The equilibrium constant and the reversibility of the reaction catalysed by nicotinamide-adenine dinucleotide kinase from pigeon liver.

The reversibility of the NAD+ kinase reaction was established, and the kinetic parameters of the rate equation in the reverse direction were determined. The equilibrium constant of the reaction was determined by using the purified pigeon liver enzyme and radioactively labelled nicotinamide nucleotides. The relationship between kinetic parameters of the forward and reverse reactions is in reasonable agreement with the measured equilibrium constant. As expected from the proposed mechanism of action, the enzyme does not catalyse isotope exchange between NAD+ and NADP+ in the absence of ADP and ATP. Although homogeneous as judged by polyacrylamide-gel electrophoresis, the enzyme preparation exhibits ADP/ATP isotope-exchange activity which could not be separated from NAD+ kinase activity, but kinetic evidence suggests that this is probably due to a contaminant.     

Testing for microscopic reversibility in the gating of maxi K+ channels using two-dimensional dwell-time distributions.

An assumption usually made when developing kinetic models for the gating of ion channels is that the transitions among the various states involved in the gating obey microscopic reversibility. If this assumption is incorrect, then the models and estimated rate constants made with the assumption would be in error. This paper examines whether the gating of a large conductance Ca-activated K+ channel in skeletal muscle is consistent with microscopic reversibility. If microscopic reversibility is obeyed, then the number of forward and backward transitions per unit time for each individual reaction step will, on average, be identical and, consequently, the gating must show time reversibility. To look for time reversibility, two-dimensional dwell-time distributions of the durations of open and closed intervals were obtained from single-channel current records analyzed in the forward and in the backward directions. Two-dimensional dwell-time distributions of pairs of open intervals and of pairs of closed intervals were also analyzed to extend the resolution of the method to special circumstances in which intervals from different closed (or open) states might have similar durations. No significant differences were observed between the forward and backward analysis of the two-dimensional dwell-time distributions, suggesting time reversibility. Thus, we find no evidence to indicate that the gating of the maxi K+ channel violates microscopic reversibility.     

Rate constants for a mechanism including intermediates in the interconversion of ternary complexes by horse liver alcohol dehydrogenase

Transient kinetic data for partial reactions of alcohol dehydrogenase and simulations of progress curves have led to estimates of rate constants for the following mechanism, at pH 8.0 and 25 degrees C: E in equilibrium E-NAD+ in equilibrium *E-NAD+ in equilibrium E-NAD(+)-RCH2OH in equilibrium E-NAD+-RCH2O- in equilibrium *E-NADH-RCHO in equilibrium E-NADH-RCHO in equilibrium E-NADH in equilibrium E. Previous results show that the E-NAD+ complex isomerizes with a forward rate constant of 620 s-1 [Sekhar, V. C., & Plapp, B. V. (1988) Biochemistry 27, 5082-5088]. The enzyme-NAD(+)-alcohol complex has a pK value of 7.2 and loses a proton rapidly (greater than 1000 s-1). The transient oxidation of ethanol is 2-fold faster in D2O, and proton inventory results suggest that the transition state has a charge of -0.3 on the substrate oxygen. Rate constants for hydride ion transfer in the forward or reverse reactions were similar for short-chain aliphatic substrates (400-600 s-1). A small deuterium isotope effect for transient oxidation of longer chain alcohols is apparently due to the isomerization of the E-NAD+ complex. The transient reduction of aliphatic aldehydes showed no primary deuterium isotope effect; thus, an isomerization of the E-NADH-aldehyde complex is postulated, as isomerization of the E-NADH complex was too fast to be detected. The estimated microscopic rate constants show that the observed transient reactions are controlled by multiple steps.     

The use of transition state acid dissociation constants in pH-dependent enzyme kinetics

It is shown that the variation of reaction rate with pH in systems where several protonated forms of the reactants appear to be kinetically active may be expressed most economically in terms of transition state acid dissociation constants. The advantages of this approach are described in relation to the formal analysis of experimental data with regard to both simple and complex reactions and the satisfaction of the principle of microscopic reversibility. Ligand binding to metmyoglobin is used to illustrate the value of the approach in searching for detailed mechanistic explanations.     

