# Simpliﬁcation of Reaction Networks, Conﬂuence and Elementary Modes

^{1}

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## Abstract

**:**

## 1. Introduction

#### Outline

## 2. Preliminaries

#### 2.1. Confluence Notions

**Definition**

**1.**

**Lemma**

**1.**

**Definition**

**2.**

#### 2.2. Multisets

#### 2.3. Commutative Semigroups

## 3. Reaction Networks without Kinetics

**Definition**

**3.**

**Example**

**1.**

**Definition**

**4.**

#### 3.1. Stoichiometry Matrices

#### 3.2. Elementary Modes

**Definition**

**5.**

**v is on an extreme ray:**there exists no ${v}^{\prime}\in {\mathit{ker}}_{+}(S)\backslash \{{0}^{{\mathbb{N}}^{n}}\}$ such that $\mathit{supp}({v}^{\prime})\u228a\mathit{supp}(v)$, and**v is factorised:**there exists no ${v}^{\u2033}\in {\mathit{ker}}_{+}(S)$ such that $v=k{v}^{\u2033}$ for some natural number $k\ge 2$.

**Theorem**

**1**

#### 3.3. Elementary Flux Modes

**Definition**

**6.**

## 4. Simplifying Reaction Networks without Kinetics

#### 4.1. Intermediate Elimination

**Example**

**2.**

#### 4.2. Eliminating Dependent Reactions

**Example**

**3.**

## 5. Simplifying Flux Networks

#### 5.1. Vector Representations of Reaction Networks

**Definition**

**7.**

#### 5.2. Simplification Rules

#### 5.3. Factorization

**Example**

**4.**

**Lemma**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 5.4. Proving Confluence via Elementary Modes

**Lemma 3**(Diamond).

**Proof.**

**Lemma**

**4.**

**Proof.**

**Case**- $V{\Rrightarrow}_{\mathrm{F}-\mathrm{FACT}}{V}^{\prime}$. Suppose that (F-Fact) replaces vector ${v}_{1}$ by vector ${v}_{1}^{\prime}$ so that ${v}_{1}={k}^{\prime}{v}_{1}^{\prime}$ for some ${k}^{\prime}\ne 0$. Hence ${n}_{1}{k}^{\prime}{\mathbf{r}}_{{v}_{1}^{\prime}}+{n}_{2}{\mathbf{r}}_{{v}_{2}}+\dots +{n}_{k}{\mathbf{r}}_{{v}_{k}}\in \mathit{inv}({\mathbf{r}}_{{V}^{\prime}})$. And thus, ${({n}_{1}{k}^{\prime}{v}_{1}^{\prime}+{n}_{2}{v}_{2}+\dots +{n}_{k}{v}_{k})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}({V}^{\prime})$, which is equivalent to ${({n}_{1}{v}_{1}+\dots +{n}_{k}{v}_{k})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}({V}^{\prime})$ as required.
**Case**- $V{\Rrightarrow}_{\mathrm{F}-\mathrm{DEP}}{V}^{\prime}$. By rule (F-Dep) there exist $k\in \mathbb{N}$, $\mathbf{v}\in {V}^{k}$ and $v\in {\mathbb{N}}^{k}$ such that $V={V}^{\prime}\uplus \{{\mathbf{v}}_{v}\}$. If all ${v}_{i}$ are distinct from ${\mathbf{v}}_{v}$ then trivially ${n}_{1}{\mathbf{r}}_{{v}_{1}}+\dots +{n}_{k}{\mathbf{r}}_{{v}_{k}}\in \mathit{inv}({\mathbf{r}}_{{V}^{\prime}})$. Otherwise, we can assume without loss of generality that ${v}_{1}={\mathbf{v}}_{v}$ with v and $\mathbf{v}$ as in rule (F-Dep). Suppose that these have the forms $v=({m}_{1},\dots ,{m}_{l})$ and $\mathbf{v}=({w}_{1},\dots ,{w}_{l})$. Since ${\mathbf{r}}_{{\mathbf{v}}_{v}}={({m}_{1}{\mathbf{r}}_{{w}_{1}}+\dots +{m}_{l}{\mathbf{r}}_{{w}_{l}})}^{\mathcal{R}}$, it follows that:$${n}_{1}{m}_{1}{\mathbf{r}}_{{w}_{1}}+\dots +{n}_{1}{m}_{l}{\mathbf{r}}_{{w}_{l}}+{n}_{2}{\mathbf{r}}_{{v}_{2}}+\dots +{n}_{k}{\mathbf{r}}_{{v}_{k}}\in \mathit{inv}({\mathbf{r}}_{{V}^{\prime}}).$$This yields ${({n}_{1}{m}_{1}{w}_{1}+\dots +{n}_{1}{m}_{l}{w}_{l}+{n}_{2}{v}_{2}+\dots +{n}_{k}{v}_{k})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}({V}^{\prime})$. Since ${v}_{1}={\mathbf{v}}_{v}={({m}_{1}{w}_{1}+\dots +{m}_{l}{w}_{l})}^{{\mathbb{N}}^{n}}$ this is is equivalent to ${({n}_{1}{v}_{1}+\dots +{n}_{k}{v}_{k})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}({V}^{\prime})$ as required.
**Case**- $V{\Rrightarrow}_{\mathrm{F}-\mathrm{INTER}}{V}^{\prime}$. Suppose that the intermediate species $X\in \mathcal{I}$ was eliminated thereby. Recall that ${\sum}_{i=1}^{k}{n}_{i}{\mathbf{r}}_{{v}_{i}}\in \mathit{inv}({\mathbf{r}}_{V})$. We can assume without loss of generality that ${n}_{i}\ne 0$ for all $1\le i\le k$. Let P, C, $\mathit{prod}$, and $\mathit{cons}$ be as introduced in the Diamond Lemma 3, where $\mathcal{G}={\mathbb{N}}^{n}$, homomorphism h the identity on ${\mathbb{N}}^{n}$, and ${g}_{i}={n}_{i}$ for all $1\le i\le k$. The lemma then yields:$$\begin{array}{c}{({\sum}_{p\in P}{\sum}_{c\in C}{n}_{p}{n}_{c}\phantom{\rule{4pt}{0ex}}({v}_{p}{\diamond}_{X}{v}_{c}))}^{{\mathbb{N}}^{n}}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{({\sum}_{p\in P}{n}_{p}\phantom{\rule{4pt}{0ex}}\mathit{cons}\phantom{\rule{4pt}{0ex}}{v}_{p}+{\sum}_{c\in C}{n}_{c}\phantom{\rule{4pt}{0ex}}\mathit{prod}\phantom{\rule{4pt}{0ex}}{v}_{c})}^{{\mathbb{N}}^{n}}\hfill \end{array}.$$Since ${({\sum}_{i=1}^{k}{n}_{i}{\mathbf{r}}_{{v}_{i}})}^{\mathcal{R}}={0}^{\mathcal{R}}$ it follows that $\mathit{prod}=\mathit{cons}$. Furthermore, $\mathit{prod}\ne 0$ since otherwise $P=C=\mathsf{\varnothing}$ so that (F-Inter) could not be applied. Since $\mathit{cons}=\mathit{prod}$, this tuple is equal to $\mathit{prod}{({\sum}_{p\in P}{n}_{p}{v}_{p}+{\sum}_{c\in C}{n}_{c}{v}_{c})}^{{\mathbb{N}}^{n}}$. With $M=\{m\in \{1\dots k\}\mid {\mathbf{r}}_{{v}_{m}}(X)=0\}$ we get:$$\begin{array}{c}{({\sum}_{p\in P}{\sum}_{q\in C}{n}_{p}{n}_{c}({v}_{p}{\diamond}_{X}{v}_{c})+{\sum}_{m\in M}\mathit{prod}\phantom{\rule{4pt}{0ex}}{n}_{m}{v}_{m})}^{{\mathbb{N}}^{n}}\mathbf{r}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{(\sum _{i=1}^{k}\mathit{prod}\phantom{\rule{4pt}{0ex}}{n}_{i}{v}_{i})}^{{\mathbb{N}}^{n}}\mathbf{r}\hfill \end{array}.$$This multiset is an invariant, since ${({\sum}_{i=1}^{k}{n}_{i}{\mathbf{r}}_{{v}_{i}})}^{\mathcal{R}}={0}^{\mathcal{R}}$. It follows that:$$\begin{array}{c}{({\sum}_{p\in P}{\sum}_{c\in C}{n}_{p}{n}_{c}({v}_{p}{\diamond}_{X}{v}_{c})+{\sum}_{m\in M}{n}_{m}\phantom{\rule{4pt}{0ex}}\mathit{prod}\phantom{\rule{4pt}{0ex}}{v}_{m})}^{{\mathbb{N}}^{n}}\mathbf{r}\in \mathit{inv}({\mathbf{r}}_{{V}^{\prime}})\end{array}.$$This implies ${({\sum}_{i=1}^{k}\mathit{prod}\phantom{\rule{4pt}{0ex}}{n}_{i}{v}_{i})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}({V}^{\prime})$. Since $\mathit{prod}\ne 0$ and since ${\mathit{inv}}_{\mathbf{r}}({V}^{\prime})$ is closed by factorization with nonzero factors, it follows that ${\sum}_{i=1}^{k}{n}_{i}{\mathbf{r}}_{{v}_{i}}\in {\mathit{inv}}_{\mathbf{r}}({V}^{\prime})$ as required.

