# Absolute Concentration Robustness in Rank-One Kinetic Systems

Eduardo R. Mendoza
Mathematics and Statistics Department
Center for Natural Sciences and Environmental Research
De La Salle University
Manila 0922, Philippines
<EMAIL_ADDRESS>
&Dylan Antonio SJ. Talabis
Institute of Mathematical Sciences and Physics
University of the Philippines
Los Banos, Laguna 4031, Philippines
<EMAIL_ADDRESS>&Editha C. Jose
Institute of Mathematical Sciences and Physics
University of the Philippines
Los Banos, Laguna 4031, Philippines
<EMAIL_ADDRESS>
&Lauro L. Fontanil
Institute of Mathematical Sciences and Physics
University of the Philippines
Los Banos, Laguna 4031, Philippines
<EMAIL_ADDRESS>
Max Planck Institute of Biochemistry, Martinsried near Munich, GermanyFaculty
of Physics, Ludwig Maximilian University, Munich 80539 GermanyCorresponding
Author

###### Abstract

A kinetic system has an absolute concentration robustness (ACR) for a
molecular species if its concentration remains the same in every positive
steady state of the system. Just recently, a condition that sufficiently
guarantees the existence of an ACR in a rank-one mass-action kinetic system
was found. In this paper, it will be shown that this ACR criterion does not
extend in general to power-law kinetic systems. Moreover, we also discussed in
this paper a necessary condition for ACR in multistationary rank-one kinetic
system which can be used in ACR analysis. Finally, a concept of equilibria
variation for kinetic systems which are based on the number of the system’s
ACR species will be introduced here.

##  1 Introduction

Robustness is the capacity of a system to maintain essential functions in the
presence of internal or external stresses [17]. It is an important
characteristic that helps biological systems adapt to environmental changes
and thrive. Several types of robustness have already been identified, but this
paper focuses on the kind that requires the positive steady states of a system
to possess certain qualities.

As defined in [24], a biological system has an absolute concentration
robustness (ACR) for a molecular species if in every positive steady state,
the system admits, the concentration of the said species is the same. The
identification of the steady states of a system is not an easy task, making
this property difficult to determine. Shinar and Feinberg provided a
sufficient condition that guarantees a deficiency-one mass-action kinetic
(MAK) system exhibits an ACR [24]. The said condition requires the
corresponding network to have two nonterminal complexes that differ only in a
specific species. This structural condition can be easily observed in most
small networks and thus offers some advantages.

In 2018, Fortun et al. showed that the result of Shinar and Feinberg can be
readily extended to deficiency-one power-law kinetic systems with reactant-
determined interactions (PL-RDK) [13]. A PL-RDK system is a kinetic system
that is more general than the MAK system which requires reactions with the
same reactant complex to have identical kinetic order vectors. In a MAK
system, the elements of a kinetic order vector are the stoichiometric
coefficients in the corresponding reactant complex.

The deficiency-one requirement of the results mentioned above is significantly
limiting when it comes to analyzing the capacity of a system to admit an ACR.
Fortunately, Fortun and Mendoza [11] and Lao et al. [18] came up with more
general results which do not require a network to have a specific deficiency.
Their results utilized the concept of network decomposition which is done by
partitioning the reaction set of the network such that each partition
generates a network (called subnetwork) smaller than the given network. The
analysis then focuses on the low-deficiency subnetworks (with a deficiency of
at most one) that contain a Shinar-Feinberg pair (SF-pair). An SF-pair is a
pair of reactions with kinetic order vectors that only differ in a particular
species. This result allows the ACR determination to be confined to the
identified subnetworks that are relatively easier to handle.

Recently, Meshkat et al. [20] provided a necessary and sufficient condition
for the existence of stable ACR in a rank-one MAK system. A system has stable
ACR if it has an ACR such that each of its positive steady states is stable.
However, this ACR criterion does not extend in general to PL-RDK systems, in
contrast to the sufficient conditions for low deficiency systems by Horn and
Jackson in 1972 [15] (as generalized by Fortun and Mendoza in 2021 [11]) and
Shinar and Feinberg in 2010 [24]. Counterexamples and examples of PL-RDK
systems for the ACR criterion are discussed in this paper.

This paper also discussed a necessary condition for ACR in multistationary
rank-one kinetic systems. The use of this condition in ACR analysis is
illustrated by examples from classes of multistationary rank-one mass action
systems introduced by Pantea and Voitiuk in 2022 [26]. In addition, a concept
of equilibria variation for kinetic systems based on the number of ACR species
is introduced here. Its basic properties are derived and a general low bound
is computed for deficiency zero PL-RDK systems. For multistationary rank-one
systems, however, the necessary condition leads to a much sharper lower bound.

This is how this paper was organized. Fundamental concepts on chemical
reaction networks, kinetic systems, and robustness are provided in Section 2.
In Section 3, the problem of extending the result of Meshkat et al. in PLK
sytem is presented. A necessary condition for ACR in rank-one multistationary
kinetic system is given in Section 4. ACR and equilibria variation are
discussed in Section 5. A summary and an outlook are provided in the last
section.

##  2 Fundamentals of chemical reaction networks and kinetic systems

###  2.1 Structure of chemical reaction networks

We review in this section some necessary concepts and results on chemical
reaction network, the details of which can be found in [6, 2, 11].

First, we introduce some notations used in this paper. The sets of real
number, nonnegative real numbers, and positive real numbers are denoted,
respectively, by $\mathbb{R},\mathbb{R}_{\geq 0}$, and $\mathbb{R}_{>0}$.
Given that $\mathscr{I}$ is a finite index set, $\mathbb{R}^{\mathscr{I}}$
denotes the usual vector space of real valued functions with domain
$\mathscr{I}$ where addition, subtraction, and scalar multiplication are
defined the usual way. For $x\in\mathbb{R}^{\mathscr{I}}$ and
$i\in\mathscr{I}$, $x_{i}$ denotes the $i^{th}$ coordinate of $x$. Lastly, if
$x\in\mathbb{R}^{\mathscr{I}}_{>0}$ and $y\in\mathbb{R}^{\mathscr{I}}$, then
$x^{y}\in\mathbb{R}^{\mathscr{I}}_{>0}$ is defined to be
$x^{y}=\displaystyle\prod_{i\in\mathscr{I}}x^{y^{i}}_{i}$.

######  Definition 2.1.1.

A chemical reaction network is a triple
$\mathscr{N}=(\mathscr{S},\mathscr{C},\mathscr{R})$ of three non-empty finite
sets:

  1. 1.

a set species $\mathscr{S}$;

  2. 2.

a set $\mathscr{C}$ of complexes, which are nonnegative integer linear
combinations of the species; and

  3. 3.

a set $\mathscr{R}\subseteq\mathscr{C}\times\mathscr{C}$ of reactions such
that

     * •

$(y,y)\notin\mathscr{R}$ for all $y\in\mathscr{C}$, and

     * •

for each $y\in\mathscr{C}$, there exists a $y^{\prime}\in\mathscr{C}$ such
that $(y,y^{\prime})\in\mathscr{R}$ or $(y^{\prime},y)\in\mathscr{R}$.

The nonnegative coefficients of the species in a complex are referred to as
stoichiometric coefficients. In this paper, a reaction
$R_{i}=(y_{i},y^{\prime}_{i})$ is also denoted by $R_{i}:y_{i}\rightarrow
y^{\prime}_{i}$ and $y_{i}$ and $y^{\prime}_{i}$ are called the reactant and
product complexes of $R_{i}$, respectively. Further, we use the symbols
$\sigma,\kappa$, and $\rho$ to denote the numbers of species, complexes, and
reactions, respectively. The following example shows that a CRN can be
represented by a digraph where the complexes and reactions serve as the
digraph’s vertices and arcs, respectively.

