Heterogeneous Graph Mining for Biological Pattern Discovery in Metabolic Pathways

Transcription

1 Heterogeneous Graph Mining for Biological Pattern Discovery in Metabolic Pathways Alexandra Zaharia, Bernard Labedan, Christine Froidevaux, Alain Denise LRI, I2BC, Université Paris Sud, CNRS, Université Paris Saclay SeqBio 2016, Nantes, November 17 th 2016

2 Biological motivation Metabolic pathway (directed graph) Vertices reactions; Arcs connect reactions sharing a metabolite. A B C D E F Gene neighboring (undirected graph) Vertices genes; Edges connect adjacent genes (same chromosome, same strand). Objective: find consecutive reactions in a metabolic pathway that are catalyzed by products of neighboring genes. 2 / 19

3 Model Metabolic pathway: D = (V, A) Gene neighboring: G = (U, E ) A B C D E 3 / 19

4 Model Metabolic pathway: D = (V, A) Gene neighboring: G = (U, E ) A B C D E Correspondence graph: G = (V, E) [Babou, 2012] Vertices reactions; Edges translate gene adjacency (in G ) into reaction connectivity. Correspondence function: f : V 2 U v f(v) C, D A, D E B C A Associates to every vertex in D a subset of vertices of G. 3 / 19

5 Model D = (V, A) G = (U, E ) A B C D E G = (V, E) f : V 2 U v f(v) C, D A, D E B C A 4 / 19

6 Model D = (V, A) G = (U, E ) A B C D E G = (V, E) f : V 2 U v f(v) C, D A, D E B C A 4 / 19

7 Model D = (V, A) G = (U, E ) A B C D E G = (V, E) f : V 2 U v f(v) C, D A, D E B C A 4 / 19

8 Model D = (V, A) G = (U, E ) A B C D E G = (V, E) f : V 2 U v f(v) C, D A, D E B C A 4 / 19

9 Problem formulation Objective: find consecutive reactions in a metabolic pathway that are catalyzed by products of neighboring genes. Walk: ordered sequence of vertices such that any two consecutive vertices of the walk are connected by an arc. Trail: walk with no repeated arcs. Path: walk with no repeated vertices. paths trails walks 5 / 19

10 Problem formulation Longest Supported Path (LSP) [Fertin et al., 2012] Input: A directed graph D = (V, A), an undirected graph G = (V, E). Output: A longest path P in D such that G[V(P)] is connected. LSP is NP-hard [Fertin et al., 2015]. Heuristic solution proposed if D is a DAG [Fertin et al., 2012]. What to do if D contains cycles? Span: the number of distinct vertices in a trail. 6 / 19

11 Problem formulation Longest Supported Path (LSP) [Fertin et al., 2012] Input: A directed graph D = (V, A), an undirected graph G = (V, E). Output: A longest path P in D such that G[V(P)] is connected. Supported Trail of Maximum Span (STMS) Input: A directed graph D = (V, A), an undirected graph G = (V, E), an arc (u, v) in D. Output: A trail of maximum span T in D passing through (u, v) such that G[V(T)] is connected. Solution can contain cycles. STMS is NP-hard. How to enumerate trails in D? Line graph 6 / 19

12 Problem formulation Let D = (V, A) be a directed graph. The line graph of D is the graph L(D) = (A, A ), where (x, y) A x A, y A, x = (r, s), y = (s, t), with r, s, t V. D ,2 2,3 8,3 L(D) ,4 4,5 5, ,6 6,7 7,5 7 / 19

13 Problem formulation Let D = (V, A) be a directed graph. The line graph of D is the graph L(D) = (A, A ), where (x, y) A x A, y A, x = (r, s), y = (s, t), with r, s, t V. Trail (= distinct arcs) Path (= distinct vertices) D ,2 2,3 8,3 L(D) ,4 4,5 5, ,6 6,7 7,5 T = T is the trail in D corresponding to path P in L(D). Notation: T = L -1 (P). P = (1,4) (4,5) (5,6) (6,7) (7,5) (5,8) (8,3) 7 / 19

