Transcript Graphs and Networks with Bioconductor
Graphs and Networks with Bioconductor
Wolfgang Huber EMBL/EBI Bioconductor Conference 2005 Based on chapters from "Bioinformatics and Computational Biology Solutions using R and Bioconductor", Gentleman, Carey, Huber, Irizarry, Dudoit, Springer Verlag.
Graphs
Set of nodes and set of edges.
Nodes: objects of interest Edges: relationships between them A useful abstraction to talk about relationships and interactions (think of integer numbers, apples and fingers) Edges may have weights, directions, types
Practicalities
As always, need to distinguish between the true, underlying property of nature that you want to measure, and the actual result of a measurement (experiment) 1.
False positive edges
2.
False negative edges
(were tested, were not found, but are there in nature) 3.
Untested edges
are there in nature) (were not tested, are not in your data, but Uncertainty is not usually considered in mainstream graph theory, but cannot be ignored in functional genomics.
Nice application of these concepts to protein interactions: Gentleman and Scholtens, SAGMB 2004
Representation
Node-edge lists Adjacency matrix (straightforward) Adjacency matrix (sparse) From-To matrix They are equivalent, but may be hugely different in performance and convenience for different applications. Can coerce between the representations
Algorithms
Bioconductor project emphasizes re-use and interfacing to existing, well-tested software implementations rather than reimplementing everything from scratch ourselves.
RBGL package: interface to B oost G raph L ibrary; started by V. Carey, R. Gentleman, now driven by Li Long.
Example: a pathway
Elementary computations on IMCA pathway
> library("graph") > data("integrinMediatedCellAdhesion") > class(IMCAGraph) > s = acc(IMCAGraph, "SOS") Ha-Ras Raf MEK 1 2 3 ERK MYLK MYO 4 5 6 F-actin cell proliferation 7 5
Machine-readable pathway databases
KEGG reactome BioCarta (biocarta.com) National Cancer Institute cMAP
Gene Ontology (GO)
A structed vocabulary to describe molecular function of gene products, biological processes, and cellular components.
Plus A set of "is a", "is part of" relationships between these terms Directed acyclic graph
GO graphs
>tfG=GOGraph("GO:0003700", GOMFPARENTS)
Gene-Literature graphs
DKC1
Graphs: vocabulary
Directed, undirected graphs Adjacent nodes Accessible nodes Self-loop Multi-edge Node degree Walk: alternating sequence of nodes and incident edges Closed walk Distance between nodes, shortest walk Trail: walk with no repeated edges Path: trail with no repeated nodes (except possibly first/last) Cycle Connected graph Weakly connected directed graph (see next page)
Strong and weak connectivity
Graphs: vocabulary
Cut: remove edges to disconnect a graph Cut-set: remove nodes - " Connectivity of a graph Cliques
Special types of graphs
Bipartite graph
Bipartite graphs
A G adjacency matrix (n x m) of a bipartite graph G with node sets U, V One mode graphs A U = A G t A G A V = A G A G t (Boolean algebra)
Multigraphs
Can have different types of edges
Hypergraphs
:= set of Nodes + set of hyperedges A hyperedge is a set of nodes (can be more than 2) A directed hyperedge: pair (tail and head) of sets of nodes
Directed acyclic graphs
Useful for representing hierarchies, partial orderings (e.g. in time, from general to special, from cause to effect) Many applications: GO MeSH Graphical models
Random Edge Graphs
n nodes, m edges p(i,j) = 1/m with high probability: m < n/2: many disconnected components m > n/2: one giant connected component : size ~ n.
(next biggest: size ~ log(n)). degrees of separation : log(n).
Erdös and Rényi 1960
Random graphs
Random edge graph: V: nodes either p: or edges: randomEGraph(V, p, edges) probability per edge number of edges Random graph with latent factor: V: nodes randomGraph(V, M, p, weights=TRUE M: p: latent factor probability For each node, generate a logical vector of length length(M) , with P(TRUE)= p . Edges are between nodes that share >= 1 elements. Weights ) can be generated according to number of shared elements.
