Michaelis-Menten Reaction Research Articles

The Michaelis–Menten Reaction at Low Substrate Concentrations: Pseudo-First-Order Kinetics and Conditions for Timescale Separation

We demonstrate that the Michaelis–Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke’s comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is s0≪K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_0 \\ll K$$\\end{document}, where K=k2/k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K=k_2/k_1$$\\end{document} is the Van Slyke–Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.

Thermodynamics for reduced models of chemical reactions by PEA and QSSA

The partial equilibrium approximation (PEA) and the quasi-steady-state approximation (QSSA) are two classical methods for reducing complex macroscopic chemical reactions into simple computable ones. Previous studies have primarily focused on the accuracy of solutions before and after applying model reduction. In this paper, however, we adopt a thermodynamic view and aim to establish a quantitative connection on the essential thermodynamic quantities, such as entropy production rate, free-energy dissipation rate, and entropy flow rate, between the original chemical mass-action equations and the reduced models obtained by PEA or QSSA. Our findings reveal that the reduced models by PEA and QSSA may not preserve the thermodynamic structure of the original full model, particularly when algebraic relations are used instead of differential equations. This is evident in the loss of non-negativity of the free-energy dissipation rate. To validate our results, we apply them to the Michaelis-Menten reactions as a prototype, both analytically and numerically. We anticipate that our study will inspire a reevaluation of the effectiveness of various model reduction or approximation methods from the perspective of nonequilibrium thermodynamics. Published by the American Physical Society 2024

The unreasonable effectiveness of the total quasi-steady state approximation, and its limitations

In this note, we discuss the range of parameters for which the total quasi-steady-state approximation of the Michaelis–Menten reaction mechanism holds validity. We challenge the prevalent notion that total quasi-steady-state approximation is “roughly valid” across all parameters, showing that its validity cannot be assumed, even roughly, across the entire parameter space — when the initial substrate concentration is high. On the positive side, we show that the linearized one-dimensional equation for total substrate is a valid approximation when the initial reduced substrate concentration s0/KM is small. Moreover, we obtain a precise picture of the long-term time course of both substrate and complex

Algorithmic criteria for the validity of quasi-steady state and partial equilibrium models: the Michaelis-Menten reaction mechanism.

We present "on the fly" algorithmic criteria for the accuracy and stability (non-stiffness) of reduced models constructed with the quasi-steady state and partial equilibrium approximations. The criteria comprise those introduced in Goussis (Combust Theor Model 16:869-926, 2012) that addressed the case where each fast time scale is due to one reaction and a new one that addresses the case where a fast time scale is due to more than one reactions. The development of these criteria is based on the ability to approximate accurately the fast and slow subspaces of the tangent space. Their validity is assessed on the basis of the Michaelis-Menten reaction mechanism, for which extensive literature is available regarding the validity of the existing various reduced models. The criteria predict correctly the regions in both the parameter and phase spaces where each of these models is valid. The findings are supported by numerical computations at indicative points in the parameter space. Due to their algorithmic character, these criteria can be readily employed for the reduction of large and complex mathematical models.

Natural Parameter Conditions for Singular Perturbations of Chemical and Biochemical Reaction Networks

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis-Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

Numerical Simulation of Temporal Variability of Methane Emissions from Mozhaysk Reservoir

Estimates of methane emission from the Mozhaysk reservoir surface were carried out using the mathematical model LAKE2.3. The average emission value is 361 tC per year, the average flux = 37.7 mgC–CH4 m–2 day–1. Comparison of the obtained estimates with in situ measurements revealed, that the methane emission and specific flux according to the model are in good agreement with the observations data. The ebullition makes the largest contribution to the total emission. During the heating period, an increase of methane emission is observed with a maximum before the autumn mixing stage. In the course of numerical experiments with the model, it was found that the amplitude of methane fluxes into the atmosphere is associated with fluctuations in atmospheric pressure, and the most significant emissions peaks associated with water level drawdowns. Effective method for calibrating the diffusion component of the methane flux into the atmosphere is the potential rate of methane oxidation in the Michaelis-Menten reaction, and for ebullition it is the methane generation parameter in bottom sediments — q10. For the described numerical experiments, the article presents the values of the annual emissions of methane into the atmosphere.

