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Polynomial superlevel set representation
ofthemultistationarity region ofchemical
reaction networks
AmirHosein Sadeghimanesh* and Matthew England*
Introduction
Many problems in applied sciences can be modelled by a parametric polynomial system,
and therefore to solve such problems we must be able to explore the properties of these
systems. In particular, we often seek to identify areas of the parameter space where a
property holds. e contribution of this paper is a new methodology for exploring these.
Motivation: multistationarity regions ofCRNs
We are motivated by the problem of understanding the multistationarity behavior of a
Chemical Reaction Network (CRN). In a CRN, variables represent the concentrations of
the species. ese change as time passes and are studied as part of the ﬁeld of dynami
cal systems. is is of polynomial type when the kinetics is assumed to follow the mass
action rules. e equilibria of such a dynamical system are therefore the solutions to a
system of polynomial equations. However, the coeﬃcients of the terms on the polynomi
als may involve some parameters. ese parameters are usually the rates under which a
reaction occurs and the total amounts (thought of as a dependency on the initial concen
tration of the species).
Both the variables and parameters can only attain nonnegative real values. A network
is called multistationary if there exists a choice of parameters for which the network has
Abstract
In this paper we introduce a new representation for the multistationarity region of a
reaction network, using polynomial superlevel sets. The advantages of using this poly
nomial superlevel set representation over the already existing representations (cylindri
cal algebraic decompositions, numeric sampling, rectangular divisions) is discussed,
and algorithms to compute this new representation are provided. The results are given
for the general mathematical formalism of a parametric system of equations and so
may be applied to other application domains.
Keywords: Polynomial superlevel set, Steady states, Multistationarity, Parameter
analysis
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RESEARCH
Sadeghimaneshand England
BMC Bioinformatics (2022) 23:391
https://doi.org/10.1186/s12859022049216
BMC Bioinformatics
*Correspondence:
Amirhossein.
Sadeghimanesh@coventry.ac.uk;
Matthew.England@coventry.
ac.uk
Research Centre
for Computational Sciences
and Mathematical Modelling,
Coventry University, Coventry, UK
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Page 2 of 26
Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
more than one equilibrium. ere are already many algorithms developed for answering
the binary question of whether a system can exhibit multistationarity [1–9]. e input of
these algorithms is a reaction network and the output is the conﬁrmation or rejection of
the possibility of exhibiting multistationary behavior.
In the case where multistationarity can exist it then becomes important to determine
the parameters where the network has this behavior. ere has been less progress in this
direction in the literature to date: the present paper oﬀers a promising new development
for this problem.
Prior work
Reviewing the state of the art in the literature, we see one vein of work focused on spe
ciﬁc reaction networks, with success following heuristic or manual calculations to ﬁnd a
suitable parameter which may not work in generality [10, 11]. en, in another vein of
work the system of equations for ﬁnding equilibria are solved for many random points
from the parameter space to approximate the region where the network is multistation
ary [12–15].
Recently in [16] a new approach to get a description of the multistationarity region is
proposed. In this method one does not need to solve the system of equations to count
the number of equilibriums. Instead one computes an integral to get the expected num
ber of equilibriums when the parameters are following a random distribution. is
method partitions the parameter region into subsets that are a Cartesian product of
intervals, called hyperrectangles. By choosing the uniform distribution and computing
the average number of equilibriums on these hyperrectangles, one can approximate the
multistationarity region as a union of subhyperrectangles. While eﬃcient and widely
applicable, this list of hyperrectangles does not allow the reader much information or
intuition about the geometry of this region, such as connectedness or convexity.
Contribution
In this work, we propose using polynomial superlevel sets to approximate the union of
the hyperrectangles from [16] as a set that can be described by one polynomial. Polyno
mial superlevel sets are already employed to approximate semialgebraic sets and have
been used in control and robust ﬁltering contexts, see [17, 18]. e polynomial super
level set representation we propose is a more compact representation of the region
compared to a list of many hyperrectangles each described as a Cartesian product of
intervals. Further, to check if a point belongs to the region given in this representation
one can easily just evaluate the polynomial in this point. Further beneﬁts of the polyno
mial superlevel set description of the region will be explored later.
Organization ofthepaper
e organization of this paper is as follows. e mathematical framework of reaction
networks and the deﬁnition of the multistationarity region is given ﬁrst followed by a
section containing the notations regarding parametric functions and deﬁnitions of the
sampling and the rectangular representations of the multistationarity region from [19,
Section2.4].
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
Wethen deﬁne polynomial superlevel sets formally and describe how one can algo
rithmically ﬁnd a polynomial superlevel set representation of a set using the sampling
and the rectangular representations. We demonstrate how to use it to ﬁnd the polyno
mial superlevel set representation of the multistationarity region of a reaction network.
In theﬁnal section we discuss methods that sometimes can speed up computation of the
polynomial superlevel set representation by the help of bisecting algorithms and where
possible, algorithms for computing the expected number of solutions independently of
solving the system itself.
