Hardy-Weinberg law



William L Hunt wrote:
> On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
> <[email protected]> wrote:
>
>
>>Tim Tyler wrote:
>>
>>>Anon. <[email protected]> wrote or
>>>quoted:
>>>
>>>
>>>>Jim McGinn wrote:
>>>>
>>>>
>>>>>[email protected] (friend) wrote in
>>>>>message news:<[email protected]>...
>>>>
>>>>>>Pardon my ignorance, but I have only just discovered
>>>>>>this law.
>>>>>
>>>[...]
>>>
>>>
>>>
>>>>>Many of the more popular myths of the current paradigm
>>>>>of evolutionary biology pivot off a kind of rhetorical
>>>>>trick. Specifically the trick involves employing a word
>>>>>that has more than one meaning in an argument (or
>>>>>special case) to achieve the illusion of scientific
>>>>>validity. This is *all* that's going on with the Hardy-
>>>>>Weinberg, socalled, Law. And you hit the nail on the
>>>>>head with respect to which word is the "pivot" with
>>>>>respect to how this rhetorical trick is manifested in
>>>>>Hardy-Weinberg: randomness.
>>>>
>>>>Wierd. The Hardy-Weinberg law is deterimistic: there is
>>>>no randomness in it.
>>>
>>>
>>>The Hardy-Weinberg law is normally stated in a form that
>>>refers to a large population where mating is random.
>>>
>>
>>The theorem was derived for an infinite population.
>>
>>
>>>E.g. see:
>>>
>>> http://library.thinkquest.org/19926/java/tour/06.htm?t-
>>> qskip1=1
>>>
>>>Alas, this page expresses the law in terms of an infinite
>>>population :-(
>>>
>>>A disasterous error - IMO - since talking about gene
>>>frequencies in an infinite population is a sign of
>>>mathematical ignorance.
>>>
>>
>>No, it's a simplifying assumption. I think to accuse Hardy
>>in particular of mathematical ignorance deserves, well, a
>>non-mathematician's apology.
>>
>>
>>>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.
>>
>>But it is - in finite populations, you get an excess of
>>homozygotes, as any student of population genetics
>>should know.
>
>
> 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)?

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

--
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
 
Hi Bill,

in article [email protected], William L Hunt
at [email protected] wrote on 6/21/04 4:44 PM:

>>> 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.
>>
>> But it is - in finite populations, you get an excess of
>> homozygotes, as any student of population genetics
>> should know.
>
> 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.

Non-random mating does not necessarily produce an excess of
homozygotes relative to HW expectations. Non-random mating
can produce any pattern you can imagine. Negative
assortative mating (e.g., rare male mating advantage), for
example generates an excess of heterozygotes. A scheme of
non-random mating can even be engineered to generate HW
genotype proportions while reducing variance in the
distribution of outcomes relative to random mating.

Cheers,

Guy
 
"Name And Address Supplied" <[email protected]> wrote in
message news:[email protected]...
> "Perplexed in Peoria" <[email protected]> wrote
> in message
news:<[email protected]>...
> > "Anon." <[email protected]> wrote
> > in message news:[email protected]...
> > > But it is - in finite populations, you get an excess
> > > of homozygotes,
as
> > > any student of population genetics should know.
> >
> > Dooohh! Ignore my other post. Reality gives an excess of
heterozygotes.
> > The Hardy-Weinberg "law" gives an excess of homozygotes.
> >
>
> To clarify, are we assuming no selfing here? If not, I'm
> struggling to see why you get the result that you do.

Whoops, I was confused as to whether I was responding to NAS
or BOH. I'll watch for BOH's answer as to why finite
populations affect HW in his response to Bill Hunt.
 
