An evolutionary approach to enzyme kinetics: Optimization of ordered mechanisms

A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.     

Studies on yeast nucleoside triphosphate-nucleoside diphosphate transphosphorylase (nucleoside diphosphokinase). IV. Steady-state kinetic properties with thymidine nucleotides (including 3'-azido-3'-deoxythymidine analogues).

A study of the steady-state kinetics of the crystalline brewer's yeast (Saccharomyces carlsbergensis) nucleoside diphosphokinase, with the magnesium complexes of the adenine and thymidine nucleotides as reactants, has led to a postulated kinetic mechanism which proceeds through a substituted enzyme. This agrees with the earlier conclusions of Garces and Cleland [Biochemistry 1969; 8:633-640] who characterized a reaction between the magnesium complexes of the adenine and uridine nucleotides. An advantage of using thymidine nucleotides as reactants is that they permit accurate, rapid and continuous assays of the enzymatic activity in coupled-enzymatic tests. Through measurements of the initial velocities and product inhibition studies, the Michaelis constants, maximum velocities, and inhibition constants could be evaluated for the individual substrates. Competitive substrate inhibition was encountered at relatively high substrate concentrations, which also permitted an evaluation of their ability to act as 'dead-end' inhibitors. The Michaelis constants for the 3'-azido-3'-deoxythymidine (AzT) analogues were also evaluated and, although these values were only somewhat higher than those of their natural substrates, the Km's for the adenine nucleotides as paired substrates were lower and the Vmax's were drastically reduced. The pharmacological implications of these observations are touched upon and extrapolated to the cases where therapeutic doses of AzT may be employed.     

Thermodynamics and kinetics of the glyoxylate cycle

Because the standard Gibbs energies of formation of all the species of reactants in the glyoxylate cycle are known at 298.15 K, it is possible to calculate the apparent equilibrium constants of the five reactions in the cycle in the pH range 5-9 and ionic strengths from 0 to approximately 0.35 M. In making calculations on such a system, it is convenient to specify concentrations of coenzymes like NADox and NADred because they are involved in many reactions and may be in steady states. Calculations are given for [NADox] = 1000[NADred] and [NADox] = 10[NADred]. Equilibrium compositions are calculated using computer programs when all the reactants are present initially and when only glyoxylate and CoA are present initially. The kinetics of the reactions in the glyoxylate cycle at specified concentrations of NADox and NADred are calculated by numerical solution of the steady-state rate equations for the case where the reactant concentrations are below their Michaelis constants and only glyoxylate and CoA are present initially.     

On true and apparent Michaelis constants in enzymology. I. Differences

Differences between both true and apparent rate constants and Michaelis constants have been examined. Rate constants of elementary stages of real mechanisms are true ones. True Michaelis constant Km is expressed by equation Km = (k(-1) + k2)/k. True constants may be determined for reliable mechanism only for which the equation of initial rate was obtained which displays physical sense of these constants and permits to find the method of their calculation. The true constant values are independent of concentration of reactants, activators, inhibitors, extraneous agents and pH. The apparent rate constants are such constants of the composite reaction which are observed when this reaction is described by the equation of simple reaction. Michaelis constant calculated by a half of the ultimate constant is an apparent constant. The apparent constants may be functions of several true rate constants and/or concentrations of reacting substances. The evident physical sense of apparent constants being absent, only formal relation between the reaction rate and reactant concentration independent of the investigated mechanism is provided.     

Optimizing the sensitivity of enzyme systems to perturbations from equilibrium

Algebraic derivations and numerical examples illustrate how metabolite pool sizes and enzyme rate constants influence the rate at which a multireactant enzyme system, initially poised in a near-equilibrium steady state, responds to small perturbations in the concentrations of the reactants. Certain enzymes, such as those employing the ordered bi bi catalytic mechanism, become relatively insensitive to perturbations when the reactants are all present at high concentrations. Other enzymes, such as those employing the ping-pong bi bi mechanism, are most sensitive to perturbations at high reactant concentrations. The ratio of the reactant concentrations to one another significantly alters sensitivity to perturbations; equations are presented for calculation of the reactant concentrations yielding maximal sensitivity to perturbations. Natural selection could choose metabolite pool sizes and enzyme rate constants which would optimize the performance of these systems, but changing metabolic loads (naturally or experimentally imposed) constantly alter the sensitivity of these systems to perturbations, changing the relative strengths of various connections in metabolic control networks.     

