1. University of Connecticut
Hidden Markov Models of Haplotype Diversity and Applications in Genetic Epidemiology
Ion Mandoiu
University of Connecticut

2. Outline HMM model of haplotype diversity Applications Conclusions
Phasing
Error detection
Imputation
Genotype calling from low-coverage sequencing data
Conclusions

3. Single Nucleotide Polymorphisms
Main form of variation between individual genomes: single nucleotide polymorphisms (SNPs)
High density in the human genome:  1  107 SNPs out of total 3  109 base pairs
… ataggtccCtatttcgcgcCgtatacacgggActata …
… ataggtccGtatttcgcgcCgtatacacgggTctata …
… ataggtccCtatttcgcgcCgtatacacgggTctata …

4. Haplotypes and Genotypes
Diploids: two homologous copies of each autosomal chromosome
One inherited from mother and one from father
Haplotype: description of SNP alleles on a chromosome
0/1 vector: 0 for major allele, 1 for minor
Genotype: description of alleles on both chromosomes
0/1/2 vector: 0 (1) - both chromosomes contain the major (minor) allele; 2 - the chromosomes contain different alleles +
two haplotypes per individual
genotype

5. Sources of Haplotype Diversity: Mutation
The International HapMap Consortium. A Haplotype Map of the Human Genome. Nature 437,

6. Sources of Haplotype Diversity: Recombination

7. Haplotype Structure in Human Populations

8. HMM Model of Haplotype FrequenciesFn … H1 H2 Hn
Fi = founder haplotype at locus i, Hi = observed allele at locus i
P(Fi), P(Fi | Fi-1) and P(Hi | Fi) estimated from reference genotype or haplotype data
For given haplotype h, P(H=h|M) can be computed in O(nK2) using forward algorithm
Similar models proposed in [Schwartz 04, Rastas et al. 05, Kimmel&Shamir 05, Scheet&Stephens 06]

9. Outline HMM model of haplotype diversity Applications Conclusions
Phasing
Error detection
Imputation
Genotype calling from low-coverage sequencing data
Conclusions

10. Genotype Phasing ? h1:0010111 h2:0010010 g: 0010212 h3:0010011

11. Maximum Likelihood Genotype Phasing… F1 F2 Fn H1 H2 Hn …
F'1
F'2
F'n
H'1
H'2
H'n G1 G2 Gn
Maximum likelihood genotype phasing: given g, find (h1,h2) = argmaxh1+h2=g P(h1|M)P(h2|M)

12. Computational Complexity
[KMP08] Cannot approximate maxh1+h2=g P(h1|M)P(h2|M) within a factor of O(n1/2 -), unless ZPP=NP
[Rastas et al.] give Viterbi and randam sampling based heuristics that yield phasing accuracy comparable to best existing methods (PHASE)

13. Outline HMM model of haplotype diversity Applications Conclusions
Phasing
Error detection
Imputation
Genotype calling from low-coverage sequencing data
Conclusions

14. Genotyping Errors
A real problem despite advances in technology & typing algorithms
1.1% of 20 million dbSNP genotypes typed multiple times are inconsistent [Zaitlen et al. 2005]
Systematic errors (e.g., assay failure) typically detected by departure from HWE [Hosking et al. 2004]
In pedigrees, some errors detected as Mendelian Inconsistencies (MIs)
Many errors remain undetected
As much as 70% of errors are Mendelian consistent for mother/father/child trios [Gordon et al. 1999]

15. Likelihood Sensitivity Approach to Error Detection in Trios
Mother
Father
Child
Likelihood of best phasing for original trio T h1 h3 h1 h2 h3 h4

16. Likelihood Sensitivity Approach to Error Detection in Trios
Mother
Father
Child
h’ 1
h’ 3
h’1
h’2
h’ 3
h’ 4
Likelihood of best phasing for modified trio T’ ?
Likelihood of best phasing for original trio T

17. Likelihood Sensitivity Approach to Error Detection in Trios
Mother
Father
Child ?
Large change in likelihood suggests likely error
Flag genotype as an error if L(T’)/L(T) > R, where R is the detection threshold (e.g., R=104)

18. Alternate Likelihood Functions
[KMP08] Cannot approximate L(T) within O(n1/4 -), unless ZPP=NP
Efficiently Computable Likelihood Functions
Viterbi probability
Probability of Viterbi Haplotypes
Total Trio Probability

19. Comparison of Alternative Likelihood Functions (1% Random Allele Errors)

20. Log-Likelihood Ratio Distribution
FPs caused by same-locus errors in parents

21. “Combined” Detection Method
Compute 4 likelihood ratios
Trio
Mother-child duo
Father-child duo
Child (unrelated)
Flag as error if all ratios are above detection threshold

22. Comparison with FAMHAP (Children)

23. Comparison with FAMHAP (Parents)

24. Outline HMM model of haplotype diversity Applications Conclusions
Phasing
Error detection
Imputation
Genotype calling from low-coverage sequencing data
Conclusions

