Mixed volume of small reaction networks
aa r X i v : . [ q - b i o . M N ] A p r Mixed volume of small reaction networks
Nida Obatake , Anne Shiu , and Dilruba Sofia Department of Mathematics, Texas A&M University Department of Mathematics, University of Massachusetts-Dartmouth
April 29, 2020
Abstract
An important invariant of a chemical reaction network is its maximum numberof positive steady states. This number, however, is in general difficult to compute.Nonetheless, there is an upper bound on this number – namely, a network’s mixedvolume – that is easy to compute. Moreover, recent work has shown that, for certainbiological signaling networks, the mixed volume does not greatly exceed the maximumnumber of positive steady states. Continuing this line of research, we further investigatethis overcount and also compute the mixed volumes of small networks, those with onlya few species or reactions.
Keywords: chemical reaction network, steady state, Newton polytope, mixed volume
MSC classes:
For chemical reaction networks, information about steady states – both their number andtheir nature (stability, etc.) – yields insight into a network’s capacity for processing infor-mation. Therefore, there have been numerous investigations into the capacity for multiplesteady states, especially for networks arising from biology (see, e.g., [2, 5, 7, 8, 12, 17]).The next step, determining the maximum number of steady states of a given network,is more difficult. Indeed, this question, mathematically, asks us to compute the maximumnumber of positive roots of a family of parametrized polynomial systems. Therefore, we areinterested in upper bounds on this maximum number that are easy to compute.One such bound, introduced in [14], is the mixed volume of a network (see also the closelyrelated definition in [10]). This bound is surprisingly good for certain biological signalingnetworks, with the “mixed-volume overcount” – the difference between the mixed volumeand the maximum number of steady states – no more than 2 or 4 [14]. Related results forthree infinite families of networks are obtained in [10].1ere we further investigate the mixed volume and the mixed-volume overcount, witha focus on small networks, those with just a few species or reactions. Our results are asfollows. First, for networks with only one species, we show how to read off the mixedvolume (and mixed-volume overcount) directly from the network (Theorems 3.2 and 3.4),and conclude that the mixed-volume overcount can be arbitrarily large (Corollary 3.3). Next,we investigate networks with two species and two reactions, and show that among those thatare at-most-bimolecular, nearly all have mixed-volume overcount 0 (Theorem 3.13). Thus,the mixed volume is an excellent bound for such networks.The outline of our work is as follows. We provide background in Section 2. Our mainresults are presented in Section 3, and we end with a discussion in Section 4.
Below, we give background on chemical reaction systems (Section 2.1), their steady states(Section 2.2), mixed volume (Section 2.3), and networks having only one species (Section 2.4).
Definition 2.1.
A reaction network G := ( S , C , R ) consists of three finite sets: (1) a set ofspecies S := { A , A , . . . , A s } ; (2) a set C := { y , y , . . . , y p } of complexes (finite nonnegative-integer combinations of the species); and (3) a set of reactions, which are ordered pairs ofcomplexes, excluding diagonal pairs: R ⊆ ( C × C ) r { ( y, y ) | y ∈ C} .Throughout our work, s and r denote the numbers of species and reactions, respectively. Areaction network is genuine if every species takes part in at least one reaction.Writing the i -th complex as y i A + y i A + · · · + y is A s (here, y ij ∈ Z ≥ is thestoichiometric coefficient of A j , for j = 1 , , . . . , s ), this complex is at-most-bimolecularif y i + y i + · · · + y is ≤
