Network Translation and Steady-State Properties of Chemical Reaction Systems

Abstract

Network translation has recently been used to establish steady-state properties of mass action systems by corresponding the given system to a generalized one which is either dynamically or steady-state equivalent. In this work, we further use network translation to identify network structures which give rise to the well-studied property of absolute concentration robustness in the corresponding mass action systems. In addition to establishing the capacity for absolute concentration robustness, we show that network translation can often provide a method for deriving the steady-state value of the robust species. We furthermore present a MILP algorithm for the identification of translated chemical reaction networks that improves on previous approaches, allowing for easier application of the theory.

Appendices

Appendix A: Proof of Theorem 2

Note that, under the hypotheses of Theorem 2, we can define a generalized mass action system $\tilde{{\mathcal {M}}}$ on $\tilde{{\mathcal {N}}}$ that is steady state equivalent to ${\mathcal {M}}$ by either Lemma 2 or Lemma 4 of Johnston (2014). Consequently, robust ratios and ACR hold in ${\mathcal {M}}$ if and only if they hold in $\tilde{{\mathcal {M}}}$. Theorem 2 is therefore a direct corollary of the following result pertaining to generalized mass action systems. As the proof is nearly identical to that of Lemma S3.20 in the Supplemental Material of Shinar and Feinberg (2010), we do not restate the background details and results.

Theorem 4

Consider a generalized chemical reaction network ${\mathcal {N}}$, with an associated generalized mass action system ${\mathcal {M}}$ that admits a positive steady state.

• 1. If ${\mathcal {N}}$ has a stoichiometric deficiency of zero and is weakly reversible, then ${\mathcal {M}}$ has a robust ratio in each pair of kinetic complexes y and $y'$ belonging to a common linkage class in ${\mathcal {N}}$.

• 2. If ${\mathcal {N}}$ has a stoichiometric deficiency of one, then ${\mathcal {M}}$ has a robust ratio in every pair of nonterminal kinetic complexes y and $y'$ in ${\mathcal {N}}$.

Proof

(1) Consider the system of ODEs (10) corresponding to the generalized mass action system ${\mathcal {M}}$ and suppose that the network is weakly reversible and has a stoichiometric deficiency of zero. Since the network has a stoichiometric deficiency of zero, it follows that $\hbox {ker}({\mathbf {A}}_\kappa )=\hbox {ker}({\mathbf {Y}}{\mathbf {A}}_\kappa )$. Since the network is weakly reversible, $\hbox {ker}({\mathbf {Y}}{\mathbf {A}}_\kappa )$ may be characterized by Theorem 6 of Johnston (2014). Furthermore, since the linkage classes provide a partition of $\hbox {ker}({\mathbf {Y}}{\mathbf {A}}_\kappa )$, by Theorem 3.3 of Millán et al. (2012) we have that the steady states are generated by binomials of the form

$$\begin{aligned} K(y'){\mathbf {x}}^{h(y)} - K(y){\mathbf {x}}^{h(y')}, \ y, y' \in {\mathcal {L}}_\theta , \ \theta = 1, \ldots , l, \end{aligned}$$

where the K(y) are the tree constants associated to the kinetic matrix of the generalized mass action system. As a consequence, at each positive steady state ${\bar{{\mathbf {x}}}} \in {\mathbb {R}}_{> 0}^n$, and for each pair of complexes y, $y'$ in the same linkage class, we have

$$\begin{aligned} \frac{{\bar{{\mathbf {x}}}}^{h(y)}}{{\bar{{\mathbf {x}}}}^{h(y')}} = \frac{K(y)}{K(y')} \end{aligned}$$

which concludes the proof.

(2): Consider the system of ODEs (10) corresponding to the generalized mass action system ${\mathcal {M}}$ and suppose that the network has a stoichiometric deficiency of one. From Lemma S3.19 of Shinar and Feinberg (2010), ker$({\mathbf {A}}_{\kappa })$ admits a basis $\{b^1, \ldots , b^t\} \subset {\mathbb {R}}_{\ge 0}^c$ with support on the t terminal strong linkage classes of $({\mathcal {S}}, {\mathcal {C}}, {\mathcal {R}})$. Lemma S3.20 in the same paper gives $\dim (\ker ({\mathbf {Y}}{\mathbf {A}}_\kappa )) \le 1 + t$.

Now consider an arbitrary positive equilibrium ${\bar{{\mathbf {x}}}} \in {\mathbb {R}}_{> 0}^n$. It follows that $\varPsi _K({\bar{{\mathbf {x}}}}) \in \text{ ker }({\mathbf {Y}}{\mathbf {A}}_\kappa ) \cap {\mathbb {R}}_{> 0}^c$. Since no vectors in $\{ b^1, \ldots b^t \}$ have support on the nonterminal complexes, we have that $\dim (\ker ({\mathbf {Y}}{\mathbf {A}}_\kappa )) = 1 + t$ so that there is a basis of $\text{ ker }({\mathbf {Y}}{\mathbf {A}}_\kappa )$ given by $\{ b^0, b^1, \ldots , b^t\}$ where only $b^0 \in {\mathbb {R}}_{\ge 0}^c$ has support on the nonterminal complexes. It follows that

$$\begin{aligned} \varPsi _K({\bar{{\mathbf {x}}}}) = \lambda _0 b^0 + \sum _{\theta = 1}^t\lambda _\nu b^\theta . \end{aligned}$$

