State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation

Abstract

The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in mesoscopic reaction networks. Since its direct solution rapidly becomes intractable due to the increasing size of the state space, truncation of the state space is necessary for solving most dCMEs. It is therefore important to assess the consequences of state space truncations so errors can be quantified and minimized. Here we describe a novel method for state space truncation. By partitioning a reaction network into multiple molecular equivalence groups (MEGs), we truncate the state space by limiting the total molecular copy numbers in each MEG. We further describe a theoretical framework for analysis of the truncation error in the steady-state probability landscape using reflecting boundaries. By aggregating the state space based on the usage of a MEG and constructing an aggregated Markov process, we show that the truncation error of a MEG can be asymptotically bounded by the probability of states on the reflecting boundary of the MEG. Furthermore, truncating states of an arbitrary MEG will not undermine the estimated error of truncating any other MEGs. We then provide an overall error estimate for networks with multiple MEGs. To rapidly determine the appropriate size of an arbitrary MEG, we also introduce an a priori method to estimate the upper bound of its truncation error. This a priori estimate can be rapidly computed from reaction rates of the network, without the need of costly trial solutions of the dCME. As examples, we show results of applying our methods to the four stochastic networks of (1) the birth and death model, (2) the single gene expression model, (3) the genetic toggle switch model, and (4) the phage lambda bistable epigenetic switch model. We demonstrate how truncation errors and steady-state probability landscapes can be computed using different sizes of the MEG(s) and how the results validate our theories. Overall, the novel state space truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks.

Appendix: Proof of Lemma 1

Proof

By sorting the state space according to the partition $\tilde{\varOmega }^{(\infty )}$ and reconstructing the transition rate matrix $\tilde{{{\varvec{A}}}}$ in Eq. (6), the dCME can be rewritten as $\frac{\hbox {d}\tilde{{{\varvec{p}}}}^{(\infty )}(t)}{\hbox {d}t} = \tilde{{{\varvec{A}}}} \tilde{{{\varvec{p}}}}^{(\infty )}(t)$, where $\tilde{{{\varvec{p}}}}^{(\infty )}$ is the probability distribution on the partitioned state space. We sum up the master equations over all microstates in each group $\mathcal {G}_i$ and obtain a separate aggregated equation for each group. As the reordered matrix $\tilde{{{\varvec{A}}}}$ is a block tridiagonal matrix, the summed discrete chemical master equation is reduced to:

$$\begin{aligned}&\frac{\hbox {d} p^{(\infty )}(\mathcal {G}_0, t)}{\hbox {d}t} = \frac{\hbox {d}\sum \nolimits _{{{\varvec{x}}}\in \mathcal {G}_0} p({{\varvec{x}}},t)}{\hbox {d}t} = \left( \mathbbm {1}^T {{\varvec{A}}}_{0,0} \right) \tilde{{{\varvec{p}}}}^{(\infty )}(\mathcal {G}_{0},t) + \left( \mathbbm {1}^T {{\varvec{A}}}_{0,1} \right) \tilde{{{\varvec{p}}}}^{(\infty )}(\mathcal {G}_{1},t), \nonumber \\&\frac{\hbox {d}p^{(\infty )}(\mathcal {G}_i, t)}{\hbox {d}t} = \frac{\hbox {d}\sum _{{{\varvec{x}}}\in \mathcal {G}_i} p({{\varvec{x}}},t)}{\hbox {d}t} = \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i-1} \right) \tilde{{{\varvec{p}}}}^{(\infty )}(\mathcal {G}_{i-1},t)\nonumber \\&\quad +\,\left( \mathbbm {1}^T {{\varvec{A}}}_{i,i} \right) \tilde{{{\varvec{p}}}}^{(\infty )}(\mathcal {G}_{i},t) + \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} \right) \tilde{{{\varvec{p}}}}^{(\infty )}(\mathcal {G}_{i+1},t), \quad \text {for}\, i=1, \ldots , \infty . \end{aligned}$$

(30)

The overall probability change of each group $\mathcal {G}_i$ depends on the probability vector $\tilde{{{\varvec{p}}}}^{(\infty )}(\mathcal {G}_{i},t)$ itself, as well as the probability vector $\tilde{{{\varvec{p}}}}^{(\infty )}(\mathcal {G}_{i-1},t)$ and the probability vector $\tilde{{{\varvec{p}}}}^{(\infty )}(\mathcal {G}_{i+1},t)$ of the immediate neighboring groups. It also depends on the rates of synthesis and degradation reactions in elements of ${{\varvec{A}}}_{i,i-1}$ and ${{\varvec{A}}}_{i,i+1}$, respectively, as well as rates of coupling reactions in ${{\varvec{A}}}_{i,i}$. From the definition of transition rate matrix given in Eq. (3), we have:

$$\begin{aligned} \mathbbm {1}^T {{\varvec{A}}}_{0,0}= & {} - \mathbbm {1}^T {{\varvec{A}}}_{1,0},\nonumber \\ \mathbbm {1}^T {{\varvec{A}}}_{i-1,i} + \mathbbm {1}^T {{\varvec{A}}}_{i,i}= & {} -\mathbbm {1}^T {{\varvec{A}}}_{i+1,i}, \quad \text {for}\;i=1, \ldots , \infty . \end{aligned}$$

(31)

At the steady state when all $\frac{\hbox {d} p^{(\infty )}(\mathcal {G}_i)}{\hbox {d}t} = 0$, we combine line 1 of Eq. (30) and line 1 of Eq. (1), and obtain:

$$\begin{aligned} \left( \mathbbm {1}^T {{\varvec{A}}}_{1,0} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{0}) = \left( \mathbbm {1}^T {{\varvec{A}}}_{0,1} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{1}). \end{aligned}$$

