Multistability of Small Reaction Networks
MMULTISTABILITY OF SMALL REACTION NETWORKS ∗ XIAOXIAN TANG † AND
HAO XU ‡ Abstract.
For three typical sets of small reaction networks (networks with two reactions, oneirreversible and one reversible reaction, or two reversible-reaction pairs), we completely answer thechallenging question: what is the smallest subset of all multistable networks such that any multistablenetwork outside of the subset contains either more species or more reactants than any network inthis subset?
Key words. chemical reaction networks, mass-action kinetics, multistationarity, multistability
AMS subject classifications.
1. Introduction.
For the dynamical systems that arise from biochemical reac-tion networks, we ask the following question:
Question
Given a class of networks with the same number of irreversible re-actions and the same number of reversible-reaction pairs, what is the smallest nonemp-ty subset of multistable networks such that any multistable network outside of the sub-set has either more species or more reactants than any network from the subset? Here,we define the number of reactants as the maximum sum of stoichiometric coefficientsin the reactant complexes (see Definition 2.3).
We will formally state this question in Section 2.3, see Question 2.4. The abovequestion is motivated by the multistationarity (multistability) problem of biochemicalreaction systems, which is crucial for understanding basic phenomena such as decision-making process in cellular signaling [3, 9, 23, 5]. Given a network, we pursue rateconstants such that the corresponding dynamical system arising under mass-actionkinetics has at least two (stable) positive steady states in the same stoichiometriccompatibility class. Mathematically, one needs to identify a value or an open regionin the parameter space for which a parametric semi-algebraic system has at least tworeal solutions, which is a fundamental problem in computational algebraic geometry[14, 6]. It is well-known that networks with only reaction admit no multistationar-ity/multistability. So, for Question 1.1, the first non-trivial case to study is the classof networks with two reactions (possibly reversible). It is implied by [13] that forthe networks with two pairs of reversible reactions, the smallest nonempty subclassof multistable networks is the set with a single network “0 ⇐⇒ X , X ⇐⇒ X ”.That means for any other network with two reversible-reaction pairs, if it admitsmultistability, either it has at least 2 species, or the number of reactants is at least4. In this paper, our main contributions are complete answers to Question 1.1 for thenetworks with exactly two reactions (see Theorem 2.5) and those with one irreversiblereaction and one reversible reaction (see Theorem 2.6).Our main focus is the multistability problem. Generally, multisability is a muchmore difficult problem than multistationarity because the standard algebraic toolfor studying stability (Routh-Hurwitz criterion [11], or alternatively Li´enard-Chipartcriterion [7]) is computationally challenging (e.g., [17], [21]). Fortunately, for the net-works with one-dimensional stoichiometric subspaces, we can determine stability bychecking the trace of the Jacobian matrix (see Lemma 3.2). Using the simpler criteria, ∗ Submitted to the editors DATE. † School of Mathematical Sciences, Beihang University, Beijing, China ([email protected],https://sites.google.com/site/rootclassification/). ‡ School of Mathematical Sciences, Beihang University, Beijing, China ([email protected]).1 a r X i v : . [ q - b i o . M N ] A ug XIAOXIAN TANG, AND HAO XU we employ elimination method (from algebraic geometry), and then we prove an upperbound (and a lower bound for the networks such that the nondegeneracy conjecture[13, Conjecture 2.3] is true) for the maximum number of stable positive steady statesin terms of the maximum number of positive steady states (see Theorems 3.14 and3.15), which shows a multistable network admits at least three positive steady states,and so the number of reactants should be at least three (in fact, for two-reactionnetworks, the number of reactants should be at least four, see Theorem 4.9). A recentstudy on at-most-bimolecular networks [16] supports our result. These results extend[13, Theorem 3.6 2(c)], which is for one-species networks, to two-reaction networksand to two-species networks with one irreversible and one reversible reaction, or withtwo pairs of reversible reactions. Remark that these results are based on a sign con-dition (see Theorem 3.5), which also provides one way to determine multistationarity(by checking if the determinant of the Jacobian matrix changes sign) for small net-works with one-dimensional stoichiometric subspaces (see Corollary 3.13). There havebeen a long list of such criterion (without or with a steady-state parametrization),see [4, 19, 22, 1, 14, 6, 8]. One criteria in the list based on degree theory [8, Theo-rem 3.12] requires the networks to admit no boundary steady states, which can notbe directly applied to two-reaction networks since if a two-reaction network admitsmultistationarity, then it must admit boundary steady states (see Theorem 4.8).This work can be viewed as one step toward an ambitious goal: a complete classifi-cation of multistable networks with one-dimensional stoichiometric subspaces. As thefirst step toward the big goal, Joshi and Shiu [13] solved the multistationarity problemfor small networks with only one species or up to two reactions (possibly reversible).Later, Shiu and de Wolff [20] extended these results to nondegenerate multistation-arity for two-species networks with two reactions (possibly reversible). The idea ofstudying small networks is inspired by the fact that nondegenerate multistationaritycan be lifted from small networks to related large networks [12, 2]. Here, our contri-bution is straightforward: one can directly read multistable networks with few speciesand few reactants from the two main results Theorem 2.5 and Theorem 2.6. For two-reaction networks with up to four reactants and up to three species, there are in factonly two kinds of networks (but infinitely many) that are multistable. For instance, byTheorem 2.5, we directly see the network “ X −→ X + X , X + X + X −→ X ”admits no multistability. And, for the networks with one irreversible and one re-versible reaction, if there are up to three reactants and up to two species, then onlyfour kinds of networks are multistable.The rest of this paper is organized as follows. In Section 2, we introduce mass-action kinetics systems arising from reaction networks. We formally state our problemand present the main results in Section 2.3. In Section 3, for the small networks withone-dimensional stoichiometric subspaces, we provide a sign condition (see Theorem3.5), which reveals the relationship between the maximum number of positive steadystates and the maximum number of stable positive stable steady states (see Theorems3.14 and 3.15). In Section 4, we study networks with exactly two reactions. We provea list of necessary conditions for a two-reaction network to admit multistability (forinstance, see Theorems 4.8 and 4.9). Based on these results, for the set of all two-reaction networks, we find the smallest subset of all multistable networks such that anymultistable network contains either more species or more reactants than any networkin this subset (see the proof of Theorem 2.5). We extend these results for networkswith reversible reactions in Section 5 (see the proof of Theorem 2.6). Finally, we endup this paper with open problems inspired by Theorem 3.15, see Section 6. ULTISTABILITY OF SMALL REACTION NETWORKS