For any nonterminal complexes $y^i, y^j \in {\mathcal {C}}$, we have that ${\bar{{\mathbf {x}}}}^{h(y^i)} = \lambda _0 b^0_i$ and ${\bar{{\mathbf {x}}}}^{h(y^j)} = \lambda _0 b^0_j$ so that, after solving for $\lambda _0$ and rewriting, we have

$$\begin{aligned} \frac{{\bar{{\mathbf {x}}}}^{h(y^i)}}{{\bar{{\mathbf {x}}}}^{h(y^j)}} = \frac{b^0_i}{b^0_j}. \end{aligned}$$

Since the positive equilibrium ${\bar{{\mathbf {x}}}}$ and nonterminal complexes $y^i$ and $y^j$ were chosen arbitrarily, we are done. $\square$

Appendix B: Proof of Theorem 3

Proof of Theorem 3

Assume that $\tilde{{\mathcal {N}}}$ is a translation of ${\mathcal {N}}$ which has a stoichiometric deficiency of zero and is weakly reversible, and that $\tilde{{\mathcal {M}}}$ is the associated translated mass action system with rate constant vector ${\tilde{k}} \in {\mathbb {R}}_{> 0}^{{\tilde{m}}}$ defined according to (16). We start by rewriting the dynamical equations (5) associated with ${\mathcal {M}}$ in a manner which emphasizes the kinetic complexes of the translation, then considering the steady equations of the mass action system ${\mathcal {M}}$ when rewritten in this manner.

By considering reactions $r \in {\mathcal {R}}_I$ and $r \notin {\mathcal {R}}_I$ separately, the system (5) associated with ${\mathcal {M}}$ may be written

$$\begin{aligned} \begin{aligned} \frac{\hbox {d}{\mathbf {x}}}{\hbox {d}t}&= \sum _{r: y \rightarrow y' \in {\mathcal {R}}} k_r (y' - y) \; {\mathbf {x}}^y,\\&= \sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}} \\ r \notin {\mathcal {R}}_I \end{array}} k_r (y' - y) \; {\mathbf {x}}^{y} + \sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}} \\ r\in {\mathcal {R}}_I \end{array}} k_r (y' - y) \; {\mathbf {x}}^{y},\\&= \sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}} \\ r\notin {\mathcal {R}}_I \end{array}} k_r (y' - y) \; {\mathbf {x}}^{y} + \sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}} \\ r\in {\mathcal {R}}_I \end{array}} k_r({\mathbf {x}}) (y' - y) \; {\mathbf {x}}^{h(g(y))}. \end{aligned} \end{aligned}$$

(32)

where $k_r({\mathbf {x}}) := \left( {\mathbf {x}}^{y}/{\mathbf {x}}^{h(g(y))} \right) k_r$ for $r: y \rightarrow y' \in {\mathcal {R}}_I$. This formulation emphasizes the dependence of (5) on the kinetic complexes of the translation, at the expense of having state-dependent rates $k_r({\mathbf {x}})$ for $r \in {\mathcal {R}}_I$.

Denote by ${\tilde{{\mathbf {Y}}}}$ the matrix corresponding to the stoichiometric complexes of the translated reaction network, and by ${\tilde{\varPsi }}_K({\mathbf {x}}) = ({\mathbf {x}}^{h({\tilde{y}}^1)}, \ldots , {\mathbf {x}}^{h({\tilde{y}}^{{\tilde{c}}})})$ the vector of monomials of the kinetic complexes. Denote by ${\tilde{{\mathbf {A}}}}_\kappa ({\mathbf {x}})$ the kinetic matrix associated to the mass action system on the translated chemical reaction network with kinetic parameters defined by

$$\begin{aligned} {\tilde{k}}_{{\tilde{r}}}({\mathbf {x}}) = \sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}}\setminus {\mathcal {R}}_I\\ f(r) ={\tilde{r}} \end{array}}k_{r} + \sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}}_I\\ f(r) = {\tilde{r}} \end{array}} k_r({\mathbf {x}}). \end{aligned}$$

(33)

We can then rewrite the dynamical equations (32) associated with ${\mathcal {M}}$ as

$$\begin{aligned} \frac{\hbox {d}{\mathbf {x}}}{\hbox {d}t} = {\tilde{{\mathbf {Y}}}} {\tilde{{\mathbf {A}}}}_{\kappa }({\mathbf {x}}) {\tilde{\varPsi }}_K({\mathbf {x}}). \end{aligned}$$

(34)

On the other hand, the dynamical equations associated with $\tilde{{\mathcal {M}}}$ can be written as

$$\begin{aligned} \frac{\hbox {d}{\mathbf {x}}}{\hbox {d}t} = {\tilde{{\mathbf {Y}}}} {\tilde{{\mathbf {A}}}}_{\kappa } {\tilde{\varPsi }}_K({\mathbf {x}}), \end{aligned}$$

(35)

where the kinetic matrix ${\tilde{{\mathbf {A}}}}_{\kappa }$ is defined using the kinetic parameters in (16). Hence the two systems (34) and (35) differ only in the rates corresponding to reactions $r \in {\mathcal {R}}_I$. Recall that we write ${\tilde{K}}(y)$, ${\tilde{y}} \in \tilde{{\mathcal {C}}}$, for the tree constants associated to ${\tilde{{\mathbf {A}}}}_\kappa$.

