Translated Chemical Reaction Networks

Abstract

Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analyzed via a variety of techniques, including stoichiometric network analysis, deficiency theory, and algebraic techniques (e.g., Gröbner bases). In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a network’s capacity to permit a particular class of steady states, called toric steady states, to topological properties of a generalized network called a translated chemical reaction network. These networks share their reaction vectors with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature.

Appendices

Appendix A: Deficiency Result

Lemma 5

The deficiency δ=n−ℓ−s of a chemical reaction network $\mathcal{N}$, where n is the number of stoichiometrically distinct complexes, ℓ is the number of linkage classes, and $s =\operatorname{dim}(S)$, also satisfies $\delta= \operatorname{dim}(\operatorname{ker}(Y) \cap \operatorname{Im}(I_{a}))$.

Proof

It follows from basic dimensional considerations that

$$\operatorname{dim}\bigl(\operatorname{ker}(Y \; I_a)\bigr) = \operatorname{dim}\bigl( \operatorname{ker} (I_a)\bigr) + \operatorname{dim}\bigl(\operatorname{ker}(Y) \cap \operatorname{Im}(I_a)\bigr). $$

From the rank–nullity theorem we have

$$\operatorname{dim}\bigl(\operatorname{ker}(Y \; I_a)\bigr) = r - \operatorname{dim}\bigl( \operatorname{Im}(Y \; I_a)\bigr) = r-s. $$

The rank of I _a corresponds to the number of complexes minus the number of linkage classes, so that $\operatorname{dim}(\operatorname{Im}(I_{a})) = n- \ell$. It follows that

$$\operatorname{dim}\bigl(\operatorname{ker}(I_a)\bigr) = r - (n - \ell) = r + \ell- n. $$

It follows that

$$\begin{aligned} \delta =& \operatorname{dim}\bigl(\operatorname{ker}(Y) \cap\operatorname{Im}(I_a)\bigr) = \operatorname{dim}\bigl(\operatorname{ker}(Y \; I_a)\bigr) - \operatorname{dim}\bigl( \operatorname{ker} (I_a)\bigr)\\ =& (r-s) - (r+\ell-n) = n - \ell- s, \end{aligned}$$

and we are done. □

Appendix B: Kernel of A _k

In this appendix, we present a more detailed characterization of ker(A _k ) for a mass action system $\mathcal{M} = (\mathcal{S},\mathcal {C},\mathcal{R},k)$.

Consider a weakly reversible chemical reaction network $\mathcal{N} =(\mathcal{S},\mathcal{C},\mathcal{R})$ and let $\mathcal{L}_{k}$, k=1,…,ℓ, denote the network’s linkage classes. Define a subgraph $\mathcal{T} \subseteq\mathcal{R}$ to be a spanning i-tree if $\mathcal{T}$ spans all of the complexes in some linkage class $\mathcal{L}_{k}$, contains no cycles, and has the unique sink $i \in\mathcal{C}$. Let $\mathcal{T}_{i}$ denote the set of all spanning i-trees for i=1,…,n. We define the following network constants.

Definition 13

Consider a weakly reversible chemical reaction network $\mathcal{N} = (\mathcal{S},\mathcal{C},\mathcal{R})$ with reaction weights k _j , j=1,…,r. Then the tree constant for i=1,…,n is given by

$$ K_i = \sum_{\mathcal{T} \in\mathcal{T}_i} \prod _{j \in\mathcal{T}_i} k_j. $$

(31)

Remark 12

To compute the tree constants K _i , we restrict ourselves to the linkage class containing the complex $i \in\mathcal {C}$. We then determine all of the spanning trees which contain this complex as the unique sink, multiply across all the weighted edges in each tree, and then sum over all such trees. The terms K _i can also be computed by computing specific minors of the kinetic matrix A _k restricted to the support of the linkage classes (Proposition 3, Craciun et al. 2009). Note that the term “tree constant” is our own.

The following result characterizes $\operatorname{ker}(A_{k})$ in terms of the tree constants (31). This result appears in various forms within the chemical reaction network literature. A basic form, just concerned with the signs of the individual components, can be found in Feinberg (1979) (Proposition 4.1) and Gatermann and Huber (2002) (Theorem 3.1). A more specific result can be obtained by the Matrix-Tree Theorem (Stanley 1999). This form is explicitly connected with the reaction graph of a chemical reaction network in Craciun et al. (2009) (Corollary 4). A direct argument is also contained in Sect. 3.4 of Johnston (2011). We defer to these references for the proof.

