Figure 2

Example of the flood network minimization.Original gene regulatory network (A) is transformed (B) to introduce saturation gadgets and potentially basal nodes (not shown). Minimized network is obtained after a flood threshold is applied to the flooded transformed network. Degree of minimization can be varied from aggressive (C) to mild (D) depending on the value of the threshold. Capacity of the edge is the upper limit for the magnitude of the flood through that edge, which depends on the upstream regulation. For example: edge 3→6 has a high capacity and a high positive flood, but the flood through edge 5→2 is low regardless its high capacity. Floods are calculated for the essential traversals from the input(s)/signal(s) to the output(s)/sink(s): 1→3→6, 1→4→6, 1→4→6→5→2→6, 1→3→6→5→2→6, 1→3→4→6, etc. The total flood through edge 2→6 is zero due to the strong negative flood 5→2 (inhibition of node 2). Therefore, nodes 2 and 5 are not part of the minimized network connecting inputs and outputs regardless of the imposed flood threshold. In contrast, node 4 may be included in the minimized network if a flood threshold is relatively low: flood 1→3 is replicated into two edges of high capacity 3→6 (positive flood, activation of node 6) and 3→4 (negative flood, inhibition of node 4). Large positive flood 1→4 in combination with a negative flood 3→4 produces a weak positive flood 4→6. Edge 4→6 is included in the minimized network only if the flood threshold is below 4→6 flood (D). If edge 4→6 is not in minimized network (C), edges 1→4 and 3→4, and node 4 are also not included, as they are disconnected from the output node(s).