Mathematical Biosciences

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1 Mathematical Biosciences 26 (2008) Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: Homotopy methos for counting reaction network equilibria Gheorghe Craciun a, *, J. William Helton b, Ruth J. Williams b a Department Mathematics an Department Biomolecular Chemistry, University Wisconsin Maison, Maison, WI 53706, USA b Mathematics Department, University California at San Diego, La Jolla, CA , USA article info abstract Article history: Receive 9 November 2007 Receive in revise form 27 August 2008 Accepte 4 September 2008 Available online 2 September 2008 Keywors: Bihemical reaction networks Multiple equilibria Homotopy Dynamical system moels complex bihemical reaction networks are usually high-imensional, nonlinear, an contain many unknown parameters. In some cases the reaction network structure ictates that positive equilibria must be unique for all values the parameters in the moel. In other cases multiple equilibria exist if an only if special relationships between these parameters are satisfie. We escribe methos base on homotopy invariance egree which allow us to etermine the number equilibria for complex bihemical reaction networks an how this number epens on parameters in the moel. Ó 2008 Elsevier Inc. All rights reserve.. Introuction Dynamical system moels complex bihemical reaction networks are usually high-imensional, non-linear, an contain many unknown parameters. As was shown recently in [0], base on the assumption mass-action kinetics, graph-theoretical properties some bihemical reaction networks can guarantee the uniqueness positive equilibrium points for any values the reaction rate parameters in the moel. On the other han, relatively simple reaction networks o amit multiple positive equilibria for some values the parameters as shown in [7,8,0]. The aforementione results o not aress the epenence the number equilibria on the parameter values unless there is a unique equilibrium for every set parameters. Also they o not aress the general problem existence positive equilibria. Here we escribe methos using egree theory to analyze general bihemical ynamics (not only mass-action kinetics). These methos allow us to etermine how the number positive equilibria for a complex bihemical reaction network epens on the parameters the moel. They will ten also imply the existence positive equilibria. Also we obtain uniqueness positive equilibria in various situations uner significantly weaker assumptions than in [7,8,0]... Overview We are intereste in equilibria for high-imensional, non-linear ynamical systems that originate from chemical ynamics. These * Corresponing author. Tel.: aresses: craciun@math.wisc.eu (G. Craciun), helton@ucs.eu (J.W. Helton), williams@eucli.ucs.eu (R.J. Williams). ynamical systems are systems orinary ifferential equations the form c t ¼ f ðcþ; ð:þ where f is a smooth function efine on a subset the orthant R n P0 vectors c in R n having non-negative components. Such ynamical systems usually have a large number state variables, i.e., n is large. In aition, the parameters efining f are ten not well known. The fus this paper is on equilibria for ynamical systems the form (.), that is on c * for which f(c * )=0. We consier the ynamical system (.) on a subset R n P0 which is the closure a omain in R n >0. We give conitions for the number equilibria (.) to remain constant as we continuously eform (homotopy) the function f through a family functions. A key assumption is that the following conition hols for all members the family: (DetSign) The eterminant the Jacobian matrix ðþ f is either strictly positive or strictly negative on. n (Recall that the o Jacobian ðcþ at c is the matrix j i ðcþ; i; j ¼ ;...; n.) What we observe is that when the conition (DetSign) hols for all f in the family an is boune, then the number equilibria for the ynamical system (.) is a constant for all f in the family, provie there are no equilibria on the bounary for any f in the family (see Theorem.). We further inicate how this result extens to unboune omains such as ¼ R n >0 uner suitable conitions (see e.g., Theorem 4.), incluing those assiate with a mass-conserving reaction network operating in a chemical reactor with inflows an outflows /$ - see front matter Ó 2008 Elsevier Inc. All rights reserve. oi:0.06/j.mbs

2 G. Craciun et al. / Mathematical Biosciences 26 (2008) This paper extens previous finings in several ways. The (DetSign) conition was introuce by Craciun an Feinberg [7,8] in the context chemistry with ¼ R n >0 an they observe that many chemical reaction networks have the property (DetSign). They gave many examples an many tests for this conition to hol in the case where f is a system polynomials an is R n >0. Then they [7,8] prove that if the components f are polynomials corresponing to mass-action kinetics (operating in a continuous flow stirre tank reactor), an if (DetSign) hols on R n >0 for all positive rate constants, then for each particular choice rate constant, when an equilibrium exists, it is unique. Here we obtain stronger conclusions with weaker assumptions. In particular the following are features our approach. () Rather than all positive rate constants we can select a (vector value) rate constant k 0 interest at which (DetSign) hols. Then one merely nees a continuous curve k(k) rate constants joining k 0 to another k at which (DetSign) hols an at which the ynamical system has a unique positive equilibrium. (2) For a mass-conserving reaction network operating in a chemical reactor with inflows an outflows, uner the (Det- Sign) assumption in (), we prove existence an uniqueness a positive equilibrium, see Theorem 5.8. (3) In () an (2), the function f nee not be polynomial, but is require only to be continuously ifferentiable. Of practical importance are rational f as one fins in Michaelis Menten or Hill type chemical moels, see Sections 5, 6. (4) We give methos, see Section 6, combining the items above to escribe large regions rate constants where a chemical reaction network has a unique positive equilibrium. We also point out in this paper that the bihemical reaction network moels introuce an analyze by Arcak an Sontag [3,4] satisfy (DetSign) an we can also rule out bounary equilibria (where they give enough ata). Consequently, uner extremely weak hypotheses, we obtain that each these moels has a unique positive equilibrium, see Section 2.2. The finings Arcak an Sontag