Positive Equilibria of
Hill-Type Kinetic Systems

We can view $\mathscr{C}$ as a subset of $\mathbb{R}^{m}_{\geq 0}$. The ordered pair $\left({{C_{i}},{C_{j}}}\right)$ corresponds to the familiar notation ${C_{i}}\to{C_{j}}$.

The reactant map $\rho$ induces a further useful mapping called reactions map. The reactions map $\rho^{\prime}$ associates the coordinate of reactant complexes to the coordinates of all reactions of which the complex is a reactant, i.e., $\rho^{\prime}:\mathbb{R}^{n}\to\mathbb{R}^{r}$ such that $f:\mathscr{C}\to\mathbb{R}$ mapped to $f\circ\rho$.

We also associate three important linear maps to a positive element $k\in\mathbb{R}^{r}$. The $k$-diagonal map ${\rm{diag}}(k)$ maps $\omega_{i}$ to $k_{i}\omega_{i}$ for $R_{i}\in\mathscr{R}$. The $k$-incidence map $I_{k}$ is defined as the composition ${\rm{diag}}(k)\circ\rho^{\prime}$. Lastly, the Laplacian map $A_{k}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is given by $A_{k}=I_{a}\circ I_{k}$.

CRNs can be seen as graphs. Complexes can be viewed as vertices and reactions as edges. If there is a path between two vertices $C_{i}$ and $C_{j}$, then they are said to be connected. If there is a directed path from vertex $C_{i}$ to vertex $C_{j}$ and vice versa, then they are said to be strongly connected. If any two vertices of a subgraph are (strongly) connected, then the subgraph is said to be a (strongly) connected component. The (strong) connected components are precisely the (strong) linkage classes of a chemical reaction network. The maximal strongly connected subgraphs where there are no edges from a complex in the subgraph to a complex outside the subgraph is said to be the terminal strong linkage classes. We denote the number of linkage classes, the number of strong linkage classes, and the number of terminal strong linkage classes by $l,sl,$ and $t$, respectively. A chemical reaction network is said to be weakly reversible if $sl=l$, and it is said to be $t$-minimal if $t=l$.

The ODE or dynamical system of a chemical kinetics system is $\dfrac{{dx}}{{dt}}=f\left(x\right)$. An equilibrium or steady state is a zero of $f$.

A chemical reaction network is said to admit multiple equilibria if there exist positive rate constants such that the ODE system admits more than one stoichiometrically compatible equilibria.

Analogously, the set of complex balanced equilibria [19] is given by

A positive vector $c\in\mathbb{R}^{m}$ is complex balanced if $K\left(c\right)$ is contained in $\ker{I_{a}}$, and a chemical kinetic system is complex balanced if it has a complex balanced equilibrium.

If the kinetic order matrix is the transpose of the molecularity matrix, then the system becomes the well-known mass action kinetics (MAK).

In [2], a discussion on several sets of kinetics of a network is given. They identified two main sets: the complex factorizable kinetics (CFK) and its complement, the non-complex factorizable kinetics (NFK).

We obtain $K={\rm{diag}}\left(k\right)\circ\rho^{\prime}\circ{\Psi_{K}}$ by the use of the definition of the $k$-incidence map $I_{k}$. This implies that a complex factorizable kinetics $K$ has a decomposition into diagonal matrix of rate constants and an interaction map $\rho^{\prime}\circ{\Psi_{K}}:R_{\geq 0}^{m}\to R_{\geq 0}^{r}$. The values of the interaction map are “reaction-determined” in the sense that they are determined by the values on the reactant complexes.

Complex factorizable kinetics generalizes the key structural property of MAK. The complex formation rate function decomposes as $g={A_{k}}\circ{\Psi_{K}}$ while the species formation rate function factorizes as $f=Y\circ{A_{k}}\circ{\Psi_{K}}$. The complex factorizable kinetic systems in the set of power-law kinetic systems are precisely the PL-RDK systems [2].