From line 2 of Eq. (30) at steady state and after incorporating line 1 of Eq. (1), we have: $( \mathbbm {1}^T {{\varvec{A}}}_{1,2}) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{2}) = (\mathbbm {1}^T {{\varvec{A}}}_{0,0}) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{0}) -( \mathbbm {1}^T {{\varvec{A}}}_{1,1} ) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{1}).$ After further incorporating line 1 of Eq. (30) at steady state, we have $( \mathbbm {1}^T {{\varvec{A}}}_{1,2}) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{2}) = - (\mathbbm {1}^T {{\varvec{A}}}_{0,1}) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{1}) - ( \mathbbm {1}^T {{\varvec{A}}}_{1,1}) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{1}).$ Incorporating line 2 of Eq. (1), we have:

$$\begin{aligned} \left( \mathbbm {1}^T {{\varvec{A}}}_{2,1} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{1}) = \left( \mathbbm {1}^T {{\varvec{A}}}_{1,2} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{2}). \end{aligned}$$

Assume $( \mathbbm {1}^T {{\varvec{A}}}_{i,\,i-1} ) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i-1}) = (\mathbbm {1}^T {{\varvec{A}}}_{i-1,\,i}) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i}),$ we have from the ith line of Eq. (30) at the steady state

$$\begin{aligned} \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1})= & {} - \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i-1} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i-1}) - \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i})\nonumber \\= & {} - \left( \mathbbm {1}^T {{\varvec{A}}}_{i-1,i} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i}) - \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i}).\qquad \end{aligned}$$

(32)

With the ith line of Eq. (1), we further have:

$$\begin{aligned} \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1}) = \left( \mathbbm {1}^T {{\varvec{A}}}_{i+1,i} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i}). \end{aligned}$$

Overall, we have:

$$\begin{aligned} \left( \mathbbm {1}^T {{\varvec{A}}}_{1,0} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{0})= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{0,1} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{1}),\nonumber \\ \left( \mathbbm {1}^T {{\varvec{A}}}_{i+1,i} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i})= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1}), \quad \text {for}\; i=1, \ldots , \infty . \end{aligned}$$

(33)

As both sides are constants, we can find $\alpha _i$ and $\beta _{i+1}$ such that:

$$\begin{aligned} \left( \mathbbm {1}^T {{\varvec{A}}}_{i+1,i} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i})= & {} \mathbbm {1}^T \alpha _{i} \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i}) = \alpha _{i} \mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i}),\nonumber \\ \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} \right) \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1})= & {} \mathbbm {1}^T \beta _{i+1} \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1}) = \beta _{i+1} \mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1}), \end{aligned}$$

(34)

for all $i = 0, 1, \ldots$, where i is the total copy number of the MEG. We obviously have:

$$\begin{aligned} \alpha _i = \left( \mathbbm {1}^T {{\varvec{A}}}_{i+1,i} \right) \cdot \frac{\tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i})}{\mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i})} \quad \text {and} \quad \beta _{i+1} = \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} \right) \cdot \frac{\tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1})}{\mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1})}, \end{aligned}$$

where $\alpha _i$ is the sum of column sums of sub-matrix ${{\varvec{A}}}_{i+1,i}$ weighted by the steady-state probability distribution $\tilde{{{\varvec{\pi }}}}^{(\infty )}$ on group $\mathcal {G}_i$, $\beta _{i+1}$ is the sum of column summation of sub-matrix ${{\varvec{A}}}_{i,i+1}$ weighted by the steady-state probability distribution on group $\mathcal {G}_{i+1}$.

As $\mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i})$ is the total steady-state probability mass over states in group $\mathcal {G}_i$, we substitute Eq. (1) back into Eq. (1) and obtain the following relationship of steady-state distribution on the partitions of $\tilde{\varOmega }^{\infty }$:

$$\begin{aligned} \alpha _{0} \mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{0})= & {} \beta _{1} \mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{1}),\nonumber \\ \alpha _{i} \mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i})= & {} \beta _{i+1} \mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_{i+1}), \quad \text {for}\;i=1, \ldots , \infty . \end{aligned}$$

(35)

The steady-state solution to Eq. (1) is equivalent to the steady-state solution of a dCME with the transition rate matrix ${{\varvec{B}}}$ defined as in Eq. (9).

1.1 Proof of Lemma 2

Proof

If $\lim \nolimits _{N \rightarrow \infty } \sup \nolimits _{i > N} \frac{\alpha ^{(\infty )}_{i}}{\beta ^{(\infty )}_{i+1}} \ge 1$ held, then there would be an infinite number of terms $\frac{\alpha ^{(\infty )}_{i}}{\beta ^{(\infty )}_{i+1}} > 1$. There should exist an integer $N^{\prime }$ such that for all $i > N^{\prime }$, we have $\beta ^{(\infty )}_{i+1} \le \alpha ^{(\infty )}_{i}$. According to Eq. (1), we would have $\tilde{\pi }^{(\infty )}_{i+1} \ge \tilde{\pi }^{(\infty )}_{i}$ in the steady state for all $i > N^{\prime }$. This contradicts with the assumption of a finite system, as the total probability mass on boundary states increases monotonically as the net molecular copy number of the network increases after $N^{\prime }$. This makes the overall system a pure-birth process. Therefore, for a finite biological system, we have Eq. (15).

1.2 Proof of Theorem 1

Proof

From Eq. (13), we can first derive an explicit expression of the true error ${{\mathrm{Err}}}^{(N)}$ using the aggregated synthesis and degradation rates $\alpha ^{(\infty )}_{k}$ and $\beta ^{(\infty )}_{k+1}$ given in Eq. (8):

$$\begin{aligned} \begin{aligned} {{\mathrm{Err}}}^{(N)}&= 1 - \sum _{{{\varvec{x}}}\in \varOmega ^{(N)}} \pi ^{(\infty )}({{\varvec{x}}}) = 1 - \sum _{i = 0}^{N} \mathbbm {1}^T \tilde{{{\varvec{\pi }}}}^{(\infty )}(\mathcal {G}_i)\\&= 1 - \tilde{\pi }_0^{(\infty )} \left( 1+\sum _{j=1}^{N} \prod _{k=0}^{j-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}\right) \\&= 1 - \frac{1+\sum \nolimits _{j=1}^{N} \prod \nolimits _{k=0}^{j-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}}{1+\sum \nolimits _{j=1}^{\infty } \prod \nolimits _{k=0}^{j-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}} = \frac{\sum \nolimits _{j=N+1}^{\infty } \prod \nolimits _{k=0}^{j-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}}{1+\sum \nolimits _{j=1}^{\infty } \prod \nolimits _{k=0}^{j-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}} . \end{aligned} \end{aligned}$$

