The PPI dataset is represented as a weighted directed graph G = (V, E, w), where nodes (V) represent proteins, edges (E) PPIs, and the scores (w) the confidence in each interaction. The scores (edge weight w) range from 0, indicating no interaction, to 1, indicating an interaction with high confidence.

The PPI dataset used in this work is obtained from the STRING database (version 7.1) 9222324. This dataset contains computationally and experimentally derived PPIs, including interactions from other databases (e.g., MINT 25, BioGRID 26, DIP 27, and Reactome 28), microarray experiments, high-throughput experiments, and a mined literature corpus. Furthermore, PPIs are transferred between orthologous pairs of proteins over different organisms. All of these datasets are combined and for each PPI a confidence score is calculated. This way the information from multiple sources is combined into a single score that expresses the overall confidence in each PPI. This score is derived by calculating the joint membership of proteins with PPI in KEGG pathways 29.

The problem posed here is similar to the problem of finding Steiner trees in graphs 30, or more specifically, vertex-weighted Steiner trees. In this problem formalization, a weighted graph G = (V, E, w) and a non-emtpy set of terminals T ⊆ V is given, with w ∈ ℝ⁺. The optimal Steiner tree is defined as the connected subgraph G' = (V', E', w') with G' ⊆ G, for which the summed weight w_sum(E') = ∑_e∈E' w_e is minimal, and T ⊆ V' holds. The Steiner tree problem on graphs was shown to be -complete 31 and, thus, is in most cases solved with heuristics. One of these heuristics is Prim's algorithm 32, which iteratively extends the subgraph G' by adding the vertex with the smallest distance until all nodes in T are connected in G'. A more recent heuristic presented by Melhorn et al. 33 proceeds by first calculating the minimal distance between all nodes in T, and then assembling the minimal Steiner tree by iteratively connecting the nodes with the smallest distance to each other.

Here, the aim is to select a subgraph of G' ⊆ G that connects a set of source proteins S to a set of target proteins T. Given a graph G = (V, E, w) with w ∈ [0, 1] and a disjoint source S ⊆ V and target set T ⊆ V (S ∩ T = ∅), the aim is to find the optimal subgraph G' ⊆ G such that for every s ∈ S and for every t ∈ T at least one path P(s, t) exists in G', whenever such a path exists in G. If either the source or the target set is empty, the problem formalization of Steiner trees is applied. Then the aim is to connect all nodes that are given either in S or in T (S ∩ T) to each other.

For any given pathway, the overall confidence is calculated by multiplying the individual confidence values of the utilized edges:

This objective is based on the assumption that the edge scores reflect independent confidence values, and implies that the resulting score gives the overall confidence in the pathway – that all contained edges are true biological interactions.

To validate the correctness of the inferred pathways we compute the recall and precision rates with respect to a specified reference pathway. These rates can be calculated with respect to PPIs or proteins. The recall rate is defined as the fraction of PPIs/proteins in the reference pathway that are inferred (Equation 2) and the precision rate is defined as the fraction of inferred PPIs/proteins that are contained in the reference pathway (Equation 3).

The topological validation is only performed for pathways that are provided by KEGG. Another possibility for testing the plausibility of inferred pathways – without the need for validation pathways – is to test if the inferred pathway can be associated with a certain biological process. To perform such an analysis, we map the proteins contained in each pathway to their 'biological process', defined by the Gene Ontology (GO) 38. The tool Term Finder 39 is used for this purpose, which calculates a p-value for each biological process using the hypergeometric distribution.

A direct validation against other methods for automatically inferring signal transduction pathways is omitted, because most of these algorithms are validated through pathways with one source and one target protein 101113. The recall and precision rates obtained by the different methods can, however, give a rough estimate of the relative performance.

BowTieBuilder is applied to several sets of source and target proteins. In principle, any type of source or target protein can be processed by BowTieBuilder; in this work, however, if not stated otherwise, the source proteins are membrane-bound proteins and the target proteins are TFs.