2. A reaction network is at-most-bimolecular if every complex in thenetwork is at-most-bimolecular.It is customary to write a reaction ( y i , y j ) as y i → y j , and y i is the called the reactant and y j is the product. Also, a reaction y i → y j is reversible if its reverse reaction y j → y i is alsoin R , and we denote such a pair by y i ⇋ y j . A reaction y i → y j defines the reaction vector y j − y i , which encodes the net change in each species resulting from the reaction. Thestoichiometric matrix Γ is the s × r matrix whose k -th column is the reaction vector of the k -th reaction. Each reaction comes with a rate constant κ ij , which is a positive parameter.Next, we let x , x , . . . , x s represent the concentrations of the s species, which we viewas functions x i ( t ) of time t . Also, we define the monomial x y i := x y i x y i · · · x y is s . A chemical reaction system is the dynamical system that arises, via mass-action kinetics,from a chemical reaction network ( S , C , R ) and a choice of rate constants ( κ ∗ ij ) ∈ R r> (recallthat r is the number of reactions), as follows: d x dt = X y i → y j is in R κ ij x y i ( y j − y i ) =: f κ ( x ) . (1)2iewing the rate constants as a vector of parameters κ = ( κ , κ , . . . , κ m ), we havepolynomials f κ,i ∈ Q [ κ, x ], for i = 1 , , . . . , s . For simplicity, we will write f i rather than f κ,i .The stoichiometric subspace, S := span ( { y j − y i | y i → y j is in R} ), is the vector sub-space of R s spanned by all reaction vectors y j − y i . Thus, S = im(Γ), where Γ is thestoichiometric matrix. Let d = s − rank(Γ). A conservation-law matrix of G , denoted by W ,is a row-reduced d × s -matrix whose rows form a basis of the orthogonal complement of S .A trajectory x ( t ) that starts at a positive vector x (0) = x ∈ R s> remains, forall positive time, in the following stoichiometric compatibility class with respect to thetotal-constant vector c := W x ∈ R d : S c := { x ∈ R s ≥ | W x = c } . (2) Example 2.2.
Consider the network G = { k −−→ , B k −−→ A } . This network has r = 2 non-reversible reactions involving p = 4 distinct complexes – represented as vectors(2 , , (0 , , (0 , , (1 ,
0) – on s = 2 species, A and B . Also, the network is genuine andat-most-bimolecular. The stochiometric matrix of G isΓ = (cid:20) − − (cid:21) . The stoichiometric subspace S , which has dimension d = 1, is spanned by (1 , − T , and aconservation-law matrix of G is W = (cid:2) (cid:3) . Let x ( t ) = ( x ( t ) , x ( t )) ∈ R ≥ denote thevector of concentrations of species A and B . A conservation law for G is x + x = c for c ∈ R ≥ . The chemical reaction system of G arising from mass-action kinetics is d x dt = (cid:18) − k x + k x k x − k x (cid:19) . For a chemical reaction system, a steady state is a nonnegative concentration vector x ∗ ∈ R s ≥ at which the right-hand side of the ODEs (1) vanish: f κ ( x ∗ ) = 0. We will focus onpositive steady states x ∗ ∈ R s> .To analyze steady states in a stoichiometric compatibility class, we use conservation lawsin place of linearly dependent steady-state equations, as follows. Let I = { i < i < · · · < i d } denote the indices of the first nonzero coordinate of the rows of conservation-law matrix W .For a total-constant vector c , define the function f c,κ : R s ≥ → R s as follows: f c,κ,i = f c,κ ( x ) i := ( f i ( x ) if i I, ( W x − c ) k if i = i k ∈ I. (3)The system (3) is called the system augmented by conservation laws. Remark 2.3.
For networks without conservation laws, the augmented system is just theoriginal system f c,κ in (1). 3 efinition 2.4.
1. A network is multistationary if there exist positive rate constants κ ij such that, for thecorresponding chemical system (1), there is some stoichiometric compatibility class (2)having two or more positive steady states.2. A network admits k positive steady states (for some k ∈ Z ≥ ) if there exists a choice ofpositive rate constants so that the resulting mass-action system has exactly k positivesteady states in some stoichiometric compatibility class.The maximum number of positive steady states of a network G is the maximum valueof k (with k ∈ Z ≥ ) for which G admits k positive steady states. Here we recall from [14] the mixed volume of a network, which is in general an upperbound on the maximum number of positive steady states. For background on convex andpolyhedral geometry, see [9, 18]. In particular, for a polynomial f = b x σ + b x σ + · · · + b ℓ x σ ℓ ∈ R [ x , x , . . . , x s ], where the exponent vectors σ i ∈ Z s are distinct and b i = 0 forall i , the Newton polytope of f is the convex hull of its exponent vectors: Newt( f ) :=conv { σ , σ , . . . , σ ℓ } ⊆ R s . Definition 2.5.
