Henri-Michaelis-Menten Research Articles

The measurement of true initial rates is not always absolutely necessary to estimate enzyme kinetic parameters

In the chapters dealing with enzyme reactions, the authors of all Biochemistry textbooks and of even more specialized texts consider that the characteristic parameters (kcat and Km) must be determined under initial or steady-state rate conditions. This implies the transformation of a very limited proportion of substrate (at most 10–20%) or a continuous recording of the product or substrate concentration vs. time. Both options can present practical difficulties. Is it possible to get around these very stringent conditions? Here we show that in the most favourable cases up to 70% of the substrate can be converted resulting in systematic errors on the parameters (that can easily be taken account of) if the simple Henri-Michaelis–Menten equation is utilised. Alternatively, the integrated form of the same equation directly yields excellent estimates of the same parameters. Our observations should greatly facilitate the task of researchers who study systems in which measurements of the reaction progress are painstaking or when substrate concentrations close to the detection limit must be used. The general conclusion is that it is not always absolutely necessary to determine initial or steady-state rates to obtain reliable estimations of the enzyme kinetic parameters..

Metabolic Reactions of Henri Type and Their Usage in Metabolic and Cell Growth Models

We study mathematically and computationally time-course experimental data of metabolic processes and batch fermentation processes. The measurement data are given together with bounds for the involved systematic errors. We describe and motivate our modelling approach on the case study with the Henri-Michaelis-Menten (HMM) biochemical reaction of the enzyme-substrate dynamics where two fractions of enzymes (free and bound) are involved. We discuss i) computational approaches and tools for the estimation of the rate parameters from time-course measurements; ii) the application of Henri reaction mechanism in metabolic networks and cell growth models. In the latter case, two fractions of the bacterial cells are considered: dividing and non-dividing cells. The Henri reaction scheme is used to describe the transition of bacterial cells from one fraction to another as well as the cell growth due to substrate uptake. Interval methods and advanced computational tools are used whenever model validation issues are discussed.

Modeling the respiratory chain complexes with biothermokinetic equations — The case of complex I

The mitochondrial respiratory chain plays a crucial role in energy metabolism and its dysfunction is implicated in a wide range of human diseases. In order to understand the global expression of local mutations in the rate of oxygen consumption or in the production of adenosine triphosphate (ATP) it is useful to have a mathematical model in which the changes in a given respiratory complex are properly modeled. Our aim in this paper is to provide thermodynamics respecting and structurally simple equations to represent the kinetics of each isolated complexes which can, assembled in a dynamical system, also simulate the behavior of the respiratory chain, as a whole, under a large set of different physiological and pathological conditions. On the example of the reduced nicotinamide adenine dinucleotide (NADH)–ubiquinol–oxidoreductase (complex I) we analyze the suitability of different types of rate equations. Based on our kinetic experiments we show that very simple rate laws, as those often used in many respiratory chain models, fail to describe the kinetic behavior when applied to a wide concentration range. This led us to adapt rate equations containing the essential parameters of enzyme kinetic, maximal velocities and Henri–Michaelis–Menten like-constants (KM and KI) to satisfactorily simulate these data.

On the Mathematical Modelling of Fermentation Processes Using Uncertain Data: Case Studies

Experimental data for the dynamics of fermentation processes including enzymes, substrate and product are obtained together with bounds for the systematic errors in the data. The availability of error bound allows us to study mathematically and computationally these experimental data using verication methods paying special attention to the errors involved. We describe and motivate our modelling approach on several case studies applying the Henri-Michaelis-Menten (HMM) biochemical reaction schemes of the enzyme-substrate dynamics where two fractions of enzymes (free and bound) are involved. We briefly demosntrate how basic ideas from these case studies can be applied to fermentation processing involving bacteria cells. The aim of this presentation is to demonstrate how verication methods can be used to conrm or reject the proposed mathematical models as possible fermentation mechanisms. Advanced computational and visualization tools are demonstrated.

Computational Studies of Some Structured Cell Growth Models

We study mathematically and computationally experimental data of batch fermentation processes. The data for the bacterial biomass, substrate and product are given together with bounds for the involved systematic errors. We first describe and motivate our modelling approach on the case study with the Henri-Michaelis-Menten (HMM) biochemical reaction of the enzyme-substrate dynamics where two fractions of enzymes (free and bound) are involved. Similarly, two phases (fractions) of the bacterial cells are considered: dividing and non-dividing cells. The HMM reaction scheme is used to describe the transitition of bacterial cells from one phase to another as well as the cell growth due to substrate uptake. The reaction schemes are formulated with a strong emphasis on the enzymatic competence of the cells as well as their reproduction abilities. The aim of this study is to confirm or reject the proposed reaction schemes as possible cell growth mechanisms. To this end we use verification interval methods and advanced computational tools.

