# Introduction

I’ve mentioned many times by now that population geneticists often look at the world backwards. To those of you who aren’t population geneticists,1 looking at the world backwards is probably as awkwards as walking backwards. Sometimes, though, it turns out that walking backwards is useful, as when you’re trying to keep an eye on where you’ve been, not just where you’re going. That’s what we’re about to do with genetic drift. So far we’ve been trying to predict what will happen in a population given a particular effective population size. But when we collect data we are often more interested in using those data to understand the processes that produced the patterns we find in them than in predicting what will happen in the future. We’re using data to provide insight about where we’ve been, not where we’re going. So let’s take a backward look at drift and see what we find.

# Reconstructing the genealogy of a sample of alleles

Specifically, let’s keep track of the genealogy of alleles. In a finite population, two randomly chosen alleles will be identical by descent with respect to the immediately preceding generation with probability $$1/2N_e$$.2 That means that there’s a chance that two alleles in generation $$t$$ are copies of the same allele in generation $$t-1$$. If the population size is constant, meaning that the number of allele copies3 is also constant, that also means that there’s a chance that some allele copies present in generation $$t-1$$ will not have descendants in generation $$t$$. Looking backward, then, the number of allele copies in generation $$t-1$$ that have descendants in generation $$t$$ is always less than or equal to the number of allele copies in generation $$t$$. That means if we trace the ancestry of allele copies in a sample back far enough, all of them will be descended from a single common ancestor.4 Figure 1 provides a simple schematic illustrating how this might happen.

Time runs from the top of Figure 1 to the bottom, i.e., the current generation is represented by the circles in the botton row of the figure. Each circle represents an allele. The eighteen alleles in our current sample are descended from only four alleles that were present in the populations ten generations ago. The other fourteen alleles present in the population ten generations ago left no descendants. How far back in time we’d have to go before all alleles are descended from a single common ancestor depends on the effective size of the population, because how frequently two (or more) alleles are descended from the same allele in the preceding generation depends on the effective size of the population, too. But in any finite population the pattern will look something like the one I’ve illustrated here.

# Mathematics of the coalescent: two alleles5

The mathematician J. F. C. Kingman developed a convenient and powerful way to describe how the time to common ancestry is related to effective population size . The process he describes is referred to as the coalescent, because it is based on describing the probability of coalescent events, i.e., those points in the genealogy of a sample of alleles where two alleles are descended from the same allele in the immediately preceding generation.6 Let’s consider a simple case, one that we’ve already seen, e.g., two alleles drawn at random from a single population.

The probability that two alleles drawn at random from a population are copies of the same allele in the preceding generation is also the probability that two alleles drawn at random from that population are identical by descent with respect to the immediately preceding generation. We know what that probability is,7 namely $\frac{1}{2N_e^{(f)}} \quad .$ I’ll just use $$N_e$$ from here on out, but keep in mind that the appropriate population size for use with the coalescent is the inbreeding effective size. Of course, this means that the probability that two alleles drawn at random from a population are not copies of the same allele in the preceding generation is $1 - \frac{1}{2N_e} \quad .$ We’d like to calculate the probability that a coalescent event happened at a particular time $$t$$, in order to figure out how far back in the ancestry of these two alleles we have to go before they have a common ancestor. How do we do that?

Well, in order for a coalescent event to occur at time $$t$$, the two alleles must have not have coalesced in the generations preceding that.8 The probability that they did not coalesce in the first $$t-1$$ generations is simply $\left(1 - \frac{1}{2N_e}\right)^{t-1} \quad .$ Then after having remained distinct for $$t-1$$ generations, they have to coalesce in generation $$t$$, which they do with probability $$1/2N_e$$. So the probability that two alleles chosen at random coalesced $$t$$ generations ago is $P(T=t) = \left(1 - \frac{1}{2N_e}\right)^{t-1}\left(\frac{1}{2N_e}\right) \quad . \label{eq:two-allele}$ It’s not too hard to show, once we know the probability distribution in equation ([eq:two-allele]), that the average time to coalescence for two randomly chosen alleles is $$2N_e$$.9

It’s also not too hard to arrive at this conclusion intuitively. If I tell you, for example, that the probability that the UConn football team will win a football game is 10 percent, you’d probably guess that, on average, you’d have to wait 10 games before they won. Ten games is just one over the probability of winning any one game. In the case of the coalescent, the probability of a coalescent event in any generation is $$1/2N_e$$, so the average time to a coalescent event is $$2N_e$$.10

