Signaling Hypergraph. Set V of nodes proteins, small molecules, etc.

Transcription

1 Signaling Hypergraph Set V of nodes proteins, small molecules, etc. Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

2 Signaling Hypergraph Set V of nodes proteins, small molecules, etc. Set V of hypernodes subsets of V Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

3 Signaling Hypergraph Set V of nodes proteins, small molecules, etc. Set V of hypernodes subsets of V Set E of signaling hyperedges e =(T (e), H(e)) Tail T (e) andheadh(e) subsets of V Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

4 Signaling Hypergraph Set V of nodes proteins, small molecules, etc. Set V of hypernodes subsets of V Set E of signaling hyperedges e =(T (e), H(e)) Tail T (e) andheadh(e) subsets of V Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

5 Signaling Hypergraph Set V of nodes proteins, small molecules, etc. Set V of hypernodes subsets of V Set E of signaling hyperedges e =(T (e), H(e)) Tail T (e) andheadh(e) subsets of V Positive regulators added to tail T (e) Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

6 Signaling Hypergraph Set V of nodes proteins, small molecules, etc. Set V of hypernodes subsets of V Set E of signaling hyperedges e =(T (e), H(e)) Tail T (e) andheadh(e) subsets of V Positive regulators added to tail T (e) Signaling hypergraph H =(V, V, E) Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

7 Signaling Hypergraph Represents Wnt Reactions Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology 2014.

8 Signaling Hypergraph Represents Wnt Reactions Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology 2014.

9 Signaling Pathway Analysis Fundamental Problem: what reactions connect two proteins/complexes?

10 Pathway Analysis with Signaling Hypergraphs Definitions and notation Reachability and hyperpaths Shortest hyperpaths problem Algorithm to compute shortest hyperpaths Constructing signaling hypergraphs Results on the Wnt Signaling pathway Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

11 Reachability

12 Reachability Rule: Hypernodes in the head of hyperedge e are reachable from s only if all hypernodes in the tail of e are reachable from s.

13 Reachability Rule: Hypernodes in the head of hyperedge e are reachable from s only if all hypernodes in the tail of e are reachable from s.

14 Reachability Rule: Hypernodes in the head of hyperedge e are reachable from s only if all hypernodes in the tail of e are reachable from s.

15 Reachability Rule: Hypernodes in the head of hyperedge e are reachable from s only if all hypernodes in the tail of e are reachable from s.

16 Reachability Rule: Hypernodes in the head of hyperedge e are reachable from s only if all hypernodes in the tail of e are reachable from s.

17 Reachability Rule: Hypernodes in the head of hyperedge e are reachable from s only if all hypernodes in the tail of e are reachable from s. Hypernodes are B-connected to s only if all hypernodes in the tail of an incoming hyperedge are B-connected to s. Ausiello et. al. Optimal traversal of directed hypergraphs. Technical Report, Gallo, Longo, and Pallottino. Directed hypergraphs and applications. Discrete Applied Mathematics 1993.

18 Reachability Rule: Hypernodes in the head of hyperedge e are reachable from s only if all hypernodes in the tail of e are reachable from s. Hypernodes are B-connected to s only if all hypernodes in the tail of an incoming hyperedge are B-connected to s. Ausiello et. al. Optimal traversal of directed hypergraphs. Technical Report, Gallo, Longo, and Pallottino. Directed hypergraphs and applications. Discrete Applied Mathematics 1993.

19 Hyperpaths Definition: An s-t B-hyperpath (s, t) is a sub-hypergraph of H where t is B-connected to s in (s, t), and (s, t) is minimal w.r.t the deletion of hypernodes and hyperedges. Ausiello et. al. Optimal traversal of directed hypergraphs. Technical Report, Gallo, Longo, and Pallottino. Directed hypergraphs and applications. Discrete Applied Mathematics 1993.

20 Pathway Analysis with Signaling Hypergraphs Definitions and notation Reachability and hyperpaths Shortest hyperpaths problem Algorithm to compute shortest hyperpaths Constructing signaling hypergraphs Results on the Wnt Signaling pathway Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

21 Problem Formulation Given H =(V, V, E) withhypernodess, t 2V, we seek to compute a B-hyperpath (s, t) of H with the smallest number of hyperedges.

22 Problem Formulation Given H =(V, V, E) withhypernodess, t 2V, we seek to compute an acyclic B-hyperpath (s, t) of H with the smallest number of hyperedges.

23 Problem Formulation Given H =(V, V, E) withhypernodess, t 2V, we seek to compute an acyclic B-hyperpath (s, t) of H with the smallest number of hyperedges.

