Ne Mo Finder Dissecting genomewide protein-protein intractions with

Ne. Mo. Finder: Dissecting genomewide protein-protein intractions with meso-scale network motifs Mike Yuan

Outline of this presentation • • Introduction to PPI Introduction to Graph Mining Related work Problem statement Details of the Ne. Mo. Finder algorithm Summary References

Protein Interactions A Protein may interact with: – Other proteins – Nucleic Acids – Small molecules

Finding Protein Partners

Motivation • Important for biological functions • To understand the function of a protein, we need to find its interacting partners

Graph Theory Vertex (node) Cycle Edge -5 Directed Edge (Arc) 10 Weighted Edge 7 Molecular interaction networks are mapped as graphs

The protein interaction network…

Graph mining • Methods for Mining Frequent Subgraphs • Mining Variant and Constrained Substructure Patterns • Applications: – Graph Indexing – Similarity Search – Classification and Clustering

Why Graph Mining? • Graphs are ubiquitous – Chemical compounds (Cheminformatics) – Protein structures, biological pathways/networks (Bioinformactics) – Program control flow, traffic flow, and workflow analysis – XML databases, Web, and social network analysis • Graph is a general model – Trees, lattices, sequences, and items are degenerated graphs • Complexity of algorithms: many problems are of high complexity

from H. Jeong et al Nature 411, 41 (2001) Graph, Everywhere Aspirin Internet Yeast protein interaction network Co-author network

Graph Pattern Mining • Frequent subgraphs – A (sub)graph is frequent if its support (occurrence frequency) in a given dataset is no less than a minimum support threshold • Applications of graph pattern mining – Mining biochemical structures – Program control flow analysis – Mining XML structures or Web communities – Building blocks for graph classification, clustering, compression, comparison, and correlation analysis

Example: Frequent Subgraphs GRAPH DATASET (A) (B) (C) FREQUENT PATTERNS (MIN SUPPORT IS 2) (1) (2)

Frequent Subgraph Mining Approaches • Apriori-based approach: if a graph is frequent, all of its subgraphs are frequent ─ the Apriori property – AGM/Ac. GM: Inokuchi, et al. (PKDD’ 00) – FSG: Kuramochi and Karypis (ICDM’ 01) – PATH#: Vanetik and Gudes (ICDM’ 02, ICDM’ 04) – FFSM: Huan, et al. (ICDM’ 03) • Pattern growth approach – Mo. Fa, Borgelt and Berthold (ICDM’ 02) – g. Span: Yan and Han (ICDM’ 02) – Gaston: Nijssen and Kok (KDD’ 04)

Problem Statement • PPI network G=(V, E) _ each vertex represents a unique protein _ each edge between v. A and v. B indicates there is an interaction between A and B • Network motif _frequently occurring subgraph pattern in a network • fg is the number of occurrences of a subgraph g, g is repeated if fg>F. • fg_randi is the frequency of g in a randomized network Grandi, for 1 ≤ i ≤ N, N is the number of the randomized networks. sg is the number of times fg ≥ fg_randi, g is unique if its sg > S. • Network motif discovery algorithm

Problem Statement (cont) • • Motivation of Ne. Mo. Finder- existing research has following limitations: _Number of network motifs candidates increases exponentially _Interesting network motifs are repeated and unique and Apirori algorithms are not applicable _The graph isomorphism problem is an NP problem Ne. Mo. Finder _ a network motif discovery algorithm to discover repeated and unique meso-scale network motifs in a large PPI network

Key procedures • Example graph G • Find repeated trees • Use repeated trees to partition a network into a set of graphs • Introduce graph cousins to facilitate the candidate generation and frequency counting processes.

Step 1. Discover Repeated Subgraphs • Step 1. 1 find repeated size-k trees • Eg. Size 2 to size 5 trees t 2 t 5_1 t 3 t 4_1 t 5_2 t 5_3 t 4_2

Step 1. discover repeated subgraphs (cont) • ft 2 = 7, ft 3 = 13, ft 4_1 = 6, ft 4_2 =17, ft 5_1=1, ft 5_2 = 5, ft 5_3 = 7. • T 2 = {t 2}, T 3 = {t 3}, T 4 ={t 4_1, t 4_2} and T 5 = {t 5_2, t 5_3}.

