Absolute Concentration Robustness and Multistationarity in Reaction Networks: Conditions for Coexistence

Many reaction networks arising in applications are multistationary, that is, they have the capacity for more than one steady state; while some networks exhibit absolute concentration robustness (ACR), which means that some species concentration is the same at all steady states. Both multistationarity and ACR are significant in biological settings, but only recently has attention focused on the possibility for these properties to coexist. Our main result states that such coexistence in at-most-bimolecular networks (which encompass most networks arising in biology) requires at least $3$ species, $5$ complexes, and $3$ reactions. We prove additional bounds on the number of reactions for general networks based on the number of linear conservation laws. Finally, we prove that, outside of a few exceptional cases, ACR is equivalent to non-multistationarity for bimolecular networks that are small (more precisely, one-dimensional or up to two species). Our proofs involve analyses of systems of sparse polynomials, and we also use classical results from chemical reaction network theory.


Introduction
A mass-action kinetics system exhibits absolute concentration robustness (ACR) if the steadystate value of at least one species is robust to fluctuations in initial concentrations of all species [28].Another biologically significant property is the existence of multiple steady states, that is, multistationarity.Significantly, this property has been linked to cellular decision-making and switch-like responses [22,34].
As both ACR and multistationarity are important properties, it is perhaps surprising that their relationship was explored only recently, when the present authors with Joshi showed that ACR and multistationarity together -or even ACR by itself -is highly atypical in randomly generated reaction networks.This result dovetails with the fact that the two properties are somewhat in opposition, as multiple steady states are not in general position in the presence of ACR.
The results of Joshi et al. are asymptotic in nature (as the number of species goes to infinity), and they pertain to networks that are at-most-bimolecular (which is typical of networks arising in biology) and reversible (which is not) [19].This naturally leads to the following question: Question 1.1.For multistationarity and ACR to coexist, how many species, reactions, and complexes are needed?Which networks (without the requirement of being reversible) of small to medium size allow such coexistence?Another motivation for Question 1.1 comes from synthetic biology.In order to design reaction networks with certain dynamical properties, we need to better understand the design principles that allow for such behaviors, as well as the constraints on the size (such as the minimum numbers of species, reaction, and complexes) of such networks.
Our work focuses on answering Question 1.1.Broadly speaking, our results fall into two categories: (i) results that give lower bounds on the dimension of a network or its number of species, reactions, or complexes; and (ii) results for certain classes of networks (one-dimensional, up to 2 species, and so on).Our primary focus is on at-most-bimolecular networks, but we also present results on general networks.
In the first category, our results are summarized in the following theorem, which gives some minimum requirements for ACR and nondegenerate multistationarity to coexist.This coexistence is typically on a nonzero-measure subset of the parameter space of reaction rate constants.
Theorem 1.2 (Main result).Let G be an at-most-bimolecular reaction network with n species such that there exists a vector of positive rate constants κ * such that the mass-action system (G, κ * ) has ACR and is nondegenerately multistationary.Then G has: (1) at least 3 species (that is, n ≥ 3), (2) at least 3 reactant complexes (and hence, at least 3 reactions) and at least 5 complexes (reactant and product complexes), and (3) dimension at least 2.
If, additionally, G is full-dimensional (that is, G has no linear conservation laws), then G has: (4) at least n + 2 reactant complexes (and hence, at least n + 2 reactions), and (5) dimension at least 3.
For the proof of Theorem 1.2, we refer the reader to Section 3 for part (3) (Lemma 3.20); Section 4 for parts (1), (2), and (5) (Theorem 4.1); and Section 5 for part (4) (Theorem 5.1).Additionally, many of the lower bounds in Theorem 1.2 are tight.Indeed, this is shown for parts (1)-(3) through the following network: {A + B → 2C → 2B, C → A} (Example 4.10).As for part (4), this bound is proven for networks that need not be at-most-bimolecular, and its tightness is shown in that context (Proposition 5.3).
While Theorem 1.2 concerns nondegenerate multistationarity, we also investigate the capacity for ACR together with degenerate multistationarity, specifically, in networks with 4 reactant complexes (Proposition 4.15).Finally, we prove two additional results in the spirit of Theorem 1.2.The first states that 3 is the minimum number of pairs of reversible reactions needed (in reversible networks) for multistationarity, even without ACR (Theorem 3.21).The second concerns networks that are not full-dimensional, and states the minimum number of reactant complexes needed for the coexistence of ACR and nondegenerate multistationarity is n − k + 1, where 1 ≤ k ≤ n − 2 is the number of linearly independent conservation laws (Theorem 5.5).
As for our second category of results, we start with one-dimensional networks, a class of networks for which ACR [24,25], multistationarity [21,29], and even multistability [32] is well studied.Such networks do not allow for the coexistence of ACR and nondegenerate multistationarity (Proposition 3.10).Moreover, one-dimensional bimolecular networks can only be multistationary if they are degenerately so (Lemma 3.20).Moreover, we explicitly characterize all such degenerate networks (Lemma 3. 19).Here our proofs make use of recent results of Lin, Tang, and Zhang [23,32].
Another class of at-most-bimolecular networks we analyze are those with exactly 2 species (Section 4.1).For such networks that are reversible, we characterize the property of unconditional ACR, which means that ACR occurs for all possible values of rate constants (Theorem 4.4).As for networks that need not be reversible, we show that ACR and multistationarity can coexist, but only in a degenerate way.Moreover, up to relabelling species, only two such networks allow such coexistence for a nonzero-measure subset of the space of reaction rate constants (Theorem 4.8).
Our works fits into a growing body of literature that explores the minimal conditions needed for various dynamical behaviors, including the two properties that are the focus of the current work: multistationarity [21,23,29] and ACR [24,25].There are additional such studies on multistability [30] and Hopf bifurcations [5,6,31,32,35] (which generate periodic orbits).For instance, in analogy to Theorem 1.2 above, the presence of Hopf bifurcations requires an at-most-bimolecular network to have at least 3 species, 4 reactions, and dimension 3 [5,35].
This article is organized as follows: Section 2 introduces reaction networks, multistationarity, and ACR.Section 3 contains several results on steady states and their nondegeneracy.We use these results in Sections 4 and 5 to prove our main results.We conclude with a discussion in Section 6.

Background
This section recalls the basic setup and definitions involving reaction networks (Section 2.1), the dynamical systems they generate (Section 2.2), absolute concentration robustness (Section 2.3), and a concept pertaining to networks with only 1 species: "arrow diagrams" (Section 2.4).

Reaction networks.
A reaction network G is a directed graph in which the vertices are nonnegative-integer linear combinations of species X 1 , X 2 , . . ., X n .Each vertex is a complex, and we denote the complex at vertex i by y i = n j=1 y ij X j (where y ij ∈ Z ≥0 ) or y i = (y i1 , y i2 , . . ., y in ).Throughout, we assume that each species X i , where i = 1, 2, . . ., n, appears in at least one complex.
Edges of a network G are reactions, and it is standard to represent a reaction (y i , y j ) by y i → y j .In such a reaction, y i is the reactant complex, and y j is the product complex.A species X k is a catalyst-only species in reaction y i → y j if y ik = y jk .In examples, it is often convenient to write species as A, B, C, . . .(rather than X 1 , X 2 , X 3 , . . . ) and also to view a network as a set of reactions, where the sets of species and complexes are implied.
Example 2.1.The reaction network {0 ← A → 2A, B ← A + B} has 2 species, 5 complexes, and 3 reactions.The species B is a catalyst-only species in the reaction B ← A + B.
A reaction network is reversible if every edge of the graph is bidirected.A reaction network is weakly reversible if every connected component of the graph is strongly connected.Every reversible network is weakly reversible.
One focus of our work is on at-most-bimolecular reaction networks (or, for short, bimolecular), which means that every complex y i satisfies y i1 + y i2 + • • • + y in ≤ 2. Equivalently, each complex has the form 0, X i , X i + X j , or 2X i (where X i and X j are species).The networks in Examples 2.1-2.2 are bimolecular.
2.2.Mass-action systems.Let r denote the number of reactions of G.We write the i-th reaction as y i → y i and assign to it a positive rate constant κ i ∈ R >0 .The mass-action system arising from a network G and a vector of positive rate constants κ = (κ 1 , κ 2 , . . ., κ r ), which we denote by (G, κ), is the following dynamical system arising from mass-action kinetics: (1) where x y i := n j=1 x y ij j .Observe that the right-hand side of the ODEs (1) consists of polynomials f κ,i , for i = 1, . . ., n.For simplicity, we often write f i instead of f κ,i .
The question of which polynomials f i can appear as right-hand side of mass-action ODEs is answered in the following result [16,Theorem 3.2].
Lemma 2.3.Let f : R n → R n be a polynomial function, that is, assume that f i ∈ R[x 1 , x 2 . . ., x n ] for i = 1, 2, . . ., n.Then f arises as the right-hand side of the differential equations (1) (for some choice of network G and vector of positive rate constants κ) if and only if, for all i = 1, 2, . . ., n, every monomial in f i with negative coefficient is divisible by x i .
Next, observe that the mass-action ODEs (1) are in the linear subspace of R n spanned by all reaction vectors y i − y i (for i = 1, 2, . . ., r).We call this the stoichiometric subspace and denote it by S. The dimension of a network is the dimension of its stoichiometric subspace.(This dimension is sometimes called the "rank" [4,30].)In particular, if dim(S) = n (that is, S = R n ), we say that G is full-dimensional.
A trajectory x(t) of (1) with initial condition x(0) = x 0 ∈ R n >0 remains, for all positive time, in the following stoichiometric compatibility class of G [12]: For full-dimensional networks, there is a unique stoichiometric compatibility class: P = R n ≥0 .For networks that are not full-dimensional, every nonzero vector w in S ⊥ yields a (linear) conservation law w, x = w, x(0) that is satisfied by every x ∈ P x(0) , where −, − denotes the usual inner product on R n .
← A + B} has a one-dimensional stoichiometric subspace (spanned by (1, 0)) and generates the following mass-action ODEs (1): Observe that the negative monomials in the first ODE are −κ 1 x 1 and −κ 3 x 1 x 2 , and each of these is divisible by x 1 , which is consistent with Lemma 2.3.Next, the stoichiometric compatibility classes (2) are rays of the following form (where T > 0): The equation x 2 = T is the unique (up to scaling) conservation law.
A steady state of a mass-action system is a non-negative vector x * ∈ R n ≥0 at which the right-hand side of the ODEs (1) vanishes: f κ (x * ) = 0. Our main interest in this work is in positive steady states x * ∈ R n >0 .The set of all positive steady states of a mass-action system can have positive dimension in R n , but this set typically intersects each stoichiometric compatibility class in finitely many points [15].Finally, a steady state x * is nondegenerate if Im(df κ (x * )| S ) = S, where df κ (x * ) is the Jacobian matrix of f κ evaluated at x * .
We consider multiple steady states at two levels: systems and networks.A mass-action system (G, κ) is multistationary (respectively, nondegenerately multistationary) if there exists a stoichiometric compatibility class having more than one positive steady state (respectively, nondegenerate positive steady state).A reaction network G is multistationary if there exists a vector of positive rate constants κ such that (G, κ) is multistationary.For a reaction network G, we let cap pos (G) (respectively, cap nondeg (G)) denote the maximum possible number of positive steady states (respectively, nondegenerate positive steady states) in a stoichiometric compatibility class.

