The algorithm for calculating MCSs was developed by S. Klamt and E.D. Gilles [12] and operates on EMs [1,2,4]. In fact, EMs and MCSs complement each other, as will be discussed later on.

The theory behind the use of EMs [1,2,4] for calculating MCSs is the fact that an EM is minimal, thus non-decomposable in terms of the reactions (enzymes) utilized; removing a reaction from an EM results in the system not being able to achieve steady state with the remaining reactions of the EM. So, if the objective reaction is identified for the network function of interest, and EMs are calculated for it, the MCSs would be the reactions that, if taken out, would result in the system not being able to achieve steady state with the remaining reactions in these particular EMs, i.e., cause the dysfunction of the system with respect to the objective reaction, so the corresponding network function is repressed.

MCSs can be used for studying the fragility of a network structure and identifying suitable targets for metabolic functionalities. For example, we have used MCSs [16] to study the functionalities of anthocyanin related genes in flowering plants.

To illustrate the MCS concept, consider the example network (NetEx) used in [11] and shown in Figure 1 below. The characteristics and hypergraphical nature of the network are important in defining its MCSs.

Features to Note about NetEx

Consider the case where the synthesis of product P is of particular interest; then, the reaction PSynth is the objective reaction in the same context as a target reaction [17], whereby all flux vectors with a non-zero flux through reaction PSynth are of importance.

The first step would be to determine the qualitatively distinct possible ways of producing P; this is equivalent to calculating EMs as illustrated in Figure 2 below.

As shown in the above figure, there are six EMs for NetEx, five of which involve the PSynth reaction (highlighted networks). In order to eliminate the production of P, all the EMs that involve PSynth need to be blocked. By definition, an EM is blocked by removing any of its constituent reactions, therefore, any combination of reactions, one taken from each EM, forms a cut set that disables flux through the EMs. For our network example, NetEx, a MCS for the objective reaction, PSynth, is a set of reactions whose knockout blocks the five EMs involving PSynth, thus disabling flux through PSynth at steady state.

The notion of MCSs does exist in other theories and research areas, particularly in relation to risk analysis. In developing the algorithm for MCSs, S. Klamt and E.D. Gilles [12] looked at previous similar definitions of MCSs that existed in other areas at the time. These included fault trees and graph theory which shall be discussed here; other similar concepts are looked at later in Section 5.

Referring to the example network NetEx in Figure 1, the MCSs for the objective reaction, PSynth, can be determined as follows:

The MCSs for a given objective reaction in a large metabolic network, however, cannot be done by a simple examination; an algorithm would be needed to compute the MCSs. The first algorithm was developed by Klamt and Gilles [12] although others have been developed since, to improve on the computational speed and efficiency; these are discussed in Section 3.2.

The MCS Algorithm

The MCS algorithm devised by Klamt and Gilles [12] relies on the fact that:

where N is the number of EMs; and

Before MCSs are computed, the set of EMs is split into two disjoint sets:

The right-hand side of Equation 2 above, illustrates, respectively, the set of EMs (e^,t,j) comprising the target modes (E^t) and the set of EMs (e^nt,k) comprising the non-target modes (E^nt) [11]. Since removing a set of MCSs ensures inactivation of all target modes E^t,j, only non-target modes E^nt,k could survive, which means that all remaining flux distributions r will show zero flux in the objective reaction, r_objR.

The pseudocode of the MCS algorithm for calculating MCSs initially developed by S. Klamt and E.D. Gilles is provided in [12] and further modified for the example network, NetEx, discussed in [11].

For the NetEx network, the algorithm calculates seven MCSs in addition to the trivial MCS (PSynth itself). To illustrate, one of the MCSs (MCS2) is shown in Figure 7 below:

The seven MCSs and the corresponding EMs are shown in the first two tables of Table 1.

S. Klamt, in 2006 [11], redefined the MCS from that of the original concept expressed under 2 earlier, to “a minimal (irreducible) set of structural interventions (removal of network elements) repressing a certain functionality specified by a deletion task”. This new definition indicates the key role that the deletion task plays in the difference between the new generalized approach and the initial MCS concept.

