Bistability and oscillations in chemical reaction networks

J. Math. Biol.DOI 10.1007/s00285-008-0234-7 Mathematical Biology

Bistability and oscillations in chemical reactionnetworks

Mirela Domijan · Markus Kirkilionis

Received: 4 April 2008© Springer-Verlag 2008

Abstract Bifurcation theory is one of the most widely used approaches for analysisof dynamical behaviour of chemical and biochemical reaction networks. Some ofthe interesting qualitative behaviour that are analyzed are oscillations and bistability(a situation where a system has at least two coexisting stable equilibria). Both phe-nomena have been identified as central features of many biological and biochemi-cal systems. This paper, using the theory of stoichiometric network analysis (SNA)and notions from algebraic geometry, presents sufficient conditions for a reaction net-work to display bifurcations associated with these phenomena. The advantage of theseconditions is that they impose fewer algebraic conditions on model parameters thanconditions associated with standard bifurcation theorems. To derive the new condi-tions, a coordinate transformation will be made that will guarantee the existence ofbranches of positive equilibria in the system. This is particularly useful in mathemat-ical biology, where only positive variable values are considered to be meaningful.The first part of the paper will be an extended introduction to SNA and algebraicgeometry-related methods which are used in the coordinate transformation and set upof the theorems. In the second part of the paper we will focus on the derivation ofbifurcation conditions using SNA and algebraic geometry. Conditions will be derivedfor three bifurcations: the saddle-node bifurcation, a simple branching point, bothlinked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatorybehaviour. The conditions derived are sufficient and they extend earlier results from

We have added a dedication of the paper to K. Gatermann.

M. Domijan (B) · M. KirkilionisMathematics Institute, University of Warwick, Coventry CV4 7AL, UKe-mail: [email protected]

M. Kirkilionise-mail: [email protected]

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M. Domijan, M. Kirkilionis

stoichiometric network analysis as can be found in (Aguda and Clarke in J ChemPhys 87:3461–3470, 1987; Clarke and Jiang in J Chem Phys 99:4464–4476, 1993;Gatermann et al. in J Symb Comput 40:1361–1382, 2005). In these papers some nec-essary conditions for two of these bifurcations were given. A set of examples willillustrate that algebraic conditions arising from given sufficient bifurcation conditionsare not more difficult to interpret nor harder to calculate than those arising from nec-essary bifurcation conditions. Hence an increasing amount of information is gainedat no extra computational cost. The theory can also be used in a second step for asystematic bifurcation analysis of larger reaction networks.

Keywords Polynomial differential equations · Bifurcation theory · Extreme currents

Mathematics Subject Classification (2000) 92C45 · 37G10 · 14A10

1 Introduction

Chemical reaction networks have been extensively studied for their interesting andcomplex dynamic behaviour [16–18,28,29,34,36]. However, often on account of theircomplex structures, analysing their dynamic behaviour via bifurcation analysis hasproven to be difficult. Hence advanced methods such as stoichiometric network analy-sis (SNA) have been introduced. In this paper the ideas from stoichiometric networkanalysis will be used to formulate sufficient conditions for certain networks display-ing some well-known dynamic behaviour. Such results are helpful in order to gaina better understanding of large reaction networks, or sub-networks that drive certainbehaviours. The results derived in this paper should also simplify the inverse process ofmodeling, i.e. the search for reaction networks of a specific class defined by exhibitinga desired qualitative behaviour.

In this paper two different dynamic behaviours are of interest: bistability and oscil-lations. The former is a situation where a system has two stable equilibria. Both arecentral features of numerous chemical and biochemical processes. The work presentedhere is motivated by several seminal papers by Clarke [7–9] where stoichiometric net-work analysis was introduced to confirm whether some well-known chemical systemshave the potential to display oscillations. The crux of Clarke’s idea was to observe thedynamics of the system in the reaction rate space, rather than in the species concen-tration space. This work was revolutionary, because it significantly simplified modelanalysis. Instead of searching for steady state solutions where all concentrations areunknown and parameters are not specified, their analysis was restricted to dealing onlywith reaction rates. Clarke and his collaborators extended this idea in [1,10] wherethey derived several conditions on the reaction rates. By assuming the existence of apositive concentration steady state, its linearisation will have either a zero eigenvalue,or a pair of purely imaginary eigenvalues. These eigenvalue conditions will then forma necessary condition for the positive concentration steady state to be a saddle-node orHopf bifurcation point. However, in this series of papers, Clarke and coauthors neverconfirmed the existence of the positive concentration steady state. This was donemuch later in a paper by Gatermann et al. in [21]. Gatermann et al. used toric geome-try theory to prove that Clarke’s analysis of the cone of reaction rates was correct, and

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Bistability and oscillations in chemical reaction networks

also derived conditions that guarantee the existence of positive concentration steadystates, when the system is analysed in the reaction rate coordinates. The theory of toricgeometry will be reviewed in Sect. 3. Gatermann et al. also mentioned the eigenvalueresults published in [10], but they went no further in the notions of bifurcation analysis.

In this paper, it will be shown that the use of SNA and the associated algebraicgeometry concepts lead to a general bifurcation analysis framework for polynomialsystems. In this situation the algebraic information can be used to derive sufficientconditions for all major types of bifurcations. At the same time they guarantee that theequilibria discussed are positive, and hence physiologically meaningful. Our resultsare illustrated with several examples, some taken from the aforementioned papers forthe sake of comparison.

Other important but not related approaches to investigate the qualitative behaviourof reaction systems are given in [12,33]. These papers also derive conditions underwhich a polynomial system might undergo certain bifurcations. But the methods arebased on other algebraic concepts, a combination of graph theory and expansionsof coefficients of the associated characteristic polynomial. An overview to differentgraph theoretical approaches to polynomial reaction systems can be found in [14].

1.1 Chemical reaction systems

A chemical reaction system with r reactions and m reacting species is described by atime-continuous dynamical system derived from reaction schemes. Each reaction canbe written in the form

α1 j S1 + · · · + αnj Snk j→ β1 j S1 + · · · + βnj Sn, j = 1, . . . , r, (1)

where the Si , 1 ≤ i ≤ n, are the chemical species and each k j is the kinetic constantof the j th reaction. Kinetic coefficients take into account all effects on the reactionrate apart from reactant concentrations, for example, temperature, light conditions, orionic strength in the reaction. The coefficients αi j and βi j represent the number ofSi molecules participating in j th reaction at reactant and product stages, respectively.The net amount of species Si produced or consumed by the reaction is named thestoichiometric coefficient and defined by ni j := βi j − αi j . These coefficients arearranged in a stoichiometric matrix, denoted by N . The rate at which the j th reactiontakes place is modeled under the assumption that reactions obey mass-action kinetics.This assumption means that the reaction rate must take the form of a monomial,

v j (x, k j ) = k j

m∏

i=1

xκi ji ,

where κi j is the molecularity of the species Si in the j th reaction. In mass-actionkinetics, the kinetic exponent κi j reduces to being simply αi j . Kinetic exponentsare arranged in a kinetic matrix, denoted by κ . The time evolution of the speciesconcentrations is described by the following initial value problem:

x = Nv(x, k), (2)

x(0) > 0, (3)

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where x(0) are initial species concentrations. Interesting qualitative behaviours suchas bistability and oscillations have been observed in chemical reaction systems ofmass-action type [12,30]. They can be interpreted to result from a bifurcation, i.e. aqualitative change in the behaviour of the system’s solutions when one or more ofthe parameters are varied. Hence, a common approach in identifying such behaviourhas been to derive conditions under which the system is able to undergo an associatedbifurcation [21,33]. In this paper, sufficient conditions for bifurcations will be derived.

The paper is split into two parts. The first part contains an overview of stoichiometricnetwork analysis (Sect. 2) and some eigenvalue results from [1,10]. It also containssome useful notions of algebraic geometry in Sect. 3. Second part of the paper isthe new derivation of sufficient conditions for the saddle-node bifurcation (Sect. 4.1),simple branching point (Sect. 4) and a simple Hopf bifurcation (Sect. 5). In Sect. 4.3, amethod for detecting bistability is discussed. Several examples in each section illustratethe theory.

Part A: algebraic geometry

2 Theory of stoichiometric network analysis

This part of the paper will show how essential algebraic concepts can be associatedto reaction networks. In a further step this will be the basis for introducing higheralgebraic structures, most importantly toric varieties. Stoichiometric Network Analy-sis (SNA), introduced in a seminal paper by Clarke [9], is one of the most prominentmethods of analysing chemical and biochemical networks. Using the assumptionsdescribed in the introduction, the reaction network dynamics takes the form shown in(2). The local existence and uniqueness of the solutions to (2) are guaranteed [46]. Alsoall species concentrations will always stay in the positive orthant of the concentrationspace, namely x(t) ≥ 0 (x(t) > 0) if x(0) ≥ 0 (x(0) > 0) [45]. If rank(N ) = r ≤ n,then the reaction system has (n − r) conservation relations [27]. A conservation rela-tion is a relation between a set of species with their total concentration being preserved.The set of conservation relations takes the form

gTl x = cl , l = 1, . . . , n − r,

where each gTl ∈ ker(N ) and cl ∈ R+ is a constant describing the total conserved con-

centration. The set of conservation relations forms an invariant space [45]. Equilibriaof the network (x0) for chosen set of parameters k0 are determined by the conditions

Nv(x0, k0) = 0, (4)

x0 > 0. (5)

Any such equilibrium clearly depends on the kinetic parameters k ∈ Rp+ , but also

on the constants cl , l = 1, . . . , n − r from the conservation relations. Due to (4) and(5), all stationary reaction rates belong to an intersection of the kernel of N and thepositive orthant of the reaction space,

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v(x0, k0) ∈ {z ∈ R

r |N z = 0, z ∈ Rr+} = K er(N ) ∩ R

nr , (6)

which form a convex polyhedral cone Kv [35]. This cone Kv is spanned by a set ofminimal generating vectors Ei ’s,

Kv ={

t∑

i=1

ji Ei : ji > 0 ∀ i

}.

