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Proc Natl Acad Sci U S A. Dec 4, 2007; 104(49): 19175–19180.
Published online Nov 27, 2007. doi:  10.1073/pnas.0705731104
PMCID: PMC2148257
Applied Mathematics, Cell Biology

Subnetwork analysis reveals dynamic features of complex (bio)chemical networks

Abstract

In analyzing and mathematical modeling of complex (bio)chemical reaction networks, formal methods that connect network structure and dynamic behavior are needed because often, quantitative knowledge of the networks is very limited. This applies to many important processes in cell biology. Chemical reaction network theory allows for the classification of the potential network behavior—for instance, with respect to the existence of multiple steady states—but is computationally limited to small systems. Here, we show that by analyzing subnetworks termed elementary flux modes, the applicability of the theory can be extended to more complex networks. For an example network inspired by cell cycle control in budding yeast, the approach allows for model discrimination, identification of key mechanisms for multistationarity, and robustness analysis. The presented methods will be helpful in modeling and analyzing other complex reaction networks.

Keywords: reaction network, structure, elementary flux mode, bistability

In building mathematical models of intracellular processes, network topology and parameter values are often uncertain. Frequently, one can constrain the set of feasible network topologies by biological intuition, but this is only of limited help when it comes to parameter estimation. Thus, the question arises naturally as to whether there exists a set of parameter values such that a given network can express a certain qualitative behavior. In analyzing biological switches in, for example, developmental processes, the existence of multiple positive steady states (multistationarity) is of primary interest (1). Therefore, the connection between network topology and multistationarity has been discussed in a variety of contributions. In refs. 2 and 3, necessary conditions for bistability—the existence of two stable steady states—are given. The so-called Thomas conjecture states that positive feedback loops are necessary but not sufficient for bistability (3). In addition, Feinberg's chemical reaction network theory (CRNT) gives necessary and sufficient conditions for multistationarity, provided that all kinetics are of the mass action form and that the network satisfies certain structural conditions.

CRNT has recently received some attention in (systems) biology (46) because it connects questions about multistationarity for a system of ordinary differential equations (ODEs) derived from a biochemical reaction network to the network structure alone. Its assertions do not depend on parameter values and it only assumes mass action kinetics for all reactions. In principle, CRNT is a powerful tool because it translates questions on multistationarity into questions about the feasibility of just one of a potentially large number of systems of inequalities that are determined only by the network structure. However, CRNT requires the analysis of potentially large numbers of inequality systems, with many inequalities and unknowns. In addition, for some network structures, nonlinear inequalities have to be considered. As a consequence, current algorithms that automate the generation and analysis of these inequality systems (7, 8) practically limit the application of CRNT to networks of at most 20 complexes where the inequality systems are linear. This is clearly insufficient for investigating multistationarity in complex cellular signaling and metabolic networks that can encompass hundreds of components.

An apparent solution to the complexity challenges in cellular networks consists of defining smaller subunits—“motifs” (9) or “modules” (10)—that one can analyze separately. How to infer behaviors at the whole-network level, however, is not always clear. We argue that the analysis of certain subnetworks can be sufficient to confirm multistationarity. The subnetworks of interest are elementary flux modes, important tools in systems biology (11, 12). Briefly, instead of using CRNT to analyze multistationarity of the complete network, we propose to analyze subnetworks defined by a special class of flux modes called stoichiometric generators. The advantages of this approach are the following. One only has to generate those inequality systems that correspond to a subnetwork defined by a stoichiometric generator. Such a subnetwork presents a powerful reduction of the overall network; thus, there are fewer CRNT inequalities to be taken into account. Moreover, we show that, under some mild conditions on the stoichiometric generator, the inequality systems for the subnetwork are guaranteed to be linear. Finally, we prove that multistationarity in a subnetwork, induced by a stoichiometric generator, is sufficient for multistationarity in the overall network, provided some (computationally) easy to verify conditions hold. Thereby, we extend the class of networks where multistationarity can be investigated with the help of linear inequality systems. If one does not find multistationarity in one of these subnetworks, however, multistationarity can not be ruled out. As a proof-of-concept, we investigate a reaction network that is inspired by the control of the G1/S transition in the cell cycle of Saccharomyces cerevisiae.

