Re: Haldane's Dilemma

From: Perplexed in Peoria (jimmenegay_at_sbcglobal.net)
Date: 02/19/05 

Date: Sat, 19 Feb 2005 01:07:09 -0500 (EST)

"Tim Tyler" <tim@tt1lock.org> wrote in message news:cv5opu$29jv$1@darwin.ediacara.org...
> Perplexed in Peoria <jimmenegay@sbcglobal.net> wrote or quoted:
>
> > My impression of Haldane's argument (I admit, I have never checked the
> > original sources) was that the population size doesn't matter. The
> > increased number of selective deaths available in a large population is
> > exactly balanced by the increased number of selective deaths needed to
> > fix a gene.
>
> A better sum:
>
> The time taken to find a particular positive mutation is proportional to
> the number of trials that can be executed in parallel - i.e. it's
> proportional to the population size [1].

But now you are addressing the other one of the two limits that ReMine
mentions. Not the limit that Haldane addresses.

Haldane considered the limit that arises from the time required for
a beneficial mutation (already present) to increase in frequency to
fixation. There is a "parallel-processing" aspect to this, as well.
Nature can be moving several alleles to fixation at the same time.
It will move those with high selective coefficients quickly, and
those with low selective coefficients more sedately. However, there
is a limit to the "total amount of selective work" that Nature can
perform in one generation. Or, so says Haldane.

> The time to fixation depends on the time taken for the beneficial
> mutation to spread through the population. That figure goes
> according to sqrt(n) in a population confined to a surface and
> cubert(n) in a population in a three dimensional medium.

You are considering only the spatial structure here. Even in such a
diffusion model, the rate of diffusion will vary proportionately with
the selection coefficient.

> So large populations reap the benefits of being able to explore more of
> their surrounding evolutionary space simultaneously, while not paying
> too great a cost in terms of the time it takes for beneficial mutations
> to reach selection.

In Haldane' argument, a panmictic population was assumed, so your sqrt(N)
factor doesn't even appear. Every individual is a "neighbor" of every
other individual. So the factor is N^(1/bigNumber) or simply 1.

> [1] Up to a point, anyway. After a while the population size exceeds
> the number of easily accessible mutations, adding more members to the
> population results in much the same mutations being tried twice -
> so after a while, diminishing returns tends to set in.

Actually, I don't think that the "rate of innovation" saturates. The
rate of explored single-point mutations will saturate, but in larger
populations you are able to explore simultaneous multi-point mutations.
As Wright pointed out, a large population extends over a greater expanse
of the fitness landscape. (Incidentally, I think you were right in
your dispute with Remine about all the single-point innovations eventually
being "used up").

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