Parametric Alignment

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Parametric Alignment

• Colin Dewey and Lior Pachter

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Biological Sequence Alignment

• What are we really doing?
• Establishing homology relationships
• Given two biological strings ?1,?2, define
  homology relation Rhom on ?1 ? ?2

-CGATT GCC-TA
Homologous positions Nucleotides evolved from a
common ancestral nucleotide
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Alignment Parameters

• Global Alignment (Needleman-Wunsch)
• Special case of alignment pair HMM
• Parameters (combinations of emission and
  transition probabilities)
• Match (m)
• Mismatch (x)
• Gap Start (g)
• Gap Extend (e)

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Example Global Alignment
?1 GCCTA, ?2 CGATT log(match) 1,
log(mismatch) -1
C
G
A
T
T
G
C
C
T
A
of possible alignments D(5,5) 1,683
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Example Global Alignment
?1 GCCTA, ?2 CGATT log(match) 1,
log(mismatch) -1
C
G
A
T
T
-GCCTA CG-ATT -GCCTA CGA-TT -GCCTA CGATT- GC-CT
A -CGATT GCC-TA -CGATT
G
C
C
T
A
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Parametric Alignment

• An alignment corresponds to a monomial
• Represented by point

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Input/Output
Newton Polytope
Parametric Aligner
?1,?2
Normal Fan
Alignments
-GCCTA CG-ATT -GCCTA CGA-TT
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Example Polytope
VERTICES 1 1 4 1 2 4 1 0 6 1 2 6 1 0
10 FACETS -4 0 1 0 1 0 -6 2 1 10 -2 -1 2 -1 0
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Normal Fan
log(mismatch)
4
3
log(match)
2
1
0
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Constrained Normal Fan
log(mismatch) lt 0
log(mismatch)
log(mismatch) lt log(match)
log(match)
1
0
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Output
Number of polytope vertices 5 Inequalities for
cone 0 -1m 0x gt 0 1m -1x gt 0 match
-0.5 mismatch -1 gap -1 Number of optimal
alignments for cone 1 Inequalities for cone
1 1m 0x gt 0 0m -1x gt 0 match
0.5 mismatch -0.5 gap -0.5 Number of optimal
alignments for cone 5 Total number of optimal
alignments 6
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Alignments
Cone 0 GCCTA CGATT
Cone 1 -GCCTA CG-ATT -GCCTA CGA-TT -GCCTA CGATT
- GC-CTA -CGATT GCC-TA -CGATT
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Random vs. Related Sequences
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Polytope vertices for different models
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Theorem

• Theorem There exist two homologous biological
  sequences ?1,?2 such that no choice of parameters
  for the 2-parameter Needleman-Wunsch algorithm
  gives the biologically correct alignment

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Proof of Theorem
dr_3 AGTGATTT----TTGCATAACAGGTCTACTT---- hs_34
AGGCATCAGAAGTTGAGAGACAACTCTCCATGCAG
TACATGAC----------ATTTTCGAGAAAAAAA
TCCACGCCCTCAGAGAAGACTTTCGGGAGAAAAA
VERTICES 1 38 33 1 37 33 1 38 45 1 0 70 1 0 121
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Implementation
templatelttypename SemiRinggt void alignGlobalLastR
ow(const string seq1, const
string seq2, const typename
SemiRingElement match, const
typename SemiRingElement mismatch,
const typename SemiRingElement gap,
vectorlttypename SemiRingElementgt
row)
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Implementation
const Element one SemiRingmultiplicativeIdenti
ty // Initialize row row.resize(seq2.size()
1) row0 one // Calculate first row for
(size_t j 1 j lt seq2.size() j) rowj
gap rowj - 1 // Calculate remaining
rows Element up, diag for (size_t i 1 i lt
seq1.size() i) diag row0 row0
gap for (size_t j 1 j lt seq2.size()
j) up rowj if (seq1i -
1 seq2j - 1) rowj match
diag gap (up rowj - 1) else
rowj mismatch diag gap (up
rowj - 1) diag up
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Next Steps

• Bayesian Alignment
• Prior probabilities on parameters
• Find best alignment given priors
• More easily