Topological Analysis in PPI Networks & Network Motif Discovery Jin Chen MSU CSE Fall 1

Layout Topological properties of real networks – Degree distribution (power-law & exponential) – Path distance (small-world, non-small-world) Network motif – Definitions – Algorithms 2

WWW has power-law degree distribution Distribution of links on the www a)Outgoing links. The tail of the distributions follows P(k)≈k -r, with r out =2.45 b)Incoming links, and r in =2.1 c)Average of the shortest path between two documents as a function of system size R. Albert, H. Jeong, A.-L. Barabási, Nature 401, 130 (1999) The degree distribution scales as a power-law 3

Power grid has exponential degree distribution R. Albert et al, Phys. Rev. E 69, (R) (2004) 4

Metabolic networks have a power-law degree distribution H. Jeong et al., Nature 407, 651 (2000) Archaeoglobus fulgidus E. coli Caenorhabditis elegans All 5

Regulatory Network of E. Coli has out-degree power- law distribution & in-degree exponential distribution Shen-Orr et al. Nature Genetics 31, (2002) from RegulonDB (Salgado et al. 2006) The distribution of the number of transcription factors controlling a gene is exponential The distribution of the number of genes regulated by a transcription factor is power-law with an average of ~5 6

Small-world networks A small-world network is a network in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps A small-world network is defined as: Small-world properties are found in many real-world phenomena where L is the distance between two randomly chosen nodes; N is the number of nodes N in the network 7

Six degrees of separation Six degrees of separation = everyone is on average approximately six steps away from any other person on Earth But if persons are linked if they knew each other, then the number of degrees of separation between Albert Einstein and Alexander the Great is almost certainly greater than

Relationship btw. power-law & small-world If a network has a degree-distribution which can be fit with a power law distribution, it is taken as a sign that the network is small-world But a small-world network is not necessary to have power-law distribution (e.g. clique) 9

Robustness Barabasi AL hypothesized that the prevalence of small world networks in biological systems may reflect an evolutionary advantage of such an architecture One possibility is that small-world networks are more robust to perturbations than other network architectures It would provide an advantage to biological systems that are subject to damage by mutation or viral infection 10

True PPIs fit small-world, false PPIs distributed randomly Hypothesis: true PPIs fit the pattern of a small-world network; false PPIs are distributed randomly in the network By studying the local cohesiveness for each PPI, true and false PPIs can be separated – Incorporate a set of clustering coefficient measures of neighborhood cohesiveness – Look for “network motifs” as an index of how well the PPIs are locally connected Debra S. Goldberg, Frederick P. Roth (2003). PNAS, 100(8) 4372–4376.

“Network Motifs: Simple Building Blocks of Complex Networks” – Focused on directed, cyclic subgraphs of 3 or 4 nodes in yeast (no self-loops) – Used exhaustive enumeration and random networks as a comparison Concept of Network Motif Milo et al. Science (2002) Vol. 298 no pp

In the 13 possible 3 node networks, one predominates in gene expression networks (Feed forward loop) In the 199 possible 4 node networks, one predominates (bi- fan) Concept of Network Motif X Z Y X Z Y W Feed Forward loop Bi-fan 13

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Efficient sampling algorithm for detecting network motifs – Focused on directed, cyclic graphs – Used a sampling approach to estimate motif frequency – Found motifs of size 6 & 7 Concept of Network Motif Kashtan et.al. Bioinformatics (2004) Volume20, Issue11 Pp

Problem Definition Given a PPI network – Unlabelled & undirected subgraphs – Find repeated and unique motifs of size 2 to K (5 to 25) Mining Maximal Frequent Subgraphs from Graph Databases (SPIN, FSSM) – Looks for frequent labelled subgraphs from a database of graphs – Counts whether a subgraph occurs at least once in a graph 16 Huan et al. SIGKDD (2004)

Tough problem 1.Number of motifs increases exponentially with size 2.Motifs frequency is not A priori 3.Graph isomorphism does not have polynomial solution 17 Concepts of frequency f1: allow arbitrary overlaps of nodes & edges---NOT DOWNWARD CLOSURE! f2: allow overlaps of nodes but edges disjoint f3: no overlap allowed (edge and node-disjoint)

