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The effective population size (''N''''e'') is the number of individuals that an
idealised populationIn population genetics an idealised population is one that can be described using a number of simplifying assumptions. Models of idealised populations are either used to make a general point, or they are fit to data on real populations for which the ...
would need to have in order for some specified quantity of interest to be the same in the idealised population as in the real population. Idealised populations are based on unrealistic but convenient simplifications such as random mating, simultaneous birth of each new generation, constant population size, and equal numbers of children per parent. In some simple scenarios, the effective population size is the number of breeding individuals in the population. However, for most quantities of interest and most real populations, the census population size ''N'' of a real population is usually larger than the effective population size ''N''''e''. The same population may have multiple effective population sizes, for different properties of interest, including for different genetic loci. 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 Genetic diversity is the total number of genetic characteristics in the genetic makeup of a species, it ranges widely from the number of species to differences within species In biology, a species is the basic unit of biological classificati ...
divided by four times the
mutation rate In genetics Genetics is a branch of biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interaction ...
$\mu$, because in such an idealised population, the
heterozygosity Zygosity (the noun, zygote A zygote (from Greek ζυγωτός ''zygōtos'' "joined" or "yoked", from ζυγοῦν ''zygoun'' "to join" or "to yoke") is a eukaryotic cell formed by a fertilization event between two gamete A gamete ( /ˈ ...
is equal to $4N\mu$. In a population with selection at many loci and abundant
linkage disequilibrium In population genetics Population genetics is a subfield of that deals with genetic differences within and between s, and is a part of . Studies in this branch of examine such phenomena as , , and . Population genetics was a vital ingredien ...
, 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 Population genetics is a subfield of that deals with genetic differences within and between s, and is a part of . Studies in this branch of examine such phenomena as , , and . Population genetics was a vital ingredient in the of the . Its pri ...
in 1931 by the
American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America (USA), commonly known as the United States (U.S. or US), or America, is ...

geneticist A geneticist is a biologist who studies genetics, the science of genes, heredity, and genetic variation, variation of organisms. A geneticist can be employed as a scientist or a lecturer. Geneticists may perform general research on genetic proce ...
Sewall Wright Sewall Green Wright FRS(For) H FRSE (December 21, 1889March 3, 1988) was an American geneticist known for his influential work on evolutionary theory Evolutionary thought, the recognition that species change over time and the perceived unde ...

.

# Overview: Types of effective population size

Depending on the quantity of interest, effective population size can be defined in several ways.
Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a subs ...
and
Sewall Wright Sewall Green Wright FRS(For) H FRSE (December 21, 1889March 3, 1988) was an American geneticist known for his influential work on evolutionary theory Evolutionary thought, the recognition that species change over time and the perceived unde ...

originally defined it as "the number of breeding individuals in an
idealised populationIn population genetics an idealised population is one that can be described using a number of simplifying assumptions. Models of idealised populations are either used to make a general point, or they are fit to data on real populations for which the ...
that would show the same amount of dispersion of allele frequencies under random
genetic drift Genetic drift (allelic drift or the Sewall Wright effect) is the change in the frequency of an existing gene In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical stru ...

or the same amount of
inbreeding Inbreeding is the production of offspring In biology, offspring are the young creation of living organisms, produced either by a Asexual reproduction, single organism or, in the case of sexual reproduction, two organisms. Collective offspri ...
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 In population genetics, ''F''-statistics (also known as fixation indices) describe the statistically expected level of Zygosity, heterozygosity in a population; more specifically the expected degree of (usually) a reduction in heterozygosity when c ...

, but they are not identical. Today, the effective population size is usually estimated empirically with respect to the sojourn or coalescence time, estimated as the within-species
genetic diversity Genetic diversity is the total number of genetic characteristics in the genetic makeup of a species, it ranges widely from the number of species to differences within species In biology, a species is the basic unit of biological classificati ...
divided by the
mutation rate In genetics Genetics is a branch of biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interaction ...
, yielding a coalescent effective population size. Another important effective population size is the selection effective population size 1/scritical, where scritical is the critical value of the
selection coefficient In population genetics, a selection coefficient, usually denoted by the letter ''s'', is a measure of differences in relative Fitness (biology), fitness. Selection coefficients are central to the quantitative description of evolution, since fitness ...
at which selection becomes more important than
genetic drift Genetic drift (allelic drift or the Sewall Wright effect) is the change in the frequency of an existing gene In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical stru ...

.

## Empirical measurements

In ''Drosophila'' populations of census size 16, the variance effective population size has been measured as equal to 11.5. 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. A genealogical analysis of human hunter-gatherers (
Eskimo Eskimo ( ) or Eskimos is a term used to refer to two closely related Indigenous peoples: The Inuit (including the Alaskan Iñupiat, the Greenlandic Inuit, and the Canadian Inuit) and the Yupik peoples, Yupik (or Siberian Yupik, Yuit) of eastern S ...

s) 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.

