ℝ is the set of real numbers, ℝ+n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFDeIudaqhaaWcbaGaey4kaScabaGaemOBa4gaaaaa@3900@ is the set of all n - dimensional vectors with real and positive components, and ℝ^{m × n} is the set of all m × n matrices with real entries. Given m, n ∈ ℕ, we use the notation M = {1, ..., m} and N = {1, ..., n}. For a set C, we use |C| to denote its cardinality. If A ∈ ℝ^{m × n} and U ⊆ M, then A_U denotes the submatrix of A containing the rows with indices in the set U. Therefore, if x ∈ ℝⁿ, i ∈ N, and U ⊂ N, then x_i and x_U ∈ ℝ^|U| denote its ith component and the vector formed by taking components with indices in set U, respectively. The inequality x ≥ 0 is interpreted componentwise, i.e., x_i ≥ 0, i = 1, ..., n. Each vector x ∈ ℝⁿ induces a ray, r = {αx| α > 0}. We denote the set of nonzero indices of a ray r ⊂ ℝⁿ as NZ(r) ⊂ N. The notation r¹ + r², where r¹, r² are rays, refers to the usual (Minkowski) set sum.

A cut set for an objective metabolic function is a set of reactions whose knockout abolishes that function 4. Formally, a set of reactions C is a cut set for an objective reaction j in metabolic network S if and only if v_j > 0 is feasible in the wild type (i.e. ∃v ∈ K|v_j > 0) and

v_C = 0 → v_j = 0, ∀v ∈ K.

where

K = {v ∈ ℝⁿ|S_v = 0, v ≥ 0}

is the feasible flux cone of S.

More generally, we call C ⊂ N a cut set for an objective reaction set J ⊂ N if the wild type is able to sustain flux through every reaction in J and Equation 1 is satisfied for at least one j ∈ J. For example if J corresponds to the set of biomass component sinks in the network, a cut set for J will be a reaction set C whose knockout disables the producibility 9 of at least one biomass component.

A cut set C is minimal if no proper subset of C is a cut set. Minimal cut sets (MCS) represent "elementary failure modes" of metabolic networks 5. MCS also capture epistatic relationships between network components that have essential roles in robust systems-level functions. In this paper, we refer to a reaction as "k-essential" (e.g. 1-essential, 2-essential etc.) for a metabolic objective (e.g. biomass production) if it belongs in a MCS of size k for that function.

Brute force LP approaches (e.g. FBA) can be applied to find MCS by exhaustively testing a collection of reaction knockouts to determine which of these "cuts" the objective reaction j. To test whether a reaction set C is a cutset for k, one employs an LP to determine the feasibility of v_j > 0 subject to v_C = 0, v ∈ K. MCS can also be identified as "minimal hitting sets" of elementary modes 10. In combinatorics, a hitting set of a collection of sets C MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NaXpeaaa@374D@, each taken from a universe of items U, is a set H ⊂ U that intersects every set in C MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NaXpeaaa@374D@. In this paper, we refer to a set H ⊂ N as a hitting set for a collection of vectors E ⊂ ℝⁿ if H intersects the nonzero components of r for every r ∈ E. H is a minimal hitting set for E if none of its subsets are hitting sets.

The results of Klamt and Gilles show that an MCS for an objective reaction j is a minimal hitting set of the collection of all elementary modes that employ j 410. This network-based cut set criterion follows from the fact that if one knocks out a set of reactions that disables all j-containing elementary modes, then one will constrain the flux through reaction j to 0 in all feasible fluxes. These results directly extend to extreme pathways, which are equivalent to the elementary modes of an irreversible metabolic network 11. Namely, an MCS for reaction j is a minimal hitting set of all the j-containing extreme pathways. We choose the extreme pathway convention in this paper because it tends to be a more compact representation of the pathway structure of metabolism 12.

Both the brute-force LP approach and the MCS algorithm of Klamt and Gilles have deep limitations in discovering genome-scale MCS. The size of most genome-scale metabolic networks (> 500 – 1000 reactions) renders brute-force LP approaches practical only for finding low cardinality MCS (k ≤ 2), though high performance computing and restriction of the search to a subset of model reactions can extend this limit to k ≤ 4 1. The MCS algorithm of Klamt and Gilles is not applicable to the genome-scale due to its dependence on elementary mode computation, which is only feasible for small metabolic networks (i.e. less than 300 reactions) 4101113.