are impressive in that uner strong hypotheses they prove global asymptotic stability equilibria, a topic that this paper oes not aress..2. More etail Now we give some formal efinitions. Let be a omain in R n, i.e., an open, connecte set in R n. We enote the closure by an the bounary by o. A function f :! R n is smooth if it is once continuously ifferentiable on.if is boune, for such a smooth function f, the following norm is finite: kf k :¼ sup kf ðcþk: c2 Here kk enotes the Eucliean norm on R n. When is boune, a family f k :! R n for k 2 [0,], is a continuously varying family functions provie each f k is smooth an the mapping k? f k is continuous on [0,] with the norm kk on the functions f k.azero f :! R n is a value c 2 such that f(c) = 0, where 0 is the zero vector in R n. A zero f is an equilibrium point for the ynamical system (.). The following is an immeiate consequence Theorem 3.2 which will be prove in Section 3. This theorem an examples given in Section 2 are esigne to illustrate our approach; then more targete theorems are given in Sections 4 an 5, followe by more examples in Section 6. Theorem.. Suppose R n is a boune omain an f k :! R n for k 2 [0,], is a continuously varying family smooth functions such that f k oes not have any zeros on the bounary for all k 2 [0,]. If k et ðcþ 0 for all c 2 whenever k = 0 an whenever k =, then the number zeros f 0 in equals the number zeros f in. The omain interest for chemical ynamics (cf. (.)) is typically the orthant R n >0, but this is not boune, so it violates the hypothesis is boune. Thus in applying Theorem. we must approximate R n >0 by a large boune omain an check for the absence bounary equilibria. One can think the bounary o in two pieces: that which intersects the bounary R n >0, calle the sies, an the outer bounary, o \ R n >0. We show that if we assume conservation mass (e.g., by atomic balance) in our moel an augment with suitable outflows, then natural boune omains can be chosen which have no equilibria on the outer bounary. An example such a natural boune omain in R 2 >0 is shown in Fig.. Also uner assumptions positive invariance an augmentation with inflows, there are no equilibria on the sies. In these cases, we conclue that there is exactly one non-negative equilibrium c * for (.) an that it is actually a positive equilibrium, i.e., it lies in R n >0. This result is escribe in etail in Sections 4 an Organization the paper In this paper, we give many examples wiely varying types to illustrate our contention that our metho applies broaly an is easy to use. Section 2 gives several examples illustrating Theorem.. Section 3 summarizes egree theory since our pro is base on this an relies on the observation that when (DetSign) is true, an there are no bounary equilibria, then the egree f with respect to 0 equals ± the number equilibria in a boune omain. Then in Section 3 we prove Theorem. an more. Section 4 escribes the mathematical benefits mass issipation (incluing mass conservation) an inflows an outflows. Section 5 escribes a chemical reaction network framework which contains in aition to mass-action kinetics, Michaelis Menten an Hill ynamics. We conclue in Section 6 with more examples an new methos presente in the context these examples. For many our examples, the eterminant the Jacobian f was compute symbolically using Mathematica. As a complement to this paper, we have establishe a webpage at containing Mathematica notebooks for many examples in this paper an a emonstration notebook that reaers may eit to run their own examples. 2. Examples Our goal in this section is to present some examples showing how to use Theorem.. In the press, we mention that all chemical reaction examples Arcak an Sontag [3,4] satisfy (DetSign) an fit well into our approach here. Later in Section 6 we give broaer categories examples. 2.. Two examples on treating bounary equilibria We start with two examples, the stuy which goes back to a class examples stuie by Thron [23,24]. Here c satisfies (.) an the Jacobian for all c has the form: 2 3 a 0 0 b n. b a ¼. 0 b 2 a. 3.. ; ð2:þ b n a n

3 42 G. Craciun et al. / Mathematical Biosciences 26 (2008) Fig.. An example a natural boune omain in R 2 >0 with the outer bounary an sies inicate. where a i P 0, b i P 0, i =,...,n, may epen on c. This cyclic feeback structure is common in gene regulation networks, cellular signaling pathways, an metabolic pathways [4]. Thron showe that all eigenvalues have non-negative real part (lal stability) if b bn < a an ðsecðp=nþþn. Arcak an Sontag showe that an equilibrium such a ynamical system is unique an globally stable uner strong global restrictions. Here we observe that our key assumption, the eterminant the Jacobian never changes sign, is met, which is a major step towar checking when a unique positive equilibrium exists for this class problem. Lemma 2.. When the Jacobian has the form (2.), we have h i et ¼ðÞ n P n j¼ a j þ P n j¼ b j ; ð2:2þ which if not zero has sign inepenent a i,b i P 0. Pro. Direct computation. h We shall now consier several examples from papers Arcak an Sontag primarily to illustrate that checking for absence bounary equilibria is straightforwar; subsequently we obtain existence an uniqueness an equilibrium. In this section, we assume that all parameters in the reactions are strictly positive. In the sequel, we shall use _c as an abbreviation for the time erivative c. Example 2.2. We start with an example from Section 6 [3] which they took from Thron [23]. For this, _c ¼ p c 0 p p 2 þ c 3 c ; 3 _c 2 ¼ p 3 c p 4 c 2 ; _c 3 ¼ p 4 c 2 p 5c 3 : p 6 þ c 3 Now we apply Theorem. to obtain the conclusion: ð2:3þ ð2:4þ ð2:5þ For each set parameters p j > 0; j ¼ ; 2;...; 6; c 0 > 0, there is a unique equilibrium point c * in R 3 >0 for the chemical reaction network with ynamics given by (2.3) (2.5) an there is no equilibrium point on the bounary R 3 P0. However, we have mae no comment on stability (even lal), while [3] gives certain conitions ensuring global stability. (In fact, one can algebraically solve for the two equilibria (2.3) (2.5) as functions the parameters. Inspection reveals that exactly one these is in R 3 >0 an neither is on the bounary R3 P0. The point us treating this example is to show in a simple context how our metho works.) Pro. We first observe that since the parameters p an c 0 only appear in the prouct term p c 0 in (2.3) (2.5), for the purposes the pro, it suffices to prove the result for 0<c 0 <. We shall