"Name And Address Supplied" <[email protected]> wrote in
message news:[email protected]...
> "Perplexed in Peoria" <[email protected]> wrote
> in message
news:<[email protected]>...
> > "Anon." <[email protected]> wrote
> > in message news:[email protected]...
> > > But it is - in finite populations, you get an excess
> > > of homozygotes,
as
> > > any student of population genetics should know.
> >
> > Dooohh! Ignore my other post. Reality gives an excess of
heterozygotes.
> > The Hardy-Weinberg "law" gives an excess of homozygotes.
> >
>
> To clarify, are we assuming no selfing here? If not, I'm
> struggling to see why you get the result that you do.
>

I am assuming no selfings. Thus, in a finite population, I
get results at variance with Hardy-Weinberg. But, if I
include the selfings, I get results identical with HW.

It might help if you would give a short explanation of
why you wrote: "in finite populations, you get an excess
of homozygotes, as any student of population genetics
should know."

Perhaps I should know, but I don't.
 
Bob O'Hara <[email protected]> wrote or quoted:
> Tim Tyler wrote:
> > Anon. <[email protected]> wrote or
> > quoted:
> >>Tim Tyler wrote:
> >>>Anon. <[email protected]> wrote or
> >>>quoted:

> >>>>Wierd. The Hardy-Weinberg law is deterimistic: there
> >>>>is no randomness in it.
> >>>
> >>>The Hardy-Weinberg law is normally stated in a form
> >>>that refers to a large population where mating is
> >>>random.
> >>
> >>The theorem was derived for an infinite population.
> >>
> >>>E.g. see:
> >>>
> >>> http://library.thinkquest.org/19926/java/tour/06.htm-
> >>> ?tqskip1=1
> >>>
> >>>Alas, this page expresses the law in terms of an
> >>>infinite population :-(
> >>>
> >>>A disasterous error - IMO - since talking about gene
> >>>frequencies in an infinite population is a sign of
> >>>mathematical ignorance.
> >>
> >>No, it's a simplifying assumption. I think to accuse
> >>Hardy in particular of mathematical ignorance deserves,
> >>well, a non-mathematician's apology.
> >
> > I never said the problem was at Hardy's end.
>
> In fairness, Hardy does suppose "that the numbers are
> fairly large". But, as we know know, in a finite
> population, there will be an excess of homozygotes
> (because of inbreeding), so H-W isn't correct (but a
> reasonable approximation if the population is large).

I suggested explaing using a limit - not a mere finite
population.

> >>>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.
> >>
> >>But it is - in finite populations, you get an excess of
> >>homozygotes, as any student of population genetics
> >>should know.
> >
> > Any mention of gene frequencies in an infinite
> > population is nonsense - as I stated originally.
> >
> > You can't talk about a fraction of an infinite
> > population having a trait. You would get different
> > results for that fraction depending on how you
> > enumerated through the population.
>
> I don't understand what you mean, but by that argument,
> you can't even define a fraction or a probability.

Fractions have nothing to do with infinite sets.

> > It's like claiming that half the integers are even.
>
> Err, they are. There are just rather a lot of them.

No, there aren't.

There are an infinite number of even numbers.

There are an infinite number of odd numbers.

Divide infinity by infinity and the result is indeterminate.

> > Such statements are total mathematical gibberish.
> >
> > What *can* be said is that the fraction of the set of
> > integers from to N that are even tends to 0.5 - as N
> > => oo.
>
> So what happens when N is infinity?

The fraction is undefined.

> > No serious mathematician can talk about fractions of
> > infinite sets and expect to be taken seriously.
>
> But they do.

No - not unless the fractions are "zero" or "one".

> It's how probability is defined as a concept.

Probability is defined as a mathematical limit, as N
approaches infinity.

That uses a limit as a finite set increases in size - not a
fraction of an infinite set.

E.g. see:

http://www.wordiq.com/definition/Probability

> I have a colleague who even wrote mathematical papers
> about fractions of uncountable sets.

If you can show me, I should be able to tell you if they
contain the fallacy under discussion.

Probably he doesn't do that at all - and instead uses a
limit.

> Infinity is a difficult concept (I know - there are lots
> of it I don't understand), so I think one should be
> cautious about making any pronouncements on it unless one
> is sure about what mathematics does and does not say on
> the subject.