Network Definitions

We consider mass action models for biochemical reaction networks in the form of ODEs and start with a brief presentation of the corresponding notation introduced in refs. 1315. Consider a simple reaction network of the form

equation image

with three chemical species A, B, and C involved in two reversible reactions. The associated kinetic parameters for the reactions are k1, …, k4. Let n be the number of species (n = 3 for the above example). Each species can be associated with a continuous variable representing its concentration. The combinations of species involved as educts or products of a reaction, the nodes of a network, are called complexes. In 1, we have complexes A + B, C, A, and the zero complex 0 to indicate that the system is open with respect to A. Let m be the number of complexes (with m = 4 for the small example). We assume that each network is displayed in the standard form defined in refs. 13 and 14: node labels are unique; that is, each complex is displayed only once. When the species concentrations x1, … for species A, … are collected in a vector x = (x1, …, xn)T, each xi can be associated to a unit vector ei of An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgn, the n-dimensional Euclidean space. Each complex can be related to a vector y corresponding to the sum of its constituent species: A + B with y1 = e1 + e2, C with y2 = e3, A with y3 = e1, and 0 with y4 = 0, the three-dimensional zero vector.

Let Y = [ y1, …, ym] be a matrix whose column vectors are the complexes of the network; i.e., Y [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgn×m. The reactions are represented by the incidence matrix Ia [set membership] {−1, 0, 1}m×r. Each column of Ia represents a reaction and has exactly one entry −1 for the educt complex and one entry 1 for the product complex. The remaining entries are zero. According to the mass-action law, each reaction has a reaction rate vi consisting of a rate constant ki [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg>0 and a monomial x yi = x1y1,i, …, xnyn,i, where x is the vector of concentrations and y is the column vector of Y corresponding to the educt complex of the reaction. Let := [ 1, …, r] [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgn×r be a collection of column vectors of Y with the following property: the i th column vector of corresponds to the educt of the i th reaction (i.e., vi = ki x yi). When several reactions share the same educt, will contain several copies of the corresponding complex vector i.

Using , we can collect the reaction rates in a function v : An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgr × An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgnAn external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgr, v(k, x) = diag(k) [var phi] (x), where k is the vector of rate constants and [var phi] (x) = (x1, …, x r)T the vector of monomials. The ODEs corresponding to a reaction network are now defined as

equation image

In general, the stoichiometric matrix N does not have maximal row rank. For s := rank (N), there exist ns conservation relations

equation image

with WT N = 0 for a W [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgn×(ns) of rank ns. Every biochemical reaction network endowed with mass action kinetics defines a system of the form presented in Eqs. 2 and 3.

For the small example in 1 we obtain

equation image

v(k, x) = (k1 x1 x2, k2x3, k3x1, k4)T. Thus, Eq. 2 takes the form

equation image

For the conservation relation in Eq. 3, we obtain x2 + x3 = c.

Chemical Reaction Network Theory

The distinguishing feature of CRNT is its ability to connect the structure of a reaction network and the existence of (multiple) equilibria for the corresponding system of ODEs. Its general idea can be summarized in the following way: for any network, a nonnegative integer δ called the deficiency can be derived from the network structure alone. For its formal definition, one more concept is needed: the linkage class. Network 1 consists of two sets of complexes: {A + B, C} and {A, 0}. Both sets are internally connected by reactions, while no reactions exist between elements of distinct sets. Sets of complexes that are internally connected by reactions are called linkage classes. Let l be the number of linkage classes in an arbitrary network. With the number of complexes, m, and the rank of N, s, the network deficiency is defined as the nonnegative integer δ = mls (13). Note that the deficiency only depends on the network structure and thus, in particular, δ is independent of parameter values. For the small example, it is easy to check that δ = 4 − 2 − 2 = 0.

