A generalization of Birch's theorem and vertex-balanced steady states for generalized mass-action systems

Mass-action kinetics and its generalizations appear in mathematical models of (bio-)chemical reaction networks, population dynamics, and epidemiology. Vertex-balanced steady states may contain information about the dynamical properties of these systems and have useful algebraic properties. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem that provides a sufficient condition for the existence and uniqueness of vertex-balanced steady states for generalized mass-action systems.


Introduction
Reaction networks are used to represent the numerous interactions of chemical species in chemistry and biochemistry. An example is the following enzymatic reaction, where a substrate S 0 is converted into a product S 1 by an enzyme E via an intermediate species ES 0 : To model the dynamics of the concentrations of the chemical species involved in a network, one typically assumes that the reactions proceed according to some specified kinetic rate functions. A well-studied model is mass-action kinetics; it assumes that the reaction rate is proportional to the concentrations of its reactants. According to mass-action kinetics, the reaction E + S 0 → ES 0 proceeds at rate κ 1 [E][S 0 ], where κ 1 > 0 is a rate constant, and [X] is the concentration of species X as a function of time t. The rates of change of the concentrations of E, S 0 and ES 0 due to this single reaction are The overall rate of change due to all reactions in the network is the sum over its individual reactions, e.g.
Mass-action systems have been studied extensively. Reaction network theory, as initially developed by Horn, Jackson and Feinberg [9,11,15], tries to conclude dynamical properties from simple characteristics of the underlying reaction network. Moreover, as the reaction rate constant is usually obtained empirically and thus subjected to uncertainty, an ideal theoretical result should not depend on the precise values of the rate constants; indeed this is the case for many classical results in reaction network theory. Much of the early work on mass-action systems can be found in the lecture notes [10,14].
Mass-action kinetics is the most common model for the dynamics of chemical concentrations when the reactions occur at dilute concentrations in a well-mixed fluid. That is not the typical environment in which biochemical reactions naturally take place. The cell is typically crowded and the intracellular matrix viscous; proteins might be embedded in membranes. Various models have been developed to account for this difference.
One proposed model is that of power-law kinetics, where the exponents (or kinetic orders) in the reaction rate functions do not necessarily follow the stoichiometric coefficients. In the catalysis example above, we may want the concentration of E to be modelled by the equation for some constants α, β, γ, δ > 0. This is an example of power-law kinetics. Classical mass-action kinetics and power-law kinetics can be incorporated into the framework of generalized mass-action kinetics introduced in [20,21]. (See Section 3 for a precise definition of generalized mass-action systems.) For classical mass-action systems, an important subset of positive steady states is the set of complex-balanced equilibria Z κ . They have been introduced also for generalized mass-action systems and are called vertex-balanced steady states in this work (Definition 4.1). Such a steady state is a concentration vector x * (one coordinate for each chemical species) such that at every vertex of the graph, the sum of the incoming rate functions evaluated at x * equals to the sum of the outgoing rate functions evaluated at x * .
The existence of a complex-balanced equilibrium of a classical mass-action system has major implications for its dynamical properties. For example, it implies that all positive steady states are complex-balanced, and that there is a unique positive steady state within every invariant affine subspace [11]; moreover, it also implies that there is a monomial parametrization for the set of complexbalanced equilibria [6]. Not surprisingly, the story for vertex-balanced steady states of a generalized mass-action system is more complicated, and less is known about these steady states. Nonetheless, the existence of a vertex-balanced steady state also implies that the set of vertex-balanced steady states admits a monomial parametrization [21].
In this paper, we are interested in how many (if any) vertex-balanced steady states there are within each invariant affine subspace of a generalized mass-action system. This paper initially aims to understand which reaction networks admit vertex-balanced steady states, and if so, how many are there. Interestingly, this question can be reformulated in two different ways, one related to a generalization of Birch's theorem in statistics [22], and the other to the bijectivity of generalized polynomial maps, similar to ones which appear in geometric modelling [7,20]. Indeed, the following questions are essentially equivalent: 1. When does a generalized mass -action system have exactly one vertex-balanced steady state  within each invariant affine subspace, for any choice of rate constants? 2. Given vector subspaces S, S ⊆ R n , when does the intersection (x 0 + S) ∩ (x * • exp S ⊥ ) consist of exactly one point, for any x 0 , x * ∈ R n >0 ? 1 3. Given vectors w 1 , . . . , w n , w 1 , . . . , w n ∈ R d , when is the generalized polynomial map on R d >0 defined by bijective onto the relative interior of the polyhedral cone generated by w 1 , . . . , w n , for any x * ∈ R n >0 ? These questions will be expanded upon and explained in detail in Section 5.
Among the questions 1 to 3 above, we initially focus on question 2, which is strongly related to Birch's theorem in statistics. One way to state Birch's theorem is: given a vector subspace S ⊆ R n , the intersection (x 0 + S) ∩ (x * • exp S ⊥ ) consists of exactly one point, for any x 0 , x * ∈ R n >0 . In question 2, we have two vector subspaces S, S, so it should not come as a surprise that an additional hypothesis is needed.
