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.     

Kinetic studies of alpha-galactosidase-containing mold pellets on PNPG hydrolysis.

Little is known about techniques for applying untreated microbial cells containing enzymes directly to industrial processes as a biocatalyst. The kinetic behavior of alpha-galactosidase-containing spherical pellets which are formed naturally under given conditions in a submerged culture of Mortierella vinacea was studied on the hydrolysis of PNPG (p-nitrophenyl-alpha-D-galactopyranoside). The effect on intraparticle diffusion on the overall reaction rate was assessed by the use of an effectiveness factor, which was calculated by the approximate solution to the equation derived from the mass balance within a pellet. The experimental effectiveness factors were found to be represented as a single function of the modified Thiele modulus, including such parameters as pellet size, enzyme concentration in the pellet, and substrate concentration. As the diffusional effect became more significant, the marked substrate inhibition as seen for a free enzyme disappeared gradually. The effect of product inhibition on the pellets was much weaker than that for a free enzyme at a given substrate concentration. In the region of diffusion controlled reaction, it was found that the rate is proportional to the square root of the enzyme concentration in the pellet. In addition, similarly to what was reported previously for a free enzyme,the reaction in a batch system was found to be approximately representable as simple first-order kinetics in which the rate constant was dependent on the initial substrate concentration.     

Optimal Design of a Series of GSTR's Performing Reversible Reactions Catalyzed by Soluble Enzymes: A Theoretical Study

The condition for the minimum overall reactor volume of a given number of CSTR's in series is theoretically determined for a reversible, single reactant-single product (Uni-Uni) enzyme catalyzed reaction. The reactor network is assumed to operate in steady-state, isothermal conditions with a single phase and a constant activity of biocatalyst. The method is based on a mathematical analysis of the discrete substrate concentration profile along the CSTR's assuming complete micromixing. The algebraic equations describing the critical loci are obtained for the general case, the mathematical proof that these equations define a minimum is presented, and an exact solution arising from an asymptotic situation is found. An approximate analytical method of optimization based on the aforementioned critical behavior is reported and its validity and usefulness discussed. The formulae introduced can be used in more general situations as tools for getting the approximate range where the optimal overall volume of the series of CSTR's lies. Hence, the reasoning developed is important for the preliminary CSTR design and relevant in the initial steps of the more involved methods of numerical optimization. Finally, the enzymatic conversion of fumarate to L-malate is examined as a model system in order to assess the usefulness and applicability of the analysis developed.     

Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios

We extend the validity of the quasi-steady state assumption for a model double intermediate enzyme-substrate reaction to include the case where the ratio of initial enzyme to substrate concentration is not necessarily small. Simple analytical solutions are obtained when the reaction rates and the initial substrate concentration satisfy a certain condition. These analytical solutions compare favourably with numerical solutions of the full system of differential equations describing the reaction. Experimental methods are suggested which might permit the application of the quasi-steady state assumption to reactions where it may not have been obviously applicable before.     

Kinetic analysis of enzyme systems with suicide substrate in the presence of a reversible competitive inhibitor, tested by simulated progress curves

The use of suicide substrates remains a very important and useful method in enzymology for studying enzyme mechanisms and designing potential drugs. Suicide substrates act as modified substrates for the target enzymes and bind to the active site. Therefore the presence of a competitive reversible inhibitor decreases the rate of substrate-induced inactivation and protects the enzyme from this inactivation. This lowering on the inactivation rate has evident physiological advantages, since it allows the easy acquisition of experimental data and facilitates kinetic data analysis by providing another variable (inhibitor concentration). However despite the importance of the simultaneous action of a suicide substrate and a competitive reversible inhibition, to date no corresponding kinetic analysis has been carried out. Therefore we present a general kinetic analysis of a Michaelis-Menten reaction mechanism with double inhibition caused by both, a suicide substrate and a competitive reversible inhibitor. We assume rapid equilibrium of the reversible reaction steps involved, while the time course equations for the reaction product have been derived with the assumption of a limiting enzyme. The goodness of the analytical solutions has been tested by comparison with the 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. In conclusion, we present a complete kinetic analysis of an enzyme reaction mechanism as described above in an attempt to fill a gap in the theoretical treatment of this type of system.     

