Re: Haldane's Dilemma - clarifications - and Felsenstein [LONG]

In article <e7jp1h$2g2m$1@xxxxxxxxxxxxxxxxxxx>,
Malcolm <regniztar@xxxxxxxxxxxxxx> wrote:
Here we go.
.....
[program itself snipped]

Programmers will note a few technical problems introduced by Prof
Felsenstein's changes. These are to do with the fact that computers cannot
represent real numbers exactly, and when you ahve lots of genes the "w" or
fitness measure gets very large.

The program is certainly much closer to the standard population genetics
model of genetic drift, mutation, recombination, and natural selection,
and looks good.


One interesting thing to note is that, in this model, when the number of
genes is large - I origin[in]ally set it to 30000 to represent the number
thought to exist in the human genome, then the number of mutations that make
it into the population is approximately 2 * SELECTIONCOEFFICIENT. When I
tried to speed the program up by a factor of 300 by reducing the numebr of
genes to 100, this relationship disappeared.

Maybe this is worth pursuing.

This relationship has been pursued: that for loci with a favorable mutation
the probability that a new mutation getting fixed is approximately 2s, where
s is the selective advantage in heterozygote, is well known. It was
discovered in 1927 by none other than ... JBS Haldane! (In part V, "Selection
and mutation" of his great series of papers on "A mathematical theory of
natural and artificial selection"). A more accurate formula, due to Motoo
Kimura in 1958 and 1963, is, with a diploid population size N:

(1-exp(-2s)) / (1-exp(-4Ns))

I don't know whether that will be helpful in this case -- the differences when
there are fewer loci may result from multiple mutations with their
substitutions overlapping in time, which goes beyond Haldane's 1927 and
Kimura's 1963 model.

For references and derivations of both of these, buy my theoretical
population genetics book (actually it's a free PDF):
http://evolution.gs.washington.edu/pgbook/pgbook.html
and see sections VII.7 and VII.8.

I suspect that in the program as you have modified it, the correct value of
the reproductive excess is something like (1+s)^n-1, where n is the
number of favorable mutations that have fixed so far. It would be
interesting to hear whether this is the figure Walter Remine would use in this
case.

----
Joe Felsenstein joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Genome Sciences and Department of Biology,
University of Washington, Box 357730, Seattle, WA 98195-7730 USA

.

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