Example 2.1 (continued). We return to the network
← A + B} and its ODEs (3).A direct computation reveals that when κ 1 ≥ κ 2 , there is no positive steady state.On the other hand, when κ 2 > κ 1 , the steady states form exactly one stoichiometric compatibility class (4) -namely, the one given by T = (κ 2 − κ 1 )/κ 3 -and all such steady states are degenerate.Hence, G is multistationary but not nondegenerately multistationary.

Example 2.2 (continued).
The following (full-dimensional) reaction network and indicated rate constants yield a mass-action system with 3 nondegenerate positive steady states [19,Remark 3.6]: Therefore, this network is nondegenerately multistationary.
2.3.Deficiency and absolute concentration robustness.The deficiency of a reaction network G is δ = m − − dim(S), where m is the number of vertices (or complexes), is the number of connected components of G (also called linkage classes), and S is the stoichiometric subspace.The deficiency is always non-negative [12], and it plays a central role in many classical results on the dynamical properties of mass-action systems [1,2,3,11,17,18].
Two such results are stated below.These results, which are due to Feinberg and Horn [13,14,17], are stated for weakly reversible networks (the setting in which we use these results later).
Lemma 2.4 (Deficiency-zero theorem).Deficiency-zero networks are not multistationary.Moreover, if G is a weakly reversible network with deficiency zero, then for every vector of positive rate constants κ, the mass-action system (G, κ) admits a unique positive steady state in every stoichiometric compatibility class.Lemma 2.5 (Deficiency-one theorem).Consider a weakly reversible network G with connected components (linkage classes) G 1 , G 2 , . . ., G .Let δ denote the deficiency of G, and (for all i = 1, 2, . . ., ) let δ i denote the deficiency of G i .Assume the following: (1) δ i ≤ 1 for all i = 1, 2, . . ., , and Then G is not multistationary: for every vector of positive rate constants κ, the mass-action system (G, κ) admits a unique positive steady state in every stoichiometric compatibility class.
Our next topic, ACR, like multistationarity, is analyzed at the level of systems and also networks.Definition 2.6 (ACR).Let X i be a species of a reaction network G with r reactions.
(1) For a fixed vector of positive rate constants κ ∈ R r >0 , the mass-action system (G, κ) has absolute concentration robustness (ACR) in X i if (G, κ) has a positive steady state and in every positive steady state x ∈ R n >0 of the system, the value of x i is the same.This value of x i is the ACR-value of X i .
(2) The reaction network G has unconditional ACR in species X i if, for every vector of positive rate constants κ ∈ R r >0 , the mass-action system (G, κ) has ACR in X i .Remark 2.7 (Existence of positive steady states).ACR requires the existence of a positive steady state (Definition 2.6(1)).This requirement is sometimes not included in definitions of ACR in the literature.However, this is not an extra requirement for some of the networks we consider, namely, weakly reversible networks, for which positive steady states are guaranteed to exist (see Deng et al. [9] and Boros [7]).
Remark 2.8.The property of unconditional ACR is often too restrictive.Thus, many of our results focus on ACR (or other properties) that hold for some full-dimensional subset of the parameter space of rate constants R r >0 (where r is the number of reactions of a given network).The Lesbesgue measure of such a subset is nonzero.For simplicity, we use "measure" to mean Lebesgue measure.
Example 2.1 (continued).We revisit the network {0 From our earlier analysis, the mass-action system has ACR in B when κ 2 > κ 1 (which defines a nonzero-measure subset of the rate-constants space R 3 >0 ), but lacks ACR when κ 2 ≤ κ 1 (as there are no positive steady states).
Example 2.9.Consider the following network G, which is bimolecular and full-dimensional: The mass-action ODEs are as follows: , there are no positive steady states and hence no ACR.Now assume κ 1 κ 2 = κ 3 κ 4 .In this case, the positive steady states are defined by the line x 1 = κ 1 κ 2 , and so the system is multistationary and has ACR in species X 1 .However, all the steady states of this system are degenerate.
Example 2.10.Consider the following network [19,Example 2.6], which we call G: The mass-action ODEs (1) are as follows: It follows that G has unconditional ACR in species A with ACR-value κ 5 /κ 6 (the existence of positive steady states comes from the fact that G is reversible; recall Remark 2.7).
The following result, which is [25,Lemma 5.1], concerns ACR in one-dimensional networks.
Lemma 2.11.Let G be a one-dimensional network with species X 1 , X 2 , . . ., X n .If G has unconditional ACR in some species X i * , then the reactant complexes of G differ only in species X i * (more precisely, if y and y are both reactant complexes of G, then y i = y i for all i ∈ {1, 2, . . ., n} {i * }).

Arrow diagrams.
In this subsection, we recall the arrow diagrams associated to one-species networks.These diagrams are useful for stating results about such networks [21,25,26].
Definition 2.12 (Arrow diagram).Let G be a reaction network with only one species X 1 .Let m denote the number of (distinct) reactant complexes of G, which we list in increasing order of molecularity: ←→} m defined by: It is often useful to consider the arrow diagrams of "embedded" one-species networks, as follows.
Definition 2.14.Let G be a reaction network with species X 1 , X 2 , . . ., X n .Given a species X i , the corresponding embedded one-species network of G is obtained by deleting some (possibly empty) subset of the reactions, replacing each remaining reaction B}, and {0 → A}.

