# Loss of allelic variation per generation of full-sib mating

If we take a male and female, cross them, and collect one son and one daughter to produce the next generation (and continually do this for many generations) - (full-sib mating design) how much genetic variance is lost per generation/how do we calculate this?

Assume all sites initially have allelic variation, and we imagine a diploid sexual species.

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A classical result from the Wright-Fisher model of genetic drift is that at each generation the heterozygosity ($2x(1-x)$) is expected to decrease by $\frac{1}{2N}$ due to genetic drift. ($x$ is the frequency of one of the two alleles). In your example the expected loss of heterozygosity is $¼$. This same result has been derived from coalescent theory and a similar result is derived from the birth-death Moran model. The issue is that this model assume bi-allelic loci only.

Let's think of 1 tetra-alellic (4 alleles) locus and let's assume that the 4 alleles in our two original parents are all different. We can draw the 16 possible pairs of offsprings and see the probability to lose 0, 1 or 2 alleles in the next generation

lose 0 allele: ¼
lose 1 allele: ½
lose 2 alleles: ¼
lose more than 2 alleles: 0
Therefore, the expected number of lost alleles is $1$!


We can then do the same (calculate the probability mass function) for two parents with 3 alleles and then 2 alleles (with various frequencies). It is not very complicated. But it would be much better to have a general model to describe the loss of heterozygosity through time for more than 2 alleles. This paper probably gives this solution but I haven't read it yet. I will probably read it once and I will improve my answer then!

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