Explicit analytic approximations for time-dependent solutions of the generalized integrated Michaelis–Menten equation

Various explicit reformulations of time-dependent solutions for the classical two-step irreversible Michaelis–Menten enzyme reaction model have been described recently. In the current study, I present further improvements in terms of a generalized integrated form of the Michaelis–Menten equation for computation of substrate or product concentrations as functions of time for more real-world, enzyme-catalyzed reactions affected by the product. The explicit equations presented here can be considered as a simpler and useful alternative to the exact solution for the generalized integrated Michaelis–Menten equation when fitted to time course data using standard curve-fitting software.     

A Kinetic Analysis of Coupled (or Auxiliary) Enzyme Reactions

As a case study, we consider a coupled (or auxiliary) enzyme assay of two reactions obeying the Michaelis–Menten mechanism. The coupled reaction consists of a single-substrate, single-enzyme non-observable reaction followed by another single-substrate, single-enzyme observable reaction (indicator reaction). In this assay, the product of the non-observable reaction is the substrate of the indicator reaction. A mathematical analysis of the reaction kinetics is performed, and it is found that after an initial fast transient, the coupled reaction is described by a pair of interacting Michaelis–Menten equations. Moreover, we show that when the indicator reaction is fast, the quasi-steady-state dynamics are governed by three fast variables and one slow variable. Timescales that approximate the respective lengths of the indicator and non-observable reactions, as well as conditions for the validity of the Michaelis–Menten equations, are derived. The theory can be extended to deal with more complex sequences of enzyme-catalyzed reactions.     

Oscillatory enzyme reactions and Michaelis–Menten kinetics

Oscillations occur in a number of enzymatic systems as a result of feedback regulation. How Michaelis–Menten kinetics influences oscillatory behavior in enzyme systems is investigated in models for oscillations in the activity of phosphofructokinase (PFK) in glycolysis and of cyclin-dependent kinases in the cell cycle. The model for the PFK reaction is based on a product-activated allosteric enzyme reaction coupled to enzymatic degradation of the reaction product. The Michaelian nature of the product decay term markedly influences the period, amplitude and waveform of the oscillations. Likewise, a model for oscillations of Cdc2 kinase in embryonic cell cycles based on Michaelis–Menten phosphorylation–dephosphorylation kinetics shows that the occurrence and amplitude of the oscillations strongly depend on the ultrasensitivity of the enzymatic cascade that controls the activity of the cyclin-dependent kinase.     

Alternative algebraic rate‐integration approach for progress‐curve analysis of enzyme kinetics

A recent article of Zavrel et al. in this journal (Eng. Life Sci. 2010, 10, 191–200) described a comparison of several computer programs for progress‐curve analysis with respect to different computational approaches for parameter estimation. The authors applied both algebraic and dynamic parameter estimations, although they omitted time‐course analysis through the integrated rate equation. Recently, it was demonstrated that progress‐curve analysis through the integrated rate equation can be considered a simple and useful alternative for enzymes that obey the generalized Michaelis–Menten reaction mechanism. To complete this gap, the time‐dependent solution of the generalized Michaelis–Menten equation is here fitted to the progress curves from the Zavrel et al. reference article. This alternative rate‐integration approach for determining the kinetics parameters of Michaelis–Menten‐type enzymes yields the values with the greatest accuracy, as compared with the results obtained by other (algebraic or dynamic) parameter estimations.     

The average enzyme principle

The Michaelis–Menten equation for an irreversible enzymatic reaction depends linearly on the enzyme concentration. Even if the enzyme concentration changes in time, this linearity implies that the amount of substrate depleted during a given time interval depends only on the average enzyme concentration. Here, we use a time re-scaling approach to generalize this result to a broad category of multi-reaction systems, whose constituent enzymes have the same dependence on time, e.g. they belong to the same regulon. This “average enzyme principle” provides a natural methodology for jointly studying metabolism and its regulation.     

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.     

A new mechanism and kinetic model for the enzymatic synthesis of short-chain fructooligosaccharides from sucrose

A kinetic model based on a ping-pong mechanism was developed under the steady-state hypothesis to account for the short-chain fructooligosaccharides (sc-FOS) synthesis using the commercial cellulolytic enzyme preparation, Rohapect CM. This new mechanism takes into account the interactions between the enzyme species and potential substrates (sucrose and sc-FOS) as a single complex reaction, allowing a better understanding of the reaction kinetics.The initial reaction rate laws appropriately describe the kinetic profiles of the examined substrates. Whereas sucrose exhibited Michaelis–Menten behavior with substrate inhibition, 1-kestose and nystose followed Michaelis–Menten and sigmoid enzyme kinetics. In addition, the enzyme was competitively inhibited by glucose and exhibited significant hydrolytic activity in the presence of nystose.The overall model was simultaneously fitted to experimental data from three initial sucrose concentrations (0.5, 1.5 and 2.1 M) using a multi-response regression with kinetic parameters that have biochemical relevance and are independent of the enzyme concentration. According to the model, sucrose acts almost exclusively as a fructosyl donor substrate. The mathematical development described herein is expected to be suitable for modeling similar enzymatic reaction systems.     

