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SVM . . . if Pr(+1|v) > 0.5 then t (v) = +1 else t(v) = -1 G - PowerPoint PPT Presentation

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G RAPHLET K ERNELS FOR V ERTEX C LASSIFICATION Presenter: Jos Lugo-Martnez Phd Candidate jlugomar@indiana.edu Jun 12, 2015 O UTLINE Overview of classification problems on graphs Graphlet kernels for vertex classification Case

  1. G RAPHLET K ERNELS FOR V ERTEX C LASSIFICATION Presenter: José Lugo-Martínez Phd Candidate jlugomar@indiana.edu Jun 12, 2015

  2. O UTLINE • Overview of classification problems on graphs • Graphlet kernels for vertex classification • Case study: Structure-based functional residue prediction – Inferring molecular mechanisms of disease (if time permits)

  3. C LASSIFICATION P ROBLEMS ON G RAPHS Graph Classification Vertex Classification Edge Classification Link Prediction Task: Classify graph as +1 or -1 Human lymphocyte kinase

  4. C LASSIFICATION P ROBLEMS ON G RAPHS Vertex (or Edge) Classification Task: Classify node (or edge) as +1 or -1 Y394 of human lymphocyte kinase Depth-3 graph neighborhood for Y394

  5. S EMI - SUPERVISED L EARNING S CENARIO Objective: Predict class label for each unlabeled node A Training Data +1 -1 B A A +1 -1 -1 A B A B B B . . . +1 -1 +1 . A B A B B A B A . … . +1 -1 -1 +1 A B A B A B A B +1 +1 B A B A B A A A Neighborhood graph Research Question How to measure similarity between rooted neighborhoods?

  6. P ROBLEM S TATEMENT Given two neighborhood graphs N(u), N(v) from a space of graphs . The problem of rooted neighborhood comparison is to find a mapping s.t. (N(u), N(v)) quantifies the similarity of N(u) & N(v) Task : Design meaningful similarity measures between vertex neighborhoods

  7. G RAPH K ERNELS • Define kernel functions on pair of graphs G and G’ vector of counts - measure of similarity between G and G’ # of data points • Kernel matrix such that • Properties of I. Symmetric II. Positive semi-definite

  8. M ETHODOLOGY O VERVIEW Test Data V Training Data +1 -1 . . . SVM . . . if Pr(+1|v) > 0.5 then t (v) = +1 else t(v) = -1

  9. G RAPH K ERNELS R ESEARCH IN A N UTSHELL • Diffusion kernels – Kondor & Lafferty (2002) • Focus on counting graph substructures • Three categories based on – walks and paths • Kashima et al. (2003), Borgwardt & Kriegel (2005) – subtree patterns • Hido & Kashima (2009), Shervashidze et al. (2011) – subgraphs • Shervashidze et al. (2009), Vacic et al. (2010) How about other factors? Image from Hido, S. & Kashima, H. ICDM 2009.

  10. Graphlet Kernels

  11. G RAPHLET K ERNEL Count non-isomorphic labeled n-graphlets An n- graphlet is a small (n ≤ 5) connected rooted subgraph 3-graphlets Vacic, V. et al. J Computational Biology 17(1): 55 (2010).

  12. B ASE G RAPHLETS Undirected: Directed: Lugo-Martinez J. and Radivojac P. Network Science, 2(2), 254-276 , (2014).

  13. L ABELED G RAPHLETS vertex labels alphabet same symmetry class

  14. G RAPHLET K ERNEL E XAMPLE 3-graphlets u v A A A A A A Graphlet kernel, N=4 A A A B AAA AAB ABA ABB BAA BAB BBA BBB AAA AAB ABB BAA BAB BBB AAA AAB ABB BAA BAB BBB 2 1 3 1 3 2 3 3 1 1 1