2. Background.2.1. Chemical reaction networks.
In this section, we briefly recall the stan-dard notions and definitions on reaction networks, see [6, 13] for more details. A re-action network G (or network for short) consists of a set of s species { X , X , . . . , X s } and a set of m reactions: α j X + · · · + α sj X s −→ β j X + · · · + β sj X s , for j = 1 , , . . . , m, (2.1)where all α ij and β ij are non-negative integers, and ( α j , . . . , α sj ) (cid:54) = ( β j , . . . , β sj ).We call the s × m matrix with ( i, j )-entry equal to β ij − α ij the stoichiometric matrix of G , denoted by N . We call the image of N the stoichiometric subspace , denoted by S . We denote by x , x , . . . , x s the concentrations of the species X , X , . . . , X s , re-spectively. Under the assumption of mass-action kinetics , we describe how theseconcentrations change in time by following system of ODEs:˙ x = f ( x ) := N · κ x α x α · · · x α s s κ x α x α · · · x α s s ... κ m x α m x α m · · · x α sm s , (2.2)where x denotes the vector ( x , x , . . . , x s ), and each κ j ∈ R > is called a rate constant .By considering the rate constants as an unknown vector κ = ( κ , κ , . . . , κ m ), we havepolynomials f i ∈ Q [ κ, x ], for i = 1 , , . . . , s .A conservation-law matrix of G , denoted by W , is any row-reduced d × s -matrixwhose rows form a basis of S ⊥ , where d := s − rank( N ) (note here, rank (W)=d).Our system (2.2) satisfies W ˙ x = 0, and both the positive orthant R s> and its closure R ≥ are forward-invariant for the dynamics. Thus, a trajectory x ( t ) beginning at anonnegative vector x (0) = x ∈ R s> remains, for all positive time, in the following stoichiometric compatibility class with respect to the total-constant vector c := W x ∈ R d : P c := { x ∈ R s ≥ | W x = c } . (2.3)That means P c is forward-invariant with respect to the dynamics (2.2).In this work, we mainly focus on the three families of small networks defined as G := { the networks with exactly two reactions , i.e. , m = 2 in (2.1) } , G := { the networks with one irreversible and one reversible reaction } , and G := { the networks with two of reversible-reaction pairs } . We denote the union ∪ i =0 G i simply by G . Also, we simplify/clarify our notation (2.1)for reversible reactions. For any G ∈ G , we denote it byΣ si =1 α i X i ⇐⇒ Σ si =1 β i X i , Σ si =1 α i X i −→ Σ si =1 β i X i , (2.4)and for any G ∈ G , we denote it byΣ si =1 α i X i ⇐⇒ Σ si =1 β i X i , Σ si =1 α i X i ⇐⇒ Σ si =1 β i X i . (2.5) XIAOXIAN TANG, AND HAO XU
Definition
For two networks G and ˆ G in G , we say the network ˆ G has theform of the network G , if there exist finitely many networks G (0) , . . . , G ( n ) such that G (0) = G , G ( n ) = ˆ G , and for any i ∈ { , . . . , n − } , we can obtain G ( i +1) from G ( i ) by switching two species in G ( i ) . Example
The network (2.6) has the form of the network (2.7) . Because wewill obtain exactly the network (2.6) if we switch the two species X and X in (2.7) . X + X −→ X , X + 2 X −→ X + 3 X . (2.6) X + X −→ X , X + X −→ X + 3 X . (2.7) A steady state of (2.2) is a concentration-vector x ∗ ∈ R s ≥ at which f ( x ) on the right-hand side of the ODEs (2.2) vanishes, i.e., f ( x ∗ ) = 0. If asteady state x ∗ has all strictly positive coordinates (i.e., x ∗ ∈ R s> ), then we call x ∗ a positive steady state . If a steady state x ∗ has zero coordinates (i.e., x ∗ ∈ R s ≥ \ R s> ),then we call x ∗ a boundary steady state . We say a steady state x ∗ is nondegenerate ifim (Jac f ( x ∗ ) | S ) = S , where Jac f ( x ∗ ) denotes the Jacobian matrix of f , with respect to( w.r.t ) x , at x ∗ . A nondegenerate steady state x ∗ is Liapunov stable if for any (cid:15) > t >
0, there exists δ > (cid:107) x ( t ) − x ∗ (cid:107) < δ implies (cid:107) x ( t ) − x ∗ (cid:107) < (cid:15) for any t ≥ t . A Liapunov stable steady state x ∗ is locally asymptotically stable if there exists δ > (cid:107) x ( t ) − x ∗ (cid:107) < δ implies lim t →∞ x ( t ) = x ∗ . Anondegenerate steady state x ∗ is exponentially stable (or, simply stable in this paper)if all non-zero eigenvalues of Jac f ( x ∗ ) have negative real parts. Note that if a steadystate is exponentially stable, then it is locally asymptotically stable [18].Suppose N ∈ Z ≥ . A network admits N (nondegenerate) positive steady states if for some rate-constant vector κ and for some total-constant vector c , it has N (nondegenerate) positive steady states at the same stoichiometric compatibility class P c . A network admits N stable positive steady states if for some rate-constant vector κ and for some total-constant vector c , it has N stable positive steady states at thesame stoichiometric compatibility class P c .The maximum number of positive steady states of a network G is cap pos ( G ) := max { N ∈ Z ≥ ∪ { + ∞}| G admits N positive steady states } . The maximum number of nondegenerate positive steady states of a network G is cap nondeg ( G ) := max { N ∈ Z ≥ ∪ { + ∞}| G admits N nondegenerate positive steady states } . The maximum number of stable positive steady states of a network G is cap stab ( G ) := max { N ∈ Z ≥ ∪ { + ∞}| G admits N stable positive steady states } . It is obvious that if ˆ G has the form of G , then cap pos ( ˆ G ) = cap pos ( G ), cap nondeg ( ˆ G ) = cap nondeg ( G ), and cap stab ( ˆ G ) = cap stab ( G ).We say a network admits multistationarity if cap pos ( G ) ≥
2. We say a networkadmits nondegenerate multistationarity if cap nondeg ( G ) ≥
2. We say a network admits multistability if cap stab ( G ) ≥ Definition
For a non-negative integer K , a network G with reactions de-fined in (2.1) is at-most- K -reactant if for all j ∈ { , . . . , m } , we have (cid:80) sk =1 α kj ≤ K ,and we say G is K -reactant (or, the number of reactants of G is K ) if G is at-most- K -reactant and there exists j ∈ { , . . . , m } such that (cid:80) sk =1 α kj = K . For i ∈ { , , } , define M i as the set of subsets of G i such that every H ∈ M i satisfies the two conditions: ULTISTABILITY OF SMALL REACTION NETWORKS G ∈ H , G admits multistability, and(ii) for any G ∈ G i \H and for any ˆ G ∈ H , if G admits multistability, then G contains either more species or more reactants than ˆ G . Question
For i ∈ { , , } , find H ∗ ∈ M i such that H ∗ (cid:54) = ∅ and for any H ∈ M i , we have H ∗ ⊂ H . We provide a complete answer to Question 2.4, see Theorem 2.5, Theorem 2.6and Theorem 2.7.
Theorem
Given G ∈ G , if G has up to species and G is at-most- -reactant, then G admits multistability if and only if G has the form of one of the twonetworks (2.8) and (2.9) below X + 3 X −→ X + X , X + X −→ X ;(2.8) X + 2 X + X −→ β X , X −→ β X + β X + β X , (2.9) where β ∈ { , } , β ∈ Z > , β = β (2 − β ) and β = β + 2 . Theorem 2.5 means for G , the answer to Question 2.4 is H ∗ := { G has the form of the network (2.8), or the network (2.9) } . In fact, Theorem 2.5 implies that the above set satisfies the conditions (i) and (ii).For any
H ∈ M i , if there exists G ∈ H ∗ such that G (cid:54)∈ H , then by the condition (ii),there exists a multistable network ˆ G ∈ H such that ˆ G has either less than 3 speciesor less than 4 reactants, which is a contradiction to Theorem 2.5. So, we definitelyhave H ∗ ⊂ H . Similarly, one can understand why Theorems 2.6 and 2.7 below answerQuestion 2.4 for G and G , respectively. Theorem
Given G ∈ G , if G has up to species and G is at-most- -reactant, then G admits multistability if and only if G has the form of one of thenetworks listed in Rows (7) – (10) of Table 4. Theorem
For G ∈ G , if G has only one species and G is at-most- -reactant,then G admits multistability if and only if G has the form of the network ⇐⇒ X , X ⇐⇒ X . (2.10)It is straightforward to prove Theorem 2.7 by Theorem 3.14 (see Section 3.3) and[13, Theorem 3.6]. We provide the details in “SM.pdf” (Table 5). Here, our maincontributions are Theorem 2.5 and Theorem 2.6. See the proofs in Section 4.2 andSection 5. Note that for each set G i , an ambitious goal is to find the subset of allmultistable network, which can be viewed as the “largest” element in M i . Our workprovides one way to achieve the goal by detecting multistable networks when thenumber of species and the number of reactants are restricted.
3. Small networks with one-dimensional stoichiometric subspaces.3.1. Stability.
Assumption
For any G ∈ G with reactions defined in (2.1) , by the defini-tion of reaction network, we know ( α , . . . , α s ) (cid:54) = ( β , . . . , β s ) . Without loss ofgenerality, we would assume β − α (cid:54) = 0 throughout this paper. Lemma
For any G ∈ G , if the stoichiometric subspace of G is one-dimension-al, then for a nondegenerate steady state x ∗ , it is stable if and only if (cid:80) si =1 ∂f i ∂x i | x = x ∗ < . XIAOXIAN TANG, AND HAO XU
Proof.
Since the stoichiometric subspace of G is one-dimensional, there exists λ ∈ R such that β − α ... β s − α s = − λ β − α ... β s − α s . (3.1)We substitute (3.1) into f ( x ) in (2.2), and we have( β i − α i ) f = ( β − α ) f i , for i = 2 , . . . , s. (3.2)(For instance, if G ∈ G , then f i = κ ( β i − α i ) Π sk =1 x α k k − λκ ( β i − α i ) Π sk =1 x α k k , i = 1 , . . . , s. (3.3)So, the equality (3.2) directly follows from (3.3). If G ∈ G or G , we can similarlyderive (3.2).) By (3.2), the matrix Jac f ( x ∗ ) has rank 1, and so, it has at most onenon-zero eigenvalue. Note also Jac f ( x ∗ ) has at least one non-zero eigenvalue since x ∗ is nondegenerate. Therefore, there is only one non-zero eigenvalue, which is equal tothe trace of the Jacobian matrix Jac f ( x ∗ ). Lemma [13, Lemma 4.1, Theorem 5.8, Theorem 5.12] For G ∈ G , if G admits multistationarity, then there exists λ ∈ R \{ } such that the equality (3.1) holds, and additionally, if G ∈ G , then the scalar λ is positive. Corollary
For G ∈ G , if G admits multistationarity, then the stoichiomet-ric subspace of G is one-dimensional. For any G ∈ G , suppose f ( x ) = ( f ( x ) , . . . , f s ( x )) isdefined as (2.2), and suppose the stoichiometric subspace of G is one-dimensional.Define the system augmented with the conservation laws: h := f , h i := ( β i − α i ) x − ( β − α ) x i − c i − , ≤ i ≤ s. (3.4) Theorem
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. If for a rate-constant vector κ ∗ and a total-constant vector c ∗ , G hasexactly N distinct positive steady states x (1) , . . . , x ( N ) , where x (1) , . . . , x ( N ) are orderedaccording to their first coordinates (i.e., x (1)1 < . . . < x ( N )1 ), and all positive steadystates are nondegenerate, then | Jac h ( x ( i ) ) || Jac h ( x ( i +1) ) | < for i ∈ { , . . . , N − } . The goal of this subsection is to prove Theorem 3.5. We first prepare somelemmas. In fact, Theorem 3.5 directly follows from Lemma 3.6, Lemma 3.8, Lemma3.10 and Lemma 3.11.