Results on steady states and nondegeneracy
This section contains results on the steady states of mass-action systems.We use these results in later sections to prove our main results.Section 3.1 analyzes the steady states of full-dimensional networks, while Section 3.2 pertains to non-full-dimensional networks.Next, Section 3.3 focuses on bimolecular networks and investigates scenarios in which the right-hand side of a mass-action ODE vanishes.Finally, Section 3.4 concerns bimolecular networks that are reversible.
3.1.Full-dimensional networks.Consider a reaction network G with n species, r reactions, and exactly j reactant complexes1 ; and let κ * ∈ R r >0 be a vector of positive rate constants.We often rewrite the mass-action ODE system (1) for (G, κ * ) as follows: where N is an (n × j)-matrix (with real entries) and m 1 , m 2 , . . ., m j are distinct monic monomials in x 1 , x 2 , . . ., x n given by the reactant complexes.
Proposition 3.1 (Nondegenerate steady states and the matrix N ).Let G be a full-dimensional reaction network with n species, and κ * be a vector of positive rate constants.Let N be a matrix defined, as in (7), by the mass-action ODE system of (G, κ * ).
(2) If rank(N ) = n and G has exactly n + 1 reactant complexes, then the positive steady states of (G, κ * ) are the positive roots of a system of binomial equations (sharing some common monomial m 0 ) of the following form: Proof.Assume (G, κ * ) is a full-dimensional mass-action system in n species, and let N be as in (7).First, we prove (1).Assume rank(N ) ≤ n − 1, and let x * be a positive steady state.It follows that the polynomials f i , as in (1), are linearly dependent (over R).Hence, the Jacobian matrixeven before evaluating at x * -has rank less than n.Thus, the image of the Jacobian matrix, after evaluating at x * , has dimension less than n, i.e, Im(df (x * )| S ) = R n = S. Hence, x * is degenerate.
Next, we prove (2).As in equation (7), we write the mass-action ODEs for (G, κ * ) as where N is n × (n + 1) and the m i 's are distinct monic monomials in x 1 , x 2 , . . ., x n .As G is full-dimensional and rank(N ) = n, we can relabel the m i 's, if needed, so that the square sub-matrix of N formed by the first n columns has rank n.Thus, by row-reducing N , we obtain a matrix of the following form (where β 1 , β 2 , . . ., β n ∈ R): We conclude from the above discussion that the positive steady states of (G, κ * ) are the positive roots of the following n binomial equations (which are in the desired form): Before moving on to part (3), we summarize what we know (so we can use it later).The positive steady states are the roots of the binomials (8), which we rewrite using Laurent monomials (our interest is in positive roots, so there is no issue of dividing by zero): We apply the natural log to (9) and obtain the following, which involves the n×n matrix A := (a ij ): Now we prove (3).Assume x * is a nondegenerate, positive steady state.(We must show that no other positive steady states exist.)By part (1), the n × (n + 1) matrix N has rank n, so the proof of part (2) above applies.Assume for contradiction that x * * is a positive steady state, with x * * = x * .Then, by (10), the linear system Ay = ln(β) has more than one solution, and so rank(A) ≤ n − 1.It follows that the set of positive steady states, {(e y 1 , e y 2 , . . ., e yn ) | Ay = ln(β)}, is positivedimensional and so (by the Inverse Function Theorem and the fact that G is full-dimensional) all positive steady states of (G, κ * ) are degenerate.This is a contradiction, as x * is nondegenerate.Remark 3.2.For algebraically inclined readers, observe that the equations in Proposition 3.1(2) define a toric variety.Additionally, every such variety has at most one irreducible component that intersects the positive orthant [8,Proposition 5.2].This fact can be used to give a more direct proof of Proposition 3.1(3).
Remark 3.3.The end of the proof of Proposition 3.1 concerns nondegenerate positive steady states and their relation to the dimension of the set of positive steady states.More ideas in this direction are explored in the recent work of Feliu, Henriksson, and Pascual-Escudero [15].Corollary 3.4 (When f i is zero).Let G be a full-dimensional reaction network with n species, let κ * be a vector of positive rate constants, and let f 1 , f 2 , . . ., f n denote the right-hand sides of the mass-action ODEs of (G, κ * ).If f i is the zero polynomial, for some i ∈ {1, . . ., n}, then every positive steady state of (G, κ * ) is degenerate.
Proof.This result follows directly from Proposition 3.1(1) and the fact that, in this case, the rank of N , as in (7), is strictly less than n.
The next two results pertain to networks with few reactant complexes (at most n, where n is the number of species) and many reactant complexes (at least n), respectively.Proposition 3.5 (Networks with few reactants).Let G be a reaction network with n species.
(1) If G has exactly 1 reactant complex, then, for every vector of positive rate constants κ * , the mass-action system (G, κ * ) has no positive steady states.(2) If G has exactly j reactant complexes, where 2 ≤ j ≤ n (in particular, n ≥ 2), and G is full-dimensional, then every positive steady state (of every mass-action system defined by G) is degenerate.
Proof.Assume G has n species, which we denote by X 1 , X 2 , . . ., X n , with exactly j reactant complexes, for some 1 ≤ j ≤ n.Let κ * be a vector of positive rate constants.As in (7), we write the mass-action ODE system arising from (G, κ * ) as follows: where N := (N ij ) is an (n × j)-matrix (with entries in R) and m 1 , . . ., m j are distinct monic monomials in x 1 , . . ., x n (as G has n species and j reactant complexes).
We first prove part (1).In this case, the right-hand sides of the ODEs have the form k , with at least one c i = 0.It follows that there are no positive steady states.We prove part (2).Assume that G is full-dimensional (the stoichiometric subspace is R n ) and that 2 ≤ j ≤ n.Let x * = (x * 1 , x * 2 , . . ., x * n ) be a positive steady state.We must show x * is degenerate.We first consider the subcase when the rank of the matrix N is at most (n − 1).By Proposition 3.1(1), every positive steady state is degenerate.Now we handle the remaining subcase, when N has rank n (and hence, N is n × n).Now, solving the steady-state equations f 1 = • • • = f n = 0 can be accomplished by multiplying the expression in (11) by N −1 , which implies that every monomial m 1 , . . ., m n evaluates to zero at steady state.Hence, no positive steady states exist.Proposition 3.6 (Networks with many reactants).If G is a full-dimensional network with n species and exactly j reactant complexes, where j ≥ n, then: (1) There exists a vector of positive rate constants κ * , such that the corresponding matrix N , as in (7), has rank n. (2) If there exists a vector of positive rate constants κ * such that the matrix N does not have rank n, then there exists a vector of positive rate constants κ * * such that (G, κ * * ) has no positive steady states.
Proof.Assume G is full-dimensional, with n species, r reactions (denoted by y 1 → y 1 , . . .y r → y r ), and exactly j reactant complexes, where j ≥ n.
We begin with part (1).Let κ = (κ 1 , . . ., κ r ) denote the vector of unknown rate constants (each κ i is a variable).Let N be the (n × j) matrix for (G, κ) in the sense of N in (7).More precisely, the entries of N are Z-linear combinations of the κ i 's, such that, for every vector of positive rate constants κ * ∈ R r >0 , the evaluation N | κ=κ * is the matrix N as in ( 7) for (G, κ * ).As G is full-dimensional, there are no R-linear relations among the n rows of N .Hence, the size-n minors of N define a (possibly empty) measure-zero subset V ⊆ R r >0 .Thus, R r >0 V is nonempty, and every κ * ∈ R r >0 V yields a matrix N = N | κ=κ * with rank n.This proves part (1).For part (2), suppose that there exists κ * ∈ R r >0 such that the resulting matrix N has rank strictly less than n.It follows that there is a linear relation: where c 1 , . . ., c n are real numbers -not all 0 -and the f κ * ,i denote the right-hand sides of the mass-action ODEs for (G, κ * ).
On the other hand, for unknown rate constants κ, as in the proof above for part (1), n is not the zero polynomial.Thus, when we rewrite this expression as a sum over r reactions y i → y i as follows: , where d i ∈ Z for all i, we conclude that d i = 0 for some i.By relabeling reactions, if needed, we may assume that i = 1.Now consider the following vector of positive rate constants κ * := (κ * 1 + , κ * 2 , . . ., κ * r ), for some > 0. Assume for contradiction that (G, κ * ) has a positive steady state x * .At steady state, f κ * ,i evaluates to 0, for all i, and this yields the first equality here: and the second and third equalities come from the fact that the mass-action ODEs are linear in the rate constants and from equation (12), respectively.We obtain x y 1 | x=x * = 0, which contradicts the fact that x * is a positive steady state.This concludes the proof.
The next proposition returns to a topic from Proposition 3.1, namely, networks with n species and n + 1 reactant complexes.Proposition 3.7 (Networks with n + 1 reactants).Assume G is a full-dimensional network, with n species and exactly n + 1 reactant complexes, which we denote as follows: Let A denote the n × n matrix obtained from the (n + 1) × n matrix Y := (y ij ) by subtracting the last row from every row and then deleting the last row.
(2) If rank(A) ≤ n − 1, then there exists a vector of positive rate constants κ * such that (G, κ * ) has no positive steady states.
Subcase: rank(N ) = n.Part (2) of Proposition 3.1 pertains to this setting, so we can follow that proof.In particular, equation ( 9) -the (n × n) matrix A there exactly matches the matrix A here -implies that the positive steady states are defined by a linear system of the form Ay = ln(β), where y = (ln(x 1 ), . . ., ln(x n )) .Hence, as rank(A) = n, we have at most one positive steady state and so (G, κ * ) is not multistationary.
Case 2: rank(A) ≤ n − 1.We must show that there exists a choice of rate constants so that the resulting system has no positive steady states.Proposition 3.6(1) implies that there exists κ * such that the following holds: the matrix N defined by (G, κ * ) has (full) rank n. (13) Fix such a choice of κ * .If (G, κ * ) has no positive steady states, then we are done.Therefore, for the rest of the proof, we assume that (G, κ * ) admits a positive steady state.
In what follows, we need to consider additional vectors of positive rate constants (besides κ * ) and their corresponding matrices N , as in (7).Therefore, as in the proof of Proposition 3.6(1), let κ = (κ 1 , . . ., κ r ) (where r is the number of reactions) denote the vector of unknown rate constants, and let N be the n × (n + 1) matrix for (G, κ) in the sense of N in (7), so that for every vector of positive rate constants κ * ∈ R r >0 , the evaluation N | κ=κ * is the matrix N as in (7).We now follow the ideas in the proof of Proposition 3.1, part (2), with the difference being that we now consider unknown rate constants κ.The mass-action ODEs for (G, κ) are given by: where m 1 , . . ., m n+1 are distinct monic monomials in x 1 , x 2 , . . ., x n .Our next aim is to row-reduce N (over the field Q(κ 1 , . . ., κ r )).Accordingly, for 1 We claim that, for all 1 ≤ k ≤ n + 1, the polynomial [B k ] is nonzero.By symmetry among the monomials m i , it suffices to show that [B n+1 ] is nonzero.To show this, assume for contradiction that [B n+1 ] = 0.Then, N can be row-reduced to a matrix in which the last row has the form (0, 0, . . ., 0, ω), where 0 = ω ∈ Q(κ 1 , . . ., κ r ).Now consider the evaluation at κ = κ * .By (13), the matrix N = N | κ=κ * has (full) rank n, so ω| κ=κ * is nonzero.However, this implies that positive steady states of (G, κ * ) satisfy ω| κ=κ * m n+1 = 0, much like in (8).Thus, (G, κ * ) has no positive steady states, which is a contradiction, and hence our claim holds.
Next, as [B n+1 ] is nonzero, we can apply a version of Cramer's rule to row-reduce N to the following matrix (where I n denotes the size-n identity matrix): Thus, as in ( 8), the positive steady states are the positive roots of the equations m i − β i m n+1 = 0 (for i = 1, 2, . . ., n), where: Thus, β i | κ=κ * > 0 (for all i = 1, 2, . . ., n), since (G, κ * ) admits a positive steady state.We conclude from this fact, plus the claim proven earlier (namely, that [B ] = 0 for all ), that the following is For the rest of the proof, we restrict our attention to rate constants, like κ * , that are in Σ.For such rate constants, like in (8-10), the positive steady states are the roots of the following equation Next, as rank(A) ≤ n − 1, there exists a nonzero vector γ ∈ R n in the orthogonal complement of the column space of A. By relabeling the m i 's (which permutes the columns of N ), if needed, we may assume that γ 1 = 0.By construction of γ and equation ( 14), we have γ, ln(β) = 0, which is readily rewritten as follows: For ε > 0, let κ * denote the vector of rate constants obtained from κ * by scaling by (1 + ε) all rate constants of reactions in which the reactant generates the monomial m 1 .As Σ is an open set, κ * ∈ Σ for ε sufficiently small.Also, by construction, the matrix Thus, by replacing κ * by κ * ε , the left-hand side of equation ( 15) is scaled by (1 + ε) −γ 1 , and so there exists ε > 0 for which equation (15) does not hold (when evaluated at κ = κ * ε ).Hence, this vector κ * ε yields a mass-action system (G, κ * ε ) with no positive steady states, as desired.Proposition 3.6 implies that for networks with at least n reactant complexes (where n is the number of species), some choice of rate constants yields a matrix N with (full) rank n.Our next result shows that when this condition holds (even for networks with fewer reactants), every species appears in at least one reactant complex.
We introduce the following shorthand (which we use in several of the next results): a complex Lemma 3.8 (Reactants involve all species).Let G be a full-dimensional reaction network with n species, let κ * be a vector of positive rate constants, and let N be the matrix for (G, κ * ), as in (7).If rank(N ) = n and (G, κ * ) has a positive steady state, then for every species X i , at least one reactant complex of G involves X i .
Proof.We prove the contrapositive.Assume that there is a species X i such that for every reactant complex a 1 X 1 + a 2 X 2 + • • • + a n X n we have a i = 0.Then, by Lemma 2.3, the right-hand side of the mass-action ODE for X i , which we denote by f i , is a sum of monomials, all of which have positive coefficients.But (G, κ * ) has a positive steady state, so f i must be 0. We conclude that the i-th row (of the n rows) of N is the zero row and so rank(N ) ≤ n − 1.