Let P , P , . . . , P s ⊆ R s be polytopes. The volume of the Minkowskisum λ P + λ P + . . . + λ s P s is a degree- s homogeneous polynomial in nonnegativevariables λ , λ , . . . , λ s . In this polynomial, the coefficient of λ λ · · · λ s , denoted byVol( P , P , . . . , P s ), is the mixed volume of P , P , ..., P s . Definition 2.6.
Let G be a network with s species, r reactions, and a d × s conservation-lawmatrix W . Let f c,κ , as in (3), denote the resulting system augmented by conservation laws.Let c ∗ ∈ R d =0 , and let κ ∗ ∈ R r> be generic. Let P , P , . . . , P s ⊂ R s be the Newton polytopesof f c ∗ ,κ ∗ , , f c ∗ ,κ ∗ , , . . . , f c ∗ ,κ ∗ ,s , respectively. The mixed volume of G (with respect to W ) isthe mixed volume of P , P , . . . , P s .The mixed volume (Definition 2.6) is well defined [14, Remark 8]. The next result followsfrom Bernstein’s theorem [3] (see [14, Proposition 8]): Proposition 2.7.
For every network, the following inequality relates the maximum numberof positive steady states and the mixed volume (with respect to any conservation-law matrix):maximum number of positive steady states ≤ mixed volume . (4)The mixed-volume overcount measures how tight the bound (4) is. Of particular interestare networks with 0 mixed-volume overcount, because for these networks, the mixed volumeprecisely and efficiently calculates the maximum number of positive steady states.4 efinition 2.8. The mixed-volume overcount of a reaction network G is(mixed volume of G ) − (maximum number of positive steady states of G ) . An example considered in earlier work is the extracellular signal-regulated kinase (ERK)network. This is an important biological signaling network known to be multistationary (andalso bistable) [14, 16]. For the ERK network and several simplified versions of the network,the mixed-volume overcount is 2 – for the fully irreversible and reduced subnetworks – or(conjectured to be) 4 - for the full network and the subnetwork obtained by removing onereaction (specifically, the reaction k on ) [14, Proposition 9 and Conjecture 1]. Here we recall some definitions from [13].
Definition 2.9.
Let G be a reaction network containing only one species A . Each reactionof G therefore has the form aA → bA , where a, b ≥ a = b . Let m be the number of(distinct) reactant complexes, and let a < a < . . . < a m be the stoichiometric coefficients.The arrow diagram of G , denoted by ρ = ( ρ , . . . , ρ m ), is the element of {→ , ← , • ←→} m with: ρ i := → if for all reactions a i A → bA in G , we have b > a i ← if for all reactions a i A → bA in G , we have b < a i • ←→ otherwise. Definition 2.10.
For nonnegative integers T ≥
0, a T -alternating network is a 1-speciesnetwork with exactly T + 1 reactions and with arrow diagram ρ ∈ {→ , ←} T +1 such that, if T ≥
1, we have ρ i = → if and only if ρ i +1 = ← for all i ∈ { , , . . . , T } . Example 2.11.
Consider the following network: G = { ← A → A ⇋ A } . Two 1-alternating subnetworks of G have arrow diagram ( → , ← ): { A → A, A ← A } and { A → A, A ← A } . On the other hand, { ← A, A → A } is not a 1-alternatingsubnetwork of G : its arrow diagram is ( • ←→ ). Finally, { ← A, A → A, A ← A } is a2-alternating subnetwork of G with arrow diagram ( ← , → , ← ).The following result follows directly from [13, Theorem 3.6] and its proof: Proposition 2.12 (Number of steady states for one-species networks) . Let G be a reactionnetwork with only one species (and at least one reaction). Then, the maximum numberof positive steady states of G equals the maximum value of T ∈ Z ≥ for which G has a T -alternating subnetwork. 5 Results
In Section 3.1, we characterize the mixed volume and mixed-volume overcount of networkswith only one reaction or one species. As a consequence, we show that the mixed-volumeovercount can be arbitrarily large (Corollary 3.3). Subsequently, in Section 3.2, we showthat nearly all (genuine) networks with two species and two reactions have mixed-volumeovercount 0 (Theorem 3.13).
