Computing Smallest Intervention Strategies for Multiple Metabolic Networks in a Boolean Model

Abstract

This article considers the problem whereby, given two metabolic networks N₁ and N₂, a set of source compounds, and a set of target compounds, we must find the minimum set of reactions whose removal (knockout) ensures that the target compounds are not producible in N₁ but are producible in N₂. Similar studies exist for the problem of finding the minimum knockout with the smallest side effect for a single network. However, if technologies of external perturbations are advanced in the near future, it may be important to develop methods of computing the minimum knockout for multiple networks (MKMN). Flux balance analysis (FBA) is efficient if a well-polished model is available. However, that is not always the case. Therefore, in this article, we study MKMN in Boolean models and an elementary mode (EM)-based model. Integer linear programming (ILP)-based methods are developed for these models, since MKMN is NP-complete for both the Boolean model and the EM-based model. Computer experiments are conducted with metabolic networks of clostridium perfringens SM101 and bifidobacterium longum DJO10A, respectively known as bad bacteria and good bacteria for the human intestine. The results show that larger networks are more likely to have MKMN solutions. However, solving for these larger networks takes a very long time, and often the computation cannot be completed. This is reasonable, because small networks do not have many alternative pathways, making it difficult to satisfy the MKMN condition, whereas in large networks the number of candidate solutions explodes. Our developed software minFvskO is available online.

1. Introduction

Metabolic networks represent relations between biochemical reactions and metabolites in living cells. The removal (or knockout) of metabolism-related genes is often simulated in metabolic networks, as the perturbation of genes frequently corresponds to the inhibition of certain reactions in metabolic networks.

Many types of mathematical models have been developed for this purpose. For small size networks, models using the ordinary differential equations (ODEs) are often used. Although ODEs have a detailed explanatory power, their applicability is limited by the difficulty of obtaining the necessary kinetic parameters, and their limited scalability. On the other hand, less detailed approaches like Boolean models and constraint-based models have been used in larger networks (Gonçalves et al., 2013).

Flux balance analysis (FBA) (Raman and Chandra, 2009; Varma and Palsson, 1994) is a constraint-based mathematical model of metabolic networks in which the stoichiometry and the biomass objective functions are used to predict cell growth rates (Kauffman et al., 2003). Although the standard FBA simply maximizes the biomass objective function, the minimization of metabolic adjustment method (MOMA) (Segre et al., 2002) seeks to minimize the difference between the wild and the knocked-out flows. Flux variability analysis (FVA) assesses the range of this difference (Shlomi et al., 2009), and the optimal knockout strategies have been developed based on such flow models. Optknock (Burgard et al., 2003) determines which reactions should be knocked-out to maximize the biomass objective function. Using bilevel programming, the biomass objective function is first maximized for each knockout(s), and then a reaction set is chosen from the resulting candidates. In contrast, although RobustKnock (Tepper and Shlomi, 2010) is also based on bilevel programming, it maximizes the minimized biomass objective functions by selecting different reactions. Optorf (Kim and Reed, 2010) integrates transcriptional regulatory networks and metabolic networks. The above-mentioned flow models need to define the biomass objective function, which often involves a linear combination of more than 100 metabolites (Raman and Chandra, 2009). Furthermore, it is difficult to define the biomass objective function for higher organisms such as humans, and the essential assumption concerning growth-optimal behavior is not always fulfilled (Schuster et al., 2008).

Another flow model approach for metabolic networks is based on the elementary mode (EM) concept, which does not require a biomass objective function. EM is a minimal set of reactions in a steady state, in which all irreversible reactions are used in the appropriate direction (Schuster and Hilgetag, 1994; Schuster et al., 2000). Stelling et al. (2002) estimated the effect of knockouts by the number of elementary modes (EMs) that include the knocked-out reaction. Based on the idea of topological flux balance (TFB) (Smart et al., 2008), the topological impact degree (TID) calculates the number of reactions that have an EM in common with the knocked-out reactions (Jiang et al., 2009; Tamura et al., 2011). The flux balance impact degree (FBID) is the number of reactions that are not included in any EMs that do not include the knocked-out reactions (Zhao et al., 2013). A minimal cut set (MCS) is the minimal set of reactions whose inactivation leads to a failure of the specified reactions (Klamt and Gilles, 2004; Klamt, 2006). Acuña et al. (2009) proved that computing the MCS is NP-hard. MCSs in a metabolic network are EMs in a dual network (Ballerstein et al., 2012). EMs and MCSs can be calculated by FluxAnalyzer (Klamt and Gilles, 2004) and CellNetAnalyzer (CNA) (Klamt et al., 2007), MATLAB-based tools that can be applied to many network-based problems. Because knockouts may induce side effects that disables the desired functionality, the constrained MCS allows additional constraints that preserve a set of desired modes (Hädicke and Klamt, 2011). Based on an EM-based analysis and the idea of minimal metabolic functionality, optimal knockout strategies were analyzed to maximize industrial production, and the effect was confirmed by biological experiments (Trinh et al., 2006; Unrean et al., 2010).

Another mathematical model of metabolic networks is the Boolean model. Although not suitable for analyzing the mass flow of metabolic networks, the logical analysis of this model is relatively solid as it needs less information than the flow-based models. The synthetic accessibility is defined as the number of reactions required to transform a set of source metabolites into a set of target metabolites (Wunderlich and Mirny, 2006). For a given set of source nodes (called seeds), the scope is defined as the set of producible compounds (Handorf et al., 2005). The damage is the number of reactions that are affected by the knocked-out reactions (Lemke et al., 2004). To take side effects into account, Sridhar et al. (2008) developed a branch-and-bound-based algorithm, OPMET, which minimizes damage to nontarget nodes. Tamura et al. (2010) developed an integer linear programming (ILP)-based method for the Boolean reaction cut (BRC) problem and analyzed the computational complexity of BRC (Tamura and Akutsu, 2010), in which the number of inhibited reactions is minimized to make target compounds nonproducible (Tamura et al., 2010). In the Boolean model of metabolic networks, reactions and compounds can be represented by “AND” and “OR” nodes, respectively, and the network can then be considered as a bipartite graph. Lu et al. (2014) considered the problem of adding a minimum reaction set so that the target compound becomes producible in a Boolean metabolic network. Although the above research focuses on metabolic networks, the basic framework of the Boolean model can be extended to other types of biological networks. For example, Flöttmann et al. formalized signal transduction networks using reaction contingency-based bipartite Boolean modeling (Flöttmann et al., 2013), and Samaga et al. (2010) studied minimal intervention sets (MISs) for Boolean signaling networks.

Thus, in many cases, the effects of reaction inhibition are first estimated in each metabolic network model, and then a certain optimization problem is defined for each estimation model. Furthermore, in addition to maximizing the desired functions, some problems minimize undesirable functions as side effects in the same network.

As a reasonable extension of the above research, this article considers the situation in which different types of cells exist in the same place. For example, many types of bacteria exist in our intestine. Some of them are good bacteria, others are bad bacteria, and they have different metabolic networks. As another example, normal cells and cancer cells may exist in the same organ. In such cases, it may be useful to compute the optimal knockout strategies that would cause the bad cells to lose some essential functionality but allow the good cells to survive. Therefore, we consider the following minimal knockout for multiple networks (MKMN) problem: given two different metabolic networks with source and target compounds, obtain the minimum number of reactions whose inhibition induces the target compounds to become nonproducible in bad cells but producible in good cells.

Although we focus in this article on the case in which two networks are given, it is straightforward to extend the proposed methods to the case in which more than two networks are given.

We analyze this problem using the Boolean model (MKMN-B) and the EM-based model (MKMN-EM). As both MKMN-B and MKMN-EM are NP-complete (as shown in the “Theoretical results” section), we develop methods based on ILP (Schrijver, 1998; Li et al., 2007) for MKMN-B and MKMN-EM. ILP is often used to formalize NP-complete problems, and there is an efficient free ILP solver called CPLEX (IBM, 2010). In MKMN-B, to properly account for the effect of cycles, we utilize the notion of the maximal valid assignment (MaxVA) (Tamura et al., 2010) and the minimal valid assignment (MinVA) (Lu et al., 2014). To obtain faster ILP-based methods by reducing the number of variables, we develop IP-FVS1 and IP-FVS2, which utilize the idea based on feedback vertex sets (FVS) (Tamura et al., 2010), MaxVA is strictly applied in IP-FVS1 but is not applied to the nodes detected by FVS in IP-FVS2. We also develop faster algorithms IP-FVS1-approx and IP-FVS2-approx by limiting the number of time steps, although the optimality of the resulting solutions is not ensured. For MKMN-EM, we develop IP-EM, in which CNA is used to obtain the EMs, and then formalize MKMN-EM using ILP, since MKMN-EM is NP-complete, even when EMs are given.

