But you’ve just described the very DFE you’re looking at!
Of course it does! Selection is the thing doing the optimizing, but can only act beyond the threshold for selection.
And what mechanism, exactly, is it that will do this optimization below the selective threshold?
This makes me think I’ve used words unfamiliar to you. Perhaps a visual aid is in order? 
This figure, adapted from Bank et al 2014 (linked below) shows what I mean by ‘lethal mode as a second exponential curve’ (as indicated by the red arrow). You may note that such a second mode is absent from your diagram.
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It’s the object of the mutations (the genome) which is optimized. The mutations are not optimized.
Selection. Because the selective threshold is for mutations, and the optimization is for the genome.
Ah, now I see what you’re saying. So somehow there’s this huge gulf where there are no deleterious mutations, and then they suddenly become strongly deleterious/lethal. On the other side, mutations that are deleterious and selectable neatly counterbalance beneficial selectable mutations. Same for e. neutrals.
Does anybody else here agree with Crispr’s suggested distribution? And what accounts for that giant gap where nothing is happening?
I would assume that @glipsnort at least must disagree with this distribution, since it doesn’t seem to account for the concept of optimization in the genome in any meaningful way. (Why would it be equally easy to make medium-sized changes that are beneficial or deleterious in an optimized genome?)
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I don’t think that [-1,1] range is a constraint on either axis…but we can move on from that point.
I agree it’s possible, and that it shows that you don’t need a link. It’s a great counter example.
But their are infinitely many other distributions that do that same, so I don’t know it is the right one.
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This figure was used to establish a conceptual model, and I believe may have been exaggerated for clarity. It was not used to represent either actual data or modeled parameters.
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And what is getting selected to optimize the genome, if not mutations?
I agree with @John_Harshman here. “No strictly neutral mutations” does not imply the distribution is asymptotic with the Y axis. In this case, it most certainly is not, because there is not infinite density at 0. Rather “no strictly neutral” mutations implies that the height of the curve at 0 is 0, which is not how Kimura drew the curve.
I think @John_Harshman was pointing out that we can’t even see most of the probability density, which is arbitrarily close to 0.
If you’re saying there’s no giant gulf, then I must say it’s not a clear demonstration at all. Because in your graph, there is a giant gulf where no mutations are happening. That appears to be very much by design.
Is it not? The x axis is supposed to be for selection coefficients. I thought the value for lethality was s = -1, but please correct me if I’m wrong.
I think the range depends on the units of the fitness function. Depending how we define the units, it could be unbounded.
Kimura stated his model contained no strictly neutral mutations [something I believe both of our resident population geneticists have also agreed with]. However, you’re right, it’s not literally asymptotic since there is not an infinite volume of mutations. It must come to an end at some point.
Not arbitrarily. It is acknowledged in the literature that the great majority of mutations are very small in effect, which would in turn mean that most of our probability should be centered near 0.
Good question, but I must qualify my response as partial speculation:
- Any beneficial mutation that is easy to acquire, quickly is. Fitness will tend to hit a “wall” which is harder to bypass.
- Mildly deleterious mutations, and perhaps only occasionally beneficial mutations don’t have the “wall effect”, and so taper off more slowly.
I base this on some tinkering with Genetic Algorithms for optimizing a configuration (non-biological computer simulation). Fitness improves swiftly and smoothly when gains are easy to make, most slowly and in steps as the optimization progresses, and further gains become harder to find. Less fit members of the population will persist depending on the rules used to thin the population. In my tests I kept the top 50% by fitness, and all for at least 3 generations, which left a lot of room for “sub-optimal” to hang around. I didn’t make any plots quite like that Kimura plot, but the shape of the curve doesn’t not surprise me.
It seems like there is some large gaps in how you are using basic statistical terms. The distribution doesn’t need to be centered at zero (median or mean), but we do expect a peak near zero (a mode).
The height, width and skew of that peak near zero is fitness essentially impossible to directly measure and any conclusion based on the skew of this peak (as is Genetic Entropy) is not well founded. The skew is a subtle third order statistic that would be difficult to measure even if we knew the height and width of the peak (but we don’t!).
See that last paragraph? It’s a clear statement of a scientific objection that many of us have.
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What does this mean? Are you suggesting there’s a different volume of effectively neutral beneficials compared to the volume of effectively neutral deleterious?
Interesting. I would not expect the volumes to be equal, but there will be an equilibrium or “balance point” where the population is stable. perhaps I am envisioning something diferent than you are asking tho …
Consider a population that has already reached its optimum. Members of the population which are at the optimum will make that “wall” on the right-hand side of the graph, and there will be no “tail” above that. As deleterious mutations accumulate in portions of the population, their fitness will be below the maximum, making a long tail to the left.
I left out an important assumption: I’m assuming that selection occurs with inverse probability relative to fitness. Less-fit --> greater probability of being selected out. If there is a “hard” or fixed selection rule we might see a wall on the left side too. It’s easy to code a hard rule in a simulation, but I doubt they occur in nature.
There is also the relative nature of fitness, where older beneficial mutations are outpaced by newer mutations, making them relatively less fit or even relatively harmful. Competition is a driving factor in selection and changing environments.
Imagine trying to balance a sharp pencil on its point. Neutral mutations are like that. Genetic drift will make the pencil fall either to loss or fixation.
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I certainly don’t know if it is the ‘right’ one either. But there is strong evidence of a disproportionate number of lethal mutations relative to expectations from any simple curve able to also model observations near neutrality. In a way, this is understandable from the theory by recognizing that the fitness effects are bounded from [-1,1], but our curves ‘want’ to extend beyond -1 in the asymptotic approach to 0 probability. So it all ‘piles up’ at -1. I’m not a mathematician.
Not a clear demonstration of what? It isn’t a clear demonstration of the actual shape of the DFE, but it does clearly illustrate the conceptual distinction between the two modes, which I suspect was their intention. It is also possible that the probability in the middle is non-zero, but is too small to see on the figure at the scale used.
I’m not sure which is true, as I have no access to the authors inner thoughts and I don’t care to ask them. And I don’t think it is relevant to this discussion anyway, because I only used that image because it was an illustration of a second mode, and it is the concept of a second mode that matters here. So rather than debating whether the specific shape of that figure is a possibly accurate description of the general DFE, why not focus on the concept of a bimodal DFE more generally?
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