How is that relevant?

Are you suggesting I don't know what I am talking about?

That is not the case.
--
__________
|im |yler http://timtyler.org/ [email protected] Remove
lock to reply.
 
Hi Bob,

I snipped the following out of a longer post.

in article [email protected], Bob O'Hara at
[email protected] wrote on 6/21/04 4:44 PM:

> But, as we know know, in a finite population, there will
> be an excess of homozygotes (because of inbreeding), so
> H-W isn't correct (but a reasonable approximation if the
> population is large).

I am not sure how extensive you intended this claim to be,
but I think it is generally false. The finiteness of
populations does not cause an excess of homozygosity
relative to HWC expectations. Similarly, there is no reason
to expect the excess of homozygosity to grow with decreasing
population size, nor is there a reason to expect the degree
of inbreeding to increase relative to that which would occur
in random mating as population size decreases. The HWC model
merely predicts the way in which existing allelic diversity
is translated into genotypic diversity. It makes no
assertions about the extent of allelic diversity you are
likely to find in a population, which is strongly influenced
by population size.

Cheers,

Guy
 
> In essence, any finite population will become inbred over
> time (at least to some extent), and this increases
> homozygosity.

Boy, are you going to feel foolish after you get a good
night's sleep and review what you have written :)
 
On Tue, 22 Jun 2004 20:16:27 +0000 (UTC), "Anon."
<[email protected]> wrote:

>William L Hunt wrote:
>> On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
>> <[email protected]> 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
>

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? Possibly what confused me is that in
large widespread populations one doesn't really expect to
see a precise Hardy-Weinberg equilibrium. I expect there
will always be some "isolation-by-distance" in the mating
choices that will actually cause an excess in the
homozygotes above that predicted by HW (not a random mating
condition). This is detectable in the human population
blood group alleles but the measurement is only slightly
more homozygotes than an HW prediction. William L Hunt

>--
>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
 
Perplexed in Peoria <[email protected]> wrote or quoted:
> In reply to a post from Bob containing:

> > 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.
>
> Boy, are you going to feel foolish after you get a good
> night's sleep and review what you have written :)

I believe it is conventional enough.

A finite population will be subject to drift - which will
remove alleles from the population - and increase
homozygosity as a result.

Ridley also refers to this effect as "inbreeding" - writing
in his textbook:

"The increase in homozygosity under drift is due to
inbreeding".

...and...

``Inbreeding can happenin any breeding system with a small
population, and becomes more likely the smaller the
population.''

- Matt Ridley, Evolution, 3rd edition, p.149

I note that Matt repeats the "infinity" error on p.148
by writing:

``If N is infinitely large, the degree of heterozygosity is
stable: there is no march to homozygosity.''

This is an unambiguous reference to the allele frequency in
an infinite population - which is a mathematical fiction :-(

A more mathematically coherent statement would have been:

``As N tends to infinity, the degree of heterozygosity tends
to stablise;
the rate of the march to homozygosity tends to become
infinitessimal.
--
__________
|im |yler http://timtyler.org/ [email protected] Remove
lock to reply.
 
Perplexed in Peoria wrote:
>>In essence, any finite population will become inbred over
>>time (at least to some extent), and this increases
>>homozygosity.
>
>
> Boy, are you going to feel foolish after you get a good
> night's sleep and review what you have written :)
>
I'm not sure I feel foolish, but I agree it would be better
to have written this:

In essence, any finite population will become inbred over
time (at least to some extent), and this means an increase
in homozygosity.

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
 
Tim Tyler wrote:
> Bob O'Hara <[email protected]> wrote or quoted:
>
>>Tim Tyler 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.
>>>>
>>>>But it is - in finite populations, you get an excess of
>>>>homozygotes, as any student of population genetics
>>>>should know.
>>>
>>>Any mention of gene frequencies in an infinite population
>>>is nonsense - as I stated originally.
>>>
>>>You can't talk about a fraction of an infinite population
>>>having a trait. You would get different results for that
>>>fraction depending on how you enumerated through the
>>>population.
>>
>>I don't understand what you mean, but by that argument,
>>you can't even define a fraction or a probability.
>
>
> Fractions have nothing to do with infinite sets.
>
But there are an infinite number of fractions, so they have
at least that to do with infinite sets.

