I've decided to jot down some simple* formalisms which I can refer new readers to on this website. So today....
You know that if you have a novel mutation within a population, its probability of fixation if it is neutral is:
1/(2N), where N = effective population, in a neutral scenario where the mutation confers neither advantage or disadvantage. So in a population of 100, a new mutant has a 1 out of 200 chance of fixation, going from 0.5% in the initial generation, to 100% in the generation of fixation. In a population of 1000, a new mutation has a .05% chance of fixation, and so forth.
In a non-neutral case, fixation probability is 2s, where s equals the positive selective advantage conferred by the allele against the population mean fitness. So if the selection coefficient is 0.01, the probability of fixation is 2%.
In regards to the time until fixation, for a neutral allele iit s 4N, where N is the effective population, and the product is generations.
For a selectively beneficial allele, it is (2/s)(ln(2N)). The "left side" of this relation is more important since the parameter N is converted into its natural log.
* In the first draft of this post I used the word "trivial," but I don't think these formalisms are necessarily clear and obvious intuitive statements, even if they are the most basic of algebras. So I've reedited it to "simple," as I think keeping in mind relations like 1/(2N) and 4N are important insights which are necessary for a gestalt comprehension of evolutionary dynamics.
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How about using "basic", reflecting the fundamental nature of these relations without implying anything about the ease of understanding?
Also, reflecting my inability to grasp simple things, doesn't the different cases for neutral and non-neutral cases imply that for sufficiently small populations or small enough positive bias, a neutral mutation would actually have an easier time getting fixated than a positive one? With a selection coefficient of 0.005, and a population of 100 (just tweaking your examples above), you'd have 2% chance for a neutral mutation, and a %1 chance for a slightly beneficial one.
What are the limitations for these relationships? is it the neutral case that falls apart for small populations (don't think that's the case), or is the estimation for beneficial mutations invalid other than for a "large enough" population (more likely)?
janne,
here is an old post that answers some of your questions. if it is unclear, feel free to follow up.
What are the limitations for these relationships? is it the neutral case that falls apart for small populations (don't think that's the case), or is the estimation for beneficial mutations invalid other than for a "large enough" population (more likely)?
again, see my above post. but, please note, the rate of neutral substitution remains constant. an old post where i elaborate. but yes, i suspect the general consensus would be that over the long term neutral non-adaptive evolution can sweep over the whole genome as the population shrinks. in contrast, in large populations neutral evolution occurs only on the neutral portions of the genome, while non-neutral loci are subject to selective constraint or being driven to fixation.
your general hunches derived from the formalisms are correct, the key is the contextual nuance of these hunches as they play out in the messy specifics of biological populations....
This is in a diallelic population, right?
This is in a diallelic population, right?
yeah. didn't make that explicit. the mutant allele is p, and every other allele is q! :) though seriously, you've seen some of the multiallelic overdominance models i'm sure...don't go confusing people.
also, feel free to point to posts of yours that you believe would be beneficial. i'm pressed for time and so i don't have time to browse your site, but it looks interesting (obviously to myself), and i'd be interesting in posting to lucid exposition of this sort of material.
I would, but I fear I'd make newbie mistakes like getting the terminology wrong and end up confusing people. For example, I just said diallelic where I meant diploid.
Incidentally:
So in a population of 100, a new mutant has a 1 out of 50 chance of fixation, going from 1% in the initial generation, to 100% in the generation of fixation.
Are you sure you mean 1 in 50 there? 1/(2*100) = 1/200. And wouldn't a single allele in a diploid population of size 100 be at 0.5% in the initial generation? Or am I just confusing myself here?
I agree, trivial would be the wrong word...the derivation of most of these relationships is rather nontrivial, though the expressions themselves have a simple form. Also, probably worth noting that you are talking about the dynamics of a single locus with 2 alleles in a Wright Fisher population...if you have different demography, or substructure (e.g. http://www.nature.com/nature/journal/v433/n7023/full/nature03204.html) or more complicated selection scenarios, these equations don't necessarily hold.
Coalescent --
I saw your webpage and noted that you thought the Gillespie book was good. I also used to think it was good maybe 2 years ago, but I've come to realize that even the second edition is actually quite misleading in many respects. It will not prepare you for the state of the art. I would recommend Jobling's Human Evolutionary Genetics and Graur & Li's Molecular Evolution instead.
There are many problems with the Gillespie book, but they are not obvious on a first reading. Some of them:
1) the 2nd edition is published in 2004, but does not mention the human genome once!
2) most of the papers which he spends some time covering are ancient history. The most recent topic which he covers is probably Tajima's D, from the late 80's...however, the treatment is so superficial that it is confusing.
3) There is an overemphasis on trivialities like the difference between heterozygosity and H (which basically reduces to sampling with and without replacement from a population of size N...a distinction which is negligible in any real life scenario)
4) I could go on and on, but the book is really a fossil which is not worth spending scarce time on. Read Jobling instead and you'll marvel at the difference. Graur and Li is also very good, though a bit older at this point.
gc: thanks for the tip - I hadn't realised Gillespie was so out of date.
yes! i goofed on the arithmetic. chalk it up to the late hour, though i should have mastered that by elementary school.
i liked the jobling book too, though if you want more theory and thickness cavalli-sforza & bodmer's book from '71 is pretty good too (it has been reissued in trade paper and is pretty affordable). i like hedrick's new edition of genetics of populations for straight pop gen myself.
coalescent,
btw, when i rec. hedrick (or hartl & clark), i think you would be well served by just skimming the text and focusing on the technical "boxes," as you know math and you probably just need to orient yourself to how it is worked out in pop gen.
sorry, yes, diploid. wright-fisher....
The Coalescent: My Favourite! If we are recommending good books on the coalescent check out "Gene Genealogies, Variation and Evolution: A Primer in Coalescent Theory" by Hein, Schierup, and Wiuf. I have enjoyed it immensely and feel it is the best book in explaining coalescent theory on the market.