This subsection introduces important definitions and earlier results from the decomposition theory of chemical reaction networks. A more detailed discussion can be found in [3].

We denote a decomposition by $\mathscr{N}=\mathscr{N}_{1}\cup\mathscr{N}_{2}\cup...\cup\mathscr{N}_{k}$ since $\mathscr{N}$ is a union of the subnetworks in the sense of [14]. It also follows immediately that, for the corresponding stoichiometric subspaces, ${S}={S}_{1}+{S}_{2}+...+{S}_{k}$.

The following important concept of independent decomposition was introduced by Feinberg in [4].

In [8], it was shown that for any independent decomposition, the following holds

We can also show incidence-independence by satisfying the equation $n-l=\sum{\left({{n_{i}}-{l_{i}}}\right)}$, where $n_{i}$ is the number of complexes and $l_{i}$ is the number of linkage classes, in each subnetwork $i$.

In [3], it was shown that for any incidence-independent decomposition,

In addition, $\mathscr{C}$-decompositions form a subset of incidence-independent decompositions.

Feinberg established the following relation between an independent decomposition and the set of positive equilibria of a kinetics on the network.

The analogue of Feinberg’s 1987 result for incidence-independent decompositions and complex balanced equilibria is shown in [3]:

The following converse of Theorem 2.19 iii holds for a subset of incidence-independent decompositions with any kinetics.

In this section, we associate a unique poly-PL system to an HTK system. Its key property, that its sets of positive equilibria and of complex balanced equilibria coincide with those of the Hill-type system, is the basis of our analysis. We compare the approach with a simpler procedure for determining the equilibria sets in order to clarify its significance for deriving kinetic system properties of the original system. We begin with a brief review of concepts and results about poly-PL kinetic (denoted by PYK) systems. For further details on these systems, we refer to [11, 30].

We recall the definition of a poly-PL kinetics from Talabis et al. [30] and introduce some additional notation:

We set ${K_{j}}\left(x\right):={k_{ij}}{x^{{F_{ij}}}}$ with ${k_{ij}}={k_{i}}{a_{ij}}$ where $i=1,...,r$ for each $j=1,...,h$. $(\mathscr{N},K_{j})$ is a PLK system with kinetic order matrix $F_{j}$. Also, we let the matrix $A:=A_{ij}=\left(a_{ij}\right)$. We denote with PY-NIK the set of poly-PL kinetics with only nonnegative kinetic orders.

It follows from the canonical PL-representation of a poly-PL kinetics that the ${\mathop{\rm span}\nolimits}\left\langle{{\mathop{\rm im}\nolimits}K}\right\rangle$ is contained in the sum $\left\langle{{\mathop{\rm im}\nolimits}{K_{1}}}\right\rangle+...+\left\langle{{\mathop{\rm im}\nolimits}{K_{h}}}\right\rangle$.

It is shown in [11] that many properties of PLK systems can be extended to the set of PL-decomposable poly-PL systems.

The $S$-invariant termwise addition of reactions via maximal stoichiometric coefficients (STAR-MSC) method is based on the idea of using the maximal stoichiometric coefficient (MSC) among the complexes in the CRN to construct reactions whose reactant complexes and product complexes are different from existing ones. This is done by uniform translation of the reactants and products to create a “replica” of the CRN. The method creates $h-1$ replicas of the original network. Hence, its transform $\mathscr{N}^{*}$ becomes the union (in the sense of [14]) of the replicas and the original CRN.

We now describe the STAR-MSC method. All $x=(x_{1},\ldots,x_{m})$ are positive vectors since the domain in the definition of a poly-PL kinetics is $\mathbb{R}^{m}_{>0}$. Let $M=1+\max\{y_{i}|y\in\mathscr{C}\}$, where the second summand is the maximal stoichiometric coefficient, which is a positive integer.

For every positive integer $z$, let $z$ be identified with the vector $(z,z,\ldots,z)$ in $\mathbb{R}^{m}$. For each complex $y\in\mathscr{C}$, form the $(h-1)$ complexes

Each of these complexes are different from all existing complexes and each other as shown in the following Proposition in [11]:

The transformation is used to extend the multistationarity algorithm in [15, 16] from power-law to poly-PL systems. Further details of the STAR-MSC transformation are provided in [21, 22].

