Theory of the kinetics of reactions catalyzed by enzymes attached to the interior surfaces of tubes

A theoretical treatment is given of the kinetics of reactions catalyzed by enzymes attached to the inner surface of a tube, through which the substrate solution passes. A utilization factor, the ratio of the actual reaction rate to that in the absence of diffusional effects, is defined. A numerical procedure is proposed and numerical and approximate solutions for the utilization factor are given for five kinetic conditions: (a) Michaelis-Menten behavior, (b) substrate inhibition, (c) product inhibition (competitive), (d) product, inhibition (non-competitive), and (e) product inhibition (anticompetitive). When the enzyme chemically attached to a tube obeys a Michaelis-Menten relationship, criteria for insignificant and significant diffusional effects are proposed.     

Transient response of encapsulated enzymes in hollow-fiber reactor

A mathematical model for the transient response of encapsulated enzymes is developed showing the effects of the outer boundary layer, the encapsulating membrane, the partition coefficient, and diffusion with reaction within the encapsulating medium. The model incorporates both first-order kinetics and Michaelis-Menten kinetics for the reaction rate. Using typical hollow-fiber or microcapsule parameters, the model shows that (a) the partition coefficient affects the overall rate only when the rate-limiting step is diffusion through the membrane, (b) the transient overall effectiveness factor rises sharply with time and approaches an asymptotic value for most situations, and (c) the first-order approximation to Michaelis-Menten kinetics is not valid when the initial outside bulk concentration is higher than the Michaelis constant and the overall rate is reaction limited. The model is compared with experimental data using uricase in a hollow-fiber enzyme reactor configuration. Batch assay and CSTUER (continuous-stirred ultrafiltration enzyme reactor) studies were conducted on the free enzyme to provide some of the parameters used in the model. The CSTUER data fit the case of substrate inhibition kinetics with the apparent Michaelis constant approaching zero. The hollow-fiber reactor was conducted with uricase dissolved in both a buffer solution and a concentrated hemoglobin solution. Diffusivities of the solute were measured in both solutions as was the osmotic pressure of the hemoglobin solution. While experimental data for uricase in buffer solution could easily be matched by the model, that in the concentrated hemoglobin solution could not.     

Importance of product inhibition in the kinetics of the acylase hydrolysis reaction by differential stopped flow microcalorimetry

The hydrolysis of N-acetyl-L-methionine, N-acetylglycine, N-acetyl-L-phenylalanine, and N-acetyl-L-alanine at 298.35K by porcine kidney acylase I (EC 3.5.1.14) was monitored by the heat released upon mixing of the substrate and enzyme in a differential stopped flow microcalorimeter. Values for the Michaelis constant (K(m)) and the catalytic constant (k(cat)) were determined from the progress of the reaction curve employing the integrated form of the Michaelis-Menten equation for each reaction mixture. When neglecting acetate product inhibition of the acylase, values for k(cat) were up to a factor of 2.3 larger than those values determined from reciprocal initial velocity-initial substrate concentration plots for at least four different reaction mixtures. In addition, values for K(m) were observed to increase linearly with an increase in the initial substrate concentration. When an acetate product inhibition constant of 600+/-31M(-1), determined by isothermal titration calorimetry, was used in the progress curve analysis, values for K(m) and k(cat) were in closer agreement with their values determined from the reciprocal initial velocity versus initial substrate concentration plots. The reaction enthalpies, Delta(r)H(cal), which were determined from the integrated heat pulse per amount of substrate in the reaction mixture, ranged from -4.69+/-0.09kJmol(-1) for N-acetyl-L-phenylalanine to -1.87+/-0.23kJmol(-1) for N-acetyl-L-methionine.     

Optimum design of a series of CSTRs performing reversible Michaelis-Menten kinetics: a rigorous mathematical study

