NeMoFinder: Dissecting genome- wide 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 NeMoFinder 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

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

The protein 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

Graph, Graph, Everywhere Aspirin Yeast protein interaction network from H. Jeong et al Nature 411, 41 (2001) Internet 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 FREQUENT PATTERNS (MIN SUPPORT IS 2) (A)(B)(C) (1)(2)

Frequent Subgraph Mining Approaches Apriori-based approach: i f a graph is frequent, all of its subgraphs are frequent ─ the Apriori property –AGM/AcGM: 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 –MoFa, Borgelt and Berthold (ICDM’02) –gSpan: 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 f g is the number of occurrences of a subgraph g, g is repeated if fg>F. f g_randi is the frequency of g in a randomized network G randi, for 1 ≤ i ≤ N, N is the number of the randomized networks. s g is the number of times f g ≥ f g_randi, g is unique if its s g > S. Network motif discovery algorithm

Problem Statement (cont) Motivation of NeMoFinder- 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 NeMoFinder _ 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.

Step1. Discover Repeated Subgraphs Step1.1 find repeated size-k trees Eg. Size 2 to size 5 trees t 2 t 3 t 4_1 t 4_2 t 5_1 t 5_2 t 5_3

Step1. discover repeated subgraphs (cont) f t2 = 7, f t3 = 13, f t4_1 = 6, f t4_2 =17, f t5_1 =1, f t5_2 = 5, f t5_3 = 7. T 2 = {t 2 }, T 3 = {t 3 }, T 4 ={t 4_1, t 4_2 } and T5 = {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.

Step1.2 Use repeated size-k trees to partition graph (cont) Set of graphs GD 4 G 4_1 G 4_2 G 4_3 G 4_4 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 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 g 1_2 t 4_2 h 3 g 1_1 f g1_1 = 2 and f g1_2 = 5

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 g 2 f(g 2 )<2, algorithm stops

Algorithm1 NeMoFinder 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 ← FindRepeatedTrees(k); 6: GD k ← GraphPartition(G, T) 7: D ← D  T; 8: D’ ← T; 9: i ← k; 10: while D’≠ ∅ and i ≤ k × (k − 1)/2 do 11: D’ ← FindRepeatedGraphs(k,i,D’); 12: D ← D  D’; 13: i ← i + 1; 14: end while 15: end for Step1: 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

Algorithm1 NeMoFinder (cont) 16: for counter i from 1 to N do 17: G rand ← RandomizedNetworkGeneration(); 18: for each g  D do 19: GetRandFrequency(g,G rand ); 20: end for 21: end for 22: U ← ∅ ; 23: for each g D do 24: s ← GetUniqunessValue(g); 25: if s ≥ S then 26: U ← U  {g}; 27: end if 28: end for 29: return U; Step 2: Determine subgraph frequency in randomized networks Step 3: Compute uniqueness of subgraphs

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

Find repeated subgraphs Algorithm 2 FindRepeatedGraphs(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 ← CandidateGeneration(k, i, D’); 4: D’’ ← FrequencyCounting(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 2,1 m 2,2 …m n,1 m n,2 …m n,n Transform adjancy matrix into canonical adjacency matrix (CAM) which has the maximal code Definition of subCAM 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 subCAM(g) = subCAM(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 t 4_1 h 1 h

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 CandidateGeneration(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 ← GetCousin(g); 6: for each h  H do 7: g’ ← join(g, h); 8: C ← C  {g}; 9: end for 10: end for 11: return C; Step 1: Find set of cousins Step2: join g with cousins to form new subgraph

Frequency counting Leveraging properties of the different types of cousins _L x : set of graphs in GD k embedding x _If type of h=type I direct cousin of g, g’ is subgraph obtained by g and h, then L g’ = L g ∩ L h, f g’= |L g ∩ L h | _if type of h = Type III distant cousin,then f g’= |L g ∩ L h | _if type of h = Type II twin cousin then f g’ =CheckAllOccurances(g) _L t4_1 ={G 4_1,G 4_2,G 4_3,G 4_5 }, L h2 = {G 4_1,G 4_2,G 4_3,G 4_4,G 4_5 } L g1_2 = L t4_1 ∩ L h2 ={G 4_1,G 4_2,G 4_3,G 4_5 }, f g1_2 =4>2

Frequency counting Algorithm 4 FrequencyCounting(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: L g ← set of graphs in GDk embedding g; 7: L h ← set of graphs in GDk embedding h; 8: if f g < F or f h < F then 9: f g’ ← 0; 10: else if type of h = Type I direct cousin then 11: f g’ ← |L g ∩ L h | 12: else if type of h = Type III distant cousin then 13: f g’ ← |L g ∩ L h | 14: else if type of h = Type II twin cousin then 15: f g’ ← CheckAllOccurances(g); 16: end if 17: if f g’ > F then 18: D’’ ← D’’  {g’}; 19: end if 20: end for 21: return D’’; Case h is direct cousin Case h is distant cousin Case h is twin cousin

Summary NemoFinder-an efficient network motif discovery algorithm to discover larger- sized 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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