**Case**- $V{\Rrightarrow}_{\mathrm{F}-\mathrm{FACT}}{V}^{\prime}$. Suppose that (F-Fact) replaces vector ${v}_{1}$ by vector ${v}_{1}^{\prime}$ so that ${v}_{1}={k}^{\prime}{v}_{1}^{\prime}$ for some ${k}^{\prime}\ne 0$. Since ${({k}^{\prime}{n}_{1}{\mathbf{r}}_{{v}_{1}^{\prime}}+\dots +{k}^{\prime}{n}_{k}{\mathbf{r}}_{{v}_{k}^{\prime}})}^{\mathcal{R}}={0}^{\mathcal{R}}$ we have ${n}_{1}{\mathbf{r}}_{{v}_{1}}+{n}_{2}{k}^{\prime}{\mathbf{r}}_{{v}_{2}^{\prime}}+\dots +{n}_{k}{k}^{\prime}{\mathbf{r}}_{{v}_{k}^{\prime}}\in \mathit{inv}({\mathbf{r}}_{V})$. And thus, ${({n}_{1}{v}_{1}+{n}_{2}{k}^{\prime}{v}_{2}^{\prime}+\dots +{n}_{k}{k}^{\prime}{v}_{k}^{\prime})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}(V)$, which is equivalent to ${({n}_{1}{k}^{\prime}{v}_{1}^{\prime}+\dots +{n}_{k}{k}^{\prime}{v}_{k}^{\prime})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}(V)$, and thus ${({n}_{1}{v}_{1}^{\prime}+\dots +{n}_{k}{v}_{k}^{\prime})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}(V)$ as required.
**Case**- $V{\Rrightarrow}_{\mathrm{F}-\mathrm{DEP}}{V}^{\prime}$. By rule (F-Dep) there exist $k\in \mathbb{N}$, $\mathbf{v}\in {V}^{k}$ and $v\in {\mathbb{N}}^{k}$ such that $V={V}^{\prime}\uplus \{{\mathbf{v}}_{v}\}$. If all ${v}_{i}^{\prime}$ are distinct from ${\mathbf{v}}_{v}$ then trivially ${n}_{1}{\mathbf{r}}_{{v}_{1}^{\prime}}+\dots +{n}_{k}{\mathbf{r}}_{{v}_{k}^{\prime}}\in \mathit{inv}({\mathbf{r}}_{V})$. Otherwise, we can assume without loss of generality that ${v}_{1}^{\prime}={\mathbf{v}}_{v}$ with v and $\mathbf{v}$ as in the rule. Suppose that these have the forms $v=({m}_{1},\dots ,{m}_{l})$ and $\mathbf{v}=({w}_{1},\dots ,{w}_{l})$. Since ${\mathbf{r}}_{{\mathbf{v}}_{v}}={({m}_{1}{\mathbf{r}}_{{w}_{1}}+\dots +{m}_{l}{\mathbf{r}}_{{w}_{l}})}^{\mathcal{R}}$, it follows that:$${n}_{1}{m}_{1}{\mathbf{r}}_{{w}_{1}}+\dots +{n}_{1}{m}_{l}{\mathbf{r}}_{{w}_{l}}+{n}_{2}{\mathbf{r}}_{{v}_{2}^{\prime}}+\dots +{n}_{k}{\mathbf{r}}_{{v}_{k}^{\prime}}\in \mathit{inv}({\mathbf{r}}_{V}).$$This yields ${({n}_{1}{m}_{1}{w}_{1}+\dots +{n}_{1}{m}_{l}{w}_{l}+{n}_{2}{v}_{2}^{\prime}+\dots +{n}_{k}{v}_{k}^{\prime})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}(V)$. Since ${v}_{1}^{\prime}={\mathbf{v}}_{v}={({m}_{1}{w}_{1}+\dots +{m}_{l}{w}_{l})}^{{\mathbb{N}}^{n}}$ this is is equivalent to ${({n}_{1}{v}_{1}^{\prime}+\dots +{n}_{k}{v}_{k}^{\prime})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}(V)$ as required.
**Case**- $V{\Rrightarrow}_{\mathrm{F}-\mathrm{INTER}}{V}^{\prime}$. Suppose that the intermediate species $X\in \mathcal{I}$ was eliminated thereby. We recall that ${\sum}_{i=1}^{k}{n}_{i}{\mathbf{r}}_{{v}_{i}^{\prime}}\in \mathit{inv}({\mathbf{r}}_{{V}^{\prime}})$. Without loss of generality, we can assume that all elements of ${V}^{\prime}$ occur exactly once in this sum. Let $V=\{{v}_{1},\dots ,{v}_{l}\}$, $P=\{p\mid {\mathbf{r}}_{{v}_{p}}(X)>0\}$, $C=\{c\mid {\mathbf{r}}_{{v}_{c}}(X)<0\}$, and $M=\{m\mid {\mathbf{r}}_{{v}_{m}}(X)=0\}$. If ${v}_{i}^{\prime}={v}_{p}{\diamond}_{X}{v}_{c}$ for $p\in P$ and $c\in C$, we note ${o}_{pc}={n}_{i}$. Otherwise, if ${v}_{i}^{\prime}={v}_{m}$ with $m\in M$, we note ${o}_{m}={n}_{i}$. By the rule (F-Inter) we have:$$\begin{array}{ccc}\hfill {({\sum}_{i=1}^{k}{n}_{i}{v}_{i}^{\prime})}^{{\mathbb{N}}^{n}}& =& {({\sum}_{p\in P}{\sum}_{c\in C}{o}_{pc}{v}_{p}{\diamond}_{X}{v}_{c}+{\sum}_{m\in M}{o}_{m}{v}_{m})}^{{\mathbb{N}}^{n}}\hfill \\ & =& {\left(\right)}^{{\sum}_{p\in P}}{\mathbb{N}}^{n}\hfill \end{array}$$Hence ${({\sum}_{i=1}^{k}{n}_{i}{v}_{i}^{\prime})}^{{\mathbb{N}}^{n}}\mathbf{r}\in {\mathit{inv}}_{\mathbf{r}}(V)$.

**Lemma**

**5.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

## 6. Reaction Networks with Deterministic Semantics

#### 6.1. Kinetic Expressions

**Definition**

**8.**

#### 6.2. Constrained Flux Networks

**Definition**

**9.**

**Definition**

**10.**

#### 6.3. Systems of Constrained Equations with ODEs

**Definition**

**11.**

#### 6.4. Deterministic Semantics

#### 6.5. Contextual Equivalence

**Definition**

**12.**

**Lemma**

**6.**

**Proof.**

## 7. Simplification of Constrained Flux Networks

#### 7.1. Linear Steadiness of Intermediate Species

**Definition**

**13.**

**Partial steady state:**the concentration of X is steady, i.e., $C\vDash \mathit{cst}(X)$.**Linear consumption:**if a reaction in V consumes X then its kinetic expression is linear in X, that is: if $v;e\in V$ such that ${\mathit{r}}_{v}(X)<0$ then $\mathit{C}\vDash e=X{e}^{\prime}$ for some expression ${e}^{\prime}$ such that $X\notin p\mathit{Specs}({e}^{\prime})$.**Independent production:**if a reaction in V produces X then its kinetic expression does not contain X except for subexpressions $X(0)$: for any $v;e\in V$, if ${\mathit{r}}_{v}(X)>0$ then $X\notin p\mathit{Specs}(e)$.**Nonzero consumption:**the consumption of X is nonzero: $\mathit{C}\vDash \sum \{e\mid v;e\in V,\phantom{\rule{4pt}{0ex}}{\mathbf{r}}_{v}(X)<0\}\ne 0$.