Notations
e cardinality of a set A is denoted by
#(A)
. Let
x∈Z
and
n∈Z\{0}
. In this paper
we deﬁne x modulo n to be n instead of 0 whenever x is a multiple of n. For a function
f:A1→A2
and a point
u∈A2
, the level set of f is denoted by
Lu(f)
and is deﬁned as
{x∈A1f(x)=u}
. For two points
a=(a1,...,an)
and
b=(b1,...,bn)
in
Rn
, the nota
tion [a,b] is used to show the hyperractangle
n
i=1[
a
i
,b
i]
. For a subset S of a hyperrec
tangle
B⊆Rn
, let
Vol(S)
denote the normalized volume of S with respect to B, i.e.
When a random vector
X=(X1,...,Xn)
is distributed by a uniform distribution on a
set
S⊆Rn
, we write
X∼U(S)
. If X is distributed by a normal distribution with mean
µ∈Rn
and variance
σ2∈R>0
, then we write
X∼N(µ
,
σ2)
and we mean that
X1,...,Xn
are identically and independently distributed by
N(µi
,
σ2)
. e expectation of g(X) when
X is distributed by a probability distribution q is denoted by
E
g(X)

X
∼
q
.
Computer information
All computations for the examples of this paper were done on a computer with the
following information. Processor: Intel(R) Core(TM) i710850H CPU
@2.70GHz 2.71 GHz. Installed memory (RAM): 64.0 GB (63.6 GB usa
ble). System type: 64bit Operating System, x64based processor.
e software and programming languages used for the computations reported in this
paper had the following version numbers: Maple 2021, Matlab R2021a, YALMIP,
SeDuMi 1.3, Julia 1.6.2, MCKR 1.0.
Multistationarity region ofchemical reaction networks
In this section, we introduce the concepts of reaction network theory that are needed
throughout the rest of the paper, with the help of a simple gene regulatory network
example.
One can think of a gene as a unit encoding information for the synthesis of a prod
uct such as a protein. First, a group of DNA binding proteins called transcription fac
tors bind a region of the gene called promoter. Now an enzyme called RNA polymerase
starts reading the gene and produces an RNA until it arrives in the terminator region of
the gene. e process until here is called the transcription step. After transcription is
completed, the resulting RNA leaves the nucleus (in eukaryotes) and reaches ribosomes.
In ribosomes, the second step, called translation, gets started. Ribosomes assemble a
Vol
(S)
=
Vol
(
S
)
Vol
(B)
.
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
protein from amino acids using the manual guide written in the RNA. A gene encod
ing of a protein recipe is said to be expressed when it gets transcribed to an RNA, and
the RNA translated to the protein. A gene is not always expressed in a constant rate.
ere might be proteins that bind the transcription factors or the promoter region and,
as a result, inhibit the RNA polymerase starting the transcription process. On the other
hand, there might be other proteins in which their binding to the transcription factors or
the promoter region enhances the transcription.
Consider a simple example from [20, Figure2], depicted here in Fig.1a. ere are three
genes with proteins A,B, and C as their ﬁnal products. Denote their concentrations at
time t by [A](t),[B](t) and [C](t) respectively. e concentration of these proteins will
not remain constant all the time, and we have an Ordinary Diﬀerential Equation (ODE)
system describing the variation of the concentrations as time passes, see Fig.1b. Each
protein is degraded with a ﬁrstorder kinetics with the reaction rate constants
kA,d,kB,d
and
kC,d
correspondingly. Protein A activates the expression of the second gene with
MichaelisMenten kinetics with the maximum rate
kB,max
and the Michaelis constant
k−1
B,A
. e third gene gets activated by both proteins A and B together with the product
of two MichaelisMenten kinetics, with maximum rate
kC,max
and Michaelis constants
k−1
C,A
and
k−1
C,B
. e ﬁrst gene gets expressed by the rate
kA,max
in the absence of protein C,
and protein C has an inhibitory eﬀect on the expression of the ﬁrst gene, captured by the
denominator
(1+kA,C[C](t))
in the rate expression.
A solution to the system
d[X
i
](t)
dt =0
(where the
Xi
s are A,B and C) is called an equilib
rium of the ODE system. Since the concentration of the proteins can only be nonnega
tive real numbers, the complex or negative real solutions are not relevant. Sometimes we
may only consider the positive solutions, for example, if a total consumption of a protein
is not possible or of no interest. erefore by steady states we mean positive solutions to
the system of equations
d[X
i
](t)
dt =0
. e equations in this system are called the steady
state equations.
Now we are ready to deﬁne a reaction network formally. A reaction network, or a net
work for short, is an ordered pair,
N=(S
,
R)
where
S
and
R
are two ﬁnite sets called
the set of species and the set of reactions. In our example,
S={A,B,C}
and
R
contains
six reactions: three gene expressions and three protein degradations. To each network,
an ODE is attached with concentration of the species as its variables and the constants
of the reaction rate expressions as its parameters. In our example, we have 3 suchvari
ables and 10 parameters. To ﬁx the notation assume
S={X1,...,Xn}
and that there are
r constants involved in the reaction rate expressions. en we use
xi
instead of
[Xi](t)
Gene 1 Gene 2 Gene 3
(a)
d
[
A
](
t
)
dt =kA,max
·
1
1+kA,C ·[C](t)
−kA,d ·[A](t)
d[B](t)
dt =kB,max
·
kB,A·[A](t)
1+kB,A·[A](t)
−kB,d·[B](t)
d[C](t)
dt =kC,max
·
kC,A·[A](t)
1+kC,A·[A](t)
·
kC,B·[B](t)
1+kC,B·[B](t)
−kC,d·[C](t)
(b)
Fig. 1 A regulatory network of 3 genes [20, Figure 2]. a This graph shows the relations between expressions
of the genes. We denote by an inhibitory relation and by
→
a positive relation. b The system of ordinary
diﬀerential equations for the network
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
and
ki
for the ith parameter. Denote by
fi
,
k(x)
the ith steady state equation where
x=(x1,...,xn)
and
k=(k1,...,kr)
.