On the Mathematical Modelling of Nitrate Reduction Using Reaction Schemes

Experimental data for nitrate reduction including bacterial biomass, substrate and product are obtained together with bounds for the systematic errors in the data [1]. We analyse the data using a modelling approach close to enzyme kinetic and more speci cally to Henri-Michaelis-Menten (HMM) biochemical reaction approach. To this end two phases (fractions) of the bacterial cells are considered: dividing and non-dividing cells and HMM-type reactions are used to decribe the transitition of bacterial cells from one phase to another as well as the product synthesis and cell growth due to substrate consumption and metabolic conversion [2]. The reaction schemes are formulated with a strong emphasis on the enzymatic competence of the cells as well as their reproduction abilities. The aim of this study is to con rm or reject the proposed reaction schemes as possible cell growth mechanisms. To this end we use use veri cation interval methods and advanced computational tools.

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 ) andS ( o ), which can be comparable or much different.

Substrate and enzyme concentration dependence of the Henri–Michaelis–Menten model probed by numerical simulation

The use of the classic Henry–Michaelis–Menten (HMM) model (or simply, Michaelis–Menten model) to study the substrate and enzyme concentration dependence of enzyme catalysis is a very important step in understanding many biochemical processes, including microbial growth. Although the HMM model has been extensively studied, the conditions in which the substrate concentration is not in excess have still not been adequately defined mathematically. This lack of definition occurs despite at the cellular and molecular levels most systems generally do not operate in a state of substrate excess. In the present work, we describe an approach for studying enzyme reactions in which substrate concentrations are not in excess. Our results show that the use of extent of reactions and numerical simulation of the velocities of reaction provides an important advance in this field and furnishes results not obtained in previous studies involving these aspects. This approach, in association with knowledge of the rate constants, provides a direct and easy means of examining the single substrate–enzyme profile during product formation at any enzyme–substrate ratio. This approach is more direct than previous models that required the use of empirical equations with arbitrary constants.

Theoretical and computational studies of some bioreactor models

We study certain classical basic models for bioreactor simulation in case of batch mode with decay. It is shown that in many cases the two-dimensional differential system describing the dynamics of the substrate and biomass concentrations can be reduced to an algebraic equation for the biomass together with a single differential equation for the substrate. Then from an analogy with the Henri–Michaelis–Menten enzyme kinetic mechanism a simple model is proposed for a bioreactor in batch mode with decay. Two more models are also proposed taking into account the phases of microbial growth. Some properties of these two models are studied and compared to classical Monod type models using computer simulations.

About and beyond the Henri-Michaelis–Menten rate equation for single-substrate enzyme kinetics

For more than a century the simple single-substrate enzyme kinetics model and related Henri-Michaelis–Menten (HMM) rate equation have been thoroughly explored in various directions. In the present paper we are concerned with a possible generalization of this rate equation recently proposed by F. Kargi (BBRC 382 (2009) 157–159), which is assumed to be valid both in the case that the total substrate or enzyme is in excess and the quasi-steady-state is achieved. We demonstrate that this generalization is grossly inadequate and propose another generalization based on application of the quasi-steady-state condition and conservation equations for both enzyme and substrate. The standard HMM equation is derived by (a) assuming the quasi-steady-state condition, (b) applying the conservation equation only for the enzyme, and (c) assuming that the substrate concentration at quasi-steady-state can be approximated by the total substrate concentration [S]0. In our formula the rate is already expressed through [S]0, and we only assume that when quasi-steady-state is achieved the amount of product formed is negligible compared to [S]0. Numerical simulations show that our formula is generally more accurate than the HMM formula and also can provide a good approximation when the enzyme is in excess, which is not the case for the HMM formula. We show that the HMM formula can be derived from our expression by further assuming that the total enzyme concentration is negligible compared to [S]0.

Multiple Conformers in Active Site of Human Dihydrofolate Reductase F31R/Q35E Double Mutant Suggest Structural Basis for Methotrexate Resistance

Methotrexate is a slow, tight-binding, competitive inhibitor of human dihydrofolate reductase (hDHFR), an enzyme that provides key metabolites for nucleotide biosynthesis. In an effort to better characterize ligand binding in drug resistance, we have previously engineered hDHFR variant F31R/Q35E. This variant displays a >650-fold decrease in methotrexate affinity, while maintaining catalytic activity comparable to the native enzyme. To elucidate the molecular basis of decreased methotrexate affinity in the doubly substituted variant, we determined kinetic and inhibitory parameters for the simple variants F31R and Q35E. This demonstrated that the important decrease of methotrexate affinity in variant F31R/Q35E is a result of synergistic effects of the combined substitutions. To better understand the structural cause of this synergy, we obtained the crystal structure of hDHFR variant F31R/Q35E complexed with methotrexate at 1.7-A resolution. The mutated residue Arg-31 was observed in multiple conformers. In addition, seven native active-site residues were observed in more than one conformation, which is not characteristic of the wild-type enzyme. This suggests that increased residue disorder underlies the observed methotrexate resistance. We observe a considerable loss of van der Waals and polar contacts with the p-aminobenzoic acid and glutamate moieties. The multiple conformers of Arg-31 further suggest that the amino acid substitutions may decrease the isomerization step required for tight binding of methotrexate. Molecular docking with folate corroborates this hypothesis.

