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Annotation and Alignment of the Drosophila Genomes

Annotation and Alignment of the Drosophila Genomes One (possibly wrong) alignment is not enough: the history of parametric inference 1992: Waterman, M., Eggert, M. & Lander, E. Parametric sequence comparisons, Proc. Natl. Acad. Sci. USA 89, 6090-6093

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Annotation and Alignment of the Drosophila Genomes

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  1. Annotation and Alignment of the Drosophila Genomes

  2. One (possibly wrong) alignment is not enough: the history of parametric inference • 1992: Waterman, M., Eggert, M. & Lander, E. • Parametric sequence comparisons, Proc. Natl. Acad. Sci. USA89, 6090-6093 • 1994: Gusfield, D., Balasubramanian, K. & Naor, D. • Parametric optimization of sequence alignment, Algorithmica12, 312-326. • 2003: Wang, L., Zhao, J. • Parametric alignment of ordered trees, Bioinformatics, 19 2237-2245. • 2004: Fernández-Baca, D., Seppäläinen, T. & Slutzki, G. • Parametric Multiple Sequence Alignment and Phylogeny Construction, Journal of Discrete Algorithms, 2 271-287. XPARAL by Kristian Stevens and Dan Gusfield

  3. Whole Genome Parametric AlignmentColin Dewey, Peter Huggins, Lior Pachter, Bernd Sturmfels and Kevin Woods • Mathematics and Computer Science • Parametric alignment in higher dimensions. • Faster new algorithms. • Deeper understanding of alignment polytopes. • Biology • Whole genome parametric alignment. • Biological implications of alignment parameters. • Alignment with biology rather than for biology.

  4. Whole Genome Parametric AlignmentColin Dewey, Peter Huggins, Lior Pachter, Bernd Sturmfels and Kevin Woods • Mathematics and Computer Science • Parametric alignment in higher dimensions. • Faster new algorithms. • Deeper understanding of alignment polytopes. • Biology • Whole genome parametric alignment. • Biological implications of alignment parameters. CTGAAGGAAT-------TCTATATT---------AAAGAAGATTTCTCATCATTGGTTG CTGCGGGATTAGGGGTCATTAGAGT---------GCCGAAAAGCGA---------GTTT CTGGAATAGTTAATTTCATTGTAACACATAAACGTTTTAAATTCTATTGAAA------- CTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGAGGAGAGGCCATCATCG---- CTGCGGGATTAGGAGTCATTAGAGT---------GCGGAAAAGCGG---------GTT- CTGCAGCAGTTAAATA-ATTGTAATAAACAATTCTCT--AATTTGGTCCAAA------- CTGCGGGATTAGCGGTCATTGGTGT---------GAAGAATAGATC---------CTTT analysis

  5. Whole Genome Parametric AlignmentColin Dewey, Peter Huggins, Lior Pachter, Bernd Sturmfels and Kevin Woods • Mathematics and Computer Science • Parametric alignment in higher dimensions. • Faster new algorithms. • Deeper understanding of alignment polytopes. • Biology • Whole genome parametric alignment. • Biological implications of alignment parameters. CTGAAGGAAT-------TCTATATT---------AAAGAAGATTTCTCATCATTGGTTG CTGCGGGATTAGGGGTCATTAGAGT---------GCCGAAAAGCGA---------GTTT CTGGAATAGTTAATTTCATTGTAACACATAAACGTTTTAAATTCTATTGAAA------- CTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGAGGAGAGGCCATCATCG---- CTGCGGGATTAGGAGTCATTAGAGT---------GCGGAAAAGCGG---------GTT- CTGCAGCAGTTAAATA-ATTGTAATAAACAATTCTCT--AATTTGGTCCAAA------- CTGCGGGATTAGCGGTCATTGGTGT---------GAAGAATAGATC---------CTTT analysis

  6. computational geometry

  7. = + A Whole Genome Parametric Alignment of D. Melanogaster and D. Pseudoobscura • Divided the genomes into 1,116,792 constrained and 877,982 unconstrained segment pairs. • 2d, 3d, 4d, and 5d alignment polytopes were constructed for each of the 877,802 unconstrained segment pairs. • Computed the Minkowski sum of the 877,802 2d polytopes.

