A transition state theory approach to the kinetics of conductance changes in excitable membranes

Summary The kinetics of ionic current mechanisms in excitable membranes are analyzed. It is assumed that there are voltage-dependent reactions occurring in the membrane which are independent of the flow of ionic current. The experimental evidence for this assumption is reviewed in the light of more recent results on the kinetics of conductance changes in cardiac membranes. Rate equations are then obtained using transition state theory and assuming that each reaction is rate limited by only one energy barrier. These equations give simple exponential functions for the voltage dependence of the rates. More complex functions may be obtained by assuming that more than one energy barrier is rate limiting. The single-barrier equations are used to estimate the energies of formation of the transition state. In most cases, the entropy of formation is positive but there is no systematic order in the estimated enthalpies. These results are contrasted with those for the ion permeation process itself which normally has a negative entropy of activation. This contrast reinforces the assumption that the reactions controlling membrane permeability are distinct from the ion permeation process itself. The significance of the positive entropy of formation of the transition state in the permeability reactions is discussed, and it is suggested that the membrane structures underlying these reactions may change their degree of hydration during the formation of the transition state.     

Structural changes in the transition state of protein folding: alternative interpretations of curved chevron plots.

The interpretation of folding rates is often rationalized within the context of transition state theory. This means that the reaction rate is linked to an activation barrier, the height of which is determined by the free energy difference between a ground state (the starting point) and an apparent transition state. Changes in the folding kinetics are thus caused by effects on either the ground state, the transition state, or both. However, structural changes of the transition state are rarely discussed in connection with experimental data, and kinetic anomalies are commonly ascribed to ground state effects alone, e.g., depletion or accumulation of structural intermediates upon addition of denaturant. In this study, we present kinetic data which are best described by transition state changes. We also show that ground state effects and transition state effects are in general difficult to distinguish kinetically. The analysis is based on the structurally homologous proteins U1A and S6. Both proteins display two-state behavior, but there is a marked difference in their kinetics. S6 exhibits a classical V-shaped chevron plot (log observed rate constant vs denaturant concentration), whereas U1A's chevron plot is symmetrically curved, like an inverted bell curve. However, S6 is readily mutated to display U1A-like kinetics. The seemingly drastic effects of these mutations are readily ascribed to transition state movements where large kinetic differences result from relatively small alterations of a common free energy profile and broad activation barriers.     

Fluctuations in the polymerization of sickle hemoglobin. A simple analytic model

Using the stochastic theory of chemical reactions and the theory of first passage times, a simple analytic expression is derived for the distribution of delay times that has been observed in studies of the polymerization kinetics of sickle hemoglobin under conditions where the polymerization progress curves exhibit stochastic variation. The rate of homogeneous nucleation can be readily extracted from such experiments using this expression. This work constitutes a significant addition to the rather limited number of examples where contact can be successfully made between the stochastic theory of chemical kinetics and experiment.     

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.     

Electron transfer between flavodoxin semiquinone and c-type cytochromes: correlations between electrostatically corrected rate constants, redox potentials, and surface topologies

We have measured the ionic strength dependence of the rate constants for electron transfer from the semiquinone of Clostridium pasteurianum flavodoxin to 12 c-type cytochromes and several inorganic oxidants using stopped-flow methodology. The experimental data were fit quite well by an electrostatic model that represents the interaction domains as parallel disks with a point charge equal to the charge within this region of the protein. The analysis provides an evaluation of the electrostatic interaction energy and the rate constant at infinite ionic strength (k affinity). The electrostatic charge on the oxidant within the interaction site can be obtained from the electrostatic energy, and for most of those reactants for which structures are available, the results are in good agreement with expectation. The k affinity values were found to correlate with redox potential differences, as expected from the theory of adiabatic (or nonadiabatic) outer-sphere electron-transfer reactions. Deviations from the theoretical curves are interpreted in terms of the influence of surface topology on reaction rate constants. In general, we find that electrostatic effects, steric influences, and redox potential all exert a much larger effect on reaction rate constants for the flavodoxin-cytochrome system than has been previously observed for free flavin-cytochrome interactions. The implications of this for determining biological specificity are discussed.     

