Qiong Cheng, Dipendra Kaur, Robert Harrison, Alexander Zelikovsky

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Homomorphism Mapping in Metabolic Pathways

• Qiong Cheng, Dipendra Kaur, Robert Harrison,
  Alexander Zelikovsky
• Computer Science in Georgia State University

Dec. 1 2007 RECOMB Satellite Conference on
Systems Biology 2007
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Outline

• Concept of Metabolic pathway comparison
• Enzyme similarity
• Graph mappings embeddings homomorphisms
• Min cost homomorphism problem for trees
• Optimal DP algorithm for trees
• Min cost homomorphism problem for arbitrary
  graphs
• Minimum Feedback vertex set (MFVS)
• Searching metabolic networks for
• pathway motifs
• pathway holes
• Web tool
• Architecture Brief interface
• Future work

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Metabolic pathway pathways model

• Metabolic pathway

• Metabolic pathways model

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Comparison of metabolic pathways

• Enzyme similarity and pathway topology together
  represent the similarity of pathway functionality.

• Enzyme Similarity

• Pathway topology
• Similarity

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Related work

• Linear topology

(Forst Schulten1999, Chen
Hofestaedt2004)

• Tree topology

(Pinter 2005 o(VG2VT/logVGVGVTlogVT
) )

• Arbitrary topology

Mapping Linear pattern ? Graph (Kelly et al
2004) ( o(VTi2VG2) )
Exhaustively search (Sharan et al 2005 ( o(i!)
o(VTi2VG2) ), Yang et al 2007 (
o(2VGVG2) )
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Enzyme mapping cost
Enzyme D d1 . d2 . d3 . d4

• EC (Enzyme Commission) notation

• Measure Enzyme similarity score ? by the lowest
  common upper class distribution

• Measure ? by tight reaction property

Enzyme X x1 . x2 . x3 . x4
Enzyme Y y1 . y2 . y3 . y4
?X, Y 1





?X, Y 10
?X, Y 8
otherwise
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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.
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Min cost homomorphism of multi source tree to
arbitrary graph

• A multi-source tree is a directed graph, whose
  underlying undirected graph is a tree.

• Given an multisource tree T ltVT, ETgt (Pattern)
  and an arbitrary graph G ltVG, EGgt (Text),
• find min cost homomorphism of multisource tree
  to arbitrary graph f T ? G

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Preprocessing of text graph
Transitive closure of G is graph G(V, E),
where E(i,j) there is i-j-path in G
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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

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DP table
min cost homomorphism mapping from Ts subgraph
induced by previous vertices in the order into G
DTa, uj
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Filling DP table

• Recursive function

?(vi , uj) if vi is a leaf in T
?(vi, uj) ?l1 to adj(vi)Minj1 to VGC(il,
j)
if vi is a leaf in T
Cil, jl DTil, jl l(h(j, jl) - 1)
l is penalty for gaps
h(j, jl) (hops between uj and ujl in G)
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Runtime Analysis for mapping trees

• Transitive closure takes O(VGEG).

• 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 Sj1 to VG Si1 to
VTtij Sj1 to VG degG(uj)Si1 to
VTdegT(vi)
2EGET
The total runtime for mapping trees is
O(VGEGVGVT).
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MFVS

• Minimum Feedback vertex set (MFVS)
• Given an undirected graph G(V,E) and a
  nonnegative weight function w on V
• Find a minimum weight subset of V whose removal
  leaves an acyclic graph.

