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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
• 1994: Gusfield, D., Balasubramanian, K. & Naor, D. • Parametric optimization of sequence alignment, Algorithmica 12, 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
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.
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---------AAAGAAGATTTCTCATCATTGGTTGCTGCGGGATTAGGGGTCATTAGAGT---------GCCGAAAAGCGA---------GTTTCTGGAATAGTTAATTTCATTGTAACACATAAACGTTTTAAATTCTATTGAAA-------CTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGAGGAGAGGCCATCATCG----CTGCGGGATTAGGAGTCATTAGAGT---------GCGGAAAAGCGG---------GTT-CTGCAGCAGTTAAATA-ATTGTAATAAACAATTCTCT--AATTTGGTCCAAA-------CTGCGGGATTAGCGGTCATTGGTGT---------GAAGAATAGATC---------CTTT
analysis
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---------AAAGAAGATTTCTCATCATTGGTTGCTGCGGGATTAGGGGTCATTAGAGT---------GCCGAAAAGCGA---------GTTTCTGGAATAGTTAATTTCATTGTAACACATAAACGTTTTAAATTCTATTGAAA-------CTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGAGGAGAGGCCATCATCG----CTGCGGGATTAGGAGTCATTAGAGT---------GCGGAAAAGCGG---------GTT-CTGCAGCAGTTAAATA-ATTGTAATAAACAATTCTCT--AATTTGGTCCAAA-------CTGCGGGATTAGCGGTCATTGGTGT---------GAAGAATAGATC---------CTTT
analysis
computational geometry
A Whole Genome Parametric Alignment ofD. 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.
+ =
A Whole Genome Parametric Alignment ofD. 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.
>melCTGCGGGATTAGGGGTCATTAGAGTGCCGAAAAGCGAGTTTATTCTATGGAC>pseCTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGAGGAGAGGCCATCATCGTGTAC
How do we build the polytope for ?
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,S Jukes-Cantor with linear gap penaltyO(n+m)2/3
M,X,S,G Jukes-Cantor with affine gap penaltyO(n+m)3/2
M,XTS,XTV,S,G K2P 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.
The algebraic statistical model for sequence alignment, known as the pair hidden Markov model, is the image of the map
The logarithms of the parameters give the edge lengths for the shortest path problem on the alignment graph.
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:
14
NPi,j = S*NPi-1,j+S*NPi,j-1+(X or M)*NPi-1,j-1
A
A
C
A
T
T
A
G
AA G A T T A C C A C A
Newton polytope forpositions [1,i] and [1,j]
in each sequence
Convex hull of union Minkowski sum
Polytope propagation
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
Drosophila DNase I Footprint Database (v2.0)
Home Search Browse by Target Browse by Factor
Target GeneChromosom
e ArmStart Stop
Transcription Factor
Pubmed ID (PMID)
Footprint ID (FPID)
Footprint Alignment
ems (CG2988) 3R 9723806 9723816Abd
-B (CG11648)9491376 003205
Abd-B->ems:003205
ems (CG2988) 3R 9723843 9723853Abd
-B (CG11648)9491376 003206
Abd-B->ems:003206
ems (CG2988) 3R 9723998 9724008Abd
-B (CG11648)9491376 003208
Abd-B->ems:003208
ems (CG2988) 3R 9724091 9724102Abd
-B (CG11648)9491376 003209
Abd-B->ems:003209
ems (CG2988) 3R 9724526 9724536Abd
-B (CG11648)9491376 003211
Abd-B->ems:003211
ems (CG2988) 3R 9724557 9724567Abd
-B (CG11648)9491376 003213
Abd-B->ems:003213
ems (CG2988) 3R 9724614 9724624Abd
-B (CG11648)9491376 003214
Abd-B->ems:003214
dpp (CG9885) 2L 2454657 2454685 Adf1 (CG15845) 7791801 003665 Adf1->dpp:003665
Adh (CG3481) 2L 14615472 14615509 Adf1 (CG15845) 2105454 005046 Adf1->Adh:005046
Ddc (CG10697) 2L 19116303 19116321 Adf1 (CG15845) 2318884 005464 Adf1->Ddc:005464
Antp (CG1028) 3R 2825018 2825059 Adf1 (CG15845) 2318884 006446 Adf1->Antp:006446
Adh (CG3481) 2L 14616171 14616209 Adf1 (CG15845) 2105454 005059 Adf1->Adh:005059
Antp (CG1028) 3R 2825117 2825144 Adf1 (CG15845) 2318884 006447 Adf1->Antp:006447
Antp (CG1028) 3R 2825151 2825174 Adf1 (CG15845) 2318884 006448 Adf1->Antp:006448
Back to Adf1
mel TGTGCGTCAGCGTCGGCCGCAACAGCG pse TGT-----------------GACTGCG *** ** ***
BLASTZ alignment
Back to Adf1
mel TGTGCGTCAGCGTCGGCCGCAACAGCG pse TGT-----------------GACTGCG *** ** ***
mel TGTG----CGTCAGC--G----TCGGCC---GC-AACAG-CG Pse TGTGACTGCG-CTGCCTGGTCCTCGGCCACAGCCAAC-GTCG **** ** * ** * ****** ** *** * **
