Calculation of steady-state rate equations and the fluxes between substrates and products in enzyme reactions.

1. Two methods are described for deriving the steady-state velocity of an enzyme reaction from a consideration of fluxes between enzyme intermediates. The equivalent-reaction technique, in which enzyme intermediates are systematically eliminated and replaced by equivalent reactions, appears the most generally useful. The methods are applicable to all enzyme mechanisms, including three-substrate and random Bi Bi Ping Pong mechanisms. Solutions are obtained in algebraic form and these are presented for the common random Bi Bi mechanisms. The steady-state quantities of the enzyme intermediates may also be calculated. Additional steps may be introduced into enzyme mechanisms for which the steady-state velocity equation is already known. 2. The calculation of fluxes between substrates and products in three-substrate and random Bi Bi Ping Pong mechanisms is described. 3. It is concluded that the new methods may offer advantages in ease of calculation and in the analysis of the effects of individual steps on the overall reaction. The methods are used to show that an ordered addition of two substrates to an enzyme which is activated by another ligand will not necessarily give hyperbolic steady-state-velocity kinetics or the flux ratios characteristic of an ordered addition, if the dissociation of the ligand from the enzyme is random.     

Kinetics and mechanism of action of muscle pyruvate kinase

1. The mechanism of rabbit muscle pyruvate kinase was investigated by measurements of fluxes, isotope trapping, steady-state velocity and binding of the substrates. All measurements were made at pH8.5 in Tris/HCl buffer and at 5mm-free Mg(2+). 2. Methods of preparing [(32)P]phosphoenolpyruvate from [(32)P]P(i) in high yield and determining [(32)P]-phosphoenolpyruvate and [8-(14)C]ADP are described. 3. The ratio Flux of ATP to ADP/Flux of ATP to phosphoenolpyruvate (measured at equilibrium) increased hyperbolically with ADP concentration from unity to about 2.1 at 2mm-ADP, but was unaffected by phosphoenolpyruvate concentration. Since the ratio is greater than unity, one pathway for the addition of substrates must involve phosphoenolpyruvate adding first to the enzyme in a rate-limiting step. However, the substrates must also add in the alternative order, because of the non-linear increase in the ratio with ADP concentration and because the rate of increase is very much less than that predicted from the steady-state velocity data for an ordered addition. The lack of influence of phosphoenolpyruvate on the ratio is consistent with the rapid addition of ADP in the alternative pathway. At low ADP concentrations the alternative pathway contributes less than 33% to the total reaction. 4. Isotope trapping was observed with [(32)P]phosphoenolpyruvate, confirming that when phosphoenolpyruvate adds first to the enzyme it is in a rate-limiting step. The release of phosphoenolpyruvate from the ternary complex must also be a slow step. Trapping was not observed with [8-(14)C]ADP, hence the addition of ADP to the free enzyme must be rapid unless its dissociation constant is very large (>20mm). 5. Binding studies showed that 4mol of [(32)P]phosphoenolpyruvate binds to 1mol of the enzyme, probably unligated to Mg(2+), with a dissociation constant appropriate to the mechanism indicated above. Binding of [8-(14)C]ADP could not be detected, and hence the binding of ADP occurs by a low-affinity step. The latter is also demanded by the steady-state velocity data. 6. The ratio Flux of phosphoenolpyruvate to ATP/Flux of phosphoenolpyruvate to pyruvate (determined from the incorporation of label into phosphoenolpyruvate from [3-(14)C]-pyruvate or [gamma-(32)P]ATP during the forward reaction) did not differ significantly from unity. Steady-state velocity data predicted grossly different flux ratios for ordered dissociations of the products, and the results indicate that the dissociation must be rapid and random. The data also exclude a Ping-Pong mechanism. 7. Permissible rate constants for the above mechanism are calculated. The results indicate a high degree of cooperativity in binding, whatever the order of addition of substrate.     

The mechanism of rabbit muscle phosphofructokinase at pH8.

