Sharing of long genomic segments:Theory and results in Ashkenazi Jews

Bar-Ilan UniversityJuly 26, 2012

Shai CarmiItsik Pe’er’s lab

Department of Computer ScienceColumbia University

Outline

• Introduction: Identity-by-descent (IBD) sharing• Theory of IBD sharing

– The Wright-Fisher model and coalescent theory– The distribution of the total sharing– The cohort-averaged sharing

• Applications– Imputation by IBD– Siblings

• Jewish genetics– Background– IBD and ancient demography– The Ashkenazi Sequencing Project

• Summary

Genetic drift

• The number of offspring of each individual is random.• All pairs of individuals descend from a common ancestor.

Identity-by-descent (IBD)• When the population is small, the common ancestors are

frequently recent.• Abundance of long haplotypes which are IBD.

A B

AB

A shared segment

IBD detection• Until last decade, IBD usually defined for single markers.• Genome-wide SNP arrays enable detection of long segments.• GERMLINE (Gusev et 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

IBD applications

• Demographic inference (Palamara et al., AJHG, 2012).• Phasing (Palin et al., Genetic Epi., 2011).• Imputation (Gusev et al., Genetics, 2012).• Positive selection detection (Albrechtsen et al., Genetics, 2010).• Disease mapping (Browning and Thompson, Genetics, 2012).• Pedigree reconstruction (Huff et al., Genome Res., 2011).

A G

C T

The cell

A,C G,T

SNP array

A G

C T

A T

C G

?

IBD in Ashkenazi Jews

• Links connect individuals with shared segments.• (Gusev et al., Mol. Biol. Evol., 2011)

Ashkenazi JewishOther European

Imputation by IBD

• 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)).

Wright-Fisher model and the coalescent

• Non-overlapping, discrete generations.• A population of constant size of N

haploid individuals.• Ignore mutations (when studying IBD).• Recombination is a Poisson process with

rate 1 per Morgan.

• The coalescent:• Each pair of individuals (linages) has

probability 1/N to coalesce in the previous generation.

• Scale time: t←g/N. • .

N=10

t

Theory: mosaic of segments

• 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.

• Segment length distribution: (derivation not shown).

ℓ1

0 Lcoordinate

ℓ2ℓ3 ℓ4

ℓ5 ℓ6ℓ7 ℓ8

ℓ9 ℓ10ℓ11

m ℓT=ℓ1+ℓ5+ℓ9

A

B

Renewal theory

τ1

0 Ttime

τ2τ3 τ4

τ5 τ6τ7 τ8

τ9 τ10τ11

m tS =τ1+τ5+τ9

A

B

• Start at and .• Draw waiting time τ from the distribution .• Set .• As long as , set .

Renewal theory: solution

• Laplace transform T→s, tS→u

Mean IBD sharing

• The average number of segments ≥m is 2NL·P(ℓ≥m).

•

•

•

• For large N, <fT>≈1/(mN).

• Alternative derivation at the end of the talk (time-permitting).

The variance of the IBD sharing• (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).

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:

For the human genome,

d≥m

The cohort-averaged sharing

• The distribution is close to normal.• With variance:

• Scales as 1/n for small n.• Approaches a constant for large samples.

• Some individuals will be in the tails of this distribution! ‘hyper sharing’.

‘hyper-sharing’

Imputation by IBD

• 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

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.

Ashkenazi Jewish brief history

• End of 1st millennium:• Small Jewish communities in the Rhineland.

• 1096: Crusades.• 12-13th centuries:

• First Jewish communities in Eastern Europe.• Few thousands of individuals.

• 16-19th centuries:• The demographic miracle: exponential growth.• Prewar: about 10 million, 90% of all Jewish people.

Ashkenazi Jewish genetics• In recent years, AJ 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)• 300 Jews in 900k SNPs.

Ashkenazi Jewish genetics• Bray et al., PNAS, 2010.• 471 AJ in 700k SNPs.

• Need et al., Genome Biology, 2009.• ~100 AJ in 550k SNPs.

• Kopelman et al., BMC Genetics, 2009.• 80 AJ in 700 microsatellites.

Ashkenazi Jewish genetics• Behar et al., Nature, 2010.• ~120 Jews in 600k SNPs.

• Khazar theory incompatible.• European admixture ~20%

(but 30-50% according to other studies).• No genetic sub-structure.• AJ diseases likely due to founder effect (no selection)

• Guha et al., Genome Biology, 2012.• ~1312 AJ in 740k SNPs.

AJ

EU

ME

AJ different countries

Ashkenazi Jewish genetics

IBD in Ashkenazi Jews• Inference of AJ history • (Palamara et al., AJHG, 2012)• 2,600 AJ, 700k SNPs.

• Detect IBD segments and calculate their distribution.

• Use IBD theory to obtain an initial guess of the demographic parameters.

• Grid search around initial guess: Compare sharing in simulations of different demographies and the mean IBD in different length ranges.

• IBD is particularly informative on recent history.