### Running example 1

Chemical reaction networks (CRNs) can be represented as a directed graph. The
vertices or nodes are the complexes and the reactions are the edges. The CRN
is not unique and might not have a physical interpretation. Let us consider
the following CRN:

$X_{1}+X_{2}+X_{3}$$X_{3}$$2X_{1}$$3X_{1}+X_{2}$$4X_{1}+2X_{2}$$4X_{1}+X_{2}$$3X_{1}$$k_{1}$$k_{2}$$k_{3}$$k_{4}$

The $k_{i}$’s are called the reaction rate constants. We have $m=3$ (species),
$n=7$ (complexes), $n_{r}=3$ (reactant complexes) and $r=4$ (reactions). Here,
we can write

$\mathscr{S}=\left\\{X_{1},X_{2},X_{3}\right\\},\quad\mathscr{C}=\left\\{X_{1}+X_{2}+X_{3},X_{3},2X_{1},3X_{1}+X_{2},4X_{1}+2X_{2},4X_{1}+X_{2},3X_{1}\right\\}.$

On the other hand, the set of reaction $\mathscr{R}$ consists of the
following:

$\begin{array}[]{l}R_{1}:X_{1}+X_{2}+X_{3}\rightarrow X_{3}\\\
R_{2}:3X_{1}+X_{2}\rightarrow 2X_{1}\\\
\end{array}\quad\begin{array}[]{l}R_{3}:3X_{1}+X_{2}\rightarrow
4X_{1}+2X_{2}\\\ R_{4}:3X_{1}\rightarrow 4X_{1}+X_{2}\\\ \end{array}$

We denote the CRN $\mathscr{N}$ as
$\mathscr{N}=(\mathscr{S},\mathscr{C},\mathscr{R})$. The linkage classes of a
CRN are the subgraphs of a reaction graph where for any complexes $C_{i}$,
$C_{j}$ of the subgraph, there is a path between them. Thus, the number of
linkage classes, denoted as $l$, of Running Example 1 is three ($l=3$). The
linkage classes are:

$\mathscr{L}_{1}=\left\\{R_{1}\right\\},\quad\mathscr{L}_{2}=\left\\{R_{2},R_{3}\right\\}\quad\mathscr{L}_{3}=\left\\{R_{4}\right\\}.$

A subset of a linkage class where any two vertices are connected by a directed
path in each direction is said to be a strong linkage class. Considering
Running Example 1, there are no strong linkage classes whose number is denoted
by $sl$. We also identify the terminal strong linkage classes, the number
denoted as $t$, to be the strong linkage classes where there is no reaction
from a complex in the strong linkage class to a complex outside the same
strong linkage class. The terminal strong linkage classes can be of two kinds:
cycles (not necessarily simple) and singletons (which we call “terminal
points”).

We now define important CRN classes. A CRN is weakly reversible if every
linkage class is a strong linkage class. A CRN is t-minimal if $t=l$, i.e.
each linkage class has only one terminal strong linkage class. Let $n_{r}$ be
the number of reactant complexes of a CRN. Then $n-n_{r}$ is the number of
terminal points. A CRN is called cycle-terminal if and only if $n-n_{r}=0$,
i.e., each complex is a reactant complex. Clearly, the CRN of the Running
Example 1 is both not t-minimal and weakly reversible.

With each reaction $y\rightarrow y^{\prime}$, we associate a reaction vector
obtained by subtracting the reactant complex $y$ from the product complex
$y^{\prime}$. The stoichiometric subspace $S$ of a CRN is the linear subspace
of $\mathbb{R}^{\mathscr{S}}$ defined by

$S:=\text{span }\left\\{y^{\prime}-y\in\mathbb{R}^{\mathscr{S}}\mid
y\rightarrow y^{\prime}\in\mathscr{R}\right\\}.$

The map of complexes
$\displaystyle{Y:\mathbb{R}^{\mathscr{C}}\rightarrow\mathbb{R}^{\mathscr{S}}_{\geq}}$
maps the basis vector $\omega_{y}$ to the complex $y\in\mathscr{C}$. The
incidence map
$\displaystyle{I_{a}:\mathbb{R}^{\mathscr{R}}\rightarrow\mathbb{R}^{\mathscr{C}}}$
is defined by mapping for each reaction $\displaystyle{R_{i}:y\rightarrow
y^{\prime}\in\mathscr{R}}$, the basis vector $\omega_{R_{i}}$ (or simply
$\omega_{i}$) to the vector $\omega_{y^{\prime}}-\omega_{y}\in\mathscr{C}$.
The stoichiometric map
$\displaystyle{N:\mathbb{R}^{\mathscr{R}}\rightarrow\mathbb{R}^{\mathscr{S}}}$
is defined as $N=Y\circ I_{a}$.

In Running Example 1, the matrices $Y$ and $I_{a}$ are

$Y=\blockarray{cccccccc}C_{1}&C_{2}C_{3}C_{4}C_{5}C_{6}C_{7}\\\
\block{[ccccccc]c}1023443X_{1}\\\ 1001210X_{2}\\\ 1100000X_{3}\\\ $

$I_{a}=\blockarray{ccccc}R_{1}&R_{2}R_{3}R_{4}\\\ \block{[cccc]c}-1000C_{1}\\\
1000C_{2}\\\ 0100C_{3}\\\ 0-1-10C_{4}\\\ 0010C_{5}\\\ 0001C_{6}\\\
000-1C_{7}\\\ .$

The deficiency $\delta$ is defined as $\delta=n-l-\dim S$. This non-negative
integer is, as Shinar and Feinberg pointed out in [25], essentially a measure
of the linear dependency of the network’s reactions. In Running Example 1, the
deficiency of the network is 3.

###  2.2 Dynamics of chemical reaction networks

A kinetics is an assignment of a rate function to each reaction in a CRN. A
network $\mathscr{N}$ together with a kinetics $K$ is called a chemical
kinetic system (CKS) and is denoted here by $(\mathscr{N},K)$. Power law
kinetics (PLK) is identified by the kinetic order matrix which is an
$\rho\times\sigma$ matrix $F=[F_{ij}]$, and vector
$k\in\mathbb{R}^{\mathscr{R}}_{>0}$, called the rate vector.

######  Definition 2.2.1.

A kinetics
$K:\mathbb{R}^{\mathscr{S}}_{>0}\rightarrow\mathbb{R}^{\mathscr{R}}$ is a
power law kinetics if

$K_{i}(x)=k_{i}x^{F_{i,\cdot}}$ for $i=1,\dots,r$

where $k_{i}\in\mathbb{R}_{>0}$, $F_{i,j}\in\mathbb{R}$, and $F_{i,\cdot}$ is
the row of $F$ associated to reaction $R_{i}$.

We can classify a PLK system based on the kinetic orders assigned to its
branching reactions (i.e., reactions sharing a common reactant complex).

######  Definition 2.2.2.

A PLK system has reactant-determined kinetics (of type PL-RDK) if for any two
branching reactions $R_{i},R_{j}\in\mathscr{R}$, the corresponding rows of
kinetic orders in $F$ are identical, i.e., $F_{ih}=F_{jh}$ for $h=1,\dots,m$.
Otherwise, a PLK system has non-reactant-determined kinetics (of type PL-NDK).

Consider the CRN in Running example 1 with the following kinetic order matrix.

$F=\blockarray{cccc}X_{1}&X_{2}X_{3}\\\ \block{[ccc]c}002R_{1}\\\ 110R_{2}\\\
110R_{3}\\\ 100R_{4}\\\ .$

Observe that $R_{2}$ and $R_{3}$ are two branching reactions whose
corresponding rows in $F$ (or kinetic order vectors) are the same. Hence, the
CRN is a PL-RDK system.

