On Sat, 17 Jun 1995, Grant Morahan wrote:
> We have been analysing genetic susceptibility to an experimentally-induced
> disease in mice. The results are surprising.
> Inbred strain A -100% susceptible;
> inbred strain B -100% resistant.
> (AxB)F1 - 50% are susceptible.
> F2 - 25% are susceptible. So far, so good.
>> But 50% of the ((AxB)xA)BC1 mice develop disease.
> (We don't have results for the reciprocal BC yet).
>> The problem is that the BC doesn't seem more susceptible than the F1.
=============================================
First question: How large are your samples?
It seems unlikely that you are getting results that are quite as
neat as stated. (i.e., is it EXACTLY 50%?) Isn't it possible that there
is sampling variation blurring your results and that a larger sample will
show a difference between BC and F1? Maybe the true population disease
rates are 60% and 40% respectively and you just didn't detect it.
The data from P, F1 and F2 seem to indicate that multiple loci are
involved. Assuming the data on susceptibility in the inbred strains are
correct, if a single locus determines susceptibility, the risk to F2
should be 1/2 the F1 risk plus 1/4. Thus, risk would not decline from the
F1 to the F2 if the F1 risk were 50%.
All mice in the F1 are identical hybrids, so what causes variation in
susceptibility among them? That is probably a very difficult question to
answer.
Here is a model that gives you the probabilities that you are observing.
I may be a bit contrived, but it isn't absurd. Suppose Strain A carries a
susceptibility allele and Strain B carries a resistance allele at an
unlinked locus.
Parental Generation:
Genotype Pr(Disease)
======== ===========
AA ++ 1 (Strain A; susceptibility allele is called A)
++ BB 0 (Strain B; resistance allele is called B)
F1 Generation:
Genotype Pr(Disease)
======== ===========
A+ B+ 1/2
F2 Generation:
Genotype Pr(Disease) Frequency
======== =========== =========
AA BB 0 1/16
A+ BB 0 1/8
++ BB 0 1/16
AA B+ 0 1/8
A+ B+ 1/2 1/4
++ B+ 0 1/8
AA ++ 1 1/16
A+ ++ 1/2 1/8
++ ++ 0 1/16
Back cross (BC):
Genotype Pr(Disease) Frequency
======== =========== =========
AA B+ 0 1/4
A+ B+ 1/2 1/4
AA ++ 1 1/4
A+ ++ 1/2 1/4
In this model, allele A causes disease and allele B prevents it. But
there is a peculiar kind of epistatic interaction. The effect of the A
allele is additive when the B allele is not present:
Genotype Pr(Disease)
======== ===========
AA ++ 1
A+ ++ 1/2
++ ++ 0
The BB genotype always prevents disease:
Genotype Pr(Disease)
======== ===========
AA BB 0
A+ BB 0
++ BB 0
But here's the part that had to be contrived to fit the data:
Genotype Pr(Disease)
======== ===========
AA B+ 0
A+ B+ 1/2
++ B+ 0
The susceptibility locus is overdominant when the resistance locus is
heterozygous. More precisely, for an A+ ++ genotype, addition of a B
allele has no protective effect, but for the even more susceptible AA ++
genotype, addition of a B allele prevents disease.
I'm sure there are infinitely many models that will fit any collection of
data. I will be surprised if the model described here turns out to be the
"true" model!
Mike
Michael B. Miller, M.S., Ph.D.
Department of Psychiatry (Box 8134)
Washington University School of Medicine
4940 Children's Place, St. Louis, MO 63110
office phone: (314) 362-9428 FAX: (314) 362-9420
WWW Homepage: ftp://sirronald.wustl.edu/pub/mbmiller/mike.htm
