the Gillespie algorithm is important when all stochastically simulated reactions have rates slower or comparable to the cellular growth rate. For an unregulated single-gene system, we illustrate our findings using recently proposed hybrid simulation techniques, and systematically compare our algorithm with analytic results obtained from the chemical master equation.     

Lysozyme-lysozyme and lysozyme-salt interactions in the aqueous saline solution: a new square-well potential

We investigate lysozyme-lysozyme and lysozyme-salt interactions in electrolyte solutions using a molecular-thermodynamic model. An equation of state based on the statistical mechanical perturbation theory is applied to describe the interactions. The perturbation term includes a new square-well potential of mean force, which implies the information about the lysozyme surface and salt type. The attractive energy of the potential of mean force is correlated with experimental cloud-point temperatures of lysozyme in various solution conditions. The same attractive energy is used to predict osmotic pressure of a given system with no additional parameters. The new potential shows a satisfactory improvement in understanding the interactions between lysozymes in aqueous salt solutions.     

Relationship between the oxidation potential of the bacteriochlorophyll dimer and electron transfer in photosynthetic reaction centers

The primary electron donor in the photosynthetic reaction center from purple bacteria is a bacteriochlorophyll dimer containing four conjugated carbonyl groups that may form hydrogen bonds with amino acid residues. Spectroscopic analyses of a set of mutant reaction centers confirm that hydrogen bonds can be formed between each of these carbonyl groups and histidine residues in the reaction center subunits. The addition of each hydrogen bond is correlated with an increase in the oxidation potential of the dimer, resulting in a 355-mV range in the midpoint potential. The resulting changes in the free-energy differences for several reactions involving the dimer are related to the electron transfer rates using the Marcus theory. These reactions include electron transfer from cytochrome c₂ to the oxidized dimer, charge recombination from the primary electron acceptor quinone, and the initial forward electron transfer.     

The use of rotating disk voltammetry for the determination of homogeneous small molecule--redox protein reaction rates

Rotating disk voltammetry was used in this work to study the rates of reaction of ferricytochrome c with two very strong reductants, methyl and benzyl viologen. The rates of reaction for these reductants were found to be 4.0 × 10⁷ and 5.4 × 10⁷m^?1s^?1 at 24°C for benzyl and methyl viologen, respectively. The versatility of this method was demonstrated by the ease with which the activation parameters were obtained. The ΔH^≠ and ΔS^≠ were found to be 4.0 kcal/mol and ?10.6 cal/mol-K, respectively, for benzyl viologen. All the observed reaction rates were corrected for coulombic effects by the method of Wherland and Gray, and the electrostatically corrected rate constants were compared with the Marcus and Hopfield theories for electron transfer. The agreement was excellent for the tunneling theory but there were some discrepancies with the absolute Marcus theory. The relative Marcus approach worked quite well and, by taking into account the nonadiabaticity of the electron transfer, reasonable values were obtained for the absolute Marcus theory when realistic values of the self-exchange constants were used.     

Kinetic theory of the positive column of a low-pressure discharge in a transverse magnetic field

The influence of a transverse magnetic field on the characteristics of the positive column of a planar low-pressure discharge is studied theoretically. The motion of magnetized electrons is described in the framework of a continuous-medium model, while the ion motion in the ambipolar electric field is described by means of a kinetic equation. Using mathematical transformations, the problem is reduced to a secondorder ordinary differential equation, from which the spatial distribution of the potential is found in an analytic form. The spatial distributions of the plasma density, mean plasma velocity, and electric potential are calculated, the ion velocity distribution function at the plasma boundary is found, and the electron energy as a function of the magnetic field is determined. It is shown that, as the magnetic field rises, the electron energy increases, the distributions of the plasma density and mean plasma velocity become asymmetric, the maximum of the plasma density is displaced in the direction of the Ampère force, and the ion flux in this direction becomes substantially larger than the counter-directed ion flux.     