Effect of color cross-correlated noise on the growth characteristics of tumor cells under immune surveillance.

Based on the Michaelis-Menten reaction model with catalytic effects, a more comprehensive one-dimensional stochastic Langevin equation with immune surveillance for a tumor cell growth system is obtained by considering the fluctuations in growth rate and mortality rate. To explore the impact of environmental fluctuations on the growth of tumor cells, the analytical solution of the steady-state probability distribution function of the system is derived using the Liouville equation and Novikov theory, and the influence of noise intensity and correlation intensity on the steady-state probability distributional function are discussed. The results show that the three extreme values of the steady-state probability distribution function exhibit a structure of two peaks and one valley. Variations of the noise intensity, cross-correlation intensity and correlation time can modulate the probability distribution of the number of tumor cells, which provides theoretical guidance for determining treatment plans in clinical treatment. Furthermore, the increase of noise intensity will inhibit the growth of tumor cells when the number of tumor cells is relatively small, while the increase in noise intensity will further promote the growth of tumor cells when the number of tumor cells is relatively large. The color cross-correlated strength and cross-correlated time between noise also have a certain impact on tumor cell proliferation. The results help people understand the growth kinetics of tumor cells, which can a provide theoretical basis for clinical research on tumor cell growth.

Michaelis-Menten Quantification of Ligand Signaling Bias Applied to the Promiscuous Vasopressin V2 Receptor.

Activation of G protein-coupled receptors by agonists may result in the activation of one or more G proteins and recruitment of arrestins. The extent of the activation of each of these pathways depends on the intrinsic efficacy of the ligand. Quantification of intrinsic efficacy relative to a reference compound is essential for the development of novel compounds. In the operational model, changes in efficacy can be compensated by changes in the "functional" affinity, resulting in poorly defined values. To separate the effects of ligand affinity from the intrinsic activity of the receptor, we developed a Michaelis-Menten based quantification of G protein activation bias that uses experimentally measured ligand affinities and provides a single measure of ligand efficacy. We used it to evaluate the signaling of a promiscuous model receptor, the Vasopressin V2 receptor (V2R). Using BRET-based biosensors, we show that the V2R engages many different G proteins across all G protein subfamilies in response to its primary endogenous agonist, arginine vasopressin, including Gs and members of the Gi/o and G12/13 families. These signaling pathways are also activated by the synthetic peptide desmopressin, oxytocin, and the nonmammalian hormone vasotocin. We compared bias quantification using the operational model with Michaelis-Menten based quantification; the latter accurately quantified ligand efficacies despite large difference in ligand affinities. Together, these results showed that the V2R is promiscuous in its ability to engage several G proteins and that its' signaling profile is biased by small structural changes in the ligand. SIGNIFICANCE STATEMENT: By modelling the G protein activation as Michaelis-Menten reaction, we developed a novel way of quantifying signalling bias. V2R activates, or at least engages, G proteins from all G protein subfamilies, including Gi2, Gz, Gq, G12, and G13. Their relative activation may explain its Gs-independent signalling.

Enzyme Kinetics and Detector Sensitivity Determine Limits of Detection of Amplification-Free CRISPR-Cas12 and CRISPR-Cas13 Diagnostics.