#### 7.2. Simplification

**Proposition**

**2.**

#### 7.3. Michaelis-Menten

## 8. Preservation of Linear Steadiness

#### 8.1. $\mathit{LinNets}$

**Definition**

**14.**

- 1.
- Either X is linearly steady in $V\&\mathit{C}$, or X is only a modifier, that is for any $v;e\in V$ we have ${\mathit{r}}_{v}(X)=0$.
- 2.
- No intermediate species different from X occurs in the kinetic expression of a reaction that consumes X: for any $v;e\in V$, if $X\in Con{s}_{\mathcal{I}}({\mathit{r}}_{v})$, then $\mathit{Specs}(e)\cap \mathcal{I}\subseteq \{X\}$.
- 3.
- The rate of a reaction that produces X but does not consume an intermediate species does not depend on the concentration of any intermediate species: for any $v;e\in V$, if $X\in Pro{d}_{\mathcal{I}}({\mathit{r}}_{v})$ and $Con{s}_{\mathcal{I}}({\mathit{r}}_{v})=\varnothing $, then $\mathcal{I}\cap \mathit{Specs}(e)=\varnothing $.
- 4.
- The total stoichiometry of the intermediate species in the reactant (resp. product) of a reaction is never greater than one: for any $v;e\in V$, $|Con{s}_{\mathcal{I}}({\mathit{r}}_{v})|\le 1$ and $|Pro{d}_{\mathcal{I}}({\mathit{r}}_{v})|\le 1$.

#### 8.2. Stability of $\mathit{LinNets}$

**Proposition**

**3.**

- there exists an index i such that $Con{s}_{\mathcal{I}}({\mathit{r}}_{{v}_{i}})=Con{s}_{\mathcal{I}}({\mathit{r}}_{v})$, $Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{i}})=Pro{d}_{\mathcal{I}}({\mathit{r}}_{v})$, and
- for any $j\ne i$, $Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{j}})=Con{s}_{\mathcal{I}}({\mathit{r}}_{{v}_{j}})=\varnothing $.

**Proposition**

**4.**

**Proof.**

## 9. Confluence of the Simplification Relation

#### 9.1. Structural Confluence

**Theorem 3**(Structural confluence).

**Proof.**

#### 9.2. Non-Confluence of the Kinetic Rates

#### 9.3. Criterion for the Full Confluence

**Definition**

**15.**

**Example**

**6.**

**Theorem 4**(Confluence).

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Lemma**

**12.**

**Proof.**

## 10. An Example from the BioModels Database

## 11. Simplification of Systems of Equations

#### 11.1. Simplification of Systems of Equations

**Lemma**

**13.**

**Theorem**

**5.**

**Proof.**

#### 11.2. Simulation

**Lemma**

**14.**

**Proof.**

$\dot{A}$ | = | $\sum _{\begin{array}{c}v;e\in {W}^{\prime}\end{array}}{\mathbf{r}}_{v}(A)e$ |

= | $\phantom{\rule{-34.14322pt}{0ex}}\sum _{\begin{array}{c}\{v;e,{v}^{\prime};{e}^{\prime}\in W\mid {\mathbf{r}}_{v}(X)>0,{\mathbf{r}}_{{v}^{\prime}}(X)<0\}\end{array}}\phantom{\rule{-34.14322pt}{0ex}}{\mathbf{r}}_{v{\diamond}_{X}{v}^{\prime}}(A){\displaystyle \frac{e{e}^{\prime}}{\mathit{cons}}}\phantom{\rule{14.22636pt}{0ex}}+\phantom{\rule{-14.22636pt}{0ex}}\sum _{\begin{array}{c}\{v;e\in W\mid {\mathbf{r}}_{v}(X)=0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{\mathbf{r}}_{v}(A)e[X:=X(0)]$ | |

= | $\phantom{\rule{-34.14322pt}{0ex}}\sum _{\begin{array}{c}\{v;e,{v}^{\prime};{e}^{\prime}\in W\mid {\mathbf{r}}_{v}(X)>0,{\mathbf{r}}_{{v}^{\prime}}(X)<0\}\end{array}}\phantom{\rule{-34.14322pt}{0ex}}({\mathbf{r}}_{v}(A)+{\mathbf{r}}_{{v}^{\prime}}(A)){\displaystyle \frac{e{e}^{\prime}}{\mathit{cons}}}$ $\phantom{\rule{14.22636pt}{0ex}}+\phantom{\rule{-14.22636pt}{0ex}}\sum _{\begin{array}{c}\{v;e\in W\mid {\mathbf{r}}_{v}(X)=0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{\mathbf{r}}_{v}(A)e[X:=X(0)]$ | |

= | $\phantom{\rule{-22.76228pt}{0ex}}\sum _{\begin{array}{c}\{v;e\in W\mid {\mathbf{r}}_{v}(X)>0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{\mathbf{r}}_{v}(A){\displaystyle \frac{e}{\mathit{cons}}}(\phantom{\rule{5.69046pt}{0ex}}\sum _{\begin{array}{c}\{{v}^{\prime};{e}^{\prime}\in W\mid {\mathbf{r}}_{{v}^{\prime}}(X)<0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{e}^{\prime}$ $\phantom{\rule{14.22636pt}{0ex}})+\phantom{\rule{-14.22636pt}{0ex}}\sum _{\begin{array}{c}\{{v}^{\prime};{e}^{\prime}\in N\mid {\mathbf{r}}_{{v}^{\prime}}(X)<0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{\mathbf{r}}_{{v}^{\prime}}(A){\displaystyle \frac{{e}^{\prime}}{\mathit{cons}}}(\phantom{\rule{5.69046pt}{0ex}}\sum _{\begin{array}{c}\{v;e\in W\mid {\mathbf{r}}_{v}(X)>0\}\end{array}}\phantom{\rule{-17.07182pt}{0ex}}e\phantom{\rule{8.5359pt}{0ex}})$ | |

$+\phantom{\rule{-14.22636pt}{0ex}}\sum _{\begin{array}{c}\{v;e\in W\mid {\mathbf{r}}_{v}(X)=0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{\mathbf{r}}_{v}(A)e[X:=X(0)]$ | ||

= | $\phantom{\rule{-17.07182pt}{0ex}}\sum _{\begin{array}{c}\{v;e\in W\mid {\mathbf{r}}_{v}(X)>0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{\mathbf{r}}_{v}(A){\displaystyle \frac{e\mathit{cons}}{\mathit{cons}}}\phantom{\rule{5.69046pt}{0ex}}+\phantom{\rule{-17.07182pt}{0ex}}\sum _{\begin{array}{c}\{{v}^{\prime};{e}^{\prime}\in W\mid {\mathbf{r}}_{{v}^{\prime}}(X)<0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{\mathbf{r}}_{{v}^{\prime}}(A){\displaystyle \frac{{e}^{\prime}\mathit{prod}}{\mathit{cons}}}$ $\phantom{\rule{14.22636pt}{0ex}})+\phantom{\rule{-14.22636pt}{0ex}}\sum _{\begin{array}{c}\{v;e\in W\mid {\mathbf{r}}_{v}(X)=0\}\end{array}}\phantom{\rule{-22.76228pt}{0ex}}{\mathbf{r}}_{v}(A)e[X:=X(0)]$. |

## 12. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Soundness of the Simplification Rules for Constrained Flux Networks