14 Problem formulation Longest Supported Path (LSP) [Fertin et al., 2012] Input: A directed graph D = (V, A), an undirected graph G = (V, E). Output: A longest path P in D such that G[V(P)] is connected. Supported Trail of Maximum Span (STMS) Input: A directed graph D = (V, A), an undirected graph G = (V, E), an arc (u, v) in D. Output: A trail of maximum span T in D passing through (u, v) such that G[V(T)] is connected. Supported Corresponding Trail of Maximum Span (CTMS) Input: A directed graph D = (V, A), an undirected graph G = (V, E), an arc (u, v) in D. Output: A path P in the line graph of D such that L -1 (P) has maximum span, passes through arc (u, v) and G[V(L -1 (P))] is connected. 8 / 19

15 Allowing for skipped vertices Objective: find consecutive reactions in a metabolic pathway that are catalyzed by products of neighboring genes. Flexibility? Should be able to skip a few reactions and/or genes [Boyer et al., 2005] Gap parameters: δ G how many genes can be skipped (default: δ G = 0) δ D how many reactions can be skipped (default: δ D = 0) G A B X C D δ G = 0 δ G = 1 δ G = 2 9 / 19

16 Graph reduction Cover set of a path P in D with respect to G [Fertin et al., 2012] Intuitively, if it exists, it is a maximal subset of vertices of D that could extend P to P such that P induces connected subgraphs in G and the undirected graph underlying D. D G / 19

17 Graph reduction Cover set of a path P in D with respect to G [Fertin et al., 2012] Intuitively, if it exists, it is a maximal subset of vertices of D that could extend P to P such that P induces connected subgraphs in G and the undirected graph underlying D. D G Let: 5 D = (V, A) directed graph G = (V, E) undirected graph (u, v) arc in D S cover set of (u, v) in D with respect to G. 5 We have shown that STMS(D, G, (u, v)) yields the same solution as SMTS(D[S], G[S], (u, v)) 10 / 19

18 Path finding in the line graph D (Directed graph) 2,1 1,2 C 1 C 3 8,6 1,3 3,4 6,4 7,8 6,7 C 2 C 4 C 5 4,5 5,3 L(D) (Line graph of D) 11 / 19

19 Path finding in the line graph D (Directed graph) 2,1 1,2 C 1 C 3 8,6 1,3 3,4 6,4 7,8 6,7 C 2 C 4 C 5 4,5 5,3 L(D) (Line graph of D) For every strongly connected component (SCC) of L(D): All possible entry and exit points are determined. Paths are enumerated between feasible pairs of entry and exit points. The best ones (in terms of span of their corresponding trail in D) are retained. 11 / 19

20 HNet HNet* provides an exact solution to CTMS. Supported Corresponding Trail of Maximum Span (CTMS) Input: A directed graph D = (V, A), an undirected graph G = (V, E), an arc (u, v) in D. Output: A path P in the line graph of D such that L -1 (P) has maximum span, passes through arc (u, v) and G[V(L -1 (P))] is connected. HNet uses: The graph reduction to the cover set of the input arc. Path finding in the line graph. * HNet stands for Heterogeneous Network Mining. 12 / 19

21 HNet 2,1 1,2 C 1 C 3 8,6 1,3 3,4 C 2 C 4 6,4 C 5 4,5 5,3 L(D) (Line graph of D) C 1 C 2 C 5 C 4 C 3 C (Condensation graph of L(D)) 7,8 6,7 All paths in C are enumerated. Every path in C is translated to paths in L(D). A path in L(D) is a candidate if: (a) It contains vertex (u, v); (b) Its corresponding trail in D induces a connected subgraph in G. Solution to CTMS: the path in L(D) that fulfills (a) and (b) and whose corresponding trail in D has maximum span. 13 / 19