>
>>>It's like claiming that half the integers are even.
>>
>>Err, they are. There are just rather a lot of them.
>
>
> No, there aren't.
>
> There are an infinite number of even numbers.
>
> There are an infinite number of odd numbers.
>
> Divide infinity by infinity and the result is
> indeterminate.
>
If there are an equal number of even and odd numbers, then
half of the numbers must be even.

This must be true because for every even number, I can add
1 and get an odd number. Conversely for every odd number I
can add 1 and get an even number. Hence, by the operation
of adding 1, I can produce an even number for every odd
number and vice versa. Ergo, half of all numbers are even,
and half are odd.

I find this sort of proof preferable to throwing my hands up
in defeat.

<snip>
>>>No serious mathematician can talk about fractions of
>>>infinite sets and expect to be taken seriously.
>>
>>But they do.
>
>
> No - not unless the fractions are "zero" or "one".
>
Rubbish, unless you're denying the existence of fractions.
Fractions are fractions of an infinite set, because there is
an infinite number of numbers between 0 and 1 (proof: take
the reciprocal of every positive integer).

>
>>It's how probability is defined as a concept.
>
>
> Probability is defined as a mathematical limit, as N
> approaches infinity.
>
> That uses a limit as a finite set increases in size - not
> a fraction of an infinite set.
>
> E.g. see:
>
> http://www.wordiq.com/definition/Probability
>
This doesn't show that probability is defined as a
limit - the nearest you get is in the section
"Probability in mathematics", where they use "one
approach" to give an interpretation - essentially, the
frequentist approach. Note that when they discuss
Kolmonogorov's definition of probability as a measure,
they make no mention of any limits.

>
>>I have a colleague who even wrote mathematical papers
>>about fractions of uncountable sets.
>
>
> If you can show me, I should be able to tell you if they
> contain the fallacy under discussion.
>
> Probably he doesn't do that at all - and instead uses
> a limit.
>
This was (I think - my copy is at home) the paper:

E. Arjas & E. Nummelin & R.L. Tweedie: Semi-Markov processes
on a general state space -theory and quasi-stationarity.
J. Aust. Math. Soc. (Series A) 30 (1980): 187 - 200.

>
>>Infinity is a difficult concept (I know - there are lots
>>of it I don't understand), so I think one should be
>>cautious about making any pronouncements on it unless one
>>is sure about what mathematics does and does not say on
>>the subject.
>
>
> How is that relevant?
>
You're trying to argue about the use of infinity. I'm
pointing out that one should be careful when doing this.
This seems relevant.

> Are you suggesting I don't know what I am talking about?
>
> That is not the case.

Your evidence for this is?

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
E.A. 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
 
Guy Hoelzer wrote:
> Hi Bob,
>
> I snipped the following out of a longer post.
>
> in article [email protected], Bob O'Hara at
> [email protected] wrote on 6/21/04 4:44 PM:
>
>
>>But, as we know know, in a finite population, there will
>>be an excess of homozygotes (because of inbreeding), so
>>H-W isn't correct (but a reasonable approximation if the
>>population is large).
>
>
> I am not sure how extensive you intended this claim to be,
> but I think it is generally false. The finiteness of
> populations does not cause an excess of homozygosity
> relative to HWC expectations. Similarly, there is no
> reason to expect the excess of homozygosity to grow with
> decreasing population size, nor is there a reason to
> expect the degree of inbreeding to increase relative to
> that which would occur in random mating as population size
> decreases.

OK, at the very least I should have specified random mating
(which is specified under H-W as well). Under these
conditions, a smaller population size means a greater chance
of mating with a close relative
(i.e. inbreeding), which leads to the increase in inbreeding
over time (at least in the simple models).

