Quasi-steady State Approximation Research Articles

Application of a Film Model to Mass Transfer and Chemical Reaction at a Plasma-Liquid Interface

Abstract Plasma electrodes provide novel ways of conducting electrochemical processes in liquids, in particular because of the ability to generate unique reactive radical species. However, the radicals injected into the liquid and their ensuing reactions are often confined to a narrow region near the interface of the plasma and the liquid. Thus, mass transfer has been found to play an important role in the observed kinetics and a modeling framework that includes both transport and kinetics is required to interpret experimental data. Here, we apply the idea of a film model for interphase mass transfer to plasma-liquid electrochemical processes, whereby transport is described by a stagnant film that is inherently linked to the concentration boundary layer and the mass transfer coefficient. Equations that govern the transport and reaction of radicals and substrates within the film are solved assuming a quasi-steady state approximation. The model is applied to specific case studies from the literature to estimate important parameters that cannot easily be measured in experiments, such as the mass transfer coefficient. Our study shows that a film model can elucidate the effect of mass transfer on observed conversion rates and allow the intrinsic kinetics to be unraveled.

Microkinetic Model to Rationalize the Lifetime of Electrocatalysis: Trade-off between Activity and Stability.

Electrocatalysts which can operate for several years are required to produce hydrogen and commodity chemicals in an environmentally friendly manner. However, designing materials with long operational lifetimes is challenging, due to the lack of a conceptual framework to predict catalytic lifetimes quantitatively. Here, we report a microkinetic equation which quantifies the lifetime of an electrocatalyst undergoing dissolution. This equation was obtained by taking advantage of the fact that catalysis is much faster than deactivation, which allows the ordinary differential equations to be solved via the quasi steady-state approximation. All chemical reactions were modeled as irreversible, first-order elementary reactions. Under this assumption, the catalytic rate correlates linearly with the deactivation rate, leading to a trade-off relationship between activity and stability. Our model was supported by the correlation between theoretical and experimental lifetimes (r2 = 0.86) of a manganese oxide electrocatalyst during the oxygen evolution reaction.

Boosting of enzymatic cascades by intermediates: Theoretical analysis and model-based optimization

A recent paper demonstrated that adding intermediates of an enzymatic cascade reaction at the start of the reaction process can increase the space-time yield significantly. This method was coined boosting by intermediates (BBI) and can potentially increase the productivity of enzymatic cascade reaction systems. BBI remains largely unexplored, and its mechanism, potential, and limitations are not fully understood. In this short communication, the theoretical background of the method is revealed by studying four chemical reaction networks that exhibit boosting through intermediates. The dynamics of these networks are studied via the quasi-steady state approximation and numerical solutions. A novel graphical method that can be used to deduce which intermediates boost linear, redox-neutral enzymatic cascade reaction networks is presented. Further, how such a reaction process can be designed using model-based optimization is demonstrated.

Beyond homogeneity: Assessing the validity of the Michaelis-Menten rate law in spatially heterogeneous environments.

The Michaelis-Menten (MM) rate law has been a fundamental tool in describing enzyme-catalyzed reactions for over a century. When substrates and enzymes are homogeneously distributed, the validity of the MM rate law can be easily assessed based on relative concentrations: the substrate is in large excess over the enzyme-substrate complex. However, the applicability of this conventional criterion remains unclear when species exhibit spatial heterogeneity, a prevailing scenario in biological systems. Here, we explore the MM rate law's applicability under spatial heterogeneity by using partial differential equations. In this study, molecules diffuse very slowly, allowing them to locally reach quasi-steady states. We find that the conventional criterion for the validity of the MM rate law cannot be readily extended to heterogeneous environments solely through spatial averages of molecular concentrations. That is, even when the conventional criterion for the spatial averages is satisfied, the MM rate law fails to capture the enzyme catalytic rate under spatial heterogeneity. In contrast, a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA), is accurate. Specifically, the tQSSA-based modified form, but not the original MM rate law, accurately predicts the drug clearance via cytochrome P450 enzymes and the ultrasensitive phosphorylation in heterogeneous environments. Our findings shed light on how to simplify spatiotemporal models for enzyme-catalyzed reactions in the right context, ensuring accurate conclusions and avoiding misinterpretations in in silico simulations.

A dynamical system framework for a mathematical model of atherosclerosis

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two-parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidized LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction–diffusion wave.

Validity conditions of approximations for a target-mediated drug disposition model: A novel first-order approximation and its comparison to other approximations.

