Effective population size

Selection effective size

In an idealised Wright-Fisher model, the fate of an allele, beginning at an intermediate frequency, is largely determined by selection if the selection coefficient s ≫ 1/N, and largely determined by neutral genetic drift if s ≪ 1/N. In real populations, the cutoff value of s may depend instead on local recombination rates. This limit to selection in a real population may be captured in a toy Wright-Fisher simulation through the appropriate choice of Ne.

The ability of a species to differentiate between nearly neutral alleles can be measured by how codon bias differs from neutral expectations. The Ka/Ks ratio is also sometimes used as a proxy.

The drift-barrier hypothesis claims that populations with different selection effective population sizes are predicted to evolve profoundly different genome architectures.

History of theory

Ronald Fisher and Sewall Wright originally defined effective population size as "the number of breeding individuals in an idealised population that would show the same amount of dispersion of allele frequencies under random genetic drift or the same amount of inbreeding as the population under consideration". This implied two potentially different effective population sizes, based either on the one-generation increase in variance across replicate populations (variance effective population size), or on the one-generation change in the inbreeding coefficient (inbreeding effective population size). These two are closely linked, and derived from F-statistics, but they are not identical.

Today, the effective population size is usually estimated empirically with respect to the amount of within-species genetic diversity divided by the mutation rate, yielding a coalescent effective population size that reflects the cumulative effects of genetic drift, background selection, and genetic hitchhiking over longer time periods. Another important effective population size is the selection effective population size 1/s_critical, where s_critical is the critical value of the selection coefficient at which selection becomes more important than genetic drift.

Variance effective size

In the Wright-Fisher idealized population model, the conditional variance of the allele frequency ${\displaystyle p^{\prime }}$, given the allele frequency ${\displaystyle p}$ in the previous generation, is

${\displaystyle \mathrm{var}(p^{\prime }\mid p)=\frac{p(1-p)}{2N}.}$

Let ${\displaystyle \widehat{\mathrm{var}}(p^{\prime }\mid p)}$ denote the same, typically larger, variance in the actual population under consideration. The variance effective population size ${\displaystyle N_e^{(v)}}$ is defined as the size of an idealized population with the same variance. This is found by substituting ${\displaystyle \widehat{\mathrm{var}}(p^{\prime }\mid p)}$ for ${\displaystyle \mathrm{var}(p^{\prime }\mid p)}$ and solving for ${\displaystyle N}$ which gives

${\displaystyle N_e^{(v)}=\frac{p(1-p)}{2\widehat{\mathrm{var}}(p)}.}$

In the following examples, one or more of the assumptions of a strictly idealised population are relaxed, while other assumptions are retained. The variance effective population size of the more relaxed population model is then calculated with respect to the strict model.

Variations in population size

Population size varies over time. Suppose there are t non-overlapping generations, then effective population size is given by the harmonic mean of the population sizes:

${\displaystyle \frac{1}{N_e}=\frac{1}{t}\sum _{i=1}^t\frac{1}{N_i}}$

For example, say the population size was N = 10, 100, 50, 80, 20, 500 for six generations (t = 6). Then the effective population size is the harmonic mean of these, giving:

Note this is less than the arithmetic mean of the population size, which in this example is 126.7. The harmonic mean tends to be dominated by the smallest bottleneck that the population goes through.

Dioeciousness

If a population is dioecious, i.e. there is no self-fertilisation then

${\displaystyle N_e=N+\begin{array}{c}\frac{1}{2}\end{array}}$

or more generally,

${\displaystyle N_e=N+\begin{array}{c}\frac{D}{2}\end{array}}$

where D represents dioeciousness and may take the value 0 (for not dioecious) or 1 for dioecious.

When N is large, N_e approximately equals N, so this is usually trivial and often ignored:

${\displaystyle N_e=N+\begin{array}{c}\frac{1}{2}\approx N\end{array}}$

Variance in reproductive success

If population size is to remain constant, each individual must contribute on average two gametes to the next generation. An idealized population assumes that this follows a Poisson distribution so that the variance of the number of gametes contributed, k is equal to the mean number contributed, i.e. 2:

${\displaystyle \mathrm{var}(k)=\overline{k}=2.}$

However, in natural populations the variance is often larger than this. The vast majority of individuals may have no offspring, and the next generation stems only from a small number of individuals, so

${\displaystyle \mathrm{var}(k)>2.}$

The effective population size is then smaller, and given by:

${\displaystyle N_e^{(v)}=\frac{4N-2D}{2+\mathrm{var}(k)}}$

Note that if the variance of k is less than 2, N_e is greater than N. In the extreme case of a population experiencing no variation in family size, in a laboratory population in which the number of offspring is artificially controlled, V_k = 0 and N_e = 2N.