The mechanism of rabbit muscle phosphofructokinase was investigated by measurement of fluxes, isotope trapping and steady-state velocities at pH8 in triethanolamine/HCl buffer with 4 mM free Mg2+. Most observations were made at I0.2. The ratio Flux of fructose 1,6-bisphosphate----fructose 6-phosphate/Flux of fructose 1,6-bisphosphate----ATP at zero ATP concentration increased hyperbolically from unity to about 3.2 as the concentration of fructose 6-phosphate was increased. Similarly, the ratio Flux of fructose 1,6-bisphosphate----ATP/Flux of fructose 1,6-bisphosphate----fructose 6-phosphate at zero fructose 6-phosphate concentration increased from unity to about 1.4 as the concentration of ATP was increased. The addition of substrates must therefore be random, whatever the other aspects of the reaction. Further, from the plateau values of the ratios, it follows that the substrates dissociate very infrequently from the ternary complex and that at a low substrate concentration 72% of the reaction follows the pathway in which ATP adds first to the enzyme. Isotope-trapping studies with [32P]ATP confirmed that ATP can bind first to the enzyme in rate-limiting step and that dissociation of ATP from the ternary complex is slow in relation to the forward reaction. No isotope trapping of [U-14C]-fructose 6-phosphate could be demonstrated. The ratios Flux of ATP----fructose 1,6-bisphosphate/Flux of ATP----ADP measured at zero ADP concentration and the reciprocal of the ratio measured at zero fructose 1,6-bisphosphate concentration did not differ significantly from unity. Calculated values for these ratios based on the kinetics of the reverse reaction and assuming ordered dissociations of products or a ping-pong mechanism gave values very significantly greater than unity. These findings exclude an ordered dissociation or a substantial contribution from a ping-pong mechanism, and it is concluded that the reaction is sequential and that dissociation of products is random. Rate constants were calculated for the steps in the enzyme reaction. The results indicate a considerable degree of co-operativity in the binding between the two substrates. The observations on phosphofructokinase are discussed in relation to methods of measurement and interpretation of flux ratios and in relation to the mechanism of other kinase enzymes.     

The mechanism of the phosphoglucomutase reaction. Studies on rabbit muscle phosphoglucomutase with flux techniques

1. The kinetics of phosphoglucomutases from different sources are discussed and it is concluded that on the available evidence there are in all cases three possible mechanisms for the reaction. These are an indirect transfer of phosphate involving the phosphoenzyme (mechanism 1), a direct transfer of phosphate (mechanism 2), and an intermolecular transfer of phosphate from glucose 1,6-diphosphate to the substrate (mechanism 3). Conventional net flux measurements are shown not to differentiate between these mechanisms. 2. Flux equations are developed and it is shown that there are three flux ratios that characterize and distinguish between the mechanisms. 3. To examine these flux ratios induced-transport tests are described with ¹⁴C- and ³²P-labelled substrates. The fluxes determined with ¹⁴C- and ³²P-labelled substrates are also compared at chemical equilibrium. 4. With rabbit muscle phosphoglucomutase the results of these tests were completely consistent with mechanism 1 and unequivocally excluded any substantial part of the reaction proceeding by mechanism 2 or mechanism 3. Evidence was also obtained for an isomerization of the phosphoenzyme with an apparent rate constant about 4·5×10⁷sec.⁻¹. Taking into account the activity coefficients of the substrates the true rate constant appears to be about one-sixth of this value. 5. Isotope effects and non-ideal behaviour of the solutions are discussed and the activity coefficients of the substrates are shown to be equal by measurement of the depression of freezing point. It is concluded that these factors do not influence the tests significantly. 6. Alternative mechanisms are considered and it is concluded that the tests show that the glucose residue is transferred directly, that the phosphate is transferred indirectly with one intermediate phosphate, and that there is an isomerization of the free phosphoenzyme without reference to any other details of the reaction. Further, no assumptions are required about the constancy of rate constants. 7. The relative merits of induced transport and product inhibition for detecting isomerization of the enzyme are discussed. It is concluded that the induced-transport test is more sensitive and that its interpretation is less equivocal. 8. The application of the tests to other enzyme systems is briefly considered.     

