Quasi-steady State Assumption Research Articles

Generalized Michaelis-Menten rate law with time-varying molecular concentrations.

The Michaelis-Menten (MM) rate law has been the dominant paradigm of modeling biochemical rate processes for over a century with applications in biochemistry, biophysics, cell biology, systems biology, and chemical engineering. The MM rate law and its remedied form stand on the assumption that the concentration of the complex of interacting molecules, at each moment, approaches an equilibrium (quasi-steady state) much faster than the molecular concentrations change. Yet, this assumption is not always justified. Here, we relax this quasi-steady state requirement and propose the generalized MM rate law for the interactions of molecules with active concentration changes over time. Our approach for time-varying molecular concentrations, termed the effective time-delay scheme (ETS), is based on rigorously estimated time-delay effects in molecular complex formation. With particularly marked improvements in protein-protein and protein-DNA interaction modeling, the ETS provides an analytical framework to interpret and predict rich transient or rhythmic dynamics (such as autogenously-regulated cellular adaptation and circadian protein turnover), which goes beyond the quasi-steady state assumption.

Autoignition Problem in Homogeneous Combustion Systems: GQL versus QSSA Combined with DRG

The global quasi-linearization (GQL) is used as a method to study and to reduce the complexity of mathematical models of mechanisms of chemical kinetics. Similar to standard methodologies, such as the quasi-steady-state assumption (QSSA), the GQL method defines the fast and slow invariant subspaces and uses slow manifolds to gain a reduced representation. It does not require empirical inputs and is based on the eigenvalue and eigenvector decomposition of a linear map approximating the nonlinear vector field of the original system. In the present work, the GQL-based slow/fast decomposition is applied for different combustion systems. The results are compared with the standard QSSA approach. For this, an implicit implementation strategy described by differential algebraic equations (DAEs) systems is suggested and used, which allows for treating both approaches within the same computational framework. Hydrogen–air (with 9 species) and ethanol–air (with 57 species) combustion systems are considered representative examples to illustrate and verify the GQL. The results show that 4D GQL for hydrogen–air and 14D GQL ethanol–air slow manifolds outperform the standard QSSA approach based on a DAE-based reduced computation model.

Viral Infection Spreading in Cell Culture with Intracellular Regulation

Virus plaque assays are conventionally used for the assessment of viral infections, including their virulence, and vaccine efficacy. These experiments can be modeled with reaction–diffusion equations, allowing the estimation of the speed of infection spread (related to virus virulence) and viral load (related to virus infectivity). In this work, we develop a multiscale model of infection progression that combines macroscopic characterization of virus plaque growth in cell culture with a reference model of intracellular virus replication. We determine the infection spreading speed and viral load in a model for the extracellular dynamics and the kinetics of the abundance of intracellular viral genomes and proteins. In particular, the spatial infection spreading speed increases if the rate of virus entry into the target cell increases, while the viral load can either increase or decrease depending on other model parameters. The reduction in the model under a quasi-steady state assumption for some intracellular reactions allows us to derive a family of reduced models and to compare the reference model with the previous model for the concentration of uninfected cells, infected cells, and total virus concentration. Overall, the combination of different scales in reaction–diffusion models opens up new perspectives on virus plaque growth models and their applications.

Generalized michaelis-menten rate law beyond steady-state assumptions.

Despite over a century's use as a dominant paradigm in the description of biochemical rate processes, the Michaelis-Menten (MM) rate law stands on the restrictive assumption that the concentration of the complex of interacting molecules, at each moment, approaches an equilibrium much faster than the molecular concentration changes. The increasingly-appreciated, remedied form of the MM rate law is also based on this quasi-steady state assumption. Although this assumption may be valid for a range of biochemical systems, the exact extent of such systems is not clear. In this study, we relax the quasi-steady state requirement and propose the revised MM rate law for the interactions of molecules with active concentration changes over time. Our revised rate law, characterized by rigorously-derived time delay effects in molecular complex formation, improves the accuracy of models especially for protein-protein and protein-DNA interactions. Our simulation and empirical data analysis show that the improvement is not limited to the quantitatively better characterization of the dynamics, but also allows the prediction for qualitatively new patterns in the systems of interest. The latter include the oscillation condition and period patterns of the mammalian circadian clock and the spontaneous rhythmicity in the degradation rates of circadian proteins, both not properly captured by the previous approaches. Moreover, our revised rate law is capable of more accurate parameter estimation. This work offers an analytical framework for understanding rich dynamics of biomolecular systems, which goes beyond the quasi-steady state assumption.

