Table 6 Network with PBK+EI continued from [1]

Network with PBK and EI continued … Lastly, it is known that concentration of D V L2 (a member of disheveled family) is inversely regulated by the expression of D A C T3 [2]. High D V L2 concentration and suppression of D A C T1 leads to increase in stabilization of β-catenin which is necessary for the Wnt pathway to be active [2]. But in a recent development [7], it has been found that expression of D A C T1 positively regulates β-catenin. Both scenarios need to be checked via inspection of the estimated probability values for β-catenin using the test data. Thus, there exists direct causal relations between parent nodes D A C T1 and D V L2 and child node, β-catenin. Influence of methylation (yellow hexagonal) nodes to their respective gene (green circular) nodes represent the effect of methylation on genes. Influence of histone modifications in H3K27m e3 and H3K4m e3 (blue octagonal) nodes to D A C T3 gene node represents the effect of histone modification on D A C T3. The β-catenin (blue square) node is influenced by concentration of D V L2 (depending on the expression state of D A C T3) and behavior of D A C T1. The aforementioned established prior causal biological knowledge is imposed in the BN model with the aim to computationally reveal unknown biological relationships. The influence diagram of this model is shown in Fig. 2 with nodes on methylation and histone modification. Another model \(\mathcal {M}_{\text {PBK}}\) (not shown here) was developed excluding the epigenetic information (i.e., removal of nodes depicting methylation and histone modification as well as the influence arcs emerging from them) with the aim to check whether inclusion of epigenetic factors increases the cancer prediction accuracy.

In order to understand indirect connections further, it is imperative to know about d-connectivity/separability. In a BN model, this connection is established via the principle of d-connectivity which states that nodes are connected in a path when there exists no node in the path that has more than one incoming influence edge or there exists nodes in the path with more than one incoming influence edge which are observed (i.e., evidence regarding such nodes is available) [50]. Conversely, via principle of d-separation, nodes are separated in a path when there exists nodes in the path that have more than one incoming influence edge or there exists nodes in the path with at most one incoming influence edge which are observed (i.e., evidence regarding such nodes is available). Figure 3 represents three different cases of connectivity and separation between nodes \(\mathcal {A}\) and \(\mathcal {C}\) when the path between them passes through node \(\mathcal {B}\). Connectivity or dependency exists between nodes \(\mathcal {A}\) and \(\mathcal {C}\) when (a) evidence is not present regarding node \(\mathcal {B}\) in the left graphs of I and II in Fig. 3 or (b) evidence is present regarding node \(\mathcal {B}\) in the right graph of III in Fig. 3.

Conversely, separation or independence exists between nodes \(\mathcal {A}\) and \(\mathcal {C}\) when (a) evidence is present regarding node \(\mathcal {B}\) in the right graphs of I and II in Fig. 3 or (b) evidence is not present regarding node \(\mathcal {B}\) in the left graph of III in Fig. 3. It would be interesting to know about the behavior of TRCMPLX, given the evidence of state of S F R P3. To reveal such information, paths must exist between these nodes. It can be seen that there are multiple paths between TRCMPLX and S F R P2 in the BN model in Fig. 2. These paths are enumerated as follows:

1. S F R P3, Sample, S F R P1, TRCMPLX

2. S F R P3, Sample, D K K1, TRCMPLX

3. S F R P3, Sample, W I F1, TRCMPLX

4. S F R P3, Sample, C D44, TRCMPLX

5. S F R P3, Sample, D K K4, TRCMPLX

6. S F R P3, Sample, C C N D1, TRCMPLX

7. S F R P3, Sample, MYC, TRCMPLX

8. S F R P3, Sample, L E F1, TRCMPLX

9. S F R P3, Sample, D A C T3, D V L2, β-catenin, TRCMPLX

10. S F R P3, Sample, D A C T1, β-catenin, TRCMPLX

Knowledge of evidence regarding nodes of S F R P1 (path 1), D K K1 (path 2), W I F1 (path 3), C D44 (path 4), D K K4 (path 5), C C N D1 (path 6), and MYC (path 7) makes Sample and TRCMPLX dependent or d-connected. Further, no evidence regarding state of Sample on these paths instigates dependency or connectivity between S F R P3 and TRCMPLX. On the contrary, evidence regarding L E F1, D A C T3, and D A C T1 makes Sample (and child nodes influenced by Sample) independent or d-separated from TRCMPLX through paths (8) to (10). Due to the dependency in paths (1) to (7) and the given state of S F R P3 (i.e., evidence regarding it being active or passive), the BN uses these paths during inference to find how TRCMPLX might behave in normal and tumorous test cases. Thus, exploiting the properties of d-connectivity/separability, imposing a biological structure via simple yet important prior causal knowledge and incorporating epigenetic information, BN helps in inferring many of the unknown relation of a certain gene expression and a transcription complex.