On the point that the expected excess of homozygosity
growing with decreasing population size, I agree. What will
happen, under random mating (in a closed population etc.
etc.), is that the rate of increase in homozygosity over
time will increase.

How all this relates to the real world is another matter -
this all started off with a discussion about the H-W model,
rather than about reality.

The HWC model merely predicts the way in which
existing allelic
> diversity is translated into genotypic diversity. It makes
> no assertions about the extent of allelic diversity you
> are likely to find in a population, which is strongly
> influenced by population size.
>
Yes, we're in agreement here - there's too many other things
that can affect the population structure (see the last 80
years or so of population genetics for references!).

And all this is _before_ we start discussing selection!

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
i.f. 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
 
"Perplexed in Peoria" <[email protected]> wrote in message news:<[email protected]>...
> "Name And Address Supplied"
> <[email protected]> wrote in message
> news:[email protected]...
> > "Perplexed in Peoria" <[email protected]> wrote
> > in message
> news:<[email protected]>...
> > > "Anon." <[email protected]> wrote
> > > in message news:[email protected]...
> > > > But it is - in finite populations, you get an excess
> > > > of homozygotes,
> as
> > > > any student of population genetics should know.
> > >
> > > Dooohh! Ignore my other post. Reality gives an
> > > excess of
> heterozygotes.
> > > The Hardy-Weinberg "law" gives an excess of
> > > homozygotes.
> > >
> >
> > To clarify, are we assuming no selfing here? If not, I'm
> > struggling to see why you get the result that you do.
> >
>
> I am assuming no selfings. Thus, in a finite population, I
> get results at variance with Hardy-Weinberg. But, if I
> include the selfings, I get results identical with HW.

H-W tells us the genotype frequencies after random union of
gametes. H-W isn't applicable in finite populations without
selfing, not because the population is finite per se, but
rather it is because we have deviated from the condition of
random union of gametes.
 
in article [email protected], Tim Tyler at [email protected]
wrote on 6/23/04 9:06 AM:

> Perplexed in Peoria <[email protected]> wrote
> or quoted:
>> In reply to a post from Bob containing:
>
>>> 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.
>>
>> Boy, are you going to feel foolish after you get a good
>> night's sleep and review what you have written :)
>
> I believe it is conventional enough.
>
> A finite population will be subject to drift - which will
> remove alleles from the population - and increase
> homozygosity as a result.
>
> Ridley also refers to this effect as "inbreeding" -
> writing in his textbook:
>
> "The increase in homozygosity under drift is due to
> inbreeding".

Yeah. This is the kind of statement that led me to stop
using this textbook in my evolution course. Inbreeding does
not cause drift, or vice versa. The strength of drift
changes inversely with effective population size (by
definition) and inbreeding tends to vary the same way.

> ...and...
>
>
> ``Inbreeding can happenin any breeding system with a small
> population, and becomes more likely the smaller the
> population.''
>
> - Matt Ridley, Evolution, 3rd edition, p.149
>
> I note that Matt repeats the "infinity" error on p.148 by
> writing:
>
> ``If N is infinitely large, the degree of heterozygosity
> is stable: there is no march to homozygosity.''

This is another one of those statements that led me to drop
this textbook.

> This is an unambiguous reference to the allele frequency
> in an infinite population - which is a mathematical
> fiction :-(
>
> A more mathematically coherent statement would have been:
>
> ``As N tends to infinity, the degree of heterozygosity
> tends to stablise;
> the rate of the march to homozygosity tends to become
> infinitessimal.

Well said.

Guy
 
Bob,

in article [email protected], Anon. at
[email protected] wrote on
6/23/04 9:06 AM:

> Perplexed in Peoria wrote:
>>> In essence, any finite population will become inbred
>>> over time (at least to some extent), and this increases
>>> homozygosity.
>>
>>
>> Boy, are you going to feel foolish after you get a good
>> night's sleep and review what you have written :)
>>
> I'm not sure I feel foolish, but I agree it would be
> better to have written this:
>
> In essence, any finite population will become inbred over
> time (at least to some extent), and this means an increase
> in homozygosity.