A network with an inﬂow (injection) or an outﬂow (extraction or degradation) for at
least one of its species is called an open network. e network in Fig.1a is an open net
work because of the presence of the degradation reactions. A network can also be fully
or partially conserved.
Consider the simple single reaction network depicted in Fig.2a. e system of its ODE
equations is given in Fig. 2b. Because
˙
x1+2˙
x3=0
, the linear combination
x1+2x3
should be constant with respect to the time. erefore there exists a positive constant
T1
such that the relation
x1+2x3=T1
holds. Similarly there exist two other positive con
stants
T2
and
T3
such that
x1+2x4=T2
and
x2+2x3=T3
. e values of
T1
,
T2
and
T3
can be determined by the initial conditions of the ODE system. ese linear invariants
imply that three of the steady state equations are linearly redundant and can be replaced
by these three linear invariants which are called conservation laws in CRN theory. e
linear subspace determined by the conservation laws is called the stoichiometric compat
ibility class. For a more detailed deﬁnition of conservation laws see Deﬁnition 1 in [19,
Chapter2]. One should note that the trajectories of the ODE system are conﬁned to
stoichiometric compatibility classes. In this case, one only cares about the steady states
in one stoichiometric compatibility class.
Now we are ready to deﬁne the main concept of interest, multistationarity.
Deﬁnition 2.1 Consider a network with n species and replace redundant steady state
equations by conservation laws if there exist any. Let k stands for the vector of constants
of both the reaction rates and conservation laws and be of the size r. A network is called
multistationary over
B⊆Rr
if there exists a
k∈B
such that
fk(x)=0
has more than
one solution in
Rn
>0
.
Remark 2.2
i) One may also consider nonlinear invariants such as ﬁrst integrals as deﬁned in [21,
Deﬁnition 11].
ii) Note that we are not concerned with the choice of the kinetics such as massaction,
MichaelisMenten, Hill function, powerlaw kinetics and Ssystems, or the form of
2O –
2+2H+k
−−→ O2+H
2O2
(a)
dx1
dt =−2kx2
1x2
2,dx3
dt =kx2
1x2
2
dx2
dt =−2kx2
1x2
2,dx4
dt =kx2
1x2
2
(b)
Fig. 2 A simple example of a closed network consisting of one reaction. a Two molecules of superoxide and
two hydron atoms react to each other and produce one molecule of dioxygen and a molecule of hydrogen
perixide. The reaction rate here follows the massaction kinetics with the reaction rate constant k. b The
system of ODE equations of the network. The concentrations of
O2
−
,
H+
,
O2
and
H2O2
are denoted by
x1
,
x2,x3
and
x4
respectively
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
the steady state equations such as polynomial or rational functions. erefore the
results of this paper will remain valid and practical for a general reaction network.
From here on the word parameters also includes the constants of the conservation
laws in addition to the reaction rate constants. To answer the question of whether a net
work is multistationary or not one can use one of many algorithms available in the litera
ture, see [1–5, 7, 9] for a few examples. However, to partition the parameter space into
two subsets, one consisting of the choices of parameters for which
fk(x)
has more than
one solution and the other comprising those parameter choices for which
fk(x)
has at
most one solution, is a more laborious task which we tackle in this paper.
Deﬁnition 2.3 Consider a reaction network with the setting and notation of Deﬁni
tion2.1. e set
is called the multistationarity region of the network.
e region B in Deﬁnition2.3 represents the regions of scientiﬁc interest. It will usually
be a hyperrectangle made by the inequality restrictions of the form
ki,min <ki<ki,max
for the parameters. is is because, for example the rate of expression of a gene can not
be any arbitrary positive number but must be limited; or the constant of conservation
laws may be limited from above due to the limitation of the materials in the lab.
Prior state‑of‑the‑art forparametric systems ofequations
Let
fk:Rn→Rm
be a parametric function with
B⊆Rr
as its parameter region and u a
point in
Rm
. For each choice of the parameters
k⋆∈B
, the system
fk
⋆
(x)=u
is a non
parametric system of equations. One can solve this system and look at the cardinality of
the solution set. For diﬀerent choices of
k⋆
, this number can be diﬀerent. erefore we
deﬁne a new function
u
f:B→Z≥
0
∪ {∞}
sending
k∈B
to
#
L
u
(f)
, i.e. the size of the
level set of
fk
(the set of points in
Rn
which
fk
maps to u). Now one can partition B into
the union of level sets of the map
u
f
. For a general form of
fk(x)
, ﬁnding
L
i
(�u
f)
is a hard
question.