Analysis of slow-binding enzyme inhibitors at elevated enzyme concentrations

The improvement in the characterization of slow-binding inhibitors achieved by performing experiments at elevated enzyme concentrations is presented. In particular, the characterization of slow-binding inhibitors conforming to a two-step mode of inhibition with a steady-state dissociation constant that is much lower than the initial dissociation constant with enzyme is discussed. For these systems, inhibition is rapid and low steady-state product concentrations are produced at saturating inhibitor concentrations. By working at elevated enzyme concentrations, improved signal-to-noise ratios are achieved and data may be collected at saturating inhibitor levels. Numerical simulations confirmed that improved parameter estimates are obtained and useful data to discern the mechanism of slow-binding inhibition are produced by working at elevated enzyme concentrations. The saturation kinetics that were unobservable in two previous studies of an enzyme inhibitor system were measured by performing experiments at an elevated enzyme concentration. These results indicate that consideration of the quality of the data acquired using a particular assay is an important factor when selecting the enzyme concentration at which to perform experiments used to characterize the class of enzyme inhibitors examined herein.     

Determination of kinetic parameters of enzyme-catalyzed reactions with a minimum number of velocity measurements

Duggleby [Duggleby, R.G., 1979. Experimental designs for estimating kinetic parameters for enzyme-catalyzed reactions. J. Theor. Biol. 81, 672-684] discussed the “design of several replicate measurements of the velocity at as many experimental conditions as there are parameters to be estimated.” He discussed the application of this method to A→products, without and with competitive inhibition, and commented briefly on A+B→products. The availability of computer applications that can solve large sets of simultaneous equations makes it possible to use this method to calculate kinetic parameters for more complicated enzyme mechanisms. This article is concerned with rapid-equilibrium rate equations, but this method can also be used with steady-state rate equations. Computer programs are provided for the calculation of the three kinetic parameters for ordered A+B→products from three velocity measurements and for the calculation of the four kinetic parameters for random A+B→products from four velocity measurements. Computer programs are also provided for competitive inhibition, uncompetitive inhibition, and mixed inhibition of ordered A+B→products.     

Rat liver cytoplasmic glucose-6-phosphate dehydrogenase. Steady-state kinetic properties and circular dichroism.

Steady-state kinetic studies including initial velocity, NADPH product inhibition, dead-end inhibition, and combined dead-end and product inhibition measurements with purified rat liver glucose-6-phosphate dehydrogenase indicate a sequential and obligatory addition of substrates in the order of NADP+, glucose-6-P for the catalytic pathway at pH 8.0. Although instability of 6-phosphoglucono-delta-lactone precluded product inhibition experiments which might directly exclude an enzyme-6-phosphoglucono-delta-lactone complex, the absence of an enzyme-glucose-6-P complex suggests that the enzyme-lactone product is unlikely and the release of products is also ordered, with NADPH released last. Consideration of the kinetic constants (Ka = 2.0 muM, Kiq = 13 muM) and cellular concentration of the substrates and products suggests extensive inhibition of the enzyme in vivo and control by the NADPH/NADP+ ratios. Circular dichroism spectra of the enzyme in 20 mM phosphate buffer at pH 7.0 and 25 degrees C indicate 51% helix and 33% pleated sheet structures which is considerably different from results (14% helix) with yeast enzymes.     

Kinetics of inhibition by 8-oxy-GTP and 8-bromo-GTP of Escherichia coli RNA-polymerase synthesis of pppApU dinucleotide on the promotor a1 of phage T7deltaD111 DNA in a limited set of substrates

Detailed analysis of the kinetics of inhibition of E. coli RNA-polymerase-catalyzed synthesis of dinucleotide pppApU by 8-oxy-GTP and 8-Br-GTP on promoter A1 of the bacteriophage T7 delta D111 with an incomplete set of substrates was carried out. In accordance with the mathematical models obtained, we calculated quantitative parameters of binding of these nucleotide analogs to the centers whose geometry is suitable for incorporation of ATP and UTP. 8-oxy-GTP and 8-Br-GTP compete with ATP for the binding center (their steady-state dissociation constant ratios are 2.1 and 2.4, respectively, whereas the constant for ATP is 0.3 mM) but, unlike ATP, they are not incorporated into the product. 8-oxy-GTP competes also with UTP (its steady-state dissociation constant ratio is 21.6, the constant for UTP is 0.03 mM). 8-Br-GTP does not interact with the binding center of UTP.     

Analysis of the mechanism of chloramphenicol acetyltransferase by steady-state kinetics. Evidence for a ternary-complex mechanism.

