Fig 1.
Representations of two toy reactions as directed graphs, compound graphs, directed hypergraphs, and bipartite graphs.
In this work, we use “directed hypergraphs” and “hypergraphs” interchangeably.
Table 1.
Reactome database representations.
Fig 2.
Reactome connectivity across pathway representations (directed graph BFS, compound graph traversal [25], bipartite graph BFS, and hypergraph B-connectivity [28]).
(A) The number of node pairs in each representation that (left) are not connected by a path, (middle) are connected by a path of distance k, and (right) are connected by a path with at most distance k. (B) Heatmaps showing the proportion of nodes that are reached with distance k for every node surveyed.
Fig 3.
B-relaxation distance survey from each node in the hypergraph.
Heatmaps show the number of nodes |B≤k| in the Bk-connected set from each source node (rows) for values of k (columns) in the hypergraph. (A) B-relaxation distance for full hypergraph, (B) B-relaxation distance for hypergraph with blacklisted nodes removed [19], and (C) B-relaxation distance for hypergraph with small molecules and three highly-connected entities (cytosolic Ubiquitin, nuclear Ubiquitin, and the Nuclear Pore Complex) removed.
Fig 4.
Overlap and influence of 34 Reactome signaling pathways.
Rows indicate the source pathway PS and columns indicate the target pathway PT. (A) Node overlap of pathway members (normalized by the size of PS). (B) Influence scores for s3. Circle size denotes the number of permutations that have scores equal to or greater than the observed influence score.
Table 2.
Largest influence scores across all B-relaxation distance values for three pathway pairs in Reactome.
The sk values in bold are the largest influence scores for all pathway pairs for all values of k.
Fig 5.
A single pathway’s influence on downstream pathways using B-connectivity.
Shown are the influence of (A) signaling by Mst1, (B) signaling by BMP, and (C) signaling by Activin. The dashed black line indicates the number of nodes in the source pathway’s B≤k for different values of k. There is one line for each of the 33 other target pathways denoting the number of members that appear in B≤k, with selected pathways highlighted in bold.
Fig 6.
Hyperedges traversed to compute B0, B1, …, B4 from source pathway Mst1.
Node colors represent B-relaxation distance from k = 0 (B0, blue) to B4 (bright green). Gray nodes are entities that are not in Bk but are involved in traversed hyperedges. Star-shaped nodes are members of the MET pathway. This network is available on GraphSpace at http://graphspace.org/graphs/26755?user_layout=6707.
Fig 7.
STRING interactions within Reactome for (A) “combined score” interactions, (B) “experimental” interactions, and (C) “textmining” interactions.
The Venn diagram shows the overlap of interactions where the nodes appear in any Reactome pathway, appear in the same Reactome pathway, or are connected in the bipartite graph. The violin plot shows the distributions of interaction scores (which range from 1 to 1000) for different sets of interactions (median, percentiles, and Kruskal-Wallis p-values shown, where p < 2.2e − 16 is below the detection limit [43]). In Panel (A), the bottom violin plot shows the distributions of interaction scores for selected B-relaxation distance thresholds.
Fig 8.
Reachable, traversable and restrictive hyperedges.
This hypergraph has eight reachable hyperedges with respect to S: five traversable hyperedges (blue) and three restrictive hyperedges (red).
Fig 9.
Computing B-relaxation distance.
Connected nodes are in blue and restrictive hyperedges are in red for each iteration k. In this example, all nodes in gray are B3-connected to S = {a, b} and node r has B-relaxation distance of three.
Fig 10.
Building pathway representations from Reactome.