You are right that inbreeding is ALWAYS happening, because
the individuals in every mating pair are ALWAYS related.
This process causes there to ALWAYS be a tendency toward
increasing degrees of both inbreeding and homozygosity
within finite (all real) populations. Population
subdivision and isolation by distance influence both of
these effects, causing the degrees of inbreeding and
homozygosity to increase at an even faster rate. These
processes conspire to drive populations on the notorious
"march toward homozygosity;" however, it is glaringly clear
from empirical observations that this is a highly
unbalanced view of nature. The data clearly demonstrate
that outbreeding and mutation, which dynamically drive
populations toward lower degrees of inbreeding and
homozygosity, are usually sufficiently strong to stop the
march well short of its endpoint.

Guy
 
Bob,

in article [email protected], Anon. at
[email protected] wrote on
6/23/04 9:06 AM:

> Guy Hoelzer wrote:
>> Hi Bob,
>>
>> I snipped the following out of a longer post.
>>
>> in article [email protected], Bob O'Hara
>> at [email protected] wrote on 6/21/04 4:44 PM:
>>
>>
>>> But, as we know know, in a finite population, there will
>>> be an excess of homozygotes (because of inbreeding), so
>>> H-W isn't correct (but a reasonable approximation if the
>>> population is large).
>>
>>
>> I am not sure how extensive you intended this claim to
>> be, but I think it is generally false. The finiteness of
>> populations does not cause an excess of homozygosity
>> relative to HWC expectations. Similarly, there is no
>> reason to expect the excess of homozygosity to grow with
>> decreasing population size, nor is there a reason to
>> expect the degree of inbreeding to increase relative to
>> that which would occur in random mating as population
>> size decreases.
>
> OK, at the very least I should have specified random
> mating (which is specified under H-W as well). Under these
> conditions, a smaller population size means a greater
> chance of mating with a close relative
> (i.e. inbreeding), which leads to the increase in
> inbreeding over time (at least in the simple
> models).

Yes. This does not contradict anything I said.

> On the point that the expected excess of homozygosity
> growing with decreasing population size, I agree. What
> will happen, under random mating (in a closed population
> etc. etc.), is that the rate of increase in homozygosity
> over time will increase.

Not quite. The rate of increase in homozygosity will also
depend on the existing allelic diversity, which will tend to
decrease as a population shrinks. Your statement is correct
assuming a constant allelic diversity.

> How all this relates to the real world is another matter -
> this all started off with a discussion about the H-W
> model, rather than about reality.
>
>> The HWC model merely predicts the way in which existing
>> allelic diversity is translated into genotypic diversity.
>> It makes no assertions about the extent of allelic
>> diversity you are likely to find in a population, which
>> is strongly influenced by population size.
>>
> Yes, we're in agreement here - there's too many other
> things that can affect the population structure (see the
> last 80 years or so of population genetics for
> references!).

Thanks for the pointer!

> And all this is _before_ we start discussing selection!

How ugly could THAT get? [I don't really want to know.]

Regards,

Guy
 
Anon. <[email protected]> wrote or quoted:
> Tim Tyler wrote:
> > Bob O'Hara <[email protected]> wrote or
> > quoted:
> >>Tim Tyler 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.
> >>>>
> >>>>But it is - in finite populations, you get an excess
> >>>>of homozygotes, as any student of population genetics
> >>>>should know.
> >>>
> >>>Any mention of gene frequencies in an infinite
> >>>population is nonsense - as I stated originally.
> >>>
> >>>You can't talk about a fraction of an infinite
> >>>population having a trait. You would get different
> >>>results for that fraction depending on how you
> >>>enumerated through the population.
> >>
> >>I don't understand what you mean, but by that argument,
> >>you can't even define a fraction or a probability.
> >
> > Fractions have nothing to do with infinite sets.
>
> But there are an infinite number of fractions, so they
> have at least that to do with infinite sets.

