1. Modularity in Biological networks

2.  Hypothesis: Biological function are carried by discrete functional modules.  Hartwell, L.-H., Hopfield, J. J., Leibler, S., & Murray, A. W., Nature, 1999.  Question: Is modularity a myth, or a structural property of biological networks? (are biological networks fundamentally modular?) Modularity in Cellular Networks  Traditional view of modularity:

4. Modularity in cell biology

5. Definition of a module Loosely linked island of densely connected nodes Groups of co-expressed genes

6. Concept of modules in a network

8. Definition of a module

9. Computational analysis of modular structures Data clustering approach

10. Concept of data clustering analysis Partitioning a data set into groups so that points in one group are similar to each other and are as different as possible from the points in other groups. The validity of a clustering is often in the eye of beholder.

11. Concept of data clustering analysis In order to describe two data points are similar or not, we need to define a similarity measure. We also need a score function for our objectives. A clustering algorithm can be used to partition the data set with optimized score function.

12. Types of clustering algorithms Partition-based clustering algorithms Hierarchical clustering algorithms Probabilistic model-based clustering algorithms

13. Partitioning problem Given the set of n nodes network D={x(1),x(2),∙∙∙,x(n)}, our task is to find K clusters C={C 1,C 2,∙∙∙,C K } such that each node x(i) is assigned to a unique cluster C k with optimized score function S(C 1,C 2,∙∙∙,C K ).

14. Community structure of biological network Community 1 Community 2 Community 3

15. Score function for network clustering To maximize the intra group connections as many as possible and to minimize the inter group connection as few as possible.

16. Spectral analysis clustering algorithm

17. Adjacency Matrix A ij = 1 if ith protein interacts with jth protein A ij =0 otherwise A ij =A ji (undirected graph) A ij is a sparse matrix, most elements of A ij are zero

18. Spectral analysis

19. Algorithm ( Spectral analysis) Randomly assign a vector X=(X1,X2,…,Xn) Iterate X(k+1)=AX(k) untill it converges Try another vector which is perpendicular to previous found eigenspace

20. Topological Structure Original Network Hidden Topological Structure

21. An example Protein-protein interaction network of Saccharomyces cerevisiae

22. Assign 80000 interactions of 5400 yeast proteins a confidence value We take 11855 interactions with high and medium confidence among 2617 proteins with 353 unknown function proteins. Data source

23. Quasi-cliqueQuasi-bipartite Positive eigenvalue negative eigenvalue

24. With the spectral analysis, we obtain 48 quasi-cliques and 6 quasi-bipartites. There are annotated proteins, unannotated and unknown proteins within a quasi-clique

25. Application—function prediction

26. Hierarchical clustering algorithm A similarity distance measure between node i and j, d(i,j) The similarity measure can be let the network to be a weighted network W ij.

27. Types of hierarchical clustering Agglomerative hierarchical clustering Divisive hierarchical clustering

28. Properties of similarity measure d(i,j)≥0 d(i,j)=d(j,i) d(i,j)≤d(i,k)+d(k,j)

29. Similarity measure for agglomerative clustering Correlation Shortest path length Edge betweenness

30. How good is agglomerative clustering ?

31. Hierarchical tree (Dendrogram) threshold

32. Cluster 1 Cluster 2 Single link Distance between clusters

33. Cluster 1 Cluster 2 Complete link Distance between clusters

34. x2x2 x 3 x1x1 x4x4 x 5 1.52.02.2 3.5 Single link

35. Divisive hierarchical clustering M.E.J., Newman and M. Girvan, Phys. Rev. E 69, 026113, (2004)

36. Definition of edge betweeness

38. Calculation of edge betweenness

39. Quantitative measurement of network modularity Modularity Q

40. Threshold selection

41. Karate club network

43. Examples of agglomerative hierarchical clustering

44. Can we identify the modules? J(i,j): # of nodes both i and j link to; +1 if there is a direct (i,j) link

45. Modules in the E. coli metabolism E. Ravasz et al., Science, 2002 Pyrimidine metabolism

46. Yeast signaling proteins in MIPS PNAS, vol.100, pp.1128, (2003).

47. Spotted microarray for Saccharomyces cerevisiae Similarity measure

48. Regulatory module network

49. Genome Biology, 9, R2, (2008).