  8. A Whole Genome Parametric Alignment of D. Melanogaster and D. Pseudoobscura • Divided the genomes into 1,116,792 constrained and 877,982 unconstrained segment pairs. • This is an orthology map of the two genomes. • 2d, 3d, 4d, and 5d alignment polytopes were constructed for each of the 877,802 unconstrained segment pairs. • For each segment pair, obtain all possible optimal summaries for all parameters in a Needleman--Wunsch scoring scheme. • Computed the Minkowski sum of the 877,802 2d polytopes. • There are only 838 optimal alignments of the two Drosophila genomes if the same match, mismatch and gap parameters are used for all the segment pair alignments.

  9. >mel CTGCGGGATTAGGGGTCATTAGAGTGCCGA AAAGCGAGTTTATTCTATGGAC >pse CTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGA GGAGAGGCCATCATCGTGTAC ? How do we build the polytope for

  10. Alignment polytopes are small Theorem: The number of vertices of an alignment polytope for two sequences of length n and m is O((n+m)d(d-1)/(d+1)) where d is the number of free parameters in the scoring scheme. Examples: Parameters Model Vertices M,X,SJukes-Cantor with linear gap penalty O(n+m)2/3 M,X,S,GJukes-Cantor with affine gap penalty O(n+m)3/2M,XTS,XTV,S,GK2P with affine gap penalty O(n+m)12/5 L. Pachter and B. Sturmfels, Parametric inference for biological sequence analysis, Proceedings of the National Academy of Sciences, Volume 101, Number 46 (2004), p 16138--16143. L. Pachter and B. Sturmfels, Tropical geometry of statistical models, Proceedings of the National Academy of Sciences, Volume 101, Number 46 (2004), p 16132--16137. L. Pachter and B. Sturmfels (eds.), Algebraic Statistics for Computational Biology, Cambridge University Press.

  11. The algebraic statisticalmodel for sequence alignment, known as the pair hidden Markov model, is the image of the map The logarithms of the parameters q give the edge lengths for the shortest path problem on the alignment graph.

  12. 14 Newton Polytope of a Polynomial Definition: The Newton polytope of a polynomial is defined to be the convex hull of the lattice points in Rd corresponding to monomials in f:

  13. Newton polytope for positions [1,i] and [1,j] in each sequence Minkowski sum Polytope propagation Convex hull of union NPi,j = S*NPi-1,j+S*NPi,j-1+(X or M)*NPi-1,j-1 A C A T T A G A A A G A T T A C C A C A

  14. Back to Adf1 BP England, U Heberlein, R Tjian. Purified Drosophila transcription factor, Adh distal factor-1 (Adf-1), binds to sites in several Drosophila promoters and activates transcription, J Biol Chem 1990.

  15. Back to Adf1 mel TGTGCGTCAGCGTCGGCCGCAACAGCG pse TGT-----------------GACTGCG *** ** *** BLASTZ alignment

  16. Back to Adf1 mel TGTGCGTCAGCGTCGGCCGCAACAGCG pse TGT-----------------GACTGCG *** ** *** mel TGTG----CGTCAGC--G----TCGGCC---GC-AACAG-CG Pse TGTGACTGCG-CTGCCTGGTCCTCGGCCACAGCCAAC-GTCG **** ** * ** * ****** ** *** * **

  17. Back to Adf1 mel TGTGCGTCAGCGTCGGCCGCAACAGCG pse TGT-----------------GACTGCG *** ** *** mel TGTG----CGTCAGC--G----TCGGCC---GC-AACAG-CG pse TGTGACTGCG-CTGCCTGGTCCTCGGCCACAGCCAAC-GTCG **** ** * ** * ****** ** *** * ** mel TGTGCGTCAGC------GTCGGCCGCAACAGCG pse TGTGACTGCGCTGCCTGGTCCTCGGCCACAGC- **** * ** *** * ** *****

  18. 80.4%

  19. 85.1%

  20. 86.5%

  21. 79.1%

  22. Applications • Conservation of cis-regulatory elements • Phylogenetics: branch length estimation Jukes-Cantor correction: This is the expected number of mutations per site in an alignment with summary (x,s).

  23. Applications • Conservation of cis-regulatory elements • Phylogenetics: branch length estimation

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