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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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one possibly wrong alignment is not enough the history of parametric inference
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

slide3
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.
slide4
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

slide5
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

slide7

=

+

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.
slide8

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.
slide14

>mel

CTGCGGGATTAGGGGTCATTAGAGTGCCGA

AAAGCGAGTTTATTCTATGGAC

>pse

CTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGA

GGAGAGGCCATCATCGTGTAC

?

How do we build the polytope for

alignment polytopes are small
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.

slide16

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.

newton polytope of a polynomial

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:

slide18

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

slide19

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.

back to adf1
Back to Adf1

mel TGTGCGTCAGCGTCGGCCGCAACAGCG

pse TGT-----------------GACTGCG

*** ** ***

BLASTZ alignment

back to adf122
Back to Adf1

mel TGTGCGTCAGCGTCGGCCGCAACAGCG

pse TGT-----------------GACTGCG

*** ** ***

mel TGTG----CGTCAGC--G----TCGGCC---GC-AACAG-CG

Pse TGTGACTGCG-CTGCCTGGTCCTCGGCCACAGCCAAC-GTCG

**** ** * ** * ****** ** *** * **

back to adf123
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-

**** * ** *** * ** *****

applications
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).

applications31
Applications
  • Conservation of cis-regulatory elements
  • Phylogenetics: branch length estimation