3.2.
Networks with conservation laws.The following result is similar to several results in the prior subsection, but pertains to networks that are not full-dimensional.Proposition 3.9 (Networks with conservation laws and few reactants).Let G be a reaction network with n ≥ 3 species.Assume that G is (n − k)-dimensional, where k ≥ 1 (so, G has k conservation laws).If G has exactly j reactant complexes, for some j ∈ {2, 3, . . ., n − k}, then every positive steady state (of every mass-action system defined by G) is degenerate.
We consider two cases.First assume that rank(N ) ≤ n − k − 1.Then the polynomials f i span a subspace of dimension ≤ n − k − 1 and hence the Jacobian matrix -even before evaluating at a positive steady state -has rank ≤ n − k − 1.Every positive steady state is therefore degenerate.
Consider the remaining case: rank(N ) = n − k (so, j = n − k).In this case, multiplication by N defines an injective map R n−k → R n .Hence, by ( 16), the steady-state equations Thus, there are no positive steady states.
The next result concerns networks with n−1 conservation laws, that is, one-dimensional networks.Proposition 3.10 (One-dimensional networks).Let G be a one-dimensional reaction network, and let κ * be a vector of positive rate constants.If (G, κ * ) has ACR, then (G, κ * ) is not nondegenerately multistationary.
Proof.Assume that G is one-dimensional, with n species.Thus, G has n − 1 linearly independent conservation laws.Let κ * be a vector of positive rate constants for which there is ACR.We may assume that the ACR species is X 1 (by relabeling species, if needed).Let f 1 , . . ., f n denote the right-hand sides of the mass-action ODEs arising from (G, κ * ).
Let x * = (x * 1 , . . ., x * n ) denote an arbitrary positive steady state of (G, κ * ).(The ACR-value is x * 1 .)Let P x * denote the (one-dimensional) stoichiometric compatibility class that contains x * .It suffices to show that (1) x * is the unique positive steady state in P x * or (2) x * is degenerate.
We consider two cases.Case (a): X 1 is not a catalyst-only species (in some reaction of G).This implies that f 2 , . . ., f n are all scalar multiples of f 1 , and that the compatibility class P x * is defined by n − 1 conservation laws of the form x j = a j x 1 + b j , where a j , b j ∈ R, for j ∈ {2, 3, . . ., n}.By substituting these n − 1 relations into f 1 , we obtain a univariate polynomial in x 1 , which we denote by h.If h has multiple positive roots, then there is no ACR, which is a contradiction.If, on the other hand, h does not have multiple positive roots, then P x * does not contain multiple positive steady states (that is, x * is the unique positive steady state in P x * ).
Case (b): X 1 is a catalyst-only species in all reactions of G.In this case, f 1 = 0, and x 1 = x * 1 is a conservation law of G, and it is one of the defining equations of the compatibility class P x * .By relabeling species X 2 , . . ., X n , if needed, we may assume that X 2 is not a catalyst-only species (as G is one-dimensional).Thus, we can "extend" the conservation law x 1 = T to a "basis" of n − 1 conservation laws that define the compatibility class P x * , by appending n − 2 conservation laws of the form x j = a j x 2 + b j , where a j , b j ∈ R, for j ∈ {3, 4, . . ., n}.
Next, we substitute these n − 2 conservation relations into f 2 , which yields a polynomial in x 1 and x 2 , which we denote by g.Consider the following set, which is the positive variety of g in R 2

>0
(the values of x 3 , . . ., x n are free, so we ignore them): By construction and the fact that there is ACR in X 1 , the set Σ is contained in the hyperplane (line) 1 , and so is either one-dimensional or zero-dimensional.We consider these two subcases separately.First, assume that Σ is one-dimensional.In this subcase, Σ equals the subset of the hyperplane x 1 = x * 1 in the positive quadrant R 2 >0 , and so the compatibility class P x * consists entirely of positive steady states.The Inverse Function Theorem now implies that every positive steady state of P x * (in particular, x * ) is degenerate.
Consider the remaining subcase, in which Σ is zero-dimensional (that is, Σ consists of finitely many points).It follows that g is either non-negative on R 2 >0 or non-positive on R 2 >0 , and so f 2 is either non-negative on P x * or non-positive on P x * .Consequently, as every f i is a scalar multiple of f 2 , the steady state x * is degenerate.
3.3.Bimolecular networks.We begin this subsection with a result that clarifies how the polynomials arising in mass-action ODEs are constrained when the network is bimolecular.Lemma 3.11 (Bimolecular networks).Consider a bimolecular mass-action system (G, κ * ) with n species.Let f i be the right-hand side of the mass-action ODE for species X i (for some Fix positive values a j > 0 for all j ∈ {1, 2, . . ., n} {i}.Let g i denote the univariate polynomial obtained by evaluating f i at x j = a j for all j ∈ {1, 2, . . ., n} {i}.If the polynomial g i is nonzero, then g i has at most one sign change and hence has at most one positive root.
Proof.Let g i denote the nonzero polynomial obtained by evaluating f i at x j = a j for all j = i.Several properties of g i arise from the fact that G is bimolecular: i is non-positive, and (3) the constant coefficient is non-negative.Thus, g i has at most one sign change, and so Descartes' rule of signs implies that g i has at most one positive root.
The next two results pertain to bimolecular mass-action systems in which the right-hand side of some ODE vanishes (Propositions 3.14) or vanishes when evaluated at an ACR-value (Proposition 3.15).We motivate these results through the following example.
Example 3.12 (Enlarged Shinar-Feinberg network).A common way to construct a network with an ACR species (e.g., A) is through the existence of an f i that becomes zero when we substitute the ACR-value in place of the species.We illustrate this idea through the following network: This network is constructed from a well-studied network first introduced by Shinar and Feinberg [28] by adding three reactions involving a new species (C).We examine the mass-action ODE for B: where g := x 2 (κ 1 x 1 − κ 2 ) (which is the right-hand side of the ODE for X 2 in the original Shinar-Feinberg network) and h := x 2 x 3 (−κ 3 + κ 4 ) (arising from the additional reactions, involving X 3 ).Assume κ 3 = κ 4 .It is easy to check that (G, κ) has a positive steady state and also has ACR in species X 1 with ACR-value α = κ 2 /κ 1 .Also, observe that f 2 | x 1 =α = 0, as a result of the equalities g| x 1 =α = 0 and h = 0 (which is due to the equality κ 3 = κ 4 ).
The next two results characterize which reactions can exist in such a situation.More precisely: • Proposition 3.14 gives conditions that hold when a mass-action ODE is zero (effectively characterizing what reactions can yield h = 0 in this case).• Proposition 3.15 gives conditions that hold when a mass-action ODE is zero when evaluated at the ACR-value (effectively characterizing what reactions can yield f 2 | x 1 =α = 0 in this case, involving a decomposition like the one we observed above: The next result uses the following notation: Notation 3.13 (Empty complex).We introduce the dummy variable X 0 := 0, so that (for instance) X 0 is the empty complex and X i + X 0 := X i for any species X i .
The following result clarifies which reactions can exist if some mass-action ODE is zero.
Proposition 3.14 (When f i is zero).Let G be a bimolecular reaction network with n species X 1 , X 2 , . . ., X n .Fix 1 ≤ i ≤ n.Let κ * denote a vector of positive rate constants for G, and let f i denote the right-hand side of the mass-action ODE for species X i in the system (G, κ * ).If f i is the zero polynomial, then the set of reactions of G in which X i is a non-catalyst-only species is a (possibly empty) subset of the reactions listed here (where our use of X 0 follows Notation 3.13): (1) the reactions of the form X i + X j → 2X i (and we denote the rate constant by κ * 1,j ), where j ∈ {0, 1, . . .n} {i}, (2) the reactions of the form X i + X j → (with rate constant κ * 2,j, , where is an index for such reactions), where j ∈ {0, 1, . . .n} {i} and is any complex that does not involve X i , and, additionally, the following relationships among the rate constants hold: where a rate constant is set to 0 if the corresponding reaction is not in G.
Proof.Let κ * be a vector of positive rate constants for a bimolecular network G with n species, and let f i be the right-hand side of (G, κ * ) for the species X i .Let Σ denote the set of reactions of G in which X i is a non-catalyst-only species.Reactions not in Σ do not contribute to f i , so we ignore them for the rest of the proof.
We claim that for all reactions in Σ, the reactant complex is not one of the following 5 types: 0, X j , X j + X j , 2X j , 2X i for any j, j ∈ {1, 2, . . ., n} {i}.Indeed, any of the first 4 types of complexes would yield a constant term in f i (when viewed as a polynomial in x i ) consisting of a sum of monomials with positive coefficients; similarly, the last type (2X i ) would yield a negative x 2 i term (the fact that G is bimolecular is used here).However, f i is zero, so the claim holds.It follows that, for every reaction in Σ, the reactant complex either is X i or has the form X i + X j for some j ∈ {1, 2, . . ., n} {i}.It is straightforward to check that all possible such reactions (in which X i is a non-catalyst-only species) are listed in the proposition.Next, reactions of type (1) in the proposition contribute positively to f i , while those of type (2) contribute negatively, as follows: As f i = 0, the coefficient of x i and the coefficient of each x ij in (19) must be 0, which yields the desired equalities (18).Proposition 3.14 concerns general (bimolecular) mass-action systems, and now we consider those with ACR.The next result characterizes which reactions can exist if some mass-action ODE becomes zero when evaluated at the ACR-value.Proposition 3.15 (When f i is zero at the ACR-value).Let G be a bimolecular reaction network with species X 1 , X 2 , . . ., X n , where n ≥ 2. Let κ * denote a vector of positive rate constants.Assume that the mass-action system (G, κ * ) has ACR in species X 1 with ACR-value α > 0. Fix 2 ≤ i ≤ n.Let f i denote the right-hand side of the mass-action ODE for species X i in the system (G, κ * ).If f i = 0 and f i | x 1 =α is the zero polynomial, then the set of reactions of G in which X i is a non-catalyst-only species is a nonempty subset of the following reactions (the same as the ones in Proposition 3.14): (1) the reactions of the form X i + X j → 2X i (and we denote the rate constant by κ * 1,j ), where j ∈ {0, 1, . . .n} {i}, (2) the reactions of the form X i + X j → (with rate constant κ * 2,j, , where is an index for such reactions), where j ∈ {0, 1, . . .n} {i} and is any complex that does not involve X i .Additionally, the following relationship between the ACR-value α and the rate constants holds: In particular, the numerator and denominator of (20) are nonzero.Finally, if n ≥ 3, then the following relationships among the rate constants hold: for all j ∈ {2, 3, . . .n} {i} .
(In equations ( 20)-( 21), a rate constant is set to 0 if the corresponding reaction is not in G.) Using properties of polynomial rings over a field, it follows that (x 1 − α) divides f i .From the fact that G is bimolecular, we conclude that: for some real numbers β, γ, δ j , at least one of which is nonzero.
Thus, using (22), we have f i = βx 1 x i − αβx i , for some β ∈ R {0}.Next, we investigate which reactions contribute to the two monomials in f i .For βx 1 x i , the contributing reactions have the form X 1 + X i → 2X i and X 1 + X i → , where does not involve X i .The first reaction contributes positively, while the second type contributes negatively.Let κ * 1,1 be the reaction rate constant for X 1 + X i → 2X i (as in the statement of the lemma) and κ * 2,1, be the rate constant for reactions of type X 1 + X i → , where is an index for all the reactions of this type.We conclude that Similarly, the monomial −αβx i in f i comes from reactions of the form X i → 2X i , which contributes positively, and X i → , which contributes negatively, where is a complex that does not involve X i .Hence, Now the equations ( 23) and (24) together imply the desired equality (20).
Next, let Σ denote the set of reactions of G in which X i is a non-catalyst-only species.We showed above that Σ contains a (nonempty) subset of reactions with rate constants labeled by κ * 1,0 , κ * 1,1 , κ * 2,0, , κ * 2,1, .Let Σ ⊆ Σ denote the remaining reactions, and let G denote the subnetwork defined by the reactions in Σ .Let κ be obtained from κ * by restricting to coordinates corresponding to reactions in Σ .By construction, the mass-action ODE of (G , κ ) for species X i has right-hand side equal to 0. So, Proposition 3.14 applies (where reactions arising from j = 0, 1 in that proposition are absent from G by construction), and yields two conclusions.First, Σ is a subset of the reactions listed in Proposition 3.15 (specifically, with j = 0, 1), and so Σ is a subset of the full list (including j = 0, 1).Second, the equations ( 18) hold (for j = 0, 1), which are the desired equalities (21).
Remark 3.16.The reactions listed in Propositions 3.14 and 3.15 (the lists are the same) are not reversible.Hence, if G is a reversible network satisfying the hypotheses of either proposition, then X i is a catalyst-only species in every reaction of G.
Example 3.12 (continued).We revisit the enlarged Shinar-Feinberg network, Recall that, when κ 3 = κ 4 , the mass-action system (G, κ) has ACR in X 1 with ACR-value α = κ 2 /κ 1 , and that f 2 | x 1 =α = 0.In the notation of Proposition 3.15, the rate constants of reactions in which X 2 is non-catalyst-only are: Remark 3.17.The formula for the ACR-value, in (20), is related to the concept of "robust ratio" introduced by Johnston and Tonello [33].
3.4.Three reversible reactions are necessary for multistationarity.Recall that, in (5), we saw an instance of a (nondegenerately) multistationary, bimolecular network that consists of 3 pairs of reversible reactions.In this subsection, we prove that bimolecular networks with fewer pairs of reversible reactions are non-multistationary (Theorem 3.21).Our proof of Theorem 3.21 requires several supporting lemmas on one-dimensional networks.
For the next lemma, recall from Section 2.2 that cap pos (G) (respectively, cap nondeg (G)) denotes the maximum possible number of positive (respectively, nondegenerate and positive) steady states of a network G.In Lemma 3.18 below, part (1) was conjectured by Joshi and Shiu [21] and then proved by Lin, Tang, and Zhang [23, Theorem 4.3] (see also [27]).Part Joshi and Shiu showed that the network G = {0 ← A → 2A} is the only one-species, bimolecular network for which cap pos (G) = ∞ [20].The following lemma generalizes this result from one-species networks to one-dimensional networks.Lemma 3.19 (One-dimensional bimolecular networks with infinitely many steady states).Let G be a one-dimensional and bimolecular reaction network with n species.The following are equivalent: (1) cap pos (G) = ∞.
(2) Up to relabeling species, G is one of the following networks: where Σ consists of at least one reaction from the following set: Additionally, for the networks listed above in 2(a) and 2(b), every positive steady state (of every mass-action system arising from the network G) is degenerate.
Having shown that the case of ( 26) is consistent with Lemma 3.19, we now consider the remaining two cases, from (25), separately.First, assume the stoichiometric subspace of G is spanned by (1, 0, 0, . . ., 0).It follows that the reactions of G form a subset of the following 2n + 4 reactions: The ODEs for species X 2 , X 3 , . . ., X n are dx i dt = 0 so, x i = T i (with T i > 0) for i = 2, 3, . . ., n are the corresponding conservation laws.We substitute these conservation laws into the ODE for X 1 : When at least one k i is positive and all other k j 's are non-negative, the right-hand side of ( 27), viewed as a polynomial in x 1 , has a nonzero constant term and, hence, is not the zero polynomial.Similarly, if 0 or 1 is positive and 0 , 1 ≥ 0, then the right-hand side of ( 27) has a nonzero coefficient of x 2 1 and is again a nonzero polynomial.We conclude that if G contains at least one of the reactions labeled by k i or i , then cap pos (G) < ∞, which is consistent with Lemma 3.19.
We now consider the case when G contains no reactions labeled by k i or i , that is, every reaction of G is one of the following n + 1 reactions: The right-hand side of the ODE for X 1 , as in (27) In order for this polynomial in x 1 to become the zero polynomial for some choice of positive rate constants of G (equivalently, cap pos (G) = ∞), we must have m 1 > 0 and m j > 0 for at least one of j = 0, 2, 3, . . ., n.This gives exactly the reactions listed in Lemma 3.19(2)(b).In this case, given m j > 0 for the reactions appearing in the network, we can always choose T j > 0, such that the right-hand side of the ODE for X 1 vanishes (i.e., cap pos (G) = ∞).Moreover, when this right-hand side vanishes is the only situation in which there are positive steady states, and an easy calculation shows that all such positive steady states are degenerate.This concludes our analysis of networks with stoichiometric subspace spanned by the vector (1, 0, 0, . . ., 0).Our final case is when the stoichiometric subspace is spanned by the vector (1, −1, 0, . . ., 0).In this case, the reactions of G form a subset of the following 2n + 4 reactions: The conservation laws are x 1 + x 2 = T 2 and x i = T i for i = 3, 4, . . ., n.The ODE for species X 1 is: Consider the subcase when at least one of the i is positive and all other j 's are non-negative.After substituting the expressions arising from the conservation laws (namely, x 2 = T 2 − x 1 and x i = T i for i = 3, 4, . . ., n) into the right-hand side of the ODE (28), we obtain a polynomial in x 1 that has a positive constant term (see the second line of the right-hand side of ( 28)).Hence, if G contains at least one of the reactions labeled by i , then cap pos (G) < ∞.
By symmetry, if G has at least one of the reactions labeled by k i , then again cap pos (G) < ∞.Hence, if G contains a reaction labeled by i or k i , then this subcase is consistent with the lemma.
Consider the remaining subcase, when G is a subnetwork of {2X 1 , and so consists of only one or two reactions.If G has only one reaction, then Proposition 3.5 implies that cap pos (G) = 0 < ∞ (which is consistent with the lemma).Now assume that G has two reactions, that is, x 1 x 2 and so there are no positive steady states.When m 1 = m 2 , the ODE for X 1 becomes dx 1 dt = 0 and it follows that cap pos (G) = ∞.Moreover, a simple computation shows that all the positive steady states are degenerate.This concludes the proof.

