Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.     

The kinetic mechanism of ubiquinol: cytochrome c reductase at steady state.

The steady-state kinetics of ubiquinol: cytochrome c reductase (cytochrome bc1 complex) is analyzed in this work. The graphical pattern of the titrations is clearly indicative of a ping-pong mechanism, but the two products ubiquinone and reduced cytochrome c behave competitively with their substrate and noncompetitively with the other substrate. Hence, the mechanism of the reductase is of a ping-pong two-site type. A minimal reaction scheme for the enzymatic mechanism is proposed and approximate values of its rate constants are deduced on the assumption that each substrate is in rapid equilibrium at its catalytic site. This has been substantiated by presteady-state measurements of the reduction and oxidation of cytochrome b by a short-chain homolog of ubiquinol. Values of the rate constants of the reaction scheme have been deduced from the steady-state titrations for a series of 2,3-dimethoxy-5-methyl quinols having different hydrophobic substituents in position 6 of the ring. The results provide a quantitative estimation of the specificity of the quinol catalytic site in the transmembrane portion of the bc1 complex. In particular, a reasonable correlation is found between the rate of the second-order reaction of quinols with the enzyme and their solubility in lipids.     

Does any enzyme follow the Michaelis—Menten equation?

Summary A literature search has been conducted to see to what extent steady-state kinetics studies in the period 1965–1976 have revealed deviations from Michaelis—Menten kinetics. It was found that over 800 enzymes have been reported as giving complex curves for a variety of reasons and a group by group classification of all these enzymes has been carried out listing all the types of variations reported and the authors' explanations. In addition, for highly complex curves, we have determined the minimum degree of the rate equation. There were very few determined attempts to demonstrate adherence to the Michaelis—Menten equation over a wide variety of experimental conditions and substrate concentration and almost invariably detailed experimental work revealed unsuspected complexities. For these reasons, it is concluded that the assumption that most enzymes follow the Michaelis—Menten equation can not be supported by an appeal to the literature.     

A steady-state kinetic method for the verification of the rapid-equilibrium assumption in allosteric enzymes.

A method for testing the validity of the rapid-equilibrium assumption as it might apply to allosteric enzymes using exclusively steady-state kinetic data is presented. The method is based upon a recognition that the ratio of apparent dissociation constants for the allosteric ligand, obtained under conditions of limiting and saturating substrate concentration, must yield the thermodynamic value for the coupling parameter between the substrate and allosteric ligand even in the general steady-state case. If this value is found to be equal to the apparent coupling parameter determined from the ratio of limiting values of the Michaelis constant for substrate obtained in the absence and saturating presence of the allosteric ligand, then the substrate can be correctly viewed as effectively achieving a binding equilibrium with the enzyme in the steady-state. The utility and limitations of this method are demonstrated by examining the ADP activation of beef heart mitochondrial NAD-dependent isocitrate dehydrogenase.     

Internal thermodynamics of enzymes determined by equilibrium quench: values of Kint for enolase and creatine kinase

The equilibrium constant (Kint) for the enzyme-bound substrate and product of a one substrate/one product enzyme (enolase) and for those of a two substrate/two product enzyme (creatine kinase) have been determined. The values of Kint were determined by the rapid quenching of equilibrium mixtures of enzyme and radiolabeled substrate and product, under conditions where all of the marker substrate and product are bound. The scope and limitations of this method are discussed. Values of Kint have been collected from the literature, and it is shown that these data are consistent with the theory for kinetically optimized enzymes that is developed in the preceding paper.     

Near-critical phenomena in intracellular metabolite pools

The supply and consumption of metabolites in living cells are catalyzed by enzymes. Here we consider two of the simplest schemes where one substrate is eliminated through Michaelis-Menten kinetics, and where two types of substrates are joined together by an enzyme. It is demonstrated how steady-state substrate concentrations can change ultrasensitively in response to changes in their supply rates and how this is coupled to slow relaxation back to steady state after a perturbation. In the one-substrate system, such near-critical behavior occurs when the supply rate approaches the maximal elimination rate, and in the two-substrate system it occurs when the rates of substrate supply are almost balanced. As systems that operate near criticality tend to display large random fluctuations, we also carried out a stochastic analysis using analytical approximations of master equations and compared the results with molecular-level Monte Carlo simulations. It was found that the significance of random fluctuations was directly coupled to the steady-state sensitivity and that the two substrates can fluctuate greatly because they are anticorrelated in such a way that the product formation rate displays only small variation. Basic relations are highlighted and biological implications are discussed.     

