Genetic identity and Kinship

1. Genetic identity and Kinship

2. Genetic Identity • G1 and G2 are identical by descent (i.b.d) if they are physical copies of the same ancestor, or one of the other. • G1 and G2 are identical by state (i.b.s) if they represent the same allele. • The kinship between two relatives fij is the probability that random gene from autosomal loci in I and j are i.b.d. • The interbreeding coefficient is the probability that his or her two genes from autosomal loci are i.b.d.

3. A1A1 A1A2 A1A1 A1A2 A1A2 A2A2 Mutation occurred once • Every mutation creates a new allele • Identity in state = identity by descent (IBD)

4. A1A1 A1A1 A1A1 A1A2 A1A2 A1A1 A1A2 A1A2 A1A2 A1A2 A2A2 A2A2 A2 A2 IBD A2 A2 IBD A2 A2 alike in state (AIS) not identical by descent The same mutation arises independently

5. Identity by descent A - B C - D | | P - Q | X Let fAC be the coancestry of A with C etc., i.e. the probability of 2 gametes taken at random, 1 from A and one from B, being IBD. Probability of taking two gametes, 1 from P and one from Q, as IBD, FX

6. Identity by descent • Example, imagine a full-sib matingA - B / \P - Q | X • Indv. X has 2 alleles, what is the probability of IBD?

7. Identity by descent • Example, imagine a half-sib matingA - B - C | | P - Q | X

8. Kinship and Interbreeding • fii=0.5(1+fi) • fi=fkl, where k and l are the parents of i. • If fi>0 then i is said to be inbred. • The question is how to compute kinship, given a pedigree. • Let us look for example at brothers and sisters:

9. 1 2 3 4 6 5 Pedigree • Let us compute the kinship coefficients for all member of this pedigree. We assume that 1 and 2 are not inbred and unrelated, we start counting from the oldest generation. • We will develop an algorithm to compute fij for all members of this pedigree.

10. 1 2 3 4 6 5 Kinship coefficient algorithm

11. 1 2 3 4 6 5 Kinship coefficient algorithm II

12. 1 2 3 4 6 5 Kinship coefficient algorithm III If i originates from k and l Fii= ½+Fkl

13. 1 2 3 4 6 5 Kinship coefficient algorithm III If i originates from k and l Fij= Fji = ½(Fjk +Fjl)

14. 1 2 3 4 6 5 Kinship coefficient algorithm IV One can now reapply the algorithm on the next generation

15. 1 2 3 4 6 5 Kinship coefficient algorithm V The final result is:

16. Identity coefficients: • We can now summarize the kinship coefficient of some basic family relations:

17. Detailed Identity States – I Allele 1 Allele 2 I J

18. Detailed Identity States – I I

19. Detailed Identity States – I I I

20. Summary • Many of these relations are redundant. If I is not inbred 1,2,3 and 4 will be zero. • One can define kinship in a condensed mater, if we can interchange the maternal and paternal genes.

21. Condensed Identity States

22. Condensed Identity States • S3=S*2S*2 • S5=S*4S*5 • S7=S*9S*12 • S8=S*10S*11 S*13S*14

23. D1 D2 D3 D4 D5 D6 D7 D8 Condensed Identity States II • D1 , D2 , D3 ,D4 are 0, when i is not inbred. • D1 , D2 , D5 ,D6 are 0, when j is not inbred. • D1 , D3 , D5 ,D7 and D8 are 0, when i and j are unrelated. • Fji=D1+1/2( D3 + D5 +D7)+1/4 D8

24. Kinship and identity coefficients

25. Genotype prediction. • What is the probability that i has a given genotype, given the genotype of j ? • For example, If my uncle has a genetic disease, what is the probability that I will also have it? • What are the probabilities of brothers from inbred parents to be homozygous for a disease causing gene? • ……

26. Genotype prediction. If I is heterozygous, with an inbreeding coefficient fi

27. Genotype prediction II If I is homozygous, with an inbreeding coefficient fi

28. Genotype prediction III When j is independent of i, it only follows the H,W equilibrium. When j is equivalent to i, the probability is one if m/n=k/l and zero otherwise. When j shares one allele with I, m/n and k/l must overlap with one allele and the other one has H.W distribution.

29. Example • What is the blood type of non-inbred siblings?

30. Example I • What is the blood type of non-inbred siblings?

31. Risk Ratios and Genetic Model Discrimination. • Let us assume that each person in the population is assigned a factor of X=1 if he/she is affected by a condition and X=0 otherwise. • The Prevalence of the condition is K=E(X). • Given two non-inbred relatives i and j and given that i is affected, what is the probability that J is affected? • KR=P(Xj=1|Xi=1( • P(Xj=1,Xi=1) = P(Xj=1|Xi=1(P(Xi=1) = KRK = E(XiXj)

32. Risk Ratios and Genetic Model Discrimination. • P(Xj=1|Xi=1) = E(XiXj)/K = (cov(Xi,Xj)+K2)/K = cov(Xi,Xj)/K+K • This result simply represents the fact that the extra risk for j results from the covariance of X between i and j. • The risk ratio can thus be defined as: • lR= cov(Xi,Xj)/K2 • Let us compute this covariance, and following it the risk ratio.

33. D1 D2 D3 D4 D5 D6 D7 D8 Covariance • In a more general way, let us assume that i and j are non-inbred relatives. The covariance between their genes is defined only by condensed identity states 7,8 and 9

34. Covariance • Let us assume that a given property is defined by a single gene with multiple alleles. For the sake of simplicity let us normalize E(x)=0, and divide:

35. Covariance

36. Risk Ratio

37. Summary Concepts Kinship coefficient. Inbreeding coefficient Identity states Condensed identity states Identity coefficients Genotype prediction