An efficient algorithm for detecting frequent subgraphs in biological networks

1. An efficient algorithm for detecting frequent subgraphs in biological networks Class presentation for CPSC 689-604 Authors: Mehmet Kouturk, Ananth Grama and Wojciech Szpankowski Presented by: Songjian Lu Professor: Jianer Chen

2. Contents • Introduction • Metabolic Pathways in detail • Mining metabolic pathways • Algorithm • Result • Some comments

3. Introduction-1 • Metabolic pathways • Model to a directed graph • Node—representing enzymes • Edge—representing the product of one enzyme is consumed by a reaction catalyzed by another enzyme enzyme enzyme enzyme enzyme

4. Introduction-2 • Protein interaction network • Node—representing Protein • Edge—representing interaction between proteins • Pairwise interactions—getting by two-hybrid experiments • Multi-way interactions—getting by mass spectrometry experiments • Database BIND(http://www.blueprint.org/bind) • Database DIP(http://dip.doe-mbi.ucla.edu/) protein protein protein

5. Metabolic pathway detail-1 • DEFINITION:A metabolic pathway P(M,Z,R) is a collection of metabolites M, enzymes Z, and reactions R, where each reaction rR is associated with a set of enzymes Z(r)Z, a set of substrates S(r)M, and a set of products T(r) M. S(r) Z(r) T(r)

6. Metabolic pathway detail-2 • DEFINITION:Given metabolic pathway P(M,Z,R), the associated directed graph G(V,E) of P is constructed as follows:for any enzyme zi  Z, there is a node vi  V. There is an edge from vi to vj, i.e. (vi,vj)  E if and only if r1,r2  R, such that zi  Z(r1), zj  Z(r2) and T(r1)S(r2)  .

7. Metabolic pathway detail-3

8. Mining Metabolic pathways-1 • DEFINITION:Given a collection of graphs G1,G2,…,Gn and support threshold , the Maximal Frequent Subgraph Discovery problem is one of finding all maximal connected subgraphs that are contained in at least n of the input graphs.

9. Mining Metabolic pathways-2

10. Algorithm

11. Result-1

12. Result-2

13. Comment-1 • Graph isomorphism problem is very hard • Given two graphs, if they are isomorphic? • Given two graphs, if one graph is isomorphic to a subgraph of another graph?

14. Comment-2 • Given two graph G1, G2, if there exists a induced subgraph of k vertices in G1 and a induced subgraph of k vertices in G2, such that these two subgraph are isomorphic?(w[1] hard) • Given two graph G1, G2, if there exists a subgraph of k edges in G1 and a subgraph of k edges in G2, such that these two subgraphs are isomorphic?(w[1] hard) • Given two graph G1, G2, if there exists a subtree of k vertices in G1 and a subtree of k vertices in G2, such that these two subtrees are isomorphic?(FPT) • Given two graph G1, G2, if there exists a subpath of k vertices in G1 and a subpath of k vertices in G2, such that these two subpaths are isomorphic?(FPT)

15. Comment-3 • Given two graph G1, G2, if there exists a induced subgraph of k vertices in G1 and a induced subgraph of k vertices in G2, such that these two subgraph are isomorphic?(w[1] hard) G1 G2 G v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1

16. Thank you very much

17. Homework • Problem 1: Given G1=(V1,E1),G2=(V2,E2), and integer k, if there exists a subgraph G1’ of k edges in G1, a subgraph G2’ of k edges in G2 such that there is an isomorphic mapping  from G1’ to G2’. Prove this problem is NP-complete. • Problem2:Given G=(V,E), IV, and integer k, if there exists a subtree T of k vertices, such that all leaves are in I. Prove this problem is NP-complete.