Qiong Cheng Georgia State University Joint work with Robert Harrison (GSU) Alexander Zelikovsky (GSU)

1. Qiong Cheng Georgia State University Joint work with Robert Harrison (GSU) Alexander Zelikovsky (GSU) Fast Alignments of Metabolic Networks

2. Outline • Metabolic pathway & pathways model • Background in metabolic network alignments • Enzyme similarity • Topology similarity • Optimal alignment problem formulation- graph homo-homeomorphism • Computation model & solution for multi-source tree pattern • Experiments and analysis • Our software • Future work

3. Metabolic pathway & pathways model Metabolic pathways model 2.7.1.13 1.1.1.34 substrate 1.1.1.49 3.1.1.31 enzyme substrate product 1.1.1.44 enzyme product 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

4. Alignments of metabolic pathways match match match Mismatch/Substitute • Pattern P : query pathway Text T : pathway in database • Enzyme similarity and pathway topology similarity together represent the similarity of pathway functionality.

5. = Δ[X, Y] log2c(X, Y) = = = Enzyme similarity Enzyme D = d1 . d2 . d3 . d4 • EC (Enzyme Commission) notation • Calculation of enzyme-to-enzyme dissimilarity score Δ[X, Y] 1) By the lowest common upper class distribution e.g. X=1.1.1.39 Y=1.1.1.44 the lowest common upper class of X and Y is 1.1.1 c(X, Y ) = #({1.1.1.*}) 2) 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

6. Topology similarity Sv in VT Δ(v, fv(v)) f Text Embedding - Subgraph isomorphism • gene duplication and function sharing • = vertex collapsing • 1+2=Graph homomophism • enzyme insertions • = edge subdividing • l -fine per insertion • 1+3=Approximate graph homeomorphism (Pinter et al 2005 ) Pattern • enzyme deletion • = bypass deletion : send vertex to b (Kelly et al 2005) • 1+3+4= graph homeomorphism • subpath deletion • = strong deletion : send vertex to d (Yang et al 2007) (1+5) 1+2+3+4+5 = graph homo-homeo morphism = l Se in ET (|fe(e)|-1)

7. A B C D A X D A A B X B C D Types of topology in alignments • 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) )

8. Optimal alignment problem formulation-graph homo-homeo morphism • Given: • a metabolic pathway P =<VP, EP> (Pattern) and • a metabolic network T =<VT, ET> (Text) • Find minimum cost alignment f : P  T so that • fv : every vertex in VP is mapped to a vertex in VT U {b,d}; • fl : every path lP across vertices in fv-1(VT) is mapped to path lT • Minimize cost(f)=∑u in VP Δ(u, fv(u))+ λ∑l (|fl(l)|-1) Efficient solution for optimal network alignment of multisource tree to arbitrary graph

9. Alignment operations and cost • Matches of enzymes between pattern and text - Cost(match of u->fv(u))=0 • Mismatches of enzymes - Cost(mismatch of u -> fv(u))=Δ(u, fv(u)) • Insertions of text enzyme to pattern - Cost(insertion of v under fl)=λ • Deletions of pattern enzyme -Cost(deletion of u under f)= Δ(u, b / d) • 1) Bypass deletion 2) Strong deletion 3) Week deletion

10. ignoring direction Notation of Computation Model • A multi-source tree is a directed graph, whose underlying undirected graph is a tree. • Insertions of pattern vertex = deletion of text vertices between v and vj h(v, vj) = #(hops between v and vj) • Cost of deletion of text vertices between v and vj= λ X h(v, vj) • Assume that child’ contribution to their parent’s mapping are independent to another child’s contribution Assume that pattern root vertex can not be deleted

11. min(cost(ui, vj)+ λh(v,vj)) vj B(ui, v) strongD(ui) min(weakD(u, ui, uik) + cost(uik, vj)+ λh(v,vj)) uik A(u, v) Computation model for multi-source tree pattern u v Three possibilities of the contribution of the child ui to the parent u’s mapping (u->v): 1. ui is mapping to vj (vj is a descendent of v) 2. ui is strong deleted: strongD(ui) 3. ui is bypass deleted: weakD(u, ui, uik) ui vj uik Text T Pattern P cost(u,v)=Δ(u,v)+∑ min ui

12. Recurrence relation for the network alignment min(A(ui, vj)+ λh(v,vj)) vj A(u,v)=Δ(u,v)+∑ B(ui, v) ui min(weakD(u, ui, uik) + λh(v,vj) + A(uik, vj)) uik Base cases: A(u,v)=min(Δ(u,v), strongD(u)) when vertex u is leaf B(u,v)= ∞ Recurrence equation: strongD(ui) B(ui, v)=min

13. Solution • Preprocessing: • Transitive closure of text T • Pattern graph ordering • Calculate the penalties of pattern vertex strong deletion • Calculate the penalties of pattern vertex weak deletion • Dynamic Programming + Adaption of Dijkstra • Runtime for DP solution with Fibonacci heaps: • O(|VP|(|ET| + |VT|log|VT|)).

