Michaelis-Menten Rate Equation Research Articles

The Removal of Time-Concentration Data Points from Progress Curves Improves the Determination of Km: The Example of Paraoxonase 1.

Several approaches for determining an enzyme’s kinetic parameter Km (Michaelis constant) from progress curves have been developed in recent decades. In the present article, we compare different approaches on a set of experimental measurements of lactonase activity of paraoxonase 1 (PON1): (1) a differential-equation-based Michaelis–Menten (MM) reaction model in the program Dynafit; (2) an integrated MM rate equation, based on an approximation of the Lambert W function, in the program GraphPad Prism; (3) various techniques based on initial rates; and (4) the novel program “iFIT”, based on a method that removes data points outside the area of maximum curvature from the progress curve, before analysis with the integrated MM rate equation. We concluded that the integrated MM rate equation alone does not determine kinetic parameters precisely enough; however, when coupled with a method that removes data points (e.g., iFIT), it is highly precise. The results of iFIT are comparable to the results of Dynafit and outperform those of the approach with initial rates or with fitting the entire progress curve in GraphPad Prism; however, iFIT is simpler to use and does not require inputting a reaction mechanism. Removing unnecessary points from progress curves and focusing on the area around the maximum curvature is highly advised for all researchers determining Km values from progress curves.

Understanding the kinetics of catalysed reactions in microheterogeneous thin film electrodes

In this paper which is dedicated to Professor Richard Compton for his 65th birthday, we examine the problem of describing the transport and kinetics of catalytic reactions in which the catalyst is immobilized within a support matrix such as for example redox enzymes immobilized in a polymer matrix, to form a chemically modified electrode. We examine a mathematical procedure which enables a full analytical solution to the Michaelis-Menten kinetic rate equation when coupled to Ficksian diffusion in thin bounded film. This governing reaction/diffusion equation is non-linear and a full analytical solution has, up until very recently, not been developed. Analytical solutions valid for low and high substrate concentrations have been previously reported. This solution for the amperometric steady state current is accurate whatever the value of substrate concentration. The analysis is applied to diffusion/reaction in a planar slab. General analytical solutions valid for steady state conditions for both the amperometric and potentiometric sensor response are provided. We then extend this useful analysis to consider the effect of concentration polarization in the solution, and to consider the effect of competitive inhibition on the amperometric current response. Finally, the analysis is extended to a polymer modified electrode when a redox mediator is used in the polymer film. General expressions for the current are obtained which were valid for any value of the substrate concentration. Kinetic case diagrams are developed and nine approximate limiting expressions for the amperometric response at steady state when the catalytic matrix is either conducting or insulating are developed.

Analysis of the Langmuir rate equation in its differential and integrated form for adsorption processes and a comparison with the pseudo first and pseudo second order models

We present a new factorized form of the differential Langmuir rate equation and exhaustively examine the conditions under which the Langmuir kinetics reduces to the widely used pseudo-first order (PFO) and pseudo-second order (PSO) models. A graphical method is proposed and tested with experimental data from the literature for assessing the PFO and PSO models as substitutes for the Langmuir equation and estimating the adsorption parameters. We show that the integrated form of the differential Langmuir rate equation, originally derived by Marczewski (Langmuir 26(19):15229–15238, 2010), can be conveniently expressed in terms of five physically interpretable parameters commonly used in adsorption studies, namely the initial solute concentration, the adsorbent dosage, the microscopic adsorption rate constant, the adsorption amount at the equilibrium and the maximum adsorption capacity of the adsorbent. The potential applicability of the integrated Langmuir equation for the determination of the kinetic parameters of the Michaelis–Menten rate equation is also briefly discussed.

Conformational Nonequilibrium Enzyme Kinetics: Generalized Michaelis-Menten Equation.

In a conformational nonequilibrium steady state (cNESS), enzyme turnover is modulated by the underlying conformational dynamics. On the basis of a discrete kinetic network model, we use an integrated probability flux balance method to derive the cNESS turnover rate for a conformation-modulated enzymatic reaction. The traditional Michaelis-Menten (MM) rate equation is extended to a generalized form, which includes non-MM corrections induced by conformational population currents within combined cyclic kinetic loops. When conformational detailed balance is satisfied, the turnover rate reduces to the MM functional form, explaining its general validity. For the first time, a one-to-one correspondence is established between non-MM terms and combined cyclic loops with unbalanced conformational currents. Cooperativity resulting from nonequilibrium conformational dynamics can be achieved in enzymatic reactions, and we provide a novel, rigorous means of predicting and characterizing such behavior. Our generalized MM equation affords a systematic approach for exploring cNESS enzyme kinetics.