A Second-Order Singular Perturbation for Model Simplification for a Microgrid

As the integration of electronic-interfaced devices have increased, microgrid models have become too complex to perform a stability analysis. Thus, an effective model simplification method keeping most dynamics of the system becomes very essential. Singular perturbation is a common way for model simplification. However, its accuracy is insufficient when nonlinear properties dominate. This is caused by the “Quasi-Steady State Assumption” that traditional singular perturbation holds. By assuming that microgrid can only be stabilized when fast variables stop variating, the traditional method ignores some common phenomena before a stabilization occurs, leading to a loss of dynamics. To improve the accuracy, this paper proposes a “second-order singular perturbation”. Here, the traditional “Quasi-Steady State” is updated to a scenario that third-order derivatives of fast variables become zero before the microgrid stabilizes. The updated assumption covers more common phenomena before a stabilization occurs. This leads to a more precise simplification. Simulation results indicate that the proposed method outperforms traditional singular perturbation in accuracy.

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.

Effects of curvature on triple flame propagation in fuel-oxidizer counterflow

In this study, we analyze the propagation of an expanding triple flame in an axisymmetric counterflow of a fuel against an oxidizer. The problem is formulated as a thermo-diffusive model with one-step global reaction. An asymptotic analysis in the limit of large activation energy and weak strain rate is conducted. The study is supported and complemented by numerical simulations carried out for arbitrary values of the strain rate. In addition to the flame front curvature 1/Rt associated with the variation in the reactants concentrations transverse to the mixing layer, the propagation of the expanding triple flame also depends on the azimuthal curvature 1/Rf where Rf is the front leading edge radial distance. As the triple flame expands to large radial distances, its propagation becomes quasi-steady. Under a quasi-steady state assumption, an explicit expression is derived for the displacement speed of the triple flame, which is found to be linearly proportional to the total curvature 1/Rf+1/Rt. Two-dimensional axisymmetric simulations are conducted to validate in particular the quasi-steady assumption. These include transient simulations of the expanding triple flame which are compared to the numerical solution of the steady eigenvalue problem obtained in a frame attached to the propagating front under a quasi-steady assumption. Following a transient ignition phase, the triple flame is found to propagate in a quasi-steady manner when Rf (measured with the stoichiometric planar flame thickness) exceeds 5, approximately. Although the theoretical analysis is performed in the weak strain limit, the linear dependence of the triple flame speed on the curvature 1/Rfis found to be applicable over a wide range of strain rates. Besides, the analysis is extended for inwardly propagating triple flames (flame holes) and similar expressions describing the relationship between displacement speed of the triple flame and curvatures are obtained.

A Skeletal Mechanism for MILD Combustion of n-Heptane/Air Mixtures

ABSTRACT Moderate, intense or low-oxygen dilution (MILD) Combustion has many attributes such as low emissions, noiseless combustion with high thermal efficiency which are attractive for greener combustion systems. Past studies investigated flow and chemical kinetics effects in establishing this combustion mode for methane-air mixtures. Many of the practical fuels are large hydrocarbons and hence this study aims to develop a skeletal mechanism suitable for turbulent MILD combustion simulations of n-heptane/air. Computer Assisted Reduction Mechanism approach is employed to develop a mechanism involving 36 species and 205 reactions, which is validated for wide range of conditions using measurements or results obtained from a comprehensive mechanism. This mechanism is found to be excellent for predicting both ignition delay times and flame speeds for MILD conditions. The OH* and CH* obtained using quasi-steady state assumptions agree well with those obtained using the comprehensive mechanism with these chemiluminescent species. The performance of the skeletal mechanism for turbulent MILD combustion simulation is tested using Unsteady Reynolds Averaged Navier-Stokes (URANS) simulations with a finite-rate chemistry model. The computed statistics of temperature and O H number density agree quite well with measurements for highly diluted combustion cases.

Rovibrational-Specific QCT and Master Equation Study on N2(X1Σg+) + O(3P) and NO(X2Π) + N(4S) Systems in High-Energy Collisions.

This work presents a detailed investigation of the energy-transfer and dissociation mechanisms in N2(X1Σg+) + O(3P) and NO(X2Π) + N(4S) systems using rovibrational-specific quasiclassical trajectory (QCT) and master equation analyses. The complete set of state-to-state kinetic data, obtained via QCT, allows for an in-depth investigation of the Zel'dovich mechanism leading to the formation of NO molecules at microscopic and macroscopic scales. The master equation analysis demonstrates that the low-lying vibrational states of N2 and NO have dominant contributions to the NO formation and the corresponding extinction of N2 through the exchange process. For the considered temperature range, it is found that nearly 50% of the dissociation processes for N2 and NO molecules occur in the quasi-steady-state (QSS) regime, while for the Zel'dovich reaction, the distribution of the reactants does not reach the QSS conditions. Furthermore, using the QSS approximation to model the Zel'dovich mechanism leads to overestimating NO production by more than a factor of 4 in the high-temperature range. The breakdown of this well-known approximation has profound consequences for the approaches that heavily rely on the validity of QSS assumption in hypersonic applications. Finally, the investigation of the rovibrational state population dynamics reveals substantial similarities among different chemical systems for the energy-transfer and the dissociation processes, providing promising physical foundations for the use of reduced-order strategies in other chemical systems without significant loss of accuracy.