The mechanism of the enzymic reaction responsible for chloramphenicol resistance in bacteria was examined by steady-state kinetic methods. The forward reaction catalysed by chloramphenicol acetyltransferase leads to inactivation of the antibiotic. Use of alternative acyl donors and acceptors, as well as the natural substrates, has yielded data that favour the view that the reaction proceeds to the formation of a ternary complex by a rapid-equilibrium mechanism wherein the addition of substrates may be random but a preference for acetyl-CoA as the leading substrate can be detected. Chloramphenicol and acetyl-CoA bind independently, but the correlation between directly determined and kinetically derived dissociation constants is imperfect because of an unreliable slope term in the rate equation. The reverse reaction, yielding acetyl-CoA and chloramphenicol, was studied in a coupled assay involving citrate synthase and malate dehydrogenase, and is best described by a rapid-equilibrium mechanism with random addition of substrates. The directly determined dissociation constant for CoA is in agreement with that derived from kinetic measurements under the assumption of an independent-sites model.     

Biosynthetic direction substrate kinetics and product inhibition studies on the first enzyme of histidine biosynthesis, adenosine triphosphate phosphoribosyltransferase.

Initial velocity steady-state substrate kinetics for the ATP phosphoribosyltransferase reaction in the biosynthetic direction were determined and are consistent with a sequential kinetic mechanism. To hold the fractions of magnesium-complexed substrates and products constant so as to avoid possible distortion of reciprocal velocity plots Mg²⁺ binding constants to the substrates ATP and phosphoribosylpyrophosphate and the product pyrophosphate were measured under assay conditions. Several conformational states of the phosphoribosyltransferase distinguishable by other criteria gave similar substrate kinetic behavior. Product inhibition studies were conducted to elucidate the binding order. Phosphoribosyl-ATP was competitive with respect to ATP and was non-competitive with respect to phosphoribosylpyrophosphate. Pyrophosphate was non-competitive with respect to both substrates. The data are consistent with the ordered Bi-Bi kinetic mechanism with ATP binding first to free enzyme and phosphoribosyl-ATP dissociating last from enzyme-product complexes.     

Steady-state kinetic mechanism of Ras farnesyl:protein transferase.

The steady-state kinetic mechanism of bovine brain farnesyl:protein transferase (FPTase) has been determined using a series of initial velocity studies, including both dead-end substrate and product inhibitor experiments. Reciprocal plots of the initial velocity data intersected on the 1/[s] axis, indicating that a ternary complex forms (sequential mechanism) and suggesting that the binding of one substrate does not affect the binding of the other. The order of substrate addition was probed by determining the patterns of dead-end substrate and product inhibition. Two nonhydrolyzable analogues of farnesyl diphosphate, (alpha-hydroxyfarnesyl)phosphonic acid (1) and [[(farnesylmethyl)hydroxyphosphinyl]methyl]phosphonic acid (2), were both shown to be competitive inhibitors of farnesyl diphosphate and noncompetitive inhibitors of Ras-CVLS. Four nonsubstrate tetrapeptides, CV[D-L]S, CVLS-NH2, N-acetyl-L-penicillamine-VIM, and CIFM, were all shown to be noncompetitive inhibitors of farnesyl diphosphate and competitive inhibitors of Ras-CVLS. These data are consistent with random order of substrate addition. Product inhibition patterns corroborated the results found with the dead-end substrate inhibitors. We conclude that bovine brain FPTase proceeds through a random order sequential mechanism. Determination of steady-state parameters for several physiological Ras-CaaX variants showed that amino acid changes affected the values of KM, but not those of kcat, suggesting that the catalytic efficiencies (kcat/KM) of Ras-CaaX substrates depend largely upon their relative binding affinity for FPTase.     

Mechanisms and rate equations for dissociating systems.

The results presented in the previous paper (Boeker, E.A. (1978), Biochemistry 17 (preceding paper in this issue) indicate that the dissociation of the decamer of arginine decarboxylase of Escherichia coli B is enhanced by Na+ and retarded by H+. In this system, substances which increase the rate of dissociation can be treated kinetically either as substrates or activators, and substances which retard dissociation can be treated as products or inhibitors. In addition, the events needed for dissociation can occur in an ordered or a random sequence, and the dissociation itself, from a decamer to five dimers, can be a sequential or a concerted process. In order to provide a framework for the experimental results, mechanisms for the dissociation of arginine decarboxylase that take all of these factors into account are described. In addition, it is shown that the usual methods of steady-state kinetics can be applied to these systems when true initial rates are measured; rate equations are presented for each mechanism. The results can be used for any dissociating of three or more subunits and will describe the dissociation of a dimer under certain conditions.     

