Effective population size

Encyclopedia

In population genetics

, the concept of

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

(

.

.

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 −

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

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

When

of a population varies from the Fisherian

1:1 ratio, effective population size is given by:

Where

Again, this results in

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,

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

variation in fecundity

, usually pushes

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

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***N*_{e}was introduced by the AmericanUnited 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 sizeSmall 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 frequency

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...

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

### Inbreeding 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*N*_{e}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 equationThe 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 meanHarmonic 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:

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:*N*_{e}= (4*N*)/(*V*_{k}+ 2) Where*V*_{k}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,*N*_{e}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

*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*.### 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 meanMean

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,*N*_{e}is greater than*N*. HeritableHeredity

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

*N*_{e}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

*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

## External links

- http://www.kursus.kvl.dk/shares/vetgen/_Popgen/genetics/3/6.htm — on Københavns Universitet.