Doubting Didymus

Ahh, but how many of those are beneficial?



[rufus] N>0 [/rufus]

pz

Let's also ask Rufus to calculate the likelihood of one of those beneficial mutations becoming fixed in the population, while we're at it.

RufusAtticus

1/(2N) < p < 1

Oolon Colluphid

Which, I think, translates as 'reasonable'.



Cheers, Oolon

The Lone Ranger

And that doesn't even take selection into account: 1/(2N) is just the probability that a neutral allele will become fixed by chance. If the allele's beneficial, selection makes it even more likely that it'll become fixed in the population.

[Of course, that's exactly what Rufus said: the minimum probability that a beneficial allele will become fixed in the population is 1/(2N); selection means that the actual probability is higher, perhaps much higher.]

Cheers,

Michael

RufusAtticus

I couldn't remember the actual forumula, and I was too lazy to look it up. So I decided to follow up my N>0 statement with similar one. Now here is an edited repost from Theology Web, where I responded to a Socrates/Sarfati claim that it was the work of Spencer who showed what the probability of fixation was, and that large populations are less likely to have fixation of beneficial mutations.

===

Spencer? That was work first done by Haldane in 1927 and elaborated by Fisher in 1930, Wright in 1931, and Kimura in 1962. And, Socrates, you have presented it inaccurately. (Surprise . . . surprise. . . .)

If the fitnesses of the genotypes AA, Aa, and aa are 1, 1+s, and 1+2s respectively, then the probability of fixation for the mutant allele, a, is u= [1-exp(-4Nsp)]/[1-exp(-4Ns)] (1), where p is the allelic frequency of a, i.e. a is a single mutant gene.

If s is small compared to 1, then this equation reduces to 2s/[1-exp(-4Ns)] (2).

As N->inf, then u(1/2N) decreases to 2s.

If 4Ns << 1, then (1) reduces to 1/2N, i.e. the selected gene is effectively neutral.

Remember now that this is only for a mutant of additive effects. In general, increasing the level of dominance of the mutant allele increases it's probability of fixation, unless u(p) is already near 1.

Remember this is only the probability for a specific mutant gene going to fixation. If you actually want to calculate the probability of substitution (a new allele replaces the common allele), you need to take into account the mutation rate and the population size, as well as the probability of fixation. This produces an equation, r = 2Nuv, where r is the probability of fixation per generation, N is the pop size, u the prob of fixation, and v the mutation rate. If the mutation is a selected allele, this becomes r = 4Nsv. If it is a neutral allele, r=v. So it is completly obvious that selection, even slight selection, can make a big difference in substitution rates over neutrality, especially in big populations.

The general conclusion from this is that if 4Ns << 1 then drift is more important than selection in determining the outcome of the population. If 4Ns >> 1, then selection is more important. For 4NS ~ 1, both forces are important. This result is exactly opposite of Socrates' claim. In fact, it is much easier (probabilistically) for large populations to substitute benefical mutations than it is from small populations.

The Lone Ranger

Exactly so. Drift is much more likely to fix neutral alleles in small populations than in large ones, but by the same token, drift is far less likely to "accidentally" eliminate beneficial alleles in a large population than a small one, thereby allowing selection to fix them.

Cheers,

Michael

Doubting Didymus

Oh, right!

So the probability of a beneficial mutation being fixed in a population is higher than the same probability for a neutral one, but lower that absoloutely certain! How informative!

1/(2N) < p < 1 !





(smartarse.)

RufusAtticus

Hmm what happened to faust?

Celsus

I don't know, but that thread is going down in II lore as the worst E/C debate ever.

Joel