What is the change of allele frequency over time?

Genetic Drift

Chance events can have a much greater effect on allele frequencies in a small population than in a large one. For example, when a new mutation occurs in a small population, its frequency is represented by only one copy among all the copies of that gene in the population. Random effects of environment or other chance occurrences that areindependent of the genotype [i.e., events thatoccur for reasons unrelated to whether an individual is carrying the mutant allele] can produce significant changes in the frequency of the disease allele when the population is small. Such chance occurrences disrupt Hardy-Weinberg equilibrium and cause the allele frequency to change from one generation to the next. This phenomenon, known asgenetic drift, can explain how allele frequencies can change as a result of chance. During the next few generations, although the population size of the new group remains small, there may be considerable fluctuation in gene frequency until allele frequencies come to a new equilibrium as the population increases in size. In contrast togene flow [see next section], in which allele frequencies change because of the mixing of previously distinct populations, the mechanism of genetic drift is simply chance operating on a small population.

Genetic Drift and Evolutionary Theory

Genetic drift is at the core of the shifting-balance theory of evolution coined by Sewall Wright where it is part of a two-phase process of adaptation of a subdivided population. In the first phase, genetic drift causes each subdivision to undergo a random walk in allele frequencies to explore new combinations of genes. In the second phase, a new favorable combination of alleles is fixed in the subpopulation by natural selection and is exported to other demes by factors like migration between populations. Much of the basic theory of genetic drift was developed in the context of understanding the shifting-balance theory of evolution. Genetic drift has also a fundamental role in the neutral theory of molecular evolution proposed by the population geneticist Motoo Kimura. In this theory, most of the genetic variation in DNA and protein sequences is explained by a balance between mutation and genetic drift. Mutation slowly creates new allelic variation in DNA and proteins, and genetic drift slowly eliminates this variability, thereby achieving a steady state. A fundamental prediction of genetic drift theory is that the substitution rate in genes is constant, and equal to the mutation rate.

Differentiation Between Genetic and Environmental Factors in Cancer

For many cancers, distinguishing between genetic and environmental causative factors is not obvious. There is usually neither a clear-cut mode of inheritance nor a clearly identified environmental cause. Historically, evidence to help distinguish environmental and genetic factors came from a combination of epidemiology, family and twin studies, disease associations and viral factors, all of which are considered briefly here. Increasingly, in the modern era, molecular analysis and/or DNA tumor profiling provide further evidence, and these are considered later.

The Fate of a Newly Arisen Mutation in a Large Population

We first examine the impact of genetic drift on the evolutionary fate of a newly arisen mutation. Let A symbolize the group of all the old alleles at an autosomal locus, and let a be a newly arisen mutation at this locus that is initially present in only a single individual with the new genotype Aa. We initially regard this individual as a self-compatible, random-mating hermaphrodite [Hardy's assumptions for the Hardy–Weinberg law] with normal meiosis and no subsequent mutations producing new a alleles; that is, the a allele is unique in its mutational origin. The first step in the survival of this new mutant allele is to be passed on to a gamete during meiosis, whose pgf has already been given in Eq. [4.1]. The chances for a surviving to the next generation also depend upon how many offspring the initial carrier, Aa, has. Suppose that the initial Aa carrier has n offspring. Then, the pgf for the total number of a alleles this individual passes on to the next generation is:

[4.5]h[z|n]=∏j=1ngj[z]=[g[z]]n

where gj[z] is the pgf for the meiotic event associated with offspring j. Eq. [4.5] reflects the fact that all meioses are independent events with the same pgf, g[z]. The problem with Eq. [4.5] is that it assumes that we know n, the number of offspring born to the initial Aa individual that, in this simple model, survive to adulthood in the next generation. At this point we encounter another level of sampling that can contribute to genetic drift at the population level—not all individuals in general will have exactly the same number of surviving offspring even if the environment is constant and every offspring has the same probability of surviving. The random sampling of the number of surviving offspring produced by an individual can also be described by a series of probabilities, say pn, that represent the probability of having n surviving offspring. Eq. [4.5] is the conditional pgf given n, but now we can define the unconditional pgf as

[4.6]h[z]=∑n=0∞pnh[z|n]=∑n=0∞pn[g[z]]n

Note that if we define a new dummy variable t = g[z], then Eq. [4.6] becomes the pgf for the random variable n, the number of surviving offspring produced by an individual. Hence, the pgf h[g[z]] incorporates the effects of sampling meiotic events and sampling the number of surviving offspring on describing the total number of a alleles that survive into the next generation. For example, let us assume that n is from a Poisson distribution, a commonly used distribution for family size in idealized populations, as mentioned in Chapter 3. The pgf for a Poisson distribution is ek[t−1] where k is the mean number of surviving offspring of an Aa individual and t is the dummy variable. In this special case of Eq. [4.6], the pgf for the number of a alleles in the next generation is