This hypothesis is given in terms of sign vectors. For a subset S ∈ R n , its set of sign vectors σ(S) is the image of vectors in S under the coordinate-wise sign function. Its closure σ(S) contains σ(S) and all sign vectors where one or more coordinates may be replaced by zeros (see Definition 6.1).
We obtain the following generalization of Birch's theorem: Theorem 6.7. Let S, S ⊆ R n be vector subspaces of equal dimension with σ(S) ⊆ σ( S). Then for any positive vectors consists of exactly one point.
As a consequence of this theorem, we provide a sufficient condition in Theorem 6.8 for question 1 above. More precisely, provided that certain conditions hold, we show that if a generalized massaction system has at least one vertex-balanced steady state, then there is exactly one vertex-balanced steady state within every invariant affine subspace.
We introduce generalized mass-action systems and vertex-balanced steady states in Section 3 and 4 respectively. We prove Theorem 6.7 in Section 6, and conclude with an example in Section 7.

Notation
There are several component-wise operations on vectors and matrices that will appear frequently. In the list below, let x, z ∈ R n with x = (x 1 , x 2 , . . . , x n ) T and z = (z 1 , z 2 , . . . , z n ) T . Let Y = (y 1 , y 2 , · · · , y m ) be a n × m matrix. The operations we will use are: x zi i , with the convention 0 0 = 1; x n z n T whenever z j = 0 for all j; exp x = (e x1 , e x2 , · · · , e xn ) T ; log x = (log x 1 , log x 2 , · · · , log x n ) T whenever x j > 0 for all j.
When the above operations are applied to a subset of R n , they are applied to elements of the set. For example, given a set S ⊆ R n , we have exp(S) = {exp(x) : x ∈ S}, and x • S = {x • z : z ∈ S}. When x is a vector, we write x ≥ 0 to mean that every component of the vector is non-negative. Similarly, x > 0 means that every component of the vector is positive. We let R n ≥0 = {x ∈ R n : x ≥ 0}, and R n >0 = {x ∈ R n : x > 0}. We denote the cardinality of a set M as |M |.

Generalized mass-action systems
Consider a simple directed graph G = (V, E) and the corresponding weighted graph G κ = (V, E, κ) with κ ∈ R E >0 providing a positive weight for each edge in E. Let V = {v 1 , v 2 , . . . , v m } be the set of vertices. Given an edge e = v i → v j ∈ E, we call v i the source of e, and v j its target. Let us denote by V s ⊆ V the set of source vertices, that is, the set of vertices that are sources of some edges. The weight κ e > 0 on the edge e = v i → v j is called a rate constant, and we refer to the vector κ = (κ e ) e∈E as the vector of rate constants, or more simply as the rate constants. Often, we use the indices of the source and target vertices as edge label, i.e., κ vi→vj = κ ij .
Let Φ : V → R n be a map assigning to each vertex v ∈ V a reaction complex Φ(v) ∈ R n , and let Φ : V s → R n be another map that assigns to each source vertex v ∈ V s a kinetic-order complex Φ(v) ∈ R n . An edge v i → v j is called a reaction, and the vector Φ(v j ) − Φ(v i ) is the reaction vector associated to the edge v i → v j . For convenience, we often write y i instead of Φ(v i ), and y i instead of Φ(v i ). The graph G and the two maps Φ, Φ on G provide all the ingredients needed to define a generalized mass-action network, while the weighted graph G κ and the maps Φ, Φ are all that is needed to define a generalized mass-action system. Definition 3.1. A generalized mass-action network is given by (G, Φ, Φ), where G = (V, E) is a simple directed graph, and Φ : V → R n , Φ : V s → R n respectively assign to each vertex a reaction complex and to each source vertex a kinetic-order complex .
Remark. We follow the definition of a generalized mass-action network given by Müller and Regensburger in [21], rather than the one given in [20]. In particular, we do not assume the maps Φ and Φ are one-to-one.
Remark. Throughout this paper, we are concerned with generalized mass-action networks where V s = V . In this case, the graphs Φ(G) and Φ(G) are two Euclidean embedded graphs [2,5,25]. One of the equivalent definitions of a (classical) reaction network is a graph G = (V, E), where the set V of vertices is a subset of R n . Using the notation above, a reaction network is given by (G, Φ), where Φ is one-to-one [21].
Example 3.2. To illustrate the terminology above, consider a directed graph G = (V, E), and its corresponding weighted graph G κ = (V, E, κ): The set of vertices is V = {v 1 , v 2 , v 3 , v 4 , v 5 }, which coincides with the set of source vertices V s . The set of edges is The maps Φ, Φ are defined by their images in R 2 , as shown in Figure 1. Note that the vertex v 3 is mapped differently by Φ and Φ. Indeed, v 3 is mapped by Φ to the reaction complex (1, 2) T and by Φ to the kinetic-order complex (3, 1) T . Moreover, the vertices v 2 and v 3 are mapped by Φ to the same reaction complex, and hence the number of connected components and the number of vertices are different in the graphs Φ(G) and Φ(G). Now we are in a position to define generalized mass-action systems and the associated dynamical systems.