The well-known mass action kinetic system (MAK) forms a subset of PL-RDK
systems. In particular, MAK is given by $K_{i}(x)=k_{i}x^{Y_{.,j}}$ for all
reactions $R_{i}:y_{i}\rightarrow y^{\prime}_{i}\in\mathscr{R}$ with
$k_{i}\in\mathbb{R}_{>0}$ (called rate constant). The vector $Y_{.,j}$
contains the stoichiometric coefficients of a reactant complex
$y_{i}\in\mathscr{C}$.

######  Definition 2.2.3.

The species formation rate function of a chemical kinetic system is the vector
field

$f(c)=NK(c)=\displaystyle\sum_{y_{i}\rightarrow
y^{\prime}_{i}\in\mathscr{R}}K_{i}(c)(y^{\prime}_{i}-y_{i})$, where
$c\in\mathbb{R}^{\mathscr{S}}_{\geq 0}$,

where $N$ is the $m\times r$ matrix, called stoichiometric matrix, whose
columns are the reaction vectors of the system. The equation $dc/dt=f(c(t))$
is the ODE or dynamical system of the chemical kinetic system. An element
$c^{*}$ of $\mathbb{R}^{\mathscr{S}}_{>0}$ such that $f(c^{*})=0$ is called a
positive equilibrium or steady state of the system. We use
$E_{+}(\mathscr{N},K)$ to denote the set of all positive equilibria of a CKS.

Deficiency is one of the important parameters in CRNT to establish claims
regarding the existence, multiplicity, finiteness and parametrization of the
set of positive steady states.

The dynamical system $f(x)$ (or species formation rate function (SFRF)) of the
Running Example 1 can be written as

$\left[\begin{array}[]{c}\dot{X_{1}}\\\ \dot{X_{2}}\\\ \dot{X_{3}}\\\
\end{array}\right]=\blockarray{cccc}R_{1}&R_{2}R_{3}R_{4}\\\
\block{[cccc]}-1-111\\\ -1-111\\\ 0000\\\
\left[\begin{array}[]{c}k_{1}X_{3}^{2}\\\ k_{2}X_{1}X_{2}\\\
k_{3}X_{1}X_{2}\\\ k_{4}X_{1}\\\ \end{array}\right]=NK(x).$

Analogous to the species formation rate function, we also have the complex
formation rate function.

######  Definition 2.2.4.

The complex formation rate function
$g:\mathbb{R}^{\mathscr{S}}_{>0}\rightarrow\mathbb{R}^{\mathscr{C}}$ of a
chemical kinetic system is the vector field

$g(c)=I_{a}K(c)=\displaystyle\sum_{y_{i}\rightarrow
y^{\prime}_{i}\in\mathscr{R}}K_{i}(c)(\omega_{y^{\prime}_{i}}-\omega_{y_{i}})$,
where $c\in\mathbb{R}^{\mathscr{S}}_{\geq 0}$.

where $I_{a}$ is the incidence map. A CKS is complex balanced if it has
complex balanced steady state, i.e., there is a composition
$c^{**}\in\mathbb{R}_{>0}^{\mathscr{S}}$ such that $g(c^{**})=0$. We denote by
$Z_{+}(\mathscr{N},K)$ the set of all complex balanced steady states of the
system.

######  Theorem 2.2.1 (Corollary 4.8, [7]).

If a CKS has deficiency 0, then its steady states are all complex balanced.

###  2.3 A brief review of concentration robustness

The concept of absolute concentration robustness (ACR) was first introduced by
Shinar and Feinberg in their well-cited paper published in Science [24]. ACR
pertains to a phenomenon in which a species in a kinetic system carries the
same value for any positive steady state the network may admit regardless of
initial conditions. In particular, a PL-RDK system $(\mathscr{N},K)$ has ACR
in a species $X\in\mathscr{S}$ if there exists $c^{*}\in E_{+}(\mathscr{N},K)$
and for every other $c^{**}\in E_{+}(\mathscr{N},K)$, we have
$c^{**}_{X}=c^{*}_{X}$ where $c^{**}_{X}$ and $c^{*}_{X}$ denote the
concentrations of $X$ in $c^{*}$ and $c^{**}$, respectively.

Fortun and Mendoza [11] emphasized that ACR as a dynamical property is
conserved under dynamic equivalence, ie., they generate the same set of
ordinary differential equations. They further investigated ACR in power law
kinetic systems and derived novel results that guarantee ACR for some classes
of PLK systems. For these PLK systems, the key property for ACR in a species
$X$ is the presence of an SF-reaction pair. A pair of reactions in a PLK
system is called a Shinar-Feinberg pair (or SF-pair) in a species $X$ if their
kinetic order vectors differ only in $X$. A subnetwork of the PLK system is of
SF-type if it contains an SF-pair in $X$.

##  3 The extension problem for the ACR criterion for rank-one systems

Just recently, Meshkat et al. gave a necessary and sufficient condition that
guarantees the existence of stable ACR in rank-one MAK systems [20]. A system
is said to have a stable ACR if it has an ACR where each of its positive
steady states is stable. The condition requires the existence of an embedded
one-species network that follows some structures described using arrow
diagrams.

###  3.1 A review of the key results of Meshkat et al.

A network $\mathscr{N}=\left(\mathscr{S},\mathscr{C},\mathscr{R}\right)$ is
called one-species network if there is a species $A_{i}$ such that
$(y,y^{\prime})\in\mathscr{R}$ implies that $y_{j}=y_{j}^{\prime}=0$ for all
species $A_{j}\in S-\\{A_{i}\\}$. In other words, every complex in the network
takes the form $kA_{i}$, where $k$ is a nonnegative integer. The following
definition and example were taken from [20].

######  Definition 3.1.1.

Let $\mathscr{N}=\left(\\{A\\},\mathscr{C},\mathscr{R}\right)$ be a one-
species network with $|\mathscr{R}|\geq 1$. Let every reaction of
$\mathscr{N}$ be of the form $aA\rightarrow bA$, where $a,b\geq 0$ and $a\neq
b$. Suppose $\mathscr{N}$ has $m$ distinct reactant complexes with
$a_{1}<a_{2}<\dots<a_{m}$ as their stoichiometric species. The arrow diagram
of $\mathscr{N}$, denoted by $\rho=(\rho_{1},\rho_{2},\dots,\rho_{m})$, is the
element of $\\{\rightarrow,\leftarrow,\longleftrightarrow\\}^{m}$ given by:

$\rho_{i}=\left\\{\begin{array}[]{clrc@{\qquad}l}\rightarrow&\textnormal{if
for all reactions\;}a_{i}A\rightarrow bA\textnormal{\;in\;}G,\textnormal{\;it
is the case that\;}b>a_{i}\\\ \leftarrow&\textnormal{if for all
reactions\;}a_{i}A\rightarrow bA\textnormal{\;in\;}G,\textnormal{\;it is the
case that\;}b<a_{i}\\\
\longleftrightarrow&\textnormal{otherwise}\end{array}\right.$

######  Example 3.1.1.

Consider the network determined by $\\{B\rightarrow A,2A+B\rightarrow
A+2B\\}$. After removing species $A$, the embedded network has the following
reaction set $\\{0\leftarrow B\rightarrow 2B\\}$ which has
$(\longleftrightarrow)$ as arrow diagram. On the other hand, $\\{0\rightarrow
A,A\rightarrow 2A\\}$ is the embedded network obtained after removing $B$ with
arrow diagram $(\rightarrow,\leftarrow)$.

Here is the main result in [20] that provides a necessary and sufficient
condition that ensure the existence of ACR in some rank-one MAK systems. Their
results are based on the idea of arrow diagrams.

