Threshold dynamics of a stochastic Keizer’s model with stochastic incidence

Abstract

In this paper, we incorporate stochastic incidence of a chemical reaction into the standard Keizer’s open chemical reaction. We prove that a positive stationary distribution (PSD) for the associated chemical master equation exists and is globally asymptotically stable. We present threshold dynamics of the stochastic Keizer’s model in term of the profile of the PSD for both finite and infinite volume size V. This establishes a sharp link between deterministic Keizer’s model and the stochastic model. In this way, we resolve Keizer’s paradox from a new perspective. This simple model reveals that such stochastic incidence incorporated, though negligible when V goes to infinity, may play an indispensable role in the stochastic formulation for irreversible biochemical reactions.

Appendix: Proofs of main results

Proof of Theorem 4.1

By (3.1),

$$\begin{aligned} \frac{p_{n+1}^s}{p_n^s}=\displaystyle \frac{R_0(\frac{n}{V}+\frac{\varepsilon _n}{k_1})}{\frac{n+1}{V}\Big (1+\frac{k_{-1}}{k_2}\cdot \frac{n}{V}\Big )}. \end{aligned}$$

(5.1)

If $R_0\leqslant 1,$ by (5.1) and assumption $\mathrm{(H1)},$

$$\begin{aligned} \frac{p_{n+1}^s}{p_n^s}<\frac{R_0}{1+\frac{k_{-1}}{k_2}\cdot \frac{n}{V}}\leqslant R_0\leqslant 1, \forall \ n\in {\mathbb {N}}, \end{aligned}$$

which proves (a).

Now suppose that $R_0>1$. From (5.1), to find $n_{\max }$, we need to locate n for $p^s_{n+1}=p^s_n$. Denote by $n_{\pm }/V$ the roots of the following quadratic equation for every $n\in {\mathbb {N}}:$

$$\begin{aligned} \frac{k_{-1}}{k_2}x^2+\left( 1-R_0+\frac{k_{-1}}{k_2V}\right) x+\frac{1}{V}-\frac{R_0\varepsilon _n}{k_1}=0. \end{aligned}$$

Notice that the limits $n_{\pm }/V$ as $V\rightarrow \infty$ satisfy

$$\begin{aligned} \frac{k_{-1}}{k_2}x^2+(1-R_0)x=0, \end{aligned}$$

i.e.,

$$\begin{aligned} n_+/V\rightarrow x_*,\ n_-/V\rightarrow 0,\ \text {as}\ V\rightarrow \infty . \end{aligned}$$

(5.2)

In fact, by regular perturbation theory and $\mathrm{(H1)},$ it is straightforward to deduce that

$$\begin{aligned} n_+= & {} x_*V-\frac{R_0}{R_0-1}+O(1/V,V\varepsilon _n),\ \forall \ n\in {\mathbb {N}},\\ n_-= & {} \frac{1}{R_0-1}+O(1/V,V\varepsilon _n),\ \forall \ n\in {\mathbb {N}}. \end{aligned}$$

Let $n_{\min }\in \left\{ \left[ \frac{1}{R_0-1}\right] ,\left[ \frac{1}{R_0-1}\right] +1\right\}$ be such that $p^s_{n_{\min }}=\min \left\{ p^s_{\left[ \frac{1}{R_0-1}\right] },p^s_{\left[ \frac{1}{R_0-1}\right] +1}\right\}$ and $n_{\max }\in \left\{ \left[ x_*V-\frac{R_0}{R_0-1}\right] ,\left[ x_*V-\frac{R_0}{R_0-1}\right] +1\right\}$ be such that $p^s_{n_{\max }}=\max \bigg \{p^s_{\left[ x_*V-\frac{R_0}{R_0-1}\right] },$ $p^s_{\left[ x_*V-\frac{R_0}{R_0-1}\right] +1}\bigg \}.$ Then the monotonicity of $p^s$ follows:

$$\begin{aligned} p^s_0\ge \cdots \ge p^s_{n_{\min }}\le p^s_{n_{\min }+1}\le \cdots \le p^s_{n_{\max }}\ge p^s_{n_{\max }+1}\ge \cdots \cdots . \end{aligned}$$

Finally, for all large V,  $p^s_{n_{\max }}>p^s_0$ follows from $\lim _{V\rightarrow \infty }\frac{p^s_{n_{\max }}}{p^s_0}=+\infty ,$ implied by Propositions 5.1 and 5.2 below. $\square$

Proposition 5.1

Let $\psi$ be defined in (4.1).