Interest in CRISPR-Cas12 and CRISPR-Cas13 detection continues to increase as these detection schemes enable the specific recognition of nucleic acids. The fundamental sensitivity limits of these schemes (and their applicability in amplification-free assays) are governed by kinetic rates. However, these kinetic rates remain poorly understood, and their reporting has been inconsistent. We quantify kinetic parameters for several enzymes (LbCas12a, AsCas12a, AapCas12b, LwaCas13a, and LbuCas13a) and their corresponding limits of detection (LoD). Collectively, we present quantification of enzyme kinetics for 14 guide RNAs (gRNAs) and nucleic acid targets for a total of 50 sets of kinetic rate parameters and 25 LoDs. We validate the self-consistency of our measurements by comparing trends and limiting behaviors with a Michaelis-Menten trans-cleavage reaction kinetics model. For our assay conditions, activated Cas12 and Cas13 enzymes exhibit trans-cleavage catalytic efficiencies between order 105 and 106 M-1 s-1. For assays that use fluorescent reporter molecules (ssDNA and ssRNA) for target detection, the kinetic rates at the current assay conditions result in an amplification-free LoD in the picomolar range. The results suggest that successful detection of target requires cleavage (by an activated CRISPR enzyme) of the order of at least 0.1% of the fluorescent reporter molecules. This fraction of reporters cleaved is required to differentiate the signal from the background, and we hypothesize that this required fraction is largely independent of the detection method (e.g., endpoint vs reaction velocity) and detector sensitivity. Our results demonstrate the fundamental nature by which kinetic rates and background signal limit LoDs and thus highlight areas of improvement for the emerging field of CRISPR diagnostics.

On the anti-quasi-steady-state conditions of enzyme kinetics

Quasi-steady state reductions for the irreversible Michaelis–Menten reaction mechanism are of interest both from a theoretical and an experimental design perspective. A number of publications have been devoted to extending the parameter range where reduction is possible, via improved sufficient conditions. In the present note, we complement these results by exhibiting local conditions that preclude quasi-steady-state reductions (anti-quasi-steady-state), in the classical as well as in a broader sense. To this end, one needs to obtain necessary (as opposed to sufficient) conditions and determine parameter regions where these do not hold. In particular, we explicitly describe parameter regions where no quasi-steady-state reduction (in any sense) is applicable (anti-quasi-steady-state conditions), and we also show that – in a well defined sense – these parameter regions are small. From another perspective, we obtain local conditions for the accuracy of standard or total quasi-steady-state. Perhaps surprisingly, our conditions do not involve initial substrate.

Reaction-diffusion in a packed-bed reactors: Enzymatic isomerization with Michaelis-Menten Kinetics

A theoretical model for the enzyme catalysed isomerisation reaction in a packed-bed reactors made up of microporous particles is discussed. The system is described in terms of finite-range Fickian diffusion and non linear Michaelis-Menten reaction kinetics to produce a non linear reaction/diffusion equation which is solved both numerically by MATLAB and analytically by hyperbolic function methods (HFM). Approximate analytical expressions of the substrate concentration within the microparticle, the related flux and the effectiveness factor for a wide range of values of pore-level Thiele modulus and kinetic parameters are reported using these simple, and robust methods. A satisfactory agreement is obtained between the estimated analytical solution and the numerical simulation.

Amperometric biosensors and coupled enzyme nonlinear reactions processes: A complete theoretical and numerical approach

The transient response of amperometric enzyme-based biosensors working in trigger mode is discussed. Nonlinear time-dependent partial differential equations for Michaelis–Menten reaction kinetics are solved analytically using a new approach of homotopy perturbation technique. The simple and closed-form analytical expression for concentration profiles are provided. Subsequently, the biosensor's current, sensitivity, resistance, and amplification are derived from the concentration profiles. The current response is predicted under steady-state conditions when T→∞, proving the validity of the mathematical analyses. The limiting situations of catalytic sites (unsaturation and saturation) are considered. The compatibility of analytical results with simulation and limiting case results can be observed from the graphs and tables presented. The existence of the moving boundary between the two categories of catalytic sites is also discussed.