AJ (genetic) history

Expansion rate ≈1.34

3,000

N

t

Effective size

60,000

300

5,000,000

Years ago

800

Present

AJ sequencing

• Why Sequencing?

• Rare variants (no ascertainment bias)• Copy-number variants• Functional variants

• Improve power of demographic inference

• Improve understanding of recent population explosion

• Natural selection (positive/negative)

• Jewish disease genes

• Higher power in disease mapping?

The Ashkenazi Genome Consortium• Labs:

• Lencz, Atzmon, Cho, Clark, Ostrer, Ozelius, Peter, Darvasi, Offit, Pe’er

• Columbia, Einstein, Mount Sinai, MSKCC, Yale, HUJI

• Phase I:• 137 healthy AJ genomes, 40 AJ

Schizophrenia patients• 25/7/2012: 77 delivered (48+29)• Samples: ~60yo, multi-disease controls• Technology: Complete Genomics• Cost: about $2500/genome

• Phase II (2013):• Sequence the entire bottleneck (300-400 individuals).

Sample selection

• Remove relatives

• Remove non-AJ individuals

• Select individuals to maximize utility for imputation.

Backup and distribution pipeline• Raw size: 300GB/genome (60TB/project).• Variant calls and summaries: 1.5GB/genome

(300GB/project).

• Pipeline:• Checksum disks• Copy entire data to a fault tolerant, network

distributed file system (MooseFS).• Checksum copy• Backup entire data also in Einstein and

Columbia Medical School.• Distribute variant calls, summaries, and new

processed files in a dedicated server.

• Combine all genomes (VCF, Plink).• Phasing (statistical + molecular).

Quality control

Property Value (exome)Fraction called 96.6% (98%)

Coverage 55x

Fraction with coverage > 20x 93% (95%)

Concordance with SNP array 99.87%

Ti/Tv ratio 2.14 (3.05)

First 48 healthy individuals

• Quality usually uniform across all individuals.• One female with triple X chromosome.• A few with likely many false CNVs.• Two inbred individuals.

• Use to calibrate error rate: 800 heterozygous variants (400 SNPs) in a 45MB homozygote region.

VariantsProperty Value (exome)

Total SNPs 3.4M (22k)

Novel SNPs 3.7% (3.9%)

Het/hom ratio 1.64 (1.67)

Insertions count 224k (243)

Deletions count 239k (219)

Substitutions count 82k (369)

Synonymous SNPs 10520

Non-synonymous SNPs 9680

Nonsense SNPs 71

Other disrupting 241

CNV count 348

SV count 1489

MEI count 3491

AJ and Europeans

• 13 Complete Genomics public genomes.• Some quality differences.• Similar number of variants of all kinds.• het/hom ratio: 1.64 vs. 1.59.

• Upcoming data from 33 Flemish genomes.

• Minor differences.• More variants in AJ.• More allele sharing.• More population

specific variants.

Summary

• Identity-by-descent (IBD) theory:• IBD is an important tool in population genetics.• We developed theory of IBD sharing and a few applications.

• Ashkenazi Jewish (AJ) genetics:• AJ are genetically distinct and homogeneous group, close to

Europeans and Middle-Easterns.• Demographic inference using IBD revealed a severe

bottleneck.• We began The Ashkenazi Genome Project to sequence the

majority of genetic variation in AJ and provide a reference panel for disease mapping.

• Initial results available for QC, variant statistics, and comparison to Europeans.

The endThanks to:• Itsik Pe’er• IBD:

• Pier Francesco Palamara• Vladimir Vacic

• AJ sequencing:• Todd Lencz (LIJMC)• Gil Atzmon, Harry Ostrer (EIN.)• Lorraine Clark (CU)

• Funding:• Human Frontiers Science program Cross-

Disciplinary Fellowship.

Identity-by-descent (IBD)

founder chromosomes

contemporary chromosomes

Identity-by-descent

Mosaic of segments

• 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.

ℓ1

0 Lcoordinate

ℓ2ℓ3 ℓ4

ℓ5 ℓ6ℓ7 ℓ8

ℓ9 ℓ10ℓ11

m ℓT=ℓ1+ℓ5+ℓ9

A B

t

A

B

AB

Mosaic of segments

• Li and Durbin (Nature, 2011) found that at the end of a segment, • Therefore,

ℓ1

0 Lcoordinate

ℓ2ℓ3 ℓ4

ℓ5 ℓ6ℓ7 ℓ8

ℓ9 ℓ10ℓ11

m ℓT=ℓ1+ℓ5+ℓ9

A

B

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.

Varying population size

• Use results of Li and Durbin (Nature, 2011).

and then proceed as before. • The mean IBD sharing:

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.

The variance of the total sharing (2)

• Express π2 in terms of the Laplace transform of Φ(t1,t2).

• π2

• Use the coalescent with recombination to find

where A-E are defined in terms of q1, q2, and the scaled recombination rate ρ.

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-value

below a threshold.• For a fixed budget, trade-off in the

number of cases/controls to sequence.

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).

IBD in AJAre `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

Complete Genomics WGS