• (a) If $R_0\leqslant 1,$ then $\psi (x)$ is decreasing and thus $\psi (x)<0,\ \forall \ x\in (0,+\infty ).$

• (b) If $R_0>1,$ then $\psi (x)$ is increasing in $(0,x_*)$ and decreasing in $(x_*,+\infty ).$

Proposition 5.2

Assume that $\mathrm{(H1)}$ holds. Then for every $\lim _{V\rightarrow \infty }\frac{n}{V}=x>0,$

$$\begin{aligned} \limsup _{V\rightarrow \infty }\frac{1}{V}\log \frac{p_n^s}{p_0^s}\leqslant \psi (x). \end{aligned}$$

(5.3)

Moreover, assume $\mathrm{(H2)}$ holds, then

$$\begin{aligned} \lim _{V\rightarrow \infty }\frac{1}{V}\log \frac{p_n^s}{p_0^s}=\psi (x); \end{aligned}$$

(5.4)

in particular, if $\underset{V\rightarrow \infty }{\lim }\frac{n_1}{V}=x_1>0$ and $\underset{V\rightarrow \infty }{\lim }\frac{n_2}{V}=x_2>0,$ then

$$\begin{aligned} \lim _{V\rightarrow \infty }\frac{1}{V}\log \frac{p_{n_1}^s}{p_{n_2}^s}=\psi (x_1)-\psi (x_2). \end{aligned}$$

(5.5)

Proof

Note that

$$\begin{aligned} \begin{aligned} \frac{1}{V}\log \frac{p_n^s}{p_0^s}=&\frac{1}{V}\sum _{j=0}^{n-1}\log \frac{R_0(\frac{j}{V}+\frac{\varepsilon _j}{k_1})}{\frac{j+1}{V}\left( 1+\frac{k_{-1}}{k_2}\cdot \frac{j}{V}\right) }\\ =&\frac{1}{V}\log \frac{R_0V\varepsilon _0}{k_1}+\frac{1}{V}\sum _{j=1}^{n-1}\left( \log \left( 1+\frac{V\varepsilon _j}{k_1 j}\right) \right. \\&\left. +\log \frac{j}{j+1}\right) +\frac{1}{V}\sum _{j=1}^{n-1}\log \frac{R_0}{1+\frac{k_{-1}}{k_2}\frac{j}{V}}\\ =&\frac{1}{V}\log \frac{R_0V\varepsilon _0}{k_1}+\frac{1}{V}\sum _{j=1}^{n-1}\log \left( 1+\frac{V\varepsilon _j}{k_1 j}\right) +\frac{1}{V}\cdot \log \frac{1}{n}\\&+\frac{1}{V}\sum _{j=1}^{n-1}\log \frac{R_0}{1+\frac{k_{-1}}{k_2}\frac{j}{V}}. \end{aligned} \end{aligned}$$

(5.6)

By $\mathrm{(H1)},$

$$\begin{aligned} \frac{1}{V}\cdot \log \frac{1}{n}\leqslant & {} \frac{1}{V}\sum _{j=1}^{n-1}\log \left( 1+\frac{V\varepsilon _j}{k_1 j}\right) +\frac{1}{V}\cdot \log \frac{1}{n}\nonumber \\\leqslant & {} \frac{1}{V}\sum _{j=1}^{n-1}\log \left( 1+\frac{1}{j}\right) +\frac{1}{V}\cdot \log \frac{1}{n}=0, \end{aligned}$$

which implies (5.3). If $\mathrm{(H2)}$ holds, then

$$\begin{aligned} \frac{1}{V}\log \frac{R_0V\varepsilon _0}{k_1}\rightarrow 0,\ \text {as}\ V\rightarrow \infty . \end{aligned}$$

Thus the first three terms of RHS of (5.6) vanish as $V\rightarrow \infty .$ The last term is the Riemann sum for the integral $\psi (x).$ This proves the proposition. $\square$

Proof of Theorem 4.2

We only prove case (b). Case (a) can be proved using similar arguments. Let X be the number of species X,  a random variable. In the following, it suffices to show that, for all sufficiently small $\delta \in (0,x_*),$

$$\begin{aligned} {\mathbb {P}}[|X/V-x_*|>\delta ]\rightarrow 0, \text {as}\ V\rightarrow \infty . \end{aligned}$$

(5.7)