Now suppose that ${\bar{{\mathbf {x}}}} \in {\mathbb {R}}_{> 0}^n$ is a positive steady state for ${\mathcal {M}}$. From (32) and (33), and noting that the stoichiometric deficiency of the translated reaction network is zero by assumption, we have that

$$\begin{aligned} \begin{aligned} {\tilde{{\mathbf {Y}}}}{\tilde{{\mathbf {A}}}}_\kappa ({\bar{{\mathbf {x}}}}){\tilde{\varPsi }}_K({\bar{{\mathbf {x}}}}) = \mathbf{0 } \; \Longleftrightarrow \; {\tilde{{\mathbf {A}}}}_{\kappa }({\bar{{\mathbf {x}}}}){\tilde{\varPsi }}_K({\bar{{\mathbf {x}}}}) = {\mathbf {0}}. \end{aligned} \end{aligned}$$

(36)

We define $K({\tilde{y}}^i;{\mathbf {x}})$ to be the tree constant (21) associated with the stoichiometric complex ${\tilde{y}}^i$ in the translated reaction network with the state-dependent rates (33). By Lemma S3.19 of Shinar and Feinberg (2010), $\text{ ker }({\tilde{{\mathbf {A}}}}_\kappa ({\bar{{\mathbf {x}}}}))$ admits a basis with support on the terminal strong linkage classes of the network, which coincide with the linkage classes of the translated reaction network by weak reversibility. By Theorem 6 of Johnston (2014), this basis has the form $\{ \tilde{{\mathbf {K}}}^1({\bar{{\mathbf {x}}}}), \ldots , \tilde{{\mathbf {K}}}^{{\tilde{l}}}({\bar{{\mathbf {x}}}}) \} \subset {\mathbb {R}}_{\ge 0}^{{\tilde{c}}}$ where $\tilde{{\mathcal {L}}}_\theta$, $\theta = 1, \ldots , {\tilde{l}},$ are the linkage classes of $\tilde{{\mathcal {N}}}$, and $\tilde{{\mathbf {K}}}^\theta _j({\bar{{\mathbf {x}}}}) = {\tilde{K}}(y^j;{\bar{{\mathbf {x}}}})$ if $y^j \in \tilde{{\mathcal {L}}}_\theta$ and $\tilde{{\mathbf {K}}}^\theta _j ({\bar{{\mathbf {x}}}})= 0$ otherwise.

Proof (1) Consider complexes y, $y'$ and ${\tilde{y}}$, ${\tilde{y}}'$ as in point (1). Since the complexes ${\tilde{y}}$, ${\tilde{y}}'$ belong to the same linkage class and $y=h({\tilde{y}})$, $y'=h({\tilde{y}}')$, we have

$$\begin{aligned} \frac{{\bar{{\mathbf {x}}}}^y}{{\bar{{\mathbf {x}}}}^{y'}} = \frac{{\bar{{\mathbf {x}}}}^{h({\tilde{y}})}}{{\bar{{\mathbf {x}}}}^{h({\tilde{y}}')}} = \frac{{\tilde{K}}({\tilde{y}};{\bar{{\mathbf {x}}}})}{{\tilde{K}}({\tilde{y}}';{\bar{{\mathbf {x}}}})}. \end{aligned}$$

(37)

Notice that the tree constants of $K(y^i;{\bar{{\mathbf {x}}}})$ of ${\tilde{{\mathbf {A}}}}_\kappa ({\bar{{\mathbf {x}}}})$ and $K(y^i)$ of ${\tilde{{\mathbf {A}}}}_\kappa$ are given by the same polynomials in the kinetic parameters. Consequently, their values differ only in the corresponding entries $k_r({\bar{{\mathbf {x}}}})$ and $k_r^*$ for $r \in {\mathcal {R}}_I$.

(a):

If the translation is proper then ${\mathcal {R}}_I = \emptyset$ and the kinetic matrices ${\tilde{{\mathbf {A}}}}_\kappa ({\mathbf {x}})$ and ${\tilde{{\mathbf {A}}}}_\kappa$ coincide. It follows trivially that the ratio (37) does not depend upon ${\bar{{\mathbf {x}}}}$ and reduces to ${\tilde{K}}({\tilde{y}})/{\tilde{K}}({\tilde{y}}')$.

(b):

Suppose that the translation is improper and resolvable with the rate constants $k^*_r$, $r \in {\mathcal {R}}_I$, in (16) chosen such that ${\mathcal {M}}$ and $\tilde{{\mathcal {M}}}$ are steady-state equivalent. Consequently, we have that ${\tilde{{\mathbf {A}}}}_\kappa {\tilde{\varPsi }}_K({\bar{{\mathbf {x}}}})=0$. It follows that, for this choice of rate constants, the ratio (37) does not depend on ${\bar{{\mathbf {x}}}}$ and resolves to ${\tilde{K}}({\tilde{y}})/{\tilde{K}}({\tilde{y}}')$.

(b):

If the translation is improper, then the systems (34) and (35) do not necessarily admit the same positive steady states. However, if the ratio (37) does not depend on any rate constant $k_r$, $r \in {\mathcal {R}}_I$, then in particular it does not depend on ${\bar{{\mathbf {x}}}}$, and is a ratio of polynomials in the $k_r$, $r \notin {\mathcal {R}}_I$, that coincides with ${\tilde{K}}({\tilde{y}})/{\tilde{K}}({\tilde{y}}')$.

Proof (2) (a) Suppose the pair of complexes y, $y'$ are translated to the same complex, and

$$\begin{aligned} y - y' = \sum _{\theta =1}^{{\tilde{l}}} \mathop {\sum _{i,j=1}^{{\tilde{c}}}}_{{\tilde{y}}^i, {\tilde{y}}^j \in \tilde{{\mathcal {L}}}_\theta } c_{ij} (h({\tilde{y}}^{i}) - h({\tilde{y}}^{j})) \end{aligned}$$

(38)

for some constants $c_{ij} \in {\mathbb {R}}$. Then, for any positive steady state ${\bar{{\mathbf {x}}}} \in {\mathbb {R}}_{> 0}^n$ of ${\mathcal {M}}$, we can write

$$\begin{aligned} \frac{{\bar{{\mathbf {x}}}}^y}{{\bar{{\mathbf {x}}}}^{y'}} =\prod _{\theta =1}^{{\tilde{l}}} \mathop {\prod _{i,j=1}^{{\tilde{c}}}}_{y_i, y_j \in {\mathcal {L}}_\theta }\left( \frac{{\bar{{\mathbf {x}}}}^{h({\tilde{y}}^i)}}{{\bar{{\mathbf {x}}}}^{h({\tilde{y}}^j)}} \right) ^{c_{ij}}. \end{aligned}$$

