Re: Hamilton's rule
- From: Guy Hoelzer <hoelzer@xxxxxxx>
- Date: Fri, 4 Nov 2005 17:51:01 -0500 (EST)
in article djf5gf$1fru$1@xxxxxxxxxxxxxxxxxxx, Perplexed in Peoria at
jimmenegay@xxxxxxxxxxxxx wrote on 10/22/05 10:02 PM:
> "Guy Hoelzer" <hoelzer@xxxxxxx> wrote in message
> news:djdra8$u9l$1@xxxxxxxxxxxxxxxxxxxxxx
>> in article dj91v8$1t9o$1@xxxxxxxxxxxxxxxxxxx, Perplexed in Peoria at
>> jimmenegay@xxxxxxxxxxxxx wrote on 10/20/05 2:24 PM:
>>
>>> "Guy Hoelzer" <hoelzer@xxxxxxx> wrote in message
>>> news:dj6jvu$tpa$1@xxxxxxxxxxxxxxxxxxxxxx
>>>> in article dj603t$kb4$1@xxxxxxxxxxxxxxxxxxx, Catherine Woodgold at
>>>> an588@xxxxxxxxxxxxxxxxxxx wrote on 10/19/05 10:34 AM:
>>>>
>>>>> Guy Hoelzer (hoelzer@xxxxxxx) writes:
>>>>>> in article dj1v9g$1p6k$1@xxxxxxxxxxxxxxxxxxx, Perplexed in Peoria at
>>>>>> jimmenegay@xxxxxxxxxxxxx wrote on 10/17/05 9:56 PM:
>>>>>> GH:-
>>>>>> In fact, I think it [selective pressure in favor of kin-selected
>>>>>> altruism]
>>>>>> would typically become negligible even at intermediate
>>>>>> frequencies of the altruism allele (say above 20 or 30%).
>>>>>
>>>>> CW:-
>>>>> I completely disagree (see two arguments above).
>>>
>>> So do I. And Catherine's arguments are entirely correct.
>>>
>>>>>>> JM:-
>>>>>>> Or are you saying that 'rb>c' remains valid in determining the direction
>>>>>>> of the evolutionary force, regardless of frequency, but that the
>>>>>>> magnitude of the force depends upon frequency? If that is your position
>>>>>>> then I apologize unconditionally.
>>>>>>
>>>>>> Oops. I should have read this first. This is indeed my position.
>>>
>>> And I have to withdraw my apology. Guy, you are very confused on HR.
>>> Pay attention to what Catherine is saying. She understands this material
>>> considerably better than you do.
>>
>> The argument I am making about the effectiveness of kin selection (the
>> expected extent of response kin selection) as the altruism allele changes in
>> frequency was never described by Hamilton, AFAIK, and I am not claiming that
>> it is part of the rule.
>
> My impression is that it was described by Hamilton in the 1964 paper. If
> you have a copy of "Narrow Roads", take a look at the first two paragraphs
> on page 36. Hamilton talks about a 'diluting effect', which I understand
> to be a factor capturing the effectiveness of selection in his model.
Thanks. I don't have a copy of "Narrow Roads", but I will go back to the
'64 paper and look for language on the diluting effect. Frankly, I'm
surprised that I don't remember it. Maybe it was a minor side comment that
he didn't expound on much. That certainly wouldn't be inappropriate in the
first paper introducing how the model works, but it would have been nice to
see this limitation on the process explored more thoroughly in subsequent
theoretical work on kin selection.
>> I am trying to extend Hamilton's model in a way
>> that is consistent with conventional selection theory by asking about
>> variation in evolutionary effect as opposed to validity of the kin selection
>> process in principle. It is analogous to making the point that natural
>> selection is completely ineffective at driving a deleterious recessive
>> allele from a gene pool. In that case, the last copy of the allele (almost)
>> inevitably exists in a heterozygous genotype and does not reduce anyone's
>> fitness. The only process that can eliminate a deleterious recessive allele
>> is drift. That does not mean that natural selection is irrelevant to
>> evolution at all frequencies of a deleterious recessive allele, but its
>> relevance varies with the frequency of the allele, as does the expected rate
>> of the response to selection.
>
> Well, I understand what you are saying about the last copy. And I also
> understand that selection is most effective when the allele under selection
> is intermediate in frequency - it is less effective when rare or near
> fixation. One sees factors like (p(1-p)), where p is the allele frequency.
> That applies to ordinary selection and kin selection both. And I understand
> that Hamilton's 'dilution factor' introduces yet another kind of reduction
> to the effect of NS which is specific to kin selection.
Good. This puts us closer to being on the same page.
> What I don't understand is how either of these factors makes selection
> somewhat ineffective when the allele frequency rises to something like
> 20-30%. ISTM that this is the level at which selection just starts
> becoming really effective.
It certainly is a level of moderate frequency when something like
directional selection with additive allele effects gains full traction. I
mentioned the 20-30% range explicitly as a speculation, and I would not try
to argue that it couldn't be 10% or 40%, or that it would be any particular
value for all contexts. I just through out this range to generate the kind
of useful response that you gave in this post.
I can't comment yet on Hamilton's "dilution effect", but here is my take.
The informativeness of "r" (the IBD version) should be non-linearly related
to the frequency of the altruism allele. The precision of the IBD rule is
never very great, because close relatives may not contain the allele, while
some of those identified as non-relatives (those more distantly related)
might. The likelihood that the allele is carried by a relative does not
change much as the frequency of the allele increases, but it goes up faster
for non-relatives. Finally, the utility of this indicator of allele sharing
rests on the ratio of the likelihood the allele is shared with a close
relative divided by the likelihood it is shared by randomly selected
individual in the population. Because I expect this ratio to decrease in a
concave form as allele frequency increases, I also expect the selective
value of using "r" (the IBD version) to guide the allocation of altruism to
drop off quickly as the allele frequency increases. I admittedly have not
done the math on this issue, which should not be too difficult, and I am
happy to be shown wrong with regard to the shape of the curve relating the
utility of "r" to allele frequency "p".
> So, I guess I simply don't understand your argument (or intuitions, or
> whatever). Are you assuming that the number of opportunities for
> altruism vary in a frequency-dependent manner?
No. See above.
> Are you saying that the
> effective values of b and c are frequency dependent?
No, although this could be a pretty interesting question.
Guy
.
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