######  Theorem 3.1.1.

Let $\mathscr{N}$ be a rank-one network with species
$A_{1},A_{2},\dots,A_{m}$. Then, the following are equivalent:

  1. 1.

$\mathscr{N}$ has stable ACR and admits a positive state.

  2. 2.

There is a species $A_{i^{*}}$ such that the following holds:

    1. (a)

for the embedded network of $\mathscr{N}$ obtained by removing all species
except $A_{i^{*}}$, the arrow diagram has one of the following forms:

$\begin{array}[]{rl}(\longleftrightarrow,\leftarrow,\leftarrow,\dots,\leftarrow),&(\rightarrow,\rightarrow,\dots,\rightarrow,\longleftrightarrow)\\\
(\rightarrow,\rightarrow,\dots,\rightarrow,\longleftrightarrow,\leftarrow,\leftarrow,\dots,\leftarrow),&(\rightarrow,\rightarrow,\dots,\rightarrow,\leftarrow,\leftarrow,\dots,\leftarrow)\end{array}$
(3.1.1)

    2. (b)

the reactant complexes of $\mathscr{N}$ differ only in species $A_{i^{*}}$
(i.e., if $y$ and $\hat{y}$ are both reactant complexes, then
$y_{i}=\hat{y_{i}}$ for all $i\neq i^{*}$).

Notice that in the above example, the embedded network obtained by removing
species $A$ from $\mathscr{N}$ has $(\longleftrightarrow)$ as arrow diagram
that falls under the enumerated arrow diagrams in the above theorem. Moreover,
the reactant complexes in $\mathscr{N}$ differ only in $A$. It follows that
$\mathscr{N}$ has a stable ACR.

###  3.2 Examples of PL-RDK systems for which the ACR critera do not hold

Here is the direct adaptation of the sufficient condition of Theorem 3.1.1 in
PLK-systems and formalism. Let $(\mathscr{N},K)$ be a rank-one PLK-system
having species $A_{1},A_{2},\dots,A_{m}$. Also, let $F$ be the kinetic order
matrix of the system. Then, $\mathscr{N}$ has an ACR if there is species
$A_{j^{*}}$ such that the following holds

  1. 1.

for the embedded network of $\mathscr{N}$ obtained by removing all species
except $A_{j^{*}}$, the arrow diagram has one of the following forms:

$\begin{array}[]{rl}(\longleftrightarrow,\leftarrow,\leftarrow,\dots,\leftarrow),&(\rightarrow,\rightarrow,\dots,\rightarrow,\longleftrightarrow)\\\
(\rightarrow,\rightarrow,\dots,\rightarrow,\longleftrightarrow,\leftarrow,\leftarrow,\dots,\leftarrow),&(\rightarrow,\rightarrow,\dots,\rightarrow,\leftarrow,\leftarrow,\dots,\leftarrow)\end{array}$
(3.2.1)

  2. 2.

the kinetic order vectors are pairwise SF-pairs in $A_{j^{*}}$ (i.e.,
$F_{ij}=F_{lj}$ for all $j\neq j^{*}$).

The following counterexample shows that the statement above is not necessarily
true.

######  Example 3.2.1.

Consider $\mathscr{N}=(\\{A,B\\},\mathscr{C},\mathscr{R})$, where
$\mathscr{R}$ consists of the following reactions and rate constants:

$\begin{array}[]{cl}R_{1}:&B{\xrightarrow[r_{1}]{}}A\\\
R_{2}:&2A+B{\xrightarrow[r_{2}]{}}3A\\\
R_{3}:&3A+B{\xrightarrow[r_{3}]{}}2A+2B\\\
R_{4}:&4A+B{\xrightarrow[r_{4}]{}}3A+2B\\\ \end{array}$

This is a rank-one network with $S=\textnormal{span\;}\\{A-B\\}$ as its
stoichiometric subspace. Suppose that it is endowed with power law kinetics
and has the following kinetic order matrix:

$F=\blockarray{ccc}A&B\\\ \block{[cc]c}p_{1}qR_{1}\\\ p_{2}qR_{2}\\\
p_{3}qR_{3}\\\ p_{4}qR_{4}\\\ $

where $p_{i}$’s are integers. We obtained the following embedded network after
removing $B$.

$\begin{array}[]{c}0\rightarrow A\\\ 2A\leftrightarrow 3A\leftarrow
4A\end{array}$

With the four distinct reactant complexes in this network, here is the
corresponding arrow diagram $(\rightarrow,\rightarrow,\leftarrow,\leftarrow)$.
Now, observe that $(\mathscr{N},K)$ has the following ODEs:

$\begin{array}[]{cc}\dot{A}=&{r_{1}}A^{p_{1}}B^{q}+{r_{2}}A^{p_{2}}B^{q}-{r_{3}}A^{p_{3}}B^{q}-{r_{4}}A^{p_{4}}B^{q}\\\
\dot{B}=&-{r_{1}}A^{p_{1}}B^{q}-{r_{2}}A^{p_{2}}B^{q}+{r_{3}}A^{p_{3}}B^{q}+{r_{4}}A^{p_{4}}B^{q}\end{array}$

Solving for the positive equilibria, one of these equations is equated to zero
and get

$\begin{array}[]{cc}B^{q}({r_{1}}A^{p_{1}}+{r_{2}}A^{p_{2}}-{r_{3}}A^{p_{3}}-{r_{4}}A^{p_{4}}&=0\\\
\Leftrightarrow{r_{1}}A^{p_{1}}+{r_{2}}A^{p_{2}}-{r_{3}}A^{p_{3}}-{r_{4}}A^{p_{4}}&=0\end{array}$

Observe that when $p_{1}<p_{2}<p_{3}<p_{4}$, the polynomial

${r_{1}}A^{p_{1}}+{r_{2}}A^{p_{2}}-{r_{3}}A^{p_{3}}-{r_{4}}A^{p_{4}}$ (3.2.2)

has exactly one positive root by the Descarte’s Rule of Signs. This means that
the system has ACR in $A$. On the other hand, suppose that

$\begin{bmatrix}p_{1}\\\ p_{2}\\\ p_{3}\\\
p_{4}\end{bmatrix}=\begin{bmatrix}0\\\ 3\\\ 1\\\ 2\end{bmatrix}\;$ and
$\begin{bmatrix}r_{1}\\\ r_{2}\\\ r_{3}\\\
r_{4}\end{bmatrix}=\begin{bmatrix}4\\\ 1\\\ 2\\\ 3\end{bmatrix}\;$

Then, the polynomial in (3.2.2) becomes

$4+A^{3}-2A-3A^{2}=4-2A-3A^{2}+A^{3}$ (3.2.3)

This polynomial has two positive roots namely, $1$ and $1+\sqrt{5}$. This
means that the system does not have an ACR in $A$ in this case.

It can be observed in the above counterexample the crucial role played by
$p_{i}$’s. If these parameters are not carefully chosen, ACR may not be
observed in the system. The problem becomes more complicated when $p_{i}$’s
are non integers since the solvability of the polynomial in (3.2.2) may not be
determined immediately. This negatively affects the possibility of extending
the result of Meshkat et al. to PLK-systems where $p_{i}$’s are allowed to be
real numbers.

###  3.3 The stable ACR criterion for homogenous PL-quotients of mass action
systems

In this section, we introduce the set of homogeneous monomial quotients of
mass action systems (PL-MMK) where the stable ACR criterion holds. To define
PL-MMK, we recall some definition and result from [16].