A versatile equation to describe reversible enzyme inhibition and activation kinetics: Modeling β-galactosidase and butyrylcholinesterase

Current treatments for Alzheimer's disease involve inhibiting cholinesterases. Conversely, cholinesterase stimulation may be deleterious. Homocysteine is a known risk factor for Alzheimer's and vascular diseases and its active metabolite, homocysteine thiolactone, stimulates butyrylcholinesterase. Considering the opposing effects on butyrylcholinesterase of homocysteine thiolactone and cholinesterase inhibitors, understanding how these molecules alter this enzyme may provide new insights in the management of dementia. Butyrylcholinesterase does not strictly adhere to Michaelis–Menten parameters since, at higher substrate concentrations, enzyme activation occurs. The substrate activation equation for butyrylcholinesterase does not describe the effects of inhibitors or non-substrate activators. To address this, global data fitting was used to generate a flexible equation based on Michaelis–Menten principles. This methodology was first tested to model complexities encountered in inhibition by imidazole of β-galactosidase, an enzyme that obeys Michaelis–Menten kinetics. The resulting equation was sufficiently flexible to permit expansion for modeling activation or inhibition of butyrylcholinesterase, while accounting for substrate activation of this enzyme. This versatile equation suggests that both the inhibitor and non-substrate activator examined here have little effect on the substrate-activated form of butyrylcholinesterase. Given that butyrylcholinesterase inhibition can antagonize stimulation of this enzyme by homocysteine thiolactone, cholinesterase inhibition may have a role in treating Alzheimer and vascular diseases related to hyperhomocysteinemia.     

Ring-opening polymerization of ɛ-caprolactone catalyzed by a novel lipase Candida sp. 99-125

The ring-opening polymerization of ?-caprolactone catalyzed by a novel lipase Candida sp. 99-125 was achieved in organic solvents. ?-Caprolactone was found to be polymerized in quantitative yield in toluene at 80?°C for 5 days. The poly (?-caprolactone) obtained had a number-average molecular weight of 14,120 (polydispersity index?=?1.161). The effect of enzyme concentration, temperature, reaction time, reaction medium, and water content on monomer conversion and product molecular weight were investigated. Kinetic analysis showed that the enzyme displayed Michaelis–Menten kinetics and showed stronger affinity for ?-caprolactone (with a K_m value of 6.1?mmol/L) than other reported lipases.     

Kinetic study of sphingomyelin hydrolysis for ceramide production

Kinetic study of sphingomyelin hydrolysis catalyzed by Clostridium perfringens phospholipase C was, at the first time, conducted for ceramide production. Ceramide has the major role in maintaining the water-retaining properties of the epidermis. Hence, it is of great commercial potential in cosmetic and pharmaceutical industries such as in hair and skin care products. The enzymatic hydrolysis of sphingomyelin has been proved to be a feasible method to produce ceramide. The kinetic performance of sphingomyelin hydrolysis in the optimal two-phase (water:organic solvent) reaction system was investigated to elucidate the possible reaction mechanism and also to further improve the hydrolysis performance. Enzyme in solution had less thermal stability than the enzyme powder and the immobilized enzyme. The thermal inactivation of phospholipase C in all the three forms did not follow the first order reaction at 65 °C. The reactions for both the soluble and immobilized enzymes followed Michaelis–Menten kinetics. K_m's for the soluble and immobilized enzymes were 1.07 ± 0.32 and 1.26 ± 0.19 mM, respectively. The value of V_max was markedly decreased by the immobilization without much change in K_m, as if the immobilization functioned as the non-competitive inhibition. Ceramide as product activated the hydrolysis reaction, however, and its addition mainly caused the increase in the affinity of the enzyme–substrate complex.     

Rational polynomial equation as an unbiased approach for the kinetic studies of Drosophila melanogaster acetylcholinesterase reaction mechanism

The hydrolysis of substrates by cholinesterases does not follow the Michaelis–Menten reaction mechanism. The well-known inhibition by excess substrate is often accompanied by an unexpectedly high activity at low substrate concentrations. It appears that these peculiarities are the consequence of an unusual architecture of the active site, which conducts the substrate molecule over many stages before it is cleaved and released. Structural and kinetic data also suggest that two substrate molecules can attach at the same time to the free, as well as to the acetylated, enzyme. We present a procedure which provides an unbiased framework for mathematical modelling of such complex reaction mechanisms. It is based on regression analysis of a rational polynomial using classical initial rate data. The determination of polynomial degree reveals the number of independent parameters that can be evaluated from the available information. Once determined, these parameters can substantially facilitate the construction and evaluation of a kinetic model reflecting the expected molecular events in an enzymic reaction. We also present practical suggestions for testing the postulated kinetic model, using an original thermodynamic approach and an isolated effect in a specifically mutated enzyme.     