The origin of reaction specificity in serine hydroxymethyltransferase

Cytosolic serine hydroxymethyltransferase has been shown previously to exhibit both broad substrate and reaction specificity. In addition to cleaving many different 3-hydroxyamino acids to glycine and an aldehyde, the enzyme also catalyzes with several amino acid substrate analogs decarboxylation, transamination, and racemization reactions. To elucidate the relationship of the structure of the substrate to reaction specificity, the interaction of both amino acid and folate substrates and substrate analogs with the enzyme has been studied by three different methods. These methods include investigating the effects of substrates and substrate analogs on the thermal denaturation properties of the enzyme by differential scanning calorimetry, determining the rate of peptide hydrogen exchange with solvent protons, and measuring the optical activity of the active site pyridoxal phosphate. All three methods suggest that the enzyme exists as an equilibrium between "open" and "closed" forms. Amino acid substrates enter and leave the active site in the open form, but catalysis occurs in the closed form. The data suggest that the amino acid analogs that undergo alternate reactions, such as racemization and transamination, bind only to the open form of the enzyme and that the alternate reactions occur in the open form. Therefore, one role for forming the closed form of the enzyme is to block side reactions and confer reaction specificity.     

Evolutionary optimization of the catalytic efficiency of enzymes.

1. The rate equation for a generalized Michaelian type of enzymic reaction mechanism has been analyzed in order to establish how the mechanism should be kinetically designed in order to optimize the catalytic efficiency of the enzyme for a given average magnitude of true and apparent first-order rate constants in the mechanism at given concentrations of enzyme, substrate and product. 2. As long as on-velocity constants for substrate and product binding to the enzyme have not reached the limiting value for a diffusion-controlled association process, the optimal state of enzyme operation will be characterized by forward (true and apparent) first-order rate constants of equal magnitude and reverse rate constants of equal magnitude. The drop in free energy driving the catalysed reaction will occur to an equal extent for each reaction step in the mechanism. All internal equilibrium constants will be of equal magnitude and reflect only the closeness of the catalysed reaction to equilibrium conditions. 3. When magnitudes of on-velocity constants for substrate and product binding have reached their upper limits, the optimal kinetic design of the reaction mechanism becomes more complex and has to be established by numerical methods. Numerical solutions, calculated for triosephosphate isomerase, indicate that this particular enzyme may or may not be considered to exhibit close to maximal efficiency, depending on what value is assigned to the upper limit for a ligand association rate constant. 4. Arguments are presented to show that no useful information on the evolutionary optimization of the catalytic efficiency of enzymes can be obtained by previously taken approaches that are based on the application of linear free-energy relationships for rate and equilibrium constants in the reaction mechanism.     

Relations between biochemical thermodynamics and biochemical kinetics

The parameters in steady-state or rapid-equilibrium rate equations for enzyme-catalyzed reactions depend on the temperature, pH, and ionic strength, and may depend on the concentrations of specific species in the buffer. When the complete rate equation (i.e. the equation with parameters for the reverse reaction as well as the forward reaction) is determined, there are one or more Haldane relations between some of the kinetic parameters and the apparent equilibrium constant for the reaction that is catalyzed. When the apparent equilibrium constant can be calculated from the kinetic parameters, the equilibrium composition can be calculated. This is remarkable because the kinetic parameters all depend on the properties of the enzymatic site, but the apparent equilibrium constant and the equilibrium composition do not. The effects of ionic strength and pH on the unoccupied enzymatic site and the occupied enzymatic site have to cancel in the Haldane relation or in the calculation of the apparent equilibrium constant using the rate constants for the steps in the mechanism. Several simple enzymatic mechanisms and their complete rate equations are discussed.     

Generalized microscopic reversibility, kinetic co-operativity of enzymes and evolution.

