At 11:28 PM 09/15/1999 -0500, Wesley wrote:
WRE>Hardy, Weinberg, and Castle all addressed the same problem and
WRE>came up with the same math to explain it. The problem,
WRE>though, was to explain why a dominant allele did not simply
WRE>and always sweep to fixation in a population, eliminating
WRE>recessive alleles. What HWC demonstrated was that given some
WRE>assumptions, a single round of reproduction resulted in a
WRE>stable set of binomial parameters. The initial case that they
WRE>showed this for was a two-allele system with one dominant and
WRE>one recessive allele. The representation of alleles was
WRE>thereafter in equilibrium, and thus one sees reference to HWC
WRE>equilibrium (well, usually just Hardy-Weinberg equilibrium,
WRE>since most people overlook Castle). But what were the
WRE>assumptions? Those were 1) an essentially infinite population
WRE>size, 2) panmixia or ranodm mating, 3) no selection, and 4) no
WRE>new alleles. (See Futuyma pp.231-233.) To claim that HWC
WRE>disproves evolutionary change is pretty funny, given that the
WRE>assumptions specifically eliminate consideration of cases
WRE>where evolutionary change would occur.
AC>These can be factored in.
Ayup. That's why I pointed to Fisher and Futuyma, which each
discuss the math that takes the two-allele model beyond the
HCW conditions. Once any of the assumptions discussed above
are violated, one is no longer discussing HCW, but rather a
different model. We don't need to consider anything more
complicated than a two-allele model just yet.
AC>No matter how you cut the cake, until a mutation is expressed,
AC>it cannot be selected for.
Ayup. That's why I had the following text in my message, to
show that these considerations have been addressed and do not
conform with Art's previous assertions:
WRE>The comments by Art on when selection can kick in for a new
WRE>mutation are among his best in this post. The case of novel
WRE>mutations becoming lost in a population has been calculated.
WRE>Fisher is, again, a good source. In the absence of selective
WRE>pressure, the odds that a mutation will be lost is about 0.37
WRE>by the second generation, and about 0.89 by the fifteenth.
WRE>One can work out, based upon the specifics of population size,
WRE>the expected number of homozygotes in the population.
WRE>Certainly, it's odds against, but not such odds that one
WRE>should consider it "rare" or "exceptional" that a beneficial
WRE>recessive mutation hangs around long enough in a population to
WRE>undergo selection. I'd rate the appopriate term to be
WRE>"sometimes".
What puzzles me is why Art would trim that section above when
it directly related to the assertion that he was making about
the rarity with which selection could act upon and change the
proportion of an allele in a population.
AC>Hardy-Weinburg enables calculations using reasonable
AC>assumptions to be done (there are lots of engines available on
AC>the internet for trying this out), and when you do the
AC>calculations, they do not favor the testing or elimination of
AC>recessive mutations, or the fixing of dominant ones. Of
AC>course, you can always change the conditions so that any
AC>particular outcome can be achieved, but I did say
AC>"reasonable".
Art is entitled to continue to believe that a model that
excludes mutation, genetic drift, and natural selection *by
definition* says something profound about the effects of
mutation, genetic drift, and natural selection in real world
populations. Art has not convinced me that I should join him,
though.
Some HW calculators labor under the same misapprehension
concerning the general applicability of HCW that Art seems to
share. This does not make them right. In some cases, the
simulation is labeled as being a "Hardy-Weinberg" model, but
the HCW assumptions are violated. These are the result of
confusing a specific instance of a model for its class. The
class of model is usually a two-allele model; HCW is the
specific instance that results when the HCW assumptions are in
force.
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.
WRE>Well, the assertion is false, whether one brandishes HWC
WRE>around or not. I recommend a 1930 book by Ronald Fisher for
WRE>the relevant math, "The Genetical Theory of Natural
WRE>Selection". While fixation is rare, having an allele that has
WRE>an extremely high proportional representation is common,
WRE>coupled with one or several recessive alleles at small
WRE>proportions. Change in proportion is most rapid when an
WRE>allele is at an intermediate proportion, but slower when the
WRE>allele is either rare or very common. (Modelling this via
WRE>Monte Carlo methods yields a curve very much like the Verhulst
WRE>logistic equation.)
