# Inbreeding Coefficient

Reference work entry

First Online:

**DOI:**https://doi.org/10.1007/978-1-4020-6754-9_8397

Denotes the probability that *two alleles at a locus in an individual* are identical by *descent* from a common ancestor, i.e., the chance that an individual is homozygous for an ancestral allele by inheritance (not by mutation). The inbreeding coefficient for second cousins is 1/64 and for third cousins it is 1/256. In case only one of the parents is common at the starting generation the inbreeding coefficient of half-sibs at uncle-niece or aunt-nephew marriage is 1/16, for first cousins 1/32, for second cousins 1/128, and for third cousins 1/512.

*Consanguinity*(coancestry) is a similar concept but the coefficient of coancestry indicates the chances that

*one allele in two individuals*would be identical by descent. F symbolizes the coefficient of inbreeding. The calculation of F is based on the fact that in a diploid at each locus there are two alleles and only one is contained in any gamete (either one in a particular egg or sperm). Thus, each individual has 0.5 chance for passing on a particular allele to a particular offspring. To illustrate the method of calculation examples will be discussed (see Fig. I16). Brother (X) and sister (Y) have two common parents (W) and (V). An offspring of the mating (X) × (Y) → (X)→(I) or (W)→(Y)→I), therefore his chances for homozygosity for one allele derived from (W) is 0.5

^{2}= 0.25. In other words, in F

_{2}the chance is 1/4 for homozygosity for any allele according to the Mendelian law. In a half-brother and half-sister progeny grandparent (A) can transmit a particular allele to (I) either through (B) or (C) parents and the inbreeding coefficient of (I) is 0.5

^{3}= 1/8 because three individuals are involved in the transmission route (A), (B) and (C). Similarly, the inbreeding coefficient of other types of matings can be calculated as indicated in the chart. In half first cousin mating individuals (C), (B), (A), (E) and (D) are involved in the transmission path, each with a 0.5 chance thus the coefficient of inbreeding becomes 0.5

^{5}= 0.03125 = 1/32. In two generations of brother-sister matings (see scheme 6), the transmission of alleles may follow the routes [E-C-F, F-D-E], {E-C-A-D-F, F-D-B-C-E, E-D-A-C-F and F-C-B-D-E}, i.e., [2] and {4} paths of [0.5]

^{3}and {0.5}

^{5}, respectively. The coefficient of inbreeding is 2[0.5]

^{3}+ 4{0.5}

^{5}= 0.375 = 3/8. If there are multiple paths through the same ancestor, all the paths through the shared ancestors must be included in the calculation with the precaution that the same ancestor must be counted only once in the same path. The method can be illustrated by another example where (Z) and (U) are the common ancestors and again the inbreeding coefficient of individual (I) is sought. There are two routes through (Z): T-Z-L-K and T-Z-M-K and also two paths through (U): K-M-U-T and K-L-U-T. Each of these four paths involves four ancestors contributing genes to (I). Therefore, the coefficient of inbreeding of (I) is 4(0.5)

^{4}= 0.25 = 1/4. Under practical conditions of breeding far more complicated schemes may be encountered yet their solution can be sought in terms of these simple examples. It is easier to determine the loops of gamete contribution by working backwards from the critical individual, (I) in this case. It is conceivable that the common ancestors are not completely unrelated, contrary to the assumption in the calculations here, but they may have some degree of relatedness and their inbreeding coefficient, F

_{A}(ancestral coefficient of inbreeding) must also be taken into account. Therefore, the general formula for the coefficient of inbreeding is F = Σ[(0.5)

^{n}(1 + F

_{A}) where Σ is the sum of the paths through which an individual can derive identical alleles from his ancestors and n = the number of individuals in the paths. 1 + F

_{A}is the correction factor for the inbreeding coefficient of the common ancestor in the path. Calculating the coefficient of inbreeding may not only be very important in a breeding project, but it may also be relevant to human families. It is assumed that the frequency of a recessive genetic disorder is q

^{2}= 1 × 10

^{−6}and if the population is in a genetic equilibrium the frequency of that allele is q = \( \sqrt{q^2}=\sqrt{0.000001}\!\!\). Then the risk related parents face of having an afflicted child is: q

^{2}(1 − F) + q(F). Since the inbreeding coefficient of the offspring of first cousins is F= 1/16 = 0.0625 (see Figs. I17 and I18), after substitutions one obtains: 0.000001 × 0.9375) +(0.001 × 0.0625) ≈ 0.000063. Since 0.000063 is ∼63 fold higher than 0.000001 (the frequency of individuals with this affliction in the general population), first cousin parents are at a >63 fold greater risk than unrelated parents to have an offspring afflicted with a hereditary disease that has a gene frequency of 0.001. In some cases inbreeding may be detected by DNA fingerprinting or by nucleotide sequence of the genome. F, coefficient of coancestry, consanguinity, relatedness degree, inbreeding progress, inbreeding rate, fixation index, genetic load, homozygosity mapping, DNA fingerprinting, inbreeding depression; Fisher RA 1965 The Theory of Inbreeding, Academic Press, New York.

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