Random graph with predefined degree distribution: randomNodeGraph(nodeDegree) nodeDegree: named integer vector sum of all node degrees must be even
100 nodes 50 edges Random edge graph degree distribution
Random graphs versus permutation graphs
For statistical inference, one can consider null hypotheses based on aforementioned random graph models; and ones based on node permutation of data graphs.
The second is often more appropriate.
Cohesive subgroups
For data graphs, the concept of clique is usually too restrictive (false negative or untested edges) n-clique: distance between all members is <=n.
(Clique: n=1)
k-plex: maximal subgraph G in which each member is neighbour of at least |G|-k others.
(Clique: k=1)
k-core: maximal subgraph G in which each member is neighbour of at least k others.
(Clique: k=|G|-1) After: Social Network Analysis, Wasserman and Faust (1994)
graph, RBGL, Rgraphviz
graph basic class definitions and functionality RBGL interface to graph algorithms Rgraphviz rendering functionality Different layout algorithms. Node plotting, line type, color etc. can be controlled by the user.
Creating our first graph
> library("graph"); library(Rgraphviz) > myNodes = c("s", "p", "q", "r") > myEdges = list( s = list(edges = c("p", "q")), p = list(edges = c("p", "q")), q = list(edges = c("p", "r")), r = list(edges = c("s"))) > g = new("graphNEL", nodes = myNodes, edgeL = myEdges, edgemode = "directed") > plot(g)
Querying nodes, edges, degree
> nodes(g) [1] "s" "p" "q" "r" > edges(g) $s [1] "p" "q" $p [1] "p" "q" $q [1] "p" "r" $r [1] "s" > degree(g) $inDegree s p q r 1 3 2 1 $outDegree s p q r 2 2 2 1
Graph manipulation
> g1 <- addNode("e", g) > g2 <- removeNode("d", g) > ## addEdge(from, to, graph, weights) > g3 <- addEdge("e", "a", g1, pi/2) > ## removeEdge(from, to, graph) > g4 <- removeEdge("e", "a", g3) > identical(g4, g1) [1] TRUE
adjacent and accessible nodes
> adj(g, c("b", "c")) $b [1] "b" "c" $c [1] "b" "d" > acc(g, c("b", "c")) $b a c d 3 1 2 $c a b d 2 1 1
Graph representations: from-to-matrix > ft [,1] [,2] [1,] 1 2 [2,] 2 3 [3,] 3 1 [4,] 4 4 > ftM2adjM(ft) 1 2 3 4 1 0 1 0 0 2 0 0 1 0 3 1 0 0 0 4 0 0 0 1
GXL: graph exchange language
GXL (www.gupro.de/GXL) is "an XML sublanguage designed to be a standard exchange format for graphs".
The graph package provides tools for im- and exporting graphs as GXL
RBGL: interface to the Boost Graph Library Connected components cc = connComp(rg) table(listLen(cc)) 1 2 3 4 15 18 36 7 3 2 1 1 Choose the largest component wh = which.max(listLen(cc)) sg = subGraph(cc[[wh]], rg) rg Depth first search dfsres = dfs(sg, node = "N14") nodes(sg)[dfsres$discovered] [1] "N14" "N94" "N40" "N69" "N02" "N67" "N45" "N53" [9] "N28" "N46" "N51" "N64" "N07" "N19" "N37" "N35" [17] "N48" "N09"
depth / breadth first search bfs(sg, "N14") dfs(sg, "N14")
connected components sc = strongComp(g2) nattrs = makeNodeAttrs(g2, fillcolor="") for(i in 1:length(sc)) nattrs$fillcolor[sc[[i]]] = myColors[i] wc = connComp(g2) plot(g2, "dot", nodeAttrs=nattrs)
shortest path algorithms Different algorithms for different types of graphs o all edge weights the same o positive edge weights o real numbers …and different settings of the problem o single pair o single source o single destination o all pairs
Functions
bfs dijkstra.sp
sp.between
johnson.all.pairs.sp
shortest path set.seed(123) rg2 = randomEGraph(nodeNames, edges = 100) fromNode = "N43" toNode = "N81" sp = sp.between(rg2, fromNode, toNode) sp[[1]]$path [1] "N43" "N08" "N88" [4] "N73" "N50" "N89" [7] "N64" "N93" "N32" [10] "N12" "N81" sp[[1]]$length [1] 10 1
shortest path ap = johnson.all.pairs.sp(rg2) hist(ap)
minimal spanning tree gr mst = mstree.kruskal(gr)
connectivity Consider graph g with single connected component.