Including Rebinding Reactions in Well-Mixed Models of Distributive Biochemical Reactions

The behavior of biochemical reactions requiring repeated enzymatic substrate modification depends critically on whether the enzymes act processively or distributively. Whereas processive enzymes bind only once to a substrate before carrying out a sequence of modifications, distributive enzymes release the substrate after each modification and thus require repeated bindings. Recent experimental and computational studies have revealed that distributive enzymes can act processively due to rapid rebindings (so-called quasi-processivity). In this study, we derive an analytical estimate of the probability of rapid rebinding and show that well-mixed ordinary differential equation models can use this probability to quantitatively replicate the behavior of spatial models. Importantly, rebinding requires that connections be added to the well-mixed reaction network; merely modifying rate constants is insufficient. We then use these well-mixed models to suggest experiments to 1) detect quasi-processivity and 2) test the theory. Finally, we show that rapid rebindings drastically alter the reaction’s Michaelis-Menten rate equations.

Finding Robust Adaptation Gene Regulatory Networks Using Multi-Objective Genetic Algorithm.

Robust adaptation plays a key role in gene regulatory networks, and it is thought to be an important attribute for the organic or cells to survive in fluctuating conditions. In this paper, a simplified three-node enzyme network is modeled by the Michaelis-Menten rate equations for all possible topologies, and a family of topologies and the corresponding parameter sets of the network with satisfactory adaptation are obtained using the multi-objective genetic algorithm. The proposed approach improves the computation efficiency significantly as compared to the time consuming exhaustive searching method. This approach provides a systemic way for searching the feasible topologies and the corresponding parameter sets to make the gene regulatory networks have robust adaptation. The proposed methodology, owing to its universality and simplicity, can be used to address more complex issues in biological networks.

Identification of robust adaptation gene regulatory network parameters using an improved particle swarm optimization algorithm.

Robust adaptation is a critical ability of gene regulatory network (GRN) to survive in a fluctuating environment, which represents the system responding to an input stimulus rapidly and then returning to its pre-stimulus steady state timely. In this paper, the GRN is modeled using the Michaelis-Menten rate equations, which are highly nonlinear differential equations containing 12 undetermined parameters. The robust adaption is quantitatively described by two conflicting indices. To identify the parameter sets in order to confer the GRNs with robust adaptation is a multi-variable, multi-objective, and multi-peak optimization problem, which is difficult to acquire satisfactory solutions especially high-quality solutions. A new best-neighbor particle swarm optimization algorithm is proposed to implement this task. The proposed algorithm employs a Latin hypercube sampling method to generate the initial population. The particle crossover operation and elitist preservation strategy are also used in the proposed algorithm. The simulation results revealed that the proposed algorithm could identify multiple solutions in one time running. Moreover, it demonstrated a superior performance as compared to the previous methods in the sense of detecting more high-quality solutions within an acceptable time. The proposed methodology, owing to its universality and simplicity, is useful for providing the guidance to design GRN with superior robust adaptation.

A Finite-Difference Solution of Solute Transport through a Membrane Bioreactor

The current paper presents a theoretical analysis of the transport of solutes through a fixed-film membrane bioreactor (MBR), immobilised with an active biocatalyst. The dimensionless convection-diffusion equation with variable coefficients was solved analytically and numerically for concentration profiles of the solutes through the MBR. The analytical solution makes use of regular perturbation and accounts for radial convective flow as well as axial diffusion of the substrate species. The Michaelis-Menten (or Monod) rate equation was assumed for the sink term, and the perturbation was extended up to second-order. In the analytical solution only the first-order limit of the Michaelis-Menten equation was considered; hence the linearized equation was solved. In the numerical solution, however, this restriction was lifted. The solution of the nonlinear, elliptic, partial differential equation was based on an implicit finite-difference method (FDM). An upwind scheme was employed for numerical stability. The resulting algebraic equations were solved simultaneously using the multivariate Newton-Raphson iteration method. The solution allows for the evaluation of the effect on the concentration profiles of (i) the radial and axial convective velocity, (ii) the convective mass transfer rates, (iii) the reaction rates, (iv) the fraction retentate, and (v) the aspect ratio.

Multiple approaches to predicting oxygen and glucose consumptions by HepG2 cells on porous scaffolds in an axial-flow bioreactor.