Diffusive motions control the folding and unfolding kinetics of the apomyoglobin pH 4 molten globule intermediate

The sperm whale apomyoglobin pH 4 folding intermediate exists in two forms, Ia and Ib, that mimic transient kinetic intermediates in the folding of the native protein at pH 6. To characterize the nature of the kinetic barrier that controls the formation of the earliest intermediate Ia, we have investigated the effects of small viscogenic cosolvents on its folding and unfolding kinetics. The kinetics are measurable by stopped-flow fluorescence and follow a cooperative two-state model in the absence and presence of cosolvents. Small cosolvents stabilize Ia, but, by applying the isostability test to separate the viscogenic effect of the cosolvent from its stabilizing effect, we found that, in both folding and unfolding conditions, the apparent rate constant decreases when solvent viscosity increases. The unitary inverse dependence of the apparent rate constant on solvent viscosity indicates a diffusion-controlled reaction. This result is consistent with the hypothesis that folding of the apomyoglobin pH 4 intermediate obeys a diffusion-collision model. Additionally, the temperature dependence of the reaction rate at constant viscosity indicates that the formation of Ia is also controlled by an energy barrier. Linear free energy relationships show that the transition state of the U <==> Ia reaction is compact and buries 45% of the surface area that is buried in native apomyoglobin. We conclude that the transition state of the U <==> Ia reaction resembles that for the formation of native proteins; namely, it is dry and its compactness is closer to that of the folded (Ia) form than of the unfolded form.     

Kinetics of the pectin methylesterase catalyzed de-esterification of pectin in frozen food model systems

The pectin methylesterase (PME) catalyzed de-esterification of pectin was studied in four frozen food model systems based on sucrose, fructose, maltodextrin, and carboxymethylcellulose (CMC) in a temperature range from -24 to 20 degrees C, with the aim of elucidating the applicability of the theory of "food polymer science" on the kinetics. The rate substantially decreased around the glass transition temperature in the case of CMC, while very low rates were observed far above the glass transition temperature in the case of maltodextrin, fructose, and sucrose model systems. In general, the kinetics of this reaction was found to be influenced more by factors such as the characteristics of the component solutes, freeze concentration, the possible viscosity enhancement due to a particular combination of solutes, and the molecular size of the substrate molecule rather than the glass transition process. The Arrhenius equation described the temperature dependence of kinetics both in the liquid state of all the systems studied (r(2) > or = 0.97) and the glassy state of CMC (r(2) = 0.95). A clear break in the Arrhenius plot was observed as the temperature decreased to subfreezing temperatures. The Arrhenius equation could describe the kinetics reasonably well in the rubbery state for fructose and sucrose model systems (r(2) > 0.992). In the case of maltodextrin and CMC, the Arrhenius plots showed a slight curvature followed by a break at the glass transition temperature for CMC. The WLF equation with system-dependent coefficients better described the kinetics in the rubbery state of the CMC and part of the maltodextrin system. A linear relationship between the logarithm of the rate and T - Tg' described the kinetics in the sucrose as well as fructose model systems (r(2) = 0.9928 and 0.993, respectively).     

Steady state kinetics of an enzyme reaction with one substrate and one modifier

This study examines the steady state kinetics of a reaction involving an enzyme, a substrate and a modifier when the reaction follows Michaelis-Menten kinetics. Conditions for Michaelis-Menten kinetics are deduced, and it is shown that an analogue of detailed balance determines the complexity of the rate equations in these cases. A scheme to distinguish many cases of Michaelis-Menten kinetics is presented. It is shown that steady state kinetics are, in general, insufficient to specify the mechanism of a reaction, since different effects of a modifier can give identical steady state kinetic data.     

Kinetic analysis of folding and unfolding the 56 amino acid IgG-binding domain of streptococcal protein G.

