Re: Date for Last Common Ancestor?

Stephen Barnard (
Thu, 22 Aug 1996 07:53:16 -0800

Geoffrey Norman Watson wrote:
> On Wed, 21 Aug 1996, Stephen Barnard wrote:
> >
> > Your definition is fine, as far as it goes. Like you say, you need further
> > assumptions to get the size to converge to one.
> >
> > The assumption that I made, which seems perfectly reasonable to me, is that the
> > probability that every mother in M_k-1 produces *exactly* one mother in M_k is
> > less than one.
> >
> > Steve Barnard
> >
> Your assumption is reasonable, but how to justify it?

It can be justified on a priori statistical grounds. For large sets it can be
justified by the law of large numbers. For small sets it can be justified for
essentially the same reason, but by looking at averages over time. I haven't
actually *done* such a thing, but it's pretty clear to me that it can be done.

> One defect of this analysis
> is that it is difficult to see how it could be used for simulations. The
> generation of the sets M_k is in the opposite direction to time, and a lot of
> what is happening depends on the rest of the population, which is not modelled
> here.

This is true, but I question the value of simulations anyway. The date of the LCA
(female) is so contingent that I don't see how a simulation could give a meaningful
estimate. Maybe a large number of simulations could narrow it down. I don't know.
It seems like you would have to simulate forward in time, but maybe not.

One thing to realize is that all of the members of a set M_k are not necessarily
alive at the same time, so k cannot be exactly correlated with the date. As k
increases you would expect the dates of the members of M_k to spread out.

> Forwards simulations from a small base population show that female-line-lineages
> die out randomly. To eliminate them rapidly you need a small population, and
> obviously the probability of losing one falls as the number of lineages left
> reduces. This supports your requirement that the probability is less than one
> but I can't see how you would derive it rigorously.
> ------------------------------------------------------------------------------
> Geoffrey Watson

Steve Barnard