The deletion task can be specified by several Boolean rules that clearly represent and describe, unambiguously, the flux patterns or the functionality to be repressed. This increases the practical applicability of MCSs because they can now be determined for a large variety of complex deletion problems and for inhibiting very special flux patterns instead of just for studying structural fragility and identifying knock-out strategies.

The refinements and extensions to the initial MCS concept offer a broader range of possible ways in which MCSs can be used to assess, manipulate and design biochemical networks. A comparison of the concept versions is covered later.

Further refinement of MCSs has also been undertaken [15] to deal with their limitation of disabling desired functionalities along with the targeted ones. To address this limitation, Hädicke and Klamt [15] generalized MCSs to Constrained MCSs (cMCSs) that take into consideration side constraints and allow for a set of desired modes, with a minimum number preserved, to be defined.

This generalization provides a flexibility for cMCSs to be applied to existing methods, for example Minimal Metabolic Functionalities [25,26], OptKnock [27], and RobustKnock [28] can be reformulated as special cases of cMCSs. As demonstrated in [15], the cMCSs approach offers great flexibility in defining and solving knock out problems.

The next section compares the three concepts, to get a better understanding of MCSs and how they have developed.

In computational complexity theory [33,34], deterministic polynomial (P) and non-deterministic polynomial time (NP) are two classes of decision problems that classify computational problems according to their inherent difficulty in terms of their solvability by a computer. The computation problem can be stated by a set of mathematical instructions consisting of problem instances and solutions to these problem instances.

A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantify the amount of resources needed to solve them, such as time and storage. One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do and the big O notation is useful for analyzing the run time for class P and NP problems.

The big O notation can analyze the efficiency of algorithms such as the time (T) (or the number of steps) it takes to complete a problem of size n. For example the time might be found to be T(n) = 6n²-2n+5. As n grows large, the n² term will come to dominate, so that all other terms can be neglected. The coefficients also become irrelevant if T(n) is compared to other orders of expression e.g., n³ or n⁴; U(n) = n³, will always exceed T(n) when n gets larger than 6. The number of steps, on the other hand, depends on the details of the machine model on which the algorithm runs, although different types of machines generally vary by only a constant factor in the number of steps needed to execute an algorithm. So the algorithm has order of n² time complexity denoted by the big O as:

Note: The following two right-hand side big O notations have dramatically different meanings:

The first case states that f(m) exhibits polynomial growth, while the second, assuming m > 1, states that g(n) exhibits exponential growth.

Class P consists of those decision problems whose solution can be obtained using a deterministic algorithm that runs in polynomial time, i.e., runs in O(n^k)steps for some non-negative integer k, where n is the input size. A deterministic algorithm only has one choice in each step taken to execute the problem, i.e., it would have the same output for every run on the same input instance for the problem.

Class NP consists of those decision problems for which there exists a nondeterministic algorithm that runs in polynomial time with two phases: a) the guessing phase where a nondeterministic algorithm is used to generate an arbitrary string of characters that may or may not correspond to a solution of the input, and b) the verification phase which uses a deterministic algorithm to check and verify that the generated string is a valid solution or reject it otherwise. Both phases need to be completed in polynomial time (O(n^j))where j is a non-negative integer.

The complexity class P is contained in NP, but NP contains many important problems, the hardest of which are called NP-complete problems, for which no polynomial-time algorithms are known for solving them (although they can be verified in polynomial time). The most important open question in complexity theory, the P=NP problem, asks whether such algorithms actually exist for NP-complete, and by corollary, all NP problems. It is widely believed that this is not the case.

The complexity of EMs and MCSs in metabolic networks is covered in [30,31] and are found to be NP-hard. A problem is NP-hard if an algorithm for solving it can be translated into one for solving any NP-problem, so NP-hard means "at least as hard as any NP-problem", although it might, in fact, be harder.

In addition to finding EMs being NP-hard, it has also been shown [30] that finding cuts of minimum size without computing all EMs is also NP-hard. This doesn’t help from the point of view of genetic intervention where it is desirable to find a MCS of minimum size, thus a need for computational methods to address the problem.

