The distribution of the IBD sharing and applications

1. The distribution of the IBD sharing and applications Shai Carmi ItsikPe’er’s lab Department of Computer Science Columbia University Tel Aviv University July 23, 2012

2. Outline • IBD: introduction • Coalescent theory of IBD • Distribution of pairwise sharing. • The variance. • The variance of the cohort-averaged sharing. • Applications • Imputation by IBD • Sequencing study design. • Siblings. • Demographic inference. • Jewish genetics. • Summary

3. Identity-by-descent (IBD) L. Macmillan, UNC • In isolated, small populations all individuals have recent common ancestor. • Abundance of long haplotypes which are IBD. B A A B A shared segment

4. IBD detection • Until last decade, IBD usually defined for single markers. • Genome-wide SNP arrays enable detection of long segments. • GERMLINE (Gusevet al., Genome Res., 2009):A fast algorithm for detection of IBD segment in large cohorts. • Divide the chromosomes into small windows. • For each window, hash the genotypes of each individual and search for perfect matches. • Extend seeds, as long as match is good enough. • Record matches longer than a cutoff m. • Other methods exist. A B

5. Questions • How much IBD is expected in model populations? • Consider the fraction of genome shared between all possible pairs. • Mean? • Variance? • Distribution? • Applications • Demographic inference • Study design • Positive selection detection • Phasing and imputation • Pedigree reconstruction

6. Sequencing study design • A large genotyped cohort. • A subset is selected for sequencing. • Look for IBD segments between sequenced and not-sequenced individuals. Select A • Impute variants along IBD segments. • To maximize utility, select individuals with most sharing (Gusev at al., Genetics, 2012 (INFOSTIP)).

7. Sequencing study design • A large genotyped cohort. • A subset is selected for sequencing. • Look for IBD segments between sequenced and not-sequenced individuals. Select A • Is the strategy useful? Is it worth prioritzing? • How is the average sharing of each individual to the rest of the cohort distributed?

8. Wright-Fisher model • Non-overlapping, discrete generations. • A population of constant size of N haploid individuals. • Ignore mutations (when studying IBD). • Recombination is a Poisson process. • Each pair of individuals (linages) has probability 1/N to coalesce in the previous generation. • In the limit of continuous-time and large population size, approximated by the coalescent. • (Scaled) Time to MRCA is exponential with rate 1. N=10

9. Mosaic of segments ℓT=ℓ1+ℓ5+ℓ9 A m B ℓ11 ℓ5 ℓ1 ℓ9 ℓ7 ℓ3 ℓ10 ℓ8 ℓ6 ℓ2 ℓ4 0 L coordinate • Consider two (unrelated) chromosomes. • The total sharing fT: The fraction of the chromosome in shared segments of length ≥m. • Observation:All sites are in shared segments, but length can be small due to ancient common ancestor.

10. Mosaic of segments ℓT=ℓ1+ℓ5+ℓ9 A m B ℓ11 ℓ5 ℓ1 ℓ9 ℓ7 ℓ3 ℓ10 ℓ8 ℓ6 ℓ2 ℓ4 0 L coordinate • Assume the (scaled) coalescence time at a site is t. • A segment of length ℓ is shared if there is no recombination event in the history of the two linages. • Number of meioses: 2Nt. t A B A B

11. Mosaic of segments ℓT=ℓ1+ℓ5+ℓ9 A m B ℓ11 ℓ5 ℓ1 ℓ9 ℓ7 ℓ3 ℓ10 ℓ8 ℓ6 ℓ2 ℓ4 0 L coordinate • Li and Durbin (Nature, 2011) found that at the end of a segment, • Therefore,

12. Renewal theory tS =τ1+τ5+τ9 A m B τ1 τ11 τ9 τ5 τ7 τ3 τ10 τ6 τ8 τ4 τ2 T 0 time • Distribution of waiting times:

13. Renewal theory: solution • Laplace transform T→s, tS→u

14. Mean IBD sharing • Can be derived in many ways. • (1) • (2) • The average number of segments ≥mis 2NL·P(ℓ≥m). • (3) Palamara, …, Pe’er, AJHG, 2012. • At the end of the talk (time-permitting).

15. Varying population size • Use results of Li and Durbin (Nature, 2011). and then proceed as before. • The mean IBD sharing:

16. The variance of the IBD sharing • Can also be calculated in a number of ways. • (1) • (2) Define I(s), the indicator, with probability π (=<fT>) , that site s is in a shared segment between two given chromosomes. • Define the number of sites as M. • The variance requires calculating two-sites probabilities. • Almost-exact solution at the end of the talk (time-permitting).