The optimum design of a given number of CSTRs in series performing reversible Michaelis-Menten kinetics in the liquid phase assuming constant activity of the enzyme is studied. In this study, the presence of product in the feed stream to the first reactor, as well as the effect of the product intermediate concentrations in the downstream reactors on the reaction rate are investigated. For a given number of N CSTRs required to perform a certain degree of substrate conversion and under steady state operation and constant volumetric flow rate, the reactor optimization problem is posed as a constrained nonlinear programming problem (NLP). The reactor optimization is based on the minimum overall residence time (volume) of N reactors in series. When all the reactors in series operate isothermally, the constrained NLP is solved as an unconstrained NLP. And an analytical expression for the optimum overall residence time is obtained. Also, the necessary and sufficient conditions for the minimum overall residence time of N CSTRs are derived analytically. In the presence of product in the feed stream, the reversible Michaelis-Menten kinetics shows competitive product inhibition. And this is, because of the increase in the apparent rate constant K^' _m that results in a reduction of the overall reaction rate. The optimum total residence time is found to increase as the ratio (‚₀) of product to substrate concentrations in the feed stream increases. The isomerization of glucose to fructose, which follows a reversible Michaelis-Menten kinetics, is chosen as a model for the numerical examples.     

Kinetics, mechanism, and time course analysis of lipase-catalyzed hydrolysis of high concentration olive oil in AOT-isooctane reversed micelles

Candida rugosa lipase has been used to investigate the hydrolysis of high concentration olive oil in the AOT-isooctane reversed micellar system at W(o) = 10, pH 7.1, and 37 degrees C. Results from this work show the hydrolytic reaction obeys Michaelis-Menten kinetics up to the initial substrate concentration of 1.37M, with turnover number k(cat) and Michaelis constant K(M) of 67.1 mumol/min mg enzyme and 0.717M, respectively. A competitive inhibition by the main product, oleic acid, has been found with a dissociation constant K(I) for the complex EP* of 0.089M. The rate equation was further analyzed in the time course reaction and was found in agreement with the experimental results for lower substrate concentrations, up to 0.341M. Large deviation occurred at high substrate concentrations, which may be due to the effects of large consumption of water on kinetics, on the formation of glycerol, and on the deactivation of lipase in the hydrolysis reaction as well.     

On the use of apparent kinetic parameters for immobilized enzyme with noncompetitive substrate inhibition

The use of a simple rate equation with apparent parameters to describe the kinetic behavior of an immobilized enzyme with noncompetitive substrate inhibition was assessed. To do so, the reaction rate was calculated as a function of the interfacial substrate concentration, and the results were used to identify the apparent kinetic parameters by nonlinear regression. This procedure was repeated for different values of the diffusional constraints and of the inhibition constant. The equation using apparent parameters can describe the global kinetic behavior, provided that the diffusional and inhibitory constraints are not too high. When the constraints are high, a Michaelis-Menten equation can be used to model the kinetics for interfacial concentrations lower than the concentration leading to the maximum reaction rate.     

Nucleotide regulation of Escherichia coli glycerol kinase: initial-velocity and substrate binding studies

Substrate binding to Escherichia coli glycerol kinase (EC 2.7.1.30; ATP-glycerol 3-phosphotransferase) was investigated by using both kinetics and binding methods. Initial-velocity studies in both reaction directions show a sequential kinetic mechanism with apparent substrate activation by ATP and substrate inhibition by ADP. In addition, the Michaelis constants differ greatly from the substrate dissociation constants. Results of product inhibition studies and dead-end inhibition studies using 5'-adenylyl imidodiphosphate show the enzyme has a random kinetic mechanism, which is consistent with the observed formation of binary complexes with all the substrates and the glycerol-independent MgATPase activity of the enzyme. Dissociation constants for substrate binding determined by using ligand protection from inactivation by N-ethylmaleimide agree with those estimated from the initial-velocity studies. Determinations of substrate binding stoichiometry by equilibrium dialysis show half-of-the-sites binding for ATP, ADP, and glycerol. Thus, the regulation by nucleotides does not appear to reflect binding at a separate regulatory site. The random kinetic mechanism obviates the need to postulate such a site to explain the formation of binary complexes with the nucleotides. The observed stoichiometry is consistent with a model for the nucleotide regulatory behavior in which the dimer is the enzyme form present in the assay and its subunits display different substrate binding affinities. Several properties of the enzyme are consistent with negative cooperativity as the basis for the difference in affinities. The possible physiological importance of the regulatory behavior with respect to ATP is considered.     

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.     