Four MCS computational methods have been developed since the first two (original and general MCS concepts [11,12]); mainly to improve the computational complexity of obtaining MCSs but they also open MCSs and EMs to a wider area of application:

The algorithm and enumeration improvement aim to avoid having to compute elementary modes in order to obtain reaction cuts; instead of a MCS that disables too many EMs, it would be desirable to find a MCS that cuts the target reaction but leaves certain reactions intact or as many EMs as possible intact. These types of MCSs are NP-hard. The developments in [30] provide the capacity to analyze the complexity of the underlying computational tasks that would assist in determining which tasks can be tackled.

One area in which MCSs have been applied is fragility. Fragility is the vulnerability of a system to failure due to external or internal perturbations. It is inversely related to robustness [39], the capacity for a system to maintain its functions despite perturbations [40]. Prior to the use of MCSs for measuring structural fragility, EMs have been used to study the robustness of networks [41,42]; they have also been used in more recent studies on pathway knockout and redundancy in metabolic networks [43].

The application of MCSs to measure fragility can be found in [11,12,16]. The fragility coefficient, F_i, defined as the reciprocal of the average size of all MCSs in which reaction i participates [12], is used as a quantitative measure for determining how essential the reactions are: the lowest value of F_i would be closest to 0 where reaction i is one of many reactions occurring in a MCS, and the highest is 1 where reaction i is the only reaction in a MCS and therefore essential for the objective function. The average fragility over all the reactions is taken as the overall structural fragility of the network.

For example, in the network example NetEx, reaction R1 has two MCSs: the first MCS is MCS2 which has 2 reactions and the second is MCS6 which consists of 3 reactions; the fragility coefficient (F₁) for R1 would therefore be 2/(2+3) which would be 2/5 or 0.4. The specific fragility coefficients of reactions in NetEx with respect to the production of P are as follows:

The above table shows that reaction R3 is essential for the production of P as is obviously the case for Psynth. This indicates that the loss of function of R3 would automatically render the other reactions meaningless for the production of P.

S. Klamt and E.D. Gilles [12] applied MCSs in their study of the central metabolic network of E.coli, earlier investigated by Stelling et al to study robustness using EMs. They found the number of MCSs to vary for different compound substrates that E.coli was growing on. For example, there were more MCSs, including the largest MCS for growth, on glucose than on acetate for which the lower number of MCSs were predominantly smaller. This indicated that E.coli growth on glucose was less fragile than on acetate.

In the generalized MCS concept [11], Klamt further discussed their work on using MCSs to measure structural fragility of a network function. The results of the previous work [12] and other work [44,45] showed that environmental conditions, such as the type of substrates or availability of oxygen, greatly affected network properties like the essentiality of a gene/reaction, so it is important for a network structural fragility analysis to clearly define environmental conditions in addition to the deletion task describing the network functionality being considered.

MCSs can be used to verify a network because the minimal sets of target reactions/genes they provide are mathematically complete in relation to the structure of the network. Thus, the simultaneous removal of genes making up each MCS should lead to the elimination of the objective function. If the prediction is incorrect in an experiment and the phenotype is still viable, it means that the network structure is incorrect or incomplete.

So, the set of MCSs could be systematically used to verify a given network structure by experimentally checking the phenotype predictions of MCSs in an organism: correct predictions would provide verification of the network whilst false predictions could be pursued to identify missing reactions/genes or compounds in the network structure. For example, say there is a reaction E = A + B in the network example NetEx (Figure 1) that has not been identified, applying MCS3 could reveal that there is a missing reaction in the network because compound E would still be formed and P synthesized.

Past work on network verification has been done using Flux Balance analysis (FBA) [46,47] and elementary mode analysis [42]. These were used to verify phenotype predictions for single mutants of E.coli, the predictions of which were found to highly agree with real mutants. In such cases the single mutation is lethal if the reaction involved is essential (a single reaction constituting a MCS) for the objective function, and depends on the chosen substrate.

