Reference Work Entry

Encyclopedia of Genetics, Genomics, Proteomics and Informatics

pp 691-692

Date:

# Fixation Index

The fixation index is the average coefficient of inbreeding in a population. In case of random mating, the probability that an offspring would have exactly the same two ancestral alleles at a locus is (1/2)N, where N is the number of diploid individuals in the population. The probability of having two different alleles at the same locus is 1 − (1/2)N. The coefficient of inbreeding of the first generation of this population is also (1/2)N by definition of inbreeding. In each succeeding generation, the non-inbred part of the population will have a chance to produce offspring with an allele pair identical by descent. Therefore, the coefficient of inbreeding in the next generations will be (1/2)N + [(1 − (1/2)N] x F, where F is the inbreeding coefficient of the preceding generation. After the gth generation the coefficient of inbreeding of this population will be: Fg = (1/2)N + [1 − (1/2)N]Fg−1 and this is called the index of fixation. Its complement is the panmictic index (Pg) that represents the average non-inbred fraction of the population:
$${\rm P}_{\rm g}= 1 - {\rm F}_{\rm g}$$
The probability for the offspring to have two identical A or a alleles is FpAA and Fqaa, respectively. Also, the probability of two alleles of a locus being non-identical by descent is 1–F, and the proportions of AA, Aa, and aa are p2, 2pq, and q2 (according to the Hardy-Weinberg theorem). Because the population will have both inbred and non-inbred components, its genetic structure will be:
$$(p_{AA} + q_{aa})\ {\rm and} \ (1 - F)(p^{2}{AA}+ 2pq_{Aa} + q^2{aa})$$

When a population is completely inbred, only homozygotes are found. There may be a change in genotypes but may not be a change in allelic frequencies if both alleles have equal fitness. The change may actually be from AA + 2Aa + aaAA + AA + aa + aa, i.e., mathematically it is the same. The ultimate probability of fixation $${\rm{P}}_{\rm{f}} = {{1 - e^{ - N_e } sp} \over {1 - e^{ - N_e } s}}$$ may be estimated also on the basis of the initial frequency of the gene (= p), the selection advantage (= s) and the effective population size (Ne). [The base of the natural logarithm = e ≈ 2.718]. inbreeding, panmixis, inbreeding rate, Hardy-Weinberg theorem, mutation neutral, mutation beneficial, hybrid vigor; Whitlock MC 2003 Genetics 164:767.