Estimation of intrinsic kinetic constants for pore diffusion-limited immobilized enzyme reactions

A simple method is presented that establishes intrinsic rate parameters when slow pore diffusion of substrate limits immobilized enzyme reactions that obey Michaelis-Menten kinetics. The Aris-Bischoff modulus is employed. Data at high substrate concentrations, where the enzyme would be saturated in the absence of diffusion limitation, and at low substrate concentrations, where effectiveness factors are inversely proportional to reaction modulus, are used to determine maximum rate and Michaelis constant, respectively. Because Michaelis-Menten and Langmuir-Hinshelwood kinetics are formally identical, this method may be used to estimate intrinsic rate parameters of many heterogeneous catalysts. The technique is demonstrated using experimental data from the hydrolysis of maize dextrin with diffusion-limited immobilized glucoamylase. This system yields a Michaelis constant of 0.14%, compared to 0.11% for soluble glucoamylase and 0.24% for immobilized glucoamylase free of diffusional effects.     

General theory of determining intraparticle active immobilized enzyme distribution and rate parameters

A general theory is presented in this article for determining the intrinsic rate constants for the main reaction and deactivation reaction, the effective diffusivity of the substrate, and the active enzyme distribution within porous solid supports from deactivation study of a continuous stirred-basket reactor (CSBR). For the parallel deactivation five reaction kinetics are considered: (a) Michaelis-Menten, (b) substrate inhibition, (c) product inhibition (competitive), (d) product inhibition (anticompetitive), and (e) zero-order kinetics. The experimental results of the system of hydrogen-peroxide-immobilized catalase on controlled-pore glass particles are analyzed to demonstrate the application of the theory developed for parallel deactivation of active immobilized enzyme (IME). For series deactivation only first-order kinetics is treated, and a numerical procedure is proposed to deter mine the rate parameters and the internal active enzyme distribution. The experimental data of the system of glucose-immobilized glucose oxidase on silica-alumina and controlled-pore glass particles are used to verify the theory.     

Characterization and inhibition study of MurA enzyme by capillary electrophoresis

A capillary electrophoresis-based enzyme assay for UDP-N-acetylglucosamine enolpyruvyl transferase (MurA) is described. This method, based on UV detection, provides baseline separation of one of the reaction products, enolpyruvyluridine 5'-diphospho-N-acetylglucosamine (EP-UDP-GlcNAc), from substrates phosphoenolpyruvate (PEP) and uridine 5'-diphospho-N-acetylglucosamine (UDP-GlcNAc) within 4 min. The other product, phosphate, is not detectable by UV at 200 nm. Quantitation of individual components, substrates or product, can be accomplished based on the separated peaks. This methodology was used to determine the Michaelis constant, Km, and product formation rate constant, Kcat, for MurA. Additionally, the CE method was used to evaluate the inhibition effects on MurA using one specific compound as an example. By following similar procedures, the apparent Km values in the presence of different inhibitor concentrations were determined. The inhibition constant, Ki, can be determined from these apparent Km values. In addition, this CE method can be used to study the inhibition mechanism. The principle of this approach is generally applicable to other enzyme studies.     

Kinetic analysis of enzyme systems with suicide substrate in the presence of a reversible, uncompetitive inhibitor.

We present a general kinetic analysis of enzyme catalyzed reactions evolving according to a Michaelis-Menten mechanism, in which an uncompetitive, reversible inhibitor acts. Simultaneously, enzyme inactivation is induced by an unstable suicide substrate, i.e. it is a Michaelis-Menten mechanism with double inhibition: one originating from the substrate and another originating from the reversible inhibitor. Rapid equilibrium of the reversible reaction steps involved is assumed and the time course equations for the reaction product have been derived under the assumption of limiting enzyme. The goodness of the analytical solutions has been tested by comparison with simulated curves obtained by numerical integration. A kinetic data analysis to determine the corresponding kinetic parameters from the time progress curve of the product is suggested.     

The catalase–hydrogen peroxide system. Kinetics of catalatic action at high substrate concentrations

1. A re-examination of the catalase-hydrogen peroxide reaction at high substrate concentrations, by using the quenched-flow technique, reveals a more complex kinetic behaviour than that previously reported. At constant reaction time the catalatic process obeys Michaelis-Menten kinetics, but the apparent Michaelis constant is markedly time-dependent, whereas the conventional catalase activity is independent of time. 2. The kinetics of the ;time effect' were analysed and it is suggested that the effect derives from the formation of an inactive species (thought to be catalase Compound II). The process shows Michaelis-Menten kinetics, with a Michaelis constant equal to that for the catalatic reaction in the limit of zero reaction time. 3. It has been confirmed that certain buffer components have marked inhibitory effects on the catalatic reaction and that, in unbuffered systems, catalatic activity is substantially independent of pH in the range 4.7-10.5.     