CAD withrespect todiscriminant variety
In the case where f
k
(x)
∈R
(k)
[
x
]m
and A and B are semialgebraic sets there are a
variety of tools which can be employed, see for example [22]. In the literature, the
approach used most commonly (e.g. [10, 23, 24]) is a Cylindrical Algebraic Decom
position (CAD) computed with respect to the discriminant variety. For a full descrip
tion of this technique we refer the reader to [25, 26] or the short sketch of the main
idea in [24, section3]. Briefly: the discriminant variety of the system
fk(x)
with the
domain and codomain restrictions on the semialgebraic sets A and B is the solution
set to a new set of (nonparametric) polynomial equations with k as its indetermi
nants. This new set of polynomials can be computed algorithmically for example
using Gröbner bases and elimination theory. Then CAD with respect to the discri
minant variety decomposes B into a finite number of connected semialgebraic sets
{
k
∈
B

#
f
−1
k
(0)
∩
R
n
>0≥
2
}
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
called cells. Each cell has intersection with only one
L
i(�
u
f)
and therefore
L
i(�
u
f)
can
be expressed as union of a finite number of cells with an exact description of their
boundaries.
As we see later in this section, in many cases one is only interested in open
cells (i.e. only those cells which have full dimension [27]). A Maple package,
RootFinding[Parametric] has implemented an algorithm to compute the
open CAD with respect to the discriminant variety of a system of parametric poly
nomial equations and inequalities [28]. From here on in this paper by CAD we mean
such an open CAD with respect to the discriminant variety.
Both the computation of the discriminant variety and the subsequent decompo
sition involve the use of algorithms with doubly exponential complexity which can
cause problems. The number of cells in the decomposition will grow doubly expo
nentially in the number of parameters of
fk(x)
[29]; and even computation of the
discriminant variety itself before any decomposition can be infeasible for moderate
examples, see e.g. [24]. This makes CAD impractical for studying parametric sys
tems of polynomial equations with more than a few variables and parameters.
Approximation bysampling
Another approach adopted by scientists is to solve the system
fk
(
x
)
=u
for many
different choices of
k∈B
[12–15].
Mathematically speaking, this means that B is replaced by a finite set. Then each
L
i
(�u
f)
is expressed as a subset of this finite set. This approach hereafter is referred
as the sampling representation approach. In contrast with the CAD approach which
provides an exact description of
L
i
(�u
f)
, the sampling representation approach pro
vides an approximation. Note that there are different ways to choose the sample
parameter points for the sampling representation. One way is to arrange all points
equally distanced like a grid, and another way is to randomly sample from a distribu
tion such as the uniform distribution on B, which is the one used in this paper. For
an example of a case where a sampling representation with gridlike parameter sam
pling is used see [10, Figures7–12].
Since we are motivated from the application, we should note that in a lab, it is usu
ally not possible to design the experiment so that the parameter values are exactly
the numbers that we decide. Therefore when the experiment is designed to have
k=k⋆
, what happens is that k is a point in a neighborhood of
k⋆
and not necessarily
k⋆
itself. This can happen for example because of errors coming from the measure
ment tools or the noise from the environment. In such cases picking a point close to
the boundaries of
L
i(�
u
f)
could lead to a different result than what the experimental
ist expects, if errors or noise push it over the boundary.
Rectangular representation
A diﬀerent discretization can be done using a rectangular division of B. For example if
B is a hyperrectangle [a,b] then a grid on B is achieved by dividing B along each axis to
equal parts. en for each subhyperrectangle of B in this rectangular division we assign
the average of the number of solutions of
fk(x)=u
for several choices of k coming from
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
the subhyperrectangle. is approach hereafter is referred as the rectangular represen
tation approach. See Fig.4 to compare the three approaches visually.
Example
Consider the gene regulatory network in [30, Figure3B], depicted here in Fig.3a with
the ODE system in Fig.3b. is network has one conservation law,
x1+x4=k8
. ere
fore we consider the system of equations obtained by the ﬁrst three steady state equa
tions in the ODE system and the conservation law to study the multistationarity of this
network. For illustration purpose we ﬁx values of all parameters other than two so we
can plot the multistationarity region in 2 dimensions. In [30, Figure4] the reaction rate
constants other than
k3
were ﬁxed to the values listed below.
We ﬁx the values of all parameters other than
k3
and
k8
to these values also.
(1)
k1
=
2.81, k2
=
1, k4
=
0.98, k5
=
2.76, k6
=
1.55, k7
=
46.9.
Xk1
−−→ X+P
Pk2
−−→ 0
2Pk3
−−
−−
k4
PP
X+PP k5
−−
−−
k6
XPP
XPPk7
−−→ XPP+P
(a)
dx1
dt =−k5x1x3+k6x4,
dx2
dt =k1x1−k2x2−2k3x2
2+2k4x3+k7x4,
dx3
dt =k3x2
2−k4x3−k5x1x3+k6x4,
dx4
dt =k5x1x3−k6x4
(b)
Fig. 3 A bistable autoregulatory motif presented in [30, Figure 3B]. a X is a gene, P is a protein that can form a
dimer PP and then binding to X. The gene X will get expressed and produce P in both forms X and XPP. Finally
there is a degradation of P. b The ODE system of the gene regulatory network in a. The variables
x1,x2,x3
and
x4
are standing for the concentration of the species X, P, PP and XPP respectively
Fig. 4 Three representations of the multistationarity region of the network in Fig. 3a after ﬁxing all parameter
values other than
k3
and
k8
to the values in (1). a CAD gives the exact boundary of
L
1(�
0
f)
and
L
3(�
0
f)
. The
ﬁrst one is colored by sky blue and the later one with yellow. b A sampling representation of the parameter
region
B=[(0.0005, 0),(0.001, 2)]
by 1000 random points sampled from uniform distribution on B. 78 of
these points belong to
L
3(�
0
f)
and are colored yellow. The other 922 points belong to
L
1(�
0
f)
and are colored
by sky blue. c A rectangular representation of B. Each subrectangle is colored with respect to the average
number of solutions for 10 random points sampled from the uniform distribution on the subrectangle. The
color bar of the ﬁgure is in the right side
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
Let B be the rectangle made by the constraints
0.0005 <k3<0.001
and
0<k8<2
.