Lemma 3.20 (One-dimensional bimolecular networks).
If G is a one-dimensional, bimolecular network, then G is not nondegenerately multistationary.
Proof.Assume that G is a one-dimensional network that is nondegenerately multistationary.We must show that G is not bimolecular.We claim that cap pos (G) is finite.Indeed, if cap pos (G) = ∞, then Lemma 3.19 implies that all positive steady states are degenerate and so G is not nondegenerately multistationary, which is a contradiction.Hence, cap pos (G) < ∞.
The hypotheses of part (1) of Lemma 3.18 are satisfied, that is, cap pos (G) < ∞, and G is onedimensional and multistationary.Therefore, G has an embedded one-species network with arrow diagram (←, →).Such an embedded network (e.g., {0 ← A, 2A → 3A}) involves at least one complex that is not bimolecular, and so G is also not bimolecular.
Theorem 3.21 (Networks with up to two reversible reactions).If G is a bimolecular reaction network that consists of one or two pairs of reversible reactions, then G is not multistationary.
Proof.Assume that G is bimolecular and consists of one or two pairs of reversible reactions.Let p denote the number of pairs of reversible reactions (so, p = 1 or p = 2), and the number of linkage classes.Let s be the dimension of the stoichiometric subspace (so, s = 1 or s = 2).