You can talk about fractions just fine without ever
mentioning infinite sets.

It is incorrect to say that you can't define fractions
without reference to infinite sets.

A fraction is just one finite number divided by another one.

> >>>It's like claiming that half the integers are even.
> >>
> >>Err, they are. There are just rather a lot of them.
> >
> > No, there aren't.
> >
> > There are an infinite number of even numbers.
> >
> > There are an infinite number of odd numbers.
> >
> > Divide infinity by infinity and the result is
> > indeterminate.
>
> If there are an equal number of even and odd numbers, then
> half of the numbers must be even.

This is not true when the sizes of the sets involved
are infinite.

The ratio of two infinities is either another infinity, an
infinitessimal, or is undefined - depending on the
infinities in question.

> This must be true because for every even number, I can add
> 1 and get an odd number. Conversely for every odd number I
> can add 1 and get an even number. Hence, by the operation
> of adding 1, I can produce an even number for every odd
> number and vice versa. Ergo, half of all numbers are even,
> and half are odd.

I can easily create a map between every even number an 5
unique odd numbers - i.e I can map from 2x to 5x+1, 5x+3,
5x+5, 5x*7 and 5x+9.

That is exactly the same sort of argument as the one you
gave - yet it indicates that there are *five* odd numbers
for every even number.

What conclusion should one draw from this?

The correct conclusion is that the argument you gave
is useless.

You simply can't argue like that about ratios between the
sizes of infinite sets.

> >>>No serious mathematician can talk about fractions of
> >>>infinite sets and expect to be taken seriously.
> >>
> >>But they do.
> >
> > No - not unless the fractions are "zero" or "one".
>
> Rubbish, unless you're denying the existence of fractions.
> Fractions are fractions of an infinite set, because there
> is an infinite number of numbers between 0 and 1 (proof:
> take the reciprocal of every positive integer).

1/3 is *not* the ratio of the size of the set of numbers
smaller than 1/3 and the size of the set of numbers
greater than 1/3. That ratio is the ratio of two infinite
numbers - and thus is not well defined.

If you ask what proprtion of rational numbers is smaller
than 1/3 the answer is *not* "one third of them". The answer
is that the question makes no sense - because it is a ratio
of two infinite numbers, and - in mathematics - oo / oo is
not defined.

> >>It's how probability is defined as a concept.
> >
> > Probability is defined as a mathematical limit, as N
> > approaches infinity.
> >
> > That uses a limit as a finite set increases in size -
> > not a fraction of an infinite set.
> >
> > E.g. see:
> >
> > http://www.wordiq.com/definition/Probability
>
> This doesn't show that probability is defined as a
> limit - the nearest you get is in the section
> "Probability in mathematics", where they use "one
> approach" to give an interpretation - essentially, the
> frequentist approach. Note that when they discuss
> Kolmonogorov's definition of probability as a measure,
> they make no mention of any limits.

You *can* define probabilities in terms of limits - without
reference to infinite sets.

Simply beacuse ratios of the sizes of infinite sets make
little mathematical sense, that does not render all notions
of probability useless.

> >>I have a colleague who even wrote mathematical papers
> >>about fractions of uncountable sets.
> >
> > If you can show me, I should be able to tell you if they
> > contain the fallacy under discussion.
> >
> > Probably he doesn't do that at all - and instead uses a
> > limit.
>
> This was (I think - my copy is at home) the paper:
>
> E. Arjas & E. Nummelin & R.L. Tweedie: Semi-Markov
> processes on a general state space -theory and quasi-
> stationarity. J. Aust. Math. Soc. (Series A) 30 (1980):
> 187 - 200.

Apparently too inaccessible for me to examine on the basis
of a esoteric point in a usenet debate - unless you know
where it is publicly accessible.

