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comments:
Drift and Draft
Good question! The stochastic dynamics of (very) rare alleles are determined
by the distribution of offspring number. Fisher modelled this with a
branching process, which does not have the population size as a parameter.
If you look at binomial sampling just right, N actually disappears for rare
alleles. For example, the variance in the number of copies of the A1 allele in
the next generation, X, given that its current frequency is p, is
Var{X} = 2Npq.
if there are, say, j copies of the A1 allele (p=j/2N must be very close to
zero), then
Var{X} approx 2N x (j/2N) = j,
which is independent of N. (Here, q approx 1, so we have ignored terms of
order 1/N^2.)
My own view is that the dynamics of rare alleles are so radically different from
those of common alleles that we should have a different name for the
stochastic dynamics of rare and common allelels.
-- John Gillespie, May 25, 2004
Drift and Draft
Do you then think that Wrightian drift, change in allele
frequencies of smallish populations due merely to chance, is
a "real" phenomenon? My thinking is this: Wrightian drift
goes hand in hand with a conceptual reification of the
mathematical model in which the relative fitnesses of the
genotypes at some locus are independent of all other loci.
Once that model is replaced by a model that is dynamically
complete (or perhaps just sufficient), Wrightian drift is
replaced by your draft. In short, I conjecture that
Wrightian drift is not a "real phenomenon.
-- Robert Skipper, May 27, 2004
Drift and Draft
I think I see your point, but let me clarify one thing: By "Once that model is
replaced by a model that is dynamically complete" do you mean that the
model woul not allow the existance of neutral alleles? That is, we could
imagine a region of the genome where most of the alleles are neutral. In this
region "the relative fitnesses of the genotypes at some locus are independent
of all other loci" wouldn't apply and we could imagine genetic drift being a
"real phenomena." Of course, we don't know if such regions exist, but many
people woud claim that the vast stretches of DNA between coding regions in
genomes like ours would fit this description.
I think that many people would be happy to say that genetic drift is a real
phenomenon if parameter values for selecton and recombination fall in
certain regions of the parameter space. For example, if r/s (recombination /
selection) were greater than one for most pairs of segregating alleles and if
Ns were not huge (say 10 or less), than genetic drift in the strictess sense of
binomial sampling may not be a real phenomenon, but it could well be that
demographic stochasticity leads to second-order moments of genotype
frequency change that are like those of genetic drift. So there would be an
"approximately real phenomenon." On the other hand, if r/s is small and Ns
is large, then, like you said, we move over to a realm where classical genetic
drift is not even "approximately real."
-- John Gillespie, June 1, 2004
Drift and Draft
Let me clarify what I was getting at regarding "dynamically
complete." Population genetics models are typically couched
with the following simplifying assumptions: there is random
union of gametes, random association of genes at different
loci, and no epistatic gene interaction. A more "dynamically
complete" model would replace these assumptions in favor of
more biologically "realistic" ones.
Clearly, this kind of move can get out of hand, so that the
models become computationally intractable. But suppose one
wants to understand some apparent biological phenomenon such
as classical random genetic drift? On the simplified models,
one might come to the conceptual reification that there are
changes in allele frequencies due purely to chance (i.e.,
classical drift). But, perhaps, on a biologically more
realistic model, such a conceptual reification would not
(could not?) result.
Given the above, the idea is that the sort of dynamics which
we capture with classical drift, and which we think are
classical drift, aren't classical drift at all, but are
instead draft.
The way I understand your remarks up to now is as follows:
Many population geneticists agree that classical drift is a
biologically real phenomenon, so long as the parameter
values for recombination and selection are in the relevant
regions of the parameter space. You, on the other hand,
suggest that at best, when the parameter values are just so,
you get allele frequency change that looks like drift, but
which may not be due to drift. Am I off track?
I wonder along with Will Provine whether classical drift is
a myth? Further, I wonder if one diagnosis of why some
population geneticists believe that classical drift is a
biologically real phenomenon is because simplifying
assumptions done in modeling have biased the way they think
about what's happening biologically?
-- Robert Skipper, June 3, 2004
Response to After a several month lapse:
"Given the above, the idea is that the sort of dynamics
which we capture with
classical drift, and which we think are classical drift,
aren't classical drift at
all, but are instead draft."
I'm not sure what "classical drift" is. As Fisher was the
first to describe drift
mathematically, and as he did it using binomial
sampling, I guess we could
call classical drift binomial sampling. Personally, I
think that that would not
be particularly useful except to historians. To me,
classical drift would be the
stochastic force due to the fact that different individuals
with the same
genotype have different numbers of offspring. We
attibute the different
numbers to random, non-genetic, causes of
unspecified origins. (Like,
getting hit by a car!) It is easy to write a computer
simulation of this sort of
drift; one that includes as much demographic detail as
desired. The
mathematics of such models can be formitable, but the
stochastic dynamics
are there for all to enjoy. When models include
additional stochastic
elements, classical drift may be obscured by these
other forces. Nonetheless,
drift is still present and I would be hesitant to say
otherwise.
