The magnitude of enzyme transition state analog binding constants

Transition state binding theory utilizes non-enzymic and enzymic rate ratios to predict the ratio of transition state analog dissociation constants to substrate dissociation constants. In this paper we show that enzyme rate accelerations due solely to lessened entropy requirements, arising from the juxtaposition of a catalytic group and a substrate binding site at an enzyme active site, will result in a ratio of transition state and substrate dissociation constants which is different, in general, from the ratio of non-enzymic and enzymic rate constants. The arguments presented in this paper provide a possible explanation for the frequently observed large discrepancy between the measured and predicted values for transition state analog dissociation constants.     

Conformational aspects of inhibitor design: enzyme-substrate interactions in the transition state.

The theory of absolute reaction rates suggests that enzymes, like other catalysts, can enhance the rate of a reaction only to the extent that they bind the altered substrate in the transition state (S++) more tightly than they bind the substrate in the ground state (S). ES dissociation constants commonly fall in the physiological range, but recent kinetic studies indicate that formal ES++ dissociation constants of less than 10(-20) M are achieved by enzymes of several classes. Studies with stable analogues suggest that these remarkable powers of discrimination involve a tendency of the enzyme to close around S++ in such a way as to maximize binding contacts; that several parts of the substrate contribute to S++ binding; and that their contributions to binding affinity can be strongly synergistic.     

Short-lived intermediates in aspartate aminotransferase systems.

The kinetics of the reaction of aspartate aminotransferase with erythro-beta-hydroxy-aspartate, in which rapid mixing is followed (upon reaching a suitable stationary state) by a very fast temperature jump, is numerically simulated. Values for rate constants are used to the extent known, otherwise estimated. It is shown that reaction steps not resolvable by rapid mixing can be resolved by subsequent chemical relaxation. Since several absorption spectra of enzyme complexes overlap, use of a pH-indicator is investigated. When the pH-indicator is coupled to the protonic dissociation of free enzyme, the fast steps are easily detected in the chemical relaxation portion of the simulation. When the pH-indicator is coupled to the protonic dissociation of the (short-lived) quinoid intermediate, protonic dissociation is easily detectable in the stopped flow phase and in the chemical relaxation phase. Such transient protonic dissociation has not been detected experimentally, but is predicted by the simulation. When natural substrates are used, the magnitude of the rate constants makes it unlikely that transient proton dissociation can be detected by stopped flow alone, but a combination of stopped flow with very fast temperature perturbation allows detection of the transient proton through use of a suitable nonbinding pH-indicator. This is demonstrated by simulation for a specific case. Finally, an alternate mechanism is introduced and distinction of its kinetics from that of the original mechanism is demonstrated.     

The effect of water on the rate of conformational change in protein allostery

The influence of solvation on the rate of quaternary structural change is investigated in human hemoglobin, an allosteric protein in which reduced water activity destabilizes the R state relative to T. Nanosecond absorption spectroscopy of the heme Soret band was used to monitor protein relaxation after photodissociation of aqueous HbCO complex under osmotic stress induced by the nonbinding cosolute poly(ethylene glycol) (PEG). Photolysis data were analyzed globally for six exponential time constants and amplitudes as a function of osmotic stress and viscosity. Increases in time constants associated with geminate rebinding, tertiary relaxation, and quaternary relaxation were observed in the presence of PEG, along with a decrease in the fraction of hemes rebinding CO with the slow rate constant characteristic of the T state. An analysis of these results along with those obtained by others for small cosolutes showed that both osmotic stress and solvent viscosity are important determinants of the microscopic R --> T rate constant. The size and direction of the osmotic stress effect suggests that at least nine additional water molecules are required to solvate the allosteric transition state relative to the R-state hydration, implying that the transition state has a greater solvent-exposed area than either end state.     