Step 1. 2 Use repeated size-k trees to partition graph • Occurrences of t 4_1 in G.

Step 1. 2 Use repeated size-k trees to partition graph (cont) • Occurrences of t 4_2 in G.

Step 1. 2 Use repeated size-k trees to partition graph (cont) • Set of graphs GD 4 G 4_1 G 4_4 G 4_2 G 4_3 G 4_5

Step 1. 3: perform graph join operation to find repeated size-k graphs • Generate 3 -edge subgraphs from size-4 trees t 4_1 h 2 t 4_2 h 3 h 4 h 5

Step 1. 3: perform graph join operation to find repeated size-k graphs (cont) • Examples for graph join operations for subgraphs t 4_1 h 2 t 4_2 h 3 • fg 1_1 = 2 and fg 1_2 = 5 g 1_2 g 1_1

Step 1. 3: perform graph join operation to find repeated size-k graphs (cont) • Use subgraphs obtained to generate subgraphs g 1_2 h 6 h 7 • Graph join operations for subgraphs g 1_2 h 6 • f(g 2)<2, algorithm stops g 2

Algorithm 1 Ne. Mo. Finder 1: Input: G - PPI network; N - Number of randomized networks; K - Maximal network motif size; F - Frequency threshold; S - Uniqueness threshold; 2: Output: U - Repeated and unique network motif set; 3: D ← ∅; 4: for motif-size k from 3 to K do 5: T ← Find. Repeated. Trees(k); 6: GDk ← Graph. Partition(G, T) 7: D ← D T; 8: D’ ← T; 9: i ← k; 10: while D’≠ ∅ and i ≤ k × (k − 1)/2 do 11: D’ ← Find. Repeated. Graphs(k, i, D’); 12: D ← D D’; 13: i ← i + 1; 14: end while 15: end for Step 1: Discover repeated subgraphs Step 1. 1: Find repeated size-k trees Step 1. 2: use repeated size-k trees to partition graph Step 1. 3: perform graph join operation to find repeated size-k graphs

Algorithm 1 Ne. Mo. Finder (cont) 16: for counter i from 1 to N do Step 2: Determine 17: Grand ← Randomized. Network. Generation(); subgraph frequency in 18: for each g D do randomized networks 19: Get. Rand. Frequency(g, Grand); 20: end for 21: end for 22: U ← ∅; 23: for each g D do 24: s ← Get. Uniquness. Value(g); Step 3: Compute uniqueness of 25: if s ≥ S then subgraphs 26: U ← U {g}; 27: end if 28: end for 29: return U;

Algorithm Steps (cont) • Step 2: Determine subgraph frequency in randomized networks _Generate randomized networks Grandi(1≤i≤N) _check the frequency of the subgraphs in each of the randomized networks Grandi • Step 3: Compute uniqueness of subgraphs _ Based on frequencies in the input PPI network and the randomized networks _fg_randi is the frequency of g in a randomized network Grandi, for 1 ≤ i ≤ N, N is the number of the randomized networks. sg is the number of times fg ≥ fg_randi, g is unique if its sg > S.

Find repeated subgraphs Algorithm 2 Find. Repeated. Graphs(k, i, D’) 1: Input: D’ - Set of repeated subgraphs with k vertices and i − 1 edges; 2: Output: D’’ - Set of repeated subgraphs with k vertices and i edges; 3: C ← Candidate. Generation(k, i, D’); 4: D’’ ← Frequency. Counting(k, i, C); 5: return D’’;

Candidate generation using graph cousins • Represent subgraphs by adjacency matrices • Code(M): a sequence formed by linking the lower triangular entries of M in the following order: m 1, 1 m 2, 2…mn, 1 mn, 2…mn, n • Transform adjancy matrix into canonical adjacency matrix (CAM) which has the maximal code • Definition of sub. CAM of a graph _ A matrix obtained by setting the last edge entry in CAM(g) to 0.

