Johnston Translated CRNs 14xx 140602

Bull Math Biol (2014) 76:10811116DOI 10.1007/s11538-014-9947-5O R I G I NA L A RT I C L E

Translated Chemical Reaction Networks

Matthew D. Johnston

Received: 23 July 2013 / Accepted: 21 February 2014 / Published online: 8 March 2014 Society for Mathematical Biology 2014

Abstract Many biochemical and industrial applications involve complicated net-works of simultaneously occurring chemical reactions. Under the assumption of massaction kinetics, the dynamics of these chemical reaction networks are governed bysystems of polynomial ordinary differential equations. The steady states of these massaction systems have been analyzed via a variety of techniques, including stoichio-metric network analysis, deficiency theory, and algebraic techniques (e.g., Grbnerbases). In this paper, we present a novel method for characterizing the steady statesof mass action systems. Our method explicitly links a networks capacity to permit aparticular class of steady states, called toric steady states, to topological properties ofa generalized network called a translated chemical reaction network. These networksshare their reaction vectors with their source network but are permitted to have differ-ent complex stoichiometries and different network topologies. We apply the resultsto examples drawn from the biochemical literature.

Keywords Chemical kinetics Steady state Mass action system Complexbalancing Weakly reversible

1 Introduction

Chemical reaction networks are given by sets of reactions which transform a set ofreactants into a set of products at a given kinetic rate. Under the simplest of kinetic as-

Electronic supplementary material The online version of this article(doi:10.1007/s11538-014-9947-5) contains supplementary material, which is available to authorizedusers.

M.D. Johnston (B)Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI53706, USAe-mail: [email protected]

1082 M.D. Johnston

sumptions, that of mass action kinetics, the dynamics of a continuously-mixed chem-ical process may be modeled as an autonomous system of polynomial ordinary dif-ferential equations called a mass action system. Despite the simplistic formulationof such systems, the resulting dynamical systems may exhibit a wide range of dy-namical behaviors, including multistationarity (Craciun and Feinberg 2005, 2006),Hopf bifurcations (Wilhelm and Heinrich 1995, 1996), periodicity and chaos (rdiand Tth 1989).

Particular attention has been given recently to the nature of the steady states ofthese mass action systems, and in particular to the positive steady states (that is tosay, steady states in Rm>0). Such analysis is complicated by two main factors: (i) thenonlinear nature of the steady state equations, and (ii) the partitioning of the positivestate space into invariant affine spaces called compatibility classes. The analysis isfurther complicated by the observation that, for applied chemical processes, manyparameter values (i.e., the rate constants associated with each reaction) are typicallyunknown or only known to a certain precision; consequently, an emphasis has beenplaced on results which characterize the steady state set regardless of the rate constantvalues.

Nevertheless, many general results about the steady states of mass action systemsare well-known. It has been known since the 1970s that two fundamental classesof mass action systemsdetailed balanced systems (Volpert and Hudjaev 1985)and complex balanced systems (Horn and Jackson 1972)possess a unique posi-tive steady state within each positive compatibility class. These results were fur-ther related to the topological structure of the networks underlying reaction graph(reversibility and weak reversibility, respectively) in Feinberg (1972), Horn (1972).This network structure approach to characterizing steady states has been continued byMartin Feinberg in a series of papers focusing on network deficiency (Feinberg 1987,1988, 1995b), network injectivity (Craciun and Feinberg 2005, 2006), and concor-dance (Shinar and Feinberg 2012). This author, together with Jian Deng, ChristopherJones, and Adrian Nachman, was also instrumental in producing a paper affirmingthe long-standing conjecture that every weakly reversible network contains a positivesteady state (Deng et al. 2011).

Beginning with a series of papers published by Karin Gatermann in the early2000s, interest arose for characterizing the steady state sets of mass action systems byusing tools from algebraic geometry (Gatermann 2001; Gatermann and Huber 2002;Gatermann and Wolfrum 2005). Other prominent algebraists, including Alicia Dick-enstein and Bernd Sturmfels, have since become involved in adapting chemical reac-tion network results and terminology to this algebraic setting. These authors, alongwith Gheorghe Craciun and Anne Shiu, were instrumental in making the connectionbetween toric varieties, Birchs theorem from algebraic statistics, and complex bal-anced steady states in Craciun et al. (2009). This paved the way for the introductionof toric steady states, a generalization of complex balanced steady states which nolonger shared any direct correspondence on the topological structure of the reactiongraph (Prez Milln et al. 2012). Other related contributions to the study of the steadystates of mass action systems have been made in Clarke (1980), Conradi et al. (2008),Dickenstein and Prez Milln (2011), Feinberg (1989), Flockerzi and Conradi (2008),Markevich et al. (2004).

Translated Chemical Reaction Networks 1083

Research on the steady states of chemical reaction systems has also been con-ducted for systems which do not possess traditional mass action kinetics. One recentexample is that of generalized mass action systems introduced by Mller and Re-gensburger (2012). Generalized mass action systems maintain the topological struc-ture of standard chemical reaction networks but allow the powers of the monomialsappearing in the steady state conditions to differ from those implied by the networkstoichiometry. The authors show that a notion of complex balancing is maintainedin this generalized setting and that steady state properties can often still be inferredfrom the topological structure of the generalized reaction graph. This work also ledto a generalization of Birchs theorem.

In this paper, we introduce a method for relating the steady states of a mass ac-tion system to those of a specially-constructed generalized mass action system. Thismethod, called network translation, allows an explicit connection to be made betweensystems with toric steady states and generalized mass action systems with complexbalanced steady states. It also allows steady state properties to be inferred from gen-eralized network parameters. As such, this paper can be seen as a step toward clos-ing the gap between the network topology approaches to characterizing steady stateschampioned by Martin Feinberg et al., and the approaches of algebraists such as KarinGatermann and Alicia Dickenstein. We apply the results to several well-studied net-works contained in the biochemical literature.

While the primary application of this paper is characterizing the steady states ofmass action systems, it will be noted that translated chemical reaction networks areinteresting objects of study in their own right. We will close with a discussion of someavenues for future research, both within the study of translated chemical reactionnetworks and generalized chemical reaction networks in general.

2 Background

In this section, we present the terminology and notation relevant for the study ofchemical reaction networks and mass action systems which will be used throughoutthis paper. We will present these concepts both in the standard and generalized setting.

2.1 Chemical Reaction Networks

A chemical reaction network is given by a triple of sets N = (S,C,R) where:1. The species set S = {A1, . . . ,Am} consists of the fundamental molecules capable

of undergoing chemical change.2. The complex set C = {C1, . . . ,Cn} consists of linear combinations of the species

of the form Ci = mj=1 yijAj , i = 1, . . . , n. The terms yij Zm0 are called sto-ichiometric coefficients and allow us to define the stoichiometric vectors yi =(yi1, . . . , yim), i = 1, . . . , n.

3. The reaction set R = {R1, . . . ,Rr} consists of interactions of the form Rk =Ci Cj for some i, j = 1, . . . , n, i = j , for k = 1, . . . , r .

1084 M.D. Johnston

It is also typically assumed in chemical reaction network theory that: (a) every speciesin S appears in at least one complex in C; (b) every complex in C appears in atleast one reaction in R; and (c) there are no self-reactions (i.e., reactions of the formCi Ci ).

Remark 1 Note on indexing It will be necessary throughout this paper to retain ele-ments of both reaction-centered indexing and complex-centered indexing. This differsfrom much of chemical reaction network theory literature. To simplify the relation-ships between these sets, when ambiguity is not a concern we will represent the setsC and R by their corresponding index sets, {1, . . . , n} and {1, . . . , r}, respectively.

To further formalize the relationship between the reactions and complexes, we de-fine the set of complexes which appear on the left (right) of at least one reaction tobe the reactant (product) complex set CR (CP). We furthermore define the mappings :R CR and :R CP so that (i) ((i)) corresponds to the reactant (prod-uct) complex of the ith reaction. The mappings and will be called the reactantprofile and product profile of N , respectively. This allows the reaction network N tobe represented in the form

N : C(i) C(i), i = 1, . . . , r.For example, consider the reaction network

N : C112

C2 3 C3 4 C4.

The reactant complex set is CR= {1,2,4} and the reaction profile is((1), (2), (3), (4)

) = (1,2,2,4).Correspondingly, the product complex set is CP = {1,2,3} and the product profile is((1), (2), (3), (4)) = (2,1,3,3).

2.2 Reaction Graph and Deficiency

Interpreting chemical reaction networks as interactions between stoichiometricallydistinct complexes naturally gives rise to their interpretation as directed graphsG(V,E) where the vertices are the complexes (i.e., V = C) and the edges are thereactions (i.e., E = R). In the literature, this graph has been termed the reactiongraph of a network (Horn and Jackson 1972).

There are several properties of a networks reaction graph of which we will needto be aware. We will say that a complex Ci is connected to Cj if there exists a se-quence of complexes {C(1), . . . ,C(l)} such that Ci = C(1), Cj = C(l), and eitherC(k) C(k+1) or C(k+1) C(k) for all k = 1, . . . , l 1. We will furthermore saythat Ci is strongly connected to Cj if there is such a set where all reactions are inthe forward direction, and a (potentially different) set where all reactions are in thebackward direction. A linkage class is a maximal set of connected complexes and a


strong linkage class is a maximal set of strongly connected complexes. For instance,consider the chemical reaction network

N : C1 C2 C3 C4 C5. (1)We have the linkage classes L1 = {1,2,3} and L2 = {4,5}, and the strong linkageclasses 1 = {1}, 2 = {2,3} and 3 = {4,5}. The number of linkage classes in anetwork will be denoted by .

A chemical reaction network is said to be weakly reversible if the linkage classesand strong linkage classes coincide. For example, we can see that the network (1) isnot weakly reversible. By contrast, the following network can easily be seen to beweakly reversible:

N :C1 C2

C3

(2)

To each reaction i R, we associate the reaction vector y(i) y(i). These vec-tors keep track of the net stoichiometric change in the individual species as the resultof the reaction. They also give rise to the stoichiometric subspace

S = span{y(i) y(i) | i = 1, . . . , r} Rm.The dimension of the stoichiometric subspace is denoted s = dim(S).