As in part (1), since $\varPsi _K({\bar{{\mathbf {x}}}}) \in \text{ ker }({\tilde{{\mathbf {A}}}}_\kappa ({\bar{{\mathbf {x}}}}))$, we have that at steady state

$$\begin{aligned} \frac{{\bar{{\mathbf {x}}}}^{h({\tilde{y}}^i)}}{{\bar{{\mathbf {x}}}}^{h({\tilde{y}}^j)}} = \frac{{\tilde{K}}({\tilde{y}}^i;{\bar{{\mathbf {x}}}})}{{\tilde{K}}({\tilde{y}}^j;{\bar{{\mathbf {x}}}})} , \end{aligned}$$

and as a consequence

$$\begin{aligned} \frac{{\bar{{\mathbf {x}}}}^y}{{\bar{{\mathbf {x}}}}^{y'}} = \prod _{\theta =1}^{{\tilde{l}}} \mathop {\prod _{i,j=1}^{{\tilde{c}}}}_{y_i, y_j \in {\mathcal {L}}_\theta } \left( \frac{{\tilde{K}}({\tilde{y}}^i;{\bar{{\mathbf {x}}}})}{{\tilde{K}}({\tilde{y}}^j;{\bar{{\mathbf {x}}}})} \right) ^{c_{ij}}. \end{aligned}$$

(39)

By hypothesis, the kinetic adjustment factor ${\tilde{K}}(y,y')$ does not depend on any rate constant $k_r$ corresponding to a reaction $r \in {\mathcal {R}}_I$. Since the only state-dependent rates in (39) are those corresponding to reactions $r \in {\mathcal {R}}_I$, we have that the ratio (39) coincides with the kinetic adjustment factor ${\tilde{K}}(y,y')$ defined by 22. It follows that and y and $y'$ have a robust ratio in ${\mathcal {M}}$ and, specifically, that ${\mathbf {x}}^{y}/{\mathbf {x}}^{y'} = {\tilde{K}}(y,y')$ at every positive steady state ${\mathbf {x}} \in {\mathbb {R}}_{> 0}^n$.

(b) Now suppose that the conditions of 2 hold for all pairs of complexes $y, y'$ such that $g(y)=g(y')$. Using (38) with $y'=h(g(y))$, we can write

$$\begin{aligned} y - h(g(y)) = \sum _{\theta =1}^{{\tilde{l}}} \mathop {\sum _{i,j=1}^{{\tilde{c}}}}_{{\tilde{y}}^i, {\tilde{y}}^j \in \tilde{{\mathcal {L}}}_\theta }c_{ij} (h({\tilde{y}}^{i}) - h({\tilde{y}}^{j})). \end{aligned}$$

Let ${\bar{{\mathbf {x}}}} \in {\mathbb {R}}_{> 0}^n$ be a positive steady state of ${\mathcal {M}}$. From part (a), we have that ${\bar{{\mathbf {x}}}}^y/{\bar{{\mathbf {x}}}}^{h(g(y))} = {\tilde{K}}(y,h(g(y)))$. Furthermore, by (32), we have that

$$\begin{aligned} \begin{aligned} {\mathbf {0}} =&\sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}} \\ r\notin {\mathcal {R}}_I \end{array}} k_r (y' - y) \; {\bar{{\mathbf {x}}}}^{y} + \sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}} \\ r\in {\mathcal {R}}_I \end{array}} k_r({\bar{{\mathbf {x}}}}) (y' - y) \; {\bar{{\mathbf {x}}}}^{h(g(y))}\\ =&\sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}} \\ r\notin {\mathcal {R}}_I \end{array}} k_r (y' - y) \; {\bar{{\mathbf {x}}}}^{y} + \sum _{\begin{array}{c} r: y \rightarrow y' \in {\mathcal {R}} \\ r\in {\mathcal {R}}_I \end{array}} \left[ {\tilde{K}}(y,h(g(y)))k_r\right] (y' - y) \; {\bar{{\mathbf {x}}}}^{h(g(y))}. \end{aligned} \end{aligned}$$

(40)

It follows that ${\bar{{\mathbf {x}}}}$ is a steady state of $\tilde{{\mathcal {M}}}$. Conversely, if ${\bar{{\mathbf {x}}}} \in {\mathbb {R}}_{> 0}^n$ is a positive steady state of $\tilde{{\mathcal {M}}}$, since ${\bar{{\mathbf {x}}}}^{y}/{\bar{{\mathbf {x}}}}^{h(g(y))} = {\tilde{K}}(y,h(g(y)))$, from (40) we can conclude that ${\bar{{\mathbf {x}}}}$ is also a positive steady state for ${\mathcal {M}}$. It follows that ${\mathcal {M}}$ and $\tilde{{\mathcal {M}}}$ are steady-state equivalent, and consequently the translation is resolvable. $\square$

Appendix C: Details of the Mixed-Integer Linear Programming Code

In this “Appendix”, we describe in more detail the method sketched in Sect. 4. We first fix some notations. We define ${\mathbb {B}} = \{0,1\}$, and denote by ${\mathbf {I}}^+$ and ${\mathbf {I}}^-$ the positive and negative part of the incidence matrix, respectively, and set $\varvec{\Gamma }^+={\mathbf {Y}}{\mathbf {I}}^+$, $\varvec{\Gamma }^-={\mathbf {Y}}{\mathbf {I}}^-\in {\mathbb {N}}^{n \times m}$. In other words, the columns of $\varvec{\Gamma }^+$ and $\varvec{\Gamma }^-$ contain the stoichiometric coefficients of the product and reactant complexes of the network, respectively. Given $\alpha$ cycles and $\beta$ stoichiometric elementary modes, we denote by $I^h \subseteq \{1,\ldots ,m\}$, $h=1,\ldots ,\alpha$, the indices of reactions in the cyclic elementary modes, and by $J^h \subseteq \{1,\ldots ,m\}$, $h=1,\ldots ,\beta$, the indices of reactions in the stoichiometric elementary modes.

