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Topological Analysis in PPI Networks & Network Motif Discovery

Topological Analysis in PPI Networks & Network Motif Discovery. Jin Chen MSU CSE891-001 2012 Fall. Layout. Topological properties of real networks Degree distribution (power-law & exponential) Path distance (small-world, non-small-world) Network motif Definitions Algorithms.

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Topological Analysis in PPI Networks & Network Motif Discovery

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  1. Topological Analysis in PPI Networks & Network Motif Discovery Jin Chen MSU CSE891-001 2012 Fall

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

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

  4. Power grid has exponential degree distribution R. Albert et al, Phys. Rev. E 69, 025103(R) (2004)

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

  6. Regulatory Network of E. Coli has out-degree power-law distribution & in-degree exponential distribution 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 from RegulonDB (Salgado et al. 2006) Shen-Orr et al. Nature Genetics31, 64 - 68 (2002)

  7. 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 whereL is the distance between two randomly chosen nodes; N is the number of nodes N in the network

  8. 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 30 http://en.wikipedia.org/wiki/Six_degrees_of_separation

  9. 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)

  10. 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

  11. 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.

  12. Concept of Network Motif • “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 Milo et al. Science (2002) Vol. 298 no. 5594 pp. 824-827

  13. Concept of Network Motif • 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) Feed Forward loop Bi-fan X Y X Z Y W Z

  14. Concept of Network Motif • 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 Kashtan et.al. Bioinformatics (2004) Volume20, Issue11 Pp. 1746-1758

  15. 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 Huan et al. SIGKDD (2004)

  16. Tough problem • Number of motifs increases exponentially with size • Motifs frequency is not A priori • Graph isomorphism does not have polynomial solution 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)

  17. 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) Chen et al SIGKDD 2006

  18. 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) 2 3 1 G 4 5

  19. Find Repeated size-k Trees Find all subgraphs of size 2 to 5. t2 t3 t4_1 t4_2 t5_1 t5_3 t5_2 Fig 2. Size 2 to 5 trees

  20. Find Repeated size-k Trees 2 Occurences of t4_1 in G. 2 2 1 3 1 3 1 3 5 4 5 4 5 4 2 2 2 1 3 1 1 3 3 5 4 5 4 5 4

  21. Find Repeated size-k Trees F = 2 t2 t3 t4_1 t4_2 t5_1 t5_3 t5_2

  22. Find Repeated size-k Trees Remaining frequent trees t2 T2 = t3 T3 = t4_1 t4_2 T4 = t5_3 t5_2 T5 =

  23. Use Repeated Size-k Trees to Partition Graph Take each graph in Tk and use it to partition G (i.e. T4) 2 2 2 3 1 3 1 3 1 5 4 5 4 5 4 2 2 3 1 3 1 GD4 5 4 5 4

  24. Perform graph join operation to find repeated size-k graphs t4_1 t4_2

  25. Perform graph join operation to find repeated size-kgraphs Generate all k-node, k-1 edge graphs from each graph in Tk. (i.e. 4-node, 3-edge subgraphs from T4) & h2 h1 t4_1 & & h4 h5 h3 t4_2

  26. Perform graph join operation to find repeated size-kgraphs Join each tree with it’s cousins to produce frequent motif candidates Ck. C4 & h2 h1 t4_1 & & h4 h5 h3 t4_2

  27. Perform graph join operation to find repeated size-kgraphs Count the frequency of each graph Ck in GDk. F = 4 2 2 3 1 3 3 1 5 5 5 4 4 g1_2 2 2 F = 2 GD4 3 1 1 5 4 4 g1_1

  28. Perform graph join operation to find repeated size-k graphs. Generate k node, k+1 edge graphs from k node, k edge graphs move edge merge g1_2 h6 g2 F = 2 in GD4

  29. 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 is a Type I cousin of because g’ h g is isomorphic to

  30. Graph Cousins h g G4_5 G4_3 G4_5 G4_1 G4_2 G4_4 G4_1 G4_2 G4_3 GD4

  31. Graph Cousins h g G4_5 G4_3 G4_5 G4_1 G4_2 G4_4 G4_1 G4_2 G4_3 GD4

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

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

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

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

  36. 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

  37. Performance Test • Uetz dataset : 957 PPIs, 104 proteins • In budding yeast • MIPS CYGD dataset : 10199 PPIs, 4341 proteins • Also in budding yeast • Compared with • Exhaustive enumeration • Sampling • FPF

  38. Performance : runtime ~2.8 hrs F = 50 U = 0.95

  39. Performance : runtime ~2.8 hrs

  40. Performance : max. motif size

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