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.     

Isothermal titration calorimetry determination of individual rate constants of trypsin catalytic activity

Determination of individual rate constants for enzyme-catalyzed reactions is central to the understanding of their mechanism of action and is commonly obtained by stopped-flow kinetic experiments. However, most natural substrates either do not fluoresce/absorb or lack a significant change in their spectra while reacting and, therefore, are frequently chemically modified to render adequate molecules for their spectroscopic detection. Here, isothermal titration calorimetry (ITC) was used to obtain Michaelis–Menten plots for the trypsin-catalyzed hydrolysis of several substrates at different temperatures (278–318 K): four spectrophotometrically blind lysine and arginine N-free esters, one N-substituted arginine ester, and one amide. A global fitting of these data provided the individual rate constants and activation energies for the acylation and deacylation reactions, and the ratio of the formation and dissociation rates of the enzyme–substrate complex, leading also to the corresponding free energies of activation. The results indicate that for lysine and arginine N-free esters deacylation is the rate-limiting step, but for the N-substituted ester and the amide acylation is the slowest step. It is shown that ITC is able to produce quality kinetic data and is particularly well suited for those enzymatic reactions that cannot be measured by absorption or fluorescence spectroscopy.     

Tris(3-hydroxypropyl)phosphine is superior to dithiothreitol for in vitro assessment of vitamin K 2,3-epoxide reductase activity

Use of the reductant dithiothreitol (DTT) as a substrate for measuring vitamin K 2,3-epoxide reductase (VKOR) activity in vitro has been reported to be problematic because it enables side reactions involving the vitamin K₁ 2,3-epoxide (K₁>O) substrate. Here we characterize specific problems when using DTT and show that tris(3-hydroxypropyl)phosphine (THPP) is a reliable alternative to DTT for in vitro assessment of VKOR enzymatic activity. In addition, the pH buffering compound imidazole was found to be problematic in enhancing DTT-dependent non-enzymatic side reactions. Using THPP and phosphate-based pH buffering, we measured apparent Michaelis–Menten constants of 1.20 μM for K₁>O and 260 μM for the active neutral form of THPP. The K_m value for K₁>O is in agreement with the value that we previously obtained using DTT (1.24 μM). Using THPP, we successfully eliminated non-enzymatic production of 3-hydroxyvitamin K₁ and its previously reported base-catalyzed conversion to K₁, both of which were shown to occur when DTT and imidazole are used as the reductant and pH buffer, respectively, in the in vitro VKOR assay. Accordingly, substitution of THPP for DTT in the in vitro VKOR assay will ensure more accurate enzymatic measurements and assessment of warfarin and other 4-hydroxycoumarin inhibition constants.     

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.     

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.     

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.     

Validation of a specific prolylcarboxypeptidase activity assay and its suitability for plasma and serum measurements

Prolylcarboxypeptidase (PRCP, EC 3.4.16.2), a lysosomal carboxypeptidase, was discovered 45 years ago. However, research has been hampered by a lack of well-validated assays that are needed to measure low activities in biological samples. Two reversed-phase high-performance liquid chromatography (RP-HPLC) methods for quantifying PRCP activity in crude homogenates and plasma samples were optimized and validated. PRCP activity was determined by measuring the hydrolysis of N-benzyloxycarbonyl-l-proline (Z-Pro)-Phe. The enzymatically formed Z-Pro and Phe were measured independently under different HPLC conditions. The in-house methods showed good precision, linearity, accuracy, and specificity. Based on Michaelis–Menten constants, Z-Pro-Phe was chosen over Z-Pro-Ala as the substrate of preference. Cross-reactivity studies with dipeptidyl peptidases (DPPs) 2, 4, and 9 and prolyl oligopeptidase (PREP) confirmed the specificity of the PRCP activity assay. The average PRCP activity in plasma and serum of 32 healthy individuals was found to be 0.65 ± 0.02 and 0.72 ± 0.03 U/L, respectively. Both methods can be used to measure PRCP activity specifically in different biological samples and are well suited to evaluate PRCP inhibitors. These well-validated methods are valuable tools for studying PRCP’s role in cardiovascular diseases, stroke, inflammation, and metabolic syndrome.     

Hybrid dynamic modeling of Escherichia coli central metabolic network combining Michaelis–Menten and approximate kinetic equations

The construction of dynamic metabolic models at reaction network level requires the use of mechanistic enzymatic rate equations that comprise a large number of parameters. The lack of knowledge on these equations and the difficulty in the experimental identification of their associated parameters, represent nowadays the limiting factor in the construction of such models. In this study, we compare four alternative modeling approaches based on Michaelis–Menten kinetics for the bi-molecular reactions and different types of simplified rate equations for the remaining reactions (generalized mass action, convenience kinetics, lin-log and power-law). Using the mechanistic model for Escherichia coli central carbon metabolism as a benchmark, we investigate the alternative modeling approaches through comparative simulations analyses. The good dynamic behavior and the powerful predictive capabilities obtained using the hybrid model composed of Michaelis–Menten and the approximate lin-log kinetics indicate that this is a possible suitable approach to model complex large-scale networks where the exact rate laws are unknown.     

