Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounding Transient Moments for a Class of Stochastic Reaction Networks Using Kolmogorov's Backward Equation

Takeyuki Iwasaki, Yutaka Hori

Abstract

Stochastic chemical reaction networks (SRNs) in cellular systems are commonly modeled as continuous-time Markov chains (CTMCs) describing the dynamics of molecular copy numbers. The exact evaluation of transient copy number statistics is, however, often hindered by a non-closed hierarchy of moment equations. In this paper, we propose a method for computing theoretically guaranteed upper and lower bounds on transient moments based on the Kolmogorov's backward equation, which provides a dual representation of the CME, the governing equation for the probability distribution of the CTMC. This dual formulation avoids the moment closure problem by shifting the source of infinite dimensionality to the dependence on the initial state. We show that, this dual formulation, combined with the monotonicity of the CTMC generator, leads to a finite-dimensional linear time-invariant system that provides bounds on transient moments. The resulting system enables efficient evaluation of moment bounds across multiple initial conditions by simple inner-product operations without recomputing the bounding system. Further, for certain classes of SRNs, the bounding ODEs admit explicit construction from the reaction model, providing a systematic and constructive framework for computing provable bounds.

Bounding Transient Moments for a Class of Stochastic Reaction Networks Using Kolmogorov's Backward Equation

Abstract

Stochastic chemical reaction networks (SRNs) in cellular systems are commonly modeled as continuous-time Markov chains (CTMCs) describing the dynamics of molecular copy numbers. The exact evaluation of transient copy number statistics is, however, often hindered by a non-closed hierarchy of moment equations. In this paper, we propose a method for computing theoretically guaranteed upper and lower bounds on transient moments based on the Kolmogorov's backward equation, which provides a dual representation of the CME, the governing equation for the probability distribution of the CTMC. This dual formulation avoids the moment closure problem by shifting the source of infinite dimensionality to the dependence on the initial state. We show that, this dual formulation, combined with the monotonicity of the CTMC generator, leads to a finite-dimensional linear time-invariant system that provides bounds on transient moments. The resulting system enables efficient evaluation of moment bounds across multiple initial conditions by simple inner-product operations without recomputing the bounding system. Further, for certain classes of SRNs, the bounding ODEs admit explicit construction from the reaction model, providing a systematic and constructive framework for computing provable bounds.

Paper Structure

This paper contains 13 sections, 4 theorems, 40 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Modeling of Stochastic Reaction Networks and Problem Statement
  3. Main Result
  4. Dual system for expectation computation
  5. Bounds on expectation using dual system
  6. Bounds on moments via input bounding
  7. Numerical Examples
  8. SRN with elementary reactions
  9. SRN with rational propensity functions
  10. Conclusion
  11. Proof of Lemma \ref{['lemma']}
  12. Proof of Lemma \ref{['lemma2']}
  13. Proof of Theorem \ref{['theorem2']}

Key Result

Lemma 1

Let $(\bm{u}_1(t), \bm{q}_1(t), y_1(t))$ and $(\bm{u}_2(t), \bm{q}_2(t), y_2(t))$ denote the input, state, and output of the state-space model $\mathcal{B}$, respectively. If $\bm q_1(0)=\bm q_2(0)$, and $\bm u_1(t)\leq \bm u_2(t)$ for all $t\in[0,T]$, then for any given $T$. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure C1: Truncated state space $\mathcal{S}$ and its boundary $\partial \mathcal{S}$. The vector $\bm{q}(t)$ of conditional expectations $\mathbb{E}[f(\bm{X}(t))|\bm{X}(0)=\bm{x}]$ is defined on $\mathcal{S}$ and $\bm{u}(t)$ is defined on $\partial \mathcal{S}$.
  • Figure D1: Upper and lower bounds on mean $\mathbb{E}[X]$ (left) and variance $\mathbb{V}[X]$ (right) of the molecular copy number for the dimerization reaction network, compared with Monte Carlo simulations.
  • Figure D2: Gap between the upper and lower bounds of the mean $\mathbb{E}[X]$ for various state space sizes $N$.
  • Figure D3: Upper and lower bounds on mean copy number of protein B in the genetic toggle switch

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Remark 1