Mechanisms for the facilitated diffusion of substrates across cell membranes

Two classes of theoretical mechanisms for protein-mediated, passive, transmembrane substrate transport (facilitated diffusion) are compared. The simple carrier describes a carrier protein that exposes substrate influx and efflux sites alternately but never both sites simultaneously. Two-site models for substrate transport describe carrier proteins containing influx and efflux sites simultaneously. Velocity equations describing transport by these mechanisms are derived. These equations take the same general form, being characterized by five experimental constants. Simple carrier-mediated transport is restricted to hyperbolic kinetics under all conditions. Two-site carrier-mediated transport may deviate from hyperbolic kinetics only under equilibrium exchange conditions. When both simple- and two-site carriers display hyperbolic kinetics under equilibrium exchange conditions, these models are indistinguishable by using steady-state transport data alone. Seven sugar transport systems are analyzed. Five of these systems are consistent with both models for sugar transport. Uridine, leucine, and cAMP transport by human red cells are consistent with both simple- and two-site models for transport. Human erythrocyte sugar transport can be modeled by simple- and two-site carrier mechanisms, allowing for compartmentalization of intracellular sugars. In this instance, resolution of the intrinsic properties of the human red cell sugar carrier at 20 degrees C requires the use of submillisecond transport measurements.     

Structure-function correlation of fatty acyl-CoA dehydrogenase and fatty acyl-CoA oxidase

We have employed a new pseudosubstrate, beta-(2-furyl)propionyl coenzyme A (FPCoA), to study the functional properties of two enzymes, fatty acyl-CoA dehydrogenase from porcine liver and fatty acyl-CoA oxidase from Candida tropicalis, involved in the oxidation of fatty acids. Previous studies from our laboratory have shown that the dehydrogenase exhibits oxidase activity at the rate of dissociation of the product charge-transfer complex. This raises the question of the difference in functionality between these two flavoproteins. To investigate these differences, we have compared the pH dependence of product formation, the isotope effects using tetradeuterio-FPCoA, and the spectral properties and chemical reactivity of the product charge-transfer complexes formed with the two enzymes. The pH dependencies of the reaction of FPCoA with electron-transfer flavoprotein (ETF) for the dehydrogenase and of the reaction of FPCoA with O2 for the oxidase are quite similar. Both reactions proceed more rapidly at basic pH values while substrate binds more tightly at acidic pH values. These data for both enzymes are consistent with a mechanism in which enzyme is involved in protonation of the carbonyl group of substrate followed by base-catalyzed removal of the C-2 proton from substrate. The C-2 anion of substrate may then serve as the active species in reduction of enzyme-bound flavin. The deuterium isotope effects for both enzyme systems are primary across the entire pH range, assuring that the chemically important step of substrate oxidation is rate limiting in these steady-state kinetic experiments. The two enzymes differ in the chemical reactivity of their product charge-transfer complexes.(ABSTRACT TRUNCATED AT 250 WORDS)     

Steady-state kinetic studies of the negative co-operativity and flip-flop mechanism for Escherichia coli alkaline phosphatase.