# Variance effective size

References missing In the Wright-Fisher idealized population model, the
conditional varianceIn probability theory and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conv ...
of the allele frequency $p\text{'}$, given the
allele frequency Allele frequency, or gene frequency, is the relative frequency of an allele An allele (, ; ; modern formation from Greek ἄλλος ''állos'', "other") is one of two, or more, forms of a given gene In biology, a gene (from ''genos'' "... ...
$p$ in the previous generation, is :$\operatorname\left(p\text{'} \mid p\right)= .$ Let $\widehat\left(p\text{'}\mid p\right)$ denote the same, typically larger, variance in the actual population under consideration. The variance effective population size $N_e^$ is defined as the size of an idealized population with the same variance. This is found by substituting $\widehat\left(p\text{'}\mid p\right)$ for $\operatorname\left(p\text{'}\mid p\right)$ and solving for $N$ which gives :$N_e^ = .$

## 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
generation A generation is "all of the people born and living Living or The Living may refer to: Common meanings *Life, a condition that distinguishes organisms from inorganic objects and dead organisms ** extant taxon, Living species, one that is not ex ...

s, then effective population size is given by the
harmonic mean In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. Sometimes it is appropriate for situations when the average rate (mathematics), rate is desired. The harmonic mean can be express ...
of the population sizes: :$= \sum_^t$ 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 In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. Sometimes it is appropriate for situations when the average rate (mathematics), rate is desired. The harmonic mean can be express ...
of these, giving: : Note this is less than the
arithmetic mean In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the population size, which in this example is 126.7. The harmonic mean tends to be dominated by the smallest
bottleneck Bottleneck literally refers to the narrowed portion (neck) of a bottle A bottle is a narrow-necked container made of an impermeable material (clay, glass Glass is a non- crystalline, often transparency and translucency, transparent amorpho ...

that the population goes through.

### Dioeciousness

If a population is
dioecious Dioecy (; Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 1 ...
, i.e. there is no
self-fertilisation Autogamy, or self-fertilization, refers to the Cell fusion, fusion of two gametes that come from one individual. Autogamy is predominantly observed in the form of self-pollination, a reproductive mechanism employed by many flowering plants. Howeve ...
then :$N_e = N + \begin \frac \end$ or more generally, :$N_e = N + \begin \frac \end$ 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: :$N_e = N + \begin \frac \approx N \end$

### Variance in reproductive success

If population size is to remain constant, each individual must contribute on average two
gamete A gamete ( /ˈɡæmiːt/; from Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply ...
s to the next generation. An idealized population assumes that this follows a
Poisson distribution In probability theory and statistics, the Poisson distribution (; ), named after France, French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a f ...
so that the
variance In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

of the number of gametes contributed, ''k'' is equal to the
mean There are several kinds of mean in mathematics, especially in statistics. For a data set, the ''arithmetic mean'', also known as arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by ...
number contributed, i.e. 2: :$\operatorname\left(k\right) = \bar = 2.$ 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 :$\operatorname\left(k\right) > 2.$ The effective population size is then smaller, and given by: :$N_e^ =$ 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'' = 2''N''.

### Non-Fisherian sex-ratios

When the
sex ratio The sex ratio is the ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, ...
of a population varies from the Fisherian 1:1 ratio, effective population size is given by: :$N_e^ = N_e^ =$ 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 Inbreeding is the production of offspring In biology, offspring are the young born of living organism, organisms, produced either by a single organism or, in the case of sexual reproduction, two organisms. Collective offspring may be known as ...
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 :$F_t = \frac\left\left(\frac\right\right)+\left\left(1-\frac\right\right)F_.$ Using Panmictic Index (1 − ''F'') instead of inbreeding coefficient, we get the approximate recurrence equation :$1-F_t = P_t = P_0\left\left(1-\frac\right\right)^t.$ The difference per generation is :$\frac = 1-\frac.$ The inbreeding effective size can be found by solving :$\frac = 1-\frac.$ This is :$N_e^ = \frac$ although researchers rarely use this equation directly.

## Theoretical A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, G ... example: overlapping generations and age-structured populations

When organisms live longer than one breeding season, effective population sizes have to take into account the
life table In actuarial science Actuarial science is the discipline that applies mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic ...
s 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: : $v_i =$ Fisher's reproductive value for age $i$, : $\ell_i =$ The chance an individual will survive to age $i$, and : $N_0 =$ The number of newborn individuals per breeding season. The
generation timeIn population biology and demography, generation time is the average time between two consecutive generations in the lineages of a population In biology, a population is a number of all the organisms of the same group or species In bio ...

is calculated as : $T = \sum_^\infty \ell_i v_i =$ average age of a reproducing individual Then, the inbreeding effective population size is :$N_e^ = \frac.$

### 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 ''N''0''ƒ'' and ''N''0''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 :$\begin \frac = \frac\left\. & \end$

# Coalescent effective size

According to the
neutral theory of molecular evolution The neutral theory of molecular evolution holds that most evolutionary changes occur at the molecular level, and most of the variation within and between species are due to random genetic drift Genetic drift (allelic drift or the Sewall Wright ...
, a neutral allele remains in a population for Ne generations, where Ne is the effective population size. An idealised diploid population will have a pairwise
nucleotide diversity Nucleotide diversity is a concept in molecular genetics Molecular genetics is a sub-field of biology that addresses how differences in the structures or expression of DNA molecules manifests as variation among organisms. Molecular genetics oft ...
equal to 4$\mu$Ne, where $\mu$ is the mutation rate. The sojourn effective population size can therefore be estimated empirically by dividing the nucleotide diversity by the mutation rate. The coalescent effective size may have little relationship to the number of individuals physically present in a population. Measured coalescent effective population sizes vary between genes in the same population, being low in genome areas of low recombination and high in genome areas of high recombination. Sojourn times are proportional to N in neutral theory, but for alleles under selection, sojourn times are proportional to log(N).
Genetic hitchhiking Genetic hitchhiking, also called genetic draft or the hitchhiking effect, is when an allele An allele (, ; ; modern formation from Greek ἄλλος ''állos'', "other") is one of two, or more, forms of a given gene In biology, a gene (from ...
can cause neutral mutations to have sojourn times proportional to log(N): this may explain the relationship between measured effective population size and the local recombination rate.

# 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 In population genetics, a selection coefficient, usually denoted by the letter ''s'', is a measure of differences in relative Fitness (biology), fitness. Selection coefficients are central to the quantitative description of evolution, since fitness ...
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. Populations with different selection effective population sizes are predicted to evolve profoundly different genome architectures.