Main results on bimolecular networks
In this section, we establish minimal conditions for a bimolecular network to admit ACR and nondegenerate multistationarity simultaneously.These minimal conditions are in terms of the numbers of species, reactions, and reactant complexes.The main result is as follows.
Theorem 4.1 (Conditions for coexistence of ACR and nondegenerate multistationarity).Let G be a bimolecular reaction network.If there exists a vector of positive rate constants κ * such that the mass-action system (G, κ * ) has ACR and also is nondegenerately multistationary, then: (1) G has at least 3 species.
(2) G has at least 3 reactant complexes (and hence at least 3 reactions) and at least 5 complexes (reactant and product complexes).( 3) If G is full-dimensional, then G has at least 5 reactant complexes (and hence at least 5 reactions).
This section is structured as follows.In Subsection 4.1, we prove part (1) of Theorem 4.1 (specifically, part (1) follows from Proposition 4.3 and Theorem 4.8).Theorem 4.8 also analyzes two-species bimolecular networks with ACR and degenerate multistationarity.Additionally, we characterize unconditional ACR in two-species bimolecular networks that are reversible (Theorem 4.4).
Subsequently, in Subsection 4.2, we prove parts (2) and (3) of Theorem 4.1 (Theorem 4.9 and Proposition 4.12).We also consider full-dimensional, 3-species, bimolecular networks with only 4 reactant complexes.By Theorem 4.1, such networks do not allow for the coexistence of ACR and nondegenerate multistationary.Nevertheless, ACR and degenerate multistationarity is possible, and we characterize the possible sets of reactant complexes of such networks (Proposition 4.15).4.1.Bimolecular networks with one or two species.This subsection characterizes unconditional ACR in reversible networks with only one or two species (Proposition 4.3 and Theorem 4.4).Notably, our results show that such networks with unconditional ACR are not multistationary.Remark 4.2 (Reversible networks).Our interest in reversible networks comes from our prior work with Joshi [19].In that article, our results on multistationarity in randomly generated reaction networks arise from "lifting" this property from the following (multistationary) motif: The question arises, Are there multistationary motifs with fewer species, reactions, or complexes than the one in (29)?Discovering more motifs might aid in analyzing the prevalence of multistationarity in random reaction networks generated by stochastic models besides the one in [19].4.1.1.Networks with one species.When there is only one species, say X 1 , and the network is bimolecular, there are only 3 possible complexes: 0, X 1 , 2X 1 .Hence, every such network is a subnetwork of the following network: Therefore, the possible reversible networks, besides G X 1 itself, are listed here: Proposition 4.5 fails for networks that are not reversible.Indeed, a network without positive steady states (such as {0 → A, 0 → B}) is not multistationary and also lacks unconditional ACR.
We end this subsection by considering two-species networks that are one-dimensional.Up to relabeling species, each such network is a subnetwork of exactly one of the following networks G i : The next result, which is part (2) of Theorem 4.4, states that among the reversible subnetworks of the networks G i listed in (32), only one has unconditional ACR (namely, {X 2 X 1 + X 2 }).
Proposition 4.6.Let G be a one-dimensional, reversible, bimolecular reaction network with exactly two species.Then the following are equivalent: Proof.Let G be a two-species, one-dimensional, reversible, bimolecular reaction network.From the list (32), we know that G is a subnetwork of one of . So, it suffices to show G does not have unconditional ACR.In networks G 1 , G 2 , and G 4 , the reactant and product complexes of every reaction differ in both species X 1 and X 2 .Also, all reactions in G are reversible, so every complex of G is a reactant complex.We conclude that G has two reactant complexes that differ in both species, and hence, Lemma 2.11 implies that G does not have unconditional ACR.
We now consider the remaining case, when G is a subnetwork of G 3 .We write , the mass-action ODEs are dx 1 /dt = κ 1 x 2 − κ 2 x 1 x 2 and dx 2 /dt = 0, and so G has unconditional ACR in species X 1 with ACR-value κ 1 κ 2 .If G is a subnetwork of N 1 , then G has only one species (recall that every species of a network must take part in at least one reaction), which is a contradiction.
Our final subcase is when G contains reactions from both N 1 and N 2 .Then, from N 2 , the complex X 2 is a reactant complex of G. Similarly, from N 1 , at least one of X 1 and 2X 1 is a reactant complex of G. Hence, G contains two reactant complexes that differ in both species, X 1 and X 2 .Therefore, Lemma 2.11 implies that G does not have unconditional ACR.
Finally, the fact that the network {X 2 X 1 + X 2 } is non-multistationary follows easily from the deficiency-zero theorem (Lemma 2.4).4.1.3.Irreversible networks with two species.In [25], the following network was called a "degenerate-ACR network," because it has unconditional ACR and yet every positive steady state is degenerate: This degeneracy arises from the fact that a single (one-dimensional) stoichiometric compatibility class consists entirely of steady states [25,Example 2.12].The main result of this subsection, Theorem 4.8 below, shows that only one additional two-species network exhibits both ACR and multistationarity for a nonzero-measure set of rate constants; this network is obtained by adding to (33) the reaction A → 0. Both networks, therefore, are one-dimensional, two-species networks.
To prove Theorem 4.8, we need the following lemma, which concerns the network in (33) (and others as well).
(2) Let Σ denote the set of vectors of positive rate constants κ for which the mass-action system (G, κ) both has ACR and is multistationary.If Σ has nonzero measure, then G is one of the following networks: Proof.This result is straightforward to check by hand, so we only outline the steps, as follows.
Assume G is a subnetwork of If G admits a positive steady state, G must contain the reaction X 1 m → 2X 1 .Hence, there are three subnetworks to consider: In cases ( 2) and ( 3), the set Σ has nonzero measure.Finally, for all three of these networks, every positive steady state is degenerate (some of these networks are also covered by Lemma 3.19).
Theorem 4.8.Let G be a bimolecular reaction network with exactly two species, X 1 and X 2 .Let Σ denote the set of vectors of positive rate constants κ for which the mass-action system (G, κ) both has ACR in species X 2 and is multistationary.Then: (1) For every κ * ∈ Σ, every positive steady state of (G, κ * ) is degenerate.
(2) If Σ has nonzero measure, then G is one of the following networks: Proof.Assume that G is bimolecular and has exactly two species.If Σ is empty (for instance, if G has no reactions), then there is nothing to prove.Accordingly, assume that Σ is nonempty (and in particular G has at least one reaction).
We first claim that G has a reaction in which X 1 is a non-catalyst-only species.To prove this claim, assume for contradiction that X 1 is a catalyst-only species.Then the stoichiometric compatibility classes are defined by the equations x 1 = T , for T > 0 (we are also using the fact that G has at least one reaction).But this does not allow for multistationarity and ACR in X 2 to coexist, because two positive steady states in the same compatibility class would have the form (T, y) and (T, z), with y = z, which contradicts the assumption of ACR in X 2 .So, the claim holds.
For an arbitrary vector κ of positive rate constants, let f κ,1 and f κ,2 denote the right-hand sides (for species X 1 and X 2 , respectively) of the mass-action ODE system of (G, κ).Consider the following partition of Σ: By construction, Σ 0 ∩ Σ 1 = ∅.We first analyze Σ 0 .If Σ 0 is empty, then skip ahead to our analysis of Σ 1 .Accordingly, assume Σ 0 is nonempty, and let κ * ∈ Σ 0 .We must show that every positive steady state of (G, κ * ) is degenerate.
We claim that G is two-dimensional (assuming that Σ 0 is nonempty).We prove this claim as follows.We saw that G contains a reaction in which X 1 is a non-catalyst-only species, so Proposition 3.14 implies that for j = 0 or j = 2 (or both, where we are using Notation 3.13) our network G contains the reaction X 1 + X j → 2X 1 and at least one reaction of the form X 1 + X j → , where is a complex not involving X 1 .Consider the subcase j = 0.If some involves X 2 , then G contains X 1 → 2X 1 and X 1 → , which yield linearly independent reaction vectors and so G is two-dimensional.If none of the complexes involve X 2 , then G must contain additional reactions in which X 2 is not a catalyst-only species (to avoid f 2 = 0), and so again G is two-dimensional.The subcase j = 2 is similar.
Next, as G is two-dimensional and f κ * ,1 = 0, Corollary 3.4 implies that every positive steady state of (G, κ * ) is degenerate, as desired.Additionally, as X 1 is a non-catalyst-only species and (for all κ ∈ Σ 0 ) f κ,1 = 0, Proposition 3.14 implies that there is a nontrivial linear relation that every κ ∈ Σ 0 satisfies.Hence, Σ 0 has zero measure.
To complete the proof, it suffices to show the following about the set Σ 1 : (1) For every κ * ∈ Σ 1 , every positive steady state of (G, κ * ) is degenerate; and (2) If Σ 1 has nonzero measure, then Assume Σ 1 is nonempty (otherwise, there is nothing to prove).We introduce the following notation: for κ ∈ Σ 1 , let β( κ) denote the ACR-value for X 2 .
We now know that for every κ * ∈ Σ 1 , we have Hence, G has at least one reaction in which X 1 is a non-catalyst-only reaction and (by Proposition 3.15) every such reaction must be one of the 8 reactions displayed here: For every κ * ∈ Σ 1 , Proposition 3.15 yields the following ACR-value formula: where For reactions in (34) that are not in G, the corresponding rate constants, κ * i or κ * ij , are set to 0. Next, the possible reactions in which X 1 is a catalyst-only species are as follows: We proceed by considering three subcases, based in part on whether f κ,2 (which is a polynomial in the unknowns x 1 , x 2 , and κ) is zero: (a) f κ,2 = 0, and X 2 is a catalyst-only species in every reaction of G, (b) f κ,2 = 0, and X 2 is a non-catalyst-only species in some reaction of G, or (c) f κ,2 = 0.
We first consider subcase (a).By inspecting reactions in (34) and (36), we conclude that G must be a subnetwork of Next, we examine subcase (b).Let . By Proposition 3.14 (and by inspecting reactions in (34) and (36)), G must be a subnetwork of G 1 ∪ G 2 ∪ G 3 with at least one reaction in G 2 ∪ G 3 .Moreover, there is a nontrivial linear relation in the rate constants that holds for all κ ∈ Σ 1 .It follows that Σ 1 is contained in the hyperplane defined by this linear relation and hence has zero measure.
Let κ * ∈ Σ 1 .By examining G 1 ∪G 2 ∪G 3 , we see that the possible reactants of G are X 1 , X 2 , X 1 + X 2 .Next, G has at least 2 reactants (as otherwise, Proposition 3.5 would imply that G admits no positive steady states).Hence, by inspection, G either is full-dimensional or is a subnetwork of {0 ← X 2 → 2X 2 , X 1 + X 2 → X 1 }, which we already saw in Example 2.1 (where A = X 2 and B = X 1 ) has ACR in X 1 but not in X 2 (and the analysis of its subnetworks is similar).Hence, G is full-dimensional, and so Corollary 3.4 (and the fact that f κ * ,2 = 0) implies that every positive steady state of (G, κ * ) is degenerate.
We now assume dim(S) = 1 (and aim to reach a contradiction).Recall that G contains at least one reaction from those in (34), so in order for dim(S) = 1 it must be that G contains no reaction from (36).Hence, from the expression for f 2 (which we know is not zero), in (37), the only possible reactions in G are the ones labeled by κ 2 , κ 3,3 , κ 4,2 , κ 4,3 .Hence, the one-dimensional network G is either the network {X 1 Now it is straightforward to check that G is not multistationary, which is a contradiction.
To complete the proof, it suffices to show that, in subcase (c), the set Σ 1 has measure zero.Accordingly, let κ ∈ Σ 1 .As noted earlier, the ACR-value of 34) and (36), the right-hand side of the mass-action ODE for (G, κ) has the following form (with rate constants set to 0 for reactions not in G): By assumption, at least one of the rate constants (the κ i and κ i,j ) in (37) is nonzero.By our earlier arguments, at the beginning of subcase (c), we conclude that f κ,2 | x 2 =β(κ) = 0. Hence, the linear and constant terms of f κ,2 | x 2 =β(κ) are both 0, which, using (37), translates as follows: It follows that Σ 1 is constrained by the equations (38), at least one of which is nontrivial.Hence, Σ 1 is contained in a hypersurface and so has measure zero.