Enzyme kinetics in reversed micelles. 2. Behaviour of enoate reductase

Enoate reductase (EC 1.3.1.31) can stereospecifically reduce a variety of alpha,beta-unsaturated carboxylates. Its use was extended to apolar media by incorporating the enzyme into a reversed micellar medium. The kinetics of the enzyme in such a medium have been investigated using 2-methylbutenoic acid as substrate and NADH as a cofactor and compared with the reaction rates in aqueous solution. In aqueous solution the enzyme obeys a ping pong mechanism [Bühler et al. (1982) Hoppe-Seyler's Z. Physiol. Chem 363, 609-625]. In 50 mM Hepes pH = 7.0 with ionic strength of 0.05 M the Michaelis constants for NADH and 2-methylbutenoic acid are 20 microM and 6.0 mM respectively. In reversed micelles the kinetics of the reaction (Michaelis constant, maximum velocity as well as inhibitory effects) were markedly different. The rate of the enzymatic reaction of enoate reductase was studied using various concentrations of 2-methylbutenoic acid and various NADH concentrations. In reversed micelles composed of the anionic detergent sodium di(ethylhexyl)sulphosuccinate, the enzymatic reaction deviates substantially from the values in aqueous solution. Using our model (see preceding paper in this issue of the journal), all kinetics could be explained as evolving from enclosure in reversed micelles without any change in the intrinsic rate parameters of the enzyme. So the enzyme itself is unaffected by incorporation in reversed micelles, but the rate of intermicellar exchange as well as the microheterogeneity of the medium, resulting in very high local concentrations of the substrate, are the most important factors altering the reaction pattern. The effect of the composition of the reversed micellar medium was also investigated using either a nonionic or a cationic surfactant. In these solutions too, exchange and microheterogeneity of the medium proved to be the most important parameters influencing the enzymatic reaction. In all reversed micellar solutions inhibition by the enoate was observed at an overall concentration of 0.5-5 mM, implying that a concentration of substrate equal to the Km value in aqueous solution may already cause inhibition in reversed micelles. At this level no inhibition by NADH was observed. The microheterogeneity of the medium also explains this inhibition of the enzyme at relatively low 2-methylbutenoic acid concentrations.     

Enzymatic assays: optimization of systems by using pyruvate kinase and lactate dehydrogenase as auxiliary enzymes

A method to optimize enzymatic assays by using pyruvate kinase and lactate dehydrogenase enzymes is presented and applied to mitochondrial ATPase as an example. Optimum amounts of auxiliary enzymes, to obtain either a 99% of the initial rate in a given time (t99) or a given lag period (L), are calculated from their apparent Michaelis constants (Kapp) in the medium used and their prices per enzymatic international unit.     

Steady state kinetics of consecutive enzyme catalysed reactions involving single substrates: procedures for the interpretation of coupled assays

Sets of differential rate equations are written describing a linear sequence of reactions occurring in solution each catalysed by a control enzyme or one of the Michaelis-Menten type. It is shown that the solutions of these equations may be formulated as a set of Maclaurin polynomials, expressing the concentration of each reactant and of final product as a function of time. From arrays of such polynomials, general expressions are induced for the first non-zero term of the series. These are used to formulate a procedure (illustrated with an example simulated by numerical integration) by which results of coupled enzymic assays may be analysed in terms of maximal velocities and apparent Michaelis constants: correlation is made with other established methods for conducting coupled assays. The present procedure assumes a steady state of enzyme-substrate complexes but not of intermediate reactants.     

Purification and properties of two glyoxylate reductases from a species of Pseudomonas

1. Two enzymes that catalyse the reduction of glyoxylate to glycollate have been separated and purified from a species of Pseudomonas. Their molecular weights were estimated as 180000. 2. Reduced nicotinamide nucleotides act as the hydrogen donators for the enzymes. The NADH-linked enzyme is entirely specific for its coenzyme but the NADPH-linked reductase shows some affinity towards NADH. 3. Both enzymes convert hydroxypyruvate into glycerate. 4. The glyoxylate reductases show maximal activity at pH6·0–6·8, are inhibited by keto acids and are strongly dependent on free thiol groups for activity. 5. The Michaelis constants for glyoxylate and hydroxypyruvate were found to be of a high order. 6. The reversibility of the reaction has been demonstrated for both glyoxylate reductases and the equilibrium constants were determined. 7. The reduction of glyoxylate and hydroxypyruvate is not stimulated by anions.     