Alternate Binding Modes for a Ubiquitin-SH3 Domain Interaction Studied by NMR Spectroscopy

Surfaces of many binding domains are plastic, enabling them to interact with multiple targets. An understanding of how they bind and recognize their partners is therefore predicated on characterizing such dynamic interfaces. Yet, these interfaces are difficult to study by standard biophysical techniques that often ‘freeze’ out conformations or that produce data averaged over an ensemble of conformers. In this study, we used NMR spectroscopy to study the interaction between the C-terminal SH3 domain of CIN85 and ubiquitin that involves the ‘classical’ binding sites of these proteins. Notably, chemical shift titration data of one target with another and relaxation dispersion data that report on millisecond time scale exchange processes are both well fit to a simple binding model in which free protein is in equilibrium with a single bound conformation. However, dissociation constants and chemical shift differences between free and bound states measured from both classes of experiment are in disagreement. It is shown that the data can be reconciled by considering three-state binding models involving two distinct bound conformations. By combining titration and dispersion data, kinetic and thermodynamic parameters of the three-state binding reaction are obtained along with chemical shifts for each state. A picture emerges in which one bound conformer has increased entropy and enthalpy relative to the second and chemical shifts similar to that of the free state, suggesting a less packed interface. This study provides an example of the interplay between entropy and enthalpy to fine-tune molecular interactions involving the same binding surfaces.     

Over-the-barrier transition state analogues and crystal structure with Mycobacterium tuberculosis purine nucleoside phosphorylase

Stable chemical analogues of enzymatic transition states are imperfect mimics since they lack the partial bond character of the transition state. We synthesized structural variants of the Immucillins as transition state analogues for purine nucleoside phosphorylase and characterized them with the enzyme from Mycobacterium tuberculosis (MtPNP). PNPs form transition states with ribooxacarbenium ion character and catalyze nucleophilic displacement reactions by migration of the cationic ribooxacarbenium carbon between the enzymatically immobilized purine and phosphate nucleophiles. As bond-breaking progresses, carbocation character builds on the ribosyl group, the distance between the purine and the carbocation increases, and the distance between carbocation and phosphate anion decreases. Transition state analogues were produced with carbocation character and increased distance between the ribooxacarbenium ion and the purine mimics by incorporating a methylene bridge between these groups. Immucillin-H (ImmH), DADMe-ImmH, and DADMe-ImmG mimic the transition state of MtPNP and are slow-onset, tight-binding inhibitors of MtPNP with equilibrium dissociation constants of 650, 42, and 24 pM. Crystal structures of MtPNP complexes with ImmH and DADMe-ImmH reveal an ion-pair between the inhibitor cation and the nucleophilic phosphoryl anion. The stronger ion-pair (2.7 A) is found with DADMe-ImmH. The position of bound ImmH resembles the substrate side of the transition state barrier, and DADMe-ImmH more closely resembles the product side of the barrier. The ability to probe both substrate and product sides of the transition state barrier provides expanded opportunities to explore transition state analogue design in N-ribosyltransferases. This approach has resulted in the highest affinity transition state analogues known for MtPNP.     

The Role of Diffusion in Bimolecular Solution Kinetics

An appropriate boundary condition is derived which permits both the bimolecular association and dissociation steps to be simultaneously treated within the framework of the theory of Smoluchowski, Debye, and Collins, and Kimball. Kinetic theory expressions are derived for the intrinsic rate constants. The transient case of the suddenly switched-on reaction is considered as well as the suddenly perturbed equilibrium, but only the time dependence of the rate constants is obtained. The frequency response spectrum for a diffusion-controlled reaction is obtained in the linear approximation and compared with the corresponding Debye relaxation spectrum.     

How do enzymes work? Effect of electron circuits on transition state acid dissociation constants

The effect of electron flow through a complete circuit on transition state acid dissociation constants is used to explain the remarkable catalysis observed in a redox reaction, the formation of compound I from native peroxidase. The explanation for the huge shift in the dissociation constant of a distal histidine residue, in going from the resting enzyme to the transition state, is a complete electron circuit through many amino acid residues and hydrogen bonds which prevents the development of localized charge. The key feature is electron flow through the circuit at the instant that proton transfer is occurring in the opposite direction. Electron flow occurs in one direction for attainment of the transition state and in the opposite direction for product formation.     

Temperature-jump studies on the interaction of benzeneboronic acid with chymotrypsinogen.

The interaction of chymotrypsinogen A with benzeneboronic acid (BBA), a transition state along inhibitor of serine proteases, was investigated by the temperature-jump method using pH indicators. It was found that l/tau is dependent on BBA concentration, in contrast to the case of the alpha-chymotrypsin [EC 3.4.21.1]-BBA system in which l/tau is independent of BBA concentration. By examination of the pH dependences of the kinetic parameters, the acid dissociation behavior of His 57 in chymotrypsinogen, chymotrypsinogen-trigonal BBA complex and chymotrypsinogen-tetrahedral BBA complex was analyzed. The kinetic deuterium isotope effect was also examined and found to occur principally on the acid dissociation constants. The state of the catalytic residues in the zymogen molecule is discussed based on these results.     

