# What is a locus, anyway?

”Locus” is one of those confusing genetics terms (its meaning, not just its pronunciation). We can probably all agree with a dictionary and with Wikipedia that it means a place in the genome, but a place of what and in what sense? We also use place-related word like ”site” and ”region” that one might think were synonymous, but don’t seem to be.

For an example, we can look at this relatively recent preprint (Chebib & Guillaume 2020) about a model of the causes of genetic correlation. They have pairs of linked loci that each affect one trait each (that’s the tight linkage condition), and also a set of loci that affect both traits (the pleiotropic condition), correlated Gaussian stabilising selection, and different levels of mutation, migration and recombination between the linked pairs. A mutation means adding a number to the effect of an allele.

This means that loci in this model can have a large number of alleles with quantitatively different effects. The alleles at a locus share a distribution of mutation effects, that can be either two-dimensional (with pleiotropy) or one-dimensional. They also share a recombination rate with all other loci, which is constant.

What kind of DNA sequences can have these properties? Single nucleotide sites are out of the question, as they can have four, or maybe five alleles if you count a deletion. Larger structural variants, such as inversions or allelic series of indels might work. A protein-coding gene taken as a unit could have a huge number of different alleles, but they would probably have different distributions of mutational effects in different sites, and (relatively small) differences in genetic distance to different sites.

It seems to me that we’re talking about an abstract group of potential alleles that have sufficiently similar effects and that are sufficiently closely linked. This is fine; I’m not saying this to criticise the model, but to explore how strange a locus really is.

They find that there is less genetic correlation with linkage than with pleiotropy, unless the mutation rate is high, which leads to a discussion about mutation rate. This reasoning about the mutation rate of a locus illustrates the issue:

A high rate of mutation (10−3) allows for multiple mutations in both loci in a tightly linked pair to accumulate and maintain levels of genetic covariance near to that of mutations in a single pleiotropic locus, but empirical estimations of mutation rates from varied species like bacteria and humans suggests that per-nucleotide mutation rates are in the order of 10−8 to 10−9 … If a polygenic locus consists of hundreds or thousands of nucleotides, as in the case of many quantitative trait loci (QTLs), then per-locus mutation rates may be as high as 10−5, but the larger the locus the higher the chance of recombination between within-locus variants that are contributing to genetic correlation. This leads us to believe that with empirically estimated levels of mutation and recombination, strong genetic correlation between traits are more likely to be maintained if there is an underlying pleiotropic architecture affecting them than will be maintained due to tight linkage.

I don’t know if it’s me or the authors who are conceptually confused here. If they are referring to QTL mapping, it is true that the quantitative trait loci that we detect in mapping studies often are huge. ”Thousands of nucleotides” is being generous to mapping studies: in many cases, we’re talking millions of them. But the size of a QTL region from a mapping experiment doesn’t tell us how many nucleotides in it that matter to the trait. It reflects our poor resolution in delineating the, one or more, causative variants that give rise to the association signal. That being said, it might be possible to use tricks like saturation mutagenesis to figure out which mutations within a relevant region that could affect a trait. Then, we could actually observe a locus in the above sense.

Another recent theoretical preprint (Chantepie & Chevin 2020) phrases it like this:

[N]ote that the nature of loci is not explicit in this model, but in any case these do not represent single nucleotides or even genes. Rather, they represent large stretches of effectively non-recombining portions of the genome, which may influence the traits by mutation. Since free recombination is also assumed across these loci (consistent with most previous studies), the latter can even be thought of as small chromosomes, for which mutation rates of the order to 10−2 seem reasonable.

Literature

Chebib and Guillaume. ”Pleiotropy or linkage? Their relative contributions to the genetic correlation of quantitative traits and detection by multi-trait GWA studies.” bioRxiv (2019): 656413.

Chantepie and Chevin. ”How does the strength of selection influence genetic correlations?” bioRxiv (2020).

# Mutation, selection, and drift (with Shiny)

Imagine a gene that comes in two variants, where one of them is deleterious to the carrier. This is not so hard to imagine, and it is often the case. Most mutations don’t matter at all. Of those that matter, most are damaging.

Next, imagine that the mutation happens over and over again with some mutation rate. This is also not so hard. After all, given enough time, every possible DNA sequence should occur, as if by monkeys and typewriters. In this case, since we’re talking about the deleterious mutation rate, we don’t even need exactly the same DNA sequence to occur; rather, what is important is how often a class of mutations with the same consequences happen.

Let’s illustrate this with a Shiny app! I made this little thing that draws graphs like this:

This is supposed to show the trajectory of a deleterious genetic variant, with sliders to decide the population size, mutation rate, selection, dominance, and starting frequency. The lines are ten replicate populations, followed for 200 generations. The red line is the estimated equilibrium frequency — where the population would end up if it was infinitely large and not subject to random chance.

The app runs here: https://mrtnj.shinyapps.io/mutation/
And the code is here: https://github.com/mrtnj/shiny_mutation

(Note: I don’t know how well this will work if every blog reader clicks on that link. Maybe it all crashes or the bandwidth runs out or whatnot. If so, you can always download the code and run in RStudio.)

We assume diploid genetics, random mating, and mutation only in one direction (broken genes never restore themselves). As in typical population genetics texts, we call the working variant ”A” and the working variant ”a”, and their frequencies p and q. The genotypes AA, Aa and aa will have frequencies $p^2$, $2 p q$ and $q^2$ before selection.

Damaging variants tend to be recessive, that is, they hurt only when you have two of them. Imagine an enzyme that makes some crucial biochemical product, that you need some but not a lot of. If you have one working copy of the enzyme, you may be perfectly fine, but if you are left without any working copy, you will have a deficit. We can describe this by a dominance coefficient called h. If the dominance coefficient is one, the variant is completely dominant, so that it damages you even if you only have one copy. If the dominance coefficient is zero, the variant is completely recessive, and having one copy of it does not affect you at all.

The average reproductive success (”fitness”) of each genotype is described in terms of selection coefficients, which tells us how much selection there is against a genotype. Selection coefficients range from 0, which means that you’re winning, to 1 which means that you’ve been completely out-competed. For a recessive damaging variant, the AA homozygotes and Aa heterozygotes are unaffected, but the aa homozygotes suffers selection coefficient s.

In the general case, fitness values for each genotype are 1 for AA, $1 - hs$ for Aa and $1 - s$ for aa. We can think of this as the probability of contributing to the next generation.

What about the red line in the graphs? If natural selection keeps removing a mutation from the gene pool, and mutation keeps adding it back in again there may be some equilibrium frequency where they cancel out, and the frequency of the damaging variant is more or less constant. This is called mutation–selection balance.

Haldane (1937) came up with an expression for the equilibrium variant frequency:

$q_{eq} = \frac {h s + \mu - \sqrt{ (hs - \mu)^2 + 4 s \mu } } {2 h s - 2 s}$

I’ve changed his notation a bit to use h and s for dominance and selection coefficient. $\mu$ is the mutation rate. It’s not easy to see what is going on here, but we can draw it in the graph, and see that it’s usually very small. In these small populations, where drift is a major player, the variants are often completely lost, or drift to higher frequency by chance.

(I don’t know if I can recommend learning by playing with an app, but I definitely learned things while making it. For instance that C++11 won’t work on shinyapps.io unless you send the compiler a flag, and that it’s important to remember that both variants in a diploid organism can mutate. So I guess what I’m saying is: don’t use my app, but make your own. Or something.)

Literature

Haldane, J. B. S. ”The effect of variation of fitness.” The American Naturalist 71.735 (1937): 337-349.