The 56 amino acid B domain of protein G (GB) is a stable globular folding unit with no disulfide cross-links. The physical properties of GB offer extraordinary flexibility for evaluating the energetics of the folding reaction. The protein is monomeric and very soluble in both folded and unfolded forms. The folding reaction has been previously examined by differential scanning calorimetry (Alexander et al., 1992) and found to exhibit two-state unfolding behavior over a wide pH range with an unfolding transition near 90 degrees C (GB1) at neutral pH. Here, the kinetics of folding and unfolding two naturally occurring versions of GB have been measured using stopped-flow mixing methods and analyzed according to transition-state theory. GB contains no prolines, and the kinetics of folding and unfolding can be fit to a single, first-order rate constant over the temperature range of 5-35 degrees C. The major thermodynamic changes going from the unfolded state to the transition state are (1) a large decrease in heat capacity (delta Cp), indicating that the transition state is compact and solvent inaccessible relative to the unfolded state; (2) a large loss of entropy; and (3) a small increase in enthalpy. The most surprising feature of the folding of GB compared to that of previously studied proteins is that its folding approximates a rapid diffusion controlled process with little increase in enthalpy going from the unfolded to the transition state.     

Chemical dynamic approach to synergetics and instability in biological systems

Based on Haken's theory, self-organization or synergetics is discussed using chemical dynamics to represent an autocatalytic reaction. In a simple case the changes in a self-organizing system are given by a set of two rate equations for a primary and a partial system. When these systems mutually form a feedback loop under the adiabatic condition, the rate equation of self-organization is described by a generalized Gibbs' free energy change delta U (delta x) followed by the reaction. The sign of the parameter k'3 (k0-kex; k0, kex: rate constants with or without an external stimulus) determines the instability of the coupled system in quasi-equilibrium (k'3 approximately greater than 0; k0 greater than kex). When the product exceeds the threshold (k'3 less than 0; k0 less than kex), the system transfers into a new state, or a phase transition appears. Considering the Boltzmann distribution, the transition parameter k'3 is evaluated by an average distribution of the states and the instability is discussed using the reaction velocities vqe and vqe in the quasi-equilibrium state. As an example of this model membrane excitation is discussed briefly.     

Equilibrium and kinetics of the allosteric transition of GroEL studied by solution X-ray scattering and fluorescence spectroscopy

We have studied the ATP-induced allosteric structural transition of GroEL using small angle X-ray scattering and fluorescence spectroscopy in combination with a stopped-flow technique. With X-ray scattering one can clearly distinguish the three allosteric states of GroEL, and the kinetics of the transition of GroEL induced by 85 microM ATP have been observed directly by stopped-flow X-ray scattering for the first time. The rate constant has been found to be 3-5s(-1) at 5 degrees C, indicating that this process corresponds to the second phase of the ATP-induced kinetics of tryptophan-inserted GroEL measured by stopped-flow fluorescence. Based on the ATP concentration dependence of the fluorescence kinetics, we conclude that the first phase represents bimolecular non-cooperative binding of ATP to GroEL with a bimolecular rate constant of 5.8 x 10(5)M(-1)s(-1) at 25 degrees C. Considering the electrostatic repulsion between negatively charged GroEL (-18 of the net charge per monomer at pH 7.5) and ATP, the rate constant is consistent with a diffusion-controlled bimolecular process. The ATP-induced fluorescence kinetics (the first and second phases) at various ATP concentrations (< 400 microM) occur before ATP hydrolysis by GroEL takes place and are well explained by a kinetic allosteric model, which is a combination of the conventional transition state theory and the Monod-Wyman-Changeux model, and we have successfully evaluated the equilibrium and kinetic parameters of the allosteric transition, including the binding constant of ATP in the transition state of GroEL.     

Stochastic and deterministic simulations of heterogeneous cell population dynamics

A Monte Carlo algorithm, which can accurately simulate the dynamics of entire heterogeneous cell populations, was developed. The algorithm takes into account the random nature of cell division as well as unequal partitioning of cellular material at cell division. Moreover, it is general in the sense that it can accommodate a variety of single-cell, deterministic reaction kinetics as well as various stochastic division and partitioning mechanisms. The validity of the algorithm was assessed through comparison of its results with those of the corresponding deterministic cell population balance model in cases where stochastic behavior is expected to be quantitatively negligible. Both algorithms were applied to study: (a) linear intracellular kinetics and (b) the expression dynamics of a genetic network with positive feedback architecture, such as the lac operon. The effects of stochastic division as well as those of different division and partitioning mechanisms were assessed in these systems, while the comparison of the stochastic model with a continuum model elucidated the significance of cell population heterogeneity even in cases where only the prediction of average properties is of primary interest.     

