Genetic drift and Wright's formula

When studying small populations it is essential to introduce the phenomenon of genetic drift. It was Sewall Wright (1931) who pointed out that in a finite population there is a decreasing trend of heterozygosity $H$ from one generation to the next. In particular, there can be loss of one allele and fixation of the other even if alleles are mutually neutral, that is even if one gene does not provide any demographic advantage over the other and thus natural selection is not operating. Fig. 17 shows, through random simulations, that the fate of two genes undergoing no selection and initially present with equal frequencies ( $p_0 = \frac {1}{2}$) in a population of 10 individuals can be completely different depending on the sequence of random events. Note that, after a sufficient number of generations, the heterozygosity of the population is reduced to zero. The Hardy-Weinberg law, instead, would predict that after a sufficient number of generations the heterozygosity would necessarily converge to $2p_0(1-p_0) = \frac {1}{2}$.

Figure 17: The effect of genetic drift in a small population consisting of 10 individuals. The two random simulations show that there can occur both the loss of gene A with the consequent fixation of gene a (black circles, after 30 generations) and the fixation of A with the consequent loss of a (white squares, after 13 generations). The two genes are initially present with equal frequency.

To analyse the phenomenon of genetic drift from a theoretical viewpoint, we again consider a population that meets all the assumptions leading to the Hardy-Weinberg law except the one about the size of the adult population, i.e. it is no longer considered to be very large. It is easier to derive how heterozygosity varies in zygotes rather than in adults by using a simplified scheme like the one described in Fig. 18. In agreement with what occurs very frequently in nature, suppose that a finite number of adults indicated by $N$ has nevertheless a large number of gametes (phase 1 in the figure) which join randomly forming an equally large number of zygotes (phase 2). However, for reasons related to resource availability and/or the characteristics of the territory in question, it is realistic to assume that at generation $t$ only a finite number $N_{t}$ of individuals survives (phase 3 in the figure).

Figure 18: Scheme for the calculation of heterozygosity variation in a finite semelparous population.

Suppose that in the large pool of gametes that eventually will give rise to the adults at generation $t$, there is a fraction $p_t$ of type A alleles. The gametes join two by two, to form the zygotes which will then become adult. Since the union of gametes is random and both gametes and zygotes are in large number the Hardy-Weinberg law applies and one can deduce that the allele frequency in the zygotes will again be $p_t$, while the frequency of heterozygotes will be ${H_t} = 2{p_t}(1 - {p_t})$. As previously stated, only a finite number $N_t$ of zygotes, from among the large number of those generated, will survive to adulthood. Since no natural selection is operating according to our assumptions, different genotypes have equal survival probability. We wonder which number $j$ of type A genes will be present in the adults. As the process of survival is random, this is equivalent to tossing up an allele $2N_t$ times with the allele A having a probability $p_t$ to be chosen, because $p_t$ is the frequency of the gene in the pool of zygotes. Therefore the probability distribution $P (j)$ of $j$ is a binomial, namely

$$P(j) = \left( {\begin{array}{*{20}{c}} {2{N_t}}\\ j \end{arr... ...rac{{(2{N_t})!}}{{j!(2{N_t} - j)!}}p_t^j{\left(1 - {p_t}\right)^{2{N_t} - j}}.$$

Recall that the average value (indicated hereafter with E$[\cdot]$), the variance (indicated with Var$[\cdot]$) and the second moment of a binomial random variable are given by

$$\begin{array}{rcl} \text{E}\left[ j \right] &=& 2{N_t}{p_t}\\ ... ...ft[ j \right]} \right)^2} = 2{N_t}{p_t}(1 - {p_t}) + 4N_t^2p_t^2 . \end{array}$$