Investigation of kinetics of immobilized liver esterase by flow calorimetry

Flow calorimetry (FC) was shown to be a powerful tool for investigation of the kinetics of phenyl acetate hydrolysis catalyzed by pig liver carboxyl esterase. The enzyme was immobilized in alginate gel particles that were placed in a calorimetric flow column and the heat effect of enzyme reaction was followed in single flow and total recirculation conditions. It was shown that the registered temperature change was proportional to molar amount of substrate transformed in the column. A mathematical model describing the enzyme reaction, mass transfer, and heat effects in the calorimetric system was developed and used for the kinetic data evaluation. By combining data from single flow and recirculation modes true kinetic parameters were evaluated by the proposed mathematical procedure based on the model solution and successive approximations.

The kinetic data for carboxyl esterase showed a slide substrate inhibition by phenyl acetate. The obtained kinetic parameters were as follows: Michaelis constant K_m=2 mmol dm⁻³ and substrate inhibition constant K_i=42 mmol dm⁻³. The method can be applied to kinetic study of immobilized enzymes directly in the flow calorimeter without any requirement of an independent analytical technique.     

Modeling, simulation, and kinetic analysis of a heterogeneous reaction system for the enzymatic conversion of poorly soluble substrate.

We developed a kinetic model that describes a heterogeneous reaction system consisting of a solid substrate suspension for the production of D-amino acid using D-hydantoinase. As a biocatalyst, mass-produced free and whole cell enzymes were used. The heterogeneous reaction system involves dissolution of a solid substrate, enzymatic conversion of the dissolved D-form substrate, spontaneous racemization of an L-form substrate to D-form, and deactivation of the enzyme. In the case of using whole cell enzymes, transfer of the dissolved substrate and product through the cell membrane was considered. The kinetic parameters were determined from experiments, literature data, and by using Marquardt's method of nonlinear regression analysis. The model was simulated using the kinetic parameters and compared with experimental data, and a good agreement was observed between the experimental results and the simulation ones. Factors affecting the kinetics of the heterogeneous reaction system were analyzed on the basis of the kinetic model, and the efficiency of the reaction systems using free and whole cell enzymes was also compared.     

Conditions for the assay of glutamate semialdehyde aminotransferase that overcome the problem of substrate instability.

Investigations into the kinetic properties of glutamate semialdehyde aminotransferase, a key enzyme in the metabolic pathway leading to chlorophyll, are made difficult by the instability of the enzyme's substrate glutamate 1-semialdehyde. The rate of spontaneous disappearance of this compound from solution is shown to vary with the square of its concentration and to be pH-dependent. Thus using conditions appropriate to the assay of the enzyme, half of the substrate is lost from solution in a few minutes. Second-order rate constants for the reaction are determined and conditions are selected whereby the effects of the spontaneous reaction are rendered insignificant. The steady-state kinetic properties of the enzyme determined using these conditions are reported.     

Optimal Design of a Series of GSTR's Performing Reversible Reactions Catalyzed by Soluble Enzymes: A Theoretical Study

The condition for the minimum overall reactor volume of a given number of CSTR's in series is theoretically determined for a reversible, single reactant-single product (Uni-Uni) enzyme catalyzed reaction. The reactor network is assumed to operate in steady-state, isothermal conditions with a single phase and a constant activity of biocatalyst. The method is based on a mathematical analysis of the discrete substrate concentration profile along the CSTR's assuming complete micromixing. The algebraic equations describing the critical loci are obtained for the general case, the mathematical proof that these equations define a minimum is presented, and an exact solution arising from an asymptotic situation is found. An approximate analytical method of optimization based on the aforementioned critical behavior is reported and its validity and usefulness discussed. The formulae introduced can be used in more general situations as tools for getting the approximate range where the optimal overall volume of the series of CSTR's lies. Hence, the reasoning developed is important for the preliminary CSTR design and relevant in the initial steps of the more involved methods of numerical optimization. Finally, the enzymatic conversion of fumarate to L-malate is examined as a model system in order to assess the usefulness and applicability of the analysis developed.     