Schematic methods for probabilistic enzyme kinetics.

The theory of steady-state enzyme processes which avoids using the mass action law of chemical kinetics and consistently describes catalytic mechanisms by probabilistic concepts has recently been proposed (Mazur, 1991, J. theor. Biol. 148, 229-242). To facilitate the analysis of complex reaction graphs by this theory the possibility of constructing schematic rules similar to those used in classical kinetics is studied. It is found that due to the similarity of algebraic procedures the popular method of King & Altman can be applied in probabilistic kinetics in addition to the earlier proposed rule based on enumeration of cycles of the reaction graph. This similarity also allows one to adapt many other shortcut methods of classical kinetics for probabilistic reaction graphs. The paper considers separately the possibility of transforming reaction mechanisms so that the initial graph is replaced by a simpler but equivalent one. It is shown that there are few cases when a group of states can be replaced by one united state, with earlier known rules such as the rule of Cha for equilibrium stages being particular cases of a more general procedure. In addition a novel method is proposed which performs step-by-step reduction of any reaction graph. All the new methods can be adapted for traditional kinetics as well. The results obtained demonstrate that many schematic rules of classical kinetics are of probabilistic origin.     

Trehalose favors a cutinase compact intermediate off-folding pathway

The folding of cutinase, an enzyme displaying lipolytic activity, has been studied in the presence of trehalose. Equilibrium unfolding data show that trehalose increases the free energy change between folded and unfolded states. Unfolding kinetics reveal the presence of an intermediate which is ca. 60% folded in terms of solvent exposure. Trehalose stabilizes this intermediate relative to the folded state. In contrast, the intermediate revealed by folding kinetics is more compact than the transition state, as shown by the positive slope observed at low denaturant concentration in the chevron plot, as well as the decrease in the observable rate constant for folding with the increase in trehalose concentration. This intermediate displays more than 50% of area buried from the solvent (relative to the native state) compared to around 40% for the transition state for folding and therefore appears to be off the folding pathway. Trehalose stabilizes and guanidine hydrochloride destabilizes this compact intermediate. Both unfolding and folding kinetics show that compact conformational states are stabilized by trehalose, in agreement with current models on the effect of compatible solutes. This effect occurs even for compact states that decelerate the folding as in the case of the intermediate revealed by folding kinetics.     

Premelting thermal fluctuational base pair opening probability of poly(dA).poly(dT) as predicted by the modified self-consistent phonon theory.

We employ a mean field, modified, self-consistent phonon theory to evaluate the single base-pair opening rate and the probability of a base pair in the amino proton exchangeable state for the homopolymer poly(dA).poly(dT) at temperatures below the helix-coil transition region. Our calculated premelting single base-pair opening probabilities are in general agreement with several available experimental estimates from imino proton exchange and formaldehyde-induced DNA melting measurements. These calculated opening probabilities, however, are in disagreement with the prediction of the helix-coil transition theory. Possible reasons for the differences are discussed, especially the possible different definition of a meaningful open state in the premelting region. The premelting open state of the modified self-consistent phonon approximation theory seems to be appropriate to describe a solvent-accessible open configuration that is sufficient to facilitate important chemical reactions such as imino proton exchange and formaldehyde reaction with the bases. This can be compared with the completely unstacked open state of the helix-coil transition theory originally defined in the helix-coil transition region. We propose that the amino proton exchangeable state is different from the open state associated with melting and only involves the breaking of the amino interbase H bond. The agreement between the calculated and experimentally estimated probability of a base pair in the amino proton exchangeable state seems to support this hypothesis.     