######  Definition 3.3.1.

Two kinetics $K,K^{\prime}$ in $\mathscr{K}_{\Omega}(\mathscr{N})$ are
positive function factor equivalent (PPF-equivalent) if for all
$x\in\mathbb{R}^{\mathscr{S}}_{>}$ and every reaction $q$,
$\dfrac{K_{q}(x)}{K^{\prime}_{q}(x)}$ is a positive function of $x$ only, i.e.
independent of $q.$

A key property of PFF-equivalence is expressed in the following proposition:

######  Proposition 3.3.1 ([16]).

If $K$, $K^{\prime}$ are PPF, then
$E_{+}(\mathscr{N},K)=E_{+}(\mathscr{N},K^{\prime})$.

Let $(\mathscr{N},K)$ be a MAK system with $m$ species and $r$ reactions and
$(\beta_{1},\cdots,\beta_{m})$ any real vector. Let $F_{MAK}$ and $F_{\beta}$
be the $r\times m$ matrices defined as follows:

  * •

$F_{MAK,q}=Y^{\top}_{\rho(q)}$, where $q$ is a reaction and $Y$ is the matrix
of complexes of $(\mathscr{N},K)$.

  * •

$F_{\beta}$ with identical rows $\beta=(\beta_{1},\cdots,\beta_{m})$.

######  Definition 3.3.2.

A homogeneous PL quotient of a MAK system $(\mathscr{N},K)$ is a PLK system
$(\mathscr{N},K_{PLK})$ with the same rate constants and a kinetic order
matrix $F_{PLK}=F_{MAK}-F_{\beta}$.

We have the following result:

######  Proposition 3.3.2.

The stable ACR criterion holds for homogeneous PL-quotients of rank-one mass
action systems.

###### Proof.

Let $(\mathscr{N},K_{PLK})$ be a homogeneous PL quotient of a rank-one MAK
system $(\mathscr{N},K)$ where $(\mathscr{N},K)$ satisfies the stable ACR
criterion. For each species, write
$X_{i}^{\alpha_{i}}=X_{i}^{\beta_{i}}(X_{i}^{\alpha_{i}-\beta_{i}})$. For a
reaction $q$, each $K_{q}(x)=k_{q}\prod
X_{i}^{\beta_{i}}\prod(X_{i}^{\alpha_{i}-\beta_{i}})$. Note that $\prod
X_{i}^{\beta_{i}}$ is independent of the reaction $q$ and hence, from
Definition 3.3.1, $K_{PLK}$ and $K$ are PFF-equivalent. From Proposition
3.3.1, $E_{+}(\mathscr{N},K)=E_{+}(\mathscr{N},K_{PLK})$. Thus, their ACR
coincide and, hence the stable ACR criterion also holds for
$(\mathscr{N},K_{PLK})$. ∎

##  4 A necessary condition for ACR in multistationary rank-one kinetic
systems

In this section, we present a necessary condition for the occurrence of ACR
species in any multistationary rank-one kinetic system. We introduce the class
of co-conservative kinetic systems and show the multistationary rank-one
systems of this kind do not have any ACR species. We further illustrate the
condition for a number of multistationary rank-one mass action systems, using
a recently presented classification of such systems by Pantea and Voitiuk
[26].

###  4.1 The necessary condition for ACR in rank-one stationary kinetic
systems

The necessary condition is the following fundamental observation:

######  Theorem 4.1.1.

If $\mathscr{N}$ has rank one and $\left(\mathscr{N},K\right)$ is
multistationary, then (the line) $S$ lies in the species hyperplane of every
ACR species $X$. In other words, for any basis vector $v$ of $S$, its
$X$-coordinate is $0$.

###### Proof.

Since $\left(\mathscr{N},K\right)$ is multistationary, there is a
stoichiometric class that contains two distinct positive equilibria $x_{1}$
and $x_{2}$. In other words, $x_{1}-x_{2}\in S$ and $x_{1}-x_{2}\neq 0$. For
any ACR species $X$ of $\left(\mathscr{N},K\right)$, it follows that the
$X$-th coordinate of $x_{1}-x_{2}$ is $0$. Since $x_{1}-x_{2}\neq 0$, it is a
basis vector for the one-dimensional subspace $S$ of the rank-one system,
proving the claim. ∎

The necessary condition immediately leads to an upper bound for the number of
ACR species in a multistationary rank-one system:

######  Corollary 4.1.1.

Let $m_{\textnormal{ACR}}$ be the number of ACR species of a rank-one
multistationary system $\left(\mathscr{N},K\right)$. Then
$m_{\textnormal{ACR}}\leq m-|\textnormal{supp\;}v|$, where
$\textnormal{supp\;}v$ is the support of any basis vector of $S$.

Note that the right-hand side of the inequality in Corollary 4.1.1 is just the
number of zeros in $v$. Now, recall that a network is called conservative if
the orthogonal complement of $S$ contains a positive vector. We hence define
the following term.

######  Definition 4.1.1.

A network is called co-conservative if $S$ contains a positive vector.

In general, a positive vector in $S$ can be a linear combination of reaction
vectors with $0$ or negative coordinates. For example, for $m=3$, the
reactions $X_{1}\rightarrow X_{1}+X_{2}$ and $2X_{1}\rightarrow 3X_{1}+2X_{3}$
have the reaction vectors $(0,1,-1)$ and $(1,0,2)$ whose sum is $(1,1,1)$. In
rank-one networks however, a positive vector requires the occurrence of a
positive reaction vector $y^{\prime}-y$. This in turn implies a reaction
$y\rightarrow y^{\prime}$ with the following characteristics:

  * •

all species occur in the product complex;

  * •

each species has a higher stoichiometric coefficient in the product complex
compared to the corresponding species in the reactant complex; and

  * •

no enzymatic regulation on a reaction.

######  Corollary 4.1.2.

If a rank-one and co-conservative network is multistationary, then it has no
ACR species.

###### Proof.

Any non-zero vector $v$ in $S$ has only positive or negative coordinates,
i.e., $|\textnormal{supp\;}v|=m$. Hence, $m_{\textnormal{ACR}}=0$. ∎

###  4.2 Examples from the multistationary rank-one mass action systems

Pantea and Voitiuk introduced a complete classification of multistationary
rank-one mass action systems in [26]. The following table provides an overview
of the eight classes that they identified.

Network |  Definition
---|---
Class 1-altc: 1-alt complete networks |  has a 1D projection containing both $(\leftarrow,\rightarrow)$ and $(\rightarrow,\leftarrow)$ patterns
Class 2-alt: 2-alternating |  has a 1D projection containing both $(\leftarrow,\rightarrow,\leftarrow)$ and $(\rightarrow,\leftarrow,\rightarrow)$ patterns
Class $Z$: zigzag network |  has a 2D projection containing a zigzag
Class $S_{1}$: one-source networks |  contains exactly one source complex and two reactions of opposite directions
Class $S^{z}_{2}$ : two-source zigzag networks |  has two species, contains exactly two source complexes and has a zigzag of slope -1
Class $L$: line networks |  has two species and at least three source complexes satisfying some properties*
Class $S^{nz}_{2}$ : two-source non-zigzag networks |  an essential network that contains exactly two source complexes and $N\in$ 1-alt${}^{c}-Z$.
Class $C$: corner networks |  an essential, 1-alt complete network that contains at least three source complexes satisfying some properties*
Table 4.2.1: Classes of rank-one networks (for details concerning the concepts
and symbols used here, the readers are referred to [26]).

Under mass-action, our running example is a rank-one zigzag network but not a
line, corner and two-source zigzag network. We illustrate Theorem 4.1.1 with
two examples from different classes of multistationary rank-one systems
identified in [26].

######  Example 4.2.1 (Class 1-altc).

The rank-one network below was the focus of Example 4.2 in [26]. It was shown
in that paper that it has the capacity for multiple positive and non-
degenerate equilibria. Moreover, the network has stoichiometric subspace
generated only by the vector $v=(1,1,1,0,-1,-1)$.