[4.7]ek[g[z]− 1]=ek[12+12z−1]=ek2[z−1]

To find the probability of survival, it is easier to first find the probability of loss; that is, the probability that there are 0 copies of a in the next generation. Recall that this is found simply by setting the dummy variable to 0 to yield the probability of loss of the a allele in the next generation as e−k/2. If the total population size is approximately stable and the a allele is neutral [that is, it has no effect on the probabilities for the number of offspring], each of the individuals, including Aa, in this idealized population has an average of k = 2 offspring, and e−1 = 0.367879. Note that over a third of all new neutral mutants are lost by the very first generation after mutation just by the sampling processes that contribute to genetic drift. The probability of surviving just a single generation is 1-Probability[loss] = 0.632121.

To find the probability of surviving for just two generations, assume that n copies survived into the first generation. Because mating is at random and if we further assume the population is very large, these copies will almost certainly all be in Aa genotypes as the frequency of a is extremely rare [recall the Hardy–Weinberg law]. Under these assumptions, each of the n copies of a that are in Aa individuals will also produce a random number of a copies in the next generation as described by the pgf given in Eq. [4.6]; that is h[z]. Because there are n carriers of a in the first generation, the total pgf for the second generation given n is [h[z]] n. However, n itself is a random variable described by pgf h[t], and we need to incorporate this fact to get the unconditional pgf for the second generation. Exactly like the derivation of Eq. [4.6], the unconditional pgf for the number of a alleles in the second generation is h[h[z]]; that is, the dummy variable for the second generation is the pgf from the first generation. For the Poisson case, the pgf for the second generation is

[4.8]ek2[ek2[z− 1]−1]

Setting z = 0 and k = 2, Eq. [4.8] yields the probability of loss by the second generation to be 0.531464, so the probability of surviving for two generations is 0.468536. Thus, by just two generations, more than half of all new mutant alleles are lost by genetic drift. The recursion used to generate Eq. [4.8] can be repeated multiple times to obtain the pgf's of later generations [Schaffer, 1970]. For example, the pgf for the third generation is h[h[h[z]]]. Table 4.1 shows the probabilities of loss of the mutant allele for the first 10 generations in our idealized population. As can be seen, very few mutants survive even just 10 generations of genetic drift.

Table 4.1. The Probabilities of a New Mutant Surviving Over the First Ten Generations After Its Occurrence as a Function of the Average Number of Offspring Produced by Individuals in the Population

The ultimate probability of survival [ups] can be found by solving the equation h[z] = z for 0 ≤ z ≤ 1, and an approximation to this solution that incorporates the impact of meiosis [Eq. 4.1] is [modified from Schaffer, 1970, which only deals with the haploid case]:

[4.9]u ps≈k−2k+v

where v is the variance in the number of offspring. For the Poisson case, k = v, as mentioned in Chapter 3. Also, if k = 2 as in our example of a neutral allele in a stable population, ups = 0. This of course, is an approximation, and we will see later that the actual probability of survival in our assumed large population is extremely small in a large population but greater than 0.

Humans are unique among the large-bodied vertebrates in that we have had sustained population growth for at least the last 10,000 years with the beginning of agriculture [Coventry et al., 2010]. To consider a growing population, Table 4.1 also presents the survival probabilities for a population in which the average number of surviving offspring per individual is 3. As can be seen, the probability of survival is consistently larger under population growth. Moreover, the approximate ultimate probability of survival is [from Eq. [4.9] with k = v = 3] 0.1667.