We apply our developed methods to the metabolic networks data of clostridium perfringens SM101 (CPR) and bifidobacterium longum DJO10A (BLJ), as downloaded from the KEGG database (Kanehisa and Goto, 2000). The CPR network is denoted by N₁, and the BLJ network is denoted by N₂. Dataset 1 consists of only the central metabolism, and N₁ and N₂ consist of 73 and 82 nodes, respectively. Dataset 2 consists of the carbon metabolism, fatty acid metabolism, and biosynthesis of amino acids, and N₁ and N₂ consist of 251 and 328 nodes, respectively. Dataset 3 consists of whole metabolic networks, and N₁ and N₂ consist of 1231 and 1881 nodes, respectively. We apply IP-FVS1, IP-FVS2, and IP-EM to datasets 1, 2, and 3, where the target compounds are pyruvate, acetyl-CoA, acetate, oxaloacetate, and phosphoenolpyruvate. For most cases in dataset 3, IP-FVS1, IP-FVS2, and IP-EM could not complete the computation in a provisional time limit of 2 hr. Hence, we applied IP-FVS1-approx and IP-FVS2-approx to this dataset and limited the number of time steps to 10. For dataset 1, IP-FVS1, IP-FVS2, and IP-EM took at most 3 sec for every target compound, but in most cases there are no solutions. For dataset 2, IP-FVS1, IP-FVS2, and IP-EM took at most 20 sec for every target compound, and solutions were obtained in about half of the cases. Finally, for dataset 3, IP-FVS1-approx(10) and IP-FVS2-approx(10) finished their calculations within 15 sec for every target compound. We examine the relations between the obtained solutions and predecessors of the target compound in N₁ and N₂.

2. Materials and Methods

2.1. Main problems

In this subsection, the main problem minimal knockout for multiple networks, (MKMN) is explained using examples. Mathematical definitions are given in the next subsection. The MKMN for the Boolean model and the EM-based model are called MKMN-B and MKMN-EM respectively.

Suppose we have two metabolic networks N₁ and N₂, as shown in Figure 1A. Rectangles and circles represent reactions and compounds, respectively. For example, reaction r₁ has the substrates (reactants) {c₁, c₂}, and products {c₄, c₅, c₆} for both N₁ and N₂. However, since the topologies of N₁ and N₂ are slightly different, N₁ does not include reaction r₄. For N₁, {c₁, c₃, c₇} are called source nodes and are assumed to be supplied by the external environment, whereas {c₁, c₂, c₃, c₇} are source nodes for N₂. The purpose of MKMN is to find the minimum number of reactions whose inhibition (deletion) induces the target compound to be nonproducible in N₁ but producible in N₂. In Figure 1A, c₉ is the target compound. This should be nonproducible in N₁ but producible in N₂ after the minimum number of reaction deletions.

FIG. 1.

An example of minimum knockout for multiple networks (MKMN) problem. MKMN is a problem to find the minimum number of reactions whose inhibition makes the target compound nonproducible in N₁ but producible in N₂. (A) The MKMN solution for this example is {r₃}, whose inhibition prevents production of v_c9 in N₁ but not in N₂; (B) r₁ of N₁ in Figure 1A is decomposed into r₁ and s₁.

MKMN-B is defined as a problem to solve MKMN in a Boolean model. In MKMN-B, if all substrates exist, the reaction occurs. For example, in Figure 1A, r₁ occurs if both c₁ and c₂ exist. Note that the existence of c₂ is not always the same in N₁ and N₂. In this example, the solution of MKMN-B is to inhibit {r₃}, as c₂ becomes nonproducible, r₁ cannot take place, and as c₄ becomes nonproducible, r₂ cannot take place, and then c₉ becomes non producible in N₁ but producible in N₂ via r₂ or r₄. Note that directed cycles often play important roles. On the other hand, the inhibition of {r₁} is not a solution as this makes c₉ non producible in both N₁ and N₂. Similarly, inhibiting {r₂} is not a solution either, since c₉ would remain producible in both N₁ and N₂.

The problem setting of MKMN-B for N₁ is the same as that for the Boolean reaction cut in Tamura et al. (2010). To properly account for the effect of cycles, Tamura et al. (2010) defined the maximal valid assignment (MaxVA). For example, although the optimal solution of MKMN-B for N₁ and N₂ as in Figure 1A is {r₂}, if neither c₂ nor c₆ initially exists in N₁, then neither r₁ nor r₃ can occur, and c₉ becomes nonproducible. Therefore, “inhibiting no reactions” looks like the optimal solution of MKMN-B if c₂ is not supplied to N₁ from the external environment. To avoid such ambiguity, MaxVA is defined as the 0-1 assignment that satisfies the Boolean constraint for each node, and the number of 1's is maximal among such 0-1 assignments. In Tamura et al. (2010), it was proved that MaxVA can be calculated by initially assigning a value of 1 to all nodes; the effect of reaction inhibition gradually reaches other nodes, and finally converges to some 0-1 assignment, which corresponds to the MaxVA. In MKMN-B, the assumption of MaxVA is applied to N₁.

However, the assumption of MaxVA is not appropriate for N₂ in MKMN-B. For example, suppose that N₁ and N₂ in Figure 2A are given. If the assumption of MaxVA is applied to both N₁ and N₂, {r₁} is obtained as the optimal solution of MKMN-B, since (c₁, c₂, c₃, c₄, r₁, r₄, r₅, r₆) = (1, 1, 1, 1, 0, 1, 1, 1) is the MaxVA of N₂ if {r₁} is inhibited. However, {r₂, r₃} seems to be a more appropriate solution of MKMN-B, as the inhibition of {r₁} disconnects the source node c₁ and the target compound c₃ in N₂. To account for the effect of cycles, and make the target compound producible, the MinVA notion has been shown to be efficient (Lu et al., 2014). Similar to MaxVA, MinVA is a valid assignment, in which the number of 1's is minimal. In MKMN-B, MaxVA and MinVA are applied to N₁ and N₂, respectively.

FIG. 2.

The minimal valid assignment (MinVA) is applied to N₂. The FVS-based operation for N₂ is different from that of N₁; (A) {r₂, r₃} is the optimal solution of MKMN-B for N₁ and N₂; (B) However, if s₄ is initially assigned a value of 1 in , {r₁} is obtained as the solution.

MKMN-EM is defined as a problem to solve MKMN in the elementary mode (EM)-based model. An EM describes a feasible and balanced (steady-state) flux distribution through the network, which is minimal with respect to the utilized reactions (enzymes) (Klamt and Gilles, 2004). Note that {r₃} is not the solution of MKMN-EM in Figure 1A, since the target compound c₉ becomes nonproducible in both N₁ and N₂. In MKMN-EM, we consider only the topology of each EM, and do not consider the magnitude of each flow. Therefore, an EM can be represented by a 0-1 assignment of reaction and compound nodes. For example, Table 1 shows the 0-1 assignments of the EMs for the example network shown in Figure 3, where {c₁, c₃, c₇, c₉} is a set of external compounds. Suppose that the target compound c₇ should become nonproducible, where {c₁, c₃, c₉} is a set of source nodes. As EM1, EM2, EM3, and EM5 can produce c₇, the set of reactions to be inhibited must include at least one from each of these four EMs. For example, inhibiting {r₃,r₅} makes c₇ nonproducible since r₃ is included in EM2 and EM5, and r₅ is included in EM1 and EM3.

Table 1.

The 0-1 Assignments Corresponding to the Elementary Modes of the Example Network of Figure 3, Where {c₁, c₃, c₇, c₉} is a Set of External Compounds

	p1	q1	p2	q2	r3	r4	r5	r6	r7	r8	c1	c2	c3	c4	c5	c6	c7	c8	c9
EM1	0	0	0	0	0	1	1	1	1	0	0	0	1	1	1	1	1	0	0
EM2	0	1	0	0	1	0	0	0	1	0	1	1	0	0	0	1	1	0	0
EM3	0	1	0	1	0	0	1	1	1	1	1	1	0	1	1	1	1	1	1
EM4	1	0	1	0	0	1	0	0	0	0	1	1	1	1	0	0	0	0	0
EM5	0	0	1	0	1	1	0	0	1	0	0	1	1	1	0	1	1	0	0

FIG. 3.