These generating vectors are unique up to scaling by a positive constant. Theymay be linearly dependent as the number of extreme currents may be greater thandim(ker(N )). Here, as in [9], these vectors will be referred to as extreme currents.Extreme currents decompose the network into minimal steady-state generatingsubnetworks [9]. In the biochemical literature they are interpreted as switching onor off different parts of a metabolic pathway. The influence of a subnetwork on thefull network dynamics (i.e., how much the given subnetwork plays a part in creatinga certain steady state) depends on the constants ji ’s, which are called convex parame-ters. In fact the Jacobian of the network evaluated at any positive steady state, x0, canbe written as a convex combination of contributions from the extreme currents:

Dx Nv(x0, k0) ∈{

M∑

i=1

ji Ndiag(Ei )κT diag(h) : ji > 0, h ∈ R

m+

}. (7)

Here h is a vector defined as the inverses of any equilibrium concentrations x0,and diag(v) is a diagonal matrix with the diagonal entries given by the vector v. Thecontribution of a single extreme current is given by the term

ji Ndiag(Ei )κT diag(h),

which represents the Jacobian of the subnetwork generated by the extreme current Ei ,evaluated at any of its positive steady states. The Jacobian description in the form of(7) comes from a simple observation that for positive steady states x0,

Dx Nv(x0, k0) = Ndiag(v(x0, k0))κT diag(x−1

0 ). (8)

Together with the newly defined parameters, hs = x−1s , s = 1, . . . , n, the convex

parameters j replace the reaction constants k and the conserved mass constants c [9].Detection of bifurcations [1,10] consists of finding regions of the parameter space( j, h) where the Jacobian written in form of (8) could have a zero eigenvalue, or a pairof purely imaginary eigenvalues. One needs to consider the characteristic polynomialdet(λI − J ), where I is the identity matrix and J is the Jacobian parametrized by( j, h):

det(λI − Jac( j, h)) = λn + αn−1( j, h)λn−1 + · · · + α1( j, h)λ + α0( j, h). (9)

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This polynomial has a zero solution (or the Jacobian has a zero eigenvalue, a neces-sary condition for a saddle-node bifurcation) when α0( j, h) = 0. Now Orlando’sFormula in [20] implies that a condition on the (n − 1)-st Hurwitz determinantHn−1( j, h) = 0 is a necessary condition for the characteristic polynomial to have apair of pure imaginary eigenvalues. In [1,10] these conditions are employed to explorethe parameter space ( j, h). The aim of the authors is to reduce the dimension of thesystem parameter space. For example, in [1] the authors perform their analysis on theconvex polytope, which is the normalized version of the convex cone Kv . The normal-ization is a restriction on Kv such that

∑rj=1 v j = 1 for every v = (v1, . . . , vr ) ∈ Kv .

In a similar way no distinction will be made in this paper between points of the coneKv and a ray passing through the point.

3 The associated toric varieties

Behaviour of a chemical reaction system depends on the parameters k’s and c’s. Forthe characterization of a bifurcation, there must exist some information about thebranches of the steady states parametrized by a chosen bifurcation parameter. Forexample, if a system undergoes a Hopf bifurcation there must be a smooth curve ofsteady states parametrized by a bifurcation parameter (we refer to Liu’s theorem inSect. 5). The achievement of this paper is to derive bifurcation conditions in termsof SNA. More precisely, a coordinate transformation will be performed such thatbifurcation conditions can be placed on rays of the convex polyhedral cone, z ∈ Kv ,and also the system parameters. It can be avoided to use any of the concentrationvariables (x). Transformation back to concentration space (and x variables) will bepossible, because given choice of z and parameters k there exists at least one positiveconcentration steady state (x) such that

z = v(x, k) = x0v(x, k), (10)

where the constant x0 is introduced because of the ambiguous length of the ray z (usingthe notation already mentioned between rays and points). This can be done usingtheory from algebraic geometry. The introduction of algebraic geometry to SNA is animportant extension to the bifurcation results formulated by Clarke et al. [1,10]. Whilethese authors derived very important conditions on the eigenvalues of the system viathe reaction rates, they did not prove that for the given combination of currents and achosen bifurcation parameter, there actually exists a positive concentration steady state.Gatermann et al. in [21] were the first to use the concepts from algebraic geometry. Theyshowed that if the ray of the cone belongs to a particular set (which shall be describednext) then there exists a positive solution x . Using algebraic geometry they gave a proofthat the mapping v : R

n+ → Rr+ from the set {x ∈ R

n+ : ∃ ki j > 0 with Nv(x, k) = 0}to the convex polyhedral cone ker(N ) ∩ R

r+ is surjective. This validated Clarke’stheory that the convex polyhedral cone is covered by the image of mapping v.

The theory presented in this section will be illustrated with an example, the modifiedSelkov model of glycolytic oscillations analyzed in [15]. Some SNA analysis andalgebraic geometry analysis has already been applied to this model. Extreme currents

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for this model were calculated in [39], and the deformed toric ideal of an extendedmodel was computed in [38]. The model describes two species interacting throughfive reactions, where S1 denotes the product Fructokinase-1,6-biphosphate (F1,6BP)and S2 denotes adenosine triphosphate (ATP):

x1 = k1x21 x2 + k2 − k3x1,

x2 = −k1x21 x2 + k4 − k5x2.

The stoichiometric matrix and vector of reaction rates are:

N =[

1 1 −1 0 0

−1 0 0 1 −1

]and v(x; k) =

⎡

⎢⎢⎢⎢⎢⎣

k1x21 x2

k2

k3x1

k4k5x2

⎤

⎥⎥⎥⎥⎥⎦. (11)

This means the model consists of three extreme currents:

E1 = (0, 1, 1, 0, 0), E2 = (0, 0, 0, 1, 1) and E3 = (1, 0, 1, 1, 0). (12)

The first two extreme currents describe subnetworks of inflow and outflow ofF1,6BP and ATP, respectively. The third current combines the autocatalytic forma-tion of F1,6BP via ATP with outflow of F1,6BP and inflow of ATP.

Consider the polynomial map v : Rn+ → R

r+. Let x ∈ Rn+ and z ∈ R

r+. The aimis to derive conditions on z that guarantee that z is in the image of the map v, namelyz ∈ I m(v). These conditions can be reinterpreted as conditions depending solely onz variables. In other words, the aim is to rewrite the system

z − v(x, k) = 0 (13)

in terms of another set of equations with solution set (z ∈ I m(v)), but only withz variable terms. For this theory from algebraic geometry is needed. Therefore weintroduce some definitions. One of the basic objects in algebra is a field. Intuitively,a field is a set where one can define addition, multiplication, division and subtractionwith usual properties. A proper definition can be found in any introductory algebraictext book, such as Cox et al. [11]. Some examples of a field are the real numbers,R, the complex numbers C and rational numbers Q. Integers Z do not form a field,because division fails (namely 1 and 2 ∈ Z but 1/2 /∈ Z).

The set of equations in (13) can be rewritten as a set of polynomials (zi − vi (x, k)

i = 1, . . . , r )) whose zero solution set implicitly defines the image of v. This can bedone once further definitions for polynomials and their zero solution sets are given.

Let K[y1, . . . , ym] denote a set of all polynomials of y1, . . . , yn with coefficientsin a field K. Similarly, a set of polynomials with real coefficients is denoted byR[y1, . . . , ym] and a set of polynomials with complex coefficients is denoted by

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C[y1, . . . , ym] and so on. Polynomials, where coefficients are in terms of parame-ters k, belong to a set K(k)[x1, . . . , xn] where K(k) is an extension of a field K. Thispaper will deal with the polynomials in the set Q(k)[x, z] with x = (x1, . . . , xn) andz = (z1, . . . , zr ). A polynomial f ∈ K[y1, . . . , ym] which is split into monomials inthe form of

f = hN + hN−1 + · · · + h0

is called homogeneous of degree l if each hi ∈ K[y1, . . . , ym] , 0 ≤ i ≤ N has totaldegree l. For a set of polynomials f1, . . . , fs in K[y1, . . . , ym], the set

〈 f1, . . . , fs〉 ={

s∑

i=1

hi fi : h1, . . . , hs ∈ K[y1, . . . , ym]}

⊆ K[y1, . . . , ym]

is an ideal generated by f1, . . . , fs . The zero set of solutions of the ideal is called anaffine variety. For 〈 f1, . . . , fs〉,

V ( f1, . . . , fs) = {(a1, . . . , an) ∈ K

n : fi (a1, . . . , an) = 0 for all 1 ≤ i ≤ n}

is the affine variety defined by f1, . . . , fs . For a set of polynomials satisfying (13) theideal can be written as

I = 〈z1 − v1(x, k), z2 − v2(x, k), . . . , zr − vr (x, k)〉 ⊆ Q(k)[x, z]. (14)

This ideal is also called an ideal passing from an image of a map v. This is becauseI is constructed in such a way that I m(V ) belongs to the zero set of I , or the varietyV (I ). More formally, the ideal can be written as

I = { f ∈ Q(k)[x, z] : ∃x ′ ∈ Rn+ with f (x ′, z′) = 0 for all z′ ∈ I m(v)}. (15)

Example 1 The image of v(x, k) is in fact the vector of reaction rates. In the modifiedSelkov model, it comes in the form shown in Eq. (11). The ideal I , with I m(v) as itsvariety is,

I = 〈z1 − k1x21 x2, z2 − k2, z3 − k3x1, z4 − k4, z5 − k5x2〉.

