CL>This is like explaining relativity in non-mathematical terms
CL>by saying 'It's Einsteinian!' I think people on this list have
CL>become inured to the tactic of throwing names around.
AC>Sorry, Cliff. I didn't know you were not a biologist. A late
AC>nineteenth -century mathematician, G. H. Hardy wrote a now
AC>famous letter to the editor of Science in the early 1900's,
AC>pointing out a problem non-mathematically trained biologists
AC>had with understanding the consequences of gene distributions
AC>in a population. He showed that a simple binary expression
AC>p^2 + 2pq + q^2 sufficed to describe the frequencies of two
AC>alleles in a population over time, where p is the initial
AC>frequency of the dominant allele, q is the frequency of the
AC>recessive allele, and 2pq is the frequency of the
AC>heterozygote. Hardy concludes that for dominant alleles,
AC>there is no tendency to spread over a population, and for a
AC>recessive allele, there is no tendency to die out or to spread
AC>in a population. His work was along the same lines as that of
AC>a German by the name of Weinberg, and this fundamental
AC>principle of population genetics came to be known as the
AC>Hardy-Weinberg Law. All of this was done with the
AC>evolutionary concept in the background, but that was not
AC>explicit in Hardy's letter. Thus natural selection was not
AC>mentioned as a contributing factor.
Hardy, Weinberg, and Castle all addressed the same problem and
came up with the same math to explain it. The problem,
though, was to explain why a dominant allele did not simply
and always sweep to fixation in a population, eliminating
recessive alleles. What HWC demonstrated was that given some
assumptions, a single round of reproduction resulted in a
stable set of binomial parameters. The initial case that they
showed this for was a two-allele system with one dominant and
one recessive allele. The representation of alleles was
thereafter in equilibrium, and thus one sees reference to HWC
equilibrium (well, usually just Hardy-Weinberg equilibrium,
since most people overlook Castle). But what were the
assumptions? Those were 1) an essentially infinite population
size, 2) panmixia or ranodm mating, 3) no selection, and 4) no
new alleles. (See Futuyma pp.231-233.) To claim that HWC
disproves evolutionary change is pretty funny, given that the
assumptions specifically eliminate consideration of cases
where evolutionary change would occur.
AC>Unless the selective advantage of a trait is nearly
AC>perfect, a condition that is unattainable by definition, it
AC>will not become fixed in the population.
CL>And why not? Hardy-Weinberg!
Well, the assertion is false, whether one brandishes HWC
around or not. I recommend a 1930 book by Ronald Fisher for
the relevant math, "The Genetical Theory of Natural
Selection". While fixation is rare, having an allele that has
an extremely high proportional representation is common,
coupled with one or several recessive alleles at small
proportions. Change in proportion is most rapid when an
allele is at an intermediate proportion, but slower when the
allele is either rare or very common. (Modelling this via
Monte Carlo methods yields a curve very much like the Verhulst
logistic equation.)
AC>One can argue that a dominant allele in theory can become
AC>fixed in a population (See the discussion in Wallace Arthur's
AC>book "The Origin of Animal Body Plans" Cambridge, beginning
AC>around page 210.), if the mutation is selectively advantageous
AC>enough. This would be an extraordinary case, though.
Huh? Fisher's analysis shows that even minute selective
advantages can drive an allele to fixation or at least very
high proportions of representation.
AC>Only dominant traits that have a sufficiently high positive effect can
AC>become fixed by Hardy Weinberg pathways.
Actually, it doesn't matter whether an allele is dominant or
recessive in HCW "pathways". The only "path" is the first
round of interbreeding; the proportions are stable and
fixed thereafter under HCW assumptions.
AC>Recessive allele frequency remains constant more or less
AC>within the population unless the same mutation is repeated
AC>enough times that the gene frequency in the population becomes
AC>high enough to produce homozygous recessives. At that point,
AC>and not before, recessive alleles can be subjected to
AC>selection, and thus to evolutionary advantage. Some have
AC>postulated small but selective effects for the recessive in
AC>heterozygous condition (such effects are known for some
AC>genes), but this would be a rare exception in any
AC>case. Another consideration not anticipated by Hardy is
AC>genetic drift (whatever that is - it can be quantified, at
AC>least), which could under the right circumstances, lead to an
AC>increase in recessive allele frequency without evident
AC>selection.
Actually, drift *was* anticipated by HCW. It was eliminated
from consideration by the assumption of essentially infinite
population size.
The comments by Art on when selection can kick in for a new
mutation are among his best in this post. The case of novel
mutations becoming lost in a population has been calculated.
Fisher is, again, a good source. In the absence of selective
pressure, the odds that a mutation will be lost is about 0.37
by the second generation, and about 0.89 by the fifteenth.
One can work out, based upon the specifics of population size,
the expected number of homozygotes in the population.
Certainly, it's odds against, but not such odds that one
should consider it "rare" or "exceptional" that a beneficial
recessive mutation hangs around long enough in a population to
undergo selection. I'd rate the appopriate term to be
"sometimes".
CL>There must be some other pathways then, for recessive traits from blue
CL>eyes to sickle cell seem well established.
I recommend Fisher.
AC>Of course, one can always posit that the whole population,
AC>except for the one individual was annihilated, and that one
AC>individual (or small related population) survived, and thus
AC>raised the frequency to 100%....
The founder effect is similar, but doesn't deal in quite so
drastic a scenario. A shift in typical gene frequencies can
happen when a small founder group is the basis for a
population.
CL>I thought I might learn something by asking for a
CL>clarification. Well, I learned not to ask.
AC>Sorry, Cliff. I hope this helps.
I think Futuyma's "Evolutionary Biology" would also be of help.
Wesley