A Random-sequential Mechanism for Nitrite Binding and Active Site Reduction in Copper-containing Nitrite Reductase

The homotrimeric copper-containing nitrite reductase (NiR) contains one type-1 and one type-2 copper center per monomer. Electrons enter through the type-1 site and are shuttled to the type-2 site where nitrite is reduced to nitric oxide. To investigate the catalytic mechanism of NiR the effects of pH and nitrite on the turnover rate in the presence of three different electron donors at saturating concentrations were measured. The activity of NiR was also measured electrochemically by exploiting direct electron transfer to the enzyme immobilized on a graphite rotating disk electrode. In all cases, the steady-state kinetics fitted excellently to a random-sequential mechanism in which electron transfer from the type-1 to the type-2 site is rate-limiting. At low [NO(-)(2)] reduction of the type-2 site precedes nitrite binding, at high [NO(-)(2)] the reverse occurs. Below pH 6.5, the catalytic activity diminished at higher nitrite concentrations, in agreement with electron transfer being slower to the nitrite-bound type-2 site than to the water-bound type-2 site. Above pH 6.5, substrate activation is observed, in agreement with electron transfer to the nitrite-bound type-2 site being faster than electron transfer to the hydroxyl-bound type-2 site. To study the effect of slower electron transfer between the type-1 and type-2 site, NiR M150T was used. It has a type-1 site with a 125-mV higher midpoint potential and a 0.3-eV higher reorganization energy leading to an approximately 50-fold slower intramolecular electron transfer to the type-2 site. The results confirm that NiR employs a random-sequential mechanism.

Substrate-dependent, non-hyperbolic kinetics of pig brain prolyl oligopeptidase and its tight binding inhibition by JTP-4819

Prolyl oligopeptidase (POP) is a cytosolic serine protease that hydrolyses small peptides at the carboxyl end of the proline residue. It has raised pharmaceutical interest, since its inhibitors have been shown to have antiamnesic properties. We studied prolyl oligopeptidase kinetics with two 7-amino-4-methylcoumarin derivatives: Z-Gly-Pro-AMC and Suc-Gly-Pro-AMC. Z-Gly-Pro-AMC was found to obey standard Henri–Michaelis–Menten kinetics with a K m of 30±3 μM, whereas Suc-Gly-Pro-AMC exhibited substrate inhibition kinetics with K m and K is of 510±150 and 270±90 μM, respectively. Autodock simulations revealed that either the succinyl or the AMC-end of Suc-Gly-Pro-AMC may bind to the S′ 1 subsite of the active site. We believe that non-specifically bound Suc-Gly-Pro-AMC allows the simultaneous binding of second substrate molecule to the active site and this leads in substrate inhibition. In addition, we demonstrated that the inhibition type of a well characterized prolyl oligopeptidase inhibitor, JTP-4819, is competitive tight binding with a K ic of 0.045±0.008 nM. We suggest that due to the high concentration of prolyl oligopeptidase in the brain (0.12 n mol/g pig brain), the tight binding nature of the inhibition should be considered when using brain homogenate as the enzyme source in prolyl oligopeptidase inhibition measurements. This is of importance in studying structure–activity relationships of potent prolyl oligopeptidase inhibitors.

Significance of the Henri–Michaelis–Menten theory in abiotic catalysis: catechol oxidation by δ-MnO 2

The Henri–Michaelis–Menten theory, for more than eight decades, was only restricted to homogeneous enzymatic catalysis. A mimic of an enzymatic kinetics based on the Henri–Michaelis–Menten concept was experimentally observed in heterogeneous catalysis in the present study with δ-MnO 2 as an abiotic catalyst in the oxidation of catechol (1,2-dihydroxybenzene). Using the derived linear forms of Lineweaver–Burk or Hofstee, the data show that similar to the enzyme tyrosinase, the kinetics of the catechol oxidation catalyzed by δ-MnO 2 can be described by the Henri–Michaelis–Menten equation, V 0= V max S/( K m+ S), where V max is the maximum velocity and K m the concentration of the substrate ( S) corresponding to an initial velocity ( V 0) half of V max. By analogy to the enzymatic kinetics, the parameters V max and K m for an heterogeneous abiotic catalysis were derived for the first time. Further, based on the concentration of the active centers of the mineral oxide, the kinetic constants k cat and k cat/ K m, respectively, representing the turnover frequency and the efficiency of the mineral catalyst, were also determined from the derived general rate equation of Briggs and Haldane. As an abiotic catalyst, δ-MnO 2 has a paramount role in the oxidation of phenolic compounds in soil, sediment and water environments. Therefore, the present observation is of fundamental and practical significance in elucidating the affinity between an abiotic catalyst and a substrate based on the Henri–Michaelis–Menten theory.

