Figure 1.
A simple example network used for illustration purposes.
The interaction graph consists of 7 nodes and 7 edges. The green nodes and
can be perturbed externally; the grey nodes
,
and
are the readouts of the network whose activation state is measured in the experiments; the white nodes
and
are latent nodes which are neither perturbed nor measured (see scenarios in Table 1). (A) The initial topology of the interaction graph representing the prior knowledge. This graph produces a total fitting error of 5 over the three scenarios in Table 1. (B) The (unique) optimal subgraph of (A) minimizing the total fitting error on the experimental scenarios to 2 (see Table 1). (C) Two optimal graphs obtained from (A) by applying OPT_GRAPH: by adding edge
and either (left) removing
or (right) removing
and
, the fitting error is eradicated completely and becomes 0 (cf. Table 1).
Table 1.
Example scenarios and optimizations for the example network in Figure 1.
Figure 2.
Basic network compression rules.
(A) Parallel edges. (B) Nodes with single input. (C) Nodes with single output. (D) Shown is the compressed version of the network in Figure 1A after applying the compression rules. For further explanations see main text.
Figure 3.
Discretized measurements of the 16 considered experimental scenarios and the resulting SCEN_FIT solutions computed from the EGFR/ErbB graph model.
Each row corresponds to one experimental scenario, each column contains the measured state changes of the readout species. The discretized measurements are mapped to the fill color of the respective fields: if a node is upregulated in the respective scenario, the corresponding field is filled green, if it is downregulated, the field is filled red, and if it shows no significant change, it is filled white. Accordingly, the color of the added circles shows the sign of the node in the closest sign-consistent node labeling derived by SCEN_FIT: green circles correspond to sign 1, red circles to sign −1 and white circles to sign 0. Note that circles only appear if the measurement is not in accordance with the respective state in the sign-consistent labeling.
Figure 4.
Interaction graph model of the EGFR/ErbB signaling network.
(A) The full network adopted from [18] after removal of non-observable and non-controllable nodes. All edges are activating edges (having positive signs). (B) The compressed model obtained after applying the compression rules to (A).
Table 2.
MCoS for scenario 14 in Figure 3.
Figure 5.
Combined view of all optimal model structures derived from the compressed EGFR/ErbB model by applying the OPT_SUBGRAPH procedure with enumeration.
Figure 6.
Discretized data and the (two) SCEN_FIT solutions that result from the optimal subgraphs given in Figure 5.
The color coding is the same as in Figure 3. All six optimal subgraphs contained in Figure 5 give rise to the same SCEN_FIT solution, except for the CREB column. Here, three subgraphs show a mismatch in scenarios 5, 10, and 15 (indicated by the left semicycles), while the other three show a mismatch in scenarios 6, 11, and 16 (indicated by the right semicycles).
Table 3.
Suggestions for new edges as computed by OPT_GRAPH.
Figure 7.
Combined view of the three optimal subgraphs resulting when adding TGFα to CREB to the initial model structure.
In all three solutions, the edges erk12→p70s6_1, tgfa→stat3, p90rsk→creb and p38→creb are removed. Edges tgfa→mek12 and rac_cdc42→mek12 represent alternative pathways; at least one of both must be contained.
Figure 8.
Comparison of the fitting errors of the initial model structure (see Figures 3 and 4) and of the optimal interaction graph shown in Figure 7.
Green fields indicate an error that has been present in the original model structure, but could be removed by optimizing the model structure. Yellow fields refer to errors that could not be resolved, and red fields indicate errors that have not been present in the original model structure, but were introduced by the optimization.