> >>Infinity is a difficult concept (I know - there are lots
> >>of it I don't understand), so I think one should be
> >>cautious about making any pronouncements on it unless
> >>one is sure about what mathematics does and does not say
> >>on the subject.
> >
> > How is that relevant?
>
> You're trying to argue about the use of infinity. I'm
> pointing out that one should be careful when doing this.
> This seems relevant.

OK.

> > Are you suggesting I don't know what I am talking about?
> >
> > That is not the case.
>
> Your evidence for this is?

My mathematical credentials may not be publicly accessible
for your inspection - but I do have a degree in mathematics.

Here is the (basically correct) answer given on mathforum
regarding ratios of infinite quantities.

http://mathforum.org/library/drmath/view/53337.html

This result is codified in things such as the IEEE 754
standard.
--
__________
|im |yler http://timtyler.org/ [email protected] Remove
lock to reply.
 
"Anon." <[email protected]> wrote in message
news:[email protected]...
> Perplexed in Peoria wrote:
> >>In essence, any finite population will become inbred
> >>over time (at least to some extent), and this increases
> >>homozygosity.
> >
> >
> > Boy, are you going to feel foolish after you get a good
> > night's sleep and review what you have written :)
> >
> I'm not sure I feel foolish, but I agree it would be
> better to have written this:
>
> In essence, any finite population will become inbred over
> time (at least to some extent), and this means an increase
> in homozygosity.

I must apologize for being unclear in indicating what was
foolish. There is nothing foolish about the quoted
statement taken in isolation (and I admit that I apparently
isolated it).

What was foolish was to make this statement in the context
of a discussion of the Hardy-Weinberg law, and in
particular the "infinite population assumption" in
derivations of the law. My point here is well made by
Hoelzer and Hunt. You are either succumb to or ignoring an
ambiguity in the word "homozygosity" in this context. Do
you mean to suggest that the kind of inbreeding that
increases in a finite population over time has anything to
do with invalidating P^2: 2PQ : Q^2 ??
 
See my reply to Anon regarding what I believed was foolish.

Regarding infinities without the use of limits:

Tim, I think you are wrong here. Yes, working with the
infinite in mathematics is subtle, and it takes some extra
care. The use of limits is the first method that
mathematicians found to handle infinite sets rigorously. It
was an invention of the nineteenth century. It is one way of
achieving rigor, but it is not the only way.

During the twentieth century, several other methodologies
have been discovered permitting rigor. One of them is non-
standard analysis, though that does not seem relevant
here. Another is "measure theory" - which is quite
relevant. Look into it.

You are just simply wrong to assert that talking about the
infinite without using limits is a sign of mathematical
ignorance. You are just as wrong as would have been an
eighteenth century person to claim that ANY talk of the
infinite is unrigorous.
 
Perplexed in Peoria <[email protected]> wrote or quoted:

> Regarding infinities without the use of limits:
>
> Tim, I think you are wrong here. Yes, working with the
> infinite in mathematics is subtle, and it takes some extra
> care. The use of limits is the first method that
> mathematicians found to handle infinite sets rigorously.
> It was an invention of the nineteenth century. It is one
> way of achieving rigor, but it is not the only way.
>
> During the twentieth century, several other methodologies
> have been discovered permitting rigor. One of them is non-
> standard analysis, though that does not seem relevant
> here. Another is "measure theory" - which is quite
> relevant. Look into it.

Basically it seems to use integration - i.e. the limit
of dx -> 0.

> You are just simply wrong to assert that talking about the
> infinite without using limits is a sign of mathematical
> ignorance.

I have looked back through what I wrote. I can see no sign
of me saying that. It seems to me that you would need to
quote me saying something along these lines to back up
your claim.

You cannot simply talk about ratios between two inifite
quantities. so - unqualified mention of allele frequency in
an infinite population makes no sense.

You *can* make coherent statements about the topic - either
by specifying a limit, by specifying a sampling stratgey
(also a form of limit) - or in a number of broadly-
equivalent ways - but unqualified discussion of frequencies
in infinite populations is still bad form.
--
__________
|im |yler http://timtyler.org/ [email protected] Remove
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