Esterification of oleic acid and ethanol in plug flow (packed bed) reactor under supercritical conditions: Investigation of kinetics

The present work provided the reaction kinetic data, required to predict the performance of packed-bed plug flow bioreactor operating under supercritical conditions. It investigated the kinetics of esterification of oleic acid and ethanol. Biocatalyst Lypozyme™ IM60, which is lipase from Rhizomucor miehei immobilised on Duolite (an ion exchange resin), was employed to synthesise ethyl oleate. Experimentally, esterification was carried out using a range of oleic acid- and ethanol concentrations, respectively. The effect of water concentration on the enzyme activity was also investigated. Analytically, the reaction mechanism was investigated by a series of kinetic analysis using the Direct Linear, the Primary and the Burke–Lineweaver (double reciprocal) plots. The reaction was modelled by the Henri–Michaelis–Menten equation and the reaction constants, namely K m and V max, were evaluated. It was found that the initial water concentration in the system effected the enzyme initial activity in a bell-like manner, giving an optimum (enzyme activity,and water concentration) value for each alcohol concentration. Furthermore, it was found that under higher alcohol concentrations, the reaction experienced substrate inhibition. The latter was identified to be competitive inhibition under moderate alcohol concentrations and non-competitive inhibitions under high alcohol concentrations. Hence, reaction inhibition constant value K i was calculated.

Enzyme organization and stability of metabolic pathways: A comparison of various approaches

The aim of this paper is to compare various methods for the quantification of metabolic pathways dynamics. A Yates-Pardee metabolic pathway with enzyme organization, i.e. with spatial localization of the enzymes in a specific cellular compartment, was studied using: (i) the classical Henri-Michaelis-Menten (HMM) equations, (ii) linearization of the HMM equations in the vicinity of a steady state (linearized formalism), and (iii) Biochemical Systems Theory formalism (BST formalism). It is shown that transient solutions computed via either the linearized formalism or the BST formalism can greatly differ from transient solutions computed with the HMM equations. However, in the studied example, results remain qualitatively the same for the three formalisms. This suggests that the study of the ‘topology’ of the system may give useful insights into the metabolic pathways dynamics.

The Almost-one Parameter Hypothesis Applied to Pharmacokinetics of Ethanol Infusion

They say that the pharmacokinetics of constant-rate intravenous infusion of ethanol obeys the Henri-Michaelis-Menten (H-M-M) model. In analysing the original data from the viewpoint of the almost-one parameter hypothesis (see box 1), we found that the whole process should be divided into four phases and the H-M-M model was valid only in the last phase.

Calcium and magnesium ions and the regulation of multiplication in normal and transformed cells

THE relative importance of Ca2+ and Mg2+ in the regulation of cellular multiplication and alteration of that regulation by viral transformation has recently become a subject of considerable interest1–5. Extracellular Ca2+ and Mg2+ both have effects on metabolic parameters and macromolecular synthesis in cultured cells. Mg2+, because of its substrate level intracellular concentration and its role in macromolecular synthesis and intracellular transphosphorylation reactions, has been suggested as a possible coordinate regulator of the positive pleiotropic response, a coordinated array of reactions associated with initiation of DNA synthesis and multiplication in cultured cells6–9. Ca2+ on the other hand, has been suggested as a short term regulator of cell growth and function, largely on the basis of its tightly regulated low cytoplasmic concentration10,11. In this report, we describe a kinetic analysis of the effects of extracellular concentrations of Ca2+ and Mg2+ on survival and multiplication of normal and transformed human cells in culture. We have found that cellular multiplication rate (number of population doublings per unit time) and also cellular ‘survival rate’ (percentage survival divided by time) can be substituted in place of reaction rate in classical enzyme kinetic formulae (Fig. 1). Both responses exhibit saturation kinetics, with a transition from first order response to zero order, as the concentration of calcium, magnesium, or any other essential component of the culture medium is increased. Rate plotted against concentration yields the expected right rectangular hyperbola described by the Henri–Michaelis–Menten formulation, and the expected linear relation when plotted by the Lineweaver–Burke or van Hofstee methods. Linear regression of the Lineweaver–Burke double reciprocal plot yields from its intercepts a maximal rate for multiplication or survival (equivalent to Vmax in classical enzyme kinetics) and a Michaelis–Menten constant (Km), equal to the substrate (or ligand) concentration yielding a half maximal rate of multiplication or survival.