Proposition 3.1 (Mixed volume of one-reaction networks) . For a network with only a singlereaction, the mixed volume is 0 and the mixed-volume overcount is 0.
Proof.
Let G be a network with only one reaction. The right-hand side of the ODE consistsof a single monomial, so the Newton polytope is just a point (the exponent vector of themonomial). Hence, the mixed volume of G is 0, and so the mixed-volume overcount is 0, byProposition 2.7. Theorem 3.2 (Mixed volume of one-species networks) . Let G be a reaction network thatcontains only one species A . Let m be the number of (distinct) reactant complexes, and let a < a < . . . < a m be their stoichiometric coefficients. Thenmixed volume of G = a m − a . Proof. As G has only one species, there are no conservation laws and only one differentialequation. In this equation, the leading monomial is x a m , and the lowest-degree monomial is x a . The Newton polytope of this single polynomial is therefore the line segment between a and a m . Thus, by definition, the mixed volume of G is a m − a . Corollary 3.3.
The mixed-volume overcount can be arbitrarily large.
Proof.
Consider the network 0 k −− ⇀↽ −− k n A, where n ∈ N . The right-hand side of the mass-action ODEs (1) is the polynomial − k a n + k , which has precisely one positive real root(namely, a = n p k /k ). However, by Theorem 3.2, the mixed volume is n . So, the mixed-volume overcount is ( n − Theorem 3.4 (One-species networks with mixed-volume overcount 0) . Let G be a reactionnetwork that contains only one species A . Let m be the number of (distinct) reactantcomplexes, and let a < a < . . . < a m be their stoichiometric coefficients. Then G hasmixed-volume overcount 0 if and only if G has an ( m − a i = a + i − i ∈ { , , . . . , m } . Proof.
This result follows directly from Proposition 2.12 and Theorem 3.2.
Example 3.5 (Example 2.11 continued) . By Theorem 3.4, the network from Example 2.11has mixed-volume overcount 0. Indeed, it is a one-species network with 3 distinct reactantcomplexes (note that 0 is not a reactant complex in this network) satisfying a i = a + i − i ∈ { , } (here the notation is as in Theorem 3.4 with a = 1), and it has a 2-alternatingsubnetwork. 6 .2 Networks with two species and two reactions Up to relabeling species, there are 210 genuine, at-most-bimolecular networks with twospecies and two reactions [1]. These networks, which were enumerated by Banaji, are listedat https://reaction-networks.net/networks/ . Here we determine that 92% of thesenetworks have mixed-volume overcount 0 (Theorem 3.13); the 16 exceptional networks arelisted in Table 1.The following result, which follows directly from [13, Lemma 2.7, Lemma 4.1, and The-orem 4.8] (also cf. [13, Corollary 4.12 and the preceding paragraph]), implies that the 210networks we consider in this subsection are not multistationary.
Proposition 3.6. If G is an at-most-bimolecular reaction network with exactly two speciesand two reactions, then the maximum number of positive steady states of G is at most 1.Moreover, this maximum number is 1 if the two reaction vectors of G are negative scalarmultiples of each other, and 0 otherwise.Proposition 3.6 and the definition of mixed-volume overcount directly yield the following: Corollary 3.7.
Let G be an at-most-bimolecular reaction network with exactly two speciesand two reactions. If the mixed volume of G is at least 2, then the mixed-volume overcountis at least 1.We use the following procedure to compute (by using PHCpack [11], as in [14]) the mixed-volume overcount of a 2-species, 2-reaction network:
Procedure 3.8.