Translation complexes are well defined We define a matrix of decision variables $\varvec{\Upsilon }\in {\mathbb {R}}^{n \times m}$ where the ${ jth}$ column of $\varvec{\Upsilon }$ corresponds to the translation complex $\upsilon ^j$ of the reaction $r_j$, i.e., $\varvec{\Upsilon }_{\cdot j} = \upsilon ^j$. Since each complex in the original network can be translated to only one stoichiometric complex in the translated network, any two reactions $r_{j_1}$ and $r_{j_2}$ that have the same reactant need to be translated by the same translation complex:

$$\begin{aligned} \varvec{\Gamma }^-_{i j_1}=\varvec{\Gamma }^-_{i j_2} \text{ for } \text{ all } i=1,\ldots ,n \Rightarrow \varvec{\Upsilon }_{ij_1} = \varvec{\Upsilon }_{ij_2} \text{ for } \text{ all } i=1,\ldots ,n. \end{aligned}$$

(React)

We can optionally require that all the translation complexes are positive, by imposing

$$\begin{aligned} \varvec{\Upsilon }_{ij}\ge 0, i=1,\ldots ,n, j=1,\ldots ,m. \end{aligned}$$

(PosT)

If we do not impose that the translation complexes have positive coefficients, we need to guarantee that all the coefficients of the complexes resulting from the translations are positive. To this end, we require

$$\begin{aligned} \varvec{\Upsilon }_{ij}+\varvec{\Gamma }^-_{ij}\ge 0, \varvec{\Upsilon }_{ij}+\varvec{\Gamma }^+_{ij}\ge 0, i=1,\ldots ,n, j=1,\ldots ,m. \end{aligned}$$

(PosC)

Stoichiometric elementary modes translated to cycles Suppose now that the reactions $r_{j_1}, \ldots , r_{j_k}$ define a cycle. To ensure that these reactions form a cycle in the translated network as well, we impose

$$\begin{aligned} \varvec{\Upsilon }_{ij_1} = \varvec{\Upsilon }_{ij_2} = \cdots = \varvec{\Upsilon }_{ij_k}, \ i=1,\ldots ,n. \end{aligned}$$

(Cycle)

Consider now the stoichiometric elementary modes. Suppose that the reactions $r_{j_1}, \ldots , r_{j_k}$ define the ${ hth}$ stoichiometric elementary mode, and ${\hat{\upsilon }}^{j_1}, \ldots , {\hat{\upsilon }}^{j_k}$ are complexes such that the reactions $r_{j_1} + {\hat{\upsilon }}^{j_1}, \ldots , r_{j_k} + {\hat{\upsilon }}^{j_k}$ define a cycle by Lemma 2, for a fixed order of the reactions. We denote by ${\hat{\varvec{\Upsilon }}}^h \in {\mathbb {Z}}^{n \times |J^h|}$ the matrix with columns the complexes ${\hat{\upsilon }}^{j_1},\ldots ,{\hat{\upsilon }}^{j_k}$. Write $j'$ for the position of the index j in $J^h$, i.e., $j=J^h_{j'}$. Then to guarantee that $r_{j_1}, \ldots , r_{j_k}$ are translated to the identified cycle we impose

$$\begin{aligned} \varvec{\Upsilon }_{ij_1} - {\hat{\varvec{\Upsilon }}}^h_{ij'_1} = \varvec{\Upsilon }_{ij_2} - {\hat{\varvec{\Upsilon }}}^h_{ij'_2} = \cdots = \varvec{\Upsilon }_{ij_k} - {\hat{\varvec{\Upsilon }}}^h_{ij'_k}, \text{ for } h = 1 , \ldots , \beta , i = 1, \ldots , n. \end{aligned}$$

(Stoich)

For each stoichiometric elementary mode we have therefore a set of constraints of the form (Stoich). These constraints might not be satisfiable at the same time for all stoichiometric elementary modes; hence we want to maximize the number of stoichiometric elementary modes for which these constraints are verified. We do so by introducing additional variables.

Minimize stoichiometric modes in translated network For the h th stoichiometric elementary mode, we introduce a binary variable $\sigma _h$ that will be equal to 1 if the elementary mode is not translated to a cycle. The restrictions on $\sigma _h$ are obtained by imposing, for each pair of indices $j_1,j_2\in J^h$ and for each species $X_i$ the constraint

$$\begin{aligned} \sigma _h \ge \varepsilon (\varvec{\Upsilon }_{ij_1} - {\hat{\varvec{\Upsilon }}}^h_{ij'_1} - \varvec{\Upsilon }_{ij_2} + {\hat{\varvec{\Upsilon }}}^h_{ij'_2}). \end{aligned}$$

(Count)

Notice that (Count) reduces to (Stoich) if $\sigma _h = 0$. In order to maximize the number of stoichiometric modes which are translated to cycles, we introduce the following objective function:

$$\begin{aligned} \mathrm {minimize} \; \; \sum _{h=1}^{\beta } \sigma _h. \end{aligned}$$

(Obj)

Permutations of stoichiometric modes (optional) As observed in Remark 2, we can identify a possible cycle for each possible order of the reactions involved in the elementary mode. In general, the existence of a solution can depend on the choice of the order. We take the order of the reactions into account by introducing variables and constraints to keep track of the possible permutations of the reactions in the elementary mode. In addition, since the order of reactions in the elementary mode is not fixed, instead of calculating the translations that convert the elementary mode to a cycle, we consider the matrices ${\hat{\varvec{\Upsilon }}}^h\in {\mathbb {Z}}^{n \times |J^h|}$ as matrices of decision variables, and impose the constraint (Count) as done previously in the case of elementary modes with a fixed order.