(36)

From Eqs. (36), (13), and Lemma 2, we have:

$$\begin{aligned} \begin{aligned} \frac{{{\mathrm{Err}}}^{(N)}}{\tilde{\pi }^{(\infty )}_N}&= \frac{\sum \nolimits _{j=N+1}^{\infty } \prod \nolimits _{k=0}^{j-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}}{\prod \nolimits _{k=0}^{N-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}} = \frac{\left( \prod \nolimits _{k=0}^{N-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}\right) \left( \sum \nolimits _{j=N+1}^{\infty } \prod \nolimits _{k=N}^{j-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}}\right) }{\prod \nolimits _{k=0}^{N-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{( \infty )}_{k+1}}} \\&= \sum _{j=N+1}^{\infty } \prod _{k=N}^{j-1} \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \le \sum _{j=N+1}^{\infty } \left[ \sup \limits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} \right] ^{j-N}\\&= \sum _{j=1}^{\infty } \left[ \sup \limits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} \right] ^{j}. \end{aligned} \end{aligned}$$

(37)

When N is sufficiently large, $\sup \nolimits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} < 1$ from Lemma 2, the terms in the infinite series $\sum \nolimits _{j=1}^{\infty } \left[ \sup \nolimits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} \right] ^{j}$ then forms a converging geometric series. Therefore, we have

$$\begin{aligned} \sum _{j=1}^{\infty } \left[ \sup \limits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} \right] ^{j} = \frac{ \sup \nolimits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} }{1- \sup \nolimits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} }, \end{aligned}$$

and the following inequality holds:

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{{{\mathrm{Err}}}^{(N)}}{\bar{\pi }^{(\infty )}_N} \le \lim _{N \rightarrow \infty } \frac{ \sup \nolimits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} }{1- \sup \nolimits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\} }. \end{aligned}$$

Let $M \in \{N, \ldots , \infty \}$ be the integer such that $\frac{\alpha ^{(\infty )}_{M}}{\beta ^{(\infty )}_{M+1}} =\mathop {\sup }\nolimits _{k \ge N} \left\{ \frac{\alpha ^{(\infty )}_{k}}{\beta ^{(\infty )}_{k+1}} \right\}$, we have the following inequality equivalent to Inequality (16):

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{{{\mathrm{Err}}}^{(N)}}{\bar{\pi }^{(\infty )}_N}\le \lim _{N \rightarrow \infty } \frac{\frac{\alpha ^{(\infty )}_{M}}{\beta ^{(\infty )}_{M+1}}}{1- \frac{\alpha ^{(\infty )}_{M}}{\beta ^{(\infty )}_{M+1}}}. \end{aligned}$$

1.3 Proof of Theorem 2

Proof

We first consider two truncated state spaces $\tilde{\varOmega }^{(N)}$ and $\tilde{\varOmega }^{(N+1)}$. Following Eq. (30), two finite sets of the block chemical master equation can be constructed for these two state spaces. The first set containing N equations is built on the state space $\tilde{\varOmega }^{(N)}$.

$$\begin{aligned} \frac{\hbox {d}p^{(N)}(\mathcal {G}_0, t)}{\hbox {d}t}= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{0,0} \right) \tilde{{{\varvec{p}}}}^{(N)}(\mathcal {G}_{0},t) + \left( \mathbbm {1}^T {{\varvec{A}}}_{0,1} \right) \tilde{{{\varvec{p}}}}^{(N)}(\mathcal {G}_{1},t),\nonumber \\ \frac{\hbox {d}p^{(N)}(\mathcal {G}_{i}, t)}{\hbox {d}t}= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i-1} \right) \tilde{{{\varvec{p}}}}^{(N)}(\mathcal {G}_{i-1},t) + \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i} \right) \tilde{{{\varvec{p}}}}^{(N)}(\mathcal {G}_{i},t)\nonumber \\&+\,\left( \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} \right) \tilde{{{\varvec{p}}}}^{(N)}(\mathcal {G}_{i+1},t), \quad \text {for}\;i=1, \ldots , N-1,\nonumber \\ \frac{\hbox {d}p^{(N)}(\mathcal {G}_N, t)}{\hbox {d}t}= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{N,N-1} \right) \tilde{{{\varvec{p}}}}^{(N)}(\mathcal {G}_{N-1},t) + \left( \mathbbm {1}^T {{\varvec{A}}}_{N,N} \right) \tilde{{{\varvec{p}}}}^{(N)}(\mathcal {G}_{N},t).\qquad \end{aligned}$$

(38)

The second set is built on the state space $\tilde{\varOmega }^{(N+1)}$ containing $N+1$ equations.

$$\begin{aligned} \frac{\hbox {d}p^{(N+1)}(\mathcal {G}_0, t)}{\hbox {d}t}= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{0,0} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{0},t) + \left( \mathbbm {1}^T {{\varvec{A}}}_{0,1} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{1},t), \nonumber \\ \frac{\hbox {d}p^{(N+1)}(\mathcal {G}_{i}, t)}{\hbox {d}t}= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i-1} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{i-1},t) + \left( \mathbbm {1}^T {{\varvec{A}}}_{i,i} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{i},t) \nonumber \\&+\,\left( \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{i+1},t), \quad \text {for}\;i=1, \ldots , N-1, \nonumber \\ \frac{\hbox {d}p^{(N+1)}(\mathcal {G}_{N}, t)}{\hbox {d}t}= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{N,N-1} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{N-1},t) + \left( \mathbbm {1}^T {{\varvec{A}}}_{N,N} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{N},t)\nonumber \\&+\,\left( \mathbbm {1}^T {{\varvec{A}}}_{N,N+1} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{N+1},t), \nonumber \\ \frac{\hbox {d}p^{(N+1)}(\mathcal {G}_{N+1}, t)}{\hbox {d}t}= & {} \left( \mathbbm {1}^T {{\varvec{A}}}_{N+1,N} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{N},t)\nonumber \\&+\,\left( \mathbbm {1}^T {{\varvec{A}}}_{N+1,N+1} \right) \tilde{{{\varvec{p}}}}^{(N+1)}(\mathcal {G}_{N+1},t). \end{aligned}$$