If the deficiency δ is zero for a particular network, then no system of ODEs endowed with mass action kinetics that can be derived from the network admits multiple steady states (or sustained oscillations), regardless of the rate constants (13, 16). If δ is 1 and the network satisfies some mild additional conditions, then the deficiency one algorithm (14, 17) can be applied to decide whether the network can admit multiple steady states. If the deficiency is greater than one, then, under certain conditions, the advanced deficiency theory and corresponding algorithm can be used to decide about multistationarity (7, 18).

For each network where such an algorithm is applicable, several systems of equalities and inequalities (inequality systems, for short) can be formulated. These inequality systems only depend on the network structure and the complexes. For the deficiency one algorithm, it is guaranteed that the inequality systems are linear (17). The advanced deficiency algorithm might have to consider nonlinear inequalities (18). If one of these systems has a solution, and if the linear subspace S = im(N) contains a vector with the same sign pattern, then multistationarity is possible. In addition, a set of rate constants together with two distinct steady states can be calculated from this solution. If no such solution exists, then multistationarity is impossible.

Positive Steady States and Elementary Fluxes

Under mass-action law, positive species concentrations x [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg>0n and positive rate constants k [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg>0r imply positive reaction rates v(k, x) > 0. Then, the steady state form of Eq. 2, Y Ia v(k, x) = 0, implies that all reaction rates lie in a pointed polyhedral cone defined by the intersection of the kernel of the stoichiometric matrix Y Ia with the nonnegative orthant of An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgn; that is, v(k, x) [set membership] ker(Y Ia) ∩ An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg≥0r. Starting with the classic work of Clarke (19), this cone has become a well studied object because it characterizes the steady state flux space for a reaction network. Each element of the cone can be interpreted as a particular flux distribution, an allocation of values vi [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg to each reaction of the network, such that the overall network is in steady state.

As a pointed polyhedral cone, the flux space can be represented by nonnegative linear combinations of a finite set of generators or extreme rays (20). An elementary flux mode is a feasible flux distribution [an element v [set membership] ker(Y Ia) ∩ An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg≥0r] with a maximum number of zero entries. If, as in our setup, we consider only positive reaction rates, elementary flux modes are equivalent to extreme rays (21). Formally, the extreme rays Ei of ker(Y Ia) ∩ An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg≥0r are defined as follows:

equation image

Given Ei, Ej with Y Ia Ei = 0 and Y Ia Ej = 0. Then

equation image

where supp(Ej) = {i [set membership] {1, …, r} | Eji > 0,} denotes the support of vector Ej; i.e., the set of indices where Ej has nonzero values (21). We call a set of nonnegative vectors {E1, …, Ep} generators of the polyhedral cone.

The importance of the generators stems from their one-to-one correspondence to the reactions: nonzero entries can be interpreted as “active” reactions, and zero entries can be interpreted as “inactive” (in steady state). In this sense, every generator defines a subnetwork of the original reaction network consisting of all “active” reactions. Depending on whether a particular generator Ei [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg≥0r is contained in the kernel of Ia (Ia Ei = 0), two kinds of generators can be distinguished: generators with Ia Ei = 0, and stoichiometric generators with Ia Ei ≠ 0 (15). In general, calculating the generators is computationally hard, but there exist a variety of corresponding algorithms and software tools (21, 22).

Multistationarity and Subnetwork Analysis

Subnetworks Defined by Stoichiometric Generators.

Applications of CRNT are limited by both the network sizes and the need to analyze nonlinear inequalities. Subnetwork analysis obviously reduces the first limitation. Here, we show that investigating subnetworks defined by stoichiometric generators E, under weak additional conditions, also enables a decision on multistationarity by analyzing linear inequality systems.