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- **** * ** *** * ** *****
Drosophila DNase I Footprint Database (v2.0)
Home Search Browse by Target Browse by Factor
Target GeneChromosom
e ArmStart Stop
Transcription Factor
Pubmed ID (PMID)
Footprint ID (FPID)
Footprint Alignment
ems (CG2988) 3R 9723806 9723816Abd-B
(CG11648)9491376 003205
Abd-B->ems:003205
ems (CG2988) 3R 9723843 9723853Abd-B
(CG11648)9491376 003206
Abd-B->ems:003206
ems (CG2988) 3R 9723998 9724008Abd
-B (CG11648)9491376 003208
Abd-B->ems:003208
ems (CG2988) 3R 9724091 9724102Abd
-B (CG11648)9491376 003209
Abd-B->ems:003209
ems (CG2988) 3R 9724526 9724536Abd
-B (CG11648)9491376 003211
Abd-B->ems:003211
ems (CG2988) 3R 9724557 9724567Abd
-B (CG11648)9491376 003213
Abd-B->ems:003213
ems (CG2988) 3R 9724614 9724624Abd
-B (CG11648)9491376 003214
Abd-B->ems:003214
dpp (CG9885) 2L 2454657 2454685 Adf1 (CG15845) 7791801 003665 Adf1->dpp:003665
Adh (CG3481) 2L 14615472 14615509 Adf1 (CG15845) 2105454 005046 Adf1->Adh:005046
Ddc (CG10697) 2L 19116303 19116321 Adf1 (CG15845) 2318884 005464 Adf1->Ddc:005464
Antp (CG1028) 3R 2825018 2825059 Adf1 (CG15845) 2318884 006446 Adf1->Antp:006446
Adh (CG3481) 2L 14616171 14616209 Adf1 (CG15845) 2105454 005059 Adf1->Adh:005059
Antp (CG1028) 3R 2825117 2825144 Adf1 (CG15845) 2318884 006447 Adf1->Antp:006447
Antp (CG1028) 3R 2825151 2825174 Adf1 (CG15845) 2318884 006448 Adf1->Antp:006448
Per site analysis Group 1 mean per site % identity 51.3% 51.3% 47.8%
Group 2 mean per site % identity 47.8% 42.9% 42.9%
Difference of means (group 1 – group 2) 3.6% 8.4% 4.9%
Difference of means resampling p-value 0.05 0.003 1E-5
Distribution comparison KS p-value 0.026 0.0016 2E-6
Per base analysis Group 1 mean per base % identity 47.8% 47.8% 46.3%
Group 2 mean per base % identity 46.3% 42.4% 42.4%
Difference of means (group 1 – group 2) 1.5% 5.4% 3.9%
Per site analysis Group 1 mean per site % identity 51.3% 51.3% 47.8%
Group 2 mean per site % identity 47.8% 42.9% 42.9%
Difference of means (group 1 – group 2) 3.6% 8.4% 4.9%
Difference of means resampling p-value 0.05 0.003 1E-5
Distribution comparison KS p-value 0.026 0.0016 2E-6
Per base analysis Group 1 mean per base % identity 47.8% 47.8% 46.3%
Group 2 mean per base % identity 46.3% 42.4% 42.4%
Difference of means (group 1 – group 2) 1.5% 5.4% 3.9%
80.4%
Per site analysis Group 1 mean per site % identity 51.3% 51.3% 47.8%
Group 2 mean per site % identity 47.8% 42.9% 42.9%
Difference of means (group 1 – group 2) 3.6% 8.4% 4.9%
Difference of means resampling p-value 0.05 0.003 1E-5
Distribution comparison KS p-value 0.026 0.0016 2E-6
Per base analysis Group 1 mean per base % identity 47.8% 47.8% 46.3%
Group 2 mean per base % identity 46.3% 42.4% 42.4%
Difference of means (group 1 – group 2) 1.5% 5.4% 3.9%
85.1%
Per site analysis Group 1 mean per site % identity 51.3% 51.3% 47.8%
Group 2 mean per site % identity 47.8% 42.9% 42.9%
Difference of means (group 1 – group 2) 3.6% 8.4% 4.9%
Difference of means resampling p-value 0.05 0.003 1E-5
Distribution comparison KS p-value 0.026 0.0016 2E-6
Per base analysis Group 1 mean per base % identity 47.8% 47.8% 46.3%
Group 2 mean per base % identity 46.3% 42.4% 42.4%
Difference of means (group 1 – group 2) 1.5% 5.4% 3.9%
86.5%
Per site analysis Group 1 mean per site % identity 51.3% 51.3% 47.8%
Group 2 mean per site % identity 47.8% 42.9% 42.9%
Difference of means (group 1 – group 2) 3.6% 8.4% 4.9%
Difference of means resampling p-value 0.05 0.003 1E-5
Distribution comparison KS p-value 0.026 0.0016 2E-6
Per base analysis Group 1 mean per base % identity 47.8% 47.8% 46.3%
Group 2 mean per base % identity 46.3% 42.4% 42.4%
Difference of means (group 1 – group 2) 1.5% 5.4% 3.9%
79.1%
Applications• Conservation of cis-regulatory elements
• Phylogenetics: branch length estimation
This is the expected number of mutations per site in an alignment with summary (x,s).
Jukes-Cantor correction:
Applications• Conservation of cis-regulatory elements
• Phylogenetics: branch length estimation