We may now define the following network parameter, which was introduced byHorn (1972) and Feinberg (1972).

Definition 1 The deficiency of a chemical reaction network N is given by = n s.

The deficiency is a nonnegative parameter which can be determined from the struc-ture of the chemical reaction network itself and has been the study of significantresearch (Feinberg 1987, 1988, 1995a, 1995b).

2.3 Mass Action Systems

In order to model how the concentrations of the chemical species evolve over time,we assume that the reaction vessel is spatially homogeneous and that the reactingspecies are in sufficient quantity to be modeled as chemical concentrations. We willfurthermore assume that the system obeys mass action kinetics, so that the rate of eachreaction is proportional to the product of concentrations of the reactant species. Thatis to say, if the ith reaction has the form A1 + A2 then we have the reactionrate = ki[A1][A2], where the proportionality constant ki is commonly called the rateconstant of the reaction. We define k Rr>0 to be the vector of rate constants.

In order to determine how the concentration vector x = (x1, x2, . . . , xm) Rm0evolves over time, it is necessary to introduce the following matrices (notationadapted from Gatermann (2001), Gatermann and Huber (2002), Gatermann and Wol-frum (2005)):

1086 M.D. Johnston

The complex matrix Y Zmn0 is the matrix where the j th column is the j th stoi-chiometric vector yj , i.e., [Y ],j = yj , j = 1, . . . , n.

The matrix Ia Znr0 is the matrix with entries [Ia]ji = 1 if (i) = j , [Ia]ji = 1if (i) = j , and [Ia]ji = 0 otherwise.

The matrix Ik Rrn0 is the matrix with entries [Ik]ij = ki if (i) = j , and[Ik]ij = 0 otherwise.We will also need the mass action vector (x) Rn0, which is the vector with entries[ (x)]j = xyj , j = 1, . . . , n.

These definitions allow us to define the mass action system M = (S,C,R, k)governed by

dx

dt= Y Ia Ik (x). (3)

It is worth noting that the stoichiometric matrix := Y Ia Zmr contains the reac-tion vectors y(i) y(i), i = 1, . . . , r , as its columns. It follows that trajectories of(3) are confined to affine translates of the stoichiometric subspace space. We there-fore define the stoichiometric compatibility classes to be Cx0 = (S + x0) Rm0 andnote that, if x(t) is a solution of (3) with x(0) = x0 Rm0, then x(t) Cx0 for allt 0 (Horn and Jackson 1972; Volpert and Hudjaev 1985).

Remark 2 The deficiency can also be defined in terms of Y and Ia . We have = dim(ker(Y ) Im(Ia)). The formula given in Definition 1 is equivalent (see Ap-pendix A) but the formula given here will be more intuitive for the results of Sect. 5.1.

2.4 Generalized Mass Action Systems

An alternative but related kinetic form to mass action kinetics is power-law formal-ism. In this formulation, the kinetic terms still have the product of concentrationsform of monomials but are permitted to take (potentially non-integer) powers whichdo not necessarily correspond to the stoichiometry of the reactant complex (Savageau1969).

This has recently been extended by Mller and Regensburger (2012) to a morenetwork-focused approach called generalized chemical reaction networks.

Definition 2 A generalized chemical reaction network N = (S,C,CK,R) is a chem-ical reaction network (S,C,R) together with a set of kinetic complexes CK which arein one-to-one correspondence with the elements of C.

The set (S,C,R) determines the reaction structure and stoichiometry of the gen-eralized chemical reaction network, just as it does for a standard chemical reactionnetwork; however, each complex in C is associated to a kinetic complex in CK . Thekinetic complexes CK do not appear directly in the reaction graph but are called uponwhen determining the kinetics (i.e., the monomials in (3)).

Remark 3 The subscript notation used here for the kinetic complex CK differs fromthat of Mller and Regensburger (2012), where the kinetic complexes are denoted C.


This modification is made to avoid confusion with translated chemical reaction net-works introduced in Sect. 4.1.

Since the kinetic complexes are in one-to-one correspondence with the stoichio-metric complexes, we may consider properties of a second reaction graph with thekinetic complexes CK in place of the regular complexes C (i.e., the reaction graphof (S,CK,R)). The hypothetical kinetic reaction graph does not determine the sto-ichiometry of the network but it does play an important role in determining wheresteady states of the corresponding kinetic model may lie. The kinetic-order subspaceSK is defined as

SK = span{(yK)(i) (yK)(i) | i = 1, . . . , r

}, (4)

where the vectors (yK)j = ((yK)j1, (yK)j2, . . . , (yK)jm), j = 1, . . . , n, correspondto the kinetic complexes (CK)j , j = 1, . . . , n. The kinetic complex matrix YK is thematrix where the j th column is the j th kinetic complex vector (yK)j , i.e., [YK ],j =(yK)j , j = 1, . . . , n. The deficiency of a generalized chemical reaction network isthe same as given by Definition 1. We further define the kinetic deficiency to be thedeficiency of the network (S,CK,R) and denote it by K .

When space is not a concern, the correspondence between the stoichiometric andkinetic complexes will be denoted by dotted lines in the reaction graph. For example,we write

7A1 +A3 A1 +A2k1k2

A3 5A2 (5)

to imply that the stoichiometric complex C1 = A1 + A2 is associated with the ki-netic complex (CK)1 = 7A1 + A3 and that the stoichiometric complex C2 = A3 isassociated with the kinetic complex (CK)2 = 5A2.

The kinetic framework for generalized chemical reaction networks is the follow-ing.

Definition 3 The generalized mass action system M= (S,C,CK,R, k) correspond-ing to the generalized chemical reaction network N = (S,C,CK,R) is given by

dx

dt= Y Ia Ik K(x), (6)

where Y , Ia , and Ik are as in (3), and the generalized mass action vector K(x) hasentries [K(x)]j = x(yK)j , j = 1, . . . , n.

In other words, a generalized mass action is the mass action system (3) with themonomials xyj replaced by the monomials x(yK)j . The generalized mass action sys-tem corresponding to network (5) is

dx1dt

= dx2dt

= dx3dt

= k1x71x3 + k2x52 ,

where the stoichiometry of the network comes from the stoichiometric complexes Cbut the monomials come from the kinetic complexes CK .

1088 M.D. Johnston

Remark 4 Mller and Regensburger also give sufficient conditions on the sign vec-tors associated with the stoichiometric and kinetic-order subspace S and SK for theexistence of a unique (or multiple) generalized complex balanced steady state (states)within compatibility classes Cx0 = (x0 + S) Rm>0 (Theorem 3.10 and Proposition3.2 of Mller and Regensburger (2012), respectively). They also give a generaliza-tion of the well-known Birchs Theorem (Proposition 3.9, Mller and Regensburger2012). These conditions will not be considered in this paper but have received furtherattention in Mller et al. (2013).

Remark 5 It is worth noting that the kinetic complexes CK are not required to havethe same support as that of the original complex set C in the generalized chemicalreaction network framework. This contrasts with some general kinetic frameworksfor chemical reaction systems, e.g., Angeli et al. (2007).

3 Steady States of Mass Action Systems

When considering the steady states of a mass action system, we are typically inter-ested in the positive steady state set given by

E = {x Rm>0 | Y Ia Ik (x) = 0}. (7)

Characterizing (7) in general is a difficult algebraic task due to the nonlinear nature ofthe equations. It is somewhat surprising, therefore, that many characterizations existwithin the literature which not only guarantee certain properties of the steady state set(7), but guarantee these properties for all compatibility classes and also for all rateconstants. We will now consider two classifications of steady states which have ap-peared prominently in the literature: complex balanced steady states (Feinberg 1972;Horn 1972; Horn and Jackson 1972) and toric steady states (Craciun et al. 2009).

3.1 Complex Balanced Steady States

The following class of steady states was introduced by Horn and Jackson (1972) as ageneralization of detailed balanced steady states.

Definition 4 A positive steady state x Rm>0 of a mass action system M =(S,C,R, k) is called a complex balanced steady state if

(x) ker(Ak),where Ak := Ia Ik Rnn. Furthermore, a mass action system will be called a com-plex balanced system if every steady state is a complex balanced steady state.

It is known that if a mass action system has a complex balanced steady state, thenall steady states are complex balanced (Lemma 5B, Horn and Jackson 1972). Con-sequently, all mass action systems with complex balanced steady states are complexbalanced systems. It is also known that every positive stoichiometric compatibility


class Cx0 of a complex balanced system contains precisely one steady state (Lemma5A, Horn and Jackson 1972), and the complex balanced steady state set is given by

E = {x Rm>0 | ln(x) ln(a) S},

where a Rm>0 is an arbitrary complex balanced steady state of the system.The main result of Feinberg (1972) and Horn (1972), popularly called the Defi-

ciency Zero Theorem, relates the capacity of a network to permit complex balancedsteady states to properties of the reaction graph.

Theorem 1 (Theorem 4A, Horn 1972) Every mass action system M = (S,C,R, k)admitted by a network N = (S,C,R) is complex balanced if and only if N is weaklyreversible and has a deficiency of zero (i.e., = 0).

This result gives computable properties, depending on the network topology alone,which are sufficient to guarantee strong restrictions on the nature, location, and num-ber of steady states of the corresponding mass action systems. Remarkably, theseresults are guaranteed to hold for all possible choices of positive rate constants andall stoichiometric compatibility classes.

A surprising observation of Mller and Regensburger (2012) is that complex bal-ancing may also be meaningfully defined for generalized mass action systems.

Definition 5 A positive steady state x Rm>0 of a generalized mass action systemM= (S,C,CK,R, k) is called a generalized complex balanced steady state ifK(x) ker(Ak),

where Ak := Ia Ik Rnn.The authors show that a generalized chemical reaction network which permits

generalized complex balanced steady states is weakly reversible (Proposition 2.18,Mller and Regensburger 2012) and that the steady state set is given by

E = {x Rm>0 | ln(x) ln(a) SK}, (8)

where a Rm>0 is an arbitrary generalized complex balanced steady state of the sys-tem and SK is the kinetic-order subspace defined by (4).