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.     

Activity of mushroom polyphenol oxidase in organic medium

A kinetic study of the activity of mushroom polyphenol oxidase in an organic system was carried out to obtain detailed enzyme kinetic data in relation to optimization of reaction conditions and substrate specificity. A simple method for consistent measurement of reaction rates in the heterogeneous enzyme/organic solvent system (consisting of immobilized polyphenol oxidase and a hydrated solution of the substrate in chloroform) was designed. The aqueous content of the system was optimized using p-cresol as the substrate. With this system, a crude extract of Agaricus bisporus was used to hydroxylate and oxidize a range of selected p-substituted phenolic substrates, yielding o-quinone products. Michaelis-Menten kinetics were used to obtain apparent K(M) and V(max) values with respect to each of these substrates. Results from this analysis indicated a correlation between the enzymic kinetic parameters obtained and the steric requirements of the substrates, which could be rationalized in terms of the restricted flexibility of the enzyme when it is in chloroform and also in terms of substrate and solvent hydrophobicity. In the course of the investigation UV molar absorption coefficients of several o-quinones were measured by a novel method: (1)H nuclear magnetic resonance (NMR) spectroscopy was employed to determine component concentrations in reaction mixtures resulting from the transformation of phenols by polyphenol oxidase in chloroform. Thus the UV molar absorption coefficients could be obtained directly, avoiding the necessity to isolate the water-sensitive, unstable o-quinones. (c) 1993 John Wiley & Sons, Inc.     

Diffusion-controlled reactions of enzymes. An approximate analytic solution of Chou's model

Based on Chou's model, a criterion was derived, by which one can judge whether or not the physical picture of the critical spherical shell described by Chou for an enzyme-substrate fast reaction system can emerge. Furthermore, for those reaction systems with such a physical picture, an approximate analytical solution was presented, which can be easily handled to calculate the upper limit of the diffusion-controlled reaction and the corresponding concentration distribution of substrate molecules on the surface of the major protein outside the active site. The results thus obtained are in good agreement with those computed by Chou et al. through the approach of numerical solution. Furthermore, the physical significance of the criterion and its relation to the critical spherical shell are substantiated during the process of derivation, which is very helpful for gaining an insight into this kind of biomolecular system with surprisingly high reaction rates.     

Comparison of two experimental methods for the determination of Michaelis-Menten kinetics of an immobilized enzyme

For the application of immobilized enzymes, the influence of immobilization on the activity of the enzyme should be Known. This influence can be obtained by determining the intrinsic kinetic parameters of the immobilized enzyme, and by comparing them with the kinetic parameters of the suspended enzyme. This article deals with the determination of the intrinsic kinetic parameters of an agarose-gel bead immobilized oxygen-consuming enzyme: L-lactate 2-monooxygenase. The reaction rate of the enzyme can be described by Michaelis-Menten kinetics. Batch conversion experiments using a biological oxygen monitor, as well as steady-state profile measurements within the biocatalyst particles using an oxygen microsensor, were performed. Two different mathematical methods were used for the batch conversion experiments, both assuming a pseudosteady-state situation with respect to the shape of the profile inside the bead. One of the methods used an approximate relation for the effectiveness factor for Michaelis-Menten kinetics which interpolates between the analytical solutions for zero- and first-order kinetics. The other mathematical method was based on a numerical solution and combined a mass balance over the reactor with a mass balance over the bead. The main difference in the application of the two methods is the computer calculation time; the completely numerical calculation procedure was about 20 times slower than the other calculation procedure.The intrinsic kinetic parameters resulting from both experimental methods were compared to check the reliability of the methods. There was no significant difference in the intrinsic kinetic parameters obtained from the two experimental methods. By comparison of the kinetic parameters for the suspended enzyme with the intrinsic kinetic parameters for the immobilized enzyme, it appeared that immobilization caused a decrease in the value of V(m) by a factor of 2, but there was no significant difference in the values obtained for K(m).     