To keep all the possible orders into account, for the h th stoichiometric elementary mode, with set of reaction indices $J^h$, we define $|J^h| \times |J^h|$ binary variables ${\mathbf {P}}^h \in {\mathbb {B}}^{|J^h| \times |J^h|}$, that will identify the position of each reaction in the possible orders: ${\mathbf {P}}^h_{tj}$ will be equal to 1 if and only if the j th reaction is in position t.

We have therefore the following constraints. To ensure that each reaction is assigned one and only one position, we impose that each row and column of ${\mathbf {P}}^h$ sums to 1:

$$\begin{aligned} \sum _{j=1}^k {\mathbf {P}}^h_{tj} = 1 \ \text {for all } t = 1, \ldots , |J^h|, \end{aligned}$$

(Perm1)

$$\begin{aligned} \sum _{t=1}^k {\mathbf {P}}^h_{tj} = 1 \ \text {for all } j = 1, \ldots , |J^h|. \end{aligned}$$

(Perm2)

Now, suppose that $j'_1$, $j'_2$, t and $t'$ are indices in $\{1,\ldots ,|J^h|\}$ such that $t'=t+1\ (\mathrm {mod}\, |J^h|)$. Write $j_1$, $j_2$ for $J^h_{j'_1}$ and $J^h_{j'_2}$, respectively. If ${\mathbf {P}}^h_{tj'_1}$ and ${\mathbf {P}}^h_{tj'_2}$ are both equal to 1, then reaction $r_{j_1}$ is followed by reaction $r_{j_2}$ in the cycle. If this is the case, we want to impose that the product of reaction $r_{j_1}$ is equal to the reactant of reaction $r_{j_2}$. For each pair of indices $J^h_{j'_1}$, $J^h_{j'_2}$ in the elementary mode we therefore add the following constraints, for each species $i=1,\ldots ,n$:

$$\begin{aligned}&-\frac{1}{\varepsilon } (2 - {\mathbf {P}}^h_{tj'_1} - {\mathbf {P}}^h_{t'j'_2}) \le \varvec{\Gamma }^+_{ij_1}-\varvec{\Gamma }^-_{ij_2} + {\hat{\varvec{\Upsilon }}}^h_{ij'_1}-{\hat{\varvec{\Upsilon }}}^h_{ij'_2}\le \frac{1}{\varepsilon } (2 - {\mathbf {P}}^h_{tj'_1} - {\mathbf {P}}^h_{t'j'_2}),\\&\quad j' = j+1\ (\mathrm {mod}\, |J^h|). \end{aligned}$$

(Perm3)

We can also fix the position of one reaction arbitrarily, and set

$$\begin{aligned} {\mathbf {P}}^h_{00}= 1 \ \text {for all } h = 1, \ldots , \beta . \\ \end{aligned}$$

(Perm4)

Proper translations (optional) We might sometimes be interested in identifying a proper translation with minimum deficiency. To impose that distinct complexes in the original network are translated to distinct complexes in the translated network, we introduce $n \times m \times m$ binary variables ${\mathbf {U}}_{ij_1j_2}$. The variable ${\mathbf {U}}_{ij_1j_2}$ encodes whether the reactant complexes of reactions $r_{j_1}$ and $r_{j_2}$ have been translated to the same complex with respect to species $X_i$, i.e.,

$$\begin{aligned} |\varvec{\Gamma }^-_{ij_1} + \varvec{\Upsilon }_{ij_1} - \varvec{\Gamma }^-_{ij_2} - \varvec{\Upsilon }_{ij_2}| = 0 \ \Rightarrow \ {\mathbf {U}}_{ij_1j_2} = 1. \end{aligned}$$

This is achieved by introducing additional $n \times m \times m$ auxiliary binary variables ${\mathbf {V}}_{ij_1j_2}$, and considering the constraints

$$\begin{aligned} \left\{ \begin{aligned} \varvec{\Gamma }^-_{ij_1} + \varvec{\Upsilon }_{ij_1} - \varvec{\Gamma }^-_{ij_2} - \varvec{\Upsilon }_{ij_2}&\ge \epsilon (1 - {\mathbf {U}}_{ij_1j_2}) - M {\mathbf {V}}_{ij_1j_2}, \\ \varvec{\Gamma }^-_{ij_1} + \varvec{\Upsilon }_{ij_1} - \varvec{\Gamma }^-_{ij_2} - \varvec{\Upsilon }_{ij_2}&\le - \epsilon (1 - {\mathbf {U}}_{ij_1j_2}) + M(1-{\mathbf {V}}_{ij_1j_2}),\end{aligned} \right. \end{aligned}$$

(Proper1)

as can be easily checked by considering the four possible cases. Finally, we have to impose that, if the reactants of reactions $r_{j_1}$ and $r_{j_2}$ differ, then their corresponding translated complexes differ in at least one species. Since the variables ${\mathbf {U}}_{ij_1j_2}$ count the number of matching species, we impose, for $j_1,j_2=1,\ldots ,m$

$$\begin{aligned} \varvec{\Gamma }^-_{ij_1} \ne \varvec{\Gamma }^-_{ij_2} \text { for any } i=1,\ldots ,n \ \Rightarrow \sum _{i=1}^{n} {\mathbf {U}}_{ij_1j_2} \le n - 1. \end{aligned}$$

(Proper2)

The variables ${\mathbf {U}}_{ij_1j_2}$ and ${\mathbf {V}}_{ij_1j_2}$ corresponding to reactions $r_{j_1}$ and $r_{j_2}$ with the same reactant can be omitted. The parameters, variables and constraints of the problem are summarized in the following tables.

