On true and apparent Michaelis constants in enzymology. III. Is it linear dependence between apparent michaelis constant and limiting rate and is it possible to determine the substrate constant value using this dependence?

The Slater-Bonner method which is used for graphic determination of substrate constant (Ks) by linear dependence of apparent Michaelis constant (Km(app)) on the limiting rate (V(app)) of enzyme-catalysed reactions with activator participation has been critically analysed. It has been shown that although it is possible to record the mechanisms of such reactions as a scheme similar to Michaelis-Menten model which allow to find correlation Km(app) and V(app) as equation Km(app) = Ks + V(app)/k1[E]0 ([E]0 is a total enzyme concentration, k1 is a rate constant of enzyme-substrate complex formation from free enzyme and substrate) in order to calculate Ks and individual rate constants (k1, k(-1)), but this approach for investigation of all reactions with activator participation ought not to be used. The above equation is not obeyed in general, it may be true for some mechanisms only or under certain ratios of kinetic parameters of enzyme-catalysed reactions.     

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.     

A CSTR-hollow fiber system for continuous hydrolysis of proteins. Performance and kinetics

The effect of four operating variables (enzyme concentration, substrate concentration, flow rate, and reaction volume) on the performance of CSTR-hollow fiber membrane reactor was studied for the continuous hydrolysis of a soy protein isolate using Pronase. Based on a residence time distribution study, the reactor system was modeled as an ideal CSTR in combination with the Michaelis-Menten equation of enzyme kinetics. This kinetic model correlated conversion with a space-time parameter modified to include all four independent variables. An empirical model based on curvilinear regression analysis was also developed. Both models predicted conversion fairly well, although the kinetic model slightly underpredicts at high conversion.     

The computation of hyperbolic dependences in enzyme kinetics.

A computer program aimed at analysing results following Michaelis-Menten kinetics can be used unmodified in the treatment of other kinetic results provided that the kinetic equations in these cases can be written in the form of the Michaelis-Menten equation. A list is presented of the parameters to be set instead of substrate concentration and reaction rate, and of constants replacing Km and V, if such a program is applied in analysing enzyme inhibitions, activations and pH-dependences.     

A semi-integrated method for the determination of enzyme kinetic parameters and graphical representation of the Michaelis-Menten equation

A semi-integrated method for the determination of the enzyme kinetics parameters (Km and V) and graphical representation of the Michaelis-Menten equation is proposed as a variation of determination of initial reaction rate (v) as a function of initial substrate concentration ([S]0). The method is based on the determination of the time required to exhaust half of the initial substrate concentration as a function of the initial substrate concentration. The advantages and limitations of this method are discussed.     

A novel thermodynamic relationship based on Kramers Theory for studying enzyme kinetics under high viscosity

In most studies of enzyme kinetics it has been found sufficient to use the classical Transition State Theory (TST) of Eyring and others. This theory was based on the solvent being an ideal dilute substance treated as a heat bath. However, enzymes found in organisms adapted to very low (psychrophiles) and very high (thermophiles) temperatures are also subjected to variable solute concentrations and viscosities. Therefore, the TST may not always be applicable to enzyme reactions carried out in various solvents with viscosities ranging from moderate to very high. There have been numerous advances in the theory of chemical reactions in realistic non-ideal solvents such as Kramers Theory. In this paper we wish to propose a modified thermodynamic equation, which have contributions from kcat, Km and the viscosity of the medium in which the enzyme reaction is occurring. These could be very useful for determining the thermodynamics of enzymes catalyzing reactions at temperature extremes in the presence of substrate solutions of different compositions and viscosities.     

Activation of plasminogen by pro-urokinase. II. Kinetics

The kinetics of the activation of plasminogen by recombinant pro-urokinase obtained by expression of human urokinase cDNA in Escherichia coli was studied. The conversion of pro-urokinase (U) and plasminogen (P) to urokinase (u) and plasmin (p) is represented by a sequence of three reactions which each obey Michaelis-Menten kinetics, i.e. (Formula: see text). In this model, pro-urokinase formally behaves as an enzyme in Reaction I and as a substrate in reaction II. The experimentally measured overall rates of formation of urokinase and plasmin are in good agreement with those calculated from the kinetic parameters and the initial concentrations of pro-urokinase and plasminogen, confirming the validity of the model. It appears that recombinant pro-urokinase is an equally potent activator of plasminogen (k2/Km = 0.05 microM-1 s-1), as in urokinase (k"2/K"m = 0.02 microM-1 s-1). This is due to the fact that the proenzyme, which is virtually inactive toward low Mr substrates for urokinase, forms an intermediate of the Michaelis-Menten type with plasminogen, with a much higher affinity than that of the active enzyme with its substrate. This is an exceptional phenomenon among the serine proteases.     

A cellular automata model of enzyme kinetics

We have developed a cellular automata model of an enzyme reaction with a substrate in water. The model produces Michaelis-Menten kinetics with good Lineweaver-Burk plots. The variation in affinity parameters predicts that, in general, hydrophobic substrates are more reactive with enzymes, this attribute being more important than the relationship between enzyme and substrate. The ease of generation and the illustrative value of the model lead us to believe that cellular automata models have a useful role in the study of dynamic phenomena such as enzyme kinetics.     