(39)

At steady state, the left-hand side of the equations are zeros. For the first N equations, the corresponding block matrices are the same for both state spaces $\tilde{\varOmega }^{(N)}$ and $\tilde{\varOmega }^{(N+1)}$. We can then subtract the right-hand side of Eq. (39) from Eq. (1) and obtain the following steady-state equations:

$$\begin{aligned} \mathbbm {1}^T {{\varvec{A}}}_{0,0} {\Delta {{\varvec{\pi }}}}_{0} + \mathbbm {1}^T {{\varvec{A}}}_{0,1} {\Delta {{\varvec{\pi }}}}_{1}= & {} 0, \nonumber \\ \mathbbm {1}^T {{\varvec{A}}}_{i,i-1} {\Delta {{\varvec{\pi }}}}_{i-1} + \mathbbm {1}^T {{\varvec{A}}}_{i,i} {\Delta {{\varvec{\pi }}}}_{i} + \mathbbm {1} {{\varvec{A}}}_{i,i+1} {\Delta {{\varvec{\pi }}}}_{i+1}= & {} 0, \quad \text {for}\;i=1, \ldots , N-1,\nonumber \\ \end{aligned}$$

(40)

where ${\Delta {{\varvec{\pi }}}}_{i}={{{\varvec{\pi }}}}^{(N)}_i - {{{\varvec{\pi }}}}^{(N+1)}_i$ is the steady-state probability difference between the state group $\mathcal {G}_i$ in the dCME on $\tilde{\varOmega }^{(N)}$ and $\tilde{\varOmega }^{(N+1)}$. However, the block sub-matrix ${{\varvec{A}}}_{N,N}$ of the boundary group $\mathcal {G}_N$ is different between the two state spaces. From the construction of the aggregated dCME matrix $\tilde{{{\varvec{A}}}}$, columns of the full matrices $\tilde{{{\varvec{A}}}}^{(N+1)}$ over $\tilde{\varOmega }^{(N+1)}$ and $\tilde{{{\varvec{A}}}}^{N}$ over $\tilde{\varOmega }^{N}$ all sum to 0 (see Eq. 1). We use ${{\varvec{A}}}^{(N)}_{i,j}$ to denote the block sub-matrix of the group $\mathcal {G}_N$ for the state space $\tilde{\varOmega }^{(N)}$ and use ${{\varvec{A}}}^{(N+1)}_{i,j}$ to denote the corresponding block sub-matrix for the state space $\tilde{\varOmega }^{(N+1)}$. From the Nth line of the truncated version of Eq. (1), we have $\mathbbm {1}^T{{{\varvec{A}}}_{N-1,N}^{(N+1)}} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N+1)}} + \mathbbm {1}^T{{{\varvec{A}}}_{N+1, N}^{(N+1)}} = 0$ for $\tilde{\varOmega }^{(N+1)}$ and $\mathbbm {1}^T{{{\varvec{A}}}_{N-1,N}^{(N)}} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N)}} = 0$ for $\tilde{\varOmega }^{(N)}$. Since ${{{\varvec{A}}}_{N-1,N}^{(N)}} = {{{\varvec{A}}}_{N-1,N}^{(N+1)}}$, we have the following property

$$\begin{aligned} \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N+1)}}=\mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N)}} -\mathbbm {1}^T{{{\varvec{A}}}_{N+1,N}^{(N+1)}}, \end{aligned}$$

(41)

We also have

$$\begin{aligned} \mathbbm {1}^T{{{\varvec{A}}}_{N+1,N+1}^{(N+1)}}=-\mathbbm {1}^T{{{\varvec{A}}}_{N,N+1}^{(N+1)}}. \end{aligned}$$

(42)

From Eq. (1), we have for the steady-state probability of the state group $\mathcal {G}_N$ over the state space $\tilde{\varOmega }^{(N)}$ as:

$$\begin{aligned} \mathbbm {1}^T{{{\varvec{A}}}_{N,N-1}^{(N)}} {{\varvec{\pi }}}_{N-1}^{(N)} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N)}} {{\varvec{\pi }}}_{N}^{(N)} = 0, \end{aligned}$$

(43)

From Eq. (39), we have for the steady-state probability of the state group $\mathcal {G}_N$ and $\mathcal {G}_{N+1}$ over the state space $\tilde{\varOmega }^{(N+1)}$ as:

$$\begin{aligned} \mathbbm {1}^T{{{\varvec{A}}}_{N,N-1}^{(N+1)}} {{\varvec{\pi }}}_{N-1}^{(N+1)} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N+1)}} {{\varvec{\pi }}}_{N}^{(N+1)} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N+1}^{(N+1)}} {{\varvec{\pi }}}_{N+1}^{(N+1)} = 0, \end{aligned}$$

(44)

and

$$\begin{aligned} \mathbbm {1}^T{{{\varvec{A}}}_{N+1,N}^{(N+1)}} {{\varvec{\pi }}}_{N}^{(N+1)} + \mathbbm {1}^T{{{\varvec{A}}}_{N+1,N+1}^{(N+1)}} {{\varvec{\pi }}}_{N+1}^{(N+1)} = 0, \end{aligned}$$

(45)

respectively.

