METHODS FOR HAPLOTYPE RECONSTRUCTION

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METHODS FOR HAPLOTYPE RECONSTRUCTION

• Andrew Morris
• Wellcome Trust Centre for Human Genetics
• March 6, 2003

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Outline

• Haplotypes and genotypes.
• Reconstruction in pedigrees.
• Reconstruction in unrelated individuals.
• Interpretation and LD assessment.
• Two stage analyses.

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Haplotypes and genotypes (1)
1 0 0 0 1
1 1 0 0 0

11 01 00 00 01

4
Haplotypes and genotypes (1)
1 0 0 0 1
1 1 0 0 0

11 01 00 00 01

5
Haplotypes and genotypes (1)
1 0 0 0 1
1 1 0 0 0

11 01 00 00 01

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Haplotypes and genotypes (1)
1 0 0 0 1
1 1 0 0 0

11 01 00 00 01

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Haplotypes and genotypes (2)

• Individuals that are homozygous at every locus,
  or heterozygous at just one locus can be
  resolved.
• Individuals that are heterozygous at k loci are
  consistent with 2k-1 configurations of haplotypes.

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Why do we need haplotypes?

• Correlation between alleles at closely linked
  loci
• Fine-scale mapping studies.
• Association studies with multiple markers in
  candidate genes.
• Investigating patterns of LD across genomic
  regions.
• Inferring population histories.

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Molecular methods

• Single molecule dilution.
• Allele specific long range PCR.
• Prone to errors.
• Expensive and inefficient low throughput.

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Simplex family data (1)

• 00 01 00 11 x 01 11 01 01
• (M) (F)
• 00 01 01 01

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Simplex family data (1)

• 00 01 00 11 x 01 11 01 01
• (M) (F)
• 00 01 01 01

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Simplex family data (1)

• 00 01 00 11 x 01 11 01 01
• (M) (F)
• 00 01 01 01
• Inferred haplotypes 0001 / 0110

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Simplex family data (2)

• 00 01 00 01 x 01 01 00 01
• (M) (F)
• 00 01 00 01
• Cannot be fully resolved

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Pedigree data (1)

• 11 01 11 01 11 x 00 00 11 11 11
• 01 01 11 11 11 x 01 00 00 01 00
• 01 01 01 01 01 11 01 01 01 01 00 00 01 11
  01

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Pedigree data (1)

• 11111 / 10101 x 00111 / 00111
• 11111 / 00111 x 00010 / 10000
• 11111 / 00000 11111 / 10000 00111 /
  00010

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Pedigree data (1)

• 11111 / 10101 x 00111 / 00111
• 11111 / 00111 x 00010 / 10000
• 11111 / 00000 11111 / 10000 00111 /
  00010

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Pedigree data (2)

• Many combinations of haplotypes may be consistent
  with pedigree genotype data.
• Complex computational problem.
• Need to make assumptions about recombination.
• SIMWALK and MERLIN.

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Statistical approaches to reconstruct haplotypes
in unrelated individuals

• Parsimony methods Clarks algorithm.
• Likelihood methods E-M algorithm.
• Bayesian methods PHASE algorithm.
• Aims reconstruct haplotypes and/or estimate
  population frequencies.

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Clarks algorithm (1)

• Reconstruct haplotypes in unresolved individuals
  via parsimony.
• Minimise number of haplotypes observed in sample.
• Microsatellite or SNP genotypes.

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Clarks algorithm (2)

• Search for resolved individuals, and record all
  recovered haplotypes.
• Compare each unresolved individual with list of
  recovered haplotypes.
• If a recovered haplotype is identified,
  individual is resolved.
• Complimentary haplotype added to list of
  recovered haplotypes.
• Repeat 2-4 until all individuals are resolved or
  no more haplotypes can be recovered.

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Example

• (A) 00 01 01 00
• (B) 00 00 00 00
• (C) 00 01 00 00
• (D) 01 11 01 11
• (E) 00 11 01 01
• (F) 01 11 11 00
• (G) 00 01 11 01
• (H) 00 01 01 11
• (I) 00 00 00 00
• (J) 00 00 00 11

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Example

• (A) 00 01 01 00
• (B) 00 00 00 00
• (C) 00 01 00 00
• (D) 01 11 01 11
• (E) 00 11 01 01
• (F) 01 11 11 00
• (G) 00 01 11 01
• (H) 00 01 01 11
• (I) 00 00 00 00
• (J) 00 00 00 11

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Example

• (A) 00 01 01 00
• (B) 0000 / 0000
• (C) 0000 / 0100
• (D) 01 11 01 11
• (E) 00 11 01 01
• (F) 0110 / 1110
• (G) 00 01 11 01
• (H) 00 01 01 11
• (I) 0000 / 0000
• (J) 0001 / 0001

• Recovered haplotypes
• 0000
• 0100
• 0110
• 1110
• 0001

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Example

• (A) 00 01 01 00
• (B) 0000 / 0000
• (C) 0000 / 0100
• (D) 01 11 01 11
• (E) 00 11 01 01
• (F) 0110 / 1110
• (G) 00 01 11 01
• (H) 00 01 01 11
• (I) 0000 / 0000
• (J) 0001 / 0001