They also prove the following result.

Theorem 2 (Proposition 2.20, Mller and Regensburger 2012) Every generalizedmass action system (S,C,CK,R, k) admitted by a generalized chemical reactionnetwork (S,C,CK,R) has at least one generalized complex balanced steady stateif the underlying reaction network is weakly reversible and the kinetic deficiency ofthe network is zero (i.e., K = 0).Remark 6 It is important to note that not all of the properties of standard complexbalanced steady states apply in the generalized setting. In particular, the generalizedcomplex balanced steady state set (8) may intersect a positive stoichiometric compat-ibility class Cx0 at a unique point, multiple times, or not at all.

1090 M.D. Johnston

3.2 Toric Steady States

Algebraic techniques have recently become prominent in the study of steady stateproperties of mass action systems (Conradi et al. 2008; Craciun et al. 2009; Prez Mil-ln et al. 2012; Shiu 2010). We omit here background on the algebraic objects of in-terest, such as varieties, ideals, and Grbner bases. The interested reader is directedto the accessible textbook of Cox, Little, and OShea (Cox et al. 2007).

The connection between toric geometry, Birchs theorem in algebraic statistics,and complex balanced steady states is made in Craciun et al. (2009). In particular,they show that the steady state ideal for any complex balanced system is a toric ideal.That is to say, it is a prime ideal which is generated by binomials. This justified theauthors choice to refashion complex balanced systems as toric dynamical systems.The authors furthermore showed the following inclusion. (For a detailed introductionto the tree constants Ki and Kj (31), and the relationship between Theorem 3 andDefinition 4, see Appendix B.)

Theorem 3 (Corollary 4, Craciun et al. 2009) The steady state ideal of a complexbalanced system contains the binomials Kixyj Kj xyi where i, j Lk for somek = 1, . . . , and Ki and Kj are the corresponding tree constants.

There are a number of desirable features which follow from a system having atoric ideal. It allows, for instance, an easy parametrization of the associated variety.

It was noted in Prez Milln et al. (2012) that many mass action systems whichdo not admit complex balanced steady states nevertheless have steady state idealswhich are generated by binomials. The authors say that such systems have toricsteady states. In order to derive sufficient conditions for a system to have toric steadystates, they introduce the complex-to-species matrix := Y Ia Ik Rmn. It is a sur-prising result that, for many non-complex balanced systems, ker() can, in fact, bedecomposed in the same way as ker(Ak) can be for complex balanced systems (seeAppendix B). The authors show the following inclusion.

Theorem 4 (Theorem 3.3, Prez Milln et al. 2012) Consider a chemical reactionnetwork N = (S,C,R). Suppose that ker() has dimension d and that there exists apartition 1,2, . . . ,d of {1, . . . , n} and a basis bk , k = 1, . . . , d , of ker() withsupp(bk) = k . Then the steady state ideal is generated by the binomials [bk]ixyj [bk]j xyi for i, j k , k = 1, . . . , d .

It is striking that the binomials in Theorem 3 and Theorem 4 are both constructedby first partitioning the complexes of the chemical reaction network into disjoint com-ponents and then computing a basis for a specific kernel restricted to the support ofthese components. Furthermore, the components in Theorem 3 have a clear interpre-tation in terms of the reaction network: they are the linkage classes of the networksreaction graph. The components in Theorem 4 are less well-understood and are leftas computational constructs in Prez Milln et al. (2012).

It is the clarification of the connection between Theorem 3 and Theorem 4, andof complex balanced steady states and toric steady states in general, which will be


the main concern of this paper. We will show that the supports of the componentsderived in Theorem 4 can often be corresponded to linkage classes just as they are inTheorem 3. These linkage classes, however, will not be those of the original reactionnetwork; rather, they will be the linkage classes of a related generalized reactionnetwork which we will call a translated chemical reaction network.

4 Main Results

In this section, we introduce the notion of network translation and show how thisconcept can be used to characterize mass action systems with toric steady states. Tomotivate the results, throughout this section we will consider the following pairs ofexample networks. In both cases, the first is a standard chemical reaction network,and the second is a generalized one.

Example 1 Consider the chemical reaction network N1 and generalized chemicalreaction network N1 given, respectively, by

A1 k1 2A1,

A1 +A2 k2 2A2,

A2 k3 0,

(Regular Network N1)

and

A1 0 k1 A1 A1 +A2k3 k2

A2...

A2

(Generalized Network N1)

The network N1 is a common network representation of the LotkaVolterrapredatorprey model, where A1 represents the prey and A2 represents the preda-tor. It can be easily checked that the governing equations (3) of the mass actionsystem M1 = (S,C,R, k) and the governing equations (6) of the generalized massaction system M1 = (S, C, CK, R, k) coincide. That is to say, they are dynamicallyequivalent. We will give general conditions for such a property in Lemma 2.

Example 2 Consider the chemical reaction network N2 and generalized chemicalreaction network N2 given, respectively, by

A1k1k2

A2 k3 A3,

A1 +A3 k4 A1 +A2,

A2 +A3 k5 2A2,

(Regular Network N2)

1092 M.D. Johnston

and

A1 A1k1k2

A2k3

k4+ k1k2 k5A3 A1 +A3

...

A2(Generalized Network N2)

While the governing equations (3) of the mass action system M2 and the governingequations (6) of the generalized mass action system M2 do not coincide, it can beeasily checked that the systems have the same steady state set. In particular, they bothyield the binomial Grbner basis

k1x1 k2x2, (k1k5 + k2k4)x1x3 k1k3x2

(9)

in the lexicographical ordering x3 > x2 > x1. We will give general conditions for thiscorrespondence of steady states in Lemma 4. In particular, we will give conditionsfor when such a correspondence can be made, and how the nontrivial rate constantk4 + (k1/k2)k5 was chosen.

4.1 Translated Chemical Reaction Networks

The following is the fundamental new concept of this paper.

Definition 6 Consider a chemical reaction network N = (S,C,R) with source com-plex set CR and a generalized chemical reaction network N = (S, C, CK, R) withsource complex set CR. We will say N is a translation of N if:1. There is a surjection h1 : R R so that y(h1(i)) y(h1(i)) = y(i) y(i) for

all i = 1, . . . , r ;2. There is a surjection h2 : CR CR so that h2((i)) = (h1(i)) for all i =

1, . . . , r ; and3. There is an injection h3 : CR CR so that h2(h3(j)) = j and (CK)j = h3(j) for

all j CR.The process of finding a generalized network N which is a translation of N will becalled network translation.

Remark 7 We will use the tilde notation exclusively to denote properties of trans-lations N . For instance, we will let C denote the stoichiometric complexes and CKdenote the kinetic complexes of the translation N . We will correspondingly let andK denote the regular and kinetic deficiency of N . This differs from the definitionsgiven in Mller and Regensburger (2012), where C was used to denote the set of ki-netic complexes and was use to represent the kinetic deficiency of a generalizedchemical reaction network.


Remark 8 The relationship between R, R, CR, and CR through the mappingsh1, h2, , and can be visualized as

R h1 RN : : N

CR h2 CR,This representation is useful when interpreting property 2 of Definition 6.

These formal conditions will be required for the technical results contained inSects. 4.2 and 4.3. It will be helpful, however, to use the following intuition about thethree properties given in Definition 6 when considering examples:

1. There is a correspondence between the reactions in N and N which preserves thereaction vectors.

2. Every reaction from a common source complex in N must be mapped to a reactionfrom a common source complex in N . In other words, reactions from a singlesource complex may not be broken apart by the translation.

3. The kinetic complexes are chosen from the set of source complexes of N whichare mapped to the corresponding source complex of N .Another useful way to think about a network translations N is that it is a network

produced by translating the reactions N by adding or subtracting species to the left-and right-hand sides of each reaction, while preserving the original complexes as thekinetic complexes of the new generalized network. This operation satisfies the threeconditions of Definition 6. To investigate some subtleties which may arise, considerthe examples introduced at the beginning of the section.

Example 1 Consider translating the individual reactions of N1 in the following way:

A1 2A1 (A1) 0 A1,A1 +A2 2A2 (A2) = A1 A2,

A2 0 (+0) A2 0.This satisfies properties 1 and 2 of Definition 6. In order to satisfy property 3, weassociate to each source complex of the post-translation network the source complexof the original reaction. That is to say, we associate A1 to 0, A1 +A2 to A1, and A2to A2. This produces the generalized network N1 so that we may say that N1 is atranslation of N .

Example 2 Consider translating the individual reactions of N2 in the following way:

A1 A2 A3 (+0) A1 A2 A3,A1 +A3 A1 +A2 (A1) = A3 A2,A2 +A3 2A2 (A2) A3 A2.

1094 M.D. Johnston

While this produces the reaction structure of N2, there are two important differences.The first is that we have two reactions in N2, A1 +A3 A1 +A2 and A2 +A3 2A2, which are associated with the reaction A3 A2 in N2. We have thereforereduced the number of reactions. This is allowed by property 1 but will require someadditional consideration when transferring rate constants between the networks.

A second important difference is that we may no longer unambiguously assignkinetic complexes to all of the source complex in N2. This can be seen by noting thatA1 +A3 and A2 +A3 are both mapped to A3. This is allowed by property 2 of Def-inition 6, and property 3 allows us to choose either complex as the kinetic complex.Choosing A1 +A3 completes the correspondence, so that N2 is a translation of N2by Definition 6.

These examples demonstrate a fundamental difference between translations,namely, whether or not there is a one-to-one correspondence between the sourcecomplexes of the two networks. We therefore introduce the following important clas-sification of translations.

Definition 7 Consider a chemical reaction network N = (S,C,R) and a translationN = (S, C, CK, R). We will say N is a proper translation of N if h2 is injective aswell as surjective. A translation N will be called improper if it is not proper.

Remark 9 If N is proper then h3 = h12 so that the kinetic complexes CK are uniquelydefined in property 3 of Definition 6. A translation is improper if the kinetic com-plexes are not uniquely defined by the mapping h2.

