Selection is random

When is the effect of a mutation neutral?

A mutation (by this I mean any change to the genotype/genome of an organism) is neutral when it does not change the fitness of the organism. This can happen in different ways:

1) A mutation (SNP) that changes one nucleotide in the protein coding sequence, but does not change the amino acid. These are known as synonymous substitutions, and (mostly*) do not affect fitness.
2) When the mutation does not change fitness, just because the genomic change makes no difference for how well the cell/organism functions.
3) ... (See comments?)

Case number two can happen when the selection coefficient, s=w'/w-1, is zero (w is fitness before the mutation, w' is the fitness after). In this case the two organisms have equal fitness. However, if s is really small, then it will also not matter. The question is then how small s needs to be before the mutation will be neutral, and the answer is that it depends on the population size, N.

So why does neutrality of a mutation depend on the value of s compared to the population size? Because selection is random! Yes, I realize many will object to putting it like this. It goes against what "we" all learn, namely that mutations are random, but that selection weeds out the bad ones and promotes the good ones. Selection is indeed directional in this way, but it is also random.

The reason that selection is random is that fitness is not a value that determines with certainty who gets to reproduce. Fitness is a probability that describes the chance of reproducing. If two organisms have fitness 1.0 and 0.80, then the first has a 25% higher chance of reproducing than the second one does. And in a simulation, the way that is (properly, but not always) determined, is by selecting at random between these two organisms, but with a probability proportional to their fitness. This can for example be done by use of the acceptance-rejection method: choose any one organism at random (all equally probable), and then generate another random number between zero and the maximum fitness in the whole population. If this random number is less than the fitness of the chosen organism, then that organisms gets to reproduce. If not, reject this organism. Repeat the algorithm until you have selected as many organisms as you need (to reach carrying capacity, for example).

As you can see, done this way, selection is random, but - crucially - the probability distribution is not uniform (unless all organisms have the same fitness, in which case evolution of this population is completely neutral).

Now, in an infinitely large population any change in fitness, however small, can be "detected" by selection. If a mutation causes any change in fitness, it will make a difference for the evolution of an infinitely large population. But, in a finite population (slightly more realistic, and much easier to simulated when using agent-based** simulations), s might be too small for selection to detect the difference. The exact threshold depends on the mode of reproduction, but as a general rule (and for asexuals), if the selection coefficient is less than one over the population size, s<1/N, then the mutation is effectively neutral, and the mutation will drift in the population, either eventually going to fixation, or, more likely, disappear.

For example, if the population size is N=100, then mutations of effect s=0.001 are neutral.

A new paper in Evolution, Genome Structure and the Benefit of Sex, describes an evolutionary simulation that suggests that the benefit of sex is that recombination can change large blocks of the genome. The idea is a very well-known one, namely that because epistasis causes the fitness landscape to be rugged, i.e. having multiple peaks separated by valleys that must be crossed to reach a higher peak, point-mutations that change just one nucleotide (SNPs) may not do the trick. If crossing the valley causes a fitness decrease too detrimental, then the chance that the other higher peak is eventually located is too low. But, with recombination, larger parts of the genome can be reorganized in one fell swoop, and with some luck, one organism may find itself on another peak.

All very trivial, and easy to show with simulation, actually. One quibble I have is about a result stated in the caption of figure 2 of that paper. Here's what it says:



The highlighted portion says that (in a population of N=10,000 individuals) organisms with fitness one millionth higher than 1.02 goes to fixation much faster than the others. In other words, in a population of N=10⁵, an organism with a mutation that increases its fitness by only 10^-6 has a huge fitness advantage. And that just can't be true. It should be effectively neutral, the difference in fitness undetectable by selection.

Kick me if I can understand what's going on. My best guess is that this is an error, even though I checked with the authors, and they tell me it's not a typo, at least. Got another suggestion, dear reader? (Kudos for making it this far.)

* It actually can. Every codon has a specific tRNA that attaches to that codon, and if this tRNA is less abundant in the cell, then the new variant of the protein might be translated at a lower rate, which might affect the organism.

** Agent-based simulations retains information about each individual in the population, whereas infinitely large populations can only, and easily, be treated mathematically.

Reference:
Watson, R., Weinreich, D., & Wakeley, J. (2011). GENOME STRUCTURE AND THE BENEFIT OF SEX Evolution, 65 (2), 523-536 DOI: 10.1111/j.1558-5646.2010.01144.x