In this study, the distribution of oxygen and glucose was evaluated along with consumption by hepatocytes using three different approaches. The methods include (i) Computational Fluid Dynamics (CFD) simulation, (ii) residence time distribution (RTD) analysis using a step-input coupled with segregation model or dispersion model, and (iii) experimentally determined consumption by HepG2 cells in an open-loop. Chitosan-gelatin (CG) scaffolds prepared by freeze-drying and polycaprolactone (PCL) scaffolds prepared by salt leaching technique were utilized for RTD analyses. The scaffold characteristics were used in CFD simulations i.e. Brinkman's equation for flow through porous medium, structural mechanics for fluid induced scaffold deformation, and advection-diffusion equation coupled with Michaelis-Menten rate equations for nutrient consumption. With the assumption that each hepatocyte behaves like a micro-batch reactor within the scaffold, segregation model was combined with RTD to determine exit concentration. A flow rate of 1 mL/min was used in the bioreactor seeded with 0.6 × 10(6) HepG2 cells/cm(3) on CG scaffolds and oxygen consumption was measured using two flow-through electrodes located at the inlet and outlet. Glucose in the spent growth medium was also analyzed. RTD results showed distribution of nutrients to depend on the surface characteristics of scaffolds. Comparisons of outlet oxygen concentrations between the simulation results, and experimental results showed good agreement with the dispersion model. Outlet oxygen concentrations from segregation model predictions were lower. Doubling the cell density showed a need for increasing the flow rate in CFD simulations. This integrated approach provide a useful strategy in designing bioreactors and monitoring tissue regeneration.

Exact and approximate solutions for the decades‐old Michaelis‐Menten equation: Progress‐curve analysis through integrated rate equations

The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate V, and the Michaelis constant K(M) ) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to understand fully, or can even be misunderstood, by students when based only on the differential form of the Michaelis-Menten equation, and the variety of methods available to calculate the kinetic constants from rate versus substrate concentration "textbook data." Consequently, enzyme kinetics can be confusing if an analytical solution of the Michaelis-Menten equation is not available. Therefore, the still rarely known exact solution to the Michaelis-Menten equation is presented here through the explicit closed-form equation in terms of the Lambert W(x) function. Unfortunately, as the W(x) is not available in standard curve-fitting computer programs, the practical use of this direct solution is limited for most life-science students. Thus, the purpose of this article is to provide analytical approximations to the equation for modeling Michaelis-Menten kinetics. The elementary and explicit nature of these approximations can provide students with direct and simple estimations of kinetic parameters from raw experimental time-course data. The Michaelis-Menten kinetics studied in the latter context can provide an ideal alternative to the 100-year-old problems of data transformation, graphical visualization, and data analysis of enzyme-catalyzed reactions. Hence, the content of the course presented here could gradually become an important component of the modern biochemistry curriculum in the 21st century.

Evaluation of enzyme kinetic parameters using explicit analytic approximations to the solution of the Michaelis–Menten equation

The time-course of product accumulation during an enzyme-catalyzed reaction that conforms to the Michaelis–Menten rate equation is expressed by an explicit closed-form equation in terms of the Lambert W(x) function. Unfortunately, the use of direct solution of the Michaelis–Menten equation is limited, because the W(x) function is not widely available in curve-fitting software. The present commentary suggests an alternative to surmount this difficulty. The Lambert W(x) function can be approximated in terms of the elementary mathematical functions that enable the fitting of particular equations on time-course data of the Michaelis–Menten enzyme reaction by any nonlinear regression computer program. Three different demonstrated approximations of the W(x) with relatively high accuracies are shown here to be appropriate for use when progress curves are analyzed by the direct solution of the integrated Michaelis–Menten equation. The elementary and precise nature of these approximations makes them the most user-friendly open candidates for simple, but operative, kinetic parameter estimations from experimental time-course data.

The Kinetics of Catalytic Oxidation of 1-Butanethiol in a Gas Liquid System

In the present study the kinetics of liquid phase oxidation of light thiols using reformulated cobalt phthalocyanine based catalyst has been investigated in a semi batch bubble column reactor. 1-Butanethiol is chosen to represent thiols in lighter range of petroleum fractions like liquefied petroleum gas and light straight run naphtha. The study has shown that rate of catalytic oxidation of 1-butanethiol is linear at lower concentrations and of nonlinear nature at higher concentrations. A mechanism based rate expression has been obtained to explain this behavior and its parameters are estimated. The derived rate law is similar to Michaelis-Menten rate equation.

Hyperbolic Saturation

Several hyperbolically saturating empirical models, such as the Michaelis-Menten rate equation, Monod's relative population growth rate, competitive inhibition, and Langmuir's adsorption, are rederived from a simple queuing relation. The resulting derivations reveal and potentially explain the underlying structure and meaning of such empirical models. This view is proposed as a unifying heuristic.