Definition 3.3.
A generalized mass-action system is given by (G κ , Φ, Φ), where (G, Φ, Φ) is a generalized mass-action network, with directed graph G = (V, E), and κ ∈ R E >0 is a vector of rate constants.
Definition 3.4. For a generalized mass-action system (G κ , Φ, Φ), the associated dynamical system on R n >0 is given by As the ODE system (1) is our main object of interest, we pause to make two observations. First, the rate of change dx dt is restricted to the stoichiometric subspace S = span R {y j − y i : v i → v j ∈ E}. Consequently, every trajectory x(t) of this dynamical system is restricted to a stoichiometric compatibility class x(0)+ S. Second, if v i → v j is a reaction and Φ(v i ) = Φ(v j ), then this particular reaction does not contribute to the dynamics.
It is sometimes more convenient to write the ODE system (1) in matrix form. Let Y ∈ R n×m be the reaction complex matrix , the j-th column of which is the reaction complex y j . Let the kinetic-order complex matrix Y ∈ R n×m be defined analogously; in particular, its j-th column is the kinetic-order complex y j if v j ∈ V s and 0 if v j ∈ V s . 2 Let A κ ∈ R m×m be the negative transpose 2 The choice of y j = 0 when v j ∈ Vs is arbitrary, since the j-th column of Y does not appear in the equations that are of interest to us. In particular, they do not contribute to the right-hand side Y Aκx Y of the system of differential equations shown in (3), because they do not affect the vector Aκx Y [21].
of the Laplacian of the weighted directed graph G κ , i.e., (2) The dynamical system (1) can be rewritten as Example 3.2. Returning to Example 3.2, the dynamical system associated to (G κ , Φ, Φ) is d dt where each term corresponds to an edge in the graph G. Expanding the equations, we recognize it to be a polynomial (more generally, a power-law) dynamical system: Its stoichiometric subspace is S = R 2 . The reaction complex matrix and kinetic-order complex matrix are Y = (y 1 , y 2 , y 3 , y 4 , y 5 ) = 0 1 1 2 4 1 2 2 0 2 and Y = ( y 1 , y 2 , y 3 , y 4 , y 5 ) = 0 1 3 2 4 1 2 1 0 2 respectively. The matrix is the negative transpose of the Laplacian of the weighted graph G κ .
The definitions above and Example 3.2 are relatively abstract; one may wonder how generalized mass-action systems show up in applications. Suppose we are interested in modelling the following chemical system: Let us assume that based on experimental data, the reaction rate functions are as shown below, with rate constants κ ij > 0: Note that the third reaction follows power-law kinetics, and not classical mass-action kinetics. The system of ordinary differential equations modelling this system of reactions is precisely (4), the dynamical system associated to (G κ , Φ, Φ) of Example 3.2.
Remark. We defined a generalized mass-action network as a triple (G, Φ, Φ). As pointed out in an earlier remark, if Φ is one-to-one, then (G, Φ) is a (classical) reaction network. A classical mass-action system can be obtained as a special case of a generalized mass-action system (G κ , Φ, Φ), where Φ is one-to-one and Φ = Φ| Vs [20]. It is thus natural to extend some of the standard definitions for classical mass-action systems to generalized mass-action systems.
We say the underlying graph G is weakly reversible if every connected component of G is strongly connected, i.e., every edge is part of a directed cycle. We have already defined the stoichiometric subspace S as the span of reaction vectors. Whenever V s = V (in particular, when G is weakly reversible), we define its kinetic analogue, the kinetic-order subspace The stoichiometric deficiency of the generalized mass-action network (G, Φ, Φ) is the nonnegative integer where |V | = m is the number of vertices in G, ℓ G is the number of connected components of G, and S is the stoichiometric subspace. From the equivalent definition δ G = dim(ker Y ∩ im I E ), where I E is the incidence matrix of G, it follows that δ G is a non-negative integer [16]. In the case when G is weakly reversible, we also have the formula [14] Whenever V s = V , the kinetic-order deficiency is defined as the non-negative integer where S is the kinetic-order subspace.
Remark. In the definitions above, |V | is the number of vertices in the underlying abstract graph G, not necessarily the number of distinct reaction complexes or the number of kinetic-order complexes; ℓ G is the number of connected components of G, not necessarily the number of connected components in Φ(G) or Φ(G).
In Example 3.2, we have a weakly reversible network with |V | = 5 vertices and ℓ G = 2 connected components. We already observed that the stoichiometric subspace S is all of R 2 . However, the kinetic-order subspace is S = span R (1, 1) T . The stoichiometric deficiency in this example is δ G = 5 − 2 − 2 = 1, but the kinetic-order deficiency is δ G = 5 − 2 − 1 = 2.