22 Application to biological data Gene neighboring information 5 3 G = (U, E ) 3 5 KEGG Metabolic pathways G = (V, E) HNet D = (V, A) For every arc (u, v) of D: trail of maximum span passing through (u, v) and inducing a connected subgraph in G 14 / 19

23 Application to biological data Metabolic pathways (on average 73) of 50 bacterial species δ G and δ D between 0 and 3 5-minute timeout => 95% of data set analyzed Run-times* for: Strict neighboring (δ G = δ D = 0): ~11 minutes Average time per organism: ~13 seconds; Median time per organism: ~8 seconds. One insertion allowed (δ G = δ D = 1): ~2h24 minutes Average time per organism: ~43 seconds; Median time per organism: ~15 seconds. * Intel Core 2.5 GHz (6 MB cache), 16 GB 1600 MHz 15 / 19

24 Application to biological data Actinobacteria sco Streptomyces coelicolor A3(2) cgl Corynebacterium glutamicum ATCC bbv Bifidobacterium breve ACS-071-V-Sch8b Firmicutes sau Staphylococcus aureus N315 lmo Listeria monocytogenes EGD-e bsu Bacillus subtilis subsp. subtilis str. 168 snd Streptococcus pneumoniae ST556 cpe Clostridium perfringens str. 13 R03504 R03503 R sco dg = 1, dd = 0 cgl dg = 0, dd = 0 sau dg = 0, dd = 0 bbv dg = 0, dd = 0 lmo dg = 0, dd = 0 cpe dg = 0, dd = 0 bsu dg = 0, dd = 0 R03504 R03503 R03067 R { , } snd dg = 1, dd = 0 R00428 R05046 R05048 R04639 [R04620] R03504 R03503 R [ ] cgl dg = 0, dd = 1 16 / 19

25 Application to biological data R05046 R05048 R04639 R00428 R03504 R03503 R03066 R03067 Firmicutes sau SA0472 SA0473 SA0474 R02237 SCO3400 SCO3401 SCO3402 SCO3403 sco lmo bsu lmo0224 lmo0225 lmo0226 BSU00770 BSU00780 BSU00790 NCgl2599 NCgl2600 NCgl2601 NCgl2602 cgl bbv snd MYY_0368 MYY_0369 MYY_0370 MYY_0371 Actinobacteria cpe CPE1019 CPE1020 CPE1021 CPE / 19

26 Conclusion & perspectives Proposed a new problem formulation for identifying consecutive reactions being catalyzed by products of neighboring genes. Proposed an exact method (HNet). Integrated useful concepts into HNet: Cover set [Fertin et al., 2012]; Gap parameters [Boyer et al., 2005]. HNet is quite fast in practice. Analyze metabolic pathway variation (phylogenetic perspective). Infer ancestral bacterial metabolism (evolutionary perspective). Perform extensive study of fungi. 18 / 19

27 Thank you! Questions? 19 / 19

28 References Babou, H. M. (2012). Comparaison de réseaux biologiques (Doctoral dissertation, Université de Nantes). Boyer, F., Morgat, A., Labarre, L., Pothier, J., & Viari, A. (2005). Syntons, metabolons and interactons: an exact graph-theoretical approach for exploring neighbourhood between genomic and functional data. Bioinformatics, 21(23), Fertin, G., Babou, H. M., & Rusu, I. (2012, June). Algorithms for subnetwork mining in heterogeneous networks. In International Symposium on Experimental Algorithms (pp ). Springer Berlin Heidelberg. Fertin, G., Komusiewicz, C., Mohamed-Babou, H., & Rusu, I. (2015). Finding Supported Paths in Heterogeneous Networks. Algorithms, 8(4),

29 Walks. Trails. Paths Walk: ordered sequence of vertices such that any two consecutive vertices of the walk are connected by an arc. Trail: walk with no repeated arcs. Path: walk with no repeated vertices paths trails walks