In brief [for details and proofs, we refer to supporting information (SI) Appendix], the deficiency one algorithm imposes five conditions on the subnetwork structure. For a stoichiometric generator, four conditions are always fulfilled. For instance, we establish that for any E, s = ml − 1, leading to δ = mls = 1. The fifth condition states that each linkage class contains only one so-called terminal strong linkage class. In graph theory, a directed graph is called strongly connected if for every pair of nodes u and v there is a path from u to v and vice versa. The strongly connected components—strong linkage classes in CRNT notation (16, 17)—are the maximal strongly connected subgraphs of a directed graph. If no edge from a node inside a strong linkage class to a node outside exists, we have a terminal strong linkage class. However, we can not make statements on the number of terminal strong linkage classes for a general E, which leads us to the following fact:

Fact:

Consider a biochemical reaction network that is a subnetwork, defined by a stoichiometric generator of the overall network. Assume that the subnetwork is displayed in the standard form of CRNT, as defined above. Then the following holds: if every linkage class of the subnetwork contains only one terminal strong linkage class, then the deficiency one algorithm is applicable. Thus, in particular, only systems of linear inequalities have to be considered to decide about multistationarity.

Note that, because a reaction network defined by a stoichiometric generator cannot contain any cycles (apart from the one induced by the generator itself; see SI Appendix), each terminal strong linkage class consists of a single complex. Thus, it is straightforward to check whether a linkage class contains more than one terminal strong linkage class.

Extension to the Overall Network.

Suppose multistationarity has been established for the subnetwork defined by E using the deficiency one algorithm. Now, we want to confirm multistationarity for the overall network. We start by noting that the ODEs defined by a subnetwork can be obtained from the ODEs of the overall biochemical reaction network by assigning zero to all of those rate constants associated with reactions not contained in the subnetwork. To see this, consider the following subnetwork of 1:

equation image

with ODEs x1 = x2 = −k1 x1 x2 + k2 x3 and x3 = −x1 and conservation relations x1 + x3 = c1 and x2 + x3 = c2 (network 5 imposes an additional conservation relation). The vector of rate constants of 5 is kE = (k1, k2)T, and the “complementary vector” of rate constants not used in the subnetwork is kc = (k3, k4)T. To obtain the ODEs directly from those of network 1, the vector kE = (k1, k2, 0, 0)T can be used to define the subnetwork x = N v(kE, x) of the overall network 1. Using kc = (0, 0, k3, k4)T, one obtains the terms not contained in the ODEs defined by the subnetwork as N v(kc, x).

Let E be a stoichiometric generator with ρ nonzero entries. As for network 5, we use the decomposition k = kE + kc [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg≥0r given by

equation image

and define the vectors of rate constants contained in the subnetwork and the complementary rate constants by kE = (ki)i[set membership]supp(E) [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg>0ρ and kc = (ki)i/[set membership]supp(E) [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg>0r−ρ, respectively.

Suppose multistationarity has been established for the subnetwork defined by the stoichiometric generator E using the deficiency one algorithm (see Fact) so that there is a vector k*E [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg>0ρ of rate constants and two positive steady states x*1, x*2. Based on the implicit function theorem and on the special structure of the ODEs defined by a biochemical reaction network, we derive computationally simple tests to decide whether the positive stationary pairs (k*E, x*i) of the subnetwork can be continued to curves of positive stationary pairs [k* (ϵ), xi (ϵ)] of the overall network. To not obscure the algorithm with more mathematical detail than necessary, the relevant results and proofs are relegated to SI Appendix (in particular Theorem 1). The following algorithm can be applied without reference to the underlying mathematical details. It is based on the data (k*E, x*i) of the subnetwork and the Jacobian J (x, k) := Dx N v(k, x) of the overall system.

To apply the algorithm, collect those columns of Ia corresponding to reactions contained in the subnetwork in a matrix Ia(E) and the remaining ones in a matrix Ia(c), and define the matrices NE := Y Ia(E) [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgn×ρ and Nc := Y Ia(c) [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpgn×(r−ρ). Let SE be the orthonormal basis for im(NE), and let Wadd be the orthonormal basis for the subspace defined by the additional conservation relations imposed by the subnetwork (so that the columns of W and Wadd form a basis of the left-kernel of NE). Let [var phi]c (x) be the vector of monomials defined by the educt complexes of those reactions not contained in the subnetwork.

Algorithm:

(1) Define for i = 1, 2

equation image

If both A*i are regular, solve the linear matrix equations A*i Xi + B*i = 0 for matrices Xi.

