The effective population size (''N''
''e'') is a number that, in some simplified scenarios, corresponds to the number of breeding individuals in the population. More generally, ''N''
''e'' is the number of individuals that an
idealised population In 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 (typically change of genetic diversity or inbreeding rates) to be the same 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. For most quantities of interest and most real populations, the effective population size ''N''
''e'' is usually smaller than the
census population size ''N'' of a real population. 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 and can be attributed to the span of survival for a species. It is dis ...
divided by four times the
mutation rate
In genetics, the mutation rate is the frequency of new mutations in a single gene or organism over time. Mutation rates are not constant and are not limited to a single type of mutation; there are many different types of mutations. Mutation rates ...
, because in such an idealised population, the
heterozygosity
Zygosity (the noun, zygote, is from the Greek "yoked," from "yoke") () is the degree to which both copies of a chromosome or gene have the same genetic sequence. In other words, it is the degree of similarity of the alleles in an organism.
Mo ...
is equal to
. In a population with selection at many loci and abundant
linkage disequilibrium
In population genetics, linkage disequilibrium (LD) is the non-random association of alleles at different loci in a given population. Loci are said to be in linkage disequilibrium when the frequency of association of their different alleles is h ...
, 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 genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and pop ...
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" or "America"
** Americans, citizens and nationals of the United States of America
** American ancestry, pe ...
geneticist
A geneticist is a biologist or physician who studies genetics, the science of genes, heredity, and variation of organisms. A geneticist can be employed as a scientist or a lecturer. Geneticists may perform general research on genetic processe ...
Sewall Wright
Sewall Green Wright FRS(For) Honorary FRSE (December 21, 1889March 3, 1988) was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. He was a founder of population genetics alongsi ...
.
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 who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
and
Sewall Wright
Sewall Green Wright FRS(For) Honorary FRSE (December 21, 1889March 3, 1988) was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. He was a founder of population genetics alongsi ...
originally defined it as "the number of breeding individuals in an
idealised population In 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
Allele frequency, or gene frequency, is the relative frequency of an allele (variant of a gene) at a particular locus in a population, expressed as a fraction or percentage. Specifically, it is the fraction of all chromosomes in the population that ...
under random
genetic drift
Genetic drift, also known as allelic drift or the Wright effect, is the change in the frequency of an existing gene variant (allele) in a population due to random chance.
Genetic drift may cause gene variants to disappear completely and there ...
or the same amount of
inbreeding
Inbreeding is the production of offspring from the mating or breeding of individuals or organisms that are closely related genetically. By analogy, the term is used in human reproduction, but more commonly refers to the genetic disorders and o ...
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 heterozygosity in a population; more specifically the expected degree of (usually) a reduction in heterozygosity when compared ...
, 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 and can be attributed to the span of survival for a species. It is dis ...
divided by the
mutation rate
In genetics, the mutation rate is the frequency of new mutations in a single gene or organism over time. Mutation rates are not constant and are not limited to a single type of mutation; there are many different types of mutations. Mutation rates ...
, yielding a coalescent effective population size.
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
In population genetics, a selection coefficient, usually denoted by the letter ''s'', is a measure of differences in relative fitness. Selection coefficients are central to the quantitative description of evolution, since fitness differences deter ...
at which selection becomes more important than
genetic drift
Genetic drift, also known as allelic drift or the Wright effect, is the change in the frequency of an existing gene variant (allele) in a population due to random chance.
Genetic drift may cause gene variants to disappear completely and there ...
.
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 () is an exonym used to refer to two closely related Indigenous peoples: the Inuit (including the Alaska Native Iñupiat, the Greenlandic Inuit, and the Canadian Inuit) and the Yupik peoples, Yupik (or Siberian Yupik, Yuit) of eastern Si ...
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
In the
Wright-Fisher idealized population model, the
conditional variance In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables.