## Appendix B. Proofs for the Stability of LinNets

**Definition**

**A1.**

- a path $\tilde{v}={v}_{1}\dots {v}_{k}$ is a (non empty) sequence of unit vectors ${v}_{i}\in {\mathbb{N}}^{n}$, such that for any $1\le i<k$, we have $Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{i}})=Con{s}_{\mathcal{I}}({\mathit{r}}_{{v}_{i+1}})\ne \varnothing $;
- we denote the vector of a path by $\sum \tilde{v}=\sum _{\begin{array}{c}1\le i\le k\end{array}}{v}_{i}$;
- a path is circular if $Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{k}})=Con{s}_{\mathcal{I}}({\mathit{r}}_{{v}_{1}})\ne \varnothing $, and non-circular otherwise;
- for a circular path $\tilde{v}$, we denote the number of intermediate species occurring in the path by $\tilde{v}(X)=\left|\{1\le i\le k\mid Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{i}})=\{X\}\}\right|$;
- for a non-circular path $\tilde{v}$, we denote its beginning and its end by $Con{s}_{\mathcal{I}}({\mathit{r}}_{\tilde{v}})=Con{s}_{\mathcal{I}}({\mathit{r}}_{{v}_{1}})$ and $Pro{d}_{\mathcal{I}}({\mathit{r}}_{\tilde{v}})=Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{k}})$. In addition, we define the multiset $\tilde{v}(X)=\left|\{1\le i<k\mid Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{i}})=Con{s}_{\mathcal{I}}({\mathit{r}}_{{v}_{i+1}})=\{X\}\}\right|$. Note that we do not count $Con{s}_{\mathcal{I}}({\mathit{r}}_{\tilde{v}})$ and $Pro{d}_{\mathcal{I}}({\mathit{r}}_{\tilde{v}})$ in this multiset.

**Example**

**A1.**

**Definition**

**A2.**

- a non-circular flux-path $\tilde{v}$ is a non-circular path in W such that for any intermediate species X with $\tilde{v}(X)>0$, we have $X\in \mathit{Spec}({W}_{0})\backslash \mathit{Spec}(W)$ (meaning that one of the simplification steps ${W}_{0}{\Rrightarrow}_{C}^{*}W$ removes X from ${W}_{0}$), and such that $Con{s}_{\mathcal{I}}({\mathit{r}}_{\tilde{v}})$ and $Pro{d}_{\mathcal{I}}({\mathit{r}}_{\tilde{v}})$ are either the empty solution ∅, or the intermediate species that are still in W,
- a circular flux-path $\tilde{v}$ is a circular path in W if there is at most one intermediate species X such that $\tilde{v}(X)>0$ and $X\in \mathit{Spec}(W)$ (i.e., X is not yet simplified),
- we call flux-path a path that is either a circular or a non-circular flux-path.
- a flux-path $\tilde{v}$ is said to correspond to a flux v in W if $\sum \tilde{v}=v$.

**Example**

**A2.**

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

- 1.
- for any flux $v;e\in W$, there is a corresponding flux-path $\tilde{v}$ for W such that $\sum \tilde{v}=v$. Moreover, if v is not dependent then, for any intermediate species X, we have $\tilde{v}(X)\le 1$;
- 2.
- for any flux-path $\tilde{v}$ for W such that for any X, $\tilde{v}(X)\le 1$, there is a corresponding flux $v;e\in W$, that is $\sum \tilde{v}=v$;
- 3.
- if $v;e\in W$ depends on ${v}_{1};{e}_{1},\dots ,{v}_{k};{e}_{k}\in W$, then
- there exists an index i such that $Con{s}_{\mathcal{I}}({\mathit{r}}_{{v}_{i}})=Con{s}_{\mathcal{I}}({\mathit{r}}_{v})$, $Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{i}})=Pro{d}_{\mathcal{I}}({\mathit{r}}_{v})$, and
- for any $j\ne i$, $Pro{d}_{\mathcal{I}}({\mathit{r}}_{{v}_{j}})=Con{s}_{\mathcal{I}}({\mathit{r}}_{{v}_{j}})=\varnothing $ and any flux-path $\tilde{{v}_{j}}$ that corresponds to v is circular.

**Proof.**

- (1)
- for flux $v;e$ in the initial network ${W}_{0}$, v is necessarily a unary vector ${v}_{i}$ for some i. So we can directly associate the flux-path $\tilde{v}={v}_{i}$ that trivially corresponds to v. Since it is a unary vector, the flux v is also necessarily not dependent. Because a flux-path of size 1 is always non-circular, we also have, for any intermediate species X, $\tilde{v}(X)=0$, and thus $\tilde{v}(X)\le 1$ as required.
- (2)
- any flux-path $\tilde{v}$ for ${W}_{0}$ is necessarily of size 1. Otherwise, for $\tilde{v}$ being a non-circular flux-path, there would exist some $X\in \mathit{Specs}({W}_{0})\backslash \mathit{Specs}({W}_{0})=\varnothing $. And for $\tilde{v}$ being a circular flux-path, there would exist at least two species X and Y such that $\tilde{v}(X)>0$ and $\tilde{v}(Y)>0$, which contradicts the definition of circular flux-path. Then there exists ${v}_{i};e\in {W}_{0}$ such that ${v}_{i}=\tilde{v}$.
- (3)
- as said above, a flux $v;e\in {W}_{0}$ v can not be dependent.

- (1)
- Let $v;e\in W$, then it is the case that $v;{e}^{\prime}\in {W}^{\prime}$ for some expression ${e}^{\prime}$ because the rule (C-Dep) only removes a dependent flux and modifies some kinetic expressions. By induction hypothesis, there is a flux-path $\tilde{v}$ for ${W}^{\prime}$ such that $\sum \tilde{v}=v$. Also, because $\mathit{Specs}(W)=\mathit{Specs}({W}^{\prime})$, any flux-path for ${W}^{\prime}$ is also a flux-path for W, which proves that $\tilde{v}$ is a flux-path for v. Finally, if v is dependent in W, it is necessarily dependent in W and satisfies $\forall X\in \mathcal{I}.\tilde{v}(X)\le 1$ by induction hypothesis.
- (2)
- Let $\tilde{v}$ be a flux-path for W such that, for any intermediate species X, $\tilde{v}(X)\le 1$. Again, because $\mathit{Specs}(W)=\mathit{Specs}({W}^{\prime})$, $\tilde{v}$ is also a flux-path for ${W}^{\prime}$. By induction hypothesis, there is a corresponding flux $v;e\in {W}^{\prime}$. If $v;e$ is not the flux that is removed by the application of (C-Dep), then this flux still occurs in W (possibly with an updated kinetic) and we conclude directly. We now show that it can not actually be otherwise, and more precisely, that assuming v removed by (C-Dep) contradicts $\tilde{v}(X)\le 1$.Suppose that v is removed by (C-Dep), then v depends on some fluxes ${v}_{1};{e}_{1},\dots {v}_{k};{e}_{k}$ in ${W}^{\prime}$. For the sake of simplicity, we only consider the case where $Con{s}_{\mathcal{I}}({\mathbf{r}}_{\tilde{v}})=X$ for some intermediate $X\in W$. The other case works similarly. We necessarily have $k>1$, because $k=1$ contradicts the linearity assumption. By induction hypothesis and Point 3, there exists in particular at least one j such that $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{j}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{j}})=\varnothing $. Again, by induction hypothesis, there exists $\tilde{{v}^{\prime}}={v}_{1}^{\prime},\dots ,{v}_{l}^{\prime}$ a circular flux-path corresponding to ${v}_{j}$ in ${W}^{\prime}$. Since it is circular, there exist some intermediate species ${X}_{1},\dots ,{X}_{l}$ such that for any $1\le i<l$, $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{i}^{\prime}})={X}_{i}$ and $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{i}^{\prime}})={X}_{i+1}$, and $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{l}^{\prime}})={X}_{l}$, $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{l}^{\prime}})={X}_{1}$. Note that we also have ${X}_{i}\ne X$, since, by definition of flux-path, ${X}_{i}\notin {W}^{\prime}$ for any i, while $X\in W$. Since $\sum \tilde{v}=v=\sum {v}_{k}$ and $v}_{j}=\sum \tilde{{v}^{\prime}}=\sum _{\begin{array}{c}i\end{array}}{v}_{i}^{\prime$, the unit vectors ${v}_{i}^{\prime}$ also appear in the flux-path $\tilde{v}$. So the intermediate species ${X}_{i}$ are present at least one time in $\tilde{v}$. Moreover, there is at least one i such that in $\tilde{v}$, the unit vector ${v}_{i}$ is preceded by a unit vector that is not one of $\tilde{{v}^{\prime}}$. Therefore, ${X}_{i}$ is produced by another flux, i.e., $\tilde{v}(X)>1$, which contradicts the hypothesis.
- (3)
- Let $v;e\in W$ be a flux dependent on ${v}_{1};{e}_{1},\dots ,{v}_{n};{e}_{n}\in W$, that is, in particular, $v={\sum}_{1\le i\le k}{n}_{i}{v}_{i}$ for some ${n}_{i}>0$. Since (C-Dep) removes one flux and possibly modifies some kinetics, there is a flux $v;{e}^{\prime}\in W$ in ${W}^{\prime}$ that either depends on ${v}_{1};{e}_{1}^{\prime},\dots ,{v}_{n};{e}_{n}^{\prime}\in W$ or on ${v}_{0};{e}_{0}^{\prime},{v}_{1};{e}_{1}^{\prime},\dots ,{v}_{n};{e}_{n}^{\prime}\in W$ where ${v}_{0};{e}_{0}^{\prime}$ is the flux removed by (C-Dep). The latter case is not possible, since it would imply that $v={\sum}_{1\le i\le k}{n}_{i}{v}_{i}={n}_{0}{v}_{0}+{\sum}_{1\le i\le k}{n}_{i}{v}_{i}$ for some ${n}_{0}>0$ and unary vector ${v}_{0}$. Thus, we conclude that $v;{e}^{\prime}\in W$ depends on ${v}_{1};{e}_{1}^{\prime},\dots ,{v}_{n};{e}_{n}^{\prime}\in W$ that, by induction hypothesis, satisfies the conditions of Point 3.

