Discovering Patterns in Haplotype Data

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Discovering Patterns in Haplotype Data

• Lipari, July 24, 2003
• Esko Ukkonen
• University of Helsinki
• Esko.Ukkonen_at_Helsinki.Fi

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only recombinations mutations not shown
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Haplotype blocks

• Seminal paper by Daly et al. (Nat Gen, 2001)
• Data 500 kb region, 103 SNPs, 258 chromosomes
  (haplotypes)
• Finds blocks with striking lack of variation
• Recombination hot spots? (physical explanation)
• or just population history? (by chance)
• Other papers
• Patil et al. (Science 2001) Gabriel et al.
  (Science 2002) Zhang et al (PNAS 2002)

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Blocks by Daly et al. (Nat Gen 2001)
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Blocks by Daly et al. (Nat Gen 2001)
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Figure (Patil al)
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Analysis results

• Seminal paper by Daly et al. (Nat Gen, 2001),
  data
• 500 kb region,
• 103 SNPs,
• 258 chromosomes (haplotypes)

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Best segmentation
Boundary strength
Segmentations with gaps
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Best segmentation
Boundary strength
Gaps allowed
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Gaps allowed
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Generalizations

• microsatellite markers
• generalization to unphased data
• distances between markers
• population mixtures / comparing several block
  structures

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Global blocks vs mosaic

• blocks convenient for describing potential
  cross-over points
• not every haplotype must have a cross-over at
  every block boundary
• the true underlying fragmentation more like a
  mosaic than simple blocks
• How to find the best mosaic?

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Other view set cover

• no a priori assumption of a block structure
• find all consistent submatrices of D
• consistent submatrix consists of (almost)
  identical (sub)haplotypes (hence, it can come
  from the same founder)
• find an optimal cover of D from the submatrices

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Each fragment of a haplotype induces a submatrix
consisting of all (almost) identical fragments
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• Generalized mosaic model
• Leave some fraction of D uncovered
• Allow some noise inside each submatrix

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Greedy set cover heuristics

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Daly et al data analyzed by the greedy set cover
algorithm
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Uncovering founder sequences

• Problem Given current sequences (haplotypes),
  construct their founders that could produce the
  sequences by recombinations
• recombination hot spots?
• visualizations of potential recombination
  structures How do the blocks look like?

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Example
0 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0
0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1
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Example
0 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0
0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1
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Example
0 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0
0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 0
0 1 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 0 1 1 0
6 cross-overs
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Example
0 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0
0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1
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Example
0 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0
0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
18 cross-overs
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Example founders recombinants (generated data)
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Founder sequence reconstruction

• Set of recombinants D D1,...,Dn, sequences of
  length p over some alphabet (0,1,
  A,C,G,T,...)
• Find founder sequences F F1,...,Fk of length
  p such that D has parse in terms of F each Di is
  a catenation of fragments of Fjs
  
  

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Founder sequence reconstruction

• Set of recombinants D D1,...,Dn, sequences of
  length p over some alphabet (0,1,
  A,C,G,T,...)
• Find founder sequences F F1,...,Fk of length
  p such that D has parse in terms of F each Di is
  a catenation of fragments of Fjs
  Di Fj

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Example founders recombinants (generated data)
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Colorings, cross-overs, fragments

• find a consistent coloring different symbols on
  the same column of D must have different color
• colors founders
• cross-over two different colors adjacent on the
  same Di
• fragment block of Di with the same color

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Optimization

• minimize number of colors ( founders)
• minimize number of fragments or number of
  cross-overs ( maximize the length of fragments)
• given an upper bound M for colors, maximize
  fragment length
• given a lower bound L for (average) fragment
  length, minimize colors

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Block driven approach

• D
  
• find the vertical segments ( haplotype blocks)
  first, then minimize colors or cross-overs

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Color propagation

• D

Y
X

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Color propagation

• D

Y
X
If we select color(Y) color(X) then w(X,Y) X
? Y crossovers are eliminated
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Color propagation

• D

Y
X
If we select color(Y) color(X) then w(X,Y) X
? Y crossovers are eliminated
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P
P
W(X,Y)
X
Y
w(X,Y) common rows of X and Y W(P,P)
maximum weight of a maching U(P,P) n W(P,P)
minimum number of cross-overs
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Color propagation (cont.)

• weighted bipartite graph whose maximum weight
  matching gives minimum number of crossovers at
  this boundary
• Hungarian algorithm O(M3)
• gt optimal coloring for given segmentation if
  crossovers only on segment boundaries and all M
  colors non-empty
• coloring time O(p(nM3))

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General case

• all segmentations, all consistent colorings
• color propagation as above
• dyn prog bipartite matching

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Dynamic programming
P, P partitions of columns j-1 and j U(P,P)
minimum possible number of cross-overs when P and
P are given colors S(j,P) smallest possible
number of cross-overs in M-colorings of D(1,j)
that have coloring-induced partition P at column j
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Dynamic programming (cont.)
S(1,P) 0 S(j,P) min S(j-1,P)U(P,P)
where the minimum is over all consistent
partitions of column j-1
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General case (cont.)

• maximize average fragment length when M colors
  available
• O(pN) where N different M partitions of n
  elements
• Slow, only small M feasible

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Optimal result for M4
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• Data Daly et al
• full cover
• Hamming 0

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• Data Daly et al
• full cover
• Hamming 1

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• Data Daly et al
• 85 cover
• - Hamming 0

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• Data Daly et al
• 85 cover
• - Hamming 1

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Implementations

• MDL block finder (M. Koivisto)
  www.cs.helsinki.fi/u/mkhkoivi/software.html
• HaploVisual (P. Rastas) www.cs.helsinki.fi/u/pras
  tas/haplovisual/

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On the MDL

• M. Hansen B. Yu Model selection and the
  principle of minimum description length. J Am
  Stat Assoc 96 (2001), 746-774.

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Papers

• M. Koivisto, M. Perola, T. Varilo, W. Hennah, J.
  Ekelund, M. Lukk, L. Peltonen, E. Ukkonen, H.
  Mannila An MDL method for finding haplotype
  blocks and for estimating the strength of
  haplotype block boundaries. Pacific Symposium on
  Biocomputing 2003.
• H. Mannila et al. Minimum description length
  block finder... . AJHG 73 (2003).
• E. Ukkonen Finding founder sequences from a set
  of recombinants. Proc. Workshop on Algorithms for
  Bioinformatics WABI-2002. LNCS, Springer 2002.
• K. Zhang et al Dynamic programming algorits for
  haplotype block ... . RECOMB 2003.