# Minimal Cut Sets and the Use of Failure Modes in Metabolic Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Defining Minimal Cut Sets

“We call a set of reactions a cut set (with respect to a defined objective reaction) if after the removal of these reactions from the network no feasible balanced flux distribution involves the objective reaction”; and “A cut set C (related to a defined objective reaction) is a minimal cut set (MCS) if no proper subset of C is a cut set.”

#### 2.1. The Initial Concept of MCSs

#### 2.2. Example Network to Illustrate MCSs

- The network consists of five internal metabolites and eight reactions, of which R4 and R5 are reversible;
- Reactions crossing the system boundaries are coming from/leading to buffered/buffer metabolites.

**Figure 2.**Elementary modes (EMs) for NetEx. The EMs are represented by the solid blue arrows. Adapted from [11].

#### 2.3. Other Definitions

#### 2.3.1. Fault Trees

- MCSs obtained from the RBD are: {1}, {7}, {5,6}, {2,3,4}, {2,3,6} and {3,4,5};
- The Fault Tree is constructed by connecting the MCSs using the OR gate. Within each set that contains multiple blocks, the multiple blocks are connected with an AND gate. The equivalent Fault Tree is shown in Figure 5 below:

**Figure 5.**Equivalent Fault Tree of RBD in Figure 4. Blocks 2a-6a, 3b are duplicates of their corresponding blocks.

#### 2.3.2. Graph Theory

**Figure 6.**Hypergraph showing reactant and product nodes of R6 and R7 of NetEx with corresponding versions of substrate and bipartite graphs. In the lower hypergraph, removing R3 means no C is formed and a consequential removal of R6 and R7, which means that PSynth cannot proceed.

#### 2.4. Determining MCSs

- (1)
- Calculate EMs [3] in NetEx and identify those that start from a buffered metabolite and lead to the formation of metabolite E or the objective reaction PSynth. Since EMs are non-decomposable, removing one of the reactions from these EM will prevent the system from producing E and subsequently achieving the PSynth.There are six EMs in total, of which five lead to the formation of metabolite X and the objective reaction.
- (2)
- Determine how to prevent PSynth from taking place, i.e. stop the five EMs that involve PSynth from being functional. This can be done in various ways e.g. inactivating one or more reactions in the EMs by deleting genes of certain enzymes or other manipulations that inhibit the enzymes. Different numbers and combination of reactions can be removed to eliminate PSynth.

- any feasible steady-state flux distribution in a given network, expressed by a vector of the net reaction rates, r, can be represented by a non-negative linear combination of elementary modes as illustrated in Equation 1 (adapted from [11]):

- the removal of reactions from the network results in a new set of EMs constituted by those EMs from the original network that do not involve the deleted reactions [24].

- the set of target modes (E
^{t}), i.e., all EMs (e^{t,j}) involving the objective reaction, t - the set of non-target modes (E
^{nt}), i.e., EMs not involving the objective reaction, nt

^{,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}.

**Figure 7.**One of the Minimal cut sets (MCSs) for objective reaction PSynth: The simultaneous blocking of reactions R1 and R7 will eliminate PSynth and block the production of P.

#### 2.5. Generalized Concept of MCSs

#### 2.6. Further Refined Concept of MCSs

#### 2.7. Comparing MCS Concepts

#### 2.7.1. Same Properties

- there will always be a trivial MCS- the objective reaction itself;
- some reactions such as the biomass synthesis, are actually pseudo-reactions that are not related to a single gene or enzyme and thus cannot be repressed by inhibitions such as gene deletions;
- the definition of the MCSs: each MCS provides a minimal (irreducible) set of deletions or EMs from the set of target modes, that will achieve the elimination of the objective reaction.

#### 2.7.2. Different Properties

- A deletion task T is a set of constraints that characterize the stationary flux patterns (reactions) r to be repressed while D, derived from T, characterizes the target modes (EMs) to be targeted by MCSs. As such, D (for the target modes) and T (for the flux vectors r) are, in most cases such as in the earlier MCS concept, identical.
- In the generalized MCS concept, however, the deletion task D can either differ from T or T must be transformed into several D
_{i}that lead to sub-tasks. So, instead of only dealing with a simple deletion task T where all non-trivial flux distributions for an objective reaction are blocked, other more complicated deletion tasks and intervention goals are possible. - The generalized MCS concept offers a wider range of capacity to assess, manipulate and design biochemical networks. MCSs are no longer restricted to the removal of reactions as shown in Figure 2 but can also contain network nodes such that more general deletion problems can be tackled. The MCSs that involve the removal of other network parameters besides reactions are shown in the lower two tables (1b and 1c) of Table 1 below.