Another use of the MCS concept is in finding the necessary information that can be used to make stationary network fluxes observable. As shown in [48], EM analysis (considering all reactions in the network as reversible) supports the identification of the set of known/measured flux measurements that would enable unknown non-measured reaction rates (r_u) to be calculated or observed in a steady state flux distribution. The process includes first calculating all EMs and selecting those where the unknown reaction rate exists (r_u≠0); the set of rates to be measured are then constructed such that they contain at least one of the reactions participating in each of the selected EMs. These sets of rates to be measured are in fact the MCSs with respect to the reaction rate (r_u is the objective reaction) so, the sets of possible measurements will be minimal and non-redundant. As such, the MCSs can be screened to determine the most appropriate sets of measurement for FBA [49,50].

Take NetEx (Figure 1) as an example: considering all reactions as reversible and r_u as PSynth, nine EMs are calculated, six of which lead to the synthesis of P. Taking PSynth as the objective reaction, there would be 10 MCSs for screening, as shown in Table 4 below:

To assist FBA, EMs have been used to place thermodynamics constraints at the pathway level [51] where a directionality criterion for net mass flux in the form of negative Gibbs free energy change (∆G) is applied to a pathway, as opposed to a reaction: the EMs matrix E is a PxN pathway composition, where P is the number of pathways. To formulate pathway level constraints G, the reaction-specific parameters are first collected into an Nx1 vector (∆g) and then an element-by-element multiplication is performed with each of the P (N-dimensional) rows of E to form G. The EMs ensure that the sequence of reactions in the entire pathway is in one direction and assist FBA in identifying the objective function(s) driving the metabolic behavior of tissue cells, especially multi-functional ones [51]. The relationship between MPA and FBA is discussed in detail in [52].

Along with their role in obtaining a deeper understanding of the structural fragility of cellular networks, MCSs can also be seen as minimal target sets for efficiently repressing cellular functions. The generalized concept [11] allows MCSs to tackle a larger variety of practical deletion problems, which include the repression of undesired metabolic functions, redirecting fluxes into a desired product, and inhibiting sub-optimal flux distributions. These in effect identify targets for metabolic interventions.

For example, as illustrated in the intervention in Table 2 of NetEx, the set of MCSs (cMCSs) can be identified that would repress the synthesis of P and redirect fluxes to maximize the production of X. MCSs provide the capacity to identify an optimal intervention strategy by providing, from a structural perspective, the most efficient set of manipulations to achieve a certain deletion task. In addition to being efficient, an ideal MCS would be one that is small and therefore does not affect or weakly affect other network functions; also an MCS that does not involve network functions that are hard to eliminate e.g., a reaction with many isozymes.

The growing importance of MCSs in metabolic engineering is evident in [11,12,15], for example, [15] presents a model that uses MCSs to search for gene deletion strategies that would increase the production of microorganisms. In their approach, Hädicke and Klamt [15] address the limitation that MCSs have of disabling desired functionalities along with the targeted functionalities, by generalizing MCSs to cMCSs that allow for a set of desired modes, with a minimum number preserved, to be defined.

This generalization can be applied to existing methods which can be reformulated as special cMCS problems, providing the capacity for systematic enumeration of all equivalent gene deletion combinations and determining robust knockout strategies for coupled product and biomass synthesis, altogether offering great flexibility in defining and solving knock out problems. Other examples of MCSs in metabolic engineering can be seen in [14,29], discussed earlier in Section 3.2.

Bottlenecks characterize a point of congestion in a system that happens when workloads arrive at a given point more quickly than can be handled at that point. In a metabolic network consisting of enzymes (nodes) and substrate-product metabolite fluxes (directional edges), three topological centralities that are used to measure the importance of nodes in controlling information transfer are: in degree which refers to the number of links forwarded to the node under consideration, out degree which refers to the number of links going out of the node, and betweenness which measures the number of “shortest paths” [53] going through the node. Bottlenecks are those nodes that have many “shortest paths” going through them, much like major bridges and tunnels on a highway map.

For example, the bottleneck nodes a and b in Figure 8 below, control most of the information flow because they form an essential highway to get information from the blue to the yellow nodes so, if either of nodes a or b is knocked out, the network would collapse. In effect, bottlenecks indicate essentiality of the nodes.