- (1)
- Let $v;e$ be in W. Either there is a corresponding flux $v;{e}^{\prime}\in {W}^{\prime}$, and we conclude directly by induction hypothesis, or $v;e$ is the result of merging some ${v}_{p};{e}_{p}\in {W}^{\prime}$ that produces X and some ${v}_{c};{e}_{c}\in {W}^{\prime}$ that consume it. In this case, by induction hypothesis, there are some corresponding flux-paths $\tilde{{v}_{p}}$ and $\tilde{{v}_{c}}$. The concatenation $\tilde{v}=\tilde{{v}_{p}}\tilde{{v}_{c}}$ of these paths is a flux-path. Indeed,
- the production of $\tilde{{v}_{p}}$ coincides with the consumption of $\tilde{{v}_{c}}$ because there is an intermediate species X such that $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{c}})=\{X\}$, $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{\tilde{{v}_{p}}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p}})$ and $Con{s}_{\mathcal{I}}({\mathbf{r}}_{\tilde{{v}_{c}}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{c}})$.
- if $\tilde{v}$ is non-circular, for any intermediate species Y such that $\tilde{v}(Y)>0$, either $Y=X$ and $Y\in \mathit{Specs}({W}_{0})\backslash \mathit{Specs}(W)$, or, $\tilde{{v}_{p}}(Y)>0$ or $\tilde{{v}_{c}}(Y)>0$ and by induction hypothesis, $Y\in \mathit{Specs}({W}_{0})\backslash \mathit{Specs}({W}^{\prime})$ that is $Y\in \mathit{Specs}({W}_{0})\backslash \mathit{Specs}(W)$.
- if $\tilde{v}$ is circular, there exists an intermediate species Y which is both consumed by ${v}_{p}$ and produced by ${v}_{c}$. The flux-path $\tilde{{v}_{c}}$ cannot be circular, as this would imply $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{c}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{c}})=\varnothing \ne X$, and similarly for $\tilde{{v}_{p}}$. By definition of non-circular flux-path, X and Y are the only two species in $\tilde{{v}_{c}}$ and $\tilde{{v}_{p}}$ such that $X\in \mathit{Specs}({W}^{\prime})$ and $Y\in \mathit{Specs}({W}^{\prime})$. Thus, Y is the only non-eliminated intermediate species in $\tilde{v}$ w.r.t. W, meaning again that $\tilde{v}$ is indeed a circular flux-path.

Moreover, $\tilde{v}$ trivially corresponds to v. We prove with Point 3 that if $\tilde{v}$ is non-dependent, then $\tilde{v}(X)\le 1$ for any X. - (2)
- Let $\tilde{v}$ be a flux-path for W such that for any Y, $\tilde{v}(Y)\le 1$. Let X be the intermediate species removed by (C-Inter), in particular $\tilde{v}(X)\le 1$, hence either $\tilde{v}(X)=0$ or $\tilde{v}(X)=1$. Since $\mathit{Specs}(W)=\mathit{Specs}({W}^{\prime})\backslash \{X\}$, if $\tilde{v}(X)=0$, then $\tilde{v}$ is also a flux-path for ${W}^{\prime}$, so by induction there is a corresponding flux $v;{e}^{\prime}\in {W}^{\prime}$. Since $\tilde{v}(X)=0$, we still have a flux $v;e\in W$, that corresponds to $\tilde{v}$. If $\tilde{v}(X)=1$ then we can decompose $\tilde{v}$ into $\tilde{{v}_{p}}$ producing X and $\tilde{{v}_{c}}$ consuming X such that $\tilde{v}=\tilde{{v}_{p}}\tilde{{v}_{c}}$. $\tilde{{v}_{p}}$ and $\tilde{{v}_{c}}$ can not be circular, as this would imply that X is both consumed and produced by $\tilde{{v}_{p}}$ and by $\tilde{{v}_{c}}$, contradicting the fact that $\tilde{v}(X)=1$. Therefore $\tilde{{v}_{p}}(X)=\tilde{{v}_{c}}(X)=0$. Again, because $\mathit{Specs}(W)=\mathit{Specs}({W}^{\prime})\backslash \{X\}$ and X is the species removed by (C-Inter) and $\tilde{v}$ is a flux-path, $\tilde{{v}_{p}}$ and $\tilde{{v}_{c}}$ are (non-circular) flux-paths for ${W}^{\prime}$. We can then apply the induction hypothesis and infer that there are some corresponding fluxes ${v}_{p},{v}_{c}\in {W}^{\prime}$, the first one that produces X and the second one that consumes it. Consequently, there is a flux $v;e\in W$ that is the merging of ${v}_{p}$ and ${v}_{c}$, and that corresponds to $\tilde{v}$.
- (3)
- Let $v;e\in W$ be a flux that depends on ${v}_{1};{e}_{1},\dots ,{v}_{k};{e}_{k}\in W$ and X be the intermediate species removed by (C-Inter). We distinguish two cases: either (case 1) $v;e$ is the simplification of some flux $v;{e}^{\prime}$ (meaning that v does neither produce nor consume X) or (case 2) it results from merging fluxes that produce and consume X.(Case 1) By induction hypothesis and Point 1, there exists a flux-path $\tilde{v}$ corresponding to v for ${W}^{\prime}$. We have $\tilde{v}(X)=0$ since X has not been removed. Suppose that there is $i\in \{1,\dots ,k\}$ such that ${v}_{i}$ is the merging, by (C-Inter), of fluxes that produce and a consume X. In this case, for any $\tilde{{v}_{i}}$ corresponding to ${v}_{i}$, $\tilde{{v}_{i}}(X)>0$ and, by the Lemma A1, we would have that $\tilde{v}(X)>0$, which contradicts $\tilde{v}(X)=0$. Therefore, none of the ${v}_{i}$s are the merging of other fluxes by (C-Inter), therefore, there are ${v}_{1};{e}_{1}^{\prime},\dots ,{v}_{k};{e}_{k}^{\prime}\in {W}^{\prime}$ such that $v;{e}^{\prime}$ depends on those fluxes. Since, by induction hypothesis, Point 3 is satisfied for $v;{e}^{\prime}$ in ${W}^{\prime}$, it is also satisfied for $v;e$ in W.(Case 2) Let ${v}_{p};{e}_{p}\in {W}^{\prime}$ be a flux that produces X, and ${v}_{c};{e}_{c}\in {W}^{\prime}$ a flux that consumes it and $v;e\in W$ their merging. Let $\tilde{{v}_{p}}$ and $\tilde{{v}_{c}}$ be flux-paths in ${W}^{\prime}$ for, respectively, ${v}_{p}$ and ${v}_{c}$ (such flux paths exist by induction hypothesis). Any corresponding path $\tilde{v}$ is then the concatenation of corresponding paths $\tilde{{v}_{p}}$ and $\tilde{{v}_{c}}$.We first prove that, if either ${v}_{p};{e}_{p}$ or ${v}_{c};{e}_{c}$ is dependent in ${W}^{\prime}$, then $v;e$ is also dependent in W. By induction, if ${v}_{p};{e}_{p}$ depends on ${v}_{1}^{\prime};{e}_{1}^{\prime},\dots ,{v}_{\ell}^{\prime};{e}_{\ell}^{\prime}$, there exists a unique i such that $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{i}^{\prime}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p}})$, $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{i}^{\prime}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p}})=\{X\}$, and, for any $j\ne i$, $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{j}^{\prime}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{j}^{\prime}})=\varnothing $. Then, there is a flux ${v}_{i}^{\u2033};{e}_{i}^{\u2033}$ in W that is the merging of ${v}_{i}^{\prime}$ and ${v}_{c}$, and there are fluxes ${v}_{j}^{\prime};{e}_{j}^{\prime}$ in W for $j\ne i$. Then $v;e$ depends on ${v}_{i}^{\u2033};{e}_{i}^{\u2033}$ and the ${v}_{j}^{\prime};{e}_{j}^{\prime}$.If both ${v}_{p};{e}_{p}$ and ${v}_{c};{e}_{c}$ are not dependent, by induction hypothesis, for any $Y\ne X$, in the corresponding flux-path, we have $\tilde{{v}_{p}}(Y)\le 1$ and $\tilde{{v}_{c}}(Y)\le 1$. If $\tilde{{v}_{p}}(Y)+\tilde{{v}_{c}}(Y)\le 1$, then $\tilde{v}(Y)\le 1$. We also have $\tilde{v}(X)=1$ (indeed, X is the removed species, so $\tilde{{v}_{p}}(X)=\tilde{{v}_{c}}(X)=0$ and the X produced by $\tilde{{v}_{p}}$ is merged with the X consumed by $\tilde{{v}_{c}}$ in $\tilde{v}$). Therefore, in this case Point 1 is satisfied. If there is a Y such that $\tilde{{v}_{p}}(Y)=\tilde{{v}_{c}}(Y)=1$, then Y occurs twice in $\tilde{v}$, and there is an intermediate species Z (that can possibly be Y if no other intermediate species occurs more than once between both occurrences of Y) such that $\tilde{v}(Z)=2$ and such that there is a circular flux-path $\tilde{{v}_{cyc}}$, subpath of $\tilde{v}$ that begins and end with Z:Then using Point 2, there is a corresponding flux ${v}_{cyc}\in W$, with $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{cyc}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{cyc}})$. We can repeat the same operation on the remaining path $\tilde{{v}_{1}}\tilde{{v}_{2}}$, and obtain at each step a new (circular) flux. We stop when we obtain a remaining path $\tilde{{v}_{rem}}$, such that for any Y, we have $\tilde{{v}_{rem}}(Y)\le 1$. Then there is a corresponding flux ${v}_{rem}$ with $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{rem}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{v})$, and $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{rem}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{v})$. Then v is dependent on ${v}_{rem}$ and the set of circular fluxes we obtained in this process. Therefore, in this case Point 3 is satisfied.Now assume that ${v}_{p};{e}_{p}$ and ${v}_{c};{e}_{c}$ are dependent, and so that $v;e$ is dependent too. By induction and using Point 3, ${v}_{p};{e}_{p}$ depends on a flux ${v}_{p,i};{e}_{p,i}$ with $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p,i}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p}})$ and $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p,i}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p}})$, and on other fluxes such that $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p,j}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{p}})=\varnothing $, and similarly for ${v}_{c};{e}_{c}$. Then any ${v}_{p,j};{e}_{p,j}$ and any ${v}_{c,j};{e}_{c,j}$ is still in W, while ${v}_{p,i};{e}_{p,i}$ is merged with ${v}_{c,i};{e}_{c,i}$, forming a new flux ${v}_{i};{e}_{i}$. Then $v;e$ depends on ${v}_{i};{e}_{i}$, the ${v}_{p,j};{e}_{p,j}$ and the ${v}_{c,j};{e}_{c,j}$. Since $Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{i}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{c,i}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{c}})=Pro{d}_{\mathcal{I}}({\mathbf{r}}_{v})$, and similarly $Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{i}})=Con{s}_{\mathcal{I}}({\mathbf{r}}_{v})$, Point 3 is satisfied.

