A constructive approach to quasi-steady state reductions

Goeke, Alexandra; Walcher, Sebastian

doi:10.1007/s10910-014-0402-5

Cited by 53 publications

(142 citation statements)

References 24 publications

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“…Thus, in the rQSSA, the underlying assumption is that there is a preliminary transient phase in which nearly all of the substrate is depleted, but a negligible amount of product is generated. As a consequence of this assumption, (23) admits an approximate conservation law, s 0 ∼ c + p, that implies dp…”

Section: The Reverse Quasi-steady-state Approximationmentioning

confidence: 99%

The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

Eilertsen

Schnell

2020

Mathematical Biosciences

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In this work, we revisit the scaling analysis and commonly accepted conditions for the validity of the standard, reverse and total quasi-steady-state approximations through the lens of dimensional Tikhonov-Fenichel small parameters and their respective critical manifolds. By combining Tikhonov-Fenichel small parameter with scaling analysis, we derive improved upper bounds on the approximation error for the standard, reverse and total quasi-steady-state approximations. We find that the condition for the validity of the reverse quasi-steady-state approximation is far less restrictive than was previously assumed, and we derive a new "small" parameter that determines the validity of this approximation. Moreover, we show for the first time that the critical manifold of the reverse quasi-steady-state approximation contains a singular point where normal hyperbolicity is lost. Associated with this singularity is a transcritical bifurcation, and the corresponding normal form of this bifurcation is recovered through scaling analysis. It is commonly accepted that the reverse quasi-steady-state approximation is only valid in regions of parameter space where the influence of the singular point can be ignored. However, by using the method of slowly-varying Lyapunov functions, we provide asymptotic estimates for the validity of the reverse quasi-steady-state approximation in parameter regions where the effect of the singular point cannot be disregarded. In doing so, we extend the established domain of validity for the reverse quasi-steady-state approximation. Previous analysis has suggested that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Consequently, this opens the possibility of utilizing the reverse quasi-steady-state approximation to model enzyme catalyzed reactions and estimate kinetic parameters in enzymatic assays at much lower enzyme to substrate ratios than was previously thought.

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Section: The Reverse Quasi-steady-state Approximationmentioning

confidence: 99%

The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

Eilertsen

Schnell

2020

Mathematical Biosciences

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“…adjusting conditions (i) and (ii) accordingly. Conditions (i) and (ii) are certainly necessary for (7) or (8) to be a transformed version of (1). The first part of the next lemma shows sufficiency.…”

Section: Coordinate-free Reductionmentioning

confidence: 99%

“…These first integrals determine slow and "very slow" variables. Parts (b) and (c) are straightforward applications of [7], Theorem 1, Remark 4 and Remark 2.…”

Section: Coordinate-free Reductionmentioning

confidence: 99%

“…In Section 2 we review the work by Cardin and Teixeira [3]. Section 3 generalizes the coordinate-independent reduction algorithm from [7] to three-timescale systems. In Section 4 we start from a general parameter dependent system and extend the work from [8] on critical parameter values (Tikhonov-Fenichel parameter values) to three time scales (resp.…”

Section: Introduction and Overviewmentioning

confidence: 99%

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Coordinate-independent singular perturbation reduction for systems with three time scales

Kruff¹,

Walcher

2019

Mathematical Biosciences and Engineering

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On the basis of recent work by Cardin and Teixeira on ordinary differential equations with more than two time scales, we devise a coordinateindependent reduction for systems with three time scales; thus no a priori separation of variables into fast, slow etc. is required. Moreover we consider arbitrary parameter dependent systems and extend earlier work on Tikhonov-Fenichel parameter values -i.e. parameter values from which singularly perturbed systems emanate upon small perturbations -to the three time-scale setting. We apply our results to two standard systems from biochemistry. MSC (2010): 92C45, 34E15, 37D10

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“…The mathematical foundation to justify such a QSS reduction is given by Tikhonov's [27] respectively Fenichel's [4] work on normally hyperbolic attracting slow manifolds in singular perturbation problems. Goeke, Noethen and Walcher [22,6] provide a comprehensive discussion on the general setup of Fenichel's geometric singular perturbation theory (GSPT) with an emphasis on explaining when a QSS reduction is justified or when it leads to erroneous results.…”

Section: Introductionmentioning

confidence: 99%

Computational Singular Perturbation Method for Nonstandard Slow-Fast Systems

Lizarraga¹,

Wechselberger²

2020

SIAM J. Appl. Dyn. Syst.

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The computational singular perturbation (CSP) method is an algorithm which iteratively approximates slow manifolds and fast fibers in multiple-timescale dynamical systems. Since its inception due to Lam and Goussis [17], the convergence of the CSP method has been explored in depth; however, rigorous applications have been confined to the standard framework, where the separation between 'slow' and 'fast' variables is made explicit in the dynamical system. This paper adapts the CSP method to nonstandard slow-fast systems having a normally hyperbolic attracting critical manifold. We give new formulas for the CSP method in this more general context, and provide the first concrete demonstrations of the method on genuinely nonstandard examples.

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A constructive approach to quasi-steady state reductions

Cited by 53 publications

References 24 publications

The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

Coordinate-independent singular perturbation reduction for systems with three time scales

Computational Singular Perturbation Method for Nonstandard Slow-Fast Systems

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