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

In article <e79ufs$l7s$1@xxxxxxxxxxxxxxxxxxx>,
Malcolm <regniztar@xxxxxxxxxxxxxx> wrote:
Here is a model, similar to one I wrote earlier, which shows beneficial
alleles spreading through a population. Can you tell me how to modify the
model to incorporate a cost of substitution to show the limit you are
interested in.

We have a population of diploid organisms, with alleles which can be
favourable ('A') or unfavourable ('a'). Initially all alleles are
unfavourable. Every generation there is a single mutation, which switches a
random locus from one state to the other.
To represent fitness, for each child slot in the next generation we take two
candidate mothers and two candidate fathers, chosen at random, and select
the one with the highest count of 'A' alleles. Alleles are thus additive and
with a fairly high selection coefficient.
The output is that, after 5000 generations, average fitness is around 2500,
substantially higher than your limit.

I appreciate that not everyone can read C code, though you ought to be able
to follow the basics. If we can build a model that shows the effect you are
interested in, we are in a position to talk. I am happy to spend as much
time as it takes testing out ideas.

I've looked at your website. You seem to accuse evolutionists of fudging
your evidence. Here's your chance.
[code snipped]

I'm not Walter, but do want to comment on the model. It's
a straightforward computer simulation of natural selection of
mutations, with a few wrinkles. I wonder if you don't want to
modify those wrinkles:
1. The alleles are either A or a. In this case the population will,
after enough generations, end with with mostly A's but some a's, as
there is back mutation at a rate equal to forward mutation. So it
will be approaching an asymptotic amount of adaptation. A useful
modification would be to have the alleles be integers which are not
just characters 'A' and 'a' but can be incremented. Thus we would have
alleles 0, 1, 2, ... and this would allow new alleles to be replaced by
even better alleles. Mutation would be incrementation of the integer.
Or if you wanted to have back mutation as well mutation could sometimes
increment and sometimes decrement. The resulting model would gain
favorable mutations (with some but not much loss as well) and this
would go on indefinitely, or at least until you got near MAXINT.
2. Recombination is once per parent, in a random location. One could have
a recombination rate R which would be the fraction of times that
we have a recombination (otherwise the chromosome would be transmitted
without recombination from one of the parent's two chromosomes.
It would also be possible to have higher amounts of recombination by using a
random number to decide whether to switch back and forth between chromosomes
in the parent, deciding at each locus. But that would be slower.
3. The selection is, as you say, choice of one of two potential parents,
with the one with more favorable alleles always winning. If you wanted to
come closer to the standard Wright-Fisher model, you could have each
parent have a fitness calculation from your 'fitness' (which is a count
of the favorable alleles, say n for some individual) using
w = (1+s)^n
where s is the selection coefficient of favorable alleles. Then
you would arrange that the probability of drawing each parent would be
proportional to its fitness w. That would be slower: you'd have to
make an array of the w values and cumulate them, and then use a
random number to draw from a random point in this cumulative distribution.
4. Why not have a mutation rate? Here there is always one mutation per
generation. Easy enough, and hardly slower, to have 2NU mutations
per generation where U is a mutation rate per chromosome. You could
generate a Poisson random variable (by taking a product of random
fractions, until the product was smaller than e^(-2NU), with the Poisson
variate being the one less than the number of fractions you draw until
you get smaller than that). Then mutate the population that many times.

All this would make the model closer to the standard ones of population
genetics.

It is hard to guess what Remine would say the reproductive excess is in such
a model. Since it is possible (though very very unlikely) for an indvidual
to have N offspring, he might argue that it is as high as N. In my
old 1971 model, the reproductive excess here would be something like 1,
I think (in the competition-between-potential-parents scheme of the program).

----
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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