Calculation of IBD probabilities

Transcription

1 Calculation of IBD probabilities David Evans University of Bristol

2 This Session Identity by Descent (IBD) vs Identity by state (IBS) Why is IBD important? Calculating IBD probabilities Lander-Green Algorithm (MERLIN) Single locus probabilities Hidden Markov Model > Multipoint IBD Other ways of calculating IBD status Elston-Stewart Algorithm MCMC approaches MERLIN Practical Example IBD determination Information content mapping SNPs vs micro-satellite markers?

3 Identity By Descent (IBD) Identical by Descent Identical by state only Two alleles are IBD if they are descended from the same ancestral allele

4 Example: IBD in Siblings Consider a mating between mother AB x father CD: Sib 2 Sib AC AD BC BD AC 2 0 AD 2 0 BC 0 2 BD 0 2 IBD 0 : : 2 25% : 50% : 25%

5 Why is IBD Sharing Important? 3/4 /2 3/4 /3 2/4 /4 4/4 Affected relatives not only share disease alleles IBD, but also tend to share marker alleles close to the disease locus IBD more often than chance 4/4 /4 IBD sharing forms the basis of nonparametric linkage statistics

6 Crossing over between homologous chromosomes

7 Cosegregation > Linkage A Q Parental genotype A Q Non-recombinant Parental genotypes (many, θ) A 2 Q 2 A Q 2 A 2 Q 2 Recombinant genotypes (few, θ) A 2 Q Alleles close together on the same chromosome tend to stay together in meiosis; therefore they tend be co-transmitted.

8 Segregating Chromosomes MARKER DISEASE GENE

9 Marker Shared Among Affecteds /2 3/4 3/4 /3 2/4 /4 4/4 4/4 /4 Genotypes for a marker with alleles {,2,3,4}

10 Linkage between QTL and marker QTL Marker IBD 0 IBD IBD 2

11 NO Linkage between QTL and marker Marker

12 IBD can be trivial / 2 / 2 IBD0 / 2 / 2

13 Two Other Simple Cases / 2 / 2 / 2 / 2 IBD2 / / 2 2 / 2 2 /

14 A little more complicated / 2 2 / 2 IBD (50% chance) / 2 / 2 IBD2 (50% chance)

15 And even more complicated IBD? / /

16 Bayes Theorem for IBD Probabilities j0,,2 j IBD G P j IBD P i IBD G P i IBD P G P i IBD G P i IBD P G P G i G i IBD P ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), P(IBD ) (

17 P(Genotype IBD State) Sib Sib 2 P(observing genotypes k alleles IBD) k0 k k2 A A A A p 4 p 3 p 2 A A A A 2 2p 3 p 2 p 2 p 2 0 A A A 2 A 2 p 2 p A A 2 A A 2p 3 p 2 p 2 p 2 0 A A 2 A A 2 4p 2 p 2 2 p p 2 2p p 2 A A 2 A 2 A 2 2p p 3 2 p p A 2 A 2 A A p 2 p A 2 A 2 A A 2 2p p 3 2 p p A 2 A 2 A 2 A 2 p 4 2 p 3 2 p 2 2

18 Worked Example p 0.5 P( G IBD 0) P( G IBD ) P( G IBD 2) P( G) P( IBD 0 G) / P( IBD G) / P( IBD 2 G)

19 Worked Example / / 9 4 ) ( 4 ) 2 ( 9 4 ) ( 2 ) ( 9 ) ( 4 ) 0 ( ) ( 4 2) ( 8 ) ( 6 0) ( G P p G IBD P G P p G IBD P G P p G IBD P p p p G P p IBD G P p IBD G P p IBD G P p

20 For ANY PEDIGREE the inheritance pattern at any point in the genome can be completely described by a binary inheritance vector of length 2n: v(x) (p, m, p 2, m 2,,p n,m n ) whose coordinates describe the outcome of the paternal and maternal meioses giving rise to the n non-founders in the pedigree p i (m i ) is 0 if the grandpaternal allele transmitted p i (m i ) is if the grandmaternal allele is transmitted a / b c / d v(x) [0,0,,] a / c b / d

21 Inheritance Vector In practice, it is not possible to determine the true inheritance vector at every point in the genome, rather we represent partial information as a probability distribution of the possible inheritance vectors a b 2 a c p m p 2 3 a b 5 a c 4 m 2 b b Inheritance vector Prior Posterior /6 /8 000 /6 /8 000 / / /6 /8 00 /6 /8 00 /6 0 0 / /6 /8 00 /6 /8 00 /6 0 0 / /6 /8 0 /6 /8 0 /6 0 /6 0