$\begin{array}[]{cl}R_{1}:&B+2C+2E\rightarrow A+2B+3C+E\\\
R_{2}:&2A+2B+C+2D+E\rightarrow A+B+2D+2E\\\ R_{3}:&A+3C+D+2E\rightarrow
2A+B+4C+D+E\\\ R_{4}:&3A+3B+C+E\rightarrow 2A+2B+2E\\\ \end{array}$

######  Example 4.2.2 (Class 2-alt).

The rank-one network below is a 2-alternating network with stoichiometric
subspace generated by the vector $v=(2,1,0)$. From Corollary 4.3 in [26], we
conclude that it has the capacity for multiple positive and non-degenerate
equilibria.

$\begin{array}[]{cl}R_{1}:&2X+2Y\rightarrow Y\\\ R_{2}:&3X+Y\rightarrow
5X+2Y\\\ R_{3}:&4X+2Y+Z\rightarrow Z\\\ R_{4}:&4X+2Y+2Z\rightarrow 2X+Y+2Z\\\
\end{array}$

Direct computations show that the preceding systems both have ACR in species
$D$ and $Z$, respectively. That is, their basis vectors of $S$ have $0$
coordinate in species $D$ and $Z$, respectively. This illustrates Theorem
4.1.1.

##  5 ACR and equilibria variation in kinetic systems

In this section, we briefly discuss the relationship between concentration
robustness and equilibria variability in general by introducing the concept of
(positive) equilibria variation. We consider several examples to illustrate
its use. We assume throughout the section that the kinetic system
$(\mathscr{N},K)$ is positively equilibrated, i.e.,
$E_{+}(\mathscr{N},K)\neq\varnothing$.

###  5.1 The equilibria variation of a kinetic system

The motivation for the following definition comes from the observation that a
kinetic system has a unique (positive) equilibrium in species space if and
only if it possesses absolute concentration robustness in each of its species.

######  Definition 5.1.1.

The (positive) equilibria variation of a kinetic system $(\mathscr{N},K)$ is
the number of non-ACR species divided by the number of species, i.e.,

$v_{+}(\mathscr{N},K)=\dfrac{m-m_{ACR}}{m}.$

Clearly, the variation values lie between $0$ and $1$, and the following
proposition characterizes the attainment of the extreme values:

######  Proposition 5.1.1.

Let $E_{+}(\mathscr{N},K)$ be a kinetic system. Then,

  1. i.

$v_{+}(\mathscr{N},K)=0\Leftrightarrow|E_{+}(\mathscr{N},K)|=1$ and

  2. ii.

$v_{+}(\mathscr{N},K)=1\Leftrightarrow(\mathscr{N},K)$ has no ACR species

The proofs follow directly from the definition. We have the following
corollary for any multistationary system:

######  Corollary 5.1.1.

Let $(\mathscr{N},K)$ be a kinetic system.

  1. a.

If $(\mathscr{N},K)$ is multistationary, then
$v_{+}(\mathscr{N},K)\geq\dfrac{1}{m}$. In particular, if
$v_{+}(\mathscr{N},K)=0$, then $(\mathscr{N},K)$ is monostationary.

  2. b.

If $\mathscr{N}$ is open and $v_{+}(\mathscr{N},K)>0$, then $(\mathscr{N},K)$
is multistationary.

###### Proof.

For $(a)$: Multistationarity implies at least two distinct positive
equilibria, hence $m_{ACR}\leq m-1$, and the claims follow. For $(b)$: if the
network is open, then there is only one stoichiometric class. Hence,
multistationarity is equivalent to the occurrence of at least two distinct
equilibria in species space. ∎

######  Example 5.1.1.

Schmitz’s model of the earth’s pre-industrial carbon cycle system was analyzed
by Fortun et al. in [12]. Below is its corresponding reaction network.

${M_{5}}$${M_{2}}$${M_{1}}$${M_{4}}$${M_{6}}$${M_{3}}$$\scriptstyle{R_{1}}$$\scriptstyle{R_{2}}$$\scriptstyle{R_{5}}$$\scriptstyle{R_{11}}$$\scriptstyle{R_{9}}$$\scriptstyle{R_{3}}$$\scriptstyle{R_{6}}$$\scriptstyle{R_{8}}$$\scriptstyle{R_{10}}$$\scriptstyle{R_{12}}$$\scriptstyle{R_{4}}$$\scriptstyle{R_{7}}$$\scriptstyle{R_{13}}$
(5.1.1)

In this network, $M_{1},M_{2},M_{3},M_{4},M_{5},$ and $M_{6}$ stand for
atmosphere, warm ocean surface waters, cool ocean surface waters, deep ocean
waters, terrestrial biota, and soil and detritus, respectively. Important
numbers of the network as well as its kinetic order matrix are given in the
following.

Figure 1: (a) Some numbers related to the reaction network of Schmitz’s model
of the earth’s pre-industrial carbon cycle system; (b) kinetic order matrix of
the network.

Fortun and Mendoza [12] showed that the system has no ACR species, i.e.,
$v_{+}(\mathscr{N},K)=1>\dfrac{1}{6}$. Since the system has a conservative,
concordant, and weakly reversible network and weakly monotonic kinetics, by
Theorem 6.6 of [23], it has a unique positive equilibrium in each
stoichiometric class and consequently monostationary. Note also that the
network has rank $5<6$, hence it is a closed network.

If the multistationary kinetic system has rank one, Theorem 4.1.1 provides a
better lower bound for the equilibria variation.

######  Proposition 5.1.2.

Let $(\mathscr{N},K)$ be a multistationary rank-one kinetic system and $v$ a
basis vector of $S$. Then, $v_{+}(\mathscr{N},K)\geq\dfrac{|supp(v)|}{m}$.

###### Proof.

According to Theorem 4.1.1, $m_{ACR}\leq
m-|supp(v)|\Leftrightarrow|supp(v)|\leq m-m_{ACR}$, leading to the new lower
bound. Note that since $v$ is a basis vector, $|supp(v)|\geq 1$, confirming
the improvement. ∎

One can derive a lower bound for equilibria variation for any kinetic system
using the general species hyperplane criterion for ACR introduced by Hernandez
and Mendoza in [14]. It is based on the following considerations:

######  Definition 5.1.2.

For any species $X$, the $(m-1)$-dimensional subspace

$H_{X}:=\\{x\in\mathbb{R}^{\mathscr{S}}|x_{X}=0\\}$

is called the species hyperplane of $X$.

For $U$ containing $\mathbb{R}_{>0}$, let $\phi:U\rightarrow\mathbb{R}$ be an
injective map, i.e., $\phi:U\rightarrow\textnormal{Im}~{}\phi$ is a bijection.
By component-wise application ($m$ times), we obtain a bijection
$U^{\mathscr{S}}\rightarrow\mathbb{R}^{\mathscr{S}}$, which we also denote
with $\phi$. We formulate our concepts for any subset $Y$ of
$E_{+}(\mathscr{N},K)$, although we are mainly interested in
$Y=E_{+}(\mathscr{N},K)$.

######  Definition 5.1.3.

For a subset $Y$ of $E_{+}(\mathscr{N},K)$, the set

$\Delta_{\phi}Y:=\\{\phi(x)-\phi(x^{\prime})|x,x^{\prime}\in Y\\}$

is called the difference set of $\phi$-transformed equilibria in $Y$, and its
span $\langle\Delta_{\phi}Y\rangle$ the difference space of $\phi$-transformed
equilibria in $Y$.

In Proposition 6 of [14], it is shown that

$m_{ACR}\leq m-\dim\langle\Delta_{\phi}E_{+}\rangle.$

It follows immediately that we have the following lower bound for the
equilibria variation.