Example 3.5. We have seen earlier that generalized mass-action systems arise naturally from powerlaw kinetics. This example illustrates how generalized mass-action systems also arise naturally in the study of mass-action systems, via a process called network translation [16,17]. Network translation produces a generalized mass-action system that has the same dynamics as the original mass-action system. We look at the n-site distributive phosphorylation-dephosphorylation system under massaction kinetics.
This example first appeared in [16]; below we consider a different translation of the same massaction system. Under the original definition of generalized mass-action system in [20], which requires the reaction complex map Φ and the kinetic-order complex map Φ to be one-to-one, the translated network presented below would not have been a well-defined generalized mass-action network. However, the later definition in [21] removes the requirements that Φ and Φ are one-to-one. As a result, many more dynamical systems can be written as a generalized mass-action system, and for this example, a more natural translation exists for the n-site distributive phosphorylation-dephosphorylation system.
Let E, F be enzymes that catalyze the phosphorylation and dephosphorlyation processes, by forming intermediates ES j and FS j respectively. The n-site distributive phosphorylation-dephosphorylation system consists of the following reactions: We assume that the reaction rates follow classical mass-action kinetics. There are 3n + 3 species involved, so the system of differential equations modelling their concentrations is defined on R 3n+3 >0 . By network translation, we create a generalized mass-action system with the same differential equations. The main step involves changing the reaction complexes: adding enzyme F to the series of reactions for phosphorylation by E; and adding enzyme E to the series of reactions for dephosphorylation by F. This process produces a weakly reversible network: To define the generalized mass-action system, we take a more top-down approach, starting from a graph G with n components and 4n vertices: Although Φ and Φ map vertices to vectors in R 3n+3 , to make this example more readable, we will specify the images of Φ and Φ in terms of formal linear combination of species.
The reaction complexes are and the kinetic-order complexes are One can check that dim S = dim S = 3n; therefore, the stoichiometric deficiency and kinetic-order

Vertex-balanced steady states
Given the dynamical system associated to a generalized mass-action system (G κ , Φ, Φ), written either it is natural to ask how many steady states there are. We define the set of positive steady states as For a classical mass-action system, an important subset of positive steady states is the set of complex-balanced equilibria [11], also known as complex-balancing equilibria or vertex-balanced equilibria [4]. Horn and Jackson introduced the idea of complex balancing at equilibrium to generalize the common physical assumption of detailed balancing at thermodynamic equilibrium [11].
We illustrate the intuition behind the definition of such a steady state before introducing its analogue for a generalized mass-action system. Consider the graph G of the reaction network, and associate to each edge v i → v j a reaction rate function κ ij x yi . A concentration vector x * ∈ R n >0 is a complex-balanced equilibrium of the classical mass-action system if at every vertex v j ∈ V of the graph, the sum of incoming fluxes balances the sum of outgoing fluxes, i.e., This occurs if and only if A κ (x * ) Y = 0 [11]. Clearly, a complex-balanced equilibrium is a positive solution to a system of polynomial equations. Surprisingly, it is also a positive solution to a system of binomial equations [6].
For a generalized mass-action system, one can define a vertex-balanced steady state analogously: it is a positive steady state at which the net flux is zero across every vertex of the graph, where the flux is given by generalized mass-action kinetics.
Note that x ∈ Z κ if and only if vi→vj ∈E Remark. What we call vertex-balanced steady state here, is also called complex balancing equilibrium [20,21] or generalized complex-balanced steady state [16]. We call such a steady state vertex-balanced instead of complex-balanced to avoid a subtle point of confusion. In the case when Φ is not one-to-one, the balancing of in-fluxes and out-fluxes occurs at each vertex v ∈ V of the underlying abstract graph G. This in turn implies the balancing of fluxes at each reaction complex Φ(v) ∈ Φ(V ); however, the converse is generally false. For example, let κ > 0, and consider the generalized mass-action system given by the weighted graph G κ . The associated dynamical system dx dt = κ − κx is the same as that of the classical mass-action system given by the The concentration x * = 1 is complex-balanced for the classical mass-action system given by G ′ κ but not vertex-balanced for the generalized mass-action system given by G κ .
However, a positive steady state x = (x 1 , x 2 ) T ∈ R 2 >0 has to satisfy only two polynomial equations: The two polynomial equations (13) are linear combinations of the five polynomial equations (12); thus Z κ ⊆ E κ . This follows from the matrix expression of the associated dynamical system dx dt = Y (A κ x Y ).