Particularly in econometrics, the conditional variance is also known as the scedastic function or ...
of the allele frequency
, given the
allele frequency
Allele frequency, or gene frequency, is the relative frequency of an allele (variant of a gene) at a particular locus in a population, expressed as a fraction or percentage. Specifically, it is the fraction of all chromosomes in the population that ...
in the previous generation, is
:
Let
denote the same, typically larger, variance in the actual population under consideration. The variance effective population size
is defined as the size of an idealized population with the same variance. This is found by substituting
for
and solving for
which gives
:
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 refers to all of the people born and living at about the same time, regarded collectively. It can also be described as, "the average period, generally considered to be about 20–30 years, during which children are born and gr ...
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. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
of the population sizes:
:
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. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
of these, giving:
:
Note this is less than the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
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 (such as glass, plastic or aluminium) in various shapes and sizes that stores and transports liquids ...
that the population goes through.
Dioeciousness
If a population is
dioecious
Dioecy (; ; adj. dioecious , ) is a characteristic of a species, meaning that it has distinct individual organisms (unisexual) that produce male or female gametes, either directly (in animals) or indirectly (in seed plants). Dioecious reproductio ...
, i.e. there is no
self-fertilisation
Autogamy, or self-fertilization, refers to the 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. However, species of ...
then
:
or more generally,
:
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:
:
Variance in reproductive success
If population size is to remain constant, each individual must contribute on average two
gamete
A gamete (; , ultimately ) is a haploid cell that fuses with another haploid cell during fertilization in organisms that reproduce sexually. Gametes are an organism's reproductive cells, also referred to as sex cells. In species that produce t ...
s to the next generation. An idealized population assumes that this follows a
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
so that the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of the number of gametes contributed, ''k'' is equal to the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
number contributed, i.e. 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
:
The effective population size is then smaller, and given by:
:
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 (or gender ratio) is usually defined as the ratio of males to females in a population. As explained by Fisher's principle, for evolutionary reasons this is typically about 1:1 in species which reproduce sexually. Many species devia ...
of a population varies from the
Fisherian 1:1 ratio, effective population size is given by:
:
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
The coefficient of inbreeding of an individual is the probability that two alleles at any locus in an individual are identical by descent from the common ancestor(s) of the two parents.
The coefficient of inbreeding is: The probability that two ...
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
:
Using Panmictic Index (1 − ''F'') instead of inbreeding coefficient, we get the approximate recurrence equation
:
The difference per generation is
:
The inbreeding effective size can be found by solving
:
This is
:
although researchers rarely use this equation directly.
Theoretical
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
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 and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, what the probability is that a person of that age will die before their next birthday ("probability of deat ...
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:
:
Fisher's reproductive value for age
,
:
The chance an individual will survive to age
, and
:
The number of newborn individuals per breeding season.
The
generation time
In population biology and demography, generation time is the average time between two consecutive generations in the lineages of a population. In human populations, generation time typically ranges from 22 to 33 years. Historians sometimes use this ...
is calculated as
:
average age of a reproducing individual
Then, the inbreeding effective population size is
:
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
:
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 of mutant alleles that are selectively neutral. The ...
, 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 which is used to measure the degree of polymorphism within a population.
One commonly used measure of nucleotide diversity was first introduced by Nei and Li in 1979. This measure is defin ...
equal to 4
Ne, where
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 may refer to:
*Genetics, in biology, the science of genes, heredity, and the variation of organisms
**Genetic, used as an adjective, refers to genes
***Genetic disorder, any disorder caused by a genetic mutation, whether inherited or de nov ...
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. Selection coefficients are central to the quantitative description of evolution, since fitness differences deter ...
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.
See also
*
Minimum viable population
Minimum viable population (MVP) is a lower bound on the population of a species, such that it can survive in the wild. This term is commonly used in the fields of biology, ecology, and conservation biology. MVP refers to the smallest possible siz ...
*
Small population size
Small populations can behave differently from larger populations. They are often the result of population bottlenecks from larger populations, leading to loss of heterozygosity and reduced genetic diversity and loss or fixation of alleles and sh ...
References
External links
*
*
* https://web.archive.org/web/20050524144622/http://www.kursus.kvl.dk/shares/vetgen/_Popgen/genetics/3/6.htm — on Københavns Universitet.
{{modelling ecosystems, expanded=none
Population genetics
Population ecology
Ecological metrics
Quantitative genetics