# Mathematics of the coalescent: multiple alleles

It’s quite easy to extend this approach to multiple alleles.11 We’re interested in seeing how far back in time we have to go before all alleles are descended from a single common ancestor. We’ll assume that we have $$m$$ alleles in our sample. The first thing we have to calculate is the probability that any two of the alleles in our sample are identical by descent from the immediately preceding generation. To make the calculation easier, we assume that the effective size of the population is large enough that the probability of two coalescent events in a single generation is vanishingly small. We already know that the probability of a coalescence in the immediately preceding generation for two randomly chosen alleles is $$1/2N_e$$. But there are $$m(m-1)/2$$ different pairs of alleles in our sample.12 So the probability that one pair of these alleles is involved in a coalescent event in the immediately preceding generation is $\left(\frac{1}{2N_e}\right)\left(\frac{m(m-1)}{2}\right) \quad . \label{eq:multi-allele-first-event}$ From this it follows13 that the probability that the first coalescent event involving this sample of alleles occurred $$t$$ generations ago is $P(T=t) = \left(1-\left(\frac{1}{2N_e}\right)\left(\frac{m(m-1)}{2}\right)\right)^{t-1} \left(\frac{1}{2N_e}\right)\left(\frac{m(m-1)}{2}\right) \quad . \label{eq:multi-allele}$ So the mean time back to the first coalescent event is $\frac{2N_e}{m(m-1)/2} = \frac{4N_e}{m(m-1)} \hbox{ generations} \quad .$ Remember, though, that most coalescent events happen before the mean coalescence time.14

But this is, of course, only the first coalescent event. We were interested in how long we have to wait until all alleles are descended from a single common ancestor. Now this is where Kingman’s sneaky trick comes in. After the first coalescent event, we have $$m-1$$ alleles in our sample, instead of $$m$$. So the whole process starts over again with $$m-1$$ alleles instead of $$m$$.15 Since the time to the first coalescence depends only on the number of alleles in the sample and not on how long the first coalescence event took, we can calculate the average time until all coalescences have happened as \begin{aligned} \bar t &=& \sum_{k=2}^m \bar t_k \\ &=& \sum_{k=2}^m \frac{4N_e}{k(k-1)} \\ && \mbox{TAMO} \\ &=& 4N_e\left(1 - \frac{1}{m}\right) \\ &\approx& 4N_e \end{aligned}

When all alleles have coalesced, there’s only one allele present. Since we haven’t introduced mutation into the coalescent process yet, that’s equivalent to saying that all of the $$m$$ alleles in our sample are identical by descent, i.e., that one particular allele that was present, on average, $$4N_e$$ generations ago is the ancestor of all of the alleles in our sample, i.e., it has been fixed. You’re unlikely to remember this, since we didn’t talk about it in lecture, but $$4N_e$$ as the time to coalescence may look vaguely familiar. Look at this formula for the time to fixation of one of two alleles present in a population from the notes on genetic drift: $\bar t \approx -4N\left(p\log p + (1-p)\log(1-p)\right) \quad .$ Does it surprise you that the average time to fixation (going forward in time) looks a lot like the average time to coalescence (looking backward in time)? It shouldn’t. They’re opposite sides of the same coin.

## A continuous time version of the coalescent

Since the effective size of a population has to be pretty big for the coalescent process to be a good representation, big enough that $$(1/2N_e)^2$$ is negligible, $$4N_e$$ is generally in the hundreds or thousands. That means that even though the coalescent as I formulated it above is a discrete time process, i.e., events happen at time 1, 2, 3, $$\dots$$, it can be convenient to think of time as continuous, which is surprisingly easy to do. We start with the “well-known fact” that if $$p$$ is “small” $\log(1-p) \approx -p \quad .$ As a result, \begin{aligned} (1 - p)^t &=& e^{t \log(1-p)} \\ &\approx& e^{-pt} \quad . \end{aligned} In our case, $p = \frac{k(k-1)}{4N_e} \quad ,$ when there are $$k$$ alleles.16 So $\mbox{P}(T = t) = \left(\frac{k(k-1)}{4N_e}\right)e^{t\frac{k(k-1)}{4N_e}} \quad .$ If you’re wondering why there’s a $$t$$ in that equation instead of the $$t-1$$ you’d get from substituting direclty into equation ([eq:multi-allele]), it’s because the exponential distribution here is the limit of the geometric distribution in ([eq:multi-allele]) as the coalescence time grows large.

## An example: Mitochondrial Eve

Cann et al.  sampled mitochondrial DNA from 147 humans of diverse racial and geographic origins.17 Based on the amount of sequence divergence they found among genomes in their sample and independent estimates of the rate of sequence evolution, they inferred that the mitochondria in their sample had their most recent common ancestor about 200,000 years ago. Because all of the most ancient lineages in their sample were from individuals of African ancestry, they also suggested that mitochondrial Eve lived in Africa. They used these arguments as evidence for the “Out of Africa” hypothesis for modern human origins, i.e., the hypothesis that anatomically modern humans arose in Africa about 200,000 years ago and displaced other members of the genus Homo in Europe and Asia as they spread. What does the coalescent tell us about their conclusion?

Well, we expect all mitochondrial genomes in the sample to have had a common ancestor about $$2N_e$$ generations ago. Why $$2N_e$$ rather than $$4N_e$$? Because mitochondrial genomes are haploid, not diploid. Furthermore, since we all get our mitochondria from our mothers,18 $$N_e$$ in this case refers to the effective number of females.