17. The variance: simplified • (3) Idea: • Two distant sites will always be on a shared segment if there was no recombination event in their history. • If there was, treat sites as independent. • Neglect some small terms. • The probability of no recombination: • The variance: d≥m For the human genome,

18. The cohort-averaged sharing • The distribution is close to normal. • But with variance that approaches a constant even for large sample size n. Why? • Scales as 1/n for small n. • Approaches a constant for large samples. • For the human genome,

19. The tail of the cohort-averaged sharing- `hyper sharing’ • Even for large cohorts, the distribution of the cohort-averaged sharing retains a constant width. • Some individuals will be in the tails of this distribution!  ‘hyper sharing’. • Can be taken advantage of in sequencing studies.

20. Imputation by IBD • Our results can be used to calculate the expected imputation power when sequencing a subset of a cohort. • Assume a cohort of size n, ns of which are sequenced. • Random selection of individuals: • Selection of highest-sharing individuals: • where

21. Increase in association power • The imputed genomes can be thought of as increasing the effective number of sequences. • A simple model (Shen et al., Bioinformatics, 2011): • Variant appears in cases only. • Carrier frequency in cases equal β. • Dominant effect. • Association detected if P-valuebelow a threshold. • For a fixed budget, trade-off in the number of cases/controls to sequence.

22. Siblings • Siblings share, on average, 50% of their genomes. • What is the variance? • A classic problem. • (Visscher et al. PLoS Genet. 2006). • Used the variance to estimate heritability from siblings studies. • Genome-wide SD 5.5%. • But what if parents are inbred? • Assume shared segments are either from parents or are more remote.

23. Estimator of population size • Given one genome, estimate the population size N. • Calculate the total sharing fT. We know that • Invert to suggest an estimator: • Not very useful: estimator is biased • and has SD • Compared to for Watterson’s estimator (based on the number of het sites).

24. Ashkenazi Jews • In recent years, shown to be a genetically distinct group. • Close to Middle-Easterns and Europeans (particularly Italians and Adygei). • (Atzmon et al., Am. J. Hum. Genet., 2010) • Very large amounts of IBD (Gusev et al., Mol. Biol. Evol., 2011), likely due to a recent, severe bottleneck.

25. IBD in Ashkenazi Jews • 2,600 Ashkenazi Jews, 1M SNP array (Guha et al., Genome Biol. 2012). • Use Germline to detect IBD segments. • Compare the total sharing to simulations of inferred demography based on mean IBD in different length ranges (Palamara et al., AJHG, 2012). Excess of `hyper sharing’ in AJ

26. Admixture in AJ • Most plausible explanation: correct for admixture. When considering only individuals with close to median AJ ancestry, most of the unexplained variance disappears. The AJ component was calculated in comparison to CEU.

27. Summary • We calculated the distribution of the total IBD sharing in the Wright-Fisher model using renewal theory. • We obtained explicit expressions for the variance of both the pairwise sharing and the cohort-averaged sharing. • We calculated the expected gain in imputation and association power if individuals at the tail of the cohort-averaged sharing distribution are selected for sequencing. • The variance/distribution of IBD has many applications, some of which we presented, some are left for future work. • In the AJ population, individuals differ in cohort-averaged sharing by up to 30%. Admixture explains some of the variance.

28. The end • Thanks to: • Pier Francesco Palamara. • Vladimir Vacic • ItsikPe’er • Todd Lencz, Ariel Darvasi(for AJ genotypes) • Human Frontiers Science program Cross-Disciplinary Fellowship.

29. Identity-by-descent Identity-by-descent (IBD) founder chromosomes contemporary chromosomes

30. Mean IBD (Palamara et al.) • See (Palamara et al., AJHG, 2012). • Assume shared segments must have length at least m. • Define I(s): the indicator, with probability π, that site s is in a shared segment between two given chromosomes. • Define fT: the mean fraction of the chromosome found in shared segments, or the total sharing. • Given g, the number of generations to the MRCA: • In the coalescent, g→Nt: • Then, <fT>=π.

31. The variance of the total sharing (1) • The variance requires calculating two-sites probabilities. • Idea: • For one site, PDF of the coalescence time is Φ(t)~Exp(1). • For two sites, calculate the joint PDF Φ(t1,t2). • Φ(t1,t2) takes into account the interaction between the sites. • Given t1, t2, calculate π2 as if sites are independent.

32. The variance of the total sharing (2) • Express π2in terms of the Laplace transform of Φ(t1,t2). • π2 • Use the coalescent with recombination to findwhere A-E are defined in terms of q1, q2, and the scaled recombination rate ρ.

33. IBD in AJ Are `hyper-sharing’ individuals sharing more with everyone else, or just with other `hyper-sharing’ individuals? Each curve represents average of 1/7 of the individuals in order of their cohort-averaged sharing. Highest sharing Lowest sharing Highest sharing Lowest sharing