Remark 10 Note that, for proper translations, h1 is necessarily injective as well assurjective. This follows from the observation that if two reactions in N are trans-lated to the same reaction in N there will necessarily be two source complexes in Nmapped to the same source complex in N .

Examples 1& 2 It can be readily seen that N1 is a proper translation of N1 while N2is an improper translation of N2.

It will also be important to understand how the stoichiometric and kinetic-ordersubspaces associated with N and its translation N are related, and in particular in thecase when N is weakly reversible.

Lemma 1 Consider a chemical reaction network N = (S,C,R) and a translationN = (S, C, CK, R). Then the stoichiometric subspaces S of N and S of N coincideand, if N is weakly reversible, the kinetic-order subspace SK of N is given by

SK = span{yh3(i) yh3(j) | i, j Lk, k = 1, . . . , }, (10)where Lk, k = 1, . . . , , are the linkage classes of N .

Proof Let N = (S,C,R) be a chemical reaction network and N = (S, C, CK, R) bea strong translation of N . By property 1 of Definition 6, N and N have the same


reaction vectors and therefore have the same stoichiometric subspace S. This provesthe first claim.

To the second claim, it is well-known that the span of the reaction vectors of achemical reaction network is the same as the span of the stoichiometric differencesof complexes on the same connected component (for example, see Feinberg 1972,p. 189). Since the kinetic complexes are drawn from CR by h3, this completes theproof.

4.2 Properly Translated Mass Action Systems

We assign a kinetics to a proper translation in the following way.

Definition 8 Suppose N = (S, C, CK, R) is a proper translation of a chemical re-action network N = (S,C,R) and M = (S,C,R, k) is a mass action system cor-responding to N . Then we define the properly translated mass action system of Mto be the generalized mass action system M = (S, C, CK, R, k) where kj = ki ifj = h1(i).

This relationship is the natural correspondence since we make the same corre-spondence for rate constants as we make for reactions. In other words, for propertranslations, we will simply transfer the rate constant along with the reaction in thetranslation process.

The following relates the dynamics of a properly translated mass action systemM to the original mass action system M.

Lemma 2 Suppose M = (S, C, CK, R, k) is a properly translated mass action sys-tem of M= (S,C,R, k). Then the generalized mass action system (6) governing Mis identical to the mass action system (3) governing M. In particular, the two systemshave the same steady states.

Proof Consider a chemical reaction network N = (S,C,R) with correspondingmass action system M = (S,C,R, k) and a proper translation N = (S, C, CK, R)of N . Let M = (S, C, CK, R, k) be a properly translated mass action system of Mdefined by Definition 8. Without loss of generality, we will index the reactions of Nso that h1 may be taken to be the identity.

Since N is a translation of N , it follows from property 1 of Definition 6 that thesystem (3) governing M is given by

dx

dt= Y Ia Ik (x) = Y Ia Ik (x),

where Y and Ia correspond to the translation N . It remains to relate the rate vectorR(x) := Ik (x) to R(x) := Ik K(x) corresponding to M. Notice that we have: [K(x)]j = xyh3(j) for all j CR by property 3 of Definition 6; ki = ki , i = 1, . . . , r , by Definition 8; h2((i)) = (i) for all i = 1, . . . , r , by property 2 of Definition 6.

1096 M.D. Johnston

h3(h2(j)) = j for all j CR by Condition 1 of Definition 7.It follows that, for all i = 1, . . . , r , Ri(x) = kixyh3((i)) = kixyh3((i)) = kixyh3(h2((i))) =kix

y(i) = Ri(x). Consequently, we havedx

dt= Y Ia Ik (x) = Y Ia Ik K(x) (11)

so that M and M have the same dynamics, and we are done.

Example 1 We can now see that the rate constant choices for the proper translationN1 yield a translated mass action system by Definition 8. We have already observedthat the mass action system (3) corresponding to M1 and the generalized mass actionsystem (6) corresponding to M1 coincide. This is exactly the conclusion guaranteedby Lemma 2.

4.3 Improperly Translated Mass Action Systems

It is more difficult to sensibly define a generalized mass action system M =(S, C, CK, R, k) for an improper translation N = (S, C, CK, R) than a proper trans-lation. This is because there is necessarily at least one source complex in N whichdoes not appear as the kinetic complex of any source complex in N . As a result, animproperly translated generalized mass action system M will necessarily have fewermonomials than the original mass action system M. A comprehensive dynamicalresult like Lemma 2 is consequently not possible.

In this section, we introduce conditions, which we call resolvability conditions,under which the steady state set of a generalized mass action system M correspond-ing to an improper translation N may still be shown to coincide with the steady stateset of the original mass action system M. We will use the networks N2 and N2 ofExample 2 to illustrate the main concepts.

We will need first of all to correspond the reactant complexes in N which are notused as kinetic complexes in N to those which are. To that end, we introduce thefollowing definitions.

Definition 9 Suppose N = (S, C, CK, R) is an improper translation of a chemicalreaction network N = (S,C,R). Then:1. The kinetically relevant complex of a reaction i R will be denoted

(i)K = h3(h2

((i)

)). (12)

2. A reaction i R will be said to be improperly translated if (i) = (i)K and theset of improperly translated reactions will be denoted

RI ={i R | (i) = (i)K

}. (13)

3. The improper subspace SI of N will be defined to beSI = span{y(i) y(i)K | i RI }. (14)


These definitions clarify several relationships unique to improper translations.A source complex is said to be kinetically relevant to a reaction if it is the kineticcomplex in the translation. A reaction will then be called improperly translated if thiskinetically relevant complex differs from its own source complex in N . Finally, thestoichiometric differences between the original and kinetically relevant complexesforms a basis of the improper subspace SI .

The following result gives conditions under which the monomials correspondingto source complexes of improper reaction may be explicitly related to the correspond-ing kinetically relevant complexes.

Lemma 3 Suppose N = (S, C, CK, R) is an improper translation of a chemical re-action network N = (S,C,R). Suppose furthermore that N is weakly reversible andthat SI SK where SK is the kinetic-order subspace of N . Then, for every i RI ,there exist constants cij and pairs pj , qj CR, j = 1, . . . , s, such that

xy(i) =[

s

j=1

(xyh3(pj )

xyh3(qj )

)cij]

xy(i)K . (15)

Proof Consider an arbitrary i RI and the difference y(i) y(i)K . Since N isweakly reversible, by Lemma 1 we can define a basis for SK by {yh3(pj ) yh3(qj )}sj=1where pj , qj CR by removing linearly dependent vectors from the generating set(10). Since SI SK , it follows from (14) that we can write

y(i) y(i)K =s

j=1cij (yh3(pj ) yh3(qj )), (16)

where cij , j = 1, . . . , s, are constants. The form (15) follows directly by rearranging(16) and raising the terms into the exponent of x Rm>0.

Lemma 3 gives conditions under which the monomials corresponding to sourcecomplexes in the original network may be explicitly related to the correspondingkinetic complexes. The result is deficient, however, in that the factors which relatethe monomials are state dependent. The following stronger condition will allow us tosimplify this dependence.

Definition 10 Suppose N = (S, C, CK, R) is an improper translation of a chemicalreaction network N = (S,C,R). Suppose that N is weakly reversible and that SI SK where SK is the kinetic-order subspace of N . Define Ki , i = 1, . . . , n, to be thetree constants (31) associated with the network (S, C, R, k) with the reaction weights

kj =

{i|h1(i)=j}ki . (17)

Then:

1098 M.D. Johnston

1. For every i RI , define the kinetic adjustment factor of (i) and (i)K

K(i),(i)K =s

j=1

(Kpj

Kqj

)cij, (18)

where the constants cij and pairs pj , qj CR are determined according to (15) ofLemma 3.

2. The translation N will be called resolvable if, for every i RI , the kinetic adjust-ment factors K(i),(i)K given in (18) do not depend explicitly on any kj , j RI .The condition of resolvability will be the key to removing the state dependence

in (15). It will allow us to define a translated mass action system for an impropertranslation in the following way.

Definition 11 Consider a chemical reaction network N = (S,C,R) and an associ-ated mass action system M = (S,C,R, k). Suppose N = (S, C, CK, R) is a resolv-able improper translation of N . Consider the adjusted rate constants

ki ={ki, for i /RI ,(K(i),(i)K )ki, for i RI . (19)

We define the improperly translated mass action system to be the generalized massaction system (S, C, CK, R, k) with rate constants

kj =

{i|h1(i)=j}ki . (20)

This defines a generalized mass action system for improper translations which arealso resolvable. In this case, we may sensibly define such a system by adjusting therate constants by a carefully constructed kinetic adjustment factor which dependson the remainder of the rate constants. In order to see how Definition 11 applies inpractice, reconsider Example 2.

Example 2 We know the N2 is an improper translation of N2. We now want to de-termine if it is resolvable and, if it is, what the corresponding improperly translatedmass action system M2 is by Definition 11.

We start by defining the concepts in Definition 9. We have that the kinetic rel-evant complex (i)K for the first four reactions of N2 coincides with their sourcecomplex (i) in N2. The fifth reaction, however, is assigned the kinetic complexA1 + A3 in the translation N2 rather than the original source complex A2 + A3 sothat (5) = (5)K . It follows that RI = {5}. The improper subspace is given bySI = span{(0,1,1) (1,0,1)} = span{(1,1,0)}.

We now want to apply Lemma 3. It is clear that N2 is weakly reversible. We havealready determined SI , so it only remains to find the kinetic-order subspace SK . Thisis given by the span of the reaction vectors of the network (S, CK, R):

A1 A2 A1 +A3.


It can be easily seen that SK = span{(1,1,0), (1,1,1)} so that SI SK . Allowingthe basis elements to be represented in terms of kinetic reaction vectors correspondingto CK , we have that (16) gives

y(5) y(5)K = (0,1,1) (1,0,1) = (1)[(0,1,0) (1,0,0)].