Optimal design for CSTR's in series using reversible Michaelis-Menten reactions

An analytical expression is derived for the optimal design of a series of CSTR's performing reversible Michaelis-Menten kinetics in the liquid phase. The optimal design is based on minimum overall volume ofN reactors in series required to achieve a certain degree of substrate conversion. The reversible Michaelis-Menten equation is also able to explain competitive product inhibition and irreversible Michaelis-Menten kinetics. The reversible Michaelis-Menten kinetics covers three types of enzymatic reactions depending on the values of the rate constant for the forward (k _s) and reverse (k _p) reactions. An optimum design is obtained in the three cases ofK _s=K_p, K_s>K_p andK _s<K_p. The minimum overall reactors volume is compared with the more convenient equal-sized CSTR's. The effect of enzyme deactivation on the minimum overall reactors volume is investigated. The performance of a series of CSTR's is compared with plug-flow reactor. Glucose isomerization which exhibits reversible Michaelis-Menten kinetics is used as a model system for optimization.     

Rapid determination of kinetic constants by competitive spectrophotometry

The use of competitive spectrophotometry to measure kinetic constants for enzyme-catalyzed reactions is described. The equation for the progress curve characterizing the kinetic behavior of an enzyme acting simultaneously on two alternative substrates is derived. By the addition of a competition term to the integrated Michaelis-Menten equation, the kinetic constants of an alternative substrate can be evaluated by measuring the competition with a substrate of known kinetic constants in a single experiment. Studies are presented involving the enzymes leucine aminopeptidase (LAP) and carboxypeptidase A (CPA). The results obtained with LAP and CPA showed that the kinetic constants determined using competitive spectrophotometry were in agreement with values cited in the literature or with values determined by single substrate enzyme kinetics.     

External and internal diffusion in heterogeneous enzymes systems

The intrusion of diffusion in heterogeneous enzyme reactions, which follow. Michaelis-Menten kinetics, is quantitatively characterized by dimensionless parameters that are independent of the substrate concentration. The effects of these parameters on the overall rate of reaction is illustrated on plots commonly employed in enzyme kinetics. The departure from Michaelis-Menten kinetics due to diffusion limitations can be best assessed by using Hofstee plots which are also suitable to distinguish between internal and external transport effects. A graphical method is described for the evaluation of the reaction rate as a function of the surface concentration of the substrate from measured data.     

A Model for Single-Substrate Trimolecular Enzymatic Kinetics

We developed a kinetic model for a single-substrate trimolecular enzymatic system, where a receptor binds and stretches a substrate to expose its cleavage site, allowing an enzyme to bind and cleave it into product. We demonstrated that the general kinetics of the trimolecular enzymatic system is more complex than the Michaelis-Menten kinetics. Under a limiting condition when the enzyme-substrate binding is in fast equilibrium, the enzymatic kinetics of the trimolecular system reduces to the Michaelis-Menten kinetics. In another limiting case when the receptor dissociates negligibly slowly from the substrate, the trimolecular system is simplified to a bimolecular system, which follows the Michaelis-Menten equation if and only if there is no enzyme-substrate complex initially. We applied this model to a particular trimolecular system important to hemostasis and thrombosis, consisting of von Willebrand factor (substrate), platelet glycoprotein Ibα (receptor), and ADAMTS13 (enzyme). Using parameters from independent experiments, our model successfully predicted published data from two single-molecule experiments and fitted/predicted published data from an ensemble experiment.     

Dynamic Disorder in Quasi-Equilibrium Enzymatic Systems

Conformations and catalytic rates of enzymes fluctuate over a wide range of timescales. Despite these fluctuations, there exist some limiting cases in which the enzymatic catalytic rate follows the macroscopic rate equation such as the Michaelis-Menten law. In this paper we investigate the applicability of macroscopic rate laws for fluctuating enzyme systems in which catalytic transitions are slower than ligand binding-dissociation reactions. In this quasi-equilibrium limit, for an arbitrary reaction scheme we show that the catalytic rate has the same dependence on ligand concentrations as obtained from mass-action kinetics even in the presence of slow conformational fluctuations. These results indicate that the timescale of conformational dynamics – no matter how slow – will not affect the enzymatic rate in quasi-equilibrium limit. Our numerical results for two enzyme-catalyzed reaction schemes involving multiple substrates and inhibitors further support our general theory.     

Computer analysis of enzyme-substrate-inhibitor kinetic data with automatic model selection using IBM-PC compatible microcomputers

A weighted nonlinear least-squares curve-fitting program, implemented in compiled BASIC for the IBM-PC is described to estimate the parameters of enzyme kinetics obeying Michaelis-Menten kinetics and seven inhibition models. The effects of the inhibitor on the maximal velocity (Vm) and the Michaelis-Menten constant (Km) are used to select automatically the most plausible model of inhibition and to calculate initial estimates of parameters. The program is used to demonstrate that the inhibition of carbamyl-phenylalanine hydrolase by the product phenylalanine is consistent with the pure mixed noncompetitive model.     

