1. Constrained Hidden Markov Models for Population-based Haplotyping
Application of Probabilistic ILP II, FP
Constrained Hidden Markov Models for Population-based Haplotyping
Niels Landwehr
Joint work with Taneli Mielikäinen, Lauri Eronen, Hannu Toivonen, Heikki Mannila
University of Freiburg / University of Helsinki

2. 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

3. Human Genome and SNPs ...GATATTCGTACGGATGTTTCCA...
(marker)
SNP
(marker)
SNP
(marker)
SNP
...GATATTCGTACGGATGTTTCCA...
...GATGTTCGTACTGATGTCTCCA...
Individuals
DNA Sequence

4. Haplotypes Haplotypes AGT A G T GTC G T C 1 2 3 4 5 6 Individuals
SNP
SNP
SNP
AGT
GTC
Haplotypes
A G T
G T C
Individuals
DNA Sequence

5. Haplotypes Haplotypes 101 1 0 1 010 0 1 0 1 2 3 4 5 6 Individuals
SNP
SNP
SNP
101
010
Haplotypes
Individuals
DNA Sequence

6. Why Haplotypes? Haplotypes Disease Association Studies (Gene Mapping):
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

7. The problem: Haplotypes not directly observable. 1 . 1
{0,1}
{0}
{1}
WetLab: only genotype information (two alleles for each SNP, but chromosome origin is unknown)
Paternal
Maternal

8. 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
0 1
1 1
0 0
{0,1}
{1}
{0}
1 0
0 1
1 1
0 0
{0,1}
{1}
{0}
0 1
1 1
0 0
{0,1}
{1}
{0}
Individual 2
Individual 3 …
Individual 1

9. Haplotype Reconstruction Problem (CS Perspective)
Input: A set G of genotypes
Output: A set H of corresponding haplotype pairs
such that

10. 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

11. 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)

12. 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)

13. Constrained HMMs for haplotyping
Path for haplotype 0,1,1,0
Modeling haplotypes
Standard markov chain
More general: order k markov chain

14. 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

15. Constrained HMMs for haplotyping
Example: paths for genotype {0,1},{1},{0,1},{0}

16. 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

17. 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

18. SpaMM Model (order 2)
Iteration: extend order of model by 1, prune unlikely paths
Avoids combinatorial explosion of model size

19. SpaMM Model (order 3)
Iteration: extend order of model by 1, prune unlikely paths
Avoids combinatorial explosion of model size

20. SpaMM Model (order 4)
Iteration: extend order of model by 1, prune unlikely paths
Avoids combinatorial explosion of model size

21. SpaMM Model (order 5)
Iteration: extend order of model by 1, prune unlikely paths
Avoids combinatorial explosion of model size

22. SpaMM Model (order 6)
Iteration: extend order of model by 1, prune unlikely paths
Avoids combinatorial explosion of model size

23. SpaMM Model (final)
Final model: Model structure encodes conserved fragments
Concise representation of all haplotypes with non-zero probability

24. 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)

25. Results: Haplotype Reconstruction
Many well-engineered systems
Smart priors, averaging over several random restarts of EM, ...
SpaMM: proof-of-concept implementation, not tuned

26. 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

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

28. Results: Genotype imputation
Most haplotyping methods can also predict missing genotype values
for SpaMM, can be read off Viterbi path

29. Results: Genotype imputation
fastPHASE best known method
Again, SpaMM beats fastPHASE without averaging

30. Conclusions SpaMM: new haplotyping method Future work
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, ...)

31. Thanks!