Candidate generation using graph cousins (cont) • Definition of cousin _ Given two subgraphs g and h, if sub. CAM(g) = sub. CAM(h), then h is a cousin of g. • Three types of cousin relationship between g and h: _ Type I: Direct Cousin h is isomorphic to a subgraph g’ which has the same number of vertices and edges as g, and g’ ≠ g; _ Type II: Twin Cousin h is isomorphic to subgraph g; _ Type III: Distant Cousin h is a disconnected subgraph.

Candidate generation using graph cousins (cont) • Adjacency matrices for the graphs in figure 6 0 1 0 0 t 4_1 0 h 1 h 2

Candidate generation using graph cousins (cont) • Adjacency matrices for the graphs in figure 6 t 4_2 h 3 h 4 h 5

Candidate generation using graph cousins (cont) • Observations of above example _h 1 is a type 1 direct cousin of t 4_1 _h 2 is a type 3 distant cousin of t 4_1 _h 3 is a type 2 twin cousin of t 4_2 _h 4 is a type 1 direct cousin of t 4_2 _h 5 is a type 3 distant cousin of t 4_2

Candidate generation using graph cousins (cont) Algorithm 3 Candidate. Generation(k, i, D’) 1: Input: D’ - Set of repeated subgraphs with k vertices and i − 1 edges; 2: Output: C - Set of candidates with k vertices and i edges; 3: C ← ∅; 4: for each g D do 5: H ← Get. Cousin(g); Step 1: Find set of cousins 6: for each h H do 7: g’ ← join(g, h); Step 2: join g with cousins to form new subgraph 8: C ← C {g}; 9: end for 10: end for 11: return C;

Frequency counting • Leveraging properties of the different types of cousins _Lx: set of graphs in GDk embedding x _If type of h=type I direct cousin of g, g’ is subgraph obtained by g and h, then Lg’= Lg ∩ Lh, fg’= |Lg ∩ Lh| _if type of h = Type III distant cousin, then fg’= |Lg ∩ Lh| _if type of h = Type II twin cousin then fg’ =Check. All. Occurances(g) _Lt 4_1 ={G 4_1, G 4_2, G 4_3, G 4_5}, Lh 2 = {G 4_1, G 4_2, G 4_3, G 4_4, G 4_5} Lg 1_2= Lt 4_1 ∩ Lh 2 ={G 4_1, G 4_2, G 4_3, G 4_5}, fg 1_2=4>2

Frequency counting Algorithm 4 Frequency. Counting(k, i, C) 1: Input: GDk - Set of graphs generated by partitioning G with size-k repeated trees; C - Set of subgraph candidates with k vertices and i edges; F - Frequency threshold; 2: Output: D’’ - Set of repeated subgraphs with k vertices and i edges; 3: D’’ ← ∅; 4: for each g’ C do 5: Get the join parameter of g’: g and h; 6: Lg ← set of graphs in GDk embedding g; 7: Lh ← set of graphs in GDk embedding h; 8: if fg < F or fh < F then 9: fg’ ← 0; Case h is direct cousin 10: else if type of h = Type I direct cousin then 11: fg’ ← |Lg ∩ Lh| 12: else if type of h = Type III distant cousin then 13: fg’ ← |Lg ∩ Lh| Case h is distant cousin 14: else if type of h = Type II twin cousin then 15: fg’ ← Check. All. Occurances(g); 16: end if Case h is twin cousin 17: if fg’ > F then 18: D’’ ← D’’ {g’}; 19: end if 20: end for 21: return D’’;

Summary • Nemo. Finder-an efficient network motif discovery algorithm to discover largersized repeated and unique network motifs in PPI networks. • Use repeated trees to partition network into graphs • Graph cousins for candidate generation and frequency counting

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