In this paper we introduce NetKO, a three-step procedure for generating genome scale MCS. In the first step, we use the tableau algorithm to generate a collection of pathway fragments for the metabolic network. In the second step, we compute a collection of cut sets for each reaction j ∈ J through the analysis of pathway fragments. In the third step, we use pairwise set comparisons and LP to generate a collection of MCS for the objective reaction set J.

The feasible flux cone K of a metabolic network S is the set sum of a finite and unique collection of extreme rays E(K), called extreme pathways 14. The standard tableau algorithm for computing extreme pathways E(K) associated with stoichiometry matrix S is an iterative procedure that computes the extreme rays E(Kⁱ) for a series of polyhedral cones Kⁱ ⊂ ℝⁿ, i ∈ {0, ..., m} given by:

K i = { v | S M i v = 0 , v ≥ 0 , M i = { 1 … i } } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4saS0aaWbaaSqabeaacqWGPbqAaaGccqGH9aqpcqGG7bWEcqWG2bGDcqGG8baFcqWGtbWudaWgaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaemODayNaeyypa0JaeGimaaJaeiilaWIaemODayNaeyyzImRaeGimaaJaeiilaWIaemyta00aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqGG7bWEcqaIXaqmcqWIMaYscqWGPbqAcqGG9bqFcqGG9bqFcqGGUaGlaaa@4E47@

The algorithm is seeded with the initial cone K⁰ = ℝ+n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFDeIudaqhaaWcbaGaey4kaScabaGaemOBa4gaaaaa@3900@. The initial collection of generators E(K⁰) consists of rays induced by the Euclidean basis vectors e^j ∈ ℝⁿ. At each iteration i ∈ {1, ..., m}, the generators E(Kⁱ) are computed from the analysis of rays in E(K^{i - 1}) in three steps.

In the first step, each ray in E^i-1 is tested to determine whether it belongs to the hyperplane S_iv = 0. Rays in E(K^{i - 1}) that belong to this hyperplane are added to the collection E(Kⁱ). In the second step of iteration i, the algorithm considers each ray pair in E(K^{i - 1}) that lies on opposite sides of the hyperplane S_iv = 0. The intersection of the set sum of each ray pair with S_iv = 0 induces a new ray that lies within the cone Kⁱ. Each such ray is added to the collection E(Kⁱ).

The third and final step of each iteration i involves removal of non-extreme rays from E(Kⁱ). This is implemented by comparing the sparsity patterns of each ray pair in E(Kⁱ) and removing any ray r for which there exists an r^' such that NZ(r^') ⊂ NZ(r). Following iteration m, the tableau algorithm terminates, having computed the extreme rays of polyhedral cone K = K^m. However, given the computational complexity of the third step in each iteration, the tableau algorithm fails to terminate (i.e. reach iteration m) for most genome-scale metabolic networks.

Each iteration i of the tableau algorithm produces a collection of rays E(Kⁱ) ⊂ ℝⁿ, which we refer to as pathway fragments. Each pathway fragment in E(Kⁱ) obeys the quasi steady state assumption for the subset M_i ⊆ M of species in the system x˙ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaiaaaaa@2D57@ = Sv, v ≥ 0. Though a collection of pathway fragments contains incomplete information about metabolic network dynamics, we will show that it offers much insight into its failure modes. We exploit this property in our network-based genome-scale knockout design approach. Each pathway fragment collection E(Kⁱ) contains the extreme rays of the cone Kⁱ in Equation 3.

Applying Klamt and Gilles' criterion to Equation 3, a minimal hitting set C for the j-containing rays in E(Kⁱ) is an MCS for reaction j in the "relaxed" system SMiv MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaaWcbeaakiabdAha2baa@3165@ = 0, v ≥ 0. This means that C is a minimal. set of reactions satisfying the relation:

v_C = 0 ⇒ v_j = 0, ∀v ∈ Kⁱ

Comparison of Equations 2 and 3 shows that the cone Kⁱgenerated by a pathway fragment collection E(Kⁱ) is an overapproximation of the final feasible ux cone K (Figure 1). Formally,

K ⊆ Kⁱ.

As a result, any flux configuration that is infeasible in Kⁱwill also be infeasible in K. Alternatively stated, if C satisfies Equation 4 then it also satisfies Equation 1. Thus, C is a cut set for j in the full system Sv = 0, v ≥ 0.