Generalized microscopic reversibility implies that the apparent rate of any catalytic process in a complex mechanism is paralleled by substrate desorption in such a way that this ratio is held constant within the reaction mechanism [Whitehead (1976) Biochem. J. 159, 449--456]. The physical and evolutionary significances of this concept, for both polymeric and monomeric enzymes, are discussed. For polymeric enzymes, generalized microscopic reversibility of necessity occurs if, within the same reaction sequence, the substrate stabilizes one type of conformation of the active site only. Generalized microscopic reversibility suppresses the kinetic co-operativity of the slow transition model [Ainslie, Shill & Neet (1972) J. Biol. Chem. 247, 7088--7096]. This situation is obtained if the free-energy difference between the corresponding transition states of the two enzyme forms is held constant along the reaction co-ordinate. This situation implies that the 'extra costs' of energy (required to pass each energy barrier) that are not covered by the corresponding binding energies of the transition states vary in a similar way along the two reaction co-ordinates. The regulatory behaviour of monomeric enzymes is discussed in the light of the concept of 'catalytic perfection' proposed by Albery & Knowles [(1976) Biochemistry 15, 5631--5640]. These authors claim that an enzyme will be catalytically 'perfect' when its catalytic efficiency is maximum. If this situation occurs for a monomeric enzyme obeying either the slow transition or the mnemonical model, it can be shown that the kinetic co-operativity disappears. In other words, kinetic co-operativity of a monomeric enzyme is 'paid for' at the expense of catalytic efficiency, and the monomeric enzyme cannot be simultaneously co-operative and catalytically very efficient. This is precisely what has been found experimentally in a number of cases.     

Evolutionary optimization of the catalytic effectiveness of an enzyme

The kinetic and thermodynamic features of reactions catalyzed by present-day enzymes appear to be the consequence of the evolution of these proteins toward maximal catalytic effectiveness. These features are identified and analyzed (in detail for one substrate-one product enzymes) by using ideas that link the energetics of the reaction catalyzed by an enzyme to the maximization of its catalytic efficiency. A catalytically optimized enzyme will have a value for the "internal" equilibrium constant (Kint, the equilibrium constant between the substrates and the products of the enzyme when all are bound productively) that depends on how close to equilibrium the enzyme maintains its reaction in vivo. Two classes are apparent. For an enzyme that operates near equilibrium, the catalytic efficiency is sensitive to the value of Kint, and the optimum value of Kint is near unity. For an enzyme that operates far from equilibrium, the catalytic efficiency is less sensitive to the value of Kint, and Kint assumes a value that ensures that the rate of the chemical transformation is equal to the rate of product release. In each of these cases, the internal thermodynamics is "dynamically matched", where the concentrations of substrate- and product-containing complexes are equal at the steady state in vivo.     

Analyzing molecular reaction networks

Pathways are typically the central concept in the analysis of biochemical reaction networks. A pathway can be interpreted as a chain of enzymatical reactions performing a specific biological function. A common way to study metabolic networks are minimal pathways that can operate at steady state called elementary modes. The theory of chemical organizations has recently been used to decompose biochemical networks into algebraically closed and self-maintaining subnetworks termed organizations. The aim of this paper is to elucidate the relation between these two concepts. Whereas elementary modes represent the boundaries of the potential behavior of the network, organizations define metabolite compositions that are likely to be present in biological feasible situations. Hence, steady state organizations consist of combinations of elementary modes. On the other hand, it is possible to assign a unique (and possibly empty) set of organizations to each elementary mode, indicating the metabolites accompanying the active pathway in a feasible steady state.     

Analysis of metabolic network based on conservation of molecular structure

Recent work has revealed much about chemical reactions inside hundreds of organisms as well as universal characteristics of metabolic networks, which shed light on the evolution of the networks. However, characteristics of individual metabolites have been neglected. For example, some carbohydrates have structures that are decomposed into small molecules by metabolic reactions, but coenzymes such as ATP are mostly preserved. Such differences in metabolite characteristics are important for understanding the universal characteristics of metabolic networks. To quantify the structure conservation of metabolites, we defined the "structure conservation index" (SCI) for each metabolite as the fraction of metabolite atoms restored to their original positions through metabolic reactions. As expected, coenzymes and coenzyme-like metabolites that have reaction loops in the network show a higher SCI. Using the index, we found that the sum of metabolic fluxes is negatively correlated with the structure preservation of metabolite. Also, we found that each reaction path around high SCI metabolites changes independently, while changes in reaction paths involving low SCI metabolites coincide through evolution processes. These correlations may provide a clue to universal properties of metabolic networks.     

