Stochastic ordering tools for continuous-time Markov chains and applications to reaction network models

Abstract

Stochastic reaction networks are mathematical models with a wide range of applications in biochemistry, ecology, and epidemiology, and are often complex to analyze. Except for some special cases, it is generally difficult to predict how the abundances of all considered species evolve over time. A possible approach to address this issue is to develop tools to compare the model under study with a similar one whose behavior is better understood. The main contribution of our work is to provide direct and computable conditions that can be used to ensure the existence of an ordered coupling between two stochastic reaction networks and to identify which parameter changes in a given model lead to an increase or decrease in the count of certain species. We also make available an algorithm that implements our theory, and we illustrate it with several applications.

Figures (14)

• Figure 1: a one-dimensional reaction network and two different associated sets of rate constants $K^X$ and $K^Y$.
• Figure 2: the rate constants satisfy $\kappa_{\textup{S} \to \textup{P}}^X \leq \kappa_{\textup{S} \to \textup{P}}^Y$ and $\kappa_{\textup{P} \to \textup{S}}^X \geq \kappa_{\textup{P} \to \textup{S}}^Y$, and by Theorem \ref{['thm:lsc']} the species counts fulfill $X_\textup{S}(t) \geq Y_\textup{S}(t)$ and $X_\textup{P}(t) \leq Y_\textup{P}(t)$ for all $t \geq 0$ almost surely.
• Figure 3: no preordering structure can be found with our algorithm for this reaction network.
• Figure 4: the rate constants satisfy $\kappa_{\textup{S}+\textup{I} \to 2\textup{I}}^X \leq \kappa_{\textup{S}+\textup{I} \to 2\textup{I}}^Y$ and $\kappa_{\textup{I} \to \textup{S}}^X \geq \kappa_{\textup{I} \to \textup{S}}^Y$, and by Theorem \ref{['thm:lsc']} the species counts fulfill $X_\textup{S}(t) \geq Y_\textup{S}(t)$ and $X_\textup{I}(t) \leq Y_\textup{I}(t)$ for all $t \geq 0$ almost surely.
• Figure 5: the rate constants satisfy $\kappa_{\textup{S}+\textup{E} \to \textup{C}}^X \leq \kappa_{\textup{S}+\textup{E} \to \textup{C}}^Y$, $\kappa_{\textup{C} \to \textup{S}+\textup{E}}^X \geq \kappa_{\textup{C} \to \textup{S}+\textup{E}}^Y$ and $\kappa_{\textup{C} \to \textup{E}+\textup{P}}^X \leq \kappa_{\textup{C} \to \textup{E}+\textup{P}}^Y$, and by Theorem \ref{['thm:lsc']} the species counts fulfill $X_\textup{S}(t) \geq Y_\textup{S}(t)$ and $X_\textup{P}(t) \leq Y_\textup{P}(t)$ for all $t \geq 0$ almost surely.

Theorems & Definitions (19)

• Example 2.1
• Remark 2.2
• Theorem 3.1
• proof
• Example 3.2
• Theorem 4.1
• Remark 4.2
• proof : Proof of Theorem \ref{['thm:lsc']}
• Remark 4.3
• Example 4.4: Reversible reaction