1. A study of variations in experimental error of velocity measurement with substrate concentration for alkaline phosphatase reveals that the standard error is not constant or strictly proportional to velocity, but obeys a more complex dependence. 2. By using an approach based on error estimates at each individual substrate concentration, we show that the double-reciprocal plots in general are curved, necessitating a high-degree rate equation. The curves are analysed according to a recent classification of possible curve shapes for the 3:3 function, which is shown to be the lowest-degree rate equation satisfying the experimental data. 4. Other workers have supposed the enzyme to follow Michaelis-Menten kinetics, and it is shown that this assumption is approximately true at low temperatures in the absence of phosphate. 5. A study of the effects of phosphate concentration, pH and temperature on the kinetics shows that there is a gradual alteration in curve shape with these experimental variables, resulting in an apparent reduction in degree under certain special conditions, and particularly at low temperature. 6. It is shown that the steady-state kinetics do not require a flip-flop or half-of-sites reactivity mechanism as claimed, and a mechanism is proposed, a rate equation calculated and an analysis attempted. 7. An analysis of the product-inhibition effects for a linked two-sited Uni Bi enzyme is given. Alterations of asymptotic double-reciprocal slopes and limiting (1/nu) intercepts with products is discussed, and it is shown how the theory of product inhibition can be extended to complex kinetic situations to extract information as to molecular mechanism. 8. Deviations from Michaelis-Menten kinetics are expressed in terms of the magnitude of the appropriate Sylvester resultants.     

Purification and kinetic analysis of the two recombinant arogenate dehydrogenase isoforms of Arabidopsis thaliana.

The present study reports the first purification and kinetic characterization of two plant arogenate dehydrogenases (EC 1.3.1.43), an enzyme that catalyses the oxidative decarboxylation of arogenate into tyrosine in presence of NADP. The two Arabidopsis thaliana arogenate dehydrogenases TyrAAT1 and TyrAAT2 were overproduced in Escherichia coli and purified to homogeneity. Biochemical comparison of the two forms revealed that at low substrate concentration TyrAAT1 is four times more efficient in catalyzing the arogenate dehydrogenase reaction than TyrAAT2. Moreover, TyrAAT2 presents a weak prephenate dehydrogenase activity whereas TyrAAT1 does not. The mechanism of the dehydrogenase reaction catalyzed by these two forms has been investigated using steady-state kinetics. For both enzymes, steady-state velocity patterns are consistent with a rapid equilibrium, random mechanism in which two dead-end complexes, E-NADPH-arogenate and E-NADP-tyrosine, are formed.     

Kinetics of diffusion-influenced multisite phosphorylation with enzyme reactivation

The simplest way to account for the influence of diffusion on the kinetics of multisite phosphorylation is to modify the rate constants in the conventional rate equations of chemical kinetics. We have previously shown that this is not enough and new transitions between the reactants must also be introduced. Here we extend our results by considering enzymes that are inactive after modifying the substrate and need time to become active again. This generalization leads to a surprising result. The introduction of enzyme reactivation results in a diffusion-modified kinetic scheme with a new transition that has a negative rate constant. The reason for this is that mapping non-Markovian rate equations onto Markovian ones with time-independent rate constants is not a good approximation at short times. We then developed a non-Markovian theory that involves memory kernels instead of rate constants. This theory is now valid at short times, but is more challenging to use. As an example, the diffusion-modified kinetic scheme with new connections was used to calculate kinetics of double phosphorylation and steady-state response in a phosphorylation-dephosphorylation cycle. We have reproduced the loss of bistability in the phosphorylation-dephosphorylation cycle when the enzyme reactivation time decreases, which was obtained by particle-based computer simulations.     

The accuracy of the rapid equilibrium assumption in steady-state enzyme kinetics in the case of a multipath arbitrary enzyme mechanism with a number of equilibrium segments

A quantitative evaluation of the accuracy of the rapid-equilibrium assumption in steady-state enzyme kinetics was obtained for a multipath arbitrary enzyme mechanism with a number of equilibrium segments. Explicit expressions for estimating the contribution of any equilibrium segment to the accuracy of the rapid-equilibrium assumption were obtained. This allowed us to determine the accuracy of the rapid-equilibrium assumption (Δ) in general: 1 + Δ = (1 + Δ₁)(1 + Δ₂)... (1 + Δ_k), where Δ₁, Δ₂,..., Δ_k is the contribution of each individual equilibrium segment. The accuracy depends only on the structure and properties of equilibrium segments, which have been accounted for in the rapid-equilibrium assumption, but it is independent of the number of paths in the mechanism of the enzymatic reaction and on the structure and properties of the remaining part (steady-state) of the kinetic scheme.     