Kinetic studies of fructokinase I of pea seeds

Fructokinase I of pea seeds has been purified to homogeneity and the enzyme shown to be monomeric, with a molecular weight of 72,000 +/- 4000. The reaction mechanism was investigated by means of initial velocity studies. Both substrates inhibited the enzyme; the inhibition caused by MgATP was linear-uncompetitive with respect to fructose whereas that caused by D-fructose was hyperbolic-noncompetitive against MgATP. The product D-fructose 6-phosphate caused hyperbolic-noncompetitive inhibition with respect to both substrates. MgADP caused noncompetitive inhibition, which gave intercept and slope replots that were linear with D-fructose but hyperbolic with MgATP. Free Mg2+ caused linear-uncompetitive inhibition when either substrate was varied. L-Sorbose and beta, gamma-methyleneadenosine 5'-triphosphate were used as analogs of D-fructose and MgATP, respectively. Inhibition experiments using these compounds indicated that substrate addition was steady-state ordered, with MgATP adding first. The product inhibition experiments were found to be consistent with a steady-state random release of products. The substrate inhibition caused by MgATP was most likely due to the formation of an enzyme-MgATP-product dead-end complex, whereas that caused by D-fructose was due to alternative pathways in the reaction mechanism. The inhibition caused by Mg2+ can be explained in terms of a dead-end complex with either a central complex or an enzyme-product complex.     

Mechanism of reaction catalyzed by RNA-ligase from bacteriophage T4

The dissociation constants of the complexes of RNA-ligase with acceptors, donors and the adenylylated donor A(5')ppAp have been determined on the basis of the inhibition of ATP-pyrophosphate exchange reaction. The dissociation constants of the complexes of the enzyme with "poor" acceptors (oligouridilates) have been shown to be slightly different from those with "good" acceptors (oligoadenylates). The dependence of the reaction velocity of the formation of ligation products on the concentration of acceptors (pA)4, (pU)4 and the adenylylated donor A(5)ppAp has been studied. On the basis of the data obtained the conclusion about the random addition mechanism has been drawn. The reaction takes place in the steady-state conditions in the case of (pA)4 and in the equilibrium conditions--in the case of (pU)4.     

Kinetic investigation of the functional role of phenylalanine-31 of recombinant human dihydrofolate reductase

The role of the active site residue phenylalanine-31 (Phe31) for recombinant human dihydrofolate reductase (rHDHFR) has been probed by comparing the kinetic behavior of wild-type enzyme (wt) with mutant in which Phe31 is replaced by leucine (F31L rHDHFR). At pH 7.65 the steady-state kcat is almost doubled, but the rate constant for hydride transfer is decreased to less than half that for wt enzyme, as is the rate of the obligatory isomerization of the substrate complex that precedes hydride transfer. Although steady-state measurements indicated that the mutation causes large increases in Km for both substrates, dissociation constants for many complexes are decreased. These apparent paradoxes are due to major mutation-induced decreases in rate constants (koff) for dissociation of folate, dihydrofolate, and tetrahydrofolate from all of their complexes. This results in a mechanism proceeding almost entirely by only one of the two pathways used by wt enzyme. Other consequences of these changes are a much altered dependence of steady-state kcat on pH, inhibition rather than activation by tetrahydrofolate, absence of hysteresis in transient-state kinetics, and a decrease in enzyme efficiency under physiological conditions. The results indicate that there is no quantitative correlation between dihydrofolate binding and the rate of hydride transfer for this enzyme.     