Parameters
$n \in {\mathbb {N}}$	Number of species
$c \in {\mathbb {N}}$	Number of complexes
$m \in {\mathbb {N}}$	Number of reactions
$\varvec{\Gamma }\in {\mathbb {N}}^{n \times m}$	Stoichiometric matrix
$\varvec{\Gamma }^- \in {\mathbb {N}}^{n \times m}$	Stoichiometric coefficients of the reactants
$\varvec{\Gamma }^+ \in {\mathbb {N}}^{n \times m}$	Stoichiometric coefficients of the products
$\alpha \in {\mathbb {N}}$	Number of cyclic elementary modes
$\beta \in {\mathbb {N}}$	Number of stoichiometric elementary modes
$I^h \subseteq \{1,\ldots ,m\}$, $h=1,\ldots ,\alpha$	Indices of reactions in cycles
$J^h \subseteq \{1,\dots ,m\}$, $h=1,\dots ,\beta$	Indices of reactions in stoichiometric elementary modes
$M \in {\mathbb {R}}_{>0}$	$M \gg 1$
$\epsilon \in {\mathbb {R}}_{>0}$	$0 < \epsilon \ll 1$
Decision variables
$\varvec{\Upsilon }\in {\mathbb {R}}^{n \times m}$	Matrix of translation complexes: $\varvec{\Upsilon }_{ij} = \upsilon ^j_i$ is the stoichiometric coefficient of species $X_i$ in the translation of the reaction $r_j$
${\hat{\varvec{\Upsilon }}}^h \in {\mathbb {Z}}^{n \times |J^h|}$, $h=1,\ldots ,\beta$	Matrix of translation complexes which convert elementary mode h to cycle: ${\hat{\varvec{\Upsilon }}}^h_{ij} = {\hat{\upsilon }}^j_i$ is the stoichiometric coefficient of species $X_i$ in the translation of the j th reaction in the h th elementary mode to a cycle. ${\hat{\varvec{\Upsilon }}}^h$ is a matrix of variables if permutations of reactions are considered, and is otherwise calculated using Lemma 2
${\mathbf {P}}^h \in {\mathbb {B}}^{|J^h| \times |J^h|}$, $h=1,\ldots ,\beta$	Orders of reactions in elementary mode h: ${\mathbf {P}}^h_{tj}=1$ iff the j th reaction in the h th stoichiometric elementary mode is in position t
$\sigma \in {\mathbb {B}}^{\beta }$	Count the elementary modes that are translated to cycles: $\sigma _h=1$ iff the h th stoichiometric elementary mode is translated to a cycle
${\mathbf {U}}\in {\mathbb {B}}^{n\times m\times m}$	(Optional for proper) count the number of matching species in the translation of the reactants: ${\mathbf {U}}_{ij_1j_2}=1$ iff the translations of the reactants of reactions $r_{j_1}$ and $r_{j_2}$ are equal in species i
${\mathbf {V}}\in {\mathbb {B}}^{n\times m\times m}$	(Optional for proper) auxiliary variables
Objective
$\mathrm {minimize} \ \sum _{h=1}^{\beta }\sigma _{h}$	Minimize stoichiometric modes in translated network (Obj)

Constraints

$\begin{array}{ll} \left\{ \begin{array}{ll} \varvec{\Gamma }^-_{i j_1}=\varvec{\Gamma }^-_{i j_2}, \forall i=1,\ldots ,n \Rightarrow \varvec{\Upsilon }_{ij_1}=\varvec{\Upsilon }_{ij_2}, \forall i=1,\ldots ,n, &{}\quad \quad \quad \quad {\hbox {Reactions with the same reactant}}\\ &{} \quad \quad \quad \quad \,\,\,\, {\hbox {are translated by the same }}\\ &{} \quad \quad \quad \quad \,\,\,\, \hbox {complex } ({\mathbf{React}}) \\ j_1, j_2=1,\ldots ,m, j_1\ne j_2 &{} \end{array}\right. \\ \left\{ \begin{array}{ll} \varvec{\Upsilon }_{ij}\ge 0, i=1,\ldots ,n, j=1,\ldots ,m&{}\quad \quad \quad \quad \,\, \hbox {Translations are positive}\\ &{}\quad \quad \quad \quad \,\,\,\,\,\, \hbox {(optional) } ({\mathbf{PosT}})\\ \varvec{\Upsilon }_{ij}+\varvec{\Gamma }^-_{ij}\ge 0, \varvec{\Upsilon }_{ij}+\varvec{\Gamma }^+_{ij}\ge 0, i=1,\ldots ,n, j=1,\ldots ,m&{}\quad \quad \quad \quad \,\, \hbox {All resulting stoichiometric }\\ &{}\quad \quad \quad \quad \,\,\,\,\,\, \hbox {coefficients are positive } ({\mathbf{PosC}})\\ \end{array}\right. \\ \left\{ \begin{array}{ll} \varvec{\Upsilon }_{ij_1}=\varvec{\Upsilon }_{ij_2}, i=1,\ldots ,n, &{}\quad \\ j_1=I^h_{j}, j_2=I^h_{j+1}, j=1,\ldots ,|I^h|-1, &{}\quad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\,\hbox {Cycles are preserved }({\mathbf{Cycle}}) \\ h=1,\ldots , \alpha &{} \end{array}\right. \end{array}$