Theoretical analysis of intrinsic reaction kinetics and the behavior of immobilized enzymes system for steady-state conditions

Mathematical modeling of immobilized enzymes under different kinetics mechanism viz. simple Michaelis–Menten, uncompetitive substrate inhibition, total competitive product inhibition, total non-competitive product inhibition and reversible Michaelis–Menten reaction are discussed. These five kinetic models are based on reaction diffusion equations containing non-linear terms related to Michaelis–Menten kinetics of the enzymatic reaction. Modified Adomian decomposition method is employed to derive the general analytical expressions of substrate and product concentration for all these five mechanisms for all possible values of the parameters Φ_S (Thiele modulus for substrate), Φ_P (Thiele modulus for product) and α (dimensionless inhibition degree). Also we have presented the general analytical expressions for the mean integrated effectiveness factor for all values of parameters. Analytical results are compared with the numerical results and also with the limiting case results, which are found to be good in agreement.     

Temperature and pH dependence of mold α-galactosidase

The effect of temperature and pH on kinetic behavior of α-galactosidase of Mortierella vinacea was investigated on the hydrolysis of p-nitrophenyl-α-D -galactopyranoside (PNPG). A very unusual kinetic behavior was observed for the soluble α-galactosidase i.e., substrate inhibition diminished gradually with increasing temperature or near the neutral pH range, and the kinetics approached the ordinary Michaelis-Menten (MM) type. On the other hand, with decreasing temperature or in acidic pH range, substrate inhibition was accelerated. Therefore, Arrhenius plots based on the initial reaction rate did not give straight lines. Furthermore, the slope in the Arrhenius plot changed with substrate concentration, which would make the determination of a characteristic value using conventional methods meaningless. However, the Arrhenius plots of individual kinetic parameters in the rate equation resulted in straight lines in the temperature range 15 to 50°C. From this, the drastic change in kinetic behavior could be explained in connection with the temperature and pH dependence of kinetic parameters in the model. For mold pellets (whole-cell enzyme), however, the influence of temperature and pH was less apparent than that of soluble enzyme because of the limitation in intraparticle diffusion. By using the rate equation that was determined for soluble enzyme and the theoretically derived effectiveness factor, the overall reaction rate for mold pellets at various temperature and pH could be predicted to some extent.     

Amylopectin Synthase of Eimeria tenella: Identification and Kinetic Characterization

ABSTRACT. A soluble enzyme amylopectin synthase (UDP-glucose-α 1,4-glucan α-4-glucosyltransferase) which transfers glucose from uridine 5'-diphosphate glucose (UDP-glucose) to a primer to form α-I,4-glucosyl linkages has been identified in the extracts of unsporulated oocysts of Eimeria tenella . UDP-glucose and not ADP-glucose was the most active glucosyl donor. Corn amylopectin, rabbit liver glycogen, oyster glycogen and corn starch served as primers; the latter two were less efficient. The enzyme has an apparent pH optimum of 7.5 and exhibited typical Michaelis-Menten kinetics with dependence on both the primer and substrate concentrations. The Michaelis constants (Km). with respect to UDP-glucose, was 0.5 mM; and 0.25 mg/ml and 1.25 mg/ml with respect to amylopectin and rabbit liver glycogen. The product formed by the reaction was predominantly a glucan containing α-1,4 linkages. The specificity of the enzyme suggests that this enzyme is similar to glycogen synthase in eukaryotes and has been designated as amylopectin synthase (UDP-glucose-α-1,4-glucosetransferase EC 2.4.1.11).     

Ethanolaminephosphate cytidylyltransferase. Purification and characterization of the enzyme from rat liver.

Ethanolaminephosphate cytidylyltransferase (EC 2.7.7.14), which catalyzes a central step in phosphatidylethanolamine synthesis, has been purified 1000-fold from a postmicrosomal supernatant from rat liver. The enzyme, which requires a reducing agent, like dithiothreitol, for activity, is stable for weeks at 0-4 degrees when stored in the presence of dithiothreitol and in the pH range 7.5 to 9.0. A molecular weight of 100 to 120 X 10(3) was estimated by gel chromatography on Sephadex G-200. Gel electrophoresis in the presence of sodium dodecyl sulfate gave only one protein band with an apparent molecular weight of 49 to 50 X 10(3). The reaction catalyzed by the enzyme is reversible with a Keq for the forward reaction of 0.46 under the assay conditions. Michaelis constants of 53 and 65 muM were determined for CTP and ethanolaminephosphate, respectively. From the product inhibition pattern an ordered sequential reaction mechanism is proposed, in which CTP is the first substrate to add to the enzyme and CDP-ethanolamine is the last product to be released. The possible role of this reaction in the regulation of phosphatidylethanolamine synthesis in liver is discussed.     