(2) Obtain a positive vector of complementary rate constants k*c by solving the linear equations

equation image

If a positive k*c [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg>0r−ρ exists, define k*c [set membership] An external file that holds a picture, illustration, etc.
Object name is zpq04607833000g1.jpg≥0r as described above (by filling in 0's).

(3) Given a positive k*c (from step 2), define C*i := WaddT J (k*c, x*i) SE, D*i := WaddT J(k*c, x*i) Wadd and D*i :=C*i Xi + D*i (using Xi obtained in step 1). If both matrices D*i are regular, then there exists a line segment of positive rate constants

equation image

and a pair of smooth one-parameter curves of positive steady states x1 (ϵ), x2 (ϵ) of Eq.2 with WTx1 (ϵ) = W T x2 (ϵ) and xi(0) = x*i as long as ϵ > 0 is sufficiently small.

In summary, we propose the following steps to extend multistationarity from a subnetwork defined by a stoichometric generator to multistationarity of the overall network. Using Fact, obtain first k*E and x*1, x*2 for the subnetwork by the CRNT toolbox. Second, use Algorithm to test the continuability of (k*E, x*i) to the overall network for positive rate constants given explicitly by Eq. 6 (cf. step 2). The regularity assumptions in steps 1 and 3 entail the continuability by the implicit function theorem. Note that SI Appendix offers an alternative set of conditions that are sufficient to extend multistationarity to the overall network.

Conditions for multistationarity in biochemical reaction networks endowed with mass action kinetics have been derived using algebraic geometry in ref. 15, where it is shown that, under genericity conditions, multistationarity in a subnetwork defined by a stoichiometric generator implies multistationarity in the overall network. As we determine whether or not multistationarity indeed takes place in the overall network, we shed some new light on those results of ref. 15 in providing a computationally simple procedure to check sufficient conditions for multistationarity.

Example: G1/S Transition in Budding Yeast

Multistationarity is exploited by cellular control circuits that act as switches. They play an important role in enabling cells to make binary decisions. The mechanisms underlying such functions have been of considerable interest for experimental as well as theoretical investigations—for instance, in intracellular signaling, control of gene expression and metabolism, differentiation or the cell cycle (23, 24). Nevertheless, it is not yet completely clear which structural network characteristics are indispensable for establishing the switch-like behavior. Therefore, we considered a switching device in yeast cell cycle regulation as an ideal test case for our approach.

In a very simple form, the biochemical network consists of two regulators that mutually inhibit each other (Fig. 1A). Cyclin-dependent kinase (CDK), when associated with the mitotic cyclin Clb2p, promotes entry into mitosis through phosphorylation of its target proteins. Simultaneously, this activity prevents the exit from mitosis and subsequent passage to the G1 phase of the cell cycle. The competitive CDK inhibitor Sic1p is one component responsible for inactivating Clb2-CDK at the end of mitosis. Mitotic kinase activity, however, can phosphorylate Sic1p, thus targeting the inhibitor for rapid, proteasome-dependent degradation. The transition to a G1 state with high Sic1p concentration and low Clb2-CDK activity therefore requires activation of the phosphatase Cdc14p and concomitant stabilization of Sic1p (25).

Fig. 1.
Network structure for the G1/S model. (A) Biochemical mechanisms where arrows indicate biochemical reactions between cell cycle regulators (boxes). We distinguish between elementary reactions (solid lines) and catalyzed reactions (composite reactions; ...

Importantly, the transition between the cell cycle phases requires only transient Cdc14 activity as an input (trigger) signal for the bistable switch. Hysteresis, which means that at least two stable steady state output signals of the system exist for an identical input signal, underlies many cellular switches (23). It depends on the system's history—whether, for instance, low or high Sic1p concentration will establish. For the G1/S system, a transient activation of the phosphatase Cdc14p should move the system from the mitotic branch of low Sic1p concentration to the upper branch representing a G1 state with high Sic1p abundance and, consequently, low mitotic kinase activity. A potentially valid biochemical network has to represent this qualitative behavior. Here, we consider two very similar possible network structures. In one alternative, Clb2-CDK phosphorylates free Sic1p (binary complex model), whereas in the alternative, the already bound inhibitor is a substrate for a second kinase molecule (ternary complex model) (Fig. 1A). Both alternatives are biologically plausible yet hard to distinguish experimentally.