22 Computer Representation At each marker location l Define inheritance vector v l Meiotic outcomes specified in index bit Likelihood for each gene flow pattern Conditional on observed genotypes at location l 2 2n elements!!! L L L L L L L L L L L L L L L L

23 a) bit-indexed array L L 2 L L 2 L L 2 L L 2 b) packed tree L L 2 L L 2 L L 2 L L 2 c) sparse tree Legend Node with zero likelihood Node identical to sibling L L 2 L L 2 Likelihood for this branch Abecasis et al (2002) Nat Genet 30:97-0

24 Multipoint IBD IBD status may not be able to be ascertained with certainty because e.g. the mating is not informative, parental information is not available IBD information at uninformative loci can be made more precise by examining nearby linked loci

25 Multipoint IBD a / b / c / d 2 / IBD 0 a / c b / d / / 2 IBD 0 or IBD?

26 Complexity of the Problem in Larger Pedigrees For each person 2n meioses in pedigree with n non-founders Each meiosis has 2 possible outcomes Therefore 2 2n possibilities for each locus For each genetic locus One location for each of m genetic markers Distinct, non-independent meiotic outcomes Up to 4 nm distinct outcomes!!!

27 Example: Sib-pair Genotyped at 0 Markers Inheritance vector P(G 0000) ( θ) m 0 Marker (2 2xn ) m (2 2 x 2 ) 0 ~ 0 2 possible paths!!!

28 P(IBD) 2 at Marker Three IBD Inheritance vector (2) () () (2) m 0 Marker (L[0000] + L[00] + L[00] + L[] ) / L[ALL]

29 P(IBD) 2 at arbitrary position on the chromosome Inheritance vector m 0 Marker (L[0000] + L[00] + L[00] + L[] ) / L[ALL]

30 Lander-Green Algorithm The inheritance vector at a locus is conditionally independent of the inheritance vectors at all preceding loci given the inheritance vector at the immediately preceding locus ( Hidden Markov chain )

31 Lander-Green Algorithm Inheritance vector m 0 Marker M(2 2n ) 2 0 x calculations

32 Lander-Green Algorithm Summary Factorize likelihood by marker Complexity m e n Strengths Large number of markers Relatively small pedigrees

33 Elston-Stewart Algorithm Factorize likelihood by individual Complexity n e m Small number of markers Large pedigrees With little inbreeding VITESSE, FASTLINK etc

34 Other methods Number of MCMC methods proposed ~Linear on # markers ~Linear on # people Hard to guarantee convergence on very large datasets Many widely separated local minima E.g. SIMWALK

35 MERLIN-- Multipoint Engine for Rapid Likelihood Inference

36 Capabilities Linkage Analysis NPL and K&C LOD Variance Components Haplotypes Most likely Sampling All IBD and info content Error Detection Most SNP typing errors are Mendelian consistent Recombination No. of recombinants per family per interval can be controlled Simulation

37 MERLIN Website Reference FAQ Source Tutorial Linkage Haplotyping Simulation Error detection IBD calculation Binaries

38 Input Files Pedigree File Relationships Genotype data Phenotype data Data File Describes contents of pedigree file Map File Records location of genetic markers

39 Example Pedigree File <contents of example.ped> 0 0 x 3 3 x x x 4 4 x x x 2 x x x 4 3 x x <end of example.ped> Encodes family relationships, marker and phenotype information

40 Data File Field Codes Code M A T C Z S[n] Description Marker Genotype Affection Status. Quantitative Trait. Covariate. Zygosity. Skip n columns.

41 Example Data File <contents of example.dat> T some_trait_of_interest M some_marker M another_marker <end of example.dat> Provides information necessary to decode pedigree file

42 Example Map File <contents of example.map> CHROMOSOME MARKER POSITION 2 D2S D2S <end of example.map> Indicates location of individual markers, necessary to derive recombination fractions between them

43 Worked Example p 0.5 P( IBD 0 G) P( IBD G) P( IBD 2 G) / / merlin d example.dat p example.ped m example.map --ibd

44 Application: Information Content Mapping Information content: Provides a measure of how well a marker set approaches the goal of completely determining the inheritance outcome Based on concept of entropy E -ΣP i log 2 P i where P i is probability of the ith outcome I E (x) E(x)/E 0 Always lies between 0 and Does not depend on test for linkage Scales linearly with power

45 Application: Information Content Mapping Simulations ABI ( micro-satellite per 0cM) decode ( microsatellite per 3cM) Illumina ( SNP per 0.5cM) Affymetrix ( SNP per 0.2 cm) Which panel performs best in terms of extracting marker information? merlin d file.dat p file.ped m file.map --information

46 SNPs vs Microsatellites Information Content Densities SNP microsat 0.2 cm 3 cm 0.5 cm 0 cm Position (cm) SNPs + parents microsat + parents