Pattern Matching in Protein-Protein Interaction Graphs Ga¨ elle Brevier ( Universit´ e de Grenoble, France ) Romeo Rizzi ( Universit` a di Udine, Italy ) St´ ephane Vialette ( Universit´ e Paris-Est, France ) Lisbon, September 19, 2008 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 1 / 38
Introduction Outline Introduction 1 Exact colorful instances 2 Hardness results 3 Approximation algorithms 4 Bounded degree graphs A randomized algorithm Linear forests Future works 5 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 2 / 38
Introduction Introduction Protein interactions identified on a genome-wide scale are commonly visualized as protein interaction graphs, where proteins are vertices and interactions are edges. Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 3 / 38
Introduction Gene or Protein Interactions Databases BioGRID - A Database of Genetic and Physical Interactions DIP - Database of Interacting Proteins MINT - A Molecular Interactions Database IntAct - EMBL-EBI Protein Interaction MIPS - Comprehensive Yeast Protein-Protein interactions Yeast Protein Interactions - Yeast two-hybrid results from Fields’ group PathCalling - A yeast protein interaction database by Curagen SPiD - Bacillus subtilis Protein Interaction Database AllFuse - Functional Associations of Proteins in Complete Genomes BRITE - Biomolecular Relations in Information Transmission and Expression ProMesh - A Protein-Protein Interaction Database The PIM Database - by Hybrigenics Mouse Protein-Protein interactions Human herpesvirus 1 Protein-Protein interactions Human Protein Reference Database BOND - The Biomolecular Object Network Databank. Former BIND MDSP - Systematic identification of protein complexes in Saccharomyces cerevisiae by mass spectrometry Protcom - Database of protein-protein complexes enriched with the domain-domain structures Proteins that interact with GroEL and factors that affect their release YPD TM - Yeast Proteome Database by Incyte . . . Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 4 / 38
Introduction Introduction Comparative analysis of protein-protein interaction graphs aims at finding complexes that are common to different species. Mounting evidence suggests that proteins that function together in a pathway or a structural complex are likely to evolve in a correlated fashion. Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 5 / 38
Introduction Intoduction Pattern matching in protein-protein interaction graphs Finding a protein complex in another protein network. Graph matching Focus on mappings that preserve adjacencies (to deal with interaction datasets that are missing many true protein interactions). Injective list homomorphisms and optimization State-of-the art approaches to identifying orthologs (genes in different species that originate from a single gene in the last common ancestor of these species). Putative orthologs are represented by colors Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 6 / 38
Introduction Introduction: Searching for an exact occurrence Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 5 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 7 / 38
Introduction Introduction: Searching for an exact occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 5 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 7 / 38
Introduction Introduction: Searching for an exact occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 5 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 7 / 38
Introduction Introduction: Searching for an exact occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 5 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 7 / 38
Introduction Introduction: Searching for an exact occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 5 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 7 / 38
Introduction Introduction: Searching for an exact occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 5 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 7 / 38
Introduction Introduction: Searching for the best occurrence Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 4 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 8 / 38
Introduction Introduction: Searching for the best occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 4 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 8 / 38
Introduction Introduction: Searching for the best occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 4 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 8 / 38
Introduction Introduction: Searching for the best occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 4 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 8 / 38
Introduction Introduction: Searching for the best occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 4 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 8 / 38
Introduction Introduction: Searching for the best occurrence λ G ,λ H θ : V ( G ) − − − → V ( H ) Pattern graph ( G , λ G ) Target graph ( H , λ H ) mult ( G , λ G ) = 2 mult ( H , λ H ) = 4 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 8 / 38
Introduction Problem Max– ( ρ, σ ) –Matching–Colors • Input : Two graphs G and H and the coloring mappings λ G : V ( G ) → C , mult ( G , λ G ) = ρ , and λ H : V ( H ) → C , mult ( H , λ H ) = σ . λ G ,λ H • Solution : An injective mapping θ : V ( G ) − − − → V ( H ) . • Measure : The number of edges of G matched by the injective mapping θ . E XACT – ( ρ, σ ) –M ATCHING –C OLORS is the extremal problem of finding λ G ,λ H an injective mapping θ : V ( G ) − − − → V ( H ) that matches all the edges of G . Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 9 / 38
Introduction Introduction Trim instance An instance of the M AX – ( ρ, σ ) –M ATCHING –C OLORS or the E XACT – ( ρ, σ ) –M ATCHING –C OLORS problem is said to be trim if the following conditions hold true: for each color c i ∈ C , # C G ( c i ) ≤ # C H ( c i ) , and 1 for each edge { u i , u j } ∈ E ( G ) , there exists an edge { v i , v j } ∈ E ( H ) 2 such that λ G ( u i ) = λ H ( v i ) and λ G ( u j ) = λ H ( v j ) . Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 10 / 38
Introduction Related works in the context List injective homomorphisms for protein graphs [Fagnot, Lelandais and V., 2007; Fertin, Rizzi and V., 2005] . Reaction motifs in metabolic networks [Lacroix, Fernandes and Sagot, 2006; Hermelin, Fellows, Fertin and V., 2007] . QPath [Shlomi, Segal, Ruppin and Sharan, 2006] . Path Matching and Graph Matching in Biological Networks [Yang and Sze, 2007] . Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 11 / 38
Exact colorful instances Outline Introduction 1 Exact colorful instances 2 Hardness results 3 Approximation algorithms 4 Bounded degree graphs A randomized algorithm Linear forests Future works 5 Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 12 / 38
Exact colorful instances Exact colorful instances Theorem ( Fagnot, Lelandais and V., 2007 ) Both the E XACT – ( 1 , σ ) –M ATCHING –C OLORS problem for ∆ ( G ) ≤ 2 and the E XACT – ( ρ, 2 ) –M ATCHING –C OLORS problem are solvable in polynomial-time for any constant ρ and σ . Theorem ( Fertin, Rizzi and V., 2005 ) The E XACT – ( 1 , 3 ) –M ATCHING –C OLORS problem for ∆ ( G ) = 3 and ∆ ( H ) = 4 is NP -complete. We focus here on the E XACT – ( 1 , σ ) –M ATCHING –C OLORS problem. Brevier, Rizzi and Vialette () Pattern Matching in Protein Graphs Lisbon, September 18, 2008 13 / 38
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