apply Theorem. to prove that there is a unique equilibrium, insie any sufficiently large box; hence there is a unique equilibrium in the orthant. The right-han sie f p;c0 ðcþ the ifferential equations (2.3) (2.5), while a function c, also epens on positive parameters (p,c 0 ). One can check that the Jacobian for any these parameters has the form in (2.) for all c 2 R 3 >0, an since n = 3 an all parameters are strictly positive, the Jacobian eterminant is strictly negative. Note that for any two values the positive parameters, ðp ; c 0 Þ an ðpy ; c y 0 Þ, f kðp ;c 0 ÞþðkÞðpy ;c y Þ; k 2½0; Š, 0 efines a continuously varying family smooth functions on any boune subset R n P0. We check below that the no equilibria (i.e., no zeros f ðp;c0 Þ) on the bounary hypothesis hols on any sufficiently large box, for all positive parameters (p,c 0 ), an thereby conclue using Theorem. that the number equilibria (2.3) (2.5) in R n >0 oes not epen on (p,c 0) provie the parameters are all strictly positive. Computing the equilibria at one simple initial value ðp ; c 0Þ then finishes the pro. No equilibria on the bounary the orthant: Suppose an equilibrium has c 2 = 0. Then equation (2.4) implies c = 0 which contraicts (2.3). Likewise if we start by assuming c = 0 we get c > 0 an a contraiction. On the other han, if c 3 = 0, then (2.5) implies c 2 = 0, which reverts to the case consiere first. Thus there are no equilibria on the bounary R 3 P0. No equilibria on the outer bounary some big box: Suppose 0 < < 2 an > p j > for all j an c 0 <. Pick to be a box :¼ fc 2 R 3 >0 : c j < 4 for j ¼ ; 2; 3g. An equilibrium on the outer bounary the box satisfies () c ¼ 4 which by (2.3) implies 2 > p c 0 p 2 þc 3 ¼ p 3 c > 3.A contraiction. OR (2) c 2 ¼ 4 which by (2.5) implies > p 5c 3 p 6 þc 3 ¼ p 4 c 2 > 3. A contraiction. OR (3) c 3 ¼ 4 which by aing (2.3) (2.5) implies that 3 ¼ P p c 0 4 A contraiction. p 2 þ c 3 ¼ p 5c 3 P 4 p 6 þ c 3 þ ¼ 4 3 þ : Initializing: Up to this point, Theorem. tells us that each choice parameters yiels the same number equilibria! It is easy to compute for oneself that there is a simple choice parameters which yiels a unique positive equilibrium, for

4 G. Craciun et al. / Mathematical Biosciences 26 (2008) example p j ¼ for all j yiels the unique equilibrium, c ¼ c 2 ¼ c 0 þc 0 ; c 3 ¼ c 0. Thus there is one an only one equilibrium in R 3 >0 for each value the positive parameters (p,c 0 ). h Example 2.3. In Example Section 4 in [4], the authors escribe a simplifie moel mitogen activate protein kinase (MAPK) cascaes with inhibitory feeback, propose in [9,2]. For this, _c ¼ b c þ ð c Þ l ; ð2:6þ c þ a e þðc Þ þ kc 3 _c 2 ¼ b 2c 2 þ 2ð c 2 Þ c 2 þ a 2 e 2 þðc 2 Þ c ; ð2:7þ _c 3 ¼ b 3c 3 þ 3ð c 3 Þ c 3 þ a 3 e 3 þðc 3 Þ c 2: ð2:8þ The variables c j 2½0; Š, j =,2,3 enote the concentrations the active forms the proteins, an the terms c j, j =,2,3, inicate the inactive forms (after non-imensionalization an assuming that the total concentration each the proteins is ). Here the parameters a ; a 2 ; a 3 ; b ; ; e ; b 2 ; 2 ; e 2 ; b 3 ; 3 ; e 3 ; l; k are strictly positive. Let :¼ fc 2 R 3 : 0 < c j < ; j ¼ ; 2; 3g enote the open unit cube, the omain where this moel hols. Now we show how to apply Theorem. on to conclue that: There is a unique equilibrium in for any choice the strictly positive parameters, a ; a 2 ; a 3 ; b ; ; e ; b 2 ; 2 ; e 2 ; b 3 ; 3 ; e 3 ; l; k. Pro. First the Jacobian has the form (2.). Thus (2.2) implies that the eterminant is strictly negative for all strictly positive parameters an concentrations c 2. The pro follows the same outline as Example 2.2. Now we check the require items: No equilibria on the bounary the unit cube: Suppose there is an equilibrium c on the bounary. Then the equilibrium equations imply that () If c = 0 then (2.6) forces c =. Contraiction. (2) If c = then (2.6) forces c = 0. Contraiction. (3) If c 2 = 0 then (2.7) forces c = 0. Contraiction as above. (4) If c 2 = then (2.7) forces c 2 = 0. Contraiction. (5) If c 3 = 0 then (2.8) forces c 2 = 0. Contraiction as above. (6) If c 3 = then (2.8) forces c 3 = 0. Contraiction. Initializing: Arcak an Sontag [4] prove that there are choices parameters compatible with this moel for which there is a unique stable equilibrium point in. Alternatively, one can compute for a simple choice parameters that there is a unique positive equilibrium. The iscussion exactly as before implies that there is a unique equilibrium for all fixe strictly positive values the parameters. h We mention here that the question how one fins goo intializations for the rate constants might be a topic for further research. The goal woul be to fin methos for systematically selecting rate constants that prouce systems whose equilibria can be etermine by analytic means. We have not explore this topic at all The theory Arcak an Sontag Now we shall make some general comments on [3,4]. There were four chemical reaction examples presente in the two papers [3,4]. So far we have treate two the four here in this section. The thir example, Example 2 in Section 4 [4], isa small variant Example 2.3 an it can be treate in a similar manner to that example. In particular, it has a Jacobian the form (2.). We now turn to the fourth example Arcak an Sontag. Example 2.4. This is Example 3 in Section 4 [4] which we o not escribe in etail, since it requires about a page. While its Jacobian oes not have the form (2.), it is easy to analyze (using Mathematica) an what we foun is that the eterminant the Jacobian f is positive at all strictly positive c. Thus the theory escribe here applies provie suitable bounary behavior hols. Bounary behavior was not possible to etermine since the example was a rather general class whose bounary behavior was not specifie. In a particular case where more information is specifie one might expect that this coul be one. Arcak an Sontag [4] present a general theory which contains the examples consiere in this section an which oes not match up simply with ours. Their theory assumes an equilibrium exists (we o not). It places global restrictions on the equilibrium which guarantee that it is a unique globally stable equilibrium (we aress uniqueness but not stability). However, while the Arcak Sontag theory is ifferent than ours, we o point out in this section that all four their chemical examples have Jacobians whose eterminant sign oes not epen on chemical concentration, so our approach applies irectly, an with a bit attention to bounary behavior, gives existence a unique positive equilibrium. However, we o not obtain the very impressive global stability in [4]. 