**Table 1.**Elementary modes and the different types of MCSs of NetEx for the objective reaction PSynth. Initial MCS concept: 1a): removing reactions only; Generalized MCS concept: 1b) removing metabolites only, and 1c) reactions and metabolites together. Note: a non-zero in the EM cell indicates the reaction occurs in the EM; a “1” in the MCS indicates the reaction constitutes the MCS. Adapted from [11].

Elementary modes EM2-EM6 (grey) involve the objective reaction PSynth. | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

R1 | R2 | R3 | R4 | R5 | R6 | R7 | PSynth | A | B | C | D | E | |

EM1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |

EM2 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |

EM3 | 1 | 0 | 1 | −1 | −1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |

EM4 | 0 | 1 | 1 | 0 | −1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |

EM5 | 0 | 0 | 1 | −1 | −2 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |

EM6 | 0 | 2 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |

MCSs of NetEx for the objective reaction PSynth | |||||||||||||

1a) Initial concept: MCSs removing reactions only | |||||||||||||

MCS0 | 1 | ||||||||||||

MCS1 | 1 | ||||||||||||

MCS2 | 1 | 1 | |||||||||||

MCS3 | 1 | 1 | |||||||||||

MCS4 | 1 | 1 | |||||||||||

MCS5 | 1 | 1 | |||||||||||

MCS6 | 1 | 1 | 1 | ||||||||||

MCS7 | 1 | 1 | 1 | ||||||||||

1b) Generalized concept: Minimal cut sets removing metabolites only | |||||||||||||

MCS8 | 1 | ||||||||||||

MCS9 | 1 | ||||||||||||

MCS10 | 1 | ||||||||||||

MCS11 | 1 | 1 | |||||||||||

1c) Generalized concept: Minimal cut sets removing reactions and metabolites | |||||||||||||

MCS12 | 1 | 1 | |||||||||||

MCS13 | 1 | 1 | 1 | ||||||||||

MCS14 | 1 | 1 | |||||||||||

MCS15 | 1 | 1 | |||||||||||

MCS16 | 1 | 1 |

Intervention Problems | Target modes T | Desired modes D1 | n1 | MCSs | |
---|---|---|---|---|---|

I1) | No synthesis of undesired product P | EM2, EM3, EM4, EM5, EM6 | MCS0={ Psynth}, MCS1={R3}, MCS2={R1,R7}, MCS3={R6,R7}, MCS4= {R2, R4}, MCS5={R2,R5}, MCS6={R1,R4,R5}, MCS7={R4,R5,R6} | ||

I2) | No synthesis of undesired product P and production of X with maximal yield possible | EM2, EM3, EM4, EM5, EM6 | EM1 | 1 | MCS0={ Psynth}, MCS1={R3}, MCS2={R1,R7}, MCS3={R6,R7}, |

## 3. Computational Complexity

#### 3.1. Deterministic and Non-Deterministic Polynomial Complexity

^{2}-2n+5. As n grows large, the n

^{2}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., n

^{3}or n

^{4}; U(n) = n

^{3}, 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 n

^{2}time complexity denoted by the big O as:

^{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.

^{j}))where j is a non-negative integer.