The Ideal-Variety Correspondence Theorem (stated in Chapter 6, Cox [11]) guaran-tees that for the image of the mapping v, with the ideal I constructed as in (15), it holdsthat (z′, x ′) ∈ V (I ) for some x ′ ∈ R

n+ if and only if z′ ∈ I m(v). It is possible to createother ideals passing from the image of I m(v). As has been mentioned before, the aimis to rewrite polynomials in (13) as a set of polynomials with the same zero solutionset, but with z terms only. Two ideals will be of use here: the deformed toric ideal andthe homogeneous deformed toric ideal. Both of these ideals can be created from theideal I in the form of (14), after a change of basis and with some elimination theory.First the ideal in (14) needs to be stated in terms of other basis elements (polynomials)

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which show explicitly the relations between the z variables. Note that a careful exten-sion of the basis of the ideal does not change the variety of the ideal. The basis whichwill be needed is a Gröbner basis (of the ideal). Gröbner bases are useful because theyusually give most insight and information on an given ideal. They can be efficientlycalculated with powerful computer algebra systems such as Magma [4], Macauley2[23], Singular [24,42], CoCoA [6], and even Mathematica or Maple. A more completelist of software is given in on website maintained by the RICAM institute (http://www.ricam.oeaw.ac.at/Groebner-Bases-Implementations/view/SystemList.php).

A Gröbner basis is a basis which depends on a specific ordering of the monomialsthat belong to the ideal. A formal definition of Gröbner basis and monomial order-ings can be found in any standard algebraic geometry book such as [11]. A Gröbnerbasis exists for any monomial ordering and is not unique. Some standard methods forcomputing them are Buchberger’s algorithm and Faugères F4 algorithm. The methodsfor computing Gröbner basis are continuously improving. One example is calculatingGröbner basis via modular methods [2]. This has been implemented in Singular, asymbolic algebra software library [26].

Some standard monomial orderings are lexicographic (lex) and graded reverselexicographic ordering (grlex). In a lexicographic ordering one writes γ >lex δ forγ = (γ1, . . . , γn) and δ = (δ1, . . . , δn) ∈ Z

n+r≥0 , if in the vector difference γ − δ ∈

Zn+r the leftmost entry is positive. For two monomials f and g with orders γ and

δ, we write f >lex g if γ >lex δ. For the graded reverse lexicographic ordering onewrites γ >grlex δ if |γ | = ∑n+r

i=1 γi > |δ| = ∑n+ri=1 δi or |γ | = |δ|, and the rightmost

nonzero entry of γ − δ ∈ Zn+r is negative.

Note that is possible to convert one Gröbener basis with respect to one monomialordering into a Gröbener basis with respect to a different ordering via algorithms suchas the FGLM algorithm or the Gröbner walk algorithm. These are often employed tocompute lexicographic Gröbner bases from grevlex Gröbner bases, because a lexico-graphic ordering can be harder to compute than a graded reverse lex ordering. Theyhave been implemented in Singular libraries [24].

The Gröbner basis used must be formed by a monomial ordering which is also anelimination ordering. Let G be a Gröbner basis of I with such an elimination ordering.The intersection I ∩ Q[z] is an ideal in Q[z] and it is called an nth elimination ideal(because all n of the x variables have been eliminated). I ∩Q[z] is a ideal of Q[z], butit is also an ideal passing from image of v. I ∩ Q[z], called the deformed toric ideal,is of the form

I deftor = { f ∈ Q[z] : f (v(x, k)) = 0} ⊆ Q(k)[z]. (16)

By elimination theory, G ∩ Q[z] (G basis elements with only z variables) are aGröbner basis of I ∩ Q[z]. So basis polynomials of G with only z variables alsohave I m(v) as their zero solution set. But elimination theory can only be used ifGröbner basis is formed with elimination monomial ordering. It must be mentionedthat the deformed toric deal is an ideal formed from I m(v). By the Ideal-VarietyCorrespondence Theorem, the variety of the deformed toric ideal is such that z ∈V (I def

tor ) if and only if z ∈ I m(v). A monomial order > on K[x1, . . . , xn] is said tobe of l-elimination type provided that any monomial involving one of x1, . . . , xl is

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http://www.ricam.oeaw.ac.at/Groebner-Bases-Implementations/view/SystemList.php

http://www.ricam.oeaw.ac.at/Groebner-Bases-Implementations/view/SystemList.php


greater than all monomials in K[xl+1, . . . , xn]. A Gröbner basis with an nth eliminationmonomial ordering is required in order for all x variables to be eliminated.

The lexicographic order is an n-elimination monomial ordering. The eliminationordering used in calculations is the pure lexicographic ordering with

x1>lex · · ·>lexxn>lexz1>lex · · · >lexzr .

A Gröbner basis with this ordering represents most information about the ideal,but usually can be hard to compute [43]. In the upcoming examples, this ordering isused nevertheless and all calculations were performed with the Gröbner package inMAPLE. As mentioned there exist other orders [3] which are more efficient and easierto compute, for example the product order that induces grevlex on both Q[x1, . . . , xn]and on Q[z1, . . . , zr ]. Writing monomials in n + r variables as xαzβ , where α ∈ Z

n≥0and β ∈ Z

r≥0, this product order is defined by

xαzβ >mixed xγ zδ ⇐⇒ xα >grlex xγ or xα =grlex xγ and zβ >grlex zδ.

From this ordering we can revert back to lexicographic ordering by using the algo-rithms as mentioned above. For larger reaction networks the aforementioned algebraicpackages would increase significantly the speed of the calculations. The above con-cepts are now illustrated by using the Selkov model example:

Example 2 The ideal of I for the Selkov model has the basis F = {z1 − k1x21 x2, z2 −

k2, z3 − k3x1, z4 − k4, z5 − k5x2}. Buchberger’s algorithm is used to derive a Gröbnerbasis of I . Some preliminary definitions are needed: for fi ∈ F denote by L M( fi ) themonomial of fi with the largest exponent, and likewise LT ( fi ) as the largest term offi , both with respect to a monomial ordering . Assume the lex ordering where x1 >lex

x2 >lex z1 >lex z2 >lex z3 >lex z4 >lex z5). For any pair fi , f j ∈ F let s be the leastcommon multiple of L M( fi ) and L M( f j ), namely s = LC M(L M( fi ), L M( f j )).Define an S-polynomial of fi and f j as the combination,

S( fi , f j ) = s

LT ( fi )· fi − s

LT ( f j )· f j .

One can write S( fi , f j )F

to denote the remainder on division of S( fi , f j ) by theelements of F . Buchberger’s algorithm is based on extending F to a Gröbner basis

by successively adding non-zero remainders S( fi , f j )F

to F . Only S( f1, f3) andS( f1, f3) are non-zero S-polynomials. In fact for S( f1, f5) we have

S( f1, f5) = 1

k1f1 − x2

1

k5f5,

= 1

k1

(z1 − k1x2

1 x2

)− x2

1

k5(z5 − k5x2),

= 1

k1z1 − x2

1

k5z5.

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Using the division algorithm it can be shown that

S( f1, f5) =(

1

k3k5x1z5 + 1

z23k5

z3z5

)f3 −

(1

k1z1 − 1

k23k5

z23z5

), (17)

where f3 = z3 − k3x1, and so S( f1, f5)F = 1

k1z1 − 1

k23k5

z23z5. Clearly S( f1, f5)

F =k2

3k5z1 − k1z23z5, once the polynomial is multiplied by the factor k1k2

3k5. Repeating

above calculations it can be shown that S( f1, f3)F = S( f1, f5)

F. Thus S( f1, f5)

F

is the only new element to be added to the basis F to form the Gröbner basis. Since

the element f1 is linearly dependent on other elements of F and S( f1, f5)F

, we knowfrom Eq. (17) that the Gröbner basis of I is the set G = {k5k2

3 z1 −k1z23z5, z2 −k2, z3 −

k3x1, z4 − k4, z5 − k5x2}. Since the respective Gröbner basis of the deformed toricideal is G ∩ Q(k)[z], we have

I deftor = 〈k5k2

3 z1 − k1z23z5, z2 − k2, z4 − k4〉.

Now, z ∈ V (I deftor ) if and only if z ∈ I m(v).

Note that the deformed toric ideal is not the only ideal passing from I m(v). In ourstudy we are interested in the analysis of the convex polyhedral cone K er(N ) ∩ R

r+.As mentioned earlier, a ray of the cone can be of any arbitrary length. In this analysis,the size of any kinetic parameter or concentration values are not necessarily known.Hence, a ray of the cone is not necessarily finite. Previous concepts can be generalizedby enlarging C

r with the addition of points at ∞ to create an n-dimensional projectivespace P

r−1(C). Working with projective spaces usually simplifies computations.Before we give a description of the ideal of I m(v) in projective space some def-

initions are needed. Define an equivalence relation ∼ on the lines of the field C byletting two lines be equivalent if they are parallel in C. An n-dimensional projectivespace P

r−1(C) over a field C is a set of equivalence classes ∼ on Cr − {0}, namely,

Pr−1(C) = (Cr − {0})/ ∼ .

Each nonzero r -tuple z1, . . . , zr ∈ Cr defines a point p ∈ P

r−1(C). z1, . . . , zr ∈C

r . They are referred to as the homogeneous coordinates of p. Geometrically, one canthink of P

r−1 as a set of lines through the origin in Cr . In fact,

Pr−1(C) ∼= {lines through the origin in C

r }.

