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Homomorphism Mapping in Metabolic Pathways

Homomorphism Mapping in Metabolic Pathways. Qiong Cheng, Robert Harrison , Alexander Zelikovsky Computer Science in Georgia State University. Oct. 17 2007 IEEE 7 th International Conference on BioInformatics & BioEngineering. Outline. Comparison of metabolic pathways

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Homomorphism Mapping in Metabolic Pathways

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  1. Homomorphism Mapping in Metabolic Pathways Qiong Cheng, Robert Harrison, Alexander Zelikovsky Computer Science in Georgia State University Oct. 17 2007 IEEE 7th International Conference on BioInformatics & BioEngineering

  2. Outline • Comparison of metabolic pathways • Graph mappings: embeddings & homomorphisms • Min cost homomorphism problem • Enzyme similarity • Optimal DP algorithm for trees • Searching metabolic networks for • pathway motifs • pathway holes • Future work

  3. 2.7.1.13 1.1.1.49 1.1.1.34 1.1.1.44 3.1.1.31 A portion of pentose phosphate pathway Metabolic pathway & pathways model • Metabolic pathways model • Metabolic pathway

  4. match match match Mismatch/Substitute Comparison of metabolic pathways • Enzyme similarity and pathway topology together represent the similarity of pathway functionality. • Enzyme Similarity • Pathway topology Similarity

  5. A B C D A’ X’ D’ A’ A B’ X’ B C D Related work • Linear topology (Forst & Schulten[1999], Chen & Hofestaedt[2004];) • Tree topology (Pinter [2005] o(|VG|2|VT|/log|VG|+|VG||VT|log|VT|) ) • Arbitrary topology Mapping : Linear pattern  Graph (Kelly et al 2004) ( o(|VT|i+2|VG|2) ) Exhaustively search (Sharan et al 2005 ( o(i!) o(|VT|i+2|VG|2) ), Yang et al 2007 ( o(2|VG||VG|2) )

  6. = Δ[X, Y] log2c(X, Y) = = = Enzyme mapping cost • EC (Enzyme Commission) notation • Enzyme similarity score Δ Enzyme D = d1 . d2 . d3 . d4 • Measure Δ by the lowest common upper class distribution • Measure Δ by tight reaction property Enzyme X = x1 . x2 . x3 . x4 Enzyme Y = y1 . y2 . y3 . y4 Δ[X, Y] = 1 = = = = = Δ[X, Y] = 10 Δ[X, Y] = +∞ otherwise

  7. Sv in T Δ(v, fv(v)) f T G Graph mappings: embeddings & homomorphisms • Isomorphism • Isomorphic embedding • Homeomorphic embedding • Homomorphism Homomorphism f : T  G: fv : VT VG fe : ET paths of G Edge-to-path cost : l (|fe(e)|-1) = Homomorphism cost + l Se in ET (|fe(e)|-1) We allow different enzymes to be mapped to the same enzyme.

  8. ignoring direction Min cost homomorphism problem • A multi-source tree is a directed graph, whose underlying undirected graph is a tree. • Given a multisource tree T =<VT, ET> (Pattern) and an arbitrary graph G =<VG, EG> (Text), find min cost homomorphism f : T  G

  9. A B D C E F D B 1 Text G 1 1 2 3 A 2 C 3 2 Transitive closure 1 1 1 F 2 E Transitive Closure of G : G* Preprocessing of text graph Transitive closure of G is graph G*=(V, E*), where E*={(i,j): there is i-j-path in G}

  10. a b c d Pattern T Ordering a c d b Pattern graph ordering • Construct ordered pattern T’ • DFS traversal • Processing order in opposite way Ordered pattern T’ • Each edge ei in T’ is the unique edge connecting vi • with the previous vertices in the order

  11. a b c Text arbitrary order d Pattern T DP table DT[a, uj] min cost homomorphism mapping from T’s subgraph induced by previous vertices in the orderinto G*

  12. vi uj h(j, j’) vil uj’ G* Pattern T DT[i, j] i<|VT| j<|VG| = Filling DP table • Recursive function Δ(vi , uj)if vi is a leaf in T Δ(vi, uj) + ∑l=1 to adj(vi)Minj’=1 to |VG|C(il, j’) if vi is a leaf in T C[il, jl] = DT[il, jl] + l(h(j, jl) - 1) lis penalty for gaps h(j, jl) = #(hops between uj and ujl in G)

  13. Runtime Analysis • Transitive closure takes O(|VG||EG|). • Pattern graph ordering takes O(|VT| + |ET|) • Dynamic programming - Calculate min contribution of all child pairs of node pair (vi∈T,uj∈G) takes tij = degT (vi)degG*(uj) - Filling DT takes Sj=1 to |VG| Si=1 to |VT|tij = Sj=1 to |VG| degG*(uj)Si=1 to |VT|degT(vi) = 2|EG*||ET| The total runtime is O(|VG||EG|+|VG*||VT|).

  14. a b a b c d c d Statistical significance • Randomized P-Value computation • Random degree-conserved graph generation: • Reshuffle nodes Reshuffle edge • Reshuffle edges

  15. Experiments & applications • Identifying conserved pathways • 24 pathways that are conserved across all 4 species • 18 more pathways that are conserved across at least three of these species • All-against-all mappings among S. cerevisiae, B. subtilis, T. thermophilus, and E.coli • Resolving ambiguity • Discovering pathways holes

  16. 2.6.1.1 6.2.1.5 1.1.1.82 1.3.99.1 4.2.1.2 1. 1.1.1.82 4.2.1.2 1.3.99.1 2.6.1.1 6.2.1.5 1.2.4.- 2.3.1.- 1.2.4.2 2.3.1.61 Observation example 1 : Resolving Ambiguity Mapping of glutamate degradation VII pathways from B. subtilis to T. thermophilus (p < 0.01).

  17. Future work • Approximation algorithm to handle with the comparison of general graphs • Mining protein interaction network • A web-oriented tool will be developed • Discovery of critical elements or modules based on graph comparison • Discovery of evolution relation of organisms by pathway comparison of different organisms at different time points • Integration with genome database

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