As ${{{\varvec{A}}}_{N,N-1}^{(N+1)}} = {{{\varvec{A}}}_{N,N-1}^{(N)}}$, we subtract Eq. (44) from Eq. (43), and obtain:

$$\begin{aligned} \mathbbm {1}^T{{{\varvec{A}}}_{N,N-1}} \Delta {{\varvec{\pi }}}_{N-1} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N)}} {{\varvec{\pi }}}_{N}^{(N)} - \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N+1)}} {{\varvec{\pi }}}_{N}^{(N+1)} - \mathbbm {1}^T{{{\varvec{A}}}_{N,N+1}^{(N+1)}} {{\varvec{\pi }}}_{N+1}^{(N+1)} = 0. \end{aligned}$$

It can be rewritten by applying the matrix property of Eq. (41) as:

$$\begin{aligned} \mathbbm {1}^T{{{\varvec{A}}}_{N,N-1}} \Delta {{\varvec{\pi }}}_{N-1} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N)}} \Delta {{\varvec{\pi }}}_{N} + \mathbbm {1}^T{{{\varvec{A}}}_{N+1,N}^{(N+1)}} {{\varvec{\pi }}}_{N}^{(N+1)} - \mathbbm {1}^T{{{\varvec{A}}}_{N,N+1}^{(N+1)}} {{\varvec{\pi }}}_{N+1}^{(N+1)} = 0. \end{aligned}$$

By using the matrix property in Eq. (42), we can further rewrite it as:

$$\begin{aligned}&\mathbbm {1}^T{{{\varvec{A}}}_{N,N-1}} \Delta {{\varvec{\pi }}}_{N-1} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N)}} \Delta {{\varvec{\pi }}}_{N} + \mathbbm {1}^T{{{\varvec{A}}}_{N+1,N}^{(N+1)}} {{\varvec{\pi }}}_{N}^{(N+1)} \\&\quad + \mathbbm {1}^T{{{\varvec{A}}}_{N+1,N+1}^{(N+1)}} {{\varvec{\pi }}}_{N+1}^{(N+1)} = 0. \end{aligned}$$

From Eq. (45), the last two terms sum to 0. Therefore, we obtain the ($N+1$)st equation of the steady-state probability difference as:

$$\begin{aligned} \mathbbm {1}^T{{{\varvec{A}}}_{N,N-1}} \Delta {{\varvec{\pi }}}_{N-1} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}^{(N)}} \Delta {{\varvec{\pi }}}_{N} = 0. \end{aligned}$$

Taken together, we have the set of equations for steady-state probability differences for all $N+1$ blocks as:

$$\begin{aligned} \mathbbm {1}^T {{\varvec{A}}}_{0,0} {\Delta {{\varvec{\pi }}}}_{0} + \mathbbm {1}^T {{\varvec{A}}}_{0,1} {\Delta {{\varvec{\pi }}}}_{1}= & {} 0, \nonumber \\ \mathbbm {1}^T {{\varvec{A}}}_{i,i-1} {\Delta {{\varvec{\pi }}}}_{i-1} + \mathbbm {1}^T {{\varvec{A}}}_{i,i} {\Delta {{\varvec{\pi }}}}_{i} + \mathbbm {1}^T {{\varvec{A}}}_{i,i+1} {\Delta {{\varvec{\pi }}}}_{i+1}= & {} 0, \quad \text {for}\; i=1, \ldots , N-1, \nonumber \\ \mathbbm {1}^T{{{\varvec{A}}}_{N,N-1}} \Delta {{\varvec{\pi }}}_{N-1} + \mathbbm {1}^T{{{\varvec{A}}}_{N,N}} \Delta {{\varvec{\pi }}}_{N}= & {} 0, \end{aligned}$$

(46)

where all block sub-matrices are identical between those over the state spaces $\tilde{\varOmega }^{(N)}$ and $\tilde{\varOmega }^{(N+1)}$. We therefore obtain the set of equations of differences in steady-state probability equivalent to Eq. (1):

$$\begin{aligned} \begin{aligned} \mathbbm {1}^T {{\varvec{A}}}_{i,i-1} {\Delta {{\varvec{\pi }}}}_{i-1} = \mathbbm {1}^T {{\varvec{A}}}_{i-1,i} {\Delta {{\varvec{\pi }}}}_{i},\quad \text {for}\;i=1, \ldots , N, \\ \end{aligned} \end{aligned}$$

(47)

which produces the same steady-state solution as that of Eq. (1) after scaling by a constant. As probability vector solution to Eq. (1) has nonnegative elements, this equivalence implies that all elements in each ${\Delta {{\varvec{\pi }}}}_{i}$ have the same sign. As the total steady-state probability mass in both state spaces sum up to 1,

$$\begin{aligned} \sum _{i=1}^N \tilde{\pi }_i^{(N)} = \sum _{i=1}^{N+1} \tilde{\pi }_i^{(N+1)} = 1, \end{aligned}$$

we therefore know that the total probability differences is nonnegative:

$$\begin{aligned} \sum _{i = 1}^N \Delta \tilde{\pi }_i = \sum _{i = 1}^N \tilde{\pi }_i^{(N)} - \sum _{i = 1}^N \tilde{\pi }_i^{(N+1)} = 1 - \left( 1 - \tilde{\pi }_{N + 1}^{(N+1)}\right) = \tilde{\pi }_{N + 1}^{(N+1)} \ge 0. \end{aligned}$$

Therefore, the probability difference of each individual $\mathcal {G}_i$ between two state spaces must be nonnegative:

$$\begin{aligned} \Delta \tilde{\pi }_i = \tilde{\pi }_i^{(N)}-\tilde{\pi }_i^{(N+1)} \ge 0, \quad i = 0, 1, \ldots , N. \end{aligned}$$

This can be generalized. As N increases to infinity, we have:

$$\begin{aligned} \tilde{\pi }_i^{(N)} \ge \tilde{\pi }_i^{(N+1)} \ge \cdots \ge \tilde{\pi }_i^{(\infty )}, \quad i = 0, 1, \ldots , N. \end{aligned}$$

1.4 Proof of Theorem 3

Proof

For convenience, we use $M = N_i$ to denote the maximum net copy number in the truncated ith MEG. We first aggregate the state space $\varOmega ^{(\mathcal {I}_j)}$ into infinitely many groups $\{ \mathcal {G}_0, \mathcal {G}_1 \cdots , \mathcal {G}_M, \mathcal {G}_{M+1}, \ldots \}$ according to the net copy number in the ith MEG. We then reconstruct the permuted matrix $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_j)}$ according to this aggregation. We have:

