Re: The Living Dead


Steve Schaffner wrote:

> > For argument's sake, lets say that no offspring with more than 97
> > mutations are able to reproduce at all. Now, instead of 5,000 couples
> > mating to populate the next generation, we only have about 3,600.
> > True, these are more fit, but there are fewer of them. If they still
> > have the same maximum reproductive rate of 4 per couple, then only
> > about 14,500 offspring will be produced instead of the 20,000 produced
> > by the original parent population. Of these, won't most go downhill
> > instead of uphill? Won't most of these cross the boundary and end up
> > below the 97 detrimental mutation threshold?
>
> If there is a hard threshold at 97, then considering a population that
> starts with exactly 97 deleterious mutations each is not a good model
> for thinking about the situation. Think about a hard limit at, say,
> 110 instead. You will have more than enough offspring surviving to
> produce the next generation, and the really highly loaded tail will be
> completely removed. The effect is to increase the overall fitness of
> the population at equilibrium, not decrease it.

You will have enough offspring to produce the next generation, but will
the number of those with fewer than 97 mutations increase or decrease
or stay the same given your reproductive rate of 4 per couple? It
seems to me that in each generation, the number of those with fewer
than 97 deleterious mutations will indeed decrease at this low rate of
reproduction regardless of where you draw the line, 110 or otherwise.

> What you describe is known as truncation selection, and is one extreme
> solution proposed to explain how humans could support a high
> deleterious mutation rate. (See the discussion section of the Nachman
> and Crowell paper, where they mention it.)

Yep . . .

> An ideal poplulation with
> 3.0 deleterious mutations per gen (2x reproductive capacity, Npop =
> 10,000, selection coeff = 0.01) comes to equilibrium at around 300
> deleterious mutation per individual. (I think I reported that number
> incorrectly previously.) With truncation selection operating, with a
> threshold of 97 mutations, the population reaches equilibrium with 87
> mutations per individual.

I'm not sure about this given a reproductive rate of only 4 per couple.

> > > 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?
> >
> > The mean number has indeed decreased, but so has the absolute number of
> > offspring at or above the 97 starting point. What happens in the
> > subsequent generations?
>
> Mating in each generation tightens the distribution (combining two
> randomly selected individuals from the population reduces the variance
> on their average by 1/sqrt(2)), as does selection (which also biases
> the distribution towards the low end). Production of new offspring
> broadens the distribution, as random numbers of new mutations and
> random sampling of existing mutations introduce scatter. Since we
> started with zero variance in our thought experiment, initially the
> variance will increase, and then stabilize. Selection keeps the high
> tail from expanding indefinitely, and new mutations keep the low tail
> from expanding.

Yes, given a high enough reproductive rate. I just don't see how a
reproductive rate of 4 is going to result in equilibrium in this
particular case.

Consider another example of a steady state population of 5,000
individuals each starting out with 7 detrimental mutations and an
average detrimental mutation rate of 3 per individual per generation
(easier on my calculator). Given a reproductive rate of 4 offspring
per each one of the 2,500 couples (10,000 offspring), in one
generation, how many offspring will have the same or fewer detrimental
mutations than the parent generation?

The Poisson approximation shows that out of 10,000 offspring, only
~2,202 of them would have the same or fewer than the original number of
detrimental mutations of the parent population. This leaves ~7,798
with more detrimental mutations than the parent population. Of course,
in order to maintain a steady state population of 5,000, natural
selection must cull out 5,000 of these 10,000 offspring before they are
able to reproduce. Given a preference, those with more detrimental
mutations will be less fit by a certain degree and will be removed from
the population before those that are more fit (less detrimental
mutations). Given strong selection pressure, the second generation
might be made up of ~2,200 more fit individuals and only ~2,800 less
fit individuals with the overall average showing a decline as compared
with the original parent generation. If selection pressure is strong,
so that the majority of those with more than 7 detrimental mutations
are removed from the population, the next generation will only have
about 1,100 mating couples as compared to 2,500 in the original
generation. With a reproductive rate of 4 per couple, only 4,400
offspring will be produced as compared to 10,000 originally. Even if
the limit were drawn at more than 7 detrimental mutations, wouldn't the
final outcome still be the same?

It seems then that in order to keep up with this loss, the reproductive
rate must be increased or the population will head toward extinction.
Given a detrimental mutation rate of Ud = 3 in a sexually reproducing
population, the average number of offspring needed to keep up would be
around 20 per breeding couple (2e^Ud/2). While this is about half that
required for an asexual population (2e^Ud), it is still quite
significant.

How is my thinking off base here?

> Steve Schaffner sfs@xxxxxxxxxxxxx
> Immediate assurance is an excellent sign of probable lack of
> insight into the topic. Josiah Royce

Sean Pitman
www.DetectingDesign.com

.

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