Modeling intrinsic kinetics of enzymatic cellulose hydrolysis

A multistep approach was taken to investigate the intrinsic kinetics of the cellulase enzyme complex as observed with hydrolysis of noncrystalline cellulose (NCC). In the first stage, published initial rate mechanistic models were built and critically evaluated for their performance in predicting time-course kinetics, using the data obtained from enzymatic hydrolysis experiments performed on two substrates: NCC and alpha-cellulose. In the second stage, assessment of the effect of reaction intermediates and products on intrinsic kinetics of enzymatic hydrolysis was performed using NCC hydrolysis experiments, isolating external factors such as mass transfer effects, physical properties of substrate, etc. In the final stage, a comprehensive intrinsic kinetics mechanism was proposed. From batch experiments using NCC, the time-course data on cellulose, cello-oligosaccharides (COS), cellobiose, and glucose were taken and used to estimate the parameters in the kinetic model. The model predictions of NCC, COS, cellobiose, and glucose profiles show a good agreement with experimental data generated from hydrolysis of different initial compositions of substrate (NCC supplemented with COS, cellobiose, and glucose). Finally, sensitivity analysis was performed on each model parameter; this analysis provides some insights into the yield of glucose in the enzymatic hydrolysis. The proposed intrinsic kinetic model parametrized for dilute cellulose systems forms a basis for modeling the complex enzymatic kinetics of cellulose hydrolysis in the presence of limiting factors offered by substrate and enzyme characteristics.     

Enzyme Kinetics for Complex System Enables Accurate Determination of Specificity Constants of Numerous Substrates in a Mixture by Proteomics Platform

Many important experiments in proteomics including protein digestion, enzyme substrate screening, enzymatic labeling, etc., involve the enzymatic reactions in a complex system where numerous substrates coexists with an enzyme. However, the enzyme kinetics in such a system remains unexplored and poorly understood. Herein, we derived and validated the kinetics equations for the enzymatic reactions in complex system. We developed an iteration approach to depict the enzymatic reactions in complex system. It was validated by 630 time-course points from 24 enzymatic reaction experiments and was demonstrated to be a powerful tool to simulate the reactions in the complex system. By applying this approach, we found that the ratio of substrate depletion is independent of other coexisted substrates under specific condition. This observation was then validated by experiments. Based on this striking observation, a simplified model was developed to determine the catalytic efficiencies of numerous competing substrates presented in the complex enzyme reaction system. When coupled with high-throughput quantitative proteomics technique, this simplified model enabled the accurate determination of catalytic efficiencies for 2369 peptide substrates of a protease by using only one enzymatic reaction experiment. Thus, this study provided, in the first time, a validated model for the large scale determination of specificity constants which could enable the enzyme substrate screening approach turned from a qualitative method of identifying substrates to a quantitative method of identifying and prioritizing substrates. Data are available via ProteomeXchange with identifier PXD004665.     

A microscopic model of enzyme kinetics.

Many in vivo enzymatic processes, such as those of the tissue factor pathway of blood coagulation, occur in environments with facilitated substrate delivery or enzymes bound to cellular or lipid surfaces, which are quite different from the ideal fluid environment for which the Michaelis-Menten equation was derived. To describe the kinetics of such reactions, we propose a microscopic model that focuses on the kinetics of a single-enzyme molecule. This model provides the foundation for macroscopic models of the system kinetics of reactions occurring in both ideal and nonideal environments. For ideal reaction systems, the corresponding macroscopic models thus derived are consistent with the Michaelis-Menten equation. It is shown that the apparent Km is in fact a function of the mechanism of substrate delivery and should be interpreted as the substrate level at which the enzyme vacancy time equals the residence time of ES-complexes; it is suggested that our microscopic model parameters characterize more accurately an enzyme and its catalytic efficiency than does the classical Km. This model can also be incorporated into computer simulations of more complex reactions as an alternative to explicit analytical formulation of a macroscopic model.     