Edge connectivity of g: minimum number of edges in g that can be cut to produce a graph with two components. Minimum disconnecting set : the set of edges in this cut.
> edgeConnectivity(g) $connectivity [1] 2 $minDisconSet $minDisconSet[[1]] [1] "D" "E" $minDisconSet[[2]] [1] "D" "H"
Rgraphviz: the different layout engines dot: directed graphs. Works best on DAGs and other graphs that can be drawn as hierarchies.
neato: undirected graphs using ’spring’ models twopi: radial layout. One node (‘root’) chosen as the center. Remaining nodes on a sequence of concentric circles about the origin, with radial distance proportional to graph distance. Root can be specified or chosen heuristically.
Rgraphviz: the different layout engines
Rgraphviz: the different layout engines
Combining R graphics and graphviz: custom node drawing functions
Combining: graphviz layout and R plot
ImageMap lg = agopen(g, …) imageMap(lg, con=file("imca-frame1.html", open="w") tags= list(HREF = href, TITLE = title, TARGET = rep("frame2", length(AgNode(nag)))), imgname=fpng, width=imw, height=imh)
Show drosophila interaction network example
Application: comparing gene co-expression and protein interaction data Nodes: all yeast genes Graph 1: co-expression clusters from yeast cell cycle microarray time course Graph 2: protein interactions reported in the literature Graph 3: protein interactions found in a yeast-two-hybrid experiment Questions: Do the graphs overlap more than random? Is there anything special about overlapping edges?
Application: comparing gene co-expression and protein interaction data
Application: comparing gene co-expression and protein interaction data nPdist: number of common edges as computed by a node label per mutation model.
Number observed in data: 42
Further questions for exploratory data analysis • Which expression clusters have intersections with which of the literature clusters?
• Are known cell-cycle regulated protein complexes indeed clustered together in both graphs?
• Are there expression clusters that have a number of literature cluster edges going between them
suggesting that expression clustering was too fine, or that literature clusters are not cell-cycle regulated.
• Is the expression behavior of genes that are involved in multiple protein complexes different from that of genes that are involved in only one complex?
Generalization Nothing in the preceding treatment was specific to physical protein interactions or microarray clustering. Can you similar reasoning for many other graphs! - e.g. genomic vicinity, domain composition similarity
Application: Using GO to interprete gene lists
Using GO to interprete gene lists Packages: Gostats, Rgraphviz
Using GO to interprete gene lists
Gene-Literature graphs
DKC1
The bipartite gene-literature graph: actor and event size adjustment actors : genes actor size : number of papers that a gene appears in event : paper event size : number of genes that appear in a paper Example: R. Strausberg et al. Generation and initial analysis of more than 15,000 full-length human and mouse cDNA sequences. PNAS 99:16899 –903, 2002 cites 15,000 genes
Are two genes remarkably often co-cited?
Note, usually one count (w.l.o.g. n 22 ) is much larger than everybody else. Test statistics that do not depend on n 22 :
Closing gene lists with literature Boundary of gene list L: set of all genes that have co-citation (above threshold weight) with genes in L. Gene 1 Gene X Gene 2 Gene 3 Gene Y Gene 4 Gene 5
A pathway graph
A pathway graph
CGH aberration data Tumours
From: B. Gunawan et al., Cancer Res. 63: 6200-6205 (2003)
Graphical model for CGH aberration data oncotree package by Anja von Heydebreck
Summary Graphs are a natural way to represent relationships, just as numbers are a natural way to represent quantities.
Three main applications: (1) to represent data (e.g. PPI) (2) to represent knowledge (e.g. GO) (3) to represent high-dimensional probability distributions Bioconductor provides a rich set of tools mainly for (1) and (2). Various parts of R for (3), see also gR project.
There are still many challenges that call for methods to model uncertainty, make inference, and predictions.
Further exercises Fine control of graph rendering GOstats example