Simulations in the previous chapter revealed complex evolutionary responses to selection. Contrary to common beliefs, selection does not always drive beneficial alleles to fixation; selection can maintain allele frequencies at intermediate equilibria, or even cause fixation of alleles that confer a fitness cost. While the outcomes of simulations align well with empirical studies of selection, the mathematical models employed in Chapter 5 made some critical assumptions that may not hold up in natural populations: we assumed that there was no mutation, an infinite population size, a single population that is not connected to others, and random mating among all genotypes. All of these assumptions relate to other evolutionary forces that can bias the frequency of particular genotypes and skew allele frequencies across generations. In this chapter, we explore how the different evolutionary forces impact allele frequencies in populations, and how they interact with selection in natural settings.
Mutation: The Force Creating Novelty
Mutations provide the raw material upon which selection can act [Chapter 4]. At any given locus, mutation can cause transitions between alleles [A1 to A2, or vice versa], or introduce a new allele [A3]. Despite the critical importance of mutation to evolutionary processes, mutation by itself is a weak evolutionary force. Because mutations rates are low, mutation at any locus only causes minute changes in allele frequency across generations.
Nonetheless, there are important interactions between mutation and selection—especially in terms of the persistence of deleterious alleles in a population. In absence of mutation, selection keeps a recessive deleterious gene at a very low frequency [black line in Figure 6.1]. But as selection removes deleterious alleles in every generation, mutation continuously reintroduces them. When the rate of elimination of deleterious alleles is equal to the rate of mutation, the frequency of an allele is at an equilibrium, called the mutation-selection balance. Assuming a dominant-recessive mode of inheritance [wAA = wAa], the frequency of a deleterious allele at equilibrium is given by the mutation rate [𝜇] and the strength of selection [s]:
\[\begin{align} q=\sqrt{𝜇/s} \tag{6.1} \end{align}\]
Consequently, the frequency of a deleterious allele depends on its mutation rate and the strength of selection against it. The equilibrium frequency of a deleterious allele increases with increasing mutation rate or with decreasing strength of selection, as illustrated in Figure 6.1.
Figure 6.1: In absence of mutation, selection maintains a recessive deleterious allele at low frequency [black line]. As mutation rate increases, the equilibrium frequency of the deleterious allele increases [blue: mu=0.001; green: mu=0.01; orange: mu=0.1]. The strength of selection was 0.5 for all simulations.
Note that Equation [6.1] can be used to make inferences about mutation rates, when equilibrium allele frequencies and the strength of selection are known [because 𝜇=q2*s]. Accordingly, the principle of mutation-selection balance is an important null model that describes the relationship between selection, mutation, and allele frequencies, and it can be applied to study the prevalence of heritable diseases [see Reflection Questions for a case study on cystic fibrosis].
Genetic Drift: The Random Force
Models of selection are completely deterministic because they assume infinite population sizes. No matter how many times you run a simulation with the same parameters, you will always get exactly the same result. In reality, however, population sizes are finite. While many species on the planet do indeed have very large populations [in the order of millions and billions and even trillions], others are comparatively rare. Some rare species have naturally low populations sizes with historically restricted distributions [Figure 6.2]; others have declined in recent decades due to anthropogenic environmental change. Species with small or declining population sizes are the focus of conservation biology, which applies evolutionary principles to develop strategies for population management.
Figure 6.2: The Devils Hole pupfish [Cyprinodon diabolis] is one of the rarest vertebrates on the planet. The species is endemic to the tiny Devil’s Hole well, which is located within the Ash Meadows National Wildlife Refuge, Nevada. Since the start of population surveys, the maximum population size recorded was 553, and the lowest population size was just 38 individuals in 2006. Photo by Olin Feuerbacher, CC BY 2.0.
When population sizes are finite—and especially when they are small—random chance affects evolutionary dynamics. These changes in allele frequencies across generations caused by random events are called genetic drift. While selection is differential reproductive success caused by differential performance of variants, genetic drift is differential reproductive success that just happens. In contrast to selection, which tends to increase average fitness across generations, genetic drift does not lead to adaptation. Due to the random nature of genetic drift, populations subject to it evolve on distinct trajectories. So, if you re-run simulations allowing for drift with the same parameters, you will get a unique evolutionary path every single time. The random nature of drift assures that no evolutionary trajectory is like another.