Lemma
Let g ( z ) := a n z n + · · · + a z + a be a univariate polynomial in R [ z ] .If the equation g ( z ) = 0 has exactly r ( r ≥ distinct real roots, say z < · · · < z r , andif g (cid:48) ( z i ) (cid:54) = 0 for i ∈ { , . . . , r } , then we have g (cid:48) ( z i ) g (cid:48) ( z i +1 ) < for i ∈ { , . . . , r − } .Proof. Fix any i ∈ { , . . . , r − } , let h ( z ) be the univariate polynomial such that g ( z ) = ( z − z i )( z − z i +1 ) h ( z ). Note that g (cid:48) ( z ) = ( z − z i +1 ) h ( z ) + ( z − z i ) h ( z ) + ( z − z i )( z − z i +1 ) h (cid:48) ( z ) . So, g (cid:48) ( z i ) g (cid:48) ( z i +1 ) = − ( z i +1 − z i ) h ( z i ) h ( z i +1 ). If g (cid:48) ( z i ) g (cid:48) ( z i +1 ) >
0, then we have h ( z i ) h ( z i +1 ) <
0. Notice that h ( z ) is a continuous function, so there exists z ∈ ( z i , z i +1 ) such that h ( z ) = 0. We know z (cid:54) = z i for i ∈ { , . . . , r } , which is acontradiction to the hypothesis that g ( z ) = 0 has exactly r distinct roots. Therefore,we definitely have g (cid:48) ( z i ) g (cid:48) ( z i +1 ) < ULTISTABILITY OF SMALL REACTION NETWORKS Lemma
The determinant of Jacobian matrix of h defined in (3.4) w.r.t x (denoted by | Jac h | ) is equal to ( α − β ) s − s (cid:88) i =1 ( α i − β i ) ∂f ∂x i , (3.5) which is also equal to ( α − β ) s − s (cid:88) i =1 ∂f i ∂x i . (3.6) Proof.
Let Jac h denote the Jacobian matrix of h w.r.t x :Jac h = ∂f ∂x ∂f ∂x ∂f ∂x · · · ∂f ∂x s β − α α − β · · · β − α α − β · · · β s − α s · · · α − β . Below, we prove (3.5) by induction. First, for s = 2, we have | Jac h | = (cid:12)(cid:12)(cid:12)(cid:12) ∂f ∂x ∂f ∂x β − α α − β (cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) i =1 ( α i − β i ) ∂f ∂x i . Second, we assume for s = n ( n ≥ s = n + 1, we applyLaplace expansion to | Jac h | w.r.t the last row, and we get | Jac h | = ( α − β ) n − ( α n +1 , − β n +1 , ) ∂f ∂x n +1 + ( α − β ) n − n (cid:88) i =1 ( α i − β i ) ∂f ∂x i = ( α − β ) n − n +1 (cid:88) i =1 ( α i − β i ) ∂f ∂x i . Finally, by Lemma 3.7 and the equalities (3.2), we have (3.6).For the system h defined in (3.4), define g ( x ) := h ( x , . . . , x s ) | x = β − α β − α x − c β − α , ..., x s = βs − αs β − α x − cs − β − α . (3.7) Lemma
For the system h ( x ) (3.4) and the polynomial g ( x ) (3.7) , if x ∗ is asolution to h ( x ∗ ) = . . . = h s ( x ∗ ) = 0 , then ( α − β ) s − g (cid:48) ( x ∗ ) = | Jac h ( x ∗ ) | . (3.8) Proof.
By (3.7), and by the long division, we have g ( x ) = h + β − α (cid:80) si =2 q i h i , XIAOXIAN TANG, AND HAO XU where q , . . . , q s are polynomials in R [ x , . . . , x s ]. So, g (cid:48) ( x ∗ ) = ∂h ∂x ( x ∗ ) + 1 β − α s (cid:88) i =2 q i ( x ∗ ) ∂h i ∂x ( x ∗ )= ∂h ∂x ( x ∗ ) + 1 β − α s (cid:88) i =2 q i ( x ∗ ) ( β i − α i ) , (3.9) 0 = ∂h ∂x i ( x ∗ ) + 1 β − α q i ( x ∗ ) ∂h i ∂x i ( x ∗ ) , = ∂h ∂x i ( x ∗ ) − q i ( x ∗ ) , i = 2 , . . . , s. (3.10)By (3.10), we have q i ( x ∗ ) = ∂h ∂x i ( x ∗ ). So, by (3.9), we have g (cid:48) ( x ∗ ) = ∂h ∂x ( x ∗ ) + 1 β − α s (cid:88) i =2 ∂h ∂x i ( x ∗ ) ( β i − α i )(3.11)Note that h = f (see (3.4)). By Lemma 3.7 and (3.11), we have (3.8). Lemma
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. If for a rate-constant vector κ ∗ and a total-constant vector c ∗ , G has apositive steady state x ∗ , then the first coordinate x ∗ is contained in the open interval I := ∩ si =2 I i where I i := ( c i − β i − α i , + ∞ ) if β i − α i β − α > , + ∞ ) if β i − α i β − α = 0(0 , c i − β i − α i ) if β i − α i β − α < . (3.12) Proof.
By (3.4), for any i (2 ≤ i ≤ s ), x ∗ i = β i − α i β − α x ∗ − c i − β − α > . So x ∗ iscontained in the interval I defined in (3.12). Lemma
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. If for a rate-constant vector κ ∗ and a total-constant vector c ∗ , G hasexactly N distinct positive steady states x (1) , . . . , x ( N ) , where x (1) , . . . , x ( N ) are orderedaccording to their first coordinates (i.e., x (1)1 < . . . < x ( N )1 ), then all x (1)1 , . . . , x ( N )1 areroots to g ( x ) = 0 , and for any ≤ i ≤ N − , there is no other real root to g ( x ) = 0 between x ( i )1 and x ( i +1)1 .Proof. By (3.4) and (3.7), all x (1)1 , . . . , x ( N )1 are roots to g ( x ) = 0. By Lemma3.9, we have x (1)1 , . . . , x ( N )1 ∈ I (see (3.12)). Hence, if g ( x ) = 0 has a real solution x ∗ between the two solutions x ( i )1 and x ( i +1)1 , then x ∗ ∈ I . For j = 2 , . . . , s , let x ∗ j = β j − α j β − α x ∗ − c j − β − α . Then x ∗ is also a positive steady state, and it is differentfrom x (1) , . . . , x ( N ) , which is a contradiction to the hypothesis that G has exactly N distinct positive steady states. Lemma
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. If a steady state x ∗ is nondegenerate, then | Jac h ( x ∗ ) |(cid:54) = 0 .Proof. Let N := ( β − α , . . . , β s − α s ) (cid:62) (recall we assume β − α (cid:54) = 0). If | Jac h ( x ∗ ) | = 0, then by (3.2) and Lemma 3.7, we have Jac f ( x ∗ ) N is the zero vector.Note that the stoichiometric subspace S is spanned by the single vector N . Soim (Jac f ( x ∗ ) | S ) is the subspace spanned by the zero vector, which is not equal to S ,and hence, this is a contradiction to the hypothesis that x ∗ is nondegenerate. ULTISTABILITY OF SMALL REACTION NETWORKS Theorem [13] Suppose G ∈ G , or, suppose G ∈ G ∪ G and G has up to species. If cap pos ( G ) < ∞ , then cap pos ( G ) = cap nondeg ( G ) . Corollary
Suppose G ∈ G , or, suppose G ∈ G ∪ G and G has up to species. Assume cap pos ( G ) < ∞ . For a rate-constant vector κ ∗ , if | Jac h ( x ∗ ) | has thesame sign for any steady state x ∗ ∈ R s> , then G has at most one positive steady statein any stoichiometric compatibility class.Proof. It directly follows from Corollary 3.4, Theorem 3.5 and Theorem 3.12.