The rate constant-interaction map decomposable kinetics (RIDK) was introduced in [26]. These are chemical kinetics that have constant rates. Formally, it is a kinetics such that for each reaction $R_{q}$, the coordinate function $K_{q}:\Omega\to\mathbb{R}$ can be written in the form $K_{q}(x)=k_{q}I_{K,q}(x)$, with $k_{q}\in\mathbb{R}_{>0}$ (called rate constant) and $\Omega\in\mathbb{R}^{m}$. We call the map $I_{K}:\Omega\to\mathbb{R}^{r}$ defined by $I_{K,q}$ as the interaction map.

A poly-PL kinetics $K$ is a rate constant-interaction decomposable (RID) kinetics, the interaction $I_{K}$ being the poly-PL function. By definition, an RID kinetics is called complex factorizable [26] if at reach branching point of the network, any branching reaction has the same interaction. The subset of such kinetics is denoted by PY-RDK. We have the following Proposition from [11]:

For a Hill type kinetics $K$ and a reaction $r_{k}$, we first write $M_{k}$ and $T_{k}$ as products of two factors each, one of which contains the expressions for the positive kinetic orders and the other for the negative ones, i.e., ${M_{k}}={M_{k}}^{+}{M_{k}}^{-}$ and ${T_{k}}={T_{k}}^{+}{T_{k}}^{-}$, respectively. We then write $\dfrac{{{M_{k}}^{-}}}{{{T_{k}}^{-}}}=\dfrac{1}{{{T_{k}}^{{}^{\prime}}}}$, where ${T_{k}}^{{}^{\prime}}=\displaystyle\prod{\left({{d_{ki}}{x_{i}}^{\left|{{F_{ki}}}\right|}+1}\right)}$. Hence, $\dfrac{{{M_{k}}}}{{{T_{k}}}}=\dfrac{{{M_{k}}^{+}}}{{{T_{k}}^{+}{T_{k}}^{{}^{\prime}}}}$. Let $T\left(x\right):=\displaystyle\prod\limits_{k}{{T_{k}}^{+}{T_{k}}^{{}^{\prime}}}$. $T(x)$ can be rewritten as a product of distinct bi-PL factors

i.e., $T\left(x\right)=\displaystyle\prod\limits_{l}{{{\left|{{T_{l}}}\right|}^{\mu\left(l\right)}}}$ where $\mu(l)=$ the number of times the bi-PL occurs in $T(x)$.

For a bi-PL ${\left|{{T_{l}}}\right|}$, we define $\omega(l):=$ maximal number of occurrence in a ${T_{k}}^{+}{T_{k}}^{{}^{\prime}}$, $k=1,\ldots,r$. Clearly, for each bi-PL, $\omega(l)\leq\mu(l)$. We can now define the central concept for the construction:

Clearly, LCD$(K)$ is a factor of $T(x)$. For each reaction $q$, ${{T_{q}}^{+}{T_{q}}^{{}^{\prime}}}$is a factor of LCD$(K)$, i.e., ${T_{q}}^{+}{T_{q}}^{{}^{\prime}}{L_{q}}={\text{LCD}}\left(K\right)$ or $\dfrac{1}{{{T_{q}}^{+}{T_{q}}^{{}^{\prime}}}}=\dfrac{{{L_{q}}}}{{{\text{LCD}}\left(K\right)}}$. Hence, ${K_{q}}\left(x\right)={k_{q}}\dfrac{{{M_{q}}^{+}\left(x\right){L_{q}}\left(x\right)}}{{{\text{LCD}}\left(K\right)}}$. We now formulate the new definition of $K_{\text{PY}}$:

The key properties of $K_{\text{PY}}$ are derived in the following theorem:

We denote the canonical PL-representation of $K_{\text{PY}}=K_{\text{PY},1}+...+K_{\text{PY},h}$, where $h\leq 2^{(r-1)d_{\rm max}}$ . The kinetic order matrix of $K_{\text{PY},j}$ with $F_{j}$. As usual, the kinetic order matrix of $M:=(M_{q})$ is $F$.