## Appendix C. Proofs of the Full Confluence of the Simplification

**Lemma**

**A3.**

**Proof.**

- the combination of ${v}_{d}$ and the consuming fluxes: $\{{v}_{d}\diamond {v}_{cons};{e}_{d}{e}_{cons}/\mathit{cons}\mid {v}_{cons};{e}_{cons}\in {V}_{cons}\}$,
- the combination of ${r}_{i}$ and the consuming reactions: $\{{v}_{i}\diamond {v}_{cons};{e}_{i}{e}_{cons}/\mathit{cons}\mid {v}_{cons};{e}_{cons}\in {V}_{cons}\}$,
- the other combined fluxes: $\{{v}_{prod}\diamond {v}_{cons};{e}_{prod}{e}_{cons}/\mathit{cons}\mid {v}_{prod};{e}_{prod}\in {V}_{prod},{v}_{cons};{e}_{cons}\in {V}_{cons}\}$,
- the remaining fluxes not combined: ${\{{v}_{j};{e}_{j}[X:=X(0)]\}}_{j\ne i}$,
- the other fluxes that are not in ${V}_{prod}$, ${V}_{cons}$, ${v}_{j}$, where we substitute X by $X(0)$.

- $\{{v}_{i}\diamond {v}_{cons};{e}_{i}{e}_{cons}/\mathit{cons}+{e}_{d}{e}_{cons}/\mathit{cons}\}$,
- $\{{v}_{prod}\diamond {v}_{cons};{e}_{prod}{e}_{cons}/\mathit{cons}\}$,
- $\{{v}_{j};{e}_{j})[X:=X(0)]+\sum _{\begin{array}{c}{e}_{cons}\end{array}}{a}_{i}{e}_{d}{e}_{cons}/\mathit{cons}\}$,
- the other fluxes that are not in ${V}_{prod}$, ${V}_{cons}$, ${v}_{j}$, where we substitute X by $X(0)$.

- ${v}_{i};{e}_{i}+{e}_{d}$
- ${\{{v}_{j};{e}_{j}+{a}_{j}{e}_{d}\}}_{j\ne i}$
- ${V}_{cons}\backslash \{{v}_{d},{v}_{i}\}$,
- ${V}_{prod}$,
- the other fluxes that are not in ${V}_{prod}$, ${V}_{cons}$, ${v}_{j}$.

- $\{{v}_{i}\diamond {v}_{cons};({e}_{i}+{e}_{d}){e}_{cons}/\mathit{cons}\}$,
- $\{{v}_{prod}\diamond {v}_{cons};{e}_{prod}{e}_{cons}/\mathit{cons}\}$,
- $\{{v}_{j};({e}_{j}+{a}_{j}{e}_{d})[X:=X(0)]\}$,
- the other fluxes that are not in ${V}_{prod}$, ${V}_{cons}$, ${v}_{j}$, where we substitute X by $X(0)$.

