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
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
is knocked out, the network would collapse. In effect, bottlenecks indicate essentiality of the nodes.
Example of a bottleneck in metabolic networks.
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
], 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:
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|
|▪ Fj ||0.33 ||0.43 ||0.5 ||0.39 ||0.39 ||0.33 ||0.33 ||0.50 |
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
]. 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
The concept of bow-ties regards the metabolic network as a directed network. As illustrated in Figure 9
below, bow-ties [7
], 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.
A simplified example of a bow tie.
As illustrated above, the bow-tie structure of a directed graph has 4 components [7
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.
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.
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.
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.
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:
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;
All product reactions(P subnet) plus GSC reactions blocking the cyclic EMs- in this case no substrate (S subnet) need to be blocked;
All GSC reactions that connect the S to the P subnet. No substrate or product reactions need to be blocked;
A combination of S reactions plus GSC reactions reached from the unblocked S reactions. P reactions don’t need to be blocked;
A combination of P reactions plus GSC reactions that could reach the unblocked P reactions. S reactions don’t need blocking.
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:
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.
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.
5.4. Flux Balance Analysis
Flux balance analysis (FBA) [49
] 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.