Characterization of human β,β-carotene-15,15′-monooxygenase (BCMO1) as a soluble monomeric enzyme

The formal first step in in vitamin A metabolism is the conversion of its natural precursor β,β-carotene (C40) to retinaldehyde (C20). This reaction is catalyzed by the enzyme β,β-carotene-15,15′-monooxygenase (BCMO1). BCMO1 has been cloned from several vertebrate species, including humans. However, knowledge about this protein’s enzymatic and structural properties is scant. Here we expressed human BCMO1 in Spodoptera frugiperda 9 insect cells. Recombinant BCMO1 is a soluble protein that displayed Michaelis–Menten kinetics with a K_M of 14 μM for β,β-carotene. Though addition of detergents failed to increase BCMO1 enzymatic activity, short chain aliphatic detergents such as C₈E₄ and C₈E₆ decreased enzymatic activity probably by interacting with the substrate binding site. Thus we purified BCMO1 in the absence of detergent. Purified BCMO1 was a monomeric enzymatically active soluble protein that did not require cofactors and displayed a turnover rate of about 8 molecules of β,β-carotene per second. The aqueous solubility of BCMO1 was confirmed in mouse liver and mammalian cells. Establishment of a protocol that yields highly active homogenous BCMO1 is an important step towards clarifying the lipophilic substrate interaction, reaction mechanism and structure of this vitamin A forming enzyme.     

A spectroscopic method to determine the activity of the restriction endonuclease EcoRV that involves a single reaction

A one-step protocol is presented to determine the activity of EcoRV as a model of restriction enzymes. The protocol involved a molecular beacon as DNA substrate, with the target sequence recognized by EcoRV in the stem. EcoRV cleaved the stem forming two fragments, one of which contained the fluorophore and quencher, initially bound by 3 bp. This shorter fragment rapidly dissociated at 37 °C, causing an increase of fluorescence intensity that was used to gauge the reaction kinetics. The reaction can be described using the Michaelis–Menten mechanism, and the kinetic parameters obtained were compared with literature values involving other protocols.     

Purification of recombinant human phosphodiesterase 7A expressed in Dictyostelium discoideum

Phosphodiesterase plays an important role in regulating inflammatory pathways and T cell function. The development of phosphodiesterase 7 inhibitor may give better efficacy profile over phosphodiesterase 4 inhibitors. However, the recombinant phosphodiesterase 7 is required in large quantity for high-throughput screening of new drugs by in vitro enzymatic assays. In the present study, recombinant human PDE7A1 was expressed in Dictyostelium discoideum under the control of constitutively active actin-15 promoter. The cytosolic localization of the expressed protein was confirmed by immunofluorescence studies. Upto 2 mg of recombinant protein was purified using His-Tag affinity column chromatography followed by ion-exchange Resource Q column purification. The recombinant protein expressed in D. discoideum followed Michaelis–Menten kinetics similar to the protein expressed in mammalian system and showed no major changes in affinity to substrate or inhibitors. Thus, our study clearly demonstrates a robust expression system for successful bulk production of pharmacologically active isoform of human PDE7A1 required for high-throughput assays.     

A capillary electrophoresis method for the characterization of ecto-nucleoside triphosphate diphosphohydrolases (NTPDases) and the analysis of inhibitors by in-capillary enzymatic microreaction

A capillary electrophoresis (CE) method for the characterization of recombinant NTPDases 1, 2, and 3, and for assaying NTPDase inhibitors has been developed performing the enzymatic reaction within the capillary. After hydrodynamic injection of plugs of substrate solution with or without inhibitor in reaction buffer, followed by a suspension of an enzyme-containing membrane preparation, and subsequent injection of another plug of substrate solution with or without inhibitor, the reaction took place close to the capillary inlet. After 5 min, the electrophoretic separation of the reaction products was initiated by applying a constant current of −60 μA. The method employing a polyacrylamide-coated capillary and reverse polarity mode provided baseline resolution of substrates and products within a short separation time of less than 7 min. A 50 mM phosphate buffer (pH 6.5) was used for the separations and the products were detected by their UV absorbance at 210 nm. The Michaelis–Menten constants (K_m) for the recombinant rat NTPDases 1, 2, and 3 obtained with this method were consistent with previously reported data. The inhibition studies revealed pronounced differences in the potency of reactive blue 2, pyridoxalphosphate-6-azophenyl-2′,4′-disulfonic acid (PPADS), suramin, and N⁶-diethyl-β,γ-dibromomethylene-ATP (ARL67156) towards the NTPDase isoforms. Notably, ARL67156 does not inhibit all NTPDases, having only a minor inhibitory effect on NTPDase2. Dipyridamole is not an inhibitor of the NTPDase isoforms investigated. The new method is fast and accurate, it requires only tiny amounts of material (nanoliter scale), no sample pretreatment and can be fully automated; thus it is clearly superior to the current standard methods.     