In this section, we have proven a "relaxed" version of Klamt and Gilles' cut set criterion. Namely, we have shown that a minimal hitting set for a collection of j-containing pathway fragments is a cut set for j in a metabolic network S. As a result, the problem of computing cut sets for an objective reaction j can be reduced to the enumeration of minimal hitting sets for the j-containing pathway fragments. The latter can be achieved by iterating through the j-containing pathway fragments via the procedure MinHit (Table 3), described in the implementation section. As we will show in the results, application of this "relaxed" pathway-fragment based cut set criteria enables network-based knockout design on the genome-scale.

Our "relaxed" cut set criterion is able to generate genuine cut sets for the entire metabolic network S despite employing only a traversal across the subset M_i of metabolites. However, the incompleteness of this traversal makes our criterion sufficient but not necessary for determining whether a reaction set C constitutes a cut set for reaction j. This results in two caveats regarding the "quality" of cut sets obtained from the analysis of pathway fragments: 1) not all cut sets are guaranteed to be found and 2) cut sets that are found are not guaranteed to be minimal. To fully reduce cut sets generated by MinHit to minimality, we apply an LP based post-processing step Reduce (Table 4) to 1) checks minimality of cut sets and 2) (if necessary) reduces them to their minimal subsets. We describe Reduce in more detail in the Methods section.

There may be additional sources of non-minimality when one pursues MCS for a set of objectives J (e.g. the set of biomass sinks). In this case a cut set obtained for one reaction j¹ ∈ J may be a subset of a cut set obtained for a second reaction j² ∈ J. Both may be MCS for their respective objectives, however one (or both) may be non-minimal with respect to the objective set J. For example, if J corresponds to the set of biomass sinks in the network, an MCS for L-isoleucine production may be built by adding reactions to an L-threonine MCS. Such non-MCS may be immediately pruned through a pair-wise comparison of cut sets generated by MinHit for the various individual sinks in J, and removal of sets for which there exists a subset.

As may be intuitively expected, the quality of cut sets obtained with our approach directly depends on the "quality" of pathway fragments. At each iteration, the tableau algorithm "learns" more about network constraints imposed by the steady state requirement for each individual species. As a result, later iterations of the tableau algorithm yield more informative pathway fragments. In other words, pathway fragments gathered from an early iteration of the algorithm are "naive" and will yield fewer cut sets that are farther from being minimal. Conversely, "mature" pathway fragments gathered from a later iteration will yield larger numbers of cut sets that are closer to being minimal.

A simple approach that we apply to maximize the iteration i reached by the tableau algorithm is the "local greedy" optimization strategy described by Bell and Palsson 15. This strategy is applied after each iteration of the tableau algorithm to choose the "cheapest" metabolite for the following iteration. The cost of each metabolite is computed as the number of pathway fragments at that iteration that consume that metabolite multiplied by the number of pathway fragments that produce that metabolite. This metabolite ordering strategy can be conceptually understood as a network traversal that begins with the least connected metabolites (e.g. network dead ends) and saves highly connected metabolites (e.g. ATP, water) for later iterations. Though we exclusively use this strategy for this study, we will mention several alternative metabolite ordering approaches that may be used for network-based genome-scale knockout design later in the discussion.

The original model annotation of E. coli iJR904 groups reactions into 30 "subsystems" 8. The MCS obtained from network analysis span 23 of these 30 reaction subsystems in the E. coli metabolic network. Superimposition of the MCS on the subsystem reaction classification yields 58 unique "subsystem signatures", which are shown in Figure 3. Analyzing these signatures shows that only 289 of the MCS discovered with our approach target a single subsystem, while the vast majority (10838) target three or four subsystems. The most prevalent (7720 MCS) of these multivalent subsystem signatures target the combination of "Cell Envelope Biosynthesis", "Threonine and Lysine Metabolism", "Alternate Carbon Metabolism", and "Extracellular transport". MCS with such signatures clearly represent complex interactions between functionally disparate and parallel portions of the E. coli metabolic network.

To demonstrate the utility of NetKO, we benchmarked it against two standard in silico metabolic knockout analysis approaches: brute-force LP and random knockout. For brute-force LP, we tested all single and double knockouts. In our random knockout approach, we sampled 250,000 random reaction sets of cardinality 10. For each such set, we determined whether its knockout cut the production of biomass and reduced it to minimality with respect to biomass production (via Reduce).