After raising this into a power of x = (x1, x2, x3) R3>0 and rearranging, we arrive atthe identity

x2x3 =(

x2x1

)

x1x3 (21)

corresponding to (15). The important feature of (21) is that it explicitly relates themonomial x2x3, which corresponds to the unused complex A2 +A3, to the monomialx1x3, which corresponds to the kinetically relevant complex A1 + A3 of the fifthreaction. The adjustment factor x2/x1 is a ratio of monomials in the remainder of theset CK (the complexes A1 and A2).

We now want to determine if the translation N2 is resolvable. We have alreadyseen that the network N2 is weakly reversible and SI SK . To compute the kineticadjustment factor (18) we consider the network (S, C, R, k) with reaction weights ki ,i = 1, . . . ,4, given by (17):

A1k1k2

A2k3

k4+k5A3. (22)

Setting C1 =A1, C2 =A2, and C3 =A3, the required tree constants (31) areK1 = k2(k4 + k5) and K2 = k1(k4 + k5)

so that K(5),(5)K = K2/K1 = k1/k2. Since this does not depend explicitly on therate constant for the improper reaction, k5, we have that N2 is a resolvable impropertranslation of N2.

We may now apply Definition 11 to construct the improperly translated mass ac-tion system M = (S, C, CK, R, k). The only difference between the rate constantsof this network and (23) is that the rate constant k5 must be adjusted by the kineticadjustment factor K(5),(5)K = k1/k2. By (19) and (20), we define k1 = k1, k2 = k2,k3 = k3, and k4 = k4 + (k1/k2)k5 to get:

A1 A1k1k2

A2k3

k4+ k1k2 k5A3 A1 +A3

...

A2This coincides with the generalized mass action system given at the beginning of thesection.

We noted in the introduction of this section that the mass action system associatedwith N2 and the generalized mass action system associated with N2 shared the same

1100 M.D. Johnston

steady states. This is a general property of improperly translated mass action systemsdefined by Definition 11 with the additional condition = 0 on the stoichiometric de-ficiency of the translation. This condition allows the steady state set to be completelycharacterized by tree constants, a requirement of the result.

The following result is proved in Appendix C.

Lemma 4 Consider an improper translation N = (S, C, CK, R) of a chemical re-action network N = (S,C,R). Suppose that N is resolvable and = 0. Let M =(S,C,R, k) be a mass action system corresponding to N and M= (S, C, CK, R, k)be an improperly translated mass action system corresponding to N and defined byDefinition 11. Then the steady states of the system (6) governing M coincide withthose of the system (3) governing M.

Example 2 We can easily determine that the structural deficiency of N2 is zero(i.e., = 0) so that the generalized mass action system M2 = (S, C, CK, R, k) de-fined by Definition 11 has the same steady state set as the mass action systemM2 = (S,C,R, k) associated with N2.

4.4 Connection with Toric Steady States

We now make the following connection between translated mass action systems,complex balanced generalized mass action systems, and systems with toric steadystates. The following is the main result of the paper and is proved in Appendix D.

Theorem 5 Suppose N = (S, C, CK, R) is a weakly reversible translation of a chem-ical reaction network N = (S,C,R) which is either proper or improper and re-solvable. Let M= (S, C, CK, R, k) denote a properly or improperly translated massaction system corresponding to M = (S,C,R, k) (Definition 8 or Definition 11).Suppose furthermore that N satisfies = K = 0. Then:1. The mass action system M has toric steady states for all rate constant vectors

k Rr>0.2. The toric steady state ideal of M is generated by the binomials

Kixyh3(j) Kj xyh3(i) ,

where i, j Lk for some k = 1, . . . , , and Ki , i = 1, . . . , n, are the tree constants(31) corresponding to the reaction graph of (S, C, R, k).

3. The toric steady states of M correspond to the complex balanced steady states ofM and can be represented by

E = {x Rm>0 | ln(x) ln(a) SK},

where a Rm>0 is any generalized complex balanced steady state of M and SK isgiven by (10).

Example 1 We have that N1 is weakly reversible and = K = 0. Since N1 is aproper translation of N1, it follows that M1 has toric steady states. To characterize


the steady state ideal, we construct the weighted reaction graph (S, C, R, k) wherethe reaction weights come from Definition 8:

0 k1 A1k3 k2

A2

Setting C1 = 0, C2 = A1, and C3 = A2 gives K1 = k2k3, K2 = k1k3, and K3 = k1k2.The monomials come from the mapping h3 which corresponds source complexesCR to their kinetic complexes CK . We have Ch3(1) = A1, Ch3(2) = A1 + A2, andCh3(3) =A2. It follows that the steady state ideal isK1x

yh3(2) K2xyh3(1) , K1xyh3(3) K3xyh3(1) = k2k3x1x2 k1k3x1, k2k3x2 k1k2x1.

Direct evaluation yields x1 = k3/k1 and x2 = k3/k2, consistent with the known equi-librium of M1.

Example 2 We have that N2 is weakly reversible and = K = 0. Since N2 is aresolvable improper translation of N2, it follows by Theorem 5 that the mass actionsystem M2 corresponding to N2 has toric steady states. In order to characterize thesteady state ideal, we consider the weighted reaction graph (S, C, R, k) where thereaction weights come from Definition 11:

A1k1k2

A2k3

k4+ k1k2 k5A3. (23)

Letting C1 = A1, C2 = A2, and C3 = A3, the tree constants can be computed to beK1 = k2k4 +k1k5, K2 = (k1/k2)(k2k4 +k1k5), and K3 = k1k3. We also have Ch3(1) =A1, Ch3(2) =A2, and Ch3(3) =A1 +A3. It follows that the ideal can be generated byK1xyh3(2) K2xyh3(1) , K1xyh3(3) K3xyh3(1), which gives

(k2k4 + k1k5)x2 k1k2

(k2k4 + k1k5)x1, (k2k4 + k1k5)x1x3 k1k3x1

.

This simplifies to (9). Also, since SK = span{(1,1,0), (1,1,1)}, it follows thatSK = span{(1,1,0)} so that the steady set can be given by

E = {x R3>0 | ln(x) ln(a) span{(1,1,0)

}}

where a R3>0 is any complex balanced steady state of M2. This can be directlyrearranged to give the parametrization

E = {(x1, x2, x3) R3>0 | x1 = eta1, x2 = eta2, x3 = a3, t R}.

1102 M.D. Johnston

5 Techniques and Applications

We would like to apply Theorem 5 to characterize mass action systems which havetoric steady states. This requires having a translationeither proper, or improper andresolvablefor which specific structural conditions are satisfied. In general, however,we do not have a translation N of N given to us; rather, we must find the translation.Although not the primary purpose of this paper, in this section we present a heuris-tic method for generating translations which will then be successfully employed onseveral models from the literature.

The method introduced here requires the introduction of a few further concepts(Clarke 1980; Gatermann and Wolfrum 2005).

Definition 12 Consider a chemical reaction network N = (S,C,R) with associatedcomplex matrix Y and matrix Ia as defined in Sect. 2.3. Then:

1. The flux cone of the network is given by Cv = {v Rm0 | v ker(Y Ia)}.2. A vector E Cv is called an elementary flux mode if E = v + (1 )w for any

(0,1) and v,w Cv , v = E, w = E.3. An elementary flux mode E Cv is called a cyclic generator if E ker(Ia).4. An elementary flux mode E Cv is called a stoichiometric generator if E /

ker(Ia).

The flux modes represent modes of positive stoichiometric flux balance in thereaction network while the cyclic generators have the additional interpretation of cor-responding to cycles in the reaction graph of N . For example, the network

A112

2A1 A1 +A234

A356

A2 (24)

considered in Feinberg (1988) and Gatermann and Wolfrum (2005) has the elemen-tary flux modes E1 = (1,1,0,0,0,0), E2 = (0,0,1,1,0,0), E3 = (0,0,0,0,1,1),and E4 = (1,0,1,0,1,0), of which E1, E2, and E3 are cyclic generators (reversiblecycles in the reaction graph), and E4 is a stoichiometric generator. Every vector in theflux cone can be represented as a convex combination of the elementary flux modes(i.e., the cyclic and stoichiometric generators).

Remark 11 Notice that the condition that a network has a deficiency of zero (i.e., = dim(ker(Y ) Im(Ia)) = 0) is equivalent to the condition that the network permitsno stoichiometric generators. That is to say, if = 0, then every elementary flux modeof the network is a cyclic generator. Notice that (24) has = 1 as a result of thestoichiometric generator E4.

5.1 Translation Algorithm

In order to apply the Theorem 5, we require that the stoichiometric deficiency ofthe translation be zero (i.e., = dim(ker(Y ) Im(Ia)) = 0). This means that thetranslation N does not have any stoichiometric generators whereas, if > 0, then


original network N necessarily has stoichiometric generators. When attempting toapply Theorem 5, therefore, we have the following critical intuition:

The stoichiometric generators of N must be mapped into cyclic generators of Nthrough the translation process.

We therefore propose the following algorithm for constructing translated chemicalreaction networks.

Translation Algorithm:

1. Identify the stoichiometric generators of the flux cone.2. Assign an ordering to the reactions on the support of each stoichiometric generator

found in part. If possible, add and/or subtract species to the left- and right-handside of each reaction so that the product complex of each reaction coincides withthe reactant complex of the next to create a cycle.

3. If possible, translate all reactions from the same source complex, or those alreadyon the support of a cyclic generator, in unison (i.e., add/subtract the same term).

4. Repeat steps 2 and 3 until either successful, or all orderings of all stoichiometricgenerators has been exhausted.

5. If successful, retain the source complex of each reaction as the kinetic complex inthe translation. In the case of multiple source complexes being assigned to a singletranslated complex, any of those source complexes may be chosen.

6. If the translation is proper, assign rate constants by Definition 8. If improper, de-termine whether it is resolvable (Definition 10) and, if so, assign rate constants byDefinition 11.

If successful, this algorithm produces a weakly reversible translated chemical reac-tion network by Definition 6 with no stoichiometric generators (i.e., = 0) and anassociated properly or improperly translated mass action system by either Definition8 or 11.