3. Degree an homotopy maps The pro Theorem. an other results in this paper is base on classical egree theory. The egree a function is invariant if we continuously eform (homotopy) the function an we use that to avantage in this paper. Now we give the setup. If R n is a boune omain, an if a smooth (once continuously ifferentiable) function f :! R n has no egenerate zeros, an has no zeros on the bounary, then the topological egree with respect to zero f (or simply the egree f) equals egðf Þ¼egðf ; Þ ¼ sgn et c2z f ðcþ ; ð3:þ where sgn : R!f; 0; g is the sign function, Z f is the set zeros f in, an c * is a egenerate point means et ðc Þ ¼ 0. The egree a map naturally extens from non-egenerate smooth functions to continuous functions f :! R n. For this, one can approximate f uniformly with smooth functions F k that have no egenerate zeros an no zeros on the bounary, an then efine the egree f as the limit the egrees F k. The key fact is: this construction the egree f is inepenent the approximates F k. Fortunately, we shall only nee to compute eg(f) on smooth non-egenerate f. For a quick account this theorem an the main properties egree, see Chapter.6A [6]. Homotopy invariance the egree is the following well known property: Theorem 3.. Consier some boune omain R n an a continuously varying family smooth functions f k :! R n for k 2 [0, ], such that f k oes not have any zeros on the bounary for all k 2 [0,]. Then egðf k Þ is constant for all k 2 [0,]. Now we give a slightly more general theorem than Theorem. state in the introuction. Theorem 3.2. Suppose an f k, k 2 [0,], are as in Theorem 3.. Then k for any k 2 [0,] such that et ðcþ 0 for all c 2, the number zeros f k in must equal the absolute value the egree f k in, which equals the absolute value the egree f k 0 for any k 0 2½0; Š. Pro. If k 2 [0,] is such that the eterminant etð k =Þ oes not vanish in, then sgnðetð k =ÞÞ is inepenent c. This implies

5 44 G. Craciun et al. / Mathematical Biosciences 26 (2008) that the zeros f k are non-egenerate an, by the formula for the egree f k, that j egðf k Þj equals the number zeros f k in. The fact that the egree oes not vary with k is immeiate from Theorem 3.. h Remark 3.3. For j egðf k Þj to count the number zeros f k in, sgn et k ðc Þ nee only be the same for all zeros c * in, not for all c 2. Saly this weakening hypotheses is har to take avantage in practice. Remark 3.4. From the viewpoint numerical calculation, Theorem 3.2 strongly suggests that if the no bounary zeros hypothesis hols, an (DetSign) hols for f ¼ f k at one value k = k, an if one can calculate all zeros f k at some other value k = k 2 where (DetSign) also hols, then we can etermine the number zeros at k = k. Inee, ten we can fin a k 2 for which f k2 is simple in the sense that all zeros for k 2 are non-egenerate an the zeros can be reaily compute along with the Jacobians there, an consequently egðf k2 Þ can be compute. The import for numerical calculation is that fining a single equilibrium is ten not so onerous. After fining one equilibrium one typically makes a new initial guess an tries to fin another. Knowing if one has foun all the equilibria is the truly aunting task, since it is nearly impossible to ensure this by experiment. Thus theoretical results (hopefully those here) help with this very ifficult computational question. 4. Mass-issipating ynamical systems In this section, we consier a general ynamical system moel which inclues the more specific ynamics conservative chemical reaction networks, augmente with inflows an outflows, as escribe in the next section. In chemical engineering, the latter is commonly refere to as ynamics that goes with a continuous flow stirre tank reactor (CFSTR). In bihemistry, one may view this as a moel for intracellular behavior with prouction an egraation, or with inflow an outflow across the cell bounary. Here all species components are subject to inflow an outflow, however, to approximate the conservation some species such as enzymes, one may take the assiate inflow rate value in c in an egraation factor in K o to be arbitrarily small, if esire. In preparation for efining a ynamical system on the orthant R n P0, we consier a smooth function g : Rn P0! Rn, where g has the property that for each j 2f;...; ng, the jth coorinate g(c) is non-negative whenever the jth coorinate c 2 R n P0 is zero. Consier the ynamical system assiate with this function given by _c ¼ gðcþ for c 2 R n P0 : ð4:þ This ynamical system (4.) is calle positive-invariant because the conition on g. This guarantees that the ynamics leaves the orthant R n P0 invariant. Given m 2 Rn >0, the ynamical system (4.) is calle mass-issipating with respect to m if m gðcþ 6 0; ð4:2þ for all c 2 R n P0 ; it is calle mass-issipating if it is mass-issipating with respect to m for some m 2 R n >0. In this case, on Rn P0, ðm cþ ¼ m gðcþ 6 0: ð4:3þ t Now we consier the ynamical system (4.) augmente with inflows an outflows: _c ¼ c in K o c þ gðcþ; ð4:4þ where K o is an n n iagonal matrix with strictly positive entries on the iagonal. We interpret the term c in 2 R n >0 as a constant inflow rate, an the term K o c as an outflow rate which for each component is proportional to the concentration that component. It is easy to check that with this augmentation, the ynamics still leaves the orthant R n P0 invariant. However, the mass-issipating property is only inherite at large values the concentration c. We are now reay to state our main theorem in this context. Theorem 4.. Let c in ; m 2 R n >0, an K o be an n n iagonal matrix with strictly positive iagonal entries. Consier a smooth function g : R n P0! Rn such that the ynamical system (4.) is positiveinvariant an mass-issipating with respect to m. Define f ðcþ :¼ c in K o c þ gðcþ for c 2 R n P0 : Then the augmente system (4.4), with inflows an outflows, has no equilibria on the bounary R n P0, an if et 0 on R n >0, then there is exactly one equilibrium point for this system in R n >0. Pro. It suffices to prove that f has no zeros on the bounary R n P0 an if et 0 onr n >0, then f has exactly one zero in Rn >0. Define f k ðcþ :¼ c in K o c þ kgðcþ; for c 2 R n P0 ; k 2½0; Š: Fix M > m c in an let M ¼ c 2 R n >0 : m ðk Þ < M : Then M is a boune omain an ff k : k 2½0; Šg is a continuously varying family smooth functions on M. For