An Experimental and Modeling Study on the Combustion of Gasoline-Ethanol Surrogates for HCCI Engines

As an effective clean fuel, ethanol has the characteristics of improving antiknock quality and reducing emissions. It is an ideal antiknock additive for Homogeneous Charge Compression Ignition (HCCI) engines. The oxidation of gasoline-ethanol surrogates in HCCI engines is a very complex process which is dominated by the reaction kinetics. This oxidation process directly determines the performance and emissions of HCCI engines. Coupling the computational fluid dynamic (CFD) model with the gasoline-ethanol surrogate mechanism can be used for fuel design, so the construction of a reduced mechanism with high accuracy is necessary. A mechanism (278 species, 1439 reactions) at medium and low temperatures and experiments in a HCCI engine for the oxidation of gasoline-ethanol surrogates were presented in this paper. Directed relation graph with error propagation (DRGEP) method and quasi-steady-state assumption (QSSA) method were used in order to get a reduced model. Then, the kinetics of the vital reactions related to the formation and consumption of H and OH were adjusted. To validate the model, the HCCI experiments for the oxidation of gasoline-ethanol surrogates were conducted under different operating conditions. The verification result indicated that the present model can predict the oxidation process of gasoline-ethanol effectively.

Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics.

The recently developed physics-informed neural network (PINN) has achieved success in many science and engineering disciplines by encoding physics laws into the loss functions of the neural network such that the network not only conforms to the measurements and initial and boundary conditions but also satisfies the governing equations. This work first investigates the performance of the PINN in solving stiff chemical kinetic problems with governing equations of stiff ordinary differential equations (ODEs). The results elucidate the challenges of utilizing the PINN in stiff ODE systems. Consequently, we employ quasi-steady-state assumption (QSSA) to reduce the stiffness of the ODE systems, and the PINN then can be successfully applied to the converted non-/mild-stiff systems. Therefore, the results suggest that stiffness could be the major reason for the failure of the regular PINN in the studied stiff chemical kinetic systems. The developed stiff-PINN approach that utilizes QSSA to enable the PINN to solve stiff chemical kinetics shall open the possibility of applying the PINN to various reaction-diffusion systems involving stiff dynamics.

Simplified modeling combustion chemistry of neat and blended large hydrocarbon fuels with different functional groups

In the present study, a simplified modeling approach based on a two-step reaction scheme (TSRS) was developed to describe the high-temperature combustion chemistry of large hydrocarbon fuels. The characteristic reaction time scale analysis indicated a distinct two-step reaction property for the conversion of large hydrocarbon molecules. Along with the quasi-steady state assumption, TSRS utilized a seven-reaction sub-model to describe the fuel decomposition to a small group of intermediate products, and a detailed foundational fuel combustion chemistry model to describe the oxidation of these decomposed products. In the first half of this paper, the TSRS model was constructed for four fuel components with different functional groups, including n-heptane, iso-octane, methylcyclohexane, and n-butylbenzene, and validated sequentially by their speciation data, and global combustion properties of ignition delay time and laminar flame speed data. Sensitivity analysis and rate of production analysis revealed the primary decomposition paths in the TSRS model. Then, in the second half of the paper, algebraically blending the TSRS models of neat fuel components enabled the construction of surrogate fuel models for real fuels, including a two-component primary reference fuel (PRF), a three-component gasoline surrogate fuel (toluene-PRF), a four-component Chinese jet fuel (RP-3) surrogate, and a four-component diesel surrogate. Predictions from the TSRS-based surrogate models for real fuels were shown to be consistent with the speciation data as well as the global combustion properties, whenever available in the literatures, indicating the robustness and broad applicability of the TSRS approach.