AC>I am not sure which assertion you are referring to, but if you
AC>are speaking of a normal recessive allele that is not being
AC>selected for, then I would like to see some (since we are
AC>speaking of math) evidence for that.
The asssertion immediately preceding my text might make
a good candidate. I'll repeat it here:
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.
This assertion says that selection must be almost completely
in favor of an allele in order to drive it to fixation, and
that in the absence of such very strong selection, fixation of
an allele is not possible. (It also says such strong
selection cannot be achieved, but we'll leave that alone.)
Using two-allele models that do allow for selective
coefficients, Fisher found that even quite small selective
advantages (IIRC, coefficients down around 10^-5) would cause
the representation of an allele to rise to high proportions.
As I noted above, fixation most often would not occur, but it
certainly is possible, even with very small selective
advantage. The relevant math is contained in the book I
referenced from Fisher. I'm not where I can copy down the
equations (in this case, from Futuyma, since I don't own a
copy of Fisher), so Art can tell me whether a specific page
reference to a book will suffice, or if he eventually wants
the equations themselves. Since I'm still hoping that Art
will honor my request for Spetner's equation for information,
I feel that I should be willing to reciprocate as best I can.
Now let's look at Art's new candidate assertion again:
AC>I am not sure which assertion you are referring to, but if you
AC>are speaking of a normal recessive allele that is not being
AC>selected for, then I would like to see some (since we are
AC>speaking of math) evidence for that.
This assertion concerns whether a rare allele can go to
fixation when selection pressure is 0. The relevant math
concerns "genetic drift", not natural selection (which was
what I was discussing by reference to Fisher). The folks who
develop the mathematics of genetic drift most authoritatively
are Sewall Wright (there are several volumes, all pretty well
laced with math), Crow, and Kimura. Bottom line: even rare
alleles *can* increase in proportion to fixation, although the
most likely fate of any particular rare allele is
disappearance from a population. Interestingly, in the case
of genetic drift, it doesn't make any difference whether the
allele is recessive, dominant, co-dominant, or otherwise.
The math even applies to non-expressed regions of the genome.
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.
WRE>Huh? Fisher's analysis shows that even minute selective
WRE>advantages can drive an allele to fixation or at least very
WRE>high proportions of representation.
AC>This can easily be tested in the engines available on the
AC>internet. Again, one must ask what assumptions he has made,
AC>and whether we judge these to be reasonable. The math is
AC>there, but so is a lot of wishful thinking. Remember, Fisher
AC>was driven to promote evolution, and was not an unbiased
AC>observer.
I don't suppose that anybody developing these models is
unbiased. Fortunately, one can evaluate the published models
where all the assumptions are laid out and the equations are
given, and come to one's own conclusion. And when discussing
these, one can do more than simply use "Is not!" argumentation
by showing specific problems in the math or unreasonableness
of the assumptions. So far, I haven't noticed Art getting
specific about what he finds wrong in Fisher's analysis.
HCW equilibrium depends upon assumptions, four of which I
listed. I will now go through them and compare them to
real-world population characteristics.
1) Essentially inifinite population size.
In the real world, populations are finite. Finite
population sizes introduce sampling effects. These sampling
effects are what drives genetic drift. The likelihood of
change given genetic drift is related to actual population
size.
2) Panmixia or random mating.
In the real world, mating can occur non-randomly. Such factors
as female choice can introduce sexual selection, which causes
change in proportion of alleles. There are other effects that
non-random mating causes, but I'll move on for the moment.
3) Absence of selective pressure.
In the real world, predation, competition, and cooperation
all introduce factors which can cause certain traits to be
favored or selected for, or other traits to be selected
against. Any of these can cause a change in proportion of
alleles. How quickly such change occurs depends upon a number
of factors, including the coefficient of selection placed upon
the trait and the population size.
4) No new alleles.
In the real world, mutation rates for point mutation are
well-characterized. Much work has established the existence
of "recurrent mutations", which can cause the re-appearance
of an allele that had been in a population previously but
had been lost at some point.
Each of the assumptions of HCW equilibrium are known to be
violated by real-world populations. The first and fourth
assumptions are always false. The degree to which the second
and third assumptions are false can vary considerably
depending upon which populations one looks at.
I went to Art's home page and tried to look at his "biology" page.
The server coughed this up.
[Quote]
File Not Found
The requested URL /BIOLOGY.html was not found on this server.
[End Quote]
Wesley