The essentiality of the bottleneck nodes is illustrated in the above graph which shows that they are “AND” nodes, traversed in series and you cannot get from the input nodes to the output except through node a “AND” node b. The in degree of node a is 4 and the out degree is 1; these centralities only consider the partners connected directly to a particular node, whereas the betweenness considers a node’s position in the network and, as shown for a, is much higher e.g. 28. Thus, bottlenecks in metabolic networks could be defined as nodes with a high betweenness centrality.

One importance of bottlenecks is in relation to whole genome duplication (WGD) [54,55], where studies have shown that genes encoding hubs and bottleneck enzymes tend to express highly and evolve conservatively and thus were preferentially retained as homeologs [56]. Other studies include identification of novel targets for metabolic engineering of microorganisms used for sustainable production of fuels and chemicals [57] where the set of hub and bottleneck genes/enzymes were found to be a better strategy than manipulation of a single gene/enzyme.

In relation to MCSs, although MCSs can similarly determine the essentiality of enzymes, they do so in terms of repressing an objective function, represented by an objective reaction(s). For example, to use MCSs to calculate the essentiality of reactions/enzymes for a whole network, the objective function to repress would be the formation of all end products in the network, which would likely lead to combinatorial problems in larger networks. For the example network, NetEx, (refer to Figure 1), the objective reactions to repress in order to block all products are R5 and PSynth. In relation to the 6 EMs shown in Table 1, there are 16 MCSs for repressing the reactions R5 and PSynth. These MCSs are shown in Table 5 below with the corresponding fragility coefficients for each reaction:

The above table shows no reaction with a fragility coefficient [12] of 1, indicating that there is no essential reactions/enzymes (bottleneck) that, when blocked, would cause a collapse of the network NetEx. Bottlenecks would require a fragility coefficient of 1 because they represent an essential reaction that forms a bridge or tunnel to get from the input side of the network to the output. MCSs don’t necessarily have to, as shown by the fragility coefficients in Table 2 above, which can be used to extract information on the relative importance of reactions/enzymes.

For example, ignoring the outermost reactions connected to the products (R5 and PSynth) in NetEx, R3 is the reaction with the highest fragility coefficient of 0.5. When we look at the corresponding EMs, R3 is also involved in the highest number of 5 EMs. Characterising that as a bottleneck does not seem unreasonable when looking at the NetEx diagram. In fact, adding the number of 1’s in the EM table is somewhat like the “betweenness” index that bottlenecks are based on.

However, there is a significant difference: EM’s are not just shortest paths in the network; they are paths that are “short” in the sense of being irreducible, but their more important feature is that they cover all the mutually independent paths from substrates to products compatible with steady state. So, they reflect a lot more about the functioning of the network, not just the topology. Such betweenness in bottlenecks or derived from EMs, is basically what the fragility coefficient [12] expresses from MCSs. In effect, the fragility coefficient serves the same purpose as betweenness from the perspective of how fragile the structure of the network is at each reaction/enzyme but in a more comprehensive manner because it takes into account all MCSs that each reaction is involved in; in this respect the betweenness derived from MCS is much more informative for metabolism than the simple bottleneck concept.

The analysis of the connectivity structure of genome-based metabolic networks of 65 fully sequenced organisms [7] revealed that the global metabolic network was organized in the form of a bow-tie [7,58]. Metabolism has also been described as several nested bow-ties and large-scale organizational frameworks such as the bow-tie were necessary starting points for higher-resolution modeling of complex biological processes [59]. Studies and detailed information on the bow-tie topological features of metabolic networks and their functional significance can be seen in [7,58,59,60,61].

The concept of bow-ties regards the metabolic network as a directed network. As illustrated in Figure 9 below, bow-ties [7,58,59,60,61], show similarity in structure to bottlenecks, except there is a difference in how the nodes are connected: the nodes that make up a bow-tie are “OR” nodes, i.e. they are traversed in parallel, while the nodes of a bottleneck are “AND” nodes, traversed in series.

As illustrated above, the bow-tie structure of a directed graph has 4 components [7,58,59,60,61]:

The bow-tie decomposition of a network can assist with the problem of combinatorial explosion encountered when calculating EMs and MCSs in large sized metabolic networks. For example, suppose that EMs are calculated separately for each of the three subnets (substrate S, GSC and product P), a typical EM for the full network can then be reconstituted by joining a substrate mode and product mode to one of their connecting GSC modes. The large number of ways in which this can be done is a manifestation of the combinatorial explosion, and demonstrates that the bow-tie splitting will substantially reduce the computational effort of calculating EMs and the resulting MCSs.