The 0-1 assignments of elementary modes (EMs) of this network are listed in Table 1.

2.2. Mathematical definitions of main problems

In this subsection, the main problems of this article are mathematically defined. A metabolic network is defined as a directed bipartite network N = (V_c, V_r, E), where V_c is a set of compound nodes, and V_r is a set of reaction nodes. E is a set of edges connecting a compound node and a reaction node. Therefore, neighbors of compound nodes are always reaction nodes, and neighbors of reaction nodes are always compound nodes. In MKMN, two metabolic networks N₁ = (, , E₁) and N₂ = (, , E₂) are given, where || = m₁, || = n₁, || = m₂, || = n₂, m = m₁ + m₂, and n = n₁ + n₂ hold. For example, in Figure 1A,  = {r₁, r₂, r₃}, m₁ = 9, n₁ = 3, m₂ = 10, and n₂ = 4 hold.

V_ex is a set of external nodes, that are affected by the external environment not described in N.  ⊂  and  ⊂  are sets of source nodes in N₁ and N₂, respectively. Similarly,  ⊂ and  ⊂  are sets of target nodes in N₁ and N₂, respectively. It holds that , , ,  ⊂ V_ex. In Figure 1A,  = {c₁, c₃, c₇},  = {c₁, c₂, c₃ c₇}, and  =  = {c₉} hold. Note that some may not be included by either V_s or V_t in the EM-based analysis. In this article, we assume that source nodes do not have incoming edges but that target nodes are allowed to have outgoing edges to take the multiple target compounds into account.

In both the Boolean model and the EM-based model, values of either “0” or “1” are assigned to each node as the magnitude of each flow is not taken into account in MKMN-EM. Suppose that “1” is assigned to a node. In the Boolean model, this means that either the reaction occurs or the compound exists. In the EM-based model, it means that the corresponding node is included in some EM. If “0” is assigned to a node, then either the reaction does not occur, or the compound does not exist in the Boolean model. In the EM-based model, it means that the corresponding node is not included in the EM. Let V_a ⊆  ∪  be perturbed reactions. If is included in V_a, v = 0 always holds whatever values are assigned to other nodes.

Let A_B be such an assignment. In the Boolean model, A_B is called a valid assignment (VA) if it satisfies each of the following:

• (i) For each , v = 1 holds;

• (ii) for each , v = 1 if and only if there is some such that and u = 1 hold;

• (iii) for each , v = 1 holds if and only if holds and u = 1 holds for all .

Therefore, in the Boolean model, each reaction node corresponds to an “AND” node, and each compound node corresponds to an “OR” node. A is the MaxVA for V_a, if A is a valid assignment for the given V_a, and the number of 1's is the maximal. Similarly, A is the MinVA for V_a, if it is a valid assignment for the given V_a and the number of 1's is minimal.

MKMN-B is mathematically defined as follows:

Problem: MKMN-B (minimum knockout for multiple networks in the Boolean model)

• • Input: Metabolic networks N₁ = (, , E₁), N₂ = (, , E₂), source nodes  ⊂   ⊂ , and target nodes  ⊂ ,  ⊂ .

• • Output: The minimum cardinality set V_a ⊆  ∪ such that MaxVA for N₁ ensures every is 0, and MinVA for N₂ ensures every is 1.

Note again that the MaxVA and the MinVA are applied to N₁ and N₂, respectively. (See also the examples of Figs. 1 and 2.)

On the other hand, an EM describes a feasible and balanced (steady-state) flux distribution through the network, which is minimal with respect to utilized reactions (enzymes) (Klamt and Gilles, 2004). An EM is said to be a relevant EM if v₁ = v₂ = 1 holds for some and in its 0-1 assignment. Then, MKMN-EM is mathematically defined as follows:

Problem: MKMN-EM (minimum knockout for multiple networks in the elementary mode model)

• • Input: Metabolic networks N₁ = (, , E₁), N₂ = (, , E₂), external nodes V_ex1 ⊂  ∪ , V_ex2 ⊂  ∪ , target nodes  ⊂ ,  ⊂ , and a stoichiometry matrix S.

• • Output: The minimum cardinality set V_a ⊆  ∪ , which satisfies each of the following:

  • –For all , v = 0 holds for any relevant EM on N₁ = (, \V_a, E₁).

  • –For all , v = 1 holds for some relevant EM on N₂ = (, \V_a, E₂).

2.3. Integer linear programming–based method for MKMN-B

Since MKMN-B is NP-hard as discussed in “Theoretical results,” we develop an integer linear programming (ILP)-based method for its solution. In ILP, every Boolean constraint must be represented by linear equations or inequalities. In this article, we use two linear representations of Boolean constraints:

LP1 (Tamura et al., 2010): Since the Boolean “AND” relation can be converted into , it can be represented by the following linear inequalities:

where all variables are binary.

Similarly, as the Boolean “OR” relation can be converted into , it can be represented by the following linear inequalities:

where all variables are binary.

LP2 (Akutsu et al., 2012): Another type of linear function representation of Boolean functions is as follows: The Boolean “AND” can be represented by the following linear inequalities:

where all variables are binary.

Similarly, the Boolean “OR” can be represented by the following linear inequalities:

where all variables are binary.

As described in Tamura et al. (2010), we should introduce notion of time for calculating the MaxVA. A naive method for calculating the MaxVA is as follows. Each node is initially assigned 1, specified reactions are inhibited, effects of the inhibition affect neighbor nodes in the next time step, and the finally converged 0-1 assignment is the MaxVA. Similarly, as described in Lu et al. (2014), the MinVA can be calculated by initially assigning 0 to each node other than the source nodes. However, in this naive method, the number of time steps should be the same as the total number of nodes to ensure that the 0-1 assignment converges. Thus, the number of variables needed for the ILP formalization is O((m + n)²).

By utilizing an FVS, the number of time steps is reduced to f, and then the number of variables in the ILP formalization becomes O(f (m + n)). [The FVS is a set of nodes whose removal makes the graph acyclic, and f is the size of the FVS (Tamura et al., 2010).] For example, r₁ of N₁ in Figure 1A can be decomposed into r₁ and s₁, as shown in of Figure 1B. Different from the naive method, the time step advances by one only when the value of the “r” node is copied to the “s” node. Suppose that r₃ is inhibited in Figure 1B. Because we consider the MaxVA for N₁ (and ), all source nodes and “s” nodes are assigned 1 at t = 0. If r₃ is not inhibited, all nodes become 1 at t = 0 since the time step advances only when the value of r₁ is copied to s₁. However, since r₃ is inhibited, (r₃, c₂, r₁) = (0, 0, 0) holds at t = 0, whereas (s₁, c₆) = (1, 1) holds at t = 0. Then, r₁ = 0 at t = 0 is copied to s₁ at t = 1. Thus, all nodes other than c₁, c₃, and c₇ become 0 when r₃ is inhibited. Note that the necessary time steps for calculating this is 2 (t = 0 and t = 1). This is the size of FVS + 1. In the naive method, the time step advances whenever 0 affects its adjacent nodes. Therefore, the number of necessary time steps is as large as the number of nodes to ensure that the 0-1 assignment converges.

Based on this idea, we formalize an ILP-based method to solve MKMK-B. In the following, we explain our proposed methods using examples. It is straightforward to extend the examples to a general case.

When N₁ and N₂ in Figure 1A are given, IP-FVS1-LP1 is formalized as follows.

Maximize

Subject to

/* Definitions of N₁

for all

/* Reactions

/* Compounds

/* SR1

/* Es

/* Source compounds

/* Definitions of N₂

for all

/* Reactions

/* Compounds

/* Es

/* Source compounds

/* SR1

where every variable takes a value of either 0 or 1. “T” and “F” stand for true (1) and false (0), respectively. “R”, “C,” and “E” stand for reaction, compound, and enzyme, respectively. For each reaction node, TRi(t) = 1 (resp., FRi(t) = 1) represents r_i = 1 (resp., r_i = 0). Therefore, TRi(t) + FRi(t) = 1 holds for any reaction node r_i at time step t; “t = 0” means the initial time step. For example, if TR2(1) = 0, which means r₂ = 0 (the reaction node r₂ does not occur) at t = 1, FR2(1) = 1 automatically holds at the same time. The use of FRi(t) simplifies this illustration. In the implementation, FRi(t) is replaced by 1-TRi(t) to reduce the number of variables. Similarly, each compound c_i is represented by TCi(t) and FCi(t), where TCi(t) + FCi(t) = 1. For example, TC3(1) = 1 means that c₃ = 1 (the compound c₃ exists) at t = 1, and FC3(1) = 1 means that c₃ = 0 (the compound c₃ does not exist) at t = 1.