Using the RootFinding[Parametric] package of Maple we get the exact
description of the multistationarity region of the network, in 0.12 s, depicted in
Fig.4a.
A sampling representation of the multistationarity region is found by solving the
system of the equations for 1000 points
(k3,k8)
sampled from the uniform distribu
tion on [(0.0005,0),(0.001,2)]. We used the vpasolve command from Matlab to
solve the system numerically. e Matlab code to generate this sampling representa
tion took 154s to run, with the output visualised in Fig.4b.
A rectangular representation is given by dividing [(0.0005, 0), (0.001, 2)] to 100
equal subrectangles and then solving the system for 10 points
(k3,k8)
sampled from
the uniform distribution on each subrectangles. e subrectangles are colored with
respect to the average number of solutions. is computation also was done by Mat
lab and took 166s, with the output visualised in Fig.4c.
Polynomial superlevel set representation
Superlevel sets
Deﬁnition 4.1 Consider
f:Rn→R
, an arbitrary function. For a given
u∈R
a super
level set of f is the set of the form
When
u=1
we drop the index and write only U(f). Naturally, a polynomial superlevel set
is a superlevel set of a polynomial.
Polynomial sublevel sets are deﬁned similarly as in Deﬁnition4.1 with the only diﬀer
ence the direction of the inequality. However, in this paper, we only focus on super
level sets. For
d∈Z≥0
let
Pd
denote the set of polynomials of total degree at most d. A
sum of squares (SOS) polynomial of degree 2d is a polynomial
p∈P2d
such that there
exist
p1,...,pm∈Pd
so that p
=m
i=1
p
2
i
. We denote the set of SOS polynomials of
degree at most 2d by
2d
.
eorem4.2 ([17, eorem2]) Let
B⊆Rn
be a compact set and K a closed subset of B.
For
d∈N
deﬁne
en there exists a polynomial
pd∈Sd
such that
Furthermore
limd→∞ Vol(U(pd)−K)=0
.
Given a pair (B,K) where
B⊆Rn
is a compact set and
K⊆B
a closed set, and given
d∈N
; we call the polynomial superlevel set U(p) (with p being the polynomial
pd∈Sd
Uu(f)={x∈Rnf(x)≥u}.
Sd={p∈Pdp≥0 on B,p≥1 on K}.
B
pd(x)dx
=
inf
B
p(x)dx

p
∈
Sd
.
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
found in eorem4.2) the PSS representation of
K⊆B
of degree d. When K is a semi
algebraic set, one can ﬁnd
pd
numerically using a minimization problem subject to some
positivity constraints [18, Equation13].
Let
B=[aB,bB]
and
Ki=[aKi,bKi],i=1, ...,m
be some hyperrectangles in
Rn
such
that
K:= ∪m
i=1Ki⊆B
. By solving a similar optimization problem it is possible to ﬁnd the
PSS representation of
K⊆B
. Let
d∈N
. e goal is to ﬁnd the coeﬃcients of a polyno
mial of degree d such that
B
p(x)
dx
becomes minimum subject to some conditions.
Before presenting the constraints, let us look at the target function. A polynomial p(x) of
degree d can be written as
α
∈
Nn
d
cαx
α
. Here
Nn
d
is the set of
α=
(α1,...,αn)
∈Zn
≥0
such that
n
i=1
α
i≤d
. Now the integral can be simpliﬁed as below.
Since
B
x
αdx
are constant real numbers independent of the coeﬃcients of the polyno
mial, the target function is a linear function on the coeﬃcients of p(x) which are the
variables of the optimization problem.
Now let us look at the constraints. First of all p(x) has to be nonnegative on B. is can
be enforced by letting
where
r=⌊
d
2⌋
the largest integer less than or equal to
d
2
. Secondly we need
p(x)≥1
on
K or in other words
p(x)−1≥0
on K. is holds if and only if
p(x)−1≥0
on each
Ki
.
erefore for every
i=1, ...,m
one more constraint of the shape (2) has to be added:
Recall Deﬁnition2.3: the multistationarity region of a network is in fact a superlevel set,
U
2(�
0
f)
. e goal is to ﬁnd a PSS representation of the set
U
2(�
0
f)
. One way to accom
plish this goal is to ﬁnd a rectangular representation of the multistationarity region and
then solve the above mentioned SOS optimization problem. e next example illustrates
this idea. To tackle it we use a Matlab toolbox called YALMIP [31, 32] which can receive
an SOS optimization problem, process it and use other solvers to solve it. For the solver
to be used by YALMIP, we chose SeDuMi [33].