Complex-balanced equilibria of classical mass-action systems have been studied extensively. Some of the classical results extend directly to the case of generalized mass-action systems, even when the maps Φ and Φ, assigning reaction complexes and kinetic-order complexes, are not one-to-one. For example, it is known that [20,21]: 3 i) If Z κ = ∅ for some κ > 0, then the underlying graph G is weakly reversible.
iii) For a weakly reversible generalized mass-action network, δ G = 0 if and only if Z κ = ∅ for any choices of rate constants κ > 0.
iv) For a weakly reversible generalized mass-action network, if δ G = 0, then for any choice of rate constants κ > 0, any positive steady state is a vertex-balanced steady state, i.e., E κ = Z κ .
In the example of the n-site phosphorlyation-dephosphorlyation system (Example 3.5), we noted that δ G = δ G = 0. By statements (iii) and (iv) above, we conclude for any rate constants κ, the set of vertex-balanced steady states Z κ is non-empty, and all positive steady states are vertex-balanced. Moreover, the set of positive steady states is given by E κ = Z κ = x * • exp S ⊥ , where x * is any positive steady state and S is the kinetic-order subspace, i.e., the vector space spanned by the differences of kinetic-order complexes according to the edges in the graph. It should be noted that the n-site phosphorlyation-dephosphorlyation system is multistationary when n ≥ 2 [3,18], i.e., the system admits multiple steady states within the same stoichiometric compatibility class. In other words, for some choices of rate constants, there are multiple vertex-balanced steady states within some stoichiometric compatibility class. This contrasts with any classical complex-balanced mass-action system, where Z κ meets every stoichiometric compatibility class at most once.
In applications, the vector of rate constants κ ∈ R E >0 is often not known precisely. Surprisingly, some important results for complex-balanced equilibria in classical mass-action systems hold irrespective of the precise values of the rate constants. We are interested in results for vertex-balanced equilibria of generalized mass-action systems that are in this sense independent of the choice of rate constants. We have observed that the solution trajectories are confined to a stoichiometric compatibility class x 0 + S, where x 0 ∈ R n >0 is an initial state and S is the stoichiometric subspace. Therefore, our object of study is the intersection (x 0 + S) ∩ Z κ for any x 0 ∈ R n >0 and any κ ∈ R E >0 .

Problem reformulations
In the introduction, we have mentioned that the following questions are essentially equivalent: 1. When does a generalized mass-action system have exactly one vertex-balanced steady state within each stoichiometric compatibility class, for any choice of rate constants?
2. Given vector subspaces S, S ⊆ R n , when does the intersection (x 0 + S) ∩ (x * • exp S ⊥ ) consist of exactly one point, for any x 0 , x * ∈ R n >0 ? 3. Given vectors w 1 , . . . , w n , w 1 , . . . , w n ∈ R d , when is the generalized polynomial map on R d >0 defined by bijective onto the relative interior of the polyhedral cone generated by w 1 , . . . , w n , for any x * ∈ R n >0 ? Before we discuss the relationship between these problems in detail, let us first make a historical note. When speaking of a weakly reversible classical mass-action system, Horn and Jackson [11] proved that if the system has at least one complex-balanced equilibrium, then every stoichiometric compatibility class has exactly one complex-balanced equilibrium. Indeed, they showed that every positive steady state of such a system is complex-balanced and locally asymptotically stable within its stoichiometric compatibility class. A complex-balanced equilibrium is globally stable within its stoichiometric compatibility class when the network has a single connected component [1], or is strongly endotactic [12], or when the system is in R 3 [8,23]. A general proof of global stability of complex-balanced equilibrium within its stoichiometric compatibility class was proposed for all complex-balanced systems in [4].
The first of the three questions above is phrased in the context of reaction networks. We start with a generalized mass-action network and suppose that for some rate constants κ, there is a vertexbalanced steady state x * ∈ Z κ . What is a condition (E) on the network (G, Φ, Φ) for the existence of a vertex-balanced steady state within every stoichiometric compatibility class? What is a condition (U) on (G, Φ, Φ) so that a vertex-balanced steady state is unique within its stoichiometric compatibility class? We would like to obtain conditions for these to hold or fail that are independent of the rate constants κ. More precisely: Recall that Z κ = x * • exp S ⊥ for any x * ∈ Z κ . Thus, the vertex-balanced steady states within any stoichiometric compatibility class x 0 + S belong to the intersection (x 0 + S) ∩ (x * • exp S ⊥ ). This leads us to the following reformulation of Problem 1: x 0 , x * ∈ R n >0 . 2. If S, S satisfy condition (U), then (x 0 + S) ∩ (x * • exp S ⊥ ) contains at most one point, for any If a generalized mass-action system happens to be a classical mass-action system, then its stoichiometric subspace S is also the kinetic-order subspace S. The existence and uniqueness of a point in the intersection (x 0 + S) ∩ (x * • exp S ⊥ ) for any x 0 , x * ∈ R n >0 is the content of Birch's theorem in algebraic statistics [22].