An analog electronic circuit model for cAMP-dependent pathway—towards creation of Silicon life

<abstract> <p>Among the most sought after breakthroughs nowadays to combat computational saturation in the electronic hardware realm, neuromorphic and cytomorphic mimetics of biological structures seem potentially promising. Biological circuits are distinguishable due to their minuscule dimensions and immensely low power consumption; yet they achieve extremely complex and magnificent tasks of life, such as, thinking, memorizing, decision making and self-regulating in response to the surroundings. Low power analog circuit solutions are edged over digital ones as they are inherently noisy and fuzzy like bio-systems. In this paper, an analog circuit equivalent for a well-known biological pathway, cyclic adenosine monophosphate (cAMP), has been proposed, exploiting the fabrication characteristics of an analog transistor. The work demonstrates an application of previously published research of the authors, where it was shown that a single transistor operating in analog mode can mimic some fundamental biological circuit processes like receptor-ligand binding, Michaelis Menten and Hill process reactions. Since biological pathways are chain connections of such reactions, same modular approach can be used to build electronic pathways using those basic transistor circuits. Although the idea of creating silicon life seems far-fetched at this stage, this work supplements the idea of cytomorphic chips which is already gaining interest of bio-engineering community.</p> </abstract>

Experimental measurements of the bacterial oxidation of HT in soils: Impact over a zone influenced by an industrial release of tritium in HT form

Tritium is a radionuclide released to the atmosphere by nuclear industries in various forms, mainly HTO and to a lesser extent HT. However, some nuclear sites may emit predominantly HT in the atmosphere. The HT is oxidized to HTO essentially in the top cm of soils, and that the formed HTO is then possibly released into the atmosphere. HTO is an assimilable form by plants. Therefore, it is important to understand the environmental behaviour of HT.In this work, we adapt the bacterial oxidation model of HT in soils of Ota et al. (2007) by laboratory experiments on soils typical of western France, and we have in particular adapted the frequency factor A and the Michaelis-Menten enzymatic reaction parameter (Km) on the basis of an Arrhenius equation in function of the porosity of the soil.We then applied this model to the environment near the reprocessing plant of Orano la Hague (France), which emits a significant amount of HT. Based on the adapted model, and knowing the atmospheric variations of HTO and HT over the period 2013–2016, we estimated that the mean HTO activity in soil due to atmospheric HT reached 0.6 Bq.L−1 (with a peak value of 5 Bq.L−1) while the mean value with all sources taken into account is 6.2 Bq.L−1. Then, in an environment such as that surrounding the Orano La Hague plant, where near-field atmospheric HT activity is very high, the bacterial oxydation contribution to produce HTO in the soil can be considered as approximately 10%. The flux to the atmosphere from these source representing approximately. 1.5 Bq.m−2.d−1. If we consider an area of 2 km around the plant (i.e. 13 km2), we estimate 218 Bq.s−1 of HTO was released by the soil, representing less than 0.1% of the direct atmospheric release of HTO around the site. From this work, it appears clear that this secondary source term from the soil is insignificant at this specific site.

On the Validity of the Stochastic Quasi-Steady-State Approximation in Open Enzyme Catalyzed Reactions: Timescale Separation or Singular Perturbation?

The quasi-steady-state approximation is widely used to develop simplified deterministic or stochastic models of enzyme catalyzed reactions. In deterministic models, the quasi-steady-state approximation can be mathematically justified from singular perturbation theory. For several closed enzymatic reactions, the homologous extension of the quasi-steady-state approximation to the stochastic regime, known as the stochastic quasi-steady-state approximation, has been shown to be accurate under the analogous conditions that permit the quasi-steady-state reduction in the deterministic counterpart. However, it was recently demonstrated that the extension of the stochastic quasi-steady-state approximation to an open Michaelis-Menten reaction mechanism is only valid under a condition that is far more restrictive than the qualifier that ensures the validity of its corresponding deterministic quasi-steady-state approximation. In this paper, we suggest a possible explanation for this discrepancy from the lens of geometric singular perturbation theory. In so doing, we illustrate a misconception in the application of the quasi-steady-state approximation: timescale separation does not imply singular perturbation.

