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Modularity in Biological networks
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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:
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Modularity in cell biology
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Definition of a module Loosely linked island of densely connected nodes Groups of co-expressed genes
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Concept of modules in a network
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Definition of a module
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Computational analysis of modular structures Data clustering approach
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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.
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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.
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Types of clustering algorithms Partition-based clustering algorithms Hierarchical clustering algorithms Probabilistic model-based clustering algorithms
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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 ).
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Community structure of biological network Community 1 Community 2 Community 3
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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.
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Spectral analysis clustering algorithm
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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
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Spectral analysis
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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
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Topological Structure Original Network Hidden Topological Structure
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An example Protein-protein interaction network of Saccharomyces cerevisiae
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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
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Quasi-cliqueQuasi-bipartite Positive eigenvalue negative eigenvalue
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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
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Application—function prediction
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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.
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Types of hierarchical clustering Agglomerative hierarchical clustering Divisive hierarchical clustering
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Properties of similarity measure d(i,j)≥0 d(i,j)=d(j,i) d(i,j)≤d(i,k)+d(k,j)
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Similarity measure for agglomerative clustering Correlation Shortest path length Edge betweenness
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How good is agglomerative clustering ?
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Hierarchical tree (Dendrogram) threshold
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Cluster 1 Cluster 2 Single link Distance between clusters
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Cluster 1 Cluster 2 Complete link Distance between clusters
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x2x2 x 3 x1x1 x4x4 x 5 1.52.02.2 3.5 Single link
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Divisive hierarchical clustering M.E.J., Newman and M. Girvan, Phys. Rev. E 69, 026113, (2004)
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Definition of edge betweeness
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Calculation of edge betweenness
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Quantitative measurement of network modularity Modularity Q
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Threshold selection
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Karate club network
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Examples of agglomerative hierarchical clustering
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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
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Modules in the E. coli metabolism E. Ravasz et al., Science, 2002 Pyrimidine metabolism
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Yeast signaling proteins in MIPS PNAS, vol.100, pp.1128, (2003).
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Spotted microarray for Saccharomyces cerevisiae Similarity measure
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Regulatory module network
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Genome Biology, 9, R2, (2008).
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