SI 614 Network subgraphs (motifs) Biological networks

1. SI 614Network subgraphs (motifs) Biological networks Lecture 11 Instructor: Lada Adamic

2. Outline • motifs • motif detection (software & Pajek) • review of network characteristics • used to compare model with real-world network • one more: degree assortativity • biological networks • types • characteristics • hierarchical modularity model

3. Schematic view of network motif detection

4. Motifs can overlap in the network motif to be found graph motif matches in the target graph http://mavisto.ipk-gatersleben.de/frequency_concepts.html

5. Examples of network motifs (3 nodes) • Feed forward loop • Found in neural networks • Seems to be used to neutralize“biological noise” • Single-Input Module • e.g. gene control networks X Y Z X a b b c c d d

6. All 3 node motifs

7. Examples of network motifs (4 nodes) • Parallel paths • Found in neural networks • Food webs W X Y Z

8. 4 node subgraphs (computational expense increases with the size of the graph!)

9. Network motif detection • Some motifs will occur more often in real world networks than random networks • Technique: • construct many random graphs with the same number of nodes and edges (same node degree distribution?) • count the number of motifs in those graphs • calculate the Z score: the probability that the given number of motifs in the real world network could have occurred by chance • Software available: • http://www.weizmann.ac.il/mcb/UriAlon/

10. What the Z score means m = mean number of times the motifappeared in the random graph the probability observing a Z score of 2 is 0.02275 In the context of motifs: Z > 0, motif occurs more often than for random graphs Z < 0, motif occurs less often than in random graphs |Z| > 1.65, only a 5% chance of random occurence s standard deviation # of times motif appeared in random graph x - mx zx = sx

11. Finding classes on graphs based on their motif “profiles”

12. Finding motifs (cliques and subgraphs) in Pajek • Create a second network that is the subgraph you are looking for e.g. an undirected triad *Vertices 3 1 "v1" 2 "v2" 3 "v3" *Arcs *Edges 2 3 1 1 2 1 1 3 1

13. finding motifs with Pajek • Use the two drop down menus in the ‘networks’ list to specify two networks: • Then run Nets>Fragment (1 in 2)>Find • under Net>Fragment (1 in 2)>Options • can select ‘induced’ subnetwork containing only overlapping fragments in

14. finding motifs with Pajek (cont’d) • Now we have just the triads: • Creates a hierarchy object with the membership of each triad listed

15. Comparing network models with the real thing • check for structural similarity between the artificial network (the model) and the real world network • degree distribution • assortativity • do high degree nodes connect to other high degree nodes? • average shortest path • dependence on size of network • clustering coefficient • compare to a randomized version conserving node degree • dependence on node degree • dependence on size of network • motif profile

16. How can we randomize a network whilepreserving the degree distribution? • Stub reconnection algorithm(M. E. Newman, et al, 2001, also known in mathematical literature since 1960s) • Break every edge in two “edge stubs”AB to AB • Randomly reconnect stubs • Problems: • Leads to multiple edges • Cannot be modified to preserve additional topological properties

17. Local rewiring algorithm • Randomly select and rewire two edges (Maslov, Sneppen, 2002, also known in mathematical literature since 1960s) • Repeat many times • Preserves both the number of upstream and downstream neighbors of each node

18. Conserving additional low-level topological properties • In addition to ki one may also conserve: • The exact numbers of loops or other motifs • The size and numbers of components: Internet – all nodes have to be connected to each other • Metropolis algorithm: two edges are rewired based on E=(Nactual-Ndesired)2/Ndesired • If E0 rewiring step is always accepted • If E>0 rewiring step is accepted with p=exp(-E/T)

19. Assortativity • Social networks are assortative: • the gregarious people associate with other gregarious people • the loners associate with other loners • The Internet is disassortative: Assortative: hubs connect to hubs Random Disassortative: hubs are in the periphery

20. Correlation profile of a network • Detects preferences in linking of nodes to each other based on their connectivity • Measure N(k0,k1) – the number of edges between nodes with connectivities k0 and k1 • Compare it to Nr(k0,k1) – the same property in a properly randomized network • Very noise-tolerant with respect to both false positives and negatives

21. Correlation profiles give complex networks unique identities 2D picture Protein interactions Internet slide by Sergei Maslov

22. Correlation profiles give complex networks unique identities Sergei Maslov: 2D histogram Protein interactions Internet

23. Correlation profiles -cont’d • Pastor-Satorras and Vespignani: 2D plot average degree of the node’s neighbors degree of node