Non-Fisherian sex-ratios

When the sex ratio of a population varies from the Fisherian 1:1 ratio, effective population size is given by:

${\displaystyle N_e^{(v)}=N_e^{(F)}=\frac{4N_m{N}_f}{N_m+N_f}}$

Where N_m is the number of males and N_f the number of females. For example, with 80 males and 20 females (an absolute population size of 100):

Again, this results in N_e being less than N.

Inbreeding effective size

Alternatively, the effective population size may be defined by noting how the average inbreeding coefficient changes from one generation to the next, and then defining N_e as the size of the idealized population that has the same change in average inbreeding coefficient as the population under consideration. The presentation follows Kempthorne (1957).

For the idealized population, the inbreeding coefficients follow the recurrence equation

${\displaystyle F_t=\frac{1}{N}\left(\frac{1+F_{t-2}}{2}\right)+\left(1-\frac{1}{N}\right)F_{t-1}.}$

Using Panmictic Index (1 − F) instead of inbreeding coefficient, we get the approximate recurrence equation

${\displaystyle 1-F_t=P_t=P_0{\left(1-\frac{1}{2N}\right)}^t.}$

The difference per generation is

${\displaystyle \frac{P_{t+1}}{P_t}=1-\frac{1}{2N}.}$

The inbreeding effective size can be found by solving

${\displaystyle \frac{P_{t+1}}{P_t}=1-\frac{1}{2N_e^{(F)}}.}$

This is

${\displaystyle N_e^{(F)}=\frac{1}{2\left(1-\frac{P_{t+1}}{P_t}\right)}}$.

Theory of overlapping generations and age-structured populations

When organisms live longer than one breeding season, effective population sizes have to take into account the life tables for the species.

Haploid

Assume a haploid population with discrete age structure. An example might be an organism that can survive several discrete breeding seasons. Further, define the following age structure characteristics:

${\displaystyle v_i=}$ Fisher's reproductive value for age ${\displaystyle i}$,

${\displaystyle {\ell }_i=}$ The chance an individual will survive to age ${\displaystyle i}$, and

${\displaystyle N_0=}$ The number of newborn individuals per breeding season.

The generation time is calculated as

${\displaystyle T=\sum _{i=0}^{\mathrm{\infty }}{\ell }_i{v}_i=}$ average age of a reproducing individual

Then, the inbreeding effective population size is

${\displaystyle N_e^{(F)}=\frac{N_{0}T}{1+\sum _i{\ell }_{i+1}^2{v}_{i+1}^2(\frac{1}{{\ell }_{i+1}}-\frac{1}{{\ell }_i})}.}$

Diploid

Similarly, the inbreeding effective number can be calculated for a diploid population with discrete age structure. This was first given by Johnson, but the notation more closely resembles Emigh and Pollak.

Assume the same basic parameters for the life table as given for the haploid case, but distinguishing between male and female, such as N₀^ƒ and N₀^m for the number of newborn females and males, respectively (notice lower case ƒ for females, compared to upper case F for inbreeding).

The inbreeding effective number is

${\displaystyle \begin{array}{cc}\frac{1}{N_e^{(F)}}=\frac{1}{4T}\{\frac{1}{N_0^f}+\frac{1}{N_0^m}+\sum _i{\left({\ell }_{i+1}^f\right)}^2{\left(v_{i+1}^f\right)}^2\left(\frac{1}{{\ell }_{i+1}^f}-\frac{1}{{\ell }_i^f}\right)& \\ +\sum _i{\left({\ell }_{i+1}^m\right)}^2{\left(v_{i+1}^m\right)}^2\left(\frac{1}{{\ell }_{i+1}^m}-\frac{1}{{\ell }_i^m}\right)\}.& \end{array}}$