Re: The Living Dead

"Seanpit" <seanpitnospam@xxxxxxxxxxxxxxxxxxxxxxxxxxx> writes:

> Reinforcing epistasis removes those with higher detrimenal loads. How
> does it help in keeping the overall ballance in equilibrium given a
> slowly reproducing population?
>
> Let's think about this a bit. You started with a steady state
> population of 10,000 diploids each having 97 mildly detrimental
> mutations. They mate to produce 20,000 offspring. Given an average Ud
> of 3, the average number of deleterious mutations, per individual, is
> 100. Of course, given the Poisson distribution, some will have more
> and some will have less. You argue that, "quite a large fraction of
> the 2nd generation population has fewer deleterious alleles than the
> 1st generation." Therefore, it is clear to see that the sexually
> reproducing population can go uphill, so to speak. Sound good so far?
>
>
> Without any additional mutations of any kind affecting this population,
> genetic recombination will divide the population about half and half,
> in a normalized binomial distribution, where one side has equal or more
> and the other has equal or less than 97 mutations. Out of the 20,000
> offspring, about 47% will have a beneficial mutational load. That's
> 9400 offspring that have moved at least 1 level uphill.
>
> Now, what happens when you bring in a detrimental mutation rate of Ud =
> 3? How many of these 9,400 stay on the plus side of the equation? The
> majority of these 9,400 have only a few beneficial mutations. The
> detrimental mutations will also be distributed in a Poisson
> distribution pattern where those with the least beneficial mutations
> will be hit with the least number of detrimental mutations and those
> with the most beneficial mutations will be hit with the most
> detrimental mutations.
>
> For example, of those with a neutral or positive mutational balance,
>
> About 1140 of them will have 0 beneficial mutations. Of these, about 56
> will not be hit by any detrimental mutations and will maintain a
> neutral mutational balance.
>
> About 1,100 of them will have 1 beneficial mutation. Of these, 54 will
> not be hit by any detrimental mutations and stay with 1 beneficial
> mutation. 164 will be hit by 1 detrimental mutation and maintain a
> neutral mutational balance and the rest will have a detrimental
> mutational balance.
>
> About 1040 of them will have 2 beneficial mutations. Of these, about 52
> will not be hit by any detrimental mutations and stay with 2 beneficial
> mutations. 162 will be hit by 1 detrimental mutation and go to the
> level of 1 beneficial mutation. 233 will be hit by 2 detrimental
> mutations and maintain a neutral mutational balance, and the rest will
> have a detrimental mutational balance.
>
> Etc . . .
>
> In short, out of the 20,000 offspring, less than 3,000 will have either
> a neutral balance or better. The rest will have a detrimental
> mutational load relative to the parent generation.

Your numbers are a little off, but they're close enough. You
shouldn't have stopped there, however. Here are the number of
offspring, classified by the number of inherited deleterious alleles,
and the number of them that have 97 or fewer after new mutations are
added:

inherited N(offspring) N(del <= 97)
97 1144 57
96 1133 226
95 1098 465
94 1043 675
93 971 792
92 885 811
91 791 764
90 692 684
89 593 591

If you keep going, you will find a total of 7285 offspring with 97
or fewer deleterious alleles, compared to 12715 with > 97 deleterious
alleles. (If you'd just used the Poisson approximation (100 +/- 10
deleterious alleles), you'd have gotten 8148 offspring with <= 97,
and a normal approximation would give 7264.)

That's the raw number of mutations. Now apply selection. How much
more likely are the 7285 offspring (or the 6265 offspring with < 97
deleterious alleles) to survive and reproduce than the ones with more
deleterious mutations (some of them with many more)? That depends on
the selection coefficient. If the coefficient is high, so that, say,
the average lightly loaded offspring is twice as likely to reproduce
as the average heavily loaded offspring, then the lightly loaded
offspring will actually contribute more to the next generation
than the heavily loaded ones, and the mean number of deleterious
alleles will decrease.

Note that this is well within the reproductive capacity that I
specified. Take the extreme case: all 7285 lightly loaded offspring
form part of the next breeding generation, along with 2715 heavily
loaded ones. Has the mean number of deleterious alleles in the
population increased or decreased?

> It seems then that a reproductive rate of just 4 is not enough to
> maintain equilibrium with a detrimental mutation rate of Ud = 3. In
> other words, the sand will roll downhill much faster than it rolls
> uphill. Overall, each level of fitness will loose much more to the
> lower levels than it contributes to the higher levels of fitness. How
> is this compensated for without significantly increasing the
> reproductive rate?

Since you haven't done the calculation correctly, and haven't
considered the effect of selection (rather an important omission when
you're studying selection itself), your conclusion is premature. More
accurately, it's wrong.

> And, for rates of Ud=5, it gets a whole lot worse.
> How does reinforcing epistasis solve the problem? Removing those with
> higher numbers of detrimental mutations doesn't seem to help replace
> the loss of those with low numbers of detrimental mutations.

You have a basic misunderstanding here. At equilibrium, half the
population will have more than the mean number of deleterious
alleles. If you have 5000 with < 97 and 500 with > 97, that doesn't
mean you're losing half of the lightly loaded ones each generation,
it means you're in equilibrium.

> Beyond this little problem, consider that the Y-chromosome in males
> does not undergo significant recombination events. It basically
> undergoes asexual replication. So, how is the male population helped
> by recombination and/or epistasis at all? Doesn't it follow that males
> are headed for extinction in slowly reproducing populations even faster
> than females?

This was exactly why it was suggested for a while that the Y
chromosome was headed for extinction. Note a couple of points.
First, there are not many genes (twenty-some, I believe) in the
non-recombining portion of the Y, so the number of deleterious
mutations there is likely to be quite small (even accounting for the
higher mutation rate in males). Second, the large palindromic repeats
on the Y enable gene conversion within the chromosome to play the
same role as recombination between autosomes, permitting the genes in
those regions to escape mutational decay.

--
Steve Schaffner
Program in Medical and Population Genetics
The Broad Institute of MIT and Harvard

.

Relevant Pages

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