j ¼ ;...; n, consier c j 2 M such that the jth coorinate c j is zero. Then the jth coorinate f k ðc j Þ must be strictly positive, because the jth coorinate c in is strictly positive, an the jth coorinate gðc j Þ is non-negative, by the positive-invariance assumption. Therefore, f k has no zeros on the sies M, i.e., on M \ or n P0. Also, we have m f k ðcþ ¼m c in m ðk o cþþkmgðcþ 6 m c in m ðk o cþ < 0; ð4:5þ for all c 2 M such that m ðk o cþ¼m. Here we have use the massissipating property m for the first inequality an the fact that M > m c in for the secon inequality. It follows that f k has no zeros on the outer bounary M, i.e., on fc 2 R n >0 : m ðk Þ ¼Mg. Thus, f k has no zeros on the bounary M for all k 2 [0,]. Then, by Theorem 3., the egree f k on M, egðf k ; M Þ, is constant for all k 2 [0,]. Next we observe that c ¼ðK o Þ c in is the unique zero f 0 an is insie M, an 0 ¼K o, an so by (3.), we obtain egðf 0 ; M Þ¼sgnðetðK o ÞÞ ¼ ðþ n. Hence, by Theorem 3.2, if et ¼ et 0on M, then f ¼ f has exactly one zero in M. Since M > m c in was arbitrary an the sets M : M > m c in fill out R n P0, it follows that f has no zeros on the bounary Rn P0.If furthermore, et 0 on all R n >0, then it follows that f has exactly one zero in R n >0. h Remark 4.2. A special case mass-issipating is mass-conserving, namely m gðcþ ¼0. For ynamical systems assiate to chemical reaction networks this has a natural interpretation. Inee, the ynamics chemical concentrations resulting from chemical interactions among several types molecules will be mass-conserving whenever there exists a mass assignment for each chemical species which is conserve by each reaction, or whenever each chemical species (or molecule) is mae up atoms that are also conserve by each reaction. More generally, the ynamics will be mass-issipating whenever no reaction prouces more mass than it consumes, respectively, prouces more ¼ 0, atoms than it consumes. Mass conservation implies et og since m og ¼ 0 when m g ¼ 0. Thus augmenting with outflows is require to make the hypothesis on the sign et in our theorems meaningful. The paper [5], which buils on the

6 G. Craciun et al. / Mathematical Biosciences 26 (2008) current one, introuces a more general eterminant that applies when there are no outflows (or only some outflows). This then helps one count equilibria in a manner generalizing what we have one here. Remark 4.3. Theorem 4. still hols with a much less restrictive efinition mass-issipating, e.g., by replacing m with the graient rl for an appropriate class functions L : R n P0! R P0. Here mass or atom count is behaving like what is calle storage function in engineering systems theory, see [8]. Inee, the inequality m f k ðcþ 6 m c in m ðk o cþ erive in (4.5) is what is calle a issipation inequality on the storage function c! m c, which in fact also plays the role a running cost. 5. Dynamics chemical reaction networks We now introuce the stanar terminology Chemical Reaction Network Theory (see [6,,4,7]). A chemical reaction network is usually specifie by a finite set reactions R involving a finite set chemical species S. For example, a chemical reaction network with two chemical species A an A 2 is schematically given in the iagram : ð5:þ The ynamics the state this chemical system is efine in terms functions c A ðtþ an c A2 ðtþ which represent the concentrations the species A an A 2 at time t. The currence a chemical reaction causes changes in concentrations; for instance, whenever the reaction A þ A 2! 2A curs, the net gain is a molecule A, whereas one molecule A 2 is lost. Similarly, the reaction 2A 2! 2A results in the creation two molecules A an the loss two molecules A 2. A common assumption is that the rate change the concentration each species is governe by mass-action kinetics [6, 4,22,7,8,0], i.e., that each reaction takes place at a rate that is proportional to the prouct the concentrations the species being consume in that reaction. For example, uner the mass-action kinetics assumption, the contribution the reaction A þ A 2! 2A to the rate change c A has the form k A þa 2!2A c A c A2, where k A þa 2!2A is a positive number calle the reaction rate constant. In the same way, the reaction 2A 2! 2A contributes the negative value 2k 2A2!2A c 2 A 2 to the rate change c A2. Collecting the contributions all the reactions, we obtain the following ynamical system assiate to the chemical reaction network epicte in (5.): _c A _c A2 ¼k 2A!A þa 2 c 2 A þ k A þa 2!2A c A c A2 k A þa 2!2A 2 c A c A2 þ k 2A2!A þa 2 c 2 A 2 2k 2A!2A 2 c 2 A þ 2k 2A2!2A c 2 A 2 ; ¼ k 2A!A þa 2 c 2 A k A þa 2!2A c A c A2 þ k A þa 2!2A 2 c A c A2 k 2A2!A þa 2 c 2 A 2 þ 2k 2A!2A 2 c 2 A 2k 2A2!2A c 2 A 2 : ð5:2þ The objects on both sies the reaction arrows (i.e., 2A ; A þ A 2, an 2A 2 ) are calle complexes the reaction network. Note that the complexes are non-negative integer combinations the species. On the other han, we will see later that it is very useful to think the complexes as (column) vectors in R n, where n is the number elements S, via an ientification the set species with the stanar basis R n, given by a fixe orering the species. For example, via this ientification, the complexes above be- come 2A ¼ 2 0, A þ A 2 ¼, an 2A 2 ¼ 0 2. We can now formulate a general setup which inclues many situations, certainly those above. 5.. The general setup Now we present basic efinitions an illustrate them. Definition 5.. A chemical reaction network is a triple ðs; C; RÞ, where S is a set n chemical species, C is a finite set vectors in R n P0 with non-negative integer entries calle the set complexes, an R C C is a finite set relations between elements C, enote y! y 0 which represents the set reactions in the network. Moreover, the set R cannot contain elements the form y! y; for any y 2 C there exists some y 0 2 C such that either y! y 0 or y 0! y; an the union the supports all y 2 C is S, where the support an element a 2 R n is suppðaþ ¼fj : a j 0g. To each reaction y! y 0 2 R, we assiate a reaction vector given by y 0 y. The last two constraints the efinition amount to requiring that each complex appears in at least one reaction, an each species appears in at least one complex. For the system (5.), the set species is S ¼fA ; A 2 g, the set complexes is C ¼f2A ; A þ A 2 ; 2A 2 g an the set reactions is R ¼f2A A þ A 2 ; A þ A 2 2A 2 ; 2A 2 2A g, an consists 6 reactions, represente as three reversible reactions. In examples we will ten refer to a chemical