1. Phasing of 2-SNP Genotypes Based on Non-Random Mating Model Dumitru Brinza joint work with Alexander Zelikovsky Department of Computer Science Georgia State University Atlanta, USA

2. Outline Molecular biology terms Motivation Problem formulation Previous work Our contribution Phasing of 2-SNP genotypes Phasing of multi-SNP genotypes Results

3. Molecular biology terms Human Genome – all the genetic material in the chromosomes, length 3×10 9 base pairs Difference between any two people occur in 0.1% of genome SNP – single nucleotide polymorphism site where two or more different nucleotides occur in a large percentage of population. Genotype – The entire genetic identity of an individual, including alleles, SNPs, or gene forms. (e.g., AC CT TG AA AC TG) Haplotype – A single set of chromosomes (half of the full set of genetic material). (e.g., A C T A A T) Genotype is a mixture of two haplotypes.

4. From ACTG to 0,1,2 notations Haplotype: Wild type SNPs are referred as 0 Mutated SNPs are referred as 1 Genotypes: Homozygous SNPs are referred as 0 (mixture of 00) or 1 (mixture of 11) Heterozygous SNPs are referred as 2 (mixture of 01,10) homozygous haplotype SNPheterozygous Two haplotypes per individual Genotype for the individual  1100 1 0 01 1101 1 1 00 1102 1 2 02

5. Motivation  Haplotype may contain large amount of genetic markers, which are responsible for human disease.  Haplotypes may increase the power of association between marker loci and phenotypic traits.  Evolutionary tree can be reconstructed based on haplotypes.  Physical phasing (haplotypes inferring) is too expensive. Great need in computational methods for extracting haplotype information from the given genotype information.  Existing methods are either extremely slow or less accurate for genome-wide study.

6. Phasing problem (Haplotype inference) Inferring haplotypes or genotype phasing is resolution of a genotype into two haplotypes 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 2 h-1 possible haplotype pairs explaining this genotype (h=20k for the genome-wide). also there are around 10% missing data. This is hopeless without genetic model

7. Previous work PHASE – Bayesian statistical method (Stephens et al., 2001, 2003) HAPLOTYPER – proposed a Monte Carlo approach (Niu et al., 2002) Phamily – phase the trio families based on PHASE (Acherman et al., 2003) GERBIL – statistical method using maximum likelihood (ML), MST and expectation-maximization (EM) (Kimmel and Shamir, 2005) SNPHAP – use ML/EM assuming Hardy-Weinberg equilibrium (Clayton et al., 2004)

8. Contribution We explore phasing of genotypes with 2 SNPs which have ambiguity when the both sites are heterozygous. There are two possible phasing and the phasing problem is reduced to inferring their frequencies. Having the phasing solution for 2-SNP genotypes, we propose an algorithm for inferring the complete haplotypes for a given genotype based on the maximum spanning tree of a complete graph with vertices corresponding to heterozygous sites and edge weights given by the inferred 2-SNP frequencies. Extensive experimental validation of proposed methods and comparison with the previously known methods

9. Phasing of 2-SNP genotypes At least one SNP is homozygous – phasing is well defined: Both SNPs are heterozygous – ambiguity  Cis- phasing  Trans- phasing 01 or Example 21 01 11 22 0 1 22 0 1 1 0

10. Odds of cis- or trans- phasing Odds ratio of being phased cis- / trans- Additive odds ratio is better (also noticed in PHASE) LD (linkage disequilibrium) between SNPs i and j

11. Confidence in cis- or trans- phasing Closer pairs of SNPs are more linked (less crossovers ) The confidence c ij in phasing 2 SNPs i and j is inverse proportional to squared distance: Logarithm is for sign-indication of cis-/trans- preference c ij ≤ 0 means cis- with certainty |c ij | c ij > 0 means trans- with certainty |c ij | 22 i j 0 1 22 i j 0 1

12. Certainty of cis- or trans- phasing n – number of genotypes F 00, F 01, F 10, F 11 – true haplotype frequencies (observed + true in 22) ? 1 0 2 1 1 0 1 0 1 1 1 0 0 1 0 0 2 0 1 0 1 2 0 1 2 0 1 0 1 2 1 1 0 1 1 0 ? 0 1 0 1 1 0 1 2 0 0 2 1 Genotypes i j  #01 + 2  #00 + 2  #11 + 2  #10 + 1, #11 + 1  (#00 + 1, #11 + 1) or (#01 + 1, #10 + 1) * 