The maximal number of time steps f₁ and f₂ can be calculated by f₁ = |F₁| +1 and f₂ = |F₂| +1, where |F₁| and |F₂| are the sizes of the FVS for N₁ and N₂, respectively. Note that “ +1” is necessary because at least 1 time step is necessary even if there is no FVS. Furthermore, since the number of time steps of N₁ and N₂ may be different, the ILP formalization is defined for each network. In the above example for IP-FVS1-LP1, (2), (4)–(19), (38), and (39) give the ILP formalization of N1, whereas (3), (20)–(37), and (40) give the ILP formalization of N2. To distinguish each node, “_N1” (resp., “_N2”) is appended to each variable name to denote which network the node belongs to. For example, TR1_N1(1) = 1 means that r₁ = 1 (the reaction node r₁ occurs) in N₁ at t = 1, and FR1_N2(3) = 1 means that r₁ = 0 (the reaction r₁ does not occur) in N₂ at t = 3. “_N1” and “_N2” are not appended to the variables of nodes that are common to both N₁ and N₂.

In IP-FVS1-LP1, the Boolean “AND” relation of a reaction node is converted into LP1 type linear inequalities as shown in (5). The relation in (5) represents the constraints between the reaction node r₂ and each incoming compound node (c₃ and c₄) in N₁. Two additional variables TE2(t₁) and FE2(t₁) are also included in (5), since we add a virtual predecessor node e_i to each reaction node r_i in both networks. TEi(t) and FEi(t) denote whether r_i is inhibited. Since r_i is represented by an “AND” node, e_i = 0 can ensure that r_i remains inactive, even if all other predecessors of r_i are 1. Furthermore, the same TEi(t) and FEi(t) are used for each common reaction node r_i between N₁ and N₂. In the above example for IP-FVS1-LP1, the common reaction nodes r₂ of the two networks are connected by e₂ [TE2 and FE2 in (5) and (21)].

The Boolean “OR” relation of a compound node is converted into LP1-type linear inequalities, as shown in (12), when the indegree of the compound node is more than 1. If the indegree of the compound is 1, we need only copy the value from the variable of the predecessor reaction node as shown in (7).

For N₁, the values of each source node and each node newly created by s_i at t = 0 are set to 1 to realize the MaxVA as shown in (17)–(19) and (38). The value of the target node is set to 0 at t = f₁ to ensure that the target node eventually cannot be produced in N₁. For N₂, only the source nodes and target node values at t = f₁ are set to 1, as shown in (3) and (34)–(37), to realize MinVA.

The MKMN result can be calculated by maximizing TEi(t), as shown in (1), since e_i represents whether the reaction node is inhibited. As mentioned above, FEi(t) is replaced by 1-TEi(t). Relation (1) implies that the minimal number of reaction knockouts is calculated by maximizing the number of reaction nodes that are not knocked out. In addition, (2) indicates that the target compound c₉ should eventually not be produced in N₁, whereas (3) suggests that the target compound c₉ should eventually be produced in N₂.

Relations (4)–(19) are the node constraints for N₁; {r₁} is chosen as the FVS F₁. The maximal time step f₁ of N₁ is 2, where f₁ = |F₁| +1 = 1 + 1 = 2. Relations (4)–(6) represent the constraints on reactions r₁–r₃, respectively, and (7)–(12) represent the constraints on compounds c₂, c₄, c₅, c₆, c₈, and c₉, respectively. In (8)–(10), the predecessor reaction node r₁ of compounds c₄–c₆ is replaced by s₁, where TSR1_N1(t₁) = 1 represents s₁ = 1 (s₁ is active) at t₁ in N₁, and TSR1_N1(t₁) = 0 represents s₁ = 0 (s₁ is inactive) at t₁ in N₁. Relation (13) increases the time step by 1 when the value of r₁ is copied to s₁. Relations (14)–(16) ensure that the state of each reaction (inhibited or uninhibited) does not change during the time transition, and (17)–(19) define c₁, c₃, and c₇ as the source nodes of N₁.

Relations (20)–(37) are the node constraints for N₂. Since the FVS F₂ = {}, the maximal time step f₂ of N₂ is 1, where f₂ = |F₂| +1 = 0 + 1 = 1. Relations (20)–(23) represent the constraints on reactions r₁–r₄, respectively, and (24)–(29) give the constraints on compounds c₄, c₅, c₆, c₈, c₉, and c₁₀, respectively. Relations (30)–(33) ensure that the state of each reaction (inhibited or uninhibited) does not change during the time transition. Reaction nodes r₁, r₂, and r₃ are common to both N₁ and N₂, whereas r₄ exists only in N₂. Relations (34)–(37) define c₁, c₂, c₃, and c₇ as the source nodes of N₂.

Relation (38) states that the initial value of s₁ is set to 1 (s₁ = 1 at t₁ = 0) for the MaxVA in N₁, and (39)–(40) give the maximal time steps of N₁ and N₂, respectively. Relation (41) denotes that “T … ” represents “true (1)” and “F … ” represents “false (0).”

Since MinVA is calculated for N₂, IP-FVS1-LP2 for MKMN-B shown in Figure 2 is as follows:

Maximize

Subject to

/* Definitions of N₁

for all

/* Reactions

/* Compounds

/* Es

/* Source compounds

/* Definitions of N₂

for all

/* Reactions

/* Compounds

/* SR4

/* Es

/* Source compounds

/* SR4

where each variable takes a value of either 0 or 1. The Boolean relations of the reaction nodes and compound nodes are converted into LP2 type linear inequalities here. As (42) maximizes the number of reactions that are not inhibited, it corresponds to minimizing the number of inhibited reactions. Relation (43) ensures that the target compound c₃ finally becomes nonproducible in N₁, whereas (44) guarantees that c₃ remains producible in N₂. Relations (45)–(47) and (54)–(57) represent the Boolean relations of reactions in N₁ and N₂, respectively. Similarly, (48)–(49) and (58)–(60) represent the Boolean relations of compounds in N₁ and N₂, respectively. Relations (50)–(52) and (62)–(65) ensure that the state of each reaction (inhibited or uninhibited) does not change in that time transition. Relations (53) and (66) mean that c₁ is the source compound in both N₁ and N₂, whereas (68)–(69) give the numbers of time steps for N₁ and N₂, which are calculated from the sizes of the FVS. Here, {r₄} is chosen as the FVS. In (60), the predecessor reaction node r₄ of the compound c₄ is replaced by s₄. Relation (67) states that the initial value of s₄ is set to 0 (s₄ = 0 at t = 0) as MinVA is applied to N₂. All variables represented by “F … ” are replaced with “T … ” variables in the implementation.

Although the above ILP-formalizations do not include cycles in both N₁ and N₂, in the general case, MaxVA and MinVA are applied to N₁ and N₂, respectively.

2.4. Exception of the maximal valid assignment

MKMN-B can be solved by IP-FVS1. However, the problem setting of MKMN-B, in which MaxVA and MinVA are applied to N₁ and N₂, respectively, may be inappropriate for some cases. For example, suppose that N₁ and N₂ are as in Figure 4A. Originally, p₂ and q₂ are from one reversible reaction (r₂), which is decomposed into two irreversible reactions. Both p₂ and q₂ can be inhibited by the inhibition for r₂. If MaxVA is assumed in N₁, neither p₂ nor q₂ becomes 0 unless r₂ is inhibited. Thus, there is a case where an original reversible reaction works as if it were a source node in N₁.

FIG. 4.

A reversible reaction r₂ decomposed into p₂ and q₂. The r₁ is decomposed in N₁, and p₂ is decomposed in N₁ and N₂ so that there is no directed cycle in the resulting networks. Additionally, s₁(0) = 1 since the detected cycle in (A) includes an irreversible reaction. However, s₂(0) = 0 since the detected cycle in (A) does not include an irreversible reaction.

To handle such reversible reactions more appropriately, we define a variant of MaxVA by modifying the FVS-based method as follows: if the cycle detected by the FVS-based method does not include an original irreversible reaction, 0 is assigned to the newly created node at t = 0.

