5.5.3 Structure of FBGs at multiple resolutions

One partition in each of the four Flux-Balance Graphs has been already analysed.

However, complex graphs such as these often have important partitions into communities at different levels of granularity (Bacik et al. 2015; Delvenne, Yaliraki, and Barahona 2010).

In the Markov Stability framework, we can explore these scales by scanning through different values of Markov time (see last subsection in Section 2.3.1).

For instance, Figure 5.6a shows the number of communities found in the graph Mglc as one scans through a range of Markov times.

Short Markov times result in fine-grained partitions (e.g., near t = 0.3 there are over 20 communities); Markov time is increased, we will find coarser partitions until (as Markov time tends to infinity) a single community containing all the nodes is found.

Five Markov times at which robust partitions into 11,7, 5, 3, and 2 communities have been selected.

To analyse the partitions (and the partition into pathways) simultaneously, a Sankey diagram is constructed (Figure 5.6b).

These diagrams allow to visualise the composition of the different partitions and how they are related in terms of their members.

In the example from Figure 5.6b, we start on the left side with the reactions of Mglc divided in metabolic pathways.

As we move to the right, we can see how the reactions in each of the pathways assemble into the partitions obtained with Markov Stability.

This figure highlights different features and properties of the E. coli metabolic network in aerobic conditions and with glucose as the sole source of carbon.

In some cases, the reactions of pathways such as the oxidative phosphorylation or glycolysis are grouped mostly together in the same community.

Interestingly, the TCA cycle, although it appears as a cohesive unit for most Markov times is split in two at t = 19.72.

The pathways with reactions in charge of exchanging substances with the exterior of the cell (exchange and transport) are spread among the partitions in all Markov times; these are pathways in which the reactions do not interact amongst themselves.

These pathways act more like ‘roles’ (i.e., importing and moving substrates) than like cohesive metabolic sub-units.

Other pathways such as the pentose phosphate pathway is divided into different communities except at t = 6.01 when, as the TCA cycle, comes together before splitting up again.

This phenomenon illustrates that some biological features may only become relevant at specific resolutions.

The question of how to deal with this resolution dependency results of vital importance.

In general there is no set rule to pick a concrete Markov Time in which to analyse the community structure of each network.

It is a case by case decision.

There are however some general guidelines that increase the usefulness or interest of particular resolution.

Times in which the Variation of Information decreases abruptly point in mayor network re-structuring moments which may be of interest.

Furthermore, long time intervals in which the VI stays very low or zero show very stable partitions which are potentially more interesting.

Finally, there is compromise that has to be reached: enough Markov Time to reach stable partitions with communities of significant size.

This depends on the size of the network too, since larger networks will need longer times.

Figure 5.6. Multiscale community structure of the graph Mgcl.

(a) Number of communities (blue line) and Variation of Information (red line) in the Mglc graph as we scan Markov times (see text and Methods section) for communities. Five Markov times in which the network can be split into 11, 7, 5, 3, and 2 communities and the VI is low or has a pronounced dip were selected.

(b) Alluvial diagram (Rosvall and Bergstrom 2010) showing how the reactions that form each of the pathways (left) assemble in communities of different size as Markov time is scanned. Note that in this graph there are no reactions (with positive flux) in the pyruvate metabolism subsystem.

Figure 5.6. Multiscale community structure of the graph Mgcl.

(c) Communnity structure of the graph Mglc in the selected Markov times.