Constrained Hidden Markov Models for Population-based Haplotyping

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Constrained Hidden Markov Models for
Population-based Haplotyping
Application of Probabilistic ILP II, FP6-508861
www.aprill.org

• Niels Landwehr
• Joint work with Taneli Mielikäinen, Lauri Eronen,
  Hannu Toivonen, Heikki Mannila
• University of Freiburg / University of Helsinki

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Outline

• Population-based haplotype reconstruction
• Infer haplotypes from genotypes reconstruct
  hidden phase of genetic data
• Important problem in biology/medicine e.g.
  disease association studies
• An approach using constrained HMMs
• Sparse markov chains to represent conserved
  haplotype fragments
• HMM model that can be learned directly from
  genotype data
• Experimental results

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Human Genome and SNPs
SNP (marker)
SNP (marker)
SNP (marker)
...GATATTCGTACGGATGTTTCCA... ...GATGTTCGTACTGATGTC
TCCA... ...GATATTCGTACGGATGTTTCCA... ...GATATTCGTA
CGGATGTTTCCA... ...GATGTTCGTACTGATGTCTCCA... ...GA
TGTTCGTACTGATGTCTCCA...
Individuals
1 2 3 4 5 6
DNA Sequence
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Haplotypes
SNP
SNP
SNP
A G T G T C A
G T A G T G T
C G T C
Individuals
1 2 3 4 5 6
DNA Sequence
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Haplotypes
SNP
SNP
SNP
1 0 1 0 1 0 1
0 1 1 0 1 0 1
0 0 1 0
Individuals
1 2 3 4 5 6
DNA Sequence
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Why Haplotypes?

• Haplotypes
• define our genetic individuality
• contribute to risk factors of complex diseases
  (e.g., diabetes)
• Disease Association Studies (Gene Mapping)
• find genetic difference between a case and a
  control population
• Identifying SNPs responsible for disease might
  help find a cure
• Also useful for
• Linkage disequilibrium studies Summarize genetic
  variation
• Understanding evolution of human populations

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The problem Haplotypes not directly observable
. 1 . . . 1 . . . 0 . . . 0 . . . 1 .
. 0 . . . 0 . . . 0 . . . 1 . . . 1 .
Paternal
Maternal
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Population-based Haplotype Reconstruction

• Given the genotypes of several individuals, infer
  for every individual the most likely underlying
  haplotype pair
• Hidden data reconstruction problem using
  probabilistic model exploit patterns in the
  haplotypes (linkage disequilibrium)

haplotype pair
genotype
1 0 1 0 0 1 1 1 0 1 1 1 0 0 1 0
0,1 0,1 0,1 1 0,1 1 0 0,1
1 0 1 0 0 1 1 1 0 1 1 1 0 0 1 1
0 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1
0,1 0,1 1 1 1 1 0 1
0,1 0,1 0,1 1 0,1 1 0 1
Individual 2
Individual 3
Individual 1
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Haplotype Reconstruction Problem (CS Perspective)
Input A set G of genotypes
Output A set H of corresponding haplotype pairs
such that
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Population-based Haplotype Reconstruction

• Given a model M for the distribution of
  haplotypes, can infer most likely resolution

Hardy-Weinberg equilibrium

• Need to estimate this model from available
  genotype data

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Prior Work on Haplotype Reconstruction

• Competitive application domain for several years
  many systems developed
• characterized by the statistical model and
  learning/reconstruction algorithms employed
• Special-purpose statistical models
• Approximate Coalescent (PHASE 2001,2003,2005)
• Block-based (Gerbil 2004,2005)
• Variable-length MC (HaploRec 2004,2006)
• Founder-based (HIT 2005)
• Local clusters (fastPHASE 2006)

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Prior Work on Haplotype Reconstruction

• Special-purpose learning/reconstruction
  algorithms
• MCMC variant
• Approximate EM partition ligation
• Our approach
• Model haplotypes using (sparse) markov chains
• Natural extension to a Hidden Markov Model on
  genotypes
• Directly learnable from genotype data (standard
  Baum-Welsh)