For example, suppose that N₁ and N₂ are as in Figure 4A. Since the cycle consisting of {r₁, c₆, r₃, c₂} includes an original irreversible reaction, after decomposing r₁ of N₁ into r₁ and s₁ (see Fig. 4B), s₁(0) is assigned a value of 1. Furthermore, cycles consisting of {c₃, p₂, c₈, q₂}, {c₃, p₂, c₉, q₂}, {c₄, p₂, c₈, q₂}, and {c₄, p₂, c₉, q₂} are also decomposed. Suppose that p₂ is decomposed as shown in N₁ and N₂ in Figure 4B. In this case, as the cycle does not include an original irreversible reaction, s₂(0) is assigned a value of 0 for both N₁ and N₂ (see also Table 2). Note that if s₂(0) = 1, the values of c₉ in N₁ are fixed to 1, and r₂ must be inhibited to make c₉ = 0, as it operates as if a reversible reaction was a kind of source node.

Table 2.

Initial Values of Nodes Included by the FVS

	IP-FVS1	IP-FVS2
s2(0) of N1	1	0
s2(0) of N2	0	0

In IP-FVS1, N₁ and N₂ are based on MaxVA and MinVA, respectively. However, in IP-FVS2, in addition to the above assumption, nodes included by the FVS consisting of only reversible reactions are assigned an initial value of 0. IP-FVS1 and IP-FVS2, integer linear programming-based methods; MaxVA, maximal valid assignment; MinVA, minimal valid assignment.

2.5. Fast approximation algorithm for MKMN-B with large networks

Although IP-FVS1 and IP-FVS2 successfully reduce the number of variables in the ILP formalization from O((m + n)²) to O(f (m + n)), O(f (m + n)) may be still too large if the network is very big. One reasonable strategy for handling this problem is to limit the number of time steps to some small constant. IP-FVS1 and IP-FVS2 need f time steps to ensure that the MaxVA and MinVA are always calculated, and hence the optimal solution of MKMN-B is always obtained. However, as the number of time steps necessary to obtain the optimal solution of MKMN-B depends on the topology of the network obtained after removing the FVS, we generally require fewer than f time steps. Although the proposed method does not ensure that the solution is optimal (as f time steps are necessary for the worst case), it is possible to confirm that MaxVA and MinVA in the obtained solution satisfy the condition of MKMN-B; that is, the target compound is not producible in N₁ but is producible in N₂. Note that this validation process can be conducted in polynomial time, since MKMN-B is NP-complete (see “Theoretical results” section). Let IP-FVS1-approx(t) and IP-FVS2-approx(t) be the IP-FVS1 and IP-FVS2 algorithms in which the number of time steps is limited to t. If t is not large, the numbers of variables used in IP-FVS1-approx(t) and IP-FVS2-approx(t) are O(m + n).

2.6. Elementary mode–based method for MKMN-EM

To solve MKMN-EM, we utilize CNA to calculate the EMs. Since MKMN-EM is NP-complete even if the EMs are given (see “Theoretical results” section), we develop an ILP-based method, IP-EM, to solve MKMN-EM for given EMs. For example, suppose that N₁ and N₂ have four and three relevant EMs, respectively, as shown in Table 3.

Table 3.

Example for MKMN-EM

N1		N2
EM1	r1, r2, r4	EM5	r1, r2, r7
EM2	r1, r3, r6	EM6	r2, r3, r4
EM3	r2, r4, r5	EM7	r4, r5
EM4	r5, r7

After obtaining the EMs that produce the target compound using CellNetAnalyzer, MKMN-EM can be formalized by ILP. MKMN-EM, minimum knockout for multiple reactions in the elementary mode model; EM, elementary mode; ILP, integer linear programming.

Then IP-EM can be formulated as follows:

maximize r₁ + r₂ + r₃ + r₄ + r₅ + r₆ + r₇

such that

where all variables are binary, and Boolean relations should be converted into linear inequalities. Constraints (70)–(73) are converted into

For example, since (70) requires that at least one of {r₁, r₂, r₄} must be 0, it can be represented by (75). Furthermore, the constraint (74) is converted into

and then converted into

using LP2, where all variables are binary. In this example, inhibiting {r₁, r₅} is an optimal solution, since either r₁ or r₅ is included in each of EM1, EM2, EM3, and EM4, and neither r₁ nor r₅ is included in EM6.

3. Computational Experiments

We conducted computational experiments on an Intel(R) Xeon(R) CPU E5-2690 0, 2.90GHz with cache size 20.48 MB, operating system on SUSE Linux Enterprise Server 11 SP3 (x86 64). CPLEX version 12.4.0.0 was used as the ILP solver to obtain the global optimality (IBM, 2010).

Metabolic networks of clostridium perfringens SM101 (CPR) and bifidobacterium longum DJO10A (BLJ) were represented by N₁ and N₂, respectively, in the MKMN. KGML KEGG Markup Language (KGML) of BLJ and CPR were downloaded from the KEGG PATHWAY database (Kanehisa and Goto, 2000). Datasets 1, 2, and 3 were extracted and used as small, medium, and large networks, respectively. Details of datasets 1, 2, and 3 are given in Tables 4, 5, and 6, respectively. We considered the target compounds to be C00022 (pyruvate), C00024 (acetyl-CoA), C00033 (acetate), C00036 (oxaloacetate), and C00074 (phosphoenolpyruvate). In addition, “5 compounds” indicates the problem in which none of the above five compounds are producible in N₁ but all are producible in N₂.

Table 4.

Details of the Networks of Dataset 1

	N1	N2	Total (N1 + N2)
#compound	36	43	79
#irreversible reaction	11	8	19
#reversible reaction	26	31	57
#node	73	82	155

N₁ and N₂ are the central metabolism of bifidobacterium longum DJO10A (BLJ) and clostridium perfringens SM101 (CPR), respectively. N₁ consists of {cpr00010.xml cpr00030.xml}, and N₂ consists of {blj00010.xml,blj00020.xml,blj00030.xml} in the KEGG database.

Table 5.

Details of the Networks of dataset 2

	N1	N2	Total (N1 + N2)
#compound	132	157	289
#irreversible reaction	86	133	219
#reversible reaction	33	38	71
#node	251	328	579

N₁ and N₂ consist of carbon metabolism, fatty acid metabolism, and biosynthesis of amino acids of BLJ and CPR, respectively. N₁ is {cpr01200.xml, cpr01212.xml cpr01230.xml}, and N₂ is {blj01200.xml,blj01212.xml,blj01230.xml} in the KEGG database.

Table 6.

Details of the Networks of Dataset 3

	N1	N2	Total (N1 + N2)
#compound	622	567	1189
#irreversible reaction	400	337	737
#reversible reaction	209	205	414
#node	1231	1881	2340

N₁ and N₂ consist of whole metabolic networks of BLJ and CPR, respectively, downloaded from KEGG markup language (KGML) in the KEGG database.

3.1. Datasets

In dataset 1, N₁ and N₂ correspond to the central metabolism of BLJ and CPR, respectively. N₁ includes 36 compounds and 37 reactions (26 reversible reactions) and consists of glycolysis, gluconeogenesis, and pentose phosphate pathway. N₂ includes 43 compounds and 39 reactions (31 reversible) and consists of glycolysis, gluconeogenesis, citrate (TCA) cycle, and pentose phosphate pathway. Thus, there are a total of 155 nodes for dataset 1; increasing to 212 nodes after decomposing each reversible reaction into two irreversible reactions.

In dataset 2, N₁ and N₂ correspond to the carbon metabolism, fatty acid metabolism, and biosynthesis of amino acids of BLJ and CPR, respectively. N₁ includes 132 compounds and 119 reactions (33 reversible), whereas N₂ includes 157 compounds and 171 reactions (38 reversible). Thus, there are a total of 579 nodes for dataset 2, rising to 650 nodes after decomposing each reversible reaction into two irreversible reactions.

In dataset 3, N₁ and N₂ correspond to the whole pathways of BLJ and CPR, respectively. N₁ includes 622 compounds and 609 reactions (209 reversible), whereas N₂ includes 1189 compounds and 1151 reactions (414 reversible). Thus, there are a total of 2340 nodes for dataset 3, increasing to 2714 nodes after decomposing each reversible reaction into two irreversible reactions.

3.2. results for dataset 1

Tables 7 and 8 show the computation time and size of the solutions obtained for each target compound by each MKMN method for dataset 1. As dataset 1 has small networks, the computation time is less than 2 sec for every case, and there is generally no MKMN solution.

Table 7.

