Re: Hamilton's rule


"Tim Tyler" <tim@xxxxxxxxxxx> wrote in message news:djj01g$2ugn$1@xxxxxxxxxxxxxxxxxxxxxx
> Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote or quoted:
> > "Tim Tyler" <tim@xxxxxxxxxxx> wrote in message news:djdrab$uea$1@xxxxxxxxxxxxxxxxxxxxxx
> > > Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote or quoted:
> > > > "Tim Tyler" <tim@xxxxxxxxxxx> wrote in message news:dj9t57$29lv$1@xxxxxxxxxxxxxxxxxxxxxx
>
> > > > > Hamilton defined r in terms of genes which are identical by
> > > > > descent in 1964. I'm using his definition of r here, r(IBD).
>
> [...]
>
> > > > > You are mistaken if you think I am talking about IIS at all here.
> > > >
> > > > Your level of confusion here is very high. You have snipped without
> > > > comment the portion of my posting which addresses your confusion.
> > > > (It contained a claim about a situation in which two individuals
> > > > are descended from the same set of 1000 individuals).
> > >
> > > It appears that you refer to:
> > >
> > > ``Your greatest confusion is in trying to relate the fact that any IIS
> > > allele is ultimately identical by descent from a long-ago common
> > > ancestor (not necessarily true, by the way, if you define a gene to
> > > be as small as a single base pair) to the measure of genealogical
> > > relatedness known as IBD. 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.''
> > >
> > > That appears to be talking about "the correct value of 'r' for use in
> > > Hamilton's rule".
> >
> > It is talking about the correct meaning of IBD 'r'. I am not saying
> > here that Hamilton made a mistake. I am saying that he got it right
> > and you are getting it wrong.
>
> My position on the latter issue: neither of us made a mistake.
>
> > > If I've been at all unclear on this point, I apologise, but I am -
> > > and have been for some time - talking about r(IBD) - which is what
> > > Hamilton defined r as being in 1964 - and considering the case of
> > > genes being defined to be very small.
> >
> > And if I have been unclear, I also apologize. My position is that
> > you have not been talking about the same thing as Hamilton. You
> > only think it is the same thing. It is very different.
>
> Right. Hamilton defines r pretty clearly - IMO - on page 3 of
> his 1964 paper. However, it is, of course possible that I have
> misinterpreted him in some way. Since that seems to be our
> primary remaining disagreement, I propose we focus on that issue -
> and to that end I've snipped a few lose ends in your reply that
> don't bear on this issue.

OK.

> > If I understand you correctly, you are claiming that (if the complete
> > genealogical history of every gene were known) one way to compute
> > IBD r between two individuals would be to compare the genomes, count
> > the number of genes which are identical by descent (regardless how
> > remote the ancestor), and then divide by the total number of genes.
>
> That is what I would claim. Hamilton uses the phrase "mean
> number of genes per locus IBD" at the previously-mentioned
> location. By "mean" I take it that he is adding up the
> numbers and dividing by the number of loci he has treated.


Well, there is another possible interpretation. You could also take the
average over 'possible universes'. If I ask you to flip a coin three
times and claim that the mean number of heads is 0.5, I am using a
mean over both flips and over universes. If you just take the mean
over flips, then the mean is either 0, 1/3, 2/3, or 1, but I don't
know which.

This distinction may lie close to the heart of our dispute.

> Let us first get the definition of IBD out of the way in
> case our difference lies there. By "IBD" I mean what it
> says on:
>
> http://darwin.eeb.uconn.edu/eeb348/lecture-notes/identity/node1.html
>
> ...namely:
>
> ``Two alleles at a single locus are identical by descent if
> the are identical copies of the same allele in some earlier
> generation, i.e., both are copies that arose by DNA
> replication from the same ancestral sequence without any
> intervening mutation.''
>
> If you have a different understanding of this term, now would
> be a good time to speak up.

I can live with that.

> > That is just wrong. It isn't what Hamilton meant, it isn't what
> > Malecot meant, and it isn't what Woodgold, Felsenstein, and I mean.
>
> Right. So what *do* you mean?
>
> > Look again at what I wrote about the 1000 shared ancestors. Don't
> > jump to the conclusion that I must be missing your point because
> > I am disagreeing with you.
>
> I have done that as requested - and it doesn't seem to help.

Doesn't help in what sense? You don't understand how I arrived at
that value of r = 0.001, or you don't think that that figure is right?

> > > Using the resulting values of r in Hamilton's rule would be highly
> > > *in*appropriate - and would produce completely the wrong answers.
> >
> > It is inappropriate to use % similarity as 'r' regardless of whether
> > you get the right or wrong answers.
>
> Actually "% similarity" would work reasonably well in
> Hamilton's rule - provided you are talking about genes of
> a reasonable size.

