Coclique level structure for stochastic chemical reaction networks

TL;DR

This work tackles the challenge of deriving explicit mean first passage times ($MFPT$) for high-dimensional stochastic chemical reaction networks (SCRNs) by introducing the concept of coclique level structures. It develops graph-based tests and an algorithm to identify all coclique level structures under a unimolecular-change assumption, enabling closed-form analytical bounds for $MFPT$s via surrogate birth-death processes. The authors prove existence criteria linked to bipartiteness, provide an efficient enumeration procedure for all coclique level functions, and derive concrete $MFPT$ bounds that quantify how reaction rates influence transition times. The framework is illustrated on chromatin modification circuits and network motifs, revealing how parameter choices modulate stochastic dynamics and providing mechanistic insight with potential broad applicability to finite-state SCRNs and non-mass-action kinetics.

Abstract

Continuous time Markov chains are commonly used as models for the stochastic behavior of chemical reaction networks. More precisely, these Stochastic Chemical Reaction Networks (SCRNs) are frequently used to gain a mechanistic understanding of how chemical reaction rate parameters impact the stochastic behavior of these systems. One property of interest is mean first passage times (MFPTs) between states. However, deriving explicit formulas for MFPTs can be highly complex. In order to address this problem, we first introduce the concept of coclique level structure and develop theorems to determine whether certain SCRNs have this feature by studying associated graphs. Additionally, we develop an algorithm to identify, under specific assumptions, all possible coclique level structures associated with a given SCRN. Finally, we demonstrate how the presence of such a structure in a SCRN allows us to derive closed form formulas for both upper and lower bounds for the MFPTs. Our methods can be applied to SCRNs taking values in a generic finite state space and can also be applied to models with non-mass-action kinetics. We illustrate our results with examples from the biological areas of epigenetics, neurobiology and ecology.

Figures (6)

• Figure 1: Three-species cascade motif: reaction diagram, graph $\mathcal{G}$ and associated Markov chain (Example \ref{['ex:illustrativeexample']}). (a) Chemical reaction system diagram. The numbers on the arrows correspond to the reactions associated with the arrows as described in (\ref{['reacs-3speciesexample']}) in the main text. (b) Graph $\mathcal{G}$ associated with the chemical reaction system in panel (a). (c) State space and transitions for the projected continuous time Markov chain $\widecheck{X}=\{ (X_1(t),X_2(t))^T:\: t \ge 0\}$, which keeps track of $(n_{\mathrm{W}},n_{\mathrm{Y}})$ through time. Here, we consider $\mathrm{N_{tot}}=2$ and we use dots to represent the states, and red double-ended arrows to represent transitions in both directions. Additionally, we use shades of blue to distinguish the level to which each state belongs. The function $L(x_1,x_2)$ associated with the coclique level structure is $L(x_1,x_2)=x_1+2x_2$. The rates associated with the one-step transitions for the projected Markov chain $\widecheck{X}$ are given in (\ref{['rates3species']}).
• Figure 2: Key steps of the algorithm for identifying all coclique level functions for the projected continuous time Markov chain $\widecheck X$ associated with a SCRN under Assumption \ref{['assumption:Unimolecular_change']} and assuming that $\mathcal{G}$ is weakly connected.
• Figure 3: Example of a SCRN whose associated graph $\mathcal{G}$ is not weakly connected (Example \ref{['ex:Gen2']}). (a) Chemical reaction system diagram. The numbers on the arrows correspond to the reactions associated with the arrows as described in (\ref{['reacs-example2']}) in the main text. (b) Graph $\mathcal{G}$ associated with the chemical reaction system in panel (a). (c) State space and transitions of the projected continuous time Markov chain $\widecheck{X}=\{ (X_1(t),X_3(t))^T:\: t \ge 0\}$, which keeps track of $(n_{\mathrm{S_1}},n_{\mathrm{S_3}})$ through time. Here, we consider $N_1=2$, $N_2=3$ and we use dots to represent the states, and red double-ended (single-ended) arrows to represent transitions in both directions (in a single direction). Additionally, we use shades of blue to distinguish the level to which each state belongs. The function $L$ associated with the coclique level structure is $L(x_1,x_3)=x_1+x_3$. On the right-hand side of the panel, we show the rates associated with the one-step transitions for the projected Markov chain $\widecheck{X}$. Note that, given an initial condition, the quantity of species $\mathrm{S}_5$ does not change over time and we denote this conserved quantity by $\mathrm{N_3}$.
• Figure 4: Chromatin modification circuit including only histone modifications: reaction diagram, graph $\mathcal{G}$ and associated Markov chain. (a) Chemical reaction system diagram. The numbers on the arrows correspond to the reactions associated with the arrows as described in (\ref{['reacs2D']}) in the main text. (b) Graph $\mathcal{G}$ associated with the chemical reaction system in panel (a). (c) State space and transitions of the projected continuous time Markov chain $\widecheck{X}=\{ (X_1(t),X_2(t))^T:\: t \ge 0\}$, which keeps track of $(n_{\mathrm{D^R}},n_{\mathrm{D^A}})$ through time. Here, we consider $\mathrm{D_{tot}}=3$ and we use dots to represent the states, and red double-ended arrows to represent transitions in both directions. Additionally, we use shades of blue to distinguish the level to which each state belongs. The function $L$ associated with the coclique level structure is $L(x_1,x_2)=x_1-x_2$. The rates associated with the one-step transitions for the projected Markov chain $\widecheck{X}$ are given in (\ref{['rates2D']}).
• Figure 5: Full chromatin modification circuit: reaction diagram, graph $\mathcal{G}$ and associated Markov chain. (a) Chemical reaction system diagram. The numbers on the arrows correspond to the reactions associated with the arrows as described in (\ref{['reacs4D']}) in the main text. (b) Graph $\mathcal{G}$ associated with the chemical reaction system in panel (a). (c) State space and transitions of the projected continuous time Markov chain $\widecheck{X}=\{ (X_1(t),X_2(t),X_3(t),X_4(t))^T:\: t \ge 0\}$, which keeps track of $(n_{\mathrm{D^R_{12}}},n_{\mathrm{D^A}},n_{\mathrm{D^R_1}},n_{\mathrm{D^R_2}})$ through time. Here, we consider $\mathrm{D_{tot}}=2$ and we use dots to represent the states, and red double-ended arrows to represent transitions in both directions. Additionally, we use shades of blue to distinguish the level to which each state belongs. The function $L$ associated with the coclique level structure is $L(x_1,x_2,x_3,x_4)=2x_1-x_2+x_3+x_4$. The rates associated with the one-step transitions for the projected Markov chain $\widecheck{X}$ are given in (\ref{['rates4D']}).

Theorems & Definitions (20)

• Lemma 3.1
• Remark 3.1
• Lemma 3.2
• proof
• Example 3.1
• Theorem 4.1
• proof
• Remark 4.1
• Theorem 4.2
• proof