More explicitly, the reactions constituting MCSs of a whole network can be classified in terms of the blocked reactions’ locations in the bow-tie decomposition:

These classifications can be used to investigate the question of whether a bow-tie decomposition can be derived from a known MCSs table. For example, a plausible strategy to identify GSC reactions is as follows:

If the bow-tie structure is pronounced, there should be a clear separation between the small, high fragility coefficient MCSs that belong to the GSC and the rest, otherwise the choice of a cutoff may be problematic. An MCS analysis may be helpful to examine if a bow-tie structure exists and partially detect members of its main components, but not to make a full partitioning.

Noting that bow-ties can assist with combinatorial explosion by decomposing large networks into subnets that can be analyzed by MCSs and EMs, we conclude that despite some overlap in the concepts and applications of bow-ties and MCSs, there is no clear cut correspondence between the two network descriptions. While bow-ties try to extract subsets of nodes that are of importance in the metabolic network, the EM and MCS approaches focus on comprehensive sets that are in different ways essential. Moreover, EMs are, by construction, the “constituents” of a steady metabolic state. So they, and MCSs, reflect the stoichiometry underlying the network and describe the metabolism, not just the topology of the network. In this respect, MCS (and EM) analysis is more powerful than bow-ties that just characterize network topology.

‘Weak nutrient sets’ is a concept analogous to MCSs, that was developed by Imielinski et al. [63] to demonstrate the duality between weak producibility and the existence of certain extreme semipositive conservation relations (ESCRs) in a media. ESCRs were defined as the simplest semi-positive linear combinations of species concentrations that were invariant to all metabolic flux configurations. A biochemical species was called producible in a constraints-based metabolic model if a feasible steady-state flux configuration existed that sustained its nonzero concentration during growth.

Weak nutrient sets are analogous to MCSs in a metabolic network in that a MCS C for an objective reaction j is a set of reactions whose elimination renders flux through j infeasible at steady state so, a necessary and sufficient condition for C to be a cut set for j is that C is a hitting set for all j-containing elementary modes. Similarly, U is a weak nutrient set for species or metabolite i if and only if U is a hitting set for all of the i-containing ESCRs.

The ‘weak nutrient sets’ algorithm identified all minimal nutrient media that left an arbitrary metabolite weakly producible with respect to a given metabolic network. Details of the concept and its application can be seen in [63].

Flux balance analysis (FBA) [49,64,65] shares a common underlying mathematical framework with MCSs and EMs except that, while EMs identify all possible and feasible non-decomposable metabolic routes for a given network at steady state, FBA derives a feasible set of steady-state fluxes optimizing a stated cellular objective e.g, optimizing the biomass production per substrate uptake. EM analysis establishes a link between structural analysis and metabolic flux analysis (MFA) where thermodynamically and stoichiometrically feasible stationary flux distributions for a network can be obtained from the linear combinations of the EMs.

Calculating EMs and MCSs for larger networks can lead to problems with combinatorial explosion. However, because they are unique for a given network structure, they provide the full range of potential functionalities of the metabolic system and are therefore useful for investigating all physiological states that are meaningful for the cell in the long term. FBA, on the other hand, is more efficient, providing good predictions of mutant phenotypes and using linear programming to obtain a single (not necessarily unique) solution to an optimization problem. However, because it focuses on a specific behavior, FBA cannot cope with cellular regulation without additional constraints; it fails whenever network flexibility has to be taken into account, e.g., in the analysis of pathway redundancy or in quantitative prediction of gene expression [42].

We conclude that MCSs and EMs offer a convenient way of interpreting metabolic functions while FBA can be used to explore the relationship between the metabolic genotype and phenotype of organisms. MCSs, EMs and FBA can also be used together to interpret shifts in metabolic routing that could occur in response to environmental and internal/genetic challenges. Because they are mathematically equivalent, the predictions from the three methods would be the same except that MCSs enable the systematic search of more than one mutation.