4.2.
Bimolecular networks with at least three species.In the previous subsection, we showed that a bimolecular network must have at least 3 species in order for ACR and nondegenerate multistationarity to coexist.Consequently, this subsection focuses on bimolecular networks with at least 3 species.We prove that the coexistence of ACR and nondegenerate multistationarity requires a minimum of 3 reactant complexes and a minimum of 5 complexes (Theorem 4.9).The remainder of this subsection focuses on a family of networks with n species and n reactants, for which ACR and nondegenerate multistationarity coexist (Section 4.2.1), and then analyzes full-dimensional networks with 3 species (Section 4.2.2).
Theorem 4.9 (Minimum number of complexes).Let G be a bimolecular reaction network with at least 3 species.If there exists a vector of positive rate constants κ * such that (G, κ * ) has ACR and is nondegenerately multistationary, then: (1) G has at least 3 reactant complexes (and hence, at least 3 reactions), and (2) G has at least 5 complexes (reactant and product complexes).
Proof.We first prove part (1).Let (G, κ * ) be as in the statement of the theorem and let n denote the number of species, where n ≥ 3.By relabeling species, if needed, we may assume that (G, κ * ) has ACR in species X 1 .Let f 1 , f 2 , . . ., f n denote the right-hand sides of the mass-action ODEs of (G, κ * ).As (G, κ * ) has ACR, we know that at least one of the right-hand sides is nonzero.Let f i denote one of these nonzero polynomials.Assume for contradiction that G has only 1 or 2 reactant complexes.Since G admits a nondegenerate positive steady state, Proposition 3.5 implies that G is not full-dimensional and G has exactly 2 reactant complexes.
We claim that all the right-hand sides f are scalar multiples of each other.More precisely, we claim that for all j ∈ {1, 2, . . ., n} {i}, there exists c j ∈ R such that f j = c j f i .Indeed, each f j has at most two monomials (because G has exactly two reactant complexes), so if f j is not a constant multiple of f i , then some R-linear combination of f i and f j is a monomial and hence (G, κ * ) has no positive steady state (which is a contradiction).
Thus the positive steady states of (G, κ * ) are precisely the positive roots of f i = 0 and the linear equations given by the conservation laws.Since X 1 is the ACR species and G is bimolecular, we must have , where α is the (positive) ACR-value and β j ∈ R for all j = 0, 1, . . ., n.
We consider several cases, based on how many of the coefficients β 2 , β 3 , . . ., β n are nonzero.We begin by considering the case when β 2 = β 3 = • • • = β n = 0.In this case, f i is a (nonzero) polynomial in x 1 only, and so has the form , where γ 1 , γ 2 ∈ R and 0 ≤ m 1 < m 2 ≤ 2. As (G, κ * ) has a positive steady state, we conclude that γ 1 and γ 2 are nonzero and have opposite signs.Now Lemma 2.3 implies that i = 1 (so, In fact, Lemma 2.3 implies that X 2 , . . ., X n are catalyst-only species of G (equivalently, the massaction ODE right-hand sides for X 2 , . . ., X n are zero for all choices of positive rate constants).Such a system is not multistatationary, which is a contradiction.Now consider the case when two or more of the β 2 , β 3 , . . ., β n are nonzero.In this case, there exist distinct j 1 , j 2 (where 2 ≤ j 1 , j 2 ≤ n) with β j 1 , β j 2 = 0. Then f i contains the monomials x j 1 , x j 2 , x 1 x j 1 , x 1 x j 2 which contradicts the fact that G has exactly two reactant complexes.
The final case is when exactly one of the β 2 , β 3 , . . ., β n is nonzero.Relabel the species, if needed, so that β 2 = 0.In this case, the two reactant complexes of G involve only species X 1 and X 2 .By using Lemma 2.3 again, much like we did for the prior case, we conclude that X 3 , . . ., X n are catalyst-only species of G and so (G, κ * ) is effectively the mass-action system of a (bimolecular) network with only two species, X 1 and X 2 .Now it follows from Theorem 4.8 that (G, κ * ) is not nondegenerately multistationarity, which contradicts our assumption.This completes part (1).
We prove part (2).Assume for contradiction that G has at most 4 complexes.By Proposition 3.10, the dimension of the stoichiometric subspace of G must be at least 2. So, the deficiency of G satisfies Hence, the deficiency of G is 0 or 1, and the latter requires G to have exactly one linkage class.Now Lemmas 2.4-2.5 imply that G is not multistationary, which is a contradiction.Theorem 4.9 gives a lower bound on the number of reactant complexes and the number of all complexes (reactants and products), and the next example shows that these bounds are tight.The example also shows the tightness of the lower bounds on the number of species and the dimension of the network (from Theorem 1.2).
Example 4.10.Consider the following bimolecular network with 3 species and 3 reactant complexes and 5 complexes: This network is two-dimensional, as the total amount of X 1 , X 2 , X 3 is conserved.For every vector of positive rate constants κ, the system (G, κ) is nondegenerately multistationary and also has ACR in X 3 with ACR-value κ 2 /(2κ 3 ).Details are given in the proof of Proposition 4.11, below, which pertains to a family of networks that includes the network G.

4.2.1.
Non-full-dimensional networks.The bimolecular network in Example 4.10 is the n = 3 case of the networks G n that we introduce in the next result.These networks have the property that every reactant complex is bimolecular, but (when n ≥ 4) one of product complexes is not.Proposition 4.11 (ACR and multistationarity for all rate constants).For all n ≥ 3, consider the following network with n species, n reactant complexes, and n reactions: Each such network G n satisfies the following: (1) there is a unique (up to scaling) conservation law, which is given by x 1 + x 2 + x 3 = T , where T represents the total concentration of species X 1 , X 2 , X 3 ; and (2) for every vector of positive rate constants κ ∈ R n >0 , the system (G n , κ) is nondegenerately multistationary and also has ACR in species X 3 , X 4 , . . ., X n .
Proof.Fix n ≥ 3. The mass-action ODEs for G n are as follows: The network G n has exactly one conservation law (up to scaling), and it is given by x 1 +x 2 +x 3 = T .Additionally, using the first two ODEs, we compute that the value of species X 3 at all positive steady states is κ 2 2κ 3 .Next, we use this steady-state value for X 3 , together with the first and fourth ODEs, to obtain the expression κ 2 2 2κ 3 κn for the steady-state value for X j , for j ≥ 4. Thus, ACR in X 3 , X 4 , . . ., X n will follow once we confirm the existence of positive steady states.
Next, we investigate the steady-state values of X 1 and X 2 .Using the steady-state value of X 3 , the conservation law, and the first ODE, we see that the steady-state values of X 1 and X 2 correspond to the intersection points of the line x 1 + x 2 + κ 2 2κ 3 = T and the curve x 1 x 2 = κ 2 2 2κ 1 κ 3 .This is depicted qualitatively below (by [green] dashed lines and a [red] solid curve, respectively).It follows that, given any vector of positive rate constants κ ∈ R n >0 , when T is sufficiently large, there are two pairs of (nondegenerate) positive steady-state values for X 1 and X 2 , and so (G n , κ) is nondegenerately multistationary (and thus admits a positive steady state, and so has ACR).4.2.2.Full-dimensional networks with 3 species.Consider a bimolecular network G that has 3 species.We saw that if G admits ACR and nondegenerate multistationarity simultaneously, then G has at least 3 reactant complexes (Theorem 4.9).If, however, G is full-dimensional, then more reactants are required, as stated in the following result.Proposition 4.12 (Minimum number of reactants for full-dimensional 3-species networks).Let G be a full-dimensional bimolecular reaction network with exactly 3 species.If there exists a vector of positive rate constants κ * such that (G, κ * ) has ACR and is nondegenerately multistationary, then G has at least 5 reactant complexes (and hence at least 5 reactions) Proposition 4.12 is a direct consequence of Propositions 3.1(3) and 3.5, and a stronger version of this result appears in the next section (Theorem 5.1).Proposition 4.12 implies that if a full-dimensional bimolecular network with 3 species and fewer than 5 reactions has both ACR and multistationarity, then this coexistence happens in a degenerate way.We illustrate this situation with two examples, and then characterize all such networks with exactly 4 reactant complexes (Proposition 4.15).For this system, the set of positive steady states is {(x, y, z) ∈ R 3 >0 | xy = z = 1}, and every positive steady state is degenerate.We conclude that this system is multistationary (but degenerately so) and has ACR in species Z (with ACR-value 1).For this system, the set of positive steady states is {(x, y, z) ∈ R 3 >0 | x = y, z = 1}, and every positive steady state is degenerate.Thus, this system is (degenerately) multistationary and has ACR in species Z (with ACR-value 1).
The next result shows that Examples 4.13 and 4.14 cover all cases of three-species, four-reactant networks with ACR and (degenerate) ACR occurring together, in the sense that these two networks represent the only two possibilities for the set of reactant complexes (when a certain full-rank condition is met, which we discuss below in Remark 4.16).Proposition 4.15 (Networks with 3 species and 4 reactants).Let G be a full-dimensional bimolecular reaction network with exactly 3 species -which we call X, Y, Z -and exactly 4 reactant complexes.If κ * is a vector of positive rate constants such that: (a) rank(N ) = 3, where N is the matrix for (G, κ * ) as in (7), (b) (G, κ * ) has ACR in species Z, and (c) (G, κ * ) is multistationary (which is degenerately so, by Proposition 4.12), then the set of reactant complexes of G is either {X, X + Z, Y, Y + Z} or {0, X + Y, Z, 2Z}.
Proof.Let G, κ * , and N be as in the statement of the proposition.In particular, G has 3 species and 4 reactants, and (G, κ * ) admits a positive steady state, which we denote by (x * , y * , α) (so α is the ACR-value of Z).Also, N has rank 3 and so Proposition 3.1(2) and its proof imply that steady-state equations can be "row-reduced" so that the positive steady states of (G, κ * ) are the roots of 3 binomial equations of the following form: where β j ∈ R (for j = 1, 2, 3) and m i = x a i y b i z c i (for i = 1, 2, 3, 4) are 4 distinct monic monomials given by the reactant complexes.Also, each m i (for i = 1, 2, 3, 4) has degree at most 2 in x, y, z (as G is bimolecular).In other words, a i , b i , c i are non-negative integers that satisfy the following: We infer that β 1 , β 2 , β 3 > 0, because otherwise h 1 = h 2 = h 3 = 0 would have no positive roots.
For i ∈ {1, 2, 3}, consider the following, where we recall that α is the ACR-value of Z: where d i := α c i > 0 and d i := β i α c 4 > 0. For i ∈ {1, 2, 3}, by construction, g i (x * , y * ) = 0 and so the subset of the positive quadrant R 2 >0 defined by g i = 0, which we denote by S i , is nonempty.There are four possible "shapes" for each set S i : (1) S i = R .By construction and the fact that α is the ACR-value, the set of all positive steady states of (G, κ * ) is the set {(x, y, α) | (x, y) ∈ S 1 ∩ S 2 ∩ S 3 }.Hence, in the case of (a), (G, κ * ) is not multistationary, which is a contradiction.
Next, we show that case (c) does not occur.On the contrary, assume that it does.Then ).Since m 1 , m 2 , m 3 , m 4 are 4 distinct monomials, it must be that c 1 , c 2 , c 3 , c 4 are 4 distinct non-negative integers.However, as noted earlier, c i ∈ {0, 1, 2} for each i, which yields a contradiction.
Finally, we consider case (b).This case happens only when one of the following subcases occur: Subcase 1: Exactly 1 of the 3 subsets S i is the positive quadrant, and the other two coincide.Without loss of generality, assume S 1 = R 2 >0 and so Finally, Lemma 3.8 implies that each of species X and Y takes part in some reactant complex, so we obtain the following (again using (a 1 , b 1 ) = (a 4 , b 4 ) and (a 2 , b 2 ) = (a 3 , b 3 )): The only non-negative solutions to the conditions in (40), (41), and (42) are as follows: In all of these solutions, the set of reactant complexes is {X, X + Z, Y, Y + Z}.
Subcase 3: None of the subsets S i are positive quadrants, and the 3 sets coincide.This implies that (a 1 , b 1 ) = (a 2 , b 2 ) = (a 3 , b 3 ) = (a 4 , b 4 ).These conditions are symmetric to those in subcase 2, and so the reactant complexes are {0, Z, 2Z, X +Y }.This completes subcase 3 (and case (b)).Remark 4.16 (Rank condition).Proposition 4.15 includes the hypothesis that the matrix N for the system (G, κ * ) has (full) rank 3.If we remove this hypothesis, we can obtain more full-dimensional networks with 3 species and 4 reactants that allow ACR and (degenerate) multistationarity to occur together.We present one such network in Example 4.17.
Example 4.17.Consider the following full-dimensional network with 3 species and 4 reactants: The system (G, κ * ) obtained by setting all the reaction rates to 1 has the following ODEs: The matrix N (defined above) has rank 2, the set of positive steady states is {(x, y, z) ∈ R 3 >0 | x = 1, y(z − 1) = 1}, and every positive steady state is degenerate.Thus, this system is (degenerately) multistationary and has ACR in species X (with ACR-value 1).
In the next section, we see that the exceptional networks in Proposition 4.15 -namely, fulldimensional, three-species networks with reactant-complex set {0, Z, 2Z, X + Y } or {X, Y, X + Z, Y + Z} -do not have unconditional ACR.Indeed, this fact is a direct consequence of a more general result concerning networks with n species and n + 1 reactants (Theorem 5.4).