Combining Michaelis-Menten theory and enzyme deactivation reactions for the kinetic study of enzymatic hydrolysis by different pretreated sugarcane bagasse

Conversion of lignocellulose to sugar depends on expensive cellulase and xylanase, hence, improvement of enzyme utilization is important. Enzymatic kinetics can guide researchers toward efficient enzymes utilization. Based on Michaelis-Menten theory and enzyme deactivation reactions, the novel kinetic models describing sugar (glucose and xylose) yield are deduced. The model based on second-order reaction is significantly superior compared to first-order reaction, which is verified by initial and maximum hydrolytic rate. Hydrolytic rate displays a linear relationship with enzyme loading. Sugar yield differs in enzymatic hydrolysis of various pretreated substrates, which is explained by the preferred model. Enzyme deactivation based on the model shows that higher enzyme loading results in reduced activity retention percentage, due to feedback inhibition of products and ineffective adsorption of substrate to enzyme during the hydrolytic process. Therefore, the developed model allows more in-depth study of enzymatic kinetics and provides a useful guide for efficient enzyme utilization.

Solving nonlinear reaction–diffusion problem in electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme systems using Taylor series method

A mathematical model of electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme is discussed. The model involves the coupled system of non-linear reaction–diffusion equations of substrate and hydrogen ion. The non-linear term in this model is related to the Michaelis–Menten reaction of the substrate and non-Michaelis–Menten kinetics of hydrogen ion. The approximate analytical expression of concentration of substrate and hydrogen ion has been derived by solving the non-linear reactions using Taylor’s series method. Reaction rate and effectiveness factor are also reported. A comparison between the analytical approximation and numerical solution is also presented. The effects of external mass transfer coefficient and the electrostatic potential on the overall reaction rate were also discussed.

Chemical Reaction Rate from a (Semiempirical) Dynamical Point of View

The aim of the present paper is to check - or better to confirm - the mathematical validity of chemical reaction rate, faced as a set of differential equations. Firstly one - way elementary reactions are considered, in the most general case. Secondly the same thing is done with two-way (opposing) elementary reactions. At this stage, we show that the two – way reaction, as we mean it, is compatible with the reduction of the total Gibbs energy as expected in every natural process. As an example of a two way elementary reaction of a completely solvable problem we give the hydrolysis of sucrose to glucose and fructose, where the “inversion” of sucrose is examined not only with the initial linear reaction of “Wilhelmy” (1850), but also with the two way nonlinear reaction introduced. Finally the validity of the mathematical model is checked for more complex cases such as the Michaelis-Menten mechanism or reactions in solution, where it is found that the two cases, apparently are four – dimensional while in reality are two – dimensional (after the “subtraction” of the constraints of “motion”) and naturally cannot exhibit chaotic behavior. In all cases the treatment is not one-hundred-percent mathematically austere but it has also arbitrary although reasonable hypotheses.

On the quasi-steady-state approximation in an open Michaelis-Menten reaction mechanism.

The conditions for the validity of the standard quasi-steady-state approximation in the Michaelis–Menten mechanism in a closed reaction vessel have been well studied, but much less so the conditions for the validity of this approximation for the system with substrate inflow. We analyze quasi-steady-state scenarios for the open system attributable to singular perturbations, as well as less restrictive conditions. For both settings we obtain distinguished invariant manifolds and time scale estimates, and we highlight the special role of singular perturbation parameters in higher order approximations of slow manifolds. We close the paper with a discussion of distinguished invariant manifolds in the global phase portrait.

A design principle for posttranslational chaotic oscillators.

SummaryChaos behavior has been observed in various cellular and molecular processes. Here, we modeled reversible phosphorylation dynamics to elucidate a design principle for autonomous chaos generation that may arise from generic enzymatic reactions. A comprehensive parameter search demonstrated that the reaction system composed of a set of kinases and phosphatases and two substrates with two modification sites exhibits chaos behavior. All reactions are described according to the Michaelis-Menten reaction scheme without exotic functions being applied to enzymes and substrates. Clustering analysis of parameter sets that can generate chaos behavior revealed the existence of motif structures. These chaos motifs allow the two-substrate species to interact via enzyme availability and constrain the two substrates' dynamic changes in phosphorylation status so that they occur at different timescales. This chaos motif structure is found in several enzymatic reactions, suggesting that chaos behavior may underlie cellular autonomy in a variety of biochemical systems.