## CALCULATION OF THE COEFFICIENT OF RELATIONSHIP (R) FOR SEX-LINKED GENES

In a situation where all the genes in an organism are sex-linked (as they are in honey bees), then the coefficient of relationship between relatives is influenced by the sexes of those relatives and of any intermediate individuals linking them. The relevance of R for sex-linked genes is that sex-linkage causes a slight inaccuracy in R for the whole individual in humans, because the measure, based on Σ(1/2)n, assumes that all the genes are autosomal.

However, any differences in R between different relatives, for sex-linked genes, are overwhelmed by the much larger numbers of autosomal genes on which R is based. Since only one out of 23 pairs of chromosomes is involved, the overall error is probably much less than 5%, and in many cases less than the sampling errors. The situation is different for some other species, e.g in Drosophila, the fruit fly, which has only 4 pairs of chromosomes, sex-linked genes make up about 1/4 of the total number.

The following discussion and calculations, which are summarised in Table 6, show how the R values between various relatives are estimated for sex-linked genes by path analysis. They indicate the likely direction and magnitude of any inaccuracies in R caused by genes on the X chromosome.

A male's sex-linked inheritance comes exclusively from his mother. Because she has two X chromosomes and he has only one, 100% of his sex-linked genes are the same as 50% of hers. Therefore, the coefficient of relationship between mother and son is the geometric mean of these two figures, i.e. 70.7% (see here in the preceding monograph for an explanation).

The sex-linked genes of a female come equally from both parents. However, since the father has only one X chromosome, it is predetermined which genes the daughter will receive from him. Since 100% of the sex-linked genes in the father are identical with 50% in the daughter, the relationship between father and daughter is also 70.7%.

As pointed out in the previous monograph, mother and daughter have a 50% relationship for both sex-linked and autosomal genes; and since a son receives no X chromosomes from his father, the father-son relationship is 0.

Thus, the relationship between parents and children is higher between opposite sexes than between the same sex. These four relationships between parents and their children, can be used to calculate coefficients of relationship between all the other relatives, using path analysis. The parent-offspring R values between each generation are multiplied together for each extended path. If there is more than one path, the total values for each path are then added together. The coefficients of relationship are summarised below for each of the four parent-offspring relationships:

Figure 7 Parent and Child Relationships for Sex-linked Genes

The diagrams for full sibs, half sibs and uncles/aunts - nephews/nieces follow, (Figures 8 to 12).

Figure 8 Full Sib Relationships for Sex-linked Genes

Figure 9 Paternal Half Sib Relationships for Sex-linked Genes

Figure 10 Maternal Half Sib Relationships for Sex-linked Genes

Figure 11 Paternal Uncle and Aunt Relationships for Sex-linked Genes

Figure 12 Maternal Uncle and Aunt Relationships for Sex-linked Genes

Coefficients of relationship for sex-linked genes can be worked out for grandparents and single first cousins in the same way. The values for several relationships are summarised in Table 6.

Table 6 Coefficients of Relationship When All the Genes Are Either Sex-linked or Autosomal

 Relatives All Genes Sex-linked All Genes Autosomal Male Probands Female Probands (Males and Females) Parents Father .000 .707 .500 Mother .707 .500 .500 Full Sibs Brother .500 .354 .500 Sister .354 .750 .500 Half Sibs Paternal Half Brother .000 .000 .250 Paternal Half Sister .000 .500 .250 Maternal Half Brother .500 .354 .250 Maternal Half Sister .354 .250 .250 Uncles and Aunts Paternal Uncle .000 .354 .250 Paternal Aunt .000 .250 .250 Maternal Uncle .250 .177 .250 Maternal Aunt .530 .375 .250 Grandparents Paternal Grandfather .000 .000 .250 Paternal Grandmother .000 .500 .250 Maternal Grandfather .500 .354 .250 Maternal Grandmother .354 .250 .250 First Cousins Paternal Uncle's Son .000 .000 .125 Paternal Uncle's Daughter .000 .250 .125 Paternal Aunt's Son .000 .177 .125 Paternal Aunt's Daughter .000 .125 .125 Maternal Uncle's Son .000 .000 .125 Maternal Uncle's Daughter .177 .125 .125 Maternal Aunt's Son .375 .265 .125 Maternal Aunt's Daughter .265 .188 .125

Conclusions

The highest value of R for sex-linked genes is 0.75 between two full sisters. That is why sterile worker bees, ants and wasps, whose genes are all sex-linked, sacrifice their own reproductive future to ensure the continuing success of their mother, the queen. The workers have a better chance of perpetuating their own genes by helping their mother to produce more of their own full sisters (R = 0.75), rather than breeding offspring of their own (R = 0.5). This strategy only works, however, when the queen is made to produce more females than males. Surveys, with ants, have shown that the female to male ratio is around 3 : 1, which turns out to be the ideal ratio from the workers' point of view, in maximising their contribution of genes to future generations, (Dawkins, 1976).

The highest value of R, relative to the one for autosomal genes, is that for a single first cousin relationship between a male and his maternal aunt's son, at .375, compared with only .125 for autosomal genes.

The lowest value is 0, which is shared by 1/3 of all the relatives. In these cases, the connecting pathways are broken by two or more males in succession. The only relatives without a 0 value are full sibs, where the lowest figure is .354, and is because they are always related through their mother.

Apart from mother and daughter, other relationships in Table 6 which are the same for both sex-linked and autosomal genes, are between: two full brothers, two maternal half sisters, paternal aunt and niece, maternal uncle and nephew, maternal grandmother and granddaughter and between the following first cousins: a paternal aunt's daughter and her female cousin; and a maternal uncle's daughter and her female cousin. Because the values are the same for both types of gene, the coefficients of relationship, based on Σ(1/2)n for autosomal genes, are more accurate for these relationships than for any others.