It follows from Proposition 5.1 that $\psi$ achieves its positive maximum at $x_*.$ By (5.5), there exists $V_0:=V_0(\delta )>0$ such that $\forall \ V>V_0,$

$$\begin{aligned} p_0^s< & {} p^s_{[x_*V]}e^{-V\psi (x_*)/2},\\ p_{[(x_*-\delta )V]}^s< & {} p^s_{[x_*V]}e^{-V(\psi (x_*)-\psi (x_*-\delta ))/2} \end{aligned}$$

and

$$\begin{aligned} p_{[(x_*+\delta )V]}^s<p^s_{[x_*V]}e^{-V(\psi (x_*)-\psi (x_*+\delta ))/2}. \end{aligned}$$

Note that $\psi (x_*)>0.$ By continuity of $\psi ,$ assume without loss of generality that $\psi (x_*\pm \delta )>0$ and $\delta <\min \left\{ x_*,\frac{k_2}{k_{-1}}\right\}$. Hence by Theorem 4.1 (b), there exists $V_1>V_0,$ such that for all $V>V_1,$

$$\begin{aligned} p_{n}^s\leqslant p_{[(x_*-\delta )V]}^s<p^s_{[x_*V]}e^{-V/2\cdot (\psi (x_*)-\psi (x_*-\delta ))},\ \forall \ 0\leqslant n\leqslant [(x_*-\delta )V] \end{aligned}$$

and

$$\begin{aligned} p_{n}^s\leqslant p_{[(x_*+\delta )V]}^s<p^s_{[x_*V]}e^{-V/2\cdot (\psi (x_*)-\psi (x_*+\delta ))},\ \forall \ n\geqslant [(x_*+\delta )V], \end{aligned}$$

By $\mathrm {(H1)},$

$$\begin{aligned} \frac{p_{n+1}^s}{p_n^s}<\frac{R_0}{1+\frac{k_{-1}}{k_2}\cdot \frac{n}{V}}<\frac{R_0}{1+R_0},\ \forall \ n\geqslant \left[ \frac{k_2R_0}{k_{-1}}V\right] +1. \end{aligned}$$

Then it follows from $\delta <\frac{k_2}{k_{-1}}$ that

$$\begin{aligned} \sum _{n=\left[ \frac{k_2R_0}{k_{-1}}V\right] +1}^{\infty }p_n^s\leqslant & {} p^s_{1+\left[ \frac{k_2R_0}{k_{-1}}V\right] }\cdot \frac{1}{1-\frac{R_0}{1+R_0}}=(R_0+1)p^s_{1+[\frac{k_2R_0}{k_{-1}}V]}\nonumber \\\leqslant & {} (R_0+1)p_{[(x_*+\delta )V]}^s,\ \forall \ V>V_1. \end{aligned}$$

Hence for all $V>V_1,$

$$\begin{aligned} \begin{aligned}&{\mathbb {P}}[|X/V-x_*|>\delta ]\\&\quad \leqslant \sum _{n=0}^{\left[ \frac{1}{R_0-1}\right] }p_n^s+\sum _{n=\left[ \frac{1}{R_0-1}\right] +1}^{[(x_*-\delta )V]}p_n^s+\sum _{n=[(x_*+\delta )V]}^ {\left[ \frac{k_2R_0}{k_{-1}}V\right] }p_n^s+\sum _{n=1+\left[ \frac{k_2R_0}{k_{-1}}V\right] }^{\infty }p_n^s\\&\quad <\left( \left[ \frac{1}{R_0-1}\right] +1\right) p^s_{[x_*V]}e^{-V\psi (x_*)/2}\\&\qquad +\left( [(x_*-\delta )V]-\left[ \frac{1}{R_0-1}\right] \right) p^s_{[x_*V]}e^{-V(\psi (x_*) -\psi (x_*-\delta ))/2}\\&\qquad +\left( \left[ \frac{k_2R_0}{k_{-1}}V\right] -[(x_*+\delta )V]+1\right) p^s_{[x_*V]}e^{-V/2\cdot (\psi (x_*)-\psi (x_*+\delta ))}\\&\qquad +(R_0+1)p^s_{[x_*V]}e^{-V(\psi (x_*) -\psi (x_*+\delta ))/2}\\&\quad \leqslant \left( \left[ \frac{k_2R_0}{k_{-1}}V\right] +R_0+3\right) e^{-V/2\cdot d}, \end{aligned} \end{aligned}$$

with $d=\min \{\psi (x_*)-\psi (x_*-\delta ),\psi (x_*)-\psi (x_*+\delta )\}>0.$ This shows that (5.7) $\square$