# On the Properties of the Reaction Counts Chemical Master Equation

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Reaction Counts CME

#### 2.1. Formulation

#### 2.2. Analytical Solutions of the Reaction Counts CME

**Example**

**1.**

**Theorem**

**1.**

- 1.
- There exists a permutation matrix P such that ${P}^{T}\phantom{\rule{0.166667em}{0ex}}{A}^{*}\phantom{\rule{0.166667em}{0ex}}P$ is lower triangular;
- 2.
- The spectrum of ${A}^{*}$ is the set:$$spec\left({A}^{*}\right)=\left\{-\sum _{n=1}^{{N}_{r}}{\alpha}_{n}({x}_{0},r):r\in \mathrm{\Lambda}\right\}.$$

**Proposition**

**1.**

## 3. Partitioning the State Space

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Paths

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

#### 4.1. Path Chains

**Definition**

**5.**

**Proposition**

**2.**

**Proof.**

#### 4.2. Gated- and Un-Gated Path Chains

**Definition**

**6.**

**Definition**

**7.**

**Lemma**

**2.**

**Proof.**

#### 4.3. Cascade of Gating and Un-Gating

**Theorem**

**3.**

**Proof.**

**Example**

**2.**

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) Cartoon showing the mapping of the reactions counts birth-death process to the species count birth-death process. (

**B**) An evaluation of the birth-death process for parameters $({x}_{0}=0,{c}_{b}=1.0,{c}_{d}=0.1).$ The plot shows: The distribution of the species having population one over time $p\left(Z\right(t)=1);$ and the distributions of reaction $(1,0),(1,2),$ and $(2,3)$ firing at time $t,$ in the reaction counts setting. (

**C**) $(\u2020)=p({R}_{0}\left(t\right)=(0,1))$, $(\u2020\u2020)=p({R}_{0}\left(t\right)=(0,1))+p({R}_{0}\left(t\right)=(1,2))$, $(\u2020\u2020\u2020)=p({R}_{0}\left(t\right)=(0,1))+p({R}_{0}\left(t\right)=(1,2))+p({R}_{0}\left(t\right)=(2,3)).$ The distribution of the species count chemical master equation (CME) in state 1 at t is approached from below by the sum of the probabilities of the reactions firing which result in being in state $1$.

**Figure 2.**The cartoon above depicts the generator of a path chain. With $\alpha (\xb7)$ being the total outward propensity and $\beta (\xb7,\xb7)$ the propensity to transition to the next state in the chain. All reactions leading away from the chain are directed into the sink state.

**Figure 3.**The cartoon above depicts the cascade of gating being performed on a path chain. When a state is gated, the propensities leaving the state are set to zero (depicted with a red cross). When the state is un-gated, the propensities are reintroduced.

**Figure 4.**(

**A**) Graph of the path probability $p({X}_{g,1}\left(t\right)=(1,3))$ and the upper bound of the path probability ${u}_{5}\left(t\right)$ for the time interval $t\in [0,10].$ (

**B**) Case 1: $({c}_{b}=1.0,{c}_{d}=0.15),$ Case 2: $({c}_{b}=1.0,{c}_{d}=0.2),$ Case 3: $({c}_{b}=2.0,{c}_{d}=0.15).$ “analytical” refers to the path probability and “approximation” refers to the upper bound of the path probability.

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Sunkara, V.
On the Properties of the Reaction Counts Chemical Master Equation. *Entropy* **2019**, *21*, 607.
https://doi.org/10.3390/e21060607

**AMA Style**

Sunkara V.
On the Properties of the Reaction Counts Chemical Master Equation. *Entropy*. 2019; 21(6):607.
https://doi.org/10.3390/e21060607

**Chicago/Turabian Style**

Sunkara, Vikram.
2019. "On the Properties of the Reaction Counts Chemical Master Equation" *Entropy* 21, no. 6: 607.
https://doi.org/10.3390/e21060607