reaction network by specifying R only, since R encompasses all the information about the network. In the sequel, we shall sometimes simply say reaction network in place chemical reaction network. Definition 5.2. A kinetics for a reaction network ðs; C; RÞ is an assignment to each reaction y! y 0 2 R a reaction rate function K y!y 0 : R n P0! Rn P0 : By a kinetic system, which we enote by ðs; C; R; KÞ, we mean a reaction network taken together with a kinetics. For each concentration c 2 R n P0, the non-negative number K y!y 0ðcÞ is interprete as the currence rate the reaction y! y 0 when the chemical mixture has concentration c. Hereafter, we suppose that reaction rate functions are smooth on R n P0, an that K y!y 0ðcÞ ¼0 whenever suppðyþ å suppðcþ. Although it will not be important in this article, it is natural to also require that, for each y! y 0 2 R the function K y!y 0 is strictly positive precisely when suppðyþ suppðcþ, i.e., precisely when the concentration c contains at non-zero concentrations those species that appear in the reactant complex y. Definition 5.3. The species formation rate function for a kinetic system ðs; C; R; KÞ is efine by r : R n P0! Rn where rðcþ ¼ K y!y 0ðcÞðy 0 yþ for c 2 R n P0 : The assiate ynamical system for the kinetic system ðs; C; R; KÞ is _c ¼ rðcþ ¼ K y!y 0ðcÞðy 0 yþ; ð5:3þ where c 2 R n P0 is the non-negative vector species concentrations. The interpretation rðþ is as follows: if the chemical concentration is c 2 R n P0, then r jðcþ is the prouction rate species j ue to the currence all chemical reactions. To see this, note that r j ðcþ ¼ K y!y 0ðcÞðy 0 j y jþ;

7 46 G. Craciun et al. / Mathematical Biosciences 26 (2008) an that y 0 j y j is the net number molecules species j prouce with each currence reaction y! y 0. Thus, the right-han sie the equation above is the sum all reaction currence rates, each weighte by the net gain in molecules species j with each currence the corresponing reaction. Note that r j ðcþ coul be less than zero, in which case jr j ðcþj represents the overall rate consumption species j Special case: mass-action kinetics Definition 5.4. A mass-action system is a quaruple ðs; C; R; kþ, where ðs; C; RÞ is a chemical reaction network an k ¼ðk y!y 0Þ is a vector reaction rate constants, so that the reaction rate function K y!y 0 : R n P0! Rn P0, for each reaction y! y0 2 R, is given by massaction kinetics: K y!y 0ðcÞ ¼k y!y 0c y where c y ¼ Yn i¼ c y i i : (Here we aopt the convention that 0 0 ¼.) The assiate mass-action ynamical system is _c ¼ k y!y 0c y ðy 0 yþ: ð5:4þ In the vector equation (5.4), the total rate change the vector concentrations c is compute by summing the contributions all the reactions in R. Each reaction y! y 0 contributes proportionally to the prouct the concentrations the species in its source y, that is, c y, an also proportional to the number molecules gaine or lost in this reaction. Finally, the proportionality factor is k y!y 0. For example, we can rewrite (5.2) in the vector form (5.4) as _c ¼ k 2A!A _c þa 2 c 2 þ k A þa 2 2!2A c c 2 þ k A þa 2!2A 2 c c 2 þ k 2A2!A þa 2 c þ k 2A!2A 2 c 2 þ k 2A2!2A 2 c 2 2 : ð5:5þ Mass conservation Now we see in terms this setup how one obtains mass conservation as efine in Section 4. Definition 5.5. The stoichiometric subspace S R n a reaction network ðs; C; RÞ is the linear subspace R n spanne by the reaction vectors y 0 y, for all reactions y! y 0 2 R. Note that, accoring to (5.3), for a given value c, the vector _c is a linear combination the reaction vectors. This implies that each stoichiometric compatibility class ðc 0 þ SÞ\R n P0 is an invariant set for the ynamical system (5.3) with initial conition c 0 2 R n P0. Definition 5.6. A reaction network ðs; C; RÞ is calle conservative if there exists some positive vector m 2 R n >0 which is orthogonal to all its reaction vectors, i.e., m ðy 0 yþ ¼0; for all reactions y! y 0 in R. Then m is calle a conserve mass vector. Remark 5.7. Each trajectory a conservative reaction network is boune. A conservative reaction network can have no inflows or outflows (see the efinition inflow an outflow in the next section) Main results We now consier conservative reaction networks augmente with inflow an outflow reactions (for each the species). An inflow reaction is a reaction the form 0! A an an outflow reaction is one the form A! 0, where A is a species. The reaction vector y 0 y assiate with an inflow reaction for species j is the vector containing all zeros, except that it has a one in the jth position. The reaction vector assiate with an outflow reaction for species j is the negative the vector assiate with an inflow reaction for that species. Here for the kinetics assiate with the inflows an outflows, we assume that the reaction rate function for each inflow reaction is a positive constant an the value the reaction rate function for each outflow reaction is a positive constant times the concentration the species flowing out. The latter correspons to egraation each species at a rate proportional to its concentration. The following theorem may be use to etermine the number equilibria for conservative reaction networks augmente by such inflows an outflows. It requires a positive eterminant conition an is the analog Theorem 4. in this context. Theorem 5.8. Consier some conservative reaction network ðs; C; RÞ with conserve mass vector m. Let K be a kinetics for this network with assiate species formation rate function g. Consier an augmente kinetic system ðs; ec; er; KÞ f obtaine by aing inflow an outflow reactions for all species so that the assiate ynamical system is: _c ¼ rðcþ :¼ c in K o c þ gðcþ; ð5:6þ where c in 2 R n >0 an K o is an n n iagonal matrix with strictly positive iagonal entries. Suppose that or et ðcþ 0; for all c 2 R n >0. Then the ynamical system (5.6) has exactly one equilibrium c in R n >0 an no equilibria on the bounary Rn P0. Pro. We want to apply Theorem 4.. The function g is given by the right member (5.3), where the functions K y!y 0 are all smooth an have the property that K y!y 0ðcÞ ¼0 whenever suppðyþ å suppðcþ. Consequently, g is smooth an, whenever c 2 R n P0 is such that c j ¼ 0 for some j, then we have g j ðcþ P K y!y 0ðcÞðy j Þ¼0; because K y!y 0ðcÞ ¼0 whenever y j > 0 an c j ¼ 0, by the support property K. It follows that the ynamical system (5.6) is positive-invariant. Furthermore, the system is mass-issipating, since m gðcþ ¼ K y!y 0ðcÞm ðyy 0 Þ¼0; by the assumption that the reaction network ðs; C; RÞ is conservative. The conclusion then follows immeiately from Theorem 4.. h Remark 5.9. The results escribe above use the assumption that all species have inflows, in orer to conclue that there are no equilibria on the bounary R n >0. On the other han, for very large classes chemical reaction networks escribe in [2], this assumption is actually not neee in orer to rule out the existence such bounary equilibria (an observation by Davi Anerson University Wisconsin []). Remark 5.0. The paper [5], for the case non-autatalytic reactions, gives a conition on the stoichiometric matrix (in our