Chemically gated electron transfer. A means of accelerating and regulating rates of biological electron transfer

Long-range protein electron transfer [ET] reactions may be relatively slow because of long ET distance and low driving force. It is possible to dramatically increase the rate of such nonadiabatic reactions by using an adiabatic chemical reaction to activate the system for rapid ET. Three such examples are discussed; nitrogenase, pyruvate:ferredoxin oxidoreductase, and the methylamine dehydrogenase-amicyanin complex. In each example, the faster activated ET reaction is gated (i.e., rate-limited) by the chemical reaction. However, the reaction rate is still orders of magnitude greater than that of the ungated true ET reaction in the absence of chemical activation. Models are presented to describe the mechanisms of activation in the context of ET theory, and the relevance of such chemically gated ET to the regulation of metabolism is discussed.     

Rff,Conceptual Development of a Full Periodic Table Force Field for Studying Reaction Potential Surfaces

The attributes of a general molecular mechanics force field needed for the study of chemical reactions are described. This attribute set is referred to as a reaction force field (RFF). The functional forms of a first generation reaction force field, RFF1, are presented along with illustrative transition state geometries, activation energies, and reaction coordinate imaginary frequencies. This revised version was published online in June 2006 with corrections to the Cover Date.     

Light-induced charge separation in ruthenium based triads - New variations on an old theme

Success with artificial photosynthesis requires control of the photoinduced electron transfer reactions leading to charge-separated states. In this review, some new ideas to optimize such charge-separated states in ruthenium(II) polypyridyl based three-component systems with respect to: (1) long lifetimes and (2) ability to store sufficient energy for catalytic water splitting, are presented. To form long-lived charge-separated states, a manganese complex as electron donor and potential catalyst for water oxidation has been used. The recombination reaction is unusually slow because it occurs deep down in the Marcus normal region as a consequence of the large bond reorganization following the manganese oxidation. For the creation of high energy charge-separated states, a strategy using bichromophoric systems is presented. By consecutive excitations of the two chromophores, the formation of charge-separated states that lie higher in energy than either of the two excited states could in theory be achieved, the first results of which will be discussed in this review.     

Application of the Marcus theory for analysis of the temperature dependence of the reactions leading to photosynthetic water oxidation: results and implications

The temperature dependence of donor side reactions was analysed within the framework of the Marcus theory of nonadiabatic electron transfer. The following results were obtained for PS II membrane fragments from spinach: (1) the reorganisation energy of P680^+? reduction by Y_Z is of the order of 0.5?eV in samples with a functionally fully competent water oxidising complex (WOC); (2) destruction of the WOC by Tris-washing gives rise to a drastic increase of λ to values of the order of 1.6?eV; (3) the reorganisation energies of the oxidation steps in the WOC are dependent, on the redox states S _i with values of about 0.6?eV for the reactions Y_Z ^OX S ₀→Y_Z S ₁ and Y_Z ^OX S ₁→Y_Z S ₂, 1.6?eV for the reaction Y_Z ^OX S ₂→Y_Z S ₃ and 1.1?eV (above a characteristic temperature u_c of about 6??°C) for the reaction Y_Z ^OX S ₃→→Y_Z S ₀+O₂. Using an empirical rate constant-distance relationship, the van der Waals distance between Y_Z and P680 was found to be about 10?Å, independent of the presence or absence of the WOC, whereas the distance between Y_Z and the manganese cluster in the WOC was ≥15?Å. Based on the calculated activation energies the environment of Y_Z is inferred to be almost "dry" and hydrophobic when the WOC is intact but becomes enriched with water molecules after WOC destruction. Furthermore, it is concluded that the transition S ₂→S ₃ is an electron transfer reaction gated by a conformational change, i.e. it comprises significant structural changes of functional relevance. Measurements of kinetic H/D isotope exchange effects support the idea that none of these reactions is gated by the break of a covalent O-H bond. The implications of these findings for the mechanism of water oxidation are discussed.