Target-mediated drug disposition (TMDD) is a phenomenon characterized by a drug's high-affinity binding to a target molecule, which significantly influences its pharmacokinetic profile within an organism. The comprehensive TMDD model delineates this interaction, yet it may become overly complex and computationally demanding in the absence of specific concentration data for the target or its complexes. Consequently, simplified TMDD models employing quasi-steady state approximations (QSSAs) have been introduced; however, the precise conditions under which these models yield accurate results require further elucidation. Here, we establish the validity of three simplified TMDD models: the Michaelis-Menten model reduced with the standard QSSA (mTMDD), the QSS model reduced with the total QSSA (qTMDD), and a first-order approximation of the total QSSA (pTMDD). Specifically, we find that mTMDD is applicable only when initial drug concentrations substantially exceed total target concentrations, while qTMDD can be used for all drug concentrations. Notably, pTMDD offers a simpler and faster alternative to qTMDD, with broader applicability than mTMDD. These findings are confirmed with antibody-drug conjugate real-world data. Our findings provide a framework for selecting appropriate simplified TMDD models while ensuring accuracy, potentially enhancing drug development and facilitating safer, more personalized treatments.

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

Population Pharmacokinetics of Subcutaneous Epcoritamab in Relapsed or Refractory B-Cell Non-Hodgkin Lymphoma

Background: The US FDA approved epcoritamab SC, a CD3xCD20 bispecific antibody developed using the DuoBody ® platform, in May 2023 for the treatment of adults with relapsed or refractory (R/R) diffuse large B-cell lymphoma (DLBCL), not otherwise specified, including DLBCL arising from indolent lymphoma, and high-grade B-cell lymphoma after at least 2 lines of systemic therapy. Here we report the results of a population pharmacokinetic (PK) model-based analysis using data from 2 phase 1/2 clinical trials (EPCORE™ NHL-1 [NCT03625037] and EPCORE NHL-3 [NCT04542824]) evaluating epcoritamab in patients with R/R B-cell non-Hodgkin lymphoma (B-NHL). Methods: In EPCORE NHL-1 dose expansion and EPCORE NHL-3, epcoritamab SC was administered QW during weeks 1-12 (cycles 1-3), Q2W during weeks 13-36 (cycles 4-9), and Q4W thereafter (cycles ≥10). In EPCORE NHL-1 dose escalation, dosing was QW in cycles 1 and 2, Q2W in cycles 3‒6, and Q4W in cycles ≥7. Patients received step-up priming (0.004-0.16 mg), intermediate (0.25-1.6 mg), and full (0.0128-60 mg) doses. A total of 6819 quantifiable PK observations were collected and analyzed from 327 patients (large B-cell lymphoma [LBCL], n=212; indolent B-NHL, n=97; and mantle cell lymphoma, n=18) who received epcoritamab SC. Nonlinear mixed-effects modeling was used for data analysis. Results: The PK of epcoritamab following SC administration in patients with R/R B-NHL were adequately characterized by a quasi-steady state approximation of a 2-compartment target-mediated drug disposition model with first-order absorption. Among patients with R/R LBCL who received the recommended full dose (48 mg) in EPCORE NHL-1, the estimated median time of maximum concentration (t max) was 4 days (range, 0.3-7) after the first full dose and 2.3 days (range, 0.3-3.2) at the end of weekly dosing (end of cycle 3). The geometric mean (coefficient of variation [CV] %) of the apparent total volume of distribution was 25.6 L (82%). Epcoritamab exposure increased more than proportionally over the range of full doses up to the recommended full dose (1.5-48 mg). For the 48-mg full dose of epcoritamab, the geometric mean (CV%) of terminal half-life at the end of cycle 3 was 22 days (58%), while the geometric mean (CV%) of apparent total clearance after the end of cycle 3 was 0.53 L/day (40%). Body weight was a statistically significant covariate on apparent clearance and central and peripheral volumes of distribution, and age was significant on absorption rate; however, no association was identified between body weight or age and clinical efficacy or safety. After accounting for body weight, no statistically significant effects on epcoritamab PK were observed for age, sex, race, renal or hepatic function, or other disease characteristics. The risk of immunogenicity was low; antidrug antibodies (ADAs) were detected in only 4 (2.6%) of 156 evaluable patients treated with the approved 0.16/0.8/48-mg dosing regimen. No clinically meaningful differences in epcoritamab exposure were observed between ADA-positive patients and ADA-negative patients. Conclusions: The population PK model appropriately characterized epcoritamab PK in R/R B-NHL. No dosage adjustments are recommended in subpopulations based on body weight, age, sex, race, mild to moderate renal impairment, or mild hepatic impairment.