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

15. a b c d Pattern P Ordering a c d b Pattern graph ordering • Construct ordered pattern P’ • DFS traversal • Processing order in opposite way Ordered pattern P’ • Each edge ei in P’ is the unique edge connecting vi • with the previous vertices in the order

16. Penalties for pattern vertex deletions strongD(u) = Δ(u,d) + ∑Δ(ui*,d) Ui* weakD(u, ui, v) =∑ (∑ strongD(ui,j)+Δ(ui,b)) Ui Ui,j

17. Dynamic programming Create two dynamic table A and B C D a b E c F d H Pattern P B’(v,f(u))=min(A(v,y)+ lh(f(u),y)) y={des(f(u)} Text T B A Arbitrary order Arbitrary order 5 4 4 • Fill A and B from bottom to up 5 13 3 3,E 3,E • Track back ∞ ∞ 2,H 1,H 3,H 10 10 10 1 A(u,f(u))=Δ(u,f(u))+ Schild v of uB’(v,f(u))

18. Adaption of dijkstra a b C D c E F A C D E F H B C D E F H d ∞ d d ∞ ∞ ∞ ∞ ∞ 10 10 1 10 Pattern P Text T H, 0 C, 9 F, 9 D, ∞ E, 9 C, 9 D, ∞ E, 2 C, 3 C, 9 F, 1 D, ∞ D, ∞ E, 9 H for each x ЄVT insert (x, A(v,x)- l ) into Q B(v,x)  ∞ while Q is not empty delete from Q item (y,k) with the minimum key k for each (x,y) ЄET if B(v,x) >k+ l B(v,x)k+ l if key of x >k+ l decrease key of x in Q with k+ l 1 3 2 1 2 3 • Runtime for priority queue Q with Fibonacci heaps: O(|ET| + |VT|log|VT|). • Total Runtime : O(|VP|(|ET| + |VT|log|VT|))

19. Handling cycles e a b e c d a b c d • DP does not work when pattern has cycles • “Fix” images for some pattern vertices and • reduce to acyclic case • Find Minimum Feedback vertex set F(P): • VP-F(P) is acyclic • NP-complete but easy to be approximate • Runtime is increased by factor O(VT |F(P)|) • Total Runtime : O(|VT||F(P)||VP|(|ET| + |VT|log|VT|))

20. 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

21. Experiments & applications • All-against-all mappings among S. cerevisiae, B. subtilis, T. thermophilus, and E.coli • 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 • Significant deletions • Resolving ambiguity • Discovering pathways holes

22. Significant deletion Pattern: Aspartate superpathway in E. coli Text: Lysine biosynthesis in T. thermophilus Mapping result: unmatched vertices are deleted. We show the solid conserved subpath and dotted deleted subpath in pattern. The dotted subpath produce biotin which is not required for text organism.

23. Resolving Ambiguity • Mapping of glutamate degradation VII pathways from B. subtilis to T. thermophilus (p<0.01). The shaded node reflects enzyme homology. • Similar corresponding enzymes 1.2.4.2 and 2.3.1.61 with the similar function to 1.2.4.- and 2.3.1.- can be found in T. thermophilus • 1.2.4.2 and 2.3.1.61 has been reported in B. subtilis

24. Pathway holes: find and fill • Hole = missing enzyme in pathway description (in database) • Finding holes is a difficult task: comparison can help Aligning of formaldehyde oxidation V pathway in B. subtilis to formy1THF biosynthesis pathway in E. coli • 3.5.1.10 is missing from B. subtilis but exists in B. clausii which is close to B. subtilis

25. Our software • http://alla.cs.gsu.edu:8080/MinePW/pages/gmapping/GMMain.html

26. Reference Q. Cheng, A. Zelikovsky, Network Mapping of Metabolic Pathways, Analysis of Complex Networks: From Biology to Linguistics, Wiley-VCH 2009 Q. Cheng, P. Berman, R. Harrison and A. Zelikovsky, "Fast Alignments of Metabolic Networks ", Proc. of IEEE International conference on Bioinformatics and Biomedicine (BIBM 2008), pp 147-152   Q. Cheng, D. Kaur, R. Harrison, and A. Zelikovsky, "Mapping and Filling Metabolic Pathways ", RECOMB Satellite Conference on Systems Biology 2007    Q. Cheng, R. Harrison, and A. Zelikovsky, "Homomorphisms of Multisource Trees into Networks with Applications to Metabolic Pathways", Proc. of IEEE 7-th International Symposium on BioInformatics and BioEngineering (BIBE'07)  Q. Cheng, R. Harrison, and A. Zelikovsky. "MetNetAligner: a web service tool for metabolic network alignments". Bioinformatics 2009 (To appear)

27. Future work Refine the method of filling pathway holes and improve the performance Discover critical metabolic elements/modules/motifs Describe evolution of metabolic pathways Integrate with genome database

28. Acknowledgments GSU Molecular Basis of Disease (MBD) fellowship Peter Karp Oleg Rokhlenko Florian Rasche Amit Sabnis, Dipendra Kaur Kelly Westbrooks, Irina Astrovskaya, Stefan Gremalschi, Jingwu He, Dumitru Brinza,Weidong Mao ,Nisar Hudewale

29. Thank you for your attention!

30. Bio-Map GENOME protein-gene interactions PROTEOME protein-protein interactions METABOLISM Bio-chemical reactions Citrate Cycle

31. Comparison on different methods Alignment of tree pathways from different species with optimal homomorphism (HM) and optimal network alignment (NA). Average number of mismatches and gaps are reported on common statistically significant matched pathways.