**Lemma**

**A4.**

**Proof.**

- ${V}_{X}=\{{v}_{X};{e}_{X}\mid X\in Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{X}}),Y\notin {v}_{X}\}$, the fluxes producing X without Y,
- ${V}_{{X}^{\prime}}=\{{v}_{{X}^{\prime}};X{e}_{{X}^{\prime}}\mid X\in Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{{X}^{\prime}}}),Y\notin {v}_{{X}^{\prime}}\}$, the fluxes consuming X without Y,
- ${V}_{mod(X)}=\{{v}_{mod(X)};{e}_{mod(X)}\mid X\notin Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{mod(X)}})\cup Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{mod(X)}}),X\in \mathit{Vars}({e}_{mod(X)}),Y\notin {v}_{mod(X)}\}$, the fluxes with modifier X and without Y,
- ${V}_{Y}=\{{v}_{Y};{e}_{Y}\mid Y\in Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{Y}}),X\notin {v}_{Y}\}$, the fluxes producing Y without X,
- ${V}_{{Y}^{\prime}}=\{{v}_{Y};Y{e}_{{Y}^{\prime}}\mid Y\in Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{Y}}),X\notin {v}_{Y}\}$, the fluxes consuming Y without X,
- ${V}_{mod(Y)}=\{{v}_{mod(Y)};{e}_{mod(Y)}\mid Y\notin Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{mod(Y)}})\cup Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{mod(Y)}}),Y\in \mathit{Vars}({e}_{mod(Y)}),X\notin {v}_{mod(Y)}\}$, the fluxes with modifier Y and without X,
- ${V}_{X{Y}^{\prime}}=\{{v}_{X{Y}^{\prime}};Y{e}_{X{Y}^{\prime}}\mid X\in Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{X{Y}^{\prime}}}),Y\in Con{s}_{\mathcal{I}}({\mathbf{r}}_{{r}_{X{Y}^{\prime}}})\}$, the fluxes producing X and consuming Y,
- ${V}_{{X}^{\prime}Y}=\{{v}_{{X}^{\prime}Y};X{e}_{{X}^{\prime}Y}\mid Y\in Pro{d}_{\mathcal{I}}({\mathbf{r}}_{{v}_{{X}^{\prime}Y}}),X\in Con{s}_{\mathcal{I}}({\mathbf{r}}_{{v}_{{X}^{\prime}Y}})\}$, the fluxes producing Y and consuming X,
- ${V}_{mod(XY)}=\{{v}_{mod(XY)};{e}_{mod(XY)}\mid X,Y\notin {v}_{mod(XY)}\}$, the fluxes with modifier X and Y.

- $T}_{X}=\sum _{\begin{array}{c}{V}_{X}\end{array}}{e}_{X$ $T}_{{X}^{\prime}}=\sum _{\begin{array}{c}{V}_{{X}^{\prime}}\end{array}}{e}_{{X}^{\prime}$
- $T}_{Y}=\sum _{\begin{array}{c}{V}_{Y}\end{array}}{e}_{Y$ $T}_{{Y}^{\prime}}=\sum _{\begin{array}{c}{V}_{{Y}^{\prime}}\end{array}}{e}_{{Y}^{\prime}$
- $T}_{{X}^{\prime}Y}=\sum _{\begin{array}{c}{V}_{{X}^{\prime}Y}\end{array}}{e}_{{X}^{\prime}Y$ $T}_{X{Y}^{\prime}}=\sum _{\begin{array}{c}{V}_{X{Y}^{\prime}}\end{array}}{e}_{X{Y}^{\prime}$

- ${V}_{X}\diamond {V}_{{X}^{\prime}}=\{{v}_{X}\diamond {v}_{{X}^{\prime}};{e}_{X}{e}_{{X}^{\prime}}/({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})\}$,
- ${V}_{X}\diamond {V}_{{X}^{\prime}Y}=\{{v}_{X}\diamond {v}_{{X}^{\prime}Y};{e}_{X}{e}_{{X}^{\prime}Y}/({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})\}$,
- ${V}_{X{Y}^{\prime}}\diamond {V}_{{X}^{\prime}}=\{{v}_{X{Y}^{\prime}}\diamond {v}_{{X}^{\prime}};{e}_{X{Y}^{\prime}}{e}_{{X}^{\prime}}Y/({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})\}$,
- ${V}_{X{Y}^{\prime}}\diamond {V}_{{X}^{\prime}Y}=\{{v}_{X{Y}^{\prime}}\diamond {v}_{{X}^{\prime}Y};{e}_{X{Y}^{\prime}}{e}_{{X}^{\prime}Y}Y/({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})\}$.

- ${V}_{mod(X)}^{\prime}=\{{v}_{mod(X)};{e}_{mod(X)}[X:=X(0)]\}$,
- ${V}_{mod(XY)}^{\prime}=\{{v}_{mod(XY)};{e}_{mod(XY)}[X:=X(0)]\}$.

- ${V}_{Y}=\{{v}_{Y};{e}_{Y}\}$,
- ${V}_{{Y}^{\prime}}=\{{v}_{Y};Y{e}_{{Y}^{\prime}}\}$,
- ${V}_{mod(Y)}=\{{v}_{mod(Y)};{e}_{mod(Y)}\}$.

- ${V}_{X{Y}^{\prime}}\diamond {V}_{{X}^{\prime}Y}=\{{v}_{X{Y}^{\prime}}\diamond {v}_{{X}^{\prime}Y};{e}_{X{Y}^{\prime}}{e}_{{X}^{\prime}Y}Y(0)/{T}_{{X}^{\prime}Y}\}$.
- ${V}_{mod(X)}^{\prime}=\{{v}_{mod(X)};{e}_{mod(X)}[X:=X(0)]\}$,
- ${V}_{mod(Y)}=\{{v}_{mod(Y)};{e}_{mod(Y)}[Y:=Y(0)]\}$,
- ${V}_{mod(XY)}^{\prime}=\{{v}_{mod(XY)};{e}_{mod(XY)}[{x}_{X}:=X(0)][Y:=Y(0)]\}$.

- $({V}_{X}\diamond {V}_{{X}^{\prime}Y})\diamond ({V}_{X{Y}^{\prime}}\diamond {V}_{{X}^{\prime}})=\{{v}_{X}\diamond {v}_{{X}^{\prime}Y}\diamond {v}_{{X}^{\prime}}\diamond {v}_{X{Y}^{\prime}};{\displaystyle \frac{{e}_{X}{e}_{{X}^{\prime}}{e}_{X{Y}^{\prime}}{e}_{{X}^{\prime}Y}}{({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})T}}\}$,
- $({V}_{X}\diamond {V}_{{X}^{\prime}Y})\diamond {V}_{{Y}^{\prime}}=\{{v}_{X}\diamond {v}_{{X}^{\prime}Y}\diamond {v}_{{Y}^{\prime}};{\displaystyle \frac{{e}_{X}{e}_{{Y}^{\prime}}{e}_{{X}^{\prime}Y}}{T}}\}$,
- ${V}_{Y}\diamond ({V}_{X{Y}^{\prime}}\diamond {V}_{{X}^{\prime}})=\{{v}_{Y}\diamond {v}_{X{Y}^{\prime}}\diamond {v}_{{X}^{\prime}};{\displaystyle \frac{{e}_{Y}{e}_{X{Y}^{\prime}}{e}_{{X}^{\prime}}}{T}}\}$,
- ${V}_{Y}\diamond {V}_{{Y}^{\prime}}=\{{v}_{Y}\diamond {v}_{{Y}^{\prime}};{\displaystyle \frac{{e}_{Y}{e}_{{Y}^{\prime}}({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})}{T}}\}$.

- ${({V}_{X{Y}^{\prime}}\diamond {V}_{{X}^{\prime}Y})}^{\prime}=\{{v}_{X{Y}^{\prime}}\diamond {v}_{{X}^{\prime}Y};{\displaystyle \frac{{T}_{{X}^{\prime}}{T}_{Y}+{T}_{Y}{T}_{{X}^{\prime}Y}+{T}_{X}{T}_{{X}^{\prime}Y}}{T({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})}}{e}_{X{Y}^{\prime}}{e}_{{X}^{\prime}Y}\}$,
- ${V}_{mod(X)}^{\u2033}=\{{v}_{mod(X)};{e}_{mod(X)}[X:={\displaystyle \frac{{T}_{X}{T}_{{Y}^{\prime}}+{T}_{X}{T}_{X{Y}^{\prime}}+{T}_{Y}{T}_{X{Y}^{\prime}}}{T}}]\}$,
- ${V}_{mod(XY)}^{\u2033}=\{{v}_{mod(XY)};{e}_{mod(XY)}[X:={\displaystyle \frac{{T}_{X}{T}_{{Y}^{\prime}}+{T}_{X}{T}_{X{Y}^{\prime}}+{T}_{Y}{T}_{X{Y}^{\prime}}}{T}}][Y:={\displaystyle \frac{{T}_{Y}{T}_{{X}^{\prime}}+{T}_{Y}{T}_{{X}^{\prime}Y}+{T}_{X}{T}_{{X}^{\prime}Y}}{T}}]\}$,
- ${V}_{mod(Y)}^{\prime}=\{{v}_{mod(Y)};{e}_{mod(Y)}[{x}_{Y}:={\displaystyle \frac{{T}_{Y}{T}_{{X}^{\prime}}+{T}_{Y}{T}_{{X}^{\prime}Y}+{T}_{X}{T}_{{X}^{\prime}Y}}{T}}]\}$.