So every ray of the convex cone, independent of its length, is a single point inprojective space. This is precisely the reason why the notation used did not distinguishbetween points and rays. A homogeneous deformed toric ideal is defined as

I deftor = { f ∈ Q[z] : f (v(x, k)) = 0, f is homogeneous} ⊆ Q(k)[z]. (18)

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The variety of the homogeneous deformed toric ideal is the so called Zariski closureof the image of the mapping v in projective space. This is the smallest variety (withrespect to set inclusion) that contains the image of v in the projective space. The varietyof I def

tor is V ( I deftor ) ⊆ P

r−1. The homogeneous deformed toric ideal is formed fromI m(v). By the Ideal-Variety Correspondence [z] ∈ V (I def

tor ) if and only if, z ∈ I m(v).Note that different choices of parameters ki j give a parametrization of the family of

projective varieties. By defining the variety of the deformed toric ideal and the varietyof the homogeneous deformed toric ideal, the conditions for z solutions in (13) havebeen rewritten in terms of equations with z and kinetic parameters k only, as given byV (I def

tor ) or V ( I deftor ). This corresponds to a coordinate transformation with potentially

less basis elements. However, it still needs to be confirmed that for a specific choiceof z and k there exists a positive steady state x such that z = x0v(x, k). But this istrue, since z ∈ V (I def

tor ) if and only if z ∈ I m(v).Since the cone is parametrized by convex parameters j , the conditions on the

z variables (z ∈ V (I deftor )) can be reformulated in terms of conditions on convex

parameters j . Take an ideal J ⊆ Q(k)[ j] by substituting z = ∑Mi=1 ji Ei into basis

elements of I deftor . Then for a chosen parameter set k, if j ∈ V (J ) ∩ R

M+ , a positive

solution x exists for∑M

i=1 ji Ei = v(x, k). We return to the example for illustration:

Example 3 As shown earlier, the Selkov model has three currents E1, E2 and E3, see(12). So, z1 = j3, z2 = j1, z3 = j1 + j3, z4 = z2 + z3, z5 = j2. Substituting jcoordinates into the deformed toric ideal yields

J = 〈k5k23 j3 − k1( j1 + j3)

2 j2, j1 − k2, j2 + j3 − k4〉.

The equations for j ∈ V (J ) are

k5k23 j3 − k1( j1 + j3)

2 j2 = 0 j1 − k2 = 0,

j2 + j3 − k4 = 0.

For j ∈ V (J ), the system has at least one positive steady state for chosen kparameters. By using the Hermite Normal Form some of these steady states can becalculated:

x1 = j1 + j3k3

, (19)

x2 = j2k5

. (20)

The above results can now be applied to formulate a general framework for a ‘non-negative’ bifurcation theory of time-continuous dynamical systems with polynomialright hand sides, such as the ones given by mass-action reaction networks. This willbe the topic of the second part of the paper.

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Part B: bifurcation analysis

In bifurcation theory the dimension of a given parameter space is obviously important.Working with the deformed toric variety allows some of the kinetic parameters to berestated in terms of convex parameters j and a subset of remaining kinetic parameters,denoted by k. We explain this construction. By using the deformed toric ideal J aparameter set k can be split into two disjoint sets k∗ and k. This can be done such thateach element of k∗ is a C1 function of k and j , namely, k∗ = k∗( j, k). The advantageof this parameter change is the following important property: if j ∈ V (J ), then forany choice of ( j, k) there exists at least one positive steady state x .

In standard bifurcation analysis additional conditions would have to be posed toguarantee such a positive steady state. Consider the well-known Brusselator system,reviewed in Example 5.2. It has four kinetic parameters, labeled k1 to k4. PerformingSNA analysis yields two convex parameters, j1 and j2. It is easy to show that k ={k1, k3}. In order for the Brusselator to have a simple Hopf bifurcation at a positivesteady state x (that will be derived later), parameters ( j, k) must satisfy the condition:

j2 =(

1 − k1 j32

k33 j1

)j1.

Standard bifurcation theory leads to two conditions:

k24k1 + k3

3 − k2k23 = 0,

k1k4

k2k3− k3

k4�= 0

and it is not guaranteed that the corresponding steady state x is positive. Obviouslyin biology and chemistry applications only positive steady states are meaningful andof interest. The Brusselator therefore highlights the real advantage of the bifurcationconditions derived via SNA and toric geometry. Moreover, because j ∈ V (J ) arezero-set solutions of polynomials in Q(k)( j), the toric variety conditions guaranteethe existence of a respective branch of equilibria on which the bifurcation occurs. Oneway of calculating this branch of equilibria is to use the Hermite Normal Form tosolve the system Nv(x, k) = ∑M

i=1 ji Ei , where j ∈ V (J )∩RM+ . From now on, when

referring to the variable h (the inverse of a steady state), the following convention shallbe assumed:

hs := hs( j, k) = 1/xs( j, k) (1 ≤ s ≤ n), (21)

with x( j, k) as the solution of

Nv(x, (k∗, k)) =M∑

i=1

ji Ei , j ∈ V (J ) ∩ RM+ . (22)

Here k∗ = k∗( j, k). The existence of a branch of equilibria is especially importantfor the Hopf bifurcation theorem. While the theorem from Liu [32] (given in Sect. 5)and all bifurcation theorems stated in [25] assume the existence of a smooth curve of

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equilibria close to the Hopf bifurcation point, the SNA conditions derived in Theorem 6will automatically guarantee the existence of a respective branch of positive equilibria.Finally, it should be noted that the change to convex parameters does not necessarilyincrease the dimension of the parameter space, as can be seen from the examplesillustrating the bifurcation theorems.

4 Branching and bifurcations from simple eigenvalues

In this section conditions for existence of saddle-node bifurcations points and simplebranching points are derived. Both bifurcations are associated with the occurrence ofmultistability.

4.1 Saddle-node bifurcation

Consider the mass-action ODE system (2) of the chemical reaction scheme (1) withM extreme currents. Assume that the chemical reaction network has no conservationrelations, namely ker(N T ) = {0}. If there are conservation relations, they can beincorporated into ODEs to generate a new set with no conservation relations, we refer toExample 4.1.2. In one parameter continuation all system parameters are kept fixed andonly one is allowed to vary. As explained earlier, the parameter set is reparametrizedas { j (µ), k(µ)}, where now µ is the free parameter. Typically one chooses one of thej’s or ks as such a µ. We state the saddle-node bifurcation theorem:

Theorem 1 Consider the mass-action chemical reaction system (2) with ker(N T ) ={0}. Assume the system has M extreme currents and parametrization { j (µ), k(µ)},where j ∈V (J )∩R

M+ . Let Hi denote the i×i Hurwitz determinants of the characteristicpolynomial (9) and define w and y to be the left and right eigenvectors of Jac( j, h),with h(µ) from (21). Let the following conditions hold for some value of µ = µ0:

(SNB1) α0( j (µ0), h(µ0))) = 0 and Hn−1( j (µ0), h(µ0)) �= 0.(SNB2) wT Ni �= 0 i = 1, . . . , r , where Ni is i th column of N, and(SNB3) wT Ndiag(

∑Mi=1 ji (µ0)Ei )Sdiag(h(µ0))y �= 0, where each column of the

matrix S is defined by Si := diag(κ i )κT diag(h(µ0))y, with κ i denoting thei th row of the kinetic matrix.

Then there exists a smooth positive curve of equilibria in Rn × R passing through

(1/h(µ0), µ0) at which the system undergoes a saddle-node bifurcation. Moreover,there exists a corresponding kinetic parameter vector k ∈ R

r+ such that system (2)undergoes the saddle-node bifurcation.

The proof will rely on establishing an equivalence with the following conditionsdescribed in [25]:

Theorem 2 Let

x = fµ(x), x ∈ Rn, µ ∈ R,

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be a system of differential equations with equilibrium p at µ = µ0. Assume that atthis equilibrium the following hypotheses are satisfied:

(SN1) Dx fµ0(p) has a simple eigenvalue 0 with right eigenvector y and left eigen-vector w. Also Dx fµ0(p) has k eigenvalues with negative real parts and(n − k − 1) eigenvalues with positive real parts (counting multiplicity),

(SN2) wT ( ∂∂µ

fµ(p, µ0)) �= 0, and

(SN3) wT (D2x fµ0(p)(y, y)) �= 0.

Then there is a smooth curve of equilibria in Rn × R passing through (p, µ0) tangent

to the hyperplane Rn ×µ0. Depending on the signs of expressions in (SN2) and (SN3)

there are no equilibria near (p, µ0) when µ < µ0 (or µ > µ0) and two equilibrianear (p, µ0) when µ > µ0 (or µ < µ0). These two equilibria near (p, µ0) arehyperbolic and have manifolds of dimensions k and k + 1, respectively.

Proof It will be shown that conditions (SNB1)–(SNB3) imply conditions (SN1)–(SN3). Since j (µ0) ∈ V (J ) ∩ R

M+ , by our choice of ( j (µ0), k(µ0)) there exists atleast one positive solution, i.e., x = p. It can be calculated by recovering k∗ parameters.If we define h(µ0) to be the inverse of the positive solution, then Jac( j (µ0), h(µ0))

is the Jacobian of the system evaluated at a positive steady state. A necessary andsufficient condition for Jac( j, h) to have a zero eigenvalue is that α0( j, h) = 0. Anecessary condition for a characteristic polynomial to have at least a pair of pureimaginary eigenvalues or multiple zero eigenvalues is Hn−1( j, h) = 0, a result fromOrlando’s formula [20]. Hence, (SNB1) implies (SN1).

Now, since each parameter ki only features in one corresponding reaction rate,vi (x, k), the partial derivative of the vector field with respect to ki is

∂

∂kiNv(x, k) = N

∂

∂kiv(x, k) = vi (x, k)

kiNi =

(n∏

l=1

xαlii

)Ni ,

where Ni is the i th column of N . We reparametrize the ODE modeling (2) as k = k(µ),by choosing any ki is as the bifurcation parameter µ and keeping the remaining k’sas constants. At the corresponding positive steady state p = (p1, . . . , pn), condition(SN2) can be rewritten as,

wT ∂

∂kiNv(p, k(µ0)) = wT

(n∏

l=1

pαlil Ni

)

=(

n∏

i=1

pαlil

)wT Ni , for any i = 1, . . . , r .