Theorem 6

Let $\mathcal{N} = (\mathcal{S},\mathcal{C},\mathcal{R})$ denote a weakly reversible chemical reaction network. Let K _i denote the tree constants (31) corresponding to i=1,…,n. Then

$$\emph{ker} (A_k) = \emph{span} \{ \mathbf{K}_1, \mathbf{K}_2, \ldots, \mathbf{K}_{\ell} \}, $$

where K _j =([K _j ]₁,[K _j ]₂,…,[K _j ] _n ) has entries

$$[K_j]_i = \left \{ \begin{array}{l@{\quad}l} K_i, & \mathit{if}\ i \in\mathcal {L}_j, \\ 0 & \mathit{otherwise}. \end{array} \right . $$

Remark 13

This theorem may be extended to networks which are not weakly reversible by considering the terminal strongly linked components of a chemical reaction network. As all the relevant networks considered in this paper are weakly reversible, however, Theorem 6 will suffice for our purposes here.

Appendix C: Proof of Lemma 4

Proof

Consider an improper translation $\tilde{\mathcal{N}} = (\tilde{\mathcal {S}},\tilde{\mathcal{C}},\tilde{\mathcal{C}}_{K},\tilde{\mathcal{R}})$ of a chemical reaction network $\mathcal{N} = (\mathcal{S},\mathcal {C},\mathcal{R})$ which is resolvable. Let $\mathcal{M} = (\mathcal {S},\mathcal{C},\mathcal{R},k)$ be a mass action system corresponding to $\mathcal{N}$ and $\tilde{\mathcal{M}} = (\tilde{\mathcal{S}},\tilde {\mathcal{C}},\tilde{\mathcal{C}}_{K},\tilde{\mathcal{R}},\tilde{k})$ be an improperly translated mass action system corresponding to $\tilde {\mathcal{N}}$ and defined by Definition 11. We will write the steady state condition for $\mathcal{M}$ as

$$ Y \; I_a \; I_k \; \varPsi(\mathbf{x}) = 0 $$

(32)

and the steady state condition for $\tilde{\mathcal{M}}$ as

$$ \tilde{Y} \; \tilde{I}_a \; \tilde{I}_{\tilde{k}} \; \tilde{\varPsi }_K(\mathbf{x}) = 0. $$

(33)

Since $\tilde{\mathcal{N}}$ is improper, the vector $\tilde{\varPsi }_{K}(\mathbf{x})$ contains a subset of the monomials in Ψ(x) by property 3 of Definition 6. Consequently, to relate (32) and (33), we need to remove explicit dependence on the monomials in Ψ(x) corresponding to complexes not in $\tilde{\mathcal{C}}_{K}$. We will accomplish this by rewriting the unused monomials in Ψ(x) in terms of the monomial in $\tilde{\varPsi}(\mathbf{x})$ corresponding to the kinetically relevant complex, and absorbing the required adjustment factor into the matrix I _k . (This will produce a state-dependent matrix $\tilde {I}_{\tilde{k}(\mathbf{x})}$. We will resolve the state-dependency at a later stage.)

Since $\tilde{\mathcal{N}}$ is resolvable, it follows that it is weakly reversible and $\tilde{S}_{I} \subseteq\tilde{S}_{K}$. Consequently, by Lemma 3, it follows that, for every $i \in\mathcal{R}_{I}$, there are constants c _ij and pairs $p_{j}, q_{j} \in\tilde{\mathcal {CR}}$, $j =1, \ldots, \tilde{s}$, such that

$$ \mathbf{x}^{y_{\rho(i)}} = \Biggl[\prod _{j=1}^{\tilde{s}} \biggl( \frac {\mathbf{x}^{y_{h_3(p_{j})}}}{\mathbf{x}^{y_{h_3(q_{j})}}} \biggr)^{c_{ij}} \Biggr] \; \mathbf{x}^{y_{\rho(i)_K}}. $$

(34)

We now introduce

$$ k'_i(\mathbf{x}) = \left\{ \begin{array}{l@{\quad}l} k_i, & \mbox{for}\ i \notin\mathcal {R}_I, \\ {}\bigl[\prod_{j=1}^{\tilde{s}} \bigl( \frac{\mathbf {x}^{y_{h_3(p_{j})}}}{\mathbf{x}^{y_{h_3(q_{j})}}} \bigr)^{c_{ij}} \bigr] k_i, & \mbox{for}\ i \in\mathcal{R}_I \end{array} \right. $$