Theorem
Suppose G ∈ G . If cap pos ( G ) = N ≥ N ∈ Z ≥ ) , then cap stab ( G ) ≤ (cid:100) N (cid:101) .Proof. The conclusion directly follows from Corollary 3.4, Lemma 3.2, Lemma3.7, Theorem 3.5.
Theorem
Suppose G ∈ G , or, suppose G ∈ G ∪ G and G has up to species. If cap pos ( G ) = N ≥ N ∈ Z ≥ ) , then (cid:98) N (cid:99) ≤ cap stab ( G ) ≤ (cid:100) N (cid:101) .Proof. The conclusion directly follows from Corollary 3.4, Lemma 3.2, Lemma3.7, Theorem 3.5 and Theorem 3.12.
4. Networks in G .4.1. Boundary steady states and multistationarity. Lemma
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. For the two systems f ( x ) (3.3) and h ( x ) (3.4) , if x ∗ ∈ R s is a solutionto f ( x ∗ ) = . . . = f s ( x ∗ ) = 0 , then | Jac h ( x ∗ ) | = κ ( α − β ) s − Π sk =1 x ∗ kα k − s (cid:88) i =1 ( β i − α i )( α i − α i )Π k (cid:54) = i x ∗ k . (4.1) Proof.
By (3.3), we have ∂f i ∂x i := κ α i ( β i − α i ) x − i Π sk =1 x kα k − λκ α i ( β i − α i ) x − i Π sk =1 x kα k , (4.2)and κ ( β i − α i )Π sk =1 x ∗ kα k = λκ ( β i − α i )Π sk =1 x ∗ kα k . (4.3)By (4.2) and (4.3), we have ∂f i ∂x i ( x ∗ ) := κ ( β i − α i )( α i − α i ) x ∗ i − Π sk =1 x ∗ kα k . (4.4)Hence, by Lemma 3.7 and (4.4), we have (4.1). Lemma
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. If G admits a nondegenerate steady state, then the numbers in thesequence ( β − α )( α − α ) , . . . , ( β s − α s )( α s − α s )(4.5) can not be all zeros.Proof. The conclusion follows from Lemma 3.11 and Lemma 4.1.
Lemma
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. If the network G admits multistationarity, then ∃ i, j ∈ { , . . . , s } s.t. ( β i − α i )( α i − α i )( β j − α j )( α j − α j ) < . (4.6)0 XIAOXIAN TANG, AND HAO XU
Proof.
By Theorem 3.12 and Lemma 4.2, if cap pos ( G ) ≥
1, then the numbers inthe sequence (4.5) can not be all zeros. Then, the conclusion follows from Lemma 4.1and Corollary 3.13.
Remark
Lemma
Given G ∈ G , if G has no boundary steady state in any stoichiomet-ric compatibility class, then for any k (1 ≤ k ≤ s ) , we have either α k = 0 or α k = 0 (i.e., the two monomials Π sk =1 x kα k and Π sk =1 x kα k have no common variables).Proof. Note that for G ∈ G , we know h defined in (3.4) is h = ( β − α ) ( κ Π sk =1 x α k k − λκ Π sk =1 x α k k ) . (4.7)Clearly, if there exists k ∈ { , . . . , s } such that α k > α k >
0, then thereexists at least one boundary steady state ( x = 0 , . . . , x s = 0) in the stoichiometriccompatibility class P c defined by c = (0 , . . . , ∈ R s − , which is a contradiction to thehypothesis that G has no boundary steady state in any stoichiometric compatibilityclass. Example
The converse of Lemma 4.5 might not be true. Consider the con-sistent network X + 2 X κ −→ X + X , X κ −→ X + X . The two monomials Π sk =1 x kα k = x x and Π sk =1 x kα k = x have no common vari-ables. For the total constants c = − , c = 1 , and for the rate constants κ = 25 , κ =4 , there is a boundary steady state (1 , , . Lemma
Given G ∈ G , if for any k (1 ≤ k ≤ s ) , we have either α k = 0 or α k = 0 , then the network G does not admit multistationarity.Proof. Assume G admits multistationarity. By Corollary 3.4, the stoichiometricsubspace of G is one-dimensional. For any k (1 ≤ k ≤ s ), we have either α k = 0 or α k = 0. If α k = 0, then ( β k − α k )( α k − α k ) = − β k α k ≤
0. If α k = 0, then byLemmas 3.3, there exists λ > β k − α k )( α k − α k ) = − λ ( β k − α k )( α k − α k ) = − λ β k α k ≤ . So, the signs of the non-zero numbers in the sequence (4.5) are all negative. ByLemma 4.3, G does not admit multistationarity, which is a contradiction. Theorem
Given G ∈ G , if G has no boundary steady state in any stoichio-metric compatibility class, then the network G does not admit multistationarity.Proof. If G has no boundary steady state in any stoichiometric compatibilityclass, by Lemma 4.5, for any k (1 ≤ k ≤ s ), we have either α k = 0 or α k = 0. ByLemma 4.7, G does not admit multistationarity. Theorem
Given G ∈ G , if G is at-most- -reactant, then G does not admitmultistability.Proof. If the two monomials in h ( x ) (see (4.7)) have no common variables, thenby Lemma 4.7, G does not admit multistationarity. So, G admits no multistability.Note the total degree of h ( x ) w.r.t x is at most 3 since G is at-most-3-reactant.Thus, if the two monomials in h ( x ) have common variables, then the equations h ( x ) = . . . = h s ( x ) = 0 have at most 2 common positive solutions. That means cap pos ( G ) ≤
2. So, by Theorem 3.14, cap stab ( G ) ≤ ULTISTABILITY OF SMALL REACTION NETWORKS Lemma
Given G ∈ G , if G has exactly species and if cap stab ( G ) ≥ ,then we have β k − α k (cid:54) = 0 , and (4.8) α k − α k (cid:54) = 0 , for any k ∈ { , , } , (4.9) and we also have β − α β − α = β − α β − α = β − α β − α < . (4.10) Proof.
Clearly, if cap stab ( G ) ≥
2, then cap nondeg ( G ) ≥ cap pos ( G ) ≥
2. So,by Corollary 3.4, the stoichiometric subspace of G is one-dimensional, and hence thesteady states are common solutions to the equations h ( x ) = h ( x ) = h ( x ) = 0(see (3.4)). Note that by [13, Theorem 5.2], there exist two distinct numbers j , j ∈{ , , } such that for any k ∈ { j , j } , ( β k − α k )( α k − α k ) (cid:54) = 0. Without loss ofgenerality, assume j = 1 and j = 2. Below, we show ( β − α )( α − α ) (cid:54) = 0.In fact, we can rewrite the equations h ( x ) = h ( x ) = h ( x ) = 0 as x = δx − ξ x − ξ := (cid:96) ( x , x ) , (4.11) x = A x − B := (cid:96) ( x ) , (4.12) x = A x − B := (cid:96) ( x ) , (4.13)where δ = ( λκ κ ) α − α > , ξ i = α i − α i α − α , ( i = 1 , A j = β j − α j β − α , and B j = c j − β − α , ( j = 2 , β − α = 0, then the equation (4.13) becomes x = − B , and so, the bivariatefunction (cid:96) ( x , x ) in (4.11) becomes ˆ (cid:96) ( x ) = δ ( − B ) − ξ x − ξ . Note that ˆ (cid:96) ( x ) is apower function, and (cid:96) ( x ) is a linear function. There are at most 2 intersection pointsof their graphs in the first quadrant. So the equations h ( x ) = h ( x ) = h ( x ) = 0 haveat most 2 common positive solutions. By Theorem 3.14, G admits no multistability,which is a contradiction. If α − α = 0, then (cid:96) ( x , x ) becomes a power function˜ (cid:96) ( x ) = δx − ξ . With a similar argument, we can deduce a contradiction.Finally, by Lemma 3.3, we have (4.10) since β k − α k (cid:54) = 0 for all k ∈ { , , } .For k = 1 , . . . , s , define γ k := min { α k , α k } , and ˜ α kj := α kj − γ k ( j = 1 , h (4.7) can be written as h = Π sk =1 x γ k k ˜ h , where˜ h := ( β − α ) (cid:16) κ Π sk =1 x ˜ α k k − λκ Π sk =1 x ˜ α k k (cid:17) . (4.14)Let ˜ g ( x ) := ˜ h ( x , . . . , x s ) | x = β − α β − α x − c β − α , ..., x s = βs − αs β − α x − cs − β − α . (4.15) Lemma
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. Let g ( x ) and ˜ g ( x ) be the polynomials respectively defined in (3.7) and (4.15) . For a fixed rate-constant vector κ ∗ ∈ R > and a total-constant vector c ∗ ∈ R s − , if G has a positive steady state x ∗ , then ˜ g (cid:48) ( x ∗ ) has the same sign with s (cid:88) i =1 ( β i − α i )( α i − α i )Π k (cid:54) = i x ∗ k , (4.16) and additionally, if x ∗ is a stable positive steady state, then ˜ g (cid:48) ( x ∗ ) < . XIAOXIAN TANG, AND HAO XU
Proof.