Input : A 2-species, 2-reaction network G . Output : the mixed-volume overcount of G .0. Compute the system augmented by conservation laws (3), denoted by f c,κ , for somechoice of conservation-law matrix W .1. Compute the mixed volume of G , as follows. Viewing the two polynomials in f c,κ aspolynomials in x and x , substitute 1 for all coefficients; let poly1 and poly2 be theresulting polynomials. Next, run the following Macaulay2 code: loadPackage "PHCpack"S = CC[x1,x2];F = {poly1 , poly2};mixedVolume(F)
2. Compute the maximum number of positive steady states:(a) If G has no linear conservation laws, the maximum number of positive steadystates is 0.(b) If G has a linear conservation law, determine the maximum number of positivesteady states of G by analyzing the possible numbers of positive roots of f c,κ = 0(or by other means, e.g., if applicable, Proposition 3.6).7. Output the difference between the mixed volume (from Step 1) and the maximumnumber of positive steady states (from Step 2). Proof of correctness of Procedure 3.8.
The correctness of Step 1 is due to the fact that mixedvolume considers only the supports of polynomials. The correctness of Step 2(a) follows from[13, Lemma 4.1]. Step 2(b) is correct by construction of f c,κ . Finally, the correctness of Step 3follows directly from the definition of mixed-volume overcount (Definition 2.8). Example 3.9.
Consider G = { A + B k −−→ k ←−− } .0. The system augmented by conservation laws is ( f ( x , x ) = x + x − c f ( x , x ) = 2 k x + k x x . (5)1. Take k = 2 k = − c = 1 in (5), and compute the mixed volume of the resultingpolynomial system. The mixed volume of the network is 1.2. We compute the maximum number of steady states:(a) There is a linear conservation law (namely, f ), so continue to Step 2(b).(b) The reaction vectors, ( − ,
1) and ( − , not negative scalar multiples of eachother. So, by Proposition 3.6, the maximum number of positive steady states is 0.Alternatively, notice that f ( x ∗ , x ∗ ) > x ∗ , x ∗ >
0, and so f c,κ = 0 neverhas positive roots.3. The mixed-volume overcount is 1 − f c,κ = 0 (Step 2(b) of Procedure 3.8) is not straightforward in general. Example 3.10 (Example 2.2 continued) . Recall the genuine 2-species, 2-reaction network { k −−→ , B k −−→ A } . Using Procedure 3.8, we show below that the mixed-volumeovercount of the network is 1.0. The system augmented by conservation laws is ( f ( x , x ) = x + x − c f ( x , x ) = 2 k x − k x . (6)1. Take 2 k = − k = − c = 1 in (6), and compute the mixed volume of the resultingpolynomial system. The mixed volume of the network is 2.2. We compute the maximum number of steady states:8a) There is a linear conservation law (namely, f ), so continue to Step 2(b).(b) The reaction vectors are ( − ,
2) and (1 , − f c,κ = 0, as follows. First, f = 0yields x = c − x , which we substitute into f = 0 to get g ( x ) = 2 k x − k ( c − x ) = 2 k x + k x − k c . This is a quadratic in x with positive leading coefficient and negative vertical in-tercept (since k , k , c > k , k , c >
0, the quadratichas a unique positive real root in x , namely, x ∗ = (cid:16) − k + p k + 8 c k k (cid:17) / (4 k ).Therefore, the maximum number of steady states is at most 1. In fact, this num-ber is 1: when k = 1 / k = 1 and c = 2, there is a unique positive steadystate, namely, ( x ∗ , x ∗ ) = (1 , − Example 3.11.
Let G = { k −−→ k −−→ A + B } .0. The system augmented by conservation laws is ( f ( x , x ) = x + x − c f ( x , x ) = 2 k x − k x . (7)1. Take 2 k = − k = − c = 1 in (6), and compute the mixed volume of the resultingpolynomial system. The mixed volume of the network is 2.2. We compute the maximum number of steady states:(a) There is a linear conservation law (namely, f ), so continue to Step 2(b).(b) The reaction vectors, ( − ,
2) and (1 , − f = 0 for x (anduse the fact that we are interested in only positive x , x ), which yields x ∗ =( p k /k ) x ∗ . Next, we substitute this expression into f = 0 and then solve toobtain x ∗ = c / (1+ p k /k ). Thus, the network always admits a unique positivesteady state ( x ∗ , x ∗ ).3. The mixed-volume overcount is 2 − Remark 3.12.
