Re: ReMine and reproductive excess

[As this post inadvertantly was made a subthread of another, I am
reposting it so that it will be its own thread and not get tangled
with the other one].

This is a new thread intended to explore how we can model substitution
in reproducing populations with finite numbers of individuals and finite
numbers of loci, to see whether, and how, Walter ReMine's arguments about
reproductive excess apply.

I want to start with neutral mutations. Suppose that we have a population
of N haploid individuals. Each has L loci at which mutation can occur.
For the moment all of these loci are completely linked -- there is no
genetic recombination between them. Each locus has n sites at which
mutants can occur. There is a mutation rate u per site.

In this simple model, all of these mutations are assumed to be neutral.
Suppose also that each haploid genome has exactly one offspring in the
next generation. Thus there is no genetic drift at all.

I choose this model because it is one in which there seems to me to be
no reproductive excess at all. Yet neutral mutations occur and the
population changes. At each of the nL sites in the genome there will
be u neutral mutations per generation (on average).

The sites have 4 possible states (the usual bases A, C, G, and T).
Whatever state we start out with, it has probability 1 - (1-u)^t
of having has at least one change in that site after t generations.
If this number is small, then almost all those changes are visible when we
sequence the genome at the end of that time. If it is substantial,
then some of the mutations may occur on top of others and not all of them
will be visible. To very very good approximation, the fraction of sites
at which a genome in the final population differs from its original
ancestor (the one in generation zero that gave rise to it) is

(3/4) (1 - exp(-(4/3)ut) )

(if needed we can work out the exact formula). So, if N = 100,000
and L = 1000, each locus having n = 100 sites, a neutral mutation rate of
10^(-8) per site would yield in 200,000 generations

(3/4) (1 - exp(-(4/3)(0.00000001)(200000) ) = 0.00199734

differences per site. (This is very close to the 200000 x 0.00000001 which
we might naively expect if all mutations were never obscured by later
mutations).

With nL sites per haploid individual in the population, we expect the
genomes after 200000 generations to each differ from their original ancestor
genomes at an average of 0.00199734 n L = 199.734 sites.

Is this substitution without reproductive excess? Or not substitution?
Or is there some reproductive excess here that I do not see? I realize
that ReMine argues that neutral mutations are not limited in their
rate of substitutions by reproductive excess, but he says that this is
so not because they require no reproductive excess, but because

> Concerning *neutral* substitutions, special mechanisms come into play
> which automatically increase the rate given by cost-payment analysis
> until it exactly matches the rate set by the neutral mutation rate.
> There are still costs, and they are still paid, but in the end the
> neutral mutation rate is what sets the limit on the neutral substitution
> rate.
(ReMine on December 8, 1997 in this newsgroup)

I don't see the "special mechanisms here". Are they present? Or is
this not a case where there is any substitution (I suspect that this is
what ReMine will argue)?

Let's explore this case, and let ReMine tell us.

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

.

Relevant Pages

  • ReMine and reproductive excess
    ... This is a new thread intended to explore how we can model substitution ... I want to start with neutral mutations. ... Suppose also that each haploid genome has exactly one offspring in the ... no reproductive excess at all. ...
    (sci.bio.evolution)
  • ReMines reproductive excess
    ... This is a new thread intended to explore how we can model substitution ... I want to start with neutral mutations. ... Suppose also that each haploid genome has exactly one offspring in the ... no reproductive excess at all. ...
    (sci.bio.evolution)
  • Re: ReMines reproductive excess
    ... > I want to start with neutral mutations. ... all of these mutations are assumed to be neutral. ... > no reproductive excess at all. ... At each of the nL sites in the genome there will ...
    (sci.bio.evolution)
  • Re: The cost of substitution
    ... It should be self-evident that neutral mutations do not ... isn't enough reproductive excess for a given observed rate of neutral ... we could express the mutation rate as per generation, ...
    (sci.bio.evolution)