Up to now we have assumed that all individuals in the population have the same average number of offspring. However, other than the assumptions that the total population size is large and capable of indefinite growth, the k in our model of offspring number only refers to the average number of offspring by bearers of the new mutant a. Suppose the overall average number of offspring in the growing population were four, then an average size of just three offspring would mean extremely strong natural selection against the Aa individuals bearing the new, mutant allele [a 25% reduction in number of expected offspring in the next generation]. As will be shown in Chapter 9, strong selection against a dominant mutant such as a would result in its rapid elimination when genetic drift and population growth are ignored. As Table 4.1 and the ups of 0.1667 show, even a strongly deleterious dominant allele can persist in the human gene pool. A recessive deleterious allele is even more sheltered against the effects of natural selection [Chapter 9], so such recessive deleterious alleles will have an even higher probability of persistence in the human gene pool. Indeed, deep sequencing studies reveal that humans have many more rare variants that appear deleterious over that expected in a constant-sized population [Coventry et al., 2010]. Recall also from Chapter 3 the large number of rare variants that individual humans carry that are loss-of-function mutations or otherwise predicted to be deleterious [Gudbjartsson et al., 2015]. The accumulation of deleterious mutations in the gene pool is sometimes called the mutational load, and humans have a uniquely high mutational load [Lynch, 2010]. The concept of mutational load was first introduced by Muller [1950], who won the Nobel Prize for his work demonstrating that radiation can increase the mutation rate. Muller was concerned about an increase in radiation levels due to nuclear testing and the threat of nuclear war increasing the mutational load in humans, and Lynch was concerned with mutation rates and relaxed selection. However, population growth, and therefore indirectly agriculture, has played a much more important role in increasing the mutational load in humans. Demography and genetic drift are major evolutionary forces that have strongly shaped the unique nature of the human gene pool with its vast excess of rare, deleterious variants.

Finite Population Size

Genetic drift can cause nonrandom associations between alleles at different loci. While the expected value of pairwise gametic disequilibrium due to drift over many generations is zero, the variance is large for closely linked loci in small populations. The demographic structure of a population will affect the amount of gametic disequilibrium observed. A small founder population or a bottleneck in the recent past can cause significant gametic disequilibrium for closely linked loci. While less gametic disequilibrium will be generated by genetic drift in a rapidly growing population, gametic disequilibrium present before or during the early phase of the expansion will persist.

Abstract

Random genetic drift describes the stochastic fluctuations of allele frequencies due to random sampling in finite populations. Over time, genetic drift can lead to fixation or loss of genetic variants, thereby systematically eliminating diversity from a population. This trend is counterbalanced by mutations that continuously produce new variants. A number of powerful frameworks, such as coalescence theory, have been developed to study how these processes interact in shaping patterns of genetic diversity in populations. Random genetic drift and mutation also lie at the foundation of Kimura's neutral theory of evolution, which constitutes the standard null model of molecular population genetics.

Modifiers of Recombination and Genetic Drift in Large Populations

Genetic drift acts in all populations, and so the stochastic effects of finite population size can play a role in large populations as well. Under Hill–Robertson interference [discussed above], genetic linkage is seen to increase the amount of genetic drift near a selected locus, thus reducing the effective population size for the locus when either a beneficial mutation arises or in the presence of purifying selection against a deleterious allele. Keightley and Otto [2006] contrasted the probability of fixation for an allele modifying recombination with a neutral allele, and showed that purifying selection against repeated deleterious mutations provided an advantage to modifier alleles, causing them to fix with a higher probability. Surprisingly, this effect increased with increasing population size.

To understand this somewhat counter-intuitive result, we note that recombination frees the focal locus from Hill–Robertson interference, allowing deleterious mutations to be purged by selection. A larger number of polymorphic loci increases the opportunity for Hill–Robertson interference, which increases the advantage seen for recombination. Larger populations [where genetic drift is overall weaker] will maintain greater polymorphism, and thus see on average a greater amount of Hill–Robertson interference, and a larger advantage to recombination. The Keightley–Otto model gives a truly synthetic treatment of the role of negative disequilibrium where both selection and drift determine how selection on a new mutation affects the fate of other loci, and recombination frees loci from these shared fates.

Genetic Drift

Genetic drift is the change in allele frequencies in a population over time due to random sampling events [e.g., differences among individuals in survival or fecundity that are unrelated to their phenotype/genotype]. Although the specific genetic consequences of genetic drift during a given demographic bottleneck are unpredictable, the overall effect of drift is to erode genetic diversity.

Effective population size, or Ne, is a measure of how sensitive a population is to genetic drift. Ne is defined as the size of a hypothetical, theoretically ideal population that would experience the same level of inbreeding, loss of heterozygosity, and genetic drift per generation as the real population in question [Kimura and Crow, 1963]. Other factors besides the census size of a population will influence the change in allele frequencies over time [e.g., an uneven sex ratio, past fluctuations in population size, nonrandom variation in family size]; by excluding these factors, Ne makes it possible to evaluate and compare measurements of drift across species with very different life histories. There are different ways to empirically estimate Ne over both short- and long-term time scales [see review by Hare et al., 2011], but Ne is virtually always smaller, and often much smaller, than the census size of a population. Frankham [1995] reviewed published estimates of Ne/N for wildlife species, and found that Ne averaged only 10–11% of total census size.

In large [unthreatened] populations, it takes a long time to see a major effect of genetic drift on allele frequencies; genetic diversity represents a balance between mutation and natural selection. However, when Nes