Since(∏n

l=1 pαlil

)> 0, it follows that (SNB2) implies (SN2). Now, with all above

conditions satisfied, we will show that they together imply (SN3). First, the tensorproduct D2

x Nv(x, k)(r, r) for r ∈ Rn , must be rewritten in form similar to (8). Let

D2x Nv(x, k)(r, r) = T (x, k, r)r (23)

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where the i th column of matrix T (x, k, r) is

Ti = ∂

∂xiDx Nv(x, k)r.

Because of the form of Jacobian Dx Nv(x, k) taken from (8), by the chain rule weget

∂

∂xiDx Nv(x, k)

= N∂

∂xi(diag(v(x, k))) κT diag(x−1) + Ndiag(v(x, k))κT ∂

∂xi

(diag(x−1)

).

Taking κ i as i th row of κ ,

∂

∂xi(diag(v(x, k))) = diag(v(x, k))diag(κ i )x−1

i ,

∂

∂xi(diag(x−1)) = −diag((x−1))�i x−1

i ,

with �i defined as a matrix of zeros with only the i × i th entry equal to 1. Consideringeach i th column of T is multiplied by factor of (x−1)i , and that each column can alsobe written as a vector difference, it holds that

T = Ndiag(v(x, k))Sdiag(x−1) − Ndiag(v(x, k))κT Udiag(x−1),

where S and U are matrices with the i th columns equal to

Si = diag(κ i )κT diag(x−1)r,

Ui = �i diag(x−1)r = diag(x−1)�i r.

But U = diag(x−1)diag(r), and hence

T = Ndiag(v(x, k))Sdiag(x−1) − Dx Nv(x, k)diag(r)diag(x−1). (24)

Using Eqs. (23) and (24), condition (SN3) can be rewritten as

wT D2x Nv(p, k(µ0))(y, y) = wT T (p, k(µ0), y)y

= wT Ndiag(v(p, k(µ0)))Sdiag(p−1)y,

since wT ∈ K er(Dx Nv(p, k(µ0))). From here it is easy to see that (SNB3) is suffi-cient for (SN3) to be true.

It must be emphasized that the chemical reaction network must be written in theform that already incorporates the conservation relations, which means K er(N T )

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must be an empty set. Assume the opposite, i.e. that the system contains r conserva-tion relations. Because of the form of the Jacobian, Dx Nv(x0, k0) in (8), K er(N T ) ⊆K er(DT

x Nv(x0, k0)). Each conservation relation implies that there is a column ofDx Nv(x0, k0) which is a linear combination of the other columns. Hence one zeroeigenvalue of DT

x Nv(x0, k0) corresponds to each conservation relation. In order for asaddle-node bifurcation to appear in a system with r conservation relations, the num-ber of zero eigenvalues of DT

x Nv(x0, k0) must be r + 1. However, the dimension ofthe eigenspace of the zero eigenvalues is always less or equal to the algebraic multi-plicity of the eigenvalue, in this case r + 1. Therefore it is possible that K er(N T ) =K er(DT

x Nv(x0, k0)), in which case the left eigenvectors w of Dx Nv(x0, k0) willnever satisfy condition (SNB3). Next the applicability of the saddle node bifurcationtheorem with be shown with two examples.

4.1.1 Subnetwork of the peroxidase–oxidase reaction system

The following model is a core bistable subnetwork of a peroxidase–oxidase reactionmodel from [5], as described in [38]. An analysis of the necessary bistability conditions(necessary saddle-node conditions) for the full PO reaction system is given in [1]. Herethe model includes also reactions between the N AD∗ free radicals and O2:

2N AD∗ k1→ ∅∅ k2→ N AD∗ k3→ N AD∗

N AD∗ + O2k4→ ∅

O2

k5�k6

∅

The first and second reaction model the inflow and outflow of N AD∗, while thethird models the autocatalytic formation of N AD∗. The fourth reaction is the oxidationof N AD∗ when in contact with O2, and the fifth and sixth reaction model the inflowand outflow of O2. Given mass-action kinetics the differential equations describingthis process are

x1 = −2k1x21 + k2 + k3x1 − k4x1x2,

x2 = k5 − k6x2 − k4x1x2.

Here x1 and x2 denote the concentrations of N AD∗ and O2, respectively. Thesystem has six kinetic parameters and the network consists of five extreme currents:

E1 = (1, 2, 0, 0, 0, 0),

E2 = (1, 0, 2, 0, 0, 0),

E3 = (0, 0, 1, 1, 1, 0),

E4 = (0, 0, 0, 0, 1, 1),

E5 = (0, 1, 0, 1, 1, 0).

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The extreme currents (in same order) represent the following reaction subnetworks,each admitting steady state solutions:

2N AD∗ k1→ ∅∅ k2→ N AD∗

}E1

2N AD∗ k1→ ∅N AD∗ k3→ 2N AD∗

}E2

N AD∗ k3→ 2N AD∗

N AD∗ + O2k4→ ∅

O2k5→ ∅

⎫⎪⎪⎬

⎪⎪⎭E3

O2

k5�k6

∅}

E4

∅ k2→ N AD∗

N AD∗ + O2k4→ ∅

O2k5→ ∅

⎫⎪⎪⎬

⎪⎪⎭E5

From the variety of the deformed toric ideal the following relations for the kineticparameters and convex parameters can be derived:

k23 ( j1 + j2) − k1 (2 j2 + j3)

2 = 0,

2 j1 + j5 − k2 = 0,(25)−k6 k3 ( j3 + j5) + k4 (2 j2 + j3) j4 = 0,

j3 + j4 + j5 − k5 = 0.

With these relations four of the kinetic parameters k1,k2, k4 and k5 can be rewritten asconvex parameters, with k3 and k6 remaining parameters in the transformed system.For the convex parameters js satisfying above relations positive species variablesexist. Their inverses are calculated via Hermite Normal form:

h1 = k32 j2+ j3

,

h2 = k6j4

.(26)

The Jacobian matrix in convex parameters reads

Jac( j, h) =[

(−4 j1 − 2 j2 − j5) h1 − ( j3 + j5) h2

− ( j3 + j5) h1 −( j3 + j4 + j5)h2

],

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and the characteristic polynomial takes the form

λ2 + α1λ + α0.

Here the coefficients are

α0 = j4h2h1(4 j1 + 2 j2 + j5) + h1h2( j3 + j5)(4 j1 + 2 j2) − h1h2 j3×( j3 + j5)( j3 + j4 + j5),

α1 = (4 j1 + 2 j2 + j5)h1 + ( j3 + j4 + j5)h2.

Since α1 is always positive, (SNB1) is satisfied when α0 = 0:

j4(4 j1 + 2 j2 + j5) + ( j3 + j5)(4 j1 + 2 j2) − j3( j3 + j5)( j3 + j4 + j5) = 0. (27)

The right and left eigenvectors of Jac( j, h) subject to (27) are

w( j) = c1

[1

− ( j3+ j5)j3+ j4+ j5

]and y( j, h) = c2

[1

− ( j3+ j5)h1( j3+ j4+ j5)h2

],

with some arbitrary lengths c1, c2 ∈ R. Then (SNB2) reduces to

j5 + j3 �= 0 and j4 �= 0, (28)

but these conditions are automatically satisfied if all convex parameters are positive.Next for (SNB3) to hold the following condition needs to be satisfied:

0 �= −(4 j3 j2 j4 + j3 j25 − 3 j2

5 j4 + 2 j2 j25 + j3

3 + 2 j23 j5 + 2 j2 j2

3 − j5 j24

+2 j2 j24 + 4 j3 j2 j5 + 4 j4 j2 j5 − 4 j3 j5 j4 − j4 j2

3 )h22( j3 + j4 + j5)h

21, (29)

where h1 and h2 are of the forms given in Eq. (26). When convex parameters jand kinetic parameters k3 and k6 satisfy conditions (27) and (29), then the systemundergoes a saddle-node bifurcation. Numerous values can be chosen in order to satisfythese conditions. All other kinetic parameters and species concentration values can beretrieved by substituting the results into Eqs. (25) and (26). Figure 1 shows a plot of theSNB curves in two-parameter space calculated via the SNA method and via XPPaut,i.e. numerically. Here the values taken are: k3 = 2, k6 = 0.14, and for the convexparameters: j1 = 0.05, j2 = 0.0337, j3 = 1.4268, j4 = 3.1732, j5 = 0.4. All kineticparameter values could be retrieved by back-substitution. Bifurcation parameters arek1 and k4, respectively (because XPPaut does not work with convex parameters).

4.1.2 A cell-cycle model

The given cell cycle is a complex network that can be described in terms of threemodules as introduced by Tyson et al. [44]. Two of these modules have been identifiedas bistablenetworks, where the levels of cyclin serve as an indicator for the cell to

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0.14

0.12

0.10

0.08

0.06

0.04

k 4

0.50.40.30.20.10.0

k2

bistableregion

SNBxpp

SNBgb

SNBxpp

SNBgb

Fig. 1 Two-parameter bifurcation diagram of the peroxidase–oxidase reaction model for k1 = 0.3k3 =2, k5 = 5, k6 = 0.14. Moreover k4 and k2 are bifurcation parameters. SNBxpp and SNBgb denote saddle-node bifurcations detected by XPPaut and the SNA method, respectively. SNA conditions cannot detect acurve for k4 < 0.056 detected by XPPaut, since for these parameter values the SNB is not a positive steadystate. A numerical survey with XPPaut shows that the model is bistable for pairs of (k4, k2) between thetwo upper SNBxpp curves

switch from one phase to another. The second module regulates the cell’s transitionfrom G2 phase to M-phase, which is mitosis. Here a model of the second moduleproposed by [41] is considered:

C + M+ k1→ C + M

M + Wk2→ W + M+

M + C+ k3→ M + C

Ck4→ C+

W + k5→ W

M + Wk6→ M + W +.