Extended Parker–Sochacki method for Michaelis–Menten enzymatic reaction model

In this article, a new approach—namely, the extended Parker–Sochacki method (EPSM)—is presented for solving the Michaelis–Menten nonlinear enzymatic reaction model. The Parker–Sochacki method (PSM) is combined with a new resummation method called the Sumudu–Padé resummation method to obtain approximate analytical solutions for the model. The obtained solutions by the proposed approach are compared with the solutions of PSM and the Runge–Kutta numerical method (RKM). The comparison proves the practicality, efficiency, and correctness of the presented approach. It serves as a basis for solving other nonlinear biochemical reaction models in the future.     

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.     

Metabolic response to point mutations reveals principles of modulation of in vivo enzyme activity and phenotype

The relationship between sequence variation and phenotype is poorly understood. Here, we use metabolomic analysis to elucidate the molecular mechanism underlying the filamentous phenotype of E. coli strains that carry destabilizing mutations in dihydrofolate reductase (DHFR). We find that partial loss of DHFR activity causes reversible filamentation despite SOS response indicative of DNA damage, in contrast to thymineless death (TLD) achieved by complete inhibition of DHFR activity by high concentrations of antibiotic trimethoprim. This phenotype is triggered by a disproportionate drop in intracellular dTTP, which could not be explained by drop in dTMP based on the Michaelis–Menten‐like in vitro activity curve of thymidylate kinase (Tmk), a downstream enzyme that phosphorylates dTMP to dTDP. Instead, we show that a highly cooperative (Hill coefficient 2.5) in vivo activity of Tmk is the cause of suboptimal dTTP levels. dTMP supplementation rescues filamentation and restores in vivo Tmk kinetics to Michaelis–Menten. Overall, this study highlights the important role of cellular environment in sculpting enzymatic kinetics with system‐level implications for bacterial phenotype.     

Structure-based design of HSPA5 inhibitors: From peptide to small molecule inhibitors

We identified nine small-molecule hit compounds of Heat shock 70 kDa protein 5 (HSPA5) from cascade in silico screening based on the binding modes of the tetrapeptides derived from the peptide substrate or inhibitors of Escherichia coli HSP70. Two compounds exhibit promising inhibition activities from cancer cell viability and tumor inhibition assays. The binding modes of the hit compounds provide a platform for development of selective small molecule inhibitors of HSPA5.     

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.     

Enzyme kinetics and distinct modulation of the protein kinase N family of kinases by lipid activators and small molecule inhibitors

The PKN (protein kinase N) family of Ser/Thr protein kinases regulates a diverse set of cellular functions, such as cell migration and cytoskeletal organization. Inhibition of tumour PKN activity has been explored as an oncology therapeutic approach, with a PKN3-targeted RNAi (RNA interference)-derived therapeutic agent in Phase I clinical trials. To better understand this important family of kinases, we performed detailed enzymatic characterization, determining the kinetic mechanism and lipid sensitivity of each PKN isoform using full-length enzymes and synthetic peptide substrate. Steady-state kinetic analysis revealed that PKN1–3 follows a sequential ordered Bi–Bi kinetic mechanism, where peptide substrate binding is preceded by ATP binding. This kinetic mechanism was confirmed by additional kinetic studies for product inhibition and affinity of small molecule inhibitors. The known lipid effector, arachidonic acid, increased the catalytic efficiency of each isoform, mainly through an increase in k_cat for PKN1 and PKN2, and a decrease in peptide K_M for PKN3. In addition, a number of PKN inhibitors with various degrees of isoform selectivity, including potent (K_i<10 nM) and selective PKN3 inhibitors, were identified by testing commercial libraries of small molecule kinase inhibitors. This study provides a kinetic framework and useful chemical probes for understanding PKN biology and the discovery of isoform-selective PKN-targeted inhibitors.     

Papain-catalysed hydrolysis of some hippuric esters. A new mechanism for papain-catalysed hydrolyses

1. The Michaelis–Menten parameters for the papain-catalysed hydrolysis of a number of alkyl, aryl and alkyl-thiol esters of hippuric acid have been determined. 2. For all the aryl esters and most of the alkyl esters studied, the catalytic constant, k₀, is 2–3sec.⁻¹ and most probably represents deacylation of the common intermediate, hippuryl-papain. 3. Two alkyl esters and hippurylamide, however, have catalytic rate constants, k₀, less than 2–3sec.⁻¹. It is possible to interpret all the available kinetic data in terms of a three-step mechanism in which an enzyme–substrate complex is first formed, followed by acylation of the enzyme through an essential thiol group, followed by deacylation of the acyl-enzyme. 4. The logarithm of the ratio of the Michaelis–Menten parameters, which reflect the acylation rate constant, for four aryl esters of hippuric acid studied give a linear Hammett plot against the substituent constant, σ. Arguments are presented that indicate acid as well as nucleophilic catalysis in the acylation process and that the most likely proton donor is an imidazolium ion. 5. It is suggested that this imidazolium ion is part of the same histidine residue that has been tentatively implicated in the deacylation process (Lowe & Williams, 1965b). 6. A new mechanism is proposed for the papain-catalysed hydrolysis of N-acyl-α-amino acid derivatives.