Relatedness 1: IBD and coefficients of relatedness

Transcription

1 Relatedness 1: IBD and coefficients of relatedness or What does it mean to be related? Magnus Dehli Vigeland NORBIS course, 8 th 12 th of January 2018, Oslo

2 Plan Introduction What does it mean to be related? Pedigree-based meassures of relatedness: The concept of IBD (identical by descent) Coefficient of kinship/inbreeding The IBD coefficients k 0, k 1, k 2 Jacquard's 9 identity coefficients Relatedness coefficients in R

3 What does it mean to be related? Sosial antropology definition: being conected through a pedigree having a common ancestor...not too far back Definition based on "genetic intuition": sharing DNA? (more than unrelated people) To make all this precise, we need some terminology...

4 IBD and autozygosity IBD = "Identical by descent" = identical alleles with a common origin in the given pedigree AB AB AB BB IBS = "Identical by state" AB AA = identical alleles autozygous = homozygous + IBD AA autosomet

5 Relatedness yellow belt: Coefficient of kinship/inbreeding P Q Wright (1921): The kinship coefficient φ between P and Q R Sewall Wright ( ) Mendel's 1. law! φ P,Q = P( random allele of P is IBD with random allele of Q ) = P( R receive IBD alleles from her parents ) = P( R is autozygous ) P and Q related = f R the inbreeding coefficient of R φ P,Q > 0

6 Examples A/B C/D A/B A A A/A Same for C and D f = P(A/A autozygous) 2 f = P(A/A autozygous) 2 2 = = 1 8 A or B = = = 1 16 A or B

7 Wright's path formula Simple form works in most cases: f R = φ P,Q = v 1 2 v +1 Translation: Find all paths v between P and Q For each path compute 0.5 Take the sum! v +1 P R Q Example: 2 paths (length = 4) (length = 4) φ = = 1 16 =

8 Wright's path formula in full generality φ P,Q = A v 1 2 v +1 (1 + f A ) sum over all common ancestors A of P and Q and all non-collapsing paths v fra P til Q via A v is the length of v f A is the inbreeding coefficient of A Applicable to any pairwise relationship, however complex!

9 Interpretations of the inbreeding coefficient autosomal f = P( random locus autozygous ) = expected fraction of the genome that is autozygous Parents father/daughter full sibs f (of child) 1/4 uncle/niece 1/8 first cousins 1/16 second cousins 1/64 third cousins 1/256 father/daughter incest: f = 0.25

10 Red belt: The IBD triangle Charles Cotterman ( ) Elisabeth Thompson ( )

11 IBD coefficients: Warm-up Summary so far: Two individuals are related if they can have IBD alleles Their kinship coefficient meassures the amount of IBD sharing Natural generalisation: How many alleles are IBD in each locus? A/B C/D Humans are diploid IBD = 0, 1 or 2 A/C A/D 1 allele IBD A/C A/C 2 alleles IBD A/C B/D 0 alleles IBD

12 IBD coefficients: Definiton Given two (non-inbred) individuals */* */* For a random autosomal locus k 0 = P(0 alleles IBD) k 1 = P(1 alleles IBD) k 2 = P(2 alleles IBD) IBS = Identical by state IBD = Identical by descent Note: k 0 + k 1 + k 2 = 1

13 Example 1: Parent vs child A/B?/? */* k 0 = 0 k 1 = 1 k 2 = 0 Note the difference between IBD and IBS: A/A A/A Don't be deceived by appearences! A/A

14 More "trivial" examples MZ twins Unrelated inviduals A/B A/B A/B C/D k 0 = 0 k 1 = 0 k 2 = 1 k 0 = 1 k 1 = 0 k 2 = 0

15 The case of full siblings A/B C/D k 0 = k 1 = k 2 = A/C

16 The relatedness triangle Impossible region! All relationships satisfy k 12 4k 0 k 2 Recall: k 0 + k 1 + k 2 = 1

17 The relatedness triangle A/B A/B A/B All have k 0 = k 1 = 0.5 Recall: k 0 + k 1 + k 2 = 1 NB: Some relationships coincide!

18 An important identity φ = 1 4 k k 2

19

20 After a short coffee break: Black belt: Jacquard's identity coefficients Albert Jacquard ( )

21

22 Black belt: Jacquard's identity coefficients Jacquard (1970): Structures Génétiques des Populations Motivation: Inbred relationships k 0, k 1, k 2 are not well defined Example: A/B A/C Albert Jacquard ( ) A/A What's the IBD status here??? 1 or 2? Cannot be summarised in one number. Configuration: A father's alleles A A child's alleles

23 Jacquard's 9 coefficients Two individuals, autosomal locus: alleles of individual 1 alleles of individual 2 9 possible IBD configurations: Any pairwise relationship can be summarised by the relative frequencies of each of these. Jacquard's condensed identity coefficients Δ 1, Δ 2,..., Δ 9

24 Very simple with non-inbred individuals A/B C/D k 2 k 1 k 0

25 Non-trivial example Suppose the parents are first cousins, but not themselves inbred The Jacquard coefficients for the father vs child:

26 Relatedness coefficients in R/paramlink Functions for computing coefficients: inbreeding(x) jacquard(x, ids) Visualisation in the IBD triangle IBDtriangle() showintriangle(k0, k2)