Examples
We continue with the example fromFig.3. Consider the rectangular representation of the
multistationarity region of that example given in Fig.4c. To ﬁnd the PSS representation of
B
p(x)dx =
B
α∈Nn
d
cαxαdx
=
α∈Nn
d
cαB
xαdx
=
α
∈
Nn
d
B
xαdxcα
.
(2)
p(x)−
n
j
=1
sB,j(x)xj−aB,jbB,j−xj∈�2r,sB,j∈�2r−2,j=1, ...,n
,
p(x)−1−
n
j
=1
sKi,j(x)xj−aKi,jbKi,j−xj∈�2r,sKi,j∈�2r−2,j=1, ...,n
.
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
this set, we let
B=[(0.0005, 0),(0.0010, 2)]
and K be the union of rectangles
Ki
s such that
their associated number is greater than or equal to 2. From the 100 subrectangles of B, 9 of
them satisfy this condition. ese subrectangles are colored orange in Fig.5. We use the
YALMIP and SeDuMi packages of Matlab to solve the SOS optimization discussed before
this example. To report the computation time we add the two times reported in the output of
YALMIP: the “yalmiptime” and “solvertime”. It takes about 2s to get the coeﬃcients
of the polynomial p of the PSS representation of degree 2. Figure5 shows the plot of U(p).
Unfortunately the same code does not produce a better approximation when we increase
d, the degree of p, from 2 to 4, 8 or 16. e output from Matlab gives similar ﬁgures in these
cases as Fig.5.
Consider another gene regulatory example from [19, Chapter2]. To avoid lengthening
the text, we only reproduce the system of equations needed to study the multistationarity of
the network:
We ﬁx all parameters other than
k7
and
k8
to the following values coming from Equation
(2.10) of [19, Chapter2]:
(3)
k
1
x
7
x
5
−k
5
x
1
=0k
2
x
8
x
6
−k
6
x
2
=0
k
3x1−k7x3=0k4x2−k8x4=0
k
9x7x4−k11x9=0k10 x8x3−k12x10 =0
k
13x9x4−k15 x11 =0k14 x10x3−k16 x12 =
0
x
5=k17 x6=k18
x7+x9+x11 =k19 x8+x10 +x12 =k20
.
(4)
k
1
=k
2
=k
3
=k
4
=1, k
5
=0.0082, k
6
=0.0149,
k
9=k10 =0.01, k11 =k12 =10000, k13 =2,
k
14 =25, k15 =1, k16 =9, k17 =k18 =k19 =1,
k20 =
4.
Fig. 5 The PSS approximation of the multistationarity region for the network in Fig. 3a of degree 2 obtained
by the information of Fig. 4c. The union of orange colored subrectangles is considered as the initial
approximation of the multistationarity region obtained by the rectangular representation and chosen as the
set K. The yellow colored area is the diﬀerence of
U(p)−K
. Remember that the PSS representation of the
multistationarity region is the yellow region which contains the orange rectangles as well
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
We reproduced the rectangular representation of the multistationarity region of this
network by Matlab, as shown in Fig.6a. From the 100 subrectangles in total, for 28 of
them the average number of steady states is greater than or equal to 2. Using YALMIP
and SeDuMi it took between 1 and 2s to get the polynomials of the PSS representations
of degrees 2, 4 and 8 represented in Fig.6b–d respectively. For this example, the PSS
approximation of degree 4 looks diﬀerent than that of degree 2, but for degree 8 the plot
looks similar to degree 4.
Advantages ofPSS representation overrectangular representation
It is natural to ask why one should ﬁnd a PSS representation of the multistationar
ity region using the rectangular representation given one already has the rectangular
representation? Let
B⊆Rr
be the parameter region of the form of a hyperrectangle,
and
K⊆B
be the multistationarity region. In the rectangular representation we have
K≃∪
m
i=1Ki
where
Ki=[aKi,bKi]
are hyperrectangles. In the PSS representation we
have
K≃U(p)
where p is a polynomial of degree d.
Fig. 6 PSS representations of diﬀerent degrees for the mulistationarity region of the LacITetR gene
regulatory network using the information we got from the rectangular representation. The union of orange
colored subrectangles is considered as the approximation of the multistationarity region obtained by the
rectangular representation and chosen as the set K. The yellow colored area is the diﬀerence of
U(p)−K
. a
The rectangular representation of multistationarity region of the network with the system of equations given
in (3) and some parameters being ﬁxed by the values in (4). b–d PSS representations of the multistationarity
region of degrees 2, 4 and 8 respectively obtained by the information of Fig. 6a
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1 When
r≥4
, plotting K is impossible. In order to save or show the rectangular repre
sentation one needs to use a matrix of size
(m)×(2r)
, where each row stands for one
Ki
and the ﬁrst r columns have the coordinates of the point
aKi
and the second r col
umns correspond to the coordinates of the point
bKi
. However for the PSS represen
tation one needs to use only a vector of size
r+d
r
=
d
i
=
0
r−1+i
r
−
1
, where
r−1+i
r
−
1
entries are coeﬃcients of the terms of degree i. e terms are ordered
from smaller total degree to larger and for the terms of the same total degree we use
the lexicographic order.
