"Guy Hoelzer"  wrote in message
news:cb7rtf$mn0$1{at}darwin.ediacara.org...
> in article car9d1$2lm6$1{at}darwin.ediacara.org, Perplexed in Peoria at
> jimmenegay{at}sbcglobal.net wrote on 6/16/04 10:15 PM:
> > Whatever method is used, the result has to be an "r"
with the following
> > property:  Randomly choose one of the two genes at any locus in the
donor.
> > Suppose that the frequency of this allele in the general population is
"p".
> > Now, randomly choose one of the two genes at the same locus in the
> > recipient.  It must be the case that the probability that the two
randomly
> > selected genes are identical is (r + (1 - r)p).  That is, there is a
> > probability r that they are identical IBD, but if not, then there is
still
> > a probability p that they are identical for other reasons - because the
> > allele is fairly common in the population.
>
> I think there is a big practical problem with applying any models relying
on
> IBD calculations, because our data on genealogical history is virtually
> always very shallow and incomplete.

No doubt.  I have no idea how field workers go about applying the model.
Though I would imagine that the problems with estimating "b" and
"c"
are much larger than the problems of estimating "r".

> For example, all individual organisms
> (across all species) are probably genealogically related, but...  There is
> also the problem of genealogical relatedness in the face of mutation
(common
> decent without identity).

If you believe that these issues are "big practical problems", then I
suspect that you don't yet understand the model.  The effect on r of
pushing the calculation back additional generations is small in a sexual
population.  And the chance that a mutation has destroyed an altruistic
allele within those few generations is also small.

> > Or, if like McGinn, you have an intuition that geneological history
cannot
> > be causal in this situation, ignore the IBD above.  "r"
is simply a
measure
> > of how much more likely than "p" it is that the two
genes are identical
> > for whatever reason.  The key thing is that the formula (r + (1-r)p)
> > gives the probability that the alleles are "shared".
>
> Hmm.  There are some things about this formulation that I like, and some
> problems I see.  Can you please save me a little research time and tell us
> where you come by the formula (r + (1-r)p)?  Is this your interpretation
of
> Hamilton, or has it been published?

It is a straightforward interpretation of the verbal explanation given in
Maynard Smith's "Evolutionary Genetics" (2nd ed. p169)

  Now we can picture the genome of the recipient as consisting of two
  parts:
  1. a fraction r containing genes IBD to genes in the actor; and
  2. a fraction (1-r) consisting of genes that are a random sample
     of genes in the population.

As a "proof" that this is the correct interpretation, you can derive
"rb>c" from this formula.
1. Assume b and c are the benefits and costs of a single altruistic
   action.
2. Assume a "penetrance factor" f (which may depend on r!) gives the
   number of times an allele causes altruism during its organism's
   lifetime.  Assume that homozygous altruists share the cost between
   alleles.  The factor f allows us to ignore whether the allele is
   dominant or recessive.  For example, if the allele is recessive,
   then f will be small when p is small.  But if the allele is
   dominant, then f will start large and then fall to half the
   original value, due to "sharing the credit" between homozygous
   gene instances.
3. Calculate the total benefit received by recipients.  Divide this
   up between carriers and non-carriers, with the benefit split in
   heterozygotes.  Now calculate the "per capita" benefit to each
   haploid genome of the two types.
4. Calculate the total and per capita costs of altruism.
5. Compare the per capita benefit for non-carriers to the per
   capita (benefit - cost) for carriers.  You will notice that
   the variable factor f cancels out, as does the allele frequency
   p.  You are left with the fact that carriers receive enough
   extra altruism to compensate for the costs of acting altruistically
   when rb>c.
Try it.  As I wrote:

> > Why is that particular formula so important?  Well, when you do the
math,
> > you will see that the average fitness of allele carriers will be greater
> > than non-carriers, as long as the carriers direct their altruism to
> > recipients of relatedness "r".  That is, average
fitness of carriers
> > will be higher as long as rb>c.  And the parameter
"p" nicely cancels
> > out of the equations.  Hamilton's rule is independent of p.  As long
> > as "r" has the meaning above.
>
> My criticism here is the same as one that I have been posting earlier in
> this thread.  Your conclusion is sensitive to the implicit assumption that
> the population is large (effectively infinite) and well mixed.  This
> combination does not generally exist in nature, because the larger a
system
> is the harder it is to mix up.

Not my conclusion.  Hamilton's.  But I think you exagerate the sensitivity.

I think that all that is required is that the "well-mixed" or effectively
random mating breeding population is larger than the local
socially-interacting clique of each individual.  And that "r" is pretty
much the same everywhere.  It is not important whether "p" is pretty
much the same everywhere.

> It also gets harder and harder to
> effectively mix the finite population when "p" is either very small or
very
> large, because in either case there are very few quanta of the rare
allele.

True enough, but now you are criticizing how deterministic Hamilton's rule
is, rather than claiming that it is incorrect (biased) at extreme
frequencies.
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