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Acceleration of Moment Bound Optimization for Stochastic Chemical Reactions Using Reaction-wise Sparsity of Moment Equations

Tomoki Sadatoshi, Antonis Papachristodoulou, Yutaka Hori

Abstract

Moment dynamics in stochastic chemical kinetics often involve an infinite chain of coupled equations, where lower-order moments depend on higher-order ones, making them analytically intractable. Moment bounding via semidefinite programming provides guaranteed upper and lower bounds on stationary moments. However, this formulation suffers from the rapidly growing size of semidefinite constraints due to the combinatorial growth of moments with the number of molecular species. In this paper, we propose a sparsity-exploiting matrix decomposition method for semidefinite constraints in stationary moment bounding problems to reduce the computational cost of the resulting semidefinite programs. Specifically, we characterize the sparsity structure of moment equations, where each reaction involves only a subset of variables determined by its reactants, and exploit this structure to decompose the semidefinite constraints into smaller ones. We demonstrate that the resulting formulation reduces the computational cost of the optimization problem while providing practically useful bounds.

Acceleration of Moment Bound Optimization for Stochastic Chemical Reactions Using Reaction-wise Sparsity of Moment Equations

Abstract

Moment dynamics in stochastic chemical kinetics often involve an infinite chain of coupled equations, where lower-order moments depend on higher-order ones, making them analytically intractable. Moment bounding via semidefinite programming provides guaranteed upper and lower bounds on stationary moments. However, this formulation suffers from the rapidly growing size of semidefinite constraints due to the combinatorial growth of moments with the number of molecular species. In this paper, we propose a sparsity-exploiting matrix decomposition method for semidefinite constraints in stationary moment bounding problems to reduce the computational cost of the resulting semidefinite programs. Specifically, we characterize the sparsity structure of moment equations, where each reaction involves only a subset of variables determined by its reactants, and exploit this structure to decompose the semidefinite constraints into smaller ones. We demonstrate that the resulting formulation reduces the computational cost of the optimization problem while providing practically useful bounds.

Paper Structure

This paper contains 13 sections, 5 theorems, 27 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Formulation and Problem Statement
  3. Moment equation for stochastic chemical reaction
  4. Semidefinite programming for bounding moments
  5. Matrix decomposition using reaction-wise sparsity
  6. Reaction-wise sparsity of moment equations
  7. Sparsity-based reformulation of the constraints
  8. Numerical Demonstration
  9. Structural Interpretation via Chordal Decomposition
  10. Conclusion
  11. Definition of $\Phi_i$
  12. Definition of $\Omega_i^{(k)}$
  13. Rate constants used in the simulation

Key Result

Theorem 1

sakurai2017convexsakurai2018optimization Consider a stochastic chemical reaction network with given stoichiometry $\{\bm{s}_i\}_{i=1}^{r}$ and propensity functions $\{w_{i}(\bm{x})\}_{i=1}^{r}$. Suppose the reaction network satisfies Assumption 1, and let $\bm{m}^*$ denote the vector of the stationa $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure C1: Detailed sparsity pattern of the coefficient matrix $\hat{A}_i$ in eq. \ref{['eq:equ_hatM_i']}
  • Figure D1: Comparison before and after matrix decomposition for each truncation order $\mu$ (a) Schematic diagram of the reactions ($\rightarrow$ : Activation, $\dashv$ : Repression) (b) the bounds of $\mathbb{E}[x_6]$ (c) computation time
  • Figure E1: Block arrow pattern of the coefficient matrices

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Theorem 3
  • Remark 2
  • Remark 3