The translation of the biology-inspired reaction scheme into the formal network notation of CRNT is shown in Fig. 1B for the ternary complex model. The network has n = 9 species, with x{1…9} corresponding to Sic1, Sic1P, Clb, Clb·Sic1, Clb·Sic1P, Cdc14, Sic1P·Cdc14, Clb·Sic1P·Cdc14, and Clb·SicClb, respectively. These species form m = 17 complexes in r = 18 reactions. Note that the zero-complex 0 is associated with a nine-dimensional zero vector y1 = (0, …, 0). It incorporates that the system is open with respect to Sic1 and its phosphorylated form Sic1P: Sic1 can enter and leave the system, Sic1P can leave the system (see Fig. 1A). The ODEs derived from the standard reaction network (Fig. 1B) using mass action kinetics are given in SI Appendix.

Subnetwork Analysis

Multistationarity of Subnetworks.

The ternary complex network as defined above has deficiency δ = 5. Hence, the advanced deficiency algorithm has to be applied for its analysis. As it turns out, the implementation in the Chemical Reaction Network Toolbox by Martin Feinberg and Phillip Ellison (see refs. 7 and 8) cannot decide about multistationarity, because nonlinear inequalities have to be considered. In contrast, using the software tool CellNetAnalyzer (21), it is easy to establish that the network contains five stoichiometric generators. Another six minimal subnetworks are trivial because they contain two-cycles (forward and backward reactions in reversible reactions). The matrix of elementary flux modes E is provided in SI Appendix and the stoichiometric generators correspond to columns E7E11 of this matrix. Fig. 2 depicts the corresponding subnetworks graphically. Visual inspection of the subnetworks confirms that each linkage class contains exactly one terminal strong linkage class. Thus, by Fact the deficiency one algorithm is applicable. Using the CRNT toolbox (8), we find that the subnetworks corresponding to generators E7, E9, and E10 (Fig. 2 A, C, and D) can have multiple steady states. In contrast, it is guaranteed that those networks defined by E8 and E11 (Fig. 2 B and E) can not admit multistationarity.

Fig. 2.
Subnetwork structures corresponding to the stoichiometric generators (E7E11) for the ternary complex model (AE, respectively). Active reactions are denoted by black arrows, whereas gray color indicates reactions not used.

To demonstrate multistationarity numerically, we analyzed the bifurcation behavior of the complete system. For example, we applied Algorithm to elementary mode E10, where variation of the total concentration of Cdc14 can lead to qualitatively different steady state solutions (see SI Appendix for all model and analysis details). Using the CRNT toolbox, we established a vector of rate constants k*E and a pair of steady states x*1, x*2 for the subnetwork (given in SI Appendix). Matrices A*i and D*i are regular and positive k*c exist. Thus, Eq. 6 can be used to obtain k* (ϵ) for the overall network. Numerical analysis shows that for ϵ [set membership] [0, 12 · 10−3], the overall network exhibits multistationarity (Fig. 3). Aside, for ϵ [set membership] (0, 3·10−3) we found a Hopf bifurcation, thus indicating oscillations for the overall network (see SI Appendix for more comments about the possibility of Hopf bifurcations). Hence, the analysis of subnetworks already indicates that the required existence of multiple steady states can be achieved with the ternary complex model. The results confirm a prior finding from bifurcation analysis of an integrated cell cycle model that the Sic1 module alone can lead to bistability (26). Subnetwork analysis highlights possible reaction mechanisms underlying the bistability, and we consider these results as a proof-of-concept for the approach proposed here.

Fig. 3.
Extension of solutions to the overall network. Bifurcation diagram for the ternary complex model for several vectors of rate constants calculated as described in Algorithm using Eq. 6 and ϵ = 4·10−3 (dashed line), 8·10 ...