Well, I guess I have to concede that point, if by 'reasonable size'
you mean something like 1-20% of a chromosome. But that is something
of a coincidence. It has nothing to do with the rationale for why
Hamilton's rule works.

> It would work /especially/ well in an
> outbred population. That is why Hamilton specifies that he
> is talking about an outbred population twice in the section
> I'm talking about.

> Under your interpretation, I believe the degree of
> inbreeding in the population would be irrelevant -
> and there would be no need for Hamilton to mention
> it twice in the section on the definition of "r" -
> where he does.

Inbreeding is a complicated issue here. Inbreeding in generations
immediately prior to the focal generation mean that Wright's coefficient
and IBD give different answers for relatedness. That is a big part
of the reason why Hamilton mentions it. IBD gives the 'right answer',
but at the time of the writing, most people were unfamiliar with IBD
but Wright's coefficient was better known.

However, you have in the past made statements to the effect that high
levels of sequence similarity are evidence of inbreeding. Since I do
not agree with this, I suspect that we are using the word differently.

In cases of kin recognition or "in the nest" behaviors, or something
of that sort, I don't think that it makes a whole lot of difference
whether the population is moderately inbred. It does seem to make a
difference if high effective r values are obtained by "viscosity".

> > But lets define a new number M (for McGinn) which is the % similarity
> > between donor and recipient measured at the base pair level. And
> > let us also define a number T (for Tyler) which is the average
> > % similarity between the donor and a random member of the population.
> >
> > Then, it is the case that (M-T)/(1-T) is pretty much
> > the same thing as Hamilton's 'r'.
>
> Not as Hamilton *defined* "r" in 1964. Look at his definition
> on the third page of his paper. There is not the slightest
> mention of "% similarity between the donor and a random
> member of the population" He is working *entirely* from
> two individuals, called A and B. There is no mention of a
> relatedness to a third "average" member - or anything
> remotely equivalent.

That is true. Though please note that this definition is
sandwiched between two warnings that it only applies when
the locus is not under selection.

> If you *did* define r like that, that figure would work fine
> in Hamilton's rule.
>
> However, the basic point is that that's not how Hamilton
> defined his "r" in 1964.
>
> > And this is true regardless of whether you measure %
> > similarity at the base pair level, at the amino acid
> > level, at the exon level, at the protein level, or at
> > the 'evolutionary gene' level.
>
> Right. Great. My point is - as usual - that is not how
> Hamilton said he was defining "r" in his 1964 paper.
>
> I expect you can get out your copy of NROGL and see for
> yourself. Page 33, near the bottom. "r" is defined
> in terms of the probability that _two_ individuals
> share genes by descent.
>
> There's no mention of modifying this probabilty according to
> what genes other members of the population may or may not have.

Nor would I suggest that it should be modified. The modification
is needed only if you are working with IIS rather than IBD.

> Averaging is done - but it is across all loci in the
> two individuals, A and B, not across other members of
> the population.

I am only suggesting that you average across members of the
population if you are using the shortcut formula for regression r.
That is r = (M-T)/(1-T) where M is percent similarity (IIS) between
donor and recipient and T is average percent similarity between
donor and a random member of the population.

> I hope all this helps clarify my position.

It does. And I am not sure exactly how to respond. It sounds
like you are saying that Hamilton got it seriously wrong in
1964 when he suggested using probability of identity by descent
as r, without also specifying that he was thinking of a gene size
much larger than most people at that time thought of when they
thought of a gene. And I am saying that you are misinterpreting
Hamilton.

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.

Do you understand this calculation? If so, what do you think is
wrong with it?


.

Relevant Pages

  • Re: Units of IBD
    ... >>> make out of your claim that all probabilities have units. ... Hamilton started with a model ... Hamilton's rationale for R = Genes IBD. ...
    (sci.bio.evolution)
  • Re: Hamiltons rule
    ... > It is talking about the correct meaning of IBD 'r'. ... >> genes being defined to be very small. ... > you have not been talking about the same thing as Hamilton. ... > It is inappropriate to use % similarity as 'r' regardless of whether ...
    (sci.bio.evolution)
  • Re: Hamiltons rule
    ... >>or common descent from a shared ancestor. ... > But of course all genes are descended from a common ancestor. ... > not IBD, for the notion of IBD to make sense. ... > The Hamilton conditions, that's what. ...
    (sci.bio.evolution)
  • Re: Hamiltons rule
    ... >>other genes in relatives as a result of descent from one another ... > But of course all genes are descended from a common ancestor. ... > not IBD, for the notion of IBD to make sense. ... Hamilton employed this technique? ...
    (sci.bio.evolution)
  • Re: Am not (was: Felsenstein is a liar)
    ... > leave more replica genes in the next generation than an allele having ... parental care is almost completely nonexistent. ... Hamilton pointing out the obvious? ...
    (sci.bio.evolution)
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