Kinetic importance of the missing step in dithiobenzoate-mediated RAFT polymerizations of acrylates

The effect of retardation exhibited in dithiobenzoate-mediated polymerizations is perhaps the most controversial kinetic phenomenon discussed in polymer chemistry in the last 20 years. However, the Missing Step reaction (MSR), which involves the termination between a propagating radical and a 3-arm star product, has emerged as a hypothesis to explain the full kinetic picture for acrylate polymerizations. Here, a kinetic model, the most detailed of its kind, is developed to describe and study the MSR. The model is based on the method of moments and includes RAFT pre-equilibrium and equilibrium, backbiting, cross-termination, MSR and chain-length dependent radical termination. After validation through comparison with experimental data available in the literature, the model is used to demonstrate that the kinetic behavior of the propagating and intermediates radicals can be described by the quasi-steady state assumption and that backbiting does not significantly influence predicted conversion profiles and product molecular weights. It is also shown that the induction period observed experimentally can be represented by either slow initiation/reinitiation of primary radicals or by assuming that radical intermediates formed by addition of primary radicals to the RAFT agent have increased stability; however, the associated rate constants for these reactions had to be adjusted from literature values. A comparison between the MSR and Intermediate Radical Termination (IRT) representations RAFT-mediated acrylate polymerization identifies chain end-group functionality (EGF) as a means to further discriminate between these theories.

Minimizing the number of optimizations for efficient community dynamic flux balance analysis.

Dynamic flux balance analysis uses a quasi-steady state assumption to calculate an organism's metabolic activity at each time-step of a dynamic simulation, using the well-known technique of flux balance analysis. For microbial communities, this calculation is especially costly and involves solving a linear constrained optimization problem for each member of the community at each time step. However, this is unnecessary and inefficient, as prior solutions can be used to inform future time steps. Here, we show that a basis for the space of internal fluxes can be chosen for each microbe in a community and this basis can be used to simulate forward by solving a relatively inexpensive system of linear equations at most time steps. We can use this solution as long as the resulting metabolic activity remains within the optimization problem's constraints (i.e. the solution to the linear system of equations remains a feasible to the linear program). As the solution becomes infeasible, it first becomes a feasible but degenerate solution to the optimization problem, and we can solve a different but related optimization problem to choose an appropriate basis to continue forward simulation. We demonstrate the efficiency and robustness of our method by comparing with currently used methods on a four species community, and show that our method requires at least 91% fewer optimizations to be solved. For reproducibility, we prototyped the method using Python. Source code is available at https://github.com/jdbrunner/surfin_fba.

원형 실린더 구조물의 풍방향 공력감쇠비 산정

In the light of its positive damping property, along-wind aerodynamic damping is recognized to have a positive role so as to stabilize the wind induced vibration of tall building structure. The theoretical model developed to predicts aerodynamic damping under the assumption of quasi-steady state demonstrates the occurrence mechanism of aerodynamic damping and describes the influence factors affecting the formation of the damping. In this study, the along-wind aerodynamic damping identified from the measured acceleration responses acquisited through the aeroelastic model test with variant wind speed was compared with the theoretical damping in order to evaluate the reliability of the model. Aerodynamic damping was evaluated based on a virtual dynamic shaker method which is system identification technique developed recently. From the results of this study, it is found that the wind aerodynamic damping has a difference from the theoretical model, which is mainly attributed to the magnitude of the turbulence in different height and mean wind speeds.

Model reduction of aerobic bioprocess models for efficient simulation

Owing to the increasing demand for large scale and high efficiency in manufacturing processes, computer aided tools for process operation and control are rapidly gaining popularity. An important state variable in aerobic processes is the dissolved oxygen, which can be easily measured online and is an important indicator of the metabolic activity. However, due to the fast kinetics of the oxygen transfer, dynamical models describing aerobic bioprocesses tend to be highly stiff. This can lead to significant numerical problems hampering its use for fixed step discretization methods and computationally costly applications such as computer fluid dynamics. In this work we use the slow-motion invariant manifold and the quasi steady state assumption methods to eliminate the differential equation describing the dissolved oxygen (the fast mode). By doing this, the tractability of the model is significantly increased with a neglectable loss in description power. The reduced model is also useful for simplifying the observer design problems, which is demonstrated by a state and parameter estimation example at the end of the work.

Modeling the Kinetics, Curing Depth, and Efficacy of Radical-Mediated Photopolymerization: The Role of Oxygen Inhibition, Viscosity, and Dynamic Light Intensity.