Theoretical and experimental study of the Norrish I photodissociation of aromatic ketones.

A set of selected acetophenone derivatives was investigated using absorption and emission spectroscopy, laser flash photolysis and DFT calculations. The triplet state lifetimes and the activation energy of the cleavage reaction were measured. Computed triplet-triplet absorption spectra were found in very good agreement with the experimental ones. Bond dissociation energies, activation energies, partial charges, ground state geometries were calculated. The transition state theory TST was successfully used to calculate the cleavage rate constants: a very good correlation was found between the experimental and the calculated values. It is found that the entropy change influences the preexponential factor. This study also points out the role of the partial charges in the transition state, although this effect alone does not account for the reaction rate constants.     

Kinetics of the co-operative and non-co-operative reaction of Helix pomatia haemocyanin with oxygen

The kinetics of the reaction of Helix pomatia haemocyanin with oxygen have been studied under conditions where ligand binding is co-operative (n = 4.5). The dissociation of oxygen from oxyhaemocyanin in the presence of sodium dithionite and the combination of deoxyhaemocyanin with oxygen were studied by the stopped-flow technique. The combination with oxygen, as well as the dissociation of oxyhaemocyanin, are clearly autocatalytic. The initial rate constant for oxygen combination to the fully deoxygenated state is 0.2 to 0.3 × 10⁶m^?1 s^?1; during the course of the reaction the rate constant increases to a value higher than 10⁶m^?1s^?1.The initial rate of oxygen dissociation from fully saturated haemocyanin is 10 s^?1, increasing to about 30 s^?1 as the reaction proceeds. Thus, both the combination and the dissociation rate constants contribute to the co-operativity of oxygen binding.Temperature-jump relaxation experiments were carried out at fractional oxygen saturations larger than 0.7. The dependence of the relaxation rate upon the concentration of the reactants indicates the presence of one principal bimolecular process. The calculated combination and dissociation rate constants for this process are: 3.8 × 10⁶m^?1 s^?1 and 10 s^?1, respectively. Evidence is presented which shows that the transition from the T-state to the R-state of the protein is relatively slow. Both the T and R-state seem to be largely stabilized at the expense of intermediate states.Under other conditions, where oxygen binding is non-co-operative, temperature-jump and stopped-flow experiments reveal considerable kinetic heterogeneity.     

Some Thoughts about Bond Energies,Bond Lengths,and Force Constants

The bond energy (BE) of a polyatomic molecule cannot be measured and, therefore, determination of BEs can only be done within a model using a set of assumptions. The bond strength is reflected by the intrinsic BE (IBE), which is related to the intrinsic atomization energy (IAE) and which represents the energy of dissociation under the provision that the degree of hybridization is maintained for all atoms of the molecule. IBE and BE differ in the case of CC and CH bonds by the promotion, the hybridization, and the charge reorganization energy of carbon. Since the latter terms differ from molecule to molecule, IBE and BE are not necessarily parallel and the use of BEs from thermochemical models can be misleading. The stretching force constant is a dynamical quantity and, therefore, it is related to the bond dissociation energy (BDE). Calculation and interpretation of stretching force constants for local internal coordinate modes are discussed and it is demonstrated that the best relationship between BDEs and stretching force constants is obtained within the model of adiabatic internal modes. The valence stretching force constants are less suitable since they are related to an artificial bond dissociation process with geometrical relaxation effects suppressed, which leads to an intrinsic BDE (IBDE). In the case of AX_n molecules, symmetric coordinates can be used to get an appropriate stretching force constant that is related to the BE. However, in general stretching force constants determined for symmetry coordinates do not reflect the strength of a particular bond since the related dissociation processes are strongly influenced by the stability of the products formed.     