######  Proposition 5.1.3.

Let $(\mathscr{N},K)$ be a kinetic system. Then,
$\dfrac{\dim\langle\Delta_{\phi}E_{+}\rangle}{m}\leq v_{+}(\mathscr{N},K)$.

In general, it is challenging to compute
$\dim\langle\Delta_{\phi}E_{+}\rangle$. However, for PLP systems, this can be
done, and in the next section, we apply it to compute equilibria variation for
weakly reversible, deficiency zero PL-RDK systems.

###  5.2 Equilibria variation in deficiency zero PL-RDK systems

A kinetic system $(\mathscr{N},K)$ is a PLP (positively equilibrated-log
parametrized) system if there is a reference equilibrium $x^{*}$ and a flux
subspace $P_{E}$ of $\mathbb{R}^{\mathscr{S}}$ such that
$E_{+}(\mathscr{N},K)=\\{x\in\mathbb{R}^{\mathscr{S}}_{>}|\log x-\log x^{*}\in
P_{E}^{\perp}\\}$. For any PLP system, Lao et al. [18] showed that ACR is
characterized by the species hyperplane criterion for PLP systems: $X$ is an
ACR species if and only if it is contained in
$H_{X}:=\\{x\in\mathbb{R}_{\mathscr{S}}|x_{X}=0\\}$. This implies that $m-\dim
P_{E}\leq m-m_{ACR}$ and, hence, $1-\dfrac{\dim P_{E}}{m}\leq
v_{+}(\mathscr{N},K)$.

Jose et al. [16] showed that a CLP (complex balanced-log parametrized) system,
i.e., $Z_{+}(\mathscr{N},K)=\\{x\in\mathbb{R}^{\mathscr{S}}_{>}|\log x-\log
x^{*}\in P_{Z}^{\perp}\\}$ is absolutely complex balanced, i.e.,
$E_{+}(\mathscr{N},K)=Z_{+}(\mathscr{N},K)$ if and only if it is both CLP and
PLP and $P_{E}=P_{Z}$. It was shown in [21] that any complex balanced PL-RDK
system is a CLP system with $P_{Z}=\tilde{S}$. Since after Feinberg, any
weakly reversible deficiency zero kinetic system is absolutely complex
balanced, then a weakly reversible deficiency zero PL-RDK system is a PLP
system with $P_{E}=\tilde{S}$. Hence, for any such system,
$1-\dfrac{\tilde{s}}{m}\leq v_{+}(\mathscr{N},K)$, where $\tilde{s}$ is the
kinetic rank of the system.

######  Example 5.2.1.

Fortun and collaborators studied variants of the Anderies et al. [1] model of
the earth’s pre-industrial carbon cycle (see the reaction network below) in
[9] and [12].

${A_{1}+2A_{2}}$${2A_{1}+A_{2}}$${A_{1}+A_{2}}$${2A_{2}}$${A_{2}}$${A_{3}}$$\scriptstyle{R_{1}}$$\scriptstyle{R_{2}}$$\scriptstyle{R_{3}}$$\scriptstyle{R_{4}}$
(5.2.1)

Note that $A_{1}$, $A_{2}$, and $A_{3}$ here stand for land, atmosphere, and
ocean, respectively. Given below are some important numbers and the kinetic
order matrix of the network in 5.2.1.

Figure 2: (a) Some numbers related to the reaction network of the model of the
earth’s pre-industrial carbon cycle system given by Anderies et al. in [1];
(b) kinetic order matrix of the network.

In [12], Fortun and Mendoza showed that the system is dynamically equivalent
to a weakly reversible, deficiency zero system with
$\tilde{S}^{\perp}=\left\langle\begin{pmatrix}-1&\dfrac{p_{2}-p_{1}}{q_{2}-q_{1}}&\dfrac{p_{2}-p_{1}}{q_{2}-q_{1}}\end{pmatrix}\right\rangle$.
Let $R$ denote the ratio $\dfrac{p_{2}-p_{1}}{q_{2}-q_{1}}$. Based on the
value of $R$, one obtains three classes of Anderies system: $AND_{>}(R>0)$
consists of multistationary systems with no ACR species, $AND_{0}(R=0)$
contains only monostationary systems with two ACR species and $AND_{<}$
contains both injective and non-injective systems but also no ACR species. The
variants studied in [10] and [14] belong to $AND_{>}$ and $AND_{0}$,
respectively.

Based on the previous results, we have $v_{+}(\mathscr{N},K)=1$ if
$(\mathscr{N},K)$ is contained in either $AND_{>}$ or $AND_{<}$, and
$v_{+}(\mathscr{N},K)=\dfrac{1}{3}$ if $(\mathscr{N},K)$ is in $AND_{0}$.

###  5.3 Equilibria variation of independent subnetworks

In this section, we discuss equilibria variation for independent subnetworks.
It is important in this regard to differentiate between the two concepts of
subnetworks that we introduced in previous work: embedded and non-embedded. We
hence begin with a review of these concepts and their properties relevant to
absolute concentration robustness (ACR), which is the basis of our concept of
equilibria variation. We then collect some relevant results from previous
publications and add some new details. We conclude by using the results of the
reaction network analysis of metabolic insulin signaling by Lubenia et al.
[19] to illustrate the different concepts and relationships. We in fact show
that the various inequalities between the different equilibria variation
values are sharp, i.e., equality is achieved in various subnetworks of the
insulin system.

####  5.3.1 Review of subnetwork properties

In a subnetwork in a decomposition
$\mathscr{N}=\mathscr{N}_{1}\cup\mathscr{N}_{2}\cup\cdots\cup\mathscr{N}_{k}$,
often, a smaller number of species occurs than in the whole network. If
$\mathscr{N}=(\mathscr{S},\mathscr{C},\mathscr{R})$, we call a subnetwork
$(\mathscr{C}^{\prime},\mathscr{R}^{\prime})$, where
$\mathscr{R}^{\prime}\subseteq\mathscr{R}$ and
$\mathscr{C}^{\prime}=\mathscr{C}|_{\mathscr{R}^{\prime}}$, embedded if its
species space is $\mathscr{S}$ and non-embedded if it has
$\mathscr{S}|_{\mathscr{C}^{\prime}}$ as species space, which we denote with
$\mathscr{S}^{\prime}$.

We use the embedded representation in a decomposition because it conveniently
allows the set operations on equilibria sets. A basic fact is the following
observation:

######  Proposition 5.3.1.

If $X$ is an ACR species of a subnetwork, then $X$ is an element of
$\mathscr{S}^{\prime}$.

This means that we need only one count of ACR species in a subnetwork,
$m^{\prime}_{ACR}$. For non-ACR species and equilibria variation, we have the
following relationships:

######  Proposition 5.3.2.

Let $m^{\prime}=|\mathscr{S}^{\prime}|$ and
$v_{+}(\mathscr{N}^{\prime},K^{\prime})$,
$\tilde{v}_{+}(\mathscr{N}^{\prime},K^{\prime})$ be the equilibria variations
of the non-embedded and embedded subnetworks. Then,

  1. i.

$m-m^{\prime}_{ACR}=(m^{\prime}-m^{\prime}_{ACR})+(m-m^{\prime})$ and

  2. ii.