#### 3.2. MCS Computational Methods

- i)
- The first method was presented by Imielinski and Belta [35] and considers obtaining cut sets from the computation of sub-EMs which are EMs of a submatrix of the stoichiometry matrix [36]; the submatrix in turn is formed by taking a subset of the rows of the stoichiometry matrix. In other words, the sub-EMs are flux configurations that place only a subset of species in the system at steady state. Because the sub-EMs naturally emerge from the intermediate steps of the tableau algorithm for EM computing [3], it means that the sub-EMs can be obtained from a network of any size, hence overcoming the problem where the metabolic network is too large and complex that it becomes NP-hard to find MCSs. A possible drawback is that there is no guarantee that all the cut sets will be found and their minimality is also not guaranteed so the cut sets would need to be checked for minimality and further reduced to MCSs where necessary. Development of this computational framework is described in detail in [35] as well as its application to a genome scale metabolic model of E.coli.
- ii)
- The second method is by Haus et al. [14] and involves modifying existing algorithms to develop more efficient methods for computing MCSs. Their first algorithm is a modification of Berge’s algorithm [37] and computes MCSs from EMs, thereby improving on the time and memory required for enumeration; the second algorithm is based on Fredman and Khachiyan [38] and directly computes MCSs from the stoichiometric matrix, with the hypergraph of EMs containing the blocked reactions being generated on the side.
- iii)
- The third method, contributed by Ballerstein et al. [29], also determines MCSs directly without knowing EMs. Their computational method is based on a duality framework for metabolic networks where the enumeration of MCSs in the original network is reduced to identifying the EMs in a dual network so both EMs and MCSs can be computed with the same algorithm. They also proposed a generalization of MCSs by allowing the combination of inhomogeneous constraints on reaction rates.
- iv)
- The fourth method includes an approximation algorithm for computing the minimum reaction cut and an improvement for enumerating MCSs, recently proposed by Acuña et al. [30]. These emerged from their systematic analysis of the complexity of the MCS concept and EMs, in which it was proved that finding a MCS, finding an EM containing a specified set of reactions, and counting EMs are all NP-hard problems.

## 4. Applications of MCSs

- i)
- If the cut occurred naturally, e.g., a reaction malfunctioning due to spontaneous mutation, the MCS would serve as an internal failure mode with respect to a certain functionality and could be applied to study structural fragility and robustness on a local and global scale.
- ii)
- If, on the other hand, the cut is a deliberate intervention e.g., gene deletion, enzyme inhibition or RNA interference, then the MCS would be seen as a target set that could, for example, be suitable for blocking metabolic functionalities, and thus have significant potential in metabolic engineering and drug discovery. These applications can be extended to enable the MCSs to be used for assessing/verifying, manipulating and designing biochemical networks.

#### 4.1. Fragility Analysis

_{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.

_{1}) 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:

R1 | R2 | R3 | R4 | R5 | R6 | R7 | Psynth | |
---|---|---|---|---|---|---|---|---|

Fi | 0.4 | 0.5 | 1 | 0.375 | 0.375 | 0.4 | 0.5 | 1 |

#### 4.2. Network Verification

#### 4.3. Observability of Reaction Rates in Metabolic Flux Analyses

_{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].

_{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:

**Table 4.**EMs and MCSs of NetEx (all reactions are reversible): A zero in an EM row indicates that the corresponding reaction is not involved in the EM corresponding to that row; a ‘1’ in a MCS row indicates that the reaction in that column constitutes the MCS corresponding to that row, e.g., R3 constitutes MCS1.

R1 | R2 | R3 | R4 | R5 | R6 | R7 | PSynth | |
---|---|---|---|---|---|---|---|---|

EM1 | 1 | 0 | 1 | -1 | -1 | 1 | 0 | 1 |

EM2 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |

EM3 | 1 | 0 | 0 | 0 | 1 | 1 | -1 | 0 |

EM4 | 0 | 2 | 1 | 1 | 0 | 0 | 1 | 1 |

EM5 | 0 | 1 | 1 | 0 | -1 | 0 | 1 | 1 |

EM6 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |

EM7 | 0 | 0 | 1 | -1 | -2 | 0 | 1 | 1 |

EM8 | 1 | -1 | 0 | -1 | 0 | 1 | -1 | 0 |

EM9 | 2 | 0 | 1 | -1 | 0 | 2 | -1 | 1 |

R1 | R2 | R3 | R4 | R5 | R6 | R7 | PSynth | |

MCS1 | 1 | |||||||

MCS2 | 1 | |||||||

MCS3 | 1 | 1 | ||||||

MCS4 | 1 | 1 | ||||||

MCS5 | 1 | 1 | ||||||

MCS6 | 1 | 1 | 1 | |||||

MCS7 | 1 | 1 | 1 | |||||

MCS8 | 1 | 1 | 1 | |||||

MCS9 | 1 | 1 | 1 | |||||

MCS10 | 1 | 1 | 1 |

#### 4.4. Pathway Energy Balance Constraints

#### 4.5. Target Identification and Metabolic Interventions

## 5. Similar concepts

#### 5.1. Bottlenecks

**Table 5.**MCSs for NetEx, where all the EMs form the objective function. A “1” in the row of a MCS indicates inclusion of that reaction in the MCS, e.g, MCS1 consists of reactions R3 and R4, which means that simultaneous blocking of R3 and R4 would collapse NetEx. Fj shows the fragility coefficients of the reactions.