In particular, we can also calculate the mean, the variance and the second moment of the allele frequency in adults, that is the random variable $\frac {j}{2 N_t}$. Of course, it turns out

$$\left[ {\frac{j}{{2{N_t}}}} \right] = \frac{{\text{E}\left[ j \right]}}{{2{N_t}}} = {p_t}$$ (2.1)

$$~ ~ ~ \notag$$ (2.2)

$$\left[ {\frac{j}{{2{N_t}}}} \right] = \frac{{\text{Var}\left[ j \right]}}{{4N_t^2}} = \frac{1}{{2{N_t}}}{p_t}(1 - {p_t})$$ (2.3)

$$~ ~ ~ \notag$$ (2.4)

$$\left[ {{{\left( {\frac{j}{{2{N_t}}}} \right)}^2}} \right] = \frac{{\text{E}\left[ {{j^2}} \right]}}{{4N_t^2}} = \frac{{\text{... ...\right]} \right)}^2}}}{{4N_t^2}} = \frac{1}{{2{N_t}}}{p_t}(1 - {p_t}) + p_t^2 .$$ (2.5)

When adults reproduce, the allele frequency in the gametes, which are large in numbers, will still be $\frac {j}{2 N_t}$. On the other hand, when the gametes of generation $t + 1$ randomly join to form a large number of zygotes, the frequency of heterozygotes $H_{t+1}$ will follow the Hardy -Weinberg law, that is

$${H_{t + 1}} = 2\frac{j}{{2{N_t}}}\left( {1 - \frac{j}{{2{N_t}}}} \right) .$$

Since $j$ is a random variable, the frequency of heterozygotes is random too. We can compute the expected frequency of heterozygotes at generation $t + 1$, using the relationships (1-3):

$$\begin{array}{rcl} \text{E}\left[ {{H_{t + 1}}} \right] & = &... ...}} \right)\left( {1 - \frac{1}{{2{N_t}}}} \right) . \end{array}$$

As $H_t=2p_t\left(1-p_t\right)$, then the resulting Wright's formula for genetic drift is

   E$$\left[ {{H_{t + 1}}} \right] = {H_t}\left( {1 - \frac{1}{{2{N_t}}}} \right) .$$

In other words, the average value of the frequency of heterozygotes decreases from one generation to the next by a factor $1 - \frac {1}{{2{N_t}}}$. Quite evidently, the smaller the population the more important this factor is. A very similar formula can be obtained considering the frequency of heterozygotes in the adult population. More precisely, since the heterozygosity of the adults at generation $t$ is a random variable too, it turns out

$$\left[ {{H_{t + 1}}} \right] = \left[ {{H_t}} \right]\left( {1 - \frac{1}{{2{N_t}}}} \right).$$ (2.6)

If the population is not made up of hermaphrodites or by an equal number of males and females, all reproductive, Wright's formula still holds provided the total number of individuals is replaced by the so-called effective population size $N_{e,t}$, which is the number of individuals that actually breed in the $t$-th breeding season. In very many populations, $N_{e,t}$ may be only a small fraction of the total population size $N_t$. If we denote by $\overline {H _t}$ the average heterozygosity at time $t$ the following formula holds

$${\overline H _{t + 1}} = {\overline H _t}\left( {1 - \frac{1}{{2{N_{e,t}}}}} \right).$$

Often, populations are considered that are at demographic equilibrium, namely in which $N_{e,t} = N_e =$   constant. In this case it turns out

$${\overline H _t} = {\overline H _0}{\left( {1 - \frac{1}{{2{N_e}}}} \right)^t}.$$

Fig. 19 graphically shows that in populations with effective size of 5-10 individuals the heterozygosity loss over 10 generations is between 40% and 60%. Populations with effective size higher than 50-100 individuals lose less than 1% of their heterozygosity from one generation to the next. After 10 generations, a population of 50 individuals still keeps 90% of its initial heterozygosity.

Figure 19: Loss of heterozygosity (ratio between the percentage $H_t$ of heterozygotes in the $t$-th generation, and the initial percentage $H_0$ of heterozygotes) vs. time as a function of different effective population sizes $N_e$, assumed to be constant over time.