The Computation Time of IP-FVS1, IP-FVS2, and IP-EM for Each Target Compound with Dataset 1

	IP-FVS1	IP-FVS1	IP-FVS2
target compound\type	(with LP1)	(with LP2)	(with LP2)	IP-EM
C00022 (Pyruvate)	2.07 sec	1.45 sec	0.65 sec	0.47 sec
C00024 (Acetyl-CoA)	<0.01 sec	<0.01 sec	<0.01 sec	<0.01 sec
C00033 (Acetate)	0.21 sec	0.18 sec	0.19 sec	0.27 sec
C00036 (Oxaloacetate)	<0.01 sec	<0.01 sec	<0.01 sec	<0.01 sec
C00074 (Phosphoenolpyruvate)	0.18 sec	0.18 sec	0.57 sec	0.49 sec
All the above 5 compounds	0.18 sec	0.18 sec	0.18 sec	0.49 sec

Table 8.

Size of the Solutions Given by IP-FVS1, IP-FVS2, and IP-EM for Each Target Compound with Dataset 1

Target compound\type	IP-FVS1	IP-FVS2	IP-EM
C00022 (Pyruvate)	no solution	2	2
C00024 (Acetyl-CoA)	no solution	no solution	no solution
C00033 (Acetate)	no solution	no solution	no solution
C00036 (Oxaloacetate)	no solution	no solution	no solution
C00074 (Phosphoenolpyruvate)	no solution	2	1
All the above 5 compounds	no solution	no solution	2

However, when the target compound is C00022 (pyruvate), {R01541, R04779} and {R04779, R05605} are solutions for IP-FVS2 and IP-EM, respectively, as shown in Table 9, where RXXXXX () are the IDs in KEGG. R04779 is a reaction from beta-D-Fructose 6-phosphate to beta-D-Fructose 1,6-bisphosphate located seven steps before pyruvate in the pathway of glycolysis. R01541 is a reaction from 2-Dehydro-3-deoxy-D-gluconate to 2-Dehydro-3-deoxy-6-phospho-D-gluconate, locating two steps before pyruvate in the pentose phosphate pathway, and R05605 is a reaction between 2-Dehydro-3-deoxy-6-phospho-D-gluconate and pyruvate in the pentose phosphate pathway.

Table 9.

Solutions Given by IP-FVS1, IP-FVS2, and IP-EM for Each Target Compound with Dataset 1

Target compound	Type	The obtained solution
C00022	IP-FVS1	no solution
	IP-FVS2	R01541(N1), R04779(N1)
	IP-EM	R04779(N1), R05605(N1)
C00074	IP-FVS1	no solution
	IP-FVS2	R04779(N1), R01541(N1)
	IP-EM	R04779(N1)
5 compounds	IP-FVS1	no solution
	IP-FVS2	no solution
	IP-EM	R04779(N1), R05605(N1)

“(N₁)” indicates that the reaction appears in N₁ but not in N₂.

When the target compound is C00074 (phosphoenolpyruvate), {R04779, R01541} and {R04779} are solutions for IP-FVS2 and IP-EM, respectively. When all five compounds are targeted, only MKMN-EM has a solution {R04779, R05605}.

Supplementary Tables S1–S3 the Supplementary Material (available online at www.liebertpub. Com/cmb) show the relation among the target compound, predecessors of the target compound in N₁ and N₂, and solutions given by each MKMN method for each target compound. For example, Supplementary Table S1 shows that R00200 and R00703 are predecessors of the target compound C00022 for both N₁ and N₂. R05605 is a predecessor of C00022 in N₁ but not in N₂. R04779 (or R01541) is not a predecessor of C00022 in either N₁ or N₂. In the tables of supporting information, “ko” indicates reactions included in the solution by each MKMN method. We can see that solutions for MKMN are not trivial, as they do not consist of only predecessors of the target compound.

3.3. Results for dataset 2

Tables 10 and 11 show the computation time and size of the solutions obtained for each target compound by each MKMN method for dataset 2. As the networks are not very large, the computation time is generally less than 20 sec. MKMN-B-FVS1 only has solutions when the target compound is C00022 or C00024, and MKMN-B-FVS2 only has solutions for C00022, C00024, or C00033; in contrast, MKMN-EM has solutions for all cases.

Table 10.

Computation Time of IP-FVS1, IP-FVS2 and IP-EM for Each Target Compound with Dataset 2

	IP-FVS1	IP-FVS1	IP-FVS2
Target compound\type	(with LP1)	(with LP2)	(with LP2)	IP-EM
C00022 (Pyruvate)	8.95 sec	11.19 sec	10.5 sec	14.59 sec
C00024 (Acetyl-CoA)	8.57 sec	11.31 sec	11.91 sec	4.62 sec
C00033 (Acetate)	0.74 sec	0.74 sec	9.82 sec	17.48 sec
C00036 (Oxaloacetate)	8.17 sec	7.05 sec	5.32 sec	10.92 sec
C00074 (Phosphoenolpyruvate)	0.72 sec	0.74 sec	0.72 sec	13.36 sec
All the above 5 compounds	0.74 sec	0.76 sec	0.74 sec	4.64 sec

Table 11.

Size of the Solutions Given by IP-FVS1, IP-FVS2, and IP-EM for Each Target Compound with Dataset 2

Target compound\Type	IP-FVS1	IP-FVS2	IP-EM
C00022 (Pyruvate)	7	5	4
C00024 (Acetyl-CoA)	3	1	1
C00033 (Acetate)	no solution	1	1
C00036 (Oxaloacetate)	no solution	no solution	1
C00074 (Phosphoenolpyruvate)	no solution	no solution	5
All the above 5 compounds	no solution	no solution	4

When the target compound is C00022 (pyruvate), the solutions obtained for IP-FVS1, IP-FVS2, and IP-EM are {R00200, R00214, R00216, R00586, R00945, R01196, R05605}, {R00200, R00214, R00216, R01196, R05605}, and {R00214, R00216, R01512, R05605}, respectively listed in Table 12. As shown in Figure 5, among the above reactions, {R00200, R00220} are predecessors of C00022 in both N₁ and N₂, whereas {R00214, R00216, R001196, R05605} are predecessors of C00022 in N₁ but not in N₂ (Supplementary Table S4). However, {R00586, R00945, R01512} are not predecessors of C00022 in either N₁ or N₂. R00586 is a reaction between L-Serine and O-Acetyl-L-serine in the pathway of cysteine and methionine metabolism. R00945 is a reaction between L-Serine and glycine in the pathway of glycine, serine, and threonine metabolism. R01512 is a reaction between 3-Phospho-D-glyceroyl phosphate and 3-Phospho-D-glycerate in the pathway of glycolysis. The relations among the reactions included in the obtained solutions and predecessors of N₁ and N₂ are described in Supplementary Table S4.

Table 12.

Solutions Given by IP-FVS1, IP-FVS2, and IP-EM for Each Target Compound with Dataset 2

Target compound	Type	The obtained solution
C00022	IP-FVS1	R00200, R00214(N1), R00216(N1), R00586(N1) R00945, R01196(N1), R05605(N1)
	IP-FVS2	R00200, R00214(N1), R00216(N1), R01196(N1), R05605(N1)
	IP-EM	R00214(N1), R00216(N1), R01512, R05605(N1)
C00024	IP-FVS1	R00230, R00238(N1), R01196(N1)
	IP-FVS2	R01196(N1)
	IP-EM	R01196(N1)
C00033	IP-FVS1	no solution
	IP-FVS2	R01196(N1)
	IP-EM	R01196(N1)
C00036	IP-FVS1	no solution
	IP-FVS2	no solution
	IP-EM	R00345
C00074	IP-FVS1	no solution
	IP-FVS2	no solution
	IP-EM	R00199(N1), R00206(N1), R01015(N1), R01829(N1), R04533(N1)
5 compounds	IP-FVS1	no solution
	IP-FVS2	no solution
	IP-EM	R00214(N1), R00216(N1), R01512, R05605(N1)

“(N₁)” indicates that the reaction appears in N₁ but not in N₂.

FIG. 5.

Solutions for dataset 2 with the target compound C00022 (Pyruvate). When the target compound is C00022 (Pyruvate), solutions of IP-FVS1, IP-FVS2, and IP-EM for dataset 2 are represented by red, blue, and green lines, respectively.

When the target compound is C00024 (Acetyl-CoA), the solutions for IP-FVS1, IP-FVS2, and IP-EM are {R00230, R00238, R01196}, {R01196}, and {R01196}, respectively (Table 12). R00230 is a predecessor of C00024 in both N₁ and N₂, whereas R00238 and R01196 are predecessor of C00024 in N₁ but not in N₂ (Supplementary Table S5).