8 G. Craciun et al. / Mathematical Biosciences 26 (2008) terminology the matrix whose columns are the vectors y 0 y for y! y 0 2 R) which is necessary an sufficient for the eterminant the Jacobian r to be one sign for all concentrations an all K y!y 0 which are monotone increasing in each variable. Our theory is less restrictive as illustrate by Example Applications In practice, most ynamical system moels bihemical reaction networks contain a large number unknown parameters. These parameters correspon to reaction rates an other chemical properties the reacting species. In this section, we treat a variety examples such moels an illustrate the use Theorems 4. an 5.8. In some these examples, the eterminant the Jacobian et or is one sign everywhere on the open orthant for all parameters an in some it is not. Even in the latter cases we escribe ways to fin classes rate functions for which there exists a unique positive equilibrium. The first subsection assumes mass-action kinetics an efines (remins) the reaer the Craciun Feinberg eterminant expansion via an example. Critical is the sign each term in the expansion an whether all terms have the same sign or miss this by a little, namely, only one or two terms in the eterminant expansion has a coefficient with an anomalous sign. Here we point out that all examples in [7,8,0] have very few anomalous signs. When there are no anomalous signs, these papers show that any positive equilibrium is unique for all parameter values, an they evelop an use graph theoretical methos for etermining when there are no anomalous signs. Here for cases few anomalous signs, we propose an illustrate a technique for ientifying parameter values for which a positive equilibrium exists an is unique. The paper, [5], subsequent to this one, gives ways counting the number anomalous signs in terms graphs assiate to a chemical reaction network. In the secon subsection, we continue with the general framework Section 5, an move beyon mass-action kinetics to rate functions satisfying certain monotonicity conitions (see Definitions 6.3 an 6.4). The weaker conition, Definition 6.4, hols for many bihemical reactions an allows one to make sense the signs which cur in the eterminant expansion. Hence one can apply the methos in this paper. The number anomalous signs can be etermine for the examples in this section using: (a) the graph-theoretic methos Craciun an Feinberg [7,8] when there are no anomalous signs an the kinetics are mass-action type, an (b) symbolic computation the eterminant the Jacobian using Mathematica when there are some anomalous signs or the kinetics are general (not necessarily mass-action). The reaer will fin Mathematica notebooks at for many the examples in this section (incluing all those that fall uner (b)), as well as a emonstration notebook that reaers may eit to run their own examples. This stware works well when the number species is small; for larger numbers, the eterminant expansion has too many terms to be hanle reaily. We conclue this preamble with some intuition unerlying the case when there are anomalous signs. In general, we expect that for very small values the parameters appearing in the reaction rate functions for a conservative reaction network (an, in the limit, for vanishing parameter values), the ynamics the system augmente by inflows an outflows will be ominate by the inflow an outflow terms, an etðor=þ will not vanish; moreover, if the inflow an (linear) outflow terms ominate the ynamics, then the equilibrium will be unique. Examples 6. an 6.6 illustrate how this observation can be mae rigorous an can be use together with Theorem 5.8 an the pro Theorem 4. to conclue the existence an uniqueness an equilibrium for a subset the Table Some examples reaction networks an the signs coefficients in their Jacobian eterminant expansion when augmente with inflows an outflows (with outflow rate constants equal to one) Reaction network (i) A + B P B+ C Q C 2A (ii) A + B P B+ C Q 0 C + D R D 2A (iii) A + B P B+ C Q C + D R D+ E S E 2A (iv) A + B P B+ C Q 0 C A (v) A + B F A+ C G C + D B C+ E D (vi) A + B 2A (vii) 2A + B 3A (viii) A +2B 3A parameter space, even if the result oes not hol for the entire parameter space. 6.. The eterminant expansion, its signs an uses Example 6.. Consier the mass-action kinetics system given by the chemical reaction network (6.), which is an irreversible version the network shown in Table.(i) [7] (see Table (i)): A þ B! P; B þ C! Q; ð6:þ C! 2A: If we a inflow an outflow reactions for all species, the assiate ynamical system moel for (6.) is _c A ¼ k 0!A k A!0 c A k AþB!P c A c B þ 2k C!2A c C ; _c B ¼ k 0!B k B!0 c B k AþB!P c A c B k BþC!Q c B c C ; _c C ¼ k 0!C k C!0 c C k BþC!Q c B c C k C!2A c C ; _c P ¼ k 0!P k P!0 c P þ k AþB!P c A c B ; _c Q ¼ k 0!Q k Q!0 c Q þ k BþC!Q c B c C : ð6:2þ Accoring to Remark 4.3 in [7] the ynamical system above oes have multiple positive equilibria for some values the reaction rate parameters. If we assume that all outflow rate constants k A!0 ;...; k Q!0 are equal to, then the eterminant the Jacobian the reaction rate function is: etðor=þ ¼k AþB!P c A k BþC!Q c C k BþC!Q c B k BþC!Q k AþB!P c A c B k C!2A k C!2A k AþB!P c A k C!2A k BþC!Q c C k AþB!P c B k AþB!P k C!2A c B k AþB!P k BþC!Q c 2 B k AþB!Pk BþC!Q c B c C þ k AþB!P k BþC!Q k C!2A c B c C : ð6:3þ Note that there is only one positive monomial in the expansion in (6.3). The concentrations in it are c B c C, but there is also a negative monomial with concentrations c B c C, an the two combine to give ½k AþB!P k BþC!Q þ k AþB!P k BþC!Q k C!2A Šc B c C : Number anomalous signe terms in et expansion Thus if k C!2A 6, then the positive monomial will be ominate by a negative monomial. Therefore, if k C!2A 6, then etðor=þ 0 for this network, everywhere on R 5 >0. Note that ðm A ; m B ; m C ; m P ; m Q Þ¼ð; ; 2; 2; 3Þ is a conserve mass vector for the reaction network (6.). It follows from Theorem

