"Anon."  wrote in message
news:cbumkp$1r7u$1{at}darwin.ediacara.org...
> William L Hunt wrote:
> > On Mon, 28 Jun 2004 16:02:21 +0000 (UTC), "Anon."
> >  wrote:
> >
> >
> >>William L Hunt wrote:
> >>
> >>>On Tue, 22 Jun 2004 20:16:27 +0000 (UTC), "Anon."
> >>> wrote:
> >>>
> >>>
> >>>
> >>>>William L Hunt wrote:
> >>>>
> >>>>
> >>>>>On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
> >>>>> wrote:
> >>>>
> >>>
> >>>>>>>Popularisers should make explicit the
behaviour is what happens as
> >>>>>>>the population size tends towards infinity
- and not attempt to
pass
> >>>>>>>it off as an effect in an infinite population.
> >>>>>>
> >>>>>>:-BOH
> >>>>>>But it is - in finite populations, you get an
excess of homozygotes,
as
> >>>>>>any student of population genetics should know.
> >>>>>
> >>>>>:-WLH
> >>>>> This above statement doesn't sound right to me?
> >>>>>It is known that if the matings are other than
random there will be
> >>>>>an excess of homozygotes (over a Hardy-Weinberg equilibrium
prediction
> >>>>>with random matings) but this is even usually best
to see when the
> >>>>>populations are large. Small populations (with
random matings) are
> >>>>>expected to diverge from a precise Hardy-Weinberg
equilibrium simply
> >>>>
> >>>>>from the sampling effects of the small size but I
don't recall any
> >>>>
> >>>>>bias to this divergence (more or fewer homozygotes than a
> >>>>>Hardy-Weinberg prediction). If the sampling
(mating) is truly random,
> >>>>>I don't see how you could predict a direction (excess of
homozygotes)?
> >>>>
> >>>>:-BOH
> >>>>I don't know which textbooks you have to hand, I have Futuyma's
> >>>>"Evolutionary Biology" (2nd ed. from 1986),
and in Chapter 5
> >>>>("Population Structure and Genetic Drift")
he has a section called
> >>>>"Population Size, Inbreeding, and Genetic
Drift" where he shows that
any
> >>>>finite population will become inbred, which means a reduction in
> >>>>heterozygosity.  I'm sure the same thing is in Hartl
& Clarke.  Look
out
> >>>>for equations like H_t = H_0 (1-1/2N)^t.
> >>>>
> >>>>In essence, any finite population will become inbred
over time (at
least
> >>>>to some extent), and this increases homozygosity.
> >>>>
> >>>>Bob
> >>>>
> >>>
> >>>:-WLH
> >>> The discussion was of a Hardy-Weinberg equilibrium and
your original
> >>>statement "in finite populations, you get an excess
of homozygotes", I
> >>>took to refer to an excess over that predicted by HW. Apparently you
> >>>were referring to something else that has nothing to do with
> >>>Hardy-Weinberg equilibrium. This threw me off and I suspect it may
> >>>have for some others also. Guy Hoelzer also responds to this in an
> >>>above thread.
> >>> Yes, if the population is small, many more loci will reach fixation
> >>>(homozygote) to one allele or the other and there is less genetic
> >>>diversity. This has nothing to do with Hardy-Weinberg.
Hardy-Weinbery
> >>>only speaks to loci where there are still two alleles present in the
> >>>population at some frequency and it predicts what the distribution
> >>>will be. It originally sounded like you were saying was
that if there
> >>>is a frequency of p=.5 for allele 'A' and q=.5 for allele
'a', that in
> >>>a large population you would expect the distribution to be a
> >>>Hardy-Weinberg AA=.25, Aa = .50, aa=.25 but for some small
population
> >>>size, "you" might expect it show an "excess
of homozygotes", such as
> >>>AA=.30, Aa=.40, aa=.30. I now am not sure what you meant?
> >>
> >>:-BOH
> >>This is what I meant.  In a finite population, the expected number of
> >>heterozygotes is less than predicted from HWE.  Gale goes through the
> >>calculations in his textbook (my edition is from the early 80s).  Most
> >>textbooks use a deterministic calculation, but get the same result.
> >>Either way, it all goes back to Wright.
> >>
> >>Bob
> >>
> >
> >  Can you give me a formula that for a particular locus, given N q and
> > p for a randomly mating population, gives the expected frequency of
> > AA,Aa, and aa similiar to HWE above but allows one to see what
> > distribution is expected for a particular finite population size(N)? I
> > don't recall seeing such a formula?
>
> I suspect you have seen it, but not expressed in this way!
>
> The expected frequency of heterozygotes is 2pq(1-1/2N), the expected
> frequency of the AA homozygote is p^2 + p(1-p)/2N.  It turns out that
> 1/2N is just the inbreeding coefficient, of course.
>
> My reference is J. S. Gale (1980) Population Genetics (Tertiary Level
> Biology), and I forgot to bring it in to work today.  But the proof of
> the heterozygote deficit follwos from calculating E[2p(1-p)] = 2E[p] -
> 2E[p^2].
>
> E[p]=p and E[p^2] is calculated from the definition of a variance:
>
> Var[p] = E[p^2] - E^2[p]
> and
> Var[p] = p(1-p)/2N,  so E[p^2] = p(1-p)/2N - p^2.
>
> Plugging this into E[2p(1-p)] we get
>
> 2E[p] - 2E[p^2] = 2p - 2( p(1-p)/2N - p^2 )
>                  = 2(p-p^2) - 2p(1-p)/2N
>                  = 2p(1-p)(1-1/2N)
>
> which is the result we want.

Ah!  I think I get it now.  To put your argument into my (probably
imprecise) words:

HW predicts expected heterozygote frequency of 2p(1-p) on the
assumption of one generation of random mating with no drift.

BOH/Gale predicts expected heterozygote frequency of 2p(1-p)(1-1/2N)
on the assumption of one generation of random mating with one
generation of drift.

Since there is really no way to have random mating without drift,
the BOH/Gale assumptions are the correct assumptions to make in
a finite population.  And that, rather than the issue of selfing
as I had assumed, is the reason why careful geneticists endorse
the HW predictions only in the limit of an infinite population.

Now I see why you don't feel foolish.  I feel a little foolish
because I had interpreted what you were saying as being an effect
of past drift, rather than a fresh effect of one generation's
worth of drift. ;-(  My apologies.

However, I still think you are wrong on "binomial/trinomial". ;-)
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