Stochastic Reaction Networks Within Interacting Compartments

Abstract

Stochastic reaction networks, which are usually modeled as continuous-time Markov chains on $\mathbb Z^d_{\ge 0}$, and simulated via a version of the “Gillespie algorithm,” have proven to be a useful tool for the understanding of processes, chemical and otherwise, in homogeneous environments. There are multiple avenues for generalizing away from the assumption that the environment is homogeneous, with the proper modeling choice dependent upon the context of the problem being considered. One such generalization was recently introduced in Duso and Zechner (Proc Nat Acad Sci 117(37):22674–22683 , Duso and Zechner (2020)), where the proposed model includes a varying number of interacting compartments, or cells, each of which contains an evolving copy of the stochastic reaction system. The novelty of the model is that these compartments also interact via the merging of two compartments (including their contents), the splitting of one compartment into two, and the appearance and destruction of compartments. In this paper we begin a systematic exploration of the mathematical properties of this model. We (i) obtain basic/foundational results pertaining to explosivity, transience, recurrence, and positive recurrence of the model, (ii) explore a number of examples demonstrating some possible non-intuitive behaviors of the model, and (iii) identify the limiting distribution of the model in a special case that generalizes three formulas from an example in Duso and Zechner (Proc Nat Acad Sci 117(37):22674–22683 , Duso and Zechner (2020)).

Appendix

The purpose of this section is to prove Theorem 4.9, which we recall says the following:

Theorem 6.1

Let X be a non-explosive continuous-time Markov chain on a countable discrete state space ${\mathbb {S}}$ with generator $\mathcal L$. Let $B\subset {\mathbb {S}}$, and let $\tau _B$ be the time for the process to enter B. Suppose there is some bounded function V such that for all $x\in B^c$,

$$\begin{aligned} {\mathcal {L}}{\mathcal {V}}(x)\ge 0. \end{aligned}$$

Then $\mathbb P_{x_0}(\tau _B<\infty )<1$ for any $x_0$ such that

$$\begin{aligned} \sup _{x\in B}V(x)<V(x_0). \end{aligned}$$

Just like the theorem itself, the proof draws heavy inspiration from Tweedie (1994). Before providing the proof, we state the following well-known result:

Lemma 6.2

(Dynkin’s Formula) Suppose X is a Markov chain with finite state space ${\mathbb {S}}$, and let ${\mathcal {L}}$ be the generator of X. Then for any a.s. bounded stopping time $\tau$ and any $x\in {\mathbb {S}}$, we have

$$\begin{aligned} \mathbb E_x[f(X_\tau )]=f(x)+\mathbb E_x\left[ \int _0^\tau {\mathcal {L}}{\mathcal {f}}(X_s)\mathrm ds\right] \end{aligned}$$

In fact Dynkin’s Formula is well-known in much greater generality than what is stated above, but as stated it is not hard to prove and is enough for our purposes.

Proof of Theorem 4.9

Define W on ${\mathbb {S}}$ via $W=V-\sup _{x\in B}V(x)$. Notice that $W(x_0)$ is strictly positive, W is nonpositive on B, and ${\mathcal {L}}{\mathcal {W}}={\mathcal {L}}{\mathcal {V}}$. Fix some enumeration of ${\mathbb {S}}$ in which $x_0$ is the first element, and for $m\in \mathbb N$ let $\mathbb S_m$ denote the first m elements of ${\mathbb {S}}$. Let $\tau _m$ be the first time X is not in ${\mathbb {S}}_m$. Let $\Delta$ be a new state not in ${\mathbb {S}}$, and for $m\in \mathbb N$ define a new Markov chain $X^m$ via

$$\begin{aligned} X^m_t= {\left\{ \begin{array}{ll} X_t &{} t<\tau _m\\ \Delta &{} t\ge \tau _m \end{array}\right. } \end{aligned}$$