A Novel Multi-Step Global Mechanism Scheme for n-Decane Combustion.

Based on the directed relation graph with error propagation (DRGEP) reduction method, a detailed mechanism consisting of 119 species and 527 reactions for n-decane was simplified. As a result, a skeletal mechanism comprising 32 species and 73 reactions was derived. Subsequently, the quasi-steady state approximation (QSSA) reduction method was employed to further simplify the skeletal mechanism, resulting in a reduced mechanism with 18 species and 14 global reactions. A comparison between the reduced mechanism, skeletal mechanism, and detailed mechanism revealed that the reduced and skeletal mechanisms successfully replicated the combustion characteristics of the detailed mechanism under a range of initial conditions. These models can be credibly incorporated into large-scale combustion simulation, serving as a solid foundation for enhancing computational efficiency.

Linear and Nonlinear Modes and Data Signatures in Dynamic Systems Biology Models

The particulars of stimulus–response experiments performed on dynamic biosystems clearly limit what one can learn and validate about their structural interconnectivity (topology), even when collected kinetic output data are perfect (noise-free). As always, available access ports and other data limitations rule. For linear systems, exponential modes, visible and hidden, play an important role in understanding data limitations, embodied in what we call dynamical signatures in the data. We show here how to circumscribe and analyze modal response data in compartmentalizing model structures—so that modal analysis can be used constructively in systems biology mechanistic model building—for some nonlinear (NL) as well as linear biosystems. We do this by developing and exploiting the modal basis for dynamical signatures in hypothetical (perfect) input–output (I-O) data associated with a (mechanistic) structural model—one that includes inputs and outputs explicitly. The methodology establishes model dimensionality (size and complexity) from particular I-O datasets; helps select among multiple candidate models (model distinguishability); helps in designing new I-O experiments to extract “hidden” structure; and helps to simplify (reduce) models to their essentials. These modal analysis tools are introduced to NL enzyme-regulated and protein–protein interaction biosystems via nonlinear normal mode (NNM) and quasi-steady state approximation (QSSA) analyses and unified with linear models on invariant 2-dimensional manifolds in phase space, with properties similarly informative about their dominant dynamical properties. Some automation of these highly technical aspects of biomodeling is also introduced.

A geometric analysis of the impact of large but finite switching rates on vaccination evolutionary games

In contemporary society, social networks accelerate decision dynamics causing a rapid switch of opinions in a number of fields, including the prevention of infectious diseases by means of vaccines. This means that opinion dynamics can nowadays be much faster than the spread of epidemics. Hence, we propose a Susceptible–Infectious–Removed epidemic model coupled with an evolutionary vaccination game embedding the public health system efforts to increase vaccine uptake. This results in a global system “epidemic model + evolutionary game”. The epidemiological novelty of this work is that we assume that the switching to the strategy “pro vaccine” depends on the incidence of the disease. As a consequence of the above-mentioned accelerated decisions, the dynamics of the system acts on two different scales: a fast scale for the vaccine decisions and a slower scale for the spread of the disease. Another, and more methodological, element of novelty is that we apply Geometrical Singular Perturbation Theory (GSPT) to such a two-scale model and we then compare the geometric analysis with the Quasi-Steady-State Approximation (QSSA) approach, showing a criticality in the latter. Later, we apply the GSPT approach to the disease prevalence-based model already studied in (Della Marca and d’Onofrio, Comm Nonl Sci Num Sim, 2021) via the QSSA approach by considering medium–large values of the strategy switching parameter.

The Potential Roles of Transacylation in Intracellular Lipolysis and Related Qssa Approximations

Fatty acids (FAs) are crucial energy metabolites, signalling molecules, and membrane building blocks for a wide range of organisms. Adipose triglyceride lipase (ATGL) is the first and presumingly most crucial regulator of FA release from triacylglycerols (TGs) stored within cytosolic lipid droplets. However, besides the function of releasing FAs by hydrolysing TGs into diacylglycerols (DGs), ATGL also promotes the transacylation reaction of two DG molecules into one TG and one monoacylglycerol molecule. To date, it is unknown whether DG transacylation is a coincidental byproduct of ATGL-mediated lipolysis or whether it is physiologically relevant. Experimental evidence is scarce since both, hydrolysis and transacylation, rely on the same active site of ATGL and always occur in parallel in an ensemble of molecules. This paper illustrates the potential roles of transacylation. It shows that, depending on the kinetic parameters but also on the state of the hydrolytic machinery, transacylation can increase or decrease downstream products up to 80% respectively 30%. We provide an extensive asymptotic analysis including quasi-steady-state approximations (QSSA) with higher order correction terms and provide numerical simulation. We also argue that when assessing the validity of QSSAs one should include parameter sensitivity derivatives. Our results suggest that the transacylation function of ATGL is of biological relevance by providing feedback options and altogether stability to the lipolytic machinery in adipocytes.