$0\leq\tilde{v}_{+}(\mathscr{N}^{\prime},K^{\prime})-v_{+}(\mathscr{N}^{\prime},K^{\prime})\leq\dfrac{m-m^{\prime}}{m}$

###### Proof.

$(i.)$ should be read as: the number of non-ACR species in an embedded network
is the number of non-ACR species in the non-embedded network plus the number
of non-occurring species, and $(ii.)$: dividing the equation in $(i.)$ by $m$
and using the inequality
$\dfrac{m^{\prime}-m^{\prime}_{ACR}}{m}\leq\dfrac{m^{\prime}-m^{\prime}_{ACR}}{m^{\prime}}$,
we obtain
$\tilde{v}_{+}(\mathscr{N}^{\prime},K^{\prime})-v_{+}(\mathscr{N}^{\prime},K^{\prime})\leq\dfrac{m-m^{\prime}}{m}$.
On the other hand, the left-hand side, after forming a common denominator, is
now equal to $\dfrac{m^{\prime}_{ACR}(m-m^{\prime})}{mm^{\prime}}\geq 0$,
since all factors are non-negative. ∎

####  5.3.2 Equilibria variation of independent subnetworks

In [18], Proposition 4.4 states that if species $X$ has ACR in
$\mathscr{N}_{i}$ and the decomposition is independent, then $X$ has ACR in
$\mathscr{N}$, i.e.,
$\displaystyle|\mathscr{S}_{ACR,i}|\leq\left|\bigcup_{i=1}^{k}\mathscr{S}_{ACR,i}\right|\leq|\mathscr{S}_{ACR}|$.
We hence have the following corollary:

######  Proposition 5.3.3.

$v_{+}(\mathscr{N},K)\leq\tilde{v}_{+}(\mathscr{N}^{\prime},K^{\prime})$ for
any independent subnetwork $\mathscr{N}^{\prime}$.

###### Proof.

It follows that $m-m_{ACR}\leq m-m^{\prime}_{ACR}$ and dividing both sides
with $m$ results in the claim. ∎

####  5.3.3 The example of metabolic insulin signaling in healthy cells

Lubenia et al. [19] constructed a mass action kinetic realization of the
widely used model of metabolic insulin signaling (in healthy cells) by
Sedaghat et al. [22]. They used the kinetic system’s finest independent
decomposition (FID) to conduct an ACR analysis and showed that $m_{ACR}\geq 8$
for all rate constants (such that the system has positive equilibria) and
$m_{ACR}=8$ for some rate constants. Hernandez et al. [3] confirmed that
$m_{ACR}=8$ for all rate constants with a positively equilibrated system.
Hence,
$v_{+}(\mathscr{N},K)=\dfrac{20-8}{20}=\dfrac{12}{20}=\dfrac{3}{5}=0.60$.

The table below (see Table 5.3.1), which was taken from [19], provides an
overview of the characteristics of the FID subnetworks. Note that, with the
exception of $\mathscr{N}_{1}$, all subnetworks are rank-one systems. Applying
the Meshkat et al. criterion to these 9 subnetworks showed that only the one-
species systems $\mathscr{N}_{2}$ and $\mathscr{N}_{10}$ had ACR for the
species $X_{6}$ and $X_{20}$, respectively. All other subnetworks had no ACR
species. Hence, we have:

  * •

For
$i=3,\dots,9,v_{+}(\mathscr{N},K)=0.60<1=v_{+}(\mathscr{N}_{i},K_{i})=\tilde{v}_{+}(\mathscr{N}_{i},K_{i})$

  * •

For
$i=2,\dots,10,v_{+}(\mathscr{N}_{i},K_{i})=0<0.60=v_{+}(\mathscr{N},K)=\tilde{v}_{+}(\mathscr{N}_{i},K_{i})=\dfrac{19}{20}=0.95$

Network numbers | $\mathscr{N}_{1}$ | $\mathscr{N}_{2}$ | $\mathscr{N}_{3}$ | $\mathscr{N}_{4}$ | $\mathscr{N}_{5}$ | $\mathscr{N}_{6}$ | $\mathscr{N}_{7}$ | $\mathscr{N}_{8}$ | $\mathscr{N}_{9}$ | $\mathscr{N}_{10}$
---|---|---|---|---|---|---|---|---|---|---
Species | 7 | 1 | 4 | 3 | 3 | 2 | 3 | 3 | 4 | 1
Complexes | 7 | 2 | 6 | 2 | 4 | 2 | 4 | 4 | 6 | 2
Reactant complexes | 7 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 4 | 2
Reversible reactions | 5 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1
Irreversible reactions | 4 | 0 | 3 | 0 | 2 | 0 | 2 | 2 | 2 | 0
Reactions | 14 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 4 | 2
Linkage classes | 1 | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 3 | 1
Strong linkage classes | 1 | 1 | 6 | 1 | 4 | 1 | 4 | 4 | 5 | 1
Terminal strong linkage classes | 1 | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 3 | 1
Rank | 6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
Reactant rank | 7 | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 4 | 1
Deficiency | 0 | 0 | 2 | 0 | 1 | 0 | 1 | 1 | 2 | 0
Reactant deficiency | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1
Table 5.3.1: Some numbers of FID subnetworks.

The authors considered also the coarsening
$\mathscr{N}_{A}\cup\mathscr{N}_{B}$, where $\mathscr{N}_{A}$ is the union of
all deficiency zero subnetworks and $\mathscr{N}_{B}$ the union of all
positive deficiency subnetworks. They also showed that all ACR species are
contained solely in $\mathscr{N}_{A}$. Hence, in this case, we have:

  * •

$v_{+}(\mathscr{N}_{A},K_{A})=\dfrac{13-8}{13}=0.38<0.6=v_{+}(\mathscr{N},K)=\tilde{v}_{+}(\mathscr{N}_{A},K_{A})$

  * •

$v_{+}(\mathscr{N},K)=0.6<1=v_{+}(\mathscr{N}_{B},K_{B})=\tilde{v}_{+}(\mathscr{N}_{B},K_{B})$

These examples show that all the inequalities in the propositions above are
sharp.

##  6 Summary and outlook

This study was motivated by Meshkat et al.’s result which allows the analysis
of rank-one MAK systems for possession of stable ACR. The result guarantees a
MAK system such robustness if the two identified structural conditions are
met. These conditions require the system to have an embedded network that
follows certain structures and reactant complexes that differ in just one
species. We attempted to extend this result to more general PLK systems but to
no avail. We found an example illustrating how these conditions do not always
work in a general setting. This means that, unlike other earlier results on
ACR, the conditions in this result do not always insure the existence of ACR
in a PLK system.

On the other hand, we found a subclass of PLK systems where the stable ACR
criterion of Meshkat et al. holds. We call this subclass homogenous monomial
quotients of mass action systems or PL-MMK for short. This subclass is
obtained by utilizing the set of rate constants of a given MAK system and its
modified kinetic order matrix.

We also discovered a property that is necessarily present in a multistationary
rank-one system that possesses an ACR. Specifically, the corresponding result
indicates that such system that has an ACR must have a basis vector generator
of the stoichiometric subspace with 0 as a coordinate in the ACR species. We
illustrated this result using a multistationary system that was given in the
paper of Pantea and Voitiuk.

Finally, we considered the concept of equilibria variation of independent
subnetworks in this paper. We discussed some mathematical relationships of the
equilibria variations of embedded and non-embedded subnetworks. These
relationships were illustrated through the data shown in [19] that Lubenia et
al. used for analyzing metabolic insulin signaling of healthy cells. It is
important to note that ACR species of a subnetwork is always contained in the
corresponding non-embedded subnetwork.

For future studies, one can look at the extension problem of the rank-one ACR
criterion, that is, the identification of further kinetic systems (beyond mass
action) for which it holds. One can also consider the exploration of
relationships between multistationarity classes and the necessary condition
for rank-one mass action systems as well as an extension of the Pantea-Voitiuk
classification beyond mass action. Further, it is also interesting to identify
further kinetics sets for which the general low bound can be computed or a
much sharper alternative as in rank-one multistationary systems can be
derived.

Acknowledgement

D. Talabis, E. Jose, and L. Fontanil acknowledge the support of the University
of the Philippines Los Baños through the Basic Research Program.

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