Variation in genetic identity within kinships

Abstract

In a recent paper of this journal, Rasmuson (1993) computed the standard deviation (s.d.) of the genetic identity of relatives, i.e. of the proportion of the genome which is identical by descent. For example, for parent and offspring the mean identity is 50 per cent and its s.d. is 0 per cent because the offspring inherits exactly one-half of the autosomal genes from each parent; whereas for grandparent and grandoffspring or for half-sibs, the mean is 25 per cent and, as there is segregation and recombination, the s.d. is not zero, Rasmuson giving a value of 2.5 per cent for each case. The variance of identity was computed byRasmuson on the assumption that the genotype comprised a number of independently segregating discrete units, the segregation index, which was set at 100, a value considered appropriate for human females (equal to the haploid chromosome number plus the total number of chiasmata). In her model it is, in effect, assumed that the genome comprises 100 chromosomes in each of which no recombination occurs, whereas recombination events are essentially continuously distributed along the chromosomes and occur at different locations in each generation. An equivalent problem to that considered byRasmuson for the case of unilateral descendants is the variance of the contribution of the non-recurrent parental line in a backcrossing programme. Coincidentally, I have recently derived a formula to compute this variance for a model that takes account of the distribution of recombinations (Hill, 1993) The variances are not the same for collateral relatives of corresponding mean identity. The reason is that, for two loci with recombination fraction c, the probability that a grandoffspring inherits genes at both loci from a grandparent is (1 -c)/2, whereas the probability that two haif-sibs inherit the same pair of genes is (1 -c)2/2 + c2/2. Only for c =0 (complete linkage) and c = 1/2 (no linkage) are these quantities equal. For halfsibs, the mean identity is E(PHS) = 1/4 and the variance isusing results of Franklin (1977). In our example s.d. (PHS) = 2.82 per cent, lower than s.d. (P2) for grandoffspring-grandparent. For full-sibs the variance is 2V(PHS), because the two parents' contributions are independent. Formulae can be computed for other relationships, but are more complicated and beyond the scope of this note. The methods of Weir and Cockerham (1969) could be employed, and also means and variances of identity of genotype derived for fullsibs, for example, rather than of genes as here.

Increased accuracy of artificial selection by using the realized relationship matrix

“…4 to 0 . 6 (Hill, 1993;Visscher et al, 2006). This variation in relationship comes about because sibs inherit large segments of chromosomes from their parents.…”

“…Formulae for the variation in realized relationship between the different types of relatives have been published by Hill (1993) and Guo (1996) and we will use their formulae.…”

“…However, the variance in G ij can also be calculated for a genome consisting of chromosomes of known length in Morgan. Hill (1993) and Guo (1996) present formulae for this. For instance, using their formula, if the genome consists of a single chromosome 35 M long, then the variance in relationship between pairs of full-sibs is 0 .…”

“…5. Although we have assumed a single chromosome here, Hill (1993) showed that the variation in relationship is not particularly affected by assumptions on the number of chromosomes, provided the total length of the genome was kept constant.…”

“…Elements of this matrix are derived as the predicted proportion of the genome that is identical by descent (IBD) among two individuals given their pedigree relationship. However, Mendelian sampling during gamete formation results in variation in the realized proportion of the genome, which is IBD between pairs of individuals with the same predicted relationship coefficients (Franklin, 1977;Hill, 1993 ;Guo, 1996). For example, between full-sib individuals the predicted proportion of the genome that is IBD is 0 .…”