Determination of kinetic constants for peptidyl prolyl cis-trans isomerases by an improved spectrophotometric assay

The kinetic properties and substrate specificity of two well-characterized peptidyl prolyl cis-trans isomerases (PPIases), cyclophilin and the FK-506 binding protein (FKBP), have been previously examined [Fischer, G., Bang, H., Berger, E., & Schellenberger, A. (1984) Biochim. Biophys. Acta 791, 87-97; Harrison, R.K., & Stein, R.L. (1990) Biochemistry 29, 1684-1689; Albers, M.W., Walsh, C.T., & Schreiber, S. L. (1990) J. Org. Chem. 55, 4984-4986]. The chymotrypsin-coupled enzymatic assay employed in these studies suffers from two serious shortcomings. Due to the low equilibrium population of the X-cis-Pro-Phe-pNA isomer (the PPIase substrate), in conjunction with the low solubility of p-nitroaniline generated by chymotrypsin hydrolysis, substrate concentrations in the saturating region are not experimentally attainable. Secondly, the uncatalyzed cis-trans isomerization obscures the interpretation of the initial velocity. As a result of these limitations, the steady-state kinetic parameters (Km,Kcat) have not been determined. Here we introduce an improved version of the spectrophotometric assay and report for the first time the Michaelis constants and turnover numbers for both PPIases with established substrates. The improvements in the experimental conditions originate in a medium-induced increase in the equilibrium population of the cis X-Pro conformer and in conducting the assay at 0 degrees C to suppress the uncatalyzed thermal isomerization. In addition, we present a rigorous mathematical model of the spectrophotometric progress curves that accounts for the contributions of the residual background rate.(ABSTRACT TRUNCATED AT 250 WORDS)     

Effect of coenzymes and temperature on the process of in vitro refolding and reassociation of lactic dehydrogenase isoenzymes.

Dissociation and deactivation of the H4 and M4 isoenzymes of lactic dehydrogenase in strong denaturants may be reversed with a yield of reactivation up to 100%. The products of reconstitution are indistinguishable from the native enzymes as far as the Michaelis constants and the dissociation constants for substrate and coenzyme as well as spectral and hydrodynamic properties are concerned. The presence of NAD+ and NADH does not affect either the conformational state of the product of reconstitution, or the kinetics of reactivation, using the pure apoenzymes as a reference. At 20 degrees C the kinetics of reactivation for LDH-M4 in the presence and absence of coenzyme may be quantitatively described by a second-order rate equation (k2 = 23.4 +/- 2.6 mM-1S-1) while LDH-H4 is characterized by a uni-bimolecular reaction sequence (k1 = 1.45 +/- 0.45 X 10(-3)-S-1, k2 = 5 +/- 1 mM-1S-1), in agreement with earlier observations (Rudolph, R., et al. (1977), Biochemistry 16, 3384-3390). Regarding the influence of temperature on the rate of reactivation no significant anomalies are detectable within the range of 0-25 degrees C. The (apparent) activation energies, taken from the linear Arrhenius plots, are 58 kcal/mol for the association reaction of LDH-M4, and 41 kcal/mol for the transconformation reaction of LDH-H4.     

Michaelis constants of mushroom tyrosinase with respect to oxygen in the presence of monophenols and diphenols

The complex reaction mechanism of tyrosinase involves three enzymatic forms, two overlapping catalytic cycles and a dead-end complex. Analytical expressions for the catalytic and Michaelis constants of tyrosinase towards phenols and oxygen were derived for both, monophenolase and diphenolase activities of the enzyme. Thus, the Michaelis constants of tyrosinase towards the oxygen (K(mO(2))) are related with the respective catalytic constants for monphenols (k(M)(cat)) and o-diphenols (k(D)(cat)), as well as with the rate constant, k(+8). We recently determined the experimental value of the rate constant for the binding of oxygen to deoxytyrosinase (k(+8)) by stopped-flow assays. In this paper, we calculate theoretical values of K(mO(2)) from the experimental values of catalytic constants and k(+8) towards several monophenols and o-diphenols. The reliability and the significance of the values of K(mO(2)) are discussed.     