▪ MCSs | R1 | R2 | R3 | R4 | R5 | R6 | R7 | PSynth | Total |
---|---|---|---|---|---|---|---|---|---|

▪ MCS1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 2 |

▪ MCS2 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 2 |

▪ MCS3 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 2 |

▪ MCS4 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 2 |

▪ MCS5 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 2 |

▪ MCS6 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 2 |

▪ MCS7 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 2 |

▪ MCS8 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |

▪ MCS9 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 3 |

▪ MCS10 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 3 |

▪ MCS11 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 3 |

▪ MCS12 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 3 |

▪ MCS13 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 3 |

▪ MCS14 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 3 |

▪ MCS15 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 3 |

▪ MCS16 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 3 |

▪ Total | 4 | 6 | 3 | 7 | 7 | 4 | 6 | 3 | |

▪ F_{j} | 0.33 | 0.43 | 0.5 | 0.39 | 0.39 | 0.33 | 0.33 | 0.50 |

#### 5.2. Bow-Ties

- (1)
- The input domain (substrate subset (S)), which contains substrates that can be converted reversibly to intermediates or directly to metabolites in the GSC, but those directly connected to the GSC cannot be produced from the GSC.
- (2)
- The knot or GSC, which is the metabolite converting hub [60], where protocols manage, organize and process inputs, and from where, in turn, the outputs get propagated. The GSC follows the graph theory definition [62] and contains metabolites that have routes (can be several) connecting them to each other; it is the most important subnet in the bow-tie structure.
- (3)
- The output domain (product subset (P)), which contains products from metabolites in the GSC and can also have intermediate metabolites but the products cannot be converted back into the GSC [7]. In other words, the reactions directly linking substrates to the GSC and the GSC to the products are irreversible.
- (4)
- The resulting metabolites that are not in the GSC, S or P subsets form an isolated subset (IS), the simplest structured of the four bow-tie components [7], which can include metabolites from the input domain S or the output domain P but those metabolites cannot reach the GSC or be reached from it.

- (1)
- All substrate reactions (S subnet) plus GSC reactions blocking any cyclic EMs that could take place without inputs from the substrate reactions. In this case, no product reactions (P subnet) need blocking;
- (2)
- All product reactions(P subnet) plus GSC reactions blocking the cyclic EMs- in this case no substrate (S subnet) need to be blocked;
- (3)
- All GSC reactions that connect the S to the P subnet. No substrate or product reactions need to be blocked;
- (4)
- A combination of S reactions plus GSC reactions reached from the unblocked S reactions. P reactions don’t need to be blocked;
- (5)
- A combination of P reactions plus GSC reactions that could reach the unblocked P reactions. S reactions don’t need blocking.

- From all MCS, eliminate any that involve reactions that are known to belong to S or P;
- Order the remainder by increasing size and/or decreasing mean fragility coefficient;
- Choose a cutoff value in this sequence, and allocate all reactions that belong to MCSs in the top section of the sequence to the GSC.

#### 5.3. Weak Nutrient Sets

#### 5.4. Flux Balance Analysis

## 6. Conclusions

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Clark, S.T.; Verwoerd, W.S.
Minimal Cut Sets and the Use of Failure Modes in Metabolic Networks. *Metabolites* **2012**, *2*, 567-595.
https://doi.org/10.3390/metabo2030567

**AMA Style**

Clark ST, Verwoerd WS.
Minimal Cut Sets and the Use of Failure Modes in Metabolic Networks. *Metabolites*. 2012; 2(3):567-595.
https://doi.org/10.3390/metabo2030567

**Chicago/Turabian Style**

Clark, Sangaalofa T., and Wynand S. Verwoerd.
2012. "Minimal Cut Sets and the Use of Failure Modes in Metabolic Networks" *Metabolites* 2, no. 3: 567-595.
https://doi.org/10.3390/metabo2030567