On Fri, 4 Jun 2004 17:14:15 +0000 (UTC), Tim Tyler 
wrote:

>r norman  wrote or quoted:
>> Tim Tyler  wrote:
>> >Larry Moran  wrote or quoted:
>
>> >> For beneficial alleles that arise in a population the
probability of
>> >> fixation can be approximated by 
>> >> 
>> >>                      P = 2s
>> >> 
>> >> where s is the selective advantage. This equation holds
for small values of
>> >> s (s < 0.05) and populations greater than about 100
individuals. It is 
>> >> pretty much independant of population size for realistic
populations.
>> >
>> >AIUI, population sizes in the real world range from one individual up
>> >to billions of them.  Probability of fixation for a particular,
>> >non-neutral allele can range from around 0.5 in small populations to 
>> >0.0001 in large ones.
>> >
>> >So - it seems rather misleasing to assert that the probability of 
>> >fixation in "realistic populations" is "pretty
much independent of 
>> >population size".
>> >
>> >I can only think that you are using one of the terms
"realistic" or 
>> >"independent" in an unorthodox manner.
>> 
>> Populations in an abstract, conceptual world range from one individual
>> on to up.  You can create artificial examples of populations in the
>> "real" world smaller than 100 individuals and you can
find instances
>> in the very short time span just before a population went extinct
>> where it passed through a range of sizes varying from one hundred down
>> through one to zero.
>> 
>> What is your favorite endangered species?   Snow Leopard, Siberian
>> Tiger, Whooping Crane, Cheetah, California Condor?  These all number
>> in excess of 100.  How long do you think a population can remain less
>> than 100, especially down in the low dozens or single digits where the
>> population size really does make a difference for fixation?  What is
>> the probability of extinction for such populations? 
>> 
>> Real life science uses weasel words like "realistic
population size"
>> and "pretty much independent" to indicate what is useful
to real field
>> biologists doing real observational studies of real populations living
>> in the real world.  And fewer-than-100 doesn't "really" occur.
>
>You are mistaken.
>
>Some species that start off with one mutant individual whose
>offspring mate with each other.
>
>Other species arise from one individual through polyploidity events.
>
>Yet more species arise from a breeding pair stranded on an island.
>
>These things really do happen - and can represent significant
>evolutionary events.
>
>Drift is an important force at such small population sizes.
>
>It is an error to assert that small population sizes do not exist
>in the real world - except during extinction events.
>
>It is also a mistake to think that the probabilty of extinction of
>such small populations is necessarily low.  Two individuals
>on an island may have few predators, plentiful food resources
>and may have little difficulty finding a mate.  Note that for
>drift to occur the population doesn't have to *stay* small for
>very long - one generation may easily be enough.
>
>So: small population sizes often occur at speciation events - a location 
>from which they can impact the path the evolutionary process takes.
>
>They often arise when when a new and isolated environment is invaded by 
>only a few members of a species.
>
>The process is usually known as "The Founder Effect".
>
>  http://encyclopedia.thefreedictionary.com/Founder%20effect
>
>...has more details about that.

My error.  I completely forgot the founder effect, something I
routinely teach.  So there are two ways a population can be less than
100 for a very brief time, when it is on the way out and when it is on
the way up.  In both cases, the situation lasts for a very brief time.

Still, neither has much to do with the real question: does population
size influence the probability of fixation of beneficial alleles?

In the founder effect, the individuals starting the new population do
carry a set of alleles with frequencies rather different from those of
the parent population and the allele frequencies might fluctuate
wildly (drift) during the short period of population growth.  However,
the population is still the same species as the ancestral form,  It is
not until enough genetic difference develops to form reproductive
isolation that you get a new species.  And during all that time during
which those changes occur, it is most unlikely that the population
remains that small.  Once the population reaches a reasonable size,
the beneficial alleles are likely to take over once more and the
actual number of individuals will not matter. 

The real question is: given a new mutation in the population, will it
become fixed (or at least form into some form of stable polymorphism)
and so remain, or will it disappear?  That situation seldom happens in
populations as small as you suggest simply because a small population
experiences fewer total number of mutations and because the population
remains small for a very short span in the total lifetime of the
species.  Given a set of existing alleles that are not yet in a stable
polymorphism, will a catastrophe producing an extreme bottleneck or a
founding event allow one or more to become fixed? Definitely.  But
what amount of genetic difference between species is accounted for by
that mechanism?
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