The approaches that we present in this section for computing the maximumnumber of steady states of a network (Steps 2(a) and 2(b) of Procedure 3.8) rely on the factthat the networks are at-most-bimolecular and have only two reactions and two species. Ingeneral, however, completing Step 2 is not straightforward: as mentioned in the Introduction,9t requires counting the number of positive real roots of a parametrized polynomial system.This complication further motivates the need for graphical, algebraic, and geometric toolsfor counting positive steady states, in order to bypass a direct analysis of the polynomialsystem f c,κ = 0.By applying Procedure 3.8, we obtain a classification of genuine, at-most-bimolecularnetworks with two species and two reactions (Theorem 3.13). Network Mixed volume (1) 2 A −−→ −−→
A + B 2(2) 2 A −−→ , B −−→ A 2(3) 2 A −−→ A , B −−→ A + B 2(4) B −−→ A , −−→ A + B 2(5) B −−→ A , −−→ A + B 1(6) 2 A −− ⇀↽ −− −−→ A + B ←−− −−→ A , −−→ −−→ , A −−→ A + B 1(10) 2 B −−→ , A −−→ A + B 2(11) A −− ⇀↽ −− −−→ ←−− −−→ A + B −−→ −−→ A , A + B −−→
B 1(15) A + B −− ⇀↽ −− −−→ A , A + B −−→
Theorem 3.13 (Mixed volume of two-species, two-reaction networks) . Let G be a genuine,at-most-bimolecular network with 2 species and 2 reactions. Then G has mixed-volumeovercount 0 if and only if G is (up to relabeling species) not one of the 16 networks listed inTable 1. Moreover, each network in Table 1 has mixed-volume overcount 1. Proof.
Using Procedure 3.8, we computed the mixed-volume overcount for all genuine 2-species, 2-reaction networks; see the supplementary file
MV-overcount-2s-2r-networks.csv in the repository https://github.com/neeedz/mixedvolume . More details are as follows.Among the 210 networks, 185 of them have mixed volume 0 and thus have mixed-volumeovercount 0. For the remaining 25 networks (see Appendix A), it is straightforward tocompute the maximum number of positive steady states using Proposition 3.6 or by directlyanalyzing the system f c,κ = 0 as in Examples 3.9–3.11.We end this section by investigating why the networks in Table 1 have nonzero mixed-volume overcount. These 16 networks fall into four classes:10. Networks (3), (9), (10), and (14) are essentially one-species networks (for each network,one of the two ODEs is 0), and so can be analyzed using the results in Section 3.1.2. Networks (6), (11), and (15) consist of a single pair of reversible reactions, so (e.g., byProposition 3.6) the maximum number of positive steady states is 1.3. Networks (5), (8), (12), (13), and (16) have one species that is consumed in everyreaction (while the other species is produced). Thus, the maximum number of positivesteady states is 0.4. Networks (1), (2), (4), and (7) (and also networks (3), (6), (10), (11), and (15)) havemixed volume 2, so, by Corollary 3.7, the mixed-volume overcount is at least 1. Remark 3.14.
In Examples 3.10 and 3.11, we computed the maximum number of positivesteady states (Step 2 of Procedure 3.8) by reducing the system f c,κ = 0 to a single univariatepolynomial, and then checking that the positive roots (which can be viewed as “partialsolutions”) can be extended to positive roots of the original system. Doing this for generalnetworks, however, is difficult. Indeed, for readers with knowledge of algebraic geometry, wenote that the Extension Theorem [6, pp. 118-120] requires an algebraically closed field andpolynomials with a certain shape. Example 3.15.
Consider the following network with 3 species and 10 reactions:0 −− ⇀↽ −− A , −− ⇀↽ −− B , −− ⇀↽ −− C2 A −− ⇀↽ −− A + B −− ⇀↽ −− B + C . This network has no conservation laws, and its augmented system is f = k − k x − k x + ( k − k ) x x + k x x f = k − k x + k x − k x x f = k − k x + k x x − k x x . Analyzing the augmented system is challenging, and determining the maximum number ofsteady states of the network is not straightforward. This number is at least 2 [12], and wecompute that its mixed volume is 6. What is the mixed-volume overcount? Our wish is toanswer this question in the future through a generalized version of Procedure 3.8.