• Recovered haplotypes
• 0000
• 0100
• 0110
• 1110
• 0001

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Example

• (A) 0000 / 0110
• (B) 0000 / 0000
• (C) 0000 / 0100
• (D) 01 11 01 11
• (E) 00 11 01 01
• (F) 0110 / 1110
• (G) 00 01 11 01
• (H) 00 01 01 11
• (I) 0000 / 0000
• (J) 0001 / 0001

• Recovered haplotypes
• 0000 0111
• 0100
• 0110
• 1110
• 0001

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Example

• (A) 0000 / 0110
• (B) 0000 / 0000
• (C) 0000 / 0100
• (D) 01 11 01 11
• (E) 0100 / 0111
• (F) 0110 / 1110
• (G) 00 01 11 01
• (H) 00 01 01 11
• (I) 0000 / 0000
• (J) 0001 / 0001

• Recovered haplotypes
• 0000 0111
• 0100 0011
• 0110
• 1110
• 0001

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Example

• (A) 0000 / 0110
• (B) 0000 / 0000
• (C) 0000 / 0100
• (D) 0111 / 1101
• (E) 0100 / 0111
• (F) 0110 / 1110
• (G) 0110 / 0011
• (H) 0001 / 0111
• (I) 0000 / 0000
• (J) 0001 / 0001

• Recovered haplotypes
• 0000 0111
• 0100 0011
• 0110 1101
• 1110
• 0001

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Example problem

• (A) 0000 / 0110
• (B) 0000 / 0000
• (C) 0000 / 0100
• (D) 01 11 01 11
• (E) 0100 / 0111
• (F) 0110 / 1110
• (G) 00 01 11 01
• (H) 00 01 01 11
• (I) 0000 / 0000
• (J) 0001 / 0001

• Recovered haplotypes
• 0000 0111
• 0100 0011
• 0110
• 1110
• 0001

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Example problem

• (A) 0000 / 0110
• (B) 0000 / 0000
• (C) 0000 / 0100
• (D) 01 11 01 11
• (E) 0100 / 0111
• (F) 0110 / 1110
• (G) 00 01 11 01
• (H) 00 01 01 11
• (I) 0000 / 0000
• (J) 0001 / 0001

• Recovered haplotypes
• 0000 0111
• 0100 0010
• 0110
• 1110
• 0001

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Clarks algorithm problems

• Multiple solutions try many different orderings
  of individuals.
• No starting point for algorithm.
• Algorithm may leave many unresolved individuals.
• How to deal with missing data?

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E-M algorithm (1)

• Maximum likelihood method for population
  haplotype frequency estimation.
• Allows for the fact that unresolved genotypes
  could be constructed from many different
  haplotype configurations.
• Microsatellite or SNP genotypes.

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E-M algorithm (2)

• Observed sample of N individuals with genotypes,
  G.
• Unobserved population haplotype frequencies, h.
• Unobserved configurations, H, consisting of a
  complimentary haplotype pairs Hi Hi1,Hi2.

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E-M algorithm (3)

• Likelihood
• f(Gh) ?k f(Gkh)
• ?k ?i f(GkHi) f(Hih)
• where f(Hih) f(Hi1h) f(Hi2h) under
  Hardy-Weinberg equilibrium.

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E-M algorithm (4)

• Numerical algorithm used to obtain maximum
  likelihood estimates of h.
• Initial set of haplotype frequencies h(0).
• Haplotype frequencies h(t) at iteration t updated
  from frequencies at iteration t-1 using
  Expectation and Maximisation steps.
• Continue until h(t) has converged.

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Expectation step

• Use haplotype frequencies, h(t), to calculate the
  probability of resolving each genotype, Gk, into
  each possible haplotype configuration, Hi.
• E(HiGk,h(t)) f(GkHi) f(Hi1h(t)) f(Hi2h(t))
• f(Gkh(t))

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Maximisation step

• Compute haplotype frequencies using procedure
  equivalent to gene counting.
• hs(t1) ?k ?i Zsi E(HiGk,h(t))
• 2N
• Zsi number of copies (0,1,2) of sth haplotype
  in configuration Hi.

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E-M algorithm comments

• Can handle missing data.
• For many loci, the number of possible haplotypes
  is large, so population frequencies are difficult
  to estimate re-parameterisation.
• Does not provide reconstructed haplotype
  configuration for unresolved individuals can use
  maximum likelihood configuration.

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PHASE algorithm (1)

• Treats haplotype configuration for each
  unresolved individual as an unobserved random
  quantity.
• Evaluate the conditional distribution, given a
  sample of unresolved genotype data.
• Microsatellite or SNP genotypes.
• Reconstruction and population haplotype frequency
  estimation.