Theoretical investigation on the kinetics and branching ratio of the gas phase reaction of sevoflurane with Cl atom

The present work deals with the theoretical investigation on the Cl initiated H-atom abstraction reaction of sevoflurane, (CF₃)₂CHOCH₂F. A dual-level procedure has been adopted for studying the kinetics of the reaction. Geometrical optimization and frequency calculation were performed at DFT(BHandHLYP)/6-311G(d,p) while single-point energy calculation was made at CCSD(T)/6-311G(d,p) level of theory. The intrinsic reaction coordinate (IRC) calculation has also been performed to confirm the smooth transition from the reactant to product through the respective transition state. The rate constants were calculated using conventional transition state theory (TST). It has been found that 99 % of the reaction proceeded via the H-atom abstraction from the –CH₂F end of the sevoflurane. The rate constant of the dominant path is found to be 1.13 × 10^?13 cm³ molecule^?1 s^?1. This is in excellent agreement with the reported experimental rate constant of 1.10 × 10^?13 cm³ molecule^?1 s^?1 obtained by relative rate method using FTIR/Smog chamber and LP/LIF techniques.     

Rate of allosteric change in hemoglobin measured by modulated excitation using fluorescence detection.

We have measured the forward and reverse rates of the allosteric transition of hemoglobin A with three CO molecules bound by using modulated excitation coupled with fluorescence quenching of the DPG analogue, PTS (8-hydroxy-1,3,6 pyrene trisulfonic acid). This dye is observed to bind to the T state with significantly larger affinity than to the R state, and thus provides an unequivocal marker for the molecule's conformational change. The allosteric rates obtained with the fluorescent dye (pH 7.0, bis-Tris buffer) are (3.4 +/- 1.0) x 10(3)s-1 for the R to T transition and (2.1 +/- 0.5) x 10(4)s-1 for the T to R transition. This gives an equilibrium constant L3 of 0.16 +/- 0.06. These results provide good agreement with modulated difference spectra calibrated from model compounds, arguing that there is little if any difference in the kinetics observed by the heme spectra and the kinetics of the full subunit motion. The equilibrium constant between structures (L3) is smaller in the absence of phosphates than observed in phosphate buffer (0.33). However, the rates of the allosteric transition increase in the absence of phosphates as compared with the corresponding rates in phosphate buffer of 1.0 x 10(3)s-1 and 3.0 x 10(3)s-1. The effects of inorganic phosphates on the equilibrium can be separated from the effects on kinetics. We find that phosphates also affect the dynamic behavior of hemoglobin, and the presence of 0.15 M phosphate can be viewed as raising the transition state energy between R and T conformations by approximately 0.5 kcal/mol exclusive of the T state stabilization. Dissociation constants for the dye were measured to be 104 +/- 25 microM for unligated T state and 930 +/- 300 microM for the fully ligated R state. The best fit equilibrium constant (125 +/- 40 microM) for three ligands bound does not differ significantly from that measured without ligands bound. Incidental to the measurement technique is the determination of the rates of binding and release of the dye. The association rate for binding to the T state is large, (at least 4 x 10(9) M-1 s-1) and may be diffusion limited, while the association and dissociation rates for R state binding, while not determined with precision, are clearly much smaller, of the scale of 10(5) M-1 s-1 for association.     

Local interactions and the optimization of protein folding

The role of local interactions in protein folding has recently been the subject of some controversy. Here we investigate an extension of Zwanzig's simple and general model of folding in which local and nonlocal interactions are represented by functions of single and multiple conformational degrees of freedom, respectively. The kinetics and thermodynamics of folding are studied for a series of energy functions in which the energy of the native structure is fixed, but the relative contributions of local and nonlocal interactions to this energy are varied over a broad range. For funnel shaped energy landscapes, we find that 1) the rate of folding increases, but the stability of the folded state decreases, as the contribution of local interactions to the energy of the native structure increases, and 2) the amount of native structure in the unfolded state and the transition state vary considerably with the local interaction strength. Simple exponential kinetics and a well-defined free energy barrier separating folded and unfolded states are observed when nonlocal interactions make an appreciable contribution to the energy of the native structure; in such cases a transition state theory type approximation yields reasonably accurate estimates of the folding rate. Bumps in the folding funnel near the native state, which could result from desolvation effects, side chain freezing, or the breaking of nonnative contacts, significantly alter the dependence of the folding rate on the local interaction strength: the rate of folding decreases when the local interaction strength is increased beyond a certain point. A survey of the distribution of strong contacts in the protein structure database suggests that evolutionary optimization has involved both kinetics and thermodynamics: strong contacts are enriched at both very short and very long sequence separations. Proteins 29:282–291, 1997. © 1997 Wiley-Liss, Inc.