Structure and conformational flexibility of Candida rugosa lipase

Anandamide, an endogenous ligand for cannabinoid receptors, loses its biological activities when it is hydrolyzed to arachidonic acid and ethanolamine by anandamide amidohydrolase. We overexpressed a recombinant rat enzyme with a hexahistidine tag in a baculovirus-insect cell expression system, and purified the enzyme with the aid of a Ni-charged resin to a specific activity as high as 5.7 micromol/min/mg protein. The purified recombinant enzyme catalyzed not only the hydrolysis of anandamide and palmitoylethanolamide, but also their reverse synthetic reactions. In order to attain an equilibrium of the anandamide hydrolysis and its reverse reaction within 10 min, we utilized a large amount of the purified enzyme. The equilibrium constant ([arachidonic acid][ethanolamine])/([anandamide][water]) was calculated as 4x10(-3) (37 degrees C, pH 9.0). These experimental results with a purified enzyme preparation quantitatively confirmed the reversibility of the enzyme reaction previously observed with crude enzyme preparations.     

Equations of substrate-limited growth: the case for blackman kinetics

A simplified model of cell metabolism, consisting of a series of linked reversible enzymatic reactions dependent on the concentration of a single external substrate has been developed. The general mathematical solution for this system of reactions is presented. This general solution confirms the concept of a rate-limiting step, or “master reaction”, in biological systems as first proposed by Blackman. The maximum rate of such a process is determined by, and equal to, the maximum rate of the slowest forward reaction in the series. Of practical interest in modeling the growth rate of cells are three cases developed from the general model. The simplest special case results in the Monod equation when the maximum forward rate of one enzymatic reaction in the cell is much less than the maximum forward rate of any other enzymatic reactions. More realistic is the case where the maximum forward rates of more than one enzymatic reaction are slow. When two slow enzymatic reactions are separated from each other by any number of fast reactions that overall can be described by a large equilibrium constant, the Blackman form results: and A third case is that in which two slow enzymatic steps are separated by an equilibrium constant that is not large. Unlike the Monod and Blackman forms, which contain only two arbitrary constants, this model contains three arbitrary constants: The Monod and Blackman forms are special cases of this three constant form. In comparing equations with two arbitrary constants the Monod equation gave poorer fit of the data in most cases than the Blackman form. It is concluded that workers modeling the growth of microorganisms should give a t least as much consideration to the Blackman form as is given to the Monod equation.     

Time hierarchy, equilibrium and non-equilibrium in metabolic systems.

A metabolic system consists of cooperating biochemical reactions. The motion is described by differential equations in the metabolites. The right-hand sides of these equations are linear combinations of the velocities of the individual reactions. These velocities depend in a non-linear manner on the metabolite concentrations (according to the law of mass action). A characteristic "metabolic" time may be defined for the motion of the whole system. It scales the essential metabolic events whose evolution time is comparable to this metabolite time unit. The constituent reactions of the metabolic system have an individual characteristic time which need not coincide with the general metabolic time. The individual time characterises the approach to the individual equilibrium of the isolated undisturbed reaction. According to the ratio of these two time scales, a single reaction may be fast, or slow, or essential, as compared with the metabolic events. Characteristic time of a single reaction and its steady-state deviation from equilibrium are closely related. It can be shown that the relative deviation from equilibrium of a reaction within the metabolic network is of the same numerical order as the ratio between individual time to metabolic time. The interaction of many reactions with different characteristic times introduces a time hierarchy into the system. This can be made transparent by appropriate scaling and by linear transformation of the system. The subsystem of fast cooperating reactions (dehydrogenases, phosphotransferases) attains a state which is near to the individual equilibrium and reestablishes this state after perturbation. The equilibration is fast; an ultrarapid phase of cofactor equilibrium can be distinguished from the fast phase of substrate equilibrium (exchange of metabolic material between different pathways). During the slower metabolic phase these near-equilibria manifest themselves as stoichiometric linkage between unrelated metabolites. The latter cease to be independent variables and combine to metabolic pools. It can be strictly shown that the essential variables at the metabolic time scale are carrier pools and the degree of occupancy of these carriers by metabolic groups. Chemically different types of carrier pools may be functionally linked together by fast reactions. A consequence of such an arrangement of reactions are distance effects: Changes at one end of a metabolic map may be directly conveyed to other pathways via stoichiometric linkage brought about by fast equilibration of cofactor reactions.     

