Inbreeding Coefficient after Selfing

Self-pollinate a heterozygote and half the offspring are homozygous. Self those and half of what remains fixes again. Each generation halves the heterozygosity that is left, which is the entire mathematics of line development in a selfing crop.

The series moves faster than intuition suggests. After three generations a line is 87.5% homozygous, after five 96.9%, after six 98.4%. This is precisely why breeding programmes in wheat, barley, soybean and tomato talk about F6 and F7 lines and then stop: the residual heterozygosity is smaller than the plot-to-plot variation the field itself contributes, so further selfing buys uniformity nobody can measure.

Two shortcuts sidestep the series entirely. Doubled haploids take a gamete, double its chromosome number, and produce a completely homozygous plant in a single step — $F = 1$ exactly, not asymptotically — which removes several years from a cycle. Single-seed descent goes the other way, advancing generations as fast as possible in a glasshouse without selection, on the reasoning that selection is wasted while the material is still segregating.

What the equation does not say is the cost. Inbreeding exposes recessive deleterious alleles as homozygotes, and in an outcrossing species such as maize the resulting inbreeding depression is severe — inbred maize lines are visibly feeble compared with the population they came from. That loss is not a side effect of the hybrid system; it is the thing that makes hybrids valuable, because crossing two unrelated inbreds restores the heterozygosity in one step.