At the most basic level, evolution by genetic drift happens as a consequence of sampling error across generations. If you imagine a locus with two alleles [A and a] of equal frequency, the theoretically predicted allele frequencies under Hardy-Weinberg conditions in the next generation are of course
p=q=0.5. However, chance might cause significant deviations from theoretical expectations in reality. Every individual in the population essentially has a 50 % chance to inherit the A-allele on its first chromosome, and a 50% chance to inherit the A-allele on its second chromosome. In other words, the genotype an individual inherits in absence of selection is equivalent to two coin tosses [one for each allele inherited], where the probability for receiving a
particular genotype is dependent on the allele frequencies in the population. We can simulate this in R using the rbinom[] function, with the allele frequency [p] and the population size [N] as input variables. So for the Devil’s Hole pupfish [Figure 6.2], with its low-bound population size of 38, the simulated
allele frequency in the next generation is:
N=38
p=0.5
rbinom[1, size=2*N, prob=p]/[2*N]## [1] 0.5526316If we repeat this simulation 1,000 times, you can see that there can be substantial deviations from the predicted allele frequency of p=0.5 [Figure 6.3]. Only about 10 % of observations fall within the predicted 0.5-bin, and the frequency of A can be as low as 0.3 and as high as 0.7 just because of random chance. That is a massive shift in allele frequency across a single generation.
As you know from experience, the number of coin tosses impacts how close a result matches the theoretical predictions. If you toss your coin ten times, you may get tails eight times, which represents a 60 % deviation from the theoretical prediction. However, the more often you toss your coin, the closer your overall frequency of tails will get to the predicted 50 %. The same principle applies to the effects of genetic drift as a function of population size. When population sizes are small, genetic drift can induce substantial deviations from theoretical predictions, but the effects of drift get smaller as populations size increases. Using the same simulation as above—but with a population size of 1,000—reveals that observed allele frequencies align much better with theoretical predictions, with a spread of observed allele frequencies of A between 0.46 and 0.54 [Figure 6.3].
So how exactly does population size impact the strength of genetic drift? If a new mutation arises in a population of diploid organisms with a population size of N, the frequency of the new allele is 1/2N. Each neutral allele has the same chance of drifting to fixation, which is equal to the allele frequency. Hence, the likelihood that the a new allele gets fixed in a population is 1/2N. Correspondingly, novel alleles are more likely to get fixed by chance in small populations.
Figure 6.3: Observed distributions of allele frequencies by randomly selecting alleles [A or a] from a pool with equal allele frequencies [p=0.5]. The deviation from theoretical expectations are much larger for the small population [N=38] than for the larger population [N=1,000]. This illustrates how the strenth of genetic drift declines as a function of population size.
Effective Population Size
The total population size [census population size] in natural populations is not the same as the effective population size [Ne], which is the size of the breeding population. Effective population size takes into consideration that many individuals that reach adulthood never breed in natural populations. Consequently, effective population size is almost always smaller than the census population size. Effective population size is particularly impacted by deviations from 1:1 sex ratios. In such cases, effective population size can be estimated as
\[\begin{align} N_e = \frac{4N_mN_f}{N_m+N_f} \tag{6.2} \end{align}\]
where Nm is the number of males and Nf the number of females. If you assume census population of 100 with equal sex ratio, Ne is 100. If you assume a sex ratio of 1:9, Ne drops to just 36. Distinguishing between N and Ne is important for conservation biology and many population genetic analyses related to genetic drift and inbreeding. For example, when Ne is significantly smaller than N, the probability of fixation of an allele in response to drift can be much higher than estimated by census population sizes.
Besides sampling error, genetic drift can also have profound impacts on allele frequencies when there are rapid reductions in population size. In general, we distinguish between two scenarios: [1] Population bottlenecks occur when catastrophic events [large-scale wild fires, floods, etc.] drastically reduce the size of a population. In such instances, survival is less dependent on individuals’ traits [that would be selection] than individuals being in the right place at the right time. Hence, the allelic composition of the generation after a bottleneck largely reflects a random subsample of the original population. [2] Founder effects occur when a small subset of a population disperses into a new area and founds a new population. In that case, only a random subset of alleles travels along with the founding individuals. Founder effects are particularly important in island populations, where species expand their range in a step-wise fashion along island chains. This can lead to serial founder effects with continuous loss of genetic diversity [Figure 6.4].
Figure 6.4: Allelic richness in populations of monarch butterflies [Danaus plexippus]. The original population from the United States exhibits the highest levels of allelic richness. Allelic richness declined in a step-wise fashion as butterflies first colonized Hawaii and then other islands throughout the Pacific. Data from Pierce et al. [2014].
Interactions Between Drift and Selection
In small populations, genetic drift affects the fate of alleles under selection. Drift can cause deleterious mutations to be more common than expected by selection alone, and it can cause beneficial alleles to disappear from the population. The fate of alleles subject to selection and drift is dependent on the product of 2Nes, as depicted in Figure 6.5. If a new mutation is neutral [s=0], the probability of fixation is 1/2Ne [𝛌=1; dotted line], as described above. If the new mutation is deleterious [s1], then the likelihood of fixation become greater than what is expected by chance. For example, for 2Nes=5, the likelihood of fixation for the new mutation is 5 times higher than what would be expected by chance.