(48)

where the subscripts m and n of each block matrix ${{\varvec{A}}}_{m,n}^{(\mathcal {I}_j)}$ indicate the actual net copy numbers of the corresponding aggregated states of the ith MEG. Next, we further partition the matrix into four blocks by truncating the ith MEG at the maximum copy number of M. Specifically, ${{\varvec{A}}}_\mathbf{1 ,\mathbf 1 }^{(\mathcal {I}_j)}$ in the right-hand side of Eq. (48) is the northwest corner sub-matrix of $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_j)}$, which contains all transitions between microstates in the state space $\varOmega ^{(\mathcal {I}_{i,j})}$:

$$\begin{aligned} {{\varvec{A}}}_\mathbf{1 ,\mathbf 1 }^{(\mathcal {I}_j)}= & {} \left( {\begin{array}{c} {{{\varvec{A}}}_{g,h}^{(\mathcal {I}_j)}} \end{array}} \right) = \{A_{{{\varvec{x}}}_m, {{\varvec{x}}}_n}\}, {{\varvec{x}}}_m, {{\varvec{x}}}_n \in \varOmega ^{(\mathcal {I}_{i,j})},\quad \text {and}\;0 \le g,h \le M. \end{aligned}$$

(49)

${{\varvec{A}}}_\mathbf{1 ,\mathbf 2 }^{(\mathcal {I}_j)}$ is the northeast corner sub-matrix of $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_j)}$, which contains all transitions from microstates in state space $\varOmega ^{(\mathcal {I}_{j})}{/}\varOmega ^{(\mathcal {I}_{i,j})}$ to microstates in state space $\varOmega ^{(\mathcal {I}_{i,j})}$:

$$\begin{aligned} {{\varvec{A}}}_\mathbf{1 ,\mathbf 2 }^{(\mathcal {I}_j)}= & {} \left( {\begin{array}{c} {{{\varvec{A}}}_{g,l}^{(\mathcal {I}_j)}} \end{array}} \right) = \{A_{{{\varvec{x}}}_m, {{\varvec{x}}}_n}\}, {{\varvec{x}}}_m \in \varOmega ^{(\mathcal {I}_{i,j})}, {{\varvec{x}}}_n \in \varOmega ^{(\mathcal {I}_{j})}{/}\varOmega ^{(\mathcal {I}_{i,j})},\nonumber \\&\quad \text {and}\;0 \le g \le M,\quad l \ge M+1. \end{aligned}$$

(50)

${{\varvec{A}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)}$ is the southwest corner sub-matrix of $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_j)}$, which contains all transitions from microstates in state space $\varOmega ^{(\mathcal {I}_{i,j})}$ to microstates in state space $\varOmega ^{(\mathcal {I}_{j})}{/}\varOmega ^{(\mathcal {I}_{i,j})}$:

$$\begin{aligned} {{\varvec{A}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)}= & {} \left( {\begin{array}{c} {{{\varvec{A}}}_{k,h}^{(\mathcal {I}_j)}} \end{array}} \right) = \{A_{{{\varvec{x}}}_m, {{\varvec{x}}}_n}\}, {{\varvec{x}}}_m \in \varOmega ^{(\mathcal {I}_{j})}{/}\varOmega ^{(\mathcal {I}_{i,j})}, {{\varvec{x}}}_n \in \varOmega ^{(\mathcal {I}_{i,j})},\nonumber \\&\quad \text {and}\; 0 \le h \le M, k \ge M+1, \end{aligned}$$

(51)

and ${{\varvec{A}}}_\mathbf{2 ,\mathbf 2 }^{(\mathcal {I}_j)}$ is the southeast corner sub-matrix of $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_j)}$, which contains all transitions between microstates in state space $\varOmega ^{(\mathcal {I}_{j})}{/}\varOmega ^{(\mathcal {I}_{i,j})}$:

$$\begin{aligned} {{\varvec{A}}}_\mathbf{2 ,\mathbf 2 }^{(\mathcal {I}_j)}= & {} \left( {\begin{array}{cc} {{{\varvec{A}}}_{k,l}^{(\mathcal {I}_j)}} \end{array}} \right) = \{A_{{{\varvec{x}}}_m, {{\varvec{x}}}_n}\}, \quad \text{ with } \;{{\varvec{x}}}_m\quad \text{ and }\quad {{\varvec{x}}}_n \in \varOmega ^{(\mathcal {I}_{j})}{/} \varOmega ^{(\mathcal {I}_{i,j})},\nonumber \\&\text {and}\quad k,l \ge M+1. \end{aligned}$$

(52)

We now truncate the state space at the maximum copy number M of the ith MEG. A matrix ${{\varvec{A}}}^{(\mathcal {I}_{i,j})}$ on the truncated state space $\varOmega ^{(\mathcal {I}_{i,j})}$ using the same partition $\{ \mathcal {G}_0, \mathcal {G}_1, \ldots , \mathcal {G}_M \}$ can be constructed as:

$$\begin{aligned} \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} = \left( {\begin{array}{c} {{{\varvec{A}}}_{g,h}^{(\mathcal {I}_{i,j})}} \end{array}} \right) ,\quad \text {and}\quad 0 \le g,h \le M. \end{aligned}$$

(53)

Similar to the matrix $\tilde{{{\varvec{A}}}}$ in Eq. (6), both matrices $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})}$ and $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{j})}$ are tridiagonal matrix with ${{{\varvec{A}}}_{m,n}^{(\mathcal {I}_{i,j})}}=0$ and ${{{\varvec{A}}}_{m,n}^{(\mathcal {I}_{j})}}=0$ for any $|m - n| > 1$.

