Neutral Expectations

Last week, I linked to an article in Seed about synonymous mutations with deleterious effects in humans. It's heavy with errors, but I didn't linger too much on them. Larry Moran, on the other hand, got a bit more riled up than I did, and John Logsdon (whose blog has the potential to be something cool) agrees with Larry. And I agree with both of them.

The issue here is with the neutral theory, which Larry describes quite well (see also the coverage in this primer to population genetics). Many people misinterpret the neutral theory and think that it claims that all mutations in some class (non-coding, synonymous, etc) have no fitness cost or benefit. It does no such thing. The neutral theory merely says that there are many mutations with absolutely no fitness cost or benefit. The neutral theory -- and every other reasonable alternative -- allows for both adaptive and purifying selection in addition to neutral evolution.

There are a few graphs below the fold to illustrate this point.

When discussing different models of molecular evolution, we must understand what they say about the frequency of different types of mutations and the frequency of different types of fixations (also known as substitutions). For simplicity's sake, we will discuss three categories of alleles: deleterious, neutral, and advantageous. Deleterious mutations will almost never fix in the population, neutral mutations will fix with a probability proportional to the inverse of the population size, and advantageous alleles will fix with a probability proportional to the fitness benefit they confer. Any and all models of molecular evolution use this paradigm, but they differ in the frequency of mutations in each category.

The fitness benefits of an advantageous mutation depends on the advantage individuals carrying the mutation have relative to individuals without the mutant allele (which we will call 's') and the population size (which we will call 'N'). A given advantageous mutation is more likely to fix in a large population than a small one because the product of the selection coefficient and population size (Ns) is greater in larger populations. The effect of population size is diminished for mutations with large fitness costs or benefits, Ns<-1 or NS>1.

A neutral mutation (one that confers no fitness cost or benefit) is more likely to fix in a small population because random fluctuations in allele frequency are greater in small populations. Please note, however, that the rate of neutral fixations is equal in all populations because large populations accumulate more mutations (because they have more individuals) than small populations -- that makes up for the lower probability of fixation of any single neutral mutation in large populations.

The neutral theory (in our simplified model with only three categories of mutations) predicts that the majority of new mutations will be either deleterious or neutral. It allows for a very small class of advantageous mutations, of which many reach fixation. The deleterious mutations are purged from the population and nearly none reach fixation. Many of the neutral mutations fix.

As you can see in the graph above, the majority of molecular evolution, according to the neutral model, occurs via non-adaptive processes. The narrow bar centered at zero indicates the frequency of neutral mutations (left) and neutral fixations (right). There are some fixations that result from Darwinian selection (adaptive evolution), shown by the small bump in the graph on the right. But the model predicts so few mutations in this class (they don't even show up on the graph on the left) that most of the divergence between species occurs at sites that confer no selective cost or benefit.

As a contrast to the neutral model, I have created a caricatured adaptive model (see below). The main difference between this adaptive model and the neutral model is in the frequency of mutations that confer a fitness benefit. The adaptive model allows for a couple orders of magnitude more beneficial mutations, which means there are substantially more fixations driven by natural selection. In both models, the majority of new mutations are either deleterious or neutral. And, in both models, the deleterious ones are purged and the neutral ones fix with a probability dependent mutation rate. But, to hammer home the point, the adaptive model leads to many more adaptive fixations than the neutral model based solely on the frequency of beneficial mutations.

Finally, I would be remiss to not mention the nearly neutral theory. In the neutral model, there is a class of mutations that confers no fitness cost or benefit (they are truly neutral). The nearly neutral model uses the idea that the fitness cost/benefit of a mutation is a product of the selection coefficient (s) and population size (N). If Ns < 1, natural selection will remove the mutation from the population because it is deleterious. If Ns > 1, a mutation has a high chance of fixing. But if Ns is in the ballpark of 1, a mutation will behave much like a neutral allele.

That means that the neutrality of a mutation depends not only the selective coefficient of a mutation, but also on the size of the population in which it arose. Small populations will have many more mutations with Ns near one than large populations because small populations have a smaller N. Because there are more mutations that are essentially neutral in a small population, small populations will experience more evolution due to drift. Large populations will have fewer mutations behaving as neutral variants -- they'll either be slightly deleterious or slightly advantageous -- so you'll see more fixations of alleles as the result of positive selection and more removal of alleles as the result of purifying selection.

The expected distribution of mutations and fixations in different fitness categories for the nearly neutral theory (shown above) resembles that of the neutral theory with one major exception: the neutral mutations in the neutral theory have been replaced by a distribution of mutations that are nearly neutral (there aren't values on the axes, but assume the distribution is in the ballpark of -1<Ns<1). The many deleterious mutations that arise in a population will still be removed by natural selection, and the few highly advantageous mutations will be fixed by selection. But the probability of fixation of the nearly neutral mutations (those in the distribution near zero) will depend on population size. In small populations, the fate of many will depend on random sampling (ie, genetic drift). In large populations, many will be removed or fixed by selection, but there are still other that will be removed or fixed by neutral processes.

In reality, we don't see three discrete fitness categories; there is actually a continuum of fitness costs and benefits for new mutations which determine their probability of loss or fixation. The simple model presented above can be extended to a more realistic one if you so desire. But the purpose of this post was to illustrate a point, so I shied away from adding unnecessary complexities.

Which is the true model for molecular evolution? None of the above, that's for sure. And, really, it depends on the locus and it depends on the time frame. Over short evolutionary time (within species or between closely related species), the many non-coding sites will evolve according to neutrality (fixed by chance, not by selection). But over longer time scales, even protein coding sequences appear to evolve neutrally -- most likely because they evolve slower so there simply aren't enough substitutions over short amounts of time to be able to observe neutral rates.

As for the utility of the neutral theory: it makes an excellent null hypothesis when searching for genomic regions under the influence of natural selection. This recent review by John Stinchcombe and Hopi Hoekstra does a good job explaining how we use the neutral theory to determine expected patterns of polymorphism and test for outlier loci that are not evolving neutrally (ie, aren't evolving according to the manner expected of mutations with no fitness cost or benefit). It's quite difficult (and impractical) to come up with a sweeping generalization regarding neutral and adaptive evolution across entire genomes. But the neutral theory has a very nice utilitarian practicality that cannot be neglected.

Getting back to the article in question (you know, the one that started this whole post), we see that the neutral theory has not been overturned, but it is constantly being overturned. Every time someone performs a test for natural selection (see here and links therein), they are testing the neutral theory. And every time they identify a non-neutral pattern of evolution, they are overturning the neutral theory . . . for that particular locus.