CALCULATION OF THE COEFFICIENT OF INBREEDING (F) FOR SEX-LINKED GENES
The effect of inbreeding, with respect to sex-linked genes, is not usually an important issue because they are in the minority in humans compared with autosomal genes. However, there may be certain situations, where the effects of inbreeding through sex-linked genes alone are important. This is certainly true of the fruit fly (Drosophila) which has only 4 pairs of chromosomes, and honey bees where all the genes are sex-linked. One possible explanation for the strange marriage laws in some countries, which discriminate against marriage between ortho-cousins, particularly those with the same family name, may be the increased incidence of harmful recessive sex-linked conditions in the progeny of these cousin marriages. (For details of ortho-cousins see here in the monograph).
As pointed out in the main text, there are no consistent patterns of sex-linked inheritance, since both sexes are present in all four types of cousins and in their parents and children. Therefore, any differences, if they exist, must be due to differences in their inbreeding coefficients for sex-linked genes. The following analysis calculates the inbreeding coefficients for the children of the four types of cousin marriage for sex-linked genes, and estimates the relative magnitude of any harmful effects expected within each type for two sex-linked loci.
Genetic Analysis of the Effects of Inbreeding on the Children of First Cousins for Sex-linked genes
With sex-linked inheritance, the coefficient of inbreeding (F), for the children of a cousin marriage, is affected by how the sexes are arranged within the marriages of the cousins and their parents. This influences the proportions of harmful genotypes produced by each kind of cousin marriage. Under sex-linkage, inbreeding only causes an increase in the frequency of homozygous recessives in females. There is no meaning to the inbreeding coefficient of a male since he is hemizygous (i.e. he has only one X chromosome) and is, therefore, not affected by inbreeding. Females can be homozygous or heterozygous. Crow and Kimura (1970) give the following method for determining F: "The rule for obtaining the inbreeding coefficient for a sex-linked locus in females is: Proceed as usual except that only females in a path are counted and any path with two successive males is omitted."
The reason for the first part of the rule is that a male passes on the single X chromosome received from his mother to his daughters unchanged. It is as though he never existed and so the relationship (R) between grandmother and granddaughter is always 50% and the male can be excluded from the path. The male parent of an inbred female child is likewise ignored and the paternal grandmother is treated as a substitute parent. Also, if the common ancestor is a male, he passes identical copies of the same X chromosome to all his daughters. Therefore any two daughters of the same male will always have the same paternal X chromosome in common, giving them a relationship (R), through their father, of 50%. (With autosomal genes, sisters only have a relationship of about 25% through their father.). This means that any male common ancestor can be disregarded in the path, provided he passes his sex-linked genes through two of his daughters.
The second part of the rule is a consequence of the fact that a male passes no X chromosomes to his sons, only a Y; so the relationship between father and son for genes on the X chromosome is 0, and the line of descent is broken. The following pages show diagrams and calculations for the eight different mating combinations involving first cousins. All males are printed in red and the proband (the nominated cousin) is arrowed in each case.
Figure 13 Patrilateral Ortho-cousins
Find FI
The two paths between G and H, through the two common ancestors A and B, are both interrupted by two or more males and are invalid. Therefore, FI = 0
Find FI
The two paths are again interrupted, so: FI = 0
Figure 14 Matrilateral Ortho-cousins
Find FI
| Common Ancestors of Parents of G and H | Paths | (1/2)n+1 | |
| A | D - E - H | 1/8 | |
| B | D - B - E - H | 1/16 | |
| Σ[(1/2)n+1(1 + FA)] | = | 3/16 Therefore FI = 3/16 | |
Find FI
| Common Ancestors (of Parents G and H) | Paths | (1/2)n+1 | |
| A | G - D - E | 1/8 | |
| B | G - D - B - E | 1/16 | |
| Σ[(1/2)n+1(1 + FA)] | = | 3/16 Therefore FI = 3/16 | |
Figure 15 Patrilateral Cross-cousins
Find FI
Since both paths between G and H are invalid, FI = 0
Find FI
| Common Ancestors (Of Parents G and H) | Paths | (1/2)n+1 | |
| A | invalid | 0 | |
| B | G - B - E | 1/8 | |
| Σ[(1/2)n+1(1 + FA)] | = | 1/8 Therefore FI = 1/8 | |
Figure 16 Matrilateral Cross-cousins
Find FI
| Common Ancestors (Of Parents G and H) | Paths | (1/2)n+1 | |
| A | invalid | 0 | |
| B | D - B - H | 1/8 | |
| Σ[(1/2)n+1(1 + FA)] | = | 1/8 Therefore FI = 1/8 | |
Find FI
Both paths are invalid, therefore, FI = 0
Summary and Conclusions
Table 7 F Values of Female Progeny for Sex-linked Genes
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It can be seen from Table 7 that patrilateral ortho-cousins, who have the same family name and whose marriage is most discriminated against, are the ones least affected by inbreeding. On the other hand, matrilateral ortho-cousins do show a higher value for F than either of the two types of cross-cousins. If we compare all ortho-cousins with all cross-cousins, the average F values are 3/32 and 1/16 respectively. To relate this to harmful sex-linked genes we can calculate the effects for a fairly common gene (sex-linked colour blindness) and a rare one (haemophilia).
Gene Frequencies:
Sex-linked colour blindness (q = 0.07). Haemophilia (q = 0.0002)
Recessive Genotype Frequencies in a Random-mating Population
Males = q Females = q2
Recessive Genotype Frequencies in Progeny of Cousin Marriages
Males = q Females = q2 + Fq(1 - q)
Tables 8 and 9 show the relative risks of inheriting colour blindness and haemophilia under random mating compared with matings between the main two types of cousins.
Table 8 Incidence of Sex-linked Colour Blindness
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Therefore, the estimated average incidence of colour-blind females in the progeny of ortho-cousins is 0.0110 and for cross-cousins 0.0089. If the frequency is calculated over all the progeny (including males) it will be as follows:
| Ortho-cousins | Cross-cousins | |
| 0.07 + 0.0110 = 0.0405 | 0.07 + 0.0089 = 0.0395 | |
| 2 | 2 | |
It is unlikely that this small difference of 0.001 (0.1%) would be detected at the family level.
Table 9 Incidence of Haemophilia
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Averaged over all the progeny:
| Ortho-cousins | Cross-cousins |
| 1/2(1/5000 + 1/53,230) = 0.000 109 | 1/2(1/5000 + 1/79,760) = 0.000 106 |
i.e. A difference of only 3 in a million progeny. With this low incidence it is even less likely that it would be noticed. The real reason for the discrimination, therefore, could be socio-economic, religious, traditional or a simple misunderstanding of the effects of inbreeding.
In those rare cases where the grandparents, A and/or B, are known to be either 'affected' or 'carriers', then the numbers of 'affected' grandchildren will increase dramatically in all four cases; although patrilateral ortho-cousins will always have the lowest number. Finally, because more genes are involved, the largest inbreeding effect overall, will be caused by autosomal genes and will take the form of general inbreeding depression as well as an increased risk of inheriting other specific abnormalities. However, this will affect all types of single first cousins equally and is irrelevant to the present comparison.