(35)

and define

$$ \tilde{k}_{j}(\mathbf{x}) = \sum _{\{ i | h_1(i)=j \}} k'_i(\mathbf{x}) $$

(36)

for $j=1, \ldots, \tilde{n}$. This gives rise to the state dependent kinetic matrix $\tilde{I}_{\tilde{k}(\mathbf{x})} \in\mathbb {R}_{\geq0}^{\tilde{r} \times\tilde{n}}$ with entries $[\tilde {I}_{\tilde{k}(\mathbf{x})}]_{ij} = \tilde{k}_{i}(\mathbf{x})$ if $\tilde {\rho}(i)=j$ and $[\tilde{I}_{\tilde{k}(\mathbf{x})}]_{ij} = 0$ otherwise.

We now will prove that $Y \; I_{a} \; I_{k} \; \varPsi(\mathbf{x}) = \tilde{Y} \; \tilde{I}_{a} \; \tilde{I}_{\tilde{k}(\mathbf{x})} \; \tilde{\varPsi }_{K}(\mathbf{x})$ by showing that, for all $j=1, \ldots, \tilde{r}$,

$$[\tilde{\varGamma}]_{\cdot,j} \cdot\tilde{R}_j(\mathbf{x}) = \sum_{\{ i | h_1(i)=j \}} [\varGamma]_{\cdot,i} \cdot R_i(\mathbf{x}), $$

where Γ:=Y I _a , $\tilde{\varGamma} := \tilde{Y} \; \tilde {I}_{a}$, R(x):=I _k Ψ(x), and $\tilde {R}(\mathbf{x}) := \tilde{I}_{\tilde{k}(\mathbf{x})} \; \tilde{\varPsi }_{K}(\mathbf{x})$. We first make several observations, listed in the order they will be used:

• By definition, [Γ]_⋅,i =y _ρ′(i)−y _ρ(i) and $[\tilde{\varGamma}]_{\cdot, j} = \tilde{y}_{\tilde{\rho}'(j)} - \tilde {y}_{\tilde{\rho}(j)}$.

• For h ₁(i)=j, we have $\tilde{\rho}(j) = h_{3}(\tilde{\rho}(h_{1}(i))) = h_{3}(h_{2}(\rho(i)))=\rho(i)_{K}$ (since $\tilde{\rho}(h_{1}(i)) = h_{2}(\rho (i))$ by property 2 of Definition 6) so that $[\tilde {\varPsi}_{K}(\mathbf{x})]_{\tilde{\rho}(j)} = \mathbf{x}^{y_{\tilde{\rho }(j)}} = \mathbf{x}^{y_{\rho(i)_{K}}}$.

• By the constructions (34) and (35), we have $k'_{i}(\mathbf {x})\:\mathbf{x}^{y_{\rho(i)_{K}}} = k_{i}\:\mathbf{x}^{y_{\rho(i)}} = R_{i}(\mathbf{x})$ for all i=1,…,r.

• For every i=1,…,r, we have $\tilde{y}_{\tilde{\rho }'(h_{1}(i))} - \tilde{y}_{\tilde{\rho}(h_{1}(i))} = y_{\rho'(i)} - y_{\rho (i)}$ by property 1 of Definition 6.

It follows that, for every $j=1, \ldots, \tilde{r}$, we have

$$\begin{aligned}{} [\tilde{\varGamma}]_{\cdot, j} \cdot \tilde{R}_j(\mathbf{x}) =& ( \tilde{y}_{\tilde{\rho}'(j)} - \tilde{y}_{\tilde{\rho}(j)}) \: \tilde{k}_j \: \mathbf{x}^{y_{\tilde{\rho}(j)}} = ( \tilde{y}_{\tilde {\rho}'(j)} - \tilde{y}_{\tilde{\rho}(j)}) \sum _{ \{ i | h_1(i)=j \}} k_i'(\mathbf{x}) \; \mathbf{x}^{y_{\rho(i)_K}} \\ =& \sum_{\{ i | h_1(i)=j \}} (y_{\rho'(i)}-y_{\rho(i)}) \: k_i \; \mathbf{x}^{y_{\rho (i)}} =\sum _{\{ i | h_1(i)=j \}} [\varGamma]_{\cdot,i} \cdot R_i( \mathbf{x}). \end{aligned}$$