Algorithm parameters Input a Protein-Protein Interaction (PPI) network G – K : maximal motif size – F : frequency threshold – S : uniqueness threshold Output set U of frequent and unique motifs of size 3 to K Since motifs are small (2 to 25 nodes), use adjacency matrices. Further, represent motifs as Canonical Adjacency Matrices (CAM) 18 Chen et al SIGKDD 2006

Find Repeated size-k Trees Given a graph G Let K = 5 (max motif size) Let F = 2 (min frequency) Let S = 0.95 (uniqueness threshold) G 19

Find Repeated size-k Trees Find all subgraphs of size 2 to 5. Fig 2. Size 2 to 5 trees t2t2 t3t3 t 4_1 t 4_2 t 5_1 t 5_2 t 5_3 20

Find Repeated size-k Trees Occurences of t 4_1 in G

Find Repeated size-k Trees Treet2t2 t3t3 t 4_1 t 4_2 t 5_1 t 5_2 t 5_3 Freq t2t2 t3t3 t 4_1 t 4_2 t 5_1 t 5_2 t 5_3 F = 2 22

Find Repeated size-k Trees Remaining frequent trees t2t2 t3t3 t 4_1 t 4_2 t 5_2 t 5_3 T 2 = T 3 = T 4 = T 5 = 23

Use Repeated Size-k Trees to Partition Graph Take each graph in Tk and use it to partition G (i.e. T4) GD4 24

Perform graph join operation to find repeated size-k graphs 25 t 4_1 t 4_2

Perform graph join operation to find repeated size-k graphs Generate all k-node, k-1 edge graphs from each graph in T k. (i.e. 4-node, 3-edge subgraphs from T 4 ) t 4_1 t 4_2 & & & h1h1 h2h2 h3h3 h4h4 h5h5 26

Perform graph join operation to find repeated size-k graphs Join each tree with it’s cousins to produce frequent motif candidates C k. t 4_1 t 4_2 & & & h1h1 h2h2 h3h3 h4h4 h5h5 C4C4 27

Perform graph join operation to find repeated size-k graphs Count the frequency of each graph C k in GD k. GD g 1_2 g 1_1 F = 4 F = 2 28

Generate k node, k+1 edge graphs from k node, k edge graphs Perform graph join operation to find repeated size-k graphs. g 1_2 h6h6 g2g2 F = 2 in GD 4 move edge merge 29

Graph Cousins Not allowed to join in this state Only consider the spanning trees and the subgraphs created from them Use the properties of the cousins to trim the number of graph isomorphism tests that need to be done Since the spanning trees partition the graph space, using these cousins saves some time 30

Graph Cousins Type I : Direct Cousin h is isomorphic to a subgraph which has the same number of nodes & edges as g and g != h g h g’ is a Type I cousin of because is isomorphic to 31

Graph Cousins GD 4 g h G 4_1 G 4_2 G 4_3 G 4_4 G 4_5 G 4_1 G 4_2 G 4_3 G 4_5 32

Graph Cousins GD 4 gh G 4_1 G 4_2 G 4_3 G 4_4 G 4_5 G 4_1 G 4_2 G 4_3 G 4_5 33

Graph Cousins Type II : Twin Cousin h is isomorphic to a subgraph g. h g is isomorphic to 34

Graph Cousins Type III : Distant Cousin h is a disconnected subgraph of g. h g is a disconnected subgraph of 35

Graph Cousins Type III : Distant Cousin h is a disconnected subgraph of g. is a disconnected subgraph of h g 36

Saves time when counting graph frequency GD k partitions the network into several subgraphs If they can limit the isomorphism search to a subset of those graphs, they can save time Graph Cousins 37

Determine subgraph frequency in random networks A frequent subgraphs may appear frequently by chance In order to determine the significance of a subgraph, generate random networks with the same number of node and the same number of edges Also impose the constraint that each node must have the same number of neighbors as it’s counterpart in the real network 38

Performance Test Uetz dataset : 957 PPIs, 104 proteins – In budding yeast MIPS CYGD dataset : PPIs, 4341 proteins – Also in budding yeast Compared with – Exhaustive enumeration – Sampling – FPF 39

Performance : runtime ~2.8 hrs F = 50 U =

Performance : runtime ~2.8 hrs 41

Performance : max. motif size 42