The Cys319 loop modulates the transition between dehydrogenase and hydrolase conformations in inosine 5'-monophosphate dehydrogenase

X-ray crystal structures of enzyme-ligand complexes are widely believed to mimic states in the catalytic cycle, but this presumption has seldom been carefully scrutinized. In the case of Tritrichomonas foetus inosine 5'-monophosphate dehydrogenase (IMPDH), 10 structures of various enzyme-substrate-inhibitor complexes have been determined. The Cys319 loop is found in at least three different conformations, suggesting that its conformation changes as the catalytic cycle progresses from the dehydrogenase step to the hydrolase reaction. Alternatively, only one conformation of the Cys319 loop may be catalytically relevant while the others are off-pathway. Here we differentiate between these two hypotheses by analyzing the effects of Ala substitutions at three residues of the Cys319 loop, Arg322, Glu323, and Gln324. These mutations have minimal effects on the value of k(cat) (≤5-fold) that obscure large effects (>10-fold) on the microscopic rate constants for individual steps. These substitutions increase the equilibrium constant for the dehydrogenase step but decrease the equilibrium between open and closed conformations of a mobile flap. More dramatic effects are observed when Arg322 is substituted with Glu, which decreases the rates of hydride transfer and hydrolysis by factors of 2000 and 130, respectively. These experiments suggest that the Cys319 loop does indeed have different conformations during the dehydrogenase and hydrolase reactions as suggested by the crystal structures. Importantly, these experiments reveal that the structure of the Cys319 loop modulates the closure of the mobile flap. This conformational change converts the enzyme from a dehydrogenase into hydrolase, suggesting that the conformation of the Cys319 loop may gate the catalytic cycle.     

Transient state kinetics of Enzyme I of the phosphoenolpyruvate:glycose phosphotransferase system of Escherichia coli: equilibrium and second-order rate constants for the phosphotransfer reactions with phosphoenolpyruvate and HPr

The first two reactions in the phosphotransfer sequence of bacterial phosphoenolpyruvate:glycose phosphotransferase systems are the autophosphorylation of Enzyme I by phosphoenolpyruvate followed by the transfer of the phospho group to the low-molecular weight protein, HPr. Transient state kinetic methods were used to estimate the second-order rate constants for both phosphotransfer reactions. These measurements support previous conclusions that only the dimer of Enzyme I, EI2, is autophosphorylated, and that the rate of formation of dimer is slow compared to the rate of its phosphorylation. The rate constants of the two autophosphorylation reactions of EI2 by PEP are 6.6 x 10(6) M(-1) s(-1), and differ from one another by a factor of less than 3. The rate constant for the transfer reaction between phospho-EI2 and HPr is unusually large for a covalent reaction between two proteins (220 x 10(6) M(-1) s(-1)), while the constant for the reverse reaction is 4.2 x 10(6) M(-1) s(-1). Using the previously reported equilibrium constant for the autophosphorylation reaction, 1.5, the overall equilibrium constant for phosphotransfer from PEP to HPr is 80, somewhat higher than that previously reported. The results also show that EI2 can phosphorylate multiple molecules of HPr without dissociating to a monomer (EI), and that EI can accept a phospho group from phospho-HPr. These results are directly applicable to predicting the rates of phosphoenolpyruvate phosphotransferase system sugar uptake in whole cells.     