It follows that we have

$$ Y \; I_a \; I_k \; \varPsi(\mathbf{x}) = \tilde{Y} \; \tilde{I}_a \; \tilde {I}_{\tilde{k}(\mathbf{x})} \; \tilde{ \varPsi}_K(\mathbf{x}) = \tilde{Y} \; \tilde{A}_{\tilde{k}(\mathbf{x})} \; \tilde{\varPsi}_K(\mathbf{x}), $$

(37)

where $\tilde{A}_{\tilde{k}(\mathbf{x})} := \tilde{I}_{a} \; \tilde{I}_{\tilde{k}(\mathbf{x})} \in\mathbb{R}_{>0}^{\tilde{n} \times \tilde{n}}$ is a state dependent kinetic matrix with positive off-diagonal entries corresponding to the structure of the translation $\tilde{\mathcal{N}}$ and rates given by (35) and (36).

Consider the reaction graph of the network $(\tilde{\mathcal{S}},\tilde {\mathcal{C}},\tilde{\mathcal{R}}, \tilde{k}(\mathbf{x}))$ with state dependent edge weights $\tilde{k}(\mathbf{x}) \in\mathbb{R}_{\geq 0}^{\tilde{r}}$ given by (35). In order to remove the state dependence in $\tilde{A}_{\tilde{k}(\mathbf{x})}$, we consider the system at steady state. Since $\tilde{\delta} = \operatorname{dim} ( \operatorname{ker}(\tilde{Y}) \cap\operatorname{Im}(\tilde{I}_{a})) = 0$, it follows that

$$ \tilde{Y} \; \tilde{A}_{\tilde{k}(\mathbf{x})} \; \tilde{ \varPsi}_K(\mathbf {x}) = 0 \quad \Longleftrightarrow\quad \tilde{A}_{\tilde {k}(\mathbf{x})} \; \tilde{\varPsi}_K(\mathbf{x}) = 0. $$

(38)

Now let $\tilde{K}_{j}(\mathbf{x})$, $j=1, \ldots, \tilde{n}$, denote the state dependent tree constants (31) of the reaction graph of $(\tilde{\mathcal{S}},\tilde{\mathcal{C}},\tilde{\mathcal {R}},\tilde{k}(\mathbf{x}))$. Since $\tilde{\mathcal{N}}$ is weakly reversible, by Theorem 6, we have that

$$\operatorname{ker} (\tilde{A}_{\tilde{k}(\mathbf{x})}) = \operatorname{span} \bigl\{ \tilde{ \mathbf{K}}_1(\mathbf{x}), \tilde{\mathbf{K}}_2( \mathbf{x}), \ldots, \tilde{\mathbf{K}}_{\tilde{\ell}}(\mathbf{x}) \bigr\} , $$

where $\tilde{\mathbf{K}}_{j}(\mathbf{x}) = ([\tilde{K}_{j}(\mathbf {x})]_{1},[\tilde{K}_{j}(\mathbf{x})]_{2},\ldots,[\tilde{K}_{j}(\mathbf {x})]_{\tilde{n}})$ has entries

$$\bigl[\tilde{K}_j(\mathbf{x})\bigr]_i = \left \{ \begin{array}{l@{\quad}l} \tilde{K}_i(\mathbf{x}) & \mbox{if } i \in\tilde{\mathcal{L}}_j, \\ 0 & \mbox{otherwise.} \end{array} \right . $$

It follows that, if $i, j \in\tilde{\mathcal{L}}_{k}$ for some $k=1, \ldots, \tilde{\ell}$, we have

$$ \frac{\mathbf{x}^{y_{h_3(i)}}}{\tilde{K}_{i}(\mathbf{x})} = \frac {\mathbf{x}^{y_{h_3(j)}}}{\tilde{K}_{j}(\mathbf{x})} \quad\Longrightarrow\quad \frac{\mathbf{x}^{y_{h_3(i)}}}{\mathbf {x}^{y_{h_3(j)}}} = \frac{\tilde{K}_{i}(\mathbf{x})}{\tilde {K}_{j}(\mathbf{x})}. $$

(39)