A unified approach to simulate thermal hydraulic instabilities and the mutual interactions using transient distributed model

In the present work, a 1-D thermal-hydraulic model is developed to simulate two phase flow boiling for a working coolant R134a circulating inside a single horizontal channel which includes a surge tank. The TH model is the first of its kind which solves the transient conservation equations based on distributive formulations without the incorporation of quasi-steady state approximation and still can simulate pure pressure drop oscillations (PDOs), Ledinegg excursion and density wave oscillations (DWOs), and the co-existence of those instabilities as well using a unified approach by employing appropriate external characteristic curves. The model is validated with the available experimental data and then, further used for intended purpose as mentioned above. Finally, the notable conclusions are drawn based on the simulations conducted. Results indicate that the initial equilibrium position in the negative slope region of the pressure drop vs. mass flux curve is unstable, leading to Ledinegg excursion or PDOs based on the steepness of the external pump characteristic curve. Increased surge tank volume extends PDOs. For Ledinegg excursion, the unstable position shifts to positive slope regions, and gets stabilized or destabilized depending on the occurrence of DWOs. DWOs are more likely at low mass flux and positive slope region. With a constant mass flow rate from an external pump, DWOs occur with a surge tank upstream of the heated section but not without it. Simulations demonstrate coexistence of PDOs, Ledinegg excursion, and DWOs in certain situations.

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.

SLOW AND FAST SUBSYSTEMS FOR COMPLEX UNCOMPETITIVE INHIBITOR MECHANISMS

To fully understand intricate enzyme reaction models, one must explore beyond the confines of chemical and biological tools and look toward mathematical modeling and model reduction techniques. Mathematical modeling and model reduction techniques have the potential to provide a vast array of analysis tools for such models. This piece of work entails a review and discussion of a complex noncompetitive inhibitor model. This model is composed of seven non-linear differential equations with constant rates. We propose two efficient model reduction techniques: quasi- steady-state approximation (QSSA) and quasi-equilibrium approximation (QEA). By utilizing the suggested methods, the model equations are segregated into slow and fast subsystems, leading to the attainment of reduced models and slow manifolds with fewer variables and parameters. The outcomes manifest some analytical approximate solutions for the proposed model and establish a profound agreement between model dynamics for both the original and the reduced models. Observing that the reduced models can accurately identify certain critical model parameters is intriguing.

Mathematical analysis of the impact of community ignorance on the population dynamics of dengue

This study proposes a dengue spread model that considers the nonlinear transmission rate to address the level of human ignorance of dengue in their environment. The SIR − UV model has been proposed, where SIR denotes the classification of the human population and UV denotes the classification of the mosquito population. Assuming that the total human population is constant, and the mosquito population is already in its steady-state condition, using the Quasi-Steady State Approximation (QSSA) method, we reduce our SIR − UV model into a more simple IR-model. Our analytical result shows that a stable disease-free equilibrium exists when the basic reproduction number is <1. Furthermore, our model also shows the possibility of a backward bifurcation. The more ignorant the society is about dengue, the higher the possibility that backward bifurcation phenomena may appear. As a result, the condition of the basic reproduction number being <1 is insufficient to guarantee the extinction of dengue in a population. Furthermore, we found that increasing the recovery rate, reducing the waning immunity rate, and mosquito life expectancy can reduce the possibility of backward bifurcation phenomena. We use dengue incidence data from Jakarta to calibrate the parameters in our model. Through the fast Fourier transform analysis, it was found that dengue incidence in Jakarta has a periodicity of 52.4, 73.4, and 146.8 weeks. This result indicates that dengue will periodically appear at least every year in Jakarta. Parameter estimation for our model parameters was carried out by assuming the infection rate of humans as a sinusoidal function by determining the three most dominant frequencies. Numerical and sensitivity analyses were conducted to observe the impact of community ignorance on dengue endemicity. From the sensitivity analysis, we found that, although a larger community ignorance can trigger a backward bifurcation, this threshold can be minimized by increasing the recovery rate, prolonging the temporal immunity, or reducing the mosquito population. Therefore, to control dengue transmission more effectively, media campaigns undertaken by the government to reduce community ignorance should be accompanied by other interventions, such as a good treatment in the hospital or vector control programs. With this combination of interventions, it will be easier to achieve a condition of dengue-free population when the basic reproduction number is less than one.

