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.

Bob

-- 
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5)
FIN-00014 University of Helsinki
Finland
Telephone: +358-9-191 23743
Mobile: +358 50 599 0540
Fax:  +358-9-191 22 779
WWW:  http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: http://www.jnr-eeb.org
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