Here M and M+ denote active and inactive concentrations of MPF (mitosis promot-ing factors). C and W with same superscripts denote inactive(+) and active concen-trations of cyclin cdc-25, and Myt-1, a member of the wee1 family of protein kinases.The model has three conservation relations. It is assumed that the total concentrationsof MPF, cdc-25 and Myt-1 (denoted by mt , ct and wt ) are constant. In [41] the authorsderive the model in terms of the active form

m = k1c(mt − m) − k2mw,

c = k3m(ct − c) − k4c,

w = k5(wt − w) − k6mw.

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Since mt , ct and wt are parameters, the system is rewritten as

m = k7c − k1mc − k2mw,

c = k8m − k3mc − k4c,

w = k9 − k5w − k6mw,

to continue with familiar labeling. We have k7 = k1mt , k8 = k3ct and k9 = k5wt .There are nine kinetic parameters in this system. It can be written again in terms ofconvex parameters associated to six extreme currents:

E1 = (1, 0, 0, 0, 0, 0, 1, 0, 0),

E2 = (0, 1, 0, 0, 0, 0, 1, 0, 0),

E3 = (0, 0, 1, 0, 0, 0, 0, 1, 0),

E4 = (0, 0, 0, 1, 0, 0, 0, 1, 0),

E5 = (0, 0, 0, 0, 1, 0, 0, 0, 1),

E6 = (0, 0, 0, 0, 0, 1, 0, 0, 1).

The variety of the deformed toric ideal for convex parameters gives rise to thefollowing six relations:

k8k7 j1 − k1 ( j1 + j2) ( j3 + j4) = 0,

−k2 j6 + k6 j2 = 0,

k8k7 j3 − ( j3 + j4) ( j1 + j2) k3 = 0,(30)−k4 ( j1 + j2) + k7 j4 = 0,

−k5k8 j6 + k6 ( j3 + j4) j5 = 0,

j5 + j6 − k9 = 0.

This implies six of the kinetic parameters (say, k3, k5, k6, k7, k8, k9) can be rewrit-ten as combinations of convex parameters and the other three kinetic parameters(k1, k2, k4). For convex parameters satisfying the relations in (30) positive valuesof m, w and c can be retrieved. Via Hermite Normal Form, their inverses are:

1/m = h1 = k1 j4k4 j1

,

1/c = h2 = k4

j4,

1/w = h3 = j1k2k4

j2 j4k1.

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The Jacobian matrix in convex parameters reads

Jac( j, h) =⎡

⎢⎣(− j1 − j2) h1 j2h2 − j2h3

j4h1 (− j3 − j4) h2 0

− j6h1 0 (− j5 − j6) h3

⎤

⎥⎦ .

Here the characteristic polynomial of the Jacobian matrix is a cubic polynomial ofthe form

λ3 + α2λ2 + α1λ + α0.

All coefficients of the characteristic polynomial are positive except α0. Hence, for(SNB1) to hold we only need to check that α0 = 0. Therefore we have

j1 j4 j6 + j1 j3 j5 + j1 j3 j6 + j1 j4 j5 + j2 j3 j5 − j2 j4 j6 = 0. (31)

The right and left eigenvectors of Jac( j, h) subject to (31) are

w( j) = c1

⎡

⎢⎣− j5+ j6

j2

− j5+ j6j3+ j41

⎤

⎥⎦ and y( j, h) = c2

⎡

⎢⎢⎣

− ( j5+ j6)h3j6h1

− h3 j4( j5+ j6)h2 j6( j3+ j4)

1

⎤

⎥⎥⎦ ,

with c1, c2 ∈ R. Then (SNB2) reduces to the condition

j5 + j6 �= 0. (32)

Obviously, this condition holds for all positive convex parameters. Finally (SNB3)takes the form

wT Ndiag(v( j))Sdiag(h)y = h23( j5 + j6)

j2 j26 ( j3 + j4)2

(−3 j5 j2 j6 j2

4 + 3 j1 j25 j2

4 + 3 j1 j26 j2

4

− j2 j26 j2

4 + 6 j1 j5 j6 j24 + 3 j2

5 j2 j3 j4 + 4 j1 j25 j3 j4

+ 8 j1 j5 j6 j3 j4 + 4 j1 j26 j3 j4 + j3 j2 j2

6 j4 + j1 j25 j2

3

− j5 j2 j6 j23 + j2

5 j2 j23 + j1 j2

6 j23 + 2 j1 j5 j6 j2

3

)

�= 0. (33)

If there are convex parameters j satisfying conditions (31) and (33), then the systemundergoes a saddle-node bifurcation. If k1, k2 and k4 are fixed the precise values ofother kinetic parameters and also the variable values (m, w, c) can be calculated suchthat a saddle-node bifurcation will occur. A comparison of these results with those ofXPPaut is given in Fig. 2. In this two-parameter (k1, k7) bifurcation plot, the full curveof XPPaut cannot be traced. This is because for k7 > 2.5 the saddle-node bifurcation

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12

10

8

6

4

2

0

k 1

70.1

2 3 4 5 6 71

2 3 4 5 6 710

k7

SNBxpp

SNBgb

Fig. 2 Two-parameter bifurcation diagram of the cell cycle model for k2 = 1, k3 = 1, k4 = 10, k5 = 2.5,k6 = 1, k8 = 1 and k9 = 2.5. k1 and k7 are bifurcation parameters. SNBxpp and SNBgb denote saddle-node bifurcations detected by XPPaut and our method, respectively. SNA conditions cannot detect SNBsfor k7 > 2.5 since for these parameter values the SNB is not a positive steady state

is not a positive steady state (and not biologically meaningful), and hence the SNAconditions cannot detect it.

4.2 Simple branching point

For completeness we also consider simple branching points. One can restate the con-dition for a curve of equilibria given by

Nv(x, k) = 0 (34)

as a curve in Rn+1 defined by n equations:

F(y) = 0, F : Rn+1 → R

n . (35)

Here the computation of the equilibria is parametrised by one parameter, say kl . Azero solution y∗ ∈ R

n+1 of F is called a branching point if there are at least two dif-ferent smooth curves satisfying (35), and passing through y∗. Some of the well knownbranching points are: The pitchfork bifurcation (a situation where a stable steady statelooses stability to two new simultaneously created steady states as the bifurcation para-meter is varied), and the transcritical bifurcation (where two steady states of differentstability types coalesce and exchange their stability types). The pitchfork bifurcationis associated with bistability. The conditions for a simple branching point to occurcan be stated with SNA theory. But for above more specific branching points someadditional conditions are needed, and this will not be covered in this paper. Denote atranspose of an extended kinetic matrix as

123


κT = [κT |el ],

where el is a normalized basis vector in Rn , with the lth entry being the only non-zero

entry. Each i th row of κ is denoted κ i . Let h = (h, k−1l ). The Jacobian matrix of (35)

in convex pair ( j, h) coordinates becomes

J ( j, h) = Ndiag

(M∑

i=1

ji Ei

)κT diag(h).

For any vectors x, w ∈ Rn+1 and u ∈ R

n , consider a matrix S(x) to be an (n + 1) ×(n + 1) matrix with each i th column defined by

S(x)i := diag(κ i )κT diag(h)x .

Consider a function b : Rn × R

m × Rm → R where

b(u, x, w) = uT Ndiag

(M∑

i=1

ji Ei

)S(x)diag(h)w.

With these notations we can state the following theorem:

Theorem 3 Consider the mass-action chemical reaction system (2) with M extremecurrents and corresponding convex parameters j1, . . . , jM . Assume the system has aparametrization { j (µ), k(µ)}, where j ∈V (J ) ∩ R

M+ . If at some µ = µ0,

(BP1) rank(J ( j (µ0), h(µ0))) = n −1, and with q1, q2 being two linearly indepen-dent left eigenvectors, and r a right eigenvector of J ( j (µ0, h(µ0)), it holdsthat

(BP2) b(r, q1, q1)b(r, q2, q2) − b(r, q1, q2)2 < 0, then there exist branches of posi-

tive equilibria locally which intersect at a simple branching point.

Proof The conditions can be derived from any standard conditions for a simple branch-ing point, see [31]. The local existence of branches follows from the use of the toricvariety V (J ). ��

4.3 Bistability

Bistability describes a situation where a network contains two stable equilibria.Recently a plethora of theoretical and numerical evidence has emerged that suggestthe importance of bistability in a number of chemical and biochemical systems, see[13]. Bistability is often characterized as a result of a bifurcation, and then most fre-quently is created by the occurrence of two saddle-node bifurcations, as a bifurcationparameter is varied. Hence for bistability to occur two saddle node bifurcations needto be identified as one of the parameters is varied. By fixing some values for all butone parameter in the peroxidase–oxidase subnetwork, it can be confirmed that there

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are two values of the bifurcation parameter where a saddle-node bifurcations occurred(Fig. 1). In the cell-cycle model, when biologically feasible parameters from [41] wereused, it only gave rise for one saddle-node bifurcation. This is to be expected, since inmodeling the cell-cycle, the authors of [41] made the simplifying assumption that oneof the saddle-node bifurcations takes place when cdc-25 and Myt are at zero concen-trations. As mentioned earlier, stoichiometric network analysis cannot be applied tonon-positive steady states, hence both saddle-node bifurcations could not be identified.

4.4 Another method for identifying bistability

The following interpretation of the bistability criteria is a small extension of themethod applied and published in [38] and [40]. The aim is to find a parametriza-tion k(µ) for which bistability occurs as µ is varied. Usually µ = ki for somei = 1, . . . , p. With the network described via extreme currents, ( j, h) have to sat-isfy condition (SNB1) in order to guarantee existence of positive concentration steadystates. Along the curve of ( j (k), h(k), k), three values of k need to be determinedwhere α0( j (k), h(k)) changes sign three times in order of increasing (or decreasing)k values, and Hn−1( j (k), h(k)) �= 0. Then a parametrization k(µ) is chosen and oneshould calculate µ values such that αn( j (k(µ)), h(k(µ))) = 0, and finally one needsto check conditions (SNB2) and (SNB3) for these points.