  15. H OW M ANY L ABELED G RAPHLETS ? Undirected Directed | ∑ | = 1 | ∑ | = 20 | ∑ | = 1 | ∑ | = 20 n 1 1 20 1 20 2 1 400 3 1,200 3 3 16,400 30 217,200 4 11 1,045,600 697 102,673,600 5 58 100,168,400 44,907 137,252,234,400 base graphlets labeled graphlets

  16. L IMITATIONS OF G RAPHLET K ERNEL • Exact matches less likely as alphabet size increases • Can’t handle misannotated labels or missing edges – e.g. protein 3D structures can be noisy and incomplete • Ineffective for evolving graph neighborhoods – e.g. closely relate protein structures Goal: Design robust kernels in the presence of noisy and incomplete data

  17. E DIT D ISTANCE G RAPHLET K ERNELS Generalize the concept of counting graphlets Incorporate flexibility in counting via edit distance Definition ( Graph Edit Distance ) Given two vertex- and/or edge-labeled graphs G and H. The edit distance between these graphs corresponds to the minimum number of edit operations necessary to transform G into H. • Allowed edit operations include insertion or deletion of vertices and edges, or in the case of labeled graphs, substitutions of vertex and edge labels • Any sequence of edit operations that transforms G into H is referred to as an edit path • Thus, the graph edit distance between G and H corresponds to the length of the shortest edit path between them Lugo-Martinez J. and Radivojac P. Network Science, 2(2), 254-276 , (2014).

  18. E DIT D ISTANCE O PERATIONS Incorporate flexibility in counting via edit distance Vertex label substitutions Edge insertions or deletions A A A A A A B A A A A A A A A A B A A A B A A A symmetric

  19. E XAMPLE R EVISITED 1-label substitution u v A A A A A A A A A B A A A A B A B A A AAA AAB ABA ABB BAA BAB BBA BBB AAA AAB ABB BAA BAB BBB AAA AAB ABB BAA BAB BBB 2 1 3 1 3 2 3 3 1 1 1

  20. L ABEL S UBSTITUTION K ERNEL 1-label substitution u v A A A A A A A A A B A A A A B A B A A AAA AAB ABA ABB BAA BAB BBA BBB AAA AAB ABB BAA BAB BBB AAA AAB ABB BAA BAB BBB 2 1 1 1 3 1 3 2 3 3 1 1 1

  21. L ABEL S UBSTITUTION K ERNEL 1-label substitution u v A A A A A A A A A B A A A A B A B A A AAA AAB ABA ABB BAA BAB BBA BBB AAA AAB ABB BAA BAB BBB AAA AAB ABB BAA BAB BBB 2 2 2 2 1 1 1 3 1 3 2 3 3 2 2 1 1 1 1 1 1 1

  22. E DGE I NDELS K ERNEL 1-edge indel u v A A A A A A A A A A A A A A A A B A A AAA AAB ABA ABB BAA BAB BBA BBB AAA AAB ABB BAA BAB BBB AAA AAB ABB BAA BAB BBB 2 1 3 1 3 2 3 3 1 1 1

  23. E DGE I NDELS K ERNEL 1-edge indel u v A A A A A A A A A A A A A A A A B A A AAA AAB ABA ABB BAA BAB BBA BBB AAA AAB ABB BAA BAB BBB AAA AAB ABB BAA BAB BBB 3 1 1 3 1 3 2 3 3 1 1 1

  24. E DGE I NDELS K ERNEL 1-edge indel u v A A A A A A A A A A A A A A A A B A A AAA AAB ABA ABB BAA BAB BBA BBB AAA AAB ABB BAA BAB BBB AAA AAB ABB BAA BAB BBB 3 1 1 3 1 3 2 3 3 2 1 1 2 1

  25. E DIT D ISTANCE K ERNELS # of edit distance operations Edit distance graphlet kernel Normalized edit distance kernel Lugo-Martinez J. and Radivojac P. Network Science, 2(2), 254-276 , (2014).