Since the stoichiometric subspace of G is one-dimensional, the steady state x ∗ is a common solution to the equations h ( x ) = . . . = h s ( x ) = 0 (see (3.4)).By (4.15),we have ˜ g ( x ∗ ) = 0. So, comparing (3.7) and (4.15), we have g (cid:48) ( x ∗ ) =Π sk =1 x ∗ kγ k ˜ g (cid:48) ( x ∗ ). By Lemma 3.8 and Lemma 4.1, g (cid:48) ( x ∗ ) has the same sign with (4.16),and so, ˜ g (cid:48) ( x ∗ ) has the same sign with (4.16). Additionally, if x ∗ is stable, by Lemma3.2, Lemma 3.7 and Lemma 4.1, we know the sign of (4.16) is negative, and hence,˜ g (cid:48) ( x ∗ ) < Lemma
Given G ∈ G , suppose the stoichiometric subspace of G is one-dimensional. Let I := ( a, A ) be the interval defined in (3.12) , where a ∈ R , and A ∈ R ∪ { + ∞} . Let ˜ g ( x ) be the polynomial defined in (4.15) . For a rate-constantvector κ ∗ and a total-constant vector c ∗ , if the degree of ˜ g ( x ) w.r.t x is , and if G has at least two stable positive steady states, then ˜ g ( x ) satisfies the three conditionsbelow:(i) ˜ g ( a ) > ,(ii) ˜ g ( A ) < (here, ˜ g (+ ∞ ) := lim x → + ∞ ˜ g ( x ) ), and(iii) there exists x ∗ ∈ ( a, A ) such that ˜ g ( x ∗ ) = 0 and ˜ g (cid:48) ( x ∗ ) > .Proof. By (4.15), we clearly see that if x ∗ is a positive steady state, then x ∗ ∈ ( a, A ) and ˜ g ( x ∗ ) = 0. If x ∗ is stable, then by Lemma 4.11, we have ˜ g (cid:48) ( x ∗ ) <
0. So,if G has at least two stable positive steady states x (1) and x (2) (here, we assume x (1)1 < x (2)1 ), then for i ∈ { , } , x ( i )1 ∈ ( a, A ), ˜ g ( x ( i )1 ) = 0 and ˜ g (cid:48) ( x ( i )1 ) <
0. Since thedegree of ˜ g ( x ) w.r.t x is 3, there exists a third simple real root x ∗ to the equation˜ g ( x ) = 0. By Lemma 3.6, we know x ∗ ∈ ( x (1)1 , x (2)1 ) ⊂ ( a, A ), and ˜ g (cid:48) ( x ∗ ) > g ( x ) as˜ g ( x ) = C ( x − x (1)1 )( x − x ∗ )( x − x (2)1 ) , where C ∈ R . So ˜ g (cid:48) ( x (1)1 ) < g (cid:48) ( x (1)1 ) = C ( x (1)1 − x ∗ )( x (1)1 − x (2)1 ) < i.e., C < . Thus, ˜ g ( a ) = C ( a − x (1)1 )( a − x ∗ )( a − x (2)1 ) > g ( A ) < Definition
Given matrices of reactant coefficients α = ( α kj ) s × and ˆ α =(ˆ α kj ) s × , which are associated with two networks G and ˆ G in G , we say α is equiv-alent to ˆ α , if there exist finitely many matrices α (0) , . . . , α ( n ) such that α (0) = α , α ( n ) = ˆ α , and for any i ∈ { , . . . , n − } , we can obtain α ( i +1) from α ( i ) by switchingtwo rows or two columns of α ( i ) . Clearly, if a network ˆ G ∈ G has the form of a network G ∈ G , then the two matrices ofreactant coefficients associated with G and ˆ G are equivalent (remark that the conversemight not be true). Recall Example 2.2. The two sets of reactant coefficients (say α and ˆ α ) of networks (2.6) and (2.7) can be written as matrices α = and ˆ α = . We can obtain α from ˆ α by first switching the two columns and then switching thelast two rows. Lemma
If a -species network G ∈ G is at-most- -reactant, and if G admitsmultistability, then G can only have the form of one of the networks listed in Table 2.Proof. If G admits multistability, then by Theorem 4.9, G must be 4-reactant, ULTISTABILITY OF SMALL REACTION NETWORKS h (4.7) w.r.t x is exactly 4, i.e.,max { (cid:88) k =1 α k , (cid:88) k =1 α k } = 4 . (4.17)By Lemma 4.7, the two monomials in h have common variables. So, the degree of˜ h (4.14) w.r.t x is at most 3. On the other hand, Theorem 3.14 implies that if G admits multistability, then cap pos ( G ) ≥
3. Note that all positive steady states of G are common solutions to the equations ˜ h ( x ) = h ( x ) = . . . = h s ( x ) = 0. So, thedegree of ˜ h w.r.t x is at least 3. Overall, the degree of ˜ h w.r.t x is exactly 3. So, bythe definition of ˜ h , we have (cid:88) k =1 min { α k , α k } = 1 . (4.18)Therefore, by Lemma 4.3 and Lemma 4.10, we know that the matrix of reactant coef-ficients α := ( α kj ) × and the matrix of product coefficients β := ( β kj ) × associatedwith G belong to the set B := { ( α, β ) ∈ Z × ≥ × Z × ≥ s.t. (4.6), (4.9), (4.10), (4.17) and (4.18) hold } . (4.19) Define a map π : Z × ≥ × Z × ≥ → Z × ≥ such that for any ( α, β ) ∈ Z × ≥ × Z × ≥ , π ( α, β ) = α . Let B α := { α ∈ Z × ≥ s.t. (4.9), (4.17) and (4.18) hold } . (4.20) Notice that B α is a finite set. For each α ∈ B α , define its equivalence class in B α as [ α ] := { ˆ α ∈ B α | ˆ α is equivalent to α } . We explicitly compute the set B α by Maple2020 [15], and it is straightforward to check by a computer program that thereare 12 equivalence classes in B α (see a supporting file “irreversible.mw” in Table 5).We pick up a representative from each equivalence class, and we present them in Table1. Note B = π − ( B α ) ∩ B = π − ( ∪ α ∈B α [ α ]) ∩ B = ∪ α ∈B α ∪ ˆ α ∈ [ α ] (cid:0) π − (ˆ α ) ∩ B (cid:1) . (4.21)By Definition 4.13, if ˆ α ∈ [ α ], then there exist two permutation matrices P and Q such that ˆ α = P αQ . Thus, there exists a bijection φ : π − ( α ) ∩ B → π − (ˆ α ) ∩ B such that for any ( α, β ) ∈ π − ( α ) ∩ B , φ ( α, β ) := (ˆ α, P βQ ). By Definition 2.1, thetwo networks associated with ( α, β ) and φ ( α, β ) have the same form. Thus, by (4.21),the multistable network G has the form of a network associated with an element in π − ( α ) ∩ B for a representative α in B α . In the rest of the proof, we explain how tocompute π − ( α ) ∩ B for each representative recorded in Table 1.For the values of α kj recorded in Table 1-Row (1), the condition (4.10) implies( β − / ( β − < , (4.22) β / ( β − < , (4.23) β / ( β − < . (4.24)Note that β kj ∈ Z ≥ . So by (4.23) and (4.24), we have β = β = 0. Also, notethat the sequence (4.5) is now β − , β − , and β − . (4.25)4 XIAOXIAN TANG, AND HAO XU α α α α α α Table 1
Representatives of equivalence classes in B α (4.20) Since both β − β − β − >
0. So, by(4.22), we have β = 0. We substitute β = β = β = 0 and the values of α kj recorded in Table 1-Row (1) into (4.10), and we get( β − β = 1 , and ( β − β = 1 . We solve β kj from these two equations over Z ≥ , and we get β = 3, β = 1 and β = 1. Above all, we conclude that π − ( α ) ∩ B = { , } . Similarly, from each α recorded in each row of Table 1, we can solve the corresponding π − ( α ) ∩ B , and we record the corresponding network in Table 2. Proof of Theorem 2.5. “ ⇐ ”: For the network (2.8), it is straightforward tocheck that the equality (3.1) holds for λ = 1. Let κ = 9, κ = 50, c = 6, and c = . By solving the equations h ( x ) = h ( x ) = h ( x ) = 0 (see (3.4) and (4.7)),we see that the network has three nondegenerate positive steady states: x (1) = ( 72 − √ ,
52 + √ ,
125 + √ , x (2) = (5 , ,
