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Global topological properties of biological networks

Global topological properties of biological networks. Protein-Protein Interaction Network. Saccharomyces cerevisiae. Node: protein Edge: protein-protein interaction. E. coli metabolic network. Basic features of a network. Degree distribution Clustering coefficients

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Global topological properties of biological networks

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  1. Global topological properties of biological networks

  2. Protein-Protein Interaction Network Saccharomyces cerevisiae Node: protein Edge: protein-protein interaction

  3. E. coli metabolic network

  4. Basic features of a network • Degree distribution • Clustering coefficients • Average shortest path length

  5. Degree of a node (k) Degree of ith node ki= number of nodes linking with it

  6. Degree of a node (k) kin= number of nodes linking in kout= number of nodes linking out

  7. Clustering Coefficient (CC) ith node has ki neighbors linking with it Ci=2Ei/ki(ki-1)=2/9 Ei is the actual number of links between ki neighbors maximal number of links between ki neighbors is ki(ki-1)/2

  8. Average shortest path length

  9. Shortest path length

  10. All pair shortest path Algorithm • Floyd Algorithm: d(k)ij: shortest path between i,j with intermediate node’s label not higher than k d(k-1)ij i j d(k-1)ik d(k-1)kj k d(k)ij=min(d(k-1)ij,d(k-1)ik+d(k-1)kj)

  11. Pseudocode • D(0)ij=Aij=adjacency matrix • For k=1 -> N • for i=1 -> N • for j=1 -> N • D(k)ij=min(D(k-1)ij,D(k-1)ik+D(k-1)kj) • Return D

  12. Small world network

  13. Three ways to generate networks

  14. Random networks • Paul Erdös & Alfréd Rényi model : Hugarian mathematicians in 1959 Paul Erdös Alfréd Rényi 1913~1996 1921~1970

  15. Poisson distribution Erdös & Rényi model Randomly connect two nodes with probability P=1/5 linking probability N=10 number of nodes <K>=NP=2 average degree Probability distribution of degree k Exponential Network

  16. Scale free network Albert-László Barabási “Statistical mechanics of complex networks” Review of Modern Physics74, 47-97 (2002)

  17. Scale free Network • A new node is added and deleted randomly to and from the network, i.e. N is not fixed • The new node preferably connects with other node with higher connections with m edges, i.e P(k)~k-γ Scale Free Network A.-L.Barabási, R. Albert, Science 286, 509 (1999)

  18. Scale free network

  19. Mean Field Theory , with initial condition degree distribution γ = 3 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)

  20. Hierarchical Networks

  21. Are biological networks random, scale free or hierarchical?

  22. Degree distribution of PPI P(k)~k-1.62 2617 proteins 11855 interactions Data from HMS-PCI, Yeast two hybrid, and TAP data Scale free

  23. Degree distribution of metabolic network a: Archaeoglobus fulgidus b: E.coli c: C. elegans d: Averaged over 43 organisms Scale free !!!

  24. Metabolic networks Protein networks Hierarchy in biological networks

  25. Many highly connected small clusters combine into few larger but less connected clusters combine into even larger and even less connected clusters Real Networks Have a Hierarchical Topology What does it mean? • The degree of clustering follows:

  26. Biological networks are hierarchical Power law degree distribution Power law clustering coefficient distribution

  27. References • Albert-László Barabási and Zoltán N. Oltvai,Network Biology: Understanding the Cells's Functional OrganizationNature Reviews Genetics 5, 101-113 (2004). • O. Mason, and M. Verwoerd, Graph Theory and Networks in Biology, IET Syst. Biol, 1, 89-119, (2007).

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