A generalized theory of the transition time for sequential enzyme reactions.

In a sequence of coupled enzyme reactions the steady-state production of product is preceded by a lag period or transition time during which the intermediates of the sequence are accumulating. Provided that a steady state is eventually reached, the magnitude of this lag may be calculated, even when the differentiation equations describing the process have no analytical solution. The calculation may be made for simple systems in which the enzymes obey Michaelis-Menten kinetics or for more complex pathways in which intermediates act as modifiers of the enzymes. The transition time associated with each intermediate in the sequence is given by the ratio of the appropriate steady-state intermediate concentration to the steady-state flux. The theory is also applicable to the transition between steady states produced by flux changes. Application of the theory to coupled enzyme assays allows a definition of the minimum requirements for successful operation of the assay. The theory can be extended to deal with sequences in which the enzyme concentration exceeds substrate concentration.     

Theoretical models of DNA topology simplification by type IIA DNA topoisomerases

It was discovered 12 years ago that type IIA topoisomerases can simplify DNA topology—the steady-state fractions of knots and links created by the enzymes are many times lower than the corresponding equilibrium fractions. Though this property of the enzymes made clear biological sense, it was not clear how small enzymes could selectively change the topology of very large DNA molecules, since topology is a global property and cannot be determined by a local DNA–protein interaction. A few models, suggested to explain the phenomenon, are analyzed in this review. We also consider experimental data that both support and contravene these models.     

A 13C nuclear magnetic resonance study of CO2/HCO-3 exchange catalyzed by human carbonic anhydrase I

Rates of CO2/HCO-3 exchange, catalyzed by human carbonic anhydrase I (or B) at chemical equilibrium, were estimated from the nuclear magnetic resonance linewidths of 13C-labeled substrates. The results show that the maximal exchange rate constant is independent of pH in the range 5.7-8.0, whereas the apparent substrate dissociation constant depends on pH. Exchange proceeds rapidly in the absence of added buffers, and the addition of buffers has negligible effects on exchange rates. Exchange is equally rapid with 1H2O or 2H2O as solvents. Chloride ions inhibit CO2/HCO-3 exchange competitively. The maximal exchange rates obtained with human carbonic anhydrase I are 50 times slower than those obtained with human isoenzyme II (or C). From a comparison of the exchange kinetics with the steady-state kinetics of CO2 hydration and HCO-3 dehydration it is tentatively concluded that the transfer of H+ between active site and medium proceeds with rates of similar magnitudes in the two isoenzymes, whereas the central catalytic step, the interconversion of enzyme-bound CO2 and HCO-3, is much slower in isoenzyme I than in isoenzyme II.     

Partition kinetics of ribulose-1,5-bisphosphate carboxylase from Rhodospirillum rubrum

When the enzymatically generated intermediate 2-carboxy-3-keto-D-arabinitol-1,5-bisphosphate (II) was used as a substrate with fresh enzyme, 70% reacted to produce 3-phosphoglycerate (3PGA). When a reaction mixture of enzyme plus [1-32P]ribulose 1,5-bisphosphate (RuBP) was quenched in the steady state with the tightly bound inhibitor 2-carboxyarabinitol-1,5-bisphosphate, 30% of the enzyme-bound species was released as 3PGA and 70% as RuBP. The major source for this partition was the ternary substrates Michaelis complex. The level of carboxylated intermediate in the steady state was determined to be 8% of active sites under the conditions of substrate saturation. No burst was seen in the appearance of product when 6.5 eq of [1-32P]RuBP was mixed with enzyme plus saturating CO2 and the reaction followed in the steady state. From these data plus the steady-state Vmax and Km of RuBP it is possible to derive the five bulk rate constants represented in the scheme ECO2 + RuBP in equilibrium ERuBPCO2 in equilibrium E X II----E + 2(3PGA).     

Evaluation of steady-state kinetic parameters for enzymes solubilized in water-in-oil microemulsion systems.