910 ) , x (3) = ( 72 + √ , − √ , − √ . It is straightforward to check by Lemma 3.2 that x (1) and x (3) are stable.For the network (2.9), if β = 0, then for any β ∈ Z > , β = 2 β and β = β + 2. It is straightforward to check that the equality (3.1) holds for λ = β > κ = 1, κ = β , c = , and c = . Then we have h = ( β − α ) ( κ Π sk =1 x α k k − λκ Π sk =1 x α k k ) = − (cid:0) x x x − x (cid:1) ,h = ( β − α ) x − ( β − α ) x − c = − x + x − , and h = ( β − α ) x − ( β − α ) x − c = − x + x − . By solving the equations h ( x ) = h ( x ) = h ( x ) = 0, the network has three nonde-generate positive steady states: x (1) = ( 198 − √ , − √ , − √
338 ) , x (2) = ( 34 , , , x (3) = ( 198 + 3 √ ,
454 + 3 √ ,
218 + 3 √
338 ) . It is straightforward to check by Lemma 3.2 that x (1) and x (3) are stable. Similarly, if β = 1, then for any β ∈ Z > , β = β and β = β + 2. Let κ = 1, κ = β , ULTISTABILITY OF SMALL REACTION NETWORKS c = , and c = . Then the network has three nondegenerate positive steadystates: x (1) = ( 198 − √ , − √ , − √
338 ) , x (2) = ( 34 , , , x (3) = ( 198 + 3 √ ,
458 + 3 √ ,
218 + 3 √
338 ) . It is straightforward to check by Lemma 3.2 that x (1) and x (3) are stable. Here, wecompute these steady states by Maple2020 [15], see “witness1.mw” in Table 5.“ ⇒ ”: By Theorem 3.15 and [13, Theorem 3.6 2(b), Theorem 4.8], if G ∈ G and G has up to 2 species, then G admits no multistability. From Table 2, it is directlyseen that the networks (2.8) and (2.9) are respectively listed in Row (3) and Row (7).By Lemma 4.14, we only need to show none of the other networks listed in Table 2admits multistability.For the network in Table 2-Row (1), the polynomial ˜ g ( x ) defined in (4.15) is˜ g ( x ) = κ x x x − λκ | x = − x − c ,x = − x − c , where λ := − β − α β − α >
0, and the interval I defined in (3.12) is (0 , min {− c , − c } ).Note that ˜ g (0) = − λκ < κ ∈ R > . So, by Lemma 4.12, this network inRow (1) does not admit multistability.For the network in Table 2-Row (2), the polynomial ˜ g ( x ) is˜ g ( x ) = κ x x − λκ x | x = β − β − x − c β − ,x = β β − x − c β − , where λ := − β − α β − α >
0, and the interval I is (max { , c β } , c β − ). From thesecond column of Row (2), we see that β − > β − − β ( β − <
0, and β = β − >
0. If c β <
0, then ˜ g (0) = λκ c β − <
0, and so, by Lemma 4.12(i), the network in Row (2) does not admit multistability. If c β ≥
0, then by Lemma4.11, for any positive steady state x ∗ of G , ˜ g (cid:48) ( x ∗ ) has the same sign with (cid:88) i =1 ( β i − α i )( α i − α i )Π k (cid:54) = i x ∗ k = ( β − x ∗ x ∗ + 2( β − x ∗ x ∗ − β x ∗ x ∗ = (( β − x ∗ − β x ∗ ) x ∗ + 2( β − x ∗ x ∗ = − c x ∗ + 2( β − x ∗ x ∗ , which is negative (Note β − < g ( x ) is˜ g ( x ) = − (cid:0) κ x x − λκ x (cid:1) | x = − ( β − x + c ,x = x + c , where λ := − β − α β − α >
0, and the interval I is (max { , c β − , − c } , + ∞ ). From thesecond column of Row (4), we see that β − <
0. If − c > − c > c β − ,then by the fact that ˜ g ( − c ) = κ c (( β − c + c ) ≤ − c ≤
0, then by Lemma6
XIAOXIAN TANG, AND HAO XU x ∗ of G , ˜ g (cid:48) ( x ∗ ) has the same sign with (cid:88) i =1 ( β i − α i )( α i − α i )Π k (cid:54) = i x ∗ k = − x ∗ x ∗ + 2( β − x ∗ x ∗ + x ∗ x ∗ = ( x ∗ − x ∗ ) x ∗ + 2( β − x ∗ x ∗ = − c x ∗ + 2( β − x ∗ x ∗ < − c ≤ c β − (i.e., c + c ( β − ≤ (cid:88) i =1 ( β i − α i )( α i − α i )Π k (cid:54) = i x ∗ k = − x ∗ x ∗ + 2( β − x ∗ x ∗ + x ∗ x ∗ = − x ∗ x ∗ + ( β − x ∗ x ∗ + x ∗ (( β − x ∗ + x ∗ )= − x ∗ x ∗ + ( β − x ∗ x ∗ + x ∗ ( c + c ( β − < β − x ∗ + x ∗ = c + c ( β −
2) above is deduced byeliminating x ∗ from the two conservation law equations ( β − x ∗ + x ∗ − c = 0and − x ∗ + x ∗ − c = 0). So by Lemma 4.12 (iii), the network in Row (4) does notadmit multistability. Similarly, we can prove the network in Row (10) does not admitmultistability.For the network in Table 2-Row (6), the polynomial ˜ g ( x ) is˜ g ( x ) = − ( κ x x − λκ x ) | x = − x + c ,x = − β x + c , where λ := − β − α β − α >
0, and the interval I is (0 , min { c , c β } ) (from the secondcolumn of Row (4), we see that β > c < c β , then by the fact that ˜ g ( c ) = λκ ( − β c + c ) ≥ c β ≤ c (i.e., − β c + c ≤ x ∗ of G , ˜ g (cid:48) ( x ∗ ) has the same sign with (cid:88) i =1 ( β i − α i )( α i − α i )Π k (cid:54) = i x ∗ k = − x ∗ x ∗ + 2 x ∗ x ∗ − β x ∗ x ∗ = − x ∗ x ∗ + 2 x ∗ ( x ∗ − β x ∗ )= − x ∗ x ∗ + 2 x ∗ ( − β c + c ) < x ∗ − β x ∗ = − β c + c above is deduced by eliminating x ∗ from the two conservation law equations x ∗ + x ∗ − c = 0 and β x ∗ + x ∗ − c = 0). Soby Lemma 4.12 (iii), the network in Row (6) does not admit multistability. Similarly,the network in Row (9) does not admit multistability.For the network in Table 2-Row (11), the polynomial ˜ g ( x ) is˜ g ( x ) = ( β − (cid:0) κ x − λκ x x (cid:1) | x = β β − x − c β − ,x = β β − x − c β − , where λ := − β − α β − α >
0, and the interval I is (max { , c β , c β } , + ∞ ). Note that˜ g (+ ∞ ) has a positive sign for any κ ∈ R > . So, by Lemma 4.12 (ii), if G admitsmultistability, then the network in Row (11) does not admit multistability. (cid:3) ULTISTABILITY OF SMALL REACTION NETWORKS Table 2
All possible multistable networks in G with 4 reactants and 3 species Network ( β , β , β , β , β , β ∈ Z ≥ X X X → X X → X X X X → β X β X β X X X → β X β , − β β −
2) + 2 , β − , , β , β ∈ { , } ; 0 < β ≤ β − X X → X X X X → X X X X → β X X → β X β X β X , β , , β , β − β , β
12 + 2) β ∈ { , } ; β > X X → β X β X β X X X → β X β X β , − β β −
2) + 2 , (2 − β β − , , β , β β ∈ { , } ; 0 < β ≤ β − β ∈ { , } (6) X X → X β X X X → X β X , , β , , , − β β ∈ { , } (7) X X X → β X X → β X β X β X , β , , β , β − β , β
12 + 3) β ∈ { , } ; β > X X → β X β X β X X X → β X β X β , − β β −
2) + 2 , (3 − β β − , , β , β β ∈ { , } ; 0 < β ≤ β − β ∈ { , , } (9) X X → X β X X X → X β X , , β , , , − β β ∈ { , , } (10) X X X → β X X → β X β X β X , β , , β , β − β , β
12 + 4) β ∈ { , } ; β > X → β X β X β X X X X → β
21 + 4 , β , β , , , β > X X → β X X X X → β X X β , , , − β , , β ∈ { , , } Remark
5. Networks in G : proof of Theorem 2.6. Lemma [20, Theorem 3.5] Given G ∈ G , if G has exactly species, then G admits nondegenerate multistationarity if and only if there exists λ ∈ R \{ } such thatthe equality (3.1) holds for s = 2 , and ∃ k ∈ { , } s.t. max { α k , β k } < α k < β k or min { α k , β k } > α k > β k . (5.1) Lemma
Suppose G ∈ G , and suppose G has exactly species. If G admitsmultistability, then we have ( β − α )( β − α ) (cid:54) = 0 , and (5.2) β − α β − α = β − α β − α (cid:54) = 0 . (5.3) Proof.