2 To test if a point
k⋆∈B
belongs to K using the rectangular representation one should
check m conditions of the form
k⋆∈Ki
which means verifying an inequality on each
coordinate of the point, i.e.
a
K
i
,j
≤
k
⋆
j≤
bK
i
,
j
. If one of the conditions
k⋆∈Ki
is posi
tive, then there is no need to check the rest, otherwise all should fail to conclude that
k⋆ ∈ K
. However, using the PSS representation one needs to check only one condi
tion of an evaluation form, i.e.
p(k⋆)≥1
.
3 Recall from the last paragraph ofthe “Approximation by sampling” Section explain
ing that parameters near the boundary of the multistationarity region are not suita
ble choices for an experimentalist. To check the distance of a point
k⋆∈B\K
to the
boundaries of K using the rectangular representation one should ﬁnd distance of
k⋆
from boundaries of each
Ki
and then taking the minimum. However, using the PSS
representation, in both cases of
k⋆∈B\K
or
k⋆∈K
, one just needs to ﬁnd the dis
tance of
k⋆
from the algebraic set deﬁned by
p(k)−1=0
, for example by Lagrange
multipliers, as in the next section.
To conclude, if
r+d
r
is considerably smaller than 2mr, then storing the PSS repre
sentation instead of the initial rectangular representation will save memory without
loosing information about the multistationarity region.
Approximating thedistance ofparameter point fromtheboundary
To illustrate how to approximate distance of a parameter point from the boundaries of
the multistationarity region using a PSS representation we continue with the example
fromFig.3.
Let p be the polynomial of degree 2 in two variables
k7
and
k8
corresponding to U(p)
in Fig.6b. We will approximate distance of the point
k⋆=(0.08, 0.02)
from the bound
ary of the multistationarity region by the distance of
k⋆
from the algebraic set deﬁned
by
p(k7,k8)−1=0
. is question is equivalent to minimizing the Euclidean distance
function of a point
(k7,k8)
from the point
k⋆
subject to the constraint
(k7,k8
)
∈L1
(
p)
.
e target function is
(k7−
0.08
)
2
+(k8−
0.02
)
2 which gets minimized if and only
if
(k7−0.08)2+(k8−0.02)2
gets minimized. An elementary way to solve this minimi
zation problem is to use the method of Lagrangian multipliers [34, Chapter7, eo
rem1.13]. Deﬁne
F
(k7,k8,)=(k7−0.08)
2
+(k8−0.02)
2
+
p(k7,k8)−1
.
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Now we must ﬁnd the critical points of
F(k7,k8,)
. So we should solve the system of
equations obtained by
∂F
∂k7
=
∂F
∂k8
=
∂F
∂
=0
. It takes 0.167s to solve this system of equa
tions by the solve command in Maple. It has 4 solutions, from which 2 belong to the
rectangle
B=[(0, 0),(0.1, 0.1)]
and the minimum of the target function is obtained
at the point (0.04499222669, 0.04161251428). e distance of this point from
k⋆
is
0.04114176669.
Constructing aPSS representation fromasampling representation
It is not necessary to have a rectangular representation to get the PSS representation.
Let
B=[aB,bB]
be a hyperrectangle and
K={a(1)
,
...
,
a(m)}
a ﬁnite set. Let
d∈N
,
and the goal be to ﬁnd the coeﬃcients of a polynomial of degree d such that
B
p(x)
dx
becomes minimum subject to some conditions. We already saw that the target func
tion is linear. e constraint
p≥1
on K can be enforced by
p(a(i))≥1
for every i,
which are linear constraints. e positivity of p on B can be enforced by Eq.(2) or by
adding a large enough number of random points from B and putting the constraint
p(a)>0
. e later idea makes the problem solvable by any common linear program
ming tool. However, here we still use Eq.(2).
Let us illustrate this with our ongoing example. Consider the sampling representa
tion of the multistationarity region of the network of Example3.4 given at Fig.4b.
To ﬁnd the PSS representation of this set, we let
B=[(0.0005, 0),(0.001, 2)]
and K to
be the set of points for which the system
fk(x)=0
had more than one positive solu
tion. ere are 1000 points from which 78 of them are parameter choices where the
network has three steady states. Using the YALMIP package of Matlab, it takes less
than a second to get the coeﬃcients of each of the polynomials p of the PSS represen
tation of degrees 2, 6 and 10. Figure7a–c show the plots of U(p) for degrees 2, 6 and
10 respectively. e plot for degree 6 actually looks worse than the plot for degree 2
(further away from the actual result in Fig.4a), although the one for degree 10 looks
a little better. In all these cases YALMIP ﬁnished the computations with a message
‘Numerical problems (SeDuMi)’ indicating that the solver found the prob
lem to be numerically illposed. Rescaling the parameter region of interest, B, to
[(0,0),(1,1)] and then transforming the PSS polynomial back to the original B allows
a better PSS approximations via YALMIP. For degrees 2 and 6 the numerical problem
message is avoided but for degree 10 it remains. e results are shown in Fig.7d–f.
