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Asymptotic analysis for stationary distributions of scaled reaction networks

Linard Hoessly, Carsten Wiuf, Panqiu Xia

Abstract

We study stationary distributions in the context of stochastic reaction networks. In particular, we are interested in complex balanced reaction networks and reduction of such networks by assuming a set of species (called non-interacting species) are degraded fast (and therefore essentially absent in the network), implying some reaction rates are large compared to others. Technically, we assume these reaction rates are scaled by a common parameter $N$ and let $N\to\infty$. The limiting stationary distribution as $N\to\infty$ is compared to the stationary distribution of the reduced reaction network obtained by algebraic elimination of the non-interacting species. In general, the limiting stationary distribution might differ from the stationary distribution of the reduced reaction network. We identify various sufficient conditions for when these two distributions are the same, including when the reaction network is detailed balanced and when the set of non-interacting species consists of intermediate species. In the latter case, the limiting stationary distribution essentially retains the form of the complex balanced distribution. This finding is particularly surprising given that the reduced reaction network might be non-weakly reversible and exhibit unconventional kinetics.

Asymptotic analysis for stationary distributions of scaled reaction networks

Abstract

We study stationary distributions in the context of stochastic reaction networks. In particular, we are interested in complex balanced reaction networks and reduction of such networks by assuming a set of species (called non-interacting species) are degraded fast (and therefore essentially absent in the network), implying some reaction rates are large compared to others. Technically, we assume these reaction rates are scaled by a common parameter and let . The limiting stationary distribution as is compared to the stationary distribution of the reduced reaction network obtained by algebraic elimination of the non-interacting species. In general, the limiting stationary distribution might differ from the stationary distribution of the reduced reaction network. We identify various sufficient conditions for when these two distributions are the same, including when the reaction network is detailed balanced and when the set of non-interacting species consists of intermediate species. In the latter case, the limiting stationary distribution essentially retains the form of the complex balanced distribution. This finding is particularly surprising given that the reduced reaction network might be non-weakly reversible and exhibit unconventional kinetics.
Paper Structure (16 sections, 15 theorems, 126 equations)

This paper contains 16 sections, 15 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Reaction networks
  5. Non-interacting species and reduction
  6. The conditional distribution as a stationary distribution
  7. Two special cases
  8. Positive recurrence of the reduced SRN
  9. A remark on Theorem \ref{['thm_sdpi0']}
  10. Convergence of complex balanced distributions
  11. Proofs
  12. Proof of Lemma \ref{['lmm_fgrn']}
  13. Proof of Theorem \ref{['thm_sdpi0']}
  14. Proof of Proposition \ref{['prop_itms']}
  15. Proof of Proposition \ref{['prop_rvsb']}
  16. ...and 1 more sections

Key Result

Lemma 2.7

Let $\mathcal{N}$ be weakly reversible and $\mathcal{U}$ a set of non-interacting species. Then, $\mathcal{U}_{\mathrm{pro}} = \mathcal{U}_{\deg}$ and $\mathcal{U}$ is eliminable.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2: Cappelletti
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Lemma 2.9
  • ...and 24 more