Linear Reduction for Haplotype Inference

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Linear Reduction forHaplotype Inference

• Alex Zelikovsky
• joint work with Jingwu He

WABI 2004
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Outline

• SNP, haplotypes and genotypes
• Haplotype Inference
• Linear reduction method
• Improvements
• Experimental results
• Conclusions future work

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Human Genome and SNP

• Length of Human Genome ? 3 ? 109 base pairs
• Difference between any two people ? 0.1 of
  genome
• ? 3 ? 106 base pairs
• Total number of single nucleotide polymorphisms
  (SNP)
• ? 1 ? 107 base pairs
• SNPs are mostly bi-allelic, e.g.,
• two variants (alleles) out of 4 possible
  (A,C,T,G) A/C
• having a nucleotide in a certain position or
  missing it A/-
• Major allele more frequent allele wild type
  vs SNP
• Minor allele (snip) frequency should be
  biologically considerable, e.g., over 1
• There are more less frequent SNP

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Haplotype and Disease Association

• Deafness inheritance ? moral problems
• SNP contribute to risk factors of complex
  diseases
• having certain SNP increases 10 times chances of
  having diabetes
• but association is too fragile for doctors 3 ?
  10-6 ? 30 ? 10-6
• combinations of SNPs haplotypes are
  responsible for diseases
• International HapMap project http//www.hapmap.or
  g
• SNP maps are constructed across the human genome
  with density of about one SNP per thousand
  nucleotides.
• HapMap tries to identify 1 million tag SNPs
  providing almost as much mapping information as
  entire 10 million SNPs
• Unfortunately, not as much known about SNP
  combinations

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Haplotypes and Genotypes

• Diploid organisms two different copies of
  each chromosome recombined copies of parents
  chromosomes
• Too expensive to examine two versions of a
  chromosome separately
• Much cheaper to obtain genotype (mixed) data
  rather than haplotype (separated) data
• Haplotype description of single copy (0wild
  type,1minor allele)
• Genotype description of mixed two copies
  (000, 111, 201)

WABI 2004
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Haplotype Inference Problem

• Haplotype Inference (HI) Problem
• Given n genotype vectors (0, 1 or 2),
• Find n pairs of haplotype vectors, one pair of
  haplotypes per each genotype explaining genotypes
  
• For individual genotype with h heterozygous sites
  there are 2h-1 possible haplotype pairs
  explaining this genotype
• This is hopeless without genetic model
• Parsimonious models ? minimize number of
  haplotypes

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Computational Haplotype Inference Problem

• Assumptions
• small number of repeated mutations
• small number of recombinations
• If data allow, then explain them only with
  mutations (perfect phylogeny)
• It is possible when there no 4-gamete rule
  violations
• for any pair of SNPs only 3 combinations out of
  4 (00/01/10/11) are present
• Fastest implemented algorithm DPPH
• Known programs for general data (with possible
  4-gamete rule violations)
• PHASE, HAPLOTYPER, HAP, Set-cover based, etc.

WABI 2004
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Reducing the Set of SNPs

• Often many columns corresponding to SNP sites are
  analogous one column can be obtained from
  another by swapping 0s and 1s
• One of such columns can be dropped same as for
  two equal columns
• What would be generalization?
• If one site is dependent (or can be
  reconstructed) from k other sites, then drop
  this dependent site it does not carry any
  useful additional information
• General reduction method
• Encoding reduce number of sites be removing
  dependent sites
• Infer site-reduced haplotypes for the
  site-reduced genotypes using known haplotype
  inference method
• Decoding reconstruct dependent SNPs from sites
  of reduced haplotypes
• Main requirement to reduction method should be
  fast

WABI 2004
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Linear Dependence of SNPs

• Consider linear dependence
• To make analogous sites linearly dependent
  change notations 0/1 ? -1/1
• Also for genotypes 0/1/2 ? -1/1/0 and genotype is
  half-sum of (linearly dependent from explaining
  haplotypes)
• Keep only linear independent SNP (tag SNPs)
  all other SNP can be reconstructed using linear
  combinations
• Equivalent factorization problem find
  representation
• G IX H

WABI 2004
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Factorization Problem

• Factorization problem
• Given a 0/1/-1 genotype matrix G
• Find representation,
• G IX H
• where IX graph incidence matrix
  (exactly two 1s in each row)
• and H -1/1 haplotype matrix
• Solution
• Factorize G T (ETC)
• T tags basis of columns of G
• - solve factorization for T T IX
  H
• - finally G (IX H) (ETC) IX
  (H (ETC)) IX H

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Linear Encoding Algorithm
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Linear Decoding Algorithm
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Graph-Based Decoding

• Extend haplotype graph Xr obtained from HI
  algorithm to Xm for all m sites
• Very often the graphs Xr and Xm are isomorphic,
  but not always
• Consider example
• g1 (1, 0, 1) and g2 (0, -1, -1)
• reduced set (1,0) and (0,-1)
• The corresponding reduced haplotype graph has 3
  vertices, while Xm has 4 vertices
• The simple way is to split the vertices if we
  find an error

WABI 2004
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Handling Imperfect Phylogeny

• The genotype data may have indications of
  inconsistency with the perfect phylogeny model, 4
  gamete rule violation
• We could choose h independent columns without
  such violation
• Algorithm in greedy manner

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Experimental Results

• In Table 1, Our Results show that the advantage
  in runtime of Linearly Reduced DPPH grow fast
  with testcase size and reaches factor of 60 for
  largest instances.
• In all testcases, if DPPH find unique solution,
  so does the LR DPPH and the solution is
  identical.
• In Table 2 and 3, we can see the running time is
  drastically reduced compared to the original
  PHASE while the quality measured is not larger.
• In Table 4 and 5, we can see same advantage by
  using Linearly Reduced HAPLOTYPER instead
  original HAPLOTYPER.
• The last two data, we work on the real data from
  the drosophila haplotypes and human chromosome.

WABI 2004
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Experimental Results
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Experimental Results
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Conclusions and Future work

• Our method significantly speed up popular
  haplotype inference tools such as DPPH,
  HAPLOTYPER and PHASE in all cases thus not
  compromising the quality.
• We ever reach 50 faster than DPPH.
• Future work includes implement handling imperfect
  phylogeny algorithm.
• We are going to investigate an application of
  suggested linear reduction to finding a small
  number of representative sites sufficient to
  distinguish all haploytpes

WABI 2004