The algorithm is deficient in several ways. Notably, even if the algorithm worksfor some ordering of the reactions of a stoichiometric generator in step 2, it may notwork for all choices. Certain orderings may allow multiple stoichiometric generatorsto fit together while certain others may not. Furthermore, even when the stoichiomet-ric generators can be translated into cycles, they may fail to construct a consistentnetwork by requirements of step 3.

We will see in this section that the algorithm may nevertheless provide a valuablemeans by which to construct translations, and for characterizing the steady states ofmass action systems by Theorem 5. The key observation is that this procedure allowsan explicit characterization of the steady set (7) of M based on knowledge of thetopological network structure of a related generalized network, N (Mller and Re-gensburger 2012). This clarifies the connection between established deficiency-basedapproaches and the newer algebraic work contained in Prez Milln et al. (2012).Further consideration of algorithmic approaches to network translation will be thesubject of future work.

1104 M.D. Johnston

5.2 Application I: Futile Cycle

Consider the enzymatic network N given by

S + Ek+1k1

SEk2 P + E,

P + Fk+3k3

PFk4 S + F.

(25)

This network describes an enzymatic mechanism where one enzyme E catalyzes thetransition of a substrate S into a product P , and a separate enzyme F catalyzes thereverse transition. Due to the appearance that the two enzymes are competing to undothe work of the other, this network is often called the futile cycle (Angeli and Sontag2008; Prez Milln et al. 2012; Wang and Sontag 2008).

The steady states of this network under mass action (and more general) kineticshas been well-studied in the mathematical literature. The most thorough discussion isgiven in Angeli and Sontag (2008), where the authors show through a monotonicityargument that, for every choice of rate constants, every stoichiometric compatibilityclass Cx0 of (25) contains precisely one positive steady state and that this steadystate is globally asymptotically stable relative to Cx0 . It has also be shown by thedeficiency one algorithm (Feinberg 1988), the main argument of Wang and Sontag(2008), and Theorem 5.5 of Prez Milln et al. (2012) that the network may not permitmultistationarity.

We show in the Supplemental Material that the Translation Algorithm can be suc-cessfully applied by using the translations

S + Ek+1k1

SEk2 P + E (+F),

P + Fk+3k3

PFk4 S + F (+E),

(26)

to give the proper translation N = (S, C, CK, R)

S + E S + E + Fk+1k1

SE + F SE,k4 k2

PF PF + Ek3k+3

P + E + F P + F

with associated mass action system M with the given rate constants. Since N satis-fies = K = 0, by claim 1 of Theorem 5 we have that M has toric steady states forall values of the rate constants. Further details characterizing the steady state set byTheorem 5 are contained in the Supplemental Material (ESM).


5.3 Application II: Multiple Futile Cycle

Now consider the multiple futile cycle N given by

S0 + Ekon0

koff0

ES0kcat0 S1 + E S1 + F

lon0loff0

FS1lcat0 S0 + F,

......

Sn1 + Ekonn1

koffn1ESn1

kcatn1 Sn + E Sn + Flonn1

loffn1FSn

lcatn1 Sn1 + F.(27)

This network is a generalization of the futile cycle analyzed in Sect. 5.2. In thisnetwork, one enzyme E facilitates a forward cascade of transitions from substrateS0 to substrate Sn while another enzyme F facilitates the reverse transitions. Thisnetwork has been frequently used in the literature to model multisite sequentiallydistributed phosphorylation networks of unspecified length (Gunawardena 2005;Holstein et al. 2013; Manrai and Gunawardena 2009).

Despite the structural similarities with (25), there are notable dynamical differ-ences in the corresponding mass action systems M. It was first shown in Markevichet al. (2004) that, even for the simple case n = 2, the system (27) admits rate constantvectors k Rr>0 for which M exhibits multistationarity. A subsequent focused studyof the case n = 2 in Conradi et al. (2008) provided a detailed characterization of therate parameters which permit multistationarity.

The network (27) has also been studied for arbitrary values of n 1 in Gunawar-dena (2005, 2007), Prez Milln et al. (2012), Wang and Sontag (2008). It is knownthat, for all n 2, the associated mass action systems M admit rate constant vectorsk Rr>0 for which multistationarity is permitted and that an upper bound on the num-ber of steady states in each compatibility class is 2n 1 (Wang and Sontag 2008).It is furthermore conjectured that this upper bound can be tightened to n + 1 for oddn, and n for even n. In Prez Milln et al. (2012), the authors prove that, for all rateconstant vectors k Rr>0, the mass action system M associated with this networkhas toric steady states. The authors then explicitly calculate the steady state ideal interms of those rate constants.

In the Supplemental Material we show that the Translation Algorithm can be ap-plied by breaking the network into subcomponents of the form

Si1 + Ekoni1

koffi1ESi1

kcati1 Si + E,

Si + Floni1

loffi1FSi

lcati1 Si1 + F,

1106 M.D. Johnston

and using the translations

Si1 + Ekoni1

koffi1ESi1

kcati1 Si + E (+iE + F),

Si + Floni1

loffi1FSi

lcati1 Si1 + F(+(i + 1)E).

This gives the proper translation N = (S, C, CK, R) = ni=1 Ni where Ni is givenby

Si1 + (i + 1)E + Fkoni1

koffi1ESi1 + iE + F,

lcati1 kcati1FSi + (i + 1)E

loni1

loffi1Si + (i + 1)E + F,

(28)

for i = 1, . . . , n. For each Ni , we associate the kinetic complexes Si1 + E, ESi1,Si + F , and FSi , respectively, to the translated complexes in (28), starting in the topleft and rotating clockwise.

It can be computed that = K = 0 for the translation N (see Supplemental Mate-rial). Since the network is clearly weakly reversible, it follows by claim 1 of Theorem5 that M has toric steady states for all rate constant vectors k Rr>0. Further detailscharacterizing the steady state set are contained in the Supplemental Material.

5.4 Application III: EnvZ/OmpR Signaling System

Consider the signaling network N given by

XDk1k2

Xk3k4

XTk5 Xp,

Xp + Yk6k7

XpYk8 X + Yp,

XT + Ypk9k10

XT Ypk11 XT + Y,

XD + Ypk12k13

XDYpk14 XD + Y.

(29)

This network was first considered in Example (S60) of the Supporting Online Mate-rial of Shinar and Feinberg (2010) as a model for the EnvZ/OmpR signaling system isEscherichia coli. The mass action systems M associated with N were shown in thatpaper to exhibit concentration robustness in the species Yp . That is to say, the steadystate value of [Yp] were shown to be independent of the overall molar concentrations


of X, Y , and their derivatives. The network was reproduced in Prez Milln et al.(2012) where the authors showed that the systems M have toric steady states for allrate constant values.

We now show that the steady states of M can be characterized by appealing tonetwork translation and Theorem 5. Details of application of the Translation Algo-rithm are provided in the Supplemental Material. We start by relabeling the speciesas in Prez Milln et al. (2012):

A1 = XD, A2 = X, A3 = XT, A4 = Xp, A5 = Y,A6 = XpY, A7 = Yp, A8 = XT Yp, A9 = XDYp.

We then translate each linkage class in the following way:

A112A2

34A3 5A4 (+A1 +A3 +A5),

A4 +A567A6 8A2 +A7 (+A1 +A3),

A3 +A7910

A8 11A3 +A5 (+A1 +A2),

A1 +A71213

A9 14A1 +A5 (+A2 +A3)

to get the following translation N = (S, C, CK, R), where we have labeled the reac-tions as they correspond to (29):

2A1 +A3 +A512A1 +A2 +A3 +A5

34A1 + 2A3 +A5

14 11 5A2 +A3 +A9 A1 +A2 +A8 A1 +A3 +A4 +A5

1213 9 10 7 6A1 +A2 +A3 +A7 8A1 +A3 +A6

(30)

For all complexes except A1 + A2 + A3 + A7, we assign the kinetic complexesassociated to each complex in N to be the pre-translation source complex in N .We notice, however, that the source complexes of R9 and R12 (A3 +A7 and A1 +A7, respectively) are both translated to A1 +A2 +A3 +A7. We may choose eitheroriginal reactant complex to be the associated kinetic complex so we will arbitrarilychoose A3 +A7.

Since A1 +A7 does not appear as the kinetic complex for any source complex inN , this is an improper translation. We show in the Supplementary Material that Nis a resolvable improper translation and that the rate constants from (19) and (20) of

1108 M.D. Johnston

Definition 11 are

ki ={

ki, for i R \ {12},(k2(k4+k5)

k1k3)k12, for i = 12.

Since N is a resolvable improper translation of N , is weakly reversible, and that =K = 0 (easily checked), by claim 1 of Theorem 5, M has toric steady states for allrate constant values. Further details characterizing the steady state set are containedin the Supplemental Material.

6 Conclusions and Future Work

In this paper, we introduced the notion of a translated chemical reaction network asa method for characterizing the steady states of mass action systems.

The method of network translation relates a chemical reaction network N =(S,C,R) to a generalized chemical reaction network N = (S, C, CK, R), called atranslation of N , which has the same reaction vectors as N but different complexesand consequently different connectivity properties in the translated reaction graph.We defined two classes of translations, proper translations (Definition 7) and resolv-able improper translations (Definition 10), which allowed a translated mass actionsystem M = (S, C, CK, R, k) to be defined (Definition 8 and Definition 11, respec-tively). We then presented conditions on the network topology of N which allowedan explicit connection to be made between complex balanced steady states of M andtoric steady states of M (Theorem 5). Finally, in Sect. 5 and the corresponding Sup-plemental Material, we applied the results to a series of examples drawn from theliterature.

The study of translated chemical reaction networks specifically, and generalizedchemical reaction networks in general, is very new and there are consequently manyaspects of the theory which have not be fully investigated. A few of the key points offuture work include:

1. The Translation Algorithm presented in Sect. 5.1 depends heavily on intuitionwhich may be lacking for large-scale biochemical networks. A stronger algorithm,and computational implementation, is required for broad-based application.

2. There is notable room for improvement in the conditions for resolvability of im-proper translations (Definition 10). In particular, computing the ratios Kpj /Kqj in(18) may be tedious. The author suspects that there are simpler sufficient condi-tions for resolvability of improper translations.