Another reformulation of the above problems was introduced by Müller and Regensburger [20], in terms of injectivity/surjectivity of an exponential map or a generalized polynomial map onto a polyhedral cone. Such polynomial maps appear in other applications; for example, a renormalized version of the generalized polynomial appears in computer graphics and geometric modelling, where the map being one-to-one implies that the curve or surface does not self-intersect [7]. Let x * ∈ R n >0 be an arbitrary vector, and S, S ⊆ R n be vector subspaces, with d = codim S, d = codim S. Choose a basis for S ⊥ and let the basis vectors be the rows of the matrix W ∈ R d×n . Similarly, choose a basis for S ⊥ and let the basis vectors be the rows of W ∈ R d×n . Write the two matrices in terms of their columns: W = (w 1 , w 2 , · · · , w n ) and W = ( w 1 , w 2 , · · · , w n ). In this manner, S ⊥ = im(W T ), S = ker W , and S ⊥ = im( W T ), S = ker W . Finally, write C 0 (W ) for the relative interior of the polyhedral cone C(W ), i.e., C 0 (W ) is the set of all positive combinations of For any x * ∈ R n >0 , define the maps x * i e w i ,λ w i . Problem 2 is equivalent to the following (see [20,21] for details): Problem 3. Let S, S ⊆ R n be vector subspaces. What are conditions (E) and (U) on S, S, so that the following statements are true?

If S, S satisfy condition (E), then the map
Müller and Regensburger characterized when the maps f x * , F x * are one-to-one, namely, if and only if σ(S) ∩ σ( S ⊥ ) = {0} [20,Theorem 3.6]. Recall that, for a subset S ⊆ R n , its set of sign vectors σ(S) is the image of vectors in S under the coordinate-wise sign function (Definition 6.1). They also provided a sufficient condition for bijectivity: if σ(S) = σ( S) and (+, +, · · · , +) T ∈ σ(S ⊥ ), then f x * , F x * are bijective (and indeed, real analytic isomorphisms) [20,Proposition 3.9]. Our main result (Theorem 6.8) can be regarded as a generalization of this result. Recently, Müller, Hofbauer, and Regensburger have used a different approach to characterize when f x * , F x * are bijective maps for arbitrary x * ∈ R n >0 [19].

Main result
In previous work as well as in ours, the conditions (E) and (U) are stated in terms of sign vectors. For a brief introduction to sign vectors of linear subspaces, we refer the reader to the appendix in [20].
The set of sign vectors for a subset S ⊆ R n is the collection σ(S) = {σ(x) : x ∈ S}. A partial order on the set {0, +, −} n is given by τ ≥ τ ′ if and only if τ j ≥ τ ′ j for all j, with the convention + > 0 and − > 0. The closure of a set Λ of sign vectors is the set We define an orthant 4 of R n to be a maximal subset of R n on which σ is constant. Geometrically, the sign vector σ(x) tells us which orthant O x the vector x lies in, while the closure σ(x) refers to the union of O x and the boundary of O x . Finally, we define an orthogonality relation on {0, +, −} n ; we say that two sign vectors τ and τ ′ are orthogonal (denoted τ ⊥ τ ′ ) if i. either τ j · τ ′ j = 0 for all 1 ≤ j ≤ n, or ii. there exist indices i, j such that τ i · τ ′ i = + and τ j · τ ′ j = −, where the product operation on signs is as one would expect: + · + = − · − = +, + · − = −, and + · 0 = − · 0 = 0 · 0 = 0.
It is easy to see that if x, y ∈ R n are orthogonal vectors, then σ(x) ⊥ σ(y).
We show in this section that if σ(S) ⊆ σ( S) and dim S = dim S, then for any x 0 , x * ∈ R n >0 , the intersection (x 0 + S) ∩ (x * • exp S ⊥ ) contains exactly one point. The intuitive idea behind our result is that the sign condition σ(S) ⊆ σ( S) is related to a transversal intersection of the two manifolds (x 0 + S) and (x * • exp S ⊥ ). If we have one intersection point, say x * ∈ (x * + S) ∩ (x * • exp S ⊥ ), we cannot lose the intersection point as we translate the affine plane from (x * + S) to (x 0 + S).
We first show in Lemma 6.2 that our sign condition σ(S) ⊆ σ( S) implies the uniqueness condition σ(S) ∩ σ( S ⊥ ) = {0} in [20]. In Lemma 6.4, we establish transversality of the manifolds (x + S) and (x * • exp S ⊥ ). Lemma 6.3 prevents our desired intersection point from escaping to the boundary of R n >0 or to infinity. Finally, these results lead to Theorem 6.7, concluding the existence and uniqueness of a point in the intersection (x 0 + S) ∩ (x * • exp S ⊥ ). In Theorem 6.8 and Corollary 6.9, we apply this result to generalized mass-action systems.
, the intersection Γ also lies in the positive orthant. We first show that Γ is bounded away from infinity and from the boundary of R n >0 . Suppose that is not the case. Let x k ∈ Γ be a sequence such that either lim sup k→∞ x k i = ∞ or lim inf k→∞ x k i = 0 for some index 1 ≤ i ≤ n. Passing to a subsequence, we may assume that where I 1 , I 2 , I 3 partition the index set {1, 2, . . . , n}, and I 1 ∪ I 2 = ∅.
