Re: Hamilton's rule


"Guy Hoelzer" <hoelzer@xxxxxxx> wrote in message news:dkun5k$qs4$1@xxxxxxxxxxxxxxxxxxxxxx
> in article dkt9ce$3ac$1@xxxxxxxxxxxxxxxxxxx, Perplexed in Peoria at
> jimmenegay@xxxxxxxxxxxxx wrote on 11/9/05 8:50 AM:
>
> > "Guy Hoelzer" <hoelzer@xxxxxxx> wrote in message
> > news:dks3mk$2km4$1@xxxxxxxxxxxxxxxxxxxxxx
> >> Jim,
> >>
> >> I am going to disagree with some of the details of your argument below, but
> >> I agree with your approach in general.
> >>
> >> in article dkjvng$1sm0$1@xxxxxxxxxxxxxxxxxxx, Perplexed in Peoria at
> >> jimmenegay@xxxxxxxxxxxxx wrote on 11/5/05 8:10 PM:
> >>
> >>> "Guy Hoelzer" <hoelzer@xxxxxxx> wrote in message
> >>> news:dkgokl$g5l$1@xxxxxxxxxxxxxxxxxxxxxx
> >>>> ... here is my take.
> >>>> The informativeness of "r" (the IBD version) should be non-linearly related
> >>>> to the frequency of the altruism allele.
> >>>
> >>> It is linear or constant, depending on how you define "informativeness.
> >>> See below.
> >>>
> >>>> 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.
> >>>
> >>> Graph frequency in focal individual (Y axis) vs frequency in population
> >>> (X axis). Two graphs. The one for a typical non-relative is simply a
> >>> straight line at 45 degrees from a frequency of zero (when the population's
> >>> frequency is zero) to a frequency of 1.0 (when the population's frequency
> >>> is 1.0). The second graph for a relative is also a straight line, though
> >>> one with a less steep slope.
> >>
> >> So far so good.
> >>
> >>> This line runs from 'r' (when the population's
> >>> frequency is zero) to 1.0 (when the population's frequency is 1).
> >>
> >> Agreed if by "zero" you mean "near zero" and allow for enough copies to
> >> exist that they can blanket close relatives in the expected frequencies.
> >> This could be a large number for fecund species.
> >
> > Ok.
> >
> >>> In the above, I am assuming either a haploid model, or that the donor is a
> >>> homozygous diploid. If the donor is heterozygous (altruism is dominant)
> >>> then the second graph starts at r/2.
> >>
> >> You lost me here. You seem to be dealing only with frequency, so I don't
> >> see how dominance relates at this point. Where in your argument so far, for
> >> example, does the altruistic phenotype come into play? I can't make sense
> >> of your claim this line should start at r/2 without this information.
> >
> > Well, if the gene is dominant, and the frequency of the gene in the population
> > as a whole is near zero, then it is overwhelmingly likely that the donor has
> > only a single copy.
>
> I think you are mixing inconsistent arguments here. The argument you were
> laying out above is about frequency only, and you haven't established how
> either dominance or phenotypes are relevant. If you make the relevance of
> phenotype clear, I think that dominance will also make the relationship in
> your graph non-linear. Either the graph includes phenotypic effects or it
> doesn't.

Well, the line in question is labeled 'Donor'. If that is not a phenotypic
effect, I don't know what is.

> >>> It is useful to draw a third line on our graph, with the focal individual
> >>> being the donor. For haploids and homozygous diploids, this is simply
> >>> a constant line at 1.0. For heterozygous diploids, this line runs from
> >>> 0.5 to 1.0.
> >>
> >> What would this line represent? The frequency of the allele in a
> >> heterozygote is fixed at 0.5, so what would it mean for this line to 'run to
> >> 1.0?' The "frequency in the focal individual" wouldn't change.
> >
> > It is the frequency of the allele in donors as a class. Donors have at least
> > one copy, but they may have two. One copy is overwhelmingly likely when
> > frequency in the population is zero. Two copies is overwhelmingly likely
> > when the frequency in the population is one.
>
> I suspect that you are often using language indicating single individuals
> when you mean to talk about classes of individuals. It would help me
> understand your argument better if you helped me to keep this straight.

Whoops, sorry about that. See my response to the other posting.

> >>> Notice that all three lines meet at 1.0 when the allele is fixed in the
> >>> population. But for all other frequencies, the donor line is on top,
> >>> the "typical non-relative" line is on the bottom, and the "typical
> >>> recipient"
> >>> line is in the middle. It lies a fraction r of the way up from the bottom
> >>> line to the top line.
> >>
> >> Why wouldn't the line for a heterozygote be below the other two lines when
> >> the allele is very common in the population?
> >
> > There is no line for heterozygotes. I don't know where you got the idea that
> > there was.
>
> You wrote: "For heterozygous diploids, this line runs from 0.5 to 1.0."

Ouch. I did say that. Which explains why we each think the other is
confused. My Bad.

Let me try again. The 'Donor' line gives the expected frequency of the
altruism allele in donors as a class. If the allele is recessive, then
the line is a constant 1. If the allele is dominant, then the line runs
(straight) from 0.5 at low population frequencies to 1.0 at high
population frequencies. It may be worth noting that at low population
frequencies, if the allele is dominant then donors will almost all be
heterozygotes.

And that is what I wrote on my second attempt:

> > The third line is for donors, who may or may not be heterozygous.
> > The donor line is in one position for dominant alleles, and in a different
> > position for recessive alleles.


.

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