Recall that we have β − α (cid:54) = 0 by Assumption 3.1. If β − α = 0, thenby (3.7), we have g ( x ) = ( β − α ) (cid:16) κ Γ α x α − κ Γ β x β + λκ Γ α x α (cid:17) , where Γ = − c β − α . So, g ( x ) has at most 3 terms, and hence, the number ofsign changes of the coefficients is at most 2. By Descartes’ rule of signs [10] (see asupporting file “SM.pdf” in Table 5), g ( x ) = 0 has at most 2 positive roots. So, the8 XIAOXIAN TANG, AND HAO XU
Table 3
Representatives of equivalence classes in C σ (5.10) see “reverse1.mw” see “reverse2.mw” see “reverse3.mw” see “reverse4.mw” (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) network G admits at most 2 positive steady states. By Theorem 3.14, the networkdoes not admit multistability, which is a contradiction. So, the inequality (5.2) holds.Finally, by Lemma 3.3, we have (5.3). Definition
Given two matrices of reactant coefficients σ = (cid:18) α β α α β α (cid:19) and ˆ α = (cid:18) ˆ α ˆ β ˆ α ˆ α ˆ β ˆ α (cid:19) , which are associated with two 2-species networks G and ˆ G in G , we say σ is stronglyequivalent to ˆ σ , if there exist finitely many matrices σ (0) , . . . , σ ( n ) such that σ (0) = σ , σ ( n ) = ˆ σ , and for any i ∈ { , . . . , n − } , we can obtain σ ( i +1) from σ ( i ) by switchingthe two rows or the first two columns of σ ( i ) . Example
Consider the two networks below. X + 2 X ⇐⇒ , X −→ X + 2 X . (5.4) 0 ⇐⇒ X + X , X −→ X + 3 X . (5.5) The two matrices of reactant coefficients of networks (5.4) and (5.5) can be rewrittenas σ = (cid:18) (cid:19) and ˆ σ = (cid:18) (cid:19) . We can obtain σ from ˆ σ by firstswitching the first two columns and then switching the two rows. So σ is stronglyequivalent to ˆ σ . Lemma
If a -species network G ∈ G is at-most- -reactant, and if G admitsmultistability, then G can only have the form of one of the networks listed in Table 4.Proof. If G admits multistability, then by Theorem 3.14, we have cap pos ( G ) ≥
3. Note that all positive steady states of G are common solutions to the equations h ( x ) = . . . = h s ( x ) = 0 (see (3.4)). So, the degree of h w.r.t x is at least 3. Since G is at most 3-reactant, the degree of h w.r.t x is at most 3. Overall, the degree h w.r.t x is exactly 3, i.e., max { (cid:88) k =1 α k , (cid:88) k =1 β k , (cid:88) k =1 α k } = 3 . (5.6)So, we have cap pos ( G ) = 3. That means G has no boundary steady states and so,min { α , β , α } = 0 , and min { α , β , α } = 0 . (5.7) ULTISTABILITY OF SMALL REACTION NETWORKS τ := (cid:18) α β α β α β α β (cid:19) associated with G belong to the set C := { (cid:18) α β α β α β α β (cid:19) ∈ Z × ≥ s.t. (5.1), (5.2), (5.3), (5.6), (5.7) hold } . (5.8) Define a map π σ : Z × ≥ → Z × ≥ such that for any (cid:18) α β α β α β α β (cid:19) ∈ Z × ≥ ,its image under π σ is (cid:18) α β α α β α (cid:19) . Note that if τ satisfies (5.1), then π σ ( τ )satisfies ∃ k ∈ { , } s.t. max { α k , β k } < α k , or min { α k , β k } > α k > . (5.9)Let C σ := { (cid:18) α β α α β α (cid:19) ∈ Z × ≥ s.t. (5.2), (5.6), (5.7), and (5.9) hold } . (5.10) For any σ ∈ C σ , define a equivalence class C σ as[ σ ] C := { ˆ σ ∈ C σ | ˆ σ is strongly equivalent to σ } . It is straightforward to check by a computer program that there are 25 equivalenceclasses in C σ (see supporting files “reverse1.mw”–“reverse4.mw” in Table 5), and wepick up one element from each equivalence class as a representative. We present the25 representatives in Table 3.In Table 3, for any representative σ recorded in a unbold/uncolored cell, the set π − σ ( σ ) ∩ C is empty. For instance, for the first column of the second row, we have σ = (cid:18) α β α α β α (cid:19) = (cid:18) (cid:19) , which satisfies the condition (5.9) because for k = 1, max { α k , β k } < α k holds. Bythe condition (5.1), we have β > α = 2. The condition (5.3) can be written as β − = β − − . So, we have β − < β − >
0. Hence, β = 0 is the onlysolution for β in Z ≥ . So we have β − , and we have no solution for β in Z ≥ . Therefore, π − σ ( σ ) ∩ C = ∅ . Similarly, we can easily verify that π − σ ( σ ) ∩ C isempty for any other σ recorded in a unbold cell (see “reverse1.mw”–“reverse4.mw”).We repeat the representatives in the bold/colored cells in the first column of Table 4,and we write down their corresponding networks in the second column.Note C = π − σ ( C σ ) ∩ C = ∪ σ ∈C σ ∪ ˆ σ ∈ [ σ ] C (cid:0) π − σ (ˆ σ ) ∩ C (cid:1) . (5.11)By Definition 5.3, if ˆ σ ∈ [ σ ] C , then there exist two permutation matrices P and Q such that ˆ σ = P σQ . Thus, there exists a bijection φ : π − σ ( σ ) ∩ C → π − σ (ˆ σ ) ∩ C such that for any τ ∈ π − σ ( σ ) ∩ C , φ ( τ ) := P τ Q . By Definition 2.1, the two networksassociated with τ and φ ( τ ) have the same form. Thus, by (5.11), any multistablenetwork G has the form of a network associated with an element in π − σ ( σ ) ∩ C for arepresentative σ recorded in the first column of Table 4. In the rest of the proof, weexplain how to compute π − σ ( σ ) ∩ C for each representative in C σ recorded in Table 4.0 XIAOXIAN TANG, AND HAO XU
For the reactant coefficients recorded in Table 4-Row (1), the matrix σ is (cid:18) α β α α β α (cid:19) = (cid:18) (cid:19) , which satisfies the condition (5.9) because for k = 1, max { α k , β k } < α k holds. Bythe condition (5.1), we have β > α = 3. The condition (5.3) can be written as β − − = β − , i.e., β = ( β − π − σ ( σ ) ∩ C = { (cid:18) β ( β − (cid:19) | β ∈ Z > } . Similarly, from each set of reactant coefficients recorded in the first column of Table4, we can solve π − σ ( σ ) ∩ C , and we record the corresponding β and β in the thirdcolumn. Proof of Theorem 2.6. “ ⇐ ”: For the network in Table 4–Row (7), it is straight-forward to check that for any β >
2, the equality (3.1) holds for λ = − ( β − < κ = , κ = 16, κ = β − and c = −
9. Then we have h = ( β − α ) (cid:0) κ x − κ x − λκ x x (cid:1) = 12 x − x + 32 x x , and h = ( β − α ) x − ( β − α ) x − c = − x − x + 9 . By solving the equations h ( x ) = h ( x ) = 0, the network has three nondegeneratepositive steady states: x (1) = (4 − √ , √ , x (2) = (1 , , x (3) = (4 + √ , − √ Itis straightforward to check by Lemma 3.2 that x (1) and x (3) are stable. Similarly,we can show the networks in Rows (8)–(10) admit multistability. We present thecomputation in a supporting file, see “witness2.mw” in Table 5.“ ⇒ ”: By Theorem 3.15 and [13, Theorem 3.6 2(b)], if G ∈ G and G has only 1species, then G admits no multistability. By Lemma 5.5, we only need to show thenetworks listed in Table 4-Rows (1)–(6) do not admit multistability.For the network in Table 4-Row (1), the polynomial g ( x ) defined in (3.7) is g ( x ) = − ( κ x x − κ − λκ x ) | x =( x + c ) / = − ( κ − λκ ) x − c κ x + κ , where λ := − β − α β − α = − β − α β − α >
0. The number of sign changes of the coefficientsis at most 2 since g ( x ) has at most 3 terms. By Descartes’ rule of signs, g ( x ) = 0has at most 2 positive roots. So, this network admits at most 2 positive steady statesand by Theorem 3.14, the network does not admit multistability.For the network in Table 4-Row (2), g ( x ) = − ( κ x x − κ − λκ x ) | x =2 x + c = ( λκ − κ ) x − c κ x − c κ x + κ , and the interval I is (max { , − c } , + ∞ ), where λ := − β − α β − α = − β − α β − α >