Matrix $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})}$ and sub-matrix ${{\varvec{A}}}_\mathbf{1 ,\mathbf 1 }^{(\mathcal {I}_j)}$ reside on the same state space $\varOmega ^{(\mathcal {I}_{i,j})}$ and have exactly the same permutation, i.e., the matrix element $A^{(\mathcal {I}_{i,j})}_{{{\varvec{x}}}_m,{{\varvec{x}}}_n} \in \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})}$ and $A^{(\mathcal {I}_{j})}_{{{\varvec{x}}}_m,{{\varvec{x}}}_n} \in {{\varvec{A}}}_\mathbf{1 ,\mathbf 1 }^{(\mathcal {I}_j)}$ describes the same transitions between microstates ${{\varvec{x}}}_m, {{\varvec{x}}}_n \in \varOmega ^{(\mathcal {I}_{i,j})} \subset \varOmega ^{(\mathcal {I}_{j})}$. Only diagonal elements in ${{{\varvec{A}}}_{M,M}^{(\mathcal {I}_{i,j})}}$ have different rates. By construction, $\mathcal {G}_M$ and $\mathcal {G}_{M+1}$ are the only two aggregated groups that are involved in transition between states across the boundary of $\varOmega ^{(\mathcal {I}_{i,j})}$. The sub-matrix ${{\varvec{A}}}_{M+1,M}^{(\mathcal {I}_j)}$ is the only nonzero sub-matrix in ${{\varvec{A}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)}$, which forms the reflection boundary and is involved in synthesis reactions from microstates in group $\mathcal {G}_M$ to microstates in $\mathcal {G}_{M+1}$. As a property of the rate matrix, we have

$$\begin{aligned} \mathbbm {1}^T {{\varvec{A}}}_{M-1,M}^{(\mathcal {I}_j)} +\mathbbm {1}^T {{\varvec{A}}}_{M,M}^{(\mathcal {I}_j)} +\mathbbm {1}^T {{\varvec{A}}}_{M+1,M}^{(\mathcal {I}_j)} = \mathbf 0 ^T, \end{aligned}$$

(54)

and

$$\begin{aligned} \mathbbm {1}^T {{\varvec{A}}}_{M-1,M}^{(\mathcal {I}_{i,j})} +\mathbbm {1}^T {{\varvec{A}}}_{M,M}^{(\mathcal {I}_{i,j})} = \mathbf 0 ^T. \end{aligned}$$

(55)

Since ${{\varvec{A}}}_{M-1,M}^{(\mathcal {I}_j)} = {{\varvec{A}}}_{M-1,M}^{(\mathcal {I}_{i,j})},$ we have from Eq. (55) $\mathbbm {1}^T {{\varvec{A}}}_{M,M}^{(\mathcal {I}_{i,j})} = - \mathbbm {1}^T {{\varvec{A}}}_{M-1,M}^{(\mathcal {I}_{i,j})} = - \mathbbm {1}^T {{\varvec{A}}}_{M-1,M}^{(\mathcal {I}_{i,j})}.$ With Eq. (54), we further have

$$\begin{aligned} \mathbbm {1}^T {{\varvec{A}}}_{M,M}^{(\mathcal {I}_{i,j})} = \mathbbm {1}^T {{\varvec{A}}}_{M,M}^{(\mathcal {I}_j)} +\mathbbm {1}^T {{\varvec{A}}}_{M+1,M}^{(\mathcal {I}_j)}. \end{aligned}$$

By construction, the only differences between the sub-matrix ${{\varvec{A}}}_{M,M}^{(\mathcal {I}_{i,j})}$ and ${{\varvec{A}}}_{M,M}^{(\mathcal {I}_j)}$ are in the diagonal elements. Therefore, we have

$$\begin{aligned} {{\varvec{A}}}_{M,M}^{(\mathcal {I}_{i,j})} = {{\varvec{A}}}_{M,M}^{(\mathcal {I}_j)} + \text {diag}\left( \mathbbm {1}^T {{\varvec{A}}}_{M+1,M}^{(\mathcal {I}_j)}\right) . \end{aligned}$$

That is:

$$\begin{aligned} \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} = {{\varvec{A}}}_\mathbf{1 ,\mathbf 1 }^{(\mathcal {I}_j)} + \text {diag}\left( \mathbbm {1}^T {{\varvec{A}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)}\right) . \end{aligned}$$

For convenience, we use the notation ${{\varvec{R}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)} = \text {diag}(\mathbbm {1}^T {{\varvec{A}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)})$, and have $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} = {{\varvec{A}}}_\mathbf{1 ,\mathbf 1 }^{(\mathcal {I}_j)} + {{\varvec{R}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)}$. We partition the steady-state vector ${{\varvec{\pi }}}^{(\mathcal {I}_{j})}$ accordingly into two sub-vectors: ${{\varvec{\pi }}}^{(\mathcal {I}_{j})} = ({{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }, {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 })$, where ${{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }$ corresponds to states in $\varOmega ^{(\mathcal {I}_{i,j})}$, and ${{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 }$ corresponds to states in $\varOmega ^{(\mathcal {I}_{j})}{/}\varOmega ^{(\mathcal {I}_{i,j})}$. As $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{j})} {{\varvec{\pi }}}^{(\mathcal {I}_{j})} = \mathbf 0$, we have:

$$\begin{aligned} {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } + {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } = \mathbf 0 , \end{aligned}$$

therefore

$$\begin{aligned} \left[ \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} - {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } \right] {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } + {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } = \mathbf 0 . \end{aligned}$$

Hence, we have:

$$\begin{aligned} \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } = {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } - {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 }. \end{aligned}$$

(56)

As all off-diagonal entries of transition rate matrix $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{j})}$ are nonnegative, we know that ${{\varvec{A}}}_\mathbf{1 ,\mathbf 2 }^{(\mathcal {I}_j)} \ge \mathbf 0$, and ${{\varvec{R}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)} \ge \mathbf 0$. Since ${{\varvec{\pi }}}^{(\mathcal {I}_{j})} = ({{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }, {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 })$ is the steady-state distribution of the rate matrix $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_j)}$ with ${{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \ge \mathbf 0$ and ${{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } \ge \mathbf 0$, we have ${{\varvec{A}}}_\mathbf{1 ,\mathbf 2 }^{(\mathcal {I}_j)} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } \ge \mathbf 0$, and ${{\varvec{R}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \ge \mathbf 0$. As all columns of matrix $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})}$ sum to zero, i.e., $\mathbbm {1}^T \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} = \mathbf 0 ^T,$ we have:

$$\begin{aligned} \mathbbm {1}^T {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } - \mathbbm {1}^T {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } = \mathbbm {1}^T \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } = \mathbf 0 ^T. \end{aligned}$$