Effect of biocatalyst swelling on the operation of packed-bed immobilized enzyme bioreactor

A general mathematical model was developed for predicting the performance and simulation of a packed-bed immobilized enzyme reactor performing a reaction that follows Michaelis–Menten kinetics with competitive product inhibition. The performance of a packed-bed immobilized enzyme reactor was analyzed taking into account the effect of bed swelling on various diffusional phenomena such as axial dispersion, internal and external mass transfer limitations. The numerical solutions were compared with experimental data obtained for a packed-bed reactor operating with β-galactosidase entrapped in Ca-alginate-K-κ-carrageenan gels for lactose hydrolysis.     

The original Michaelis constant: translation of the 1913 Michaelis-Menten paper

Nearly 100 years ago Michaelis and Menten published their now classic paper [Michaelis, L., and Menten, M. L. (1913) Die Kinetik der Invertinwirkung. Biochem. Z. 49, 333-369] in which they showed that the rate of an enzyme-catalyzed reaction is proportional to the concentration of the enzyme-substrate complex predicted by the Michaelis-Menten equation. Because the original text was written in German yet is often quoted by English-speaking authors, we undertook a complete translation of the 1913 publication, which we provide as Supporting Information . Here we introduce the translation, describe the historical context of the work, and show a new analysis of the original data. In doing so, we uncovered several surprises that reveal an interesting glimpse into the early history of enzymology. In particular, our reanalysis of Michaelis and Menten's data using modern computational methods revealed an unanticipated rigor and precision in the original publication and uncovered a sophisticated, comprehensive analysis that has been overlooked in the century since their work was published. Michaelis and Menten not only analyzed initial velocity measurements but also fit their full time course data to the integrated form of the rate equations, including product inhibition, and derived a single global constant to represent all of their data. That constant was not the Michaelis constant, but rather V(max)/K(m), the specificity constant times the enzyme concentration (k(cat)/K(m) × E(0)).     

Lipase-catalyzed synthesis of palmitanilide: Kinetic model and antimicrobial activity study

Enzymatic syntheses of fatty acid anilides are important owing to their wide range of industrial applications in detergents, shampoo, cosmetics, and surfactant formulations. The amidation reaction of Mucor miehei lipase Lipozyme IM20 was investigated for direct amidation of triacylglycerol in organic solvents. The process parameters (reaction temperature, substrate molar ratio, enzyme amount) were optimized to achieve the highest yield of anilide. The maximum yield of palmitanilide (88.9%) was achieved after 24 h of reaction at 40 °C at an enzyme concentration of 1.4% (70 mg). Kinetics of lipase-catalyzed amidation of aniline with tripalmitin has been investigated. The reaction rate could be described in terms of the Michaelis–Menten equation with a Ping–Pong Bi–Bi mechanism and competitive inhibition by both the substrates. The kinetic constants were estimated by using non-linear regression method using enzyme kinetic modules. The enzyme operational stability study showed that Lipozyme IM20 retained 38.1% of the initial activity for the synthesis of palmitanilide (even after repeated use for 48 h). Palmitanilide, a fatty acid amide, exhibited potent antimicrobial activity toward Bacillus cereus.     

A method to describe enzyme-catalyzed reactions by combining steady state and time course enzyme kinetic parameters

Background

Complete analysis of single substrate enzyme-catalyzed reactions has required a separate use of two distinct approaches. Steady state approximations are employed to obtain substrate affinity and initial velocity information. Alternatively, first order exponential decay models permit simulation of the time course data for the reactions. Attempts to use integrals of steady state equations to describe reaction time courses have so far met with little success.

Methods

Here we use equations based on steady state approximations to directly model time course plots.

Results

Testing these expressions with the enzyme β-galactosidase, which adheres to classical Michaelis–Menten kinetics, produced a good fit between observed and calculated values.

General significance

This study indicates that, in addition to providing information on initial kinetic parameters, steady state approximations can be employed to directly model time course kinetics.Integrated forms of the Michaelis–Menten equation have previously been reported in the literature. Here we describe a method to directly apply steady state approximations to time course analysis for predicting product formation and simultaneously obtain multiple kinetic parameters.     