Thermodynamics of a diffusional protein folding reaction

The folding reactions of several proteins are well described as diffusional barrier crossing processes, which suggests that they should be analyzed by Kramers' rate theory rather than by transition state theory. For the cold shock protein Bc-Csp from Bacillus caldolyticus, we measured stability and folding kinetics, as well as solvent viscosity as a function of temperature and denaturant concentration. Our analysis indicates that diffusional folding reactions can be treated by transition state theory, provided that the temperature and denaturant dependence of the solvent viscosity is properly accounted for, either at the level of the measured rate constants or of the calculated activation parameters. After viscosity correction the activation barriers for folding become less enthalpic and more entropic. The transition from an enthalpic to an entropic folding barrier with increasing temperature is, however, apparent in the data before and after this correction. It is a consequence of the negative activation heat capacity of refolding, which is independent of solvent viscosity. Bc-Csp and its mesophilic homolog Bs-CspB from Bacillus subtilis differ strongly in stability but show identical enthalpic and entropic barriers to refolding. The increased stability of Bc-Csp originates from additional enthalpic interactions that are established after passage through the activated state. As a consequence, the activation enthalpy of unfolding is increased relative to Bs-CspB.     

Kinetics of the co-operative reaction of Helix pomatia hemocyanin with oxygen. Oxygen binding at low and intermediate oxygen saturations

The kinetics of oxygen binding of Helix pomatia α-hemocyanin has been studied at low and intermediate levels of ligand saturation, under conditions in which oxygen binding is highly co-operative. Temperature-jump relaxation spectra are heterogeneous and can be resolved into a slow and a fast phase. The latter is related to a bimolecular reaction, i.e. the binding of oxygen. At very low degrees of fractional saturation (<0.15) the reactant concentration-dependence of the faster relaxation rate allows the combination and dissociation rate constants of the low affinity or T-state to be estimated as 1.3 × 10⁶m^?1 s^?1 and 300 s^?1, respectively. A possible interpretation of the slow component in the relaxation spectrum is discussed.In stopped-flow experiments, after mixing deoxyhemocyanin with oxygen-containing buffer, most of the binding process to the T-state is lost in the dead time. The observed initial rates of oxygen binding are between 15 and 120 s^?1. depending on the oxygen concentration, and may reflect the rate of the allosteric change from a low to a high affinity state (T→R transition), which is slower than oxygen binding.Similarities and differences in the overall kinetic properties of small and giant respiratory proteins, i.e. hemoglobin and hemocyanin, are discussed.     

Kinetic investigations on the phase transition of phospholipid bilayers

Pressure-jump experiments were performed on vesicles and liposomes of dimyristoyl phosphatidylcholine and dipalmitoyl phosphatidylcholine following the time course of solution turbidity. For both lipids two relaxation effects were evaluated the time constants of which exhibit clear maxima at the midpoint of the phase transition. The time constants lie for vesicles in the 100 μs and 1 ms ranges and for liposomes in the 1 ms and 10 ms ranges. The processes are slightly faster for dimyristoyl phosphatidylcholine than for dipalmitoyl phosphatidylcholine. All relaxation times are concentration-independent. The time constant and amplitude behaviours indicate that all processes are cooperative in agreement with previous interpretations. It is demonstrated that cooperative units can be evaluated from the relaxation amplitudes. These are of the same order of magnitude as those obtained from static experiments. On the grounds of the present kinetic investigation we can state that the application of the linear Ising model to two-dimensional processes as attempted for the static lipid phase transition is inadequate.     

Phase transition kinetics of phosphatidic acid bilayers A stopped-flow study of the electrostatically induced transition

The kinetics of the electrostatically induced phase transition of dimyristoyl phosphatidic acid bilayers was followed using the stopped-flow technique. The phase transition was triggered by a fast change in the pH or the magnesium ion concentration and followed by recording the time dependence of the absorbance. When the phase transition was induced by a pH jump the time course of the absorbance could be described by two exponentials, their time constants displaying the for cooperative processes characteristic maximum at the transition midpoint. The time constants are in the 10 and 100 ms range for the H⁺ triggered transition from the fluid to the ordered state. A third slower process shows no appreciable temperature dependence and is probably caused by vesicle aggregation. For the OH^--induced transition fron the ordered to the fluid state the time constants are in the 100 and 1000 ms range. The fluid-ordered transition could also be triggered by addition of magnesium ions. Of the several observed processes only the fastest in the 10–100 ms time range could definitely be assigned to the fluid-ordered transition while the others are due to aggregation phenomena. The experimental data were compared with results obtained from pressure jump experiments and could be interpreted on the basis of theories for non-equilibrium relaxation.     