Comparing the results (Table 1) we found that NetKO was able to yield a larger number of MCS that targeted more reactions in the network than either brute force or random knockout approaches. Table 1 shows that our network-based genome-scale knockout design yielded over 50 times as many MCS as both brute-force and random KO approaches. Furthermore, MCS obtained using our network-based approach implicate a larger number of reactions as k < 10-essential for biomass production for rich media E. coli. Though we employ different cardinality limits for the network based (k ≤ 10) and brute force (k ≤ 2) approaches in this benchmark, we note that these results reflect a similar order of magnitude of LP operations (115,243 vs 487,671), resulting in higher return for a similar expenditure of computational resources. In comparison to the random approach, which also tests reaction sets of cardinality 10, we note that NetKO employs two orders of magnitude fewer linear programs than a random knockout approach (113,989 vs 7,221,149) while yielding many more high-order interactions.

Comparison of brute KO and NetKO results show that in many ways the two approaches are complementary (Table 1). For example, NetKO is not guaranteed to find all cut sets of a given cardinality: it misses 99 of the 215 MCS found with brute-force LP. Furthermore, MCS obtained using brute-force LP cover an alternate set of reactions and biomass components than MCS discovered using NetKO. This occurs because NetKO is biased towards finding high-cardinality MCS. Though the number of reactions belonging to Brute-force LP and NetKO derived MCS are the same, we note that NetKO establishes k < 10-essential roles for 184 E. coli reactions that have no 1- or 2- essential role in biomass production. The differences in biomass targets also reflect the fact that brute force LP is biased towards biomass components targeted by low-cardinality cut sets (e.g. LPS). Because of these differences, one can use "brute force LP" results to determine general network properties regarding the rate of interaction between genes for a given degree of epistasis k 2, which cannot be gleaned from the results of NetKO. Conversely, unlike brute force LP, one can use NetKO to uncover very high-order epistatic relationships for robust network functions.

Our random KO results demonstrate quite strikingly how difficult it is to find high order MCS in the E. coli metabolic network (Table 1). Though many (52%) of the 250,000 random reaction sets of size 10 cut biomass production, all of these cut sets are built from a relatively small number (223) of size 1 to 3 MCS. The rarity of high cardinality MCS in these results is a testament to the efficacy of NetKO in exploiting network structure for knockout design. Additionally, the difference between the size of these cut sets and the size of MCS they contain shows why it is so expensive (requiring over 7 million LP's) to reduce these random high-cardinality cut sets to minimality. In contrast, cut sets generated through pathway fragment analysis in NetKO are much closer to minimality, requiring much fewer LP's in the Reduce computational step.

In E. coli iJR904, L-threonine production is invulnerable to all single or double knockouts. However, NetKO discovers 10655 MCS that target the production of this amino acid: all but 10 of these L-threonine targeting MCS employ more than 5 reactions, and 9725 involve 8 or more reactions.

Interestingly, these 10655 genome-scale MCS arise exclusively from a 41 reaction subnetwork of E. coli, which is depicted in Figure 4. Furthermore, only 7 of the reactions in this subnetwork (ASAD, ASPK, DAPabc, HSDy, METabc, 26dap-M and met-L nutrient fluxes) have 1- or 2-essential roles in rich media biomass production; however, low cardinality MCS containing these reactions target peptidoglycan and L-methionine rather than L-threonine production. Our MCS results thus rigorously establish novel systems-level roles for all 41 of these reactions. These results link the systems-level process of L-threonine production in E. coli to the functioning of a highly parallelized biosynthetic subnetwork. Epistasis between components of this subnetwork only become apparent through the analysis of high-order knockouts. The L-threonine subnetwork shown in Figure 4 is composed of reactions from the 'Alternate Carbon Metabolism', 'Cell Envelope Biosynthesis', 'Putative Transporters', 'Threonine and Lysine Metabolism', and 'Transport, Extracellular' subsystems. Though it is responsible for a large number of MCS, this subnetwork can be easily understood as the convergence of several pathways linking extracellular carbon sources, cell membrane components, amino acids, and other intracellular species to L-threonine. As we show below, analysis of these pathways yields biochemically intuitive rationale for why these reaction subsets emerge as biomass MCS.

The simplest of the 10655 L-threonine targeting MCS discovered using our approach consist of reactions immediately upstream of L-threonine in this subnetwork. As shown in Figure 4, the most direct sources of L-threonine in the intracellular environment are three transport reactions that couple the influx of L-threonine to the import of sodium (THRt4), the import of proton (THRt2r), and the hydrolysis of ATP (THRabc). L-threonine is also synthesized from L-aspartate via a pathway that employs L-homoserine as an intermediate. The third major source of L-threonine is via the threonine aldolase reaction (THRAr), whose substrates are acetaldehyde and glycine.