Main results on general networks
The results in the prior section pertain to networks that are bimolecular, while here we analyze networks that need not be bimolecular.We consider full-dimensional networks (Section 5.1) and non-full-dimensional networks (Section 5.2) separately.
5.1.Full-dimensional networks.In Proposition 4.11, we saw a family of networks that admit ACR and nondegenerate multistationarity together.These networks have n reactants (where n is the number of species), but are not full-dimensional.In this subsection, we show that for fulldimensional networks, the coexistence of ACR and nondegenerate multistationarity requires at least n + 2 reactants (Theorem 5.1).We also show that this lower bound is tight (Proposition 5.3).Additionally, we consider full-dimensional networks with only n + 1 reactants and show that if such a network is multistationary (even if only degenerately so), then the network can not have unconditional ACR (Theorem 5.4).
Theorem 5.1 (Minimum number of reactants for full-dimensional networks).Let G be a fulldimensional reaction network with n species.If there exists a vector of positive rate constants κ * such that the mass-action system (G, κ * ) has ACR and also is nondegenerately multistationary, then n ≥ 2 and G has at least n + 2 reactant complexes and hence, at least n + 2 reactions.
Proof.It follows readily from definitions that ACR and multistationarity do not coexist in networks with only one species, so n ≥ 2. We proceed by contrapositive.We consider two cases.If G has at most n reactant complexes, then Proposition 3.5 (which requires n ≥ 2) implies that every positive steady state of (G, κ * ) is degenerate and so (G, κ * ) is not nondegenerately multistationary.In the remaining case, when G has n + 1 reactant complexes, Proposition 3.1(3) implies that (G, κ * ) is not nondegenerately multistationary.
The next example shows that the bound in Theorem 5.1 is tight for n = 2.
Example 5.2.The following network is full-dimensional and has 2 species, 4 reactant complexes, and 4 reactions (the out-of-order labeling of the rate constants is to be consistent with Proposition 5.3, which appears later): Observe that all reactant complexes are bimolecular, but one of the product complexes is not.In the next result, we show that this network exhibits ACR (in species A with ACR-value κ 2 /κ 1 ) and nondegenerate multistationarity when κ 2 2 > 4κ 3 κ 4 .(Proposition 5.3).Among full-dimensional networks for which ACR and nondegenerate multistationarity coexist, this network is optimal in the sense that it has the fewest possible species, reactant complexes, and reactions (by Theorem 5.1).
In the next result, we generalize the network in Example 5.2 to a family of networks that show that the lower bound on the number of reactions in Theorem 5.1 is tight for all n.The networks in the following result are also optimal in terms of the molecularity of the reactant complexes (they are bimolecular), although two of the product complexes have high molecularity.
The steady-state equation for X 2 implies that x 1 = κ 2 /κ 1 at all positive steady states, so there is ACR in X 1 (whenever positive steady states exist).Next, the steady-state equations for X 1 and X 2 imply that the steady state values of X 2 are x ± 2 := . Both of these steady state values are positive precisely when the discriminant k 2 2 − 4k 3 k 4 is positive (this is a straightforward computation; alternatively, see [10,Proposition 2.3]).Now we use the steady-state equation for X j , where j ≥ 3, to compute the two positive steady states that exist whenever (κ * 2 ) 2 > 4κ * 3 κ * 4 : x , where x * 1 := κ 2 /κ 1 .Finally, nondegeneracy can be checked by a computing the Jacobian matrix.Our next result concerns full-dimensional networks with one more reactant than the number of species, as follows.
Theorem 5.4 (Networks with n + 1 reactants).Let G be a full-dimensional network, with n species and exactly n + 1 reactant complexes.If G is multistationary, then there exists a vector of positive rate constants κ such that (G, κ) has no positive steady states, and hence G does not have unconditional ACR.
Proof.Assume that G is full-dimensional, has exactly n + 1 reactant complexes (where n is the number of species), and is multistationarity.By definition, there exists κ * ∈ R r >0 , where r is the number of reactions, such that (G, κ * ) is multistationary.Let N be the n × (n + 1) matrix arising from (G, κ * ), as in (7); and let A be the n × n matrix defined by G, as in Proposition 3.7.
We claim that rank(N ) ≤ n − 1 or rank(A) ≤ n − 1.Indeed, if rank(N ) = n and rank(A) = n, then the proof of Proposition 3.7 shows that (G, κ * ) is not multistationary, which is a contradiction.
If rank(N ) ≤ n − 1, then Proposition 3.6(2) implies that there exists κ * * ∈ R r >0 such that (G, κ * * ) has no positive steady states.Similarly, in the remaining case, when rank(N ) = n and rank(A) ≤ n − 1, the desired result follows directly from Proposition 3.7(2).5.2.Non-full-dimensional networks.In an earlier section, we saw a family of networks with n species, n reactant complexes, and exactly one conservation law, for which ACR and nondegenerate multistationarity coexist (Proposition 4.11).Our next result shows that this n is the minimum number of reactant complexes (when there is one conservation law), and, furthermore, as the number of conservation laws increases, the minimum number of reactant complexes required decreases.
Theorem 5.5 (Minimum number of reactants).Let G be a reaction network with n ≥ 3 species and k ≥ 1 conservation laws (more precisely, G has dimension n − k).If there exists a vector of positive rate constants κ * such that the system (G, κ * ) is nondegenerately multistationary and has ACR in some species, then G has at least n − k + 1 reactant complexes.
Proof.If G has k ≥ 1 conservation laws and at most n − k reactant complexes, then Propositions 3.5(1) and 3.9 together imply that G is not nondegenerate multistationarity.
As noted earlier, the bound in Theorem 5.5 is tight for k = 1, due to Proposition 4.11.We also know that, for k = n − 1, the bound holds vacuously (Proposition 3.10).Our next result shows that the bound is also tight for all remaining values of k (namely, 2 ≤ k ≤ n − 2).
On the other hand, if k = n − 2, consider the following network: Each such network G n,k satisfies the following: (1) G n,k has n species, n − k + 1 reactants, and n − k + 1 reactions; (2) G n,k has dimension n − k, and the following are k linearly independent conservation laws: x 1 + x 2 + x 3 = T and x j = T j for j ∈ {n − k + 2, . . ., n}. (3) for every vector of positive rate constants κ, the system (G n , κ) is nondegenerately multistationary and also has ACR in species X 3 , X 4 , . . ., X n−k+1 .
Proof.This result can be checked directly, in a manner similar to the proof of Proposition 4.11.Indeed, for every vector of positive rate constants, there is ACR in species X 3 , X 4 , . . ., X n−k+1 and exactly two nondegenerate positive steady states when T is large enough.
The reaction networks in Proposition 5.6 are not bimolecular, and they contain reactions with many catalyst-only species (namely X n−k+2 , . . ., X n ).We do not know whether there exist reaction networks that are bimolecular and do not contain reactions with catalyst-only species, and yet (like the networks in Proposition 5.6) show that the lower bound in Theorem 5.5 is tight.

Discussion
In this article, we proved lower bounds in terms of the dimension and the numbers of species, reactant complexes (and thus reactions), and all complexes (both reactant and product complexes) needed for the coexistence of ACR and nondegenerate multistationarity.Additionally, we showed that these bounds are tight, via the network {A + B → 2C → 2B, C → A} (Example 4.10).
Networks like the one in Example 4.10 contain special structures that may be biologically significant.Exploring such structures will aid in establishing design principles for creating networks with ACR and multistationarity.We plan to explore such networks and their architecture in the future.
In the present work, our interest in multistationarity comes from the fact that it is a necessary condition for multistability.Another interesting direction, therefore, is to investigate conditions for coexistence of ACR and multistability, rather than multistationarity.The "minimal" networks in the current work admit only two positive steady states and are not multistable.Hence, we conjecture that the lower bounds (on dimension and the numbers of species, reactant complexes, and all complexes) for the coexistence of ACR and multistability are strictly larger than the bounds proven here for ACR and multistationarity.
Finally, we are interested in the conditions for the coexistence of other combinations of biologically significant dynamical properties, such as ACR and oscillations.In addition to the minimum requirements for their coexistence, we also hope to discover new network architectures or motifs that can be used to design synthetic networks possessing these dynamical properties.

Example 4 . 14 .
Consider the following network:{X + Z → Z, Y + Z Y → 0, 2X ← X → X + Y }.Like the network in Example 4.13, this network is full-dimensional and has 3 species, 4 reactions, and 4 reactant complexes; however, the set of reactant complexes differs.When all rate constants are 1, the mass-action ODEs are as follows:

Proposition 5 . 6 .
Let n ≥ 3, and let k ∈ {2, 3, . . ., n − 2}.If k = n − 2, consider the following network: and then deleting any trivial reactions (i.e, reactions of the form a i X i → a i X i , in which the reactant and product complexes are equal) and keeping only one copy of duplicate reactions.