On one hand, x k ∈ K + S, so decompose it as x k = v k + s k , where v k ∈ K and s k ∈ S. Since K ⊆ R n >0 is compact, each component of v k is uniformly bounded from above and below from zero. Thus for i ∈ I 1 , we have s k i → ∞; in particular, s k i > 0 for sufficiently large k. Similarly, if i ∈ I 2 , then s k i < 0 for sufficiently large k, because s k i + v k i → 0 and v k i > 0 is bounded away from zero. Hence the sign of s k i is constant for any i ∈ I 1 ∪ I 2 for sufficiently large k. Because σ(s k ) ∈ σ(S) ⊆ σ( S), there is a vector u ∈ S such that u i > 0 if i ∈ I 1 and u i < 0 if i ∈ I 2 .
On the other hand, for all k, and we have The sum over I 3 is uniformly bounded for all k. Now let k → ∞. For i ∈ I 1 , we know u i > 0 and x k i → ∞, so the sum over I 1 is positive and unbounded. For i ∈ I 2 , we know u i < 0 and x k i → 0, so log → −∞, so the sum over I 2 is also positive and unbounded. Consequently, 0 = lim k→∞ u, log x k x * = ∞, a contradiction. Hence, Γ ⊆ R n >0 is bounded away from infinity and away from the boundary of the positive orthant.
Next, we want to show that Γ ⊆ R n >0 is a closed subset. Let us fix ε > 0 such that Γ lies inside the hypercube Q = [ε, ε −1 ] n ⊆ R n >0 . Being the intersection of two closed sets, , which is again a closed set. Therefore, the set ( is the intersection of two closed sets, and thus it is closed in R n >0 . We say two manifolds X and Y of R n intersect transversally if at each point p ∈ X ∩ Y , their tangent spaces span the entire Euclidean space, i.e., T p (X) + T p (Y ) = R n . We refer the reader to [13,24] for the theory of transversality and intersection.
Again, let x 0 , x * ∈ R n >0 be two arbitrary vectors in what follows. In Lemma 6.2, we showed that our sign condition σ(S) ⊆ σ( S) implies σ(S) ∩ σ( S ⊥ ) = {0}, which is equivalent to the intersection (x 0 + S) ∩ (x * • exp S ⊥ ) containing at most one point. Indeed, this weaker sign condition together with dim S = dim S is enough to conclude that the two manifolds x 0 + S and x * • exp S ⊥ intersect transversally. This is the content of the follow lemma.

Proof. For any intersection point
If we further assume that dim S = dim S, we note that T p (x 0 +S)+T p (x * •exp S ⊥ ) is of dimension n. In other words, the manifolds x 0 + S and x * • exp S ⊥ intersect transversally. Now we are ready to state and prove our main result. The proof starts with a known intersection point, x * ∈ (x * + S) ∩ (x * • exp S ⊥ ). Next, we translate the affine space (x * + S) to (x 0 + S), creating a (d + 1)-dimensional strip of the form K + S, where d = dim S and K is a compact subset of R n >0 . This strip intersects x * • exp S ⊥ transversally, and we use Corollary 6.6 below to conclude that the intersection (K + S) ∩ (x * • exp S ⊥ ) is a one-dimensional manifold, whose boundary lies on the boundary of the affine strip K + S. Finally, we conclude the existence of a boundary point on x 0 + S by the uniqueness condition.
Consider the following differential topology result: Consider the setting where the ambient manifold is Y = R n >0 . If f is the inclusion map of a submanifold X into R n >0 , to say that the maps f and f | ∂X intersect the manifold Z transversally is equivalent to the manifolds X and ∂X intersect Z transversally. The preimage f −1 (Z) is the submanifold X ∩ Z. Moreover, dimension of the the intersection X ∩ Z is given by the equation We arrive at the following corollary: Corollary 6.6. Let X, Z ⊆ R n >0 be submanifolds, where Z is boundaryless. Suppose X intersects Z transversally and ∂X also intersects Z transversally. Then X ∩ Z is a manifold with boundary Our main result is: Theorem 6.7. Let S, S ⊆ R n be vector subspaces of equal dimension with σ(S) ⊆ σ( S). Then for any positive vectors x 0 , x * ∈ R n >0 , the intersection (x 0 + S) ∩ (x * • exp S ⊥ ) consists of exactly one point.