When the target compound is C00033 (acetate), IP-FVS1 does not obtain a solution, but IP-FVS2 and IP-EM both give the solution {R00196}. As described in Supplementary Table S6, R00196 is not the predecessor of either N₁ or N₂. Next, when the target compound is C00036 (oxaloacetate), only MKMN-EM has a solution, {R00345}. R00345 is a predecessor of C00036 in both N₁ and N₂ as shown in Table S7 of supplemental data When the target compound is C00074(phosphoenolpyruvate), only MKMN-EM has a solution {R00199, R00206, R01015, R01829, R04533}. As described in Supplementary Table S8, R00199 and R00206 are predecessors of C00074 in N₁ but not in N₂. However, none of R01015, R01829, or R04533 are predecessors of C00074 in either N₁ or N₂.

Finally, when all five compounds are targeted, MKMN-B does not have the solution, but MKMN-EM has the solution {R00214, R00216, R01512, R05605}. As described in Supplementary Table S9, R00214, R00216 and R05605 are predecessors of at least one of the five target compounds in N₁ but not in N₂. R01512 is not the predecessor of any of the five target compounds in either N₁ or N₂, and is a reaction between 3-Phospho-D-glyceroyl phosphate and 3-Phospho-D-glycerate locating two steps before C00074 (phosphoenolpyruvate) in the glycolysis pathway.

3.4. Results for dataset 3

Tables 13 and 14 show the computation time and size of the solutions obtained for each target compound by each MKMN method for dataset 3. For IP-FVS1, although the calculation was completed for every target compound, no solutions were found when the target compound was either C00033 (acetate), C00074 (phosphoenolpyruvate), or “5 compounds.” For IP-FVS2, the calculation only completed when the target compound was C00074 (phosphoenolpyruvate) or “5 compounds.” CNA could not calculate the EMs for the EM-based method, as the networks are too large.

Table 13.

The Computation Time by IP-FVS1, IP-FVS2, and IP-EM for Each Target Compound with Dataset 3

	IP-FVS1	IP-FVS1	LP-type2	IP-EM
Target compound\type	(with LP1)	(with LP2)	IP-FVS2
C00022 (Pyruvate)	26 min 37 sec	54 min 56 sec	>2 hours	NA
C00024 (Acetyl-CoA)	32 min 6 sec	56 min 19 sec	>2 hours	NA
C00033 (Acetate)	16 min 17 sec	12 min 14 sec	>2 hours	NA
C00036 (Oxaloacetate)	32 min 38 sec	1 hr 8 min 35 sec	>2 hours	NA
C00074 (Phosphoenolpyruvate)	21 min 1 sec	12 min 34 sec	5 min 29 sec	NA
All the above 5 compounds	21 min 44 sec	12 min 15 sec	7 min 2 sec	NA

NA: Not available.

Table 14.

Size of the Solutions Given by IP-FVS1, IP-FVS2, and IP-EM for Each Target Compound with Dataset 3

Target compound\type	IP-FVS1	IP-FVS2	IP-EM
C00022 (Pyruvate)	12	NA	NA
C00024 (Acetyl-CoA)	7	NA	NA
C00033 (Acetate)	no solution	NA	NA
C00036 (Oxaloacetate)	3	NA	NA
C00074 (Phosphoenolpyruvate)	no solution	no solution	NA
All the above 5 compounds	no solution	no solution	NA

As listed in Table 13, IP-FVS1 can compute solutions with either LP1 or LP2 within 70 min for each target compound. However, IP-FVS2 requires a computation time of over 2 hr, and frequently cannot obtain a solution.

As the computation time is large and solutions are not always obtained, we applied IP-FVS1-approx(10) and IP-FVS2-approx(10) to dataset 3. Tables 15 and 16 show the computation time and size of the solutions given by IP-FVS1-approx(10) and IP-FVS2-approx(10) for each target compound for MKMN-B for dataset 3. As can be seen in Table 15, the computation time of IP-FVS1-approx(10) and IP-FVS2-approx(10) is at most 15sec, and computation is always completed although there is no solution for some cases. The relations among solutions of IP-FVS1, IP-FVS1-approx(10), IP-FVS2, and IP-FVS2-approx(10) are summarized in Table 17.

Table 15.

Computation Time of IP-FVS1-approx(10) and IP-FVS2-approx(10) for Each Target Compound with Dataset 3

	IP-FVS1-approx(10)	IP-FVS1-approx(10)	IP-FVS2-approx(10)
Target compound \ type	(with LP1)	(with LP2)	(with LP2)
C00022 (Pyruvate)	9.43 sec	6.92 sec	12.05 sec
C00024 (Acetyl-CoA)	9.18 sec	6.71 sec	14.95 sec
C00033 (Acetate)	4.46 sec	1.22 sec	14.01 sec
C00036 (Oxaloacetate)	8.78 sec	7.81 sec	10.66 sec
C00074 (Phosphoenolpyruvate)	2.81 sec	0.97 sec	1.32 sec
All the above 5 compounds	2.72 sec	0.96 sec	1.37 sec

Table 16.

The Size of the Obtained Solutions by IP-FVS1-approx(10) and IP-FVS2-approx(10) for Each Target Compound with Dataset 3

Target compound\type	IP-FVS1-approx(10)	IP-FVS2-approx(10)	IP-EM
C00022 (Pyruvate)	12	3	NA
C00024 (Acetyl-CoA)	7	3	NA
C00033 (Acetate)	no solution	3	NA
C00036 (Oxaloacetate)	3	2	NA
C00074 (Phosphoenolpyruvate)	no solution	no solution	NA
All the above 5 compounds	no solution	no solution	NA

Table 17.

Solutions Given by IP-FVS1, IP-FVS1-approx(10), IP-FVS2, IP-FVS2-approx(10), and IP-EM for Each Target Compound with Dataset 3

Target compound	Type	The obtained solution
C00022	IP-FVS1	R00200, R00212, R00214(N1), R00220, R00470(N1), R00703, R00704(N1), R00782, R00896, R02320, R05605(N1), R05636
	IP-FVS1-approx(10)	R03105 is chosen instead of R00896.
	IP-FVS2	NA
	IP-FVS2-approx(10)	R00200, R00470(N1), R01220
	IP-EM	NA
C00024	IP-FVS1	R00212, R00228, R00230, R00238(N1), R01177(N1), R04386(N1), R05351
	IP-FVS1-approx(10)	the same as the above
	IP-FVS2	NA
	IP-FVS2-approx(10)	R00212, R03026(N1), R05351
	IP-EM	NA
C00033	IP-FVS1	no solution
	IP-FVS2	NA
	IP-FVS2-approx(10)	R00212, R00238(N1), R05351
	IP-EM	NA
C00036	IP-FVS1	R00345, R00355, R00483(N1)
	IP-FVS1-approx(10)	R00345, R00355, R00357
	IP-FVS2	NA
	IP-FVS2-approx(10)	R00345, R00485(N1)
	IP-EM	NA

“(N₁)” indicates that the reaction appears in N₁ but not in N₂.

When the target compound is C00022 (pyruvate), the solution obtained by IP-FVS1 is {R00200, R00212, R00214, R00220, R00470, R00703, R00704, R00782, R00896, R02320, R05605, R05636}; the solution given by IP-FVS1-approx(10) is almost the same, but R00896 is replaced by R03105. Although IP-FVS2 cannot obtain a solution, IP-FVS2-approx(10) obtains the solution {R00200, R00470, R01220}. The relations among solution obtained by each method and the predecessors of the target compound in N₁ and N₂ are described in Supplementary Table S10.

When the target compound is C00024 (acetyl-CoA), IP-FVS1 and IP-FVS1-approx(10) obtain the same solution {R00212, R00228, R00230, R00238, R01177, R04386, R05351}. Although IP-FVS2 cannot obtain a solution, IP-FVS2-approx(10) gives {R00212, R03026, R05351}. The relations among the selected reactions and predecessors of the target compound are described in Supplementary Table S11.

Next, when the target compound is C00033 (acetate), FVS-type1 has no solution and IP-FVS2 cannot complete the calculation within 2 hr, whereas IP-FVS2-approx(10) obtains the solution {R00212, R00238, R05351} (see also Supplementary Table S12). When the target compound is C00036 (oxaloacetate), the solutions given by IP-FVS1 and IP-FVS1-approx(10) are {R00345, R00355, R00483} and {R00345, R00355, R00357}, respectively. Although IP-FVS2 cannot complete the calculation, IP-FVS2-approx(10) obtains the solution {R00345, R00485}. The relation is summarized in Supplementary Table S13.

Finally, when the target compound is C00074 (phosphoenolpyruvate) or“5 compounds,” neither FVS-type1 nor FVS-type2 has solutions.

