Re: Date for Last Common Ancestor?

John Hawks (hawksj@umich.edu)
Fri, 16 Aug 1996 11:42:39 -0400

CHESSONP wrote:

> The point of the example is to illustrate the fault in your logic. A
> decreasing sequence of numbers may not converge to anything. A decreasing
> sequence of numbers that is bounded will converge to its greatest lower
> bound which need not be zero. A decreasing sequence of positive whole
> numbers will converge to zero, but any finite segment of this sequence
> will not. A non-increasing sequence of positive whole numbers will
> converge to its greatest lower bound, which may not be zero.
>
> You cannot claim that the size of your set of great .... grand mothers
> reaches one at some point simply because each set (going backward by
> generation) is not larger than the previous one. On logical grounds alone
> the number of females in this "lca founding population" could be any
> number no larger than the smallest known human population size at some
> time.

You are right about Steve's proof. In fact, to be accurate, Steve's proof must be
expressed in terms of probability. In each generation, each mother will, with a
certain probability, have one daughter, more than one daughter, or no daughters. The
use of probability is what introduces *stochasticity* (of which you will probably
have heard) into the proof. It cannot be demonstrated in this way that there *must*
be a single mitochondrial ancestor. However, it can be demonstrated with
*overwhelming* likelihood that such an ancestor existed.

In addition, we can evaluate the likelihood that such an ancestor existed during any
period of time, making whatever assumptions about population demography that we
want. In this way, hypotheses about past population history can be tested with
reference to the *actual* mtDNA distribution of living humans. Such a process of
inference can, with some alterations, be applied to alleles of other genes.

I will point out that a non-probabilistic demonstration of the existence of a single
mtDNA ancestor is a corollary of the assumption that the mitochondria of animal and
plant cells must have had a single origin as an independent mitochondrial organism.

John Hawks