Mathematical investigations of growth of microorganisms in the gradostat

Microbial ecosystems with spatial distribution of substrate (nutrient) supply in the form of gradients have been studied in a laboratory system called the gradostat, which is a series of coupled chemostats. We investigate analytically and numerically a mathematical model, similar to the one for a single chemostat based on Michaelis-Menten kinetics, of the growth of one species of microorganisms in the gradostat in the presence of one limiting substrate and two limiting complementary substrates. Our analysis predicts various patterns of spatial distribution of microorganisms at steady state and suggests further experiments to be performed with the gradostat.This work was supported by the Deutsche Forschungsgemeinschaft     

Dynamic modeling of an enzymatic membrane reactor for the treatment of xenobiotic compounds

A membrane enzymatic reactor, consisting of a stirred tank coupled to an ultrafiltration membrane was set up for the enzymatic oxidation of xenobiotic compounds. The azo dye Orange II was selected for the model compound and manganese peroxidase for the oxidative enzyme. The ligninolytic cycle was initiated and maintained by the controlled addition of all factors (reactants, mediators, and stabilizers) at suitable rates. Considering the distinctiveness of this process, in which the substrate to be oxidized is not the primary substrate for the enzyme, a kinetic model was developed. The azo dye concentration and hydrogen peroxide addition rate were found to be the main factors affecting the process. The reaction kinetics was defined using a Michaelis-Menten model with respect to the Orange II concentration and a first-order linear dependence relative to the H(2)O(2) addition rate. The dynamic model, which takes into account both the kinetics and the hydraulics of the system, was validated by comparing the experimental results in continuous operation under steady and non-steady state to model predictions. In particular, the model predicted the behavior of the system when unexpected alterations in steady-state operation occurred. Furthermore, the model allowed us to obtain the most appropriate H(2)O(2)/Orange II ratio in the feed to maximize the process efficiency.     

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.     

Strategies in the Preparation of Homochiral Compounds Using Combined Enantioselective Enzymes

In order to obtain a homochiral product from a racemic substrate, different strategies can be followed using a moderately enantioselective enzymatic catalyst. Two new strategies are presented, involving the simultaneous use of two enzymes, parallel or consecutive. In the parallel system, the substrate enantiomer yielding the unwanted product enantiomer is enantioselectively converted by the second enzyme. In the consecutive system, the substrate enantiomer yielding the desired product enantiomer is itself the preferred product of another enantioselective enzymatic reaction.

For irreversible pseudo-first order enzyme kinetics, a relationship was found which describes the dependency of the yield and enantiomeric excess for these systems on the E-values of the separate enzymes and on the ratio of their concentrations. For Michaelis-Menten kinetics, these relationships usually give good approximations.

According to these calculations, the yield and enantiomeric excess obtainable with the concepts of combined enzymes exceed significantly those obtainable with the separate enzymes, and also those obtainable with the strategy of product recirculation.     

Isolation and incorporation of rabbit liver epoxide hydrase into phospholipid vesicles.

Methods are described for the incorporation into phospholipid vesicles of epoxide hydrase isolated from liver microsomes of phenobarbital-treated rabbits. Chromatography on a Sephadex G-50 column of epoxide hydrase and egg yolk phosphatidylcholine treated with sodium cholate yielded homogeneous vesicles with a diameter of about 25 nm and containing 80 to 85% of the protein applied. At high substrate concentrations, the vesicles catalyzed the hydration of benzo(a)pyrene-4,5-oxide and styrene-7,8-epoxide at a rate similar to that obtained with the enzyme in a soluble form. However, the kinetics of styrene glycol formation catalyzed by the vesicular or microsomal preparations were complex. Convex Lineweaver-Burk plots and concave Hill plots were obtained, whereas normal Michaelis-Menten kinetics characterized the hydration catalyzed by the enzyme in a soluble form. The results could be explained if reconstitution of the enzyme into the vesicles gives rise to low affinity high capacity sites for the substrate on the enzyme, or alternatively facilitates the interaction of the substrate with such sites already present. It is suggested that reconstituted liposomes containing both the liver microsomal hydroxylase system and epoxide hydrase may prove to be a good model system for evaluating substrate specificity and factors of importance in the formation of toxic and carcinogenic metabolites by these enzymes.     

