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The effective population size (N_e) is size of an idealised population would experience the same rate of genetic drift or increase in inbreeding as in the real population. Idealised populations are based on unrealistic but convenient assumptions including random mating, simultaneous birth of each new generation, constant population size. For most quantities of interest and most real populations, N_e is smaller than the census population size N of a real population.^[1] The same population may have multiple effective population sizes for different properties of interest, including genetic drift and inbreeding.

The effective population size is most commonly measured with respect to the coalescence time. In an idealised diploid population with no selection at any locus, the expectation of the coalescence time in generations is equal to twice the census population size. The effective population size is measured as within-species genetic diversity divided by four times the mutation rate ${\displaystyle \mu }$, because in such an idealised population, the heterozygosity is equal to ${\displaystyle 4N\mu }$. In a population with selection at many loci and abundant linkage disequilibrium, the coalescent effective population size may not reflect the census population size at all, or may reflect its logarithm.

The concept of effective population size was introduced in the field of population genetics in 1931 by the American geneticist Sewall Wright.^[2][3]

Overview: Types of effective population size

Depending on the quantity of interest, effective population size can be defined in several ways. Ronald Fisher and Sewall Wright originally defined it 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". More generally, an effective population size may be defined as the number of individuals in an idealised population that has a value of any given population genetic quantity that is equal to the value of that quantity in the population of interest. The two population genetic quantities identified by Wright were the one-generation increase in variance across replicate populations (variance effective population size) and 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.^[4]

Today, the effective population size is usually estimated empirically with respect to the sojourn or coalescence time, estimated as the within-species genetic diversity divided by the mutation rate, yielding a coalescent effective population size.^[5] 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.^[6]

Empirical measurements

In Drosophila populations of census size 16, the variance effective population size has been measured as equal to 11.5.^[7] This measurement was achieved through studying changes in the frequency of a neutral allele from one generation to another in over 100 replicate populations.

For coalescent effective population sizes, a survey of publications on 102 mostly wildlife animal and plant species yielded 192 N_e/N ratios. Seven different estimation methods were used in the surveyed studies. Accordingly, the ratios ranged widely from 10^-6 for Pacific oysters to 0.994 for humans, with an average of 0.34 across the examined species. Based on these data they subsequently estimated more comprehensive ratios, accounting for fluctuations in population size, variance in family size and unequal sex-ratio. These ratios average to only 0.10-0.11.^[8]

A genealogical analysis of human hunter-gatherers (Eskimos) determined the effective-to-census population size ratio for haploid (mitochondrial DNA, Y chromosomal DNA), and diploid (autosomal DNA) loci separately: the ratio of the effective to the census population size was estimated as 0.6–0.7 for autosomal and X-chromosomal DNA, 0.7–0.9 for mitochondrial DNA and 0.5 for Y-chromosomal DNA.^[9]

Variance effective size

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

${\displaystyle \operatorname {var} (p'\mid p)={p(1-p) \over 2N}.}$

Let ${\displaystyle {\widehat {\operatorname {var} }}(p'\mid p)}$ 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 ${\displaystyle {\widehat {\operatorname {var} }}(p'\mid p)}$ for ${\displaystyle \operatorname {var} (p'\mid p)}$ and solving for ${\displaystyle N}$ which gives

${\displaystyle N_{e}^{(v)}={p(1-p) \over 2{\widehat {\operatorname {var} }}(p)}.}$

Theoretical examples

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:^[10]

${\displaystyle {1 \over N_{e}}={1 \over t}\sum _{i=1}^{t}{1 \over N_{i}}}$

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:

${\displaystyle {1 \over N_{e}}}$	${\displaystyle ={{\begin{matrix}{\frac {1}{10}}\end{matrix}}+{\begin{matrix}{\frac {1}{100}}\end{matrix}}+{\begin{matrix}{\frac {1}{50}}\end{matrix}}+{\begin{matrix}{\frac {1}{80}}\end{matrix}}+{\begin{matrix}{\frac {1}{20}}\end{matrix}}+{\begin{matrix}{\frac {1}{500}}\end{matrix}} \over 6}}$
	${\displaystyle ={0.1945 \over 6}}$
	${\displaystyle =0.032416667}$
${\displaystyle N_{e}}$	${\displaystyle =30.8}$

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

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {1}{2}}\end{matrix}}}$

or more generally,

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {D}{2}}\end{matrix}}}$

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:

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {1}{2}}\approx N\end{matrix}}}$

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:

${\displaystyle \operatorname {var} (k)={\bar {k}}=2.}$

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

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

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

${\displaystyle N_{e}^{(v)}={4N-2D \over 2+\operatorname {var} (k)}}$

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:

${\displaystyle N_e^{(v)}}$Zdroj:https://en.wikipedia.org?pojem=Effective_population_size
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