Statistical methods in genetic relatedness and pedigree analysis Oslo, January 2018 Magnus Dehli Vigeland and Thore Egeland Exercise set III: Coecients of pairwise relatedness Exercise III-1. Use Wright's path formula to compute the kinship coecient of a) Uncle - niece b) First cousins once removed Exercise III-2. Use the inbreeding function of paramlink to compute the inbreeding coecient of a child whose parents are a) Uncle - niece. b) First cousins once removed. (Hint: cousinsped(degree=1, removal=1, child=true).) Explain why the answers are the same as the kinship coecients you computed in Exercise III-1. Exercise III-3. In a case of incest a man had a child by his own granddaughter. Use paramlink to draw the pedigree and compute the inbreeding coecient of the child. Exercise III-4. a) What is the kinship coecient between monozygotic twins? b) Can you think of a relationship with kinship coecient ϕ = 1? Exercise III-5. Consider the following pedigree: a) Describe the relationship between the children in the last generation. Are the children inbred? b) Use Wright's path formula to compute the kinship coecient between the children. c) Show that the IBD coecients between the children are k = (3/8, 1/2, 1/8). (Hint: Do k 2 rst, then k 0, and nally k 1.) d) Use part c) and the formula ϕ = 1 4 k 1 + 1 2 k 2 to check your answer in b). e) Place the relation in the IBD triangle.
Exercise III-6. Consider the following pedigree: a) Describe the relationship between the children in the last generation. b) Use Wright's path formula to compute the kinship coecient between the children. c) One can show that the IBD coecients between the children are k = (7/16, 1/2, 1/16). Place the relation in the IBD triangle. Exercise III-7. The pedigree shows two half siblings whose mother has inbreeding coecient f. a) Show that the IBD coecients of the half siblings are k = ( 1 2 1 2 f, 1 2 + 1 2f, 0). b) Where in the IBD triangle do such relationships belong? c) Suppose the mother is 100% inbred (e.g. from an inbred mouse strain). Place the resulting half-sibling relationship in the IBD triangle, and comment on the result. Exercise III-8. In the pedigree below the alleles have the same origin, but a mutation has happened on the right hand side, turning the A into a B. Two scientists argue about the IBD status of the alleles in this case. Scientist 1: The alleles are clearly not IBD. They are aren't even IBS! Scientist 2: You are deceived by appearance. IBD is not dened pointwise, but for segments! Since the alleles have the same origin, their immediate surroundings are IBD, and that's what counts. Mutations are just ignorable noise. Discuss the statements. Scientist 1 is a medical geneticist, while Scientist 2 is a relatedness researcher. How is this reected in their point-of-views?
Exercise III-9. There is no function in paramlink for computing the IBD coecients (k 0, k 1, k 2 ) directly, but we can easily make one by exploiting their connection to the Jacquard coecients. Recall that for any relationship of two non-inbred individuals we have k 0 = 9, k 1 = 8 and k 2 = 7. a) Create a function called ibd.coeffs as follows. (Note: This requires the identity package.) ibd.coeffs = function(x, ids) jacquard(x, ids)[9:7] b) Test the function on various relationship where you already know the answer. For example full siblings: x = nuclearped(2) ibd.coeffs(x, 3:4) c) Use the function to compute the IBD coecients of double second cousins. Show the result in the IBD triangle. You can use the following code: x = doublecousins(2, 2) plot(x) k = ibd.coeffs(x, 17:18) k showintriangle(k[1], k[3], label="double 2nd cousins", pos=3) d) Show that k 2 1 = 4k 0k 2, i.e., that the point lies on the border of the unattainable region. Jacquard coecients Exercise III-10. Figure 1 shows the nine possible identity states for pairwise relatedness. For each pair of genotypes below, what is the corresponding state? We assume here that alleles have the same name only if they are IBD. a) A/B A/B b) A/A A/B c) 13/17 15/15 d) C/D A/B Exercise III-11. The assumption that IBS IBD in the previous exercise is rarely fulllled in practical applications. (When observing two "equal" alleles, we usually don't know if they derive from an ancestral allele or have arisen from independent mutation events.) Hence a more appropriate question is: What are the possible Jacquard states for each genotype pair? a) A/B A/B b) A/A A/B c) 13/17 15/15 d) C/D A/B Exercise III-12. Consider the relationship between a pair of siblings whose parents are rst cousins. a) Exactly one of the nine identity states is impossible for this relationship. Which one? b) Conrm the answer in a) by computing the Jacquard coecients in paramlink.
Figure 1: Jacquard's 9 condensed identity states Exercise III-13. Find the Jacquard coecients of a pair of siblings whose parents are full siblings. Verify that all 9 coecients are non-zero. (This is the simplest example of a relationship with this property.) Exercise III-14. This exercise examines the Jacquard coecients of various parent-child relationships. a) Recall the Jacquard coecients between a parent and his/her child, in the absence of inbreeding. b) Suppose a child is inbred (i.e. the parents are related), but the parents are not themselves inbred. Which of the nine identity states are now possible between either parent and the child? (Hint: No calculations are needed. For each of the nine states, either show that it is possible, or nd a simple argument that it is not.) c) A man has a child by his own daughter. Compute the Jacquard coecients of the relationship between the mother and the child. d) Suppose a woman has inbreeding coecient f and has a child with a man unrelated to her. What are the Jacquard coecients between the mother and child?
The IBD triangle