9 48 G. Craciun et al. / Mathematical Biosciences 26 (2008) that (6.2), the ynamical system for (6.), augmente with inflows an outflows (with outflow rate constants equal to one), has a unique positive equilibrium for all positive values the reaction rates such that k C!2A 6. Note that this uniqueness conclusion woul not follow irectly from the theory [7,8] nor from that in [5], since these works pertain only when the eterminant has the same sign for all rate constants an species concentrations. The same metho can be applie to conclue that the reversible version the reaction network (6.), augmente with inflows an outflows (with outflow rate constants set equal to one), also has a unique positive equilibrium for all positive values the reaction rates such that k C!2A 6 ; moreover, even if the (positive) outflow rate constants k A!0 ;...; k Q!0 are not necessarily equal to, the same conclusion hols if we know that k C!2A 6 k C!0. Example 6.2. Here we summarize several examples with massaction kinetics (in the next subsection we consier some these reactions with more general kinetics). Of the eight examples in [7,8], which are chemical reaction networks augmente with inflows an outflows (with outflow rate constants equal to one), two have the property that the coefficients the terms in their Jacobian eterminant expansion all have the same sign, an the other six have all but one sign the same. The first observation is from [7] an the secon observation, emphasizing that there is only one anomalous sign, is new here. An analysis as in Example 6. can be applie here. Table is a list the examples showing how many anomalous signs each eterminant expansion has. A similar accounting hols for examples reaction networks in [0], see Table, page These reactions involve enzymes which [0] treat with mass-action kinetic moels. They fin reaction networks, 2, 3, 5, 7, 9 in this table o not have any anomalous signs. Here we point out that the remaining reaction networks, 4, 6 an 8 have very few anomalous signs. The reaction network 4 is S þ E ES! E þ P; I þ E EI; I þ ES ESI EI þ S; an has only anomalous sign, an the reaction network 6 is S þ E ES; S2 þ E ES2; S2 þ ES ESS2 S þ ES2; ESS2! E þ P; an has only two anomalous signs. Reaction network 8 has four anomalous signs out a total over 3000 terms. Here all reactions are augmente by outflows with outflow rate constants set to one. (For the cases no anomalous signs these outflow rate constants can be taken to be arbitrarily small without changing the answer. ) The theory Sections 4 an 5 applies, if there are (arbitrarily small) inflows an outflows, to yiel that there is a unique positive equilibrium, for reaction networks, 2, 3, 5, 7, 9. It also leaves open the possibility that one can apply the technique in Example 6. to get a unique positive equilibrium for certain rate constants in reaction networks 4, 6, 8. These applications our theory require fining a conserve mass for the system without inflow an outflow, which is easy to o in all cases General reaction rate functions In this section, we follow the setup in Section 5 an move beyon mass-action kinetics to a very general class rate functions. See [9] for how one can eliminate outflows for the enzymes. Definition 6.3. We say that a reaction rate function K y!y 0 is consumptively increasing, if for each species i belonging to the support the vector y, the partial erivative the reaction rate function, ok y!y 0= i, is strictly positive on the open orthant. It is very common to assume that the reaction rate functions K y!y 0 are consumptively increasing, since this simply means that the rate a reaction increases whenever the concentration a consume species is increase unilaterally. In particular, the consumptively increasing property is true for many common chemical reaction rate laws, such as many Michaelis Menten an Hill laws, as well as for all mass-action kinetics [7]. All examples in this section entail consumptively increasing reaction rates. The consumptively increasing property can fail to hol for some classes reactions incluing those involving inhibitory enzymes an for those in which a Michaelis Menten rate epens on the proucts the reaction [20]. However, the next more lenient assumption hanles these an many aitional bihemical situations. Definition 6.4. We say that a reaction rate function K y!y 0 is strictly monotone, if for each species i on which the function K y!y 0 actually epens, the partial erivative the reaction rate function, ok y!y 0= i, has one strict sign on the open orthant. More generally, the main technique use in this section is to compute the eterminant expansion the Jacobian as a sum terms, each which is a prouct partial erivatives ok y!y 0= i where i belongs to the support y. For strictly monotone rate functions we can assign a ± to each term accoring to whether that term is everywhere positive or negative on the omain R n >0. That is, strict monotonicity guarantees the technique tracking anomalous signs in the eterminant expansion applies. Examples which involve inhibitory feeback can be written in the form (5.3) with strictly monotone rate functions. or As we observe in Section 2 the eterminant the Jacobian,, is positive for all these situations. However, at this point the terminology is in place so that we can mention the more refine property that each these examples has no anomalous signs. Example 6.5. Consier the chemical reaction network (6.4), which is the reversible network shown in (ii) in Table but, unlike in [7], in this example we o not assume that the kinetics is mass-action. A þ B P; B þ C Q; ð6:4þ C þ D R; D 2A: We augment this reaction with inflows an outflows where the outflow matrix K o is normalize to be the ientity. We suppose that each the reaction rate functions K y!y 0 is consumptively increasing as in Definition 6.3. We compute the expansion etðor=þ in terms the partial erivatives ok y!y 0ðcÞ= i, for i belonging to the support y. It is a sum coefficients times monomials in these partial erivatives; the set coefficients is shown in (6.5). f; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 3; ; ; ; ; ; ; ; 3; ; ; ; ; ; ; ; ; ; ; ; ; ; 2; 2; 2; 2; 2; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 2; 2; 2; 2; 2; 2; ; ; ; ; ; ; ; ; 2; 2; ; ; ; ; ; ; ; ; ; ; ; 2; 2; 2; 2; 2; 2; 2; 2; ; ; ; ; ; ; ; ; 2; 2; 2; ; ; ; ; ; ; 2; 2; 2; ; ; ; 2; g : ð6:5þ