Effective population size
Encyclopedia
In population genetics
, the concept of effective population size Ne was introduced by the American
geneticist
Sewall Wright
, who wrote two landmark papers on it (Wright 1931, 1938). He defined it as "the number of breeding individuals in an idealized 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". It is a basic parameter in many models in population genetics
. The effective population size is always either equal to or less than than the absolute population size
(N). See also small population size
.
.
of the allele frequency
, given the allele frequency
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 equating 
with 
and solving for 
which gives
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.
s, then effective population size is given by the harmonic mean
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
of these, giving:
Note this is less than the arithmetic mean
of the population size, which in this example is 126.7.
Of particular concern is the effect of a population bottleneck
.
, i.e. there is no self-fertilisation 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, Ne approximately equals N, so this is usually trivial and often ignored:
of a population varies from the Fisherian
1:1 ratio, effective population size is given by:
Where Nm is the number of males and Nf the number of females. For example, with 80 males and 20 females (an absolute population size of 100):
Again, this results in Ne being less than N.
s 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:
However, in natural populations the variance is larger than this, i.e.
The effective population size is then given by:
Note that if the variance of k is less than 2, Ne is greater than N. Heritable
variation in fecundity
, usually pushes Ne lower.
s for the species.
The generation time is calculated as
Then, the inbreeding effective population size is (Felsenstein 1971)
Assume the same basic parameters for the life table as given for the haploid case, but distinguishing between male and female, such as N0ƒ and N0m 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
Population genetics
Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
, the concept of effective population size Ne was introduced by the American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
geneticist
Geneticist
A geneticist is a biologist who studies genetics, the science of genes, heredity, and variation of organisms. A geneticist can be employed as a researcher or lecturer. Some geneticists perform experiments and analyze data to interpret the inheritance of skills. A geneticist is also a Consultant or...
Sewall Wright
Sewall Wright
Sewall Green Wright was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. With R. A. Fisher and J.B.S. Haldane, he was a founder of theoretical population genetics. He is the discoverer of the inbreeding coefficient and of...
, who wrote two landmark papers on it (Wright 1931, 1938). He defined it as "the number of breeding individuals in an idealized population that would show the same amount of dispersion of allele frequencies
Allele frequency
Allele frequency or Gene frequency is the proportion of all copies of a gene that is made up of a particular gene variant . In other words, it is the number of copies of a particular allele divided by the number of copies of all alleles at the genetic place in a population. It can be expressed for...
under random genetic drift
Genetic drift
Genetic drift or allelic drift is the change in the frequency of a gene variant in a population due to random sampling.The alleles in the offspring are a sample of those in the parents, and chance has a role in determining whether a given individual survives and reproduces...
or the same amount of inbreeding
Inbreeding
Inbreeding is the reproduction from the mating of two genetically related parents. Inbreeding results in increased homozygosity, which can increase the chances of offspring being affected by recessive or deleterious traits. This generally leads to a decreased fitness of a population, which is...
as the population under consideration". It is a basic parameter in many models in population genetics
Population genetics
Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
. The effective population size is always either equal to or less than than the absolute population size
Population size
In population genetics and population ecology, population size is the number of individual organisms in a population.The effective population size is defined as "the number of breeding individuals in an idealized population that would show the same amount of dispersion of allele frequencies under...
(N). See also small population size
Small population size
Small populations behave differently from larger populations. They often result in population bottlenecks, which have harmful consequences for the survival of that population.-Demographic effects:...
.
Definitions
Effective population size may be defined in two ways, variance effective size and inbreeding effective size. These are closely linked, and derived from F-statisticsF-statistics
In population genetics, F-statistics describe the level of heterozygosity in a population; more specifically the degree of a reduction in heterozygosity when compared to Hardy–Weinberg expectation...
.
Variance effective size
In the Wright-Fisher idealized population model, the conditional varianceConditional variance
In probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function...
of the allele frequency
, given the allele frequencyAllele frequency
Allele frequency or Gene frequency is the proportion of all copies of a gene that is made up of a particular gene variant . In other words, it is the number of copies of a particular allele divided by the number of copies of all alleles at the genetic place in a population. It can be expressed for...
in the previous generation, isLet
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 equating with and solving for which givesInbreeding effective size
Alternatively, the effective population size may be defined by noting how the inbreeding coefficient changes from one generation to the next, and then defining Ne as the size of the idealized population that has the same change in inbreeding. 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.