The most intuitive of these MCS consists of the three L-threonine transporters (THRt4, THRt2r, THRabc) in combination with threonine synthetase (THRS) and threonine aldolase (THRAr). Variants of this MCS replace THRS with aspartate-semialdehyde dehydrogenase (ASAD), homoserine kinase (HSK), homoserine kinase (HSK), or aspartate kinase (ASPK). A second set of variants replaces the three L-threonine transporters with the L-threonine nutrient flux. 5 MCS apply an interesting variation upon this theme by replacing the sodium dependent THRabc transporter with either the sodium nutrient flux or inhibition of the three sodium proton antiporters (NAt3_2, NAt3_1.5, NAt3_1). Because of their minimality, the knockout of any subset of these MCS will allow L-threonine production to be maintained in this silico strain. For example, knockout of L-threonine transport and threonine-aldolase will allow L-threonine to be produced from L-aspartate via the pathway shown in the bottom left of Figure 4. Alternatively stated, the minimality of these cut sets attests to the fact that the wild type in silico can employ any one of these parallel pathways for L-threonine biosynthesis.

A large number of MCS discovered using our approach target L-threonine biosynthesis by disabling pathways upstream of acetaldehyde. As shown in Figure 4, acetaldehyde is directly brought into the cell from the nutrient media via a transport reaction. Acetaldehyde is also supplied by the breakdown of phosphatidylethanolamine and phosphatidylglycerol, which are cell membrane components. Acetaldehyde is also formed from extracellular phenylpropanoate, 3-(3-hydroxy-phenyl)propionate, and 3-hydroxycinnamic acid, which are substrates for the synthesis of intracellular 2-Oxopent-4-enoate. The latter is transformed into acetaldehyde via 2-oxopent-4-enoate hydratase (OP4ENH) and 4-hydroxy-2-oxopentanoate aldolase (HOPNTAL), producing pyruvate as a byproduct. Another source of acetaldehyde is via the breakdown of deoxyribose sugars via the reactions deoxyribose phosphate aldolase (DRPA) and phosphopentomutase 2 (PPM2). The more complex L-threonine targeting MCS replace the THRAr reaction in the simple MCS described above with a combination of the reactions responsible for acetaldehyde biosynthesis. One such combination is DRPA, HOPNTAL, ACALDt, and ethanolamine ammonia-lyase (ETHAAL). A variation of this MCS interchanges ETHAAL in the latter combination with Phospholipase A2 (PLIPA2). Another combination replaces HOPNTAL with the combination of phenylpropanoate, 3-(3-hydroxy-phenyl)propionate, and 3-hydroxycinnamic acid transporters. 160 MCS target L-threonine production through a unique mechanism that involves the repression of the oxygen nutrient flux. Like the MCS discussed in the previous paragraph, these sets include reactions that involve acetaldehyde formation from cell membrane phospholipids, ribose sugar breakdown, and acetaldehyde transport. However unlike the previously described sets, these MCS exploit the oxygen-dependence of the pathways facilitating acetaldehyde production from extracellular phenylpropanoate, 3-(3-hydroxy-phenyl)propionate, and 3-hydroxycinnamic acid. As shown in Figure 4, the reactions 3-(3-hydroxy-phenyl)propionate hydroxylase (3HPPPNH), Phenylpropanoate Dioxygenase (PPPNDO), 3-hydroxycinnamate hydroxylase (3HCINNMH), 2,3-dihydroxycinnamate 1,2-dioxygenase (DHCINDO), and 2,3-dihydroxyphenylpropionate 1,2-dioxygenase (HPPPNDO) all depend on the presence of oxygen as a substrate.

The L-threonine subnetwork described in this section demonstrates a biologically intuitive pathway-like structure. The ease with this network can be understood testifies to the ability of NetKO to unravel the link between network components and systems-level functions. We further note that unlike manual pathway annotations, this subnetwork is derived purely through the analysis of flux dynamics in the genome-scale metabolic model. As a result, it serves as a rigorous and objective distillation of E. coli iJR904's behavior with respect to a particular function, i.e. L-threonine production.

MCS for essential metabolic objectives (e.g. biomass) represent potential targets for drug development. Though the model employed in this study represents a non-pathogenic strain of E. coli, there are numerous genome-scale metabolic reconstructions of pathogenic microbes in which such an approach could be used to reveal drug targets. Such organisms include S. aureus, H. pylori, and B. anthracis 16171819. Furthermore, high-order MCS could be applied to models of human metabolic networks to define novel anti-proliferative chemotherapeutic regimens 20.