1. Equations are derived for the steady-state kinetics of substrate conversion by enzymes confined within the water-droplets of water-in-oil microemulsion systems. 2. Water-soluble substrates initially confined within droplets that do not contain enzyme are assumed to be converted into product only after they enter enzyme-containing droplets via the inter-droplet exchange process. 3. Hyperbolic (Michaelis-Menten) kinetics are predicted when the substrate concentration is varied in microemulsions of fixed composition. Both kcat. and Km are predicted to be dependent on the size and concentration of the water-droplets in the microemulsion. 4. The predicted behaviour is shown to be supported by published experimental data. A physical interpretation of the form of the rate equation is presented. 5. The rate equation for an oil-soluble substrate was derived assuming a pseudo-two-phase (oil & water) model for the microemulsion. Both kcat. and Km are shown to be independent of phi aq. Km is larger than the aqueous solution value by a factor approximately equal to the oil/water partition coefficient of the substrate. The validity of the rate equation is confirmed by published data.     

Subunit interactions in enzyme transition states--antagonism between substrate binding and reaction rate

The principles of structural kinetics as applied to polymeric enzymes have been reinvestigated in order to take account of the probable existence of subunit interactions in the enzyme transition states. On the basis of simple and plausible postulates, structural rate equations have been derived for dimeric enzymes and compared to substrate binding isotherms. It then becomes possible to understand how subunit interactions affect substrate affinity and enzyme reaction rate. There exists an antagonism between substrate binding to the enzyme and the steady state rate of product appearance. If subunit interactions increase the rate of product appearance, they decrease the fractional saturation of the enzyme by the substrate. Alternatively, if they decrease the reaction velocity they increase the fractional saturation. This seemingly paradoxical effect is the direct consequence of subunit interactions occurring in both the ground and the transition states.     

Unusual transient- and steady-state kinetic behavior is predicted by the kinetic scheme operational for recombinant human dihydrofolate reductase

Association and dissociation rate constants obtained by stopped-flow spectroscopy have permitted definition of a kinetic scheme for recombinant human dihydrofolate reductase that correctly predicts full time course kinetics of the enzymatic reaction over a wide range of substrate and product concentrations. The scheme is complex compared with that for the bacterial enzyme and involves branched pathways. It successfully accounts for observed rapid hysteresis preceding steady state and for the nonhyperbolic dependence of steady-state rate on substrate and product concentrations. The major branch point in the catalytic cycle occurs at E.NADP.H4folate because either NADP or H4folate can dissociate from the ternary product complex (koff = 84 s-1 and 46 s-1, respectively). The rate of conversion of enzyme-bound substrates to products is very fast (k = 1360 s-1) and nearly unidirectional (Kequ = 37) so that other steps limit the catalytic rate. At saturating substrate concentrations these steps include release of NADP and H4folate from E.NADP.H4folate and release of products from the two abortive complexes E.NADPH.H4folate (koff = 225 s-1) and E.NADP.H4folate (koff = 4.6 s-1). Since NADP dissociates slowly from E.NADP.H2folate nearly 90% of the enzyme accumulates as this complex at steady state. Nonetheless, the catalytic rate is maintained at 12 s-1 by rapid flux of a small portion of the enzyme through an alternate branch. At physiological concentrations of substrates and products the steady-state rate is limited primarily by the rate of H2folate binding to E.NADPH so that the enzyme is extremely efficient.     

Distributions of Topological States in Circular DNA

A distinctive feature of closed circular DNA molecules is their particular topological state, which cannot be altered by any conformational rearrangement short of breaking at least one strand. This topological constraint opens unique possibilities for experimental studies of the distributions of topological states created in different ways. Primarily, the equilibrium distributions of topological properties are considered in the review. It is described how such distributions can be obtained and measured experimentally, and how they can be computed. Comparison of the calculated and measured equilibrium distributions over the linking number of complementary strands, equilibrium fractions of knots and links formed by circular molecules has provided much valuable information about the properties of the double helix. Study of the steady-state fraction of knots and links created by type II DNA topoisomerases has revealed a surprising property of the enzymes: their ability to reduce these fractions considerably below the equilibrium level.