It is obvious, but important, to note that the level of heterozygosity in a population is anyway linked to the initial heterozygosity ${\overline H _0}$. This simple remark is the essence of the so-called founder effect. It occurs whenever a few individuals leave a large population to create a new one, by immigrating to a new suitable habitat. These founders, by chance, may not be representative of the genetic variability of the whole species. The initial heterozygosity can thus be very low and genetic drift further depresses the genetic variability in the new population. One famous case is that of the lion population inhabiting the Ngorongoro crater in Tanzania (Primack, 2000). In 1962, the original population was decimated by an explosion of blood-sucking insects and reduced to 9 females and 1 male. Two years later another 7 males immigrated bringing the number of founders of the new population to 17. Compared to the nearby much larger lion population in the Serengeti, Ngorongoro's lions display a very low genetic variability which is reflected in their difficulty to grow demographically: the population after reaching a size of more than 100 individuals was then reduced to about 40 individuals in 2000.

One of the causes of reduced demographic growth in small populations with little genetic variability is the so-called inbreeding, namely the phenomenon of mating between related individuals (mother with son, uncle with niece, self-pollination in hermaphrodite flowers). We previously stated that there exists biological mechanisms that hinder inbreeding, such as blooming of male and female flowers at different times or the emigration of young males far away from their family. In small populations, though, these mechanisms are no longer so effective, for example due to the lack of reproductive partners that are not consanguineous. Therefore, there is a higher likelihood for the fixation of recessive deleterious alleles, which make offspring little viable or infertile. Considering again the example of Ngorongoro's lions, it is worth remarking that the population experienced, in addition to the founder effect, the phenomenon of inbreeding: as a result sperm abnormalities were observed in many males (Fig. 20) with a consequent decrease in fertility.

The effective population size is crucially influenced by the sex ratio and the mating mode. We denote by $N_m$ and $N_f$ the number of males and females in the population, so that $N_m + N_f = N$. If we are considering a monogamous species (like e.g. many birds) with a different number of males and females, evidently the number of pairs that are formed is determined by the sex represented by the smaller number of individuals, or

$${N_e} = 2\min ({N_m},{N_f}).$$

If instead we are considering a polygamous species, there can be different situations: sometimes a male monopolises most females (polygyny, like e.g. in the elephant seal and many marine mammals of big size), while more rarely a female monopolises most males (polyandry, for example in some monkeys). In many cases, on the contrary, it is reasonable to assume that mating is random: a good model is the one in which the probability for a female (respectively for a male) to mate increases linearly with the percentage of males (respectively females) in the population and is equal to 1 when the sex ratio is 1:1. It follows that

$${N_e} = {N_f}\frac{{2{N_m}}}{{{N_m} + {N_f}}} + {N_m}\frac{{2{N_f}}}{{{N_m} + {N_f}}} = \frac{{4{N_f}{N_m}}}{{{N_m} + {N_f}}}.$$

The loss of genetic diversity due to random drift can be countered by mutations and gene flow, phenomena that we assumed to be absent in our simplifying assumptions. The mutation rate is, however, very small, as we have seen. Quantitative analyses, not reported here, show that in populations with fewer than 100 individuals mutations cannot at all counteract genetic drift in a significant way. Much more effective is the phenomenon of gene flow (due to migration between different populations of the same species). Just one new individual per generation immigrating into a population is sufficient to effectively counter genetic drift. Immigration of 5-10 individuals per generation make the effect of drift negligible. One should not forget, however, that the reduction in the number of individuals in a population is often accompanied by the fragmentation and insularisation of habitats (see the later chapter on metapopulations). Populations become not only smaller but also more and more isolated, which greatly reduces the likelihood of gene flow.

Often, the size of many populations fluctuate over time for reasons unrelated to the genetic deterioration but due to exogenous causes, such as changes in resources availability or climatic conditions. In such a case the trend of heterozygosity in time is given by

$${\overline H _t} = {\overline H _0}\left( {1 - \frac{1}{{2{N_{e,0... ...1}{{2{N_{e,1}}}}} \right)...\left( {1 - \frac{1}{{2{N_{e,t - 1}}}}} \right) .$$

Therefore, if for some reason the effective population size is very small in a given year, Wright's factor is particularly small and the frequency of heterozygotes is drastically reduced, thus affecting the heterozygosity in all subsequent generations. This phenomenon is called a bottleneck effect.

Figure 20: Abnormalities in Ngorongoro lions' sperm. (A) normal spermatozoon, (B) two-headed spermatozoon, (C) spermatozoon with nonfunctional flagellum (from Primack, 2000).