5 The Hopf bifurcation

In this section we will use SNA theory to derive sufficient conditions for a reactionnetwork to enable having a simple Hopf bifurcation. Similar results for identifyinga Hopf bifurcation in chemical reaction networks via SNA have been published bySensse et al. [21,37,39] on which we base our results. We will rely on the RouthHurwitz theorem which states

Theorem 4 (Routh–Hurwitz) The number of eigenvalues of Dx Nv(x; k) with positivereal part equals the sum of the number of sign changes in the two sequences

{1, H1, H3, . . .} and {1, H2, H4, . . .}

where each Hi is the i ×i th Hurwitz determinant derived from (9) in the form describedin [20].

Gatermann et al. [21] and Sensse and Eiswirth [39] were the first to employ use ofthe deformed toric ideal in SNA and suggest a method to search for the Hopf bifur-cation. Their suggestion is to follow a path of steady states (x(k), k) and search fora point where the number of sign changes in the sequences of Hurwitz determinantsjumps by two. Then, via Routh–Hurwitz theorem, there is a point on the curve (x(k), k)

where two eigenvalues cross the imaginary axis. However, it is possible that these twoeigenvalues at the bifurcation point are not purely imaginary. They could have multi-ple zeros. The authors suggest monitoring the values of a0, or the determinant of the

123


Jacobian. If the determinant does not change sign, then the pair of eigenvalues cannotbe a pair of zeros. In [21,39]. it is also suggested to monitor the derivative of thesign changing Hurwitz determinants vanishing, to make sure the pair of eigenvaluescrosses the imaginary axis. However, these conditions rely on the explicit reformu-lation of conditions in terms of the concentration steady states. This paper presentsa mathematical formulation for a simple Hopf bifurcation and several examples. Thetheorem derived relies on an application of conditions from the following ‘local’ Hopfbifurcation theorem stated in terms of Hurwitz determinants:

Theorem 5 [32] Assume that for a system

x = f (x, k) x ∈ Rn, k ∈ R

there exists a smooth curve of equilibria (x(k), k) with x(k0) = k0. This systemhas a simple Hopf bifurcation if the following conditions on the coefficients of thecharacteristic polynomial of the Jacobian Dx f (x0, k0) hold:

(CH1) a0(k0) > 0, H1(k0) > 0, . . . , Hn−2(k0) > 0, Hn−1(k0) = 0; and(CH2) d

dk Hn−1(k0) �= 0.

where each Hi is the i th Hurwitz determinant.

This theorem assumes the existence of a smooth curve of equilibria close to a Hopfbifurcation point. Based on the condition stated we will derive conditions for a simpleHopf bifurcation in terms of convex parameters (extending previous work in SNA), butwill also exploit algebraic geometry: in addition conditions can be given that guaranteethe existence of a respective branch of positive equilibria on which such a simple Hopfbifurcation occurs. The existence of such a local branch of equilibria follows from theusage of the toric variety V (J ). One way of calculating the branch of equilibria isto use the Hermite Normal Form to solve the system Nv(x, k) = ∑M

i=1 ji Ei , wherej ∈ V (J ) ∩ R

M+ .In case that we interpret the extreme currents as subnetworks in equilibrium, the

theorem also gives sufficient conditions such that by variation of the strength of thesubnetworks, measured in terms of j , the total network is able to oscillate. The potentialof such findings is that biologists might have an indication of which parts of the networkmay be more important to generate oscillatory behaviour. For example, if there exists areaction rate parametrized only by a current which does not even occur in the sufficientconditions stated below, then one would expect that the corresponding reaction is notcrucial to generate any oscillations.

Theorem 6 Consider the mass-action chemical reaction system (2) with M extremecurrents and parametrization { j (µ), k(µ)}, where j ∈V (J ) ∩ R

M+ . Let Hi denote thei × i Hurwitz determinants of the characteristic polynomial (9) with h(µ) from (21).Let for some value of µ = µ0 the following conditions hold:

(HB1) α0( j (µ0), h(µ0))>0, H1( j (µ0), h(µ0))>0, . . . , Hn−2( j (µ0), h(µ0))>0,Hn−1( j (µ0), h(µ0)) = 0.

(HB2)∑M

s=1∂ Hn−1

∂ js∂ js∂µ

�= 0.

123


Then there exist kinetic parameter vectors k ∈ Rp+ and a local smooth curve of positive

equilibria parametrised by k such that the reaction network undergoes a simple Hopfbifurcation.

Proof For every j ∈ V (J ) ∩ RM+ , there exists a curve of concentration steady states

parametrized by k, namely (x(k), k) such that Nv(x(k), k) = 0, see [21]. Moreover,there exists at least one k0 such that the corresponding steady state x(k0) = x0 ∈ R

n+[21]. The choice of h is in fact 1/x0. Jac( j, h) is the Jacobian evaluated at this steadystate x0. (HB1) implies condition (CH1) and by the chain rule, (HB2) implies (CH2).

��Remarks 1 (i) Theorem 6 makes no distinction of the type of simple Hopf bifur-

cation: whether it is supercritical, subcritical or degenerate. This would requireadditional conditions on the higher order terms of the system. Such effortrequires a lengthy computation of the dynamics restricted to the centre manifoldand involves coordinate changes that have no clear interpretation in terms ofextreme currents. This is a limitation of the method.

(ii) A good future direction would be to extend the sufficient conditions for a sim-ple Hopf bifurcation to a general Hopf bifurcation. Here the problem lies infinding sufficient conditions for the existence of two eigenvalues with non-zeroimaginary part that will cross the imaginary axis along the curve of equilibria(x(k), k), while all other eigenvalues can have positive or negative real parts.

The following is a list of prominent examples that have been studied in the liter-ature. It will be shown that the conditions of the Hopf bifurcation theorem in low-dimensional examples (with up to 4 variables) can be computed relatively easily. Forhigher-dimensional models, more sophisticated symbolic computations will have tobe implemented, as discussed in Sect. 3.

5.1 Calcium dynamics

This example is a completion of an extensive analysis of the model of intracellularcalcium oscillations published in [21]. The model describes enzymatic transfer ofcalcium ions Ca+2 across the cell membrane and is a modification of model of Ca+2-induced oscillations in muscle cells proposed by Dupont and Goldbeter [22].

Four species take part in the reactions. Variables x1 and x2 denote the concentrationsof Ca+2 ions in the cytosol and the endoplasmic reticulum, respectively. Variable x3is the concentration of the enzyme catalyzing the absorption of Ca+2 back into theendoplasmic reticulum, and x4 is the concentration of the complex formed whenenzyme binds to cytosolic calcium. Influx and efflux of cytosolic calcium across thecell membrane are modeled by reaction constants k12 and k21 respectively. Calcium-induced-calcium-release (CICR) is modeled as an autocatalytic production of cytosoliccalcium with rate constant k43. Absorption of calcium back into the endoplasmicreticulum is an enzymatic reaction, involving binding and dissociation of the enzymewith cytosolic calcium, at rate constants k56 and k65 respectively, and conversion of thecalcium–enzyme complex back to endoplasmic calcium at rate k76. The differentialequations describing this process are

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x1 = −k12x1 + k21 + k43x1x2 + k56x4 − k65x1x3,

x2 = −k43x1x2 + k76x4,

x3 = k56x4 − k65x1x3 + k76x4,

x4 = −k56x4 + k65x1x3 − k76x4.

The total concentration of enzyme (free enzyme x3 and bound enzyme in the com-plex x4) is conserved:

x3 + x4 = c. (36)

The model has seven parameters: six kinetic parameters and one conservation para-meter. This model contains three extreme currents (three steady-state generating sub-networks):

E1 = (1, 1, 0, 0, 0, 0), E2 = (0, 0, 0, 1, 1, 0), and E3 = (0, 0, 1, 0, 1, 1).

In given order, they correspond to influx and efflux of cytosolic calcium, associ-ation and dissociation of the enzyme–substrate binding complex, and autocatalyticrelease of endoplasmic calcium into the cytosol offset by enzymatic absorption ofcytosolic calcium into the endoplasmic reticulum. The deformed toric ideal of convexparameters is

J = 〈k76 j2 − k56 j3, j1 − k21〉.

Reparametrizing several kinetic constants as combinations of convex parametersleads to

k21 = j1, (37)

k56 = k76 j2j3

. (38)

Via Hermite Normal Form, with restrictions from (37) and (38) positive speciesconcentrations can be calculated. Their inverses come in the form

(h1(k), h2(k), h3(k), h4(k)) =(

k12

j1,

j1k43

k12 j3,

j1k65

(j2j3

+ 1)k12 j3,

k76

j3

). (39)

Substituting them back into (36) gives a restriction on c parameter,

c = k12k56 + k12k76k21k65

j1k76k65 j3. (40)

The model in convex parameters has seven parameters: three convex parametersand four kinetic parameters, k12, k43, k65, k76. All other parameters and variable valuescan be retrieved using previous equations. The convex Jacobian is

123


Jac( j, h) =

⎡

⎢⎢⎢⎢⎢⎣

(− j1 − j2)h1 j3h2 (− j2 − j3) h3 j2h4

− j3h1 − j3h2 0 j3h4

(− j2 − j3) h1 0 (− j2 − j3) h3 ( j2 + j3) h4

( j2 + j3) h1 0 ( j2 + j3) h3 (− j2 − j3) h4

⎤

⎥⎥⎥⎥⎥⎦,

where h are of the form (39). One eigenvalue of this Jacobian is always zero (becauseof the conservation relation) and hence the part of characteristic polynomial that is ofinterest becomes

λ3 + α3λ2 + α2λ + α1.