4. Theoretical Results

Solving ILP is NP-complete. However, it does not mean that a problem that can be formalized as ILP is NP-complete. Therefore, in this section, we prove that MKMN-B and MKMN-EM are NP-complete and show the appropriateness of formalizing MKMN-B and MKMN-EM as ILP.

Theorem 1

MKMN-B is NP-complete even if the maximum degree is limited to 2, and the substrates and products of a reaction in N₁ are the same as those in N₂.

Proof: The proof of the NP-hardness is similar to that of Theorem 1 in Tamura and Akutsu (2010). The NP-completeness of the decision version of MKMN-B is shown by the polynomial time reduction from the decision version of the hitting set problem (HSP). The former aims to determine whether there exists |V_a| ≤ z, whose MaxVA satisfies that v₁ = 0 for all and v₂ = 1 for all hold. Given a set of elements , a set of subsets , and an integer z, the HSP determines whether Y (Y ⊂ X) exists such that S_i ∩ Y ≠ φ for any and |Y| ≤ z (Ausiello, 1999).

When an instance of HSP with X, S, and z is given, N₁ and N₂ of MKMN-B are constructed as follows.

For example, if an instance of the HSP is given as X = {1, 2, 3, 4} and S = {{1, 2}, {1, 3}, {2, 3}, {1, 4}, {3, 4}}, then N₁ and N₂ of MKMN-B are constructed as shown in Figure 6A. This conversion can be conducted in polynomial time.

FIG. 6.

Example of the polynomial time reduction from the HSP. (A) N₁ and N₂ of MKMN-B are constructed from an instance of the HSP with X = {1, 2, 3, 4} and S = {{1, 2}, {1, 3}, {2, 3}, {1, 4}, {3, 4}}. (B) Compound nodes with outdegree greater than 2 can be converted to nodes with outdegree at most 2. (C) Reaction nodes with indegree greater than 2 can be converted to nodes with indegree at most 2.

In the following, we show that HSP has a solution with |Y| = z if and only if MKMN-B has a solution with |V_a| = z. Suppose that HSP has a solution . Then, is a solution of MKMN-B since |Y| = |V_a| holds. Next, suppose that MKMN-B has a solution V_a. If holds, then , where is obtained by replacing with satisfying , is also a solution. Therefore, we can assume, without generality that V_a does not include . Then, is a solution of the HSP. Since the decision version of MKMN-B is clearly in NP, it belongs to NP-complete.

Each node with degree greater than 2 can be converted into nodes with degree at most 2 by the methods shown in Figure 6B and C. Since reactions in N₁ and N₂ created by this reduction do not intersect, MKMN-B is NP-hard, even when reactions common to both N₁ and N₂ have the same substrates and the same products.   ■

Theorem 2

MKMN-EM is NP-hard even if the maximum degree is limited to 2, and the substrates and products of a reaction in N₁ are the same as those in N₂.

Proof: In N₁ of the proof of Theorem 1 with , there are m EMs with c_t = 1 where  = 1 holds only for . If the HSP has a solution , then for does not have any EM with c_t = 1. On the other hand, suppose that does not have any EM with c_t = 1. If V_a includes some , then can be replaced with some satisfying since N₁ = {, \{V _a\{ }}\ {}, E₁} does not have any EM with c_t = 1 either. Therefore we can assume without generality that V_a does not include any . Then, is the solution of HSP. The other parts of the proof are similar to Theorem 1 other than is assumed.   ■

Theorem 3

MKMN-EM is NP-complete even if EMs of N₁ and N₂ are given.

Proof: If the sets of reactions of N₁ and N₂ have no intersections, then the problem can clearly be reduced to the HSP for EMs of N₁. Note that the completeness holds for the number of EMs but not for the number of nodes since the number of EMs may be exponential to the number of nodes.   ■

5. Discussion

In this article, we have studied the MKMN problem to determine the minimum set of reactions whose inhibition induces that the target compounds are not producible in N₁ but are producible in N₂. MKMN-B and MKMN-EM are the Boolean version and elementary mode (EM)-based version of MKMN, respectively.

For MKMN-B, we developed integer linear programming (ILP)-based methods, IP-FVS1 and IP-FVS2, utilizing the idea of feedback vertex sets (FVS) to reduce the number of variables present in the ILP formalizations. In IP-FVS1, MaxVA and MinVA are strictly applied to N₁ and N₂, respectively, to appropriately take the effect of cycles into account. However, since each node in FVS works as if it were a source node in the MaxVA, the solution obtained by IP-FVS1 is not always realistic. To avoid this problem, IP-FVS2 does not apply MaxVA to cycles consisting only of original reversible reactions in N₁. To solve MKMN-B for large networks, we developed IP-FVS1-approx and IP-FVS2-approx. These algorithms are fast since the number of time steps is limited to a small constant, but the optimality of their solutions is not ensured. We also developed IP-EM, an ILP-based method for MKMN-EM in which every EM including a source node and a target compound is inhibited in N₁, but at least one such EM remains in N₂.

We implemented IP-FVS1, IP-FVS2, IP-FVS1-approx, IP-FVS2-approx, and IP-EM in computational experiments, using metabolic networks of bifidobacterium longum DJO10A (BLJ) and clostridium perfringens SM101 (CPR) for N₁ and N₂, respectively. The datasets 1, 2, and 3 include 155, 650, and 2340 nodes in total, respectively. From Tables 8, 11, 14, and 16, it is seen that MKMN tends to have no solution for smaller networks but has solutions for larger networks. However, Tables 13 and 14 show that solving MKMN for large networks is computationally very expensive. In particular, IP-FVS2 and the IP-EM could not finish the computation within 2 hr for most cases. Tables 15 and 16 show that IP-FVS1-approx and IP-FVS2-approx can solve MKMN-B very efficiently even for large networks. Although optimality is not ensured by IP-FVS1-approx and IP-FVS2-approx, the solutions they produced are optimal for most cases in the dataset 3 experiment as shown in Table 17. Furthermore, since MKMN-B belongs to NP, it is not difficult to confirm that the obtained solutions by IP-FVS1-approx and IP-FVS2-approx satisfy that the target compound is not producible in N₁ but producible in N₂ even for nonrestricted time steps. It remains as a future work to develop a method of finding the smallest number of the time steps to ensure the optimality of IP-FVS1-approx and IP-FVS2-approx. It is to be noted that some bilevel programming-based method seems to be necessary to define MKMN in an FBA model since FBA needs another objective function in addition to minimizing the number of reactions. Applying Petri-net-based methods (Jin et al., 2011) is also interesting since they may extract the good points of both Boolean-based methods and FBA-based methods.

We also analyzed the obtained solutions by checking the relation among the predecessors of N₁ and N₂, and the selected reactions. From Supplementary Tables S5, S10, and S11, it may be thought that the solutions given by IP-FVS1 and IP-FVS1-approx are trivial when the predecessors of N₂ are not a subset of those of N₁, since choosing all predecessors of N₁ clearly satisfies the condition of MKMN-B. However, Supplementary Tables S3 and S7 show that this method cannot always obtain the correct solution. On the other hand, when the predecessors of N₂ are a subset of those of N₁, the obtained solution is complex as shown in Supplementary Table S4. It seems that the solutions given by IP-FVS2, IP-FVS2-approx, and IP-EM are somewhat involved, and it is difficult to infer solutions from these tables. Since a smaller set of inhibited reactions is preferable, solutions of IP-FVS2 and IP-EM are considered to be better than those from IP-FVS1. Moreover, for large networks, since IP-FVS2 and IP-EM are generally unable to complete the calculation, IP-FVS2-approx is considered to be the best method, although the optimality of its solution is not ensured in the worst-case scenario.

As theoretical results, we proved that MKMN-B is NP-complete for the number of nodes, MKMN-EM is NP-hard for the number of nodes, and MKMN-EM is NP-complete for the number of EMs. The software developed in this study and reported here is available online (minFvskO, 2015).

Acknowledgments

T.T. was partially supported by JSPS, Japan [Grant-in-Aid for Young Scientists (B) 25730005 and Grant-in-Aid for Scientific Research (A) 25250028]. J.S. is an Australian National Health and Medical Research Council (NHMRC) Peter Doherty fellow and a recipient of the Hundred Talents Program of the Chinese Academy of Sciences (CAS). T.A. was partly supported by the Chinese Academy of Sciences Visiting Professorship for Senior International Scientists, China, and Grant-in-Aid #22240009 from JSPS, Japan.

Disclosure Statement

No competing financial interests exist.