Variations in population size
Population size varies over time. Suppose there are t non-overlapping generationGeneration
Generation , also known as procreation in biological sciences, is the act of producing offspring....
s, then effective population size is given by the harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
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
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
of these, giving:
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Note this is less than the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
of the population size, which in this example is 126.7.
Of particular concern is the effect of a population bottleneck
Population bottleneck
A population bottleneck is an evolutionary event in which a significant percentage of a population or species is killed or otherwise prevented from reproducing....
.
Variance in reproductive success
With increased variation in family size, Ne is reduced: Ne = (4N)/(Vk + 2) Where Vk is the variance in family size.Dioeciousness
If a population is dioeciousDioecious
Dioecy is the property of a group of biological organisms that have males and females, but not members that have organs of both sexes at the same time. I.e., those whose individual members can usually produce only one type of gamete; each individual organism is thus distinctly female or male...
, i.e. there is no self-fertilisation 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, Ne approximately equals N, so this is usually trivial and often ignored:
Non-Fisherian sex-ratios
When the sex ratioSex ratio
Sex ratio is the ratio of males to females in a population. The primary sex ratio is the ratio at the time of conception, secondary sex ratio is the ratio at time of birth, and tertiary sex ratio is the ratio of mature organisms....
of a population varies from the Fisherian
Ronald Fisher
Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation...
1:1 ratio, effective population size is given by:
Where Nm is the number of males and Nf the number of females. For example, with 80 males and 20 females (an absolute population size of 100):
 |
 |
 |
|
 |
Again, this results in Ne being less than N.
Unequal contributions to the next generation
If population size is to remain constant, each individual must contribute on average two gameteGamete
A gamete is a cell that fuses with another cell during fertilization in organisms that reproduce sexually...
s to the next generation. An idealized population assumes that this follows a Poisson distribution
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 and/or space if these events occur with a known average rate and independently of the time since...
so that the variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of the number of gametes contributed, k is equal to the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
number contributed, i.e. 2:
However, in natural populations the variance is larger than this, i.e.
The effective population size is then given by:
Note that if the variance of k is less than 2, Ne is greater than N. Heritable
Heredity
Heredity is the passing of traits to offspring . This is the process by which an offspring cell or organism acquires or becomes predisposed to the characteristics of its parent cell or organism. Through heredity, variations exhibited by individuals can accumulate and cause some species to evolve...
variation in fecundity
Fecundity
Fecundity, derived from the word fecund, generally refers to the ability to reproduce. In demography, fecundity is the potential reproductive capacity of an individual or population. In biology, the definition is more equivalent to fertility, or the actual reproductive rate of an organism or...
, usually pushes Ne lower.
Overlapping generations and age-structured populations
When organisms live longer than one breeding season, effective population sizes have to take into account the life tableLife table
In actuarial science, a life table is a table which shows, for each age, what the probability is that a person of that age will die before his or her next birthday...
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 valueFisher's reproductive valueFisher's reproductive value was defined by R. A. Fisher in his 1930 book The Genetical Theory of Natural Selection as the expected reproduction of an individual from their current age onward, given that they have survived to their current age...
for age
,
-
The chance an individual will survive to age 
, and
-
The number of newborn individuals per breeding season.
The generation time is calculated as
-
average age of a reproducing individual
Then, the inbreeding effective population size is (Felsenstein 1971)
Diploid
Similarly, the inbreeding effective number can be calculated for a diploid population with discrete age structure. This was first given by Johnson (1977), but the notation more closely resembles Emigh and Pollak (1979).Assume the same basic parameters for the life table as given for the haploid case, but distinguishing between male and female, such as N0ƒ and N0m 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
External links
- http://www.kursus.kvl.dk/shares/vetgen/_Popgen/genetics/3/6.htm — on Københavns Universitet.