Metabolic modeling of polyhydroxybutyrate biosynthesis

A mathematical model describing intracellular polyhydroxybutyrate (PHB) synthesis in Alcaligenes eutrophus has been constructed. The model allows investigation of issues such as the existence of rate-limiting enzymatic steps, possible regulatory mechanisms in PHB synthesis, and the effects different types of rate expressions have on model behavior. Simulations with the model indicate that activities of all PHB pathway enzymes influence overall PHB flux and that no single enzymatic step can easily be identified as rate limiting. Simulations also support regulatory roles for both thiolase and reductase, mediated through AcCoA/CoASH and NADPH/NADP+ ratios, respectively. To make the model more realistic, complex rate expressions for enzyme-catalyzed reactions were used which reflect both the reversibility of the reactions and the reaction mechanisms. Use of the complex kinetic expressions dramatically changed the behavior of the system compared to a simple model containing only Michaelis-Menten kinetic expressions; the more complicated model displayed different responses to changes in enzyme activities as well as inhibition of flux by the reaction products CoASH and NADP+. These effects can be attributed to reversible rate expressions, which allow prediction of reaction rates under conditions both near and far from equilibrium.     

Fluorescence polarization and intensity kinetic studies of antifluorescein antibody obtained at different stages of the immune response.

Kinetic studies of reactions between fluorescein and antifluorescein antibody produced during early, intermediate, and late stages of the immune response have been carried out utilizing both fluorescence intensity and polarization measurements in the static (time constant similar to 5 sec) and in the stopped-flow modes (time constant similar to 5 msec). During maturation of the immune response, it was found that the "on" second-order association rate constant increased its value only by a factor of three, whereas the "off" dissociation first-order rate constant decreased by a factor of over 1000. Hence, it is the rate of dissociation which largely determines the stability of the hapten-antihapten complex. Furthermore, since second-order rate behavior was found for even heterogeneous antibody, most of the heterogeneity with respect to binding affinity occurs as a result of the heterogeneity in the rate of dissociation of the hapten-antihapten complex and not from the primary combination of hapten and antibody. Antifluorescein antibody which exhibits both high binding affinity (K similar to 5 x 10(11) M-1) and homogeneity with respect to equilibrium binding has been shown to obey second-order association kinetics over wide ranges in concentration. Despite the fact that the value of the second-order rate constant for this fluorescein-antifluorescein reaction is as large as that for most other hapten-antihapten reactions (1.4 x 10(8) M-1 sec-1), the binding reaction has an appreciable activation energy (7 kcal/mol). This is true for both divalent and univalent antibody. Furthermore, the reaction rate parameters are markedly affected by specific anions. The value of the second-order rate constant (18.5 degrees) increases according to the following scheme: salicylate less than trichloroacetate less than SCN- less than ClO4- less than Cl- less than F- less than phosphate. The activation energy increases as follows: trichloroacetate less than phosphate less than F- less than Cl- less than ClO4- less than SCN- less than salicylate, whereas estimates of the entropy of activation indicate that deltaS++ increases as follows: tricholroacetate less than phosphate similar to F- less than Cl- less than ClO4- less than SCN- less than salicylate. The same mechanism which was previously proposed by us for the antigen-antibody reaction is also consistent with the kinetics of the fluorescein-antifluorescein reaction. This mechanism postulates a bimolecular process with structural rearrangements (conformational changes and/or the loss of water) in the formation of the transition state complex. The reaction between the fluorescein hapten and its antibody hence is not diffusion limited.     

Kinetic characteristics of biochemical cyclic reaction systems: Amplification of substrate cycle system

The amplification of a substrate cycle system with reversible closed reaction of two substrates was represented by mathematical equations. The results are summarized as follows: the amplification was affected especially by the affinity of enzyme and substrate, by the rate constant in rate-limiting reaction step, and by the saturation degree of enzyme by substrate. These amplifications were not simply determined by the values of K(m) and V(max), because each rate parameter in the system can affect the degree of amplification independently. The conclusion is that the "apparent" equilibrium constant of this system cannot be uniquely estimated from only data of K(m) and V(max) even if the reaction occurs in a closed system.