Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coordinate-independent model reductions of chemical reaction networks based on geometric singular perturbation theory

Timothy Earl Figueroa Lapuz, Martin Wechselberger

TL;DR

The paper tackles the limitation and ambiguity of quasi-steady-state approximations in chemical reaction networks by developing a coordinate-free reduction framework based on ci-GSPT and the parametrization method. By focusing on normally hyperbolic slow manifolds $S_0$ and their slow flows, the authors construct parameter-specific reduced models without enforcing explicit timescale separation, and provide a geometric classification of reductions. They apply the framework to the Michaelis–Menten system (yielding 14 irreversible and 25 reversible reductions) and demonstrate scalability with a Kim–Forger oscillator reduction, illustrating both the method and its breadth. The work offers a rigorous, constructive alternative to QSSA that can adapt to larger CRNs and clarifies when and how reductions diverge from classical QSSA predictions, with potential impact on model accuracy and interpretability in systems biology.

Abstract

The quasi-steady-state approximation (QSSA) is a standard technique for reducing the complexity of chemical reaction networks (CRNs). The validity of any QSSA-based model is restricted to specific parameter regimes. Selecting the appropriate reduction is not always straightforward. At times, QSSAs are misused outside of their validity regions and, even when a particular QSSA is considered valid in a given parameter regime, other QSSAs may be simultaneously valid, creating ambiguity. Here, we employ a more powerful alternative: a constructive model reduction framework based on coordinate-independent geometric singular perturbation theory (ci-GSPT) and the parametrization method. A key advantage of this approach is its ability to derive reduced models independent of a clear timescale separation in the variables for a specific parameter configuration. We demonstrate our approach on two benchmark systems. For the Michaelis-Menten (MM) reaction, we show that the framework provides a systematic approach by exploring parameter configurations across three orders of magnitude: asymptotically large, small, and `order one'. A consequence of this systematic analysis is a geometric classification that categorizes the resulting model reductions and provides a point of comparison between our approach and common QSSA variants in the literature. For the more complex Kim-Forger model, we show that this approach successfully produces a reduction without the need for a coordinate transformation, showcasing its applicability to larger systems.

Coordinate-independent model reductions of chemical reaction networks based on geometric singular perturbation theory

TL;DR

The paper tackles the limitation and ambiguity of quasi-steady-state approximations in chemical reaction networks by developing a coordinate-free reduction framework based on ci-GSPT and the parametrization method. By focusing on normally hyperbolic slow manifolds and their slow flows, the authors construct parameter-specific reduced models without enforcing explicit timescale separation, and provide a geometric classification of reductions. They apply the framework to the Michaelis–Menten system (yielding 14 irreversible and 25 reversible reductions) and demonstrate scalability with a Kim–Forger oscillator reduction, illustrating both the method and its breadth. The work offers a rigorous, constructive alternative to QSSA that can adapt to larger CRNs and clarifies when and how reductions diverge from classical QSSA predictions, with potential impact on model accuracy and interpretability in systems biology.

Abstract

The quasi-steady-state approximation (QSSA) is a standard technique for reducing the complexity of chemical reaction networks (CRNs). The validity of any QSSA-based model is restricted to specific parameter regimes. Selecting the appropriate reduction is not always straightforward. At times, QSSAs are misused outside of their validity regions and, even when a particular QSSA is considered valid in a given parameter regime, other QSSAs may be simultaneously valid, creating ambiguity. Here, we employ a more powerful alternative: a constructive model reduction framework based on coordinate-independent geometric singular perturbation theory (ci-GSPT) and the parametrization method. A key advantage of this approach is its ability to derive reduced models independent of a clear timescale separation in the variables for a specific parameter configuration. We demonstrate our approach on two benchmark systems. For the Michaelis-Menten (MM) reaction, we show that the framework provides a systematic approach by exploring parameter configurations across three orders of magnitude: asymptotically large, small, and `order one'. A consequence of this systematic analysis is a geometric classification that categorizes the resulting model reductions and provides a point of comparison between our approach and common QSSA variants in the literature. For the more complex Kim-Forger model, we show that this approach successfully produces a reduction without the need for a coordinate transformation, showcasing its applicability to larger systems.

