1. Evaluating the Significance of Max-gap Clusters Rose Hoberman David Sankoff Dannie Durand

2. Gene content and order are preserved rearrangement, mutation Similarity in gene content large scale duplication or speciation event original genome

3. TBOX Cryb, Lhx, Nos, Tbx, Tcf, Prkar MHC Abc, C3/4/5, Col, Hsp, Notch, Pbx, Psmb, Rxr, Ten HOX Achr, Ccnd, Cdc, Cdk, Dlx, En, Evx, Gli, Hh, Hox, If, Inhb, Nhr, Npy/Ppy, Wnt FGR Adr, Ank, Egr, Fgfr, Pa, Vmat, Lpl MATN Eya, Hck, Matn, Myb, Myc, Sdc, Src … … Local Spatial Evidence of Duplication

4. Gene Clustering for Functional Inference in Bacterial Genomes The Use of Gene Clusters to Infer Functional Coupling, Overbeek et al., PNAS 96: 2896-2901, 1999.

5. Clusters are Commonly Used in Genomic Analyses Genomic Analyses Blanc et al 2003, recent polyploidy in Arabidopsis Venter et al 2001, sequence of the human genome Overbeek et al 1999, inferring functional coupling of genes in bacteria Vandepoele et al 2002, duplications in Arabidopsis through comparison with rice Vision et al 2000, duplications in Eukaryotes Lawrence and Roth 1996, identification of horizontal transfers Tamames 2001, evolution of gene order conservaiton in prokaryotes Wolfe and Shields 1997, ancient yeast duplication McLysaght02, genomic duplication during early chordate evolution Coghlan and Wolfe 2002, comparing rates of rearrangements Seoighe and Wolfe 1998, genome rearrangements after duplication in yeast Chen et al 2004, operon prediction in newly sequenced bacteria Blanchette et al 1999, breakpoints as phylogenetic features...

6. Clusters are Commonly Used in Genomic Analyses Genomic Analyses Blanc et al 2003, recent polyploidy in Arabidopsis Venter et al 2001, sequence of the human genome Overbeek et al 1999, inferring functional coupling of genes in bacteria Vandepoele et al 2002, duplications in Arabidopsis through comparison with rice Vision et al 2000, duplications in Eukaryotes Lawrence and Roth 1996, identification of horizontal transfers Tamames 2001, evolution of gene order conservaiton in prokaryotes Wolfe and Shields 1997, ancient yeast duplication McLysaght02, genomic duplication during early chordate evolution Coghlan and Wolfe 2002, comparing rates of rearrangements Seoighe and Wolfe 1998, genome rearrangements after duplication in yeast Chen et al 2004, operon prediction in newly sequenced bacteria Blanchette et al 1999, breakpoints as phylogenetic features... Need to assess cluster significance

7. Given: a genome: G = 1, …, n unique genes a set of m special genes Can we find a significant cluster of (a subset of) the m homologs?

8. Can we find a significant cluster of the red genes ? How do we formally define a cluster? Given: a genome: G = 1, …, n unique genes a set of m special genes

9. Possible Cluster Parameters –size: number of red genes in the cluster Example: cluster size ≥ 3 size = 3 genes

10. Possible Cluster Parameters –size: number of red genes in the cluster Example: cluster size ≥ 3 –length: number of genes between first and last red gene Example: cluster length ≤ 6 length = 6

11. Possible Cluster Parameters –size: number of red genes in the cluster Example: cluster size ≥ 3 –length: number of genes between first and last red gene Example: cluster length ≤ 6 length = 6

12. Possible Cluster Parameters –size: number of red genes in the cluster Example: cluster size ≥ 3 –length: number of genes between first and last red genes Example: cluster length ≤ 6 –density: proportion of red genes (size/length) Example: density ≥ 0.5 density = 6/11

13. Possible Cluster Parameters –size: number of red genes in the cluster Example: cluster size ≥ 3 –length: number of genes between first and last red genes Example: cluster length ≤ 6 –density: proportion of red genes (size/length) Example: density ≥ 0.5 density = 6/11