3. Translated chemical reaction networks are generalized chemical reaction net-works, and consequently conclusions may only be drawn as far as they are jus-tified by this underlying theory. The author suspects that, as this nascent theorybecomes more fully developed, there will be increased application for the processof network translations in characterizing the steady states of mass action systems.


Acknowledgements The author is supported by NSF grant DMS-1009275 and NIH grant R01-GM086881. The author is also grateful for the numerous constructive conversations with Anne Shiu,Carsten Conradi, Casian Pantea, Stefan Mller, and others, over email and at the AIM workshop Math-ematical problems arising from biochemical reaction networks, which pointed him toward the strongconnection between toric steady states and complex balancing in generalized mass action systems. Theauthor also thanks the two anonymous referees whose suggestions have significantly improved the paper.

Appendix A: Deficiency Result

Lemma 5 The deficiency = n s of a chemical reaction network N , wheren is the number of stoichiometrically distinct complexes, is the number of linkageclasses, and s = dim(S), also satisfies = dim(ker(Y ) Im(Ia)).Proof It follows from basic dimensional considerations that

dim(ker(Y Ia)

) = dim(ker(Ia)) + dim(ker(Y ) Im(Ia)

).

From the ranknullity theorem we have

dim(ker(Y Ia)

) = r dim(Im(Y Ia)) = r s.

The rank of Ia corresponds to the number of complexes minus the number of linkageclasses, so that dim(Im(Ia)) = n . It follows that

dim(ker(Ia)

) = r (n ) = r + n.It follows that

= dim(ker(Y ) Im(Ia)) = dim(ker(Y Ia)

) dim(ker(Ia))

= (r s) (r + n) = n s,and we are done.

Appendix B: Kernel of Ak

In this appendix, we present a more detailed characterization of ker(Ak) for a massaction system M= (S,C,R, k).

Consider a weakly reversible chemical reaction network N = (S,C,R) and letLk , k = 1, . . . , , denote the networks linkage classes. Define a subgraph T R tobe a spanning i-tree if T spans all of the complexes in some linkage class Lk , containsno cycles, and has the unique sink i C. Let Ti denote the set of all spanning i-treesfor i = 1, . . . , n. We define the following network constants.Definition 13 Consider a weakly reversible chemical reaction network N =(S,C,R) with reaction weights kj , j = 1, . . . , r . Then the tree constant for i =1, . . . , n is given by

Ki =

T Ti

jTikj . (31)

1110 M.D. Johnston

Remark 12 To compute the tree constants Ki , we restrict ourselves to the linkageclass containing the complex i C. We then determine all of the spanning trees whichcontain this complex as the unique sink, multiply across all the weighted edges ineach tree, and then sum over all such trees. The terms Ki can also be computedby computing specific minors of the kinetic matrix Ak restricted to the support of thelinkage classes (Proposition 3, Craciun et al. 2009). Note that the term tree constantis our own.

The following result characterizes ker(Ak) in terms of the tree constants (31). Thisresult appears in various forms within the chemical reaction network literature. A ba-sic form, just concerned with the signs of the individual components, can be found inFeinberg (1979) (Proposition 4.1) and Gatermann and Huber (2002) (Theorem 3.1).A more specific result can be obtained by the Matrix-Tree Theorem (Stanley 1999).This form is explicitly connected with the reaction graph of a chemical reaction net-work in Craciun et al. (2009) (Corollary 4). A direct argument is also contained inSect. 3.4 of Johnston (2011). We defer to these references for the proof.

Theorem 6 Let N = (S,C,R) denote a weakly reversible chemical reaction net-work. Let Ki denote the tree constants (31) corresponding to i = 1, . . . , n. Then

ker(Ak) = span{K1,K2, . . . ,K},where Kj = ([Kj ]1, [Kj ]2, . . . , [Kj ]n) has entries

[Kj ]i ={Ki, if i Lj ,0 otherwise.

Remark 13 This theorem may be extended to networks which are not weakly re-versible by considering the terminal strongly linked components of a chemical re-action network. As all the relevant networks considered in this paper are weaklyreversible, however, Theorem 6 will suffice for our purposes here.

Appendix C: Proof of Lemma 4

Proof Consider an improper translation N = (S, C, CK, R) of a chemical reactionnetwork N = (S,C,R) which is resolvable. Let M= (S,C,R, k) be a mass actionsystem corresponding to N and M = (S, C, CK, R, k) be an improperly translatedmass action system corresponding to N and defined by Definition 11. We will writethe steady state condition for M as

Y Ia Ik (x) = 0 (32)and the steady state condition for M as

Y Ia Ik K(x) = 0. (33)


Since N is improper, the vector K(x) contains a subset of the monomials in (x) by property 3 of Definition 6. Consequently, to relate (32) and (33), we need toremove explicit dependence on the monomials in (x) corresponding to complexesnot in CK . We will accomplish this by rewriting the unused monomials in (x) interms of the monomial in (x) corresponding to the kinetically relevant complex,and absorbing the required adjustment factor into the matrix Ik . (This will produce astate-dependent matrix I

k(x). We will resolve the state-dependency at a later stage.)Since N is resolvable, it follows that it is weakly reversible and SI SK . Con-

sequently, by Lemma 3, it follows that, for every i RI , there are constants cij andpairs pj , qj CR, j = 1, . . . , s, such that

xy(i) =[

s

j=1

(xyh3(pj )

xyh3(qj )

)cij]

xy(i)K . (34)

We now introduce

ki (x) ={

ki, for i /RI ,[s

j=1(

xyh3(pj )

xyh3(qj )

)cij ]ki, for i RI (35)

and define

kj (x) =

{i|h1(i)=j}ki (x) (36)

for j = 1, . . . , n. This gives rise to the state dependent kinetic matrix Ik(x) Rrn0

with entries [Ik(x)]ij = ki (x) if (i) = j and [Ik(x)]ij = 0 otherwise.

We now will prove that Y Ia Ik (x) = Y Ia Ik(x) K(x) by showing that, for allj = 1, . . . , r ,

[ ],j Rj (x) =

{i|h1(i)=j}[ ],i Ri(x),

where := Y Ia , := Y Ia , R(x) := Ik (x), and R(x) := Ik(x) K(x). We firstmake several observations, listed in the order they will be used:

By definition, [ ],i = y(i) y(i) and [ ],j = y(j) y(j). For h1(i) = j , we have (j) = h3((h1(i))) = h3(h2((i))) = (i)K (since

(h1(i)) = h2((i)) by property 2 of Definition 6) so that [K(x)](j) = xy(j) =xy(i)K .

By the constructions (34) and (35), we have ki (x) xy(i)K = ki xy(i) = Ri(x) for alli = 1, . . . , r .

For every i = 1, . . . , r , we have y(h1(i)) y(h1(i)) = y(i) y(i) by property 1of Definition 6.

It follows that, for every j = 1, . . . , r , we have

1112 M.D. Johnston

[ ],j Rj (x) = (y(j) y(j)) kj xy(j) = (y(j) y(j))

{i|h1(i)=j}ki (x) x

y(i)K

=

{i|h1(i)=j}(y(i) y(i)) ki xy(i) =

{i|h1(i)=j}[ ],i Ri(x).

It follows that we have

Y Ia Ik (x) = Y Ia Ik(x) K(x) = Y Ak(x) K(x), (37)

where Ak(x) := Ia Ik(x) Rnn>0 is a state dependent kinetic matrix with positive off-

diagonal entries corresponding to the structure of the translation N and rates givenby (35) and (36).

Consider the reaction graph of the network (S, C, R, k(x)) with state dependentedge weights k(x) Rr0 given by (35). In order to remove the state dependence inA

k(x), we consider the system at steady state. Since = dim(ker(Y ) Im(Ia)) = 0, itfollows that

Y Ak(x) K(x) = 0 Ak(x) K(x) = 0. (38)

Now let Kj (x), j = 1, . . . , n, denote the state dependent tree constants (31) of thereaction graph of (S, C, R, k(x)). Since N is weakly reversible, by Theorem 6, wehave that

ker(Ak(x)) = span

{K1(x), K2(x), . . . , K(x)

},

where Kj (x) = ([Kj (x)]1, [Kj (x)]2, . . . , [Kj (x)]n) has entries[Kj (x)

]i=

{Ki(x) if i Lj ,0 otherwise.

It follows that, if i, j Lk for some k = 1, . . . , , we havexyh3(i)

Ki (x)= x

yh3(j)

Kj (x)= x

yh3(i)

xyh3(j)= Ki(x)

Kj (x). (39)

Now consider i RI and let cij and pj , qj , j = 1, . . . , s, denote the values guaran-teed by Lemma 3. By (39), for every i RI , we have

s

j=1

(xyh3(pj )

xyh3(qj )

)cij=

s

j=1

(Kpj (x)

Kqj (x)

)cij. (40)

It follows by (35) and the assumption that N is resolvable that (40) only dependson the state-independent rates ki (x) = ki , i /RI . It follows that (40) may be written

s

j=1

(xyh3(pj )

xyh3(qj )

)cij=

s

j=1

(Kpj

Kqj

)cij, (41)


where the tree constants Ki are determined with respect to the reaction graph of(S, C, R, k) with the rate constants given by (35) and (36) for ki = ki for i /RI andki arbitrary for i RI (since the product (40) does not depend on these rates). Wemay now substitute (41) into (35) to get

ki (x) = ki =

ki, for i /RI ,(s

j=1( KpjKqj

)cij )ki, for i RI (42)

and

kj (x) = kj =

{i|h1(i)=j}ki . (43)

Notice that, while the tree constant pairs Kpi and Kqi depend upon a choice forthe rate constants ki for i RI , their ratios do not so that (42) has been definedconsistently.

It follows from (18) that (42) and (43) correspond to the choice of rate constantsfor the improperly translated mass action system M = (S, C, CK, R, k) defined byDefinition 11. Consequently, from (37) we have that

Y Ia Ik (x) = 0 Y Ia Ik K(x) = 0 (44)

so that the steady states of the system (3) governing M and the steady states of thesystem (6) governing M defined by Definition 11 coincide, and we are done.

