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The Architecture of Complexity: Structure and Modularity in Cellular Networks Albert-Lszl Barabsi University of Notre Dame www.nd.edu/~networks title Slide 2 Erds-Rnyi model (1960) - Democratic - Random Pl Erds Pl Erds (1913-1996) Connect with probability p p=1/6 N=10 k ~ 1.5 Poisson distribution Slide 3 World Wide Web Over 3 billion documents ROBOT: collects all URLs found in a document and follows them recursively Nodes: WWW documents Links: URL links R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999). WWW Expected P(k) ~ k - Found Scale-free Network Exponential Network Slide 4 Internet-Map Slide 5 protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map Slide 6 Citrate Cycle METABOLISM Bio-chemical reactions Slide 7 Boehring-Mennheim Slide 8 Metabolic Network Nodes : chemicals (substrates) Links : bio-chemical reactions Metab-movie Slide 9 Metabolic network Organisms from all three domains of life are scale-free networks! H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000) ArchaeaBacteriaEukaryotes Meta-P(k) Slide 10 protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map Slide 11 protein-protein interactions PROTEOME Slide 12 Topology of the protein network H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001) Prot P(k) Nodes : proteins Links : physical interactions (binding) Slide 13 Scale-free model Barabsi & Albert, Science 286, 509 (1999) P(k) ~k -3 BA model (1) Networks continuously expand by the addition of new nodes WWW : addition of new documents GROWTH: add a new node with m links PREFERENTIAL ATTACHMENT: the probability that a node connects to a node with k links is proportional to k. (2) New nodes prefer to link to highly connected nodes. WWW : linking to well known sites Slide 14 Can Latecomers Make It? Fitness Model SF model: k(t)~t (f irst mover advantage) Real systems: nodes compete for links -- fitness Fitness Model: fitness ( k( ,t)~t where = C G. Bianconi and A.-L. Barabsi, Europhyics Letters. 54, 436 (2001). Slide 15 Bose-Einstein Condensation in Evolving Networks G. Bianconi and A.-L. Barabsi, Physical Review Letters 2001; cond-mat/0011029 NetworkBose gas Fit-gets-richBose-Einstein condensation Slide 16 Robustness Complex systems maintain their basic functions even under errors and failures (cell mutations; Internet router breakdowns) node failure fcfc 01 Fraction of removed nodes, f 1 S Robustness Slide 17 Robustness of scale-free networks 1 S 0 1 f fcfc Attacks 3 : f c =1 (R. Cohen et al PRL, 2000) Failures Robust-SF Albert, Jeong, Barabasi, Nature 406 378 (2000) Slide 18 Real networks are fragmented into group or modules Society: Granovetter, M. S. (1973) ; Girvan, M., & Newman, M.E.J. (2001); Watts, D. J., Dodds, P. S., & Newman, M. E. J. (2002). WWW: Flake, G. W., Lawrence, S., & Giles. C. L. (2000). Biology: Hartwell, L.-H., Hopfield, J. J., Leibler, S., & Murray, A. W. (1999). Internet: Vasquez, Pastor-Satorras, Vespignani(2001). Modularity Traditional view of modularity: Ravasz, Somera, Mongru, Oltvai, A-L. B, Science 297, 1551 (2002). Slide 19 Modular vs. Scale-free Topology Scale-free (a) Modular (b) Slide 20 Hierarchical Networks T number of links between the neighbors k number of neighbors (connectivity) Clustering coefficient the measure of a vertex neighborhood interconnectivity. Slide 21 Real Networks HollywoodLanguage Internet (AS) Vaquez et al,'01 WWW Eckmann & Moses, 02 Slide 22 Hierarchy in biological systems Metabolic networks Protein networks Slide 23 Population density Router density Spatial Distributions Yook, Jeong and A.-L.B, PNAS 2002 Slide 24 Spatial Distribution of Routers Fractal set Box counting: N( ) No. of boxes of size that contain routers N( ) ~ -D f D f =1.5 Slide 25 Preferential Attachment Compare maps taken at different times ( t = 6 months) Measure k ( k ), increase in No. of links for a node with k links Preferential Attachment: k ( k ) ~ k Slide 26 Modeling the effect of spatial distribution Model place nodes on a two dimensional space, forming a fractal D f Preferential attachment: Question: Could the distance dependence kill the power law? Numerical results: when 2 then P(k) is exponential For the Internet: =1 (!) thus the Internet could be scale-free. Slide 27 Phase Diagram N ( ) ~ -D f D f =1.5 k(k) ~ k =1 P(d) ~ d - =1 Slide 28 Subgraphs Subgraph: a connected graph consisting of a subset of the nodes and links of a network Subgraph properties: n: number of nodes m: number of links (n=3,m=3) (n=3,m=2) (n=4,m=4) (n=4,m=5). Slide 29 Intracellular molecular interaction sub-networks Protein-protein interaction network N=5068 E=15117 Transcriptional- regulatory network N=688 E=1078 Metabolic network N=551 E=1698 Slide 30 Subgraph density in biological networks (n,m)TranscriptionMetabolicProtein E. coliS. cerevisiaeE. coliS. cerevisiae (3,2)12191017270 (3,3)0.300.315.05.84.1 (4,3)169220441220412395 (4,6)0.00 0.440.770.97 (5,4)249225872.1x10 5 5.9x10 4 1.2x10 5 (5,10)0.00 0.0550.200.66 (6,5)3.2x10 4 2.8x10 4 8.8x10 6 1.5x10 6 5.7x10 6 (6,15)0.00 0.030.36 (7,6)3.4x10 5 2.7x10 5 3.5x10 8 3.7x10 7 2.4x10 8 (7,21)0.00 # of subgraphs/# of nodes Slide 31 Motifs R. Milo et al., Science 298, 824 (2002) Motifs: Subgraphs that have a significantly higher density in the real network than in the randomized version of the studied network Randomized networks: Ensemble of maximally random networks preserving the degree distribution of the original network Slide 32 R Milo et al., Science 298, 824-827 (2002). Slide 33 R. Milo et al., Science 298, 824 (2002) Hypothesis: desirable dynamical and signal processing features. Evolutionary selection of dynamically desirable building blocks. Feed-Forward (FF) Motif is a noise filter. Why do we have motifs? Slide 34 Do motifs have biological relevance? -high degree of evolutionary conservation of the motifs within the protein interaction network (Wuchty et al., Nat. Genet. 2003) -convergent evolution in the transcriptional network of diverse species toward the same motif types (Canant and Wagner, Nat. Genet. 2003) Slide 35 What determines the number of subgraphs in biological networks? (n,m)TranscriptionMetabolicProtein E. coliS. cerevisiaeE. coliS. cerevisiae (3,2)12191017270 (3,3)0.300.315.05.84.1 (4,3)169220441220412395 (4,6)0.00 0.440.770.97 (5,4)249225872.1x10 5 5.9x10 4 1.2x10 5 (5,10)0.00 0.0550.200.66 (6,5)3.2x10 4 2.8x10 4 8.8x10 6 1.5x10 6 5.7x10 6 (6,15)0.00 0.030.36 (7,6)3.4x10 5 2.7x10 5 3.5x10 8 3.7x10 7 2.4x10 8 (7,21)0.00 Slide 36 Global network properties A.-L. B. and Z.N. Oltvai, Nat. Rev. Gen.(2004) Slide 37 Average number of subgraphs passing by a node with degree k Average number of (n,m) subgraphs in the network Number of subgraphs with n nodes Probability of having m-(n-1) extra links Calculating the number of subgraphs: I Slide 38 P(k)~k -,C(k)~k -,n