14. Possible Cluster Parameters –size: number of red genes in the cluster Example: cluster size ≥ 3 –length: number of genes between first and last red genes Example: cluster length ≤ 6 –density: proportion of red genes (size/length) –compactness: maximum gap between adjacent red genes gap ≤ 4 genes

15. Max-Gap Cluster Commonly used in genomic analyses Expandable, ensures minimum local and global density Efficient algorithm to find them (Bergeron, Corteel, Raffinot 2002) gap  g

16. Max-Gap Clusters are Commonly Used in Genomic Analyses Genomic Analyses Blanc et al 2003, recent polyploidy in Arabidopsis Venter et al 2001, sequence of the human genome Overbeek et al 1999, inferring functional coupling of genes in bacteria Vandepoele et al 2002, duplications in Arabidopsis through comparison with rice Vision et al 2000, duplications in Eukaryotes Lawrence and Roth 1996, identification of horizontal transfers Tamames 2001, evolution of gene order conservation in prokaryotes Wolfe and Shields 1997, ancient yeast duplication McLysaght02, genomic duplication during early chordate evolution Coghlan and Wolfe 2002, comparing rates of rearrangements Seoighe and Wolfe 1998, genome rearrangements after duplication in yeast Chen et al 2004, operon prediction in newly sequenced bacteria Blanchette et al 1999, breakpoints as phylogenetic features...

17. Statistical models Statistical Models Calabrese03 Danchin03 Durand03 Ehrlich97 Trachtulec01 HumanGenomeScience

18. A gap between statistical tests and cluster definitions used in practice Genomic Analyses Blanc et al 2003 Venter et al 2001 Overbeek et al 1999 Vandepoele et al 2002 Vision et al 2000 Lawrence and Roth 1996 Tamames 2001 Wolfe and Shields 1997 McLysaght02 Coghlan and Wolfe 2002 Seoighe and Wolfe 1998 Chen et al 2004 Blanchette et al 1999... Statistical Models Calabrese03 Danchin03 Durand03b Ehrlich97 Trachtulec01 HumanGenomeScience no analytical statistical model for max-gap clusters statistical significance assessed through Monte Carlo randomization

19. A gap between statistical tests and cluster definitions used in practice Genomic Analyses Blanc et al 2003 Venter et al 2001 Overbeek et al 1999 Vandepoele et al 2002 Vision et al 2000 Lawrence and Roth 1996 Tamames 2001 Wolfe and Shields 1997 McLysaght02 Coghlan and Wolfe 2002 Seoighe and Wolfe 1998 Chen et al 2004 Blanchette et al 1999... Statistical Models Calabrese03 Danchin03 Durand03b Ehrlich97 Trachtulec01 HumanGenomeScience

20. This Talk Analytical statistical tests –reference region model with m genes of interest –complete and incomplete clusters Relationship between cluster parameters and significance Future extension to whole-genome comparison

21. Outline Introduction Complete Clusters Incomplete Clusters Genome Comparison Open problems

22. Complete Clusters Given –a genome: G = 1, …, n unique genes –a set of m special genes (the “red” genes) –a maximum-gap size g Null hypothesis: Random gene order Alternate hypotheses: Evolutionary history Functional selection Test statistic –the probability of observing all m genes in a max-gap cluster in G ?

23. Probability of a Complete Cluster Count how many of the permutations of m red genes and n-m blue genes contain a max-gap cluster 1.At how many positions in the genome can we locate a cluster (e.g. place the leftmost red gene)? 2.Given the location of the first red gene, how many ways are there to place the remaining m-1 red genes so that they form a max-gap cluster?

24. w = (m-1)g + m ways to choose m-1 gaps ways to place the first gene and still have w-1 slots left edge effects

25. w-1 For clusters at the end of the genome: –Length of the cluster is constrained –Sum of the gaps is constrained: Counting clusters at the end of the genome

26. How many ways can we form a max-gap cluster of size m with length ≤ l ? g1g1 g2g2 g3g3 g m-1 l < w

27. g1g1 g2g2 g3g3 g m-1 l < w = Number of ways of choosing m-1 gaps between 0 and g so sum ≤ l-m Number of ways of rolling m-1 dice with faces labeled 0 to g so faces total ≤ l-m How many ways can we form a max-gap cluster of size m with length ≤ l ?