0. If − c <
0, then c >
0, and so the number of sign changes of the coefficients is at most2 since g ( x ) has at most 3 terms. By Descartes’ rule of signs, g ( x ) = 0 has at most2 positive roots. Similarly, if − c > λκ − κ >
0, then by Descartes’ rule ofsigns, g ( x ) = 0 has at most 2 positive roots. If − c > λκ − κ <
0, then g (cid:48) ( − c λκ − κ ) x − c κ x − c κ | x = − c = 34 λκ c > . ULTISTABILITY OF SMALL REACTION NETWORKS Table 4
All possible multistable networks in G with 3 reactants and 2 species ( α , α , β , β , α , α
22) Network β
12 and β
22 in Z ≥ , , , , ,
0) 2 X X ⇔ X → β X β X β
22 = 12 ( β − β − > , , , , , X X ⇔ X → β X β X β
22 = 2( β − β − > , , , , , X X ⇔ X X → β X β X β
22 = 2( β −
2) + 1 β − > , , , , , X X ⇔ X X → β X β X β
22 = β − β − > , , , , ,
0) 0 ⇔ X X X → β X β X β
22 = β − β − > , , , , ,
1) 0 ⇔ X X X X → β X β X β
22 = ( β −
2) + 1 β − > , , , , , X ⇔ X X X → β X β X β
22 = − ( β −
2) + 1 β − > , , , , , X ⇔ X X X → β X β X β
22 = − β −
2) + 1 β − > , , , , , X X ⇔ X → β X β X β
22 = 2( β − β − > , , , , , X X ⇔ X X → β X β X β
22 = β − β − > So g (cid:48) ( x ) = 0 has at most 1 root over the interval I , and hence g ( x ) = 0 has at most2 roots over I . Above all, the network admits at most 2 positive steady states, andso, by Theorem 3.14, the network does not admit multistability. Similarly, we canshow that the network in Table 4-Row (3) does not admit multistability.For the network in Table 4-Row (4), the polynomial g ( x ) is g ( x ) = − ( κ x x − κ x − λκ x ) | x = x + c = − ( κ − λκ ) x − c κ x − ( c κ − κ ) x + c κ , where λ := − β − α β − α = − β − α β − α >
0. Note that for any κ > κ > c ∈ R , − c κ and c κ have different signs if c (cid:54) = 0. So the number of sign changesof the coefficients of g ( x ) is at most 2. By Descartes’ rule of signs, g ( x ) = 0 has atmost 2 positive roots. So, this network has at most 2 positive steady states and byTheorem 3.14, the network does not admit multistability.For the network in Table 4-Row (5), the polynomial g ( x ) is g ( x ) = κ − κ x x − λκ x | x = x − c , and the interval I is (max { , c } , + ∞ ), where λ := − β − α β − α = − β − α β − α <
0. So thenumber of sign changes of the coefficients of g ( x ) is at most 2. By Descartes’ rule ofsigns, g ( x ) = 0 has at most 2 positive roots. So, by Theorem 3.14, the network doesnot admit multistability. Similarly, we can show that the network in Table 4-Row (6)does not admit multistability. (cid:3)
6. Discussion.
For the future work, we propose the problems below.(1) Does there exist a network G in G such that cap pos ( G ) = 3 but cap stab ( G ) <
2? Remark that for all small networks we have studied (see Table 2 and Table4), if a network admits three positive steady states, then there are two stableones.2
XIAOXIAN TANG, AND HAO XU (2) Under which conditions does a network in G admit strictly more than 3 pos-itive steady states?(3) For the set of networks G i ( i ∈ { , , } ), which subset is the smallest suchthat any network in this subset admits strictly more than 3 positive steadystates? REFERENCES[1] Murad Banaji and Casian Pantea. Some results on injectivity and multistationarity in chemicalreaction networks.
SIAM J. Appl. Dyn. Syst. , 15(2):807–869, 2016.[2] Murad Banaji and Casian Pantea. The inheritance of nondegenerate multistationarity in chem-ical reaction networks.
SIAM J. Appl. Math. , 78:1105–1130, 2018.[3] Christoph Bagowski and James Ferrell Jr. Bistability in the JNK cascade.
Curr. Biol. ,11(15):1176–82, 2001.[4] Gheorghe Craciun, and Martin Feinberg. Multiple equilibria in complex chemical reactionnetworks: I. the injectivity property.
SIAM J. Appl. Math. , 65:1526–1546, 2005.[5] Gheorghe Craciun, Yangzhong Tang, and Martin Feinberg. Understanding bistability in com-plex enzyme-driven reaction networks.
PNAS , 103(23):8697–8702, 2006.[6] Carsten Conradi, Elisenda Feliu, Maya Mincheva, and Carsten Wiuf. Identifying parameterregions for multistationarity.
PLoS Comput. Biol. , 13(10):e1005751, 2017.[7] Biswa Nath Datta. An elementary proof of the stability criterion of Li´enard and Chipart.
Linear Algebra Appl. , 22:89–96, 1978.[8] Alicia Dickenstein, Mercedes Perez Millan, Anne Shiu and Xiaoxian Tang. Multistationarity inStructured Reaction Networks.
Bull. Math. Biol. . 81(5):1527–1581, 2019.[9] James Ferrell Jr., Eric Machleder. The biochemical basis of an all-or-none cell fate switch inXenopus oocytes.
Science , 280(5365):895–898, 1998[10] David J. Grabiner. Descartes’ rule of signs: another construction.
Amer. Math. Monthly ,106(9):854–856, 1999.[11] Hoon Hong, Xiaoxian Tang, and Bican Xia, Special algorithm for stability analysis of multi-stable biological regulatory systems.
J. Symbolic Comput. , 70:112–135, 2015.[12] Badal Joshi and Anne Shiu. Atoms of multistationarity in chemical reaction networks,
J. Math.Chem. , 51(1):153–178, 2013.[13] Badal Joshi and Anne Shiu. Which small reaction networks are multistationary?
SIAM J.Appl. Dyn. Syst. , 16(2):802–833, 2017.[14] Stefan M¨uller, Elisenda Feliu, Georg Regensburger, Carsten Conradi, Anne Shiu, and AliciaDickenstein. Sign conditions for injectivity of generalized polynomial maps with applica-tions to chemical reaction networks and real algebraic geometry.
Found. Comput. Math. ,16(1):69–97, 2016.[15] Maple (2020) Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.[16] Nida Obatake, Anne Shiu, and Dilruba Sofia, Mixed volume of small reaction networks.
Preprint , arXiv:2004.14467.[17] Nida Obatake, Anne Shiu, Xiaoxian Tang, and Angelica Torres, Oscillations and bistability ina model of ERK regulation.
J. Math. Biol. , 79:1515–1549, 2019.[18] Lawrence Perko. Differential equations and dynamical systems,
Volume 7 of Texts in AppliedMathematics. Springer-Verlag, New York, third edition , 2001.[19] Guy Shinar and Martin Feinberg. Concordant chemical reaction networks.
Math. Biosci. ,240(2):92–113, 2012.[20] Anne Shiu and Timo de Wolff. Nondegenerate multistationarity in small reaction networks.
Discrete Contin. Dyn. Syst. B , 24(6):2683–2700, 2019.[21] Ang´elica Torres and Elisenda Feliu. Detecting parameter regions for bistability in reactionnetworks.
Accepted by SIAM J. Appl. Dyn. Syst. , arXiv:1909.13608.[22] Carsten Wiuf and Elisenda Feliu. Power-law kinetics and determinant criteria for the preclusionof multistationarity in networks of interacting species.
SIAM J. Appl. Dyn. Syst. , 12:1685–1721, 2013.[23] Wen Xiong, James Ferrell Jr. A positive-feedback-based bistable ‘memory module’ that governsa cell fate decision.
Nature , 426:460–465, 2003.ULTISTABILITY OF SMALL REACTION NETWORKS SUPPLEMENTARY MATERIAL
Table 5 lists all files at the online repository: https://github.com/HaoXUCode/MSRN-Supplement
Table 5
Supporting Information Files
Name File Type Results
SM.pdf PDF
Descartes’ rule and Theorem 2.7 witness2.mw/.pdf Maple/PDF
Theorem 2.6 reverse4.mw/.pdf Maple/PDF
Lemma 5.5 reverse3.mw/.pdf Maple/PDF
Lemma 5.5 reverse2.mw/.pdf Maple/PDF
Lemma 5.5 reverse1.mw/.pdf Maple/PDF
Lemma 5.5 witness1.mw/.pdf Maple/PDF
Theorem 2.5 irreversible.mw/.pdf Maple/PDFirreversible.mw/.pdf Maple/PDF