24 Problem Formulation Given H =(V, V, E) withhypernodess, t 2V, we seek to compute an acyclic B-hyperpath (s, t) of H with the smallest number of hyperedges. Theorem Finding the shortest acyclic B-hyperpath (s, t) of H is NP-hard. Ausiello et. al. Optimal traversal of directed hypergraphs. Technical report, 1992.

25 Problem Formulation Given H =(V, V, E) withhypernodess, t 2V, we seek to compute an acyclic B-hyperpath (s, t) of H with the smallest number of hyperedges. Theorem Finding the shortest acyclic B-hyperpath (s, t) of H is NP-hard, evenwhen the number of hypernodes in T (e) and H(e) are bounded by k, for k 3. Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

26 Computing the Shortest Acyclic s-t B-Hyperpath Binary variables v for hypernodes v 2V and e for hyperedges e 2E X min e such that the following constraints hold: : t =1 e2e Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

27 Computing the Shortest Acyclic s-t B-Hyperpath Binary variables v for hypernodes v 2V and e for hyperedges e 2E X min e such that the following constraints hold: : t =1 e2e B-Connection: Hypernodes have an incoming hyperedge (except for s) Sub-Hypergraph: Hyperedges have all incident hypernodes Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

28 Computing the Shortest Acyclic s-t B-Hyperpath Binary variables v for hypernodes v 2V and e for hyperedges e 2E X min e such that the following constraints hold: : t =1 e2e B-Connection: Hypernodes have an incoming hyperedge (except for s) Sub-Hypergraph: Hyperedges have all incident hypernodes Acyclicity: Topological ordering of hypernodes (real-valued variables o u ) Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

29 Computing the Shortest Acyclic s-t B-Hyperpath Binary variables v for hypernodes v 2V and e for hyperedges e 2E X min e such that the following constraints hold: : t =1 e2e B-Connection: + Sub-Hypergraph: Linearize constraints! Mixed Integer Linear Program Acyclicity: Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

30 Pathway Analysis with Signaling Hypergraphs Definitions and notation Reachability and hyperpaths Shortest hyperpaths problem Algorithm to compute shortest hyperpaths Constructing signaling hypergraphs Results on the Wnt Signaling pathway Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

31 National Cancer Institute s Pathway Interaction Database Schaefer et. al. PID: the pathway interaction database. Nucleic Acids Research 2009.

32 Constructing Signaling Hypergraphs from NCI-PID Signaling Hypergraph #Nodes 6,793 #Hypernodes 8,779 # (Hyper)edges 7,735 Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

33 Constructing Signaling Hypergraphs from NCI-PID Signaling Graph with Hypergraph Complexes Graph #Nodes 6,793 6,793 6,793 #Hypernodes 8,779 8,779 n/a # (Hyper)edges 7,735 15,622 40,346 Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

34 Pathway Analysis with Signaling Hypergraphs Definitions and notation Reachability and hyperpaths Shortest hyperpaths problem Algorithm to compute shortest hyperpaths Constructing signaling hypergraphs Results on the Wnt Signaling pathway Ritz, Tegge, Kim, Poirel, and Murali. Signaling hypergraphs. Trends in Biotechnology Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

35

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37 Question: What Wnt signaling reactions regulate both TCF and LEF?

38 Wnt Signaling Hypergraph Consider a sub-hypergraph of NCI-PID that corresponds to Wnt signaling 356 hypernodes and 374 hyperedges Introduce source s and target t hypernodes s TCF LEF t Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

39 Shortest Acyclic B-Hyperpath Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

40 Shortest Acyclic B-Hyperpath Cell Membrane WNT3A FZD5 LRP6 APC CTNNB1 Axin1 CK1 // WNT3A FZD5 LRP6 PP2A-B56 CTNNB1 APC Axin1 GSK3 GSK3 APC CTNNB1 Axin1 Nucleus PP2A-B56 CTNNB1 APC Axin1 GSK3 CTNNB1 TLE4 TCF4 TLE4 CTNNB1 CTNNB1 TCF4 APC TLE2 TCF1E Axin1 GSK3 TCF1 TLE2 CTNNB1 TCF1E LEF1 t Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

41 Shortest Acyclic B-Hyperpath Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

42 Shortest Acyclic B-Hyperpath Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

43 Shortest Acyclic B-Hyperpath Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

44 Comparison to Shortest Paths Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

45 Comparison to Shortest Paths Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

46 Performance Wnt Signaling Hypergraph: < 2 seconds for each experiment Full NCI-PID Signaling Hypergraph: < 40 seconds for each experiment Ritz and Murali. Pathway analysis with signaling hypergraphs. Proceedings of the Fifth ACM Conference on Bioinformatics, Computational Biology, and Health Informatics 2014.