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PHASE algorithm (2)

• Bayesian framework goal is to approximate
  posterior distribution of haplotype
  configurations f(HG).
• Implements Markov chain Monte Carlo (MCMC)
  methods to sample from f(HG) Gibbs sampling.
• Start at random configuration.
• Repeatedly select unresolved individuals at
  random, and sample from their possible haplotype
  configurations, assuming all other individuals to
  be correctly resolved.

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PHASE algorithm (3)

• Initial haplotype configuration H(0).
• Subsequent iterations obtain H(t1) from H(t)
  using the following steps
• Select an unresolved individual,i, at random.
• Sample Hi(t1) from f(HiG,H-i(t)).
• Set Hk(t1) Hk(t) for all k ? i.
• On convergence, each sampled configuration
  represents random draw from f(HG).

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PHASE algorithm (4)

• How to obtain f(HiG,H-i)?
• Base directly on sample frequency of observed
  haplotypes in configuration H-i.
• Better to introduce prior model for population
  haplotype frequencies, f(h).
• Coalescent process used to predict likely
  patterns of haplotypes occurring in populations.

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PHASE algorithm (5)

• Key principle
• Configuration Hi is more likely to consist of
  haplotypes Hi1 and Hi2 that are exactly the same
  as, or similar to, haplotypes in the
  configuration H-i.

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Example (1)

• Resolved haplotypes 22544 and 22334.
• Unresolved individual 22 22 35 34 44.

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Example (1)

• Resolved haplotypes 22544 and 22334.
• Unresolved individual 22 22 35 34 44.
• Possible configurations
• (1) 22334 / 22544
• (2) 22534 / 22344

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Example (1)

• Resolved haplotypes 22544 and 22334.
• Unresolved individual 22 22 35 34 44.
• Possible configurations
• (1) 22334 / 22544
• (2) 22534 / 22344 -1 and 1

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Example (1)

• Resolved haplotypes 22544 and 22334.
• Unresolved individual 22 22 35 34 44.
• Possible configurations
• (1) 22334 / 22544
• (2) 22534 / 22344 -1 and 1
• Assign high probability to sampling configuration
  (1).

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Example (2)

• Resolved haplotypes 22544 and 22334.
• Unresolved individual 22 22 46 34 44.

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Example (2)

• Resolved haplotypes 22544 and 22334.
• Unresolved individual 22 22 46 34 44.
• Possible configurations
• (1) 22434 / 22644
• (2) 22634 / 22444

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Example (2)

• Resolved haplotypes 22544 and 22334.
• Unresolved individual 22 22 46 34 44.
• Possible configurations
• (1) 22434 / 22644 1 and 1
• (2) 22634 / 22444 3 and -1
• Assign high probability to sampling configuration
  (1).

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PHASE algorithm comments

• Allows for uncertainty in haplotype
  reconstruction in Bayesian framework.
• Can handle missing data.
• Coalescent process does not explicitly allow for
  recombination, but performs well even when
  cross-over events occur (up to 0.1cM).
• Up to 50 more efficient than Clarks algorithm
  or the E-M algorithm.

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PHASE algorithm output

• Best reconstruction output for each individual.
• Uncertainty in reconstruction indicated by system
  of brackets
• inferred missing genotype uncertain with
  posterior probability less than specified
  threshold
• () inferred phase assignment uncertain with
  posterior probability less than specified
  threshold.
• 0 (1) 0 0 1 (0)
• 0 (0) 1 0 1 (1)

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PHASE algorithm interpretation

• Best reconstruction not necessarily correct.
• Uncertain haplotype configurations should be
  investigated further.
• Effective targeting of additional genotyping
  costs.

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Other Bayesian MCMC algorithms

• HAPLOTYPER
• Prior model for haplotype frequencies given by
  Dirichelet distribution.
• Deals with large number of SNPs by partition
  ligation.
• Outputs best reconstruction with uncertainty
  measured by posterior probability.
• HAPMCMC
• Log-linear prior model for haplotype frequencies
  incorporating interactions corresponding to first
  order LD between SNPs.
• Designed specifically for investigating LD across
  small genomic regions.

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Dont ignore uncertainty

• Tempting to treat best configuration as correct
  in subsequent analyses.
• Can seriously affect inferences
• Example LD across small genomic regions

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False positive error rates
Level E-M PHASE (BEST) HAPMCMC (BEST) HAPMCMC (POST)
0.01 0.008 0.204 0.060 0.009
0.05 0.045 0.362 0.183 0.056
0.1 0.103 0.446 0.247 0.091
0.2 0.212 0.566 0.392 0.201
0.5 0.524 0.767 0.640 0.527
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Two stage analyses

• For many studies, haplotype reconstruction is
  just an intermediate step required for a second
  stage of analysis.
• We dont actually care what the haplotypes are
  missing data that we must account for in second
  stage of analysis.
• Better to develop methods that allow for
  uncertainty in haplotype configuration in
  unresolved individuals, rather than haplotype
  reconstruction methods per-se.

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Summary

• Haplotype information required for analysis of
  high-density marker data.
• Many algorithms available.
• Interpret output with care.
• Reconstructed haplotypes often not of interest,
  but uncertainty must be accounted for in
  subsequent analyses.