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Stochastic Averaging and Statistical Inference of Glycolytic Pathway

Arnab Ganguly, Hye-Won Kang

Abstract

Many biological processes exhibit oscillatory behavior. Among these, glycolytic oscillations have been extensively studied due to their well-characterized biochemical reaction networks. However, the complexity of these networks necessitates low-dimensional ordinary differential equation (ODE) models to identify core mechanisms and perform stability analysis. While previous studies proposed reduced ODE models, these were typically introduced from deterministic descriptions rather than the underlying stochastic dynamics, which more accurately represent discrete reaction events occurring at random times. In this paper, we develop a rigorous probabilistic framework for deriving a reduced Othmer-Aldridge model of the glycolytic pathway from its stochastic formulation. The full system is modeled as a multiscale continuous-time Markov chain with different time and abundance scales. Under an appropriate scaling regime and specific structural conditions, we prove that the dynamics of the slow components are approximated by a two-dimensional ODE. The proof is technically involved due to the network's complexity and strong coupling between its components. We further consider the problem of parameter estimation when observations are limited to the slow species: fructose-6-phosphate and ADP. The reduced system yields a tractable loss function depending solely on these variables. We prove that the resulting estimators are statistically consistent when the data originate from the full stochastic reaction network. Together, these results provide a mathematically rigorous framework linking stochastic biochemical reaction networks, reduced deterministic dynamics, and statistically reliable parameter estimation.

Stochastic Averaging and Statistical Inference of Glycolytic Pathway

Abstract

Many biological processes exhibit oscillatory behavior. Among these, glycolytic oscillations have been extensively studied due to their well-characterized biochemical reaction networks. However, the complexity of these networks necessitates low-dimensional ordinary differential equation (ODE) models to identify core mechanisms and perform stability analysis. While previous studies proposed reduced ODE models, these were typically introduced from deterministic descriptions rather than the underlying stochastic dynamics, which more accurately represent discrete reaction events occurring at random times. In this paper, we develop a rigorous probabilistic framework for deriving a reduced Othmer-Aldridge model of the glycolytic pathway from its stochastic formulation. The full system is modeled as a multiscale continuous-time Markov chain with different time and abundance scales. Under an appropriate scaling regime and specific structural conditions, we prove that the dynamics of the slow components are approximated by a two-dimensional ODE. The proof is technically involved due to the network's complexity and strong coupling between its components. We further consider the problem of parameter estimation when observations are limited to the slow species: fructose-6-phosphate and ADP. The reduced system yields a tractable loss function depending solely on these variables. We prove that the resulting estimators are statistically consistent when the data originate from the full stochastic reaction network. Together, these results provide a mathematically rigorous framework linking stochastic biochemical reaction networks, reduced deterministic dynamics, and statistically reliable parameter estimation.
Paper Structure (10 sections, 13 theorems, 154 equations, 2 figures, 1 table)

This paper contains 10 sections, 13 theorems, 154 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction.
  2. Notational conventions
  3. Mathematical Framework
  4. Stochastic Description
  5. Stochastic Averaging
  6. Proof of \ref{['thm:FLLN-Glyco']}
  7. Discussion of the reduced order ODE system
  8. Parameter estimation
  9. Numerical Experiments
  10. Appendix

Key Result

Proposition 3.1

Suppose that assum:main holds. Then for any $T>0$, the sequence $(\Gamma^{(n)}_F,Z^{(n)}_S)$ is relatively compact as $\mathcal{M}_T([0,\infty)^{7}\times[0,T])\times D([0, T], [0,\infty)^3)$-valued random variables. Furthermore, the limit points of $Z^{(n)}_S$ are almost surely in $C([0, T], [0,\in

Figures (2)

  • Figure 1: Glycolytic Pathway
  • Figure 2: Comparison between the behavior of the normalize molecular counts for species $A_2$ in the full CTMC model and the reduced ODE model with varying scaling parameter values $n$.

Theorems & Definitions (31)

  • Proposition 3.1
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.2
  • Remark 3.1
  • proof : Proof of \ref{['prop:A3-tight-prop']}
  • Corollary 3.1
  • ...and 21 more