Kinetic equations for a modeling system with type-I radical-mediated and type-II oxygen-mediated pathways are derived and numerically solved for the photopolymerization efficacy and curing depth, under the quasi-steady state assumption, and bimolecular termination. We show that photopolymerization efficacy is an increasing function of photosensitizer (PS) concentration (C0) and the light dose at transient state, but it is a decreasing function of the light intensity, scaled by [C0/I0]0.5 at steady state. The curing (or cross-link) depth is an increasing function of C0 and light dose (time × intensity), but it is a decreasing function of the oxygen concentration, viscosity effect, and oxygen external supply rate. Higher intensity results in a faster depletion of PS and oxygen. For optically thick polymers (>100 um), light intensity is an increasing function of time due to PS depletion, which cannot be neglected. With oxygen inhibition effect, the efficacy temporal profile has an induction time defined by the oxygen depletion rate. Efficacy is also an increasing function of the effective rate constant, K = k′/, defined by the radical producing rate (k′) and the bimolecular termination rate (kT). In conclusion, the curing depth has a non-linear dependence on the PS concentration, light intensity, and dose and a decreasing function of the oxygen inhibition effect. Efficacy is scaled by [C0/I0]0.5 at steady state. Analytic formulas for the efficacy and curing depth are derived, for the first time, and utilized to analyze the measured pillar height in microfabrication. Finally, various strategies for improved efficacy and curing depth are discussed.

Model-free and kinetic modelling approaches for characterising non-equilibrium pharmacological pathway activity: Internalisation of cannabinoid CB1 receptors.

Receptor internalisation is by nature kinetic. Application of a standard equilibrium dose response model to describe the properties of a ligand inducing internalisation, while commonly used, are therefore problematic. Here, we propose two quantitative approaches to address this issue-(a) a model-free method and (b) a kinetic modelling approach-and systematically evaluate the performance of these methods against traditional equilibrium methods to characterise the internalisation profiles of cannabinoid CB1 receptor agonists. Kinetic internalisation assays were conducted using a concentration series of six CB1 receptor ligands. Internalisation rate analysis and snapshot equilibrium analysis were performed. A model-free method was developed based on the mean residence time of internalisation. A kinetic internalisation model was developed under the quasi-steady state assumption. Rates of receptor internalisation depended on both agonist and concentration. Agonist potencies from snapshot equilibrium analysis increased with stimulation time, and there was no single time point at which internalisation profiles could infer agonist properties in a comparative manner. The model-free method yielded a time-invariant measure of potency/efficacy for internalisation. The kinetic model adequately described the internalisation of CB1 receptors over time and provided robust estimates of both potency and efficacy. Applying equilibrium analysis to a non-equilibrium pathway cannot provide a reliable estimate of agonist potency. Both the model-free and kinetic modelling approaches characterised the internalisation profiles of CB1 receptor agonists. The kinetic model provides additional advantages as a method to capture changes in receptor number during other functional assays.

A reduced mechanism for 2,5-dimetylfuran with assembled mechanism reduction methods

2,5-dimethylfuran (DMF), a promising alternative bio-fuel, has received enormous kinetical investigations. In this study, a new simplified mechanism was developed to describe the oxidation characteristics of DMF over wide conditions. Starting with a detailed mechanism of 545 species and 2874 reactions, a two-stage DRG-related method was adopted to create a skeletal mechanism. Then species sensitivity analyses were conducted to eliminate remained unimportant species. Finally, quasi-steady state assumption was applied to sub-products of DMF consumption, leading to a more compact reduced mechanism of 73 species and 419 reactions. It was found that the reproduction of species profiles for fuel-lean mixture at temperatures of 800–950 K was inadequate. Some modifications and rate constant optimization based on sensitivity analysis were executed to fix this issue. The reliable predictive ability of the reduced mechanism at high temperature was validated against the ignition delay times at pressures of 1.2 – 16 atm, equivalence ratios of 0.5 – 2.0 ranging temperatures of 1100 – 1900 K. The low temperature modeling performance was verified with ignition delay times measured in a rapid compression machine over temperatures of 740 – 1100 K at pressures of 15 and 30 bar as well as mole fraction profiles of reactants, products and major intermediate species measured in a jet-stirred reactor at 10 atm. The proposed model was also proved to reproduce laminar flame velocities at various initial temperatures fairly well.

Optimal operation of dynamic (energy) systems: When are quasi-steady models adequate?

Since design optimization faces the challenge of solving inherently large optimization problems, the complexity of underlying dynamic systems is often reduced by applying quasi-steady state assumptions. It is thereby indispensable to identify the components whose transient behavior is essential to ensure meaningful results. In this study, we discuss a dynamic model for an illustrative hybrid energy system, which extends the quasi-steady models of Voll et al. (2013). Based on optimal operation with fixed design, we underline the importance of the relationship between the dynamics of the model and of the input data for the adequateness of quasi-steady operation. Our results emphasize the need for suitable ramp constraints in quasi-steady models of dynamic systems within operational optimization. Moreover, the existence of a storage unit is no sufficient justification for quasi-steady state assumptions.