The PCSF is a well-known problem in graph theory. Given an undirected graph G = (V, E), where the vertices are labeled with prizes p_v and the edges are labeled with costs c_e > 0, the goal is to identify a subnetwork G′ = (V′, E′) with a forest structure. The target is to minimize the total edge costs in E′, the total node prizes left out of V′, and the number of trees in G′. This is equivalent to minimization of the following objective function: F ( G ′ ) = Minimize ∑ e ∈ E ′ c e + β · ∑ v ∉ V ′ p v + w · k (1) where k is the number of trees in the forest, and it is regulated by parameter w. The parameter β is used to tune the prizes of nodes relative to edge costs.

Recently, we have applied PCSF to biological networks in the Forest module of the Omics Integrator software [4]. In biological networks such as protein-protein interaction (PPI) networks, every vertex represents a biomolecule, and every edge corresponds to the cellular interaction between two biomolecules. Edges of the network are given costs which correspond to confidence or frequency of that interaction. The vertices of the network are given prizes according to the measurements of differential expression, copy number, or number of mutation for that gene or protein. The set of vertices that are assigned a prize are referred to as terminal nodes. Non-terminal nodes, which were not observed in the experimental data, may appear in the solution and are called Steiner nodes. After scoring the interactome, the PCSF is used to detect a relevant subnetwork (forest). The PCSF aims to identify neighborhoods in interaction networks potentially belonging to the key dysregulated pathways of a disease or experiment.

In order to avoid a bias for the hub nodes of PPI networks to appear in solution of PCSF, we use the method introduced in Forest [4], which penalizes the prizes of nodes according to their degree in the PPI. Within the implementation, we use the parameter μ to fine-tune the penalties as: p v ′ = p v - μ * degree ( v ) . (2) The parameter μ also affects the total number of Steiner nodes in the solution. The higher the value of μ, the smaller the number of Steiner nodes in the subnetwork, and vice-versa. The recommended range of μ for biological networks is between 1e-4 and 5e-2 to fine-tune the Steiner/terminal node ratio in the subnetwork and average Steiner/terminal in-degree ratio of the corresponding nodes in the original template network.

The software was implemented in R environment, and easily can be installed within the R terminal. As input, the package requires a template network such as protein-protein interaction, protein-metabolite interaction or any other correlation-based interaction network, and it maps differentially expressed genes/proteins/metabolites from the high-throughput data as vertex prizes into the template network. Then, it computes and returns high-scoring neighborhoods to identify functional modules in the interactome. Required parameters are: β—for tuning the vertex prizes, ω—for regulating the number of distinct components in the subnetwork, and μ—for hub penalization.

The package has the following R-package dependencies:

The dependencies are automatically installed along with the PCSF package. For more details about the package dependencies and installation we refer the reader to see the supplementary documents.

A software was successfully installed and tested in the following environments: Mac OS X (10.12.4) R 3.4.0, Ubuntu (16.04) R 3.2.3, Windows 7 R 3.4.1. We have analyzed over 100 biological network instances within the computational performance comparison, and illustrated our method in a biological application to interpret the phosphoproteomic data derived from H358 cells, a model of lung cancer. Few dozens of people have tested the package on installation, debugging, parameter setting, and own data. We provided the interaction network and the phosphoproteomic data from the lung cancer within the package. Any bugs, suggestions and request related to the package can be reported throughout its GitHub repository (https://github.com/IOR-Bioinformatics/PCSF).

In this section, we compare the computational performance of our method with the message passing (MSGP) [3] algorithm. The belief propagation has been used in similar biological application such as identification of unknown protein associations [3], prediction of hidden components in regulatory networks [7], and reconstruction of multiple dysregulated pathways [8]. As a template network, we used an integrated interactome of proteins [9] and metabolites [10], which composed of 36892 nodes and 1016411 edges. We employed the phosphoproteomic data from the Breast Cancer patients in [11]. A network instances are generated by mapping each patient’s differential phosphoproteomic data as terminal nodes onto template interactome. We tested the methods for ω = {1, 2}, and provided the average statistics of 10 runs to maintain a fair comparison baseline.

The comparison statistics of the methods are reported in Table 1. The performance of the MSGP [3] algorithm [3] and our method are provided under the MSGP and PCSF columns, respectively. The μ value was set to 0 for both methods to be comparable, which removes its impact on the objective function values. The solution qualities and running times of the approaches are displayed in the table. For these large network instances, the PCSF provides comparable quality solutions to the belief propagation algorithm for both values of ω. On other hand, the PCSF significantly outperformed the belief propagation in terms of running times. There is approximately ten times speed up on average, and it can be useful to analyze large biological networks in a reasonable time.

For more information about the PCSF algorithm and rigorous performance comparisons, the interested readers are referred to [6].

In this section, we demonstrate the usage of the PCSF package on biological data, discuss the package functionalities and its output within the R environment. We analyze differential phosphoproteomic data derived from H358 cells, a model of lung cancer, that were stimulated with TGF-β. These data were previously published in [12], and were also used to demonstrate Omics Integrator [4]. We construct a template PPI network from the STRING database (version 13) [9] and apply some filtering steps [6]. A named vector containing the proteomic data and a data frame containing the filtered PPI is available within the package. The template network and data are loaded into R environment as follows:

The resulting PPI network consists of 17581 edges and 15405 nodes, out of which 58 terminal nodes correspond to differentially phosphorylated proteins. After loading the PPI network and assigning the terminal prizes, we use PCSF() to find high-confidence subnetworks by providing parameters β, ω and μ as it is discussed in Section. The dynamic and interactive output subnetwork can be plotted with plot() function.

Given that the edge weights and input data were often derived from high-throughput data and are necessarily noisy, it is recommended to test the robustness of the solution. One way to do this is to solve the PCSF several times while adding noise to edge costs, and combine all results in the final subnetwork. Edges and nodes are then given scores indicating how many times they appeared in the solutions with varying edge costs.

Next, an enrichment analysis of the final subnetwork is performed for functional interpretation. The subnetwork is clustered using the edge betweenness clustering algorithm from the igraph R-package, and for each cluster, functional enrichment is done by employing either EnrichR [13] API or topGO [14] R-package that can be specified by the user. Note that EnrichR API requires a working internet connection to perform the enrichment. If not specified, the package defaults to EnrichR, it uses topGO if there is no internet connection.

An interactive version of the annotated subnetwork can be visualized as in Fig 1. In the case of the Tgf-β stimulation data, the subnetwork provided by PCSF was enriched for relevant Gene Ontology terms like “mesenchymal-epithelial cell signaling” and “EGFR downregulation”. We also see Steiner nodes such as CBL and ITGB5, which have been shown to be involved in several models of non-small cell lung cancers [15], like the H358 cells. Therefore, we see that the PSCF algorithm points out proteins and pathways that are highly relevant to the system under study.