$\begin{array}{l} \left\{ \begin{array}{ll} \left\{ \begin{array}{l} \sigma _h \ge \varepsilon (\varvec{\Upsilon }_{ij_1} - {\hat{\varvec{\Upsilon }}}^h_{ij'_1} - \varvec{\Upsilon }_{ij_2} + {\hat{\varvec{\Upsilon }}}^h_{ij'_2}),\\ i=1,\ldots ,n, j_1=J_{j'_1}^h, j_2=J_{j'_2}^h,\\ j'_1=1,\ldots ,|J^h|, j'_2=1,\ldots ,|J^h|, j'_1\ne j'_2\\ \end{array}\right. &{}\quad \quad \quad \quad \qquad \sigma _h=1\hbox { if the }h{th} \hbox { stoichiometric }\\ &{} \quad \quad \quad \quad \qquad \,\,\,\hbox {elementary mode is not }\\ &{} \quad \quad \quad \quad \qquad \,\,\, \hbox {translated to a cycle }({\mathbf{Count}})\\ \\ \displaystyle {\sum _{j=1}^{|J^h|}}{\mathbf {P}}_{tj}^h=1, t=1,\ldots ,|J^h| &{}\quad \quad \quad \quad \quad \,\,\, \hbox { Each index in the order }\\ &{}\quad \quad \quad \quad \quad \,\,\,\,\,\,\,\, \hbox {corresponds to only one }\\ &{}\quad \quad \quad \quad \quad \,\,\,\,\,\,\,\, \hbox {reaction }({\mathbf{Perm1}})\\ \displaystyle {\sum _{t=1}^{|J^h|}}{\mathbf {P}}_{tj}^h=1, j=1,\ldots ,|J^h| &{}\quad \quad \quad \quad \quad \,\,\, \hbox { Each reaction is assigned only }\\ &{}\quad \quad \quad \quad \quad \,\,\,\,\,\,\,\, \hbox {one index in the order } ({\mathbf{Perm2}})\\ \left\{ \begin{array}{l} \varvec{\Gamma }^+_{ij_1}-\varvec{\Gamma }^-_{ij_2} + {\hat{\varvec{\Upsilon }}}^h_{ij'_1}-{\hat{\varvec{\Upsilon }}}^h_{ij'_2}\le \frac{1}{\varepsilon } (2 - {\mathbf {P}}^h_{tj'_1} - {\mathbf {P}}^h_{t'j'_2}),\\ \varvec{\Gamma }^+_{ij_1}-\varvec{\Gamma }^-_{ij_2} + {\hat{\varvec{\Upsilon }}}^h_{ij'_1}-{\hat{\varvec{\Upsilon }}}^h_{ij'_2}\ge -\frac{1}{\varepsilon } (2 - {\mathbf {P}}^h_{tj'_1} - {\mathbf {P}}^h_{t'j'_2}),\\ t=1,\ldots ,|J^h|,t' = t+1\ (\mathrm {mod}\, |J^h|),\\ i=1,\ldots ,n, j_1=J^h_{j'_1}, j_2=J^h_{j'_2},\\ j'_1=1,\ldots ,|J^h|, j'_2=1,\ldots ,|J^h|, j'_1\ne j'_2 \end{array}\right. &{}\quad \quad \quad \quad \quad \,\, \hbox { The product of the reaction with }\\ &{}\quad \quad \quad \quad \quad \,\,\,\,\,\,\, \, \hbox {index t is aligned with the}\\ &{}\quad \quad \quad \quad \quad \,\,\,\,\,\,\,\, \hbox {reactant of reaction with index }\\ &{}\quad \quad \quad \quad \quad \,\,\,\,\,\,\, \,t+1\ (\mathrm {mod}\, |J^h|)~ ({\mathbf{Perm3}})\\ {\mathbf {P}}^h_{00}=1 &{}\quad \quad \quad \quad \quad \,\,\, \hbox { First reaction is in first position }\\ &{}\quad \quad \quad \quad \quad \,\,\,\,\,\, \,\, ({\mathbf{Perm4}})\\ h=1,\ldots ,\beta &{} \end{array}\right. \end{array}$

$\begin{array}{l} \left\{ \begin{array}{ll} \left\{ \begin{array}{l} \varvec{\Gamma }^-_{ij_1} + \varvec{\Upsilon }_{ij_1} - \varvec{\Gamma }^-_{ij_2} - \varvec{\Upsilon }_{ij_2} \ge \epsilon (1 - {\mathbf {U}}_{ij_1j_2}) - M {\mathbf {V}}_{ij_1j_2},\\ \varvec{\Gamma }^-_{ij_1} + \varvec{\Upsilon }_{ij_1} - \varvec{\Gamma }^-_{ij_2} - \varvec{\Upsilon }_{ij_2} \le - \epsilon (1 - {\mathbf {U}}_{ij_1j_2}) + M(1-{\mathbf {V}}_{ij_1j_2}),\\ i=1,\ldots ,n \end{array}\right. &{} \hbox {Proper translation (optional) }\\ &{}\quad ({\mathbf{Proper1}})\\ \varvec{\Gamma }^-_{ij_1} \ne \varvec{\Gamma }^-_{ij_2} \text { for any } i=1,\ldots ,n \ \Rightarrow \sum _{i=1}^{n} {\mathbf {U}}_{ij_1j_2} \le n - 1 &{} ({\mathbf{Proper2}})\\ j_1,j_2=1,\ldots ,m. &{} \end{array}\right. \end{array}$