24. Correlation profiles -cont’d • Newman: single number -0.189 internet degree correlation coefficient The Pearson correlation coefficient of nodes on each side on an edge

25. Other examples of assortative mixing • Assortativity is not limited to degree-degree correlations other attributes • social networks: race, income, gender, age • food webs: herbivores, carnivores • internet: high level connectivity providers, ISPs, consumers • Tendency of like individuals to associate: ‘homophily’ • Scott Feld paper

26. Biological networks • In biological systems nodes and edges can represent different things • nodes • protein, gene, chemical • edges • mass transfer, regulation • Can construct bipartite or tripartite networks: • e.g. genes and proteins

27. GENOME protein-gene interactions PROTEOME protein-protein interactions METABOLISM bio-chemical reactions slide after Reka Albert

28. Cellular processes form networks on many levels • metabolic reaction networks (tri-partite) • Node types: • metabolites (substrates or products), open rectangles • metabolite-enzyme complexes (black rectangles) • enzymes (open ovals) • Edges • substrate to complex or complex to product • symmetrical edges slide after Reka Albert

29. regulatory networks nodes: genes, proteins edges: translation regulation: activating inhibiting slide after Reka Albert

30. the yeast two-hybrid method • Activation and binding domains are separated and each attached to a different protein • If the proteins interact, the two domains will be brought together and activate the transcription of a reporter gene • Can do simultaneous genome-wide experiments slide after Reka Albert

31. Resulting interaction network slide after Reka Albert

32. Properties and problems of resulting networks • Properties • giant component exists • power law distribution with an exponential cutoff • longer path length than randomized • higher incidence of short loops than randomized • Problems • false positives • false negatives • only 20% overlap between different studies

33. Implications • Robustness • resilient to random breakdowns • mutations in hubs can be deadly • Evolution • most connected hubs conserved across organisms (important) • gene duplication hypothesis • new gene still has same output protein, but no selection pressure because the original gene is still present. So some interactions can be added or dropped • leads to scale free topology

34. Metabolic networks: how to represent them • Can consider the one-mode projection of substrate interactions (undirected) slide after Reka Albert

35. Metabolic networks are scale-free • In the bi-partite graph: • the probability that a given substrate participates in k reactions is k-a • indegree:a = 2.2 • outdegree: a = 2.2 (a) A. fulgidus (Archae) (b) E. coli (Bacterium) (c) C. elegans (Eukaryote), (d) averaged over 43 organisms

36. Modularity • No modularity • Modularity • Hierarchical modularity (Pajek!) E. Ravasz et al., Science 297, 1551 -1555 (2002)

37. How do we know that metabolic networks are modular? clustering decreases with degree as C(k)~ k-1 randomized networks (which preserve the power law degree distribution) have a clustering coefficient independent of degree

38. How do we know that metabolic networks are modular? • clustering coefficient is the same across metabolic networks in different species with the same substrate • corresponding randomized scale free network:C(N) ~ N-0.75 (simulation, no analytical result) bacteria archaea (extreme-environment single cell organisms) eukaryotes (plants, animals, fungi, protists) scale free network of the same size

39. review: what would the clustering coefficient of a random network be • assume average degree of node is k • probability of one neighbor linking to another is ~ k/N • scales as N-1

40. Constructing a hierarchically modular network RSMOB model • Start from a fully connected cluster of nodes • Create 4 identical replicas of the cluster, linking the outside nodes of the replicas to the center node of the original (N = 25 nodes) • This process can repeated indefinitely • (initial number of nodes can be different than 5)

41. Properties of the hierarchically modular model RSMOB model • Power law exponent g = 2.26 (in agreement with real world metabolic networks) • C ≈ 0.6, independent of network size (also comparable with observed real-world values) • C(k) ≈ k-1, as in real world network • How to test for hierarchically arranged modules in real world networks • perform hierarchical clustering on the topological overlap map (we’ll cover hierarchical clustering in a few weeks…) • can be done with Pajek

42. Topological overlap • A: Network consisting of nested modules • B: Topological overlap matrix hierarchical clustering

43. Hubs may act within a module, or connect modules • Party hub: • simultaneous interactions • tends to be within the same module • Date hub: • sequential interactions • connect different modules Han et al, Nature 443, 88 (2004) slide after Reka Albert

44. some matching motifs frequently overlap (e.g. feed forward loop) Zhang et al, J. Biol 4, 6 (2005)