To infer signaling pathways for different biological processes, we collect several sets of membrane-bound proteins and TFs. To infer signaling pathways that control the yeast cell cycle, we collect membrane-TF sets for the yeast cell cycle phases G1 and S from the respective KEGG pathway (KEGG identifier: sce04111). For the analysis of the yeast MAPK pathway, the membrane and TF sets are obtained from the KEGG MAPK pathway (KEGG identifier: hsa04010). In addition, the human membrane and TF sets of the Erb pathway are collected from KEGG (KEGG identifier: hsa04012), and the human membrane and TF sets related to the TLR-mediated innate immune pathway are collected from a publication of Kitano et al. 18.

To combine signal transduction pathways with gene regulatory networks, all TFs that were inferred as regulators in a previous study 21 are used here as the target list. In this study, TFs were inferred to have a regulatory effect from two gene expression datasets 4041 and known cis-regulatory elements. In addition to these TFs a list of membrane proteins was collected from the Yeast Membrane Protein Library (YMPL). Based on these TFs and membrane proteins, a signaling pathway is inferred that potentially explains the higher-level regulation of these TFs in the respective gene regulatory network. All source and target proteins are provided in Additional File 1.

As mentioned earlier, BowTieBuilder favors signaling pathways that are structured like a bow tie, but it does not demand such a structure. Thus, it is of interest to quantify to what extent signaling pathways follow the bow tie structure and, in addition, to determine the core proteins. For this purpose, we provide a bow tie score (b(p) ∈ [0, 1]) that determines how 'central' a protein p is. This score is also used to determine the bow tie score of the complete pathway. This score is related to the 'betweenness' measure, in which the number of shortest paths that include the core protein determines the centrality.

To calculate this score, the possible number of connecting paths between the source S and target T proteins is first determined, which is simply the number of source proteins multiplied with the number of target proteins |S|·|T|. Then the number of source and target proteins that can be connected by a path containing p is calculated. This is given by the number of target proteins from which p can be reached (|T_p|) multiplied by the number of source proteins that can be reached from p (|S_p|). Thereby, every edge can only be traversed in one direction, since the signaling pathway is a directed graph that is traversed from the source to the target proteins. The corresponding bow tie score for any protein p reads:

To determine the core elements of any signaling pathway, b(p) is calculated for every intermediate protein p. Given these scores, the core component is defined by the set of proteins with the maximal b(p) score. This also gives the overall score of the signaling pathway. In some cases it is helpful to distill the subnetwork that constitutes the bow tie structure by removing all paths that do not pass through the core component. We refer to such signaling pathways as 'core bow tie'.

To evaluate all heuristics, they are applied to the G1-phase cell cycle, S-phase cell cycle, MAPK pathways of yeast, and the human Erb pathway. The resulting recall and precision rates are provided in Table 1. In comparison to other heuristics, BowTieBuilder has the highest average precision with respect to proteins and PPIs. This could be expected since BowTieBuilder aims at finding the minimal pathway P, whereas the other methods add additional PPIs or proteins to the pathway. The shortest paths heuristic has the highest protein recall rate, whereas the all interactions heuristic has the highest PPI recall rate.

Depending on the type of validation (protein or PPI), the performance of some heuristics varies strongly. The all interactions heuristic, for instance, has high precision when inferring proteins, although the precision for inferring edges is significantly lower in comparison to BowTieBuilder.

In summary, BowTieBuilder has the highest average precision but the lowest average recall. The all interactions heuristic, on the other hand, has the lowest precision and highest recall rate. Thus, the average precision decreases and the average recall increases in the following order: BowTieBuilder, shortest paths, all shortest paths, and all interactions. Several of the inferred pathways are provided in Additional File 2.