Now consider $i \in\mathcal{R}_{I}$ and let c _ij and p _j ,q _j , $j=1, \ldots, \tilde{s}$, denote the values guaranteed by Lemma 3. By (39), for every $i \in\mathcal{R}_{I}$, we have

$$ \prod_{j=1}^{\tilde{s}} \biggl( \frac{\mathbf{x}^{y_{h_3(p_j)}}}{\mathbf {x}^{y_{h_3(q_j)}}} \biggr)^{c_{ij}} = \prod_{j=1}^{\tilde{s}} \biggl( \frac{\tilde{K}_{p_j}(\mathbf{x})}{\tilde{K}_{q_j}(\mathbf{x})} \biggr)^{c_{ij}}. $$

(40)

It follows by (35) and the assumption that $\tilde{\mathcal{N}}$ is resolvable that (40) only depends on the state-independent rates $k'_{i}(\mathbf{x})=k_{i}$, $i \notin\mathcal{R}_{I}$. It follows that (40) may be written

$$ \prod_{j=1}^{\tilde{s}} \biggl( \frac{\mathbf{x}^{y_{h_3(p_j)}}}{\mathbf {x}^{y_{h_3(q_j)}}} \biggr)^{c_{ij}} = \prod_{j=1}^{\tilde{s}} \biggl( \frac{\tilde{K}_{p_j}}{\tilde{K}_{q_j}} \biggr)^{c_{ij}}, $$

(41)

where the tree constants $\tilde{K}_{i}$ are determined with respect to the reaction graph of $(\tilde{\mathcal{S}},\tilde{\mathcal{C}},\tilde {\mathcal{R}},\tilde{k})$ with the rate constants given by (35) and (36) for $k'_{i} = k_{i}$ for $i \notin\mathcal{R}_{I}$ and $k'_{i}$ arbitrary for $i \in\mathcal{R}_{I}$ (since the product (40) does not depend on these rates). We may now substitute (41) into (35) to get

$$ k'_i(\mathbf{x})=k'_i = \left \{ \begin{array}{l@{\quad}l} k_i, & \mbox{for}\ i \notin\mathcal {R}_I, \\ \bigl( \prod_{j=1}^{\tilde{s}} \bigl( \frac{\tilde {K}_{p_j}}{\tilde{K}_{q_j}} \bigr)^{c_{ij}} \bigr) k_i, & \mbox{for}\ i \in\mathcal{R}_I \end{array} \right . $$

(42)

and

$$ \tilde{k}_{j}(\mathbf{x})=\tilde{k}_j = \sum_{\{ i | h_1(i)=j \}} k'_i. $$

(43)

Notice that, while the tree constant pairs $\tilde{K}_{p_{i}}$ and $\tilde {K}_{q_{i}}$ depend upon a choice for the rate constants $k'_{i}$ for $i \in\mathcal{R}_{I}$, their ratios do not so that (42) has been defined consistently.

It follows from (18) that (42) and (43) correspond to the choice of rate constants for the improperly translated mass action system $\tilde{\mathcal{M}} = (\tilde{\mathcal{S}},\tilde{\mathcal{C}},\tilde{\mathcal{C}}_{K},\tilde {\mathcal{R}},\tilde{k})$ defined by Definition 11. Consequently, from (37) we have that

$$ Y \; I_a \; I_k \; \varPsi(\mathbf{x}) = 0 \quad \Longleftrightarrow\quad \tilde{Y} \; \tilde{I}_a \; \tilde{I}_{\tilde{k}} \; \tilde{\varPsi }_K(\mathbf{x}) =0 $$

(44)

so that the steady states of the system (3) governing $\mathcal {M}$ and the steady states of the system (6) governing $\tilde {\mathcal{M}}$ defined by Definition 11 coincide, and we are done. □

Appendix D: Proof of Theorem 5

Proof

Let $\mathcal{N} = (\mathcal{S},\mathcal{C},\mathcal{R})$ be a chemical reaction network and $\tilde{\mathcal{N}} = (\tilde{\mathcal{S}},\tilde{\mathcal{C}},\tilde{\mathcal{C}}_{K},\tilde {\mathcal{R}})$ be a weakly reversible translation of $\mathcal{N}$ which is either proper or improper and resolvable. Suppose $\mathcal{M} = (\mathcal{S},\mathcal{C},\mathcal{R},k)$ is a mass action system corresponding to $\mathcal{N}$. We define the translated mass action system $\tilde{\mathcal{M}} = (\tilde{\mathcal{S}},\tilde{\mathcal{C}},\tilde {\mathcal{C}}_{K},\tilde{\mathcal{R}},\tilde{k})$ according to Definition 8 if $\tilde{\mathcal{N}}$ is proper and by Definition 11 if $\tilde{\mathcal{N}}$ is resolvable and improper.