Ring-opening polymerization of ɛ-caprolactone catalyzed by a novel thermophilic esterase from the archaeon Archaeoglobus fulgidus

The ring-opening polymerization of ?-caprolactone catalyzed by a novel thermophilic esterase from the archaeon Archaeoglobus fulgidus was successfully conducted in organic solvents. The effects of enzyme concentration, temperature, reaction time, reaction medium, and water activity on monomer conversion and product molecular weight were investigated. Poly(?-caprolactone) was obtained in almost 100% of the monomer conversion, with a number-average molecular weight of 1400 in toluene at 80 °C for 72 h. Furthermore, the Michaelis–Menten kinetic analysis showed that the enzyme had the highest affinity for ?-caprolactone, with a K_m value of 0.093 mol/l compared with other reported lipases. The possible structural and energetic effects of the enzyme on the K_m value were investigated, using molecular docking studies.     

Occurrence of dead core in catalytic particles containing immobilized enzymes: analysis for the Michaelis–Menten kinetics and assessment of numerical methods

In this article, the occurrence of dead core in catalytic particles containing immobilized enzymes is analyzed for the Michaelis–Menten kinetics. An assessment of numerical methods is performed to solve the boundary value problem generated by the mathematical modeling of diffusion and reaction processes under steady state and isothermal conditions. Two classes of numerical methods were employed: shooting and collocation. The shooting method used the ode function from Scilab software. The collocation methods included: that implemented by the bvode function of Scilab, the orthogonal collocation, and the orthogonal collocation on finite elements. The methods were validated for simplified forms of the Michaelis–Menten equation (zero-order and first-order kinetics), for which analytical solutions are available. Among the methods covered in this article, the orthogonal collocation on finite elements proved to be the most robust and efficient method to solve the boundary value problem concerning Michaelis–Menten kinetics. For this enzyme kinetics, it was found that the dead core can occur when verified certain conditions of diffusion–reaction within the catalytic particle. The application of the concepts and methods presented in this study will allow for a more generalized analysis and more accurate designs of heterogeneous enzymatic reactors.     

Perturbation Theory in the Catalytic Rate Constant of the Henri–Michaelis–Menten Enzymatic Reaction

The Henry–Michaelis–Menten (HMM) mechanism of enzymatic reaction is studied by means of perturbation theory in the reaction rate constant k ₂ of product formation. We present analytical solutions that provide the concentrations of the enzyme (E), the substrate (S), as well as those of the enzyme-substrate complex (C), and the product (P) as functions of time. For k ₂ small compared to k _?1, we properly describe the entire enzymatic activity from the beginning of the reaction up to longer times without imposing extra conditions on the initial concentrations E _o and?S _o , which can be comparable or much different.     

Evolutionary optimization of the catalytic efficiency of enzymes.

1. The rate equation for a generalized Michaelian type of enzymic reaction mechanism has been analyzed in order to establish how the mechanism should be kinetically designed in order to optimize the catalytic efficiency of the enzyme for a given average magnitude of true and apparent first-order rate constants in the mechanism at given concentrations of enzyme, substrate and product. 2. As long as on-velocity constants for substrate and product binding to the enzyme have not reached the limiting value for a diffusion-controlled association process, the optimal state of enzyme operation will be characterized by forward (true and apparent) first-order rate constants of equal magnitude and reverse rate constants of equal magnitude. The drop in free energy driving the catalysed reaction will occur to an equal extent for each reaction step in the mechanism. All internal equilibrium constants will be of equal magnitude and reflect only the closeness of the catalysed reaction to equilibrium conditions. 3. When magnitudes of on-velocity constants for substrate and product binding have reached their upper limits, the optimal kinetic design of the reaction mechanism becomes more complex and has to be established by numerical methods. Numerical solutions, calculated for triosephosphate isomerase, indicate that this particular enzyme may or may not be considered to exhibit close to maximal efficiency, depending on what value is assigned to the upper limit for a ligand association rate constant. 4. Arguments are presented to show that no useful information on the evolutionary optimization of the catalytic efficiency of enzymes can be obtained by previously taken approaches that are based on the application of linear free-energy relationships for rate and equilibrium constants in the reaction mechanism.     

A note on the kinetics of enzyme action: A decomposition that highlights thermodynamic effects

Michaelis and Menten’s mechanism for enzymatic catalysis is remarkable both in its simplicity and its wide applicability. The extension for reversible processes, as done by Haldane, makes it even more relevant as most enzymes catalyze reactions that are reversible in nature and carry in vivo flux in both directions. Here, we decompose the reversible Michaelis–Menten equation into three terms, each with a clear physical meaning: catalytic capacity, substrate saturation and thermodynamic driving force. This decomposition facilitates a better understanding of enzyme kinetics and highlights the relationship between thermodynamics and kinetics, a relationship which is often neglected. We further demonstrate how our separable rate law can be understood from different points of view, shedding light on factors shaping enzyme catalysis.