Kinetic studies on chorismate mutase-prephenate dehydrogenase from Escherichia coli: models for the feedback inhibition of prephenate dehydrogenase by L-tyrosine

Kinetic studies have been undertaken to elucidate the mechanism of the allosteric inhibition by tyrosine of the prephenate dehydrogenase activity of the bifunctional dimeric enzyme chorismate mutase-prephenate dehydrogenase. The effect of tyrosine on the initial velocity of the reactions in the presence of both prephenate and the alternative substrate, 1-carboxy-4-hydroxy-2-cyclohexene-1-propanoate, have been determined. In addition, investigations have been made of the effect of tyrosine on the inhibition of the reaction by the inhibitory analogues of prephenate, (4-hydroxyphenyl)pyruvate, and (carboxyethyl)-1,4-dihydrobenzoate. The results of the double inhibition experiments indicate clearly that the enzyme possesses a distinct allosteric site for the binding of tyrosine. The initial velocity data obtained with both substrates have been fitted to the rate equations that describe a wide range of models. From a comparison of the results obtained from studies with the two substrates, and with a knowledge of the value for the dissociation constant of the tyrosine-enzyme complex, definitive conclusions have been reached about the mechanism of the allosteric inhibition. It is concluded that tyrosine combines twice at allosteric sites and in an antisynergistic fashion, while prephenate reacts at both active sites of the dimeric enzyme as well as weakly at one of the allosteric sites. It appears that the latter is simple competition reaction that affects neither the binding of prephenate at the active site nor the rate of product formation. The model also predicts the formation of an active tyrosine-enzyme-prephenate complex that yields product at a much slower rate than does the enzyme-prephenate complex.(ABSTRACT TRUNCATED AT 250 WORDS)     

Purine nucleoside phosphorylase of rabbit liver. Mechanism of catalysis.

Initial velocity studies and product inhibition patterns for purine nucleoside phosphorylase from rabbit liver were examined in order to determine the predominant catalytic mechanism for the synthetic (forward) and phosphorolytic (reverse) reactions of the enzyme. Initial velocity studies in the absence of products gave intersecting or converging linear double reciprocal plots of the kinetic data for both the synthetic and phosphorolytic reactions of the enzyme. The observed kinetic pattern was consistent with a sequential mechanism, requiring that both substrates add to the enzyme before products may be released. The product inhibition patterns showed mutual competitive inhibition between guanine and guanosine as variable substrates and inhibitors. Ribose 1-phosphate and inorganic orthophosphate were also mutually competitive toward each other. Other combinations of substrates and products gave noncompetitive inhibition. Apparent inhibition constants calculated for guanine as competitive inhibitor and for ribose 1-phosphate as noncompetitive inhibitor of the enzyme, with guanosine as variable substrate, did not vary significantly with increasing concentrations of inorganic orthophosphate as fixed substrate. These results suggest that the mechanism was order and that substrates add to the enzyme in an obligatory order. Dead end inhibition studies carried out in the presence of the products guanine and ribose 1-phosphate, respectively, showed that the kinetically significant abortive ternary complexes of enzyme-guanine-inorganic orthophosphate (EQB) and enzyme-guanose-ribose 1-phosphate (EAP) are formed. The results of dead end inhibition studies are consistent with an obligatory order of substrate addition to the enzyme. The nucleoside or purine is probably the first substrate to form a binary complex with the enzyme, and with which inorganic orthophosphate or ribose 1-phosphate may interact as secondary substrates. The evidences presented in this investigation support an Ordered Theorell-Chance mechanism for the enzyme.     

The kinetic mechanism of S-adenosyl-L-methionine: glutamylmethyltransferase from Salmonella typhimurium

The kinetic mechanism of the CheR methyltransferase, S-adenosyl-L-methionine (AdoMet): protein-L-glutamate O-methyltransferase (EC 2.1.1.24), from Salmonella typhimurium was investigated. Initial velocity, product inhibition, and binding studies were performed, and from the data obtained, it was determined that the mechanism of the reaction catalyzed by the enzyme is random. Initial velocity rates were measured with varied amounts of both substrates, and double-reciprocal plots gave patterns which converged on or near the abscissa. The products, S-adenosyl-L-homocysteine and methylated receptor, were found to be competitive inhibitors with respect to both AdoMet and receptor. Equilibrium dialysis and immunoprecipitation studies indicated that the two substrates can bind to the enzyme independent of each other. These results are consistent with a random mechanism with no abortive complexes being formed. The Michaelis constants calculated for AdoMet and receptor were 8.62 microM and 2.03 mg/ml total membrane protein (approximately 2.10 microM Tar protein), and the apparent dissociation constants of AdoMet and the receptor were 16.8 microM and 4.07 mg/ml total membrane protein (approximately 4.2 microM Tar protein), respectively. The Kd of AdoMet for the enzyme was 10.9 microM as determined by binding studies.