Introduction

Graph metrics are quantitative measures that provide insights into the structural properties of pathway graphs, playing a crucial role in understanding the topology of biological networks and revealing key characteristics. Through the integration of WikiPathways and R’s igraph package, WayFindR provides a suite of functions that enables researchers to compute graph metrics on biological pathways. In this vignette, we demonstrate how to compute some of these metrics on the pathway named “Factors and pathways influencing insulin-like growth factor (IGF1)-Akt signaling (WP3850)” accessible at https://www.wikipathways.org/pathways/WP3850.html. This GPML file is included in the package as a system file.

Data Preparation

First, we load the required libraries:

library(WayFindR)
suppressMessages( library(igraph) )

Now we are ready to load our GPML file and convert it into an igraph object:

xmlfile <- system.file("pathways/WP3850.gpml", package = "WayFindR")
G <- GPMLtoIgraph(xmlfile)
class(G)

## [1] "igraph"

Computing graph metrics

After obtaining an igraph object, we can use functions from the igraph package to compute its structural properties.

Users have the flexibility to choose which metrics to calculate for their research purposes. However, for the exploration of cycles in graphs, we will concentrate on a selection of global metrics that are potentially intriguing:

1. Number of vertices
2. Number of edges
3. Number of negative interactions (inhibition processes or edges)
4. Presence/absence of loops
5. Presence/absence of multiple edges
6. Presence/absence of Eulerian path or cycle in the input graph
7. Number of clusters
8. Density
9. Radius of the graph
10. Diameter of the graph
11. Girth
12. Global efficiency of the graph
13. Average path length in the graph
14. Number of cliques
15. Reciprocity

We refer readers to the igraph package tutorial for more detailed explanations of these metrics.

Next, let’s create a table summarizing all the metrics of interest.

# Calculate metrics
metrics <- data.frame(nVertices = length(V(G)),
                   nEdges = length(E(G)),
                   nNegative = sum(edge_attr(G, "MIM") == "mim-inhibition"),
                   hasLoop = any_loop(G),
                   hasMultiple = any_multiple(G),
                   hasEuler = has_eulerian_cycle(G) | has_eulerian_path(G),
                   nComponents = count_components(G),
                   density = edge_density(G),
                   diameter = diameter(G),
                   radius = radius(G),
                   girth = ifelse(is.null(girth(G)), NA, girth(G)$girth),
                   nTriangles = sum(count_triangles(G)),
                   efficiency = global_efficiency(G),
                   meanDistance = mean_distance(G),
                   cliques = clique_num(G),
                   reciprocity = reciprocity(G))

## Warning in clique_num(G): At
## vendor/cigraph/src/cliques/maximal_cliques_template.h:219 : Edge directions are
## ignored for maximal clique calculation.

metrics

##   nVertices nEdges nNegative hasLoop hasMultiple hasEuler nComponents
## 1        55     61        14   FALSE       FALSE    FALSE           1
##      density diameter radius girth nTriangles efficiency meanDistance cliques
## 1 0.02053872       10      6     2          5 0.08133571     4.666256       3
##   reciprocity
## 1  0.03278689

We can find cycles and analyze cycle subgraph (i.e., the subgraph defined by including only the nodes that re presen tin at least one cycle. Here, nCyVert is the number of vertices in the cycle subgraph, nCyEdge is the number of edges in the cycle subgraph, nCyNeg is the number of edges in the cycle subgraph with the attribute “MIM” equal to “mim-inhibition”. You can visually confirm the cunts in the plot below.

cy <- findCycles(G)
length(cy)

## [1] 5

S <- cycleSubgraph(G, cy)
cymetrics <- data.frame(nCycles = length(cy),
                         nCyVert = length(V(S)),
                         nCyEdge = length(E(S)),
                         nCyNeg = sum(edge_attr(S, "MIM") == "mim-inhibition"))
cymetrics

##   nCycles nCyVert nCyEdge nCyNeg
## 1       5       9      12      6

set.seed(93217)
plot(S)
nodeLegend("topleft", S)
edgeLegend("bottomright", S)

Figure 1: Example pathway (WP3850) from WikiPathways.

Degrees and Hubs

In addition to numerous “global” graph metrics, the igraph package includes tools to compute numerous “local” metrics, which describe the properties of individual nodes or edges. In many networks (which, like pathways, can be represented by mathematical graphs), highly connected nodes are often viewed as “hubs” that play a more important role. The simplest such metric is the “degree”, which counts the number of edges connected to the node. (In directed graphs, which we use to instantiate pathway, one can talk about both inbound and outbound edges and degrees.)

Here we compute the total degree of each edge

deg <- degree(G)
summary(deg)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.000   1.000   2.000   2.218   3.000  12.000

tail(sort(deg))

## ceb96 b5c6b d51fd c0520 b2305 b3356 
##     4     5     5     6     7    12

We see that the largest degree is equal to 12. To find our which node that is, we peek into the graph.

w <- which(deg == 12)
V(G)[w]$label

## [1] "mTORC1 complex"

So, the highest degree belongs to the group representing the mTORC1 complex. We know that the in-degree for this graph node is artificially inflated by the “contained” arrows defining the group members. So, it may be worth exploring how many such arrows there are, and how many are actual interactions.

Here are the genes that are the source of inbound arrows.

A <- adjacent_vertices(G, w, "in")
A

## $b3356
## + 9/55 vertices, named, from 498de96:
## [1] b2305 b62b2 bb118 c7b3c ca5fb f8fb6 fad48 fb887 c2a5c

By default, we only see the cryptic alphanumeric identifiers. By extracting the IDs, we can find the gene names.

ids <- as_ids(A[[1]])
V(G)$label[as_ids(V(G)) %in% ids]

## [1] "FoxO "       "AMPK"        "MLST8"       "MAFbx"       "AKT1S1"     
## [6] "RPTOR"       "DEPTOR"      "amino acids" "MTOR"

Knowing the IDs, we can also determine the edge type.

Earg <- as.vector(t(as.matrix(data.frame(Source = ids, Target = names(w)))))
E(G, P = Earg)$MIM

## [1] "mim-inhibition"  "mim-inhibition"  "contained"       "mim-inhibition" 
## [5] "contained"       "contained"       "contained"       "mim-stimulation"
## [9] "contained"

Now we can plot the subgraph that connects directly to the mTORC1 complex.

B <- adjacent_vertices(G, w, "out")
subg <- subgraph(G, c(names(w), ids, as_ids(B[[1]])))
plot(subg, lwd=3)

Figure 2: Immediate portion of the pathway aronud the mTORC1 complex.

We see that five of the inbound arrows are for the genes that are “contained” in the complex, leaving seven arrows rhat have biological meaning for the pathway.

We saw above that there is another node in this pathway that has 7 connected edges. It may be worth looking more closely at that node.

w <- which(deg == 7)
V(G)[w]$label

## [1] "FoxO "

This time, we get an actual gene, FoxO.

B <- adjacent_vertices(G, w, "all")
subg <- subgraph(G, c(names(w), as_ids(B[[1]])))
plot(subg, lwd=3)

Figure 3: Immediate portion of the pathway around `FoxO.