As an aside, it strikes me that discussion of models
should be focued on
particular computer
simulations (or their underlying algorithms) rather than
the mathematics of
old. In the past, we had to make all sorts of
simplifications in order to get
mathematically-tractable models. Today, that really
isn't necessary. Perhaps
people who are interested in the models of population
genetics could start a
catalog of models written in some simple language like
python. Then, not
only would there be something concrete to discuss, but
anyone could explore
the models by simply downloading and running the
models.
-- John Gillespie, September 16, 2004
Drift and Draft
I think I may be coming to a better understanding of your
draft papers, which I've found quite challenging (no fault
of yours, to be sure).
Nevertheless, I still want to push on the relative
significance of drift versus draft. But let me be clear: I
don't want to push for the elimination of drift. Your
comments here and my own thinking have cleared things up (at
least as I see them): Drift, understood as
indiscriminate sampling, is quite real. The better questions
are when, where, and how drift and draft might be said to
operate.
In the discussion section of your paper, "The Neutral Theory
in an Infinite Population," (in Gene, vol. 261, 2000,
pp. 11-18), you say that what's important is determining
whether genetic draft is a more "important" force in natural
populations than drift. You provide a clear mathematical
expression indicating when draft is more important than
drift, but I'm keen to understand the motivation of the
relative significance claim more
intuitively.
Might one say the following? Pseudohitchiking is a solution
to the problem of the apparent disconnection between levels
of genetic variation and levels of population size variation
(in part?) because pseudohitchhiking provides an account of
stochastic dynamics of populations that's not tied to
population size. More generally: Since pseudohitchiking is
not tied to population size, it's probable that it operates
more generally than a force, such as drift, that is tied to
population size. (I think what I've said here is
suggested in the discussion section of your paper, "Genetic
Drift in an Infinite Population: The Pseudohitchiking Model"
(in Genetics, vol. 155, 2000, pp. 909-919).)
-- Robert Skipper, October 8, 2004
Yes, and Yes
"Pseudohitchiking is a solution to the problem..."
Yes, I would agree with this. (I would, however, use genetic draft rather than
pseudohitchhiking. The latter is an imperfect model of the former, which is
the real process.) You state this in a way that Maynard Smith would have
appreciated. Let's not forget that he was the one who first suggested that
hitchhiking might solve "Lewontin's Paradox."
"More generally: Since..."
Again, yes. I might state it more quantitatively and comparatively. Genetic
draft should be a fairly constant forces irrespective of population size, while
the strength of drift will fluctuate with N. Obviously, in small populations,
drift can overwhelm draft.
-- John Gillespie, October 14, 2004
Drift and Draft
Against the background of the following three long quotes
from your Genetics, Gene, and Evolution papers on draft and
population size in evolution, I interpret an evolution in
your thinking about the significance or importance of draft
relative to drift. That is, you seem to make stronger claims
about the importance of draft in subsequent publications. So....
In 2000 (Genetics 155, p. 918) you say:
"Is linked selection a more important force than drift? In
regions of low recombination, including mitochondria, the
answer is quite possibly in the affirmative. What about
regions of the genome with "normal" levels of recombination?
.... [S]ome refinements of both the models and the
parameters are needed before we accept the notion that
linked selection may be a more important force than drift."
In 2000 (Gene 261, pp. 16-17) you say:
"The important question concerns whether genetic draft is a
more important force than is genetic drift. In terms of the
variance in the change of linked alleles, genetic draft will
be more important if. Hitchhiking will have an impact on a
linked locus of r/s is less than or equal to about 0.1. For
example, if a selected allele enjoys a one percent
advantage, that it will influence the dynamics of loci
within about 0.1 centimorgans, which encompasses many loci
in most genomes.
The insensitivity of genetic draft to the population size is
directly traceable to the strongly concave increase in the
rate of substitution at the selected locus, rho, with
population size. If small regions of the genome (say chunks
of 10-100 loci) evolve as if there were effectively no
recombination, and if the rate of selective substitution in
these regions were a strongly concave increasing function of
population size, then genetic draft will interact with
mutation in such a way that genetic variation will be
insensitive to a species population size.
Molecular population genetics has been plagued by its
failure to find significant correlations between population
size and genetic variation.... Faced with these compelling
observations [ratio of protein electrophoretic
heterozygosities in very large populations, nucleotide
diversities in mice, humans, and Drosophila], genetic draft
seems to be a vastly more powerful force than is genetic drift."
And in 2001 (Evolution 55, p. 2168) you say:
"Taken together, the observations presented above [evidence
that draft may be important in regions of the Drosophila
genome and the main arguments of the paper concerning N]
suggest a radical new view of the stochastic forces at work
in natural populations. The major stochastic force acting on
common alleles is due to linked selection; this force is
called genetic draft."
It seems to me that at least part of what accounts for this
apparent evolution in your thinking is improvements in the
modeling from the Genetics paper to the Evolution paper, and
recognition of relevant observations (particularly in
Drosophila) in the Gene and Evolution papers.
Am I right in my (abstractly stated) evolution of your
thinking? Or is there more to the story?
-- Robert Skipper, December 2, 2004
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