• Bad news MFVS problem is NP-complete.
• Good news 2-approximation

• Greedy Algorithm
• Delete degree 1/0 vertices from V and set
  remaining vertices to V
• MFVSlt- f
• while V ? f do
• pick up the set S of maximal degree
  vertices
• MFVS lt- MFVS U S
• Delete degree 1/0 vertices from V

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Min cost homomorphism of arbitrary graphs

• Given an arbitrary graph P ltVP, EPgt (Pattern)
  and an arbitrary graph G ltVG, EGgt (Text),
• find min cost homomorphism f P ? G

• Algorithm
• Find minimum feedback vertex set F(P) of P
• Construct a multi source tree P ltVp-F(P),
  Ep(Vp-F(P))gt
• for every sub mapping fv F(P) ?VG do
• obtain min cost homomorphism of multi source
  tree P to arbitrary graph G under sub mapping
  fv
• choose min cost homomorphism for all sub mappings

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Runtime Analysis for mapping arbitrary graphs

• Finding min feedback vertex set takes O(VP
  ET)

• O(VG F(P)) possible mappings for MFVS

• Finding min homomorphism mapping of multi source
  tree to arbitrary graph takes O(VGEGVGVT)
  .

The total runtime is O(VG F(P)(VGEGVGVT
)).
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Statistical significance

• Random degree-conserved graph generation
• Reshuffle nodes

Reshuffle edge

• Reshuffle edges

• Randomized P-Value computation

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Experiments applications

• All-against-all mappings among S. cerevisiae, B.
  subtilis, T. thermophilus, and E.coli
  Hallobacterium

• 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

• Resolving ambiguity

• Discovering pathways holes

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Mappings with cycles
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Resolving Ambiguity
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Pathway holes

• Check if there is such enzyme in pattern
• Find the closest protein in the same group
• If identity is too high gt 80 then we expect good
  filling
• Align to previous and next enzyme the functions
  may be taken over

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Filling pathways holes
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Web Service Architecture
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Web Interface
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Future work

• Approximation algorithm to handle with the
  comparison of general graphs
• Mining protein interaction network
• 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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Reference

• Ron Y Pinter, Oleg Rokhlenko, Esti Yeger-Lotem,
  Michal Ziv-Ukelson Alignment of metabolic
  pathways. Bioinformatics. LNCS 3109.
  Springer-Verlag.(Aug 2005)21(16) 3401-8
• Sebastian Wernicke Combinatorial Algorithms to
  Cope with the Complexity of Biological Networks.
  Dissertation (December 2006)
• J. Ellson, E. Gansner, E. Koutsofios, S. North,
  and G. Woodhull. Graphviz and dynagraph - static
  and dynamic graph drawing tools. In M. Junger and
  P. Mutzel, editors, Graph Drawing Software, pages
  127-148. Springer-Verlag, 2003
• Yan and J. Han. gspan Graph-based substructure
  pattern mining. In ICDM, pages 721-724, 2002.
• N. Ketkar, L. Holder, D. Cook, R. Shah and J.
  Coble, Subdue Compression-based Frequent Pattern
  Discovery in Graph Data, Proceedings of the ACM
  KDD Workshop on Open-Source Data Mining, August
  2005.
• K, Borgwardt, S. Bottger, H. Kriegel, VGM visual
  graph mining, International Conference on
  Management of Data archive Proceedings of the
  2006 ACM SIGMOD international conference on
  Management of data
• 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,"Homomor
  phisms of Multisource Trees into Networks with
  Applications to Metabolic Pathways", Proc. of
  IEEE 7-th International Symposium on
  BioInformatics and BioEngineering (BIBE'07)

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Question?
Thanks!
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Handling Cycles

• Sorting of the pattern such that children can
  communicate only through parent
• Fix images for some pattern vertices gt
  interrupt communication through cycles
• Feedback vertex set F(T) VT-F(T) is acyclic
• Runtime is increased by factor O(VG F(T))
• t(v) of reasonable text images of v
• ? t(v) -gt min ? log(t(v)) -gtmin
• 2-approximation algo

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Software architecture of service-oriented pathway
mining tool
Services Container
Ambiguity pairs
AI
Potential holes
Rule based mining
DB
Pathway Modeling Comparison
Storage Indexing

Data-Control-View
Browsers
PDC
SW
Visualized Outputs
Additional Value Service
Simulation