Given that a human generation is about 20 years, a coalescence time of 200,000 years implies that the mitochondrial genomes in the Cann et al. sample have their most recent common ancestor about 10,000 generations ago. If the effective number of females in the human populations is 5000, that’s exactly what we’d expect. While 5000 may sound awfully small, given that there are more than 3 billion women on the planet now, remember that until the recent historical past (no more than 500 generations ago) the human population was small and humans lived in small hunter-gatherer groups, so an effective number of females of 5000 and a total effective size of 10,000 may not be unreasonable. If that’s true, then the geographical location of mitochondrial Eve need not tell us anything about the origin of modern human populations, because there had to be a coalescence somewhere. There’s no guarantee, from this evidence alone, that the Y-chromosome Adam would have lived in Africa, too. Having said that, my limited reading of the literature suggests that more extensive recent data are consistent with the “Out of Africa” scenario. Y-chromosome polymorphisms, for example, are also consistent with the “Out of Africa” hypothesis . Interestingly, dating of Y-chromosome polymorphisms suggests that Y-chromosome Adam left Africa only 35,000 – 89,000 years ago.

# The coalescent and $$F$$-statistics

Suppose we have a sample of alleles from a structured population. For alleles chosen randomly within populations, let the average time to coalescence be $$\bar t_0$$. For alleles chosen randomly from different populations, let the average time to coalescence be $$\bar t_1$$. If there are $$k$$ populations in our sample, the average time to coalescence for two alleles drawn at random without respect to population is19 \begin{aligned} \bar t &=& \frac{1}{k}\bar t_0 + \frac{k-1}{k}\bar t_1 \\ &=& \frac{\bar t_0 + (k-1)\bar t_1} {k} \quad . \end{aligned} Slatkin  pointed out that $$F_{st}$$ bears a simple relationship to average coalescence times within and among populations. Given these definitions of $$\bar t$$ and $$\bar t_0$$, \begin{aligned} F_{st} &=& \frac{\bar t - \bar t_0}{\bar t} \\ &=& \frac{\left(\frac{k-1}{k}\right)\bar t_1}{\bar t} \\ &\approx& \frac{\bar t_1}{\bar t} \quad . \end{aligned} So another way to think about $$F_{st}$$ is as a measure of the proportional increase in coalescence time that is due to populations being separate from one another. One way to think about that relationship is this: the longer it has been, on average, since alleles in different populations diverged from a common ancestor, the greater the chance that they have become different. An implication of this relationship is that $$F$$-statistics, by themselves, can tell us something about how recently populations have been connected, relative to the within-population coalescence time, but they can’t distinguish between recent common ancestry that is due to migration among populations and recent common ancestry that is due to a split between populations.

A given pattern of among-population relationships might reflect a migration-drift equilibrium, a sequence of population splits followed by genetic isolation, or any combination of the two. If we are willing to assume that populations in our sample have been exchanging genes long enough to reach stationarity in the drift-migration process, then $$F_{st}$$ may tell us something about migration. If we are willing to assume that there’s been no gene exchange among our populations, we can infer something about how recently they’ve diverged from one another. But unless we’re willing to make one of those assumptions, we can’t really say anything.20

# The coalescent and natural selection

It shouldn’t surprise you that if we can study some of the properties of drift and selection, we can also use the coalescent to understand how natural selection works in a finite population. Even though the mathematics of the coalescent are ususally simpler than the older diffusion approach for studying allele frequency changes in a finite population, they are still very complicated. I’ll simply outline one approach here known as the structured coalescent.

The idea is reasonably simple, especially if we think about selection involving only two alleles.21 When you start to think about it, you should realize two things pretty quickly:

1. Coalescent events will happen only within each of the two allele classes. If we were to trace the history back far enough, to the point where the mutation leading to a second allele occurred, then there might be coalescence involving the two classesexcept that there wouldn’t be two classes, only one.

2. The allele copies22 within one of the two allele classes will all have the same fitness properties. That means that the genealogy within each allele classs will behave just like the coalescent you’ve already seen.

There are a couple of further complications. The first one is that the probability of a coalescent event between two alleles belonging to an allele class whose frequency is $$p_t$$ is $\frac{\frac{m(m-1)}{2}}{2N_ep_t} \quad .$ If you think about it a bit, that may look reasonably familiar. If it doesn’t, look back at equation ([eq:multi-allele-first-event]). All we’ve done is to reduce the effective size of the population by a factor $$p_t$$, which is the fraction of total allele copies that belong to the allele class we’re focusing on.

The second complication is hidden in the first one. Notice that subscript on $$p_t$$. Since we’re assuming that natural selection is going on, we expect the allele frequencies to change over time. This is where the mathematics get really complicated. Since the population is finite, we can’t simply calculate the trajectory. We have to simulate it. That’s OK because when applying coalescent ideas to make inferences from data, we’re always simulating anyway. It’s just that simulating a sample when there is selection is a bit more complicated.

1. We first simulate the allele frequency trajectory, typically using our estimate of the current allele frequency as a starting point.

2. Then we simulate the coalescent history within each allele class.

3. The result is a structured coalescent sample that we can use for further analyses. We’ll talk more about how to use these simulated samples when we get to phylogeography.