Modelling amperometric enzyme electrode with substrate cyclic conversion

A mathematical model of amperometric enzyme electrodes in which chemical amplification by cyclic substrate conversion takes place in a single enzyme membrane has been developed. The model is based on non-stationary diffusion equations containing a non-linear term related to Michaelis-Menten kinetic of the enzymatic reaction. The digital simulation was carried out using the finite difference technique. The influence of the substrate concentration, the maximal enzymatic rate as well as the membrane thickness on the biosensor response was investigated. The numerical experiments demonstrate significant (up to dozens of times) gain in biosensor sensitivity at low concentrations of substrate when the biosensor response is under diffusion control.     

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.     

Michaelis-Menten kinetics under spatially constrained conditions: application to mibefradil pharmacokinetics

Two different approaches were used to study the kinetics of the enzymatic reaction under heterogeneous conditions to interpret the unusual nonlinear pharmacokinetics of mibefradil. Firstly, a detailed model based on the kinetic differential equations is proposed to study the enzymatic reaction under spatial constraints and in vivo conditions. Secondly, Monte Carlo simulations of the enzyme reaction in a two-dimensional square lattice, placing special emphasis on the input and output of the substrate were applied to mimic in vivo conditions. Both the mathematical model and the Monte Carlo simulations for the enzymatic reaction reproduced the classical Michaelis-Menten (MM) kinetics in homogeneous media and unusual kinetics in fractal media. Based on these findings, a time-dependent version of the classic MM equation was developed for the rate of change of the substrate concentration in disordered media and was successfully used to describe the experimental plasma concentration-time data of mibefradil and derive estimates for the model parameters. The unusual nonlinear pharmacokinetics of mibefradil originates from the heterogeneous conditions in the reaction space of the enzymatic reaction. The modified MM equation can describe the pharmacokinetics of mibefradil as it is able to capture the heterogeneity of the enzymatic reaction in disordered media.     

An enzyme rate equation for the overall rate of reaction of gel-immobilized glucose oxidase particles under buffered conditions. I. Pseudo-one substrate conditions

The overall rate of reaction of buffered gel-immobilized glucose oxidase particles is described by means of an enzyme rate equation which relates the overall reaction rate of a particle to the free solution characteristics of the enzyme, the effective diffusivity of the limiting substrate in the gel, the characteristic particle size, and the limiting substrate concentration adjacent to the gel surface. This equation accounts quantitatively for the limitation of the overall rate of reaction by substrate diffusion, and it is used to illustrate the influence of the system parameters, i. e., particle size, enzyme concentration, and pH, on the extent of the diffusional resistance associated with gel-immobilized glucose oxidase particles. The enzyme rate equation is generally applicable to those enzymes whose kinetics approximately follow Michaelis-Menten form when in free solution.     

Strategies in the Preparation of Homochiral Compounds Using Combined Enantioselective Enzymes

In order to obtain a homochiral product from a racemic substrate, different strategies can be followed using a moderately enantioselective enzymatic catalyst. Two new strategies are presented, involving the simultaneous use of two enzymes, parallel or consecutive. In the parallel system, the substrate enantiomer yielding the unwanted product enantiomer is enantioselectively converted by the second enzyme. In the consecutive system, the substrate enantiomer yielding the desired product enantiomer is itself the preferred product of another enantioselective enzymatic reaction.

For irreversible pseudo-first order enzyme kinetics, a relationship was found which describes the dependency of the yield and enantiomeric excess for these systems on the E-values of the separate enzymes and on the ratio of their concentrations. For Michaelis-Menten kinetics, these relationships usually give good approximations.