Defining Network Topologies that Can Achieve Biochemical Adaptation

Many signaling systems show adaptation-the ability to reset themselves after responding to a stimulus. We computationally searched all possible three-node enzyme network topologies to identify those that could perform adaptation. Only two major core topologies emerge as robust solutions: a negative feedback loop with a buffering node and an incoherent feedforward loop with a proportioner node. Minimal circuits containing these topologies are, within proper regions of parameter space, sufficient to achieve adaptation. More complex circuits that robustly perform adaptation all contain at least one of these topologies at their core. This analysis yields a design table highlighting a finite set of adaptive circuits. Despite the diversity of possible biochemical networks, it may be common to find that only a finite set of core topologies can execute a particular function. These design rules provide a framework for functionally classifying complex natural networks and a manual for engineering networks. For a video summary of this article, see the PaperFlick file with the Supplemental Data available online.

Determinação das propriedades catalíticas em meio aquoso e orgânico da lipase de Candida rugosa imobilizada em celulignina quimicamente modificada por carbonildiimidazol

Microbial lipase from Candida rugosa was immobilized by covalent binding on wood cellulignin (Eucaliptus grandis) chemically modified with carbonyldiimidazole. The immobilized system was fully evaluated in aqueous (olive oil hydrolysis) and organic (ester synthesis) media. A comparative study between free and immobilized lipase was carried out in terms of pH, temperature and thermal stability. A higher pH value (8.0) was found optimal for the immobilized lipase. The optimal reaction temperature shifted from 37 °C for the free lipase to 45 °C for the immobilized lipase. The pattern of heat stability indicated that the immobilization process tends to stabilize the enzyme. Kinetics tests at 37 °C following the hydrolysis of olive oil obeyed the Michaelis-Menten rate equation. Values for Km = 924.9 mM and Vmax = 198.3 U/mg were lower than for free lipase, suggesting that the affinity towards the substrate changed and the activity of the immobilized lipase decreased during the course of immobilization. The immobilized derivative was also tested in the ester synthesis from several alcohols and carboxylic acids.

Kinetics of microbial sulfate reduction in estuarine sediments

Kinetic parameters of microbial sulfate reduction in intertidal sediments from a freshwater, brackish and marine site of the Scheldt estuary (Belgium, the Netherlands) were determined. Sulfate reduction rates (SRR) were measured at 10, 21, and 30 °C, using both flow-through reactors containing intact sediment slices and conventional sediment slurries. At the three sites, and for all depth intervals studied (0–2, 2–4, 4–6 and 6–8 cm), the dependence of potential SRR on the sulfate concentration followed the Michaelis–Menten rate equation. Apparent sulfate half-saturation concentrations, K m, measured in the flow-through reactor experiments were comparable at the freshwater and marine sites (0.1–0.3 mM), but somewhat higher at the brackish site (0.4–0.9 mM). Maximum potential SRR, R max, in the 0–4 cm depth interval of the freshwater sediments were similar to those in the 0–6 cm interval of the marine sediments (10–46 nmol cm −3 h −1 at 21 °C), despite much lower in situ sulfate availability and order-of-magnitude lower densities of sulfate-reducing bacteria (SRB), at the freshwater site. Values of R max in the brackish sediments were lower (3.7–7.6 nmol cm −3 h −1 at 21 °C), probably due to less labile organic matter, as inferred from higher C org/N ratios. Inflow solutions supplemented with lactate enhanced potential SRR at all three sites. Slurry incubations systematically yielded higher R max values than flow-through reactor experiments for the freshwater and brackish sediments, but similar values for the marine sediments. Transport limitation of potential SRR at the freshwater and brackish sites may be related to the lower sediment porosities and SRB densities compared to the marine site. Multiple rate controls, including sulfate availability, organic matter quality, temperature, and SRB abundance, modulate in situ sulfate-reducing activity along the estuarine salinity gradient.