Generalized rate equation for single-substrate enzyme catalyzed reactions

The most widely used rate expression for single-substrate enzyme catalyzed reactions, namely the Michaelis-Menten kinetics is based upon the assumption that enzyme concentration is in excess of the substrate in the medium and the rate is mainly limited by the substrate concentration according to saturation kinetics. However, this is only a special case and the actual rate expression varies depending on the initial enzyme/substrate ratio (E₀/S₀). When the substrate concentration exceeds the enzyme concentration the limitation is due to low enzyme concentration and the rate increases with the enzyme concentration according to saturation kinetics. The maximum rate is obtained when the initial concentrations of the enzyme and the substrate are equal. A generalized rate equation was developed in this study and special cases were discussed for enzyme catalyzed reactions.     

Study of microencapsulated urease in a continuous feed,stirred tank reactor

Urease, (urea amidohydrolase, EC 3.5.1.5) co-encapsulated with haemoglobin in cellulose nitrate membranes was found to exhibit apparent Michaelis-Menten kinetics; however, a steadily increasing apparent Michaelis-Menten constant over the lifetime of the preparation was observed. The activity of the enzyme in a continuous feed stirred tank reactor (CSTR) was investigated and correlated with a mathematical model derived from basic Michaelis-Menten kinetics. Plots relating substrate conversion to feed substrate concentration and tank reactor capacity were constructed and found to be accurate to less than 15% error under the experimental conditions studied.     

Kinetic analysis of the Ca2+-dependent, membrane-bound, macrophage phospholipase A2 and the effects of arachidonic acid

The kinetics of the Ca2+-dependent, alkaline pH optimum, membrane-bound phospholipase A2 from the P388D1 macrophage-like cell line were studied using various phosphatidylcholine (PC) and phosphatidylethanolamine (PE) substrates. This enzyme exhibits "surface dilution kinetics" toward PC in Triton X-100 mixed micelles, and the "dual phospholipid model" was found to adequately describe its kinetic behavior. With substrate in the form of sonicated vesicles, the dual phospholipid model should give rise to Michaelis-Menten type kinetics. However, the hydrolysis of dipalmitoyl-PC, 1-palmitoyl-2-oleoyl-PC, and 1-stearoyl-2-arachidonoyl-PC vesicles exhibited two distinct activities. Below 10 microM, the data appeared to follow Michaelis-Menten behavior, while at higher concentrations, the data could best be fit to a Hill equation with a Hill coefficient of 2. These PCs had Vmax values for the low substrate concentration range of 0.2-0.6 nmol min-1 mg-1 and Km values of 1-2 microM. At the high substrate concentration range, the Vmax values were between 5 and 7 nmol min-1 mg-1. PC containing unsaturated fatty acids had an apparent Km, determined from the Hill equation, of about 15 microM, while the apparent Km of dipalmitoyl-PC was 0.6 microM. When 70% glycerol was included in the assays, a single Michaelis-Menten curve was obtained for both dipalmitoyl-PC and 1-stearoyl,2-arachidonoyl-PC. Possible explanations for these kinetic results include reconstitution of the membrane-bound phospholipase A2 in the phospholipid vesicle or the enzyme has tow distinct phospholipid binding function. The kinetics for both dipalmitoyl-PC and dipalmitoyl-PE hydrolysis in vesicles was very similar, indicating that the enzyme does not greatly prefer one of these head groups over the other. The enzyme also showed no preference for arachidonoyl containing phospholipid. Enzymatic activity toward PC containing saturated fatty acids was linear to about 15% hydrolysis while the hydrolysis of PC containing unsaturated fatty acids was linear to only about 5%. This loss of linearity was due to inhibition by released unsaturated fatty acids. Arachidonic acid was found to be a competitive inhibitor of dipalmitoyl PC hydrolysis with a K1 of 5 microM. This tight binding suggests a possible in vivo regulatory role for arachidonic acid. Three compounds of the arachidonic acid cascade, prostaglandin F2 alpha, 6-keto-prostaglandin F1 alpha, and thromboxane B2, showed no inhibition of enzymatic activity.     

Exponential function of chymotrypsin action

Enzyme kinetics are usually described by the hyperbolic Michaelis-Menten equation, but they can also be described by the following exponential function: -dS/dt = Vm [1 - exp (-S/Km)]. The time-dependent decrease of the substrate (-dS/dt) is an exponential function of maximal velocity (Vm), the Michaelis constant (Km) and the actual substrate value (S). This exponential function is based on the assumption that the association of the substrate-enzyme complex is a concentration-dependent process, whereas the transformation of the substrate-enzyme complex is time-dependent. It can be shown that this exponential function is a more general solution of which the hyperbolic Michaelis-Menten equation is a special derivative under the conditions of low substrate (S) and high constant (Km) values. If the association process is time-dependent, the decline in substrate values will show a more concave curve. However, exponential functions in general are more concave than hyperbolic functions. Probably, therefore, the enzyme action of chymotrypsin could be described more appropriately by the present exponential function than by the conventional hyperbolic function.