Notice that $X^m$ has finite state space $\mathbb S_m\cup \{\Delta \}$. Notice that W is bounded since V is, let $C=\sup _{x\in {\mathbb {S}}} W(x)$, and extend W to a function on ${\mathbb {S}}\cup \{\Delta \}$ by setting $W(\Delta )=C$. Let $\mathcal L_m$ denote the generator of the process $X^m$; we claim that ${\mathcal {L}}{\mathcal {W}}(x)\le {\mathcal {L}}_mW(x)$ whenever $x\in {\mathbb {S}}_m$. Indeed, notice that

$$\begin{aligned} \mathbb E_x[W(X_t)]&=\sum _{y\in {\mathbb {S}}}W(y)\mathbb P_x(X_t=y)\\&=\sum _{y\in {\mathbb {S}}}W(y)\mathbb P_x(X_t=y,t<\tau _m)+\sum _{y\in {\mathbb {S}}}W(y)\mathbb P_x(X_t=y,t\ge \tau _m)\\&\le \sum _{y\in {\mathbb {S}}}W(y)\mathbb P_x(X_t=y,t<\tau _m)+\sum _{y\in {\mathbb {S}}}C\mathbb P_x(X_t=y,t\ge \tau _m)\\&=\sum _{y\in {\mathbb {S}}_m}W(y)\mathbb P_x(X_t^m=y)+W(\Delta )\mathbb P_x(X_t^m=\Delta )\\&=\mathbb E_x[W(X^m_t)], \end{aligned}$$

and hence

$$\begin{aligned} {\mathcal {L}}{\mathcal {W}}(x) =\lim _{t\searrow 0}\frac{\mathbb E_x[W(X_t)]-W(x)}{t} \le \lim _{t\searrow 0}\frac{\mathbb E_x[W(X^m_t)]-W(x)}{t} ={\mathcal {L}}_mW(x), \end{aligned}$$

as claimed. Now for any m, applying Dynkin’s Formula to the chain $X^m$ with finite stopping time $\tau _B\wedge \tau _m\wedge m$ yields

$$\begin{aligned} \mathbb E_{x_0}[W(X^m_{\tau _B\wedge \tau _m\wedge m})]=W(x_0)+\mathbb E_{x_0}\left[ \int _0^{\tau _B\wedge \tau _m\wedge m} \mathcal L_mW(X^m_s)\mathrm ds\right] . \end{aligned}$$

But for $s<\tau _B\wedge \tau _m$ we have $X^m_s=X_s\in B^c\cap \mathbb S_m$ and hence

$$\begin{aligned} {\mathcal {L}}_mW(X^m_s) ={\mathcal {L}}_mW(X_s) \ge {\mathcal {L}}{\mathcal {W}}(X_s) ={\mathcal {L}}{\mathcal {V}}(X_s) \ge 0. \end{aligned}$$

So the integrand in Dynkin’s Formula is non-negative, and

$$\begin{aligned} W(x_0)&\le \mathbb E_{x_0}[W(X^m_{\tau _B\wedge \tau _m\wedge m})]\\&=\mathbb E_{x_0}[W(X^m_{\tau _B})\mathbb I_{\tau _B<\tau _m\wedge m}]+\mathbb E_{x_0}[W(X^m_{\tau _m\wedge m})\mathbb I_{\tau _B\ge \tau _m\wedge m}]\\&\le \mathbb E_{x_0}[W(X^m_{\tau _B})\mathbb I_{\tau _B<\tau _m\wedge m}]+C\mathbb P_{x_0}(\tau _B\ge \tau _m\wedge m). \end{aligned}$$

Note that $X_{\tau _B}^m\in B$ on the event $\tau _B <\tau _m\wedge m$. Hence $W(X_{\tau _B}^m) \mathbb I_{\tau _B<\tau _m\wedge m} \le 0$, and

$$\begin{aligned} W(x_0)\le C\mathbb P_{x_0}(\tau _B\ge \tau _m\wedge m) \end{aligned}$$

Since X is assumed to be non-explosive, $\tau _m\rightarrow \infty$ as $m\rightarrow \infty$, so taking $m\rightarrow \infty$ above gives

$$\begin{aligned} W(x_0)\le C\mathbb P_{x_0}(\tau _B=\infty ). \end{aligned}$$

But $W(x_0)$ is strictly positive and $0<W(x_0)\le C<\infty$, so $\mathbb P_{x_0}(\tau _B = \infty ) \ne 0$. That is, $\mathbb P_{x_0}(\tau _B<\infty )<1$, as desired. $\square$

Remark 6.3

Note that the proof above gives us a lower bound for the probability that the process never returns to the set B:

$$\begin{aligned} \frac{W(x_0)}{C}\le \mathbb P_{x_0}(\tau _B=\infty ), \end{aligned}$$

where $C = \sup _{x\in {\mathbb {S}}} W(x)$ and $W=V-\sup _{x\in B}V(x)$. We do not make use of this fact.