Kinetcs and decay of fumarase activity of immobilized Brevibacterium ammoniagenes cells for continuous production of L-malic acid

The kinetics of the reversible fumarase reaction of immobilized Brevibacterium ammoniagenes cells and the decay behavior of enzyme activity were investigated in a plug flow system. The time course of the reaction in the immobilized cell column was well explained by the time-conversion equation including the apparent kinetic constants of the immobilized cell enzyme. The decay rate of fumarase activity was faster in the upper sections of the column (inlet side of the substrate solution) compared with the lower sections when 1M sodium fumarate (pH 7.0) was continuously passed through the column at 37°C. It was shown that the decay rate of the fumarase activity in the immobilized cell column depends on the flow rate of the substrate solution. The effect of flow rate on the decay rate of enzyme activity was considered to be related to the rate of contamination of enzyme with poisonous substances derived from the substrate solution or to the rate of leakage of enzyme stabilizers and/or enzyme itself from the immobilized cells.     

The nonallosteric mechanism of enzyme activity regulation. Is it the only true mechanism?

The nonallosteric regulation mechanism of enzyme reaction velocity assumes that the substrate and enzyme interact via a metal cation and form simple and mixed, mono- and multi-nuclear complexes. A solution of equations for individual cases gives a function of initial reaction velocity at any given substrate or modifier concentration. This function can describe kinetic effects that are considered allosteric, as well as phenomena omitted by commonly-accepted models.     

Kinetic analysis of the general modifier mechanism of Botts and Morales involving a suicide substrate

Suicide substrates are widely used in enzymology for studying enzyme mechanisms and designing potential drugs. The presence of a reversible modifier decreases or increases the rate of substrate-induced inactivation, with evident physiological and experimental consequences. To date, only the action of a competitive or uncompetitive inhibitor of an enzyme system involving suicide substrate has been reported. In this paper, we analyse the kinetics of enzyme-catalysed reactions which evolve in accordance with the general modifier mechanisms of Botts and Morales in which enzyme inactivation is induced by suicide substrate. Rapid equilibrium of all of the reversible reaction steps involved is assumed and the time course equations for the residual enzyme activity, the inactive enzyme forms and the reaction product are derived. Partition ratios giving the relative weight of the product and inactive enzyme concentrations, and the relative contribution to the product formation of each of the unmodified and modified catalytic routes, are studied. New indices pointing to the conditions under which the modifier acts as inhibitor or as activator are suggested. The goodness of the analytical solutions is tested by comparison with the simulated curves obtained by numerical integration. An experimental design and kinetic data analysis to evaluate the kinetic parameters from the time progress curves of the product are proposed. From these results, those corresponding to several reaction mechanisms involving both a suicide substrate and a modifier, and which can be regarded as particular cases of the general case analysed here, can be directly and easily derived.     

Glucoamylase and glucose oxidase preparations and their combined application for conversion of maltose to gluconic acid

The kinetic behavior of a system of multiple enzyme in solution has been studied in a variable volume batch reactor at pH 5, controlled dissolved oxygen concentration, and T = 30°C. The enzymes used were glucoamylase (R. delemar), glucose oxidase (A. niger), and gluconolactonase (A. niger), all of which are important commercial biocatalysts, and a disaccharide was employed as the starting substrate. This study includes the basic kinetic properties of individual enzymes and interactions between components of the reaction mixture. Classical Michaelis–Menten single substrate or two substrate kinetic with parameters based on initial rate data predict correctly the batch time course of the sequential reaction network.