Re: Felsenstein and reproductive excess
- From: joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx (Joe Felsenstein)
- Date: Tue, 31 May 2005 23:39:28 -0400 (EDT)
In article <d7a6u9$1pbb$1@xxxxxxxxxxxxxxxxxxx>,
Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote:
>
>"Joe Felsenstein" <joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in
>message news:d78kfg$16l0$1@xxxxxxxxxxxxxxxxxxxxxx
>[snip]
>> ReMine has been told many times that my calculation of cost is not
>> the reproductive excess required to allow substitution (William Morse,
>> you are wrong about that). He has even occasionally acknowledged that
>> this is so (and immediately declared my cost useless). The cost I
>> define is the reproductive excess necessary to avoid extinction.
>> And for that it is simple and obvious that a beneficial mutation
>> creates no risk of extinction. [snip]
>
>It is not obvious to me that a beneficial mutation creates no risk of
>extinction.
>
>Some, perhaps most, beneficial mutations make the possessor a better
>competitor against conspecifics. In a density dependent model, these
>beneficial mutations will result in a lower population count after
>fixation than before. You might claim that this is not a risk of
>extinction, but then you are not using a density dependent model in
>your own model of a deteriorating environment.
In that 1971 model I was investigating whether Haldane's cost could be
thought of as a cost which, if too large, led to extinction. It turned
out that it could, and the formulas were very close to Haldane's,
if one assumed that the substitutions were responses to environmental
deteriorations, and the cost was calculated as the reproductive excess needed
to prevent extinction.
The condition needed is the reproductive excess needed to allow the population
to grow when its density is low. That is the very situation when density-
dependence is not relevant. So my discussion left out density-dependence,
deliberately. It might occur when the population sizes got larger.
Your more realistic model allows a class of substitutions whose effect is
on the intensity of density dependence. First, note that if the density
is driven down to a low value, substitutions whose only effect is to
compete more effectively with the other members of the population will
have very little effect. As density comes to be low, these substitutions
would basically stall out, and their rate would drop to near zero.
Only the beneficial mutations affecting the growth rate would still be
occurring (among all beneficial mutations).
Second, we have to distinguish between mutations that make an individual
lower its conspecifics' survival or reproduction, and those that instead
more effectively resist competition. Here is the standard Beverton-Holt
discrete-generations density-dependent growth equation:
1
N' = N R -------------
1 + (R-1) N/K
(use a constant-width font to display that)
so that R is the total rate of reproduction at low densities, and K is
the carrying capacity of the population (the reproductive excess is (R-1)).
In that model, a mutation can either alter R, or K, or both (or neither, in
which case it is neutral). In the case of R-selection, a beneficial mutation
affects growth rates at low densities. A mutation that is purely K-selected
has greater resistance to density-dependent effects, and will achieve a higher
carrying capacity once it is frequent. In this model, your class of
mutations would be these latter, and their substitution would stall out
once low density was reached as a result of other events.
The class of mutations you posit may not actually fit into this
equation. Does it decrease everyone's K? Or just the K's of those who
do not have the mutation?
> ... a certain portion of the limit for favored
>alleles is "wasted" in removing unfavorable mutations from the
>population. The entire limit can not be used for fixing new beneficial
>mutations.
In my scheme that is certainly true. Deleterious mutations are of course
held to low frequencies in the population. Their presence is one of the
factors that reduce reproductive excess, by causing not all of the
offspring produced to survive to adulthood. If the beneficial mutations
are mutations that increase R, they do make it easier for the population
to survive deteriorations of the environment, even though deleterious
mutations are still occurring. If one wanted to know the net effect of
an increase in overall mutation rate, one has to balance this effect of
beneficial mutations with the mutational load due to deleterious mutations
to figure out the overall effect. But if you just add beneficial
mutations, it doesn't lower the probability of survival of the population.
--
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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