All coefficients except α2 are positive:

α1 = h1 j1 j3h2h3 j2 + h1 j1 j23 h2h3 + h1 j1 j3h2 j2h4 + h1 j1 j2

3 h2h4,

α2 = j23 h2h4 + j2

3 h2h3 + j23 h1h2 − h1h3 j2

3 + h1 j1h3 j3 + h1 j1 j3h2 − h1 j2h3 j3,

+ h1 j2 j3h2 + j3h2 j2h4 + j3h2h3 j2 + h1 j1 j2h4 + h1 j1 j3h4 + h1 j1h3 j2

α3 = j2h4 + j3h4 + h3 j2 + h3 j3 + j3h2 + h1 j1 + h1 j2.

Hence, (HB1) reduces to condition

H2 = α2α3 − α1 = 0, (41)

with αi being the coefficients given above. If we choose k43 (reaction constant ofCICR) as our bifurcation parameter, condition (HB2) involves checking

∂ H2

∂k43= − j3

j3k12

k21k243

�= 0. (42)

Because all parameters are assumed to be positive, Eq. (42) is automatically satisfied.The calcium model has at least one simple Hopf bifurcation if convex parameters andkinetic parameters k12, k43, k65, k76 can be chosen so that they satisfy Equation (41).One possible choice is to have j1 = 1, j2 = 1.4786, j3 = 2.9567, k43 = 1, k76 =1, k12 = 2.4, k65 = 5.211. The values for species concentrations and other parameterscan be retrieved via Eqs. (37)–(40). Figure 3 shows results of Theorem 6 plottedagainst those reproduced by the numerical bifurcation package XPPaut. The plot is atwo-parameter bifurcation plot with bifurcation parameters k12 and k65. It shows goodagreement for these two parameters. All other values of parameters and variables werechecked and showed good agreement.

In passing, a comparison can be made with the analysis of necessary conditionsfrom [21]. Since all coefficients of the characteristic polynomial, except α2 are positive,the necessary conditions for a Hopf bifurcation to occur reduce to finding two valuesof k parameters where H2 is of different sign. This is just a different formulation of

123


Fig. 3 Two-parameter bifurcation diagram of the calcium model for k21 = 1.0, k43 = 1.0, k56 =0.5, k76 = 1.0 and c = 5. k12 and k65 are bifurcation parameters. HBxpp and HBgb denote Hopf bifurca-tions detected by XPPaut and conditions of Theorem 6, respectively

the second sufficient condition, (HB1). The only additional computation for sufficientconditions, not needed in [21], is one extra algebraic conditions coming from (HB2).This condition is automatically satisfied for any positive parameter values, hence thecalculation of sufficient Hopf conditions is not more computationally expensive.

5.2 Brusselator

Now conditions from Theorem 6 will be applied to the Brusselator, a well-knownexample of an autocatalytic oscillating chemical reaction system. Its mechanism relieson interaction of two species whose dynamics are described by following ODEs:

x1 = k1x21 x2 − k2x1 − k3x1 + k4,

x2 = −k1x21 x2 + k2x1,

where x1 and x2 are the species concentrations, and k’s denote the kinetic constants.The search for a simple Hopf bifurcation (and ultimately oscillations) without anyadditional knowledge about the system depends on four kinetic parameters. In stan-dard bifurcation theory, conditions from the Hopf bifurcation theorem could be used.Numerical analysis via XPPaut could be laborious exercise if parameters are choseninappropriately. The Brusselator can be written as product of a stoichiometric matrixand velocity vector:

x = Nv(x, k),

123


where x = (x1, x2) and

N =[

1 −1 −1 1

−1 1 0 0

]and v(x; k) =

⎡

⎢⎢⎢⎣

k1x21 x2

k2x1k3x1

k4

⎤

⎥⎥⎥⎦ .

Two extreme currents are identified in the model,

E1 = (1, 1, 0, 0) and E2 = (0, 0, 1, 1).

These decompose the system into two subnetworks: the first is an autocatalytic pro-duction of x1 with consumption of x2, as well as conversion of x1 into x2. The secondextreme current models the subnetwork of inflow and outflow of x1. The deformedtoric ideal of the system in convex parameters is

J = 〈k3 j1 − k2 j2, j2 − k4〉.

To find the variety of J (V (J )) one must solve

k3 j1 − k2 j2 = 0,(43)

j2 − k4 = 0.

If relations in (43) hold for chosen j and k, for these j the system has positive con-centration steady states. Via Hermit Normal Form, their inverses can be calculated,

h1 = k3j2

and h2 = k1 j22

k23 j1

. Moreover, some of the kinetic parameters ks can be rewrit-

ten in terms of other kinetic parameters or convex parameters using equations in (43).Namely, one can substitute k4 = j2 and k2 = k3

j1j2

. We have four parameters: two con-vex parameters j1 and j2 and two kinetic parameters k1 and k3. The convex Jacobianof the Brusselator is

Jac( j) =[

( j1 − j2)h1 j1h2

− j1h1 − j1h2

],

where h are defined as above, h1 = k3j2

and h2 = k1 j22

k23 j1

. The characteristic polynomial is

λ2 + ( j1 h2 − j1 h1 + h1 j2) λ + h1 j2 j1 h2.

Finally the Hurwitz determinants follow as

a0 = h1 j2 j1h2,

H1 = j1h2 − j1h1 + h1 j2.

123


Condition (HB2) is satisfied if j2 = (1 − k1 j32

k33 j1

) j1. Also ∂ H1∂k3

= − k1 j22

k23

− j1j2

− 1 < 0,

so (HB3) holds independently of parameter values chosen. The conditions derivedfrom Theorem 6 state that one only needs to choose positive parameters k2, k3, j1and j2 that satisfy (HB1) in order to guarantee that Brusselator has a simple Hopfbifurcation. Obviously, it is possible to find at least one set of parameters so that thiscondition is true. Hence the Brusselator has at least one simple Hopf bifurcation. Notethat working backwards, one can retrieve all kinetic parameters and concentrationvalues (k1, k4, x1, x2) where the Brusselator will undergo a simple Hopf bifurcation.The standard simple Hopf bifurcation theorem would give the same conditions forkinetic parameters, but it would also need to be checked that concentration steadystates (corresponding to these parameters) are positive. Using the conditions fromTheorem 6 this is not necessary.

5.3 Model of glycolytic oscillations

Conditions from Theorem 6 can be applied to the model of glycolytic oscillationsfrom [15]. Most calculations were already preformed in Sect. 3. The convex Jacobianwith substituted values for h1 and h2 is

Jac( j) =[

( j3 − j1)k3

j1+ j3j3

k5j2

−2 j3k3

j1+ j3−( j2 + j3)

k5j2

],

where Hurwitz determinants follow as

a0 = j3k5k3

j2( j1 + j3)

(j23 − j3 j2 + j1 j3 + j1 j2

),

H1 = k5 + j3k5

j2+ k3( j1 − j3)

j1 + j3.

Condition (HB1) is satisfied if a0 > 0 and H1 = 0. It only needs to be checked thatthere exist j’s such that H1 = 0, because if this is true, then a0 > 0. (HB2) is alwayssatisfied, because ∂ H1

∂k5= 1 + j3

j2is positive for any positive parameter values. Hence,

for the model to have simple Hopf bifurcation, one only needs to find values of convexparameters and k1, k3 and k5 such that H1 = 0 and ( j2

3 − j3 j2 + j1 j3 + j1 j2) > 0 aresatisfied. Numerous choices are available. One possible choice is to take, j1 = 1, j2 =2.85, j3 = 1.15 and k3 = 10, k5 = 0.5. Figure 4 compares the Hopf bifurcationcalculated via conditions from Theorem 6 and a Hopf bifurcation calculated by XPPaut.There is good agreement between the two curves of bifurcations. In order to makethis comparison values for all kinetic parameters k and concentrations x have to becalculated, because XPPaut handles the original equations.

6 Discussion

This paper describes sufficient conditions for three important types of bifurcations, asaddle-node bifurcation, the simple branching point and a simple Hopf bifurcation.

123


0.1

2

3

4

5

678

1

2

k2

6040200

k1

HBxpp

oscillatoryregion

HBgb

HCxpp

Fig. 4 Two-parameter bifurcation diagram of the glycolytic model for k3 = 10, k4 = 4, k5 = 0.5. k1and k2 are bifurcation parameters. HBxpp and HBgb denote Hopf bifurcations detected by XPPaut andconditions from Theorem 6, respectively. XPPaut also detects a line of homoclinic bifurcations (HC)

Restricting to reaction systems with polynomial right hand sides theorems can bederived that state the possibility of such bifurcations in terms of contributions ofsubnetworks represented by extreme currents. In case of the saddle-node bifurcationthe approach has been used to describe a method for detecting bistability in reactionnetwork. The conditions allow the calculation of the exact parameter values wherebifurcations occur. Also these conditions are not always more complicated to analysethan the necessary conditions found in [21]. In general the work shows how abstractmathematics, in this case Algebraic Geometry (applied to reaction systems in thepioneering work of Gatermann), is immensely useful in real applications.

Several extensions of this work are possible. As mentioned in an earlier commentary,extension of the sufficient conditions from a simple Hopf bifurcation to a general Hopfbifurcation is a possible direction. In more general terms also other branching andbifurcation conditions could be transcribed along the lines of this paper.

Finally it should be said the conditions derived can easily be applied to smallchemical and biochemical networks. Their implementation into existing software forextreme currents such as Fluxanalyzer [19] or Copasi, would be useful for analysisof larger systems and hence another interesting direction. The discussion in Sect. 3shows that many necessary computational tools are already available.

Acknowledgments MirelaDomijan acknowledges the support from Commonwealth Scholarship. Thispaper is part of the research activities also supported by UniNet contract 12990 funded by the EuropeanCommission in the context of the VI Framework Programme. The authors wish to thank the Editor for thecare and effort in the shaping the presentation of this paper.

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