  26. Case Study: Structure-based functional residue prediction Joint work with Vikas Pejaver, Matthew Mort, David N. Cooper, Sean D. Mooney and Predrag Radivojac

  27. P REDICTION OF F UNCTIONAL S ITES FROM P ROTEIN S TRUCTURES Xin & Radivojac. Curr Prot Pept Sci 12: 456 (2011).

  28. R ESULTS : M ETAL B INDING R ESIDUES Iron (Fe) Copper (Cu)

  29. M ULTIPLE F UNCTIONAL R ESIDUE P REDICTORS AUC measured via per chain 10-fold cross-validation

  30. Q UICK D IGRESSION • Unprecedented growth of human genetic variant data – e.g. HGMD, dbSNP • In particular, amino acid substitutions (AAS) • Focus on tools that predict effects of AAS (deleterious vs neutral) – e.g. MutPred, SIFT, PolyPhen, SNPs3D, SNAP

  31. M OTIVATION : M OLECULAR M ECHANISMS OF D ISEASE E6V 4hhb 2hbs Sickle Cell Disease • Autosomal recessive disorder • E6V in HBB causes interaction w/ F85 and L88 http://gingi.uchicago.edu/hbs2.html • Formation of amyloid fibrils • Abnormally shaped red blood cells, leads to sickle cell anemia • Manifestation of disease vastly different over patients Pauling, L. et al. Science (1949) 110: 543-548; Chui, D.H. and Dover, G.J. Curr Opin Pediatr (2001) 13: 22-27.

  32. I NFERRING M OLECULAR M ECHANISMS OF D ISEASE • Most of these tools do not predict biochemical cause of disease – In particular, molecular function alterations • Lack of comprehensive studies using protein 3D structure data Goal: Exploit the structural environment of a residue of interest to hypothesize specific molecular effects of AAS and to statistically attribute these effects to genetic disease Idea: • Develop methods to predict specific function • e.g. zinc-binding site or phosphorylation site • Apply to amino acid substitution data • Provide probabilistic estimates of molecular mechanisms of disease

  33. A PPROACH Consider: phosphorylation in structure 𝑡 occurs at position 𝑗 • residue 𝑦 is mutated to 𝑧 , at position 𝑘 ( 𝑦𝑘𝑧 ) • phosphorylation site variant position (C46W) i = 4 5 j = 46 s: … LAGDKMGMGQSCVGALFNDVQ … Loss of phosphorylation: Gain of phosphorylation: Radivojac et al. Bioinformatics 24: i241 (2008). Image from Capriotti and Altman. BMC Bioinformatics 12 (Suppl4): S3 (2011).

  34. I DENTIFYING A CTIVE M ECHANISMS OF D ISEASE Density fpr cutoff Probability of loss of property Data set Total # of AAS # of AAS # of genes # of PDB entries # of chains mapped to PDB Neutral 282,625 8,049 2,095 3,047 3,500 Disease 52,406 10,629 583 1,177 1,387

  35. L OSS AND G AIN OF F UNCTIONAL S ITES IS AN A CTIVE M ECHANISM OF D ISEASE

  36. V ALIDATION OF L OSS OF F UNCTION P REDICTIONS • Mutagenesis experimental data (UniProt) – 3,356 amino acid substitutions mapped to PDB (880 distinct proteins) • Feasibility of computationally predicting loss of functional sites

  37. L OSS OF Z INC B INDING C AUSES D ISEASE D83G Amyotrophic lateral sclerosis (ALS) • D83G in superoxide dismutase (SOD1) causes:  Loss of zinc-binding that destabilizes native structure  Leads to protein aggregation that forms amyloid-like fibrils Joyce, P.I. et al. Hum. Mol. Genet . (2014); Seetharaman, S.V. et al. Arch. Biochem. Biophys . (2010); Krishnan, U. et al. Mol Cell Biochem (2006)

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