Therefore, we have:

$$\begin{aligned} \mathbbm {1}^T {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } = \mathbbm {1}^T {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 }. \end{aligned}$$

As all entries in vector ${{\varvec{R}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }$ and ${{\varvec{A}}}_\mathbf{1 ,\mathbf 2 }^{(\mathcal {I}_j)} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 }$ are nonnegative, we have the following equality of 1-norms, i.e., the summation of absolute values of vector elements:

$$\begin{aligned} \left\| {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \right\| _1 = \mathbbm {1}^T {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } = \mathbbm {1}^T {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } = \left\| {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } \right\| _1. \end{aligned}$$

(57)

From Minkowski inequality of vector norm and Eq. (56), we have:

$$\begin{aligned} \left\| \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \right\| _1 = \left\| {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } - {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } \right\| _1 \le \left\| {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \right\| _1 + \left\| {{\varvec{A}}}^{(\mathcal {I}_{j})}_\mathbf{1 ,\mathbf 2 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{2 } \right\| _1. \nonumber \\ \end{aligned}$$

(58)

From Eq. (57), we have:

$$\begin{aligned} \left\| \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \right\| _1 \le 2 \left\| {{\varvec{R}}}^{(\mathcal {I}_{j})}_\mathbf{2 ,\mathbf 1 } {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \right\| _1. \end{aligned}$$

(59)

Now we show that the norm of $\Vert {{\varvec{R}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \Vert _1$ converges to zero when the maximum copy number M of the ith MEG goes to infinity. In the block tridiagonal matrix $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_j)}$, only the boundary block ${{\varvec{A}}}_{M+1,M}^{(\mathcal {I}_j)}$ contains nonzero elements in sub-matrix ${{\varvec{A}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)}$, and all other blocks in ${{\varvec{A}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)}$ contain only zero entries. From Cauchy–Schwarz inequality, we have:

$$\begin{aligned} \left\| {{\varvec{R}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \right\| _1= & {} \left\| \left[ \text {diag}\left( \mathbbm {1}^T {{\varvec{A}}}_{M+1,M}^{(\mathcal {I}_j)}\right) \right] \left[ {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }(\mathcal {G}_M)\right] \right\| _1\\\le & {} \left\| \text {diag}(\mathbbm {1}^T {{\varvec{A}}}_{M+1,M}^{(\mathcal {I}_j)}) \right\| _1 \cdot \left\| {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }(\mathcal {G}_M) \right\| _1, \end{aligned}$$

where ${{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }(\mathcal {G}_M)$ is the sub-vector corresponding to the state partition $\mathcal {G}_M$. Furthermore, according to Lemma 2 and Eq. (13) after replacing the subscript i in $\tilde{\pi }_i^{(\infty )}$ with M and taking into consideration of the equivalence of the infinite space $\varOmega ^{(\mathcal {I}_{j})}$ and $\varOmega ^{(\infty )}$ in regard to truncation at $\mathcal {I}_{i}$, we have the probability of the boundary block $\mathcal {G}_M$: $\left\| {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } (\mathcal {G}_M) \right\| _1 \rightarrow 0$ when $M \rightarrow \infty$. When synthesis reactions are concentration independent (zero-order reactions) as usually the case (Nelson 2015), the norm $\Vert \text {diag}(\mathbbm {1}^T {{\varvec{A}}}_{M+1,M}^{(\mathcal {I}_j)}) \Vert _1$ is a constant representing the total synthesis rates over states in $\mathcal {G}_M$. We have: $\Vert {{\varvec{R}}}_\mathbf{2 ,\mathbf 1 }^{(\mathcal {I}_j)} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \Vert _1 \rightarrow 0$ when $M \rightarrow \infty$. Therefore, with Eq. (59), we have:

$$\begin{aligned} \lim _{M \rightarrow \infty } \left\| \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \right\| _1 = 0. \end{aligned}$$

Hence,

$$\begin{aligned} \lim _{M \rightarrow \infty } \tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } = \mathbf 0 . \end{aligned}$$

That is, when the maximum copy number limit of the ith MEG is sufficiently large, both ${{\varvec{\pi }}}^{(\mathcal {I}_{i,j})}$ and ${{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }$ are the steady-state solutions of $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})} {{\varvec{y}}}= 0$. According to Perron–Frobenius theorem for the transition rate matrix of continuous-time Markov chains (Meyer 2000), the dCME governed by $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})}$ has a globally unique stationary distribution. In addition, by construction of the matrix, via enumeration of the state space, matrix $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})}$ is irreducible, as all microstates in the state space can be reached from the initial state. Therefore, the matrix $\tilde{{{\varvec{A}}}}^{(\mathcal {I}_{i,j})}$ has only one zero eigenvalue (Meyer 2000), both ${{\varvec{\pi }}}^{(\mathcal {I}_{i,j})}$ and ${{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }$ are eigenvectors corresponding to the eigenvalue 0. Therefore, we have the relationship ${{\varvec{\pi }}}^{(\mathcal {I}_{i,j})} = c {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }$, where c is an arbitrary real number. As both vectors are nonnegative, and $\mathbbm {1}^T {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \le 1 = \mathbbm {1}^T {{\varvec{\pi }}}^{(\mathcal {I}_{i,j})}$, there must exist an $\epsilon = 1 - \mathbbm {1}^T {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \ge 0$, such that ${{\varvec{\pi }}}^{(\mathcal {I}_{i,j})} = (1 + \epsilon ) {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }$. According to Lemma 2, $\mathbbm {1}^T {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 } \rightarrow 1$, when the maximum copy number limit of the ith MEG goes to infinity. Therefore, we have $\epsilon \rightarrow 0$ when $M \rightarrow \infty$. Therefore, we have shown both ${{\varvec{\pi }}}^{(\mathcal {I}_{i,j})} \ge {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }$ and ${{\varvec{\pi }}}^{(\mathcal {I}_{i,j})} \rightarrow {{\varvec{\pi }}}^{(\mathcal {I}_{j})}_\mathbf{1 }$ component-wise, when the maximum copy number limit of the ith MEG goes to infinity.