Kinetic control of co-operativity in the oxygen binding of Panulirus interruptus hemocyanin.

The kinetics of the oxygen reaction of Panulirus interruptus hemocyanin have been studied at pH 9.6 under conditions where the protein exists in the undissociated, co-operative state and in the dissociated, non-co-operative state.Temperature-jump relaxation measurements of the undissociated protein at high oxygen saturation levels show one relaxation process which has been assigned to the high oxygen affinity (R) state, the on and off kinetic constants being 3.1 × 10⁷m^?1s^?1 and 60 s^?1, respectively. Stopped-flow measurements of the oxygen dissociation reaction show (1) an autocatalytic time-course of the reaction at pH 9.6 and (2) an increase in the overall oxygen dissociation rate constant, as the pH is decreased from 9.6 to 7.0.Temperature-jump relaxation measurements of the dissociated protein show one relaxation process characterized by a very high oxygen dissociation rate constant (1500 s^?1) and a combination constant which is of the same order of magnitude as reported for undissociated protein (k_on = 4.6 × 10⁷m^?1s${}^{\mathrm{?1}}$). The behaviour of dissociated protein can be considered as characteristic of the low oxygen affinity (T) state.The results presented in this paper, together with data available for other hemocyanins as well as hemoglobins, lead to the conclusion that respiratory proteins show a common feature in the kinetic control of co-operative oxygen binding: the stability of the oxygen-protein complex is largely determined by the value of the dissociation rate constant, the oxygen combination process very often appearing to be diffusion controlled.     

Relaxation Kinetics of Lipid Membranes and Its Relation to the Heat Capacity

We investigated the relaxation behavior of lipid membranes close to the chain-melting transition using pressure jump calorimetry with a temperature accuracy of ∼10^-3 K. We found relaxation times in the range from seconds up to about a minute, depending on vesicular state. The relaxation times are within error proportional to the heat capacity. We provide a statistical thermodynamics theory that rationalizes the close relation between heat capacity and relaxation times. It is based on our recent finding that enthalpy and volume changes close to the melting transition are proportional functions.     

Merged mechanisms for hydride transfer from 1,4-dihydronicotinamides

Recent work on the reduction of heteroaromatic cations by 1,4-dihydronicotinamides and related reducing agents is reviewed. Extensive correlations are presented between the second-order rate constants (k₂) for these reactions and the second-order rate constants (k_OH) and equilibrium constants (pK_R⁺) for hydroxide ion attack on these cations. Close correlations of log k₂ with the electron affinities and one-electron reduction potentials of these cations are also presented. These relationships are considered in the context of a direct hydride transfer from donor to acceptor and also in terms of SET mechanisms which are also commonly discussed for such reactions. It is shown that the interpretation of these formal hydride transfer reactions in terms of an imbalanced development of electronic charge and C---H bond fission within the transition state species leads to a rational merging of the single-step hydride transfer mechanism and the SET mechanisms. The structures of the transition state species are expected to be highly variable and quite dependent upon the nature of the hydride donor and acceptor species, with considerable contribution from charge-transfer interactions. Such imbalanced transition state species are analyzed in terms of two different types of reaction coordinate diagrams and also in terms of the valence bond configuration mixing theory.     

Duration time of a one-dimensional random walk as a function of the energies of the intermediate states: application for dissociation and relaxation processes in DNA hybrids.

Kinetic parameters of macromolecular systems are important for their function in vitro and in vivo. These parameters describe how fast the system dissociates (the characteristic dissociation time), and how fast the system reaches equilibrium (characteristic relaxation time). For many macromolecular systems, the transitions within the systems are described as a random walk through a number of states with various free energies. The rate of transition between two given states within the system is characterized by the average time which passes between starting the movement from one state, and reaching the other state. This time is referred to as the mean first-passage time between two given states. The characteristic dissociation and relaxation times of the system depend on the first-passages times between the states within the system. Here, for a one-dimensional random walk we derived an equation, which connects the mean first-passage time between two states with the free energies of the states within the system. We also derived the general equation, which is not restricted to one-dimensional systems, connecting the relaxation time of the system with the first-passage times between states. The application of these equations to DNA branch migration, DNA structural transitions and other processes is discussed.