Molecular Evolution Activities

Drift and Draft
Near the end of your paper, "Is Population Size of a Species Relevant to its Evolution?" (in Evolution, vol. 55, 2001, pp. 2161-2169), you say that alleles whose frequency are close to 1/N are subject to the combined stochastic forces of drift and draft, but that the action of drift does not depend on population size and that it does not involve binomial sampling. In that case, what does drift involve?
-- Robert Skipper

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