Effects of Spontaneous Deamidation on the Cytotoxic Activity of the Bacillus anthracis Protective Antigen

Protective antigen (PA) is a central virulence factor of Bacillus anthracis and a key component in anthrax vaccines. PA binds to target cell receptors, is cleaved by the furin protease, self-aggregates to heptamers, and finally internalizes as a complex with either lethal or edema factors. Under mild room temperature storage conditions, PA cytotoxicity decreased (t(1/2) approximately 7 days) concomitant with the generation of new acidic isoforms, probably through deamidation of Asn residues. Ranking all 68 Asn residues in PA based on their predicted deamidation rates revealed five residues with half-lives of <60 days, and these residues were further analyzed: Asn10 in the 20-kDa region, Asn162 at P6 vicinal to the furin cleavage site, Asn306 in the pro-pore translocation loop, and both Asn713 and Asn719 in the receptor-binding domain. We found that PA underwent spontaneous deamidation at Asn162 upon storage concomitant with decreased susceptibility to furin. A panel of model synthetic furin substrates was used to demonstrate that Asn162 deamidation led to a 20-fold decrease in the bimolecular rate constant (k(cat)/Km) of proteolysis due to the new negatively charged residue at P6 in the furin recognition sequence. Furthermore, reduced PA cytotoxicity correlated with a decrease in PA cell binding and also with deamidation of Asn713 and Asn719. On the other hand, neither deamidation of Asn10 or Asn306 nor impairment of heptamerization could be observed upon prolonged PA storage. We suggest that PA inactivation during storage is associated with susceptible deamidation sites, which are intimately involved in both mechanisms of PA cleavage by furin and PA-receptor binding.

Kinetic models of phosphorylation cycles: A systematic approach using the rapid-equilibrium approximation for protein–protein interactions

Activation–inactivation cycles of signalling proteins and transcription factors catalysed by kinases and phosphatases are a core component of cellular signal transduction. We present a systematic kinetic analysis of a phosphorylation cycle that starts from the description of elementary protein–protein interaction and catalytic steps. A rapid-equilibrium approximation for protein interactions is used to reduce the set of parameters. The resulting description consists of a kinetic equation for the phosphorylation of the target and a set of conservation conditions for kinase and phosphatase. Generally no explicit rate laws exist for the two enzymes; linear or Michaelis–Menten rate equations can be obtained in special cases. Key parameters that determine the stimulus–response curve and the response time of the cycle are the concentrations of kinase and phosphatase relative to the target protein and the affinities of the two enzymes for the different phosphorylation states of the target. Characterizing the response curve by the global response coefficient, we obtain a phase diagram that shows the existence of three kinds of behaviours: graded, ultrasensitive, and a previously undescribed biphasic response. Two kinds of competition effect turn out to shape the behaviour: (1) the degree of product inhibition of each enzyme, and (2) the competition between kinase and phosphatase to bind the target protein, as determined by their relative target affinities. The approach outlined here may be useful also for analysing more complex systems, including multiple phosphorylation and kinase cascades.

Topochemical kinetics of xylanase action on kraft pulp

The fibre structure and xylan distribution of wood pulp make the classical Michaelis-Menten rate equation inappropriate for describing the enzyme kinetics of decomposition. Therefore a topochemical modified Prout-Tompkins equation has been analysed and applied to the action of xylanase on kraft pulp.The power coefficient in this kinetic equation provides information about the influence of the amount of product formed and residual substrate on the process rate. It was found that the latter determined the rate while the influence of the product is limited only at the begining of the process.

The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables

The estimation of enzyme kinetic parameters by nonlinear fitting reaction curve to the integrated Michaelis–Menten rate equation ln( S 0/ S)+( S 0− S)/ K m=( V m/ K m)× twas investigated and compared to that by fitting to ( S 0− S)/ t= V m− K m×[ln( S 0/ S)/ t] (Atkins GL, Nimmo IA. The reliability of Michaelis-Menten constants and maximum velocities estimated by using the integrated Michaelis-Menten equation. Biochem J 1973;135:779-84) with uricase as the model. Uricase reaction curve was simulated with random absorbance error of 0.001 at 0.075 mmol/l uric acid. Experimental reaction curve was monitored by absorbance at 293 nm. For both CV and deviation <20% by simulation, K m from 5 to 100 μmol/l was estimated with Eq. (1) while K m from 5 to 50 μmol/l was estimated with Eq. (2). The background absorbance and the error in the lag time of steady-state reaction resulted in negative K m with Eq. (2), but did not affect K m estimated with Eq. (1). Both equations gave better estimation of V m. The computation time and the goodness of fit with Eq. (1) were 40-fold greater than those with Eq. (2). By experimentation, Eq. (1) yielded K m consistent with the Lineweaver–Burk plot analysis, but Eq. (2) gave many negative parameters. Apparent K m by Eq. (1) linearly increased, while V m were constant, vs. xanthine concentrations, and the inhibition constant was consistent with the Lineweaver–Burk plot analysis. These results suggested that the integrated rate equation that uses the predictor variable of reaction time was reliable for the estimation of enzyme kinetic parameters and applicable for the characterization of enzyme inhibitors.