To demonstrate that the number of free parameters need not be 2 to be able to com
pute the PSS representation, we repeated the above process with all 8 parameters of the
system being free in the following hyperrectangle:
Solving the system at 1000 random points uniformly chosen from B takes the same
amount of time as solving the system at 1000 random points with only 2 of their
B=[(1, 0, 0.0005, 0, 1, 1, 40, 0),(4, 2, 0.001, 2, 4, 3, 50, 2)].
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
coordinates varying took time in the previous case. It took about 1.5s to compute the
45 coeﬃcients of the polynomial of the PSS representation of degree 2. e polynomial
found is the following.
p=0.0232593410k
2
1−0.2965863774k
2
2+559.9760177000k
2
3
−0.0618678914k2
4+0.0098179292k2
5−0.0061375489k2
6
−0.0036871825k2
7−0.0716829116k2
8−0.0110701085k1k2
−5.3373104650k1k3+0.0186292240k1k4+12.5539028600k2k
3
+0.0051935911k1k5−0.0820767495k2k4+0.0250525965k1k6
−0.0649618436k2k5−8.2165668850k3k4+0.0092355627k1k7
−0.0310047257k2k6−4.2926137530k3k5−0.0527315151k1k8
+0.0017547696k2k7−2.7647028150k3k6+0.0288291263k4k5
+0.0785940967k2k8−0.4049844472k3k7+0.0536360086k4k6
+12.6014929300k3k8−0.0091147734k4k7+0.0198800552k5k
6
−0.0567174819k4k8+0.0009448114k5k7−0.0190068615k5k8
+0.0102920949k6k7+0.0316830973k6k8+0.0022468043k7k8
−0.6043381476k1+0.7152344469k2+0.3287484744k4
+36.3811699100k3−0.0617318336k5−0.5940226892k6
+
0.2987521874
k7+
0.2558883329
k8−
5.0441247030.
Fig. 7 PSS representation of diﬀerent degrees of the mulistationarity region of the network of Example 3.4
inside the hyperrectangle
B=[(0.0005, 0),(0.001, 2)]
using the information we got from the sampling
representation of the multistationairy region. The orange colored points are the points with three steady
states and their union is considered as approximation of K. The yellow colored area is the diﬀerence of
U(p)−K
. One expects to see that this diﬀerence is getting smaller as the degree increases. However, the
Matlab code that we wrote using YALMIP and SeDuMi does not behave as expected. a–c gives the PSS
representation of the original problem of degrees 2, 6 and 10 respectively. d–f gives the PSS representation of
those degrees for the problem after after rescaling the parameters for better numerical behavior via YALMIP
and SeDuMi
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Advantages ofPSS representation overthesampling representation
Let us addresswhy oneshould ﬁnd a PSS representation of the multistationarity region
using the sampling representation if one already has a sampling representation. Let
B⊆Rr
be the parameter region of the form of a hyperrectangle,
K⊆B
be the multista
tionarity region. In the sampling representation we have
{a(1),...,a(m)}⊆K
. In the PSS
representation we have
K≃U(p)
where p is a polynomial of degree d.
1 When
r≥4
, plotting K is impossible. In order to save or show the sampling repre
sentation one needs to use a matrix of the size
m×r
, where each row stands for one
point
a(i)
and the columns correspond to the coordinates of the points. However, for
the PSS representation one needs to use a vector of the size
r+d
r
, as explained
inthe ﬁrst item of the “Advantages of PSS representation over rectangular represen
tation” section1.
2 To test if a point
k⋆∈B
belongs to K using the sampling representation is not a
straightforward task. However, using the PSS representation one needs to verify only
one condition of the evaluation form,
p(k⋆)≥1
.
3 To compute the distance of a point
k⋆∈B
to the boundaries of K, using the sampling
representation, if
k⋆ ∈ K
, one should compute the distance of
k⋆
from each point in
the sampling representation of K and then take the minimum. Using the PSS repre
sentation, whether
k ∈ K
or not, one just needs to ﬁnd the distance of
k⋆
from the
algebraic set deﬁned by
p(k)−1=0
.
In a typical example from CRN theory, r is usually much higher than 2, and therefore
item 1 is really important. When
r+d
r
is lower than rm, one can use less memory
by saving the PSS representation instead of keeping all the points of the sampling rep
resentation in the memory. Further, as items 2 and 3 show, this will not cause a loss of
information about the multistationarity region.
More involved example
We showed earlierthat the PSS representation can be generated for examples with
a higher number of parameters than two. ere, we let all 8 parameters of the net
work in Fig.3a to be free and found the PSS representation of degree two in 8 inde
terminants. Now we bring another such example which also serves to emphasize
Remark2.2 item (ii): that to have a PSS representation of the multistationarity region,
one does not need to have the right hand side functions of the ODE system to be of
polynomial or even rational functions.
Consider a gene expression system with 4 species
Xi
,
i=1, ...,4
where these species
can be mRNA or protein molecules or other relevant factors, with the ODE system
as in Fig.8a which was introduced in [35, Figure2]. As one can see the right hand side
functions involve at least a square root, and as a result this system is not polynomial, or
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Sadeghimaneshand England BMC Bioinformatics (2022) 23:391
even deﬁned by rational functions. Let us ﬁx the parameter values other than the three
degradation rates
βi
,
i=2, 3, 4
to the following values, chosen the same as in [35]:
In [35] it is shown that for
(β2,β3,β4)=(5,