13. Haplotype frequencies in 22 Random mating model => Hardy-Weinberg Equilibrium (HWE): (F 00 +F 01 +F 10 +F 11 ) 2 = F 00 2 + F 01 2 + F 10 2 + F 11 2 + 2F 00 F 01 + 2F 00 F 10 + 2F 00 F 11 + 2F 01 F 10 + 2F 01 F 11 + 2F 10 F 11 G 00 G 01 G 10 G 11 G 02 G 20 G 22 G 21 G 12 Even single-SNP haplotype frequencies may deviate from HWE (F 0 +F 1 )(F 0 +F 1 -2x)= (F 0 +x) 2 + (F 1 +x) 2 + 2(F 0 F 1 -x 2 ) xG 0 yG 1 zG 2 Accordingly we adjust expectation of 2-SNP haplotype frequencies (F 00 +F 01 +F 10 +F 11 ) 2 = F 00 2 + F 01 2 + F 10 2 + F 11 2 + 2F 00 F 01 + 2F 00 F 10 + 2F 00 F 11 + 2F 01 F 10 + 2F 01 F 11 + 2F 10 F 11 xx G 00 xy G 01 yx G 10 yy G 11 xz G 02 zx G 20 zz G 22 zy G 21 yz G 12 Compute expected haplotype frequencies in 22 as best fitting to observed deviation in single-site haplotype frequencies

14. Phasing of multi-SNP genotypes Genotype graph for genotype g is a weighted complete graph G(g ) where:  Vertices = 2’s i.e., heterozygous SNPs in g  Weight w(i,j)= | c ij | confidence in phasing 2 SNPs i and j Phasing of 2 heterozygous SNPs  c ij > 0  cis-edge 22 = 00 + 11  c ij < 0  trans-edge 22 = 01 + 10 Phasing = Genotype graph coloring Color all vertices in two colors such that  any 2 vertices connected with a cis-edge have the same color, and  any 2 vertices connected with a trans-edge have opposite colors 2 1 2 0 1 2 0 2 0 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 Genotype Haplotype #1 Haplotype #2 a b c d a b c d

15. Genotype graph coloring Exact solution: ILP – slow and not accurate Heuristic solution: Find maximum spanning tree (MST) of G and color MST instead of G 1 2 1 1 3 2 1 2 1 1 3 2 Frequent conflicts when coloring genotype graph G since it has cycles Genotype Graph Coloring Problem: Find coloring with total weight (number) of conflicting edges minimized

16. 2SNP algorithm For each pair of SNPs do Collect statistics on haplotype/genotype frequencies Compute weights reflecting likelihood of trans-/cis- For each genotype g do Find MST for the complete graph G(g ) where vertices are heterozygous sites Color G(g ) vertices and phase based on coloring For each haplotype h with ?’s (missing SNP values) do Find a haplotype h’ closest to h (with minimum number of mismatches) Replace ?’s in h with the known SNP value in h’ Runtime (two bottlenecks) O(nm) – computing haplotype frequencies for 20×m pairs of SNPs in each genotype, n is number of genotypes, m number of SNP’s. O(n 2 m) – missing data recovery, finding number of mismatches for any two haplotypes

17. Datasets Chromosome 5q31: 129 genotypes with 103 SNPs derived from the 616 KB region of human Chromosome 5q31 (Daly et al., 2001). Yoruba population (D): 30 genotypes with SNPs from 51 various genomic regions, with number of SNPs per region ranging from 13 to 114 (Gabriel et al., 2002). Random matching 5q31: 128 genotypes each with 89 SNPs from 5q31 cytokine gene generated by random matching from 64 haplotypes of 32 West African Hull et al. (2004). HapMap datasets: 30 genotypes of Utah residents and Yoruba residents available on HapMap by Dec 2005. The number of SNPs varies from 52 to 1381 across 40 regions including ENm010, ENm013, ENr112, ENr113 and ENr123 spanning 500 KB regions of chromosome bands 7p15:2, 7q21:13, 2p16:3, 4q26 and 12q12 respectively, and two regions spanning the gene STEAP and TRPM8 plus 10 KB upstream and downstream.

18. Unrelated individuals phasing validation Phasing methods can be validated on simulated data (haplotypes are known) The validation on real data is usually performed on the trio data  Offspring haplotypes are mostly known (inferred from parents haplotypes) Error types Single-Site error  Number of SNPs in offspring phased haplotypes which differ from SNPs inferred from trio data, divide by (total number of SNPs) x (total number of haplotypes) Individual error  Number of correctly phased offspring genotypes (no Single-Site errors) divide by total number of genotypes Switching error  Minimum number of switches which should be done in pair of haplotypes of offspring phased genotype such that both haplotypes will coincide with haplotypes inferred from trio data, divide by total number of heterozygous positions in offspring genotypes.

19. Results

20. Chromosome-Wide Phasing Entire chromosomes for 30 Trios from Hapmap Average Errors: Single-site: 3.3% Switching: 8.8% #SNPs 1.5K runtime 2 sec 2.5K8 sec 5.0K25 sec 10.0K55 sec 20.0K220 sec 40.0K17 min 60.0K35 min 80.0K70 min

21. Conclusion 2SNP method Several orders of magnitude faster Scalable for genome-wide study Phase 10000 SNPs in less than one hour Same accuracy as PHASE and Gerbil