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Asymptotic Analysis of the Total Quasi-Steady State Approximation for the Michaelis--Menten Enzyme Kinetic Reactions

Arnab Ganguly, Wasiur R. KhudaBukhsh

TL;DR

This work rigorously justifies the total QSSA (tQSSA) for stochastic Michaelis–Menten kinetics by developing a stochastic averaging principle. It derives a Functional Law of Large Numbers, showing the slow variable Z_V converges to a random-ODE driven by the fast equilibrium of the complex, and proves a Functional Central Limit Theorem that characterizes fluctuations around this limit through an explicit Itô SDE obtained via a Poisson equation analysis. The results are obtained in a Poisson random-measure framework, avoiding generator-based methods and yielding a random-ODE limit for the reduced model, with a detailed description of the associated fluctuations. The findings provide rigorous justification and quantitative fluctuation descriptions for tQSSA-based model reduction in multiscale chemical reaction networks, with direct implications for efficient simulation and reliable parameter inference.

Abstract

We consider a stochastic model of the Michaelis-Menten (MM) enzyme kinetic reactions in terms of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures (PRMs). It has been argued that among various Quasi-Steady State Approximations (QSSAs) for the deterministic model of such chemical reactions, the total QSSA (tQSSA) is the most accurate approximation, and it is valid for a wider range of parameter values than the standard QSSA (sQSSA). While the sQSSA for this model has been rigorously derived from a probabilistic perspective at least as early as 2006 in Ball et al. (2006), a rigorous study of the tQSSA for the stochastic model appears missing. We fill in this gap by deriving it as a Functional Law of Large Numbers (FLLN), and also studying the fluctuations around this approximation as a Functional Central Limit Theorem (FCLT).

Asymptotic Analysis of the Total Quasi-Steady State Approximation for the Michaelis--Menten Enzyme Kinetic Reactions

TL;DR

This work rigorously justifies the total QSSA (tQSSA) for stochastic Michaelis–Menten kinetics by developing a stochastic averaging principle. It derives a Functional Law of Large Numbers, showing the slow variable Z_V converges to a random-ODE driven by the fast equilibrium of the complex, and proves a Functional Central Limit Theorem that characterizes fluctuations around this limit through an explicit Itô SDE obtained via a Poisson equation analysis. The results are obtained in a Poisson random-measure framework, avoiding generator-based methods and yielding a random-ODE limit for the reduced model, with a detailed description of the associated fluctuations. The findings provide rigorous justification and quantitative fluctuation descriptions for tQSSA-based model reduction in multiscale chemical reaction networks, with direct implications for efficient simulation and reliable parameter inference.

Abstract

We consider a stochastic model of the Michaelis-Menten (MM) enzyme kinetic reactions in terms of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures (PRMs). It has been argued that among various Quasi-Steady State Approximations (QSSAs) for the deterministic model of such chemical reactions, the total QSSA (tQSSA) is the most accurate approximation, and it is valid for a wider range of parameter values than the standard QSSA (sQSSA). While the sQSSA for this model has been rigorously derived from a probabilistic perspective at least as early as 2006 in Ball et al. (2006), a rigorous study of the tQSSA for the stochastic model appears missing. We fill in this gap by deriving it as a Functional Law of Large Numbers (FLLN), and also studying the fluctuations around this approximation as a Functional Central Limit Theorem (FCLT).

Paper Structure

This paper contains 13 sections, 10 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The MM enzyme kinetic reaction system
  3. Our contributions in the context of relevant literature
  4. Notational conventions
  5. Stochastic model
  6. The scaling regime for the tQSSA
  7. The FLLN
  8. Relative compactness
  9. A first order differential operator
  10. The FCLT
  11. Solution of a Poisson equation and its properties
  12. Auxiliary results
  13. Acronyms

Key Result

Proposition 4.1

Assume $\{K^{(n)}_1 : n\ge 1\}$ is a tight sequence of random variables. Then, for any $T>0$, the sequence $\{(\Gamma^{(n)},Z^{(n)}_V) : n\ge 1\}$ is relatively compact as $\mathcal{M}_{T}( {\mathbb{R}_{+}}\times {\mathbb{R}_{+}} \times [0, T])\times D([0, T], {\mathbb{R}_{+}})$-valued random vari

Figures (1)

  • Figure 1: Accuracy of the tQSSA for the MM enzyme kinetic model. The plot shows that the exact Doob--Gillespie simulations Wilkinson2018SMS and the tQSSA solution for the MM enzyme kinetic model are very close. The parameters used in the simulations are $\kappa_1 = 1$, $\kappa_{-1} = 1$, $\kappa_2 = 0.75$, $K_2 = 0.1$, $K_1=1.0$, and $n=1000$.

Theorems & Definitions (22)

  • Proposition 4.1
  • proof : Proof of \ref{['prop:rel_compactness']}
  • Lemma 4.1
  • proof : Proof of \ref{['lem:conv-occ-ZV']}
  • Lemma 4.2
  • proof
  • Theorem 4.1
  • Remark 4.1
  • proof : Proof of \ref{['thm:tQSSA']}
  • Theorem 5.1
  • ...and 12 more