28. g1g1 g2g2 g3g3 g m-1 = Number of ways of choosing m-1 gaps between 0 and g so sum ≤ l-m Number of ways of rolling m-1 dice with faces labeled 0 to g so faces total ≤ l-m How many ways can we form a max-gap cluster of size m with length ≤ l ?

29. Accounting for edge effects is the least efficient part of calculation –Eliminate for faster approximation when w << n Now we can calculate probabilities for various parameter values Final Probability

30. Probability of Observing a Complete Cluster g m n = 500

31. Significant Parameter Values (α = 0.0001) n = 500

32. Significant Parameter Values (α = 0.0001) n = 500

33. Outline Motivation Complete Clusters Incomplete Clusters Genome Comparison Open problems

34. Incomplete clusters Is it significant to find a max-gap cluster of size h < m ? Test statistic: probability of finding at least one cluster Given: m genes of interest h=3 m=5, g=1

35. Enumerating clusters by starting position will lead to overcounting of permutations that include more than one cluster Probability of incomplete gene clusters m = 6, h = 3, g = 1

36. Alternative Approach Enumerate all permutations that do not contain any clusters of size h or larger Dynamic programming –Iteratively place “red” or “blue” genes making sure not to create any cluster of size h or larger by judicious placement of “blue genes”

37. k = 4 c = 0, j = 8 r = 6 Dynamic programming Algorithm r total number of genes to place c size of cluster so far k number of special genes j distance to previous cluster n = 6, m = 4, h = 3, g=1

38. k = 4 c = 0, j = 8 r = 6 c = 1, j = 0 c = 0, j = 8 k = 4 r = 5 k = 3 r = 5 Dynamic programming Algorithm r total number of genes to place c size of cluster so far k number of special genes j distance to previous cluster n = 6, m = 4, h = 3, g=1

39. k = 2 r = 4 k = 4 c = 0, j = 8 r = 6 c = 1, j = 0 c = 0, j = 8 k = 4 r = 5 k = 3 r = 5 c = 1, j = 1 k = 3 r = 4 c = 1, j = 0 Dynamic programming Algorithm r total number of genes to place c size of cluster so far k number of special genes j distance to previous cluster n = 6, m = 4, h = 3, g=1

40. k = 2 r = 4 k = 4 c = 0, j = 8 r = 6 c = 1, j = 0 c = 0, j = 8 k = 4 r = 5 k = 3 r = 5 c = 1, j = 1 k = 3 r = 4 c = 1, j = 0 k = 2 r = 3 c = 2, j = 0 Dynamic programming Algorithm r total number of genes to place c size of cluster so far k number of special genes j distance to previous cluster X n = 6, m = 4, h = 3, g=1

41. Incomplete cluster significance  h  g m ( ) significant region of parameter space shown in paper n = 1000 m = 50 n = 500

42. Outline Motivation Complete Clusters Incomplete Clusters Extensions Genome Comparison

43. Possible Extensions Tandem duplications Gene families Extensions for prokaryotic genomes –Gene orientation –Circular genomes –Physical distance between genes Whole genome comparison

44. Assuming identical gene content … What is the probability that at least k genes form a max-gap cluster in both genomes? g  30

45. The probability of finding a max-gap cluster of size at least k is always one An Odd Property Example: g =1

46. The probability of finding a max-gap cluster of size at least k is always one Example: g =1 An Odd Property

47. The probability of finding a max-gap cluster of size at least k is always one Example: g =1 gap An Odd Property A cluster of size k does not necessarily contain a cluster of size k-1! gap

48. The probability of finding a max-gap cluster of size at least k is always one There will always be a cluster of size n Example: g =1 An Odd Property

49. Conclusions Presented statistical tests for max-gap clusters –Evaluate the significance of clusters of a pre- specified set of genes –Choose parameters effectively –Understand trends

50. Thank You