From either Lemma 2 and Lemma 4 we have that the steady state set of $\tilde{\mathcal{M}}$ corresponds to the steady state set of $\mathcal{M}$. Correspondingly, by either (11) or (44) we have that

$$Y \; I_a \; I_k \; \varPsi(\mathbf{x}) = 0 \quad\Longleftrightarrow\quad \tilde{Y} \; \tilde{A}_{\tilde{k}} \; \tilde{ \varPsi}_K(\mathbf{x}) = 0, $$

where $\tilde{A}_{\tilde{k}}:= \tilde{I}_{a} \; \tilde{I}_{\tilde{k}}$ and $\tilde{\varPsi}_{K}(\mathbf{x})$ has entries $[\tilde{\varPsi}_{K}(\mathbf {x})]_{j} = \mathbf{x}^{y_{h_{3}(j)}}$ for $j \in\tilde{\mathcal{CR}}$.

Since $\tilde{\delta}_{K} = 0$, we may conclude by Proposition 2.20 of (Müller and Regensburger 2012) that the translated mass action system $\tilde{\mathcal{M}}$ has a complex balanced steady state. That is to say, there is a point $\mathbf{a} \in\mathbb{R}_{>0}^{m}$ which satisfies

$$ \tilde{\varPsi}_{K}(\mathbf{a}) \in\operatorname{ker}( \tilde{A}_k). $$

(45)

Furthermore, since $\tilde{\delta} = \operatorname{dim}(\operatorname{ker}(\tilde{Y}) \cap \operatorname{Im}(\tilde{I}_{a}))=0$, we have from (38) that all steady states are complex balanced steady states. It follows from Proposition 2.21 of Müller and Regensburger (2012) that the set of such steady states may be parametrized by

$$E = \bigl\{ \mathbf{x} \in\mathbb{R}_{>0}^m \; | \; \ln( \mathbf{x}) - \ln(\mathbf{a}) \in\tilde{S}^{\perp}_K \bigr\} , $$

where $\tilde{S}_{K}$ is given by (10) of Lemma 1. This is sufficient to prove claim 3.

Now consider claims 1 and 2. Since $\tilde{\mathcal{N}}$ is weakly reversible it follows by Theorem 6 that

$$ \operatorname{ker} (\tilde{A}_k) = \operatorname{span} \{ \tilde{ \mathbf{K}}_1, \tilde{\mathbf{K}}_2, \ldots, \tilde{ \mathbf{K}}_{\tilde{\ell}} \}, $$

(46)

where $\tilde{\mathbf{K}}_{j} = ([\tilde{K}_{j}]_{1},[\tilde{K}_{j}]_{2},\ldots ,[\tilde{K}_{j}]_{\tilde{n}})$ has entries

$$ [\tilde{K}_j]_i = \left \{ \begin{array}{l@{\quad}l} \tilde{K}_i & \mbox{if}\ i \in \mathcal{L}_j, \\ 0 & \mbox{otherwise} \end{array} \right . $$

(47)

where $\tilde{K}_{i}$, $i =1, \ldots, \tilde{n}$, are the tree constants corresponding to the reaction graph $(\tilde{\mathcal{S}},\tilde {\mathcal{C}},\tilde{\mathcal{R}}, \tilde{k})$.

It follows from (45), (46), and (47) that, for every $i, j \in\tilde{\mathcal{L}}_{k}$ for some $k=1, \ldots, \tilde {\ell}$, the steady states $\mathbf{x} \in\mathbb{R}_{>0}^{m}$ satisfy

$$\frac{\mathbf{x}^{y_{h_3(i)}}}{\tilde{K}_{i}} = \frac{\mathbf {x}^{y_{h_3(j)}}}{\tilde{K}_{j}} \; \; \; \Longleftrightarrow\; \; \; \tilde{K}_{j} \mathbf{x}^{y_{h_3(i)}} - \tilde{K}_{i} \mathbf {x}^{y_{h_3(j)}} = 0. $$

Since this set corresponds to the steady states of $\mathcal{M}$ by either Lemma 2 or Lemma 4, we have shown that $\mathcal{M}$ has toric steady states generated by binomials of the form required by claim 2. Since the choice of rate constants in the definition of $\mathcal{M}$ was arbitrary, claim 1 follows and we are done. □