The influence of temperature upon the hydrolysis of cellobiose by beta-1,4-glucosidases from Aspergillus niger

We have made experimental studies into the enzymatic hydrolysis of cellobiose within the temperature range of 40 degrees C to 70 degrees C at pH 4.9, by using beta-1,4-glucosidase from Aspergillus niger. At 70 degrees C there was significant enzyme deactivation, which could be fitted to a potential deactivation model with values of n equal to 1.09 and k(d) to 0.1564 (g/l)(-0.09) min(-1), whereas the rate of hydrolysis could be fitted to the Michaelis-Menten equation. Between 40 degrees C and 60 degrees C we noted a substrate inhibition and that the CEC compound formed contributed to glucose production. The apparent activation energies had values of 4.66, 8.45, 4.82, and 3.99 kJ/mol for the kinetic constants k(a) and k(a2) the Michaelis constant and the substrate inhibition constant, respectively.     

A Fortran Program for Evaluation of the Effectiveness Factor of Biocatalysts in the Presence of External and Internal Diffusional Limitations

A Fortran program called SPEFF for evaluation of the effectiveness factor of immobilized enzyme preparations of spherical form in the presence of external and internal mass transfer resistances is described, and a listing of the program is given. Enzyme distribution in the bioparticle may be uniform or nonuniform. In the latter case the enzyme distribution is approximated by fifth-order polynomial. In the program differential equations are replaced by the system of non-linear algebraic equations, and the latter are solved by Newton iteration technique. The program is developed for Michaelis-Menten kinetics with allowance for competitive product inhibition and substrate inhibition. After slight modifications the program can be used for computation of the effectiveness factor of a membrane with an immobilized enzyme, or in the case when the enzyme kinetics are more complex. A typical run on a PDP-11/45 computer took 10-20 seconds. A typical computation time in the case of IBM-compatible TURBO PC was 15-30 seconds.     

Properties of halophil nicotinamide–adenine dinucleotide phosphate-specific isocitrate dehydrogenase. True Michaelis constants, reaction mechanisms and molecular weights

True values of Michaelis constants of the NADP(+)-specific isocitrate dehydrogenase from Halobacterium salinarium were not very different from those of the apparent constants reported by Aitken et al. (1970). The true constants were affected by salt in a similar manner to that of the apparent constants obtained with NADP(+) at fixed concentrations of 1.0-0.2mm and threo-d(s)-(+)-isocitrate at fixed concentrations of 2.0-0.125mm. The response of apparent V(max.) to salt concentration was highly dependent on fixed substrate concentration in solutions of sodium chloride but much less so in solutions of potassium chloride. At several levels the results emphasize the difficulty of generalizing about the salt relations of a halophil enzyme without adequate attention to substrate concentration. The enzyme has at least two different reaction mechanisms depending on salt concentration. In its ;physiological' form (i.e. in 1.0m-potassium chloride), and also in 1.0m-sodium chloride, the reaction mechanism is ordered with NADP(+) the first substrate added and NADPH the last product released. In 0.25m-sodium chloride, however, the mechanism is different and is probably non-sequential. In 4.0m-sodium chloride with low concentrations of either fixed substrate, there was evidence of a co-operative action of the variable substrate. The evidence suggests that salt participates in the reaction mechanism in two ways: one is the reversible addition to the enzyme in a manner analogous to that of a substrate; the other is dead-end complex-formation. The relative contributions of these two types of reaction determine whether salt activates or inhibits the enzyme. In addition, the inhibition caused by high concentrations of sodium chloride is more complex than the corresponding inhibition by potassium chloride. Gel-filtration experiments indicated that at very low salt concentrations the enzyme has an apparent molecular weight of about 70800. In ;physiological' concentrations of potassium chloride the enzyme appears to be a dimer (mol.wt. 122000-135000) and, in 1.0-4.0m-sodium chloride, it behaves as a trimer or tetramer (mol.wt. 224000-251000). A preliminary method of purifying the enzyme is described.