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Constrained HMMs for haplotyping
Path for haplotype 0,1,1,0

• Modeling haplotypes
• Standard markov chain
• More general order k markov chain

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Constrained HMMs for haplotyping

• Modeling genotypes
• Hidden phase (order of pair) Hidden Markov Model
• States pairs of states of the underlying markov
  chain (state of the maternal/paternal sequence)
• Output symbol unordered pair
• Path in the model sample two haplotypes, output
  corresponding genotype
• Have to enforce Hardy-Weinberg equilibrium
• Parameter tying constraints on transition
  probabilities
• Algorithms
• Learning standard Baum-Welsh
• Reconstruction of most likely haplotype pair
  Viterbi

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Constrained HMMs for haplotyping

• Example paths for genotype 0,1,1,0,1,0

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Sparse Markov Modeling (SpaMM)

• Higher-order models (long history) needed
  exponential size of model
• However, out of the possible history
  blocks, only few occur in data (conserved
  fragments)
• Idea Sparse model, iterative structure learning
  algorithm to identify conserved fragments
  (Apriori-style)

Initialize first-order-model()
em-training( ) repeat
regularize-and-extend( ) em-training(
) until
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SpaMM Model (order 1)

• Initial model standard markov chain of order 1

• Iteration extend order of model by 1, prune
  unlikely parts
• Avoids combinatorial explosion of model size

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SpaMM Model (order 2)

• Iteration extend order of model by 1, prune
  unlikely paths
• Avoids combinatorial explosion of model size

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SpaMM Model (order 3)

• Iteration extend order of model by 1, prune
  unlikely paths
• Avoids combinatorial explosion of model size

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SpaMM Model (order 4)

• Iteration extend order of model by 1, prune
  unlikely paths
• Avoids combinatorial explosion of model size

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SpaMM Model (order 5)

• Iteration extend order of model by 1, prune
  unlikely paths
• Avoids combinatorial explosion of model size

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SpaMM Model (order 6)

• Iteration extend order of model by 1, prune
  unlikely paths
• Avoids combinatorial explosion of model size

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SpaMM Model (final)

• Final model Model structure encodes conserved
  fragments
• Concise representation of all haplotypes with
  non-zero probability

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Experimental Evaluation

• Real world population data
• Correct haplotypes have been inferred from trios
• Daly dataset 103 SNP markers for 174 individuals
• Yoruba population 100 datasets, 500 SNP markers
  each, 60 individuals
• Problem Setting
• Given the set of genotypes, algorithm outputs
  most likely haplotype pairs
• Difference to real haplotype pairs is measured in
  switch distance ( recombinations needed to
  transform pairs, normalized)

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Results Haplotype Reconstruction

• Many well-engineered systems
• Smart priors, averaging over several random
  restarts of EM, ...
• SpaMM proof-of-concept implementation, not tuned

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Results Haplotype Reconstruction

• PHASE most accurate, then fastPHASE, then SpaMM
• however, PHASE too slow for long maps
• SpaMM beats fastPHASE without averaging
• overall, competitive accuracy

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Results Runtime

• Runtime in seconds for phasing 100 markers (log.
  scale)
• SpaMM scales linearly in markers
• like fastPHASE, HaploRec, HIT
• unlike PHASE, Gerbil

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Results Genotype imputation

• Most haplotyping methods can also predict missing
  genotype values
• for SpaMM, can be read off Viterbi path

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Results Genotype imputation

• fastPHASE best known method
• Again, SpaMM beats fastPHASE without averaging

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Conclusions

• SpaMM new haplotyping method
• sparse Markov chains to encode conserved
  haplotype fragments
• Constrained HMM for modeling genotypes
• Apriori-style structure learning algorithm
• Simple, accurate, interpretable output
• Future work
• Accuracy can probably be improved using standard
  techniques (EM random restarts, averaging, ...)

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Thanks!