We apply Theorem 6.7 to show the existence and uniqueness of vertex-balanced steady state for a generalized mass-action system. Theorem 6.8 (Vertex-balanced steady states of a generalized mass-action system). Let (G, Φ, Φ) be a generalized mass-action network, with stoichiometric subspace S, and suppose that every vertex of G is the source of some edge, so that the kinetic-order subspace S is well-defined. Assume that dim S = dim S and σ(S) ⊆ σ( S). Then the following statements hold: i) Suppose for some rate constants κ, the generalized mass-action system (G κ , Φ, Φ) admits a vertex-balanced steady state x * . Then every stoichiometric compatibility class contains exactly one vertex-balanced steady state.
ii) Suppose G is weakly reversible and δ G = 0. Then for all rate constants κ, the generalized mass-action system (G κ , Φ, Φ) admits a vertex-balanced steady state x * . Moreover, every stoichiometric compatibility class contains exactly one vertex-balanced steady state.
iii) Under the premises of i), additionally suppose δ G = 0. Then every stoichiometric compatibility class contains exactly one positive steady state, which is vertex-balanced.
Proof. As x * is a vertex-balanced steady state for (G κ , Φ, Φ), the set of vertex-balanced steady state is Z κ = x * • exp S ⊥ . By Theorem 6.7, Z κ intersects the stoichiometric compatibility class x 0 + S exactly once for any x 0 ∈ R n >0 . This proves the statement i). If G is weakly reversible and δ G = 0, then the set of vertex-balanced steady states Z κ = ∅ for any κ > 0 [21]. By statement i), we conclude that every stoichiometric compatibility class contains exactly one vertex-balanced steady state.
If in addition, δ G = 0, then E κ = Z κ , i.e., there are no positive steady states that are not vertexbalanced. Consequently, there exists a unique steady state within each stoichiometric compatibility class, which is vertex-balanced.
We state a simpler version of iii) in the theorem above. Corollary 6.9. Let (G, Φ, Φ) be a weakly reversible generalized mass-action network, with stoichiometric subspace S and kinetic-order subspace S. Suppose that dim S = dim S, σ(S) ⊆ σ( S), and δ G = δ G = 0. Then for any choice of rate constants, every stoichiometric compatibility class contains exactly one positive steady state, which is vertex-balanced.
We have focused almost exclusively on the existence and uniqueness of vertex-balanced steady states for generalized mass-action systems. For complex-balanced equilibria of classical mass-action systems, much more is known. For example, complex-balanced equilibria are locally asymptotically stable within their stoichiometric compatibility classes. They are conjectured to be globally stable in their stoichiometric compatibility classes; this is known as the global attractor conjecture [4,6]. In particular, it has been shown that a complex-balanced equilibrium of a mass-action system is globally stable within its stoichiometric compatibility class if the network has a single connected component [1], or is strongly endotactic [8,23], or if the system is in R 3 [8,23]. A proof of the global attractor conjecture in full generality has been proposed in [4].
Not much is known about the local stability of vertex-balanced steady states of generalized massaction systems. It is, however, not true that a vertex-balanced steady state is always globally stable within its stoichiometric compatibility class, since it is possible for a generalized mass-action system to have multiple vertex-balanced steady states within the same stoichiometric compatibility class. Consider, for example, the following generalized mass-action system: where each box is a vertex of the graph; the top entry in each box is the reaction complex of that vertex (0 and X 1 + X 2 ), and the bottom entry in the parentheses is the kinetic-order complex (2X 1 and X 1 + 2X 2 ). The associated dynamical system of this generalized mass-action system is given by One can check that the set of vertex-balanced steady states is Z κ = {(t 2 , t) : t > 0}. If x 0 = (0, ε) T where 0 < ε < 1 4 , then there are two vertex-balanced steady states in x 0 + S = {(r, ε + r) : r ∈ R}. In particular, this implies that these vertex-balanced steady states cannot be globally stable in their stoichiometric compatibility class. Moreover, it is also possible for a unique vertex-balanced steady state (within its stoichiometric compatibility class) to be unstable. Consider the generalized mass-action system: κ κ Its associated dynamical system is This is an example of a reversible generalized mass-action system with δ G = δ G = 0, and its stoichiometric subspace S and its kinetic-order subspace S are R 2 . There is a unique positive steady state, which is vertex-balanced; nonetheless, it can be shown that this steady state is unstable. Moreover, the system is neither permanent nor persistent because any non-constant solution either converges to the origin or escapes to infinity.

An illustrative example
We conclude by applying Theorem 6.8 to the following example of a family of generalized mass-action systems. Let a, b, κ i > 0. Consider the generalized mass-action system (G κ , Φ, Φ) 0 (0) X 1 + X 2 (X 1 + aX 2 ) X 3 + X 4 (bX 1 + X 3 + X 4 ) The associated dynamical system is Another way to write the system of differential equations is The stoichiometric subspace and the kinetic-order subspace are where the dots indicate the negatives of the listed sign vectors. By visual inspection, we find that σ(S) ⊆ σ( S). Moreover, one can check that the deficiency δ G and the kinetic-order deficiency δ G are zero. Therefore, Corollary 6.9 applies and we conclude that, for any choice of rate constants, every stoichiometric compatibility class contains exactly one positive steady state, which is vertex-balanced.