For the inferred G1-phase cell cycle pathway, the PPI precision rates range from 48% to 77%, whereas the PPI recall is 77% in all cases (Table 1). The most significant biological process for the inferred proteins is 'cell cycle' (p-value: 4.00·10^-5). Four of six proteins from the KEGG pathway are contained in all inferred pathways (Figure 2), thus the protein recall and precision rates are 67%. The two proteins (Sic1 and Clb5) that constitute alternative paths through the signaling pathways in KEGG are not considered by any inference method. In all cases the core protein is Cdc28, however, its bow tie score ranges from 0.20 (KEGG pathway) to 1.00 ('all interactions'). Cdc28 is reported to be the central coordinator of the major events of the yeast cell division cycle 42.

In the case of the S-phase pathways, Cdc28 is also contained in all core components, except in the BowTieBuilder pathway which is not bow tie structured. The KEGG core component containing Cdc28 and Cdc6 with a bow tie score of 0.75 is also found in the 'all shortest paths' pathway (see Figure 3). The inferred S-phase pathways lack only the protein Clb5 in all cases. This protein binds to Cdc28 and Sic1, and thereby introduces a cycle into the signaling pathway. Hence, this protein is not considered by approaches searching for minimal graphs. Accordingly, the protein recall and precision rates are 86% in all cases, whereas the PPI recall and precision rates range from 50% to 73%. The proteins inferred in the case of the S-phase cell cycle show a significant enrichment for 'G1/S transition of mitotic cell cycle' (p-value: 9.78·10^-13).

The MAPK pathway inferred with BowTieBuilder contains several 'shortcuts' with respect to the original pathway in KEGG (Figure 4). For instance, the inferred pathway connects FAR1 directly to Cdc24 (STRING score: 0.99) and FUS3 directly to STE11 (STRING score: 0.99), whereas in KEGG they are connected through intermediate proteins. These 'shortcuts' are high-confidence PPIs in STRING and experimentally verified, thus allowing inference of shorter pathways than those given in KEGG. At this point it is unclear which connections are actually utilized in the cell, or even if this utilization depends on the specific environmental conditions. Overall, the PPI recall and precision rates are rather low, whereas the protein recall and precision rates are up to 79% (in the case of BowTieBuilder and the all interactions heuristic). The MAPK pathway responds to different external stimuli, such as pheromones and osmolarity. In accordance with these known MAPK stimuli, the GO processes 'osmotic stress' (p-value: 2.57·10^-12) and 'response to pheromone' (p-value: 1.36·10^-11) are the most significant. The core proteins that are contained in the KEGG pathway are: Ste20, Ste11, Ste7 and Fus3. Together with Ste5 these proteins form a scaffolding complex. In the inferred networks this complex is not present because of the '|shortcut' from Ste11 to Fus3. Nonetheless, Fus3, the endpoint of this scaffolding complex, is the core protein in both inferred pathways with a bow tie score similar to the KEGG pathway.

The inferred human Erb pathway is mapped to the GO term 'erb signaling pathway' (p-value: 2.51·10^-30) as the most significant biological process. Thereby, several structures also found in the KEGG pathway could be observed (Figure 5), however, with rather low recall and precision rates (Table 1). Several proteins are skipped by 'shortcut' PPI as already observed for the MAPK pathway. For the inferred Erb pathway no bow tie structure could be found.

The TLR-mediated innate immune system of humans is known to have a bow tie architecture in which eleven TLRs respond to a wide variety of pathogens, capturing so-called pathogen-associated molecular patterns. MyD88 is responsible for the activation of TLR-mediated responses. For this pathway no applicable validation pathway is available in KEGG. Nonetheless, the inference of this pathway revealed the general structure of the TLR-mediated innate immune pathway (Figure 6). Furthermore, a clear bow tie structure can be observed with MyD88 and Fadd as core elements, which is also reported in the publication of Oda et al. 17.

For a set of TFs obtained by inferring a gene regulatory network 21, BowTieBuilder was applied to infer the corresponding signaling pathway. This inference leds to several distinct signaling pathways, and the one with the highest bow tie score is depicted in Figure 7. The core component of this pathway contains several proteins, including the exocytic complex, that are related to excocytosis. These core proteins connect various membrane-bound proteins and TFs, and can be divided into different subpathways that are related to different biological processes (Figure 7).