Paper Structure

This paper contains 14 sections, 2 theorems, 77 equations, 6 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Pre-processing CRN Models
  3. Pre-processing the reversible MM reaction scheme
  4. Pre-processing the KF reaction scheme
  5. Model Reductions of CRNs Based on ci-GSPT
  6. Asymptotic assumptions on dimensionless parameters
  7. The tools of ci-GSPT
  8. ci-GSPT model reduction of the MM reaction model \ref{['eq:dimless-EK-2D']}
  9. The classical tQSSA: $\varepsilon := \gamma$, $\alpha,\beta = \mathcal{O}(1)$ and $\delta = 0$
  10. ci-GSPT model reduction of the KF reaction model \ref{['KF-full-nondim']}
  11. Geometric Classification of the MM Reaction Scheme
  12. Irreversible MM: Literature Comparison and Model Reductions
  13. Reversible MM: Relevant Cases and Model Reductions
  14. Conclusions

Key Result

Lemma 1

The non-trivial eigenvalues of the linearization of system eq:dimless-general-layer along the $k$-dimensional critical manifold $S_0$ are given by the eigenvalues of the $(r-k)\times (r-k)$ matrix $D\!f_0 N_0|_{S_0}$.

Figures (6)

  • Figure 1: A sketch of the graph embedding $\phi_0$ of the critical manifold $S_0$ and the projection of the vector field $F_1(y)$ from $T_y\mathbb{R}^r$ onto $T_yS_0$.
  • Figure 2: (a) A diagram of the ci-GSPT based model reduction of system \ref{['eq:dimless-EK-2Dirr']} for $\gamma = \mathcal{O}(\varepsilon), \alpha,\beta = \mathcal{O}(1)$. (b) A comparison of the simulation of the full system \ref{['eq:dimless-EK-2Dirr']} and the tQSSA \ref{['tQSSA_model']} on its critical manifold. Parameter values are $\alpha = 0.75, \beta = 1, \gamma = 0.005$ and the IC for the model reduction is $s(0) = 0.5687$.
  • Figure 3: Numerical validation of the model reduction for the Kim-Forger (KF) reaction model. A comparison of numerical simulations of the 4D full system \ref{['KF-4D-nondim']} (solid blue lines) and the ci-GSPT model reduction \ref{['goodwin_eq3']} (dashed green lines). (a) $\gamma = \rho_6 = 10^{-6}$ shows non-oscillatory dynamics. (b) $\gamma = 1.5 \rho_6$, where $\rho_6 = 10^{-6}$, shows oscillatory dynamics. Other parameter values and ICs are given in Table \ref{['KF_params']}. The dashed green reduction solution is visible on top of the solid blue full solution, demonstrating the high accuracy of the reduction.
  • Figure 4: Sketches of critical manifold branches (blue) arising from the MM reaction scheme, with some important features highlighted such as horizontal asymptotes and $s$-intercepts. For cases with Form 4 and $g(s) = s - \delta s$ (see Definition \ref{['criticalmanifold_def']}), $\delta > 1$ results in Form 4$^*$, $\delta < 1$ results in Form 4$^{**}$, and $\delta = 1$ is discussed in Supplementary Material I. Remaining cases where $g(s) = - \delta s$ result in Form 4$^*$ and $s_{int} = 1$.
  • Figure 5: A diagram of all 67 singularly perturbed MM cases, where $\delta = \mathcal{O}(\varepsilon)$ shows also the irreversible cases. Critical manifold(s) $S_0$ (blue) and a fast fiber $\mathcal{W}^S(y)$ where $y \in S_0$ (red: linear, purple: non-linear) are shown. Gray boxes: non-singularly perturbed cases. Green boxes: the critical manifold has degeneracy. Dotted lines and circled points denote location. Red boxes: cases with a single critical manifold and it is repelling. Supplementary Material I provides more information on parameter configuration.
  • ...and 1 more figures

Theorems & Definitions (42)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 2
  • Remark 6
  • Definition 3
  • Lemma 1
  • ...and 32 more