The binary complex model, which differs from the ternary complex model only in the mechanism for Sic1 phosphorylation by the cyclin-dependent kinase, can be decomposed into 12 elementary subnetworks (see SI Appendix for details). Among these subnetworks, 6 are stoichiometric generators. However, none of the generators admits multiple steady states according to CRNT. Nevertheless, analysis of the overall network reveals multistationarity for the overall system. Thus, it might operate as a proper switch for cell cycle control. This underlines that finding subnetworks with steady state multiplicity is sufficient but not necessary for multistationarity at the network level.

Robustness and Model Discrimination.

In contrast to the ternary complex model, where individual subunits can establish the desired switch-like behavior, the integrity of the binary complex model needs to be preserved for this function (see SI Appendix for detailed analysis). This has important consequences for the robustness of the two models. Robustness, in general, is defined as the resistance of qualitative network behavior to perturbations—for instance, in network structure or parameter values (27). Clearly, the ternary model is more robust in this sense than the binary model, although capturing this effect quantitatively is difficult. Previously, it has been proposed that robustness can also serve as a measure of plausibility for biological network models because, in fluctuating environments, robustness of functions should have been key for evolutionary selection (28). In this regard, subnetwork analysis could be used for model discrimination—the ternary complex mechanism would be considered more plausible.

Identification of Critical Reactions.

For the analysis of metabolic networks, elementary flux modes are often employed to identify essential reactions or subsystems—for example, for cell growth (12). Reactions are essential when they are used in all (relevant) flux modes because any steady state flux solution in the network can be constructed from a linear combination of the subnetworks. We can follow a similar line of argument for the analysis of dynamic features such as multistationarity. In this case, reaction participation in all elementary subnetworks that, on their own, have the potential to perform a certain function may identify critical reactions. For the ternary complex model, all subnetworks with potential multistationarity contain the formation of the trimeric complex between Sic1 and cyclin-dependent kinase, and subsequent phosphorylation of Sic1 (Fig. 2 A, C, and D). This indicates that for a robust switch-like function of the system, these reactions are important. However, they might not be essential, as analysis of the binary complex model demonstrates. Comparison of these three subnetworks also reveals that action of the phosphatase Cdc14 is not necessary for multistability, which can arise from cyclin–inhibitor interactions alone (Fig. 2 A and C). Yet, to accommodate the experimental observation that transient activation of Cdc14 is critical in vivo, one needs to consider the third subnetwork (Fig. 2D) as well.

Discussion

Establishing relations between network structure and possible behavior is an important issue for the analysis of complex (bio)chemical reaction networks. CRNT is a well founded theory for tackling these key questions—for instance, regarding the existence of multiple steady state solutions, which in biology is critical for complex processes such as the cell cycle and development. To make the approach more scalable in practice, we here presented the idea of subnetwork analysis, which combines CRNT with the well established theory of elementary flux modes primarily used for metabolic network analysis.

Our analysis of a subsystem in yeast cell cycle control showed that, in principle, this approach allows for dealing with more complex networks than CRNT is able to handle. The subnetwork approach is not always successful—sometimes an analysis of the entire network cannot be avoided to investigate multistationarity. Thus, we need a better understanding for which classes of networks the proposed reduction to subnetworks is impossible. In addition, further efforts in the algorithmic aspects of CRNT appear essential for an extension to larger systems. In general, however, subnetwork analysis can be used to investigate potential network behaviors based on the reaction network structure alone. This can help in analyzing robustness properties, identifying critical reaction mechanisms, and discriminating between competing hypotheses on network structures. Therefore, the modular analysis approach presented here is relevant for the modeling and analysis of larger classes of (bio)chemical reaction networks.

Supplementary Material

Supporting Information:

Acknowledgments

This article is dedicated to the memory of Karin Gatermann. She brought the exciting connection between algebraic geometry and (bio)chemical reaction networks to the attention of C.C. This article was inspired by numerous discussions with her. C.C. was supported by the Saxony-Anhalt Ministry of Education (Research Focus Dynamical Systems).

Footnotes

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0705731104/DC1.

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