- ${R}_{X}\diamond {R}_{{X}^{\prime}}=\{vec({r}_{X})+vec({r}_{{X}^{\prime}});{e}_{X}{e}_{{X}^{\prime}}/({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})\}$

- ${V}_{X}\diamond {V}_{{X}^{\prime}}=\{{v}_{X}\diamond {v}_{{X}^{\prime}};{\displaystyle \frac{{e}_{X}{e}_{{X}^{\prime}}({T}_{{Y}^{\prime}}+{T}_{X{Y}^{\prime}})}{T}}\}$,
- ${V}_{Y}\diamond {V}_{{Y}^{\prime}}=\{{v}_{Y}\diamond {v}_{{Y}^{\prime}};{\displaystyle \frac{{e}_{Y}{e}_{{Y}^{\prime}}({T}_{{X}^{\prime}}+{T}_{{X}^{\prime}Y})}{T}}\}$,
- $({V}_{X}\diamond {V}_{{X}^{\prime}Y})\diamond {V}_{{Y}^{\prime}}=\{{v}_{X}\diamond {v}_{{X}^{\prime}Y}\diamond {v}_{{Y}^{\prime}};{\displaystyle \frac{{e}_{X}{e}_{{Y}^{\prime}}{e}_{{X}^{\prime}Y}}{T}}\}$,
- ${V}_{Y}\diamond ({V}_{X{Y}^{\prime}}\diamond {V}_{{X}^{\prime}})=\{{v}_{Y}\diamond {v}_{X{Y}^{\prime}}\diamond {v}_{{X}^{\prime}};{\displaystyle \frac{{e}_{Y}{e}_{X{Y}^{\prime}}{e}_{{X}^{\prime}}}{T}}\}$,
- ${V}_{mod(X)}^{\u2033}=\{{v}_{mod(X)};{e}_{mod(X)}[{x}_{X}:={\displaystyle \frac{{T}_{X}{T}_{{Y}^{\prime}}+{T}_{X}{T}_{X{Y}^{\prime}}+{T}_{Y}{T}_{X{Y}^{\prime}}}{T}}]\}$,
- ${V}_{mod(Y)}^{\prime}=\{{v}_{mod(Y)};{e}_{mod(Y)}[{x}_{Y}:={\displaystyle \frac{{T}_{Y}{T}_{{X}^{\prime}}+{T}_{Y}{T}_{{X}^{\prime}Y}+{T}_{X}{T}_{{X}^{\prime}Y}}{T}}]\}$,
- ${({V}_{X{Y}^{\prime}}\diamond {V}_{{X}^{\prime}Y})}^{\prime}=\{{v}_{X{Y}^{\prime}}\diamond {v}_{{X}^{\prime}Y};{\displaystyle \frac{({T}_{Y}+{T}_{X}){e}_{X{Y}^{\prime}}{e}_{{X}^{\prime}Y}}{T}}\}$,
- ${V}_{mod(XY)}^{\u2033}=\{{v}_{mod(XY)};{e}_{mod(XY)}[{x}_{X}:={\displaystyle \frac{{T}_{X}{T}_{{Y}^{\prime}}+{T}_{X}{T}_{X{Y}^{\prime}}+{T}_{Y}{T}_{X{Y}^{\prime}}}{T}}][{x}_{Y}:={\displaystyle \frac{{T}_{Y}{T}_{{X}^{\prime}}+{T}_{Y}{T}_{{X}^{\prime}Y}+{T}_{X}{T}_{{X}^{\prime}Y}}{T}}]\}$,

- the 2 first sets are symmetric to each other, in the sense that if we switch X and Y in the first set, we obtain the second one,
- the 2 following sets are symmetric to each other,
- the 2 following sets are symmetric to each other too,
- the following set is symmetric in X and Y,
- the last set is symmetric in X and Y (since the substitutions commute).

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**Figure 1.**Evolution of the concentration of ${S}$, ${E}$, $\textcolor[rgb]{}{C}$ and ${P}$ in enzymatic network with mass-action kinetics with the parameters ${k}_{1}={k}_{2}={k}_{3}=1$, the initial concentrations $E(0)=1$, $C(0)=2$, $S(0)=4$, $P(0)=0$, and in the context of the network with a reaction $\mathsf{\varnothing}\stackrel{{k}_{4}}{\to}S$ which produces S with constant speed ${k}_{4}=2$, and a reaction $P\stackrel{{k}_{5}P}{\to}\mathsf{\varnothing}$ which degrades P with parameter ${k}_{5}=0.2$.

**Figure 5.**The elementary modes of the reaction network in Figure 4.

**Figure 6.**Simplification of reaction networks without kinetics with respect to a set $\mathcal{I}$ of intermediate species.

**Figure 7.**Elimination of intermediates X and Y in reaction network $N$ in both possible orders, leading to two different final results ${N}_{XY}$ and ${N}_{YX}$.

**Figure 9.**Simplifying flux networks for an initial n-tuple of reactions $\mathbf{r}$ and a set of intermediate species $\mathcal{I}$.

**Figure 10.**Elimination of intermediate species from flux networks in different orders is not confluent without factorization.

**Figure 11.**Expressions where $A\in \mathit{Spec},k\in \mathit{Param},c\in \mathbb{R},\text{}\mathrm{and}\text{}n\in \mathbb{N}$.

**Figure 16.**Simplification rules for n-ary constrained flux networks, with $\mathcal{I}$ the set of intermediate species and $\mathbf{r}$ the n-tuple of initial reactions.

**Figure 17.**Reaction networks for the Michaelis-Menten example. $MMne{t}_{E}$ and $MMne{t}_{C}$ are obtained from the initial network $MMnet$ after removing E and C respectively. $MMne{t}_{CE}$ is obtained after removing both C and then E in this order. $MMne{t}_{EC}$ is obtained by inverting the order of elimination.

**Figure 22.**Network W and its simplifications. (

**top left**) Network W. (

**top right**) Network ${W}_{XYZ}$ after eliminating X, Y and Z (in this order). (

**bottom left**) Network ${W}_{XZY}$ after eliminating X, Z and Y. (

**bottom right**) Network ${W}_{XYZd}$ after eliminating X, Y, Z, and the dependent reaction. The new parameter is $K={k}_{2}{k}_{3}+{k}_{3}{k}_{4}+{k}_{4}{k}_{5}$.

**Figure 25.**Simplified networks from W. Both networks have the same structure, and the kinetic expressions are defined in the table. The network ${W}_{1}$ is obtained by removing in order $S{\mathit{4}}_{n}$, $S{\mathit{24}}_{n}$, $S{\mathit{24}}_{c}$, $S{\mathit{4}}_{c}$ and the dependent fluxes. The network ${W}_{2}$ is obtained by removing in order $S{\mathit{4}}_{n}$, $S{\mathit{24}}_{n}$, $S{\mathit{4}}_{c}$, $S{\mathit{24}}_{c}$ and the dependent fluxes.

**Figure 26.**In red the elementary mode ${{v}}_{{r}{e}{d}}$, in blue ${{v}}_{{b}{l}{u}{e}}$, in green ${{v}}_{{g}{r}{e}{e}{n}}$, and in magenta ${\textcolor[rgb]{}{v}}_{\textcolor[rgb]{}{m}\textcolor[rgb]{}{a}\textcolor[rgb]{}{g}\textcolor[rgb]{}{e}\textcolor[rgb]{}{n}\textcolor[rgb]{}{t}\textcolor[rgb]{}{a}}$.

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**MDPI and ACS Style**

Madelaine, G.; Tonello, E.; Lhoussaine, C.; Niehren, J.
Simpliﬁcation of Reaction Networks, Conﬂuence and Elementary Modes. *Computation* **2017**, *5*, 14.
https://doi.org/10.3390/computation5010014

**AMA Style**

Madelaine G, Tonello E, Lhoussaine C, Niehren J.
Simpliﬁcation of Reaction Networks, Conﬂuence and Elementary Modes. *Computation*. 2017; 5(1):14.
https://doi.org/10.3390/computation5010014

**Chicago/Turabian Style**

Madelaine, Guillaume, Elisa Tonello, Cédric Lhoussaine, and Joachim Niehren.
2017. "Simpliﬁcation of Reaction Networks, Conﬂuence and Elementary Modes" *Computation* 5, no. 1: 14.
https://doi.org/10.3390/computation5010014