47 Conclusions

48 Related Research: Signaling Pathway Reconstruction Can we reconstruct the pathway given only the receptors, transcriptional regulators, and a background interactome? Ritz et. al. Pathways on Demand: Automated Reconstruction of Human Signaling Networks. Oral presentation at the RECOMB/ISCB Conference on Regulatory and Systems Genomics, November Manuscript under review.

49 Related Research: Signaling Pathway Reconstruction Can we reconstruct the pathway given only the receptors, transcriptional regulators, and a background interactome? Ritz et. al. Pathways on Demand: Automated Reconstruction of Human Signaling Networks. Oral presentation at the RECOMB/ISCB Conference on Regulatory and Systems Genomics, November Manuscript under review.

50 Related Research: Signaling Pathway Reconstruction Can we reconstruct the pathway given only the receptors, transcriptional regulators, and a background interactome? PathLinker computes the k shortest loopless paths from sources to targets. Can we compute the k shortest B-hyperpaths? Ritz et. al. Pathways on Demand: Automated Reconstruction of Human Signaling Networks. Oral presentation at the RECOMB/ISCB Conference on Regulatory and Systems Genomics, November Manuscript under review.

51 Future Work on Signaling Hypergraphs Extend signaling hypergraphs to handle negative regulators and cycles Develop algorithms for hypertrees, hypercycles, random walks, etc. Build signaling hypergraphs from other pathway databases Apply experimental data to signaling hypergraphs

52 halp: Hypergraph ALgorithms Package Algorithms for Directed Hypergraphs and Undirected Hypergraphs B-Connectedness, F -Connectedness, BF -Connectedness Min-cost hyperpaths for a special class of cost functions Random walks and min-cut algorithms Currently being added: Signaling Hypergraph class Mixed integer linear program to compute shortest B-hyperpaths

53 Acknowledgements T. M. Murali and his research group: Allison Tegge Ahsanur Rahman Brendan Avent Nick Sharp Peter Burnham NIH NIGMS R01-GM NSF DBI John Tyson (Department of Biological Sciences) Shiv Kale (Virginia Bioinformatics Institute) EPA RD

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55 Related Work: Compound Graphs and Metagraphs Limitation: pairwise connections between elements. Fukuda and Takagi. Knowledge representation of signal transduction pathways. Bioinformatics Hu et. al. Towards zoomable multidimensional maps of the cell. Nature Biotechnology 2007.

56 Motivation Wnt Signaling Computable Representations Signaling Hypergraphs Conclusions Related Work: Factor Graphs and Petri Nets Limitation: difficult to generalize common graph-theoretic concepts. Vaske et. al. Inference of patient-specific pathway activities from multi-dimensional cancer genomics data using PARADIGM. Bioinformatics Ruths et. al. The Signaling Petri Net-Based Simulator: A Non-Parametric Strategy for Characterizing the Dynamics of Cell-Specific Signaling Networks. PLOS Computational Biology Anna Ritz (Virginia Tech) Signaling Hypergraphs SIAM Talk 3/19/2015

57 Related Work: Influence Graphs and Logic Networks Limitation: unclear how to represent complex disassembly. Samaga and Klamt. Modeling approaches for qualitative and semi-quantitative analysis of cellular signaling networks. Cell Communication and Signaling 2013.

58 (A) A signaling hypergraph. Dashed portions of hyperedges denote positive regulation and are for visualization purposes only. (B) An acyclic B-hyperpath and (C) a cyclic B-hyperpath from {a,b} to {c,d} (red hyperedges).

59 A hypergraph that satisfies the B-connectedness and the B-hyperpath constraints where t is not B-connected to s.

60 Proof sketch for shortest B-hyperpaths (left) and shortest k-bounded B-hyperpaths (right).

61 Hyperedges from sets s "Dummy" hyperedges A 1 A 2 A 3 A 4 A 5 A 6 A 7 D 1 D 2 B 1 B 2 B 3 k-way tree t Proof sketch for shortest k-bounded B-hyperpaths (TCBB paper).

62 Converting a hypergraph H (A) into a graph with complexes (B) and a graph (C).

63 Optimal B-hyperpaths in the small Wnt signaling pathway to (a)-(c) ubiquitinated -catenin, denoted by U.

64 Optimal B-hyperpaths in the small Wnt signaling pathway to (d)-(f) nuclear -catenin.

65 Two B-hyperpaths computed in the full NCI-PID signaling pathway. The B-hyperpath from s 1 to t 1 is the optimal result for New Sources to Known Targets, and the B-hyperpath from s 2 to t 2 is the optimal result for Known Sources to New Targets.