The important point here is that the likelihood of fixation is dependent on both the strength of selection and the effective population size. 2Nes can be large when selection is strong or when populations are large. When populations are small, selection on novel alleles needs to be comparatively strong for them to have a high likelihood of fixation. If a mutation only provides a minor fitness benefit, drift might cause its loss from the population before it ever has a chance to become common. Conversely, when population sizes are very large, novel alleles with minute fitness benefits can have a high likelihood of fixation. As a rule of thumb, evolution of novel alleles is primarily governed by genetic drift for 2Nes-values between -1 and 1 [gray shaded area in 6.5]. Beyond that range, selection has the upper hand.
Figure 6.5: The likelihood for fixation of a novel allele increases with increasing values for Ne * s [Charlesworth 2009]. The dotted line indicates perfect neutrality; the gray-shaded area corresponds to Ne * s values for which novel mutations evolve largely by genetic drift. Note that the relationship depicted assumes N = Ne.
In case you are interested in the math underlying Figure 6.5: The probability [Q] that a new mutation spreads in a population and eventually becomes fixed is dependent on the effective population size [Ne], the census population size [N], and the selection coefficient [s]. 𝛌 is the fixation probability relative to neutral evolution [1/2Ne].
\[\begin{align} Q = \frac{N_es}{N} \frac{1}{1-exp[-2N_es]} \tag{6.3} \\ 𝛌 = \frac{Q}{1/[2N_e]} \tag{6.4} \end{align}\]
Note that this relationship assumes that the fitness of the heterozygotes is intermediate between the two homozygotes:
\[\begin{align} w_{Aa} = \frac{w_{AA}+w_{aa}}{2} \tag{6.5} \end{align}\]
Migration: The Homogenizing Force
Our view of evolutionary processes so far has assumed that populations are relatively homogenous, with random mating among all individuals contained within [i.e., panmixia]. More often than not, however, species consist of many populations that inhabit suitable habitat patches and are separated by less favorable environmental conditions [Figure 6.6A-B]. Such partial isolation can cause differentiation among populations, either because genetic drift impacts allele frequencies differently across populations, or because variation in environmental conditions among populations favors different genotypes. But despite some degree of isolation, populations within a species are typically connected through migration. Migration can be unidirectional or bidirectional, and can vary in strength [i.e., the number of migrating individuals relative to the population size]. Migration rates are typically higher between proximate populations than between populations that are far apart—a phenomenon known as isolation by distance.
Definition: Gene Flow
Population geneticists usually refer to migration as “gene flow”. Gene flow is simply the transfer of genetic material among populations.
Figure 6.6: A. Species are often assumed to be relatively homogenous units with panmixia. B. However, species can also consist of structured populations that are somewhat differentiated but still connected by migration. Variation in color indicates population differentiation; arrows represent patterns of migration among populations. C. Schematic of the one-island migration model.
Migration is an evolutionary force because it can impact the genetic composition of populations. Migrants may carry novel alleles from one population to another, acting similar to mutation in terms of introducing new genetic variation. Even in absence of novel alleles, migration between differentiated population causes changes in allele frequencies. In the absence of other evolutionary forces, it homogenizes the genetic composition of different populations. To illustrate this, let’s consider a simple scenario known as the one-island migration model [Figure 6.6C]. The model assumes two populations: a large mainland population and a small island population. Even if the number of individuals migrating in either direction is the same, the input of island individuals arriving in the mainland population is negligible because of its large size. In contrast, because of the small island population, individuals from the mainland arriving on the island can significantly impact allele frequencies, if allele frequencies differ between populations. In this case, island allele frequencies after a migration event [pi’] can be described as a function of island allele frequencies before a migration event [pi], mainland allele frequencies [pm], and the migration rate [m]:
\[\begin{align} p_i' = [1-m]p_i+mp_m \tag{6.6} \\ 𝚫p_i=p_i'-p_i=m[p_m-p_i] \tag{6.7} \end{align}\]
Applying Equation [6.6] and calculating island allele frequencies across multiple generations reveals the genetic effect of migration [Figure 6.7]: migration from the mainland to the island changes pi until it is equal to pm, and the rate of migration dictates the speed at which this conversion happens. In other words, migration homogenizes the allele frequencies across populations.
One-Island Migration Model
If you want to conduct your own simulations of migration using the one-island model, you can use the code adopted from Dyer [2017] displayed below. You can vary migration rates [migration_rates; numbers between 0 and 1] as well as the starting allele frequencies on the mainland [pm0] and the island [pi0].
migration_rates