Appendix D: Proof of Theorem 5

Proof Let N = (S,C,R) be a chemical reaction network and N = (S, C, CK, R) bea weakly reversible translation of N which is either proper or improper and resolv-able. Suppose M = (S,C,R, k) is a mass action system corresponding to N . Wedefine the translated mass action system M = (S, C, CK, R, k) according to Defini-tion 8 if N is proper and by Definition 11 if N is resolvable and improper.

From either Lemma 2 and Lemma 4 we have that the steady state set of M corre-sponds to the steady state set of M. Correspondingly, by either (11) or (44) we havethat

Y Ia Ik (x) = 0 Y Ak K(x) = 0,where A

k:= Ia Ik and K(x) has entries [K(x)]j = xyh3(j) for j CR.

Since K = 0, we may conclude by Proposition 2.20 of (Mller and Regensburger2012) that the translated mass action system M has a complex balanced steady state.That is to say, there is a point a Rm>0 which satisfies

K(a) ker(Ak). (45)

1114 M.D. Johnston

Furthermore, since = dim(ker(Y ) Im(Ia)) = 0, we have from (38) that all steadystates are complex balanced steady states. It follows from Proposition 2.21 of Mllerand Regensburger (2012) that the set of such steady states may be parametrized by

E = {x Rm>0 | ln(x) ln(a) SK},

where SK is given by (10) of Lemma 1. This is sufficient to prove claim 3.Now consider claims 1 and 2. Since N is weakly reversible it follows by Theorem

6 that

ker(Ak) = span{K1, K2, . . . , K}, (46)where Kj = ([Kj ]1, [Kj ]2, . . . , [Kj ]n) has entries

[Kj ]i ={Ki if i Lj ,0 otherwise (47)

where Ki , i = 1, . . . , n, are the tree constants corresponding to the reaction graph(S, C, R, k).

It follows from (45), (46), and (47) that, for every i, j Lk for some k = 1, . . . , ,the steady states x Rm>0 satisfy

xyh3(i)

Ki= x

yh3(j)

Kj Kj xyh3(i) Kixyh3(j) = 0.

Since this set corresponds to the steady states of M by either Lemma 2 or Lemma4, we have shown that M has toric steady states generated by binomials of the formrequired by claim 2. Since the choice of rate constants in the definition of M wasarbitrary, claim 1 follows and we are done.

References

Angeli, D., & Sontag, E. (2008). Translation-invariant monotone systems, and a global convergence resultfor enzymatic futile cycles. Nonlinear Anal., Real World Appl., 9, 128140.

Angeli, D., Leenheer, P., & Sontag, E. (2007). A Petri net approach to the study of persistence in chemicalreaction networks. Math. Biosci., 210(2), 598618.

Clarke, B. L. (1980). Stability of complex reaction networks. Adv. Chem. Phys., 43, 1215.Conradi, C., Flockerzi, D., & Raisch, J. (2008). Multistationarity in the activation of a MAPK: parametriz-

ing the relevant region in parameter space. Math. Biosci., 211, 105131.Cox, D., Little, J., & OShea, D. (2007). Undergraduate texts in mathematics. Ideals, varieties and algo-

rithms (3rd ed.). Berlin: Springer.Craciun, G., & Feinberg, M. (2005). Multiple equilibria in complex chemical reaction networks: I. The

injectivity property. SIAM J. Appl. Math., 65(5), 15261546.Craciun, G., & Feinberg, M. (2006). Multiple equilibria in complex chemical reaction networks: II. The

species-reaction graph. SIAM J. Appl. Math., 66(4), 13211338.Craciun, G., Dickenstein, A., Shiu, A., & Sturmfels, B. (2009). Toric dynamical systems. J. Symb. Comput.,

44(11), 15511565.Deng, J., Feinberg, M., Jones, C., & Nachman, A. (2011). On the steady states of weakly reversible chem-

ical reaction networks. Preprint available on the arXiv:1111.2386.Dickenstein, A., & Prez Milln, M. (2011). How far is complex balancing from detailed balancing? Bull.

Math. Biol., 73, 811828.


rdi, P., & Tth, J. (1989). Mathematical models of chemical reactions. Princeton: Princeton UniversityPress.

Feinberg, M. (1972). Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal., 49, 187194.

Feinberg, M. (1979). Lectures on chemical reaction networks. Unpublished written versions of lec-tures given at the Mathematics Research Center, University of Wisconsin. Available at http://www.chbmeng.ohio-state.edu/~feinberg/LecturesOnReactionNetworks/.

Feinberg, M. (1987). Chemical reaction network structure and the stability of complex isothermal reactors:I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci., 42(10), 22292268.

Feinberg, M. (1988). Chemical reaction network structure and the stability of complex isothermal reactors:II. Multiple steady states for networks of deficiency one. Chem. Eng. Sci., 43(1), 125.

Feinberg, M. (1989). Necessary and sufficient conditions for detailed balancing in mass action systems ofarbitrary complexity. Chem. Eng. Sci., 44(9), 18191827.

Feinberg, M. (1995a). The existence and uniqueness of steady states for a class of chemical reactionnetworks. Arch. Ration. Mech. Anal., 132, 311370.

Feinberg, M. (1995b). Multiple steady states for chemical reaction networks of deficiency one. Arch. Ra-tion. Mech. Anal., 132, 371406.

Flockerzi, D., & Conradi, C. (2008). Subnetwork analysis for multistationarity in mass-action kinetics. J.Phys. Conf. Ser., 138(1).

Gatermann, K. (2001). Counting stable solutions of sparse polynomial systems in chemistry. In E. L.Green, S. Hosten, R. C. Laubenbacher, & V. A. Powers (Eds.), Contemporary math: Vol. 286. Sym-bolic computation: solving equations in algebra, geometry and engineering (pp. 5369).

Gatermann, K., & Huber, B. (2002). A family of sparse polynomial systems arising in chemical reactionsystems. J. Symb. Comput., 33(3), 275305.

Gatermann, K., & Wolfrum, M. (2005). Bernsteins second theorem and Viros method for sparse polyno-mial systems in chemistry. Adv. Appl. Math., 34(2), 252294.

Gunawardena, J. (2005). Multisite protein phosphorylation makes a good threshold but can be a poorswitch. Proc. Natl. Acad. Sci. USA, 102, 1461714622.

Gunawardena, J. (2007). Distributivity and processivity in multisite phosphorylation can be distinguishedthrough steady-state invariants. Biophys. J., 93, 38283834.

Holstein, K., Flockerzi, D., & Conradi, C. (2013). Multistationarity in sequentially distributed multisitephosphorylation networks. Bull. Math. Biol., 75, 20282058.

Horn, F. (1972). Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch.Ration. Mech. Anal., 49, 172186.

Horn, F., & Jackson, R. (1972). General mass action kinetics. Arch. Ration. Mech. Anal., 47, 187194.Johnston, M. D. (2011). Topics in chemical reaction network theory. PhD thesis, University of Waterloo.Manrai, A., & Gunawardena, J. (2009). The geometry of multisite phosphorylation. Biophys. J., 95, 5533

5543.Markevich, N. I., Hoek, J. B., & Kholodenko, B. N. (2004). Signaling switches and bistability arising from

multisite phosphorylation in protein kinase cascades. J. Cell Biol., 164(3), 353359.Mller, S., & Regensburger, G. (2012). Generalized mass action systems: complex balancing equilibria

and sign vectors of the stoichiometric and kinetic-order subspaces. SIAM J. Appl. Math., 72(6), 19261947.

Mller, S., Feliu, E., Regensburger, G., Conradi, C., Shiu, A., & Dickenstein, A. (2013). Sign conditionsfor injectivity of generalized polynomial maps with applications to chemical reaction networks andreal algebraic geometry. Preprint available on the arXiv:1311.5492.

Prez Milln, M., Dickenstein, A., Shiu, A., & Conradi, C. (2012). Chemical reaction systems with toricsteady states. Bull. Math. Biol., 74(5), 10271065.

Savageau, M. A. (1969). Biochemical systems analysis II. The steady-state solutions for an n-pool systemusing a power-law approximation. J. Theor. Biol., 25, 370379.

Shinar, G., & Feinberg, M. (2010). Structural sources of robustness in biochemical reaction networks.Science, 327(5971), 13891391.

Shinar, G., & Feinberg, M. (2012). Concordant chemical reaction networks. Math. Biosci., 240(2), 92113.

Shiu, A. J. (2010). Algebraic methods for biochemical reaction network theory. PhD thesis, University ofCalifornia, Berkeley.

Stanley, R. (1999). Enumerative combinatorics (Vol. 2). Cambridge: Cambridge University Press.Volpert, A. I., & Hudjaev, S. I. (1985). Analysis in classes of discontinuous functions and equations of

mathematical physics. Dordrecht: Martinus Nijhoff.

1116 M.D. Johnston

Wang, L., & Sontag, E. (2008). On the number of steady states in a multiple futile cycle. J. Math. Biol.,57(1), 2552.

Wilhelm, T., & Heinrich, R. (1995). Smallest chemical reaction system with Hopf bifurcations. J. Math.Chem., 17(1), 114.

Wilhelm, T., & Heinrich, R. (1996). Mathematical analysis of the smallest chemical reaction system withHopf bifurcation. J. Math. Chem., 19(2), 111130.

Translated Chemical Reaction NetworksAbstractIntroductionBackgroundChemical Reaction NetworksReaction Graph and DeciencyMass Action SystemsGeneralized Mass Action Systems

Steady States of Mass Action SystemsComplex Balanced Steady StatesToric Steady States

Main ResultsTranslated Chemical Reaction NetworksProperly Translated Mass Action SystemsImproperly Translated Mass Action SystemsConnection with Toric Steady States

Techniques and ApplicationsTranslation AlgorithmApplication I: Futile CycleApplication II: Multiple Futile CycleApplication III: EnvZ/OmpR Signaling System

Conclusions and Future WorkAcknowledgementsAppendix A: Deciency ResultAppendix B: Kernel of AkAppendix C: Proof of Lemma 4Appendix D: Proof of Theorem 5References