Nonfixed relationship of the Michaelis constant and maximum velocity with their corresponding rate constants

The Michaelis constant (K(m)) and V(mas) (E0k(cat)) values for two mutant sets of enzymes were studied from the viewpoint of their definition in a rapid equilibrium reaction model and in a steady state reaction model. The "AMP set enzyme" had a mutation at the AMP-binding site (Y95F, V67I, and V67I/L76V), and the "ATP set enzyme" had a mutation at a possible ATP-binding region (Y32F, Y34F, and Y32A/Y34A). Reaction rate constants obtained using steady state model analysis explained discrepancies found by the rapid equilibrium model analysis. (i) The unchanged number of bound AMPs for Y95F and the wild type despite the markedly increased K(m) values for AMP of the AMP set of enzymes was explained by alteration of the rate constants of the AMP step (k(+2), k(-2)) to retain the ratio k(+2)/k(-2). (ii) A 100 times weakened selectivity of ATP for Y34F in contrast to no marked changes in K(m) values for both ATP and AMP for the ATP set of enzymes was explained by the alteration of the rate constants of the ATP steps. A similar alteration of the K(m) and k(cat) values of these enzymes resulted from distinctive alterations of their rate constants. The pattern of alteration was highly suggestive. The most interesting finding was that the rate constants that decided the K(m) and k(cat) values were replaced by the mutation, and the simple relationships between K(m), k(cat), and the rate constants of K(m)1 = k(+1)/k(-1) and k(cat) = k(f) were not valid. The nature of the K(m) and k(cat) alterations was discussed.     

Levels of thermodynamic treatment of biochemical reaction systems.

Equilibrium calculations on biochemical reaction systems can be made at three levels. Level 1 is the usual chemical calculation with species at specified temperature and pressure using standard Gibbs energies of formation of species or equilibrium constants K. Level 2 utilizes reactants such as ATP (a sum of species) at specified T, P, pH, and pMg with standard transformed Gibbs energies of formation of reactants or apparent equilibrium constants K'. Calculations at this level can also be made on the enzymatic mechanism for a biochemical reaction. Level 3 utilizes reactants at specified T, P, pH, and pMg, but the equilibrium concentrations of certain reactants are also specified. The fundamental equation of thermodynamics is derived here for Level 3. Equilibrium calculations at this level use standard transformed Gibbs energies of formation of reactants at specified concentrations of certain reactants or apparent equilibrium constants K". Level 3 is useful in calculating equilibrium concentrations of reactants that can be reached in a living cell when some of the reactants are available at steady-state concentrations. Calculations at all three levels are facilitated by the use of conservation matrices and stoichiometric number matrices for systems. Three cases involving glucokinase, glucose-6-phosphatase, and ATPase are discussed.     

Comparison of rate constants for (PO3-) transfer by the Mg(II), Cd(II), and Li(I) forms of phosphoglucomutase

Net rate constants that define the steady-state rate through a sequence of steps and the corresponding effective energy barriers for two (PO3-)-transfer steps in the phosphoglucomutase reaction were compared as a function of metal ion, M, where M = Mg2+ and Cd2+. These steps involve the reaction of either the 1-phosphate or the 6-phosphate of glucose 1,6-bisphosphate (Glc-P2) bound to the dephosphoenzyme (ED) to produce the phosphoenzyme (EP) and the free monophosphates, glucose 1-phosphate (Glc-1-P) or glucose 6-phosphate (Glc-6-P): EP.M + Glc-1-P----ED.M.Glc-P2----EP.M.Glc-6-P6. Before this comparison was made, net rate constants for the Cd2+ enzyme, obtained at high enzyme concentration via 31P NMR saturation-transfer studies [Post, C. B., Ray, W. J., Jr., & Gorenstein, D. G. (1989) Biochemistry (preceding paper in this issue)], were appropriately scaled by using the observed constants to calculate both the expected isotope-transfer rate at equilibrium and the steady-state rate under initial velocity conditions and comparing the calculated values with those measured in dilute solution. For the Mg2+ enzyme, narrow limits on possible values of the corresponding net rate constants were imposed on the basis of initial velocity rate constants for the forward and reverse directions plus values for the equilibrium distribution of central complexes, since direct measurement is not feasible. The effective energy barriers for both the Mg2+ and Cd2+ enzymes, calculated from the respective net rate constants, together with previously values for the equilibrium distribution of complexes in both enzymic systems [Ray, W. J., Jr., & Long, J. W. (1976) Biochemistry 15, 4018-4025], show that the 100-fold decrease in the kappa cat for the Cd2+ relative to the Mg2+ enzyme is caused by two factors: the increased stability of the intermediate bisphosphate complex and the decreased ability to cope with the phosphate ester involving the 1-hydroxyl group of the glucose ring. In fact, it is unlikely that the efficiency of (PO3-) transfer to the 6-hydroxyl group of bound Glc-1-P (thermodynamically favorable direction) is reduced by more than an order of magnitude in the Cd2+ enzyme. By contrast, the efficiency of the Li+ enzyme in the same (PO3-)-transfer step is less than 4 x 10(-8) that of the Mg2+ enzyme.(ABSTRACT TRUNCATED AT 400 WORDS)     

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.