Recall that our interest in the mixed volume of a reaction network comes from the fact thatit bounds the maximum number of positive steady states. We saw in previous work that thisbound is surprisingly good for certain signaling networks, and here we again found that thisbound performs well for small networks that are at-most-bimolecular. As networks arising inbiological applications are typically at-most-bimolecular, we might expect the mixed-volumeovercount to be low for biological networks of small to medium size.11nother future research direction pertains to one aim of this work, which is to read offthe mixed volume directly from a network. We now can do this for networks with justone reaction or one species (Section 3.1). As for at-most-bimolecular networks with tworeactions and two species, the mixed volume is (with the exception of the 16 networks inTable 1) exactly the maximum number of positive steady states, which can be ascertainedusing results in [13]. We would like similar results for networks with more reactions or morespecies.Continuing this line of investigation, we ask,
How do operations on networks affect themixed volume (and thus the mixed-volume overcount)?
For instance, in Table 1, networks(1) and (7) can be obtained from each other by “stretching” one reaction (without changingthe reactant or reaction vector); and similarly for networks (2) and (4). Moreover, thisoperation does not affect the mixed volume or the overcount. (This line of investigationtherefore would be somewhat similar in spirit to the work of Rojas [15] and Bihan andSoprunov [4].) Indeed, having a list of operations and their effect on the mixed volumewould greatly aid our classification of networks.
Acknowledgements
This research was initiated by DS in the 2019 REU in the Department of Mathematics atTexas A&M University, supported by the NSF (DMS-1757872). NO and AS were partiallysupported by the NSF (DMS-1752672). We thank Taylor Brysiewicz for helpful discussions.
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Networks with nonzero mixed volume
Below, we list the 25 genuine 2-species, 2-reaction networks with nonzero mixed volume,together with their maximum number of positive steady states and their augmented systems.The first 16 networks here coincide with those listed in Table 1.
Network Mixed volume Max
System (1) 2 A −−→ −−→
A + B 2 1 ( a + b − c k a − k b (2) 2 A −−→ , B −−→ A 2 1 ( a + b − c k a − k b (3) 2 A −−→ A , B −−→ A + B 2 1 ( − k a + k b −−→ A , −−→ A + B 2 1 ( a + b − c k a − k b (5) B −−→ A , −−→ A + B 1 0 ( a + b − c − k b − k b (6) 2 A −− ⇀↽ −− ( a + b − c k a − k b (7) 2 A −−→ A + B ←−− ( a + b − c k a − k b (8) B −−→ A , −−→ ( a + b − c − k b − k b (9) B −−→ , A −−→ A + B 1 0 ( k b + k a (10) 2 B −−→ , A −−→ A + B 2 1 ( − k b + k a (11) A −− ⇀↽ −− ( a + b − c − k b + k a (12) A + B −−→ ←−− ( a + b − c k a + k ab (13) 2 A −−→ A + B −−→ ( a + b − c k a + k ab (14) 2 A −−→ A , A + B −−→
B 1 0 ( − k a − k ab −− ⇀↽ −− ( − k ab + k a − b (16) B −−→ A , A + B −−→ ( a + b − c − k b − k ab (17) 0 −−→ , A + B −−→
A 1 1 ( − k ab + 2 k (18) 2 B −−→ , A + B −−→
A 1 1 ( − k b − k ab (19) A + B −−→ −−→ ( a + b − c k a − k ab (20) A + B −−→ −−→ A + B 1 1 ( a + b − c k ab − k b (21) A + B −−→ , B −−→ A 1 1 ( a + b − c k ab − k b (22) A −−→ , B −−→ A + B 1 1 ( − k a + k b −− ⇀↽ −− B 1 1 ( a + b − c k a − k b (24) A + B −−→ A , −−→ B 1 1 ( − k ab + k (25) A + B −− ⇀↽ −− A 1 1 ( − k ab + k aa