How quickly does inbreeding produce homozygosity?

What is the probability that locus is homozygous due to inbreeding after a given number of generations? The answer should be an equation expressing this probability as a function of the number of generations.

This question was inspired by a recent review of mouse genetics (Peters et al. 2007. "The mouse as a model for human biology: a resource guide for complex trait analysis" Nature Reviews Genetics 8, 58-69. doi:10.1038/nrg2025), which states, without qualification, that "Inbred mouse strains are derived from a single parental mating pair with subsequent repeated brother–sister intercrosses and no breeding from non-sibling mice. After 20 generations of inbreeding, the mice are genetically identical and homozygous at all loci." 20 generations is an accepted benchmark that leaves a very low probability of homozygosity, but that probability is not zero. What is it?

This is really three questions (or more).

1) Selfing organisms like Arabidopsis thaliana or Caenorhabditis elegans present a simple case. Here, the probability that a given locus remains heterozygous is simply (1/2)ⁿ, where n is the number of generations.

2) The mouse, which represents all diploid species where crosses between full siblings is possible. Brothers and sisters that share the same two parents. I suspect the equation for this case has been worked out. What is the answer?

3) Species where females store sperm. In this case one can isolate a female each generation. She will have mated with her (possibly half-) brothers prior to isolation. Here the probability of homozygosity is a function not only of the number of generations, but also the probability of the female having mated with a full brother vs. a half-brother.

What do we really know about earlobe genetics?

My husband and I both have attached earlobes. Our newly born son has detached earlobes. How is this possible since attached earlobes are recessive?

- Curious

That is a great question. Many of us learned in high school that attached earlobes are recessive, and it is easy to find that information repeated on the web as an illustration of dominance patterns. An example is the detailed treatment at the Singapore Science Center, which explores various possible parental combinations and concludes that "if both parents were homozygous recessive, they could not have a child with the dominant allele." That is certainly true, and it seems to make a liar out of Curious. Was her child switched in hospital? Should Curious become Alarmed or Suspicious?

I don't think so. I could not find the source of the "fact" that attached earlobes are recessive. I consulted OMIM (the Online Mendelian Inheritance in Man), and they have very little (OMIM 128900). What they do have calls into question the assumption that "attached earlobes are recessive."

Free earlobes are dominant in the view of some. Dutta and Ganguly (1965) suggested polygenic inheritance. There is a variety that is perhaps better classified as 'lobeless' than 'attached.' Lai and Walsh (1966) concluded that 'a simple Mendelian gene effect is unlikely to be responsible for the earlobe types.'

The other issue raised here is whether or not free earlobes is even a bivariate trait (in which case everyone's earlobes could be categorized as either attached or detached). That is necessary for simple dominance to make sense. In the words of students from Willowbrook high school who did a study of this, "it's harder to tell earlobes, so the count on these could be less accurate." OMIM states that "there is a variety that is perhaps better classified as 'lobeless' than 'attached.'" Different forms might show different patterns of inheritance.

I suspect that this, along with a large number of ordinary traits, will be examined anew in the era of personal genomics. For example, the people at deCODE genetics just published a report on the genetics of hair and skin pigmentation (Sulem et al., 2007: Nature Genetics 39:1443, PMID 17952075). Perhaps earlobes will be next.

This question may be most important as a cautionary tale about genetics education. Teaching students about earlobes seems very innocent, but this case illustrates why oversimplification might not be prudent.