MCS correspond to subsets of genes whose perturbation or knockout is predicted to effect a network-level phenotype. Such a gene subset provides a reasonable "epistatic" candidate hypothesis for a genetic association study that explore links between genotype and phenotype, e.g. for human disease. Current approaches to whole genome association studies employ a unsupervised approach that test all possible loci for association with a phenotype. Given the high dimensionality of genotypic (e.g. SNP) data, these studies are limited by computational resources and statistical power to searching for binary gene-gene interactions. A "systems-based candidate-gene" approach based on NetKO could provide a powerful approach to apply "a posteriori" biological knowledge to study the role of epistasis in disease. Namely, one could employ NetKO to generate hypothesis in the form of MCS on a given pathway model and test such hypotheses against genotype-phenotype data.

MCS discovered using our approach cluster into subnetworks that link individual reactions to their systems-level roles. In contrast to the daunting complexity of the entire metabolic network, these subnetworks have an intuitive pathway structure that can be understood by simple inspection. The close dual relationship between the failure and function of of metabolism has been previously outlined by Klamt 5. This dual relationship suggests that these local pathways are likely "projections" of genome-scale extreme pathways/elementary modes onto these important reaction subsets. This hypothesis could be verified by applying the method of Urbanczik and Wagner to compute generators for the projection of the flux cone K onto these reaction subsets 21.

Many of our MCS reveal essential roles for reactions whose function is normally obscured by metabolic network redundancy. Deutscher et al. refer to this generalized concept of essentiality as "k-robustness" in their investigations of multiple knockouts in yeast metabolism 1. Each complex MCS of cardinality k identified by our approach is conceptually equivalent to Deutscher et al.'s "essential set". High cardinality MCS represents an instance of a higher-order genetic or biochemical interaction between several reactions, enzymes, and genes that is mediated by the metabolic network. Similar to the results of Deutscher et al., we find that application of this generalized notion of essentiality significantly increases the number of E. coli reactions that can be thought of as participating in "essential processes" (as defined by biomass production).

The use of a genome-scale model to find essential links between genes and systems-level processes exemplifies a novel "network-based" approach to functional gene annotation. Shlomi et al. show that such an approach may be better equipped to capture functional features such as condition-dependence when compared to traditional structured vocabulary-based annotation schemes like Gene Ontology 22. NetKO provides a powerful tool for this kind of annotation by helping uncover functional links that are normally buried by redundant pathways.

Our approach can be applied in an iterative model building protocol that combines in vivo experiment and in silico analysis.

One specific application of NetKO is the simple validation of model completeness and accuracy on the genome-scale. A more profound application is to unravel the systems-level roles of individual reactions, in particular those participating in robust and redundant network functions.

Fundamentally, each MCS can be understood to suggest several important in vivo model validation experiments. The first corresponds to the knockout of that MCS and the subsequent lethality prediction. If the in vivo knockout fails to be lethal, then this suggests that either there is additional redundancy in the pathway structure of the network or the specific biomass target of that MCS is non-essential for growth 9. Once an MCS is verified as being in vivo lethal, an interesting set of experiments can be performed that test the contributions of individual reactions to the function disabled by a given MCS. To do so, one would design knockouts that disable all but one of the reactions in that MCS. Such a perturbation would "pinch" the network, allowing only extreme pathways that utilize the remaining reaction in that MCS. As a result, the flux through that reaction would be predicted to become a simple linear multiple of the growth rate (determined only by the steady state intracellular concentration of the target biomass component and its fractional contribution to biomass).

For example, the cysteine transporter (CYSabc), taurine transporter (TAURabc), biotin synthase (BTS2), and the sulfate transporter (SULabc) form an MCS for biomass by targeting L-cysteine production. Knockout of CYSabc, TAURabc, and BTS2 would be predicted by this network model to force all fluxes destined for L-cysteine production through the sulfate transporter (SULabc). According to the model, the rate of uptake of sulfate in this triple knockout will be in direct proportion to the flux through the L-cysteine sink, which corresponds to the growth-mediated dilution of L-cysteine and its consumption by protein synthesis. Such a discrete "pinching" experiment would determine whether the sulfate transporter is sufficient to sustain L-cysteine production. Extension of such an experimental paradigm to analyze further subsets of complex MCS at intermediate levels of inhibition could be used to quantitatively identify the "marginal" contributions of individual reactions/pathways to systems-level functions.