Pharmacodynamic-Mediated Drug Disposition (PDMDD) Model of Daratumumab Monotherapy in Patients with Multiple Myeloma.

We aimed to quantify the daratumumabconcentration- and CD38 dynamics-dependent pharmacokinetics using a pharmacodynamic mediated disposition model (PDMDD) in patients with multiple myeloma (MMY) following daratumumab IV or SC monotherapy. Daratumumab, a human IgG monoclonal antibody targeting CD38 with a direct on-tumor and immunomodulatory mechanism of action, has been approved to treat patients with multiple myeloma (MM). In total, 7788 daratumumab plasma samples from 850 patients with diagnosis of MMY were used. The serum concentration-time data of daratumumab were analysed using nonlinear mixed-effects modeling with NONMEM®. The PDMDD with quasi steady-state approximation (QSS) was compared to the previously developed Michaelis-Menten (MM) approximation with respect to the parameter estimates, the goodness-of-fit plots and prediction-corrected visual predictive check, as well as model-based simulations. The effect of patients' covariates on daratumumab pharmacokinetics was also investigated. The QSS approximation characterized the concentration- and CD38 dynamics-dependency of daratumumab pharmacokinetics within the doses ranging from 0.1 to 24 mg/kg after IV administration and 1200 and 1800 mg after SC administration in patients with MMY, mechanistically describing the binding of daratumumab and CD38, the internalization of the daratumumab-CD38 complex and the CD38 turnover. Compared to the previously developed MM approximation, the MM approximation with the non-constant total target and dose-correction provided substantial improvement of the model fit, but it was still not as good as the QSS approximation. The effect of the previously identified covariates as well as the newly identified covariate (baseline M protein) on daratumumab pharmacokinetics was confirmed, but the magnitude of the effect was deemed not clinically relevant. Accounting for the CD38 turnoverand its binding capacity to daratumumab, the QSS approximation provided a mechanistic interpretation of daratumumab PK parameters and consequently well described the concentration- and CD38 dynamics-dependency of daratumumab pharmacokinetics. CLINICAL STUDIES INCLUDED IN THE ANALYSIS WERE REGISTERED WITH THE NCT NUMBER BELOW AT HTTP://WWW. GOV : MMY1002 (ClinicalTrials.gov: NCT02116569), MMY1003 (NCT02852837), MMY1004 (NCT02519452), MMY1008 (NCT03242889), GEN501 (NCT00574288), MMY2002 (NCT01985126), MMY3012 (NCT03277105).

Demystification of entangled Mass Action Law

Recently, Gorban (2021) analysed some kinetic paradoxes of the transition state theory and proposed its revision that gave the “entangled mass action law”, in which new reactions were generated as an addition to the reaction mechanism under consideration. These paradoxes arose due to the assumption of quasi-equilibrium between reactants and transition states.In this paper, we provided a brief introduction to this theory, demonstrating how the entangled mass action law equations can be derived in the framework of the standard quasi steady state approximation in combination with the quasi-equilibrium generalized mass action law for an auxiliary reaction network including reactants and intermediates. We also proved the basic physical property (positivity) for these new equations, which was not obvious in the original approach.

VALIDITY OF THE QUASI STEADY STATE ASSUMPTION FOR ENZYME-CATALYSED REACTIONS WITH COMPETITIVE INHIBITION AND SUBSTRATE INPUT

Enzyme-catalysed reactions are chemical reactions within cells in which the rate of the reaction is significantly increased through the action of enzymes. They are usually part of large and complex bio¬chemical networks, which form the central processing units of the living cell. Enzymatic reactions often operate on multiple time scales, which can be characterized as being either fast or slow. The quasi steady¬state approximation (QSSA) utilizes time scale separation to pro ject these complex models onto lower-dimensional slow manifolds. In this paper, we investigate the validity of a quasi steady-state assumption for enzyme-catalysed biochemical reactions with competitive inhibition that are subject to a constant substrate input. Necessary and sufficient conditions for the validity of these assumptions were derived and were shown to be dependent, among others, on the substrate input. The validity conditions are numerically verified using the classical Runge- Kutta method.