Using $K_{\text{PY}}$, we obtain an analogue of a result in Section 15 of [32] concerning the relationship of multistationarity in a Hill-type system and in an associated power-law system:

We say that the result is only analogous since $\mathscr{N}^{*}$ is formally not identical to $\mathscr{N}$, though “essentially” it is $\mathscr{N}$, being the union of $\mathscr{N}$ and $(h-1)$ replicas.

We now introduce the following definition:

A further important property of the associated poly-PL kinetics is the following:

In [1], Arceo et al. defined HT-RDKD as the set of HT kinetics such that for any two reactions $q$ and $q^{\prime}$ with the same reactant, $F_{q}=F_{q^{\prime}}$ and $D_{q}=D_{q^{\prime}}$, where $F$ and $D$ are the kinetic order matrix (of the numerator) and the dissociation matrix respectively. In the same paper, it was shown that HT-RDKD is contained in HTK $\cap$ CFK. Without the assumption of ${\rm supp}(D_{q})={\rm supp}(F_{q})$, we would have obtained the following as an example of HTK $\cap$ CFK $\backslash$ HT-RDKD.

It is instructive to compare the association of the poly-PL kinetic system $(\mathscr{N},K_{\text{PY}})$ to an HTK system $(\mathscr{N},K)$ for any set of rate constants with a procedure for determining the equilibria set of $(\mathscr{N},K)$ for a given set of rate constants. This procedure also uses factoring out common denominators, but on a species-dependent basis.

For each species $X_{i}$, for $i=1,...,m$, we define

i.e., the set of reactions where $X_{i}$ occurs in the reactant or product complex. Furthermore, define ${T^{\left(i\right)}}\left(x\right)=\prod\limits_{k\in{\mathscr{R}_{i}}}{{T_{k}}\left(x\right)}$ and for a reaction $q$, ${\mathscr{R}_{i,q}}={\mathscr{R}_{i}}\backslash\left\{q\right\}$, ${T^{\left(i\right)}}_{\left(q\right)}\left(x\right)=\prod\limits_{k\in{\mathscr{R}_{i,q}}}{{T_{k}}\left(x\right)}$, ${P^{\left(i\right)}}_{q}\left(x\right)={M_{q}}\left(x\right){T^{\left(i\right)}}_{\left(q\right)}\left(x\right)$. The $i$-th coordinate of the SFRF (= RHS of $i$-th ODE) is given by

Writing ${\widetilde{f}_{i}}\left(x\right)$ for the last sum, since $\dfrac{1}{{{T^{\left(i\right)}}\left(x\right)}}>0$ for $x>0$, we have

Since only the reactions in $\mathscr{R}_{i}$ occur in them, the ${P^{\left(i\right)}}_{q}\left(x\right)$ expressions are generally much shorter than those in $K_{\text{PY}}(x)$, so they are much more practical for computing the equilibria sets. However, since a reaction will in general occur in more than one coordinate function $\left({\Leftrightarrow{\mathscr{R}_{i}}\cap{\mathscr{R}_{i^{\prime}}}\neq\varnothing{\text{ for }}i\neq i^{\prime}}\right)$, it will be assigned different ${{P^{\left(i\right)}}_{q}\left(x\right)}$ expressions. This means that the ODE system $\dfrac{{d{x_{i}}}}{{dt}}=$ ${\widetilde{f}_{i}}\left(x\right)$ is not generated by a chemical kinetic system since a kinetics assigns only one rate function to a reaction. Hence, this set of ODEs cannot be used to derive kinetic system properties for the HTK system $(\mathscr{N},K)$.

The following are examples of Hill-type kinetic systems and their associated poly-PL kinetic (PYK) systems. Note that the equilibria sets of these systems coincide.