Re: Hamilton's rule

Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote or quoted:
> "Tim Tyler" <tim@xxxxxxxxxxx> wrote in message news:djltdm$12fa$1@xxxxxxxxxxxxxxxxxxxxxx
> > Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote or quoted:

> > > > > > If you and I have exactly the same set
> > > > > > of 1000 ancestors ten generations back, and if 98% of our genes are
> > > > > > identical and each of them can be traced back to exactly one of our
> > > > > > shared ancestors (not necessarily the same ancestor for each gene),
> > > > > > then the correct value of 'r' for use in Hamilton's rule is still
> > > > > > arrived at by using Malecot's rules and the value of 'r' will be
> > > > > > something like 0.001.''
> > > [snip]
> > > And I think that the key to resolving this, and to your understanding
> > > how I interpret Hamilton, is to work through my example. I had
> > > all (or almost all) genes identical by descent from an ancestor only
> > > 10 generations back, yet I still claimed that the proper calculation
> > > of r as probability of genes being identical by descent was only about
> > > 0.001. I got that figure by noting that each shared ancestor was
> > > 10 generations back, which means 20 generations up and back, which
> > > means we have a 1/(2^20) of sharing an allele through any one of
> > > those 1000 shared ancestors, which gives us 1000 * 1/(2^20) total,
> > > which is roughly 0.001.

[...]

> > None of that makes much sense to me: if you are going up the generations
> > you also need to multiply the initial probability by the number of
> > possible paths the gene could have taken between the offspring and the
> > ancestor.
>
> Hmmm. I intended to have only one path to each shared ancestor. But it
> appears that I didn't provide enough ancestors to do the job. So, let
> me revise my model to have 1024 ancestors ten generations back, who
> form 512 mating pairs. Each pair has two children, one of which is an
> ancestor of the donor, and the other of which is an ancestor of the
> recipient. Absolutely no inbreeding takes place from that stage forward.

That's a minor change, not really much to do with the point I raised.
A side issue - but if you intend "absolutely no inbreeding" to refer
to no mating between relatives, then I hope you realise that's pretty
difficult to keep up for very long in a finite population ;-)

> > Since there are many possible paths of descent via which a gene from
> > an ancestor can arrive in an individual ten generations later - that
> > is going to be a very large figure.
>
> No. Now there are only two possible paths.

There are many possible ways in which a gene from the ancestor
can travel down ten generations.

You started by dealing with probabilities - talking about the
*chance* of the gene being received from either parent. Now it
seems you are considering a specific instance of one path of
descent. You can't have this both ways, either you need to
consider the probability of a particular gene travelling down
ten generations (in which case you need to consider all the
possible paths it might travel down and sum the probabilities
of each one occurring) or you need to concentrate on one
allele travelling down one particular path of descent - in
which case the only probability you can talk about is
the probability of the allele taking that one particular
path from ancestor to the present day, and - while that
is the probability you seem to have calculated - it doesn't
appear to be of any theoretical interest - and multiplying
that figure by the population size produces another pretty
meaningless figure.

> > Genes are not just diluted by a factor of two in each generation
> > (because each offspring has two parents). They are also multiplied
> > by a factor of two (because each parent has around two offspring).
>
> That makes no sense to me. We are only concerned with the genes that
> make it to the donor, and the genes which make it to the recipient.

If you are talking about a *particular* path of inheritance, then
where do probabilites come in? What does your "1/2" represent?

If you are talking about probabilites, you need to add up the
possible combinations that lead to the outcome you are interested in.

Your sum is a lot like saying there's a 1/36 chance of throwing a 9 with
two dice - because look, here's a 4 and here's a 5 and each had a 1/6
chance of occurring. Right - but that's not the *only* way of getting
a 9 from two dice.

Similarly there's more than one way in which a allele in an ancestor can
cross ten generations - and you have to add up the probabilities of each
one of those occurring - if what you are trying to arrive at is the
probability of that gene coming from that ancestor.

> > Since this aspect appears to be missing from your calculation, I
> > can't make much sense out of it.
>
> Hopefully, I have now clarified sufficiently. The point is this -
> I say that r=0.002. You say that it is more like r=0.980. At least
> one of us is wrong. Which one, and why?

I don't say r=0.980. AFAICS, you haven't said anything about the pair
of current-day individuals you are calculating relatedness between. Are
they strangers - or sisters - or what? Without some information about
that, I can't see any way of calculating their relatedness.
--
__________
|im |yler http://timtyler.org/ tim@xxxxxxxxxxx Remove lock to reply.

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Relevant Pages

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