According to these calculations, the yield and enantiomeric excess obtainable with the concepts of combined enzymes exceed significantly those obtainable with the separate enzymes, and also those obtainable with the strategy of product recirculation.     

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.     

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.     

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.     

Mathematical modelling of dynamics and control in metabolic networks. II. Simple dimeric enzymes

The dynamics of enzyme cooperativity are examined by studying a homotropic dimeric enzyme with identical reaction sites, both of which follow irreversible Michaelis-Menten kinetics. The problem is approached via scaling and linearization of the governing mass action kinetic equations. Homotropic interaction between the two sites are found to depend on three dimensionless groups, two for the substrate binding step and one for the chemical transformation. The interaction between the two reaction sites is shown capable of producing dynamic behavior qualitatively different from that of a simple Michaelis-Menten system; when the two sites interact to increase enzymatic activity over that of two independent monomeric enzymes (positive cooperativity) damped oscillatory behavior is possible, and for negative cooperativity in the chemical transformation step a multiplicity of steady states can occur, with one state unstable and leading to runaway behavior. Linear analysis gives significant insight into system dynamics, and their parametric sensitivity, and a way to identify regions of the parameter space where the approximate quasi-stationary and quasi-equilibrium analyses are appropriate.     

Kinetic analysis and modeling of the liquid-liquid conversion of emulsified di-rhamnolipids by Naringinase from Penicillium decumbens

The enzymatic conversion of an aggregate-forming substrate was kinetically analyzed and a model was applied for the prediction of reaction-time courses. An L-rhamnose molecule from a di-rhamnolipid is cleaved by Naringinase from Penicillium decumbens leading to a mono-rhamnolipid. Optimal reaction rates were found when both, substrate and product build large co-aggregates in a slightly acidic aqueous phase. On the other hand, reaction rates were independent of initial di-rhamnolipid concentration and this was interpreted by assuming that the reaction occurs in the aqueous phase according to Michaelis-Menten kinetics in combination with competitive L-rhamnose inhibition. Rhamnolipids were therefore assumed to be highly concentrated in aggregates, a second liquid phase, whereas diffusive rhamnolipid transport from and to the aqueous phase occurs due to the enzymatic reaction. Furthermore, ideal surfactant mixing between di- and mono-rhamnolipid was assumed for interpretation of the negative effect of the last on the reaction rate. A model was created that describes the system accordingly. The comparison of the experimental data, were in excellent agreement with the predicted values. The findings of this study may beneficially be adapted for any bioconversion involving aggregate-forming substrate and/or product being catalyzed by hydrophilic enzymes.     

Mathematical model for internal pH control in immobilized enzyme particles

A mathematical model has been developed for the internal pH control in immobilized enzyme particles. This model describes the kinetics of a coupled system of two enzymes, immobilized in particles of either planar, cylindrical, or spherical shape. The enzyme kinetics are assumed to be of a mixed type, including Michaelis-Menten kinetics, uncompetitive substrate inhibition, and competitive and noncompetitive product inhibition. In a case study we have considered the enzyme combination urease and penicillin acylase, whose kinetics are coupled through the pH dependence of the kinetic parameters. The hydrolysis of urea by urease yields ammonia and carbon dioxide, whereas benzylpenicillin (Pen-G) is converted to 6-amino penicillanic acid and phenyl acetic acid by penicillin acylase. The production of acids by the latter enzyme will cause a decrease in pH. Because of the presence of the ammonia-carbon dioxide system, however, the pH may be kept under control. In order to obtain information about the optimum performance of this enzymatic pH controller, we have computed the effectiveness factor and the conversion in a CSTR at different enzyme loadings. The results of the computer simulations indicate that a high conversion of Pen-G may be achieved (80-90%) at bulk pH values of about 7.5-8.