25. Genome-Wide Association Studies
Powerful method for finding genes associated with complex human diseases
Large number of markers (SNPs) typed in cases and controls
Disease causal SNPs unlikely to be typed directly
Significant statistical power gained by performing imputation of untyped Hapmap genotypes [WTCCC’07]

26. HMM Based Genotype Imputation
Train HMM using the haplotypes from related Hapmap or small cohor typed at high density
Probability of missing genotypes given the typed genotype data
 gi is imputed as

27. Experimental Results
Estimates of the allele 0 frequency based on Imputation vs. Illumina 15k

28. Experimental Results
Accuracy and missing data rate for imputed genotypes at different thresholds

29. Outline HMM model of haplotype diversity Applications Conclusions
Phasing
Error detection
Imputation
Genotype calling from low-coverage sequencing data
Conclusions

30. Ultra-High Throughput Sequencing
New massively parallel sequencing technologies deliver orders of magnitude higher throughput compared to Sanger sequencing
-SBS: Sequencing by Synthesis
-SBL: Sequencing by Ligation
-Challenges in Genome Assembly: The short read lengths and absence of paired ends make it difficult for assembly software to disambiguate repeat regions, therefore resulting in fragmented assemblies.
-New Type of sequencing error: in 454 including incorrect estimates of homopolymer lengths,
‘transposition-like’ insertions (a base identical to a nearby homopolymer is inserted in a nearby nonadjacent location) and errors caused by multiple templates attached to the same bead
Roche / 454
Genome Sequencer FLX
100 Mb/run, 400bp reads
Illumina / Solexa
Genetic Analyzer 1G
1000 Mb/run, 35bp reads
Applied Biosystems
SOLiD
3000 Mb/run, 25-35bp reads 30

31. Probabilistic Model … F1 F2 Fn H1 H2 Hn … F'1 F'2 F'n H'1 H'2 H'n G1
The Hierarchical-Factorial HMM we used to describe haplotype sequences is similar to models recently proposed by others.
At the core of the model are 2 regular HMM representing haplotype frequencies in the populations of origin of the sequenced individual’s parents.
Under this model each haplotype in the population is viewed as a mosaic formed as a result of historical recombination among a set of K founder haplotypes.
Increasing the number of founders allows more paths to describe a haplotype sequence, but at a cost of a longer run time.
Training & the estimation of this HMM is done via a Baum-Welch algorithm based on haplotype inferred from a panel representing the population of origin of each parent. G1 G2 Gn
R1,1 …
R1,c
R2,1 …
R2,c
Rn,1 …
Rn,c 1 2 n 31

32. Model Training
Initial founder probabilities P(f1), P(f’1), transition probabilities P(fi+1|fi), P(f’i+1|f’i), and emission probabilities P(hi|fi), P(h’i|f’i) trained using the Baum-Welch algorithm from haplotypes inferred from the populations of origin for mother/father
P(gi|hi,h’i) set to 1 if h+h’i=gi and to 0 otherwise
where is the probability that read r has an error at locus I
 Conditional probabilities for sets of reads are given by:
The Hierarchical-Factorial HMM we used to describe haplotype sequences is similar to models recently proposed by others.
At the core of the model are 2 regular HMM representing haplotype frequencies in the populations of origin of the sequenced individual’s parents.
Under this model each haplotype in the population is viewed as a mosaic formed as a result of historical recombination among a set of K founder haplotypes.
Increasing the number of founders allows more paths to describe a haplotype sequence, but at a cost of a longer run time.
Training & the estimation of this HMM is done via a Baum-Welch algorithm based on haplotype inferred from a panel representing the population of origin of each parent. 32

33. Multilocus Genotyping Problem
GIVEN:
Shotgun read sets r=(r1, r2, … , rn)
Base quality scores
HMMs for populations of origin for mother/father
FIND:
Multilocus genotype g*=(g*1,g*2,…,g*n) with maximum posterior probability, i.e., g*=argmaxg P(g | r)
NOTE: P(g|r) is NP-Hard… 33

34. Posterior Decoding Algorithm
For each i = 1..n, compute
Return
Joint probabilities can be computed using a forward-backward algorithm:
Direct implementation gives O(m+nK4) time, where
m = number of reads
n = number of SNPs
K = number of founder haplotypes in HMMs
Runtime reduced to O(m+nK3) using speed-up idea similar to [Rastas et al. 08, Kennedy et al. 08] 34

35. Genotyping Accuracy on Watson Reads

36. Outline HMM model of haplotype diversity Applications Conclusions
Phasing
Error detection
Imputation
Genotype calling from low-coverage sequencing data
Conclusions

37. Conclusions
HMM model of haplotype diversity provides a powerful framework for addressing central problems in population genetics & genetic epidemiology
Enables significant improvements in accuracy by exploiting the high amount of linkage disequilibrium in human populations
Despite hardness results, heuristics such as posterior or Viterbi decoding perform well in practice
Highly scalable runtime (linear in #SNPs and #individuals/reads)
Software available at

38. Acknowledgements
Sanjiv Dinakar, Jorge Duitama, Yözen Hernández, Justin Kennedy, Bogdan Pasaniuc
NSF funding (awards IIS and DBI )